Solid State Physics Advances In Research And Applications

Founding Editors FREDERICK SEITZ DAVID TURNBULL

SOLID STATE PHYSICS Advances in Research and Applications

Editors

HENRY EHRENREICH

DAVID TURNBULL Division of Applied Sciences Harvard University, Cambridge, Massachusetts VOLUME 43

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8

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90919i93

9 8 7 6 5 4 3 2 1

Contributors to Volume 43

Numbers in parentheses indicate the pages on which the authors’ contributions begin.

A. E . CARLSSON (l), Department of Physics, Washington University, St. Louis, Missouri 63130

P. M. ECHENIQUE (230), Departamento de Fisica de Materiales, Facultad de Ciencias Quimicas, Universidad del Pais Vasco/Euskal Herriko Unibertsitatea, Apdo. 1072, San Sebasticin 20080, Spain F. FLORES (230), Departamento de Fiiica del Estado Sblido, Universidad A utbnoma de Madrid, Cantoblanco, Madrid 28049, Spain M . RASOLT(94), Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831

R. H . RITCHIE(230), Health and Safety Research Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831

Preface

This volume deals with three diverse topics, all of which are of current importance and interest. They concern, respectively, the latest developments in interatomic potentials and their applications to transition metals and semiconductors, an examination of the coherent electronic states that can arise as a consequence of the conduction band electronic structure in silicon and other multivalley semiconductors, and the dynamic screening and energy loss of energetic ions as they traverse condensed matter. Interatomic potentials have assumed increasing importance as experimental data and computer simulations involving such complex phenomena as dislocation motion, surface phase stability, and grain boundary structure have become more available. The extensive survey by A. E. Carlsson of the various approaches useful in constructing energy functionals and the resulting potentials, which opens the volume, is therefore particularly welcome. The interatomic interactions discussed extend from pair potentials and functionals to cluster potentials and functionals with increasingly sophisticated ingredients. The article focuses on two very different classes of solids, transition metals and semiconductors, because their technological importance has led to many atomic simulation studies and consequently to a need for simplified descriptions based on interatomic potentials. It is shown here that pair potentials cannot by themselves describe the broad range of atomic configurations that must be included in order to understand most atomistic properties of these materials. Indeed, such potentials do not even represent a good starting point for understanding the bonding energetics of solids. The article therefore emphasizes more general effective interactomic potentials, for example, those derived from pair functionals utilizing tight-binding models of the electronic structure and the embedded atom picture. Several schemes, which fit simplified energy functionals to properties obtained from experiment or ab initio calculations, are also described, as are their applications to illustrative bulk and defect properties. These descriptions include relaxation effects. Cluster potentials are illustrated using silicon surfaces and defects. The article presents a systematic account of these approaches by suggesting how they are theoretically connected. A coherent picture is indeed beginning to emerge. Fitted schemes, based on different analyses of the electronic structure, result in functionals that are quite similar and whose physical predictions are consistent. However, the form of cluster potentials and functionals necessary for describing structural energy differences in transition metals and semiconductors is less well established

vi i

...

Vlll

PREFACE

at present. This article, together with its copious references, will be useful to anyone involved in understanding the physical phenomena resulting from the atomic complexities that are present in real solids. The title of Rasolt’s article, “Continuous Symmetries and Broken Symmetries in Multivalley Semiconductors and Semimetals,” refers to a kind of symmetry, labelled isospin symmetry, that is present in some solids as a consequence of their band structure. This symmetry can be broken, thereby theoretically giving rise to a host of new physical phenomena. It is well known from symmetry considerations of electronic wave functions in the presence of a periodic potential that there are 230 space groups. In addition, there is the full SU(2) symmetry of the spins in the absence of spin-orbit interactions and time reversal symmetry in the absence of magnetic fields. However, because of its conduction band structure, n-type silicon, for example, contains six equivalent electron valleys in the first Brillouin zone. What has been overlooked until recently is that when the effective mass approximation is valid there are continuous symmetries resulting from the valley degeneracy that are not at all implied by the symmetry of the space groups. The space of these continuous symmetries is called isospin space. The silicon effective mass Hamiltonian is invariant under the SU(n) symmetry of the valley or isospin index t (n = 6), and the SU(2) symmetry of the spin index 0.In this nomenclature, the n = 2 case refers either to a band structure having only two equivalent valleys, or, as already mentioned, to real spin degeneracy. The interesting possibility that is suggested in Rasolt’s article is the existence of a wide variety of isospin-spin broken symmetry ground states and their associated Goldstone modes. Suppose, Rasolt suggests, that it turned out to be energetically favorable to place more electrons in the isospin t = 1 valley than in the remaining equally populated valleys z = 2, . . . , n. Suppose, in addition, that the spins in the z = 1 valley are polarized so that there are more spin-up than spin-down electrons. This ground state has a broken symmetry and is ferromagnetic isospin-spin polarized. It is a broken symmetry ground state because its Hamiltonian is not invariant under some of the symmetry operations of the SU(n) x SU(2) group. There can also be antiferromagnetic and other coherent states in isospin-spin space. The associated Josephson modes can be valley density waves which, for silicon, are incommensurate with the lattice. Broken symmetry of isospin space is not likely in multivalley semiconductors in the absence of a magnetic field because of the small effective electron masses and high associated kinetic energies. A magnetic field quenches the kinetic energy effectively in two-dimensional systems. In the quantum limit, a two-dimensional electron gas in a high

PREFACE

ix

magnetic field exhibits a tendency toward instabilities in isospin space because the repulsive electron-electron interactions can be minimized without expense of kinetic energy. Such instabilities would be expected to affect the quantum Hall effect. Topics such as the nature of the isospin polarized quantum Hall effect ground state in a Si[llO] inversion layer are therefore discussed in considerable detail. The mapping of multivalley systems on isospin space has been proposed only recently, and the number of applications is still limited. The experimental observations of isospin properties are similarly limited, and in many cases ambiguous. However, the formalism has broad applicability. Because of its potential usefulness, a detailed exposition given here will be of importance to those who wish to acquaint themselves with it. The article is quite formal. However, it is also accessible to the general solid state theorist or student because many of the derivations are limited to two-valley systems for which the familiar SU(2) group, whose generators are the Pauli matrices, suffices. Nevertheless, the reader interested in the detailed property of the SU(n) continuous group may wish to refer to Reference 3 of the article. The third review, by Echenique, Flores, and Ritchie, summarizes the concepts associated with the passage of ions through condensed matter. The theory, pioneered by Bohr, Bethe, and Fermi among others, has been developed extensively since the introduction of the frequency- and wave number-dependent dielectric constant and response function theory. Density fluctuations induced by swift ions and the many physical phenomena associated with the wake are discussed in detail. The local density formalism is appropriate for describing the many-electron response to slow ions. However, core excitations and electron capture by the ions must be considered in this case. The charge states of ions are inferred from numerical calculations. The article is both didactic and comprehensive. It summarizes a field that has attracted many of the outstanding twentieth century physicists and that continues to be explored. Indeed, an article by David Pines in Volume 1 of this series, concerning electron interactions in solids, is in part concerned with some of the issues under consideration here and treats them from a perspective that is similar in some respects. The present self-contained account represents a most useful update. Henry Ehrenreich David Turnbull

SOLID STATE PHYSICS, VOLUME 43

Beyond Pair Potentials in Elemental Transition Metals and Semiconductors A. E. CARLSSON Department of Physics Washington University St. Louis. Missouri

I. Introduction .......................................................... 1. Motivation and Overview ...... ................................. 2. Difficulties with Global Interato Potential Descriptions of Transition Metals and Semiconductors .................... 11. Theoretical Justification and Physical Properties of Interatomic .................... Potentials and Functionals ............. 3. Introduction.. ............................................ 4. Structural Energies: “Constant-Volume” Interatomic Poten..................... tials . . 5. Broken Bonds: Pair Functional Approaches ... . . . . . . . . . . . . . . . . 6. Cluster Functionals.. .............................................. 7. Relation Between “Constant-Volume’’ Potentials and Cluster Functionals ................................................... 8. Interatomic Potentials as Statistical Correlation Functions. ............. 9. The Solid Solution Analogy ........................ 111. Fitted Potentials and Functionals: Implementations and Appl 10. Ab Initio versus Fitted Descriptions ................................. 11. Pair Functionals .......................................... 12. Angular Forces ................... ............................. IV. Concluding Comments .................................................

1 1

17 23 42 53 57 64

78

90

1. Introduction

1. MOTIVATION AND OVERVIEW A central theme of condensed matter theory is the effort to translate well-established microscopic interactions, in many-body systems, into higher-level descriptions containing a smaller number of degrees of freedom. At one extreme, such an effort can involve calculating a 1 Copyright @ 1990 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-607743-6

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A . E. CARLSSON

macroscopic property, such as an equation of state, directly from the underlying point-particle interactions. On the other hand, it is often useful to take an intermediate route by “integrating out” a number of degrees of freedom to obtain effective interactions that still apply at microscopic length scales but express the basic physics more clearly than the complete set of interactions. The effective interactions are typically obtained by a low-order expansion in powers of one or more “bare” interactions, which are small in magnitude. Celebrated examples of this approach are the Heisenberg-type models’ for insulating magnets, which subsume the electronic degrees of freedom in effective spin-spin interactions, and the BCS-type models’ for superconductivity, which include the phonon degrees of freedom in effective electron-electron interactions. This review will discuss several aspects of the origin and form of the effective interactions that occur between ions in condensed matter as a result of their interaction with the electron gas. Because the relevant “bare” interactions are generally quite strong, particularly in transition metals and semiconductors, a precise decomposition of the interactions into useful two-body and higher-order effective interactions is generally hard to achieve. Nevertheless, for reasons to be discussed below, such decompositions have been actively sought practically since the advent of quantum mechanics3 One seeks a simple description of the total configurational energy of a condensed matter system as a functional of the positions Ri of the atomic nuclei. The validity of the Born-Oppenheimer appr~ximation,~ which allows us to treat the electrons as if they were always in their ground state, implies that an energy functional must exist in most5 cases, although its form may be very complex. The simplest type of description would have the form 1 1 E , = - C V’(R;, R;) + - C, V,(R;, R,, Rk) + . . . , 2! 3! i j , k I,;

(1.1)

IN. W. Ashcroft and N. D. Mermin, “Solid State Physics,” Chapter 32. Holt, Rinehart, and Winston, New York, 1976. ’C. Kittel, “Quantum Theory of Solids,” second edition, Chapter 8. Wiley, New York, 1987. 3A very early attempt was made by K. Fuchs, Proc. R. SOC. London, Ser. A 153, 622 (1936). 4See J. M. Ziman, “Electrons and Phonons,” Chapter 5. Oxford University Press, Oxford, 1960. ’However, in semiconductors and insulators, electrons at isolated defects can sometimes remain in excited-state configurations for times long in comparison with typical time scales for ionic motion, invalidating a ground-state description. This issue will be discussed in Section 12.

BEYOND PAIR POTENTIALS

3

where E, is the configurational energy of the solid (relative to isolated atoms), and the V, are interatomic potential functions. We shall see that a global description of exactly this form is usually impractical. However, closely related energy functionals, of comparable simplicity, can be considerably more accurate. A real-space description of bonding energetics by such functionals is useful in material physics for several reasons. They often provide the simplest understanding of the origin of structural features in both crystalline solids and at defects. For example, the favorability of one closely packed structure over another can often be traced back to the positions of the minima in the effective interatomic pair potential, relative to the separations of the various neighbor shells in these structures.6 As will be discussed later, relaxations around defects can often be explained by modifications of the effective pair potentials around the defects. Such a direct understanding of these relatively simple features is hard to obtain from quantitative quantum-mechanical calculations, but is a necessary prerequisite to understanding the much more complex phenomena that can occur in materials science problems. In addition to providing improved interpretability, interatomic potentials and related energy functionals provide computational speed essential for computer simulations of complex materials-science problems. Some such problems, including the direct simulation of fully three-dimensional fracture dynamics, cannot at present be treated reliably even with pair potentials. However, many important problems are simple enough to be treated with interatomic potentials, but too complex to be treated directly with fully quantum-mechanical methods. Examples of the latter type are dislocation motion, surface phase transitions, and (in most cases) grain boundary structure. The range of problems treatable by fully quantummechanical methods will certainly continue to grow with expanding computing ~apabilities,~ but in the foreseeable future most materialsscience problems will require the use of interatomic potentials or schemes of comparable simplicity. This review will describe the main approaches to transcending pair potential descriptions of transition metals and semiconductors. We focus on these technologically important materials because the desire to understand their materials properties has led to a large number of atomistic simulation studies, and consequently a great need for simplified

9.Heine

and D. Weaire, in “Solid State Physics, Advances in Research and Applications” (H. Ehrenreich, F. Seitz, and D. Turnbull, eds.), Vol. 24, p. 250. Academic, New York, 1970. 7rr Simulated annealing” is a promising technique for nb inirio materials studies. See R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985).

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A. E. CARLSSON

energy functionals. At first glance the bonding mechanisms in these two classes of materials would seem to be quite distinct. One naively thinks of metallic atoms as closely packed hard balls with relatively weak attractive forces. In semiconductors one generally focuses on strong bonds with charge accumulations between the atoms. The open structures of diamond structure semiconductors immediately suggest the presence of strong angular forces; if radial pair forces dominated, one would expect that the energy could be lowered by filling in the holes in the structure, thereby increasing the coordination number. In contrast, the closely packed structures of transition metals are at least consistent with a description based on radial forces. However, to obtain a more sophisticated picture of transition metals, including structural energy differences, we shall see that angular forces are necessary, Furthermore, much of the understanding that has recently been gained of transition metal bonding is also applicable to semiconductors, and is being incorporated in the most recent semiconductor energy functionals. The topics to be treated here have only been briefly reviewed: see Refs. 8-13. The theory of interatomic potentials and related functionals for transition metals and semiconductors is considerably less well established than it is for simple metals, ionic solids, and rare gases.14 In simple metals the uniform electron gas serves as a useful starting point for a perturbative expansion in powers of the electron-ion pseudopotential. Such an expansion is substantially harder to achieve for transition metals, and probably even more so for semiconductors. For reviews of interatomic potentials in simple metals, see Refs. 6, 11, 13, 15, and 16. In *A widely ranging collection of articles on interatomic forces in condensed matter is found in Philos. Mug. A 58, No. 1 (1988). ’Simplified energy functionals for transition metals are discussed in M. W. Finnis, A. T. Paxton, D. G. Pettifor, A. P. Sutton, and Y. Ohta, Philos. Mug. A 58, 143 (1988). “Angular forces in semiconductors are reviewed by A. M. Stoneham, V. T. B. Torres, P. M. Masri, and H. R. Schober, Philos. Mug. A 58, 93 (1988) and T. Haliocioglu, H. 0. Pamuk, and S. Erkoc, Phys. Star. Sol. B 149, 81 (1988). “An extensive treatment of bonding in metals is given by D. G. Pettifor, in “Physical Metallurgy,” (R. W. Cahn and P. Haasen, eds.), p. 73, North-Holland, New York, 1983. ”Some closely related aspects of transition metal bonding are discussed by V. Heine, in “Solid State Physics, Advances in Research and Applications” (H. Ehrenreich, F. Seitz, and D. Turnbull, eds.), Vol. 35, p. 1. Academic, New York 1980. I3R. Taylor, Physicu WlB, 103 (1985). I4A broadly based discussion of interatomic potentials is given in I. M. Torrens, “Interatomic Potentials.” Academic, New York, 1972. ”J. Hafner, “From Hamiltonians to Phase Diagram-The Electronic and Statistical Mechanical Theory of Metals and Alloys,” Solid State Sciences, Vol. 70. Springer, Berlin, 1987. ‘6“Interatomic Potentials and Lattice Defects” (J. K. Lee, ed.). Metallurgical Society of AIME, Warrendale, PA, 1981. See, in particular, articles by J. Th. M. de Hosson (p. 3), E. S. Machlin (p. 33), D. M. Esterling (p. 53), and R. Taylor (p. 71).

BEYOND PAIR POTENTIALS

S

ionic ~ r y s t a l s ' ~ -separated '~ ions form a useful starting point; their attractive Coulomb interaction is countered by a short-ranged repulsive term which can be described either by density-based theories" or by pair potentials. In rare gases, the dominant attractive term results from the Van der Waals fluctuating dipole interaction;2" the repulsive interaction is similar in character to that in ionic crystals. We treat only energy functionals which are general enough to use in computer simulations of defect structures or geometrically (as opposed to chemically) disordered bulk phases. In addition, to focus the discussion, and to display the basic physics in a simple fashion, we treat only elemental systems. There are several closely related problems which are left out or only briefly discussed. We do not treat the calculation of interatomic force constants for phonons, which describe only infinitesimal displacements from a perfect crystal. Descriptions of cohesive energies and related properties of bulk semiconductors in terms of pair and higher-order potentials have been obtained recently via "bond-orbital'' methods.*',** However, since they have not been generalized to arbitrary geometries, they are not treated here. Finally, simple Ising-type models are often used to deal with solid solution ordering energies and surface phase transitions. These do not have the generality of the schemes considered here, but solid solutions will serve as a useful analogy in our discussion. We focus on the higher-order interactions because, as will be seen below, they are essential in describing the broad range of atomic environments that typically occur in materials science problems. The treatment of most structural energy differences requires the higher-order interactions as well. A very large number of atomistic simulations for transition metals have, however, been performed using only pair potentials. Typically the pair potentials have an assumed analytic form, with parameters fitted to various known properties of the metal. These calculations are not reviewed here; see, however, Refs. 16, 23, and 24. The energy functionals which we treat can be divided into the four 17

W. A . Harrison, "Electronic Structure and the Properties of Solids," W. H. Freeman, San Francisco, 1980, Chap. 13. 18C. R. A. Catlow, C. M. Freeman, M. S. Islam, R. A. Jackson, M. Leslie, and S. M. Tomlinson, Philos. Mag. A 58, 123 (1988). 19R. G . Gordon and Y. S. Kim, J . Chem. Phys. 56, 3122 (1972). "A Dalgarno and W. D . Davison, Adu. At. Mol. Phys. 2, 1 (1966). 'lRef. 17, Chap. 17. uM.van Schilfgaarde and W. A. Harrison, Phys. Rev. B 33, 26.53 (1986). U"Interatomic Potentials and the Simulation of Lattice Defects" (P. C. Gehlen, J. R. Beeler, and R. I. Jaffee, eds.). Plenum, New York, 1972. 24R. A . Johnson, J. Phys. F3, 29.5 (1973).

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A. E. CARLSSON

J FIG. 1. Diagram of types of simplified energy functionals to be treated. Arrows point in direction of increasing generality.

categories indicated in Fig. 1. These correspond to four levels of sophistication in the description of the configurational energy E , , with pair potentials providing the simplest treatment. A pictorial representation of the accuracy with which the various types of functionals treat E, is given in Fig. 2. Here, in each frame, we compare the estimates of the relative energies of various atomic configurations obtained by one type of energy functional (solid curve), with the exact physical energies (dashed curve). The “configuration” axis is meant to symbolize a variety of possible changes in configuration, such as defect formation, changes in crystal structure, or symmetry-breaking distortions. The cusp in the exact curve could, for example, be taken to correspond to the singular behavior of the electronic band energy under some of these distortions. The four categories are defined as follows: (a) Pair Potentials. One has a description of the configurational energy in the form

E , = Eo + 4

i,i

Vzff(Ri,R,),

where Eo is a reference energy and V‘,”(R,, R,) is an effective pair potential. Thus one treats only part of E , (sometimes called the “structure-dependent’’ part) with the pair potentials. These are typically designed to model a small class of atomic rearrangements. There are thus many types of effective pair potentials, which treat different properties.

BEYOND PAIR POTENTIALS

7

L

t i

\I

.--

PAIR FUNCTIONALS

CLUSTER FUNCTIONALS

CONFIGURATION

FIG. 2. Schematic illustration of accuracy with which configurational energies are obtained by simplified energy functionals. Solid curves denote energies obtained by functionals; dashed curve in each frame denotes exact energy.

This simple type of description provides great interpretability but can easily lead to large errors if it is used beyond its limited range of applicability. (b) Cluster Potentials. These improve on the pair potentials by the addition of higher-order interactions:

The three-body and higher-order terms contain information about the angles between the bonds, which is absent from a pair potential description. Typically the series is truncated at the three-body level. As illustrated in Fig. 2, inclusion of higher-order effective potentials provides a more accurate treatment of a particular range of phenomena than is given by effective pair potentials alone. In addition, a scheme containing both pair and triplet terms, for example, can treat a broader range of atomic configurations than a pair-potential scheme. Nevertheless, such schemes are far from providing a global description of E,. Cluster

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A. E. CARLSSON

potentials involve significantly more computation time than pair potentials, but can sometimes be written in a form (cf. Sec. 12) such that the computation time scales with the same power of the number of atoms as for pair potentials. (c) Pair Functionals. Instead of expressing the configurational energy of a particular atom via pair interactions, one expresses part of it as a functional of an intermediate quantity which is given as a sum of pair terms. The form of the configurational energy is the following:

i.;

i

\;

where V2 is a pair potential and g , is a pair function describing the local environment of atom i in terms of contributions from its neighbors. This local environment is associated with a single physical parameter, such as the local electron bandwidth or background electron density. The energy function U describes how the part of Ec associated with atom i depends on the local environment. It is usually thought of as describing the bonding energy coming from the valence electrons. In the special case that U is a linear function, one obtains immediately from Eq. (1.4) a pair potential, with an attractive part proportional to g,. In general U is far from linear, and forms such as a square-root behavior are commonly used. Thus the pair functional generalizes the notion of a pair potential. Pair functional approaches have roughly the same computational speed as pair potential descriptions, but provide a much improved description of a broad range of inhomogeneous environments. As illustrated in Fig. 2, the global fit to the correct energy is much better than that obtained by pair potentials, although a narrow class of properties is often described better by a carefully chosen pair potential. Pair functionals effectively contain very-high-order potentials (>lo atoms), but can in some cases be approximated by low-order environmentally dependent potentials. (d) Cluster Functionals. Several treatments have attempted to combine the virtues of cluster potentials and pair functionals by expressing the configurational energy as a functional involving cluster terms. Cluster functionals generalize cluster potentials in precisely the same fashion that pair functionals generalize pair potentials. The configurational energy has the form

BEYOND PAIR POTENTIALS

9

where the functions {g,} provide a more sophisticated description of the local environment than g , by itself. For example, g 3 and the higher-order terms can provide a measure of the bond angles, which is not possible with only radial terms. By explicitly including angular forces, cluster functionals improve on the accuracy of pair functionals; they can also handle a broader range of inhomogeneous environments than the cluster potentials. However, unless very-high-order functionals are used, one cannot obtain energies associated with fine structure in the electronic density of states, which are symbolized by the cusp in Fig. 2. Using tight-binding models for the electronic structure, it is in fact possible to evaluate such higher-order functionals without explicitly performing the sums containing the g , . In this fashion the electronic contribution to E , is obtained essentially exactly. We emphasize treatments at the three- and four-body levels, because at these levels the atomic geometries can still be visualized, and the computation time is significantly shorter than for the higher-order treatments. Reviews of the higher-order treatments can be found in Ref. 25. We will focus on the following two questions: (1) What is the theoretical basis for the use of the simplified energy functionals? Since the direct ion-ion Coulomb interaction is straightforward, the difficult task in devising energy functionals is to determine the quantum-mechanical behavior of the electrons. Thus the question can be rephrased as “To what type of approximate solution of the electronic Schrodinger equation does the use of the energy functional correspond?” An answer to this question establishes the use of the functional as a well-defined approximation, rather than simply a convenient guess. The approximate solution of the Schrodinger equation usually corresponds roughly to the perturbative treatment of several types of electron-ion interactions, or to the neglect (to varying extends) of effects due to fine structure in the electronic density of states. Information of this type gives one some u priori insight into what types of problems can be treated by a given type of energy functional. (2) Given a particular atomic rearrangement, what type and order of energy functional describes the energy associated with this rearrangement? Obviously, given almost any single atomic rearrangement, one can find many interatomic potentials and functionals which will reproduce its energy perfectly. However, this is only useful if the chosen description accurately treats a sufficiently large category of other energy changes as %ke articles by R. Haydock (p. 216) and M. J . Kelley (p. 296), in Ref. 12. Simplified versions of these methods are described in D. M. Esterling, D. K. Som, A . K. Chatterjee, and I. M. Boswarva, J . Phys. F 17, 87 (1987).

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A . E. CARLSSON

well. Three such categories which play a major role in our analysis are the following: (a) Broken-bond energies. These are typically the energies associated with the formation of defects, such as vacancies and surfaces. For most defects in metals, they can be fairly accurately described with pair functionals or with effective pair potentials derived from the properties of the perfect crystal. (b) Relaxation geometries at broken-bond defects. This category includes, for example, the changes in the interlayer spacing that occur near surfaces. In many simple models, the relaxations vanish identically at the pair potential level, but are obtained more accurately by pair functionals. The improvement can also be understood via effective pair potentials depending on the local environment. (c) Crystal structure energy differences. These correspond to processes in which the density of other atoms around a particular atom does not change appreciably, but the bond angles and dihedral angles to change significantly. For example, in transitions between the fcc and bcc structures, volume changes are typically very small, but the bond angles change by more than 15%. In transition metals and semiconductors the treatment of such structural energy differences requires the explicit inclusion of angular forces. The organization of the remainder of this review is as follows: Section 2 motivates the subsequent discussion by pointing out the difficulties associated with simple interatomic potential descriptions of the form (1.1). The most serious difficulty is that the bare interatomic pair potential, defined as the interaction energy between two atoms in a vacuum, is not even a good starting point for understanding the bonding energetics of the solid. To the extent that interatomic potentials can be used at all, they must be viewed as effective potentials which are useful only in describing a limited category of the possible atomic configurations, such as those discussed above. Part 11, introduced in Section 3, describes the theoretical analysis underlying the types of energy functionals defined above. This part is focused almost entirely on transition metals, reflecting the weight of the existing literature. Section 4 discusses a class of effective interatomic potentials obtained by perturbative treatment of various interactions between the electrons and the ions. These potentials treat structural rearrangements at constant volume but often fail to describe broken-

BEYOND PAIR POTENTIALS

11

bond energies. Section 5 describes the derivation of pair functionals from tight-binding models of the electronic structure, and from an “embeddedatom” picture. Effective interatomic potentials are also derived from these functionals. Section 6 treats the derivation of cluster functionals from tight-binding models. These provide a rudimentary description of some features of the shape of the electronic density of states, and thereby improve on pair functionals in the description of crystal structure energy differences. Section 7 analyzes the connection between the “constantvolume” potentials and the effective potentials derived from the pair and cluster functionals. An explicit linear transformation between the two series of potentials is obtained. Section 8 discusses the precise definition of effective interatomic potentials. Expressions for the potentials are given as statistical correlation functions involving the exact binding energy of the system, rather than approximate forms based on simplified models of the electronic structure. Section 9 elucidates the preceding analysis by describing the analogy between effective interatomic potentials, and the simplified Ising-type models used to treat solid solutions. Part I11 describes several schemes which fit simplified energy functionals to properties obtained from experiment or ab initio calculations, and the application of these schemes to illustrative bulk and defect properties. Section 10 briefly discusses the rationale for using such fitted schemes. Section 11 treats the application of fitted pair functionals to vacancy and surface properties of transition metals. Substantial improvement over pair potential results is seen, particularly for relaxation effects. Section 12 discusses the forms of several fitted cluster potentials and functionals for Si. Applications to surface reconstruction, small clusters, and amorphous Si are discussed. Finally, Part IV summarizes the preceding discussion, and points out gaps in our present understanding of interatomic potentials and related energy functionals.

2. DIFFICULTIES WITH GLOBAL INTERATOMIC POTENTIAL DESCRIPTIONS OF TRANSITION METALSAND SEMICONDUCTORS In this section we argue that no single, environmentally independent set of low-order interatomic potentials can describe all of the properties of a transition metal or a semiconductor. By a “global” description, we mean one in which the entire configurational energy is contained in the interatomic potentials, as in Eq. (1.1). Several empirical observations demonstrate the well-known inaccuracy of global pair potential descriptions. For example, the bare potential between two atoms in vacuum,

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A. E. CARLSSON

Ni

10

0

t 0

2

4

6

8 1 0 1 2 1 4

COORDINATION 2

FIG.3. (a) Configurational energy E, of Ni vs. coordination number 2. Dotted and dashed curves denote fits to E,, proportional to Z and fi,respectively. (b) Cohesive energy of Si vs. coordination number, taken from J. Tersoff, Phys. Rev. B 37, 6991 (1988).

corresponding to a dimer molecule, is a very poor starting point for studying the solid state. This point is illustrated in Fig. 3, which shows the measured configurational energy of Ni and Si (relative to isolated atoms) versus coordination number. The points in Fig. 3a are obtained from the cohesive energy26of bulk fcc Ni (2 = 12), the vacancy formation energy2’ of Ni (2 = l l ) , and the binding energy of the Ni2 dime?’ (2 = 1). The 26C. Kittel, “Introduction to Solid State Physics,” sixth edition, p. 55. Wiley, New York, 1986. 27A. Seeger and H. Mehrer, in “Vacancies and Interstitials in Metals” (A. Seeger, D. Schumacher, W. Schilling, and J. Diehl, eds.), p. 1. Amsterdam, North-Holland, 1970. 28‘‘CRC Handbook of Chemistry and Physics” (R. C. Weast, M. J . Astle, and W. H. Beyer, eds.), p. F-172. CRC Press, Boca Raton, Florida, 1987.

BEYOND PAIR POTENTIALS

13

points in Fig. 3b are obtained29 from the cohesive energies of the fcc (2 = 12), simple cubic (Z = 6), diamond (Z = 4), and graphitic structures of Si, and the binding energy of the Si, dimer (Z = 1). A description via radial pair potentials would have the form

where E, is the solid’s configurational energy and V2is the pair potential. If only nearest-neighbor interactions are included, and bond lengths are fixed, then (2.1) predicts a linear dependence of E , on the coordination number Z. As seen in the figure, the observed dependence in both cases is far from linear, and shows that the configurational energy per bond, or bond strength, decreases with increasing Z. The inclusion of the bond length difference between the molecule and the solid does not appreciably change the results, and the inclusion of second and further neighbor contributions actually increases the discrepancy. The measured values of E, for Ni are, in fact, fitted much better by a 2”’ dependence than by a linear dependence. This result seems surprising at first, but is consistent with the simple theoretical models to be developed in Part 11. Such large discrepancies between the behavior expected from a pair potential model and the observed behavior are characteristic of both metallic and covalent bonding. For example, in the absence of lattice relaxation effects, the magnitude of the vacancy formation energy in a pair potential model must equal that of the cohesive energy per atom.3” In contrast, all measured vacancy formation energies in metals are less than half of the cohesive energy per atom. It is unlikely, in general, that the lattice relaxation effects can account for much more than 15% of the discrepancy. In addition, radial pair potential models for cubic materials at zero pressure must rigorously satisfy the Cauchy relations31 Clz= Ca, which are often violated by 30% or more in both transition metals and semiconductors. Finally, as mentioned in Section 1, the typically open structures of semiconductors are unlikely to be stabilized by radial pair potentials alone. It is thus impossible to model all aspects of any transition metal or semiconductor with a single pair potential. However, one may still hope to model certain classes of properties of these materials with pair 29J. Tersoff, Phys. Rev. B 37, 6991 (1988). ”C. P. Flynn, “Point Defects and Diffusion,” p. 6. Clarendon, Oxford, 1972. 31“Extended”Cauchy relations can also be derived from radial pair potentials. See R. A. Johnson, Phys. Rev. B 6, 2094 (1972); C. S. G. Cousins and J . W. Martin, . I Phys. . FS, 2279 (1978).

14

A . E. CARLSSON

FIG. 4. Effective pair potentials for Cu derived from various properties. “Molecule” curve taken from N. h u n d , R. F. Barrow, W. G. Richards, and D. N. Travis, Ark. Fys. 30, 171 (1965); “equation of state” curve taken from A. E. Carlsson, C. D. Gelatt, and H. Ehrenreich, Philos. Mag. A 41, 241 (1980); “defects” curve taken from R. A. Johnson, Radiaf. Efi. 2, 1 (1969); “phonons” curve taken from D. M. Esterling and A. Swaroop, Phys. Sfatus. Solidi B %, 401 (1979). R,, denotes nearest-neighbor distance in fcc structure at equilibrium.

potentials. It is instructive to compare several such potentials for Cu (cf. Fig. 4). Here, ‘‘molecule’’ denotes the Cu2 binding energy curve,32 the potential labeled “equation of state” is obtained33 by forcing it to fit a quantum-mechanically obtained equation of state for the fcc structure, ranging from highly compressed lattice constants to isolated atoms, the “defects” curve34 is obtained from the vacancy formation energy and other defect properties, and “phonon” denotes a fit3s to the phonon spectrum. None of these potentials are similar to each other. The “defect,” “equation of state ,” and “molecule” potentials all measure broken-bond energies but have very different depths and shapes. This difference appears to be a consequence of screening effects, which allow the effects of a small number of broken bonds to be “healed” by the strengthening of other bonds. These effects are present in the “defects” potential, but are less important in the “equation of state” potential 32N. Aslund, R. F. Barrow, W. G. Richards, and D. N. Travis, Ark. Fys. 30, 171 (1965). 33A.E. Carlsson, C. D. Gelatt, and H. Ehrenreich, Philos. Mag. 41, 241 (1980). %R. A. Johnson, Radiaf. Efl.2, 1 (1969). 3sD. M. Esterling and A. Swaroop, Phys. Sfafw Solidi %, 401 (1979).

BEYOND PAIR POTENTIALS

15

because all the bonds are broken at the same time. Furthermore, the “phonon” potential is at least five times smaller than the other three, and displays Iong-ranged oscillations. Thus pair potentials which describe different types of properties of the same metal can in some cases bear almost no resemblance to each other. An obvious cure to this problem would seem to be the inclusion of triplet, and possibly higher-order, terms. One could, for example, define the pair potential to be the interaction between two atoms in vacuum (cf. “molecule” curve, Fig. 4). Then the triplet potential could be obtained by calculating the energy of a three-atom system, and subtracting off the two-body contributions to obtain the three-body potential. The four-body and higher-order interactions could be obtained in a similar fashion. Model calculation^,^^ however, suggest that this approach does not yield a useful interatomic potential series. These calculations utilize a nearestneighbor tight-binding description of the electronic structure of a metal with one electron per site to evaluate the utility of the above scheme in calculating the cohesive properties of a model fcc lattice. The exact configurational energy per atom E, (relative to isolated atoms), obtained via k-space treatment, is -2.62 (hl, where h is the interatomic coupling strength. The pair potential at the nearest-neighbor distance is simply -2 lhl, while the triplet potentials are 3 (hi and 1.18 1/1 , according to whether the triangle has three or two nearest-neighbor bonds. The pair potential contribution to E, in the fcc structure is thus -12 lhl, while the triplet contribution is 56.8 Ihl. Inclusion of both pair and triplet terms then results in a configurational energy having the wrong sign and a magnitude fifteen times too large. Even if inclusion of still higher-order potentials results in a convergent series, this series is probably not very useful, since calculations with very high-order potentials are timeconsuming and unwieldy. In diamond-structure semiconductors potentials obtained in this fashion may be a better starting point. The binding energy of the Si, dimer is 3.4 eV,28 which yields roughly 7 eV as the pair estimate of the cohesive energy per atom in the diamond structure. The observed cohesive energy is 4.6 eV,26 so that the overestimate resulting from the pair approximation is only 30-40%. This overestimate could easily be corrected by three-body terms, without having unduly large cancellation effects. Nevertheless, the pair estimate of the fcc structure cohesive energy (over 15 eV) is much too high in magnitude, and it is unlikely that a single set of practically useful low-order interatomic potentials can 36A. E. Carlsson and N. W. Ashcroft, Phys. Rev. B 27,2101 (1983).

16

A . E. CARLSSON

accurately describe both the diamond and fcc structures, and the dimer molecule.

II. Theoretical Justification and Physical Properties of Interatomic Potentials and Functionals 3. INTRODUCTION Because of the vast simplifications obtained by interatomic potentials and related energy functionals, a very large body of theoretical analysis has been devoted to the derivation of such descriptions of transition metals from ab initio starting points via systematic sequences of approximations. Comparatively little progress has been made in deriving energy functionals for semiconductors, although many parametrized fitting schemes have been proposed (cf. Sec. 12). Therefore, this part is mainly focused on transition metals. As mentioned in Section 1, derivations of simplified energy functionals for transition metals have generally been based3’ on one of two assumptions: (I) that some of the electron-ion interactions are sufficiently weak to be accurately treated in low-order perturbation theory, or (2) that the effects of fine structure in the electronic density of states can be safely neglected. In the former case (Section 4) one typically uses a uniform electron gas as a starting point, and obtains the interatomic potentials in terms of the ionic pseudopotential, the couplings between the atomic d orbitals and the electron gas, and the response function of the electron gas. As we shall see, such potentials can describe only rearrangements at constant average atomic volume. “Constant-volume’’ potentials have provided useful insights into chemical trends in structural energy differences, phonon spectra, liquid structure, and many other properties not involving large atomic or electronic density changes. In treating defect problems involving large density changes, approaches using the second assumption (Sections 5 and 6) are more applicable. Such approaches are typically based on tight-binding models or local atomic embedding functions, and naturally give rise to pair and cluster functionals which do not have the form of interatomic ”Another underlying physical picture that has been used to generate interatomic potentials is the “bond charge” model. It has been applied to both transition metals [C. C. Matthai, P. J. Grout, and N. H. March, J . Phys. Chern. Solids 42, 317 (1981)] and semiconductors [Ref. 17, Chap. 9; J. C. Phillips, Phys. Rev. 166, 832 (1968); R. M. Martin, Phys. Rev. 186, 871 (1969)l. We do not treat these methods here because they have not been developed to the point that they can be used in atomistic simulations.

BEYOND PAIR POTENTIALS

17

potentials. These describe broken-bond energies well and, if four-body terms are included, obtain some of the observed chemical trends in crystal-structure energy differences. In systems without gross variations in the local environment the pair functionals can be approximated by effective, state-dependent interatomic potentials, which are very different in shape from the “constant-volume’’ potentials. Although these two approaches have essentially complementary ranges of applicability, they can be related to each other by an explicit linear transformation (Section 7). This transformation is connected with a rigorous definition of interatomic potentials (Section 8), which is expressed in terms of the exact energy of the system rather than approximate forms based on particular simplified models of the electronic structure. The “problem dependence” of the types of effective potentials discussed here is also seen in the simpler problem of describing alloy ordering energies by effective Ising parameters. Explicit description of the analogy (Section 9) casts a useful light on the results obtained in the earlier sections.

4. STRUCTURAL ENERGIES: “CONSTANT-VOLUME” INTERATOMIC POTENTIALS It was realized early in the effort to calculate properties of metals that the band structures of simple metals are generally quite free-electronlike. This realization led to a description of the electron-ion interaction in terms of pseudopotentials considerably weaker than the true ionic p~tential.~’ The valence-band structure obtained by a pseudopotential is very close to that obtained by the true potential, but the core states are absent. If the pseudopotential V,, is sufficiently weak, the dimensionless ratio V , J E ~ (where eFis the Fermi energy of the electron gas) constitutes a natural expansion parameter for obtaining a rapidly convergent series of interatomic potentials. One starts with a uniform electron gas having a density equal to the average valence electron density of the system of interest. One then expands in powers of the non uniform part of the ionic p ~ e u d o p o t e n t i a l . ~To , ~first ~ ~ order one obtains only density-dependent one-body terms describing the interaction of the uniform electron gas with the individual ionic pseudopotentials. In second order one obtains terms which describe the interaction of the pseudopotential on one ion 38W. A . Harrison, “Pseudopotentials in the Theory of Metals,” Benjamin, New York, 1966. 39Ref. 38, Section 2.7. 40N. W. Ashcroft, in Ref. 23, p. 91. 41M.W. Finnis, J . Phys. F4, 1645 (1974).

18

A. E. CARLSSON

x--o

FIG. 5. Schematic illustration of electronic density in a uniform system (A), and an inhomogeneous system consisting of slabs (B). Position coordinate is n ; ii,! denotes globally averaged electronic density.

with the electronic charge cloud induced by another. The resulting effective pair potentials4’ have the form

where Q is the ionic charge, V,,(q) is the Fourier transform of the ionic potential, x is the electron-gas response function, and Gel is the average density of conduction electrons. The first term gives the direct repulsive ion-ion interaction, while the second term contains the “induced-charge’’ contributions; x(q, Gel)Vps(q) is by definition the electronic charge density induced to linear order by the ionic pseudopotential. The response function x depends on the density of valence electrons, with the induced charge cloud typically becoming shorter-ranged at higher densities. Thus Vgff is density dependent as well. The nomenclature for potentials of this type results from the constraint that they can treat only atomic rearrangements occurring at constant average volume per atom. For other types of rearrangements, terms involving changes in x with electronic density must be included as well, or the electron-gas energy must be recalculated. For the practical purpose of obtaining a rapidly convergent interatomic potential series, the category of “constant-volume’’ problems is actually even more restrictive than constant average volume per atom: volume must be preserved on a more local scale as well. For example, consider the two systems shown in Fig. 5. System A is a uniform crystal of a metal at its equilibrium density, while system B consists of uniform slabs at twice this density. The spacing of the slabs is equal to their thickness, so that both systems have the same globally averaged electron density fiel. However, if the slab thickness is 42Some higher-order contributions can also be included in pair schemes such as these. See M. Rasolt and R. Taylor, Phys. Rev. B 11, 2717 (1975).

BEYOND PAIR POTENTIALS

19

large in comparison with the electronic screening length, a pair potential evaluated at the density ii,, does not have much relevance to the properties of system B. For most of the pairs of interacting atoms inside a slab, the appropriate density is twice ti,,. In general, one expects43 the pair potentials in strongly inhomogeneous systems to depend on a local volume per atom. At present there is no universally accepted defiinition of this local volume. However, it should be determined mainly by contributions from a region of size comparable to the electronic screening length. If density variations occur on a length scale greater than this size, the difference between the local and average atomic volume will be important. The range of applicability of constant-volume potentials, which are based on the average atomic volume, thus excludes systems having large density variations on length scales exceeding the electronic screening length. The boundaries of this range can be roughly mapped out as follows. Crystal-structure energy differences (at fixed average volume), phonon spectra, and the structure of liquids are “constant-volume’’ problems and are in the range of applicability. The formation of large voids, and surfaces, is not. Vacancies are intermediate cases,44 falling into the constant-volume category for some low-valence simple metals. In general, the constant-volume category is smaller the higher the valence of the metal under consideration. “Constant-volume’’ pair potentials in simple metals, such as that for Al shown in Fig. 6, typically have the form of a screened Coulomb repulsion with oscillations superimposed. At large separations these are the Friedel or Ruderman-Kittel oscillation^,^^ due to the sharpness of the electron Fermi surface. These are the asymptotic form of Eq. (4.1). There is, in general, no deep well in the potential corresponding to our intuitive notion of a chemical bond. This surface and cohesive energies are not accurately obtained by constant-volume pair potentials, and vacancy formation energies cannot be naively described in terms of broken-bond contributions. However, a variety of other types of properties of simple metals, including phonon spectra, structural energy differences, the structure of liquids, and defect migration barriers, have been successfully treated by constant-volume potential^.^^^^^'^"^^^^ Because our focus is on transition metals and semiconductors, we do not describe these applications here. 43R.J. Harrison, Surf. Sci. 144, 215 (1984). “Even for Al, generally regarded as a very free-electron-like metal, vacancy formation energies are obtained poorly by constant-volume potentials. See M. Manninen, P. Jena, R. M. Nieminen, and J. K. Lee, Phys. Rev. B 24,7507 (1981). 45Ref. 1, Chap. 17.

20

A. E. CARLSSON

2 004

A1

r s = 2 069

v

a =4033A

2 003

e-

y

002

-001-

Rm

FIG. 6. "Constant-volume" effective pair potential for Al, taken from L. Dagens, M. Rasolt, and R. Taylor, Phys. Rev. B 11,2726 (1975). R,, denotes nearest-neighbor distance in fcc structure, r, is radius of sphere containing the volume per electron (in units of the Bohr radius), and a is lattice constant.

Pseudopotentials in transition metals and semiconductors are unfortunately not weak. This is apparent from their non-free-electron-like band structures, which contain relatively narrow d bands or gaps in the density of states. However, the "constant-volume'' concept has been generalized to d-band metals via the perturbative treatment& of various couplings involving the d orbitals. Initially, the s-d coupling K d was used as a small ~ a r a m e t e r , ~ resulting ~-~' in a perturbation expansion in powers of V,d/(&F - Ed), where E~ is the Fermi level and Ed is the center of the d band. Such an expansion is fairly well behaved in noble metals, where Ed is well below E ~ It. cannot, however, accurately treat transition metals with partly filled d bands, since E~ and Ed can be quite close. In these materials, it is necessary to explicitly include the strong bonding contribution arising from the preferential population of the lower energy &W. A. Hamson, Phys. Rev. 181, 1036 (1969).

47J. A. Moriarty, Phys. Rev. B 5 , 2066 (1972). 48J. A. Moriarty, Phys. Rev. B 16, 2537 (1977). 49J. A . Moriarty, Phys. Rev. B 26, 1754 (1982).

ML. Dagens, J . Phys. F 6 , 1801 (1976). 5'L. Dagens, Phys. Status Solidi 84, 311 (1977). 52L. Dagens, 1. Phys. F7, 1167 (1977). 53J. C. Upadhyaya and L.Dagens, J . Phys. F 8, L21 (1978). 54J. C. Upadhyaya and L. Dagens, J . Phys. F 9, 2177 (1979). 55N.Q. Lam, L. Dagens, and N. V. Doan, J . Phys. F 13,2503 (1983).

BEYOND PAIR POTENTIALS

21

part of the d-band complex. A simplified treatment56includes this via the use of a rectangular model d-band density of states. This form for the density of states can probably not account for transition-metal structural energy differences (cf. Section 6), but has been successfully applied” to the structure of liquid metals. A more quantitative treatment5x359 uses as a starting point an electron gas containing d impurities which are coupled to the electron gas but not to each other. A t this stage one has only volume-dependent one-body terms in the energy. One then expands in powers of the interatomic d-d couplings (in addition to the usual s-p pseudopotential). Thus, even if E~ is very close to E d , the relevant dimensionless parameter is the ratio of the interatomic d-d coupling strengths to the intra-atomic d-band width due to s-d hybridization. Unlike & / ( E ~ - E d ) ,this ratio is not singular. Such “constant-volume’’ potentialssx for V are shown in Fig. 7. The pair potential is compared to those for Ca and Cu, which have nearly empty and nearly filled d bands, respectively. The greater depth of the V pair potential is striking. It results from the strong interatomic d-d interaction: nearby placement of atomic d shells results in an increased d-band width and thus increased bonding (cf. Section 5). The magnitude of the three-body terms is also quite large; their strong angular dependence is due to that of the d orbitals. There are large cancellations between the two- and three-body terms; for example, in the energy difference between bcc and fcc V, the three-body contribution cancels roughly 80% of the two-body contribution. The shape of the phonon dispersion relation is also qualitatively changed by the three-body terms. They remove spurious instabilities in the long-wavelength part of the [lo01 longitudinal branch and also yield a more complex behavior at shorter wavelengths. Potentials of this type have not yet been used in atomistic simulation studies. However, the calibrations which have been performed” indicate that properties in the “constant-volume’’ category are treated quite well. Calculated phonon spectra for V and Cr are in excellent agreement with experimental measurements; the bcc-fcc energy difference for V has the correct sign and a magnitude consistent with the fairly uncertain experimental value. However, to treat problems with large density variations, such as the broken-bond energies associated with

?. M. Wills and W. A . Harrison, Phys. Rev. B 28, 4363 (1983); Phys. Rev. B 29, 5486 ( 1984). ”C. Hausleitner and J. Hafner, J . Phys. F 18, 1025 (1988). ”J. A. Moriarty, Phys. Rev. Lett. 55, 1502 (1985). ’9J. A. Moriarty, Phys. Rev. B 38, 3199 (1988).

22

A . E. CARLSSON

30

60

90

120

150

180”

angle 8 FIG. 7. “Constant-volume” pair potentials for Ca, Cu and V, and triplet potential for V, taken from J. A . Moriarty, Phys. Rev. Lett. 55, 1502 (1985). R , is atomic-sphere radius. Numbers of two-ion neighbors and three-ion triangles in fcc and bcc structures is indicated.

BEYOND PAIR POTENTIALS

23

surface formation, it will probably be necessary to generalize this scheme by using an inhomogeneous system as a starting point.

5. BROKEN BONDS:PAIRFUNCTIONAL APPROACHES These approaches focus on the effects of large atomic and electronic density variations at defects. Large reductions in these densities would naively be associated with broken bonds. Such effects are often treated poorly by constant-volume potentials. The theme underlying pair functional treatments is the description of the local electronic environment of a particular atom in terms of a single parameter. This parameter is given as an additive function involving short-ranged radial contributions from the neighboring atoms. The configurational energy associated with a particular site is then given simply as a readily evaluated function of the environmental parameter. Because of the pair form of the parameters describing the local environment, these methods provide computational speed essentially equivalent to that of pair potentials. As we shall see, however, the accuracy is considerably improved. a. Derivation of Functionals Justification for pair functional approaches come from two distinct treatments of the electron bonding energy, which use the local electron valence bandwidth and the local “background” electron density as environmental parameters. (1) Tight-binding analysis12,-2 This treatment is based on two assumptions: (i) That the attractive part of the solid’s bonding energy is dominated by the broadening of the partly filled valence shells of the atoms into bands when the solid is formed. (ii) That the width of the projection of the valence band electronic density of states on a particular site can be accurately given in terms of radial contributions from the neighboring atoms. Assumption (i) appears to be quite well justified for transition metals. A large body of empirical, analytical, and numerically quantitative

@‘F. Cyrot-Lackmann, J. Phys. Chem. Solids 29, 1235 (1968). 6’F. Cyrot-Lackmann, Surf. Sci. 15, 535 (1969). 62F. Ducastelle, J . Phys. (Paris)31, 1055 (1970).

24

A . E. CARLSSON ATOM

I BULK

I

SURFACE OR VACANCY

> c3 [L

W

z W

DENSITY OF STATES

FIG.8. Sketch of changes in electronic density of states occurring upon formation of solid, and subsequent formation of surface or vacancy. Arrows indicate lowering of electronic energy (exothermic) in solid and rise when surface or vacancy is formed.

analy~is~--~ has shown that, for transition metals with partly filled d shells, the dominant contribution to cohesion is due to the formation of the d band from the atomic d shells. As indicated in Fig. 8, the electrons occupying the atomic d levels can move to levels in the solid having, on the average, bonding character and lower energy. A description of the bonding entirely in terms of the d-band width and filling reproduces the observed parabolic band-filling dependence of the cohesive energy closely,63 and, except for nearly empty or nearly filled d bands, obtains the magnitude of the cohesive energy fairly accurately as well.@ A similar band-broadening effect is also present in semiconductors. Here the primary driving force behind the bonding is the formation of filled bonding levels and empty antibonding levels from atomic sp3 hybrid orbitals.*’YM If one treats the bonding and antibonding levels together as one complex, the splitting of the bonding and antibonding levels simply corresponds to the broadening of this complex. However, bandwidth effects cannot explain crystal structure preferences in transition metals and semiconductors. These require more sophisticated analyses treating the band shape, as well as its width. As we shall see in Section 6, such analyses automatically include angular forces. To justify assumption (ii) one needs a particular model for the electronic structure. The bandwidth effects are conveniently analyzed within a tight-binding model. Such models are useful for the relatively localized d orbitals of transition metals.”*63 Neglecting, for simplicity, Friedel, in “The Physics of Metals” (J. Ziman, ed.), Chap. 8. Cambridge University Press, 1969. @C. D. Gelatt, Jr., H. Ehrenreich, and R. E. Watson, Phys. Rev. B 15, 1613 (1977). “D. G. Pettifor, J . Phys. F 8 , 219 (1978). ?I. C. Phillips, “Bonds and Bonds in Semiconductors.” Academic, New York, 1973.

h3J.

BEYOND PAIR POTENTIALS

25

orbital degeneracy, one considers a one-electron Hamiltonian of the form

here the kets li) denote localized orbitals centered on sites {Ri}, for simplicity assumed orthogonal. For each electron spin the li) form a single band. The couplings h, = h(lRi - Rjl) = h(R,), where R , = lRi-R,l, are assumed to depend only on the interatomic separation. Electron-electron Coulomb interactions are ignored. The energy zero is chosen such that the i = j terms vanish, which implies that the contribution of H to the total energy vanishes for atoms too far apart to interact. Given the Hamiltonian (5.1), one aims to evaluate the total electronic band energy, which is given as the sum of the energies of the occupied one-electron states:

U g T= 2

\

EF

Ep(E) dE,

J-CC

where p ( E ) is the electron density of states, and the factor of 2 accounts for spin degeneracy. The subscript “TB” denotes “tight-binding.” U g T is an attractive contribution to the configurational energy, containing the “band-broadening’’ physics discussed above. It is convenient to divide (5.2) into individual atomic contributions:

I

EF

U E T=

U& = 1

I

2

Epi(E) dE,

(5.3)

-m

where

are the is the projected density of states on site i, and the IT), eigenfunctions of H . To obtain p , ( E ) exactly, it is in principle necessary to know the positions of all the atoms in the crystal. Furthermore, p , ( E ) is a very complicated functional of these positions. However, for our purposes we often do not need to calculate the detailed structure of p , ( E ) . To obtain an approximate evaluation of quantities such as U& which involves integrals over p , ( E ) , we need only information about its width and gross features of its shape. This information is conveniently summarized in the

26

A . E. CARLSSON

moments of p , ( E ) , defined by

(5.4) The crucial observation, which allows a description of simplicity comparable to that of interatomic potentials, is that these moments are rigorously determined by the local environment. One has the exact relationsm

hr]h]khklhlr

9

i.kJ

with similar expressions for p i as a sum of n-atom paths. (The lower-order moments provide no useful information about the structure; ph = 1 for all i, and pi = 0 since the i = j terms in H vanish by assumption.) Thus if one has an approximate expression for U& in terms of the first few p i , the electronic band energy can be evaluated with essentially the same machinery used to evaluate interatomic potential sums. Although exact evaluation of UT, requires the values of all the moments on site i, a great deal can already be learned from a description based only on p i . This moment provides a measure of the squared valence-band width and thus sets a basic energy scale for the problem; as will be discussed in Section 6, the higher moments describe the band shape. Thus a description using only p i assumes that the effects of the structure of p , ( E ) can be safely ignored. Since U;, has units of energy and p i has units of (energy)2, one has61,62767

This part of the configurational energy is thus given as a pair functional; in the notation of Eq. (1.4), g2(Rj, Ri) = h;. A simple illustration is 67T0obtain Eq. (5.6) for arbitrary band fillings, it is necessary to assume a fixed electronic charge per atom. See G . J . Ackland, M. W. Finnis, and V . Vitek, 1. Phys. F 18, L153 (1988).

BEYOND PAIR POTENTIALS

27

provided by a rectangular band,h3368having 1 p,(E) = -

2wi

IEls

w.

From Eq. (5.4) one readily obtains p i = 5W:. If one considers a model transition-metal d band with Nd electrons per site, Eq. (5.3) yields

This result accounts well for the parabolic shape of transition-metal cohesive energiesh3 when plotted versus ~d . The physics contained in Eq. (5.6) also includes a major contribution to broken-bond defect energies in transition metals. 12m,61 From Eq. (5.5) it is clear that in a nearest-neighbor model with fixed interatomic spacing, p i is simply proportional to the local coordination number. Thus the reduced coordination number at a surface or vacancy will result in a smaller bandwidth. As seen in Fig. 8, this results in the occupied states being shifted to higher energies. The contribution of the electronic band energy to the total energy then rises, as expected from Eq. (5.6). This basic physics is displayed by many defect calculations. For example, Fig. 9 shows surface (solid lines) and bulk (dashed lines) densities of states for (110) and (100) surfaces of a model bcc transition metal, treated with a d-band Hamiltonian. The (110) surface is close-packed and has two broken nearest-neighbor bonds per surface atom; the (100) surface has four broken nearest-neighbor bonds per surface atom, and is quite open. In both cases the transfer of weight from the band edges to the center of the band is readily apparent. The effect is more prominent for the (100) surface, since the coordination number loss is greater and thus the bandwidth reduction is more pronounced. In this case the shape of the band is strongly affected by the surface as well. Such effects are not included in pair functional treatments; we shall see in Section 6 that they can be very important in determining structural preferences. To evaluate the accuracy of the approximate descriptions (5.6) of Z?k, a useful model test case is provided by an unrelaxed vacancy in a simple cubic lattice (cf. Table I). The electronic structure is described by a nearest-neighbor tight-binding model, with one electron per site. The =D. G . Pettifor, Phys. Rev. Lett. 42, 846 (1979).

A. E. CARLSSON

28

FIG.9. Electronic densities of states for (110) surface (a) and (100) surface (b) of model bcc transition metal within tight-binding model. Taken from J. Friedel, Ann. Phys. ( P a r k ) 1, 257 (1976). Dashed lines denote bulk density of states, included for comparison.

TABLE I. APPROXIMATEAND EXACT ELECTRONIC BAND CONTRIBUTIONS TO FORMATIONENERGY FOR UNRELAXED VACANCK IN MODELSIMPLECUBICL A ~ I C E . ENERGIESGIVEN IN UNITS OF ABSOLUTE VALUEOF COHESIVE ENERGYPER ATOM, ICJy-J. ~~

TIGHT-BINDING Pair Functional V;fi Exacta

0.52 0.50 0.48

"G.Allan, Ann. Phys. (Park) 5, 169 (1970).

BEYOND PAIR POTENTIALS

29

electronic band contribution to the vacancy formation energy, assuming the vacant site is at atom i, is given by

~ f =auk,, z jti

where A U L denotes the change in UqB resulting from the formation of the vacancy. In a second-moment treatment, only the nearest neighbors are affected by the vacancy. Denoting the ideal sixfold coordinated lattice electronic band energy by v,, one obtains UGB= (2)InUR for the vacancy nearest neighbors. Thus

-Ef

I UFdl

- -6[(2)ll2 - 11 = 0.52,

which is within 10% of the exact quantum-mechanical resuk6' The errors obtained by pair functionals in treating the electronic band contribution to broken-bond energies are typically within a factor of two of this magnitude. Thus the bandwidth contribution to these energies dominates the effects of the structure in the density of states. A considerable improvement over simple pair potential calculations is obtained. As mentioned above, for example, a pair potential which obtains the correct value of l.J$" must3' also obtain Ef = I UpAl, which is too large by more than a factor of 2. Equation (5.6) shows why environmentally independent pair potentials describe metals and semiconductors poorly. Since p i is proportional to the coordination number Z, it follows'2 that U, a (-Z112). This is consistent with the results for Ni shown in Fig. 3. The Z'12 dependence of U, leads immediately to a bond strength proportional to 2-"2. Thus increased coordination leads to a weaker bond. The bond weakening may be viewed as a frustration effect due to the Pauli exclusion For example, in the case of a dimer molecule with one electron per atom, both electrons can be in the ground-state wave function, which has bonding character with respect to the center of the molecule. However, in the solid environment, the exclusion principle forces most of the electrons to occupy states having energies above the ground-state energy. These have some antibonding character with respect to the bond centers, weakening the bonds relative to the bond strength of the dimer. In fact, 69G. Allan, Ann. Phys. (Park) 5, 169 (1970). '@Theweakening of the bond strength with increasing coordination number has also been discussed by G . C. Abell, Phys. Rev. B 31, 6184 (1985).

30

A. E. CARLSSON

for plausible models involving only the one-electron energy it has been proved36that the sum of all the pair energies in a solid (evaluated using the dimer bond strength) provides a lower bound to the total energy of the system. Thus the system is frustrated in the sense that not all bonds can be completely satisfied at the same time. In order to treat defect problems with tight-binding pair functionals, it is necessary to augment the attractive electronic band energy Urn by a repulsive force which prevents the lattice from collapsing. The physics underlying this force is simply the Pauli repulsion resulting from the compression of the valence electron gas. However, this physics has generally not been explicitly incorporated into the expressions which have been used. Instead, simple pair potential descriptions of the repulsion energy have usually been employed. One thus obtains an energy functional of the form

Justification for expressing E , as the sum of a pair potential term and a tight-binding electronic band term has recently been given by the “tight-binding bond” This model uses the quantummechanical ground-state variational principle to justify a treatment based on frozen atomic charge densities. It is hoped that future pair and cluster functional treatments will incorporate the basic physics of this model. We note that a linear dependence of U, on p i , U, = const X pi, would by Eq. (5.5) result in

E, =d

V2(Rii)+ const x ij,

C h;. i.i

(5.9)

Thus the configurational energy would be described completely in terms of pair potentials. For metals, the square-root behavior given in Eq. (5.6) is more appropriate than the linear dependence, and nonpair terms result from the nonlinear behavior of U , . (2) “Embedded-atom” analysis7380 Although the physical picture 71A.P. Sutton, M. W. Finnis, D. G. Pettifor, and Y. Ohta, J. Phys. C21, 35 (1988). 72Y. Ohta, M. W. Finnis, D. G. Pettifor, and A. P. Sutton, J. Phys. F 17, L273 (1987). 73J. K. N ~ r s k o vand N. D. Lang, Phys. Rev. B 21, 2131 (1980). “M. Stott and E. Zaremba, Phys. Rev. B 22, 1564 (1980). 75M. J. Puska, R. M. Nieminen, and M. Manninen, Phys. Rev. B 24, 3037 (1982). 76J. K. N ~ r s k o v Phys. , Rev. 26, 2875 (1982). 77M. Manninen, Phys. Rev. B 34, 8486 (1986). 78K. W. Jacobsen, J. K. Nerskov, and M. J. Puska, Phys. Rev. B 35,7423 (1987). 79M. S. Daw and M. I. Baskes, Phys. Rev. Lett. 50, 1285 (1983). 8oM. S. Daw and M. I. Baskes, Phys. Rev. B 29,6443 (1984).

31

BEYOND PAIR POTENTIALS

TABLE 11. PARALLELS BETWEEN TIGHT-BINDING AND EMUEDDED-ATOM PAIR ON UT, AND (IEA DENOTE DERIVATIVES FUNCTIONALS. PRIMES TIGHTBINDING

EMBEDDED-ATOM ~

~~~~

Environmental Parameter

Squared local electron bandwidth &

Local electron density n,

Pair Assumption

&=

n; =

Functional form of u Effective Pair Potential Source of Nonpair Terms Higher-Order Descriptions

h (lR; - RJOZ J

u,.,= -(const) V, + 2hZU!,.,

x (pL;)”*

U,

c

n,t(lR, - RJI)

J

= numerical

function

v* + 2n,, UkA

underlying this approach is quite distinct from the tight-binding picture, there are strong parallels’ between the two approaches. These are summarized in Table 11. In the “embedded-atom’’ treatment one imagines the solid being assembled, atom by atom. The basic energy parameter is the bonding energy gained by embedding an atom in the background electronic charge density n(r) due to all of the other atoms. This energy is in principle known, provided that n(r) is known everywhere. In this case, the positions of the nuclei are determined by {n(r)}, and the configurational energy is in turn determined by these positions. To obtain simple schemes suitable for atomistic simulations, two simplifying assumptions closely analogous to those of the tightbinding approach are typically made: (i) That the bonding energy associated with a particular atom is determined by the local background electron density, due to the other atoms, at the site of the embedded atom. The background density can be taken either as the density at the nucleus, or as a weighted average over the charge density of the embedded atom. The local background electron density is analogous to the local bandwidth in the tight-binding approach (cf. Table 11). In a nearly-free-electron system, the electron density determines the Fermi energy. This, in turn, is directly related to the electron bandwidth since it measures the difference between the highest occupied states and the bottom of the band. Furthermore, band-theoretic total energy calculations have displayed close correlationsx1 between the *lV. L. Moruzzi, J . F. Janak, and A . R. Williams, “Calculated Electronic Properties of Metals,” Chapter 3 , Pergamon, New York, 1978.

32

A. E. CARLSSON

electron density and properties such as the bulk modulus and cohesive energy, which are also closely connected to the bandwidth. It should be noted that the local assumption for the atomic embedding energy is closely analogous to the “local-density’’ approximationx2 typically used in implementations of density functional theory for electrons. (ii) That the local background electron density can be given as a superposition of radial functions centered on the other Typically this radial function is taken to be the atomic charge density. Variations of this approximation are sometimes used,7x such as taking the background charge density to consist of superposed induced charge densities.

A b initio support for assumption (i) is scarce. Since screening lengths in metals are typically shorter than the interatomic spacing, one might reasonably expect the nonlocal corrections to be small. However, since these are difficult to. evaluate, their smallness has not been explicitly demonstrated. Furthermore, completely a b initio calculations using the local assumption often provide inaccurate results for transition metals unless supplemented by additional electronic terms.7x Nevertheless, one may expect an empirical pair functional form based on the local assumption (i) to provide considerable improvements over pair potential approximations, because the pair functional form has more flexibility than the pair potential form. Assumption (ii) is supported by quantummechanical electronic structure calculation^,^^ which show that for metals even in very inhomogeneous geometries the charge density is often fairly close to that obtained by superposing atomic charge densities. Given these assumptions, the mathematical treatment closely parallels that of the tight-binding approach (cf. Table 11). The expression for the configurational energy corresponding to (5 .S) is Ec = 1

c V2(RI,) + x i,i

&4(4)*

(5.10)

I

where

(5.11) ‘*W. Kohn and L. J . Sham, Phys. Rev. 140, A1133 (1965).

R3Forexample, quantum-mechanicalcalculations for a variety of metal and semiconductor surfaces [J. Tersoff, M. J . Cardillo, and D. R. Hamann, Phys. Rev. B 32, 5044 (1985)] have shown that corrugations in the charge density are well reproduced by superposing atomic charge densities, at least if the surface layer atoms are equivalent. However, asymmetric Si reconstructions are treated poorly [A. Sakai, M. J. Cardillo, and D. R. Hamann (unpublished), quoted by Tersoff et al. (above)].

33

BEYOND PAIR POTENTIALS

is the background electron density for atom i, and n,, is the radial function which determines ni. As in Eq. (5.8), V, is a radial pair potential. The “embedding function” U,,(n,) gives the gain in bonding energy resulting from embedding. the atom i in the background charge density ni. Theoretical UEAcurves for 3d transition metals are shown in Fig. 10. These curves are obtained7’ by immersing a single impurity in a uniform electron gas of density n, and evaluating the total energy changes using standard density functional techniques. An electrostatic correction involving an artificial positive background is also included. The embeddedatom functions display a rapid initial drop and a subsequent upwards curvature. The initial drop may be partially understood via the broadening of the atomic d levels into resonances, by their interaction with the electron gas. This effect is illustrated in Fig. 11, which shows the electronic densities of states entering the calculated values of UEA. The “induced” density of states is the extra density of states due to the impurity. For partly filled d-shell systems, the low-energy part of the resonance is occupied preferentially to the higher-energy part, resulting in a net attractive contribution. The physics of this contribution is closely I

I

0

I

I

I

I

I

I

0.01

1

I

0.02

I

I

1

I

0.03

I

DENSITY (a.u.)

FIG. 10. Embedding energy for transition metal atoms in uniform electron gas vs. background electron density, adapted from K. W. Jacobsen, J. K. Nerskov, and M. J . Puska, Phys. Rev. B 35, 7423 (1987).

34

A. E. CARLSSON I6 0

9V

3

4.0

n

z

00 00

20

40

80

60

ENERGY

100

120

(eV)

FIG. 1 1 . Density of states induced by transition metal impurities in uniform electron gas, adapted from K . W . Jacobsen, J . K. Nbrskov, and M. J. Puska, Phys. Rev. B 35, 7423 (1987). r, denotes radius (in units of the Bohr radius) of sphere having volume equal to the average volume per electron.

parallel to that in the tight-binding approach discussed above, with the band broadening due to the electron gas replacing that due to the orbitals on the surrounding atoms. In the “embedded-atom’’ energies UEA, there are, of course, additional attractive terms, such as contributions due to charge transfer between the electron gas and the d shells. These are not explicitly accounted for in tight-binding pair functionals. The rise in U E A with increasing density may be understood via a two-step of the embedding process. In the first step one separates the valence electrons from the ion core of the embedded atom, and places the electrons and the ion in the electron gas, not allowing the electron gas to respond to the ionic potential. In the second step the electron gas responds by screening the ion. The energy of the first step is simply N,,,,p + E ~ where ~ ~ NVal , is the number of valence electrons of the embedded atom, p is the electron gas chemical potential (measured relative to the electrostatic potential in the electron gas), and qonis the energy require to ionize the valence shell. For large n, p is dominated by the Pauli kinetic energy ( a k , ) and is thus proportional to n2’3. The screening energy in the second step is negative and grows in magnitude with increasing n. However, for large n, the screening energy is roughly proportional to -kv (where kv is the Thomas-Fermi wave vector), which grows only as -n1’6. Therefore the repulsive n2’3contribution from the Nvalpterm dominates, resulting in an upward slope. Thus a simple analysis of UEA leads naturally to a non-linear behavior with a positive curvature. By reasoning completely parallel to that used to obtain Eq.

BEYOND PAIR POTENTIALS

35

(5.9), one shows that a linear behavior of UEA would correspond to a pair potential description. The nonlinear behavior of the calculated UEA results in a description which is more general than that given by pair potentials. Theoretically and empirically obtained embedding functions U,,(n) for Ni are compared with an empirical estimate of U,, in Fig. 12. The empirical curves are obtained by fitting to cohesive and elastic properties of Ni. The empirical U , and UEA curves are in very close agreement with each other. There are, however, considerable differences between the theoretical and empirical UE, curves. The minimum for the theoretical curve occurs at a value of n considerably smaller that the value expected for Ni at its equilibrium lattice constant. At this minimum, the empirical curve is still dropping rapidly with increasing n. The discrepancy is not completely understood. However, one contribution to it comes from the fact that the theoretical U,, does not include the interactions between d shells on different sites, while the effects of these interactions are included in the inputs for the empirical scheme. In fact, to obtain accurate lattice constants for transition metals, it is n e ~ e s s a ry '~

FIG.12. Electronic band energy UTB and embedding energy U,, in three treatments of Ni. no and pi"' denote background electron density, and second moment of electronic density of states, in equilibrium fcc structure. Theoretical U,, curve taken from K. W. Jacobsen, J . K. Narskov, and M. J . Puska, Phys. Rev. B 35, 7423 (1987); empirical U,, curve taken from M. S. Daw and M. I. Baskes, Phys. Rev. B 29, 6443 (1984); empirical U, curve evaluated from parameters given in G . J. Ackland, G . Tichy, V. Vitek, and M. W. Finnis, Philos. Mag. A 56, 735 (1987).

36

A . E. CARLSSON

to supplement the theoretical UEA with an approximate correcfion treating the d-d interaction terms. This has been evaluated7’ using an analysis similar to that which generated Urn. It should also be noted that the pair potentials in Eq. (5.10) can differ in sign according to whether they are obtained t h e ~ r e t i c a l l yor ~ ~empiri~ally.’~ However, the distribution of the bonding energy between the first and second terms of Eq. (5.10) is not unique, since any part of u E A ( n ) that is linear in n can be described by a pair potential. Thus this discrepancy between the theoretical and empirical embedded-atom functionals may be a less serious problem than it appears at first. The range of applicability of pair functionals includes a broad range of broken-bond problems, such as vacancy and surface formation. As we shall see in Section 11, empirical schemes using the cohesive energy as input obtain fairly accurate vacancy formation energies; in contrast, pair potentials based on the cohesive energy yield poor vacancy formation energies. The estimates of relaxation effects obtained by pair functionals improve considerably on the pair potential estimates. In addition, pair functional schemes can obtain6* nonzero values of the Cauchy pressure C12- C U , which is not possible with equilibrium radial pair potentials (cf. Section 2). The primary disadvantage is that one cannot accurately obtain crystal-structure energy differences, or even their chemical trends; as will be seen in Section 6 these properties require four-body or higher-order functionals. In this sense the pair functionals are complementary to the “constant-volume’’ potentials, which treat brokenbond energies poorly but describe crystal-structure energy differences better. These energy differences are mainly due to changes in the shape of the electronic density of states, rather than its width. In the tight-binding picture (cf. Table 11) this shape is related to the higher moments of the electronic density of states; in transition metals and semiconductors these result in angular terms which cannot be included in a radial pair functional. In the embedded-atom approach, structural energies are probably related to terms that are nonlocal in the background electronic charge density. A practical formalism for evaluating these terms has not yet been developed. b. Effective Interatomic Potentials: The Reference Environment Concept

As mentioned in the preceding subsection, a linear dependence of the electronic energy U on either the local squared bandwidth or the %. M. Foiles, M. I. Baskes, and M. S. Daw, Phys. Rev. B 33,7983 (1986); Phys. Rev. B 37, 10378 (1988).

37

BEYOND PAIR POTENTIALS

background electron density would result in a pair potential description. Although the true dependence (cf. Fig. 12) is strongly nonlinear, a linearized expansion provides an approximation of the pair functionals via environmentally dependent pair potentials. Approximations involving triplet and higher-order potentials result from quadratic and higher-order descriptions of E,. Central to this description in terms of interatomic potentials is the notion of a reference environment. This is a particular (possibly hypothetical) environment from which the various local environments do not vary drastically. For a given problem, such a reference environment may not always exist. In a fracture simulation, for example, one can have grossly varying coordination numbers which lead to large fluctuations in both the electron bandwidth and electron density. In such cases, there may be no satisfactory description in terms of low-order interatomic potentials. However, if a suitable reference environment can be found, the configurational energy can be e~ p a n d e d ~ ~ .~ '-'' in powers of the deviations of the local bandwidths or electron densities from their values in the reference environment: '

(We give explicit expressions only for the tight-binding analysis; those for the embedded-atom analysis are completely parallel. The primes on Urn denote derivatives; unless otherwise stated these are evaluated at Y2 = YY'.) If one retains only the linear terms in Eq. (5.12), then using Eqs. (5.5) and (5.8) one has

Since the value of the first sum is unaffected by atomic rearrangements, the energy changes caused by these rearrangements are determined by the effective pair potential

Vqff(Ri,R,; pyf) = V2(Rjj)+ 2h;Uk.

(5.13)

This type of potential contains an attractive contribution from the UiB term, which is countered at short distances by the repulsive contribution 85M.W. Finnis and J . M. Sinclair, Philos. Mag. A 50, 45 (1984); Philos. Mug. A 53, 161 (1986). %S. M. Foiles, Phys. Rev. B 32, 3409 (1985). 87A.E. Carlsson, Phys. Rev. B 32, 4866 (1985).

38

A. E. CARLSSON

from V,. Examples of such effective pair potentials for Cu, derived from both the TB and E A analyses, are shown in Fig. 13, along with a widely employed type of empirical pair potential based on defect properties (also shown in Fig. 4). The reference environment for the TB and E A curves is fcc Cu at its equilibrium lattice constant. All of the potentials have a similar shape: they are short ranged, have a substantial minimum, and grow rapidly at small separations. The agreement between the potentials is encouraging, since the fitting procedures used in the three cases were quite different. However, their shape is very different from 0.4

0.2

-2

v

L

0

O N

>

-0.2

FIG. 13. Effective pair potentials for Cu, obtained from embedded-atom (EA) and tight-binding (TB) treatments, along with empirical potential (“Johnson”) derived from defect properties. R,, denotes nearest-neighbor separation in fcc structure at equilibrium volume. “Johnson,” E A and TB curves use fcc Cu at equilibrium volume as reference environment; TB (expanded) curve uses fcc Cu at volume expanded by 20%. “Johnson” curve taken from R. A. Johnson, Rudiur. Ef. 2, 1 (1969); EA curve taken from S. M. Foiles, Phys. Rev. B 32, 3409 (1985); TB and TB (expanded) curves taken from parameters given in G. J . Ackland, G. Tichy, V. Vitek, and M. W. Finnis, Philos. Mag. A 56, 735 (1987).

BEYOND PAIR POTENTIALS

39

that of the Cu “constant-volume’’ potential in Fig. 7. This difference is to be expected from the preceding theoretical analysis, since the approximations made in their derivations are appropriate for very distinct types of problems. The range of applicability of effective pair potentials of the form (5.13) is a subset of that of the pair functionals from which they are derived. It includes broken-bond energies only for defects in which the fractional change in the coordination number is small, such as vacancy formation. In this case the use of a perfect crystal reference environment is appropriate. For example, in the simple cubic vacancy model discussed above [neglecting the V, term in Eq. (5.13)], one obtainsxs

Ef= -4

X

6 X Vgff=$ IUFJl,

where Vgff is non-zero only at the nearest-neighbor distance. This estimate is within 5% of the full pair functional value (cf. Table I). Thus V;ff may be thought of as an environmentally dependent bond strength. The environmental dependence of the potential is due to the environmental dependence of UkB, or equivalently the curvature of UTR. From Eq. (5.6) one has U&Ba:p;1/2. Since p2 drops under expansion, the magnitude of UkB grows, and the effective pair potential (5.13) for an expanded reference medium is more attractive. This effect is illustrated by the “TB (expanded)” curve in Fig. 13. The reference environment for this curve is fcc Cu at a volume per atom 20% greater than the equilibrium volume. As seen in the figure, the expansion results in an increase of roughly 50% in the well depth. Even if there is no uniform reference environment which treats all of the local environments accurately, it is sometimes useful to define an effective potential associated with the local environment:

This type of potential describes the energy required to break a bond in a system which is already inhomogeneous. It could, for example, treat the formation of a vacancy at a surface or the relaxations at a broken-bond defect (cf. Section 11). A reduced coordination number results in smaller values of p i and pi, and correspondingly increased values of U;,, so that the strength of the remaining bonds increases with dropping coordination number. As shown above, in a nearest-neighbor model with equal bond lengths, p; is proportional to Z j , the local coordination number. Thus U;13a:Z;1‘2, and the attractive part of V P is proportional to (Z;1’2+ 27’”). The coordination-number effects on Vzff are illustrated in Fig. 14

A. E. CARLSSON

40

Bulk

Vacancy “back bond“

Surface Bond

Molecule

FIG. 14. Effective pair potentials for simple models of Mo, taken from A . E. Carlsson, in “Alloy Theory and Phase Equilibria” (D. Farkas and F. Dyment, eds.), p. 103. American Society for Metals, Metals Park, Ohio, 1986.

for a simple of Mo having a 10-eV d-band width and a repulsive term V, proportional to R;”. The figure shows evaluated at the nearest-neighbor distance, for several local coordination geometries. The surface bond is along a (110) surface, and is roughly 20% stronger than the bulk bond. The vacancy “back bond” (directed away from the vacancy) is almost indistinguishable from the bulk bond. These are all, however, more than four times weaker than the dimer molecule (Z = 1) bond. The above effects are also contained in the effective pair potentials derived from U,, (cf. Table II), in an entirely parallel fashion. Since the curvature of UEA(n)(cf. Figs. 10 and 12) is positive, it follows that ULA becomes more negative with decreasing a. The background electron density n, in turn, drops with decreasing atomic density and coordination number. Thus either of these factors will cause the effective pair potential to become more attractive. Going beyond the linear terms in Eq. (5.12) produces effective triplet and higher-order interactions in the pair functional schemes. The triplet potential,X6for example, is given by

Since, as mentioned above, both U, and U,, have positive curvatures, the triplet potentials are positive. This is closely related to the weakening of the bond strength with increasing coordination number and density; under these conditions the repulsive triplet contributions become progressively more and more important, and cancel part of the attractive portion of the effective pair potential. A three-body effective potential for Ni, obtained via the tight-binding analysis, is shown in Fig. 15. Here Rjj and Rik are fixed and equal, and 8 is the angle subtended by atoms j -A. E. Carlsson, in “Alloy Theory and Phase Equilibria,” (D. Farkas and F. Dyment, eds.), p. 103. American Society for Metals, Metals Park, Ohio, 1986.

41

BEYOND PAIR POTENTIALS

90

I80

8 (degrees) FIG. 15. Effective three-body potential for Ni, for isosceles triangle having two bond lengths equal to the bulk nearest-neighbor distance. 6 denotes angle between the two nearest-neighbor bonds. Evaluated using parameters given in G. J . Ackland, G . Tichy, V. Vitek, and M. W. Finnis, Philos. Mag. A 56, 735 (1987).

and k at the site i. Vg" contains a rapidly decaying repulsive contribution, Coming from the h f h t and h!kh;k terms, and a constant part coming from the h%h$ term. Thus there is essentially no angular force for 8 > 100". In the empirical angular potentials to be treated in Section 12, we shall see that appreciable forces are found at all angular separations. The relation between the pair and triplet effective potentials, and the full pair functional description, is summarized in Fig. 16. It shows the accuracy with which Urn for Ni is treated by models containing (1) effective pair potentials and (2) both effective pair and triplet potentials, evaluated at yyf. The changes in y, along the horizontal axis could be envisaged as resulting from, for example, compression or expansion of the crystal, or progressively removing the neighbors of a particular atom. As suggested earlier in Fig. 2, we see that both of the interatomic potential schemes provide an excellent description over a sufficiently narrow range of environments. The accuracy range, including the triplet potentials, is larger than that for a pair potential description. However, for gross changes in the local environment, large errors are obtained even when triplet potentials are included.

42

A . E. CARLSSON

Ni

I I

I

2

FIG.16. Accuracy of descriptions of electronic band energy U,, given by an effective pair potential description (dotted curve) and one including both pair and triplet terms (dot-dash curve). “Exact” curve is full pair functional expression (5.8). All potentials evaluated for a reference environment having second moment pL;ef, using parameters given in G . J . Ackland, G . Tichy, V. Vitek, and M. W. Finnis, Philos. Mag. A 56, 735 (1987).

6. CLUSTER FUNCTIONALS We treat the derivation of cluster functionals only within the tightbinding framework of the preceding section, since such functionals have not been derived within the “embedded-atom’’ framework. As was seen above, a pair functional description can be obtained by a tight-binding analysis involving only the second moment p i [cf. Eq. ( 5 . 8 ) ] of the electronic density of states. In this subsection we describe how more sophisticated descriptions can be obtained by treating higher moments, as given by Eq. (5.5). We describe the application of these schemes to the energy difference between the bcc and fcc structures in transition metals. In addition, we discuss a result which provides some guidance in deciding what order of interaction (pair, three-body, etc.) describes a particular energy difference. The result involves the dependence of the energy difference on the filling of the tight-binding band. It states, roughly, that if the energy difference oscillates rapidly with the band filling, or is appreciable only for a small range of band fillings, then it must involve contributions from high-order moments or, equivalently, large atomic clusters.

BEYOND PAIR POTENTIALS

43

FIG.17. Sketch of geometry of four-atom cluster contributing to p4.

In transition metals and semiconductors, the higher moments [given by the multiband form of ( 5 . 5 ) ] have a strong angular dependence. This is typically due to the angular variations of the d orbitals in transition metals and the p orbitals in semiconductors. A simplified expression8’ for this angular dependence results if one retains only orbitals of one angular momentum quantum number 1 (such as 1 = 2 for a transition metal), and in addition includes only interatomic couplings having u symmetry (1 = 0 character with respect to the bond axis). In this case the contribution to p,, from a path involving n jumps is simply proportional to Wm=, P,(cos Om), where m specifies the atoms along the path and 8, denotes the angle between the bonds originating on atom n. For example, in a d-band model, the contribution to p4 from the four-atom path shown in Fig. 17 is proportional to P,(cos 8,)P2(cos Oi)PZ(cos 8,)P2(cos el). For 1 > 0, the PI are oscillatory functions, having 1zeros in the interval 0 < 8 < 180”. Thus a rapid angular dependence results. Although realistic tight-binding models have significant couplings of other than u symmetry, the angular dependence persists in these models. In contrast to the second moment, which describes the width of the band, the higher moments describe its shape. For example, a large negative value of p3 corresponds to an electron density of states (DOS) having a long tail at negative energies and a more compressed peak at positive energies. Many structural energy differences in transition metals and semiconductors receive a major contribution from p4, which we will consider in more detail. Model densities of states for different values of p4 are shown in Fig. 18. Here po and p2 are fixed and p1 = p 3 = 0; the band shape is actually determined by the dimensionless parameter p4po/(pJ2. The densities of states are generated by the “maximumR9K.Hirai and J. Kanamori, J. Phys. Soc. Jpn. 50, 2265 (1981).

44

A. E. CARLSSON

ENERGY

FIG. 18. Model electronic densities of states evaluated for different values of p4: (a) p4= 1.3, (b) p4 = 1.9, and (c) p4= 2.5. Energy units are such that p2 = 1 in all cases; in addition po = 1, p , = p3 = 0. Curves (b) and (c) adapted from R. H. Brown, Ph.D. thesis, Washington University, 1986 (unpublished). Dashed line indicates approximate limit of band for p,, = 2.5. Shaded regions denote occupied states for half-filled band.

entropy” method.*92 As seen in the figure, examination of p4 allows one to begin to discern the presence of a gap, or “quasigap”, in the density of states. A small value of p4 corresponds to a DOS dominated by two narrow peaks, while a large value corresponds to a DOS having a central peak and appreciable tails. However, subtle details of the density of states cannot be obtained even at this level. For example, if one of the peaks in Fig. 18a were split, a much higher-order treatment would be required to obtain this effect correctly. The features seen in these model state densities are reflected by more elaborate calculations, both for transition metals and semiconductors. Calculated “canonical” transition-metal d-band densities of states (cf. Fig. 19a) for the bcc structure display a pronounced dip near the center of the band. The fcc and hcp densities of states are considerably flatter. The difference in shape is reflected in smaller values of p4 for the bcc T. Jaynes, Phys. Rev. 106,620 (1957); Phys. Rev. 108, 171 (1957).

91L.R. Mead and N. Papanicolaou, J . Math. Phys. 25, 2404 (1984); R. Collins and A.

Wragg, J. Phys. A 10, 1441 (1977). 92R. H. Brown and A. E. Carlsson, Phys. Rev. B 32, 6125 (1985).

BEYOND PAIR POTENTIALS

45

cn W

I-

2 cn LL

0

t

t

cn

Z W

n

DIAMOND

I

(b

FCC

ENERGY FIG. 19. (a) “Canonical” (d)-band densities of states for bcc, fcc, and hcp structures, adapted from 0. K. Andersen, J. Madsen, U. K. Poulsen, 0. Jepsen, and J. Kollar, Physica 86-88B,249 (1977). (b) Densities of states for Si in the diamond and fcc structures. Volume per atom in both cases equals equilibrium volume per atom in diamond structure. State densities obtained via “augmented-spherical-wave” method [A. R. Williams, J . Kubler, and C. D. Gelatt, Phys. Rev. B 19, 6094 (1979)l. Band code supplied by A. R. Williams and V. L. Moruzzi.

structure in tight-binding model calculation^.^^^^^^^ The diamond and (hypothetical) fcc densities of states for Si (cf. Fig. 19b), display a similar difference, which is again reflected in a smaller value of p4 for the y3F. Ducastelle and F. Cyrot-Lackmann, J. Phys. Chem. Solids 31, 1295 (1970). y4F. Ducastelle and F. Cyrot-Lackmann, J. Phys. Chem. Solids 32,285 (1971). ”P. Turchi and F. Ducastelle, in “The Recursion Method and Its Applications” (D. G. Pettifor and D. L. Weaire, eds.), p. 104. Springer, New York, 1985.

46

A . E. CARLSSON

diamond structure.’6 Thus one can expect that the lowest-order descrip-

tion of the energy differences between these structures should include the effects of p 4 , at least The effect of p4 on U, depends strongly on the fractional filling N, of the relevant valence band. This effect has been evaluated via exact calculation^^^ of U, as a function of N, for a large number of local site environments in both crystalline and defect geometries, shown in Fig. 20. For each of the environments, p4 was evaluated via Eq. (5.5). The electronic structure was modeled using a tight-binding model with one orbital per site. In all of the cases considered, p o = 1, and p u l= p 3 = 0; the energy units are such that p 2 = 1. We consider two extreme cases:

(1) N, = 0.1. The band is nearly empty. A density of states with larger

p4is preferred, so that \UTB\ increases with p4. This density of states has

a greater absolute width, so its lower limit lies at deeper binding energies (cf. Fig. 18). The few electrons that are present can therefore occupy states at lower energies, lowering U, .

(2) N v = 0 . 5 . The band is half-filled. Now a small value of p4 is preferred, and 1 Urn[ decreases with increasing p4. For a half-filled band the Fermi level lies inside the gap or “quasigap”, as shown in the density of states (DOS) labeled “a” in Fig. 18. Going from the DOS “c” to the DOS “a”, reducing p4, one sees that occupied electron states near the Fermi level can move to lower energies, lowering UTB. The latter effect, and the shapes of the bcc and diamond structure densities of states in Fig. 19, are in accord with the relative stability of these structures for nearly half-filled bands (the “band” for Si consists of the electron states derived from the atomic 3s and 3p orbitals). The physics underlying the stabilization is essentially that a gap at the Fermi level should lead to structural stability. In the transition metal case it is not actually possible to obtain a gap. However, the bcc structure has a closer approximation to a gap than the fcc structure, and is thus preferred for appropriate band fillings. Cluster functionals do not in general, have forms as simple as those of the pair functionals, such as Eq. (5.6). For a functional of order M, U& = UTB(p;, p i , . . . , p&) is obtained in two steps.60 First one chooses a density of states pi(,?) which has the correct moments through order M. This density of states is not uniquely determined, so one guesses a parametrized analytic form for pi(,?) and uses the moments to fix the parameters. Typical analytic forms include continued fractionsz5 and %A. E. Carlsson (unpublished)

47

BEYOND PAIR POTENTIALS Fractional Band-Filling 0

N, = 0.1

A N, = 0.3

0

N,=o.?

o N,=o~ N, = 0.5

0.9

m

A

m

A

A

“1.

%A

e f %? 1 1

~

aa;a

k

0.

,

I

0;

0.3

0.2

1.0

1.5

2.0

2.5

3.0

3.5

1 4.0

FIG. 20. Dependence of exact electronic band energy Un on p4 for various local environments, in tight-binding models with nearest-neighbor couplings. Energy units are such that p 2 = 1. Environments are as follows: a = dimer molecule, b = bulk atom in linear chain, c = vacancy nearest neighbor in linear chain, d = vacancy second neighbor in linear chain, e = bulk atom in square lattice, f = vacancy nearest neighbor in square lattice, g = bulk atom in diamond lattice, h = vacancy nearest neighbor in diamond lattice, i = bulk atom in simple cubic lattice, j = vacancy nearest neighbor in simple cubic lattice, k = bulk atom in body-centered cubic lattice, and I = vacancy nearest neighbor in body-centered cubic lattice. Adapted from R. H. Brown and A. E. Carlsson, Phys. Rev. B 32, 6125 (1985).

exponentials of polynomial^^^ in E (the “maximum-entropy’’ form). via a numerical integration of Eq. (5.3). Given pi(E) one obtains One thus obtains a cluster functional description of the form given by Eq. (1.5):

UA

For example, from Eq. (5.5) one has g,(Ri, R, , R k )= h,hjkhk,, with

A. E. CARLSSON

48

1

FIG. 21. Electronic band energy difference A&,

between bcc and fcc structures, for “canonical” d bands, vs. number of electrons in d band. M = 2, 4, and 6 curves correspond to low-order cluster functionals; “exact” curve [H. L. Skriver, Phys. Rev. B 31, 1909 (1985)] is obtained by a k-space treatment with no approximations. Unit of energy is maximal value of lUrBl for the bcc structure. Negative sign corresponds to preference for bcc structure.

analogous expressions for the higher-order terms (in the single-band case). For low values of M this procedure can be streamlined, and one can obtain an accurate analytic approximation to the dependence of U , on the p i . The range of applicability of cluster functionals is larger than that of pair functionals, because they treat the electronic structure more accurately. Low-order cluster functionals yield improved accuracy in calculations of broken-bond e n e r g i e ~ ~ .and ~ ’ in some cases begin to discern chemical trends in structural energy difference^.^^.^^'^^ As an example of the rate of convergence of cluster functionals in describing the latter, we choose the bcc-fcc energy difference in transition metals. The basic trends in this quantity, shown in Fig. 21, are obtained well by tight-binding models?’ and have been analyzed by several cluster functional treatment^.'^'^' The electronic band energy U, is obtained from a “canonical” tight-binding model9* with five d orbitals per site, and couplings decaying as l/r5. The figure shows the dependence of A U - = U,,(bcc) - U,(fcc) on the number of d electrons, essentially scanning a 97D. G. Pettifor, J . Phys. C 3, 367 (1970).

’‘0. K. Andersen, J. Madsen, U. K. Poulsen, 0. Jepsen, and J . Kollar, Physica 86-88B, 249 (1977).

BEYOND PAIR POTENTIALS

49

transition metal row. In addition to results from cluster functionals of order M = 2, 4, and 6, exact results are shown. The density of states in the cluster functionals has the "maximum-entropy'' form.92 For M = 2 one has a pair functional description, which obtains much too small values of AU, and fails to obtain the sign changes seen in the exact curve. Thus pair functionals model this structural energy difference poorly. The four-body functional, M = 4, obtains significantly better results. The bcc structure is correctly favored close to Nd = 5, and the fcc structure for Nd 5 2. The maximal value of IAU,l is roughly 60% of the exact value, which is much closer than the pair functional result. However the small region of bcc preferability for Nd 2 8 . 5 (absent in more elaborate calculations) is not obtained correctly. For M = 6 the regions of fcc and bcc preferability are obtained well. The maximal value of AU, is 75% of the correct value. The magnitude of AU, for Nd 2 7 is still, however, much too small. Thus with a four-body functional, one could begin to model the stability of the bcc structure for nearly half-filled bands, but even with six-body terms one does not obtain quantitative values of AU, for all Nd. For the purposes of evaluating the range of applicability of a cluster functional, or to obtain a cluster functional to describe a given set of properties, it is useful to have a method for establishing the order of the moments that determine the band energy difference AU, for a particular property. A rigorous r e ~ u l t ~enables ~ ~ ~ one * ' ~to obtain useful information regarding this question without explicitly performing the moment calculations. Instead one examines the dependence of AUm on the position of the Fermi level E~ in the band (or equivalently, the band filling). Let p , ( E ) and p 2 ( E ) be the densities of states corresponding to the two atomic configurations which determine AU, , so that

(Here the inclusion of the E~ term inside the integrals account a p p r ~ x i m a t e l yfor ~ ~the difference between the Fermi levels corresponding to p , and p 2 . ) The result states that if

I-_ m

Ap,, =

E " A p ( E )d E = 0

99V.Heine and J . H. Samson, J . Phys. F 10, 2609 (1980).

'9 Heine . and J. H. Samson, 1. Phys. F W, 2155 (1983).

50

A. E. CARLSSON

for all n less than some integer N , then AU,, must change sign at least N - 2 times as a function of E ~ For . example, if AU, is associated with changes only in p4 and higher moments ( N = 4 ) , the AU, must change sign at least twice. On the other hand, if AU, does not change sign at all, then Ap2 must be nonzero. A partial converse95 to this results states that

for any n. (Here it is assumed that Apo = Apl = 0). For example, if the average of AUTB over Fermi levels vanishes (n = 0), then Ap2 vanishes as well, and AUTBis determined by higher moments of Ap. The content of these results may be summarized as follows: rapid oscillations in A UTB(EF) point to contributions from high-order moments, and a smooth behavior suggests that the low-order contributions dominate. In addition, we note that a very localized feature in AU,(EF) must come from a localized feature in Ap(E), since, by Eq. (6.2),

Because the moments are smooth averages over p(E), a large number of moments is necessary to describe such a feature. These principles conveniently encapsule some of the main results we have obtained regarding the ranges of applicability of pair and cluster functionals. Figure 22 shows schematic illustrations of AU, , p1 (dashed line), and p2 (solid line), for representative cases: vacancy formation energies, the bcc-fcc energy difference in transition metals, and, as an extreme example, the energy stabilizing a Fermi surface-driven Peierls distortion"" with small amplitude. Although these properties contain contributions from the pair potential part V, of E, [cf. Eq. (6.1)], which is not included here, the major trends as functions of E~ are obtained by AU, in all three cases. Observed vacancy formation energies in transition metals are always positive, and vary fairly smoothly with band filling. If one assumes the same characteristics for AU, , as shown in Fig. 22, it is likely that AU, has a major contribution from changes in p 2 . A pair functional description may thus be viable. The explicit model results presented above in Table I confirm this assertion. In contrast, the bcc structure in transition metals is favored only for nearly half-filled d bands '"'Ref. 26, p. 284.

BEYOND PAIR POTENTIALS

bcc

VACANCY FORMATION

ENERGY

'TB

fcc

- 'TB

51

PEIRLS DISTORTION

-

PAIR FUNCTIONAL

CLUSTER FUNCTIONAL

EXACT TREATMENT

FIG. 22. Schematic illustration of dependence of AUTB on E ~ and , associated changes in electronic DOS. Dashed and solid curves indicate p , ( E ) and p , ( E ) [cf. Eq. (6.2)] respectively. Level of sophistication needed to obtain AUTBindicated at bottom.

(except for the 3d row, where magnetic effects complicate the analysis). Thus AU, likely crosses zero at least twice, suggesting that p4 and higher-order moments are important. A pair functional cannot describe AUm in this case, but a low-order cluster functional description involving at least four-body terms may be considerably better. These contentions are consistent with the model results presented in Fig. 21. The analysis of the stability of the diamond structure in semiconductors probably requires at least four-body terms as well, since it occurs only for half-filled s-p complexes. Finally, a Peierls distortion of small magnitude causes a small gap in the density of states, but otherwise leaves it fairly unperturbed. The distorted structure is favored only if E~ is inside the gap, or very close to it. A description via low-order functionals is thus impossible. The expansion procedure described in Section 5b can also be used to obtain effective cluster potentials from cluster functional^.^^^^^^^^ In a description involving M moments, the n-body effective potential is given87by

v ; ~ ( R ~R,, ,

akl3 . . .) = C. ,&. . . -,

M

m=n

3Ptn

(6.3)

52

A. E. CARLSSON

where p:.. denotes the contribution to p, from paths containing all of the atoms i , j , . . . , and no others. dU,/dp4 is obtained by numerical differentiation of the U , generated by the chosen model density of states, or by simplifying analytic relation^.^' For example, in a fourmoment description of the electronic band energy, one obtains up to four-body potentials. The four-body potential is given by

dU,/dp4 corresponds roughly to the slope of the plots in Fig. 20. As discussed above, pik' is strongly angle-dependent in d-band models, resulting in a four-body potential with rapid angular variations. Figure 23

0.1 n

1 1

45

I

I

I

I

I

J

60 75 90 105 120 135

e

FIG.23. Four-body effective potentials for a model transition metal with a nearly-half-filled band (for which the bcc structure is favored). Geometry, which is illustrated, has R , = Rj, = R,, = R,, . (After R.B. Phillips and A. E. Carlsson, unpublished.)

BEYOND PAIR POTENTIALS

53

shows the four-body potential"' for a transition metal described by a tight-binding model with five d orbitals per site, using a "maximumentropy" model9* density of states. The d band is nearly half-filled, so that the bcc structure is preferred over the fcc structure. The two curves correspond to different values of the angle @ between the planes determined by the quadrilateral. For simplicity, only configurations with equal bond lengths are considered. The angular dependence of Vq" is very rapid, with a pronounced maximum at 8=90" and minima at 6 = 70" and 110". These minima are very close to the angles observed in nearest-neighbor quadrilaterials in the bcc structure, which are 71" and 109"; by contrast, the angles observed in nearest-neighbor quadrilaterals in the fcc structure are 60°, 90", and 120", which have positive values of V.,tf.This type of potential thus provides the rudiments of a real-space picture of the favorability of the bcc structure. As was illustrated in Fig. 20, aU,/ap, changes in sign and magnitude with varying band filling. Thus V:ff exhibits these changes as well. However, the functional form of its angular dependence is determined by the angular and radial variation of the atomic d orbitals and thus depends more smoothly on band filling. The range of applicability of the potentials given by (6.3) is, of course, more restricted than that of the cluster functionals which they approximate. For calculating structural energy differences at constant volume the expansion linear of U, is quite a ~ c u r a t e . ~However, ' for calculating structural energy differences at different volumes, or for calculating structural energies around defects, it appears essential to use the full cluster functional description.

7. RELATION BETWEEN "CONSTANT-VOLUME" POTENTIALS AND CLUSTERFUNCTIONALS As mentioned above, the two types of approaches described in the preceding subsections treat complementary classes of problems. The constant-volume potentials naturally treat crystal structure energy differences and other problems in which the local atomic or electronic density does not change greatly. The pair and cluster functionals and their associated potentials, on the other hand, treat the "broken-bond'' energies accompanying these local density variations well, but cannot treat structural energy differences unless one utilizes high-order descriptions. Thus there are two distinct series of interatomic potentials. Both of these give exact answers, in principle, if the terms of all orders are kept. This situation is closely analogous to the expansion of a function on the "*R. B. Phillips and A. E. Carlsson (unpublished).

54

A . E. CARLSSON

real line in terms of two different basis sets, such as plane waves and Hermite functions. One expects that the nature of these two types of interatomic potential expansions can be clarified by explicitly establishing the connection between them. Some progress in this direction” has been made using the single-band tight-binding model given in Eq. (5.1). Within this model an explicit linear relation was obtained between the two types of potential series. The content of this relation is that the “constant-volume’’ effective pair potential contains contributions from the higher-order “bond-breaking’’ cluster potentials, averaged over the positions of the atoms in the cluster. We denote the “constant-volume’’ potentials by VZff(Ri,Rj,. . .; CV), and the “bond-breaking” potentials by VEff(Ri,R,,. . -;BB). The latter are given by Eq. (6.3). The values of the “bond-breaking” potentials, evaluated using nearest-neighbor clusters in a simple cubic lattice as an example, are shown in Fig. 24. The derivatives aE,/dp,, entering the effective potentials were evaluated using a half-filled band, via the “maximum-entropy” truncation scheme.92 For a half-filled even band the contributions from odd moments vanish. We see that the signs of the Vzff alternate as functions of n. Since potentials with different values of n sample clusters with different spatial extent, this is not surprising in view of the radial oscillations that are typically seen in “constant-volume’’ effective pair potentials for simple metals (cf. Fig. 6). This connection is clarified below. Although the VEff decay quite rapidly with increasing n, calculations of structural properties using the VEff often converge slowly. The integral defining U,, Eq. (5.3), contains a sharp cutoff at the Fermi level. A

2

4

n

6

8

FIG. 24. Effective cluster potentials obtained through moment analysis of density of states in a single-band tight-binding model. Geometries are nearest-neighbor ring clusters in a simple cubic lattice. M = 8 indicates that moments through p8 are retained in the expansion. Nearest-neighbor coupling strength in electronic Hamiltonian denoted by h. [Taken from A. E. Carlsson, Phys. Rev. B 32, 4866 (1985).]

BEYOND PAIR POTENTIALS

55

moment description, on the other hand, utilizes only smooth averages of the electronic density of states [cf. Eq. (5.4)]. Therefore, although reasonable estimates of Urn can often be obtained by low order expansions, quantitative descriptions require very high-order treatments. As was discussed in Section 6, for example, the bcc-fcc energy difference in transition metals (cf. Fig. 21) has significant errors even in a description retaining up to six-body terms. We shall see that the contributions from the higher-order BB potentials are reflected in the comparatively long range of the CV pair potential. To obtain the "constant-volume'' potential series from the "bondbreaking" series, the configurational energy is first written as follows:

E, = E:')(BB) = E:')(BB)

+ C"

1

-

C

n = 2 n ! i , j , ...

V;"(R,, R ~ .,. . ;BB)

c / Vz"(R,, Rj, . . .; BB)p,,(R,, R,, . . .) dR, dRj

1 + ,,=2" n!

. ...

Here E:') is a structure-independent term and ~n = p(Ri)p(Rj).

..

(7.2)

is the n-body atomic density. This expression holds if the changes in the moments for the process of interest are small, so that a linearized description of Urnis accurate. One then reexpresses the energy in terms of the density fluctuations Ap,, = [p(Ri) - P][p(Rj) - PI. . . , where P is the average density. By straightforward algebraic manipulations one obtains

E, = E:"(CV) x

/

+ n2"= 21n ! -

v~(R,R , ~ .,. .;CV)A~,,(R,,R ~ .,. .) d ~d~~ , . . . , (7.3)

where E:')(CV) is a structure-independent term and the "constantvolume" pair potential is given in terms of the "bond-breaking'' potentials by

VX"(R,, Rj ;CV) = VX"(Ri, Rj ;BB)

+

m=l

/

P" 7 m.

V;?,(Ri, Rj, R k , . . . ;BB) dRk dR, . . . .

(7.4)

56

A. E. CARLSSON

Similar expressions are obtained for the higher-order “constant-volume” potentials. Equation (7.4) is the central result of this section. It gives V;ff(Ri, R, ; CV) explicitly in terms of the “bond-breaking’’ cluster potentials. The contribution from the cluster potential of a given order is obtained by averaging over the positions of all but two atoms in the cluster. It is likely87 that the potentials VZff(Ri, Rj, . .;CV) thus obtained are the analogues, within the tight-binding model, of the pseudopotential-derived “constant-volume’’ potentials discussed in Section 4. The strongest evidence in favor of this assertion is that V5”(Ri, R, ; CV) exhibits the large-r cos(2k,r)/r3 behavior of the pseudopotential-derived potentials. The presence of a single dominant wavelength at large r points to a unique density for which V5ff(R,,R,: CV) is applicable. In addition, the criteria for the applicability of the two types of potentials are parallel. The criterion for the pseudopotential-derived potentials is,6 roughly, that the density of atoms contain large components only at spatial wavelengths at which the pseudopotential V,, is small. The criterion for the applicability of ~

FIG. 25. Effective pair potentials obtained by averaged treatment of cluster potentials in moment analysis. Reference environment is simple cubic structure. Dashed curve is “bond-breaking’’ potential; solid curve is “constant-volume’’ potential; dotted curve is intermediate case containing terms up to m = 8 in Eq. (7.4). Arrows denote neighbor shells in simple cubic structure. rs denotes radius (in units of the Bohr radius) of sphere having volume equal to the average volume per electron. [Adapted from A. E. Carlsson, in “Alloy Theory and Phase Equilibria” (D. Farkas and F. Dyment, eds.), p. 103, American Society for Metals, Metals Park, Ohio, 1986; A. E. Carlsson, Phys. Rev. B 32, 4866 (1985).]

BEYOND PAIR POTENTIALS

57

Vzff(Ri,Rj; CV) is the same, except that V,, is replaced by an integral involving the interatomic coupling strengths. This “change of basis” is illustrated in Fig. 25. Here the dashed curve is the “bond-breaking” potential, having its characteristic deep well, and the solid curve is the “constant-volume’’ potential. The dotted curve is an intermediate case, retaining only the m 5 8 terms in Eq. (7.4). This curve includes averaged contributions from up to eight-atom clusters and is thus considerably longer-ranged than the “bond-breaking’’ potential. Oscillations also begin to appear, which are in turn due to the changing sign of the cluster potentials with cluster size (cf. Fig. 24). The first visible minimum, for example, is due to six-atom clusters. The deep minimum in the “bond-breaking’’ potential has also been canceled by the repulsive four-body terms, so that the broken-bond energies can no longer be described accurately by V;”. Finally, as mentioned above, the “constantvolume” potential displays the asymptotic cos(2k,r)/r3 behavior seen in “constant-volume” potentials for simple metals, such as that shown in Fig. 6. The above analysis provides a new perspective on the limits of the “constant-volume’’ range of problems, discussed in Section 4. This range consists of those problems described accurately by low-order “constantvolume” potentials. The averaged treatment of the higher-order terms, used to obtain these potentials [cf. Eq. (7.4)], is expected to be accurate if the VZff(R,, Rj, . . . ;BB) are slowly varying in comparison with the changes in the p n in the process of interest. In transition metals the VZff(Ri,Rj, . . . ;BB) (cf. Fig. 23) are more rapidly varying than in simple metals, because of the angular variations and quick radial decay of the d orbitals. Therefore the category of “constant-volume’’ problems should be smaller for transition metals than for simple metals. If the process under study lies outside this category, it is likely that using V;ff(Ri, Rj ;BB), which neglects the higher-order terms, is preferable to using VSff(Ri,Rj;CV), which treats them inaccurately.

8. INTERATOMIC POTENTIALS AS STATISTICAL CORRELATION FUNCTIONS Much unease in the materials science and physics communities about the use of interatomic potentials has resulted from the absence of a precise, universally accepted definition of an effective interatomic potential in a condensed matter environment. There have been many definitions of interatomic potentials, but they have usually been based on a particular approximate model for the electronic structure, such as nearly-free-electron perturbation theory or tight-binding Hamiltonians.

58

A . E. CARLSSON

In this section we discuss approaches’”*’”’ to defining interatomic potentials which are unambiguous and are based on the exact configurational energy of the system. These approaches generalize the “constant-volume” potentials discussed in Sections 4 and 7. We have deferred the discussion of this work until now because it leans rather heavily on the preceding material. The basic idea is to use the interatomic potentials to measure the energies of virtual deviations from the structure of a “reference environment”, as was done in Section 5b. However, this environment is defined not by a particular structure, but by a probability distribution P({R,}) for a large number of atomic configurations. It could, for example, correspond to a statistically averaged liquid. The effective pair potential is defined as the functional derivative

where p, is defined by Eq. (7.2), and the brackets denote a weighted statistical average over the configurations in the reference environment. Analogous expressions define the higher-order interactions. In a system which contains only pair interactions V,, this is clearly the correct definition, since in such a system

If higher-order interactions are present, then the definition (8.1) is not unique, since in general ( E , ) is not uniquely determined by ( p2(Ri,R,)), but depends on th higher-order correlation functions as well. Thus evaluating the derivative in Eq. (8.1), to obtain the effective pair potential, can be formulated as a missing information problem: given a virtual change in the pair correlation function, one must guess the changes in the higher-order correlation functions before evaluating the change in (I&). A possible way of accomplishing this is suggested by the expression (7.3) for Ec in terms of the “constant-volume’’ potentials. One can assume that ( A p n ) = 0 for n 2 3, obtaining the truncated form

(p,(Ri, Rj, Rk)) = P [ ( P ~ ( R Rj) ~ , + p,(Rj, Rk) + PZ(R~,Ri)) - 2 ~ ~ 1 , (8-2) ‘03L. Dagens, Phys. Left. 101A,283 (1984). ‘04L. Dagens, J . Phys. F 16, 1705 (1986). ‘‘’A. E. Carlsson, Phys. Rev. L e f t 59, 1108 (1987).

BEYOND PAIR POTENTIALS

59

with similar forms for the higher-order densities. If one makes this assumption, then Eq. (8.1) provides'05 a definition of "constant-volume" potentials in terms of the exact energy E,, without the use of pseudopotential expansions or tight-binding theory. However, the above truncated form is inaccurate for condensed systems, in which short interatomic separations are essentially forbidden. If, for example, R;, Rj, and Rk are sufficiently close to each other that p2(Ri, Rj), p,(R,, Rk), and p2(Rk,RJ vanish, then using the truncated form one finds that p 3 ( R j ,Rj , R k )= (- 2 p 3 ) is negative in these "forbidden" regions; the correct result is, of course, p3 = 0. Thus the truncated form is not physically realizable, in the sense that it cannot be obtained from any positive definite probability distribution. This can lead to considerable errors, since the effective pair potential contains contributions to (E,) from forbidden configurations. The use of a physically realizable ansatz for the higher-order correlation functions should improve the accuracy of the calculated interatomic potentials. One such ansatz is provided by the "maximum-entropy'' Briefly, one uses the pair correlation function to generate a probability distribution P({R,}) which has the maximal information-theoretic entropy, subject to the constraints imposed by this pair correlation function. One then obtains the higher-order correlation functions from P( { R;}). The resulting pair potential has the form of a statistical correlation function between fluctuations in the total energy and those in the static structure factor:

where Q is the volume of the system, S,. = IJ p(R)e-'q'Rd3RI2 is the static structure factor (not yet statistically averaged), and

describes the fluctuations in the structure factor. Closely related effective potentials have been obtained'"3*"''' via a least-squares fit to the exact bonding energies in the reference medium, and to various parts of these bonding energies. One then obtains the effective pair potential as

V;"(R,, Rj) = (Ez)j j ,

(8.4)

where Ez is the part of E, that actually depends on the positions Rj and

60

A. E. CARLSSON

R,, and ( )ij denotes an average over the reference environment, has used crystalline keeping atoms i and j fixed. A related configurations treated by a tight-binding model as a reference environment. The pair potentials were defined directly in terms of the components of the electronic “density matrix” containing the pair of sites. One easily shows for both of the statistical approaches that if a system is described rigorously by a pair potential V 2 , then Vqe = V,. The definitions (8.3) and (8.4) generalize both the “equation of state” potential (cf. Fig. 4) and the “constant-volume” potentials. The former is obtained by using a very restricted reference environment, containing the energies of the material of interest only in a particular crystal structure, but at all lattice constants. The “constant-volume” potentials are obtained, in simplified models, by using a reference environment with no correlations at all. The use of a correlated reference medium, in the statistically derived potentials, may well be a significant improvement, since the pair correlation function places severe constraints on the higher-order correlation functions. These constraints are more severe the stronger the pair correlations are. As an extreme example, if it is given that the system of interest has exactly the same pair correlation function as the fcc structure, then it must have the fcc structure and has the higher-order correlation functions of this structure as well. Such constraints are not included in the “constant-volume” potentials but can be included in the statistically derived potentials. No complete implementations of Eq. (8.3) exist, but an approximate irnplementati~n’~~ has utilized a model transition metal having two- and three-body interactions (cf. Fig. 26). The curve labeled “bare” includes only the two-body terms. The curve labeled “truncated” utilizes the form (8.2) to model the triplet correlations, and thus corresponds roughly to the “constant-volume” potential. The curve labeled “excluded” utilizes a more sophisticated treatment of the higher-order correlations, in which close approaches are excluded and the triplet correlation function is everywhere positive. Clearly, the “truncated” curve overestimates the effect of the three-body interactions significantly. This probably results from the lack of exclusion at short distances, at which the three-body terms are quite large. In addition, a calculation of point defect properties in Ni has been performed’” with an effective pair potential obtained via (8.4), using a hard-sphere gas for a reference medium. The initial results are quite encouraging. Vacancy formation and migration energies of 1.33 eV and 0.90 eV, respectively, were obtained, which compare well with the experimental values of 1.80eV and 1.08eV. The shear elastic IWA.H. Macdonald and R. Taylor, Cun. 1. Phys. 62, 796 (1984).

61

BEYOND PAIR POTENTIALS 0 . 1

0.:

-> W

v ’c Lc

3.0

O.(

-

‘uN

>

2

v Lc c

-1.0”>”

-0.:

-0.4

R

-2.0

FIG. 26. Effective pair potentials obtained by different types of statistical averaging of three-body terms, in a model transition metal containing two- and three-body potentials. “Bare” curve contains no three-body contribution; “truncated” curve (note change of scale) uses decoupled form (8.2) for three-body correlation function to average three-body potentials; “excluded” curve uses improved form for three-body correlation function which eliminates three-body collisions. [Taken from A . E. Carlsson, Phys. Rev. Lett. 59, 1108 (1987).]

constants were also obtained’” to within 40% of the experimental values. The statistical approaches described here are still in the developmental stage, but appear to contain several ingredients necessary for an accurate description of bonding energetics via effective interatomic potentials.

9. THESOLID

SOLUTION

ANALOGY

The conceptual framework introduced above can be clarified by consideration of a closely related but simpler problem, which is that of describing the configurational energetics of binary solid solutions with ““N. Q. Lam and L. Dagens, J. Phys. F 16, 1373 (1986).

62

A. E . CARLSSON

TABLE111.

PARALLELS

BETWEENCONFIGURATIONAL ENERGETICS OF ELEMENTAL METALS HEATSOF SOLUTION ARE PER IMPURITY.

AND SOLID SOLUTION ORDERING ENERGIES.

ELEMENTAL METALS

SOLIDSOLUTIONS

Atomic Density p ( r )

Site “spin” a,

E , = const + i

E , = const + 4

I

dR, R, V;ff(R,, R,)Ap,(R,, R,)

Vj;)*‘*Sa,Sa, 1.1

Constant-Volume Potentials Vacancy Formation Energy - (Cohesive Energy)

Constant-Concentration Interactions Heat of Solution at ( a ) = 1 Heat of Solution at ( a ) = 0

effective Ising parameters. In this case one treats only atomic rearrangements occurring on a fixed underlying lattice. One seeks a description of the configurational energy having the form

where a,= 0 or 1 according to what type of atom is at the site i. The V(O) and V ( I )terms are structure-independent at fixed concentration; V @ ) , V ( 3 )and the higher-order terms are effective Ising interaction parameters. This type of model is simply the discrete analogue of Eq. (1.1). It can crudely model some of the problems considered above if one makes the lattice gas approximation, treating the system as an “alloy” of atoms (a, = 1) and vacancies (ai = 0). A formal proof’”’ has been given that in the absence of long-ranged interactions, the energy of a solid solution can be described exactly by an expression of the form (9.1); however, pathologies in the thermodynamic limit have not been definitely ruled out. Even if such a description exists, it may well contain very high-order interactions. It is therefore useful to define effective king parameters analogous to the “constant-volume’’ potentials of Section 4. These describe rearrangements of the alloy constituents at constant concentration (cf. Table 111). In analogy with the analysis of Section 7, the effective Ising parameters are obtainedlW by transforming Eq. (9.1) to the concentration fluctuation variables 60, = a, - (a):

C

1 E , = Eio)((a))+ - V f ) . e f f 6 ~ ; S ~.j. . , 2! ,,j

+

M. Sanchez, F. Ducastelle, and D . Gratias, Physica 128A,334 (1984) ‘“A. E. Carlsson, Phys. Rev. B 35, 4858 (1987).

loSJ.

BEYOND PAIR POTENTIALS

63

where ELo) is the energy of the completely random medium, (9.3)

f f given by similar expressions. For and the higher-order V ( n ) * eare describing many types of rearrangements at constant alloy concentration, such an expansion results in quicker convergence than that of Eq. (9.1). Because the discrete set of configurations treated by the alloy model is much more restricted than that required to model structural rearrangements, the calculation of the interaction parameters is in some ways more straightforward. Two types of schemes have been used to obtain “constant-concentration” effective interactions: (1) Mean-field multiple scattering schemes.11(~’12 Starting with a completely random medium, one expands E , in powers of the difference in the atomic scattering matrices. The coefficients in the expansion are determined by the response function of the completely random alloy, which in turn is usually calculated in a self-consistent mean-field theory. More sophisticated appro ache^"^ have calculated the interaction energy of two fixed atoms in the random alloy.

(2) “Matching schemes.” The “bare” interactions V‘“) are first ~ b t a i n e d ”by ~ choosing their values to fit the total energies of several hypothetical ordered structures, obtained via a6 initio quantummechanical band-structure calculations. These are then resummedl(’ginto effective concentration-dependent interactions via Eq. (9.3). The “constant-concentration’’ pair interactions, like the “constantvolume” pair potentials, have a restricted range of applicability. They are in most cases strongly concentration-dependent. They rigorously describe the pair correlation function at high temperatures,”’ and often obtain accurate energy differences between different ordered structures at zero temperature. ‘15 However, the “constant-concentration” pair interactions

‘‘Ducastelle 9. and F. Gautier, J .

Phys. F 6 , 2039 (1976). A . Bieber and F. Gautier, Acta Metall. 35, 1839 (1987). 112 G . M. Stocks, D . M. Nicholson, F. J . Pinski, W. H. Butler, P. Sterne, W. M. Temmerman, B. L. Gyorffy, D. D. Johnson, A . Gonis, X.-G. Zhang, and P. E. A. Turchi, in “High-Temperature Ordered Alloys” (N. S. Stoloff, C. C. Koch, C. T. Liu, and 0. Izumi, eds.), p. 15. Materials Research Society, Pittsburgh, 1987. 113 A . Gonis, G . M. Stocks, W. H. Butler, and H. Winter, Phys. Rev. B 29, 555 (1984). ‘I4J. W. D. Connolly and A . R. Williams, Phys. Rev. B 27, 5168 (1983). 11%‘ . IS IS readily shown by evaluation of the lowest order term in a high-temperature expansion of the short-range order parameters. 111

64

A. E. CARLSSON

usually treat the heat of solution poorly.”’ This phenomenon is easily understood in the solid solution context, since the effective pair interaction that describes the heat of solution contains the higher-order terms with multiplicative factors different from those in the “constantconcentration” pair interaction.“” The failure of “constantconcentration” pair interactions to obtain accurate heats of solution is analogous to the failures of “constant-volume” pair potentials in treating “broken-bond’’ energies, such as cohesive and vacancy formation energies (cf. Table 111). Both of the latter, in fact, correspond to the heat of solution in the alloy model, but at different average concentrations. If one defines the atoms as “A” sites, and the vacancies as “V” sites, then the ordered solid corresponds to a pure A lattice. Creating vacancies in the solid corresponds to dissolving a small amount of pure V in this lattice. On the other hand, separating the solid into isolated atoms (which measures the cohesive energy) corresponds to dissolving a piece of the A lattice in a host of pure V. Heats of solution at different ends of an alloy phase diagram are usually quite distinct and cannot be described by the same pair interaction. Thus, by analogy, it is not surprising that a pair potential which reproduces the cohesive energy of a solid usually obtains a poor vacancy formation energy (cf. Section 2), and vice versa. In treating both configurational energies in elemental metals, and the simplified solid solution case, a pair description applies only to a narrowly delimited range of phenomena. 111. Fitted Potentials and Functionals: Implementations and Applications

10. AE I N I T ~ Oversus FITTED DESCRIPTIONS We have so far treated several types of simplified energy functionals, whose applicability depends on the type of geometric problem under consideration as well as the chemical nature of the interacting atoms. Here we discuss another feature that differentiates between different functionals, which is the extent of their reliance on and incorporation of empirical data, or data obtained by more sophisticated theoretical techniques. Roughly, the total energy schemes may be divided into two categories, “ab inifio” and “fitted.” In ab initio schemes, one begins with an idealized system which is for all practical purposes treated exactly. As discussed above, such systems can be defined by a uniform electron gas or by an impurity atom in such an electron gas. One then uses this idealized system to describe the

BEYOND PAIR POTENTIALS

65

system under study, in some cases including, in an approximate fashion, terms which describe the deviation from the idealized system. Such terms are, for example, the s-p pseudopotential and the interatomic d-d couplings (cf. Section 4) in the “constant-volume potentials.” Although ab initio potentials and functionals do not usually obtain quantitatively accurate results, we have seen in Part I1 that the underlying theoretical analysis is very useful in explaining the systematics of defect energetics and structural energy differences. An important advantage of ab initio approaches is that one knows what is left out of the simplified description, such as, for example, higher-order terms in perturbation theory. In addition, the ambiguities that occur in the choice of database for empirical schemes are not present. However, other ambiguities often crop up, such as the choice of pseudopotential in “constant-volume’’ potentials; different choices can lead to quite different distributions of the configurational energy between one-body, two-body, and higher-order terms. l6 In fitted schemes, the focus of this Part, one begins with a functional form for the configurational energy, and obtains the values of the parameters in this form by fitting experimental measurements or accurate theoretical calculations. The functional form can be taken from theoretical analysis; empirical pair functionals can, for example, be obtained from the “embedded-atom’’ and tight-binding analyses. On the other hand, one can use an arbitrary functional form, or assume that the input database is sufficiently large to essentially determine the functional form. The fitted schemes have had many more applications to defect problems than the ab initio methods. They have the important advantage that the underlying theory is not required to make absolute estimates of the various energy changes that occur as a result of atomic rearrangements, but only to estimate the relative magnitudes of energy changes resulting from different types of rearrangements; the absolute values are taken from the theoretical or experimental input database. If the functional form of the atomic interaction is sufficiently well established, then the relative magnitudes of energy changes will be obtained accurately. Neutron scattering from atomic nuclei in solids is a useful parallel here. Although ab initio calculation of the absolute magnitude of the interaction between a neutron and a nucleus is a difficult many-body problem, simple theoretical analysis tells one that on the length scale of the interatomic spacing, this interaction has the functional form of a S function. Once the magnitude has been obtained empirically, one can accurately predict neutron scattering cross sections from arbitrary arrangements of atoms. 116

Ref. 38, Chap. 8.

66

A . E. CARLSSON

The functional form of the atomic interaction energies in condensed matter is certainly not as accurately established as that for neutron scattering, but, as seen above, considerable progress has been achieved in this direction. The dichotomy between a6 initio and fitted schemes is also present in another problem of simplification in condensed matter physics, which is the description of electronic band structures in solids with tight-binding models."' At first, this approach to band structure was based on a literal picture involving atomic orbital basis sets. It was quickly realized that this type of "ab initio" tight-binding theory gave poor results. However, band structures obtained by more sophisticated means can often be described very well by fitted tight-binding parameters. Many useful results have been obtained by using the empirical tight-binding models to treat systems beyond the scope of state-of-the-art band theories, even though it has usually not been clear exactly what the tight-binding parameters mean. As we shall see, a number of useful results have also been obtained by fitted energy functionals, containing parameters whose meaning is not at present clearly established. A new derivation and justification of tight-binding theory has recently been obtained,118 which dispenses with the simplistic atomic orbital treatment altogether. Instead, the tight-binding model is obtained by systematic identification and exploitation of small parameters in the full electronic Schrodinger equation. Such models have now been derived for both transition metals118and semiconductors. '19 Thus a major step toward obtaining an ab initio tight-binding description of electronic structure has been taken. It is to be hoped that similar conceptual advances will improve the accuracy of ab initio interatomic potentials and functionals.

11.

PAlR

FUNCTIONALS

Fitted versions of the pair functionals described in Section 5 have by now been applied to the energy and structure of a number of point and extended defects in transition metals; semiconductors cannot be treated with pair functionals because they require angular forces. In many cases considerable improvement over pair potential calculations has been achieved. In the following we discuss a subset of these applications. This subset is chosen to demonstrate in a simple fashion which properties require the higher-order terms implicitly contained in the pair function"'Ref. 17, Chap. 2. "'0.K. Andersen and 0. Jepsen, Phys. Rev. Left. 53, 257 (1984). Ii9O. K. Andersen, Z . Pawlowska, and 0. Jepsen, Phys. Rev. B 34, 5253 (1986).

BEYOND PAIR POTENTIALS

67

als, and to illustrate, as clearly as possible, the effects of these terms. We consider two distinct types of empirical fitting schemes, obtained by the tight-binding859’20(TB) and e m b e d d e d - a t ~ m ’(EA) ~ ~ ~ ~approaches”’ ~~ [cf. Section 5, Eqs. (5.8) and (5.10)]. Both of these use as input the equilibrium lattice constants, the cohesive energy per atom, and the elastic constants. The vacancy formation energy is also used as input for the fcc metals, but not for the bcc metals (for which we will discuss only TB results). In both approaches, the repulsive term V2 is obtained empirically. The approaches differ, however, in their treatment of the attractive terms U , and UEA. In the TB method, U, is taken to be proportional to ( p ~ ) ” ’ and the radial function h(R,)’ is fitted [cf. Eqs. (5.1) and (5.6)]. In contrast, the EA method fixes the radial dependence of n,,(Rij) [cf. Eq. (5.11)] to be that of the free-atom charge density, while the dependence of U,,(n,) on n, is fitted. In many cases, this difference in fitting procedures does not have large effects. However, we shall see that if the property of interest involves a competition between two- and three-body terms, the difference can lead to large discrepancies in physical predictions. a. Vacancy Properties Pair functional approaches have been used to calculate vacancy properties in several bcc and fcc transition metals.84~85~’2’-’28 We focus on the formation energy and the relaxation volume. The vacancy formation energy is a very important metallurgical parameter, since it determines the equilibrium vacancy concentration and thus enters the diffusion constant. The relaxation volume is the change in the volume of the sample due to the relaxations around the vacancy; contraction corresponds to a negative relaxation volume. The relaxation volume affects the strain field of the vacancy and thus its interaction with point and extended defects. Table IV shows results by the TB method for vacancy formation energies Ef:in bcc transition metals. The results labeled “V;” 12’G. J . Ackland, G. Tichy, V. Vitek, and M. W. Finnis, Philos. Mug. A 56, 735 (1987). 121Thedetails of the implementation in the applications we mention vary in some cases from these original papers, but retain the same physical assumptions. 122C.C. Matthai and D. J. Bacon, Philos. Mug. A 52, 1 (1985). 123J. M. Harder and D. J. Bacon, Philos. Mag. A 54, 651 (1986). 124D.J. Oh and R. A. Johnson, J. Muter. Res. 3, 471 (1988). 125R.A. Johnson, Phys. Rev. B 37, 3924 (1988); Phys. Rev. B 37, 6121 (1988). 126K. Masuda, J . Phys. (Paris) 43,921 (1982). 12’W. Maysenholder, Philos. Mug. A 53, 783 (1986). Iz8R. Rebonato, D. 0. Welch, R. D. Hatcher, and J. C. Bilello, Philos. Mug. A 55, 655 (1987).

68

A. E. CARLSSON

TABLE IV. VACANCYFORMATION ENERGIES (IN eV) OBTAINED BY PAIR FUNCTIONAL, EFFECTIVE PAIR POTENTIAL v;*,AND PAIR POTENTIAL v; MATCHED TO COHESIVE ENERGY E', (RELAXED)

E', (UNRELAXED)

V Nb Ta Cr Mo W Fe

TIGHT-BINDING PAIRFUNCTIONAL

TIGHT-BINDING PAIRFUNCTIONAL EXPERIMENT ~~~

1.91" 2.64" 3.13" 1.97" 2.58" 3.71" 2.05"

1.89" 2.53" 3.02" 1.93" 2.49" 3.61" 2.01"

5.31" 7.51" 8.10" 4.10" 6.82" 8.90" 4.28"

~

1.83' 2.48' 2. 87' 1.91b 2.54' 3.62' 1.81'

2.1 f 0.2' 2.6 f 0.3' 2.8 f 0.6' 3.0 f 0.2' 4.0 f 0.2' 1.6 f 0.2d

"M. W. Finnis and J. E. Sinclair, Philos. Mag. A 50,45 (1984); Philos. Mag. 53, 161 (1986). (Values of V;' and V z obtained using inputs supplied in this reference.) bM. J. Harder and D. J. Bacon, Philos. Mag. A 54, 651 (1986). 'K. Maier, M. Peo, B. Saile, H . E. Schaefer, and A . Seeger, Philos. Mag. A 40,701 (1979). dH. E. Schaefer, K. Maier, M. Weller, D. Herlach, A . Seeger, and J . Diehl, Scriptu Mefull. 11, 803 (1977).

are obtained for unrelaxed defects, using a pair potential fitted to the values of the cohesive energies used as input for the pair functional. These results are independent of the form of the potential; as mentioned above, if only pair potentials are included, one has3"

where ECoHis the input cohesive energy. The curve labeled ''V;ffr' is obtained by the expansion procedure described in Section 5b [cf. Eq. (5.13)], using the perfect crystal as a reference environment. We see that the pair functional results, which typically display agreement with the experimental values at the 20% level, are considerably improved over the V ; values, which in all cases are high by more than a factor of 2. These values are unlikely to be significantly improved by the inclusion of lattice relaxation effects. The improvement obtained by the pair functional can be understood by using the effective pair potential Vzff derived from the solid environment. Use of this pair potential, instead of one matched to the cohesive energy, yields the following approximate result: 12"

69

BEYOND PAIR POTENTIALS

where E C o H is the input cohesive energy, and U$% is the value of U,, in the perfect crystal. Since the repulsive contribution to E C o H cancels part of the attractive contribution from U?!, I UFdI > lEcoHl, and more than half of the first term in (11.2) is canceled by the second term. As seen above, this improves the agreement with experiment considerably. The Vgff results are, in fact, within 5% of the full pair functional results. Thus, for the purpose of calculating vacancy formation energies, the effective potentials are essentially as accurate as the full pair functional description. Table V shows relaxation volumes for vacancies in Ni, Cu, Ag, and Au. Here the pair potential V;ff3nnis taken to produce the correct lattice constant and to vanish at the second neighbor distance and beyond. Under these assumptions it is easily seen that the pair potential calculations will obtain zero relaxations, at least in the harmonic approximation. Both of the full pair functional treatments, in contrast, obtain negative relaxation volumes. Thus effective pair potentials based on bulk properties cannot treat this property. The sign of the calculated relaxations can be understood via the dependence of the effective two-body potentials for the system containing the defect [cf. Eq. (5.14)], on the local coordination number. In the notation of Eq. (5.14), these TABLE v. VACANCY RELAXATION VOLUMES (IN UNITS OF BULK ATOMICVOLUME) BY PAIRFUNCTIONALS, EFFECTIVEPAIRPOTENTIAL, AND EXPERIMENT OBTAINED - AQf

EMBEDDEDATOM TIGHT-BINDING PAIRFUNCTIONAL PAIRFUNCTIONAL Ni cu Ag Au

0.12" 0.27" 0.18" 0.41"

0.12b 0.23b 0.22h 0.27b

wnn 0 0 0 0

EXPERIMENT -

0.15-0.25' 0.06d 0. 15-0.48e,f.g

3 . M. Foiles, M. I. Baskes, and M. S. Daw, Phys. Rev. B 37, 10378 (1988). bG.J. Ackland, G. Tichy, V. Vitek, and M. W. Finnis, Philos. Mag. A 56, 735 (1987). T. Ehrhart, J . Nucl. Muter. 69-70, 200 (1978). (Quoted in Ref. b.) dH. Wenzl, in "Vacancies and Interstitials in Metals" (A. Seeger, D . Schumacher, W. Schilling, and J. Diehl, eds.), p. 363. North-Holland, Amsterdam, 1970. "M. Charles, C. Mairy, J. Hillairet, and V. Levy, J . Phys. F 6, 979 (1976). 'P. Ehrhart, H. D. Carstanen, A. M. Fattah, and J. B. Roberto, Philos. Mag. A 40, 843 (1979). gR. M. Emrick, Phys. Rev. B 22, 3563 (1980).

70

A. E. CARLSSON

potentials are given by

(11.3)

where V;’ on the right-hand side is the bulk effective pair potential, 2; and 2, are the coordination numbers of atoms i and j, and Zrefis the bulk coordination number. The increased strength of V;‘ in the reduced coordination number regions around the vacancy is clearly seen in this formula. Figure 27 illustrates this effect in a simplified two-dimensional geometry, with double lines denoting the strengthened bonds. Both the bonds parallel to the vacancy “surface,” and the “back” bonds, pointing away from the vacancy, are stronger than the bulk bonds. The strengthened bonds prefer a shorter bond length than the bulk bonds. This preference can be satisfied by a contraction of the lattice constant, resulting in a reduction of the volume of the system, and a negative relaxation volume. To lowest order in the reciprocal coordination number, neglecting nonuniform relaxations and using Eq. (11.3), one finds in the TB implementation a negative vacancy p r e ~ s u r e ”propor~ tional to - d U g / d & , where & is the volume per atom. Although few vacancy relaxation volumes are well established, the sign of this result is consistent with the rough values shown in Table V. However, one does not expect pair functionals to be accurate for the anisotropic part of the vacancy relaxations, since these should be sensitive to the angular part of the forces. For vacancies in bcc transition metals, comparison of pair functional treatments with higher-order treatments shows large discrepancies in the anisotropic relaxations.’*

SECOND NEIGHBOR

FIG. 27. Sketch of bond-strengthening effects occurring around a vacancy, in simplified two-dimensional geometry. Double lines denote bonds that are strengthened by creation of vacancy.

BEYOND PAIR POTENTIALS

71

Vacancy-vacancy interactions have also been c a l c ~ l a t e d ' ~ " ~ ' ~ ~ ~ ' ~ ~ ~ ' ~ within both the E A and TB schemes. A calculation for unrelaxed vacancies'25 in fcc Cu showed a negligible change in the attractive nearest-neighbor interaction resulting from the nonpair terms. The dominant effect remains the reduced number of broken bonds (or reduced vacancy surface area) resulting from the proximity of the vacancies. The relative effect on the farther-neighbor interactions can be much larger. To understand this effect is is convenient to think of the vacancies as being created in succession. In the simplified twodimensional geometry shown in Fig. 27, creating a second vacancy at the site labeled "second neighbor" involves breaking bonds A and B, both of which are already stronger than bulk bonds. Thus the energy to create the second vacancy is increased and a repulsive contribution to the vacancy-vacancy interaction results. Because the second-neighbor interaction is usually weaker than the nearest-neighbor interaction, this contribution can have a large relative effect. An analysislZ3of vacancyvacancy interactions in bcc transition metals showed that, in comparison with pair potential calculations, the TB approach gives higher energies for the second-neighbor divacancy relative to the nearest-neighbor divacancy. The effect is large enough to render the nearest-neighbor pair energetically favorable with respect to the second-neighbor pair, which is not'23 the case in pair potential calculations including relaxation effects. This result is consistent with the physics described above: in the bcc nearest-neighbor case (including only nearest-neighbour couplings), none of the bonds strengthened by the first vacancy are broken by the creation of the second vacancy, while in the second-neighbor case, four such bonds are broken. b. Surface Properties We consider surface energies, stresses, relaxations, and reconstructions. The surface energy cr is the work required, per unit area, to create a surface, and thus affects the work of fracture. In addition, the orientation dependence of the surface energy determines the equilibrium shapes of small crystals and voids. As has been emphasized by several the surface stress y for a solid differs from the surface lZ9G.J. Ackland and M. W. Finnis, Philos. Mag. A 54, 301 (1986). Shuttleworth, Proc. Phys. SOC.A 62, 167 (1949); Proc. R. SOC. London, Ser. A 63, 444 (1950). 13'J. J. Bikerman, Phys. Status Solidi 10, 3 (1965). I3'A. Zangwill, "Physics at Surfaces," Chap. 1. Cambridge, New York, 1988. 133S.P. Chen, A. F. Voter, and D . J. Srolovitz, Phys. Rev. Lett. 57, 1308 (1986).

13%.

A. E. CARLSSON

72 TABLEVI.

ENERGIES FOR Ni (IN ergs/cm*) OBTAINED AND EFFECTIVE PAIR POTENTIAL. FUNCTIONALS

SURFACE

BY

PAIR

~

FACE

(100) unrelaxed (100) relaxed (1 10) unrelaxed (110) relaxed (111) unrelaxed (111) relaxed

EMBEDDEDATOM TIGHT-BINDING PAIRFUNCTIONAL PAIRFUNCTIONAL 1580" 1730" 1450"

1449h 1444' 1557' 1548b 1156' 1153'

VF"" 1200 1280 1040

"S. M. Foiles, M. I. Baskes, and M. S. Daw, Phys. Rev. B 33,7893 (1986); Phys. Rev. B 37, 10378 (1988). 'G. J. Ackland, G. Tichy, V. Vitek, and M. W. Finnis, Philos. Mag. A 56, 735 (1987).

energy, although the two concepts are equivalent for a liquid. The surface stress is the average tensile stress in the surface. It affects the internal stress, and size, of small particles and whiskers. Surface relaxations are defined as the part of the atomic relaxations at the surface which preserve the symmetry of the ideal surface; surface reconstructions break this symmetry. Both of these quantities affect the surface energy and stress, and also influence the interaction of the surface with adsorbed and physisorbed atoms. The above properties have been evaluated for several metals using both the TB and EA approaches.84,12071299133-14" Calculated surface energies u for unreconstructed Ni surfaces are given in Table VI. We do not quote experimental values because accurate values are not available. However, one would expect the surface energies to be within a factor of 2 of the surface stress of the liquid, which is'41 roughly 1800 ergs/cm2. This holds for the surface energies obtained by the pair functionals. In contrast, a pair potential using the cohesive energy as input obtains14' 134K,Masuda, Phys. Srafus Solidi B 105, K119 (1981). I3'S. M. Foiles, Surf. Sci. 191, L779 (1987), and references therein. 136M. S. Daw and S. M. Foiles, Phys. Rev. Left. 59, 2756 (1987). I3'B. W. Dodson, Phys. Rev. B 35, 880 (1987). I3*B. W. Dodson, Phys. Rev. Left. 60,2288 (1988). 13L)F. Ercolessi and E. Tosatti, Phys. Rev. Left. 57, 719 (1986); F. Ercolessi, M. Parrinello, and E. Tosatti, Philos. Mag. A 58, 213 (1988). 14('H.Gollisch, Surf. Sci. 166, 87 (1986). "lJ. N. Schmit, Ph.D. Thesis, University of Liege (unpublished). (Quoted in Ref. 132.) I4*A. E. Carlsson (unpublished).

73

BEYOND PAIR POTENTIALS

surface energies of 4000-5000 ergs/cm2, much higher than the liquid surface tension. The effective potential V;ff,nnis a radial nearest-neighbor equilibrium pair potential, with its well depth fitted to an unrelaxed vacancy formation energy of 1.4eV, which is close to those obtained by the pair functionals. The V;ff3nnresults are independent of the particular form of the potential, as long as it satisfies the above constraints. Both the TB and E A results are significantly higher than the effective pair potential results. As in the case of the vacancy properties, this can be understood through the increase in bond strength with decreasing coordination number. An atom at a fcc (111) surface has nine nearest neighbors, while a vacancy nearest neighbor has eleven. Therefore the average strength of the broken bonds at the surface will be greater than at the vacancy, and the surface energy will exceed an estimate based on a pair potential derived from the vacancy formation energy. Some calibration of the reliability of the pair functional methods can be obtained by comparison of the TB and E A results with each other. The root-meansquare difference between these two sets of results is 16%; the RMS difference between the average of the TB and E A values, and the pair potential values, is 21%, only 30% larger. Thus, although both the pair functional schemes obtain corrections of the same sign, the accuracy of their determination of the corrections to the pair potential descriptions is probably not much better than 50%. Table VII shows T B results for the surface stress y. V;ff*nnis as in Table VI. The surface stress measures the change in the surface energy TABLE VII. SURFACESTRESSES FOR (100) SURFACES (IN ergs/crn*) BY PAIR FUNCTIONAL AND EFFECTIVE PAIRPOTENTIAL, AND OBTAINED EXPERIMENTAL VALUES FOR LIQUIDS. Y

TIGHT-BINDING PAIRFUNCTIONAL Ni CU Ag Au

977" 2030" 1625" 2997"

V;ff.nn

0 0 0 0

LIQUID SURFACE STRESS(EXIT.) 18Wb 1300' 900'

1 looh

"G.J . Ackland, G . Tichy, V. Vitek, and M. W. Finnis, Philos. Mag. A 56, 735 (1987). J . N. Schmit, Ph.D. thesis, University of Liege (unpublished). (Quoted in A . Zangwill, "Physics at Surfaces," Chap. 1, Cambridge University, New York, 1988.)

74

A. E. CARLSSON

under expansion of the area of the surface: (11.4) where E~~ and E , , ~ are tensile strains parallel to the surface, which is chosen to lie in the x-y plane. Thus, in a bond picture, y is due to the strain dependence of the strength of the bonds broken to form the surface. In a nearest-neighbor equilibrium pair potential (V;',"") model, this strain dependence vanishes, and y = O (cf. Table VII). In pair functional descriptions, however, the bond strength across the surface plane increases under tensile strain parallel to the surface. This is because the effective pair potential becomes deeper with decreasing atomic density [cf. Section 5b]. (In some types of surfaces the bond length across the surface plane also increases under strain parallel to the surface. However, this effect is dominated by the environmental change in the bond strength.) Thus the surface stress is positive. One can also understand the positive value of the surface stress by considering the unstrained surface. As illustrated in the simplified two-dimensional geometry of Fig. 28, the bonds in the surface (as well as the "back" bonds) are strengthened as a result of the reduced local coordination number. Consequently, these bonds prefer a shorter bond length, and a positive tensile stress is obtained. The underlying physics is thus similar to that responsible for the vacancy relaxations. As seen in Table VII, the calculated stress for Ni is comparable in overall magnitude to the surface energy (Table VI), but can in other cases differ from it by more than a factor of 3. Comparison with the liquid surface stress shows agreement within a factor of 2, except for the case of Au. However, the experimental chemical trends, which follow the cohesive energy, are not reproduced. It is not clear at present how much of this discrepancy results

m ---

---_____

SURFACE PLANE

+"BACK-ROND"

FIG. 28. Sketch of bond-strengthening effects occurring at a surface, in simplified two-dimensional geometry. Bonds denoted by triple lines are strengthened more than those denoted by double lines.

75

BEYOND PAIR POTENTIALS

FOR Ni OBTAINED BY PAIRFUNCTIONALS AND TABLEVIII. (110) SURFACERELAXATIONS EFFECTIVEPAIRPOTENTIAL. Ad,,, Ad,,, AND Ad,, DENOTEPERCENTAGE CHANGES IN LAYERSPACINGS FOR FIRSTTHREELAYERS.

EMBEDDED-ATOM TIGHT-BINDING PAIRFUNCTIONAL PAIRFUNCTIONAL A42 A43

-2.3" 0.1" -2.2"

-3.0b -0.3b -3.3b

"CR,""

2

0 0 0

EXPERIMENT -5 to - 9 4 2.5 to 3.5c.d -2.5'.d

"S. M. Foiles, M. I. Baskes, and M. S. Daw, Phys. Rev. B 33, 7983 (1986); Phys. Rev. B 37, 10378 (1988). bG. J. Ackland, G. Tichy, V. Vitek, and M. W. Finnis, Philos. Mag. A 56, 735 (1986). 'R. Feidenhans'l, J. E. Sorensen, and I. Stensgaard, Surf. Sci. W, 229 (1983). (Quoted in Ref. b.) dS. M. Yalisove, W. R. Granham, E. D. Adams, M. Copel, and T. Gustafsson, Surf. Sci. 17

from the difference between the solid and liquid surface stress, and how much is due to flaws in the pair functional model. We now turn to the atomic relaxations in the layers near the surface, using the Ni(ll0) surface as a representative example (cf. Table VIII). A nearest-neighbor equilibrium pair potential (Vq",nn) produces no relaxations. However, if further-neighbor interactions are included, the calculated relaxations typically are outward,'43 in contradiction with the experimental measurements. As seen in Table VIII, the pair functional approaches considerably improve the results, obtaining a shortening of the surface layer back bonds. The magnitude of the shortening is smaller than the rather uncertain experimental estimates. The underlying physics here (cf. Fig. 28) is the same as for the vacancy relaxations, namely the increased back bond strength due to the reduced coordination number at the ~ u r f a c e . ' ~ ~The , ' ~ second ~ layer relaxations are much smaller than those of the first layer, and in some cases, can correspond to an increased bond length. Part of this effect is due precisely to the reduction in the bond length between the first and second layers. 144,145 The reduced bond length increases the values of the squared electron bandwidth p2 or the background electron density n (cf. Section 5) in the second layer and thereby weakens the bonds [cf. Eq. (5.14)] between the second and third layers. This type of interaction between bond lengths can give rise to alternating relaxations penetrating deep into the bulk. 133 I4,R. Benedek, J . Phys. F S , 1119 (1978). lURef. 132, p. 62. 14'D. Tomanek and K. H. Bennemann, Sug. Sci. 163,503 (1985).

76

A . E. CARLSSON ENERGIES (IN ergS/Cm*) TABLEIx. RECONSTRUCTION (110) SURFACES OF fcc METALS. _

_

_

_

_

FOR

~

E(1 x 2) - E ( l x 1) EMBEDDED-ATOM PAIRFUNCTIONAL

TIGHT-BINDING PAIRFUNCTIONAL

~~

Ni Pd Pt cu Ag Au

22" - 10" -46" 19" -8" -37"

-53' -37' -25' -31h

"S. M. Foiles, Surf. Sci. 191, L779 (1987). 'G. J . Ackland, G . Tichy, V. Vitek, and M. W. Finnis, Philos. Mag. A 56, 735 (1987).

Surface reconstructions are often due to much more subtle effects than those treated here, but their trends are still in some cases obtained correctly by pair functional calculations. Table IX shows the EA and TB values for the energy differences between reconstructed and unreconstructed noble and near-noble metal (110) surfaces. The reconstruction treated is a (2 X 1) missing-row nod el,^^,'^^ and both unreconstructed and reconstructed surfaces are fully relaxed. In the EA results a strong trend favoring the reconstructed surface is seen with increasing atomic number, both for the noble and near-noble metals. The trends in these results are consistent with experimental findings, which show unreconstructed surfaces for Ni and Cu, but display missing-row type reconstruction^'^^^^^^ for Pt and Au. However, the favorability of the reconstruction is overestimated somewhat by the theoretical calculations, which predict Pd and Ag to reconstruct, in disagreement with experiment. The EA results can be simply interpreted as resulting from the competition between two- and three-body interactions. 135 The number of nearest-neighbor bonds is the same for the reconstructed and unreconstructed surfaces, but the number of second-neighbor bonds is greater for the unreconstructed case. The EA effective pair potential is attractive at the second-neighbor distance, so the pair terms favor the unreconstructed surface. There are, however, fewer nearest-neighbor triplets in the reconstructed structure. Since the nearest-neighbor effective triplet interaction is positive in the EA method, it follows that the triplet terms '&W. Moritz and D. Wolf, Surf. Sci. 163, L655 (1985). 14'C. M. Chan and M. A. Van Hove, Surf. Sci. 171, 226 (1986).

BEYOND PAIR POTENTIALS

77

favor the reconstructed structure. The trend seen in Table IX results from the fact that these terms grow in relative importance with increasing atomic number. From the EA analogues of Eqs. (5.13) and (5.15), one sees that a measure of the relative strengths of the three- and two-body terms is n r e f ( & A / u g A ) , where nref is the background density in the perfect crystal. The curvature ULA, in turn, is related8" to the Cauchy discrepancy by (11.5) where the sum is over neighbors to a particular atom, and Q0 is the volume per atom. The Cauchy pressure increases with increasing atomic number. This increase is reflected in relatively larger values of which in turn result in larger three-body contributions, favoring the reconstruction. A closely related interpretation13x has attributed surface reconstructions to the surface stress. As discussed above, this stress is due to the environmental dependence of the bond strength, which in turn increases with UAA. The above trend in reconstruction energies is, however, quite sensitive to the particular implementation of the pair functional. For example, in the TB implementation it is not observed (cf. Table IX). This is probably because the ratio pyf(U ! B / U & ) ,which determines the relative strengths of the two- and three-body terms, is fixed at - 4 by the square-root behavior of UTB; in contrast, in the E A scheme nr,f(&,/&,) varies freely. Thus in the TB implementation, the increasing values of C,*- C44 are not necessarily associated with increases in the magnitude of the three-body terms. It is not a priori clear which of the two implementations is more reliable, and further analysis is needed in order to develop the optimal pair functional description of surface reconstructions. A c a l ~ u l a t i o n 'via ~ ~ the TB scheme has also been performed for the W( 100) surface, which has a (2 X 2) reconstruction. 14' The calculations found no such reconstruction. This result is not surprising in view of the quantum-mechanical electronic structure c a l ~ u l a t i o n s which ' ~ ~ have been performed for the W( 100) surface. These calculations have attributed the driving force for the reconstructions to splittings of surface-state bands and consequent reductions of the electronic density of states near the Fermi level. Since these effects involve the shape of the density of states rather than its width, it is also necessary to employ descriptions more sophisticated than pair functional treatments. Felter, R. A. Barker, and P. J. Estrup, Phys. Rev. Left. 38, 1138 (1977). 149C.L. Fu, A. J. Freeman, E. Wimmer, and M. Weinert, Phys. Rev. Lett. 54,2261 (1985).

I4&I..

78

A . E. CARLSSON

Other applications of pair functionals to elemental metals include phonon~,'~' ion bombardment sim~lations,'~' d i s l ~ c a t i o n s , ' ~grain ~ boundarie~,"~interstitial^,'^^^"^"^^ and liquid The calculations of properties not involving large density changes in general show less dramatic effects due to the nonpair potential terms. However, an important improvement of the pair functionals for such properties is that the observed Cauchy discrepancies in the elastic constants can be reproduced, which as mentioned above is impossible with equilibrium pair potential schemes. Having the correct elastic constants in the pair functional schemes must certainly improve the long-wavelength part of the calculated phonon spectrum, although drastic improvements over the whole spectrum are not seen. For example, the (001) zone edge phonon frequencies for Ni obtained by a nearest-neighbor equilibrium pair potential matched to the bulk modulus'57 are within 3% of the E A results. Calculations86 of the equilibrium structure of liquid Cu via the EA also show only small differences between the effective pair potential results and those obtained from the full pair functional theory.

12. ANGULAR FORCES As mentioned in Section 1, the necessity of angular forces in stabilizing most semiconductor structures has led to a much larger effort to obtain angular forces for these materials than for metals. The physics underlying the angular forces in transition metals, discussed in Section 6, has been incorporated in few low-order fitted functionals and potential^'^'^ comparable to those discussed in Section 11. Most of the effort in semiconductors has been directed toward Si. We will maintain this emphasis, because it is thus possible to compare several different types of schemes, and their implications for physical properties. Some extensions of the schemes described here to other materials are described in Refs. ''OM. S. Daw and R . L. Hatcher, Solid State Commun. 56,697 (1985). '"B. J. Garrison, N. Winograd, D . M. Deaven, C. T. Reimann, D . Y . Lo, T. A. Tombrello, D. E. Harrison, Jr., and M. H. Shapiro, Phys. Rev. B 37,7197 (1988). lS2A. K. Sat0 and K. Masuda, Philos. Mug. B 43, 1 (1981). ls3Y. Oh and V. Vitek, Acta. Mefull. 34, 1941 (1986). Is4J. M. Harder and D . J. Bacon, Philos. Mag. A 58, 165 (1988). "'G. J. Ackland and R. Thetford, Philos. Mug. A 56, 15 (1987). lS6F. Aryasetiawan, M. Silbert, and M. J. Stott, 1. Phys. F 16, 1419 (1986). ls7A. E. Carlsson (unpublished). 157aSee,however, K. Masuda, R. Yamamoto, and M. Doyama, J . Phys. F13, 1407 (1983).

BEYOND PAIR POTENTIALS

79

158-161. The physical assumptions underlying the angular forces are less well established than for the pair functionals discussed in the preceding section. Therefore we will refer to the angular force schemes by the names of their originators rather than by their physical content. The organization of this section follows that of Fig. 1. Applications of pair functionals having been treated in Section 11, we now treat separately cluster potentials and cluster functionals. These are obtained by matching an assumed form for the interactions to an experimental or theoretical database. Because of the vast number of parameters needed to describe an arbitrary four-body or higher-order interaction, the cluster potentials and functionals have so far been obtained mainly at the three-body level. In addition, only assumed functional forms for the angular dependence have been used; none have been derived from theoretical analysis. Before describing the interatomic potentials and functionals, we note that there is a fundamental limitation16’ to their use in semiconductors. This is the assumption that the state of the electrons is determined uniquely by the positions of the ions. Such an assumption is valid provided that the ions move slowly enough for the electrons to always be in their equilibrium state, and that external electron sources or sinks are absent. Some defects in Si (and other semiconductors and insulators) can have several long-lived charge ~ t a t e s , ’ ~ which ~ * ~ lead ~ to differing configurational energies. The population of these charge states at a given time is dependent on the history of the system and on external electron radiation conditions. Such dependences are not included in descriptions based on interatomic potentials and functionah. Therefore these schemes cannot reliably treat defects with multiple long-lived charge states. a, Cluster Potentials The use of angular forces to model properties of semiconductors in which bond lengths and angles deviate only slightly from their ideal ”‘F. H. Stillinger, T. A . Weber, and R. A . LaViolette, J . Chem. Phys. 85, 6460 (1986). lS9E.Pearson, T. Takai, T. Halicioglu, and W. A . Tiller, J . Cryst. Growth 70, 33 (1984). ImD. K. Choi, T. Sakai, S. Erkoc, T. Halicioglu, and W. A . Tiller, J . Cryst. Growth 85, 9 (1987). 16’K. Ding and H. C. Andersen, Phys. Rev. B 34, 6987 (1986). 16*R.Biswas and D . R. Hamann, Phys. Rev. B 36, 6434 (1987). 163M. Lannoo and J . Bourgoin, “Point Defects in Semiconductors I,” Chap. 4. Springer, New York, 1981. J . Bourgoin and M. Lannoo, “Point Defects in Semiconductors 11,” Chapters 2 and 3. Springer, New York, 1983. ‘@A. Zunger, in “Solid State Physics: Advances in Research and Applications” (H. Ehrenreich and D. Turnbull, eds.), p. 275. Academic, New York, 1986.

80

A. E. CARLSSON

TABLE x. FUNCTIONAL FORMS OF THREE-BODY POTENTIALS FOR SI. IMPLEMENTATION

FORMOF V ; " ( R i , R), R , )

"F. H. Stillinger and T. A. Weber, Phys. Rev. B 31, 5262 (1985). bR.Biswas and D. R. Hamann, Phys. Rev. Let!. 55, 2001 (1985); Phys. Rev. B 36, 6434 (1987). 'E. Pearson, T. Takai, T. Haliocioglu, and W. A. Tiller, J . Cryst. Growth 70, 33 (1984).

crystal values is by now quite familiar. 16s16R Typically the potentials are matched to bulk elastic constants. Three recently developed types of SchemeS159.162,169,170for Si have attempted to expand the range of applicability of such potentials to point and extended defects by including an appropriate decaying radial dependence171 and using a larger input database. The development of these potentials has already stimulated a large number of atomic simulations of materials properties of Si. Stillinger and Weber'69 (SW) have developed a three-body potential for simulating local order in liquid Si. The functional form given in Table X was assumed for the three-body interaction. Here V is a positive constant having units of energy, g is a radial function cut off at a finite separation, and ei is the angle at atom i between the bonds directed at atoms j and k. The primary justification for this form is that if V is sufficiently large it guarantees the stability of the diamond structure relative to more closely-packed structures. The three-body terms vanish for tetrahedral environments, in which cos 8i = -4, but are usually positive. Biswas and Hamann'62*'7"(BH) subsequently developed a more elaborate scheme, which has more flexibility. The form of the angular forces was chosen to optimize computational speed without having to truncate the potentials at short interatomic spacings. The BH form, which generalizes the separable SW potential, is shown in Table X. The Ih5P. N. Keating, Phys. Rev. 145, 637 (1966). "E. 0. Kane, Phys. Rev. B 31, 7865 (1985). '07R. Jones, J . Phys. C 20, L271 (1987). '68Ref. 17, Chap. 8. 16%. H. Stillinger and T. A. Weber, Phys. Rev. B 31, 5262 (1985). 170R,Biswas and D. R. Hamann, Phys. Rev. Len. 55, 2001 (1985). I7'See also D. W. Brenner and B. J. Garrison, Phys. Rev. B 34, 1304 (1986)

BEYOND PAIR POTENTIALS

81

V, are constants having units of energy and the PI are Legendre polynomials. The SW form, for example, is obtained for go = g , = g,, V, = V, = $V,. The BH form is very convenient because it allow^'^" the three-body energy associated with a particular site to be expressed in terms of quantities which have a pair form:

where the

are “structural moments,” ni, is the unit vector in the direction of R , , and the are spherical harmonics. The structural moments can also be used to generate interactions of higher order than three-body, although this possibility has to the author’s knowledge not yet been implemented. Finally, Pearson et al. 159 (PTHT) have developed a three-body potential for Si which uses the functional form of the Axilrod-Teller triple fluctuating dipole intera~tion’~’(cf. Table X). It is instructive to compare these assumed forms with the functional forms that are obtained via a moment a n a l y ~ i s ”of ~ the electronic density of states. As was shown in Section 6, there is a direct connection between the angular dependence of these moments and the angular dependence of the associated interatomic potentials [cf. Eq. (6.3)]. In keeping with the goal of obtaining descriptions with low-order interatomic potentials, we consider only the moments up through p4. As was discussed in Section 6, a description at this level can begin to discern the effects of a gap in the density of states, via reduced values of p4. We treat only the three-body terms, since none of the empirical schemes we discuss have included four-body terms. The results which we describe are obtained for a tight-binding model containing an sp3 orbital basis set on each site. This model is a multiband form of Eq. (5.1), with i = j terms included to account for the difference between the s and p energy levels. We treat angular contributions to p4 first; as will be seen below, the angular contributions to p3 are strongly reduced in the diamond structure. In the diamond structure, the largest three-body contribution to p4involves a self-retracing path (cf. Fig. 29). The angular dependence

xm

M. Axilrod and E. Teller, J . Chem. Phys. 11, 299 (1943). A. E. Carlsson , in Atomistic Modeling of Materials: Beyond Pair Potentials, edited by V. Vitek and D. R. Srolovitz (Plenum, New York, in press).

17’B. 173

82

A. E. CARLSSON

TYPE OF PATH

Ai

MOMENT

FUNCTIONAL FORM

I

k

!4

g(Rij)g(R,,)g(R,,) x [a0 + a l cos ei + a? cos2 8, + a’* cos ej cos 8, + a3 cos 8, cos 0, cosOk]

FIG. 29. Sketch of three-body contributions to p4 and p 3 . Functions g, and g are radial functions determined by interatomic couplings. Oidenotes angle between i-j bond and i-k bond. aiare constants. In p3 contribution and in 1 = 0 p4 contribution, all s and p couplings are assumed to have same radial dependence.

of this contribution has a separable form very similar to that of the BH potential, except that the replacement P,(cos Oi) (cos @ ) I is made. The radial functions g, are simply bilinear combinations of the interatomic couplings. The only angular contribution to p3 comes from the triangular path shown in Fig. 29. The functional form of this contribution differs from that of the SW and B H potentials in that it contains a product of three radial functions instead of two. In the diamond structure one of these functions must involve at least a second-neighbor hop, reducing the contribution of this term. However, it can be important in more closely packed structures. The angular part contains products of cosines of different angles in the triangle. One of the terms, in fact, gives the PTHT form. Terms of similar form also contribute to p4, if one of the “hops” is on site. In the diamond structure these again involve second-neighbor hops and therefore make significantly smaller contributions than the self-retracing paths. Thus some of the types of terms which appear in the empirical forms can be obtained from a quantum-mechanical analysis. This analysis may then be useful in determining the relative weights of these terms, and in discriminating between the various types of empirical forms. However, there is at present no quantum-mechanical justification for neglecting four-body terms. Although nearest-neighbor four-atom paths are absent in the diamond structure, they are present in the competing crystal structures, and thus cannot be ignored in calculating structural energy differences. All of these schemes contain a radial pair potential in addition to the angular terms. This potential, and the radial functions g and g , (cf. Table

+

BEYOND PAIR POTENTIALS

83

X), have assumed analytic forms. The parameters in these forms, as well as V and V, in Table X, are determined by criteria which vary considerably between the schemes. The criteria used for the SW potential were that the diamond structure should be favored over more closely packed structures, and that the melting point and structure factor of the liquid should be obtained reasonably accurately. Two different data sets were used as input for determining the parameters in the BH potentials, and were found to result in very distinct potentials. The first potential, (BH-1) was fitted to the volume-dependent energies of nine hypothetical and physical bulk structures, along with the layer separation-dependent energies of two- and four-layer Si(ll1) slabs. The second potential (BH-2) was fitted to a smaller subset of bulk structural energies, augmented by energies of diamond structure and simple cubic slabs and the interstitial-vacancy pair formation energy. In addition, the three-body term was constrained to be a sum of terms having the SW form (cf. Table X), so that it is manifestly positive if R , = Rik. The PTHT potential was fitted to the binding energies and bond lengths of the Si, and Sig molecules, and the diamond-structure crystal. The resulting two-body potentials are compared in Fig. 30a. They display large variations in depth and shape. The shape of the BH-2 pair potential is fairly similar to that of the SW potential, but the BH-1 potential is much longer ranged and considerably shallower. The PTHT potential is deeper and has a larger curvature at the minimum than the other three. The triplet potentials are shown in Fig. 30b for an isosceles triangle with Rij = Rjk = 2.35 A. These potentials display even more variation than the pair potentials. The BH-2 triplet potential is constrained to have the same shape as the SW potential, having a well-defined minimum at 8 = 109". However, the BH-1 potential deviates strongly from this form, becoming a negative at 8 = 65" and remaining roughly constant for 8>90". The PTHT potential, like the BH-1 potential, decreases monotonically up to 6, = 180°, but is much smaller in magnitude. The discrepancies between potentials obtained by using different input data sets and assumed functional forms is reminiscent of the spread seen in Fig. 4, for pair potentials in Cu. These discrepancies suggest that even the inclusion of three-body interactions does not provide enough flexibility to describe all possible configurations of Si atoms, and that four-body or higher-order terms are needed. In fact, the SW and BH-1 schemes, and most likely the others as well, make significant errors in treating one or more properties that could be used as ~ a l i b r a t i o n , 'such ~~ 174E.R. Cowley, Phys. Rev. Lett. 60,2379 (1988).

A. E. CARLSSON

84

-6

- >

, ....... ... .......... ., ,., .......

0

30

60 90 120 150 180

8

(degrees)

FIG. 30. Two- and three-body empirical effective potentials for Si. R,, denotes nearestneighbor distance in diamond structure. Triplet potentials evaluated for isosceles triangles having two bond lengths equal to Rnn;6 is angle between these bonds. SW curves obtained from parameters given by F. H. Stillinger and T. A. Weber, Phys. Rev. B 31, 5262 (1985); BH-1 and BH-2 curves taken from R. Biswas and D. R. Hamann, Phys. Rev. B 36, 6434 (1987); PTHT curves obtained from parameters given in E. Pearson, T. Takai, T. Haliocioglu, and W. A. Tiller, J . Cryst. Growrh 70, 33 (1984). Note change of scale for BH-1 V;' curve.

as phonon spectra, point defect energies, or crystal-structure energy differences. This is certainly consistent with the analysis of Section 6, which suggests that the lowest-order description of many structural energy differences is obtained only at the four-atom level. Nevertheless, one hoped6* that the triplet potentials can at least describe classes of problems which are large enough to be useful in simulation studies. Such

85

BEYOND PAIR POTENTIALS

classes could, for example, be analogous to the "constant-volume" and "broken-bond'' classes discussed above for transition metals. These potential schemes have been used to simulate a large number of nonideal geometries, including point defect^'^"'^^ small clusters, surfaces, 170.176,17%18S interfaces,'8c188 strained layer^,'^^,'^' finite temperature elastic constant^,'^^ the a m o r p h o ~ s ' ~ ~and ' " l i q ~ i d ' ~states, ~.'~~ and phase transition^.'^^,'^^ Here we treat surface reconstructions, small clusters, and the amorphous state, because these properties allow the most straightforward comparison between the results of different schemes. Si surface reconstructions have been treated with all of the schemes described above. In all cases the (100) is found, correctly, to exhibit a (2 x 1) reconstruction. The physics behind this appears to be contained in the short range of the pair potentials. For all the potential schemes, V, is several times smaller in magnitude at the second-neighbor distance than at the nearest-neighbor distance. In the ideal (100) surface, each surface atom has two broken bonds, and the shortest separations are at the second-neighbor distance. The (2 X 1) reconstruction allows pairs of atoms to move close to the nearest-neighbor distance, sampling the '62,177317H

P. Batra and F. F. Abraham, Phys. Rev. B 35,9552 (1987). 176B.W. Dodson, Phys. Rev. B 33, 7361 (1986). 177E.Blaisen-Baroja and D. Levesque, Phys. Rev. B 34, 3910 (1986). 178B.P. Feuston, R. K. Kalia, and P. Vashishta, Phys. Rev. B 35, 6222 (1987). 17%. F. Abraham and I. P. Batra, Surf. Sci. 163, L752 (1985). Is?. Takai, T. Halicioglu, and W. A. Tiller, Surf Sci. 164, 341 (1985). I8IM. Schneider, I. K. Schuller, and A. Rahman, Phys. Rev. B 36, 1340 (1987). IR2E.T. Gawlinski and J. D. Gunton, Phys. Rev. B 36, 4774 (1987). IE3K.E. Khor and S. Das Sarrna, Phys. Rev. B 36, 7733 (1987). IwH. Balamane, T. Halicioglu, and W. A. Tiller, J. Cryst. Growth 85, 16 (1987). 185E.M. Pearson, T . Halicioglu, and W. A. Tiller, J. Cryst. Growth 83, 502 (1987). 186F.F. Abraham and J. 0. Broughton, Phys. Rev. Lett. 56, 734 (1986). 187J. Q. Broughton and F. F. Abraham, 1. Cryst. Growth 75, 613 (1986). I8'U. Landman, W. D. Luedtke, C. L. Cleveland, M. W. Ribarsky, E. Arnold, S. Ramesh, H. Baumgart, A. Martinez, and B. Khan, Phys. Rev. Lett. 56, 155 (1986). I8'U. Landman, W. D. Luedtke, M. W. Ribarsky, R. N. Barnett, and C. L. Cleveland, Phys. Rev. B 37,4637 (1988). '9. D. Luedtke, U. Landman, M. W. Ribarsky, R. N. Barnett, and C. L. Cleveland, Phys. Rev. B 37, 4647 (1988). 19'B. W. Dodson and P. A. Taylor, Appl. Phys. Lett. 49, 642 (1986). Iy2M. D. Kluge and J . R. Ray, J. Chem. Phys. 85, 4028 (1986). 193J. Q. Broughton and X. P. Li, Phys. Rev. B 35, 9120 (1987). 194M. D. Kluge, J. R. Ray, and A. Rahman, Phys. Rev. B 36, 4234 (1987). ly5R. Biswas, G. S. Grest, and C. M. Soukoulis, Phys. Rev. B 36, 7437 (1987). '%W. D. Luedtke and U. Landman, Phys. Rev. B 37,4656 (1988). I9'T. Takai, T. Halicioglu, and W. A. Tiller, Scr. Metall. 19, 709 (1985).

I7'I.

86

A. E. CARLSSON

deeper part of V, and thus reducing the total energy. The calculated reconstruction energies range from -0.61 eV to -0.84 eV per surface atom, with the surface bond lengths ranging from 2.41 A to 2.50A; in comparison, fully quantum-mechanical c a l c u l a t i ~ n s ' ~give ~ ~ 'reconstruc~ tion energies of -0.85 eV and -1.03 eV, and bond lengths of 2.25 A and 2.22A. Thus the gross rebonding effects occurring at this surface are treated fairly well even by low-order interatomic potentials. At the Si(ll1) surface each surface atom has only one broken bond, and the reconstruction energetics are more subtle. Here the PTHT potential produces a reconstr~ction'~~ but the SW potential does not.'76 It is not clear whether or not this surface can be treated by schemes truncated at the three-body level. The structure of sTall Si clusters has been treated by the SW, BH-1, and BH-2 potentials,'using both static and dynamic methods. For three-, four- and five-atom clusters, the lowest energy configurations obtained by the SW potential were triangular (equilateral), square, and pentagonal, r e ~ p e c t i v e l y ' ~ ~ .The ' ~ ~ . BH-2 potential obtained four- and five-atom structures in agreement with the SW results, but the three-atom minimum-energy structure was found to be an isosceles triangle rather than an equilateral one162.The BH-1 potential, in contrast, found the equilateral triangle as the lowest energy structure'62. For four and five atoms, however, it gave pyramidal shapes as minimum-energy structures. Although there are no direct experimental measurements of the structures of these clusters, fully quantum-mechanical total energy calculations have been performed.2w.20'The three-atom minimum-energy structure is found to be an isosceles triangle, in agreement with the BH-2 results. However, the four-atom structure, a flat rhombus, and the five-atom structure, a trigonal bipyramid, are in disagreement with all of the cluster potential calculations. This is not surprising in view of the fact that the structural energy differences used as input are obtained for bulk structures; the reduced coordination at the cluster surfaces undoubtedly has strong effects on structural energies. These are not included in the input database, although the broken-bond energies themselves are. We do not expect interatomic potential schemes to reliably treat geometries grossly different from those in the input database. 6 BH-l'95 potentials have been applied to the Both the ~ ~ 1 9 3 , 1 9 4 , 1 9and structure of amorphous Si. The calculations have matched the ex19'M. T. Yin and M. L. Cohen, Phys. Rev. E 26, 5668 (1982); Phys. Rev. Let?. 45, 1004 (1980). IWK. C. Pandey, Phys. Rev. Let?. 49, 223 (1982). zmK. Raghavachari and V. Logovinsky, Phys. Rev. Lett. 55, 2853 (1985). '''D. Tomanek and M. Schliiter, Phys. Rev. Len. 56, 1055 (1986).

BEYOND PAIR POTENTIALS

87

perimental structure factor well. All of the calculations have, however, produced a large density (210%) of coordination defects. For the SW potential, overcoordination (fivefold) defects dominate. For the BH-1 potential, roughly equal numbers of threefold and fivefold defects are found. Defect fractions as high as 10% are probably much too A calculation'61 for amorphous Ge, using a modified version of the SW potential, has obtained a much smaller density of coordination defects. The likely reason16' for this difference is that the angular three-body terms are stronger, relative to the two-body terms, in Ge than Si. This causes a larger energy penalty for overcoordination, resulting in a structure with closer to ideal coordination. The large defect densities obtained for amorphous Si do not necessarily imply defects in the potential schemes, since the simulations d o not directly model the actual growth process. In one calculation'"' the structure was obtained by creating topological defects in the crystal; in the rest it was obtained by quenching the liquid. It is not clear whether the simulations have treated a a long enough period of time to reach the amorphous structure, nor whether this structure would ever be reached from the starting points that were used. However, one can compare to ab initio molecular dynamicszo3studies of quenched liquid Si, using a small sample of 54 atoms (the interaction potential simulations typically use several hundred atoms). Only two of the atoms (or 4%) were under- or overcoordinates, a lower fraction than was obtained by the interatomic potential simulations. It is thus likely that the interatomic potential schemes underestimate the energy of the coordination defects in amorphous Si. The above results reveal strengths and weaknesses quite similar to those of the pair functionals discussed in Section 11. Properties associated with gross coordination number changes, such as the (100) surface energy and reconstruction, are handled fairly well. More subtle structural energies are not treated reliably. This is not surprising, since the structural energies are strongly dependent on the angular part of the forces, whose functional form is assumed, rather than derived. The use of angular forms based on a quantum-mechanical analysis would probably improve the reliability, but it is by no means definitely established that low-order potentials of any form can accurately treat the structural energies of interest. *''An estimate of 0.1%, on the basis of electron-spin-resonance experiments, is probably more accurate. See R. Zallen, "The Physics of Amorphous Solids," Chap. 6. Wiley, New York, 1983. '03R, Car and M. Parrinello, Phys. Rev. Lett. 60, 204 (1988).

88

A. E. CARLSSON

b. Cluster Functionals As discussed in Parts I and 11, there are strong reasons for abandoning a description based entirely on interatomic potentials, and instead using functionals which automatically take into account the environmental dependence of the effective potentials that emerge as approximations. Two classes of cluster functionals have been developed for Si, based on approaches due to Tersoffz9~zo4~z05 (subsequently modified by DodsonZM and Khor and Das Sarma'") and Baskes.'08 The following energy functional was developed by Ter~ o ff:* ~

E, =4

C [Vy'(Ri,) - Bi,({R})Vp)(Rij)].

(12.2)

i,i

Here VV) and V p ) are repulsive and attractive radial pair potentials. The Bij factor is determined by the positions of all the other atoms, { R } :

B , ( { R } ) = (1 + P"cG)-$", where

and fc is a cutoff function. The functions V p ) , V p ) , q , and Y are given assumed parametrized analytic forms: the angular dependence, for example, has the form Y ( ei)= c + e P d'OS . Thus the functional form of the angular dependence is not derived from a quantum-mechanical analysis, but is assumed. The input database used to fit the parameters, as in the BH a p p r ~ a c h e s , ' ~ ~includes ~"" a large number of ab initio total energies for bulk and surface structures. The Baskes schemezo8 utilizes the following modification of the EA pair functional form: *I

2"J. Tersoff, Phys. Rev. Lett 56, 632 (1986). Tersoff, Phys. Rev. B 38, 9902 (1988). *06B. W. Dodson, Phys. Rev. B 35, 2795 (1987). 2"7K. E. Khor and S. Das Sarma, Phys. Rev. B 38, 3318 (1988) '"'M. I. Baskes, Phys. Rev. Lett. 59, 2666 (1987).

.J'"'

BEYOND PAIR POTENTIALS

89

where

Here the functions V2, UEA, and nat, and the parameter A, are empirically adjusted. As in the Tersoff scheme, the angular dependence is empirically obtained. The input database includes the energy, bond length, and elastic constants in the diamond structure, in addition to the bond lengths of other structures having both larger and smaller coordination numbers. Both of the cluster functional approaches generalize the pair and triplet potential descriptions by including the environmental dependence of the effective bond strength, which was discussed in Section 5b. In the Tersoff scheme this is due to the B, term [cf. Eq. (12.2)]. With increasing coordination number Z, for example, the r;i, term increases and B, thus decreases, reducing the bond strength. For very large Z, the attractive part of the bond strength is proportional to Z-"' if the angular terms are neglected. This is exactly the result obtained by the tight-binding pair functional analysis of Section 5. In the Baskes approach, the environmental dependence of the bond strength is automatically included via the curvature of UEA. However, no theoretical analysis has yet shown that the assumed angular dependence, in either of the cluster functionals, is appropriate for describing structural energy differences. As discussed in the preceding subsection, four-body angular terms may well contribute significantly to those differences; no existing cluster functionals for Si include such terms. The cluster functional schemes provide promising results for some bulk and defect properties. The Baskes scheme obtains excellent structural energies.2o8Defect energies are obtained in both schemes to within 30% of the ab initio theoretical values (which are themselves quite uncertain). The Tersoff scheme and modified versions of it treat Si surface reconstruction^^^ with roughly the same accuracy as the cluster potentials discussed above; the (2 x 1) reconstruction energy and bond length for the (100) surface are in good agreement with the quantum-mechanical results, but the presence or absence of the (2 x 1) reconstruction on the (111) surface depends on the details of the implementation. The amorphous Si structure209obtained by the Tersoff scheme has a smaller density of defects than those obtained by the SW or BH schemes, and may thus be more accurate. The calculated structures and defect energies are, however, as in the case of cluster potentials, quite sensitive to the '09P. Kelires and J. Tersoff, Phys. Rev. Lett. 61, 562 (1988).

90

A. E. CARLSSON

details of the implementation. For example, an early version204 of the Tersoff method led to bcc and hcp energies too high and too low, respectively,206 in addition to an unrealistically small hexagonal-site interstitial formation energy.29 Since the form of the angular dependence has been assumed, rather than derived from explicitly quantummechanical considerations, these variations are not surprising. Comparison with a very broad database will be necessary to establish the classes of properties for which the cluster functionals are useful. IV. Concluding Comments We have seen that pair potentials cannot by themselves describe the broad range of atomic configurations that must be treated to understand most materials properties of transition metals and semiconductors. Nevertheless, there is substantial theoretical justification for the description of a narrowly delimited range of processes by effective interatomic potentials, and for the use of pair and cluster functionals to treat a broader range of atomic configurations. Such descriptions, which involve an approximate treatment of the quantum mechanical electronic bonding energy, are of course less accurate than fully quantum-mechanical treatments. This loss of accuracy is compensated for both by the enormous reduction in computing time that is obtained, and by the interpretability of the results obtained in calculations of bulk and defect properties. Thus calculations with such simplified schemes will, for the foreseeable future, provide a useful complement to fully quantummechanical calculations. The contribution of theoretical analysis in this field has been the development of both ab initio potentials and functionals, and improved functional forms for empirical fitting schemes. The latter contribution is particularly important, because of the very large number of calculations that have been performed with such schemes. The basic form of the radial pair functionals, which describe broken-bond energies and associated relaxations in transition metals, is fairly well established. Fitted schemes which are based on different analyses of the electronic structure result in functionals which are quite similar to each other, and whose predictions for many physical properties are consistent. The form of cluster potentials and cluster functionals, which are necessary for describing structural energy differences in transition metals and semiconductors, is less well established. Quantum-mechanical treatments based on both free-electron and tight-binding analysis have provided some understanding of the form of low-order cluster potentials and functionals.

BEYOND PAIR POTENTIALS

91

It is hoped that this understanding will be deepened by further theoretical analysis, and that it will be incorporated into the fitted schemes used in atomistic simulations. Although low-order potentials and functionals can never provide the quantitative accuracy of fully quantum-mechanical calculations, increasingly sophisticated descriptions of this type will surely reveal new physical effects which transcend present treatments. ACKNOWLEDGMENTS I am especially grateful to Henry Ehrenreich, who introduced me to the complexities of this topic, and to Neil Ashcroft, who for several years encouraged my interest. Robert Phillips has considerably improved this article through several patient and critical readings. I would also like to deeply thank Pranoat Suntharothok-Priesmeyer for her efforts in the preparation of the manuscript and the figures. This work was supported by the Department of Energy under Grant Number DE-FG02-84ER45130.

SOLID STATE PHYSICS, V O L U M E 4.3

Continuous Symmetries and Broken Symmetries in MuItivalley Semiconductors and Semimetals M. RASOLT Solid State Division Oak Ridge National Laboratory Oak Ridge, Tennessee

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . .

......................... Band-Structure Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isospin and the Effective-Mass Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . Ferromagnetic Alignment of the Isospin-Spin Ground State . . . . . . . . . . . Antiferromagnetic Alignment in Isospin-Spin Space . . . . . . . . . . . . . . . . . . Other Broken Symmetries in Isospin-Spin Space. . . . . . . . . . . . . . . . . . . . . .

11. Isospin Space and the Effective-Mass Ha

1. 2. 3. 4. 5. 6. Isospin-Spin Symmetry in the Presence of a Uniform Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Symmetry-Breaking Terms in the Effective-Mass Hamiltonian . . . . . . . . . . . . . . . 7. The Energy of H in the R P A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Effective Hamiltonian for the Valley Electrons . . . . . ...... 9. The Effective-Mass Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Symmetry-Breaking Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Effect of an External Magnetic Field ..... IV. Isospin Space and the Integer Quantum Hall Effect.. . . . . . . . . . . . . . . . . . . . . . . 12. Isospin Polarized Ground State in the (110) Silicon Inversion .................. Layer 13. Effect of the Isospin Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. Effect of Disorder on the Nature of the Isospin Polarization . . . . . . . . . . . 15. Valley Waves in the 110 Inversion Layer of S i . . . . . . . . . . . . . . . . . . . . . . . . 16. Dissipation in Isospin Space.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Isospin Space and the Fractional Quantum Hall Effect 17. The Fractional QHE (FQHE) Ground State i Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18. The Single Mode Approximation (SMA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19. Collective Modes in the Fully Isospin Polarized FQHE . . . . . . . . . . . . . . . . 20. Collective Modes in the Isospin Unpolarized FQHE . 21. Isospin Representation of the QHE in a Two-Layer Heterojunction . . . . .

94 95 95 98 102 105 109 113 114 114 116 122 125 130 132 133 138 142 148 151 156 156 165 168 175 178

93 Copyright 01990 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-607743-6

94

M. RASOLT

VI. Isospin in Three Dimensions . . .

......................

......

182

23. Isospin BS in Three Dimensions in the Presence of an External Magnetic Field . . . . . . . . . . . . . . . .................

187

25. Effect of Quantum Fluctuations. ....................

..........

209

........... ...........................

223 226

1. Introduction

It has been long recognized that, for example, silicon (when donor doped) contains six pockets of electrons in the first Brillouin zone. Groups of two pockets, along 100, 010, and 001 are related by inversion symmetry. Of course, these pockets of electrons are connected as well by the various other point-symmetry operations. What has been so far largely overlooked is that to a very high degree of accuracy the two pockets related by inversion symmetry satisfy the continuous SU(2)symmetry Lie group. The consequences of this continuous internal symmetry (which we will refer to as isospin space) is the subject of this review article. Having recognized this continuous symmetry, the analogy between isospin and spin space is immediate. For example, if the free energy predicts broken symmetry configuration in spin space (e.g., a ferromagnet) the corresponding SU(2) symmetry then predicts the existence of low-lying Goldstone modes (Le., spin waves). Similarly, if the free energy predicts a preference for more electrons in one of the valleys (i.e., isospin polarization) then the corresponding SU(2) symmetry predicts another low-lying Goldstone mode which we refer to as valley waves. The internal SU(2) symmetry in isospin space is apparent immediately when we approximate the different valleys by an effective mass Hamiltonian; this we do in Section 11. It is, however, very important to determine the accuracy of this isospin description when we go beyond the effective mass Hamiltonian. Corrections, beyond the effective mass approximation, will break the perfect isospin SU(2) symmetry; we discuss these symmetry-breaking terms in Section 111. In Sections IV-VI we discuss some applications of isospin space to a variety of properties in condensed-matter physics. Sections IV and V look at isospin space properties in two dimensions; such as the quantum Hall effect in a silicon

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

95

110 surface. The large magnetic field perpendicular to the twodimensional plane will be seen to ensure an isospin broken symmetry. The ensuing low-lying valley waves will provide a dissipation channel in both the integral and fractional quantum Hall effect , whose implications will be discussed in these sections. The coupling of the donor impurities to the isospin polarization is also fascinating and will also be covered in these two sections. In Section VI we discuss isospin properties in three dimensions. The presence of a magnetic field now implies isospin enhanced instabilities along the magnetic field, resulting in isospin density waves (i.e., valley density waves) just like spin density waves in spin space. Isospin degrees of freedom can also lead to superconducting instabilities along the magnetic field. In connection to the mapping of spin to isospin in the presence of a magnetic field, we note that the Zeeman splitting does not exist in isospin space. This makes the above instabilities much more favorable. Finally, we discuss the Fermi liquid properties of these multivalley systems; specifically the interplay of the metal-insulator transition and isospin space. In the various applications we will cover we rely on many other disciplines which individually have been discussed much more thoroughly elsewhere. In such cases our emphasis will be almost exclusively directed to the new features introduced by the presence of the additional isospin components.

II. lsospin Space and the Effective-Mass Hamiltonian

1. BAND-STRUCTURE SYMMETRY The symmetry considerations for the wave function of one electron, or for that matter many electrons, in the presence of a periodic potential generally involves the 32 point groups, and when translations are included the number of possible groups increases to the 230 space groups.’ In addition there exists the full SU(2) symmetry of the spins provided spin-orbit interaction is neglected, and a time reversal symmetry provided there is no external magnetic field. For example, in Figs. 1-3 we show three illustrative band structures. Figures 1 and 2 are the band structures for the semiconductors of silicon and germanium, respectively, and Fig. 3 for the semimetal of bismuth. The dispersion of the single-particle states &i0 (where n is the band index, p the reduced wave ‘M.Tinkham, “Group Theory and Quantum Mechanics”, Chapter 8. McGraw-Hill, 1964.

96

M. RASOLT

6

r

X

4

2

E o c-x w

-2

-A

-6 0

0.2

0.4 0.6 p/(2 n /a)

0.8

1.0

FIG. 1 . The band structure of silicon along the 100 direction. The pockets of donor-doped electrons are centered at crystal momentum p = Q/2. The shaded area represents these valley electrons which carry the isospin quantum numbers.

vector, and u the electron spin) are displayed along various symmetry directions in the first Brillouin zone (BZ). These symmetry directions and the various degeneracies of the band indices n (at special p points like r) are a consequence of these 230 space-group operations and time reversal. Such band structures are at the heart of many solid-state properties. The band structures of, for example, Figs. 1-3 are of course an approximation to the full many-body Hamiltonian. They are generated almost always from a mean-field type of approximation to the exact nonrelativistic Hamiltonian

where V(ri) is the potential corresponding to some fixed periodic arrangement of the nuclear charges (here we ignore classical and quantum lattice displacements) leading to one of the 230 space groups.

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

I

97

I

L1

-2

-4

r

L

X

-t

P

FIG. 2. The same as Fig 1 in the case of -germanium. The pockets of the valley electrons are now at the zone boundary (at the L point) along the 111 direction 0.6

0.3 C ._ v

w

0.1

-0.1

u

x

r

L U T P FIG.3. The band structure of bismuth along various directions in the first Brillouin zone (BZ). The low concentration of electron carriers, in this semimetal, are at the L point and the low concentration of holes are at the T point of the BZ.

98

M. RASOLT

Also in Eq. (1.1) v(ri - rj) is the interparticle Coulomb interactior. e2/lri- r,l and the indices i , j run over the number of electrons N. In terms of the second-quantized field operators vL(r) and vu(r), where vL(r) creates an electron of spin u at site r, A can be also written as2

++

c

UI.U2

(we consider fi in arbitrary d dimensions). Any kind of mean-field approximation reduces fi to a quadratic form in the field operators; i .e. a “single-particle-like” effective Hamiltonian fi0(&J (the dependence of fiO on the ground state I&) illustrates the self-consistent nature of 13,; see Section 11). We can then expand the field operators in terms of specific set of Bloch states @i(r), i.e.,

c CLpU@;m

(1-3)

CL.,.uI=

(1.4)

vLW =

P.”

such that [fiO(@G)?

E;,C:,o

and E ; ~is the dispersion, or band structure illustrated in Figs. 1-3. For a large class of problems this is the end of symmetry considerations. Here, this is where the story begins. The point we develop in this review article is that hidden in the band structures of Figs. 1-3 are “almost perfect” continuous symmetries not at all implied by the spatial symmetries of the 230 space groups. We will refer to the space of these continuous symmetries as “isospin space.” As we will see in the latter sections, IV-VI, this isospin space will lead to a host of new possible phenomena heretofore largely ignored. 2. ISOSPINAND

THE

EFFECTIVE-MASS HAMILTONIAN

To see what we have in mind, let’s look at the band structure of Fig. 1 and concentrate on the small pockets of electrons in the shaded region .L.Fetter and J . D. Walecka, “Quantum Theory of Many Particle Systems.” McGraw-Hill, 1971.

’A.

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

99

when Si is n doped. Of course, there are six equivalent such T - X directions in Si and the electron pockets have the configuration of Fig. 4. Imagine that all principal axes of the n pockets are the same [note that this n has no relation to the band index n in Eq. (1.4)]. (From Fig. 4, we see that in silicon only groups of two valleys are related by time-reversal symmetry and therefore n = 2). Then we can capture much of the physics (see Section 111) with the following effective-mass Harniltonian fi.

where (m-')mois the effective mass,

with a and

p the Cartesian coordinates in d dimensions. The qrUare the

FIG. 4. The Fermi energy surface of the six equivalent valleys of electrons in silicon along the 100, 010, and 001 directions.

100

M. RASOLT

field operators of the t = 1-+ n identical valleys with two electron spins o = 1-+ 2, fi(r, - r2) is the interparticle Coulomb repulsion screened by the dielectric constant E (see Section 111)

Finally v(r) is the potential of some random defects. Unlike the previous wave vector p, the wave vector k is measured from the center of each valley (see Figs. 1 and 5 ) . fi is clearly invariant under SU(n) symmetry of the valley index t (i.e., isospin) and SU(2) of the spin index o. [f? is SU(n) X SU(2) invariant.] When in the ground state I&) all the valleys and spins are equally occupied, i.e., when I&) is invariant under SU(n) x SU(2), the system is in the paraisospin-spin state. In this state the Fermi liquid parameters could be strongly modified via the isospin fluctuations (see Section 26). In addition a variety of new paraisospin excitations can be predicted; particularly in high magnetic fields and low dimensionality. This all will be discussed in Subsection 20. We end with a few comments concerning the SU(n) invariance of 8. As will be seen, however, the most general properties of the SU(n)

2=3

5=2

2=4

(a)

i;

i;

7=1

T=2 (b)

FIG. 5 . (a) An example of the band structure around four equivalent valleys (i.e., n = 4) with a broken symmetry (BS) in both isospin and spin space. In other words there are more electrons in isospin component t = 1 than the three other isospin components z = 2, 3, 4, which are equally occupied. Isospin t = 1 is also spin polarized; this is illustrated by the reverse shaded area in t = 1. (b) An example of a two-component isospin system where the isospin component is fully polarized and the spin Component remains unpolarized. Note that actually in both (a) and (b) the t = 1 valley should be shifted downward so that the tops of all shaded areas are the same, corresponding to the same chemical potential.

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

101

continuous groups will not be important to the interests here and things will remain quite simple. The SU(n) internal symmetry of fi in isospin space corresponds to any n X n unitary matrix (Iwhich transforms the v- 9’according to

These unitary matrices, under matrix multiplication, form the so-called SU(n) continuous group.3 Inserting any such unitary transformation into Eq. (2.1) clearly leaves fi invariant; hence fi is SU(n) invariant. A very familiar example is the case of n = 2, where the corresponding SU(2) group is the collection of all 2 X 2 unitary mat rice^.^ This n = 2 case corresponds to for example, real spin or to the case of only two equivalent valleys. It is useful to generate all such 2 X 2 matrices using for the generators the well-known Pauli matrices [see Eqs. (3.2a)l. These generators are also used to construct higher representations from the basic 2 x 2 SU(2) representation and to describe a general rotation of a many-body wave function [see Eq. (3.6)]. The extension from n = 2 to larger n (i.e., more equivalent valleys) becomes more complicated. For example, for n = 3 there are eight generators3 from which all 3 X 3 unitary matrices can be generated. A general rotation (in isospin space) of a many-body wave function, whose basic representation has three isospin components (i.e., n = 3), is an extension3 from Eq. (3.6) to exp(i&A,. 2/2) I&) where ri is now an eight-component real vector [rather than the three for SU(2); see Eq. (3.6)] and Aj are the eight generators. For the purpose of this article, however, nothing as general as that will be required. First, almost all of our continuous symmetries, in isospin space, will be confined to a two-valley system so that the more familiar SU(2) group is all that is needed. In the few examples where arbitrary n valleys are discussed we will only consider the 2 X 2 subgroups3 of the full SU(n) symmetry. These 2 x 2 subgroups are rotations within each pair of two valleys and again correspond to the more familiar SU(2) group. Such rotations will play a crucial role in describing low-lying collective modes within different pairs of valleys.

3M. Hamermesh, “Group Theory and Its Application to Physical Problems,” Chapters 7 and 11. Addison Wesley, 1964.

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M. RASOLT

3. FERROMAGNETIC ALIGNMENT OF THE ISOSPIN-SPIN GROUND STATE Perhaps even more exciting is the possibility of a wide variety of isospin-spin broken-symmetry ground states, and their corresponding Goldstone modes. Let's see what we have in mind. Suppose it turned out favorable energetically to put more electrons in one of the valleys (say isospin t = 1) than all the rest of z = 2, . . . , n valleys, which we leave all equally occupied. Suppose in addition we polarize the two spin components in the t = 1 valley so that u = 1 up spin state has more electrons than u = 2 down spin (see Fig. 5). This is then a ferromagnetic isospin-spin polarized broken-symmetry ground state. It is a broken symmetry (BS) ground state because it is not invariant under some of the symmetry operations of the SU(n) x SU(2) group and also not under the timereversal symmetry operation. The full isospin SU(n) group contains many continuous subgroups. For example, fi is invariant under n(n - 1)/2 two-dimensional SU(2) subgroups. Said differently, we can take in H any two qr (say q1and q2or q1and q3or q2and q 3 ,etc.) and perform any two-dimensional unitary transformations U and leave fi invariant. These transformations are

and

where la12 + 1612= 1. [There are, of course, n(n - 1)/2 such possible pairs.] In order to know how the many-body isospin t = 1 and spin u = 1 BS ground state IGG) transforms under the operations of Eqs. (3.1) we need to construct the three generators of these SU(2) isospin subgroups. These generators are

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

103

and

[As already said, there are n ( n - 1)/2 such Z generators.]

"). Itisalso

= (-oi )~, a n d d = ( ' InEqs. (3.2) d = (0 l 1>,u ~ 0 0 0 -1 useful to define the isospin raising and lowering operators

Z,

I 4I

=$

and 2- =

ddr[Zx(r)+ i?,(r)] = ?,

+ iZy

(3.3a)

ddr[Zx(r)- izy(r)]= 2, - i?y.

(3.3b)

Any unitary transformation of Eqs. (3.1) is accompanied by a corresponding unitary transformation of the ground state For example, a rotation of i$ around a three-dimensional (3D) unitary vector 2, for a particular pair of valleys is

The same is true for the spin space with the generators

and the spin raising lowering operators are 6+ =ax+iay and 6- = 6, - i s y . The rotation in both isospin and spin spaces leads to

where fiS is another 3D unit vector and @s the rotation around it in spin space. Incidentally it is important to recognize that while the rotation in

104

M. RASOLT

-31: :x: +

+

T=2

r=2

....

r=2

FIG. 6 . (a) The integral equation for the full cross section r of an isospin t = 1 electron, with an isospin t = 2 hole. The difference in the occupation of t = 1 and t = 2 will lead to a valley wave resonance in r. (b) Example of low order Feynman graphs for the irreducible cross section y in terms of the interparticle interaction given by the wiggly line. (c) An example of an interaction of two Goldstone modes not considered in the text.

Eqs. (3.5) is in a particular pair of valleys (say tl = 1 and z2 = 2 ) the rotation of the spin exp i(fis . &)@,/2is over the spins in every valley. Of course, we can increase the number of rotations over more pairs of valleys trivially by writing

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

105

The choice of the broken symmetry considered so far (Fig. 5) is a which is an eigenfunction of the isospin operators ground state I@G) 2* = ?; + 2; 2; and 2, and the spin operators 6’ = 6; + 6; + 6: and 6=. Accordingly, the ground state I@G) is a product of a coordinate wave function multiplied by an isospin wave function multiplied by a spin wave function (i.e., Young type3; see also Section 17). The Goldstone modes (GM) of such a I&) are isospin waves (or “valley waves”) and spin waves. There are (n - 1) SU(2) subgroups which couple to the t = 1 valley and in addition there is the one spin wave from the broken spin symmetry (see Fig. 6). This BS IGG) also implies that these GM go to zero quadratically with their momentum q provided the excitation does not carry longitudinal charge fluctuations. We shall see in Section IV that such valley waves could play an important role in the quantum Hall effect of a silicon 110 surface or of a multilayer heterojunction. There are other extensions of this type BS ground state in which the ground-state wave function I@G) is a product of space, isospin and spin components. For example, we could have filled two valleys t, and t2 differently from the remaining n - 2 valleys and in addition split the spin components u1 and u2 differently in each of the valleys (etc., etc.). Each class of such BS ( G G ) carry along a new host of GM. It is clear that the is endless. For each system under number of valley-spin BS I@G) consideration theoretical estimates for the lowest ground-state energy consideration, and of course experimental observation, provide an important guide. There are, however, other possible BS ground states which are not of the class considered above and these should also be mentioned in this section.

+

4. ANTIFERROMAGNETIC ALIGNMENT IN ISOSPIN-SPIN SPACE

Let’s then return to Eq. (2.1) and, to simplify matters a bit, consider the case of only two valleys z = 1,2 and unpolarized spins u = 1,2 (Fig. 5b). Again we polarize the isospin space by putting more electrons in valley t = 1 (actually we put all the electrons in valley t = 1; it does not matter). Now the set of rotation in Eq. (3.6) is actually too restrictive since the rotation in spin space is the same for all valleys. We can find additional continuous subgroups which leave fi [Eq. (2.1)] invariant by performing separate rotations, of the two spin components u = 1,2 in each valley. For t = 1,2 and u = 1,2 we get four continuous subgroups. First there are continuous SU(2) subgroups of the spin for each t; for

106 t=

M. RASOLT

1,2

This gives two such subgroups. Second, there are continuous SU(2) groups of the isospin for each u = 1,2

and this gives the other two subgroups. Let's see the consequences of such transformations on the valley polarized ground state I@G). Suppose we approximate by an uncorrelated product state (we will discuss the case of correlation in Section 111).

where 10) is the vacuum, c:&are the creation operators of states with isospin r, spin u and momentum k measured from the center of each valley (see Fig. 5). Now first apply the isospin rotation of Eq. (3.1) and (4.2) to to give

Next apply the spin rotation of Eq. (4.1) to I@&) to get the new state

I@&):

Ikl=(kkF

Of course since H [Eq. (2.1)) is invariant under these transformations I@&), and I@%) all have the same energy. We next wish the states to investigate the nature of the ground state I@$). To start we can write any 2 x 2 unitary matrix U as

U = exp i@[lcos 8

+ ifi

B sin 61.

We now want to consider the properties of such a state

(4.6)

I@&).

To do so

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

107

we must for the first time depart from the strict definition of isospin space. It is important for later discussion that we understand precisely what is meant by this. It all depends on the effective-mass approximation fi of Eq. (2.1) which we saw above rigorously permitted the description of the valleys in terms of a totally orthogonal space to coordinate space. However, unlike the spin space of a, which is truly orthogonal to coordinate space, isospin space is not. It will be the major task of Section 111 to demonstrate that while this description is not exact and while there must, of course, always be coupling between the valley indices and spatial coordinates, which will break the true SU(2) symmetry, the isospin description is still extremely useful and for many properties very accurate; but not always. For example, to understand the spatial structure of I@&)on the scale of roughly lattice spacing isospin formalism alone would be clearly inadequate. We can see that at once when we recognize that I@&) is a coherent superposition of Bloch states from the two different valleys. If we are to calculate the charge density, spin density, current density and spin current of I@’&) we must expect beats between the Q vectors separating the two valleys (see Fig. 1). In the true isospin formalism all the single particle states $Sa(r> [Eq. (1.3)J are plane waves, i.e.,

where xu is the function of the spin and where 52 is the volume of the system. There can, therefore, be no such important beats in any spatial properties. In short we shall see that the isospin formalism provides an excellent description of the energetics and collective excitations but is inadequate for describing the spatial properties of the BS state on a scale of a lattice spacing. Let’s then see what I$&) gives for the spatial properties: For the density profile

For the spin-density profile

108

M. RASOLT

[real spin density is, of course, S(r) = (W2)o(r)]. For the current density

and for the spin current density

where the field operators VL, qjU are those of Eq. (1.3). Using Eqs. (4.5) and (4.6) and the properties of the Pauli matrices, we find that the BS density component of p(r) is

c 4 c Re[a*bei@cos kr;

p(r) =

Ik(=O

Re[a*b~,=l,k(r)~:=Z,k(r)

kr;

=

lkl=0

&#&~,k(~)&&)]

tr(U)l (4.12)

where @a=l,k is the Bloch state [Eq. (1.3)] from valley 1 and momentum k (measured from the center of the valley) and &.=Z,k is the Bloch state from valley 2. From the Bloch symmetry we can write that @s=I,k(f)

= ei(k-Q/Z)

'

rul,k(r),

(4.13)

[where U l , k ( r ) has the translational periodicity of the lattice] and that

Therefore, p(r) has a periodicity in Q, i.e., a spatial variation which is not the periodicity of the lattice. We refer to it as a valley-density wave (VDW). We, of course, never would have seen this Q modulation had we remained in the strictly isospin space of fi [Eq. (2.1)]. The same is true of the current density j(r) which is

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

109

and for the spin-density wave (SDW)

and spin current density

2iih k p j"(r) = __ Re{a*be'@sin e[Vt$T=,,,(r)Q,=,.,(r) m lkl=0 For a particular choice of direction in isospin-spin space 8 = n,the state has a nonvanishing VDW [Eq. (4.12)] and a corresponding orbital antiferromagnetic current flow j(r) [Eq. (4.15)]. For 8 = n/2 the state is a SDW with an antiferromagnetic modulation Q and with the spin pointing along the fixed ii direction [Eq. (4.16)] and a corresponding spin current j'(r) [Eq. (4.17)]. What can we expect for the corresponding GM of this type, BS It$&)? For the GM (unlike the above spatial description) we can return to the isospin description of fi [Eq. (2.1) to very high accuracy (see Sections I11 and IV). From the SU(2) subgroups [Eqs. (4.1) and (4.2)] we in general can expect four GM. The two associated with valley density fluctuations (isospin) are expected to go to zero quadratically in their momentum q. The two associated with the antiferromagnetic spin structure should go linearly in q. We could have generalized the BS of ( G G ) (Fig. 5b) even further by not only polarizing one of the valleys but also one of the spins. Performing the symmetry operations of Eqs. (4.4) and (4.5) would for example add a SDW of variable spin direction to the collection. Again the possibilities of BS are endless.

5.

OTHER

BROKEN SYMMETRIES

IN ISOSPIN-SPIN

SPACE

Our discussion has not specified the physical dimension d of the system. It will, however, be argued in Sections IV and V that the type of BS states considered so far are more likely to occur in two dimensions and in the presence of a strong magnetic field. These types of BS states share a common origin. They constitute initially a polarization of the valleys (i.e., do not have equal numbers of electrons in all the valleys) followed by rotation in the various SU(2) groups. It is generally very expensive in kinetic energy (but not always) to achieve such a polariza-

110

M. RASOLT

tion. Lower dimensions and strong magnetic field however makes this much more likely. There are other BS states in isospin-spin space which are more likely to occur in three dimensions (in particular for a strong magnetic field; see Section VI). Such states are not accessible via rotations of a polarized state in isospin space. What we have in mind is illustrated in Fig. 7. In Fig. 7a we demonstrate a BS VDW whose configuration is made up of two independent charge density waves one for each valley (t= 1 and t = 2 ) . Such a ground state is given in the unrestricted HF as the following product state I@G)

=

n

(U:l,t=lCI=l,o,kl

lkil=O a=1,2

fl

+ UL*~,s=lC~=l,o,-kl)

+ U(T,,~=2C1=2,a,-k2) 10)

(U&,~=2Ca=2,0.k2

lkzI=O o=1,2

I

I

I

I

9

(5.1)

3-,

FIG. 7. (a) An illustration of a VDW made up of two CDW. Each of the CDW are structured from states on opposite sides of the Fermi surface within each valley (or isospin). (b) An illustration of adding a spin variation. (c) A VDW made up of states on opposite side of the F e d surface of the different isospins.

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

111

where the product over kl illustrates domains of possible nesting across the Fermi surface. We can extend the state to include SDW in each valley (Fig. 7b) by performing a spin transformation to give

o=1,2

Note that in these products we implicitly assume the contributions as well from the orthogonal states, where u + -v and v + u. We can also create a VDW from the two different valleys (Fig. 7c)

We can add a SDW component by another unitary spin transformation

How do these states differ from Eqs. (4.3)-(4.5)? First, the states do not have the exact same periodicity as Eqs. (4.4), (4.5).Second, once the energetics favor a polarized state such as Eq. (4.3), the coherence between the states of the two valleys are then given by two single coefficients a and b of the isospin rotation [Eqs. (4.4)and (4.5)];not so here. The coefficients vk, Uk, and the matrix U:o, have to be established by minimizing for the lowest energy. In short, while both classes of BS states provide SDW and VDW, they are truly d i ~ t i n c t We . ~ will see in Section VI that these latter groups discussed in this section are likely to occur in three dimensions and strong magnetic field. Concerning the GM for these BS states, since such modes depend on the internal symmetry of H , we expect our previous discussion to be relevant. Finally, let's turn to the band structure of bismuth in Fig. 3. While our interest in this review article is exclusive to equivalent valleys of electrons we like to consider very briefly the excitonic insulator BS state of electron 41n no way have we exhausted all the possibilities of BS; just as an example, an isospin glass state cannot be ruled out.

112

M. RASOLT

r-

T

-DW

t - t

L

--u

FIG. 8. The relevant detailed part of the bismuth band structure presented in Fig. 3. The electrons at point L , and the holes at point T, participate in the BS excitonic insulator ground state.’

and hole pockets. Halperin and Rice5 have written a very comprehensive review of this BS state. In fact, many of their concerns are equally important to our case. We therefore give a few comments that make contact with their discussion and our isospin formalism. In Fig. 8 we focus on the relevant part of the band structure of Fig. 3. In the spirit of the effective-mass Hamiltonian we can write for the electron and holes an H like Eq. (2.1), except that the interaction between two different combinations of isospin z of the field operators qz0,in the quartic interaction, have now a negative sign. (We assume that the electrons and holes have the same mass; it does not matter.) What does this change in sign mean in isospin language? It means that H is no longer invariant under SU(2) symmetry in isospin space. In fact the two subgroups of Eq. (4.2) become Abelian (or XY in isospin space). In short fi is invariant under

The spin subgroups remain the same as (4.1). [Note that A remains invariant under Eq. (5.5) when the electron hole masses are totally different]. An XY BS of the isospin space and an antiferromagnetic alignment of the spins will lead to four GM all going to zero linearly in q. ‘B. I. Halperin and T. M. Rice, in “Solid State Physics” (F. Seitz, D. Turnbull, and H. Ehrenreich, eds.), Vol. 21. Academic, New York, 1968.

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

6. ISOSPIN-SPIN SYMMETRY IN THE PRESENCE OF MAGNETIC FIELD

A

113

UNIFORM

The presence of a magnetic field H(r) and a corresponding vector potential A(r) [where H(r) = V x A(r)] changes the effective-mass Hamiltonian fi [Eq. (2.1)J to

+ !2g mc

1

ddr 6(r) * H(r),

where 6(r) is defined in Eq. (3.5). From Eqs. (3.5) and (6.1) it is clear that in the effective-mass Hamiltonian, even though time-reversal symmetry is broken, the magnetic field does leave the SU(2) continuous isospin symmetry intact. Of course, the last term in Eq. (6.1) does break the SU(2) symmetry of spin space. Therefore the BS states, associated with spin space, will be strongly modified as well as their corresponding GM. The exciting aspect of the isospin space is that, contrary to spin space, the energetics for BS states will be strongly enhanced by the quasi-one-dimensionality of the magnetic field. We will see that not only diagonal long-range-order BS ground states, of the type discussed so far, are enhanced in the presence of a magnetic field. Off-diagonal long-range order can also be strongly enhanced in the presence of H(r) leading to a possibility of a totally new type of superconductivity. In Sections IV-VI we will study a variety of phenomena associated with some of the choices of BS states discussed above. Before we do so, however, we must justify the effective-mass Hamiltonian H [Eq. (2.1)]. Looking back at Figs. 1-3 it is indeed surprising that in this rigid looking

114

M. RASOLT

band structure the electron pockets can exhibit such continuous symmetries, to a high degree of accuracy. We must then set, with great care, the energy scales for which these continuous symmetries break down. This we do in the next section.

111. Symmetry-Breaking Terms in the Effective-Mass Hamiltonian

At the heart of Section 11, and in fact largely in what will follow in [of Eq. (2-1)]. Here Sections IV-VI, is the effective-mass Hamiltonian we want to derive corrections to H starting with the full fi of Eq. (1.2). Two issues have to be addressed: first, we need to separate the relatively few electrons in the valleys from the many valence electrons and write an effective Hamiltonian, only for these valleys. Second, we need to rewrite this effective Hamiltonian in isospin form as an effective-mass Hamiltonian with symmetry-breaking corrections. (We note that with the effective Hamiltonian and effective-mass Hamiltonian we imply two distinctly different Hamiltonians.) We will confine our discussion to the random-phase approximation (RPA). 7. THEENERGY OF H

IN THE

RPA

Let’s return to Eq. (1.2) and add and subtract to it a self-consistent Hartree potential.

where

The self-consistent one-particle Hamiltonian in Eq. (1.4) is

[We note that other choices could certainly have been made for Ho(&)

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

OR

115

Q (C)

FIG. 9. (a) The ring diagrams which lead to the RPA ground-state energy. The arrowed lines are the electron propagators, which at this stage include both the valence and conduction electron states. The dashed line is the unscreened Coulomb interparticle interaction. The frequency o is placed there as a reminder of the dynamic fluctuations entering the RPA ground-state calculation. (b) The Hartree electrostatic energy. (c) Example of self-energy insertions in the single-particle Green’s function, which renormalize the arrowed lines and reduce back to a and b through the counterterm in Eq. (7.1).

like the density-functional theory (DFT) one-particle Hamilt~nian.~] We imagine that the band structure of Fig. 1 is a solution of this Z&. (Actually it is the DFT ~;1,.’) We analyze the structure of the energy E = (@GI fi I@G) using the Feynman graphical representation. In Fig. 9a we show the ring diagrams leading to the RPA interaction energy. We also show the Hartree contribution in Fig. 9b. [Note that any additional terms like Fig. 9c get canceled by the counterterm of Eq. (7.1) to reduce to Figs. 9a and 9b.l The dashed lines are the Coulomb interactions e2/lr, - r21. The solid arrowed lines are the single-particle propagators of 6W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). ’5. R. Chelikowsky and M. L. Cohen, Phys. Rev. B 14, 556 (1976)

116

M. RASOLT

(7.4)

where p is the noninteracting chemical potential. Incidentally, the common self-consistent screened exchange approximation' can be obtained from these graphs by: (1) treating Go(r, r') (or more precisely the wave-functions @&) as self-consistent quantities, with of course the , : 1 @;?) = 1; (2) ignoring the frequency dependence constraint that ( @ w , passing through the pair interactions (see Fig. 9). This leads at once to the screened exchange approximation for the energy and by differentiating with respect to @'s to the self-consistent screened HF set of equations. 8. EFFECTIVE HAMILTONIAN FOR

THE

VALLEYELECTRONS

The discussion so far treated both the many valence electrons and the very few valley electrons on equal footing. We want to take advantage of the small number of valley electrons and separate them from the rest. To do that, we manipulate the RPA Feynman graphs of Fig. 9. Let us first examine the set of Fig. 9a. We decompose the full Go of Eq. (7.4) to a valence part G, and a conduction part G,, where

(the sum over k is strictly inside the valleys) and

where the sum over p and n are within all the rest. We consider the contributions of Fig. 9a to second order in G,; these are shown in Fig. 10. Now, we are not concerned with absolute energies, certainly not the very big energy which is independent of G, . Actually we are not even concerned about energies within the valleys which are independent under isospin rotation of Eq. (4.4).For example, from the Bloch symmetry of @:(r) we can show that Fig. 10a is independent of 'L. Hedin and S. Lundqist, in "Solid State Physics" (F. Seitz, D. Turnbull, and H. Ehrenreich, eds.), Vol. 23. Academic, New York, 1969.

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

117

FIG. 10. Example of Feynman graphs which evolve from Fig. 9a when the electron propagators are separated into a conduction band G, and valence band G,. (a) An example of a single conduction-band Green's function (double solid line) interacting with the valence electrons. (b) Conduction electrons (i.e., valley electrons) to second order in G, interacting via the screened interparticle interaction, screened by the valence electron o d y (see Fig. 10d). (c) Other examples of conduction-conduction electron interaction to second order in G,. (d) The screened interaction V(r, r', y); see text.

isospin rotation. To see this we write the Feynman graphs of Fig. 10a in the form

F(r, r',y)Gc(r, r', y ) .

(8.3)

where to avoid working with imaginary contributions of F we rotated w to the imaginary axis, i.e., w = iy. The function F represents all the contributions multiplying the double solid arrowed lines in Fig. 10a. Now F(r, r', y) is made up entirely of Bloch states and has therefore Bloch symmetry, i.e.,

F(r, r', y ) = F(r + R,r' + R,y ) ,

(8.4)

118

M. RASOLT

where R is a lattice vector. It then follows at once that F i n Fourier space can be written as

where G is a reciprocal lattice vector. Now we perform the isospin rotation of Eq. (4.4) on the valley electrons. This translates to writing G, of Eq. (8.1) as

Turning back to Eq. (8.3), we need to show that terms like

I I 1,:)

EA = ddr ddr’

~

F(r, r‘, y ) b ~ * @ : , ~ = ~ ( r ) @ ~ , ~ =(8.7) ~(r’)

vanish. This follows from Eqs. (4.13), (4.14), and (8.5) to give

I

E i = ddq F’(q),,,,G(q

+ k + G - Q/2)S(q + k + G’ + Q/2)

(8.8)

and Eq. (8.7) vanishes. In short the contribution of Fig. 10a is isospin invariant. The same applies to the quadratic contributions, in G,, of Fig. 1Oc. (Note that the effective Hamiltonian f i e , and the effective mass Hamiltonian H are not to be confused as the same; they are two distinct Hamiltonians .) The contribution from Fig. 10b can be summed into the form

where the V(r, r‘, y) is the sum of the ring diagrams of Fig. 10d. It is important to note that V(r, r’, y) is given entirely (to second order in G,) by the valence electrons alone. It therefore can be considered as an external dynamically screened interaction between two isospin (or valley) electrons only. It also of course has the periodicity of Eq. (8.5). The sum of these ring diagrams is recognized in yet another way as the

119

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

dynamical screening of the bare Coulomb interaction v(r - r‘), i.e., V(r, r’, y ) = &-‘(r, r’, y ) v ( r - r‘),

(8.10)

where E(r, r’, y ) is the dielectric function for the valence electrons only and again has the periodicity of Eq. (8.5). The Hartree contribution of Fig. 9b can also be opened up in terms G, and G,. The result to quadratic order in G, is

E where

-

H - 2 1

I

ddr ddr’ [V(r, r‘, y = O)]pc(r)pc(rr),

(8.11)

(8.12)

If we ignore the frequency dependence in Eq. (8.10) then we can write an effective Hamiltonian, equivalent to quadratic order in G , , as the noninteracting H , ( + G ) of Eq. (7.3), plus the two interaction contributions of Eqs. (8.9) and (8.11), i.e.,

where the field operators of the valleys are

This is then the final form for the effective Hamiltonian within the valleys. What have we sidestepped in deriving Eq. (8.13)? The effect of the frequency dependence in Eq. (8.9). The effect of higher orders in G,. These, as we now show, are much smaller corrections to Eq. (8.13); nonetheless these are nontrivial. The reason is that (as will be seen in Section IV) some important properties, of for example silicon, will depend on symmetry-breaking terms as small as lop6 (or even smaller) relative to the first-order contributions coming from the effective-mass

120

M. RASOLT

Hamiltonian [Eq. (2.1)]. Our careful assessment of such corrections is therefore very relevant. a. Frequency -Dependent Corrections We return to Eq. (8.9). The frequency dependence in V(r, r‘, y ) will modify the static results in Eq. (8.13). To estimate the amount we consider the frequency dependence of the valence electron response. A rough treatment of the Lindhard dielectric function for these electrons introduces corrections at small frequencies w of order w 2 / E ; (where E , is roughly the gap). As a characteristic frequency of the valley electrons we take w=h2kt/2m (we could have taken the plasma frequency), and E,= h2Q2/2m. The correction to Eq. (8.9) we estimate as AE,,, =

($r

ddr ddr’V’(r,r’, y

= 0)

(8.15) b. Higher-Order Corrections from G,

In Fig. l l a we illustrate corrections from higher orders in G, in the RPA. In Fig. l l b we rewrite the same contributions in terms of the screened interaction V(r, r’, y = 0) (screened by the valley electrons). Therefore, a many-body calculation, within each valley, will include such higher-order corrections. At very small momentum q transfers (see Fig. 11) such corrections are very important. Higher corrections from G, are not then necessarily small due to the l / q 2 behavior of the Fourier transform of V(r, r’,y = 0). This of course will lead to important screening within the valley electrons. In any event a many-body calculation using BCff includes these terms. For isospin rotation, where q -- Q, correction from higher powers of G, will be small; of the order of Finally, we remark on another derivation of fits [Eq. (8.13)] which starts by considering the valley electrons as external particles. If we fix two particles at position rl and r2 in the lattice, then their pair interaction energy, in linear response, is given by

E = V ( r , , r2, y = 0)

(8.16)

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

+

m

+

121

...

...

(b)

FIG. 11. (a) Examples of higher-order in G,; in this case fourth order. (b) The interaction

between two conduction electrons, which was previously screened by the valence electrons only [i.e., Eq. (&lo)] will now be, in addition, screened by the conduction electrons among themselves.

with V(r, r’, y = 0 ) given in Eq. (8.10). From this, fieffof Eq. (8.13) follows at once. But electrons in the valleys are not external charges, they share statistics with the valence electrons and they are not fixed. Equation (8.16) does not provide the kind of estimate for the corrections to fieffdiscussed above.

122

M. RASOLT

9. THEEFFECTIVE-MASS HAMILTONIAN So where does the SU(2)-invariant effective-mass Hamiltonian 8, of Eq. (2.1), reside in fieffof Eq. (8.13)? and where are the symmetrybreaking (SB) corrections to H? We can get access to these terms by turning first to an uncorrelated product ground state of Eq. (4.3) rotated in isospin space to Eq. (4.4). [We ignore the SU(2) rotations in spin space; spin rotations are totally uninteresting here since they leave fieffinvariant.] So we want to study the expectation of {@&I fieBI@&).fie,, in Eq. (8.13) is written in a mixed representation. We write it next entirely in terms of the c L , k , i.e.,

where

@ ~ i , k i ( r l )@ :*,

k ~ ( ~ @rg, 2 ) k3(r2) @ ~ 4 , 4 ( ~ 1 ) .

(9.2)

Now it is not difficult to see that the quartic interaction Equar can be written as

So the quartic interactions can be written as products of two quadratic expectation values (for such a rotated I@&)). The quadratic expectation values can be readily evaluated to give

and Eq. (9.3) can be then calculated to give for the quartic interaction

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

123

the form EQuar

= EQuar,l

+ EQuar,2 + EQuar.3

7

(9.5)

where

and

To start we neglect the terms EQuar,2 and EQuar,3 (we shall see that these terms will only play a role when we consider correlated ground states I@&); i.e., states which go beyond the product state of Eqs. (4.3)

124

M. RASOLT

and (4.4)). There are symmetry relations between the various

Fk,,k2,k,,k4 T,,t2J3.54

which follow from the definition of F in Eq. (9.2). That is from the relation V(rl, r2,y = 0) = V(r2, rl ,y = 0) (which follows from time reversal symmetry) and the properties of the Bloch states @r,k(r). We list these symmetry relations: Fk’,k,k’.k

2 12 1

‘k’,k,k‘,k

1 1 1 1

(9.9)

= Fk,k’,k,k’, 12 1 2

= F-k’,-k.-kr.-k9 2

’k’,k,k’,k 2 1 1 2

2

2

(9.10) 2

= ‘k,k’,k,k‘?

(9.11)

12 2 1

and (9.12)

Using these symmetry relations we can then simplify EQ,,r,lto

(9.13) We can now, for the first time, make contact with the effectivemass Hamiltonian fi of Eq. (2.1) and also identify some of the symmetry breaking terms. To do this we need to examine the form of the F’,s, in the limit of small k and k’. At first sight the expansion in power of k‘ and k of Eq. (9.13) seems to give a leading value for EQuar,l going like second powers of lk’I2 and lkI2. Actually such terms will be very small and will precisely correspond to the symmetry-breaking terms of I? in Eq. (2.1). The point is that the F’s are singular at small k’ and k and these singular contributions lead to the effective-mass 8.To see this we need to use Eq. (9.2) for the F’s and for V(r, r’, y = 0) we need to use the periodic form of Eq. (8.5). This leads at once to, for example,

Fk‘,k,k’,k

1 1 1 1

1

=-

c v(G+

Q G

- k’)G,G

(9.14)

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

125

and from Eq. (8.10)

v ( G + k - k’)G,G = K 1 ( G + k - k’)G,GU(k- k’ + G).

(9.15)

Now the dielectric function E - ’ of the valence band is nonsingular and the leading singularity comes strictly from v(k - k‘ G) (where, G = 0). So the dominant contribution to F is

+

‘k’,k,k’,k 1 . 1.1, 1

=

v(k - k’) Q&

(9.16)

It is not difficult to show that the dominant contribution to F’kz,k,k,.k is 2, 1.1. 2

also (l/Q)v(k-k’)/&. The third term in Eq. (9.13) is, however, nonsingular and represents a contribution to the symmetry-breaking terms which we turn to shortly. Finally, we get the following dominant contribution to Eouar,l: (9.17)

+

where u2 b2 = 1 and v(k - k’) = 4ne2/lk’ - kI2, in d = 3. This result is, of course, invariant under isospin rotation and corresponds to the isospin of Eq. exchange contribution from the effective-mass Hamiltonian (2.1). All the rest coming from the power expansion in k,k‘ are the symmetry-breaking terms which are not isospin invariant. These will turn out to be very small but still very important to understand. We turn to these terms next.

10. SYMMETRY-BREAKING TERMS We carefully developed the effective-mass Hamiltonian from the rigid looking band structure of Fig. 1 and showed that its SU(2) symmetry in isospin space represents the dominant part of the interactions. In this subsection we want to discuss various aspects of the symmetry breaking terms. It is important both conceptually and for applications, we will make in the following sections, to understand these corrections and we discuss it at some length. a. SU(2) Symmetry-Breaking Terms Beyond the dominant contribution of Eq. (9.17) the F’s in Eouar,lare nonsingular and can be expanded in powers of k and k’. The sums over k

126

M. RASOLT

and k' in Eq. (9.13) permits only even powers in k and k' and so the leading symmetry breaking terms go like lkI2 or (k'I2. Such SB terms are specific to the particular system of interest; we will actually calculate these terms for a silicon (110) surface. Here we only estimate their relative magnitude to the effective-mass Hamiltonian. We take the last term in EQuar,l [Eq. (9.13)]. It represents the scattering o f an electron from valley 2 to valley 1. The momenta running across V(r, r', y = 0) are, therefore, either Q or Q + G. It follows at once that

(10.1)

the ratio R of Eq. (10.1) to Eq. (9.17) is then

(10.2)

The same is true for the other SB contributions of Eq. (9.13). Since the size of the valleys (i.e., k F ) is much smaller than their separation, we already see that SB corrections to fi are indeed very small.

b. xy or Equivalentiy U(1) Symmetry-Breaking Terms

If we restrict the isospin rotation [Eq. (4.4)] to the U ( l ) or equivalently the xy continuous group we get the transformations of Eq. (5.5). Putting it another way we can write a and b of Eq. (4.4) as a =sin Be'@ and b = cos $eCi9. It is then clear that [Eq. (9.13)] is invariant under xy rotation in isospin space [i.e., Eq. (9.13) is not dependent on @]. Is that a perfect continuous symmetry in isospin space for a two-valley system where k F<< Q? The answer is yes for an uncorrelated ground state [i.e., a product state of Eq. (4.3)] and there is no external magnetic field. To see why this is so we turn to Eqs. (9.7) and (9.8). Provided the F's are nonzero, these terms are not xy invariant (these terms do depend on @). For a product ground state, however, all the terms do vanish in Eqs. (9.7) and (9.8). To show this we take one random illustration from Eq. (9.7) and one from Eq. (9.8). Now using Eq. (8.5) for V(r, r', y ) and

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

Eqs. (4.13) and (4.14) for the

e i ( Q+G) . r,

=I

$br,k,

127

we can write, for example

e -i(q+G’). r z e i ( k - k ’ + Q ) .

r I e i ( k ’ - k ) . r2

u ~ , k ’ ( r 1 ) U 2 , k ( r 1 ) U ?.k(r2)ul,k’(r2)

d d q S ( q + G + k - k’ + Q)S(q+ G’ - k’ + k) = O .

(10.3)

For Eq. (9.8) we take for example FkP,k,kf,k and get 1 . 1.2. 2

ei(q+G). rI -i(q+G‘).

e

-I=

12

e i ( k - k ’ + Q ) . rl ei ( k ‘ - k + Q ) . r2

T,k‘(rl)U2,k(r1)U ?,k(r2)U2,k‘(r2)

ddq S(q

+G +Q +k-

k’)S(-q- G‘ + Q + k’ - k) = O . (10.4)

All other terms in Eqs. (9.7) and (9.8) can be also shown to vanish. For a product ground state the effective Hamiltonian of Eq. (9.1) has a perfect xy symmetry. c. The Effect of a Correlated Ground State on the xy Symmetry We saw that for an uncorrelated single Slater determinant, He, [Eq. (9.1)] has a perfect xy symmetry. But, we can never expect total decoupling between isospin space and spatial coordinates. In fact, for a correlated ground state there are very small SB contributions; it is important to estimate their relative magnitude to the SU(2) SB terms discussed previously. A correlated I@G) is some linear superposition of Slater determinants (or product states) of the type given in Eq. (4.3), i.e., (10.5)

128

M. RASOLT

0

e

'-- * k

P(k)

FIG.12. The electron occupation number n(k) as a function of momentum k. The solid step is for noninteracting and the dashed curve for the correlated ground state of the electron gas (taken from ref. 9).

where CY,is intended to reflect mixing coefficients of various higher-lying electron states of momentum k of Slater determinants I&). Now the expectation value of (@JfieE I@c) will no longer take the simple form of Eqs. (9.5)-(9.8). Four different momenta can enter the F functions. It is not difficult to see [(by similar considerations taken in Eqs. (10.3) and (10.4)] that crystal momentum conservation can be found when some of the excited momenti in Eq. (10.3) get close to Q/2. It then follows that Eqs. (9.7) and (9.8) no longer vanish and xy is no longer a perfect symmetry. To estimate the magnitude of such SB contributions is nontrivial; we present the following crude estimate. We ask, how many of k electrons occupy such high momentum states? This we can estimate by looking at the momentum occupation number of the valley electrons. For a noninteracting uniform electron gas this is given by

(10.6) where G,,(k, w ) is the noninteracting one-particle Green's function. This noninteracting n k along with the interacting n k , calculated in the PPA,9 are displayed in Fig. 12. It is also known that nk goes like (k,/k)' at large k in the RPA. For k = Q / 2 where the xy SB can occur, the SB terms are, therefore, down by -(kF/Q)' from the SU(2) SB terms considered in Eq. (10.2). So xy symmetry is not perfect but it is very close to it. 'E. Daniel and S. H. Vosko, Phys. Rev. 120, 2041 (1960).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

129

d. Symmetry-Breaking Contributions from the Noninteracting Part Let's turn back to fieEof Eq. (9.1). The quadratic part is clearly invariant under xy isospin rotation. It, therefore, will not modify our discussion of the xy symmetry. For SU(2) rotation, however, the quadratic (i.e., noninteracting) part EQ is not invariant in isospin space. It will, therefore, contribute under SU(2) rotations. Taking the expectation value of (10.7) with

I@&) given in Eq. (8.6) we get

The last term in Eq. (10.8) is the SB term. Now we can make a power expansion in k, i.e., (10.9) and from time reversal symmetry

&-k,2

= &k,1

so that (10.10)

Therefore, the SB term involves only the cubic terms of Eqs. (10.9) and (10.10). The volume of the summation over k, in Eq. (10.7), can be estimated to be over Bk$"/(AQ) where A and B are some average of A, and Bapy.This leads to the following estimate of the ratio R of this SB term to the leading contribution (i.e., effective-mass contribution): (10.11) Incidentally, for Eq. (10.11) it is not only the power of k F / Q which is decisive. The size of the cubic coefficient B relative to A (which, e.g., for silicon, Fig. 1, turns out to be very small) is very important.

130

M. RASOLT

To cover all ground let us also consider the relative magnitudes (to the leading effective-mass Hamiltonian contribution) of the static approximation [i.e., Eq. (8.15)] and the higher orders of G,. From Eq. (8.15) dynamic corrections to SU(2) lead to a ratio R of

R --

(2) 3td

(10.12)

(the xy symmetry remains unaffected) and for higher G, to a ratio 2d- 1

R=($)

.

(10.13)

Finally, the case of where the center of valley Q/2 is at the zone boundary (like germanium in Fig. 2) deserves the following observation. First for such a system time reveral symmetry does not relate the various valleys. The consequence is that SU(2) symmetry-breaking terms in the noninteracting part of Beffare dominant (said another way, there is no SU(2) isospin invariance even to leading order). However, xy symmetry does remain totally intact (for a product ground state) and is no different than the case of silicon. 11. EFFECTOF

AN

EXTERNAL MAGNETIC FIELD

The effective-mass Hamiltonian 8, in the presence of an external magnetic field is presented in Eq. (6.1). To discuss the SB terms in the presence of an external magnetic field, we need to add to Eq. (6.1) the contribution from the periodic crystal potential [i.e., Eq. (7.3)]. To reduce it to the Heffof Eq. (9.1) in the presence of a uniform external magnetic field, we can use the elegant k q representation of Zak." The upshot is that k, in the quadratic term of Eq. (9.1), is replaced

-

e 2c

hk+ hk + - H

X

i-

d

dk

(11.1)

''5. Zak, in "Solid State Physics" (H. Ehrenreich, F. Seitz, and D. Turnbull, eds.), Vol.

27. Academic, New York, 1972. "W. Kohn, Phys. Rev. 115, 1460 (1959). "E. I Blount, Phys. Rev. U6,1636 (1962). "L. M. Roth, 1. Phys. Chem. Solids 23, 433 (1962).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

131

The energy eigenvalues and eigenfunctions are given by

(11.2) There are off-diagonal corrections to Eq. (11.2) which couple higher bands (around the same k vectors in the Brillouin zone) but these are very small corrections in the expansion of something like (heH/mc)/(h2Q2/2m) and these will be ignored. There are also corrections from the coupling of the two isospin components which are extremely small (see Section 13b) and will also be ignored. Now the kinetic energy gets quenched perpendicular to the magnetic field. If the magnetic field is strong enough to put all the electrons in the lowest Landau level, and if we align the magnetic field perpendicular to the Q vector connecting the two valleys, then as we now show the SB contribution from the cubic terms [i.e., Eq. (10.11) will be largely eliminated.] We concentrate on one isospin component and take the dispersion in (10.9). The principal axes are along the kx and ky so that from (11.1) and (11.2) the quadratic term in (10.9), in the Landau gauge leads to,

(the z direction is uninteresting for the SB for our choice of field alignment). The eigenfunctions in (11.3) are given at once by @k$kx

k y )= 6 ( k y- ky)e 0 ikOk, y @’(kx),

(11.4)

where

and where I = (hc/eH)”2. What is important is that all the states for any k;, have the same energy heH/(mxmy)”2. The SB contribution [Eq. (lO.ll)] originates, however, from the cubic term in (10.9). But the same treatment of Eq. (11.3), with these cubic contributions included, e.g.,

(11.6)

132

M. RASOLT

continues to lead to this energy degeneracy, although the eigenfunctions +k:(kx, k,) change; i.e., the cubic contributions to SB are largely quenched in the presence of a large magnetic field. Finally, it is useful to write the corresponding function, to +k:(kx, k,), in coordinate space. It is given by (11.7) k

with +k(ri) defined in Eq. (8.14). The ri are the lattice sites, but since the envelope of an electron wave function, in each of the pockets, varies on a much slower scale than the lattice spacing, we can replace ri by the continuous r (i.e., we can ignore the length scale q in the k q representation).

-

IV. lsospin Space and the Integer Quantum Hall Effect

The additional spin degrees of freedom (additional to coordinate space) account for many important properties in condensed matter physics. The additional isospin degrees of freedom extend the scope of such possible phenomenas even further. In the normal Fermi liquid state both spin and isospin play very similar roles (see Section 26). Broken symmetry states, however, are generally quite different. Broken symmetry of isospin space (see Figs. 5 and 7) in silicon, and other multivalley systems, are not likely in the absence of a large external magnetic field (see Section VI). The reason is that the effective masses are generally quite small; i.e., the kinetic energy is relatively high. Combined with the interparticle screening [Eq. (8.lO)l it leads to a lack of tendency for broken symmetry. The presence of an external magnetic field quenches the kinetic energy by putting electrons in cyclotron orbits. This reduction in kinetic energy occurs in all dimensions but is particularly pronounced in two dimensions in the quantum limit. In the high quantum limit all electrons actually can occupy the lowest Landau level and the effect of the kinetic energy is eliminated entirely. In that limit the two-dimensional electron gas must show strong tendency for instabilities in isospin space which minimize the repulsive electron-electron interaction without expense in kinetic energy. As already mentioned in Section 11, the case of real spin differs from isospin in the presence of the Pauli term; last term in Eq. (6.1). Therefore, unless the g factor in Eq. (6.1) is very small, spin and isospin spaces act differently in the presence of an external magnetic

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

133

field. We consider BS states in isospin space only in the presence of such a magnetic field. The mapping of multivalley systems to isospin space has been proposed only recently and the number of its applications is relatively small. We believe that this formalism has a very large scope of possibilities and one purpose of this review is to motivate more such studies. In this and the following two sections we will review what has been done so far (relying heavily on what has been covered in Sections I1 and 111) starting with the simplest and most theoretically conclusive case; the integer quantum Hall effect of a (110) silicon surface.

12. ISOSPINPOLARIZED GROUND STATEI N THE (110) SILICON INVERSIONLAYER

The 110 inversion layer of Si is an excellent candidate for the isospin space formalism in two dimensions. To see this we align i perpendicular to the surface [i.e., i 11 (110)] and 2 11 (1iO) and 9 11 (OOl)]. It is then not difficult to show that the pockets of the two-dimensional wave vector p (along the surface) centered at p = Q,/2 and p = -Q,/2 are related by time reversal symmetry. [Recall that Q = (Q,, Q,) where Q is defined in Eq. (4.7).] There are, of course, two pairs of such pockets and we choose the pair which has the lowest energy.14 The third pair lies along the 9 axis and is found to have still higher energy14 and can also be ignored. [We are left then with an SU(2) situation.] The electrons from the two valleys (i.e., two isospins) are confined to the two dimensional region as described in Fig. 13. (Figure 13 actually describes the inversion layer for GaAs; the form for the inversion layer of Si is very similar.) Ignoring first the symmetry breaking terms, these two pockets along the 110 surface of Si are described by the effective-mass Hamiltonian of Eq. (2.1) with n = 2 . We still have to account for the width of the inversion layer perpendicular to the 110 surface. This we do with an envelope wave f~nction'~ (12.1)

The effective-mass Hamiltonian & [Eq. (2.1)] can then be written in two

'9.Ando,

A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982)

134

M. RASOLT

Ga As

CONDUCTIONBAND

I I I

h

Al

,Gal., As

I I I

Ga As

FIG. 13. A heterojunction of GaAs and AIGaAs. The electrons are trapped in the well, , the two-dimensional plane is perpendicular to the z below the Fermi energy E ~ and direction. We have also drawn the mirror reflection to illustrate the case of isospin representation of a two-layer heterojunction, discussed in Section 21.

dimensions as

(12.2)

where

(12.3)

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

135

where14

(12.4) The F ( q ) corrects the bare two dimensional interparticle interaction

2ne2f&q due to the presence of a width and a boundary of the

two-dimensional inversion layer. This two-dimensional t? is not likely to show polarization in isospin space. We next then apply a large uniform magnetic field perpendicular to the 110 surface; large enough to quench the kinetic energy to the lowest Landau level. The Hamiltonian ff, in second quantized representation, is then given bylS

(12.5) where m, and my are the effective masses in the x and y direction, L is the dimension of the inversion layer and

(12.6) [in Eq. (12.5), we left out the uninteresting cyclotron zero point energy.] For very large external magnetic field H the Pauli term will freeze out the spin components and we consider only the isospin degrees of freedom and p. Also, since the electrons interact via the Coulomb repulsion, e 2 / u and since in the lowest Landau level the interparticle distance is the magnetic length I = (hc/eH)'" we require that ho,>> e 2 / d for the many body-Hamiltonian in Eq. (12.5) to be valid. We now want to study the nature of the ground state of fi in isospin space. We first try a fully isospin polarized state by putting all the electrons in one valley. Since we are considering the integer Quantum hall effect, the ground state is then a single Slater determinant and the "H. Fukuyama, P. M. Platzman, and P. W. Anderson, Phys. Rev. B 19, 5211 (1979).

136

M. RASOLT

energy per particle for the case of isotropic electron mass m, = my= m* is rigorously equal to -0.6267e2/d [with F ( q ) = 1 in Eq. (12.3)]. Is that the lowest ground-state energy configuration in isospin space? We try next the normal state; i.e., with equal number of electrons in each valley.I6 The Hartree-Fock (HF) calculation is now a generalization of the single component H F on a triangular lattice to two component^.'^-'^ In the Landau gauge A(r) = Hxy, we can expand the field operators in the lowest Landau level as

(12.7) where

(12.8) where X, is the isospin function and the sum over X is restricted by 0 s X = (2n12/L)j =sL. The isospin density operator p,(q) is then given by

(12.9) where X, = X f $‘qY. We obtain the Hartree-Fock Hamiltonian assuming the following triangular charge-density wave”

(12.10) with

(12.11)

[This Q should not be confused with the Q in Eq. (4.7)]. The mean-field I6M. Rasolt and A. H. MacDonald, Phys. Rev. E 34, 5530 (1986). I’D. Yoshioka and H. Fukuyama, J. Phys. SOC.Jpn. 47, 394 (1979). I’D. Yoshioka and P. A . Lee, Phys. Rev. B 27, 4986 (1983). I9D. Yoshioka, Phys. Rev. E 29, 6833 (1984).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

137

(quadratic) HF form of I? is thenla

(12.12) where

(12.13) We next solve Eqs. (12.11) and (12.12) self-consistently for A,(Q) and the corresponding HF ground-state energy.I6 If we assume that the densities of each isospin component [Eq. (12.10)] are in phase we get an energy of -0.4154e2/d per particle. Actually if we allow the two isospin density components to have different phases we can then have an antiferromagnetic isospin state which partially compensates for the reduction in the exchange energy, when all the electrons are in a single valley, by displacing the center of up and down isospin charge density to reduce the Coulomb repulsion. We find for this isospin ground state an energy of -0.4693e2/d per particle. In short, the BS “ferromagnetic” isospin ground state is by far the lowest energy state. In conclusion, two things should be mentioned: (1) the HF solution for the fully polarized integer quantum Hall effect ground state is exact. This is not true for the unpolarized state. Nevertheless, the large energy difference between them makes it extremely likely that the integer quantum Hall effect in silicon 110 surface is isospin polarized. (2) One cannot of course examine all possible broken-symmetry states but one possibility of particular interest would be three triangular isospin sublattices rotated 120” relative to each other. On a triangular lattice this would reduce frustration and possibly lower the antiferromagnetic state further. This would require expectation values of ( v:vs) (i.e., off-diagonal isospin components) to appear in the H F equations. Again we don’t expect this state to replace the ferromagnetic isospin configuration. According to Eq. (3.4) we can rotate the polarized ground state to generate infinite number of equivalent ground states. The choice will be made by the SB terms [Eq. (9.5)] which we turn to next.

138

M. RASOLT

13. EFFECT OF SYMMETRY-BREAKING TERMS ON THE NATURE OF THE ISOSPIN POLARIZATION The isospin polarized ground state is not unique but is infinitely degenerate in isospin space. The SB terms of Eq. (9.5) remove this degeneracy. We have in Section I11 roughly estimated their relative magnitude to the leading contribution coming from the effective-mass Hamiltonian. We need to make, however, a more serious estimate of their magnitude, specifically in the 110 inversion layer of Si. Our calculation will implicitly assume (as will be found a posteriori) that these SB terms are much smaller than the dominant terms and therefore will not disrupt the stiffness of the groundstate but only create global rotations. We return then to Eq. (9.5). We neglect the contributions of EQuar,2 and EQuar.3, which precisely vanish for this single Slater determinant ground state, and turn to Eq. (9.13). Since Eq. (9.13) is x y invariant, we can write it as a function of only 8, i.e.,

+

ESB= [12(sin4e + C O S ~e) 2 sin2 e cos20(1, + 13)]

(13.1)

where (13.2)

and

We note that we are ignoring here, in the symmetry-breaking energy ESB, the dominant contributions (i.e., effective mass) in Eq. (9.16) which of course are isospin invariant. a. ESBWithout an External Magnetic Field In the absence of an external magnetic field, the wave functions of a Si 110 inversion layer entering Eq. (9.2) are written aszo

'OM. Rasolt, B. I. Halperin, and D. Vanderbilt, Phys. Rev. Len. 57, 126 (1986).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

139

where R e (2,r); g(z) is defined in Eq. (12.1). For z = 1 (i.e., Q/2 projected along the 110 surface) is the bulk state along (loo), when we set i = 1 and along (OiO), when we set i =2. The k vector is the projected 3D k onto the 110 2D surface. For z = 2 ( i e . , -Q/2 projected along the 110 surface) Q:,=, is the bulk state along (OlO), when we set i = 1 and along the (00i),when we set i = 2. We choose the ai to be real (it does not matter) and without an external magnetic field, time reversal symmetry leads to a, = a2 = l / a . In next evaluating the forms of Eq. (9.2) we assume that

&=,

V(Rl ,R2 ,y = 0)

-L

V((R, - R21) = e 2 / &IR, - R21.

(13.6)

Then we can write, for example,

'k',k,k',k 1 2 2 1

=

I

d3q F1 1 (k', k)F&(k, k')V(q

+ G),

(13.7)

where q is now restricted to the first BZ and (13.8) and the 9 k . T are given in Eq. (13.5). All the other forms can be also Using Eq. (13.5) we can decomposed to products of the quadratic FtllS2. then write

The four Bloch functions can be written in terms of the periodic part u:,,(R) and &(R), i.e.,

#;,(R) = &Q&/2

iQ,rR ik.r

u:1(R)7

(13.10)

+;,(R) = eIQ&/2e-iQzr/2

ik.r ui1(R)7

(13.11)

#$(R) = eiQ&2e-iQ,z12

ik.r Uk2(R)7 1

(13.12)

92kz(R) = e-iQ&/2 e -iQ,12eik.r

ukzW7 2

(13.13)

where Q, connects 100 and OTO and Q x , connects 100 and 010. Next we express these periodic parts in terms of the center of the four pockets

140

M. RASOLT

using p * k perturbation. Then,

+ 2

eHo) Jui;(R)) . . .)

(2k. enm)(2k.

n.mf0

(13.14)

and

c -

G(

1 u t ( ~ =) - I ~ ? ~ ( R-) ) (2k

1uy2(~))

n 20

+

n,m#O

(2k * enm)(Zk*

eno) ~U:~(R)) . . .) ,

(13.15)

where the normalization constants are Al = 1

+ C (2k - e n o ) 2 + O(k3),

(13.16)

A2 = 1

+ 2 (2k . &)’

(13.17)

nfO

nf O

- O(k3),

and I U ~ ~ ( Ris) )the periodic part at the center of the pocket of band index n. Finally en,nr

= (Cinrt C*,n,)t

(13.18)

where (U%l

a

- 17.2)

,

h2 ay c;,. = 2m E o - E n r

’

(13.19)

and (UY21

a

1.72)

h2 c;,,= 2m E o - E n .

, (13.20) ’

We used the symmetries (13.21) (13.22)

CONTINUOUS SYMMETRIES A N D BROKEN SYMMETRIES

141

We can now evaluate Eq. (13.9) and insert it in (13.7).'l (See Appendix A.) Our final result for E,, is written in terms of the dimensionless functions rj defined as (13.24)

where N is the number of electrons. These functions have been evaluated for the Si 110 surface using the pseudopotential representation for the basis set @&(R). The results are rl = +1.87, Tz= -13.47, and r3= -12.90. Incidentally, the result of Eq. (13.24) is consistent with Eq. (10.2). In other words, if the envelope function was very narrow, e.g., if b Q, then Eq. (13.24) corresponds to Eq. (10.2) with d = 2. (The finite width of b, i.e., b << Q, corrects for the somewhat 3D character of the inversion layer). Placing these results in Eq. (13.1) leads to a minimum in ESB at 8 = 0. In other words, the infinitely degenerate ground state of the integral quantum Hall effect of a Si 110 inversion layer is stabilized in an king configuration by the SB terms (i.e., with all the electrons in one valley). Of course, these results must be taken with caution, particularly due to the approximation made in Eq. 13.6. There are, however, other reasons why this system should be Ising-like, which we will discuss in the next subsection. Our discussion also did not include the effect of the magnetic field on EsB , the presence of which greatly complicates matters. -L

b. EsB with an External Magnetic Field In view of our discussion in Section 11 we know that the presence of the magnetic field greatly complicates the calculation of EsB. There are two questions to be addressed. (1) To what extent the coupling of the two valleys, due to the presence of H, modify the SU(2) and xy symmetries? (2) To what extent the change in the form of the states within each valley, due to the presence of H, modify the result of ESB? The mixing of the two valleys will effect the perfect xy symmetry since EQuar,2and EQuar,3, in Eqs. (9.7) and (9.8), no longer vanish. The magnitude of the mixing, however, is very small. It is basically the amplitude for transition from one valley to the other, which is roughly exp(-const X Qz12). Its value is much, much smaller than even the effect "M. Rasolt, B. I. Halperin, and D. Vanderbilt (unpublished).

142

M. RASOLT

of a correlated ground-state wave function on the x y symmetry, discussed in Section 111. It can be totally ignored. The change of the states within each valley will certainly modify the result of E S B . A rigorous calculation to our knowledge, for this effect has not been carried out. What is needed is to repeat the calculation of Z, , Z,, Z3 in Section 13a with a new see Eq. basis set @Fk written as an envelope of the old Bloch state (11.7), i.e.,

@:c",

(13.25) The calculation using for fk,r(q) the form in Eqs. (11.4) and (11.5) is difficult. To get a very rough estimate of the effect on EsB we write

We find that the new EkB changes by a prefactor

where u=

(J

from Eq. (13.1), i.e.,

I I I

d2hl d2h2 d2h3f(hl)f(h2)f(h2+ h3- hl).

(13.28)

A logical candidate for f(h) would, for example, be f(h) = Ae-'h'z'z'z.In any event this rough estimate shows a nonnegligible decrease in E s , from the presence of H. This then concludes the study of the nature of the isospin polarized integral quantum Hall effect ground state of a Si 110 inversion layer in a perfect world without disorder. Disorder, either due to static imperfections or thermal fluctuation, must play an important role, particularly in lower dimensions. ON THE NATURE OF THE ISOSPIN 14. EFFECTOF DISORDER POLARIZATION

The electron in the inversion layer of Fig. 13 cannot possibly be confined to a potential well which has perfect crystal symmetry along the xy plane; the situation we have considered so far. Even if the ionic arrangement within the well has such a perfect symmetry, the donor atoms, which are responsible for the 2D electron gas, will introduce

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

143

disorder in the plane. It is generally desirable to place these donors outside the well. Even then the long-range Coulomb donor potential will disrupt this translational symmetry. For other purposes we might want to place some neutral impurities in the plane, thus introducing further disorder in the 2D electron gas. It is clear that we need to understand the effect of disorder on the nature of the isospin ground state. a. Single Impurity In Eq. (6.1) we have included the effect of the impurity potential V,(R) which is made up of single impurities b(R), i.e., V,(R) =

ii(R - R i ) ,

(14.1)

R,

where Ri = (Ti,zi). In terms of the expansion of Eq. (8.14) the single impurity operator Hiis (14.2) where bkl,k2= T1.Q

[dz

I

d2r$(r - ri ;z - zi)g2(z) (14.3)

and where the @La are defined in Eq. (13.5). Actually they correspond to the envelope function in Eq. (13.25), appropriate to the lowest Landau level. We can carry out the z integration; see Appendix A. Next we write ii as

where q is restricted to the first BZ. We can then use the wave functions described in Eqs. (13.10)-(13.15) to calculate Eq. (14.3). When we consider the situation where tl = t2we can show that the contribution of Eq. (14.3) to the ground state energy can be neglected. The situation where t , # t 2is, however, very interesting. In that case, Eq. (14.3)

144

M. RASOLT

leads to

c c "'(Q = 2

'k,,kZ 2, I

G

G, L2

1.1

er(k,-kZ).r,er(Q+G)r,

(14.5a)

and where u&, is defined in Eq. (A6). In Eqs. (14.5) when l = j , 21

Q = (Q,, 0) and when 1 # j , Q = (Q,, Q,). In terms of the isospin generators [Eqs. (3.2)] and using Eqs(14.5) we can rewrite Eq. (14.2) as

l?;= 4

c c d'(Q + G, z,){cos[(Q + G) - rj]?x(r,) 2

G

I,i

- sin[(Q

+ G ) - ri]2y(rj)}.

(14.6)

It is interesting to convert Eq. (14.6) to a magnetic analog. If we define cell, i.e.,

S(rj) as the isospin in a unit

h 2

S(r,) = - (2~61~)2(r;).

(14.7)

Hj = h(rj, z;) . S(rj),

(14.8)

Then H; is where

c c &(Q + G,

1 h,(r;, zi)= 2n12h G

and

2

[,j

2 ; ) cos[(Q

c c dj(Q + G,

1 h y ( r , ,2;) = - 2nL2h G

2

1.j

2,) sin[(Q

+ G) - ri]

(14.9)

+ G) - r;].

(14.10)

b. Many Impurities and the Random Field The inversion layer, of course, contains a macroscopic number of impurities distributed randomly both along the plane and perpendicular to it. Equation (14.8) has to be summed over r;, z,. But because of the cos and sin terms in Eqs. (14.9), (14.10) the direction of h(r,, zi) varies randomly from one impurity site to another. Clearly Eq. (14.8) intro-

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

145

duces a random field in isospin space of a Si inversion layer. Such a field can have profound implication on the nature of the BS ground state.22-2' In the case where the BS ground state is Ising-like (this seems to be the case in Si 110-see Section 13-but this is far from certain; in any event it will certainly be important in a multilayer heterojunction; see Section V) a random field will be relatively unimportant. But if the BS is xy-like the random field will have two profound effects; (1) it will attempt to push the xy BS to an Ising BS;"V*~(2) in the xy state it will destroy long-range order at all temperatures and even at very low impurity d e n ~ i t y . ~To ~,'~ see this we first want to map the problem onto a lattice; at small q vectors this is adequate. The king or xy symmetries can be captured by the anisotropic Heisenberg Hamiltonian

(14.11) For nearest-neighbor interactions Ail = A. is given by (14.12) where the Z, are defined in Eq. (13.24). From the isospin susceptibility calculations (at small q ) within the inversion layer [see next subsection, Eq. (15.11)] we get for J nearest neighbor (14.13) when A,, is positive we have the king BS, when A,, is negative we have the xy BS. First, concerning point (l),we can return to Eq. (14.2). Now the reason favors the Ising BS is the following. The correction to the ground-state energy to second order in f i i can be written in terms of transverse and longitudinal isospin susceptibilities xT and xL. We assume hard isospins so that XT(q)>> xL(q).For the king BS the system responds to both the hx(r,,zi) and h,,(ri, zi) via xT. For the xy BS the same holds

a

*'A. B. Harris, J . Phys. C 7 , 1671 (1974).

23Y. Imry and S. K. Ma, Phys. Rev. Lett. 35, 1399 (1975).

24A.Aharoni, Phys. Rev. B 18, 3328 (1978).

"B. J. Minchau and R. A. Pelcovits, Phys. Rev. B 32, 3081 (1985)

146

M. RASOLT

for only one of the two components. Since second-order perturbation theory lowers the energy the effect of the impurities is to lower the Ising energy relative to the xy ground state (14.14) The form of XT(q,o)is given in the next subsection (see also Fig. 14). Now for the Ising ground state Eq. (14.11) predicts a transverse susceptibility xT(q, w ) with a gap A(,, i.e., (14.15)

-=-+

-

I

-

I

--

FIG. 14. (a) The integral equation for the isospin and spin response functions; the (Y and are either isospin or spin indices. The dashed line is the interparticle interaction and the arrowed lines are the electron propagators. (b) The electron propagator in the HF approximation.

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

147

[see the next subsection for the definition of ho+-(q)]. Using Eq. (14.14) and (15) it is not difficult to see that for sufficiently large impurity density the difference between the Ising and xy can change sign flipping xy BS to an king BS. This is the reason why impurities favor an king ground state. For very large ni or strong the hard isospin orientations are decoupled and the isospin will follow the local direction of h(rj, zi). The random field will destroy long-range order in the isospin alignment of the xy BS state in two dimensions no matter how weak the impurity potential and how low the concentration is.22923To see this we realize that while the average of the random field h(r,, z,) over a macroscopic volume Ld is zero, within domains of, say radius R, there are fluctuations in h(rj, z j ) which favor one particular direction. The size R of these “domains,” (Fig. 15) can be calculated from the following energy balance. First, within the domain R the fluctuation in number of impurities is Rd”n,‘/3. From Eq. (14.11) the energy of creating such a domain, in a system with continuous symmetry, is given by RdP2Jln(R) while the energy gained by alignment with the fluctuation in impurity is fiRd’2n,“3Ihl. The size of R is then given by

J In R = hnf”R Ih(

(14.16)

and long-range order will be always destroyed. The size of the domains will be given by the solution of Eq. (14.16). Finally, temperature

\

t t t / J

FIG. 15. An arrangement of an isospin domain, in which the isospin adjusts for the impurity field, using the xy continuous symmetry to minimize the energy. At finite temperature such geometrical arrangements (called vortices) will be excited in pairs but of opposite “charge.” (1.e. the arrows in the vortex with opposite “charge” correspond to a reflection through, say, the y axis of the center of each of the arrows.)

148

M. RASOLT

fluctuations will destroy long-range order in two dimensions for xy continuous symmetry by creating a pair of isospin vortices. We will discuss the implication of such disorders, due to impurities and temperature, in Section 16. 15. VALLEY WAVESIN

THE

110 INVERSION LAYEROF SI

In the preceeding subsections we have examined the nature of the ground state for the integral quantum Hall effect in isospin space. In the absence of SB terms we found it to favor a ferromagnetic isospin alignment. In view of our discussion in Section 11, this state will have a low-lying isospin excitation "valley wave" which could play an interesting role in the quantum Hall effect of a Si 110 inversion layer (see following subsections). Here we discuss the dispersion of such a mode; we assume for simplicity an isotropic effective mass m * in Eq. (12.5). This dispersion can be calculated by considering the isospin flip response function

where (15.2) and Z, is defined in Eqs. (3.3). The poles in frequency of the Fourier transform of K - + ( q , t), i.e., (15.3) give the dispersion o+-(q) of these modes as a function of momentum q. Of course in a magnetic field the generators of the translation group arez6 Q=

j

h a e (- :--A(rj)) arj c 1

-

e

(15.4)

Q can be shown to commute with fi [in Eq. (6.1)]. Its eigenvalues k, i.e., Q I&) = k I+k), can also be shown to correspond to q of Eqs. (15.1)-(15.3). The calculation of w+-(q) can be done exactly to first 26L.P. Gorkov and I. E. Dzyaloshinskii, Zh. Eksp. Teor. Fiz. 53, 717 (1967) [Sou. P~Ys.-JETP 26, 449 (1968)l.

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

149

order in e2/(elohwc),27*28 by considering the integral equation in Fig. 14 (by choosing the transverse component of the response function, we can neglect the double ring diagram in Fig. 14a). Here is a sketch of the calculation. First K-+(q, w ) can be written as

I

(15.6)

(r k l , a)= e i k l y @ , ( x ) .

The single-particle Green's function G,(w) is shown in Fig. 14b and the vertex function is the shaded part of Fig. 14a. We can write G,(o)as G,(w) =

1 w - C, - ie,

(15.7)

where E,=O+ for states which are occupied and E,=Ofor the unoccupied states of the individual two isospins. the self energy C, is given by

x G,(w - wf)ei ( o - 0 ' ) s

9

(15.8)

where

Since we are considering a fully polarized lowest Landau level Z, is finite only for one of the isospin components. It is easy to check that for the "Yu. A. Bychkov, S. V. Iordanskii, and G. M. Eliashberg, Pis'ma Zh. Eksp. Teor. Fiz. 33, 152 (1981) [JETP Left. 33, 143 (1981)l. "C. Kallin and B. I. Halpenn, Phys. Rev. B 30, 5655 (1984).

150

M. RASOLT

states of Eq. (12.7) 2, is indeed independent of k and since the dashed interaction in Fig. 14b is static Z, must be independent of w. Finally, the vertex function satisfies the equation

By shifting2' to a coordinate system Ak = k , - k2 and k = ( k , + k 2 ) / 2 , Eq. (15.10) can be diagonalized and solved for the vertex function I'. Equation (15.10) then yields the following form for K - + ( q , u ) : ~ ' (15.11) where A(q) = Ne-42'2/4L&212/2)

(15.12)

and (15.13) 1.5

I

I

u,=l 1.0

I

I

I

q=

u,=o

r

asymptote----__-

-

U

v

3

-

0

1

2

3

4

5

6

7

qe

FIG. 16. The valley-wave dispersion w ( q ) as a function of momentum q, for the fully isospin polarized integer QHE (taken from Ref. 28).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

151

and where L: and 1, are the Laguere polynomials and the modified Bessel function of the first kind, respectively. In Fig. 16 we display the dispersion of Eq. (15.13). [Incidentally, in the following section we give a more general derivation of K-+(q, w ) which encompasses the fractional quantum Hall effect as well.] We next turn to the effect of SB terms on this dispersion and conclude with the implication of the existence of such low-lying valley waves to the quantum Hall effect in the 110 inversion layer.

16. DISSIPATION IN ISOSPINSPACE So far we have not discussed any implication of the results to experimental measurements. In this subsection we make some concrete predictions concerning dissipation in the integer quantum Hall effect of a Si 110 inverse layer, via excitations in isospin space. The quantum Hall effect, integer or fractional, is believed to depend on a gap in the excitation i.e., the 2D electron fluid is incompressible. In subsection 12 we have conclusively shown the ground state to be isospin polarized and in subsection 15 to carry low-lying goldstone modes, “valley waves” (VW) (see Fig. 16). There is then no doubt that the gap in the integer quantum Hall effect (QHE) of a Si 110 inversion layer will be closed. What then will this do to the zero resistivity, at zero temperature, associated with the QHE? What should the longitudinal resistivity pxxlook like as a function of current density j? It turns out that dissipation does not commence at j = 0; it would only do so if the gap due to SB [i.e., (14.11)] was neglected. Therefore, as stressed before, though the gap is very small it has important implications on the experimental observations. a. Influence of the Gap on the VW

To find the dispersion of the VW in the presence of a gap we return to the magnetic analog in Eq . (14.11). We consider the case of a low level of impurity concentration and ignore the effect of the random field [last 29K. V. Klitzing, G . Dora, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980). 30D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982). 31R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). 32R. B. Laughlin, Phys. Rev. B 23, 5632 (1981). 33D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Left. 49, 405 (1982). ”D. Yoshioka, B. I. Halperin, and P. A. Lee, Phys. Rev. Lett. 50, 1219 (1983). 35B.I. Halperin, Helv. Phys. Actu 56, 75 (1983).

152

M. RASOLT

-q-

FIG. 17. A sketch of the valley-wave dispersion for the king and x-y ground states at small ql. The effect of the symmetry-breaking terms is to introduce a gap A in the Ising case and a linear dispersion in the x-y case. The slope of the dashed tangent line is the critical velocity, in each case. Inset: Sketch of the longitudinal resistivity vs the current density j in the Ising case.

term in Eq. (14.11)] on the dispersion. Both the dispersion when A,>O (Ising) and when A, 0 the dispersion develops a gap in the VW at small q, i.e.,

hw(q) = hw+-(q)

+ A,.

(16.1)

For A. < 0 the dispersion is linear in q, i.e.

where w+-(q) is given in Eq. (15.13). At intermediate q the dispersion returns to Fig. 16. b. Dissipation via the VW When the 2D electron fluid begins to move, at some drift velocity u d , the impurities will start to Cherenkov radiate VW at random sites. But because of the gap A, no VW can be radiated until some critical velocity u, is reached. By the Landau construction this u, is given by the slope of the dashed tangent line (Fig. 17) to the Ising or xy dispersions [Eqs. 36B. I. Halperin and P. C. Hohenberg, Phys. Rev. B 188, 898 (1969). 37C.Kittel, “Quantum Theory of Solids,” Chapter 4. Wiley, 1963.

CONTINUOUS SYMMETRIES A N D BROKEN SYMMETRIES

153

(16.1), (16.2)], i.e., for the Ising

v, = (8nJ12A0)1/2

(16.3)

and for the xy V, = (2nJ12 IA,I)”’.

(16.4)

To estimate this critical velocity, we take a carrier density n = 1.5 x 10” cmP2,which corresponds to magnetic length of 1 = 100A. We use the value of the r, in Eq. (13.24) and Eq. (14.12) and we choose b-’ = 20 A. This gives Ao= 1OP6meV. The critical velocity v,, for the Chernkov radiation to start, is then v, 3 X lo3cm/sec. The corresponding critical current is j , = 7 X lop5A . Above this j c , dissipation will no longer vanish. The level of dissipation in the Ising case is given by -L

where from Eq. (15.11)

In Eqs. (16.5) and (16.6) u(zi) is the matrix element of the impurity potential between two Bloch states. Again following the calculation in Appendix A we can take advantage of the relative slow variation of b, in the envelope functions, to explicitly calculate the average over the impurity sites of lu(zi)I2 along the z direction in terms of the Fourier transforms 6(q), of B(R) given in Eq. (14.5b). From Eq. (16.6), the final form for the resistivity pxxin the king model is21

C

3a b I - i i (Q + 6)12e>(v:- vf) 2 5 6 6 n2he2J215G.ii ~

pxx=

(16.7)

where a = n i / n with n, and n the 3D impurity and carrier concentration, respectively. The results in Eq. (16.7) assume, on the average, a uniform distribution of impurities. If we specifically excluded impurities from the inversion layer the effect would be greatly reduced.21 In the xy case, the matrix element for the valley-wave creation depends

154

M. RASOLT

on q through a Bogoliubov transformation which relates s(r) to the valley-wave creation operator^.^' If we ignore this q dependence then for the xy case

=zi3Tin3heJ1c.ij c IEi'(Q + G)I2 3a

fxx

1

b

112

Jd

+ - - sin-'(:)]O>(~~:

- v:).

2

(16.8)

We choose an impurity potential ~7 such that2' ( @ ] , k I ~7 1@2,k) is 1 eV (here @k is normalized to unity in one atomic volume). This scattering strength is intermediate between our estimated values for substitutional Ge and C impurities.20 We find in the Ising case, for impurity ~ IJ,.We concentration of ni = 10'' cmP3, that pxx= 2.8 Q when I J > should also remark that since the matrix elements for Cherenkov radiation of the VW's involve only large momentum transfers [see Eqs. (16.7) and (16.8)] neutral impurities, in the inversion layer, are as efficient as charged impurities in this dissipation process. We can then, if we so wish, use such neutral impurities to enhance this process (i.e., increase pxx by several orders of magnitude) with only a mild effect on the intravalley transport. In Fig. 17 we sketch, in the inset, the behavior of pxxas a function of j. Why is the inset not a step function as predicted by Eq. (16.8)? c. Effect of the Random Field on the Dissipation The form of pxx in Eqs. (16.7) and (16.8) corresponds to the dispersions of Eqs. (16.1) and (16.2) which do not include the effect of the impurities. We saw in Section 14 that even a very low level of impurities will destroy long-range order,23 even at zero temperature in the xy case. Therefore, the dispersion in Eq. (16.2) cannot be sharp (since q is no longer a good quantum number) and pxxis going to round off (Fig. 17). To give a rough description of this xy BS ground state, in the presence of impurities, we estimate roughly the size R of the domains in Eq. (14.16). We need only to average Eqs. (14.9) and (14.10) [or more precisely (14.5b)l over z i , i.e.,

[dzi fi"(Q + G, zi)

n:13

= 2n,"'ii(Q

+ G)u$,,(G).

(16.9)

21

Using this result in Eq. (14.16) we estimate, for densities corresponding

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

155

to 1 = 100Ao, that (16.10) Clearly, the domains are very large in comparison to the magnetic length. We can then imagine that within these domains our previous discussion remains accurate. More precisely, for the xy case, V W with qR >> 1 are not expected to be significantly modified even though q is no longer a good quantum number. For qR << 1 we expect some of the excitations to remain continuous while others to turn to a discrete spectrum (i.e., bound state inside the domain). The precise V W spectrum at these very small q depends, however, on the detail coupling between the isospins and the domain structure. For the Ising BS, with impurities, or an x y with an intermediate impurity concentration ni (which as we saw, flips into an I ~ i n g ) ~ ~ , ~ ~ hw+-(q) is expected to remain unmodified for q such that hw+-(q) >> A. and to again form a complex broadened spectrum when ho+-(q) << Ao. In both cases (xy and Ising) this broadening at small q will round off the step function in Eqs. (16.7) and (16.8) (see Fig. 17). In other words, when terms beyond lowest order in the impurity concentration are taken into account, the dissipation will not be precisely zero for drift velocity vd below the critical velocity v, . There is, of course also, extremely small dissipation which is always present in principle at all vd, which comes from the intersection of the dashed tangent lines in Fig. 17 at very large q (see Fig. 16). In He4 this is very important,2 but here due to the exponential factors, in Eq. (16.6), this dissipation is negligible. Experiments are always done at some finite temperature and not T=O. In the xy case, even without impurities, finite temperature will create a pair of vortices (below the Kosterlitz-Thouless3~' transition; see Fig. 15). However, we believe that for any reasonable impurity concentration, the effects of the random field, is much more important. Concerning finite temperature in the Ising case, it should be remarked, that the critical temperature T,, where long range order is destroyed, need not be very low, i.e., need not be T, < hA. Actually T, is given by T, hJ/[k, 1n(J/Ao)],41r42which could be relatively quite high. -L

38J. M. Kosterlitz and D . J. Thouless, J . Phys. C 5, L124 (1972). 39J. M. Kosterlitz and D. J. Thouless, J . Phys. C 6, 1181 (1973). 4oJ. M. Kosterlitz and D. J. Thouless, Prog. in Low Temp. Phys. 7B, 371 (1978). 41J. Jose, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Phys. Rev. B 16, 1217 (1977). 4ZD.R. Nelson and R. A. Pelcovits, Phys. Rev. B 16, 2191 (1977).

156

M. RASOLT

In this subsection we saw how isospin space formalism can be applied to the calculation of dissipation in the integer quantum Hall effect of a Si 110 inverse layer. There are clearly many other properties, of this specific system, which have not as yet been explored and in which isospin space can play a very important role. Perhaps even more interesting is the interplay between isospin space and the fractional quantum Hall effect. We discuss this in the next section.

V. lsospin Space and the Fractional Quantum Hall Effect

The ground state of the integer quantum Hall effect (QHE) is a single Slater determinant. As we saw in the previous section, the presence of two valleys does not change this; the ground state is a fully isospin polarized single Slater determinant. The ground state of the fractional QHE is a much more complicated many-body wave function. The addition of an isospin degree of freedom further enriches the possible ground states and collective modes of this system at various filling factors. (The filling factor Y is the ratio of the number electrons N, in the 2D plane, to the number of available states N+ in the lowest Landau level; i.e., Y = N / N + ) . Isospin space then creates new possibilities, both experimentally and theoretically, which further test our understanding of this remarkable many-body phenomena. In any event, the interplay of isospin space and the fractional QHE is a challenging formal problem which we explore in this section. 17. THEFRACTIONAL QHE (FQHE) GROUND STATEIN ISOSPIN SPACE

First then what is the form of the ground state, in isospin space, when N is less than N,? We need to know this, before we can turn to the various possible excitations in isospin space of the FQHE. On the basis of the discussion in Section IV of the integer QHE, we would expect a strong tendency for an isospin polarized ground state in the FQHE as well. Things, however, are not as simple as that. Even for the single valley FQHE many candidates for the ground state are possible. It is, of course, now no longer true that the single Slater determinant ground state, appropriate to the fully polarized integer QHE is the only choice. The two most examined are the CDW state, already discussed in Section IV, and the Laughlin state. There are restrictions on

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

157

the filling factor Y appropriate to the fully polarized Laughlin state; only Y = l / m with m odd can exist (we will not consider higher hierarchy of fully polarized state^).^^,^ For these filling factors the Laughlin state gives the lower ground state energy and a gap in the excitation spectrum and is therefore believed to describe the single valley FQHE. There is further support, beyond such variational considerations, which strongly suggests that such a Laughlin state is a rigorous ground state for the single valley FQHE. It comes from the study of an appropriately truncated p s e ~ d o p o t e n t i a l for , ~ ~the interparticle interaction, in which the Laughlin state is indeed rigorous (see below). When the isospin degree of freedom is added, the nature of the ground state could become quite complicated. In addition, some subtle symmetry considerations restrict the type of states appropriate for particular filling factors Y. The type of states which are possible also dictates the nature of the isospin excitation. We need to say something about all of this. a. Variational Considerations We consider the ground-state energy of the Hamiltonian in Eq. (12.5) (with m, = m y = m*) for a CDW and a Laughlin-type ground state. The CDW ground-state energy calculation in isospin space for the FQHE proceed, identically to the discussion in Section 12. The results“ are presented in Table I for a range of filling factors Y. The polarized state contains all electrons in one valley and its energy is compared with an unpolarized CDW and unpolarized valley density wave (VDW) which are the two CDW, one in each valley, displaced out of phase to minimize the Hartree energy. For the unpolarized CDW the electrons are distributed equally between the two valleys and the charge densities of the two isospins are in phase. The same is true for the VDW but with the two isospin charge densities out of phase (see Section 12). From Table I we see that down to Y = + a polarized CDW ground state is preferred. However, unlike the integer QHE, where this state is by far the most stable, for the FQHE at small Y, the VDW is not that far off. In this context the three triangular sublattice ground-state structure discussed in Section 12 would be even more interesting at small Y. Sometimes it is better to work in the cylindrical gauge% A = 4H x r. 43F. D. M. Haldane, Phys. Rev. Len. 51, 605 (1983). “B. I. Halperin, Phys. Rev. Lett. 52, 1583 (1984). 45F.D. M. Haldane, in “The Quantum Hall Effect,” (R. E. Prange and S. M . Girvin, eds.). Springer, 1987. &R. B. Laughlin, Phys. Rev. B 27, 3383 (1983).

158

M. RASOLT TABLE I. ENERGIESPER ELECTRON IN THE HARTREE-FOCK APPROXIMATION FOR DENSITY W A V E STATES OF A TWO-DIMENSIONAL ELECTRON GAS. ALLENERGIES ARE IN UNITS OF e2/&[.IN THE SECOND COLUMN THE TWO CDW ARE IN PHASE;IN THE VDW COLUMN THE Two CDW ARE OUT OF PHASE. POLARIZED V

CDW

1

-0.3220

5

1

3

2

3

z

4

1

-0.3885 -0.4123 -0.4838 -0.5076 -0.5505 -0.6267

UNPOLARIZED CDW -0.1636 -0.2204 -0.2491 -0.3 162 -0.3354 -0.3526 -0.4154

POLARIZED VD W -0.3195 -0.3823 -0.4013 -0.4332 -0.4393 -0.4798 -0.4693

The basis set of lowest Landau levels changes from Eq. (12.8) to (17.1) where z = (x + i y ) / l . The Laughlin state is constructed from such a basis set. The extension of the Laughlin state to two components has been first suggested by Halperin. It takes the form35 (17.2) where with mo + no odd and mo even and where tiand rj are the isospins for the group of electrons in valley i = 1 and j = 2, respectively. Therefore, is antisymmetric for interchange of electron coordinate with the same isospin and symmetric for opposite isospins. Of course, the true must be antisymmetric under the interchange of all electron coordinates. must be then viewed as a part of perhaps a much more complicated fully antisymmetric wave function, in isospin space, whose expectation value with the Hamiltonian in Eq. (12.5) is the same as for I@G) of Eq. (17.2) and (17.3). It is not surprising that such a mapping is possible for such a Hamiltonian, even when the isospin independent interaction u(q) is made to be an isospin dependent umP(q)(see Section

159

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

21). This is so because such a Hamiltonian does not scatter electrons between the two valleys. It then follows at once that the anticommutator between two different isospins { qG(r), qs2(r’)}= 0 need not be respected as far as the thermodynamics is concerned; like the ground state energy at T = 0. Actually, things are a bit more subtle than that and not all values of no turn out to be acceptible; we discuss this shortly. From Eqs. (12.5) and (17.2) the ground-state energy is

where

where from Eq. (17.2)

p@(zl. . . z N )= -2(mo

+ no)2 In lzi - zjl - 2(mo+ no)2 In 12,-

C In lzi - z k l + 4 c lzi12+ 4 2I 1z,12. Nl2

- 2mo

i .k

zkl

I
i<j

N/2

(17.6)

i

From Eq. (17.6) the structure factors S m p ( q )then clearly correspond to a two-dimensional two-component classical plasma (i.e., the twocomponent Laughlin state, contrary to the CDW state, is fluid-like). Such structure factors have been evaluated using the extension of the hypernated chain t e c h n i q ~ e(including ~ ~ . ~ ~ bridge-function corrections for correlation effects at short distances) to two components, for mO= 2, no= 1 in Eq. (17.6).16,49The Laughlin state is stable for a filling factor Y = 2/(2mo no) (see below). For those values of mo and no ( Y = g), we get a ground-state energy of -0.438e2/d. [Incidentally the onecomponent classical plasma was solved, essentially exactly, by Monte Carlo methods.47 In that case the hypernated chain technique (with bridge functions) gives -0.409e2/&l as compared to -0.4100e2/d in the Monte Carlo and -0.327e2/d as compared to -0.3277e2/d, for Y = f and Y = + , respectively; the two are very close.] Comparison of these

+

47J. M. Caillol, 0. Levesque, J . J. Weis, and J . P. Hansen, J .

Stat. Phys. 28, 325 (1982). 48D. Levesque, J. J. Weis, and A. H. MacDonald, Phys. Rev. B 30, 1056 (1984). 49M. Rasolt, F. Perrot, and A. H. MacDonald, Phys. Rev. Lett. 55, 433 (1985).

160

M. RASOLT

results with Table I shows that for both the fully polarized v = f , i , and the unpolarized v = $ isospin states the Laughlin state has the lower energy. Other choices of m , and no and some symmetry restrictions on the values of n,, will be discussed shortly. There are also numerical studies, on small clusters, which have been applied to two component systems. In these calculations a finite number of particles (something like 4 or 6 particles) are placed in a Cartesian g e ~ m e t r y , ' ~or . ~ a~ spherical geometry,'* in the presence of a magnetic field. The magnetic field is varied to produce a range of filling factors v. The Hamiltonian is then diagonalized exactly with no prejudice to the nature of the ground state, including its isospin structure. The results of these numerical studies uniformly support the previous discussion; i.e., for v = f and the ground state is fully isospin polarized and for v = 3 it is isospin unpolarized and both are of the Laughlin form [Eq. (17.2)]. b. Symmetry Considerations The Laughlin wave function (PG in Eq. (17.2) is not totally antisymmetric (it, however, should be). We argued before that this does not rule out the validity of such a state when the thermodynamics of fi in Eq. (12.5) are calculated. Things, however, are not that simple. There is in addition cyclic symmetry, the Fock cyclic condition,3 which forbids arbitrary values of no in Eq. (17.2) for an isospin singlet ( P G . In short, not all values of n, can evolve from a fully antisymmetric isospin singlet ground state. There is a very suggestive reason for this. We can construct (PG by starting with the Slater determinant qf(z), in which the Landau levels of both isospin components are completely full, and multiplying it by [Pu(z)lZp( p an integer) where

and where here i , j run over both isospins. In other words

Now we know that the form of Eq. (17.2) with no = 1, m, = 0 certainly gives the same ground-state energy as the full Slater determinant,

'9Chakraborty . and F. C. Zhang, Phys. Rev.

B 29,7032 (1984). "F. C. Zhang and T. Chakraborty, Phys. Rev. B 30, 7320 (1984). "F. D. M. Haldane and E. H. Rezayi, Phys. Rev. Leu. 60,956 (1988).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

161

properly antisymmetrized between the two isospin components. Replacing then qjf(z) by Eq. (17.2) with no = 1, m, = 0 leads to a restriction on n o , in Eq. (17.2), such that no = +l. Combinations like, e.g., mo = 2 and no = 3, with a corresponding Y = +,cannot be constructed in this way and will turn out not to be valid. To see this we envoke the Fock cyclic ~ o n d i t i o n If . ~ the ground state factorizes to a spatial part and an isospin part (i.e., if it is of a definite Young symmetry) then for a singlet ground state not only does GG have to be antisymmetric separately in the two isospin groups, i.e.,

where e ( i , j ) is the permutation operator that exchanges the values of the coordinates i and j in G G . It also must not be possible to further antisymmetrize GG between the isospin coordinates and one of the isospin coordinates, i.e.,

I

(17.10) It is not difficult to inspect Eq. (17.2) and see that only no = 1 satisfy Eqs. (17.9) and (17.10). Actually, many of such symmetry considerations including more complicated ground states, with no definite Young-type symmetry, become more transparent using Haldane's truncated pseudopotential m e t h ~ d . ' ~ -This ' ~ method is based on Haldane's observation that within eact Landau level the Hamiltonian 8, in Eq. (12.5), can be reduced basically to the physics of the relative angular moment of only two particles. To see this consider fi of Eq. (12.5) extended to all Landau level manifolds. Now since

this

8 can be rewritten as H = X Hij, i<j

53F.D. M. Haldane and E. H. Rezayi, Phys. Rev. Lett. 54, 237 (1985). 54F.D. M. Haldane and E. H. Rezayi, Phys. Rev. B 31, 2529 (1985).

(17.12)

162

M. RASOLT

where (17.13) We want to rewrite this in terms of the guiding centers Ri (centers of the cyclotron orbits)

Ri = ri + Z x ni

(17.14)

and SC;

= -ih

a

e + - A(c;). ar; c

(17.15)

-

The commutation relation of Ri is (17.16) Hij can then be written as

The n is the Landau level index and each manifold has iV+ degenerate angular momentum states [e.g., for n = 0 the m manifold is given see in Eq. (17.1) with m going from l-+N,]. The reason we can pull out the matrix element squared in Eq. (17.17) is that it is independent of m. The remaining e'q'(R8-R ) operates on these angular momentum states m within this single manifold n. In fact, this matrix element can be calculated by writing

+

+

where q = qx iq, and n = nx in,. (Note all lengths are scaled by I). From Eqs. (17.15) and (17.16) n and n* are raising and lowering operators for In ) , respectively. Therefore,

F ( ~=) (01 = (01

n*ne(q'n-qn*)/2 n*neq*n/2e-qn*/2

= eqZ/4(01

nn 10) e +q2[n,n1l2nn10)

n*neq'n/2e-qn'/2

Jdn

10)

(17.19)

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

163

or finally

= e-q2'4~,(q2/2),

(17.20)

where L,(x) are the Laguere polynomials. Now Hij depends only on lRi-Rjl, after integration over q. Hjj then commutes with the relative angular momentum

We can then write45 (17.22) where Pm(Mi,) is the projection operator on states with M i j = r n . The parameters V, are the energies of pairs of particles with relative angular momentum rn. A calculation similar to Eqs. (17.18)-(17.20) gives

How do we then use Eqs. (17.12) and (17.22) to learn about the mossaic of possible ground states of the FQHE in isospin space? We can build such ground states from the basis set

Irn) = (rn!)-'"(z:)" lo),

(17.24)

where Z' = (R; f iR3/2'/'.

(17.25)

Clearly from Eqs. (17.16) and (17.21) Z' are the raising and lowering operators for the angular momentum Mi and ( Z i - Zj)* for M i j . [Incidentally, Eq. (17.24) in coordinate representation is just Eq. (17.1).] Suppose we choose a truncated pseudopotential so that only V, and V, are finite. Let us then see what happens when we write the ground state

164

M. RASOLT

in the form

&(Z:. . . z ;)10) = p ( Z T , z : . . .z;)

where r n ( t i , zj) = rn, + U , S ~ , , [see ~ , Eq. (17.3)] and p ( Z : . . . Z;) is some symmetric polynomial in 2 : . . . Z;. Now we cannot exceed the highest available Landau state rn = N+ - 1. Accordingly for N particles Eq. (17.26) says that (17.27)

Suppose we choose m , r 4 and r n o + n o ~ 5and then choose Eq. (17.22), with only V, and V, finite, and such that the inequality of Eq. (17.27) is satisfied. Such a choice implies that Eq. (17.26) has a degeneracy of ground states all with zero energy corresponding to a range of different polynomials P (in Eq. (17.26)] and a range of values of rn, and n o . This is so since such polynomials can be written in terms of functions of (Z: - ZT) and (Z' Z:) and since Z+ ZT commutes with Mi,the polynomials leave the zero energy of these ground states unchanged. However, for the situation when rno = 2 and no = 1 and the equality in Eq. (17.27) is satisfied, this degeneracy is removed, leaving only one zero-energy ground state with p ( Z : . . . Z z ) = 1. It is, therefore, the exact ground state for the truncated pseudopotential of H in Eq. (17.22) with only V, and V, finite. From Eq. (17.27) it has a filling factor (at large N + ) of $. It is precisely the Laughlin state in Eq. (17.2). Since this state is an eigenstate of a permutation symmetric Hamiltonian, we are guaranteed that it is a valid state in agreement with previous consideration. We see that the truncated pseudopotential is indeed a very powerful method for extracting the proper ground states. For example, following the above discussion, such an isospin singlet state could not be generated for rno= 2, no= 3, i.e., the Y = 3 (the only possibility is n, = 1). If we choose only V, and V, to be finite similar arguments lead to a nondegenerate ground state with rn, = 1 and no = 2. A quick inspection of this state shows that it does not satisfy the Fock cyclic condition [Eq. (17.10)] and is therefore not an isospin singlet; it has BS in isospin space. (Finite-size cluster calculations confirm this.) This is a particularly interesting ground state since it carries a filling factor Y with even

+

+

CONTINUOUS SYMMETRIES AND B R O K E N SYMMETRZES

165

denominator; i.e., Y = 4. It seems to be stable for multilayer heterojunction (Fig. 13) to be discussed in Section 21.

18. THESINGLE MODEAPPROXIMATION (SMA) The various isospin ground states, at different Y, of the FQHE lead to a variety of collective excitations which carry along the signature of this fascinating quantum liquid. The forms for these collective modes are, therefore, very interesting both formally and as perhaps a way of penetrating these isospin ground-state configurations. In Section 15 we briefly described the calculation of the GM(VW) in the integer QHE. Here we derive such modes for the FQHE. Of course, the complexity of the FQHE ground states (vis-u-vis the integer QHE) does not permit a rigorous calculation of these modes for all q; we present the SMA appropriate at small wave vectors q.5s57 How do we then search for these collective modes? To do that we need to consider the various response functions of the appropriate symmetry in isospin space. These are (18.1) (18.2)

(18.3)

(18.4) (18.5) (18.6) and S+(q,t) = ST(q, t ) , F ( t )= 1 when t > 0 and e ) ( t )= 0 when t < 0. In Eqs. (18.1)-(18.3) (see next subsection) the eigenstates) . 1 are all in the lowest Landau level and the time evolution also proceeds in the lowest "R. P. Feynman, Phys. Rev. 94, 267 (1954). 56R.P. Feynman and M. Cohen, Phys. Rev. 102, 1189 (1956). 57A.Millers, D. Pines, and P. Nozitres, Phys. Rev. 127, 1452 (1962).

166

M. RASOLT

Landau level. [Note that for the fully isospin polarized case Kpp(q,t) and K,,(q, t) are the same.] The operators evolve as SJq, t ) = e'.Fi'S-(q, 0)e-'fit

(18.7)

or

We consider, €or example, the propagator K-+(q, t). Its symmetry is the same as for the circular spin response functions and its excitations are therefore the VW modes (see Section 15). To extract the dispersion of these modes, we take the time derivative of Eq. (18.3),

(18.12) where

CONTINUOUS SYMMETRIES A N D BROKEN SYMMETRIES

167

From the Fourier transform of Eq. (18.14) combined with Eq. (18.12) we get that (18.16) where

I

K2(q, w ) = dteiwrK2(q, t ) = /dre'""i(OI

[J-(q, t ) , J+(-q)] lO)e'(t).

(18.17)

Now for any Hamiltonian invariant under continuous rotations and whose ground state is polarized, the electron-hole excitations (e-h) from one isospin index to the other isospin index 2 must have a gap. The only gapless excitations possible are the GM. The following form for K-+(q, w ) [Eq. (18.12)] follows at small q and w lim lim K-+(q, w ) =

-0

q-0

A (9)

w - w+-(9)

+iE.

(18.18)

Expanding Eq. (18.15) we get

Comparison with Eq. (18.16) gives

and for the GM, w+-(q)

What we have achieved in casting the GMw-+(q) in the form of Eq. (18.21), is to transfer the calculation from the function K-+(q, w ) , which we do not know, to another function K2(q, w ) which we also do not

168

M. RASOLT

know. To get the behavior of w+-(q) at small q we need to show that

Equation (18.21) is the so-called SMA; if we can prove Eq. (18.22) then it is not even an approximation but it is exact at small q. The same calculation can be carried out with the other two response functions. For the longitudinal isospin response function K,,(q, t), [Eq. (18.2)] we write the SMA form (18.23)

where the collective mode is ozz(q).For the density-density response function Kpp(q,t) [Eq. (18.1)] we write the SMA form (18.24)

where the collective mode is wpp(q). Equation (18.21) now becomes F,(q)

+ lim q-o lim Kz(q,w ) ] b l ( q ) , 0-0

(18.25)

where

[Obviously o(q) is equal to ozz(q)for the minus sign and o(q) is equal to w,,(q) for the plus sign in Eqs. (18.26) and (18.27).] 19. COLLECTIVE MODESIN

FQHE

THE

FULLYISOSPIN POLARIZED

The effective-mass Hamiltonian in Eq. (12.5) has no “kinetic” energy contributions because we have projected the full H [Eq. (2.1)] onto the

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

169

lowest Landau level. Therefore, the various response functions discussed above, and their time dependence, must all be restricted to the lowest Landau level [i.e., in Eqs. (18.1)-(18.3) all intermediate states In) and, of course, the groundstate 10) must be constructed from the basis set in Eq. (17.1)]. A very elegant way to capture all of this was first proposed by Girvin and J a ~ h . ~The ’ idea is the following. Consider, for example, the density operator of isospin component a.It can be written in terms of the z’s in Eq. (17.1) as

where

q = qx + iq, and again zi = x i + iy, [recall we scale the unit of length by 1 = (h~/eH)’’~]. Now suppose we take the matrix element of (19.2) where Inl) is any state in the manifold of Landau level n l . It is then straightforward [from Eq. (17.1)] to check that

for any two states n‘ and n” in the manifold n, = 0 and

(n‘rrl

e2iq(a/ar)

In‘) = O

(19.4)

for any state nf” in the manifold nI > 0. The extension to many particles (z+zj) is immediate. The projected pIy(q)is then p a ( q ) given by (19.5)

But when In) steps out of the lowest Landau level (nI & ( q ) 10) = 0 the M. Girvin and T. Jach, Phys. Rev. B 29, 5617 (1984).

170

M. RASOLT

is then

corresponding projected Hamiltonian

fi given by

(We are ignoring a constant term in R.) We can now drop the intermediate states in Eqs. (18.1)-(18.3) so that

The operators now evolve in time like (19.11)

S-(q, t ) = eiHrS-(q,O)e-'" or

-i

asat

(9, t ) = [S-(q, t ) , HI.

(19.12)

We can now return to the key Eq. (18.21) [we ignore for the moment K 2 ( q ,o)]. We then first calculate F,(q)/F,(q) for K-+(q, w). From Eq. (18.11) with 3- and 3, written in the projected form we get'6749

x exp( -iq -

t)

f

exp( - q * z i )

o+ju-jexp( -iq

2)exp( - 5i q * z i ) azi

Only i = j terms survive the commutator in Eq. (19.13) and these terms are easily calculated using the translational properties of exp[iq( a/ az,)].

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

171

We get'6,49

2 (01

F,(q) =e-q2t2

[ a - i ,~

10)

+ i ]

1

C (01

uz,10) .

(19.14)

F , ( q ) = - N e P 42t2 .

(19.15)

- -e-qzt2

1

For a fully polarized system.

The term F,(q) is considerably more difficult to calculate; we sketch the calculation in Appendix B. The final result for F,(q) is

Here Sl,(ql) is the structure factor for the one component (or equivalently fully polarized) Fermions. From Eqs. (19.16) it follows that F,(q)lF,(q) = q2. We next return to Eq. (18.21) and show that lim,,K2(q, w)/F,(q) E > 0). To do this we write the spectral decomposition of goes like q2+&( Eq. (18.17)'6*49

From Appendix B and the inversion symmetry of H it is not difficult to show that

summed over the degenerate eigenstates In') goes like q2+&,with E > 0. This is so because of the prefactors 1 - e(q*q1-q*,q)'2and 1- e-(q*41-4T4)R, in Eq. (B2), which yield a linear dependence [in J-(q)] at small q going as (q*ql - qTq). The inversion symmetry of H in conjunction with the

172

M. RASOLT

integral over q1 [in Eq. (BZ)] yields additional powers of q at small q. In addition the e-h excitation spectrum has a gap so En - Eo is finite. Therefore, the limW+"limp" K2(q, w ) = q2+'. Of course the GM w-+(q) does introduce a singular behavior in Eq. (19.17) but its residue goes again like q2+&.Our final result is that for the VW at small q

We conclude by noting that this result is a consequence of the structure for the spin current operator J-(q) in the lowest Landau level. For the full spin current L(q) (for example an itinerant ferromagnet) corrections from K2(q, w ) must be included. We next consider the propagator K,,(q,t) of Eq. (18.1) [or equivalently K,,(q, t) of Eq. (18.2); the two are the same in the fully polarized case]. We follow the same calculation as for K - + ( q , t ) and list only the final result. We assume that only a single mode dominates the small q limit of K,,(q, w ) , i.e., Eq. (18.24). [Notice such that such an assumption (or probably approximation) was not used for K-+(q, w ) , where both q + 0 followed by w +0 limits were considered, and where from the SU(2) symmetry only one soft mode is expected (Fig. 18).] Then

FIG. 18. A sketch of the dispersion for a general Goldstone mode in a BS ground state of a Hamiltonian with SU(2) symmetry. The shaded area is where the mode decays into single-particle excitations; (a) strong coupling, (b) weak coupling. The GM is the solid curve which enters the shaded continuum.

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

o'20

173

rn

0

0.5

1 .o

1.5

2.0

q

19. Dispersion of the Goldstone modes w + _ ( q ) of Eq. (19.18) (solid curve) and the [Eq. dispersion of the density-density modes, or, equivalently, the isospin rotons o,,,(q) (19.24)) (dashed curve), as a function of q for filling factor v = f.

FIG.

Both F,(q) and F2(q) have been evaluated by Girvin et ul.sy-61The results are

F,(q) = -N(S,,(q)

+ e-yz12- 11,

(19.23)

and off is given by Eq. (18.25). This completes our derivation of the two excitation modes at arbitrary filling factor Y and full polarization in the FQHE. These are the low-lying GM w+-(q) [Eq. (19.18)] and the density-density mode (or in the fully polarized case equivalently the magneto roton modes) w f f(q) of Girvin et These two modes of the fully isospin polarized ground state are displayed in Figs. 19 and 20 for filling factors Y = 4 and Y = 4. At these "S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Phys. Rev. Lett. 54,581 (1985). M. Girvin, A. H. MacDonald, and P. M. Platzman, J . Magn. Magn. Mnter. 54-57, 1428 (1986a). 61S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Phys. Rev. B 33, 2481 (1986).

M. RASOLT

174

0'08

1 t

I

u

=

I

I

115

9

FIG. 20. Same as Fig. 19 for v =

4.

L

0

1.o

2.0

e

FIG. 21. Excitation spectrum of the two component fermion system at v = f. Closed squares (n, = 4) and closed circles (n, = 5) show the density-wave mode energy calculated for finite systems. Open squares (n, = 4) and open circles (n, = 5) are the corresponding results for the isospin wave mode (the Goldstone mode). These results are extrapolated to 9l = m as shown by the dashed lines at 91 > 2. The other dashed lines are the results by Rasolt and MacDonald the higher branch at lower ql being the density-wave mode and the lower being the isospin-wave mode. (Taken from Ref. 62.)

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

175

filling factors the states were found to be stable in the fully polarized Laughlin ground state (see Section 17). S,,(q) in Eqs. (19.16) and (19.24) is therefore the corresponding structure factor [see (17..5)]. Consequently the collective modes carry the signature of the type of ground state appropriate at different filling factors Y. In Fig. 21 we show a cluster calculation, by Yoshioka,6’ of these two modes, at v = f . As discussed in Section 17, such calculations are not prejudiced to the nature of the isospin FQHE ground state. The agreement at small q with the SMA is excellent and it supports our previous discussion of the nature of the isospin FQHE ground state at these filling factors. Finally, the discussion of the effect of SB and disorder and its relation to the dissipation in the FQHE, which was covered in great detail in Sections 13 and 14, will to a large extent apply here as well. The precise magnitudes of these effects are, however, much more difficult to calculate.

20. COLLECTIVE MODESIN FQHE

THE

ISOSPIN UNPOLARIZED

As we saw in the fully isospin polarized FQHE, the gap in the excitation spectrum is closed, by the GM, and dissipation is possible. Does the isospin degree of freedom close the gap for the unpolarized Y = 3 state? For the isospin unpolarized ground state the circular response (or equivalently the transverse response) and the longitudinal isospin response are all equal and we therefore need to consider the two propagators in Eqs. (18.1) and (18.2), i.e.,

and

To extract the spin excitation mode o,,(q) and the density-density mode opp(q) we manipulate K , , ( q 7 t ) and K,,(q,t) in a similar way to the 62D. Yoshioka, J . Phys.

Soc. Jpn. 55, 3960 (1986).

176

M. RASOLT

polarized case. We list only the final results. Again we assume that only a single mode dominates the small q limit, i.e., lim K,,(q, w ) =

q-0

4(q)

w - w,,(q)

+ i&

'

(20.3)

and lim Kpp(q, w ) =

gto

0-

where

Fdq) wpp(q) + i& '

(20.4)

and &(q) + lim lim K2(q, w)]/Fl(q), w-0

-0

(20.6)

[Obviously w(q) is equal to w,,(q) for the minus sign and w(q) is equal to wpp(q) for the plus sign in Eqs. (20.5), (20.7), and (20.8).] The calculation of F2(q) (in particular) is a bit lengthy but follows closely the previous calculation of K,,(q, t) in the polarized The final results are

and

[Note that the absence of f sign in front of the 2S12(ql)in Eq. (20.10) is not an error.] Equations (20.6), (20.9) and (20.10) provide the excitation modes in the isospin unpolarized ground state of the FQHE.

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

177

We conclude with a remark about the case of an arbitrary polarization. We recall that in the fully polarized system the longitudinal isospin mode is degenerate with the density-density mode leaving only two independent excitations (the VW and the density-density modes or equivalently magneto roton modes). In the unpolarized case the transverse modes are degenerate with the longitudinal isospin mode, leaving again only two independent excitations (the isospin modes and density-density modes). For the general polarization this degeneracy will split into three excitations: a soft GM, an isospin mode, and a density-density mode. Equations (20.9) and (20.10) predict that the collective modes of the unpolarized isospin ground state will have a gap. This follows from the In conservation of total angular momentum for the isotropic fluid."' Fig. 22 these two collective modes are presented, showing that at Y = the FQHE is incompressible and can have no dissipation. In Fig. 23 we plot the single-mode approximations for the static density and spin susceptibilities for the Y = $ state. These are given by setting w = 0 in Eqs. (18.24) and (18.23), i.e., (20.11)

0.0 0.0

'

I

'

0.5

I

I

I

I

'

1 .o

I

I

I

I

'

1.5

I

I

I

' 1

2.0

I

I

I

I

2.5

q

FIG. 22. Dispersion of the isospin wave w,,(q) (dashed curve) and the dispersion of the density-density mode (solid curve) as a function of q for filling factor v = $.

178

M. RASOLT

q

I

10

5

I

I

I I

I/ I

I

i

\

\

-

-

I I

\ '\

I

I

'\

8. '\

I

q

FIG. 23. Density-density susceptibility (solid curve) and isospin-isospin susceptibility (dashed curve) as a function of q for filling factor Y = 5.

and (20.12) Note that Xzz(q) is sharply peaked for q near the magnetic reciprocal lattice vector of the CDW or VDW lattice state which appeared in the Hartree-Fock calculations (see, e.g., Section 12). This reflects the incipient instability of the unpolarized incompressible fluid toward such a state which will occur for sufficiently small filling factor. OF THE QHE 21. ISOSPINREPRESENTATION HETEROJUNCTION

IN A

TWO-LAYER

Multivalleys in semiconductors and semimetals are not the only candidates for isospin representation. Two-layer heterojunctions in GaAs (see Fig. 13) also provide exciting possibilities in isospin space. Particularly, since this isospin space is man made, the nature of the interparticle interaction can be easily modified. The effective-mass Hamiltonian 8, in

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

179

Eq. (12.5), is now generalized to

(21.1) ull(q) = ~ ~ ~are ( the q ) interaction for two electrons within the same layer (i.e., the same isospin) and u12(q)is the interaction between electrons in different layers (different isospins). umf(q)is then an isospin dependent interaction which, however, is isospin conserving. In terms of the envelope functions discussed in Section 12 [Eq. (12.4)] we can write (21.2) where FnB(q)is a trivial extension of Eq. (12.4) to two layers. The SB of SU(2) isospin space, due to the isospin dependent interaction unB(q), changes the nature of the collective excitations in the FQHE derived in Section 20. a. Collective Excitations in the Fully Zsospin Polarized Ground State of a Two-Layer Heterojunction We can follow the discussion in Section 19. The calculation in the SMA is somewhat more complicated. The previous VW, which went to zero at small q, develop a gap due to the difference in uI2(q) and ul,(q). Following our previous notation the same dispersion of the VW is (21.3) where (21.4) 63A. H. MacDonald and M. Rasolt (unpublished).

180

M. RASOLT

and

The density-density mode, which for the fully isospin polarized ground state is equivalent to the isospin-roton modes, remain unchanged from Eq. (19.24). b. Collective Excitations in the Unpolarized Isospin Ground State of a Two-Layer Heterojunction

In Section 19 the transverse isospin modes o+-(q) and the longitudinal isospin modes w,,(q) were the same. Here they split due to the SB terms. We get that w(q) = -&(q)/e(q) where63

+ (eq.91-

e+q*q1)(Sll(q1+ q) - I)]

The minus sign corresponds to the longitudinal isospin mode o,,(q) [Eq. (20.3)] and the plus sign corresponds to the density-density mode [Eq. (20.4)]. Fl(q) is given in Eq. (20.9). The contribution of F2(q) to the transverse mode w+-(q) is

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

181

c. xy Symmetry in the Isospin Polarized Ground State of a Two-Layer Heterojunction We already mentioned, in Section 14b, that in this system the symmetry of the ground state is going to be xy-like; let’s see why. Equation (21.1) can be rewritten in another way which exhibits the isospin anisotropy along the isospin z directions more clearly, i.e., (21.8) where and

The first term in Eq. (21.8) is isospin invariant; the second term is not. Now, for the two-layer heterojunction there is little doube that u,,(q) > ulz(q). Therefore, any isospin polarized ground state, such as the integer QHE or the Laughlin Y = f state (see Section 17) will lower its energy by rotating from an Ising-like isospin configuration to an xy. [Very similar to the discussion in Sections 9 and 13, we assume here as well that the anisotropy is small (i.e., u z / u p<< 1) so that we can stiffly rotate the fully polarized isospin state from the z to the xy isospin direction.] (As a side remark, we note that such a rotation does not entirely eliminate the energy expense from the SB. For example, when the state is fully isospin polarized, the energy of the symmetry breaking goes like the number of particles N. When the isospin is aligned along the xy plane it is of order 1. In other words, the isospin Ising-like state rotated to the xy isospin plane is not an eigenstate of Zz (or pz in Eq. (21.8)). The eigenstates of Eq. (21.8), where up >> u, are eigenstates of Zz Im) = m Im) [Zzdefined in Eq. (3.2c)l m going from N + -N. The ( m = 0 ) state does totally eliminate the expense from the SB. This state is invariant under rotations in the xy isospin plane. However, in the usual way, the presence of an infinitesimal uniform “magnetic field” along the xy isospin plane will favor the BS ground state discussed above.) The xy continuous symmetry will change the nature of the isospin ground state and the excitation spectrum. Just as in the discussion of Section IV, the isospin ground state will lose its long range order in the presence of a random field of the impurities. For the pure state at T = 0 the VW, which in this two-layer heterojunction, developed a gap [see Eq.

182

M. RASOLT

(21.7)] will now go soft at small q. Therefore, at filling factors Y = 1, 4, $ (see Section 17) the system will show similar dissipation already discussed in Section 16. The dispersion of the VW at small q can also be calculated by mapping the isospin properties to the anisotropic Heisenberg Hamiltonian [Eq. (14.11)]. Since limqZou,(q) = ne2d/&(where d is the separation of the layers) then Eq. (21.8) predicts the following susceptibility xZZalong the isospin z axis: (21.11) where M , = N ( h / 2 ) . The susceptibility given by

h2

xxy

=

xxy along

the easy xy plane is

N

2 ho+-(q) .

(21.12)

If we map these susceptibilities onto Eq. (14.11) we can then at once [using Eq. (16.2)] write for the VW in the xy plane

+

hw(q) = [ h o + - ( q ) d e 2 / ~ 2 ~ ] ” ’ [ h o + ~ ( q ) ] ’ / 2 .(21.13) Finally, concerning the exciting possibility of the FQHE with even denominator. As we already saw in Section 17, the isospin configuration of the Laughlin state, with m,,= 1 and no = 2, was not at all clear, but one thing certain is that this is not an isospin singlet. It is then not unlikely that its BS isospin configuration would profit from the SB isospin nature of the two layer heterojunction. Numerical cluster calculations by Rezayi and HaIdanew and Yoshioka and MacDonald6’ suggest that such a state is indeed stable in this situation. Our application of isosopin was so far restricted to two dimensions, where BS isospin configurations, in high magnetic fields, are very likely. In the following section we extend these applications to three dimensions. VI. lsospin in Three Dimensions

There is no doubt that in two dimension the quenching of the kinetic energy is much more dramatic than in three dimensions. Consequently E. H. Rezayi and F. D. M. Haldane, Bull. Am. Phys. Soc. 32, 892 (1987).

64

65A.H. MacDonald (private communication).

CONTINUOUS SYMMETRIES A N D BROKEN SYMMETRIES

183

BS in isospin space is more likely in two dimensions. Nevertheless strong magnetic fields applied to 3D systems with multivalleys (like bulk Ge or Si) populated with relatively low carrier concentration (something like n = 1-3) can exhibit some fascinating BS isospin configurations. We discuss some of these here.

22. ISOSPINBS I N THREE DIMENSIONS I N THE ABSENCE OF EXTERNAL MAGNETIC FIELD

AN

We have been so far primarily concerned, in this review article, with isospin space in the presence of an external magnetic field. However, even in the absence of a magnetic field isospin BS ground state have beenM proposed. Certainly, isospin fluctuations, in the normal electron Fermi liquid, can profoundly change the Fermi liquid parameters (see Section 26). Even BS isospin states are not entirely ruled out in surface or bulk G e or Si."71 There are some Raman cross sections for n-doped Si, in the low-frequency where previous theoretical interpretation, based on free-particle intervalley density fluctuations, were found to be inadequate. Maybe the low-lying VW, which would accompany such BS, could account for this. In any event, let's see what a ground-state energy calculation may say about such 3D BS. Unlike the FQHE which is a 2D system, in three dimensions with or without a magnetic field, there is no "exact" isospin ground state. We must, therefore, resort to an approximate calculation for the correlation energy. For calculating the correIation energy E,, we consider the effective mass Hamiltonian H in Eq. (2.1) with an isotropic effective mass.73 We take the case of Ge where n = 4 and calculate the energy of three isospin configurations: (I) unpolarized isospin configuration N 1 = N2 = N3 = N 4 , where N, are the number of electrons in each pocket. (2) N , = N3 ; N2 = N 4 , with AI3 = (N,/N,)'" = 0.306. ( 3 ) A,, = 0.009. The

@M. J. Kelly and L. M. Falicov, I.Phys. C 10, 1203 (1977). 67M. J. Kelly and L. M. Falicov, Phys. Rev. B 15, 1974 (1977). 6nM. J. Kelly and L. M. Falicov, Phys. Rev. B 15, 1983 (1977). "W. L. Bloss, L. J . Sham, and B . Vinter, Phys. Rev. Lett. 43, 1529 (1979). "'W. L. Bloss, L. J. Sham, and B. Vinter, Surf. Sci. 98, 250 (1980). 71 M. Nakayama and L. J. Sham, Solid State Commun.28, 393 (1978). 72M. Chandrasekhar, M. Cardona, and E. 0. Kane, Phys. Rev. B 16, 3579 (1977). 73M. Rasolt, Phys. Rev. Lett. 50, 778 (1983).

184

M. RASOLT

correlation ene~-gy’~.~’

is calculated within the Hubbard appr~ximation.~”’~ In other words the dynamic dielectric function,

is given by

+

and f ( k ) = k 2 / [ 2 ( k 2 k i , ) ] . (The Fermi momenta of the two valleys are related by kF, = A13k,, .) The Hubbard approximation includes some of the particle-hole rescattering which become more important at lower densities. The transverse mass rn, and longitudinal mass rn,, entering Ma,, are grouped to a spherical mass rno,=3/(2rn;1+rn;’).78 The interaction u(r, - r,) is set = e 2 / ( r , r,l. Length is measured in effective and energy in Ry* = rno,e4/2h2.The r: is given Bohr radii a* = &h2/rnoee2 by (47~/3)r,*~ =(a/N)l/~;~. The kinetic and exchange energies are given essentially exactly

and @ ( p )= p”6 sin-’[(l - ~ ) ” ~ ] / -( 1p)’12, pe = rn,/rnl, rnde = (rn:rn,)1’3 and /3 = 2’13(1 + A:3)113. We take rn, = 0.082rn, rnl = 1.58rn and E = 15.36 (the appropriate values for Ge). The band-structure energy [V(r) in Eq. (2. l)] is calculated in second-order perturbation theory using the static 74

D. Pines, “Elementary Excitations in Solids.” Benjamin, 1964. 75D.Pines, “The Many Body Problem.” Benjamin, 1962. 76 J. Hubbard, Proc. R . Soc. London, Ser. A 240, 539 (1957). 77J. Hubbard, Proc. R . Soc. London, Ser. A 243, 336 (1958). 78W. F. Brinkman and T. M. Rice, Phys. Rev. B 7 , 1508 (1973). 79M. Combescot and P. Nozitres, J . Phys. C 5, 2369 (1972).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

(x!o-2)

185

(x~~-3)

5

2 2

I

I I

*x

*a

u o

(L

Y

W

W

-1

-I

-3 -2

-5

-3

FIG. 24. The AE’s represent the differences between two polarizations A,3 = 0.009 and A,3 = 0.306, for the solid and dashed curves, respectively, and the unpolarized (A,3 = 1) ground state. AEk, AE,, AEc, and AEb are the kinetic, exchange, correlation and band-structure contributions. AEt is the sum of the four and is the only one measured on the right axis.

186

M. RASOLT

(a)

(b)

FIG. 25. (a) Example of an absorption process of light (with momentum q) followed by emission of two VW of momentum q, and q, (q = q, + q3). (b) An example of Raman scattering process followed by emission of two VW.

Hubbard screening (see above), i.e., (22.3) We have considered only the contribution from donor impurities, for which V ( q ) = 4 n e 2 / q 2 , and placed them in a fcc lattice [i.e., S(q)= CG 6(q - GI]. In Fig. 24 we present the various contributions to the ground-state energy, relative to the isospin unpolarized ground state, as a function of r,* for the two different polarizations. the calculation suggests a possible isospin polarization at r,* > 7. Note also that the contribution for the impurity energy [Eq. (22.3)] shows preference to isospin polarization. We perhaps can induce isospin polarization by introducing neutral impurities. Incidentally, other isospin BS states can be generated by the rotations in isospin space discussed in Sections 3 and 4. If indeed such isospin polarization occurs it can be observed by light absorption or resonant Raman (of frequency of the gap at L , see Fig. 2)80781with the creation of two VW of opposite momentum (see Fig. 25). The appearance and disappearance of such V W as a function of neutral impurities would be very interesting8’ so“Light Scattering in Solids” (M. Balkanski, ed.). Flammarian, Paris, 1977. *“‘Theory of Light Scattering in Condensed Matter” (B. Bendow, J . L. Birman, and V. M. Agrenovich, eds.). Plenum, New York, 1976. *’Incidentally, atomic nuclei have different occupations of neutrons and protons whose isospin symmetry is broken by the electromagnetic forces. Isospin waves should reside in such nuclei; of course, corrected for the finite volume.

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

187

The various electron pockets, e.g., in Si, are related by symmetry. There are many other systems with multipockets of electrons not related by any symmetry operations with totally different masses. If the masses are relatively light (Le., comparable to the free electron mass) then generally the expense in the kinetic energy [last term of Eq. (10.8)] forbids any rotation in isospin space. However, if the masses are very large (orders of magnitude larger than the free electron mass) and if the initial state is isospin-spin polarized (see Section 3) , then such rotations are possible, provided the SB terms of Eqs. (9.5)-(9.7) compensate. Such isospin-spin rotations (Section 4) generate a host of magnetic structures. One naturally thinks of heavy Fermion systems, e.g., UPt3, as such candidates.83784We should, however, be cautious. In such systems electron-electron screening is very large.8-5-88The interparticle interaction in Eq. (9.13) becomes very weakly dependent on momentum. [With no momentum dependence the SB terms in Eqs. (9.5)-(9.7) (see also Appendix A) are identically zero.] Within each isospin component this is certainly true. But SB involves large momenta, Q and G, and we cannot rule out a preference for isospin-spin rotations. Maybe many of such instabilities, in heavy Fermion systems, can be understood in isospin space? 23. ISOSPIN BS IN THREE DIMENSIONS IN THE PRESENCE OF AN EXTERNAL MAGNETIC FIELD Even in the absence of an external magnetic field some 3 D systems were predicted, e.g., by O v e r h a u ~ e r ~ ’to ~ ~potentially exhibit SDW or CDW instabilities. In the presence of an external magnetic field some of the kinetic energy, perpendicular to the field, is quenched. This makes the formation of BS ground states more likely. In fact, for a field strong enough to place all electrons in the lowest Landau level, there is only kinetic energy expense parallel to the field, making the system quasi-one83

A . I. Goldman, G . Shirane, G . Aeppli, B. Batlogg, and E. Bucher, Phys. Rev. E 34, 6564 (1986). MR. C. Alberts, A. M. Boring, and N. E. Christensen, Phys. Rev. E 33, 8116 (1986). =P. A. Lee, T. M. Rice, J. W. Serene, L. J . Sham, and J. W. Wilkins, Comments Condensed Mutter Phys. U,99 (1986). %D. M. Newns and N. Read, Adv. Phys. 36, 799 (1987). 87P. Coleman, Phys. Rev. E 29, 3035 (1984). -P. Coleman, in “Theory of Heavy Fermions and Valence Fluctuations” (T. Kasuya and T. Saso, eds.), p. 163. Springer, Berlin, 1985. 89A. W. Overhauser, Phys. Rev. Lett. 4,462 (1960). 90A. W. Overhauser, Phys. Rev. U8, 1437 (1962). 91A.W. Overhauser, Phys. Rev. 167, 691 (1968).

188

M. RASOLT

dimensional, and making instabilities very likely. What are some of the possible instabilities? If the magnetic field is so strong that the kinetic energy hk2,/2mz, parallel to the field, is much smaller than ho,, then the type of instabilities discussed in two dimensions (see Section 12) can occur, the system can become isospin polarized. If the g factor [in eq. (6.1)] is not too small, this very high magnetic field will also align all spins parallel to the field. In the case of the single valley, recent studies ~ u g g e s t ~ ’that - ~ ~in spite of the expense in electrostatic energy, the BS ground state is a charge density wave (CDW) of several different orientations; such as a Wigner crystal. The presence of n valleys further enhance such instabilities by coherently displacing the CDW from each isospin component to minimize the expensive electrostatic energy; very much like the situation found in two dimensions perpendicular to the magnetic field (Section 12 and 17). Such a coherent isospin state is the VDW presented in Section 5 and Fig. 7a. By proper choice of the coefficients U k r and Ukr (ignoring at these high fields the spin 0)such out of phase CDW (i.e., VDW) can be created. Another related state is presented in Eq. (5.3) and Fig. 7c. Here a VDW is created from coherent combination of states from the two different isospins. From the kind of discussion presented in Section 3 , it follows that this VDW has a varying isospin polarization in the xy isospin plane. The previous VDW has a fixed isospin polarization along the isospin z direction. Ignoring SB contributions we can create infinite number of equivalent degenerate BS ground states using the SU(2) rotations of Eq. (4.2). When the magnetic field is not extremely large (i.e., when h2k$/2m* ho,)the gap introduced by the order parameter of the CDW, is expected to be small compared to h2k~/2m: (i.e., weak coupling limit). In that case the expense in electrostatic energy makes it unlikely that a CDW within a single valley would be formed. Also, both spin components are going to now be occupied and a combination of BS VDW and S D W 5 states can form. Turning back to Section 11, we are now looking at the situation described in Eqs. (5.1), (5.2). When we make the coefficients Uka and Ukr in Eq. (5.1) spin dependent then two CDW are formed for valley one (one for each spin component) and two CDW are formed for valley two (again one for each spin). Because of the Zeeman splitting the Fermi momentii k“, of each spin component u, are different. Therefore, -‘I

92A.H. MacDonald and G . W. Bryant (unpublished). 93D. Yoshioka and H. Fukuyama, J . Phys. SOC.Jpn. 50, 725 (1981). %R. Gerharts and P. Schlottmann, Z . Phys. B 34, 349 (1979). 95V.Celli and N. D . Mermin, Phys. Rev. A 140, 839 (1965).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

1/2 n “left”valleys

189

1/2 n “right“ valleys

FIG.26. The model of the multivalley Fermi surface. All valleys are identical except for the orientation of the principal axis, which is directed either at the 8 = 8, o r - 0 = Br. with respect to the magnetic field H . There are $I “left” and t n “right” valleys. (Taken from Ref. 96.)

no choice of coefficients vk,s,o and u ~ ,[in~ Eq. , ~ (5.1)] can make the density of the two CDW, within each valley, cancel. However, no such Zeeman splitting occurs in isospin space and an instability to a VDW, for each spin component, is possible. Equation (5.2), on the other hand, describes combination of spin-up and spin-down states within each valley (see Fig. 7b). Such SDW states do not couple to the direct term. Nevertheless there still is an expense in Zeeman energy in creating such SDW, which the VDW does not have, and which will always favor the formation of a VDW. Again many equivalent BS SDW and VDW can be generated using the SU(2) rotations of Eqs. (4.1) and (4.2). TesanoviC and H a l ~ e r i nstudied ~~ these instabilities in isospin-spin space within the HF approximation. They considered the more general model in which in valleys (with anisotropic masses), are tilted relative to the other in (see Fig. 26). Here we take the simplest case of an isotropic mass. Tesanovie and HalperinY6did not minimize the expectation values for the ground state energy (at T = 0) or free energy (at T # 0) of the various isospin-spin configurations in Eqs. (5.1)-(5.4). Rather they calculated the critical temperature T, of various instabilities (in the normal state) whose spin-isospin configuration corresponds to these ground states (i.e., they solve the integral equation of Fig. 14). Here we do the same but for the corresponding electron-hole cross section (see Fig. 27). An instability towards a particular ground-state broken symmetry is reflected in a singularity in the linear response functions or the %Z.TesanoviC and B. I. Halperin, Phys. Rev. B 36, 4888 (1987).

190

M. RASOLT

7=2

7 =2

2=2

z=2

7=2

7=2

FIG. 27. The electron-hole scattering amplitude r of two different isospin states in the ladder approximation. The wiggly line is the interparticle interaction. The arrowed lines are the single-particle propagators.

particle-hole cross section of the appropriate symmetry. For example, we already mentioned in Section 20, that the peaks in the xzz(q) (Fig. 23) reflected an incipient instability towards a CDW in two dimensions (of course perpendicular to the magnetic field). Tesanovit and Halperin looked for such instabilities in spin-isospin space in three dimensions parallel to the magnetic field. The cross sections for electron-hole scattering are given in Fig. 27. The arrow corresponds to either isospin or spin components. We restrict our cross sections to spin-up/spin-down to avoid any coupling with the direct term (see Fig. 14a, second term) which will always lower the critical temperature of the SDW. (For the VDW the isospin-up/isospin-up or isospin-up/isospin-down cross section will give the same TJ. The Fermi momentum along the magnetic field, Icy of spin up, is different than the Fermi momentum Ic; of spin down (due to the Zeeman splitting). The Fermi momenta of the two isospins are the same for each spin component but are different for different spins. The integral equation for r can then be written at once as

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

191

(23.2) If we restrict the calculation to the lowest Landau level then in the Landau Gauge

where @(x) is defined in Eq. (12.8). The E, are the Matsubara frequencies [& = i(n/P)(21+ l ) ] , where P = l / k g T . We next look for an instability towards a SDW within each valley of modulation q2 = kr + k; along the magnetic field. We set then k z , ,= kz,3+ q2 and consider any choice of guiding centers for the two Landau orbitals; i.e., any choice of ky,l and ky,3.(We also set El, = El?) From Eq. (23.3), Eq. (23.2) can be calculated to give

If we sum both sides of Eq. (23.2) over ky,l and ky,3using Eq. (23.4) and perform the Matsubara sum over EI2, we can reduce Eq. (23.1) to the following one-dimensional form:

192

M. RASOLT

where

and where

h2kE h2(kf")2 Ei(kz)= y 2m 2m* ~

(i = 1 or 2) and f(x) is the Fermi-Dirac occupation function. [Inciden-

tally, while Eq. (23.5) is l D , it does not imply that the physics is truly 1D. In the initial scattering cross section [Eq. (23.1)] the particle-hole scatters to the phase space of k, and k4 which is not 1D; we will discuss it further in Section 25.1 From Eq. (23.4) the function W ( p , , p y , k2,1kz,2)is given by

Following TesanoviC and H a l ~ e r i nwe ,~~ decouple Eq. (23.5) by using the Fukuyama a p p r ~ x i m a t i o nIn .~~ this approximation one replaces Eq. (23.7) by its average over the relevant momentum range in the z direction. (This range extends from - k r to +kF.) We then get an average W given by

where p ~ ( l=, i k T ( ~ ) .Finally we can evaluate the sum over k, to logarithmic accuracy in T, (in the weak coupling limit this is sufficient), i.e.,

D ( q , = k:

+ k:) = kz D ( k , , q2 = k: + k!) -

2L3 m + In( '.14 fi'k'k') (27~1)~ ky + k y 2mkBT,

97H. Fukuyama, Solid State Commun. 26, 783 (1978).

.

(23.9)

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

193

0.4

A VDW spin down VDW spin up 0 SDW

0.3

3"

c

0.2

0.1

0

0.8

0.6

0.4

1 .o

H 0.4

0.3

0.2

0.1

0 0.6

0.8

0.7

0.9

1 .o

H

FIG. 28. VDW and SDW transition temperatures measured in units of the cyclotron frequency for the isotropic mass case for the two values of the reduced g factor; g = 1 and 0.5. The valley degeneracy was chosen at n = 4 and the range of the magnetic field H was from H, , in which only the lowest Landau level with both spin states remained occupied, to H2where only the spin-down band is populated ( H i s in units of HJ. (Taken from Ref. 96.)

194

M. RASOLT

Putting Eqs. (23.8) and (23.9) in Eq. (23.5), we find an instability at a critical temperature (23.10) where

A=

2 ~* 3 ~ w. fi'(2~rl)~(k7k:)

+

(23.11)

For the VDW of the same spin component u = 1 or 2 the two isospin Fermi momenta are the same and equal to k7 or k : , respectively. The solution for T, takes again the general form of Eqs. (23.10) and (23.11). The only change is to set, in Eqs. (23.8)-(23.11), k: = kp = k ; for one of the VDW and k,"= kp = kp for the other. The final numerical integration of Eq. (23.8) and the corresponding critical temperatures of the two VDW (one for spin up and one for spin down) and the SDW are presented in Fig. 28 for n = 4. The electron density p was set to loL7cm-3, the effective mass was taken as m* = 0 . h and the dielectric constant used was E = 16. In accordance with our discussion at the start of this subsection, isospin space indeed is found to have an instability (towards a VDW with spin down) with the highest T,. As mentioned previously, TesanoviC and Halperin also study the effect of mass anisotropy on these results as well as the effect of such instabilities on the transport properties. These authors also observe the possibility that graphite under strong magnetic field might be showing such isospin instabilities. IN A STRONG MAGNETIC FIELD 24. SUPERCONDUCTIVITY

a. Instability above T, We saw that diagonal long-range order can be enhanced in isospin space in the presence of a strong magnetic field; can this also occur in the off-diagonal long-range order? There have been studies of superconductivity in the absence of an external field in multivalley semiconductors.98 The emphasis here is the interplay of a very strong magnetic field and the isospin components. The Zeeman splitting is certainly unfavorable to the BS of an s wave (i.e., spin singlet) superconducting condensate. However, the isospin index 9sM. L. Cohen, Phys. Rev. W, A511 (1963).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

T=2

2=2

2=2

2=2

2=2

195

2=2

FIG. 29. The electron-electron scattering cross section r of two different isospins in the ladder approximation. The wiggly line is the effective e-e interaction and the arrowed lines are the electron propagators.

does not couple to a magnetic field and, therefore, even at very strong magnetic fields an isospin singlet Cooper pair can remain stable under Pauli pair breaking. Of course, the orbital pair breaking still remains a problem. We will turn to orbital breaking when we discuss the superconducting condensate below T, (see below). Again, to see the effect of the magnetic field on the BS ground statew we consider the instability of the appropriate cross section in the normal state; this time in the electron-electron scattering amplitude (see Fig. 29). (We assume a high enough magnetic field and ignore real spin indices.) The integral equation for r is now

r(&,E l ,

n , > k, 7 n3 k3) = +rn(&, El3 ni ki n3 k3) 1 -V(kl,nl,k2,n2,k,,n3,k,,n,, 9

7

9

c

7

P kz,nz,k4,n4

‘(E/z

+

9

51,

J

It27k2

7

n4

t

E/2-E/,)

k4)?

(24.1)

with El, = El, E/3 - E12. We will take here the phonons as the mechanism for the e-e attraction. The electron-phonon interaction is given byloo,’nl

99M.Rasolt, Phys. Rev. Len. 58, 1482 (1987).

‘9 Morel . and P. W. Anderson, Phys. Rev. 125, 1263 (1962). ‘O‘J.

M. Ziman, “Electrons and Phonons.” Oxford University Press, 1960.

196

M. RASOLT

where

In Eq. (24.3) N is the number of ions, M the ion mass, V(p) the screened electron-ion potential, op,sthe phonon frequencies of polarization ~ ~ (s),p and , bp,s, bL,s the phonon creation-annihilation operators. From Eqs. (24.2), (24.3) V(kl, n l , k2, n 2 , k 3 , n 3 , k4, n 4 ) is given by

(24.4)

where E ( P ) is the screening of the Coulomb e-e repulsion via the same valley electrons. Again Eqs. (24.1) and (24.4) can be solved in closed form with the following approximations: Restrict all the states c$n,k to the lowest landau level, i.e., n , = n 2 = n 3 = n 4 = 0 , assume a weak p dependence of the term in Eq. (24.4), and ignore the frequency dependence of V on (El* - Ell). The solution follows closely the discussion in the preceding subsection and will not be detailed here. Our final result for the critical temperature T, is

where

and

where

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

197

when h o D > h2kg/2rn, and e2 = E~ = h o D / k B c when h o D < h2k2,/2m,. oD is a characteristic phonon frequency of w ~ ,we ~ set ; it to w D = v , ~ Q D where vS is the sound velocity. QD is the corresponding wave vector; we set it to Q D = - \ l z / l . This cutoff corresponds to the Landau-orbital Gaussian functions which have the width 1. Because all the electrons are in the lowest Landau level kF= pn212/nwith p the carrier density and n the number of valleys; here n = 2. Finally Q0 is the volume of the unit cell per ion. Equation (24.5) is a slight generalization of the generic form for T, in Eqs. (23.10) and (23.11) to include the possibility where hwD is comparable to the Fermi energy. [In any event, in such a case the static approximation of V in Eqs. (24.1) and (24.4) is not valid.] To get a very rough sense of the variation of T, with H we choose''9 to look at a simple model for donor-doped silicon. The v(Q)in Eq. (24.6) is given by

(24.7)

v(Q)

where a, = 0.342, a2 = 2.221, a3 = 0.86334, and a4 = 1.5345, and is equal to V(Q)lQ,."" An estimate of the two contributions in Eq. (24.6) shows the attractive part to be considerably larger than the electronelectron repulsion (particularly since the Coulomb repulsion is less effective than the phonon induced attraction on account of its instantaneous character).'"'-' Equation (24.5) was solved for T, numerically as a function of the magnetic lengthw 1 (i.e., magnetic field) and the results are displayed in Fig. 30. A t magnetic fields accessible presently we get a very small critical temperature of Tc lop3K. A sizable T, commences at very high magnetic fields (i.e., 1 = 30 A or =1 MG). Of course T, is extremely sensitive to the electron-phonon coupling; increasing it by a factor of 2 varies the critical temperature at 1 =5OA to 0.3K. This suggests that perhaps many of the other multivalley semiconductors, insulators, ferroelectrics, or even semimetals are preferable. In particular, among the systems those which exhibit superconductivity to begin with might be the best candidates. From Fig. 30 we see a strong enhancement of T, as a function of H. It originates from two effects. First from the number of Landau levels which participate in the electron-electron coupling of momentum p (basically the density of states). A less important effect, but still favorable for pair formation, is the reduction of the repulsive e-e terms. The physical origin L-

Io2S. G. Louie, M. Schliiter, J . R. Chelikowsky, and M. L. Cohen, Phys. Rev. B 13, 1654 ( 1976).

198

M. RASOLT

1 t

0.6

0.5

\

\

9 0.4 ._

2

0.3 0.2

"

0

\.

\.

10

20

30

40

50

I (in A) FIG.30. Critical temperature T, vs the magnetic length I at carrier density 10i7/cm3.

is the following. Even including screening the e-e interaction has a much larger range than the phonon part. The e-e interaction drops rapidly for Q larger than some thing like the Thomas-Fermi screening wave vector of the free carriers. Now for two carriers without a magnetic field the states are very extended and the long-range e-e repulsion matrix element is large. The presence of the magnetic field squeezes these states perpendicular to H and reduces this repulsion; the phonon part remains largely unchanged. This all condenses into the Q,, dependence of Eq. (24.6). We like to conclude with a few remarks concerning the actual observation of such a new superconducting isospin singlet state in, e.g., Si: (1) Donors in the conduction region will contribute to pair breaking and, even more seriously, to isospin magnetic freezeout. It is therefore crucial to keep the impurities out of the conducting region. Modulation doping gives promising possibilities for producing regions (approximately 500 8, wide) of high mobility where the electron density is fairly uniform. Nonuniformity in the electron density will of course mix in higher Landau levels. Multivalley semimetals are another possibility where the impurity problem is negligible in comparison. (2) Eventually at very high magnetic fields the ground state will isospin polarize or form other broken symmetries. This could destroy the singlet isospin Cooper pair. (3)

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

199

Neglecting higher Landau levels is appropriate when hw, > EF. For carrier density p = lo+’*cmP3, this leads to 1 = 40 A; however, including several higher levels does not drastically change our conclusions. The fact that Pauli breaking is absent, in isospin space, is only part of the story. Orbital pair breaking and its close relation to the Meissner expulsion, of the magnetic field, continues to be relevant in isospin space. Therefore, another crucial question to understand is, in which way the Meissner effect is circumvented in these very high magnetic fields? The answer depends on the nature of the superconducting order parameter, in the quantum limit, and its relation to the induced diamagnetic currents. b. Nature of the Isospin Superconducting Condensate Below T, and close to the critical lines Hc2 or H,, assuming second order phase transition, the order parameter, A(r) for the isospin condensate is small. One can then develop a Landau Ginsburg free energy density expansion in A(r).lo3In Fig. 31 we list the two graphs which correspond to such an expansion. This free energy expansion F, is given by F=4+Fh (24.8) where

I

Fs = a ( T ) d3rld3r2A*(r1)K2(rl, r2)A(r2)

I

+PO 2 F h = l d r (H

d3r,d3r2d3r3d3r4A*(rl)A*(r2)

+ 6)’ 8n

’

(24.10)

200

M. RASOLT

+ FIG. 31. Free energy contribution of Eq. (24.9) to fourth order in the A(r). The filled squares are the A(r) and the arrowed lines the electron propagators.

and where the noninteracting single-particle Green’s function Go is defined in Eq. (7.4) (without the spin index o and with o+&). Of must be calculated in the course, the single particle states +,Jr) presence of the external vector potential A(r) which is the sum of the potential &,(r) (which contains the external vector potential and the average field which arises from superconductivity) and an internally induced fluctuating a(r) [b(r) = V x a(r)]. To appreciate the nature of the superconducting condensate in the high field limit (around Zi, in Fig. 32) we first recall the situation in the low field limit where the magnetic field bending of the semiclassical electron

H

FIG. 32. A sketch of the phase diagram for the superconducting condensate in the T-H plane. Assuming the isospin singlet state is superconducting at zero field; as the field increases one crosses over from a Meissner state to the 1D high quantum limit state at H,, . The step-like crossover reflects the effect of the population of higher Landau levels. Such oscillations will continue along Hc2 with smaller amplitudes and periods. (2. Tesanovie and M. Rasolt, unpublished.) The Hc2line comes close but never touches the vertical axis.

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

20 1

path varies very slowly compared with the single-particle Green's function. Thenlo3

cO(~/, rl ; r2) +,

eieN4+-r~)/fic

COG

9

rl

7

r2),

(24.13)

where Go is the single-particle Green's function with A(r) = 0. Substituting Eq. (24.13) in Eqs. (24.11) and (24.12) and minimizing F, with respect to A(r) and a(r) one gets the standard two equation^"'^

and

where

(24.17) Equation (24.15) relates the induced current j(r) to the order parameter v(r) [which is simply related to A(r) by a constant]. Equation (24.15) is the London (or the Meissner effect) relation and it excludes the magnetic field from the condensate. Clearly in the weak magnetic field limit (around Hc2 in Fig. 32) no enhancement of T, with H (see preceding subsection) is possible. The structure of the order parameter is made up of vortices whose form is i l l ~ s t r a t e d 'in~ ~Fig. 33 and these vortices form an Abrikosov (see Fig. 34). When we cross over to the quantum limit where the interparticle spacing is comparable to the magnetic length things are quite different. lo6 In this limit (with all electrons in the lowest Landau level) the Green's function Go can be written in the symmetric gauge [i.e., Eq. (17.1)] as (24.18) IMA. A. Abrikosov, Zh. Eksp. Teor. Fiz. 32, 1442 (1957). lo5W. H. Kleiner, L. M. Roth, and S. H. Autler, Phys. Rev. 133, A1226 (1964). '062. Tesanovii: and M. Rasolt, Phys. Rev. B 39, 2718 (1989).

202

M. RASOLT

h

i

I

-t .

*r

“s

2F3

r

FIG. 33. Structure of one vortex line in a Type-I1 superconductor. The magnetic field is maximum near the center of the line. Going outwards, h decreases because of the screening in an “electromagnetic region” or radius -A (the penetration depth). (Taken from Ref. 103.)

VORTEX CORES

FIG. 34. A triangular lattice of vortex line (after Kleiner, Roth, and Autler, Phys. Rev. 133, A1226 (1964)). The plane of the figure is normal to the field direction. The contours give the lines of constant lA(r)l. (Taken from Ref. 103.)

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

203

[we have omitted the dependence parallel to the field which simply multiplies Eq. (24.18) by exp{i[k(n, - x 2 ) ] } ; again the other z = ( x i y ) / l . ] Using Eq. (24.18) in Eqs. (24.11) and (24.12) we find

+

-Zl*Z1

z;z2

+

~ ~ r2) ( r= ~ 1, e x p ( 7 - - zlz;) , (2n12)2 2 ~

(24.19)

and

There is again the induced magnetic field b(r) = V x a(r) which must be accounted for. The Green's function G to first order in a(r) is given by

where h(z, z * )= (p

+

C

-

-

A(r)) a(r) + a(r) (p + f A(r))

+ a2(r).

Adding this correction to Eq. (24.9) adds the following term (FJ

(24.22) to F,

(The r integrations are now only in the plane perpendicular to H;i.e., two dimensions.) In Eq. (24.23)

x exp( -

);" 2

+

'

(24.24)

204

M. RASOLT

e +

e2 +[A*(z,z*)u(z, z * ) + A ( z , z * ) u * ( z , z * ) ] , 2mc2

(24.25) (24.26)

+

where p is the density of electrons and a(z, z * ) = a,(z, z * ) ia,(z, z * ) . We can now minimize Eqs. (24.9), (24.10), and (24.23)-(24.25) with respect to the order parameter A(z, z * ) and the induced field b(z, z * ) . We get two relations:'" the first is

where a ( T ) and P ( T ) are given in Eqs. (24.16) and (24.17) but the k sum is now 1D. The second relation is

x

I I

d2rl d2r2A*(z1, zT)A(z2, z ; )

2

2

2

+ "); 2

(24.28)

with

These two relations replace Eqs. (24.14) and (24.15). In fact Eq. (24.28) which replaces Eq. (24.15) looks nothing like the Meissner relation. One should appreciate that the physical situation here is very different than if one expands the Ginsburg-Landau free energy in weak magnetic field. In

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

205

our case the field is very strong and penetrates everywhere into the superconductor; indeed, the superconductivity is due to such a strong field. Consequently, the relationship between supercurrent and the fluctuating part of the vector potential is entirely different from that given by the London equation. The role of K 2 is simple; as opposed to two-particle kernel in standard type-I1 superconductors, which measures the cost in “kinetic” energy arising from a nonuniform order parameter, our K 2 projects A(r) to the “lowest Landau level,” i.e., all A(r) of the form f(z)exp(-z*z/2), where f ( z ) is an arbitrary analytic function, have the same T, given by Eq. (24.5), while functions orthogonal to this form do not contribute at all. Note that the magnetic length in such states is I* = ( l / l h ) l and it corresponds to particles of charge e* = 2e. We expect that K , and FS.,,will select those particular configurations which have favorably low free energy. Finding such configurations involves in principal a minimization of F, with respect to A(r) and a(r). This is a very nontrivial problem which we can avoid here by simply noting that from (24.26) it follows that the coupling between A(r) and a(r) is of the order T,/E,, which translates to order (C/&)’ for &-b. Therefore, in the weak coupling approximation (T, << E F ) we can neglect the coupling between the order parameter and the magnetic field [i.e., between Eqs. (24.27) and (24.28)] and the minimization of the free energy reduces to the solution of Eq. (24.27) at fixed magnetic field. This is similar to the case K >> 1 in the Abrikosov theory where K is the Ginsburg-Landau parameter. We observe here that A(r) = A exp(-z*z/2), where A is a constant, is an exact solution of Eq. (24.27). It describes a superconducting order parameter localized to a region of size -1 around the origin, and extending to infinity with constant magnitude along the magnetic field. This is our superconducting “tube.” It is interesting to sketch the expected field distribution of b ( z , z * ) [i.e., Eq. (24.28)] which corresponds to this order parameter. This we do in Fig. 35. The situation here is entirely different than Fig. 33. The radius of the tube is determined by the magnetic length I and is not related to the penetration depth A. The order parameter (or superfluid density ns), for a free electron dispersion E(k), also follows, in phase, the amplitude of the local variation in b ( z , z * ) ; i.e., is enhanced by the local magnetic field. In short, there are no superconducting diamagnetic currents circling around the magnetic field. The local variation in b ( z , z * ) corresponds to microscopic currents of the type we saw, e.g., in the QHE (see Section 12). Superflow will occur in this quantum limit only parallel to the field. This single tube is not particularly useful since it leads to an order parameter which is not intensive quantity. But it suggests that we should

206

M. RASOLT

FIG. 35. The superconducting “tube” in the quantum limit. Notice that unlike Fig. 33 the radius of the tube is governed by the magnetic length I* and is not related to the

penetration depth A. The order parameter (c) increases for increasing local field (b). Note also that there are no diamagnetic superconducting currents (a) circling in the tube.

search for solution of the form

(24.29) i

where R j is a set of the “guiding” centers of superconducting tubes. The function f has to be chosen in such a way that A(r) remains in the form imposed by K , . This implies that f(z, Ri)has to be analytic function of z. With this form of the order parameter we will have to resort to an106 approximate calculation. We first choose the magnitude A so that 6F,/dA = 0, without varying f. The overall magnitude A is now eliminated from the problem, and the free energy becomes simply F , = -a(T)/(2/3(T)K), where K is given by

(24.30) The problem is now reduced to finding the set of guiding centers Ri,set of coefficients c j , and the form off which gives the smallest K.

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

207

Even within the above approximation, the minimization of K is a very complex problem. However, we can show that K4 in (24.30) can be transformed in the form analogous to the fourth-order kernel of the Abrikosov theory.104 This is due to the fact that K , acts as a delta function on the part of the Hilbert space consisting of analytic functions. Therefore, the problem of minimization of K is identical to the corresponding problem in the Abrikosov theory near Hc2, although the form of the free energy and the physics are quite different. Having made this observation, we expect that the lowest free energy state arises for the choice of real vectors Rj’s forming a hexagonal lattice in the x-y plane. For the choice of cj’s and f ( z , R;) we rely on the work of Kleiner, Roth, and Autler ,Io5 who have determined the minimum free energy configuration of the order parameter in the case of the Abrikosov flux lattice. We can adopt their results, but the scale of variation of the order parameter has to be changed from the coherence length, in their paper, to the magnetic length in the present work. The zeros in the order parameter form a triangular lattice dual to the hexagonal lattice of tubes. The area of the elementary hexagon is 2nP2;it is very important to emphasize that this quantity, at least in principle, can be arbitrarily small. In type-I1 superconductors the unit cell for flux lattice diverges near T, and decreases as one lowers the temperature but it can not be “compressed” beyond the value 2x520, where is the coherence length at zero temperature. This smallest unit cell defines the maximum upper critical field, through Hc2(0)= @0/2&, Qo being the elementary flux. In the high field case, however, the area of the unit cell scales with the magnetic field, decreasing and becoming very small with increasing H, always containing the elementary flux, and being basically independent of temperature! Therefore, there is no limit on the strength of the magnetic field arising from nonuniformity of the order parameter. If one ignores the effect of the magnetic field on the coupling constant, the transition temperature would increase very strongly with increasing magnetic field, as long as one remains within the weak coupling approximation. We have now obtained a stable, stationary solution for the order parameter which represents the local minimum of the free energy (of course, one cannot exclude the possibility that a solution exists which has a lower free energy). Making use of these result^,"^ we have K = 1.16. How do we interpret these results? The free energy is minimized by the configuration describing a Wigner lattice of superconducting tubes. It is important to note however, that the lateral size of these tubes is essentially the same as the separation between the electrons. The extremely anisotropic Cooper pair wavefunction reaches its molecular limit in the x-y plane. We expect, therefore, that quantum fluctuations

208

M. RASOLT

will be important for the ultimate form of the order parameter. These fluctuations will lead to tunneling of the Cooper pairs between superconducting tubes and the system could gain further energy by the hopping from site to site. While the quantum fluctuations will have a quantitative effect on our results (for example, they will renormalize the mean field transition temperature) we do not expect any qualitative changes in properties obtained from the mean field theory (see following subsection). In this respect, the role of quantum fluctuations in our problem is similar to that in the system of coupled superconducting chains. Qualitatively new effects may arise, however, if the motion of the electrons is restricted along the direction of the magnetic field. They would be particularly pronounced in a film geometry, with H perpendicular to the film and the thickness comparable to the coherence length. The situation is then somewhat similar to the one suggested by Kivelson et al. '07 for the quantum Hall effect. There the Wigner lattice of electrons melts into the incompressible quantum liquid via the cooperative ring exchanges. In our case, we can consider the Cooper pairs to behave like bosons of charge 2e; Ri are the coordinates describing the positions of these bosons in the x-y plane. By analogy, we would expect that our Wigner lattice of superconducting tubes may melt via cooperative ring exchanges into a highly correlated bosonic quantum liquid, once the superconducting density in the tubes is sufficiently large. One, however, does not expect such a liquid to be strictly incompressible, as it is the case for purely two-dimensional fermionic system. Whether such a state would be realized in some portion of the T-H phase diagram is clearly a question that deserves further study. We also expect that near T, the thermal fluctuations of the Wigner lattice will be very important. As pointed out by BrCzin et a1."* the fluctuations drive the transition of the Abrikosov flux lattice first order below six dimensions. We expect that a similar effect takes place in our case and the superconducting transition in very high magnetic field may be discontinuous. Finally, we should note that the cross over to the quantum limit does not need to be reserved to isospin space (see Fig. 32). As long as the g factor is less than 2, we find that the Zeeman splitting will not entirely remove the possibility of such a superconducting instability towards a spin singlet state, in relatively high magnetic fields; very much like what we saw in the e-h channel in Section

23.

As we saw in our preceding discussion, the cross sections for e-h or e-e

107

S . Kivelson, C. Kallin, D. P. Arovas, and J . R . Schrieffer, Phys. Rev. Leu. 56, 873 (1986). '"8E. Brtzin, D. R . Nelson, and A . Thiaville, Phys. Rev. B 31, 7124 (1985).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

209

scattering amplitudes are completely 1D. But in truly 1D fluctuations are so strong that no long-range order can exist at finite temperature; we need to therefore say something about the effect of fluctuations on our above discussions.

25. EFFECTOF

QUANTUM

FLUCTUATIONS

Both Eq. (23.1) for SDW or VDW instabilities and Eq. (24.1) for the superconducting instability, were reduced to a one dimensional form [see Eq. (23.5)]. The remaining one dimensional variable is the momentum k, along the magnetic field. Does that imply that a 3D electron gas, with multi-isospin components, in the presence of a very strong magnetic field (where all the electrons occupy the lowest Landau level) is truly one dimensional? Of course not. The one-dimensional form in the case of the SDW and VDW instabilities [i.e. , Eq. (23.5)] and similarly for the off-diagonal superconducting instability is a consequence of the degeneracy in the eigenstates of different momenta k,. This permitted the elimination of ky by summing over this variable (i.e., summing over the phase space perpendicular to the magnetic field) in Eqs. (23.1) and (24.1). Such a sum, however, does not make the system truly one dimensional. In fact it is precisely this degeneracy which was resolved below T, to give the Abrikosov lattice discussed in Section 24, which is a three-dimensional configuration. Above T, the three-dimensional nature of the phase transition will appear, at once, the moment we consider contributions beyond the ladder graphs of Fig. 27 (for the SDW, VDW) or Fig. 29 (for superconductivity). For example, contributions from cross terms in the irreducible vertex (like the last term in the RHS of Fig. 6b) will not permit the transformation of Eqs. (23.1) and (24.1) to one dimension. We can reduce Eqs. (23.1) and (24.1) to a truly 1D problem by replacing the Coulomb interparticle interaction 4ne2/ep2, in Eq. (23.2) or the electron-phonon mediated interaction, in Eq. (24.4) by a delta function interaction w (p): lo9

This interaction is a constant in real space, perpendicular to the magnetic field, i.e., very long range and highly unphysical. In any event according 'OSH. Schulz and H. Keiter, J . Low Temp. Phys. 11, 181 (1973).

210

M. RASOLT

to Eq. (23.4) the initial momentum k,, and k, now remain the same across any intermediate states for any scattering processes (e.g., last term of RHS of Fig. 6b). The problem is now truly 1D. In one dimension there is strong competition between VDW, superconductivity, SDW, CDW. (For simplicity we will freeze out the real spins and also consider the simpler case of two components isospin). The phase diagram for this 1D case has been studied"6118 previously. We discuss briefly the conclusions of these studies with particular attention to their implication for the mean field predictions made in Sections 23 and 24 in the case of a realistic interparticle interaction (i.e., short range in real space). The interaction of Eq. (25.1) leads to scattering only along the z direction. Now in the weak coupling limit the singular nature in the electron-electron or electron-hole scattering occurs around the Fermi surface. [See, e.g., Eqs. (23.6) and (23.9).] In one dimension, then, there are only two possible scattering processes. Either the two electrons scatter in the forward direction [i.e., g(pz = 0) = g,] or in the backward direction [i.e., g(pz = 2kF) -g,]. Of course, there is a small range of intermediate momentum lkol (or energy Eo = h2kFlkol m) around kF, whose contribution to the singular scattering amplitude is important. In this small range, however, we expect g ( p z )to remain relatively constant. In addition to the electron and holes being close to k F , both in the e-e channel and the e-h channel, the states must lie on opposite sides of the Fermi surface. [Incidentally, we are not interested in the pathological case of a half filled band in which case an Umklapp process is possible and an additional g(pz = 4kF) must be included]. The phase transition, in one dimension, is then described by collecting all the singular contributions, to say, the electron-electron scattering amplitude (see Fig. 36). In Fig. 36a g , and g, are defined. The arrowed line is an electron state with momentum kF and the dashed arrowed line is an electron with momentum -kF. The contributions to the e-e scattering amplitude r, to second order in the couplings g,, and g,, are presented in Fig. 36b and 36c. ""P. W. Anderson, J . Phys. C 3, 2346 (1971). "IN. Meynhdrd and J . Solyom, J . Low Temp. Phys. 12, 529 (1973). "*N. Meynhird and J. Solyom, J . Low Temp. Phys. 21, 431 (1975). 'I3J. Soylom, J . Low Temp. Phys. 12, 547 (1973). 'I4J. Soylom, J . Phys. F 4, 2269 (1974). .5'" Soylom, Solid State Commun. 17, 63 (1975). 'l6H. Fukuyama, T. M. Rice, C. M. Varma, and B. I. Halperin, Phys. Rev. 10, 3775 (1974). '"C. S. Ting, Phys. Reu. E 13, 4029 (1976). '"5. Solyom, Adu. Phys. 28, 201 (1979).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

211

FIG. 36. Low-order scattering processes (to second order) (b, c) in 1D Fermi systems, in terms of the backward ( g , ) and forward (g2) matrix elements. (Taken from Ref. 118.)

Note that this collection of terms allows for only forward and backward momentum exchange. Also, the intermediate electron-electron and electron-hole lines are always on opposite sides of the Fermi surface in accordance with the above discussion. Suppose we consider two electrons with different isospins a and 6. Two different cross sections can then be identified. F2(w, E,) where the two electrons emerge with the same initial isospins and the other process Tl(w, E,) where the isospin is interchanged, i.e.,

It is easy to identify these two distinct cross sections in the collection of terms in Fig. 36b and 36c. [The reason for the dependence of r(o,E,) on Eo can be found in Eqs. (25.3) and (25.4).] Each one of these diagrams contain logarithmic singularities in two dimensions. These arise from the

212

M. RASOLT

product of the e-e Green's functions, i.e.,

(25.3)

and the e-h, i.e.,

(25.4)

where G+ and G- are the 1D noninteracting Green's functions for k F and - kF, respectively. These singularities are no different than already discussed previously [e.g., Eqs. (23.6) and (23.9)] with the temperature k B T replaced by hw. One can continue to write infinite sets of such diagrams, with leading logarithmic singularities; so-called parquet graphs."""' Such sets of graphs have great resemblance to the ladder graphs already considered in Section 23 and 24. However, what is crucial to one dimension is that the e-h products [Eq. (25.4)] and e-e products be maintained consistently in both the CDW, VDW, and superconductivity response functions; this we did not do in Sections 23 and 24. When this is done correctly, no phase transition can occur in one dimension. It is instructive to see how this happens in one dimension and then to return and make contact with the implication to the short range interparticle interaction (i.e., the physically meaningful situation) discussed in Sections 23 and 24. A very elegant way to sum this set of graphs is the multiplicative renormalization group developed by Menyhfird, Solyom, and others. 11s118 The idea is to demand scale invariance of the two amplitudes Tl(w, E,) and r2(w, E,) when Eo (or equivalently k,) is scaled from Eo to Eh. The result is the usual flow equations for the couplings g , and g , , as a 'I9I. T. Diatlov, V. V. Sudakov, and K. A . Ter Matriosian, Zh. Eksp. Teor. Fiz. 32, 767 (1957) [SOU.Phys.-JETP 5, 631 (1957)l. 'zOA.A. Abrikosov, Physics 2, 5 (1965). '"B. Roulet, J. Gavoret, and P. NoziCres, Phys. Rev. 178, 1072 (1969). "'P. NoziCres, J. Gavoret, and B . Roulet, Phys. Rev. 178, 1081 (1969).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

213

function of x (where x = EA/Eo)."G1'8The two flow equations are

and

In Eqs. (25.5) and (25.6) we have included third order contributions in the couplings, which are not shown in Figs. 36. The flow equations can be easily analyzed. If we neglect the third-order terms, then

(25.7) = g2 - t g , + tg1(x).

(25.8)

To that order, there is a singularity in the renormalized coupling for g , < 0, corresponding to a phase transition at finite temperature. If we replace EA by kBT, then the critical temperature, in the weak coupling limit, is given by k,T, = Eoe+l'gl.This, of course, is closely related to the phase transitions discussed in Section 24. The magnitude of the cutoff Eo is the starting point of the recursion and must be chosen appropriately to match the T, of Eq. (24.5). If we include the third-order terms in Eqs. (25.5), (25.6), then the singularities in g ( x ) are removed. Therefore, as should be the case in one dimension, there can be a phase transition only at T = 0. The point is that special to one dimension both the e-e and e-h products must be consistently maintained to give the correct physics; this we certainly did not do in Sections 23 and 24. One can also use the flow equations to construct the response function of appropriate symmetry; just as was done in Sections 23 and 24 for the VDW and isospin singlet superconductivity. The well-known phase diagram, at T = 0, is presented in Fig. 37. Clearly the competition between the different phases in one dimension is very strong. What does this all mean to the case of short-range interparticle interactions? As already discussed above, for such physical interparticle interaction many of the terms (e.g., the cross terms in Fig. 36c) cannot be reduced to one dimension. On the other hand, the ladder type contributions are 1D-like and are singular both in the e-e and e-h channels. Therefore, the situation in Sections 23 and 24 is a kind of mixture of 3D

214

M. RASOLT

TS

ss

92

CDW

(ISsi FIG. 37. Phase diagram of the 1D Fermi gas obtained in the second-order scaling approximation. The response functions corresponding to the phases indicated in parentheses have a lower degree of divergence than the others. (Taken from Ref. 118.)

with strong 1D features. Certainly the precise interplay between e-e and e-h contributions, which led to nonsingular behavior in the renormalized couplings is not expected to occur for the case of a physical interparticle interaction in the presence of high magnetic fields. Phase transitions to VDW, CDW, and superconductivity are expected at finite T but the phase diagram is expected to be modified considerably from the mean field results presented in Sections 23 and 24. Unfortunately, a rigorous study of this phase diagram to our knowledge has not been carried out. The only serious study of fluctuation has been restricted to the effect of classical fluctuations (by Brezin et al. lox) on the superconducting phase transition. These fluctuations correspond to long-wavelength fluctuations of the vortices, in type-I1 superconductors, parallel to the field. Such fluctuations are found to drive the transition first order. We expect these conclusions to apply also to the superconducting tubes discussed in Section 24.

26. ISOSPINFLUCTUATIONS IN THE FERMI LIQUIDSTATE We, so far, primarily discussed properties of broken symmetry states, in isospin space, in the presence of a strong magnetic field. Of course, the additional isospin degrees of freedom (additional to spin space) must also lead to new effects in the normal (i.e., Fermi liquid) state. This has been recognized by F ~ k u y a m a ” and ~ more recently by S a ~ h d e v . ”These ~ lZ3H.Fukuyama, J . Phys. SOC.Jpn. 50, 3562 (1981). Iz4S. Sachdev, Phys. Rev. Leu. 58, 2590 (1987).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

215

authors have concentrated primarily on the isospin Fermi liquid properties in the presence of disorder, where very dramatic effects can occur; particularly close to the metal-insulator transition. Before we review their work we briefly point out the equivalent situation in spin space. The generalization to include the isospin components is then relatively straightforward. Most properties of a two-spin-component Fermi liquid involves small displacements of the Fermi surface and a corresponding change in the quasiparticle occupation &zko (k and u are the quasiparticle momentum and spin, respectively). For such a Fermi liquid with time reversal symmetry (i.e., without an external magnetic field), and in the absence of disorder the free energy F, for example, can be written as’25127

(26.1) where the two functions fit(k, k‘) and fTl(k, k’) are the appropriate functional derivatives of the free energy. These two functions characterize this two-spin-component Fermi liquid. As is well known, these two functions are closely related to the two vertex functions Art(k, k’) and AlT(k, k‘) given by the following two coupled integral equations:”‘ (26.2) and At1 = f‘, + G+G-[f,[ATl+ At?] - ~ ‘ ~ A T I ] .

(26.3)

f‘, and f, are defined in Fig. 38 by ignoring for the moment the isospin labels i a ndj . The wavy line in Fig. 38 corresponds to the collection of all skeleton graphs. In Eqs. (26.2) and (26.3) we have suppressed the intermediate momentum summations; i.e., we have written the integral equations for ATt(k, k’) and AtL(k, k’) in matrix form. Equations (26.2) and (26.3) can be diagonalized at once by choosing As = Aft + ATLwe get “’D. Pines and P. NoziCres, “The Theory of Quantum Liquids.” Benjamin, 1966. lZ6P.NoziCres, “Theory of Interacting Fermi Systems.” Benjamin, 1964. 127A.A . Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, “Methods in Quantum Field Theory in Statistical Physics.” Prentice Hall, 1963.

216

M. RASOLT

r,.

FIG.38. The interaction parameters I=, , r2,and They are represented by double lines to emphasize that they are effective parameters. Valleys j and i are collinear while valleys i and k are not. (Taken from Ref. 124.)

the equation

A, = 2F1- F2 + G+G-(2F, - F2)As

(26.4)

and with A, = AT1- At? we get

Aa = F 2 - G+G-F2Aa.

(26.5)

These two decoupled channels A, and A, are closely related to the spin symmetric and spin antisymmetric parts of the quasiparticle interactions (frr and f?~).We will consider the quasiparticle interaction to be s wave and ignore the momentum dependence in A, and A,. The alignment of the spins was chosen along the spin z axis. The structure of the interaction can be made rotationally invariant by choosing

= 2r1 - F2 and I?', = - F2. Incidentally, A, describes densitywhere density fluctuations and A, describes spin-spin fluctuations. To describe the Fermi liquid in spin-isospin space we need to add the

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

217

additional isospin degrees of freedom to Eq. (26.1), i.e.,

(26.8) SachdevlZ4applies Eq. (26.8) to the six pockets of silicon. In addition to the two components of spins the Fermi liquid parameters must differ for various combination of the six isospin components. This is described in Fig. 38. Fl is now the skelton part of the Fermi liquid parameter with i and j in colinear valleys; or colinear isospin components. In other words j is either equal to i or to the valley which is related to i by inversion symmetry [recall that the six valleys of Si are at ( x , - x ) , ( y , - y ) , and (2, -2) directions]. f, is the exchange part of F l . The other four valleys, indexed by k, are not colinear to i and j . Clearly, the Fermi liquid parameters between i and k must be different; they are given by f,. (Sachdev ignored the exchange process for f, because of the difference in the energy denominators of the G+G- product). The corresponding equations for the vertex functions in spin-isospin space are more complicated than the two-spin components [Eqs. (26.2) and (26.3)]. They are

(26.9) (26.10) (26.11) (26.12) (26.13) and

(26.14)

218

M. RASOLT

where K 1 = A l l+ A i , + A l l + A l l , and K 2 = A , , + A l k . These equations TT

T1

tl

TT

?T

TI

can be diagonalized with the following combinations of the Fermi liquid parameters in spin-isospin:

where A, satisfies A, = I?,

+ raA, with f a= 4 f , - r2+ S f , , (26.16)

where Ab satisfies Ah = f'b

+ rb&,with rb= -r2, (26.17)

A c = K, -2K2,

r, +

rc

r2

where A, satisfies A, = rCAcwith = 4F, - 4 r 3 . The diagonal vertex functions A,, Ab, and A, (or Fermi liquid parameters) represent three distinct properties in spin-isospin space. A, is the density-density, the spin-spin and Ac the isospin-isospin fluctuation, respectively. &, which is unique to isospin space, corresponds directly to the self-energy of an elastic longitudinal wave (i.e., a phonon) propagating along the x axis.lZ4It is not hard to see that a strain field will lift the degeneracy of the various isospin components just like a magnetic field does for the two spins,101.128 The usual spin response (i.e., A,) will now be replaced by A,. The skeleton vertex and f, are irreducible graphs in the expansion of the interparticle interaction. For a long-range interaction the direct Coulomb interaction (in f , and f.,) must be treated ~ e p a r a t e l y . ' ~ ~ Direct '" calculations of the l='s is usually carried out for some special set of graphs. It should, however, be mentioned that large numbers of valleys (n = 6 in silicon) can be helpful as a l / n expansion parameter. Actually, in the presence of disorder these parameters renormalize to some fixed point values, around the metal insulator transition, and we can say a lot without directly calculating these T's. We next then briefly discuss the effect of disorder, on the Fermi liquid, in spin-isospin space. The main point is to isolate and emphasize the new features that originate from the isospin component. In the case of spin space only, disorder modifies the interaction

rl, r2,

'"C. Herring, Bell System Tech. J . 34, 237 (19%). Iz9Z. Tesanovit and M. Rasolt (unpublished).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

219

-

IT+

En + 01 (+)

:v

i!V

E"(-1

(c)

FIG. 39. (a) Single-particle propagator in weak disorder; (b) bare diffusion vertex, and (c) bare diffusion propagator.

processes, of the pure Fermi liquid, by introducing the famous130 diffusion ladders of Fig. 39 (for weak disorder, that's all one needs). The two dashed lines, coupled with a cross, are the electron impurity scattering, averaged over the impurity sites, i.e., (26.18) where v(q) is the Fourier transform of the which is taken to be q independent. In noninteracting density of states per spin t is the impurity concentration. The solution for tor (Fig. 39a) is then G(k,

E,) = [ie, - h2k2/2m

electron impurity potential Eq. (26.18) N ( 0 ) is the the electron lifetime and ni the single particle propaga-

+ ,u + sgn(en)i/2z]-',

(26.19)

where E, = nkgT(2n + 1) and w, = 2nkBT1. As is well known, when the electron and holes are on the opposite sides of the Fermi energy [i.e., E,(E, + w l )< 0; incidentally, this is also the meaning of the plus and minus signs in the G's of Eqs. (26.2), (26.3), etc.] the solution for L in I3OP.

A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57,287 (1985).

220

M. RASOLT

Fig. 39c is diffusive, i.e.,

(26.20) where L = 1/(Dq2+ I w , ~ ) and D = z(hkF)2/(drn2)(where d is the spatial dimension). When isospin degrees of freedom are included, there are a number of possible diffusive propagators. The point is that impurity scattering couples the different isospin components; this we already saw in Section 14. An electron can scatter within the same valley with a lifetime l/zO= 2xN(O)niv2 (ni is the impurity concentration) it can scatter to a collinear (z, j ) valley with l / z , = 2~cN(O)n~v:~ and it can scatter to a noncollinear valley k with l/t2= 2nN(0)niv$. (Note that l/z, >> l/t, and l/z2.) The lifetime z in Eq. (26.19) changes to l / z = l/t,+ l/t,+ 4/z2. The diffusion propagator L now satisfies a set of four equations. In terms of the various combinations of electron hole scattering, illustrated in Fig. 40, these equations are, in matrix notation, [see Eqs. (26.9)(26.14]),

(26.21)

I

I!

I _ I I Ti Ti

1 ! 1 Ti

Ti

A Ti

Tk

!

f I

FIG. 40. (a) Various possible low-order electron-hole scattering in the presence of disorder, for the isospin space of Si. Again, i and j are collinear valleys and i and k are not. (b) The case where the two entering propagators are different; there is, of course, the exchange process as well (where i+ j in one of the pair of crossed lines).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

22 1 (26.22)

and (26.23) The term in Fig. 40b leads to a separate ladder summation. The solution of these equations lead to four forms of the diffusion propagators given by

(26.24) with i = 1-4 and where mi take different combinations of l / t , and 1 / t 2 . All the various combinations of the required diffusion propagators can be written as linear combinations of the L; . The response of the pure Fermi liquid is now modified to couple both the diffusive propagators L and the vertex functions A. Three types of response functions are illustrated in Fig. 41. The last response function X is the isospin response function to a longitudinal phonon (discussed previously) in the presence of disorder (just like the spin response function x, Fig. 41b, couples to a magnetic field). The singular nature of the L's, at small q and [Eq. (26.20)], actually helps to make some general statements about the Fermi liquid in spin-isospin space. Integrands over the small momentum q in the diffusion propagators L, in various electron hole cross sections lead to a

(c)

FIG. 41. (a) Skeleton structure of the density-density correlation in Si. (b) Skeleton structure of the spin susceptibility in Si. (c) Skeleton structure of the isospin susceptibility in Si.

222

M. RASOLT

logarithmic expansion, in two dimensions, whose leading terms in the impurity scattering can be identified and summed. The various leading order terms have been summed by Castellani et al. 131-135 for spin space and extended to spin-isospin space by Sachdev.lZ4 This logarithmic structure (as we saw already in Section 25) permits a scaling description of this system (see also F i n k e l ~ t e i n and ' ~ ~ Altshuler and A r ~ n o v ' ~ The ~). analysis is complicated and is not essential to this review. The main point is that in this theory the Fermi liquid parameters (A) enter as scaling variables in the renormalization equations. The addition then of isospin space changes somewhat the structure of the R G equation to accommodate the multivalley nature of Si (or other systems). What S a ~ h d e v ' * ~ then found is that these R G equations lead to a singularity in z ( q , w ) and ~ ( qw, ) on the metal insulator critical line. It can be interpreted as the formation of localized clusters of spins and isospins and, therefore, reflects the metal insulator transition. (Incidentally, in noninteracting disordered Fermi liquid the maximally crossed diagrams lead to localization. 13(') Sachdev also applied the R G results to make predictions about the temperature dependence of X(q, w ) . His predictions are therefore relevant to ultrasonic attenuation in Si due to spin-isospin fluctuations. In conclusion we have shown how the extra isospin degrees of freedom predict a host of new possibilities in multivalley semiconductors. We have reviewed only several applications of this formalism. Clearly just as the spin degrees of freedom played a crucial role in condensed matter physics, we believe, in time, so will isospin. The number of applications reviewed in Sections IV-VI is not large, but this is just a reflection of the relative novelty of this approach. Similarly, the number of experimental observations of isospin properties is small. Again, a key purpose of this review was to extend our discussion to a much wider scope of theoretical as well as experimental studies. 13'C. Castellani, C. DiCastro, P. A. Lee, and M. Ma, Phys. Rev. B 30, 527 (1984). 132C.Castellani, C. DiCastro, P. A. Lee, M. Ma, S. Sorella, and E. Tabet, Phys. Rev. B 30, 1596 (1984). 133C.Castellani, C. DiCastro, P. A. Lee, M. Ma, S. Sorella, and E. Tabet, Phys. Rev. B 33, 6169 (1986). 134C.Castellani, C. DiCastro, H. Fukuyarna, P. A. Lee, and M. Ma, Phys. Rev. B 33, 7277 (1986). '35C.Castellani, C. DiCastro, G. Kotliar, and P. A. Lee, Phys. Rev. Lett. 56, 1179 (1986). 136A.M. Finkelstein, Zh. Eksp. Teor. Fiz. 84, 168 (1983) [Sou. Phys.-JETP 57, 97 (1983)l. I3'B. L. Altshuler, and A. G. Aronov, Solid State Commum. 46, 429 (1983).

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

223

ACKNOWLEDGMENTS

I wish to acknowledge many stimulating discussions with B. I. Halperin, A. H. MacDonald, and Z. TesanoviL; to Velma Hendrix for her typing of the manuscript; to CECAM at the University of Paris XI, Orsay, where much of the work was written. This research was also sponsored in part by the Division of Materials Sciences, U.S. Department of Energy, under Contract No. DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc.

Appendix A

To evaluate Eq. (13.9) we first need to integrate the expression over z. To do so we take advantage of the relative slow variation in z , of g ( z ) ; as compared to &( R). Now

We next write the periodic

Uk,s(R)

as

Using Eq. (Al) one gets

+

+

where Gf = G: G, - G: Q: and where Q: = 0 when i = j and Q : = QZ when i # j . Taking advantage of the slow variation in g ( z ) (i.e., b << Gf), we can insert this result in Eq. (13.7) and integrate over qz to

224

M. RASOLT

give for example 'k',k,k'.k 1. 2.2, 1

-3b I - 16 'k',k,k'.k> 1, 2.2, I

where F' is identical to Eq. (9.2) except with d strictly equal to 2. The effect of the width of the inversion layer along z has been entirely absorbed in the prefactor 3b/16. After a somewhat tedious calculation, ZI is given by

225

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

i#j i'#j'

x [-(2k

+ (2k

- C n . , ) * ~ & ( G ) ~ ~ ~-' ,(2k' * ( G. e) n 0 ) ~ & , ( G ) ~ 2 ) ' * ( G ) 22

11

*

22

11

C n , O ) u ~ n , ( G ) u ~+~ (2k' * ( G*)C n O ) * ~ , " o ( G ) ~ ~ ~ ) ' * ( G ) ] 11

22

22

11

2

1 d2V(G) 3 (k' - k),(k' - k)pu&(G)uG(G) --b 64 G#=O4aGwaGf3 11 22

3

--

64

b

2 G i#j i'#j'

a2V(G+") aG,aGp

( k ' - k),(k' - k)pu&(G)u;iij'(G),(A5) 11

22

where

U;~,(G =)(uy21eiG.RIuyl'). 21

In Eq. (A5) Q = (Q,, Q,). The sum over k and k' over occupied states then gives Z, = N e2k,(5)3 rl,

Q Q

4n

where T1=-b-3 Q3Q 64 e2 b

C

A,

GZo ,=1,2 i,j=1,2

V(G) [+-21 (aaG, [R,(c:~~;~(G)~~(G) ~

11

22

+ C:,,U&(G)&,(G))]) 11

22

+ V(G)[C:&:&::~(G)U~,,(G) 22

11

+ C:OC:,~uId(G)ul,,(G)]] 22

11

a=1,2 i#j,i#j'

+ C:OU &(G ) u ~ ) G' *)() ] ) 11

22

226

M. RASOLT

- V(G

+ Q,)[C,~.,C~,&~:,,(G)U~(G) 22

11

A, = (mp/mLu)1/2(kF,/kF)4rwhere kF is the Fermi momentum assuming an isotropic mass and kFnare the Fermi momenta along the x and y axes and where V(q) is the 3D Fourier transform of Eq. (13.6). We will not detail the calculations of Z, and Z, which follow closely the above calculation.

Appendix B From Eqs. (18.6), (18.(10), and (18.15) (written in the projected form), we can calculate &(q). First from Eq. (18.10) (at t = 0 ) and Eq. (19.7) j-(q) is given by

After some algebra using the translational properties of eiq(a'az)etc., we get that

CONTINUOUS SYMMETRIES AND BROKEN SYMMETRIES

Using Eq. (B2) in (18.15) then leads to

227

228

M. RASOLT

A very important simplification occurs when we consider a fully isospin polarized ground state. Then the only I = i, in the sum of Eqs. (B4) and (B5), survives. We then get

But the structure factor S(q) is

[Note from Eqs. (19.2)-(19.4) all the z* must be placed to the left of the

z and then replaced by 2(d/dz).]

Equation (B8) can be written as

NS(q)=N

+ ePq2

c I#(.

d2r, . . . d2rN

x eiq*/2(z,--f) q(.. .(Zi

. . zi . . . zj . . .)

i#j

+ iq) . . . (Zj - i q ) . . .).

039)

and

Using Eqs. (B10) and (B11) in Eq. (B3) gives the &(q) in Eq. (19.16).

SOLID STATE PHYSICS, V O L U M E 43

Dynamic Screening of Ions in Condensed Matter P. M. ECHENIQUE Departamento de Fiuica de Materiales Facultad de Ciencias Quimicas Uniuersidad del Pais Vasco/Euskal Herriko Unibertsitatea San Sebasticin, Spain

F. FLORES Departamento de Fiiica del Estado Sdlido Uniuersidad Auto'noma de Madrid Cantoblanco, Madrid, Spain

R. H. RITCHIE Health and Safety Research Division Oak Ridge National Laboratory Oak Ridge, Tennessee

...........................

1. Background.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

...........................

230 23 1 23 1 236 236 236 239

6. Quanta1 Self-Energy for an External Electron in an Elec............ tron G a s . . ............................

tron Gas ......................

...........................

242 246 248

..................................... 10. Plasmon Dispersion . . . . . . . . . . . . 13. Damping Effects.. .....................

...................

250 250 252 253 256 259

229 Copyright 01990 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-607743-6

230

P . M . ECHENIQUE et a1. 14. Quantum Theory of the Wake .....................

IV .

V.

VI .

VII .

.................

15. The Nonlinear Wake ............................. . . . . . . . . . . . . . . . . . 16. Stopping Power . Straggling ........................ . . . . . . . . . . . . . . . . . 17. Energy Loss Fluctuations in the Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wake Effects on Ion and Molecular Ion Penetration in Condensed Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................. 18. Wake Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................. 19. Vicinage Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................. 20 . Dynamic Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21. Alignment-Ring Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 . Resonant Coherent Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23. Wake-Field Accelerators . . . . .. ... 24 . The Surface Vicinage Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 . The Surface Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ions Moving with Velocity LessThan uo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 . Linear Response Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27. Phase Shift Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28. Density Functional Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29. Protons and He Nuclei in an Electron Gas ........................... 30 . Effective Charge for Ions with Z, > 2 . . . . . . . . . . 31. Straggling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charge States of Ions in a Solid ......................................... 32 . Introduction .. ....... ... 33. Electronic Exchange Processes ..................................... 34. Auger Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35. Coherent Resonant Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36. Shell Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37. Results for H , He, B, C, N and 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38. Charge States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . ...

259 262 263 264 267 267 269 271 272 273 273 274 274 275 275 277 278 279 283 286 287 287 288 289 292 295 296 303 306 308

.

1 Introduction

When a swift ion penetrates condensed matter the medium responds to the presence of the added charge . Initial entry causes transient readjustment of electrons and changes in the charge state of the ion . Electrons may be stripped from the ion or captured from valence states or inner shells. Polarization of the medium occurs . Dynamic screening by valence electrons originates in perturbation of electron orbits by the Coulomb field of the moving ion as well as to repeated captures and losses to bound states on the ion . An intricate spatial and temporal pattern is created in the medium . Coherent polarization structures, tied to the instantaneous location of the ion, may be generated if the ion speed ZT > ZT,, = e 2 / h= 2.19 X 10' cmlsec . An oscillatory disturbance associated

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

231

with the entrant surface is also expected. A wake of electron density fluctuation trails the ion in direct analogy with the wake in water caused by a moving boat. Close collisions create showers of secondary electrons as well as Auger cascades in inner shells and electron-hole cascades in the valence electron gas. Exit into a low-density medium gives rise to equally complex processes: again surface oscillatory modes are possible and convoy electrons may accompany the outgoing ion. Capture of electrons at o r near the surface into Rydberg states may give rise to Auger cascades on the receding ion. A charged particle moving through condensed matter represents a complex coupled system presenting a profound challenge to the physicist's understanding and theoretical interpretation. Observation of such systems reveals much about the static and dynamic properties of the medium as well as those of the particle.

1. BACKGROUND Here we will be concerned primarily with processes in the bulk. Physical aspects of the wake will be discussed and related to energy loss and dynamic screening of the ion. Processes characteristic of various speed regimes are analyzed. Near-adiabatic interactions in the regime u << uo involve mainly electrons at the top of the valence electron sea. The density functional method allows accurate evaluation of the energy loss rate and the effective charge of the ion. When v > > v , electrons respond to the impulse of the swiftly moving ion as if they were nearly free. Perturbation theory is then valid for the evaluation of energy loss or effective charge, but corrections must be made for ions with large atomic number or for many-body effects appearing as u + uo . Empirical theory has been developed by several workers to describe phenomena occurring in this regime. Here we develop a many-body self-energy approach to obtain a priori probabilities for energy loss and for capture and loss. Results that have been obtained for low atomic number ions penetrating several elemental solids are described.

2. HISTORICAL SKETCH The use of swift ions as probes of the static and dynamic properties of matter dates from the earliest days of modern physics. Rutherford employed alpha particles to establish his nuclear model of the atom.' In a 'E. Rutherford, Phil. Mag. 21, 669 (1911).

232

P. M. ECHENIQUE ef al.

pioneering paper2 Bohr invoked concepts familiar from the theory of photon excitation of atoms to calculate the slowing-down of swift alpha particles in matter. It is interesting that this paper was published slightly before his explanation of the spectrum of the hydrogen atom.3 A great deal of published research in the intervening years has dealt with the energy loss and range of swift ions in matter.4 Bohr’s theory was of necessity based on semiclassical ideas. His deeply intuitive treatment was appropriate for slowly moving ions with fixed charge. With the advent of quantum mechanics Gaunt’ attacked the problem of energy losses of an ion incident on a hydrogen atom using the then newlydevised Born approximation of collision theory. He worked in the impact parameter representation and in the dipole approximation for the atomic transition matrix elements but did not obtain results of general applicability. Subsequently Bethe6 made a thorough analysis of the energy transfer from a swift charged particle to an atomic or molecular system, in the process generalizing the dipole sum rule of atomic physics. He showed that in the high-speed regime, u >> V g , energy loss by a swift charged particle depends primarily on u, its charge, Zle, the mean density of electrons in the undisturbed medium, no, and an important quantity, I, the mean excitation energy of electrons making up the stopping medium. Thus the energy loss per unit path length, or, in common terminology, the stopping power of the medium for the particle, may be written as dW 4nZie4 2mv2 _ dR mu2 noln(k) The mean excitation potential, I , is a logarithmic average of possible ’N. Bohr, Philos Mag. 25, 16 (1913). 3N. Bohr, Phifos. Mag. 26, 1, 476, 857 (1913). 4A great many reviews of this area of physics are available. See, e.g., N. Bohr, K. D a m . Vidensk. Selsk. Mat.-Fys. Medd. 18, No. 8 (1948); H. A. Bethe and J. Ashkin in “Experimental Nuclear Physics,” Vol. I, pp. 166-252; (E. Segre, ed.). Wiley, New York 1983. M. Inokuti, Rev. Mod. phys. 43, 297 (1971); addenda, 50, 23 (1978); S. P. Ahlen, Rev. Mod. Phys. 52, 121 (1980); Recent compilations of experimental results include those by J. F. Ziegler and colleagues: the most recent of these is J. F. Ziegler, “Handbook of Stopping Cross-Sections for Energetic Ions in all Elements,” Vol. 5 the “Stopping and Ranges of Ions in Matter.” Pergamon, New York, 1980. A comprehensive treatment is given in the monograph by M. A. Kumakhov and F. F. Komarov, “Energy Loss and Ion Ranges in Solids.” Gordon and Breach, New York, 1981. See also Y. H. Ohtsuki, “Charged Beam Interactions with Solids.” Taylor and Francis, London, 1983. ‘5. A. Grant, Proc. Cambridge Philos. SOC. 23, 732 (1927). 6H. A. Bethe, Ann. Phys. (Leipzig) 5 , 325 (1930).

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

233

transition eigenenergies, each weighted with the appropriate optical oscillator strength. It is known accurately theoretically for only a few media, but the analytical form of Eq. (2.1) is of great utility in the analysis of experiment. Equation (2.1) is valid when v,, << v << c, where ve, is a mean orbital speed of electrons in the stopping medium or Z l v o , whichever is larger, and c is the speed of light in vacuum. The original approach of Boh? yielded a formula that turns out to be valid for speeds v << Z, v o . A synthesis of the Bohr and Bethe formulas was given later by Bloch . For speeds comparable with v,,, bound electrons are less effective in taking energy from the moving ion than implied by Eq. (2.1). Then the curve of d W l d R versus ion energy becomes less than predicted by Eq. (2.1) as the energy decreases. Finally the “Bragg maximum” in dWldR is reached at a speed-v, for projectiles with Z, not much greater than one. The picture becomes still more complex when Z, >> 1. Then capture and loss of electrons by the projectile must be considered. A useful criterion for estimating the number of electrons bound to the projectile was proposed by B ~ h r He . ~ pointed out that an effective statistical equilibrium between capture and loss obtains at constant and that the speed of the most energetic electron that may remain bound to the ion is -v. Brandt8 has used this criterion together with the Thomas-Fermi statistical model for the coterie of electrons associated with the projectile to write the expression

’

Z: = Z,[I

- e~p(-v/v,Z:’~)]

(2.2)

for the effective charge Z: of the ion. This resembles closely empirical equations proposed earlier by several different experimental groups’ to fit experimental data. This definition of the effective charge for energy loss ’F. Bloch, Ann. Phys. (Leipzig) 16, 285 (1933); Z. Phys. 81, 363, (1933). Recent considerations of the Bloch correction have been given by J . A. Golovchenko, D. Cox and A. Goland, Phys. Rev. B 26, 2335 (1982) and extended to the relativistic range to yield a much simpler result than that found by Bloch by V. E. Anderson, R. H. Ritchie, C. C. Sung and P. B. Eby, Phys. Rev. A 31, 2244 (1985). ‘W. Brandt, in “Atomic Collisions in Solids”, (S. Datz, B. R. Appleton, and C. D. Moak, eds.). Plenum, New York, 1975. %. H. Barkas, “Nuclear Research Emulsions: Techniques and Theory.” Academic, New York, 1963; T. E. Pierce and M. Blann, Phys. Rev. 173, 390 (1968); L. C. Northcliffe “Passage of Heavy Ions Through Matter,” in National Academy of Sciences, National Research Council Publication 1133, Washington, D.C., 1964; H. D. Betz, Rev. Mod. Phys. 44, 465 (1972).

234

P. M. ECHENIQUE el af.

is taken as

where (dW/dR),,, is the experimentally determined stopping power of the medium for the ion and (dWldR),,,,, is the measured stopping power of the same medium for a proton at the same speed. (ZT),,, is found to depend primarily on the ion speed and to be rather insensitive to the atomic number of the medium. In condensed matter with many mobile, non-localized electrons, the charge of the ion becomes screened by the motion of the valence electron gas. Residual interactions at particle speeds v << vo are such as to expel electrons to a thin cap just outside the Fermi surface. Fermi and Teller" were the first to analyze the energy loss of a heavy charged particle in this regime. They predicted that d W / d R should be proportional to the ion speed and only weakly dependent on density of the electron gas that it penetrates. Energy loss of ions colliding with single atoms was calculated by averaging over the space-varying density of electrons in the spirit of the Thomas-Fermi treatment of the static electronic structure of a many-electron atom. Improvements on the Fermi-Teller treatment are described below. When Z1>> 1 and at very low velocities nonlinear effects come into play. The theory appropriate to this regime is detailed in Section V. At projectile speeds such that v , < v , ~ v , Z : ' ~an ion may lose electrons to, and capture electrons from, the medium. A complete a priori theory of effective charge and the dynamic screening that an ion experiences in condensed matter must describe these charge-changing processes in detail, as well as their contribution to the energy loss of the ion. Such a theoretical approach is described in Section VI. In a separate development Fermi" early realized that the response of an electronic system to the passage of a charged particle could be couched in terms of its response to an equivalent spectrum of electromagnetic radiation. This idea was later developed by Williams12 and Weizsacker" and has been used to treat many different aspects of the interaction of radiation with matter. Fermi13 analyzed the generation of Cherenkov radiation in an electrodynamic approach by utilizing the frequency-dependent dielectric func'OE. Fermi and E. Teller, Phys. Rev. 72, 399 (1947). "E. Fermi, Z . Phys. 29, 315 (1927). '?See, e.g., E. J. Williams, Rev. Mod. Phys. 17,217 (1945); C. F. v. Weizsacker, Z . Phys. 88, 612 (1934). I3E. Fermi, Phys. Rev. 57, 485 (1940).

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

235

tion of the medium. Subsequently the dielectric function was generalized to become an operator depending upon space, or equivalently, to be a function of both wave vector and frequency, in order to treat the stopping of charged particles in a plasma.I4 Following this, the random phase approximation (RPA) dielectric function of an isotropic, nonrelativistic electron gas was derived by Linhard,” building on the work of Klein.’’ This enabled a unified description of single-particle and collective aspects of a model system that has had wide utility in radiation physics and solid state physics. Before this the pioneering work of Bohm and Pines16 laid the foundations of a quantum theory of the electron gas. Their work was based on ideas about collective oscillations in classical plasmas developed from the experimental work of Tonks and Langmuir” and from the theoretical interpretations of Vlasov. l7 They quantized the equation of motion of p s , the qth Fourier component of the electron density operator in an electron gas. This was made possible by separating fourier space into two parts; for 191 < a cutoff wave number, q c , collective effects dominate single-particle interactions, while for 191 > qc the reverse is true. This division is made for mathematical convenience and permits the use of the Random Phase Approximation, RPA, to demonstrate the existence of plasmons, the quanta of plasma oscillations, and to treat many properties of the electron gas. The dielectric treatment of the electron gas does not require this artificial division in wave vector space. However, the general RPA method has had wide application to the treatment of plasma oscillations in finite geometried8 and in special problems in which a linear approximation is not applicable.”

14

See, e.g., M. E. Gertsenshtein, Zh. Eksp. Teor. Fiz. 22, 303 (1952); M. E. Gertsenshtein, Zh. Eksp. Teor. Fiz. 23, 678 (1952); A . T. Akhiezer and A . G . Sitenko, Zh. Eksp. Teor. Fiz. 23, 161 (1953); J. Neufeld and R. H. Ritchie, Phys. Rev. 98, 1632 (1955). ”J. Lindhard, K. D a m . Videmk. Selrk. Mat.-Fys. Medd. 28, No. 8 (1954); see also 0. Klein, Arkiv. Mat., Astron. Fys. 31A,No. 12 (1945). I6D. Bohn and D . Pines, Phys. Rev. 82, 625 (1951); 92, 609 (1953); See also D . Pines and D . Bohm, Phys. Rev. 85, 338 (1952), and D . Pines, Phys. Rev. 92, 626 (1954). ”L. Tonks and I. Langmuir, Phys. Rev. 33, 195 (1929); A . A. Vlasov, Zh. Eksp. Teor. Fiz. 8, 291 (1938); See also A. A . Vlasov, “Many-Particle Theory and its Application to Plasma”. State Publishing House for Technical and Theoretical Literature, Moscow (reprinted by Gordon and Breach, New York), 1961. “See, e.g., B. B . Dasgupta and D. E. Beck, in “Electromagnetic Surface Modes.” ( A . Boardman, ed.). Wiley, New York, 1982. See also P. A . Feibelman, Prog. Surf. Sci. U, No. 4 (1982). I9J. C. Ashley and R. H. Ritchie, Phys. Status Solidi 38, 425 (1970).

236

P. M. ECHENIQUE e6 af.

2. APPROACH

Much experimental work and theoretical analysis have gone into the study of energy losses and ranges corresponding to various ion-target combination^.^ However, it is only relatively recently that detailed descriptions of the spatial patterns of excitation engendered in condensed matter have been given.” A great deal of experimental data relevant to the state of swift ions in condensed matter has been accumulated but a priori theoretical attack on this problem has been carried out only since reasonably accurate models of condensed matter response functions have become available. In this review we endeavour to present a coherent account of the detailed interaction dynamics of an ionic projectile proceeding through condensed matter. We employ concepts familiar from solid state physics. The response function of the stopping medium is chosen to be generically related to that for an electron gas. The central theoretical concept is the quanta1 self-energy function of this complex composite system, developed in Section 11. Section I11 is devoted to the theory of the wake; physical properties are emphasized by using increasingly accurate representations of the medium’s response. Connection with the self-energy approach of Section I1 is made. The stopping power of the medium is related to a property of the wake and energy loss fluctuations induced by the wake are evaluated. In Section IV experimental consequences of the wake are described. Section V details physical aspects of the interaction of ions with speed v << vo and develops density functional theory as applied to this phenomenon. Charge states of ions in several different solids are inferred in Section VI by numerical realization of the theory of Section 11. A summary and overview is presented in Section VII. II. Quanta1 Self-energy of a Projectile in an Electron Gas 4. SEMICLASSICAL SELF-ENERGY FOR A CHARGE MOVING IN A UNIFORM ELECTRON GAS

Semiclassical arguments provide a simple and useful way to describe the concept of the self-energy of a charge interacting with a polarizable ”J. Neufeld and R. H. Ritchie, Phys. Rev. 98, 1632 (1955); 99, 1125 (1955); See also R. H. Ritchie, Ph. D. thesis, University of Tennessee, 1959 (unpublished); V. N. Neelavathi, R. H. Ritchie, and W. Brandt, Phys. Rev. Lett. 33, 302, 33, 670(E) (1974); 34, 560(E) (1975); see also V. N. Neelavathi and R. H. Ritchie in “Atomic Collisions in Solids,” Ed. (S. Datz, B. R. Appleton and C. D . Moat, eds.). Plenum, New York, 1975.

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

237

medium.20 We work in atomic units throughout, e2 = h = m = 1. The energy is measured in Hartrees (1 Hartree = 2 Rydbergs = 27.2 eV) and the Bohr radius, an, is the unit of length a,=0.529A. We begin by writing Poisson's equation for the scalar potential @(r, t) generated at r and at time t by an applied charge density p"(r, t) in a medium characterized by a causal dielectric constant E,

Expressing all quantities as Fourier integrals of the form

and considering

E = E(q,

w ) , Eq. (4.1) leads to

A bare particle with charge Z1 may be considered to give rise to a density po(r, t) =Z16(r-vt) if it moves with constant velocity 21. Then 2 n Z 1 8 ( w - q - v ) and the Fourier component of the induced scalar electric potential, @?., may be written

822, @b:"o= 6( w - q . v)( q2

1 ~

4% w>

- 1) .

(4.4)

The rate of energy loss per unit time, W , i.e., the power loss, is now obtained from the induced electric field Eind= -V qjind at the projectile position, as

-

W = -Zlv

. Eind(r= vt,t).

(4.5)

Using the even and odd character of the real and imaginary parts of the retarded response functions as they depend on w , W can be written as

238

P. M. ECHENIQUE et al.

From now on, we shall denote

as J dx. Equation (4.6) allows us to define P(q, w ) as the probability per unit time of losing energy w and momentum q, thus creating a real excitation in the electron gas as

(4.7) where

which can be used to define a mean inelastic collision time

l/t =

i

dx P(q, 0).

t as

(4.9)

Note that the effect of inelastic scattering can be represented by introducing an imaginary optical potential C, in the one-particle Schrodinger equation.*l This leads to a temporal decay in the probability density of the incident particle as e2’Ir, where (4.10) and where r is the energy width of the level. The real part of the self-energy is found from elementary electrostatics to be22,23

zR= tz,$ind((r= vt,t) =

I

dx Z , Re($F,).

(4.11)

”L.I. Schiff, “Quantum Mechanics”, 3rd edition. McGraw-Hill, 1985; A . Galindo and P. Pascual, “Mecinica Cuintica.” Editorial Alharnbra, Madrid, 1978. 22 J. C. Inkson, “Many-Body Theory of Solids, An Introduction,” Plenum, 1984. 23J. D. Jackson, “Classical Electrodynamics, 2nd edition, Wiley, 1975.

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

239

The complex self-energy I: = I:R+ iXI of the moving charge is then given by

This equation applies to any polarizable medium. In particular, it is appropriate to the important case of the uniform electron gas to which we shall restrict our attention from now on.

5.

QUANTAL SELF-ENERGY OF A

MOVING CHARGE

The Hamiltonian describing an external charge of mass M and charge Z1 interacting with an electron gas is written

Here Ho is the Hamiltonian of the homogeneous electron gas, T,= -V?/2M is the kinetic energy operator for the external charge and HI represents the interaction Hamiltonian between the external charge and the medium electrons, i.e.,

The coordinate of the jth electron in the electron gas is described by rj and RI denotes the coordinate of the incident charge. p(r) is the particle density operator of the electron gas p(r) = C j 6(r - Ti). The state of the charged particle is described by a plane wave li) = e““ R I normalized to unit volume, where kiis the initial momentum of the particle. We denote by {In)} the many-body eigenvectors of the homogeneous electron gas such that Ho In) = E n In). From first-order perturbation theory the probability per unit time for a transition from the initial state li) 10) to a 24D. Pines, “The Many-Body Problem.” Benjamin, New York, 1961. 25D. Pines, “Elementary Excitations in Solids.” Benjamin, New York, 1964 ’%. Galitskii and A . Migdal, Sou. Phys. JETP 7, 96 (1958). ”P. Nozieres and D . Pines, Nuouo Cirnenro 9, 470 (1958).

240

P. M. ECHENIQUE

final state If)) . 1

el

al.

= eikfRlIn) is

If we denote q = ki - kf and wnO= En - E0 and realizing that the matrix elements of ZJ(r - R,I are given by uqe-iq.r,where uq = 4nZl/q2, Eq. (5.3) becomes

where p(q) = Cj e--Iq.rjis the Fourier transform of p(r), i.e., represents fluctuations about the average particle density. The symbol + denotes complex conjugation. Thus Pi.+, is proportional to the Fourier transform of the density-density correlation function I(nl p ( q ) 10)12 a result first derived by Van Hove.28 By noting that the imaginary part of the inverse of the electron gas response function can be written, for positive w,

as24,25,27

Pi-t can be written as

where P'(q, w ) is given by

28L. Van Hove, Phys. Rev. 95,249 (1954).

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

241

which is very similar to Eq. (4.8). Now

-

For high incident momenta, A can be replaced by -q v, then P’(q, w ) given by Eq. (5.7) coincides with the classical expression of Eq. (4.8), the classical limit. Equation (5.7) can then be used to define an imaginary part of a self-energy, in the same way as we have already done in Eqs. (4.9) and (4.10), thus 1

J

z*= - 2 du P’(q, w ) .

(5.9)

The real part of the self-energy XR is given by the energy shift, AE, in the energy of the incoming particle caused by its interaction with the electron gas.

where P stands for principal part. With the use of Eq. ( 5 . 9 , we can rewrite (5.10) as (5.11)

with

8n W(q, o)=,Im 4

(5.12)

Using 1 1 = P(-) x-afiq x-a

Tin+

-a),

(5.13)

where q is a positive infinitesimal constant, we can write for the complex

242

P. M. ECHENIQUE et al.

self-energy 1

(5.14)

w-iq+A w-iq+A

(5.15)

-

Equations (5.14) and (5.15) can be shown for A = -q v to be equivalent by taking the imaginary parts in each equation and using the KramersKronig relations for the retarded response function. Equations (5.14) and (5.15) are, indeed, equivalent to the ones which can be obtained by the GW approximati~n".~%~~ to the self-energy of an external charge interacting with an electron gas. FOR 6. QUANTAL SELF-ENERGY ELECTRONGAS

AN

EXTERNAL ELECTRON IN AN

When the external particle is an electron an important difference arises with respect to the above treatment3'-33 because the incoming particle is indistinguishable from the electrons of the medium. Consider an electron in the excited state k (Fig. 1). Due to its interaction with the electron gas, it can jump to a state k - q, creating an excitation pair of energy w and momentum q. As a first approximation, we could proceed as in Section 2 by writing a Hamiltonian equivalent to the one leading to (5.7) for P'(q, w), with ki = k and kf= k - q, but now subject to the important restriction that kf has to be greater than k F ,the Fermi wave number of the electron gas. Thus PLk-,

16n2

-1

q2

E(Q,W)

--Im(-)b(w -

+ ~ [ k - q ] * - k 2 ) 8 ( l k - q I -kF).

(6.1)

One word of caution should be added here. In obtained Eq. (6.1) we have neglected exchange p r o c e s ~ e s ,interfering ~ ~ . ~ ~ with the direct process 29L. Hedin and S. Lundqvist, in "Solid State Physics, Vol. 23, p. 1 (H. Ehrenreich and D. Turbull, eds.). Academic, New York, 1969. MJ. J . Quinn and R. A . Ferrell, Phys. Rev. 1l2, 812 (1958). 31J. Hubbard, Proc. R. Soc. (London) A243,336 (1957); A240, 539 (1957). "J. Goldstone, Proc. R. Soc (London) A239, 267 (1957). 33V. Galitskii, Sou. Phys. JETP 7, 104 (1958). "P. Nozieres and D. Pines, Phys. Rev. 111, 442 (1958); see also R. H. Ritchie and J. C. Ashley, J . Phys. Chern. Solids 26, 1689 (1965).

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

243

5- 9

FIG. 1. External electron of momentum k interacting with an electron of momentum k‘ inside the Fermi sea. The final states are denoted by k - q and k’+ q. Such a final state can be reached in two different ways. In the direct process (wavy line) the momentum transfer is q. The lines - - - represent an exchange process associated with momentum transfer k’+ q - k. This second process, and therefore its interference with the direct one, has been neglected in the derivation of Eq. (6.2).

shown in Fig. 1. In an exchange process, the transition from the initial state Ik) 10) can be performed in a different way from the direct one described previously: the electron in state Ik) can make a transition to the state k’ q while the electron in k’ goes to the state k - q (- - - lines in Fig. 1). These two processes interfere, modifying our previous result of Eq. (6.1). We shall neglect this interference, however. This approximation is equivalent to the GW approximation in the language of Feynman diagram^.^^,^^,^^ The imaginary part of the self-energy is then given by Eqs. (5.9) and (6.1). The second-order energy correction, EI2)(k),can be calculated in the same way as it was done to derive Eq. (5.11), but now with restriction over final states,

+

On the other hand, a different contribution to the second-order energy

244

P. M. ECHENIQUE et al. k

#-

FIG. 2. An external electron filling a state of momentum k inhibits the corresponding vacuum fluctuation contribution to the zero-point energy of the electron gas shown in the figure. A k - q electron, cannot jump to a k state having the same spin.

shift arises in the case where the incident charge is an electron, and it is associated with the fluctuation of the vacuum state of the electron gas.35 Due to the external electron filling up the k level, fluctuations around the vacuum level such as the ones depicted in Fig. 2, which are present in the calculation of the ground state energy of the interacting electron gas, are now forbidden and have to be subtracted from the term considered above. This contribution for a given q is given, with the appropriate sign, as to be added to the direct term

Notice two differences with the expression for Ei2)(k);first the state

k - q is inside the Fermi sphere, and second k - q is the initial state while

k is the final one. We also neglect exchange processes here, as mentioned above. Before writing the final expression for the self-energy, we write down the first-order perturbation theory contribution to the exchange correction29

355. R. Schrieffer, “Theory of Superconductivity.” Benjamin, New York, 1964.

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

245

Equations (6.1), (6.2), and (6.3) can be combined to define the second order contribution to the correlation self-energy, viz.,

Note that the second term in the brackets does not contribute to the imaginary part of Cc since for the incoming electron k > k F . We should also mention that Cc usually is split into two terms, the Coulomb hole (ch) and the induced screened exchange ~ o n t r i b u t i o n s ~ ~ ~ ~ ~ CCh=

CiSX=

-1

-1

dx W(q, w ) (

dx W(q, w )

1

o - iq

+ i ( ( k - 41’-

k 2 )’

1 o - iq + i ( k 2- Ik - 41’)

The Coulomb term, Eq. (6.6) is identical to the self-energy of an external charge given by Eq. (5.15). On the other hand, the w-integration of the induced screened exchange term can be done using the Kramers-Kronig relations to give

Combining this contribution with the first order term of Eq. (6.4) we find the total screened exchange term29’3638

36B. I. Lundqvist, Phys. Kondens. Muter. 6, 206 (1967); see also B. I. Lundqvist, Phys. Status Sofidi 32, 273 (1969); M. Ichikawa and Y. H. Ohtsuki, J . Phys. Soc. Jpn. 27, 953 (1969). 37J. C. Inkson, “Many-Body Theory of Solids.” New York, 1984. 38L. Hedin, Phys. Rev. W9, A796 (1965).

246

P. M. ECHENIQUE et al.

and then the total self-energy C is given by

C = zsx+ zch. 7. BOUND ION-ELECTRON COMPOSITE INTERACTING WITH ELECTRON GAS

(6.10) AN

In this section, we shall address the problem of an ion-electron composite3M1 moving inside an electron gas. We begin by writing the Hamiltonian of the problem

where Ho is the Hamiltonian of the electron gas, rj denotes the coordinates of the jth electron in the electron gas; R,the coordinates of the ion and re denotes the coordinates of the electron in the ion-electron composite. HI is the Hamiltonian of the ion-electron composite.

The term

describes the interaction between the electron gas and the ion electron system. We shall be concerned mainly with one-electron ions. The eigenfunctions of HI are given by li)

39P. M. Echenique and R . H. Ritchie, EIhuyar 7, 1 (1979). 40 R. H. Ritchie, W. Brandt, and P. M. Echenique, Phys. Rev. B 14, 4808 (1976); R. H. Ritchie and P. M. Echenique, Israeli Phys. SOC. 4, 245 (1981); Philos. Mag. 45, 347 (1982); P. M. Echenique, R. H. Ritchie and W. Brandt, Phys. Rev. 33, 43 (1986). 41 F. Guinea, F. Flores, and P. M. Echenique, Phys. Rev. Left. 47, 604 (1981); Phys. Rev. B 25, 6209 (1982).

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

247

and Ei =

k; + on, 2(M + 1)

where

R=

(7.5)

+

re MRI 1+M

represents the coordinates of the center of mass, the total momentum of the composite, and uo(p) the wave function describing the relative motion of the electron in the composite with respect to the ion, i.e., p = re - R, with energy w o . The ion-electron composite can lose its electron, then the new composite wave function is given by

where k is the final momentum of the composite and kf is the momentum of the relative motion of the ion and the ejected electron. The corresponding eigenvalue is

1 1 M E k: f - 2 ( M + 1 ) k2+-2M+1

(7.7)

The Fermi golden rule gives us the transition probability per unit time of a transition from the initial state li) 10) of energy Ei + En to the final state of If) In) of energy Ef En;lo), In), Eo and En refer to many-body eigenvectors and eigenenergies of the homogeneous electron gas. Following the lines of previous analysis one finds

+

Several comments about this equation are worthwhile. First of all, kf is the momentum of the ejected electron referred to the moving ion; thus the momentum with respect to the electron gas is kf v. In obtaining Eq. (7.8), we have taken v to be a constant before and after the collision. Comparing Eqs. (7.8) and (6.1), one should notice that k, v has to be replaced by k - q in (6.1). That this is the case is easily seen in Eq. (7.8),

+

+

248

P. M. ECHENIQUE et al.

if we chose the initial state u,,(p) to be a plane wave ei(k-v)'p , then k, = k - v - q, the final state having a momentum k - q referred to the electron gas. Finally, the energy conservation law, Eq. ( 7 . 8 ) says that the difference between initial and final states energies (k = ko- q)

k:, ( 2 ( M 1)

+

(7.9)

goes to excitation of the electron gas. Notice that k,: (ko-d2 2(M 1) - 2(M 1)

+

(7.10)

+

is replaced in Eq. (7.8) by q - v , neglecting recoil effects assuming q v >> q'/(M 1). We also assume M / ( M 1) = I. A similar argument can be used to calculate the capture probabilities. Instead of Eq. ( 7 . 8 ) we find

-

+

pcapture

=

+

I

dx

k,

8 ( k F - 1 ki+ vl)

q2

k: 6( w - 2

- 9 -v

+ wo (7.11)

Here, the ion captures an electron filling a state ki + v located below the electron-gas Fermi level.

8 . INTERACTIONS IN WHICH THE INTERNAL STATEOF THE COMPOSITE IS UNCHANGED In the above, we have assumed that the composite changes its internal state. It may also be scattered without change.3M1 This case can be analyzed following the lines developed in Section 5: the only difference is the simultaneous interaction of the ion and the electron with the homogeneous medium. In our approach, we also neglect exchange effects and only consider a kind of Hartree approach. Then following the lines of Section 5, we obtain the following result for the probability of creating an

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

249

excitation (q, w ) in the electron gas:

This is equivalent to Eq. (4.8), replacing 2: (ion charge) by 12, (uo(p)l e-i'pluo(p))12. We take M + a and write: 1 2(M + 1) (k -

1 k2 - 2(M 1)

+

-9

*

V.

It is now interacting to realize that previous results for the composite with or without change of internal state can be cast together in a suggestive expression for the total self-energy of the composite. At this point, we do not discuss details, but appeal to the results obtained in previous sections for single particles. Referring to the results found in Eqs. (6.5), (7.8), (7.11) and (8.1), we write the self-energy for the composite as

In this equation, the different terms inside the parentheses represent the capture process, the loss process and the scattering process with no change in the internal structure of the composite, respectively. We note that the electronic states of the homogeneous electron gas are modified due to the presence of the localized state41 uo(p). A better approximation to Z, Eq. (8.3), can be obtained if we choose as starting

250

P. M. ECHENIOUE et a1

point the orthogonalized plane wave (OPW) functions,

Ik)

= eik’p - (uo(p)

I eik’”)uo(p),

(8-4)

to describe the electrons of the conduction band. Using this new basis set, the modified results may be obtained by merely replacing eik’p by (k) given above in the matrix elements of Eq. (8.3). 111. Linear Theory of the Interaction of a Probe with Condensed Matter

9. WAKECONCEPT Wake theory has been discussed in the literature on the basis of classical electrodynamics.” The induced scalar electric potential Gind(r,t) may be written (see Eqs. 4.2, 4.4)

The induced density fluctuation is given by

By using the RPA dielectric response function of the electron gas, one can evaluate Gindand anindn ~ m e r i c a l l y . ~A~convenient ~~’ and instructive first approximation to Gind and anindcan be found with the aid of the classical frequency-dependent dielectric function E(Q,

w)=1-

4

o(o + i y ) ’

(9-3)

where op= ( 4 ~ n ) ”is~ the classical plasma frequency of a free electron gas with number density n and y is an effective damping constant. With this approximation the total potential is given by2’

+(b, 5)= Zlho(b,5)+ G1(r, t ) = -+ +w(r,t ) . R 2 1

42A.Mazarro, Ph.D. thesis, University of Barcelona, 1980 (unpublished). 43A.Mazarro, P. M. Echenique, and R. H. Ritchie, Phys. Rev. B 27,4117 (1983).

(9.4)

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

251

The first term gives a f-symmetric screened Coulomb potential which for y = 0 can be written in the form

The first term is the bare Coulomb potential per unit charge of the ion. The second has a factor

and accounts for steady-state screening of the ion charge by the electrons of the medium. The second term in Eq. 9.4, @,, describes the oscillatory potential due to the wake of electron-density fluctuations trailing the ion. In this simplest, approximate form, this oscillatory portion may be written'"

(9.7) In Eqs (9.4)-(9.7) the particle is assumed to be located at the point (0, 0, vt) at time t ; f = z - vt, b = (x' +y')"2 and R = (b2+ i')"', Jo is the Bessel function of first kind and zero order. K Ois the modified Bessel function of the second kind and zero order. @,, usually is referred to as the wake potential. The induced density fluctuation is given by44

where J1 is the Bessel function of first kind and order 1. In this approximation, one obtains44

for the potential of the polarized medium at the site of the ion. In this "P. M. Echenique, R. H. Ritchie, and W. Brandt, Phys. Rev. B 20, 2567 (1979); See also R. H. Ritchie, P. M. Echenique, W. Brandt, and G . Basbas, IEEE Tram. Nucf. Sci. NS-26,1001 (1979).

252

P. M. ECHENIQUE

et

al.

approximation, therefore, the self-energy Z, of a massive point charge is given by -Z :n w p /4 v, an estimate which agrees, when electron-recoil effects are neglected, with the self-energy calculated for fast electrons in an electron gas.36 It is instructive to compare the picture of the wake as presented above with the model of a stationary charge in an electron gas. For a negative stationary point charge one speaks of a correlation hole around it, and for a positive charge one speaks of a density enhancement at the charge. In either case, the response of the medium can be represented as a screened Coulomb potential centered at the charge, giving rise to a negative self-energy with a null imaginary part. In comparison, a moving ion gives rise to a cylindrically symmetric potential with a part centered at the charge, even in z, plus an oscillatory part stretching behind the particle. The first part gives rise to a real part of the self-energy which is negative, which originates from the correlation domain of dimension u / w p around the charge. The oscillatory portion contributes to the imaginary part of the self-energy which describes the slowing down of the charge.

10.

PLASMON DISPERSION

The polarization density retains a tractable analytical form if one includes dispersion effects in the medium response. This is easily done by inserting into Eqs. (9.1) and (9.2) the hydrodynamic dielectric function4s E(q, w)=1

+ p2q2- ww',( w + iy)

(10.1)

The constant p = ($)lRvFis the speed of propagation of density disturbances in an electron gas, vF being the Fermi speed of electrons in the medium. Neglecting y as well as terms of order O(B") the expression for the density fluctuations becomes4

where the variable A = (A2Z2 - b2)lL2 with A2 = p2/(v2- B'). An equation equivalent to (10.2) was derived by Neufeld and Ritchie2' and again by 45F.Bloch, 2. Phys. 81, 363 (1933); Helu. Phys. Acta 7, 385 (1934).

253

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

Fetter.& A singularity in the polarization density occurs at A = 0, corresponding to the radial distance from the particle trajectory of b = ( p / v ) T (v >> p), thus defining the envelope of the collective wake of half-angle -/3/v. The singularity of 6n at the cone is, of course, an artifact of the model dielectric function of Eq. (10.1), and may be eliminated by introducing a cutoff in momentum space. In fact, if the full dielectric function is employed, the singularity is replaced by a series of wavelike fluctuations as described in the next section.

11. SINGLE-PARTICLE EFFECTS. PLASMON POLEAPPROXIMATION The salient features of the full quantum expression for E ( q , w ) in metals and semiconductors are retained by setting22.36 E(q,

w)=1

+ w;

+

4

q4 p2q2+-w(w 4

+ iy)

(11.1)

As in Eq. (10.1) plasmon dispersion is included through the term containing p’. Single-particle effects now are accounted for, in an approximate manner by the term equal to the square of the kinetic energy q 2 / 2 of a free electron of momentum q, since at high enough momentum transfer the electron’s behaviour is essentially free-electronlike. The energy w g may be taken to account approximately for an effective band gap in materials like semiconductors, to give a collective resonance at the frequency B, = [ ( m i + w31’2]47s48 or can be taken to be zero in metals. Substitution of Eq. (11.1) into (9.1), (9.2) yields the potential and charge density fluctuations. The induced potential at the origin is still given, except for a minor correction due to the single particle term of Eq. ( l l . l ) , by (-nZlwp/2v), but now the induced density at the origin is different from zero and is given by (11.2) This agrees with the result of an exact calculation of the probability of &A. L. Fetter, Ann. Phys. 81, 367 (1973). 47D. R. Penn, Phys. Rev. 128, 2093 (1962). -W. Brandt and J . Reinheimer, Phys. Rev. B 2, 3104 (1970).

,“.32 LOCAL DIELECTRIC,

,/’

0 01

FIG.3. induced density fluctuations along the trajectory of a projectile located at i = 0, moving with velocity v = 4 a.u. = 8.76X 10xcni/sec. i n the positive i direction through a medium characterized by wp = 0.919 a.u. = 25 eV, 0 = 0.974 a.u. = 2 13 x 10’cmlsec and o, = y = 0. Thc solid curve was calculated using Eq. (11.1). I t shows a collective wake trailing the particle and the electron dcnsity cnhanccmcnt [Eq. (11.2)] at the origin. The dashed curve displays the results of calculating 6n(0,i) from Eq. (9.3). using a dielectric function without dispersion. When dispersion is included according to Eq. (10. l), without single-particle response, 6n follows the dash-dot curve labeled “hydrodynamic approximation.”

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

.-,

Z

5

255

15

FIG. 4. Wake potential surface &(p, f ) / Z e , as calculated for a projectile moving with velocity u = 2 a.u. = 4.38 X 10' cm/sec in a medium characterized by wp= 0.919 a.u. = 25 eV; /I= 0.974 a.u. = 2.13 X 108cm/sec, and wg= y = 0. This potential surface was computed with the use of the dielectric function of Eq. (11.1). This particle is located at the coordinate system as indicated by the dot and the arrow points in the origin of the f, p. direction of v.

finding a particle at the ion normalized to the probability of finding it in the incident beam, in the limit of v >> 2,.49 Figure 3 exhibits a plot of bn(0, Z)/Z, versus Z for b = 0. The prominent peak at Z = 0 is predominantly due to a single-particle density enhancement at the ion. At distances large compared to v / o p behind the ion, &(0, Z ) oscillates with wavelength -2nv/op. The dashed curve represents the results of computing bn(0, 5) from Eq. (9.2) using the local dielectric approximation (Eq. 9.3) for E , with a cutoff of w,/vF in momentum space to avoid the unphysical divergence at b = 0. The resulting curve oscillates for all Z < O but out of phase by n / 2 compared to the solid curve. When plasmon dispersion is included according to Eq. (10.1) bn(0, 2 ) follows the dash-dot curve referred to as the hydrodynamic approximation. Figures 4 and 5 display the wake potential @,, and the induced density fluctuation for v = 2. The rest of the parameters are those of Fig. 3. 49N. F. Mott and H. S. W. Massey, "The Theory of Atomic Collisions," 3rd edition. Oxford University, London, 1965.

256

P. M. ECHENIQUE et

nl.

O

2

FIG. 5. Surfaces depicting electron density Ructions 6n(p, f) induced by a moving ion under conditions of Fig. 3. The density enhancement at the origin is clearly visible. The single-particle bow wave and the crispations superimposed on the collective wake have wavelength -2n/u. A series of wave fronts generated by the particle motion appear ahead of the particle. The crests of these waves extend to the side of, and behind, the particle, widening and blending smoothly to constitute disturbances with the limiting wavelength 2 n u / w , characteristic of collective motion. Note that the scales of f and p have been expanded as compared to those of Fig. 4 to exhibit details.

One sees the distinctive oscillations of the wake potential in the region behind the particle Z 0, of wavelength -2nlv. The bow waves, here clearly visible, form periodic paraboloidal disturbances with apexes slightly in front of the projectile and extending behind it with half-angle of opening -arc sin(p/v). They signify the emission of plasmons by the ion, that eventually are damped into electron-hole pairs.

When the quanta1 dielectric response function, originally derived by Lindhard,22 is used, the wake potential and density fluctuations cannot

DYNAMIC SCREENING OF IONS IN CONDENSED MA7TER I

-8

1

-6

I

-4

I

@w

-2

;

I

4

257 I

I

6

8

-

b- 0 rs 2 v - 0 5

-0 8-I

-8

I

-6

-4

1

-2

-0 -0

-6

-4

-2

I

I

4

6

4

6

8

I

1

8

-

b - 0 rs

2

v - 1

FIG. 6. Wake potential in the neighborhood of a proton moving in an electron gas at the aluminium density (r* = 2). The potential is plotted in (a) as a function of distance along the track for three different velocities less than the Fermi velocity. The development of the oscillatory portion of the wake is clearly seen in (b) as the velocity increases. All quantities are measured in atomic units. The RPA dielectric function has been used in the calculations.

258

N

a

P. M. ECHENIQUE

(u

0

I

ru

el

al.

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

259

any longer be reduced to one-dimensional integrals, as in the case of the plasmon-pole approximation. The calculations must be carried out numerically by evaluating two-dimensional integrals. Mazarro et al. 42,43 were the first to perform such calculations. In Fig. 6 we show the way the wake potential changes as the ion velocity increases. The potential remains nearly constant when u 5 u F ( u F = 0 . 9 6 for r, = 2 in our example). Only a slightly growing asymmetry and a slighter well-depth growth are observed. However, once the plasmon threshold is exceeded (v 2 1.32 in our case), oscillatory behaviour appears. The results of the full RPA calculations are in general agreement with the ones obtained by Echenique, Ritchie and Brandt44 with the use of the simpler plasmonpole form for E ( q , 0 ) . These results do not appear to corroborate those of Stachowiak” in that he obtains a screening until the ion velocity reaches approximately 0 . 8 with ~ ~ no screening at all for high ion velocities. In Fig. 7 we display the electron density fluctuation function 6n(0,Z ) vs. Z for different velocities of a proton in an electron gas for which r, = 2. No dramatic changes are involved for Y < u F , and for small velocities the results agree well with the static calculation of Langer and Vosko.” Oscillatory behaviour appears for u > u F . 13. DAMPING EFFECTS Ashley and Echenique5’ have evaluated the effect of damping of the electron gas, using the Mermin dielectric response f ~ n c t i o n . ’They ~ found that the magnitude of the wake potential decreases faster than the exponential dependence exp( - yZ/2u) predicted by NRB, which was obtained using a “classical” dielectric function under the assumption of very small damping. The results of a calculation illustrating the effect of damping on the wake potential are shown in Fig. 8. 14.

QUANTUM

THEORY OF THE WAKE

The wake of an ion in a condensed medium is treated semiclassically above. Although a quantal function is taken to describe the response of the medium, the ion is assumed to be described by classical mechanics. A completely quantal treatment is possible. Ritchie and coworkers4’ have ’OH. Stachowiak, Bull. Acad. Pol. Sci. 23, 1213 (1975). ”J. S. Langer and S. H. Vosko, J . Phys. Chem. Solids 12, 196 (1960). 52J. C. Ashley and P. M. Echenique, Phys. Rev. E 31, 4655 (1985). 53D. W. Mermin, Phys. Rev. E 1, 2362 (1970).

260 P. M. ECHENIQUE et al.

"

riFIG.7. Variation of the electron-density-fluctuation function with position along the track for several different proton velocities for a proton moving under the conditions of Fig. 6.

DYNAMIC SCREENING OF IONS IN CONDENSED MA7TER

261

b- 0 lr = 2 - v = 1 5

___

v-

2

FIG.7. (continued). 0.4

rs= 1.60

v.4

A

-0.4

-0.8 FIG.8. Variation of the wake potential with i at b = 0 around a proton located at f = 0 for four values of y. The Mermin dielectric response function has been used in the calculations.

262

P. M. ECHENIQUE et al.

studied the self-energy of a composite entity that interacts with condensed matter. By employing an unfolding techniques4 the self-energy of a heavy ion with a single bound electron was written as a function of space, i.e.,

Here r is measured from the position of the ion on,. = on- o n , ,Un(r)is a single-electron orbital for the bound electron, w, is its binding energy, and p n n , is the matrix element of the density operator for the electron. The real part of the term in 2; that is proportional to 2, is exactly half the classical expression for the wake potential of Eq. (9.1). The other term in Eq. (14.1) arises from the action of the electron's wake on the electron itself and cannot be simplified without further approximation. Ritchie and Echenique4" also derive an expression for the spatial dependence of the self-energy of a bound complex of two particles with arbitrary masses and charges interacting with condensed matter. This is of interest in connection with the wake potential of a swift positronium atom.

15. NONLINEAR WAKE The dielectric theory is a linear theory, thus only exact in the limit of very high velocities. In the static limit, the differences in the screening of a static proton between the nonlinear, density functional calculations and the linear calculations are dramatic for the charge densities, i.e., ratios between the nonlinear and linear results varying from 2 to 34 when rs=2-6 for the induced charge density at the origin. These differences are not so great for the potential, at most a factor of 2 (See Section V). No first-principles calculation of a nonlinear wake exists in the literature. A phenomenological way to introduce nonlinearity in the calculation of the wake potential was suggested by Vager and Gemme11.s-5-57Their approach is based on the use of the scattered Coulomb wave function to describe the induced electron density. 54J. R. Manson and R . H . Ritchie, Phys. Rev B 24, 4867 (1981). " Z . Vager and D. S. Gemmell, Phys. Rev. Lett. 37, 1352 (1976). 56D.S. Gemmell, J. Remillieux, J. C. Poizat, M. J. Gaillard, R. E. Holland and Z . Vager, Phys. Rev. Lett. 34, 1420 (1975); 2.Vager, D. S. Gemmell, and B. J . Zabransky, Phys. Rev. A 14, 638 (1976). 57D. S. Gemmell and Z. Vager, "Treatise on Heavy-Ion Science," vol. 6 , p. 243, (Allan Bromley, ed.). Plenum, 1985.

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

263

16. STOPPING POWER.STRAGGLING The energy lost by the particle per unit path in the medium may be computed either by using electromagnetic theory for the energy deposited in the medium, or more easily by calculating the retarding force with which the medium acts on the particle. The latter is given by the electric field times the charge evaluated at the position of the charge [see Eqs. (4.5)- (4.8)] dW F = -= Z1 . v . V$ind(vt, t ) / v . (16.1) dR It is not clear to us who first used the fact that the stopping power of a medium for a swift charged particle may be obtained from the gradient along the direction v/v of the self-consistent potential set up by the particle in the medium and evaluated at the position of the particle. Early workers who employed this approach are Frank and Tamm," Fermi," and Kronig and Korringa.H' In detail, from Eq. (16.1) one finds for the stopping power of the medium

dW dR

1 u

d o , (16.2)

where P ( q , o)is given in Eq. (4.8). Inclusion of damping, via the Mermin reponse function, leads to an increase of about 20% in the stopping power for small velocities6' and for damping constants appropriate to carbon. The asymptotic form of the stopping power formula may be obtained by using the classical dielectric function of Eq. (9.3). With this substitution in Eq. (16.2) one cuts off the integration over q at the value corresponding to the maximum momentum deliverable to a free electron and finds

dW dR

Z o2

-- A In(

u2

2v2 G)

(16.3)

which is directly comparable with the Bethe6 form, Eq. (2.1), if I is replaced by the plasma energy up. "1. M. Frank and I. Tamm, Dokl. Akad. Nauk. SSSR 14, 109 (1937). 59E.Fermi, Phys. Rev. 56, 1242 (1939); 57, 485 (1940).

60R.Kronig and J. Korringa, Physica (Ufrechf)10, 406 (1943). 61J. C. Ashley; Nucl Instrum. Methods 170, 197 (1980).

264

P. M. ECHENIQUE el al.

The stopping power represents a statistical average of a quantity that is subject to fluctuations. A measure of these variations requires also the mean square deviation Q2=

( ( A E - ( A E ) ) ’ ) =nW,dR.

(16.4)

Equation (16.4) defines the straggling parameter W,. As Bohr4 has shown, one may obtain a simple representation of energy loss straggling in the limit where the distribution function approaches the Gaussian form

f ( E , d R ) = (2nQ2)-1’2exp[-(E

-

l?)2/2Q2].

(16.5)

In this equation E is the mean energy loss in the path length dR and we take

the P(q, w ) is given by Eq. (4.8). Eq. (16.5) may be used when E is 10-20% of the initial energy.62

17. ENERGY FLUCTUATIONS IN THE WAKE

As discussed above the stopping power and wake potential described by Eqs. (16.2) and (9.4), respectively, represent a statistical average of a quantity that is subject to fluctuations. A measure of these variations may be obtained by comparing Q;, and Q$, i.e. the straggling of energy loss of the ion Q;, , and the straggling of energy loss (or gain) of a test charge Z located at an arbitrary position in t h e wake, with - ( d E ) , , and - ( d E ) , . The latter quantities are the expected value of the energy loss of the ion, due to its self-wake, and of the test probe, due to its self-wake plus the wake of the leading ion, respectively.44 In this section to avoid confusion we denote the energy by E. We assume that the test charge Z proceeds with the same velocity as that of the leading ion. The differential energy loss may be written in terms of the force on the test charge exerted through a path length dR as - ( d E ), = - ( Z 2 (dE )o

62

+ Z Z 1( d E )J,

(17.1)

J. Lindhard and M. Scharft, K . Dan. Vidensk. Selsk. Mat. Fys. Medd. 27, No. 15 (1953).

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

265

where l / Z : ( d E ) , is given by Eq. (16.2) and ( d E ) wis given by

+

where q2 = Q 2 w 2 / v 2 .The total force ( d E / d R ) is due to the action of electronic motion induced in the medium and may be expressed through the scalar electric potential set up by the test charge itself and through the wake potential r$w trailing the swift ion. To obtain Q 2 , the total straggling of the test charge, we need only multiply the integrands of Eqs. (16.2) and (17.2) by o,as was shown on general grounds by B ~ h rThus .~

Q2(b,Z ) = [ ( ( d E ) 2 )- ( d E ) 2 ]= 1 ZQg

+ Z Z I Q i ( b ,z ) ! ,

(17.3)

where 52; is given in Eq. (16.6). The straggling of the incident ion alone is Z:Q;. The contribution of the wake induced by the leading ion to the straggling in energy loss of the test charge at (b, z ) becomes, for 2 = 1,

It is understood that Q2 by definition must be positive definite, hence the absolute value sign on the right-hand side of Eq. (17.3). In the limiting case 2 << Z , , both the energy loss and the straggling of the test charge calculated from Eq. (17.3) without the absolute value sign may become negative if it is located in any of the regions in which the gradient of the wake potential of the leading ion subtracts from the gradient of the self-wake of the test charge evaluated at its own position. Figure 8 displays Eq. (17.3) for a test charge Z = 1 along the trajectory (b = 0) of a proton (2,= 1). The difference between Q 2 / d R (solid curve) and the f independent Q i / d R (dashed line) is seen to be
2.0

0.5

P

3

0

0.3 0.2 0.1

0 - 20

-15

-10

-5

0

? (a. u.) FIG.9. The total energy straggling per unit pathlength Q2(0, i ) / d R (solid curve) of a charge Z = 1 behind a moving proton is only slightly larger than the constant quantity, ( Q 2 / d R ) , [(a) dotted line] due to the test charge wake in the absence of the leading proton. The total stopping power, - (dE(0, f ) / d R ) [(b) solid curve] of a charge Z = 1 in the proton wake varies strongly with 2 and can approach zero, as compared to the constant ( - d E / d R ) , [(b) dashed line] of the test charge without a leading proton. The parameters chosen were the same as those used in Fig. 3.

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

267

of electrons and hence carries small momenta (<
18. WAKEBINDING In their pioneering paper, Neelavathi, Ritchie and Brandt'" (NRB) pointed out that the oscillatory wake may give rise to wake-bound electrons, i.e., electrons trapped in regions of energy minima, corresponding to a depletion in the wake electron density. Several calculations of wake-state binding energies have been made.20,40,42,43,6~70 Usually, in these calculations the potential energy of an electron is fitted in the neighborhood of the minimum by a harmonic-like potential and the electron energy levels and wave functions are found from standard 63M. H. Day, Phys. Rev. B 12, 514 (1975). #M. H. Day and M. Ebel, Phys. Rev. B 19, 3434 (1979). 65P.M. Echenique and R. H. Ritchie, Phys. Rev. B 21, 5854 (1980). "V. N. Neelavathi and R. H. Kehr, Phys. Rev. B 14, 4229 (1976). 67P. M. Echenique, Ph.D. thesis, Universidad Aut6noma de Barcelona, 1977 (unpublished). bsP. M. Echenique , Proc. Workshop on Physics with Fast Molecular Ion Beams, Argonne I l l , August 1979. 69P. M. Echenique, R. H. Ritchie, and W. Brandt, Phys. Rev. B 33, 43 (1986). "A. Rivacoba and P. M. Echenique, Phys. Rev. B 36, 2277 (1987).

268

P. M. ECHENIQUE et al.

quantum theory. The early work of NRB2' used, for emphasis and simplicity, the classical frequency-dependent dielectric function of an electron gas given by Eq. (9.3). Day63 and Day and EbeI@ employed Lindhard's classical dielectric function. Ritchie et al.40 and Echenique and ~ o w o r k e r s ~utilized . ~ ~ ,the ~ ~plasmon-pole ~ approximation. Results with the full RPA dielectric response function have been presented.4243 Behind protons and in the energy range under discussion Y = 2-10 a.u. , the binding energy is of the order of 2-6 eV and the wave function has a spatial extension of 2-5 a.u. The effect of electron-gas damping has been ~ t u d i e d , ~using ' the Mermin response function, and the binding energy is shown to decrease with increasing damping. Echenique et al.69 have evaluated the wake binding energies including the effect of the self-wake of the trapped electron on its binding energy. The self-wake effect may be regarded as a nonlocal, polaron-like phenomenon that tends to increase the binding further. Up to now, we have concentrated on the possibility of binding in the first trough of potential energy. In principle, binding could occur in any of the troughs of electron potential energy behind the leading ion. However, even for small electron damping, plasmon dispersion and single-particle effects (see Fig. 9) will greatly reduce the magnitude of the wake potential at such distances, usually strongly lessening the likelihood of binding. An interesting possibility arises when the leading ion is a negative ion, i.e., a p meson. Then binding will occur closer to the ion than for a positive ion. Rivacoba and Echenique7" have evaluated the binding energy in the first trough of the wake created by a p meson. An experiment71 using such negative probes is under way and could bring much light to the question of energetic electrons emerging from the solid at the ion velocity. Echenique and Ritchie7' have estimated the probability of capture and loss, from a free electron gas, via a third-body process, into and from a wake bound state, respectively, using the plasmon pole approximation for the dielectric response function. Their results for the mean free path associated with loss qualitatively agree with the simple formula derived Ritchie et a1.,40 viz, I , =YR-;.~. The mean free path varies from 3.5 to 33 A when the velocity varies from 2 to 8 while in the approximate formulae it goes from 5.65 to 28.3 A. The equivalent mean free path for capture is much larger, of the order of 7500 A for r, = 2 and Y = 2. This is to be compared with about 50 8, for capture into a hydrogenic state "Y. Yarnazaki (to be published). "P. M. Echenique and R . H . Ritchie, Phys. Left. 111A,310 (1985)

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

269

directly bound to the ion. Yamazaki and Oda73 have calculated, in the Brinkman-Kramer approximation, the cross sections for electron capture from a bound state of a lattice atom to a wake bound state.

19. VICINAGE EFFECT The vicinage effect refers to the fact that the wake of a leading charged particle may modify the retarding force acting on a trailing one so that the energy loss of the pair may be different from the sum of the losses that would be experienced by their individual constituents if proceeding at large distances from one another. Ion clusters may be created by injecting swift molecular ions such as H: into a solid target. The ions are stripped of their electrons after penetrating only a few atomic layers. The initial separation of the constituent ions is -1A. The dynamically modified Coulomb repulsion between its constituents causes the cluster to explode. The first experimental evidence and theoretical analysis of this vicinage effect in the energy loss of ions clusters was given by Brandt, Ratkowski and R i t ~ h i e .In~ ~their experiment, swift H: and H: ions bombarded thin foils. They found that the cluster energy loss was larger than that which would have been experienced by the constituent ions if isolated from one another. Considerable experimental and theoretical work in this area has been done s u b ~ e q u e n t l y . ~ ~ ’ ~ To make the vicinage idea more quantitative, suppose that two point charges 2, and Z 2 proceed with velocity v in a medium characterized by the dielectric function E ( q , w ) . Separation between the charges is specified by the vector R with components D and B in directions parallel with and perpendicular to v, respectively. In linear response theory and the first Born approximation, the energy loss of the cluster per unit

73Y.Yamazaki and N. Oda, Nucl. Instrum. Methods 194, 415 (1982). 74W. Brandt, A . Ratkowski and R. H. Ritchie, Phys. Rev. Lett. 33, 1329 (1974); 35, 130E (1975). 75N.R. Arista and V. H. Ponce, J . Phys. C 8, L188 (1975). 76N. R. Arista, Phys. Rev. B 18, 1 (1978). nJ. W. Tape, W. M. Gibson, J . Remillieux, R. Laubert and H. E. Wegner, Nucl. Instrum. Methodr 132, 75 (1976); R. Laubert, IEEE Trans. Nucl. Sci. NS-26,1020 (1979). 78J. C. Eckardt, G . Lantschner, N. R. Arista, and R. A . Baragiola, J . Phys. C 11, L851 (1978); M. F. Steuer, D. S. Gemmell, E . P. Kanter, E. A . Johnson, and B. J . Zabransky, Nucl. Instrum. Methods 194, 277 (1982). 79G. Basbas and R. H. Ritchie, Phys. Rev. A 25,2014 (1982).

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FIG.10. A vicinage function, g(B, D ) = SJB, D)/S,, plotted as a function of B and D for stopping of a dicluster proceeding with velocity v = 2 a.u. in a medium characterized by plasma frequency op= 0.55 a.u. = 15 eV. The response of the medium is taken to be given by the plasmon-pole dielectric function of Eq. (11.1).

length to electron excitation in the medium may be written

x [Z: + Z ; + 2Z,Z2J0(QB)cos(oD/v)] = (ZT

+ Zt)S, + 2Z,Z,Sv(B, D).

(19.1)

Here S, is the energy loss per unit path length of a single proton having the same velocity as the cluster. In Eq. (19.1) q2 = Q 2 + 0 2 / v 2 In . Fig. 10 we show a vicinage function g ( B , D) = S,(B, D ) / S , plotted as a function of B and D for stopping of a discluster proceedings with velocity u = 3 a . u . in a medium characterized by plasma frequency w, = 0.55 a.u. = 15 eV. The response of the medium is taken to be given by the plasmon-pole dielectric function. Ashley and Echenique" have w'J. C. Ashley and P. M. Echenique, Phys. Rev. B 35, 1521 (1987).

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

271

estimate the effect of plasmon damping on the vicinage function of a dicluster moving in a free electron gas. Most studies of the vicinage effect have been directed toward the excitation of valence electron. Vicinage effects are also present on inner-shell excitation processes. Lurio and coworkers” have reported vicinage effects in the x-ray yield of thin solid films under bombardment by HZ projectiles. A theoretical analysis of their experiment has been made by Basbas and R i t ~ h i eThese . ~ ~ authors use first-order perturbation theory to calculate the probability of excitation by an ionic dicluster of an electron from a bound state to a final state, that they represent approximately as a momentum eigenfunction. Recent experimentaln2 work has been reported in this problem. A study of the effects associated with the orthogonalization in the final state has been made by Ugalde et a1.83

20. DYNAMIC SCREENING We have seen that when a fast charged particle moves through a solid polarization of the medium leads to dynamical screening of the nearby ions. Electrons bound to the ion may be affected by this ~ c r e e n i n g . ~ ’ - * ~ This means that transition energies between well-defined electronic states will depend on whether the transition occurs inside or outside a solid target. In the experiment of Bell and coworker^,'^ 95 MeV sulphur ions were passed through aluminum foils with thicknesses x ranging from 40 pg cm2 to 100 pg cmP2 and the collision-induced projectile K x-rays were measured. Due to the high projectile velocity, electron stripping is extensive and, on the average, only a few electrons remain on the ion. They are effectively He-like sulphur ions. The experiment measures energies of the transition ‘P,+’S, and 3 P , - + 1 S 0These . were selected because of their different lifetimes. The transition lPI+’So has a lifetimen6 of t = 1.5 x s whereas the 3P, state is metastable (t> “A. Lurio, H. H. Andersen, and L. C. Feldman, Phys. Rev. A 17, 90 (1978). *’A. Outuka, K . Komaki, and F. Fujimoto. “Proceedings of the USA-Japan Seminar on Charge States and Dynamic Screening of Swift Ions in Solids,” Honolulu, Hawaii, 1985, p. 21 (unpublished). See also A . Ooutuka, K. Kawatsura, K. Komaki, F. Fujimoto, N. Kouchi, and H. Shibata, “Proceedings of the 12th International Conference on Atomic Collisions in Solids, Okayama, Japan, 1987” [Nucl. Instrum. Methods B 33, 304 (1988). 83J. Ugalde, C. Sarasola, P. M. Echenique, and R. H. Ritchie, Phys. Rev. B 38, 735 (1988). @D.H. Jakubassa, J. Phys. C 10 4491 (1977). =F. Bell, H. D . Betz, H. Panke, and W. Stehling, J . Phys. B 9, L443 (1976). %H. Panke, F. Bell, H . D. Betz, W. Stehling, E . Spindler, and R. Laubert, Phys. Lett. 43A,457 (1975).

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lO-"s). The dominant part of the decays 3P1-+ 'So occur outside the foil, and is therefore unaffected by film thickness; while for thin enough films there is an effect on the transition 'PI+ 'So. After correcting for energy loss and Doppler-shift, Bell et al., found that an extra reduction of about 1 eV remained to be accounted for and attributed it to dynamical screening. The energy shift for the composite of a hydrogenic-like projectile is readily found from the theory given in Section 8, and is given as

where v is the velocity of the center of mass. In obtaining Eq. (20.1) it has been assumed that M >> 1. This formula can easily be extended to the case of He-like ions if one uses variationally determined products of linear wave functions to describe the electrons. An approximate theoretical analysis of the Bell experiment leading to good agreement has been made by Tejada et aL8' The agreement is also good when a more elaborate theoryss is made, but perhaps other effects, besides dynamic screening, such as the presence of the static lattice potential, have to be taken into account.

21. ALIGNMENT-RING PATTERNS An ion trailing another at fixed separation and at the same velocity will experience a force tending to move it toward the track of the first if it is in the region -nv/wP< Z < m / w , and within a radial distance 5 u / w pof the track. This force arises because there tends to be an excess of electrons immediately behind a moving ion. The reality of this aligning force was first demonstrated by Gemmell, Remillieux and coworkers5657 in the angular distribution of protons emerging from Au single crystal foils bombarded by (HeH)+ ions in planar channeling directions. Observations of the explosion process have been made in a series of high-resolution studies of the distribution in energy and angle of the fragments produced when a variety of fast molecular ions bombard thin polycrystalline foils.57 When H: ions are used, e.g., a bimodal distribution in energy is found at a direction nearly parallel to that of the incident "J. Tejada, P. M. Echenique, 0. H. Crawford, and R . H. Ritchie, Nucl. Instrum. Methods 170, 249 (1980). KxJ. Ugalde, C. Sarasola, P. M. Echenique, and R. H . Ritchie, J . Phys. B 21, L415 (1988).

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

273

ion. An observed asymmetry corresponding to larger numbers of trailing ions than leading ones is consistent with an aligning wake. Valuable, and in some cases unique, steric information about polyatomic ions has been gained in experiments of this kind.89

22. RESONANT COHERENT EXCITATION Okorokovgo suggested that channeled He+ ions may become excited due to periodic perturbations by atoms in the bounding rows of the crystal. He postulated that photons from subsequent radiative deexcitation might be detected when the speed u of the ion satisfies the equation u = dAE/2nn, where d is the lattice spacing, A E is the He+ excitation energy and n = 1,2. . . . Repeated efforts to observe such photons were unsuccessful. This was probably due to the short lifetime of the excited state, which is expected to have a radius comparable with the lattic spacing. More recently, experiments” have been carried out with swift channeled ions having atomic numbers ranging from 5 to 9 and carrying a single K-shell electron. Resonant coherent excitation in this work is signaled by the existence of pronounced dips in the curves of survival probability of the one-electron ion as it depends upon ion speed. Theory9* shows that the wake of the ion has an important effect on the state of the bound electron. This is shown experimentally in a pronounced splitting in the curves of surviving fraction vs ion speed.

23. WAKE-FIELD ACCELERATORS It is interesting that, among the several acceleration schemes using intense electron beams that have been proposed in recent years, one of 89D. S. Gemmell, Chem. Rev. 80, 301 (1980); J. Plesser, Z. Vager, and R. Naaman, Phys. Rev. Lett. 56, 1559 (1986); Z. Vager, E. P. Kanter, G. Both, P. J. Cooney, A. Faibis, W. Koening, B. J. Zabransky, and D. Zajfman, Phys. Rev. Lett. 57, 2793 (1986). wV. V. Okorokov, D. L. Tolchenkev, I. S. Khizhnyakov, Yu. N. Cheblukov, Y. Y. Lapitski, G. F. Iferov, and Yu. N. Zhukova, Phys. Lett. A 43, 485 (1973). 91S. Datz, C. D. Moak, 0. H. Crawford, 0. H. Krause, P. F. Dittner, J. G. del Campo, J . G. Biggerstaff, J. A. Miller, P. D. Hvelplund, P. Hvelplund, and H. Knudsen, Phys. Rev. Lett. 40, 843 (1978); C. D. Moak, S. Datz, 0. H. Crawford, H. F. Krause, D. F. Dittner, Gomez del Campo, J. A. Biggerstaff, P. D. Hvelplund, and H. Knudsen, Phys. Rev. A 19, 977 (1979); S. Datz, C. D. Moak, 0. H. Crawford, H. F. Krause, P. D. Miller, P. F. Dittmer, J. G. del Campo, J. A. Biggerstaff, H. Knudsen, and P. Hvelplund, Nucl. Instrum. Methods 170, 15 (1980). =O. H. Crawford and R. H. Ritchie, Phys. Rev. A 20, 1848 (1979); 0. H. Crawford, Nucl. Instrum. Methods 170, 21 (1980).

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the most ingenious seems to be the wake-field a ~ c e l e r a t o r .In ~ ~this concept, one bunch of relativistic electrons is used to accelerate another to higher energy through the plasma wake set up by the leading bunch. This was apparently first suggested93 in a paper employing the electrostatic approximation but was later generalized in a fully relativistic, electromagnetic, numerical s t ~ d y . ’ ~It, ~appears ~ that no experimental tests of this concept have been made to date.

24. THESURFACE VICINAGE EFFECT. It has been suggested” that what we would term here the “surface vicinage effect” may be useful in the operation of induction linear accelerators for intense, high-energy electron beams. The conventional method of preventing space charge forces from disrupting such beams is to use solenoidal focusing, or to use gas cells between accelerating gaps. Cost effectiveness questions arise in connection with the former, while possible instabilities and vacuum problems may plague the latter. Theory shows that the use of conducting foils96 can provide a net focusing force in the drift tube and a net inward impulse to all particles in the electron beam. Experimental data96have been adduced in support of this concept.

25. THESURFACE WAKE Echenique6’ has studied the evolution of the wake potential as an ion emerges from a solid. It was assumed that the solid is semi-infinite, homogeneous and plane-bounded and that the ion exits the solid with constant velocity and perpendicular to the surface. He found, using a local dielectric function, that the interplay between surface plasmon excitation and bulk plasmon excitation results in flattening of the first electron-binding trough as the particle crosses the surface. It is possible to find an analytical representation for the wake potential for the case in which the charged particle moves with constant velocity 93H.Dehne, A. Febel, M. Lenke, H. Munsfeldt, J. Rossbach, F. Rossmainth, G.-A. Voss Tiweiland, and F. Willeke, “A Wake-Field Transformer Experiment,” Proc. 12th Intl. Conf. on High-Energy Accelerators, Fermilab, p. 454 (1983). Ed. F. T. Cole and R. Donaldson, Published by Universities Research Association. wP. Chen, J. M. Dawson, R. W. Huff, and T. Katsouleas, Phys. Rev. Left. 54,643 (1985). 95P.Chen, J.‘J. Su, J. M. Dawson, K. L. F. Bane, and P. B . Wilson, Phys. Rev. Lett. 56, 1252 (1986). %R. J. Adler, Particle Accelerators 12, 39 (1982); P. J. Adler, IEEE Trans. Nucl. Sci. NS-30, 3198 (1983); R. J. Adler and R. E. Miller, J . Appl. Phys. 53, 6015 (1982).

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

275

parallel to the surface and outside the ~ o l i d . ’Some ~ work along these lines has been done recently by Gumbs and coworkers.’* Surface wake effects turn out to be important even when the fast particle is moving at a distance from the surface of the order of the ratio of the fast-particle velocity to the plasma freq~ency.~’ Ohtsuki and coworkersw have analyzed the vicinage effect that takes place when two ions travel at constant separation and with constant velocity parallel to a surface that supports surface collective modes. V. Ions Moving with Velocity Less Than v,

26. LINEARRESPONSE THEORY When the velocity of the particle is small, v << v o , Eq. (16.2) can be used, together with an appropriate dielectric response function to evaluate the stopping power. A simple expression can be derived by using the small w expansion of the RPA response function, as was shown by Ritchie.IooHe obtained for the stopping power dW _ dR

3JG

(26.1)

where cy = ( 4 / 9 ~ ~ ) In ” ~the . limit of very high densities (r, << 1) Eq. (26.1) agrees with the pioneering results of Fermi and Teller.’” The approximate form for c(q, w ) used by Ritchie to derive Eq. (26.1) is equivalent to assuming that the potential around the ion is exponentially screened and therefore neglects the long oscillatory tail of the potential corresponding to the Friedel oscillations. The results of a calculation of v - ’ d W / d R for a proton moving in an electron gas of density parameter r, are shown in Fig. 11 using the full RPA response function, and using the exponential screening one. The Fermi-Teller result is also shown. A good approximation to the values obtained numerically by using the exact RPA dielectric response function for an electron gas in Eq. (16.2) has been y7Yu. A. Romanov and V. Ya Aleshkin, Sou. Phys. Solid Stare 23, 147 (1981); B. N. Libenson and V. V. Rumyantsev, Sou. Phys. JETP 59, 999 (1984). ’*M. L. Glasser and G. Gumbs, Phys. R e v . B 35, 7490 (1987); N. J. M. Horing, H. C. Tso and G. Gumbs, Phys. Rev. B 36, 1588 (1987). 99K.Suzuki, M. Kitagawa and Y. H. Otsuki, Phys. Srarus Solidi B 82, 643 (1977). lWR. H. Ritchie, Phys. R e v . 114, 644 (1959).

276

P. M. ECHENIQUE et al.

-

0.

I

I

I

I

I 1 1 1 1

I

I

I I I I '

I

I

I

THEORY A

I

0.c

I

I

I11111

r,-ONE-ELECTRON

I I I I I 1

RADIUS IN ELECTRON GAS

FIG. 11. Comparison of calculations of stopping power of an electron gas for a slow proton in the form l / v ( d W / d R ) versus the one-electron radius r,.

proposed by Lindhard and Winther"' (LW)

where x2 = 0 . 1 6 6 ~ ~ . Ferrell and Ritchielo2 have calculated the stopping power of an electron gas for slow, singly ionized He atoms, in linear response theory, ~-~~ the , paramusing a hydrogenic wave function u,,(r) = ( u ~ / J T ) " ~ with eter a determined variationally by minimizing the total energy of the ion plus a bound electron in the electron gas. The stopping power is given by Eq. (16.2) but with the substitution (26.3) inside the integrand. J. Lindhard and A. Winther, K . Dan. Vidensk. Selsk. Mat. Fys. Medd. 34, No. 4 (1964).

101

"9. L. Ferrell and R. H. Ritchie, Phys. Rev. B 16, 115 (1977).

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

277

27. PHASESHIITANALYSIS An improvement over the linear-theory result can be achieved using Feynman diagram methods. Including the effect of the Pauli principle by restricting electron states to those outside the occupied Fermi sphere only in the last transition, one can express the stopping power in terms of the transport cross section. 1ns1n5 One finds (27.1) Here n is the electron density and a, is the transport cross section for electrons at the Fermi level, which in turn can be written in terms of the scattering cross section, a(@),defined in terms of the phase shifts, 6,(EF),describing scattering by a spherically symmetric potential. Then

dW

I-, +n

dR = n v ~ v 2 ~ c

d o a ( 0 )sin(e)(l- cos e).

(27.2)

Mott and Joneslw have employed this model in the Born approximation to estimate the contribution of impurity scattering to the resistivity of metals. A similar analysis was used by Josephson and Lekner"" to study the scattering of ions by 3He quasiparticles. One finds

Analogously from Eq. (27.2) one can also express the straggling parameter W, in terms of the scattering cross section at the Fermi level. One find^"^-"'^

Finneman, Ph.d. dissertation, The Institute of Physics, Aarhus University, 1968 (unpublished). IMN. F. Mott and H. Jones, "The Theory and Properties of Metals and Alloys." Dover, New York, 1958. 'O'P. Sigmund (unpublished work). IMB. D. Josephson and J . Lekner, Phys. Rev. Lett. 23, 111 (1969). Imp. Sigmund, Phys. Rev. A 26, 2497 (1982). 1°*J. C. Ashley, A. Gras-Marti, and P. M. Echenique, Phys. Rev. A .M, 2495 (1986).

Io3J.

278

P. M. ECHENIQUE et al.

The result in terms of the phase shifts at the Fermi level is"' 3nv2 w,= c 4V3

c (21 + 1)(2m + 1)

I=Om=O

x { 1- C O S ( ~-~co~(26,) /) + C O S [ ~-( ~6 m, ) ] } J , m , (27.5)

where

and the symbol ( u ) ~is defined in terms of gamma functions by ( u )= ~ r(a k ) / T ( a ) . Because of the complete screening of the ionic charge by the medium electrons, the phase shifts satisfy the Friedel sum

+

ru~e109.11"

z 1

L =-

c (21+ 1)6,(&).

n /

(27.7)

Ferrell and Ritchie"* used Eq. (27.3) to calculate the stopping power, assuming a linear-response Yukawa type potential V ( r )= Z l e - A r / rfor the ion-interaction potential. The value of h was determined by making the phase shifts satisfy the sum rule of Eq. (27.7).

FUNCTIONAL APPROACH 28. DENSITY It is well appreciated that linear response theory, for low velocities, is suspect at real metallic densities, since even though self-consistent, it is essentially based on the first Born approximation. At real metallic densities, calculations using the density functional method to solve approximately the many-body problem show great differences with the results of linear theory. 111-113 At real metallic densities, the ground state of a static proton has two bound electrons, a feature which cannot be included consistently in any linear theory. The density fluctuations and lWC. Kittel, "Quantum Theory of Solids" Wiley, New York, 1963. "oJ. Friedel, Nuouo Cimenfo Suppl. 7 , 287 (1958). '"C. 0.Almbladh, U. von Barth, Z . D. Popovic, and M. J. Stott, Phys. Rev. B 14, 2250 (1976). 'l2P.

Jena and K. S. Singwi, Phys. Rev. B 17, 3518 (1978). Puska and R. M. Nieminen, Phys. Rev. B 27, 6121 (1983).

l L 3 M .J .

DYNAMIC SCREENING OF IONS IN CONDENSED MA'ITER

279

induced potentials induced by a static proton in an electron gas were first evaluated by Almbladh er al. 'I' The ratio of the induced electron density calculated in the nonlinear theory to that obtained in linear theory using the RPA varies over the range of metallic densities from 1.93 for ry = 1 to 33.7 for r, = 6. This last number is within 10% of the ones which will be obtained from the 1s atomic orbital, showing the atomic nature of the bound state in a dilute electron gas. The results show that linear theory grossly underestimates the charge pile-up, particularly for lower mean densities. The nonlinear results for the induced Hartree potential at the proton position, V,(r = 0) and the screening length, are practically constant over the metallic range, V,(O) 1.2 and h = 0.65. In contrast the corresponding quantities from linear theory vary from 0.8 to 0.5 and from 0.9 to 1.5, respectively, when r, varies from 2.0 to 6.0. Echenique, Nieminen and Ritchie'I4 (ENR) have evaluated the self-consistent potential around a proton and an alpha particle (He) using the density functional formalism of Hohenberg and Kohn"' and Kohn and Sham1I6 with the ansatz for exchange and correlation suggested by Gunnarsson and Lundqvist. 'I7 The phase shifts at the Fermi level calculated from this self-consistent potential satisfy the Friedel sum rule to a 2% accuracy.

-

29. PROTONS AND HE NUCLEI IN A N ELECTRON GAS In Fig. 12 we show, for protons and alpha particles, the comparison between the results of the nonlinear theory of stopping, first obtained by ENR,'I4 with the ones obtained by linear response theory (full RPA response function) and with the ones obtained by the method of Ferrell and Ritchie.'"' As r, decreases toward values much less than 1 our results tend toward agreement with linear theory, i.e., proportional to 2:. This is easily visualized when one considers that for large electron-gas densities the screening of the ion is so strong that bound states cannot exist. The electrons are scattered by a screened potential with screening length approaching zero as r, goes to zero. As r, increases, the energy loss for both H and He decreases more rapidly than predicted by linear theory due to the fact that bound states of atomic character develop, thereby tending to screen out interactions with the electron gas. The 'I4P. M. Echenique, R. M. Nieminen, and R . H . Ritchie, Solid Stare Commun. 37, 779 (1981). '"P. Hohenberg and W. Kohn, Phys. Rev. B 864, 136 (1964). '16W. Kohn and L. J. Sham, Phys. Rev. A 1133, 140 (1965). '"0. Gunnarsson and B. I. Lundqvist, Phys. Rev. B W, 4272 (1976).

280

P. M. ECHENIQUE et al.

r

S

d

FIG. 12. Stopping powers as functions of r s . Curve A is calculated in linear response theory, Eq. (26.1), for Z, = 1; curve B from Eq. (26.1) with Z, = 2. Curve C is the result Ferrell and Ritchie from (Ref. 102) for a slow, singly ionized H e atom. Curves D and E are the density functional results for a proton and a helium nucleus, respectively. In all cases v

<< V F .

energy loss of a He nucleus at large r, is smaller than that of a proton at the same velocity. This is qualitatively different from any linear theory in which the energy loss scales as the square of the ionic charge and can be understood in terms of the atomic character of the scattering process in a very dilute electron gas. The Fermi energy is also very small so the

281

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

I

2

r

3

4

5

6

7

FIG. 13. Comparison of the stopping power for a proton calculated using the densityfunctional approach to the Lindhard-Winther predictions of Eq. (26.2).

scattering is essentially that of a very low energy electron from a H and a He atom. In Fig. 13 comparison between the ENR results and those obtained with the LW formula is shown. For many solids used in experiments the density functional results show increases of 65% over those calculated from the LW formula. A comparison of experimental data for protons with the density functional predictions of ENR was made by Mann and Brandt.”’ They collected data on targets covering a wide range of atomic numbers and plotted reduced stopping powers

dWldR as a function of v/vF. The values of r, were taken from Isaacson’s table^."^ The generic data shown in Fig. 14, described in detail in the paper of Mann and Brandt, represent values for 20 different elemental solids, in the range 4 < Z2 < 83. Mann and Brandt conclude from these comparisons that within the uncertainties of the data, (a) the density “‘A. Mann and W. Brandt, Phys. Rev. B 24,4999 (1981). ‘19D. Isaacson, “Compilation of ‘5 Values,” Internal Report, Radiation and Solid State

Laboratory, New York University, (1975).

P. M. ECHENIQUE et al.

282

2.o

1.8

16

1.4

v

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1.0

1.2

V/VF

FIG. 14. Comparisons of theoretical stopping powers for protons with experimental data. The curve LW calculated from Eq. (26.2); FR is the Ferrell-Ritchie prediction from Ref. 102; ENR is the density functional result from Ref. 114. The short-dashed lines give a f15% variation about the line ENR. This figure was first drawn by Mann and Brandt. Reference 118 gives the identity of, and sources for, the generic data shown as solid circles in the figure.

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

283

functional predictions give good agreement with the data, and (b) the linear dependence on velocity holds up to v = vF. 30. EFFECTIVE CHARGE FOR IONS WITH Z, > 2 Brandt and c o ~ o r k e r s were ~ ~ ~able ~ ' to~ condense ~ a great amount of data by introducing the concept of an effective ionic charge. In the nonlinear framework bound states appear in a natural way. In fact the total charge is sometimes zero because all bound states are occupied. Thus Echenique et af.lZ4 (ENAR) define the effective charge in an operational manner as

zy=

(d: d R w I -(ZJ

-(Z1=1))

1I2

,

(30.1)

where d W / d R is the stopping-power calculated from Eq. (27.3). In Fig. 15, the effective charge calculated from Eq. (30.1) is shown as a function gf the bare charge of the incident ion for three electron gas d e n s i t i e ~ . 'Fluctuations ~~ in Z y as a function of Z , occur naturally, since they are related to the appearance of resonances (maxima) or closed shells (minima). For a dilute electron gas the screening cloud approaches the free-atom structure, and the minima appear at the formation of closed atomic shells. As the screening increases due to increasing electronic density this minimum shifts to higher ionic charges since stronger ionic potential is necessary to compensate the electronic screening and so have the strength to bind an extra electron. This is clearly shown in the graphs as displacement of the minimum from the atomic value ( Z , = 10) as the electron density increases (rs decreases). For very small r, no bound states are formed and linear response theory should be valid, then Z: approaches Z , . In Fig. 16 we compare the effective charge, calculated with the model of Brandt and c o ~ o r k e r s ~ ~ ' ~ ~ ~ with the result of the nonlinear theory.12' As expected from the "9. Schulz and W. Brandt, Phys. Rev. B 26, 4864 (1982).

"'W. Brandt, Nucf. Instrum. Methods 194, 13 (1982); see also B. S. Yarlagadda, J . E. Robinson and W. Brandt, Phys. Rev. B 17, 3473 (1978). 122 S. Kruessler, C. Varelas and W. Brandt, Phys. Rev. B 23, 821 (1981). Iz3W. Brandt and M. Kitagawa, Phys. Rev. B 25, 5631 (1982); see also F. Schultz and W. Brandt, Phys. Rev. B 26, 4864 (1982). Iz4P. M. Echenique, R. M. Nieminen, J. C. Ashley, and R. H. Ritchie, Phys. Rev. A 33, 897 (1986). 125N.Barberan and P. M. Echenique, J . Phys. B 19, L81 (1986).

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P. M. ECHENIQUE et al.

3.0

,’ z;=z,

I

I

2.5

2 .o

Z:

1.5

1 .o

0.5

0

0

2

4

6

8

10

12

14

16

18

FIG. 15. Effective charge Z ; , defined by Eq. (30.1), as a function of Z , for r,= 1.5, 2 and 4.

preceding discussions, the theory of Brandt and Kitagawa, essentially a linear theory, does not show any oscillations as a function of Z I . However, it gives a reasonable average description of Z : as a function of Z1 for different r, values. The most reasonable comparison of experimental data and theory is obtained from data taken for best-channeled ions. In Fig. 17 we compare the results of the calculation of Ashley et ~ 1 . “ ~ with Eisen’s data’” on the energy loss of “best-channeled’’ ions in the (110) axial channel in Si. The electron density in a channel increases C. Ashley, R. H. Ritchie, P. M. Echenique, and R. M. Nieminen, Nucl Instrum. Methods B 15, 11 (1986). IZ7F.M. Eisen, Can. 1. Phys. 46,561 (1968).

Iz6J.

285

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

0

2

6

4

€3

10

12

44

16

18

20

FIG.16. Effective charge Z ; plotted against the atomic number of the ion Z , , for rs = 2. BK, theory of Brandt and Kitigawa (Ref. 123); NL, nonlinear density functional result of Barberan and Echenique (Ref. 125).

from a small value along the axis to values an order of magnitude larger near the strings of atoms defining the channel. Since a range of electron densities (or impact parameters) is sampled in the energy-loss process, Ashley and coworkers’26 made their comparison with the data using an average electron gas density determined by making theoretical and

0

-FIG. I /.

.

0

2

4

,,.

6

€3

10 2,

.

. ... .

12

.

14

.

18

16

.

..

20

.

”

..

btopping power tor ”best-channeled” ions in the \ 1 IU) axial channel 01 silicon as a function of Z,. The chained curve gives the theoretical predictions for rs=2.38. The experimental points are from Ref. 127. ”

,*A,.\

286

P. M. ECHENIQUE et al.

100 rs = 2 0'

80

?

m

'*

0

0

i

i

60

i

C

.N

>

2

/

40 0'

/

20

0

.o

\

0

I

0' I

0

0 ' 0

i

f

I

I

i

0

0

I

I

1

10

20

30

40

FIG. 18. Predicted variation of the straggling parameter W / u 2 with 2 , from density functional calculations. These results illustrate the significant Z , oscillations in straggling for r, = 2.

experimental values equal at Z1= 5. The overall trend in the data is closely reproduced by the theoretical calculations. 3 1.

STRAGGLING

The energy loss per unit path length is an average quantity. The characterization of the distribution of energy losses suffered by energetic charged particles in their interaction with matter requires also the mean square deviation as discussed in Section 16. Ashley, Gras-Marti and Echenique"' and Arnau et al. 12' have calculated the energy-loss straggling of slow ions moving in an electron gas in terms of the phase shifts determined from nonlinear, density functional calculations. The result for the straggling parameter, [Eq. (16.4)] for r, = 2 is shown in Fig. 18 in the form W / u 2as a function of Z , . It shows strong Z1 fluctuations analogous to the ones appearing in the stopping power. Using the results of the '"A. Amau, P. M. Echenique ad R. H. Ritchie, Nucl. Instrum Methods B 33, 138 (1988).

DYNAMIC SCREENING OF IONS IN CONDENSED MAlTER

287

nonlinear calculations for stopping powers, Ashley et al. find W/(vZT)z = 10.5 to within -15% for 1< ZI < 40 at r, = 2. That is, most of the fluctuations in straggling shown in Fig. 18 are due to variations in ion effective charge. VI. Charge States of Ions in a Solid

32. INTRODUCTION In previous sections, we have discussed the energy loss of ions moving at low and at high velocities in condensed matter. For high velocities, v >> Zl v o , the ion is stripped of its electronic charge and the interaction with the medium is described adequately by linear theory. Sections 111 and IV have dealt with this regime where other effects such as dynamic screening, vicinage, stopping and straggling are covered. When v << u o , the ion is dressed with quasi particles which may strongly perturb electron states associated with the moving ion. In Section V we showed how the local density functional method can be used to calculate the charge states and stopping powers for different ions; in particular, it was shown how the reaction of the medium to an ion for v < vo is similar to that for a static impurity. The condition v = ZT3vomay be taken to define a regime of intermediate velocities. Zy3vois the mean velocity of the electrons filling the levels of a neutral atom with nuclear charge Z1; this result is obtained using Thomas-Fermi statistical theory.129-'30Brandt's mode18.120-'2' for the effective charge of a moving ion involves velocities of the same order of magnitude. Note that the velocity Zlv, refers to the mean velocity of the electrons most strongly bound to the ion, while Z:'3vo is the statistical mean velocity of all the electrons of the neutral atom. We take Z l v o to define the upper limit of the intermediate velocity regime, which extends from u, to Z1vo , with a mean value given by Z:/'v,, . In this section, we shall concentrate on discussing the different effects controlling the charge states of ions moving in solids. Specific calculations for light ions will be presented, showing the behaviour of different cross sections in the limit of intermediate and high velocities. The cases of H and He projectiles will be fully discussed, while only the most tightly bound electronic states of B, C, N and 0 will be considered. Thus for B, C, N and 0, we will show results corresponding to only the upper limit of the intermediate regime of velocities. Iz9L. Thomas,

Proc. Cambridge Philos. SOC. 23, 542 (1927).

I3'E. Fermi, Z . Phys. 48, 73 (1928).

288

P. M. ECHENIQUE et al.

33. ELECTRONIC EXCHANGE PROCESSES Several different mechanisms have been proposed as being responsible for electronic exchange processes of ions moving in condensed matter. The first mechanism plays an important role at low velocities. In the Auger process41 illustrated in Fig. 19a, an electron is captured (or lost) by the ion to (or from) a bound state assisted by a third body that may be a plasmon or an electron-hole pair. Condensed matter effects are important here since electrons in valence band states are involved. Other Auger processes associated with interactions between localized states of the target and bound states of the ion could contribute to the electronic exchange mechanism. These do not seem to play an important role here, however. Their corresponding cross sections are very small compared with those involving valence band electrons in the region of velocities where Auger processes are significant. A second mechanism leading to capture and loss is the coherent CAPTURE

LOSS

G V

FIG. 19. (a) Shows an Auger process for an ion moving in a uniform electron gas: (i) a conduction electron jumps to a bound state on the ion, (ii) an electron bound to the ion is excited to the conduction band. (b) Shows the coherent resonant process for an ion moving in a crystal: (i) in a capture process, the crystal pseudopotential induces transitions between a conduction and a bound state; (ii) in a loss process, an electron bound to the ion is excited to a level of the conduction band. (c) Illustrates the shell process whereby an ion moving in crystal captures one electron from an inner level of the target.

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

289

resonant interaction. 131-133 Electronic exchange processes are induced by the time-dependent crystal potential as seen from the fixed ion (Fig. 19b). The crystal atoms may be assumed to remain in their ground states in these interactions. For high ion velocities, the effective potential is mostly due to the core electrons; for low velocities, the interaction is mainly determined by the valence electrons. The coherent resonant interaction resembles a close atom-atom collision at high velocities. At low velocities condensed matter effects come into play. The third mechanism we consider is the shell process (Fig. 19c). In this case, an inner electron of a target atom can be captured by the moving ion.I34l36 Small impact parameters collisions dominate in such interactions, thus the condensed matter properties of the medium are not very important here. 34. AUGER PROCESSES

We begin by considering the Auger processes41*137-L38 associated with the interaction between the conduction band and the ion-bound states. In Section 11, we analyzed the probabilities of electronic capture and loss processes and obtained the following results

16n2 x -6 ( 0 T k 2 / 2- q ‘ V f COO) q2 x Im[-c-’(q,

o)]l(kl e*jqPluo(p))12.

(34.1)

In this equation, we have introduced the OPW wave function, Ik) = - (uo(p)l eik.p)luo(p)), instead of the plane wave lei”“) as discussed at the end of Chapter 11. This change is related to interaction 131T. Kaneko

and Y. H . Otsuki, Phys. Stat. Solidi B 14, 491 (1982). I3*F. Sols and F. Flores, Phys. Rev. B 30, 4878 (1984). 133J. D . Jackson and P. M. Platzman, Phys. Rev. B 22, 88 (1979). 134M.C. Cross in “Inelastic Ion-Surface Collisions” (N. H. Tolk, J . C. Tully, N. Hieland and C. W. White, eds.). Academic, New York, 1977; see also M. E. Cross, Phys. Rev. B 15, 602 (1977). I3’V. P. Shevelko, 2.Phys. A 287, 19 (1978). 136T.. Kaneko, Nucl. Instrum. Methods B 2, 491 (1984). 13’F. Guinea, Ph.D. thesis, Universidad Aut6noma de Madrid, 1980 (unpublished). I3*F. Sols, Ph.D. thesis, Universidad AutBnoma de Madrid, 1985 (unpublished).

290

P. M. ECHENIQUE

er al.

between the ion and electrons in the conduction band, which may amount to a strongly non-linear effect. As described there, the ion may bind electrons and the conduction-band wave functions may be strongly affected. The local density formalism provides an appropriate way of calculating the effect of the electron gas on the ion bound states for v << uo. At low and intermediate velocities, Guinea ef a1.41.139 have used a variational method to calculate the wave functions and the binding energy, w o , of the first and second electrons bound to H or He. The bound state wave function is assumed to be of the form (34.2) Tables I and I1 give values of a and wo as calculated by Guinea et aL41 as a function of the ion velocity for an electron gas with density parameter r, = 2. First of all, notice that for two electrons bound to H, the electron binding energy for v = 0 is about 1eV, not far from the results obtained within the local density formalism. On the other hand, in the limit of very high velocities, we expect to recover the wave function and the binding energies of the free atom, since in that limit the effect of the electron gas on the moving atom is negligible. For H and He these free Hartree-Fock atom values are the following:

H First electron: Second electron: He First electron: Second electron:

a = 1a.u., a = 0.687 a.u.,

wo = -0.5 a.u.

a = 2 a.u., a = 1.687 a.u.,

oo= -2 a.u. oo= -0.986a.u.

oo= 0.021 a.u.

Comparing the free atom values with those of Table I, we find that the maximum effect of the electron gas on the bound state of H appears for v = 1. In the case of He, maximum effects seem to appear, however, for the static limit v = 0. On the other hand, notice that the electron gas affects the binding energy more strongly than the wave function of the state; for H, for instance, at v = 2, the parameter a defining the bound state of a single electron is close to 1, while its binding energy is -0.184, much less than the limit value of -0.5. This shows that, for high velocities, the atomic wave functions are not too much deformed by the Is%.

Guinea and F. Flores, J . Phys. C U,4137 (1980).

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

291

TABLEI. BINDING ENERGIES (IN a . U . ) AND VALUES OF THE PARAMETER a DEFINING THE BOUND WAVE FUNCTIONFOR H AS A FUNCTION OF THE VELOCITY, v, FOR THE FIRSTAND SECOND ELECTRONS. BINDING ENERGIES ARE REFERRED TO THE BOTTOM OF THE CONDUCTION BAND.(TAKEN FROM REF.41.)

H

v

FIRSTELECTRON

SECOND ELECTRON a = 0.76

0

a = 0.90 wn = -0.106

= -0.046

0.2

a = 0.88 on=-0.096

w 0 = -0.036

0.4

a = 0.86 wn = -0.075

a = 0.76 wn = -0.012

0.6

a = 0.80

wn = -0.048

a

a

= 0.72

= 0.54

wo = -0.004 = 0.52

a = 0.72 = -0.027

wo = -0.009

a = 0.74 an= -0.027

6J(,= 0.010

1.2

a = 0.82 on= -0.041

a = 0.70 0,=0.011

1.4

a = 0.86 w0 = -0.063

W g = 0.01 1

0.8 1.0

1.6

a = 0.90

a

a = 0.58

a = .I4

a = 0.74

on= -0.096

w, = 0.006

1.8

a = 0.92 w g= -0.125

a = 0.72 w,=0.005

2.0

a = 0.94 wo = -0.184

a = 0.72 wo = 0.001

electron gas, and we can calculate the binding energy shifts using second order perturbation theory. In particular, notice that the reduction in the binding energy can be easily understood as a screening effect of the conduction band on the ion-electron interaction. This point of view has been used by Sols and Flore~'~'to calculate the shifts in the binding energies of the most tightly bound states of heavier ions such as B, C, N and 0 moving in an electron gas with velocities around Z , uo.

'9 Sols .and F. Flores, Nucl. Instrum. Methods U,171 (1986)

292

P. M. ECHENIQUE et al. TABLE11. As TABLEI FOR He. He

v

0

FIRSTELECIXON a = 1.88 0" =

0.5

1.0

a = 1.88 o,,= -0.874

2.0 2.5 3.0

a = 1.60

w O= -0.509 a = 1.60

o"= -0.450

a = 1.72

a = 1.90 =

1.5

-0.914

SECOND ELECTRON

-0.897

O"

a = 1.67

a = 1.93

w,,= -1.098

a = 1.96 w 0 = -1.252

= -0.439 = -0.540 a = 1.70

COO=

a = 1.97

-0.621

a = 1.70

w"= -1.327

O"

= -0.658

a = 1.98 w 0 = -1.456

COO=

a = 1.70 -0.682

The probabilities of capture and loss [Eq. (34.1)] have been calculated for H and He using the wave functions and binding energies of Tables I and 11. For heavier atoms and deep levels, u&) has been taken as appropriate to the free atom, while oo has been calculated using perturbation theory. 141~142

35. COHERENT RESONANT PROCESSES Auger processes are the only ones appearing in a uniform electron gas. New effects come from the crystal structure. Fig. 19b illustrates the coherent resonant process we now discuss. The resonant processes are From the point of view due to the potential seen by the moving of the ion, there appears a moving potential which gives rise to transitions between bound states of the composite and free electron states. An equivalent point of view is the one taken by Kaneko and O h t s ~ k i . ' ~These ' authors analyzed the dynamical processes created by the electrostatic potential of a single crystal atom acting on the moving 14'Y. H. Ohtsuki, "Charge Beam Interaction with Solids." Taylor, London, 1983. 14'M. A. Kurnakhov and F. F. Kornarov, "Energy Loss and Ion Ranges in Solids." Gordon and Breach, 1981.

293

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

ion; this approach neglects, however, the interference effects associated with the periodicity of the crystal. Coherent resonant processes are best understood by assuming the ion to be at rest;13' then, the effective pseudopotential seen by the electron of the composite can be written

V(r, t ) =

c V(G)eiG'('-"'), G

(35.1)

G being a reciprocal vector of the lattice. The potential (35.1) may be considered to induce excitations between a bound state of the composite described by the orbital, uo(r), and a valence electron state described by an OPW wave function Iki). Notice that in the rest frame of the ion, the momentum kf of the valence electron is given by k' - v, k' being the momentum in the laboratory frame. Using standard perturbation theory, it is straightforward to calculate the probability amplitude, a,,f, for a loss process, u,(p)-+ Ikf):

The probability per unit time for the resonant loss process is given by (35.3) Summing over all the empty final states, kf, we obtain the total probability, P k , associated with the coherent resonant process:

In this equation, we have interference effects associated with those reciprocal vectors satisfying G v = G' v; for a direction such that this condition is never satisfied (irrational angles) we are left with the

-

-

294

P. M. ECHENIQUE er al.

following result:

X

6(O, - 4k:

+G

*

v).

(35.5)

The same result is obtained from Eq. (35.4) if P k is averaged over all the possible directions of the moving ion. Eq. (35.4) includes effects associated with ion channeling that are reflected in the interference between different reciprocal vectors. If the ion charge states are not analyzed for specific directions of the fixed ion, Eq. (35.5) seems to be more appropriate for discussing coherent resonant processes. A similar argument yields the following result for the probability per unit time for a capture process due to resonant effects:

X

6 ( ~ ,-, ik? - G * V)

(35.6)

if we neglect interference effects. It is of interest to compare Eqs. (35.5), (35.6) and (34.1). In Eqs. (35.1) and (35.6), IV(G)l' appears instead of the square of the Fourier transform of the Coulomb potential, 4n/q', the dynamical factor

is replaced by 1, and inside the 6 factor, w is taken to be zero. Differences between the two results are easy to understand. The dynamical factor disappears in the resonant process due to the absence of any internal change in the structure of the lattice ions creating the moving potential. This also explains why w must be taken to be zero inside the 6 factor. Finally, 4n/q2 must obviously be replaced by V(G). Note that in Auger processes the momentum transfer may assume any value, while in resonant processes only reciprocal momenta are involved, G. In the approximation used here this is the only effect of crystal structure. If instead of CG we employ J d3q/(2n)3, we recover the results associated with the interaction with independent target atoms. The difference is only important when small momentum transfers are impor-

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

295

tant. This occurs at low ion velocities. At high velocities the incoming ion in effect suffers collisions with only a single target atom at a time. A final point to consider about the resonant results of Eqs. (35.5) and (35.6) is the actual value of V ( G ) to be used in those equations. For t~ Iu0 the crystal atoms present an effective potential to the electrons of the moving ion that is similar to the one seen by electrons of low energy in a crystal. Pseudopotentials abundantly used in band structure theory can then be introduced. For much higher energies, electrons bound to the ion can sense the core of the crystal atoms, and a stronger potential must be used. Kaneko and 0htsukil3' and Sols and Flores13*, have proposed that in the high velocity limit, the electrostatic (Hartree) potential of the crystal be used. The two different limits may then be represented adequately for resonant processes. Intermediate velocities are more difficult to analyze. Echenique and F l o r e ~ ' have ~ ~ suggested that for this regime of velocities to calculate V ( G ) one may use the theoretical techniques developed in LEED.'44 Knowing q r ( E ) the phase-shifts of electrons in the crystalline potential at a given energy one can estimate V ( G ) using the well-known expansion of eiC" in spherical harmonics. In practice, at high energies this method yields the LEED cross-sections associated with the scattering of an electron by an atom of the crystal, while at low energies, the cross sections found by using a crystal pseudopotential are recovered.

36. SHELLPROCESSES Figure (19c) shows the interaction in which the moving ion captures an '~~ electron from the deep levels of the target. Cross'34 and S h e ~ e l k ohave analyzed this process using an Oppenheimer-Brinkman-Kramers (OBK) approach. It is well-known'45 that this method yields capture cross sections that are too large, but gives, at high enough velocities, a good description of the cross-section behaviour as a function of the ion velocity. The OBK approach is based on the Born approximation, and different results may be obtained depending on the approximation used for the wave functions and the energy levels involved in the process. Assuming '43P.M. Echenique and F. Flores, Phys. Rev. B 35, 8249 (1987).

IuJ. B. Pendry, "Low Energy Electron Diffraction." Academic, London (1974); M. A.

Van Hove and S. Y. Tong, "Surface Crystallography by LEED." Springer, 1979. 145H.C. Brinkman and H. A . Kramers, K . Wel. Amsterdam 33, 973 (1930); see also p. 620 of Ref. 47.

296

P. M. ECHENIQUE

el

al.

hydrogenic wave functions and energy levels E,)= -Zi/2nt and El = -Z:/2n:, for the electron in the incoming ion and in the target, respectively, the capture cross section for the transfer 1+0 is given by (36.1)

An improved cross section within the OBK approach can ,e obtained by taking more appropriate values for Eo and El.'40 Equation (36.1) shows that outside the resonance condition the shell cross section decreases quickly with increasing n l . In practice, only those capture processes associated with the deepest levels of the target are dominant. At high velocities Eq. (36.1) overestimates the shell cross sections; it is too large by a factor of 3. At intermediate velocities, this factor increases depending on the atomic number of the incoming ion; for a proton or He, that factor can 4-5 if u > u o . This result is estimated from calculations of protons colliding with hydrogen. For heavier atoms, that factor increases to 10 or more for B, N, C and 0; these results have been obtained by Sols and F10res'~~ who have calculated the corresponding shell cross sections (see below) using an eikonal method, as proposed by Chan and Eichler.14' This should be a good approximation for the total cross sections.

RESULTS FOR H, He, B, C , N 37. CROSSSECTIONS.

AND

0

Cross sections can be obtained from the transitions probabilities, Pc and PLby defining mean transition times tCand zL for the processes as (37.1) Cross sections are related to these mean times by (37.2) where n is the target atomic density and Y a factor taking into account the '&F. Sols and F. Flores, Phys. Rev. A 37, 1469 (1988). F. T. Chan and T. Eichler, Phys. Rev. Lett. 42, 58 (1979); Phys. Rev. A 20, 1841 (1979).

147

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

297

number of final states associated with spin. For example, for the process PA'' H+, Y = 1, since only one electron jumps from H" to the conduction band while for the process H+ & p, Y = 2, since electrons of both spins may participate in the process. Figures 20a-20d show the capture cross sections as functions of v for the following processes: H+ -H A1

He2+

(Fig. 20a), He+

(Fig. 20b),

B5+ & B4+

(Fig. 2Oc) ,

08+

A 07+

(Fig. 20d),

Cross sections associated with shell, resonant and Auger processes are included. For H + and He2+, the shell cross section is calculated using an OBK approximation, reducing the results by a factor of 4.5; this factor is suggested by atomic collision calculations. 145 As regards B5+ and 08+, the shell cross sections are calculated using the eikonal appr~xirnation'~~ as discussed above; this can be expected to yield accurate results for v l Z , v o . In the same Figs. 20c and 20d, the OBK results are shown after reduction by a factor to get adequate agreement with the eikonal results:147OBK results are found to be quite good for v 2 1.5Zlv0. Figures 21a-21d show similar results for the processes: 135~1432146 H

*

He+*

H-

(Fig. 21a),

He

(Fig. 21b),

B4+ & B3+

(Fig. 21c),

07+

(Fig. 21d).

06+

R e s ~ l t s ' ~ for ~ ~the ' ~ loss ~ ~ cross ' ~ ~ sections are shown in Figs. 22 and 23, which refer to the following processes:

HA H+

(Fig. 22a),

He+ 3 He2+

(Fig. 22b),

B4+ & B5+

(Fig. 22c),

07+

508+

(Fig. 22d),

298

P. M. ECHENIOUE et a/.

10 0 (A021

10-1

10-2

I

I

10-3

1

2

IV/V.I

1

IV/V.I

3

4

2

(A'?

(Ao2)

10 O

10- 1

10-1

10-2

10-2

10-4

10-3

10-4

10-5

1

V/(!iVOl

2

1

2

V / ( 8 Vol

FIG.20. Capture cross sections associated with the following processes: (a) H + H; (b) He++ He+; (c) B5+A B4+; (d) Ox+A 07+. Shell (a,) Auger (uA) and resonant (uR)processes are shown. For H , the resonant process is unimportant; also shown is the cross section for the atomic collision H+ 3 H, normalized to one N atom. For B and 0, the shell cross sections calculated using an OBK approximation and an eikonal (E) approach are given.

299

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

1

'

1

'

1

'

cap ure H"/AI -

-

-

capture He +/A1

3 IA T

2

-

-

1

I

1

2 Vf

t l

2

1 I

"

'

"

I

'

I

capture

lAo21

V/(5

vo

4

v.

I

2

1

3

2

1

1

FIG. 21. As in Fig. 20 for (a) H-H-; 07+ & 0 6 + .

V/(8Vo) He*&He,

(c) B4*&

B3+ and (d)

300

P. M. ECHENIQUE

el

al.

loss

-

loss He+/AI

,

1.5

L

‘3 TOTAL

1

0.5

I 0.2

I

1

0.4

2

v/ v.

LL I

L UA QA r.

1

2 V/

1 I

v.

I

I

3

4

2

I

loss

lAo21

10-2

10-3

10-4

10-5

1

2 v/(5vo)

1

2 V/(8Vo)

FIG. 22. Loss cross sections associated with the following processes: (a) H a H + , (b) He+& H e + + , (c) B4+ 4 B5+,(d) 07+ OR+. Auger ( u A ) and resonant (uR) processes are shown. For H, the cross sections for the atomic collisions H+ and H“” H + are also shown.

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

301

2-4

10”

d 2 V/(5Vo)

FIG. 23. As in Fig. 22 for (a) H - & H , (d) 06+ 07+.

(b) Heo&He+,

(c) B”*B4+,

and

302

P. M. ECHENIQUE

el

al.

and

H-%H

(Fig. 23a),

He A He+

(Fig. 23b),

B3+

B4+

(Fig. 23c),

06+

07+

(Fig. 23d).

Now, consider the first electron captured by the ion. From Figs. 20 and 22 we find that the loss and capture cross sections are roughly equal for =1 . 2 ~ ~

(H++ H"),

v = 2.5v0

( H e 2 + - +He+),

v -- 6 . 5 ~ "

(B5++B4+),

= 11.2~"

(ox++07+).

These values scale with Z,; thus, aL- ac for V

1.2 - 1.4ZiVo.

In the case of H , the capture cross sections (H+ % H") are controlled by the Auger processes for v < 1.5vo, while the shell processes are dominant for v > 3v,,. In the intermediate region around 2v0 the two cross sections are comparable. For He, the Auger processes control the capture cross section (He++ A1 He+) for v < 0.5v0, while the shell processes dominate for v > 0.811,. For B (B5+%B4+), the capture cross section is controlled by shell processes for the calculated velocities, v > 0 . 5 Z l v 0 ; we can expect from the results of Fig. 20c that Auger 07+) processes become dominant for v I0.25Zlv,. For 0 (Ox+ similar results are found; the capture cross section is dominated by shell processes in the whole region of calculated velocities. Comparing the capture cross section results for the first electron of the different ions, we find H different from the other cases considered. Only for H do we find that the capture cross section around the velocity for which aL ac is controlled by the Auger processes; in all the other cases, shell processes dominate. This shows that for He, B and 0, capture of the first electron is an atomic-like process, in which condensed matter effects are unimportant. For H we find, however, that solid state effects are significant. This point can be checked by comparing cross sections for the processes H + t)H, with the corresponding atomic cross

-

-

DYNAMIC SCREENING OF IONS IN CONDENSED MA'ITER

303

sections. This is done in Figs. 20 and 21: the comparison suggests that condensed matter effects appear for v < 2v0, as previously concluded. For electron loss in H ( H a H+), the resonant and the Auger process are comparable. For He, B and 0, the results of Figs. 22b-22d show, however, that resonant processes dominate for the velocities of interest. This illustrates the same effect as before: in heavier ions, atomic-like processes associated with deeper levels are dominant in the region of velocities we are interested in, i.e., v ZlvO.Auger processes in heavy atoms only become important at velocities v IZ , v , , . In this regime, we can expect Auger processes associated with deep levels to yield a very small cross section. It is worth noting that for high velocities the capture cross sections decrease much faster with increasing v than the loss cross sections. Indeed, Echenique et have shown that as v + m , all the different capture cross sections behave as l/v'*, while all the loss cross sections only decrease like l/v2. can be discussed in similar terms The case of the second using Figs. 21 and 23. Crystal effects are even more prominent for this second electron than for the first one in H, due to the shallow character of the second bound level. Also, crystal effects are unimportant for the second electron of the other heavier atoms studied.

-

38. CHARGE STATES Since the pioneeering work by phi lip^'^^ the charge states of moving ions have been experimentally measured over a wide range of energy. 15c1s8 Using Figs. 20-23 the fraction of different charged ions inside the target can be calculated as a function of velocity. For the case of protons I48

P. M. Echenique, F. Flores, and R. H. Ritchie, Nucl. Instrum. Methods B 33, 91 (1988). A . Phillips, Phys. Rev. 97, 404 (1955). ""S. K. Allison, Rev. Mod. Phys. 30, 1137 (1958). "'A. Chateau-Thierry and A . Gladieux, in "Atomic Collisions in Solids," p. 307 (S. Datz, B. R. Appleton, and C. D . Moak, eds.). Plenum, New York, 1975. "*A. Chateau-Thierry, A. Gladieux, and B. Delaunay, Nucl Instrum. Methods l32, 553 (1976). Is3R. Behrisch, W. Eckstein, P. Meischner, BMV Scherzer, and M. Verbeek, Ref. 151, p. 315. Is4B. T. Meggitt, K. G. Harrison, and M. W. Lucas, J . Phys. B 6, L362 (1973). "'S. Kreussler and R. Sizmann, Phys. Rev. B 26, 520 (1982). 156N.V. de Castro Faria et al., Nucl. Instrum. Methods B 17, 327 (1986). ls7C. G. Ross and B. Terreault, Nucl. Instrum. Methods B 15, 146 (1986). ISsY. Yaruyama, Y. Karamori, T. Kido, and F. Fukuzawa, J . Phys. B 15, 779 (1982). I4'J.

304

P. M. ECHENIQUE

al.

el

v/ vo 3

0,6

I

0

1

1

I

I I I I I

I

I

I

1

I

I

I l l

1

1

I

I I l l

100

10

n(H+)/n(Ho) FIG. 24. (Number of bare protons)/(number of neutral hydrogens) as a function of atom velocity. (Experimental curve - - -, Ref. 149.)

the ratio of the probability of finding the bare proton to that of finding the neutral atom can be obtained from the total cross sections as (38.1)

In Figs. 24, 25 and 26, we show charge fractions of different ions as a function of ion velocity. In Figs. 24 and 25, we show results for H and He, which are in good agreement with experiment. In Fig. 26, we collect the results calculated for B, C, N and 0. The cases of C and N were calculated using an OBK approach with an appropriate reduction factor. For these cases, the results give a fair agreement with experiment for 11/21 > 1.5110. Finally, we comment that the absolute cross sections can be checked by comparing with the measured values for He and 0 at certain velocities. Thus, for He++/Al, Itoh et ~ 1 . lhave ~ ~ obtained the following cross sections: a L ( l MeV)- 1

oc(l MeV)=0.47

A2, A2,

uL(2 MeV)=0.9

oC(2 MeV)=0.12

A' A2,

Is9A. Itoh, Y. Haruyama, Y. Karamuri, T. Kid0 and F. Fukuzawa, Bull. h t . Chern. Res., Kyoto University 60,289 (1983).

305

DYNAMIC SCREENING OF IONS IN CONDENSED MATTER

3L

-2 -

1

I

I

I

IIIII

1

I

1

I

I

I I I I I I

I 1 1 1 1

I

-

-

-

-

1,4-

-

-

1-

0.6

I

I

I

I I I

I L I

I

I

I

I

I

I

1 1 1 1

I

I

while Figs. 20b and 22b yield the following values:

A*, A’,

aL(l MeV)=1.13 aC(l MeV)=0.44

A2, A,

aL(2 MeV)=0.92 aC(2 MeV)=0.10

in very good agreement with the experimental evidence.

1

1.5

2

2.5 3/

1

1.5

2

2.5

3

v / Iz v o 1 FIG.26. Fraction of charged ions (B, C, N and 0) as a function of u / Z v , .

I I l l

306

P. M. ECHENIQUE er al.

On the other hand, Sofield et al. sections for 08+ and 07+: ac(lOvo; 08’/A1) = 0.025 ac(iov,; o’+/AI) = 0.011

A’, A,

have measured the following cross aL(lOvo;07+/Al) 0.025 -L

~ ~ ~ ( oi ~o +t / ~A ~ )0.050 ; -L

A’, A,

while from Figs. 20d, 21d, 22d and 23d, the following theoretical results are surmised: a c ( i o ~ , ;o ~ + / A ~ =)0.010

ac(lOvo; 07+/Al)= 0.006

A2, A’,

~ ~ ( i07+/~1) o ~ ~= ;0.013 A2, aL(lOvo;06+/A1)= 0.028 A’.

In this case, the agreement is only reasonable. Notice that in all cases, the calculations are obtained using a first principle approach without any adjustable parameters. VII. Summary In this review we have developed theoretical concepts associated with the passage of charged particles through matter that were pioneered by Bohr, Bethe, Fermi and others. Interactions with condensed matter have been emphasized. A complete discussion of the real and imaginary part of the self-energy of an external charge interacting with a medium characterized by a q and w dependent dielectric constant has been included. Detailed discussions of the induced density fluctuation and scalar potential patterns induced by the passage of swift ions throughout condensed matter have been given. A brief comprehensive description of many physical phenomena, associated with the wake has been included. These include wake binding, vicinage effects, dynamic screening and the Okorokov effect among many others. The first Born approximations considered are most reliable at ion speeds much greater than v, . We have discussed recent theoretical work, based on the local density formalism to treat the electron-gas many-body problem appropriate when v << vo that combines ideas from atomic physics and solid state physics. This work has led to results for energy loss rates that should be highly accurate within the context of the electron-gas model since nonlinear effects, electron correlation, and exchange are included. When the 160

C. J. Sofield, N. E. B. Cowern, T. Draper, L. Bridwell, J. M. Freeman, C. J. Woods, and M. Spenner-Harper, Nucl. Instrum. Methods 170, 257 (1980).

DYNAMIC SCREENING OF IONS I N CONDENSED MATTER

307

velocity of the ion is in the range vo 5 v 5 z~”Zf/~, complicated dynamical processes occur. Many-body techniques applied recently to this problem have been discussed. These results represent an important first step toward a comprehensive theoretical study of the charge states of ions in condensed matter. A great many topics relevant to the passage of ions through matter have not been included here. At ion speeds>>v, the first Born approximation should yield reliable theoretical results. As v decreases toward V o nonlinear effects appear that are proportional to Z : in lowest order. The theory of the Z : effect has been analyzed in semiclassical approximation16’ and has led to extensive theoretical and experimental studies.I6* Application of many-body perturbation theory to this nonlinear effect has been made,’63 but further work is indicated.’@ Similar nonlinear corrections to the wake potential and density fluctuation function become important in the regime v v o as indicated above, but have not yet been studied systematically in a many-body approach. Surface effects have been covered only briefly. The surface plasmon model has been employed for estimates of energy loss by a particle crossing a surface in the speed regime v >> 21, ,67 while for v << v, the specular reflection model for the bounded g a ~ ’ ~has ~ , been ’ ~ More comprehensive treatments of energy loss, wake effects and capture and loss near surfaces are needed. Convoy electrons accompany swift ions exiting from condensed matter.’68 Experimental and theoretical work on this phenomenon has been e ~ t e n s i v ebut ’ ~ ~much remains to be done. Coherent effects on energy losses and the wake of a charged particle

-

C. Ashley, R. H. Ritchie, and W. Brandt, Phys. Rev. B 5 , 2393 (1972). ‘”See, e.g., J. Lindhard, ffucl.Instr. Methods 132, 1 (1976); R. H. Ritchie and W. Brandt, Phys. Rev. A 17, 2102 (1978); J. F. Ziegler, “Handbook of Stopping Cross Sections for Energetic Ions in All Elements.” Pergamon, New York, 1980. 163C.C. Sung and R. H. Ritchie, Phys. Rev. A 28, 674 (1983). lWC. D. Hu and E. Zaremba Phys. Rev. B 37, 9268 (1988) have analyzed this effect for u << v,, in the electron-gas model. I6’See, e.g., R. H. Ritchie, Phys. Rev. 106, 874 (1957); R. H. Ritchie, Surf. Sci. 34, 1 (1973); H. Raether, in “Physics of Thin Films,” Vol. 9, pp. 145-261 (G. Hass, ed.). Academic, (1977). ’*R. H. Ritchie and A. Marusak, Surf. Sci. 4,234 (1966). I6’See, e.g., A. Gras Marti, P. M. Echenique, and R. H. Ritchie, Surf. Sci. 173, 310 (1986). Brandt and R. H. Ritchie, Phys. Lett. A 62, 374 (1977). 1691. A. Sellin et al., in “Forward Electron Ejection in Ion Collisions,” p. 109. K. 0. Groeneveld, W. Meckbach, and I. A. Sellin, eds.). Springer, Heidelberg 1984; M. Breinig et al., Phys. Rev. A 25, 3015 (1982).

I6’J.

308

P. M. ECHENIQUE el af

due to multiple real emission of plasmons have been c o n ~ i d e r e d ’ ~ ” ~ ’ ~ ~ using an explicit solution of the time evolution operator for a simplified plasmon-charged particle interaction Hamiltonian. It would be interesting to employ path-integral methods’72 and a more realistic representation of the interaction in this connection. When slow ions collide with surfaces, or move in condensed matter where the static electron density varies appreciably over distances of the order of a few h;, Pauli excitation of electron-hole pairs becomes important. The semi-empirical treatments of F i r ~ o v ’and ~ ~ Lindha~-d’~~ have been used extensively in analysis of experimental data on atomatom collisions in this velocity range. Brandt’75 applied a phenomenological model introduced by to treat the related problem of inner shell vacancy production in atomic collisions. Further theoretical exploration of this phenomenon presents a difficult, but important task. ACKNOWLEDGMENTS Part of the research described in this review was sponsored jointly by the Office of Health and Environmental Research, U.S. Department of Energy under Contract No. DE-ACOS840FU1400, the Spanish CAYCIT, Eusko Jaurlaritza, Gipuzkoako Foru Aldundia, the USA Spain Joint Committee for Scientific and Technological Cooperation. Two of the authors, P. M. Echenique and F. Flores, gratefully acknowledge Iberduero S.A. for help and support.

I7OC. C. Sung and R. H. Ritchie, I. Phys C 14, 2409 (1981). I7’P. M. Echenique, J. C. Ashley, and R. H. Ritchie, Europhys. I. Phys. 3, 25 (1982). I7’R. P. Feynman and A. R. Hibbs, “Quantum Mechanics and Path Integrals.” McGraw Hill, New York, 1965. 1730. B. Firsov, Zh. Eksp. Teor. Fir. 36, 1517 (1959). 174J.Lindhard and M. Sharff, Phys. Rev. W, 128 (1961); see also M. A. Khumakov and F. F. Komarov “Energy Loss and Ion Ranges in Solids,” pp. 109ff. Gordon and Breach, New York, 1981. I7’W. Brandt, IEEE Trans Nucl. Sci. NS-26, 1179 (1979). 176L.Wilets, Phys. Rev. 116, 372 (1959).

Author Index

Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although his name is not cited in the text.

A

Ashley, J. C., 235(19), 242(34), 259(52), 263(61), 268(52), 270(80), 277278(108), 283( 124), 284-285( 126), 286(128), 307(161), 308(171) h u n d , N., 14(32) Astle, M. J., 12(28), 15(28) Autler, S. H., 201(105), 202,207(105) Axilrod, B. M., 81(172)

Abell, G. C., 29(70) Abraham, F. F., 85(175, 179, 186-187) Abrikosov, A. A., 201(104), 207(104), 212(120), 215(127) Ackland, G. J., 26(67), 35, 38, 41-42, 67-68(120), 69(b), 71(120, 129), 72(b, 120, 129), 73(a), 75-76(b), 77(129), 78(129, 155) Adams, E. D., 75(d) Adler, R. J., 274(96) Aeppli, G., 187(83) Agrenovich, V. M., 186(81) Aharoni, A., 145(24), 155(24) Ahlen, S. P., 232(4), 236(4), 264-265(4) Akheizer, A. T., 235(14) Alberts, R. C., 187(84) Aleshkin, V. Y.,275(97) Allan, G., 28,29(69) Allison, S. K., 303(150) Almbladh, C. O., 278-279(111) Altshuler, B. L., 222(137) Andersen, H. C., 79(161), 87(161) Andersen, H. H., 271(81) Andersen, 0. K., 45,48(98), 66(118-119) Anderson, P. W., 135(15), 195(100), 210(110), 212(110) Anderson, V. E., 233(7), 295(7) Ando, T., 133(14), 135(14) Appleton, B. R., 233(8), 236-237(20), 250-252(20), 267-268(20), 283(8), 287(8), 303(151) Arista, N. R., 269(75-76,78) Arnold, E., 85(118) Aronov, A. G., 222(137) Arovas, D. P., 208( 107) Aryasetiawan, M., 78( 155) Ashcroft, N. W., 2(1),15(36), 17(40), 19(45), 29-30(36), 51(36), 91 Ashkin, J . , 232(4), 236(4), 264-265(4)

Bacon, D. J., 67(122-123), 68(b), 71(123), 78( 154) Balamane, H . , 85-86( 184) Ballanski, M., 186(80) Bane, K. L. F., 274(95) Baragiola, R. A ,, 269(78) Barberin, N., 283(125), 285 Barkas, W. H., 233(9) Barker, R. A., 77(148) Barnett, R. N., 85(189-190) Barrow, R. F., 14(32) Basbas, G., 251(44), 268(44), 269(79) Baskes, M. I., 30(79-80), 32(79-80), 35, 36(84), 67(79-80, 84), 69(a), 72(a, 84). 75(a), 77(80), 88(208), 89 Batlogg, B., 187(83) Batra, I. P., 85(175, 179) Baumgart, H., 85(188) Beck, D. E., 235(18) Beeler, J. R., 5(23) Behrisch, R., 303(53) Bell, F., 271(85-86) Bendow, B., 186(81) Benedek, R., 75(143) Bennemann, K. H., 75(145) Bethe, H. A ,, 232(4,6), 236(4), 264-265(4) Betz, H. D., 233(9), 271(85-86) Beyer, W. H., 12(28), 15(28) Beiber, A ,, 63-64(111)

309

310

AUTHOR INDEX

Biggerstaff, J. A , , 273(91) Bikerman, J . J., 71(131) Bilello, J. C., 67(128) Birman, J . L., 186(81) Biswas, R., 79(162), 80(6, 162, 170), 84(162), 85(162, 170, 195), 86-87(195), 88(162, 170) Blaisen-Baroja, E., 85(177) Blann, M., 233(9) Bloch, F., 233(7), 252(45), 295(7) BIOSS,W. L., 183(69-70) Blount, E. I., 130(12) Boardman, A , , 235(18) Bohm, D., 235(16) Bohr, N., 232(2-4), 233(3-4), 236(4), 264-265 (4) Boring, A. M., 187(84) Boswarva, I. M., 9(25), 46(25) Both, G., 273(89) Bourgoin, J., 79(163) Brandt, W., 233(8), 246(40), 250(20), 251-252(20,44), 253(48), 259(40,44), 262(40), 264(44), 267(20,40,69), 268(20,40,44,69), 269(74), 281(118), 282,283(8, 120-123), 284-285, 287(8, 120-123). 307(161-162, 168), 308( 175) Breinig, M.,307(169) Brenner, D. W., 80(171) BrCzin, E., 208(108), 214(108) Bridwell, L., 306( 160) Brinkman, H. C., 295(145-146) Brinkman, W. F., 184(78) Bromley, A . , 262(57), 272(57) Broughton, J . Q., 85(186-187, 193). 86( 193) Brown, R. H., 44(92), 46(92), 47.4849(92), 53-54(92) Bryant, G. W., 188(92) Bucher, E., 187(83) Butler, W. H . , 63(112-113) Bychkov, Y. A , , 149(27)

Cardillo, M. J., 32(83) Cardona, M., 183(72) Carlsson, A. E., 14(33), 15(36), 29-30(36), 37(87), 40(88), 44(92), 46(92,96), 47, 48-49(92), 51(36,87), 52,53(92, 102), 54(87,92), 56(87), 58-60(105), 61, 62(109), 64(105), 72(142), 78(157), 81(173) Carstanen, H. D., 69(f) Castellani, C., 222(131-135) Catlow, C. R. A . , 5(18) Celli, V., 188(95) Chan, C. M., 76(147) Chan, F. T., 296(147), 297(147) Chakraborty, T., 160(50-51) Chandrasekhar, M., 183(72) Charles, M., 69(e) Chateau-Thierry. A . , 303(151-152) Chatterjee, A. K., 9(25), 46(25) Cheblukov, Y. N., 273(90) Chelikowsky, J. R., 115(7), 197(102) Chen, P., 274(94-95) Chen, S. P., 71-72(133), 75(133) Choi, D. K., 79(160) Christensen, N. E., 187(84) Cleveland, C. C., 85(188-190) Cohen, M., 165(56) Cohen, M. L., 86(198), 115(7), 194(98), 197(98, 102) Coleman, P., 187(87-88) Collins, R., 44(91) Combescot, M., 184(79) Connolly, J . W. D., 63(114) Cooney, P. J., 273(89) Copel, M., 75(d) Cousins, C. S. G., 13(31) Cowern, N. E. B., 306(160) Cowley, E. R . , 83(174) Cox, D., 233(7), 295(7) Crawford, 0. H., 272(87), 273(91-92) Cross, M. C., 289(134), 295(134) Cyrot-Lackmann, F., 23(60-61), 26-27(60-61), 45( 93-94), 46(60), 48(60,94), 49(94)

C Cahn, R. W., 4(11), 19(11), 24(11) Caillol, J. M., 159(47) Car, R., 3(7), 87(203)

Dagens, L., 20(50-55), 58-59(103-104), 60-61( 107)

311

AUTHOR INDEX

Dalgarno, A . , 5(20) Daniel, E., 128(9) Dasgupta, B. B., 235(18) Das Sarma, S., 85(183), 88(207) Datz, S., 233(8), 236-237(20), 250-252(20), 267-268(20), 273(91), 283(8), 287(8), 303(151) Davison, W. D., 5(20) Daw, M. S., 30(79-SO), 32(79-80), 35, 36(84), 67(79-80,84), 69(a), 72(a, 84, 136), 75(a), 77(80), 78(150) Dawson, J. M., 274(94-95) Day, M. H., 267-268(63-64) Deaven, D. M., 78(151) de Castro Faria, N. V., 303(156) DeGennes, P. G., 199(103), 201(103), 202 Dehne, H., 274(93) de Hosson, J. T. M., 4(16) Delaunay, B., 303(152) del Campo, P. F., 273(91) den Nijs, M., 151(33) Diatlov, I. T., 212(119) DiCastro, C., 222(131-135) Diehl, J., 12(27), 68-69(d) Ding, K., 79(161), 87(161) Dittmer, D. F., 273(91) Doan, N. V., 20(55) Dodson, B. W., 72(137-138), 77(138), 85(176, 191), 86(176), 88(206), 90(206) Dora, G., 151(29) Draper, T., 306( 160) Ducastelle, F., 23(62), 26(62), 36-37(62), 45(93-95), 48(94-95), 49(94), 50-53(95), 62(108), 63(110) Dyment, F., 40(88), 56 Dzyaloshinski, I. E., 148(26), 215(127)

E Ebel, M., 267-268(64) Eby, P. B., 233(7), 295(7) Echenique, P. M., 246(39-41), 248(39-41), 249(41), 250(43), 251-252(44), 259(40, 43-44,52), 262(40), 264(44), 267(40, 65,67-70), 268(40,44,52,65,68-70, 72), 270(80), 271(83), 272(87-88). 274(67), 277-278(108), 279(114), 283(124-125), 284-285(126), 286(128), 288-290(41), 295(143), 297(143), 303(148), 307(67, 167), 308(171)

Eckardt, J . C., 269(78) Eckstein, W., 303(153) Ehrenreich, H., 3(6), 4(12), 14(33), 23(12), 24(64), 27(12), 56(6), 79(164), 91, 112(5), 116(8), 130(10), 242-243(29), 245(29) Ehrhart, P., 69(c, f) Eichler, T., 296-297(147) Eisen, F. M., 284(127), 285 Eliashberg, G. M., 149(27) Emrick, R. M., 69(g) Ercolessi, F., 72(139) Erkoc, S., 78(160) Esterling, D. M., 4(16), 9(25), 14(35), 46(25) Estrup, P. J., 77(148)

F Faibis, A , , 273(89) Falicov, L. M., 183(66-68) Farkas, D., 40(88), 56 Fattah, A . M., 69(f) Febel, A , , 274(93) Feidenhans’l, R . , 75(c) Feldman, L. C., 271(81) Felter, T., 77( 148) Fermi, E., 217,219,221,234(10-11, 13), 263(59), 275,287(130) Ferrell, R. A . , 242-243(30) Ferrell, T. L., 276(102), 278(102) Fetter, A. L., 98(2), 155(2), 243(46) Feuston, B. P., 85(178) Feynman, R. P., 165(55-56), 308(172) Finkelstein, A . M., 222(136) Finneman, J., 277(103) Finnis, M. W., 4(9), 17(41), 26(67), 30(71-72), 31(9), 35,37(85), 38, 39(85), 41-42,67(85, 120), 68(a, 120), 69(b), 70(72), 71(120, 129), 72(a, 120, 129), 73(a), 75-76(b), 77-78(129) Firsov, 0. B., 308(173) Flores, F., 246(41), 248-249(41), 288(41), 289(41, 132), 290(41, 139), 291(140), 295(132, 143), 296(140, 146), 297(143, 146), 303(146, 148) Flynn, C. P., 13(20), 29(30), 68(30) Foiles, S. M., 36-37(84), 38, 40(86), 69(84), 69(a), 72(a, 84, 135-136), 75(a), 76(a, 135), 78(86)

312

AUTHOR INDEX

Fowler, A. B., 133(14), 135(14) Frank, I. M., 263(58), 272(58) Freeman, A. J., 77(149) Freeman, C. M., 5(18) Freeman, J. M., 306(160) Friedel, J., 17(63), 24(63), 27(63), 28,275,

277,278(110)

Fu, C. L., 77(149) Fujimoto, F., 271(82) Fukhs, K., 2(3) Fukuyama, H., 135(15), 136(17), 188(93),

194(97), 210(116), 212(116), 214(123), 222( 134) Fukuzawa, F., 303(158), 304(159), 305

G Galindo, A,, 238(21) Galitski, V., 239(26), 242(33) Garrison, B. J., 78(151), 80(171) Gautier, F., 63(110-lll), 64(111) Gavoret, J., 212(121-122) Gawlinski, E. T., 85(182) Gehlen, P. C., 5(23) Gelatt, Jr., C. D., 14(33), 24(64), 45 Gemmell, D. S., 262(55-57). 269(78),

272(56-57), 273(89)

Gerharts, R., 188(94) Gertsenshtein, M. E., 235(14) Gibson, W. M., 269(77) Girvin, S. M., 157(45), 169(58),

173(59-61), 177(59-61)

Gladieux, A,, 303(151-152) Glasser, M. L., 275(98) Goeneveld, K. O . , 307( 169) Goland, A., 233(7), 295(7) Goldman, A. I., 187(83) Goldstone, J., 242(32) Gollish, H., 72(140) Golovchenko, J. A,, 233(7), 295(7) Gonis, A,, 63(112-113) Gordon, R. G., 5(19) Gorkov, L. P., 148(26), 215(127) Gossard, A. C., 151(30) Granham, W. R., 75(d) Grant, J. A., 232(5) Gras-Marti, A , , 277-278(108), 286(128),

307( 167) Gratias, O . , 62(108)

Grest, G. S., 85-87(195) Grout, P. J., 16(37) Guinea, F., 246(41), 248-249(41),

288(41), 289(41, 137),290(41, 139), 291 Gurnbs, G., 275(98) Gunnarsson, O., 279(117) Gunton, J. D., 85(182) Gustafsson, T., 75(d) Gyorffy, B. L., 63(112)

Haasen, P., 4(11), 19(11), 24(11) Hafner, J., 4(15), 19(15), 21(57) Haldane, F. D. M., 157(43, 45), 160(52),

162-163(45), 182(64)

Halicioglu, T., 79(159-160), 80(c, 159),

81(159), 84,85(180,184-185,197), 86( 184) Halperin, B. I., 112(5), 138(20), 141(21), 149-150(28), 151(34-35), 152(36), 153(21), 154(20), 157(44), 158(35), 189(96), 190, 192-193(96), 210(116), 212(116), 223 Hamann, D. R., 32(83), 79(162), 80(b, 162, 170),84(162), 85(162, 170) Hamermesh, M., 101(3), 105(3), 160(3) Hansen, J . P., 159(48) Harder, J. M., 67(123), 68(b), 71(123), 78(154) Harris, A. B., 145(22), 147(22) Harrison, Jr., D. E., 78(151) Harrison, K. G., 303(154) Harrison, R. J., 19(43) Harrison, W. A., 5(17,22), 17(38-39), 20(46), 21(56), 24(21), 65(116), 66(117), 80(168) Haruyama, Y.,304(159), 305 Hatcher, R. D., 67(128) Hatcher, R. L., 78(150) Hausleitner, C., 21(57) Haydock, R., 9(25), 46(26) Hedin, L., 116(8), 242-243(29), 245(29,38) Heine, V., 3(6), 4(12), 19(6), 23(12), 27(12), 49(99-loo), 56(6) Herlach, D., 68(d) Herring, C., 218(128) Hibbs, A. R., 308(172)

AUTHOR INDEX Hieland, N., 289(134), 295(134) Hillairet, J., 69(e) Hirai, K., 43(89), 45(89), 48(89) Hohenberg, P. C., 152(36), 279(115) Hu, C. D., 307(164) Hubbard, J., 184(76-77). 242(31) Huff, R. W., 274(94) Hvelplund, P. D., 273(91)

I Ichikawa, M., 245(36), 252-253(36) Iferov, G. F., 273(90) Imry, Y., 145(23), 147(23), 154(23) Inkson, J . C., 245(37) Inokuti, M., 232(4), 236(4), 264-265(4) Iovdanskii, S. V., 149(27) Isaacson, D., 281(119) Islam, M. S., 5(18) Itoh, A , , 304(159) Izurni, O., 63(112)

J Jach, T., 169(58) Jackson, J. D., 238(23), 289(133) Jackson, R. A., 5(18) Jacobsen, K. W., 30(78), 32-33(78), 34, 35-36(78) Jaffee, R. I., 5(23) Jakubassa, D. H., 271(84) Janak, J. F., 31(81) Jaynes, E. T., 44(90), 59(90) Jena, P., 19(44), 278(112) Jepsen, O., 45,48(98), 66(118-119) Johnson, D. D., 63(112) Johnson, E. A,, 269(78) Johnson, R. A., 5(24), 13(31), 14(34), 38, 67( 124-125), 71(125) Jones, H., 277(104) Jones, R., 80(167) Jose, J., 155(41) Josephson, B. D., 277(106)

Kadanoff, L. P., 155(41) Kalia, R. K., 85(188)

313

Kallin, C., 149-150(28), 208(107) Kanamori, J., 43(89), 45(89), 48(89) Kane, E. O., 80(160), 183(72) Kaneko, T., 289(131, 136), 292(136), 295(131) Kanter, E. P., 269(78), 273(89) Kararnori, Y., 303(158), 305 Kasuya, T., 187(88) Katsouleas, T., 274(94) Kawatsura, K., 271(82) Keating, P. N., SO(165) Kehr, R. H., 267(66) Keiter, H., 209(109) Kelires, P., 89(209) Kelly, M. J., 183(66-68) Kelley, M. J., 9(25), 46(25) Khan, B., 85(188) Khizhnyakov, I. S . , 273(90) Khor, K. E., 85(183), 86(207) Khurnakov, M. A,, 232(4), 236(4), 264-265(4), 292(142), 308( 174) Kido, T., 303(158), 304(159), 305 Kim, Y. S., 5(19) Kirkpatrick, S., 155(41) Kitagawa, M., 275(99), 283(122), 284-285, 287( 122) Kittel, C., 2(2), 12(26), 15(26), 26(2), 152(37), 278(109) Kittner, M. F., 273(91) Kivelson, S., 208(107) Klein, O., 235(15), 296(15) Kleiner, W. H., 201(105), 202,207(105) Klitzing, K. V., 151(29) Kluge, M. D., 85(192, 194), 86(194) Knudsen, H., 273(91) Koch, C. C., 63(112) Koening, W., 273(89) Kohmoto, M., 151(33) Kohn, W., 32(82), 115(6), 130(11),279(16) Kollar, J., 45, 48(98) Kornaki, K., 271(82) Komarov, F. F., 232(4), 236(4), 264-265(4), 292(142), 308(174) Korringa, J., 263(60) Kosterlitz, J. M., 155(38-40) Kotliar, G., 222(135) Kouchi, N., 271(82) Krarners, H. A,, 295(145-146) Krause, H. F . , 273(91) Kronig, R., 263(60)

314

AUTHOR INDEX

Kruessler, S., 283(122), 287(122), 303(155) Kubler, J., 45

L Lam, N. Q., 20(55), 60-61(107) Landman, U., 85(188-190,196), 86(196) Lang, N. D., 30(73) Langer, J. S., 259(51) Langmuir, I., 235(17) Lannoo, M., 79( 163) Lantshner, G., 269(78) Lapitski, Y. Y., 273(90) Laubert, R., 269(77), 271(86) Laughlin, R. B., 151(31-32). 156, 157(46), 159-160 LaViolette, R. A,, 79(158) Lee, D. A,, 219(130), 222(130) Lee, J. K., 4(16), 19(16,44) Lee, P. A., 136-137(18), 151(34), 187(85), 222( 131- 135) Lekner, J., 277(106) Lenke, M., 274(93) Leslie, M., 5(18) Levesque, D., 85(177) Levesque, R., 159(47-48) Levy, V., 69(e) Li, X.P., 85-86(193) Libenson, B. N., 275(97) Lindhard, J., 235(15), 238(22), 253(22), 256(22), 264(62), 276(101), 296(15), 307(162), 308( 174) Liu, C. T., 63(112) Lo, D. Y., 78(151) Logovinsky, V., 86(200) Louie, S. G., 197(102) Lucas, M. W., 303(154) Luedtke, W. D., 85(188-190, 196), 86(196) Lundquist, S., 116(8) Lundqvist, B. I., 245(36), 252-253(36), 279( 117) Lundqvist, S., 242-243(29), 245(29) Lurio, A., 271(81)

M Ma, M., 222(131-134) Ma, S. K., 145(23), 147(23), 154(23) MacDonald, A. H., 60(106), 136-137(16),

157(16), 159(16,48-49), 170(16,49), 171(16,50), 173(59-61), 176(16,49), 177(59-61), 179- 180(63), 182(65), 188(92), 223 Machlin, E. S., 4(16) Madsen, J., 45,48(98) Maier, K., 68(c, d) Mairy, C., 69(e) Mann, A,, 281(118), 282 Manninen, M., 19(44), 30(75-76) Manson, J. R., 262(54) March, N. H., 16(37) Martin, J. W., 13(31) Martin, R. M., 16(37) Martinez, A., 85(188) Marusak, A,, 307(166) Masri, P. M., 4(10) Massey, H. S. W., 255(49) Masuda, K., 67(126), 71(126), 72(134), 78( 152) Matthai, C. C., 16(37), 67(122) Maysenholder, W., 67(127), 70(127) Mazzarro, A,, 250(42-43), 259(42-43), 267-268(42-43) Mead, L. R., 44(91) Meckbach, W., 307(169) Meggitt, B. T., 303(154) Mehrer, H., 12(27) Meischner, P., 303( 153) Mermin, D. W., 259(53), 263 Mermin, N. D., 2(1), 19(45), 188(95) Meynhhrd, N., 210( 111-112), 212( 111-112) Migdal, A., 239(26) Miller, J . A,, 273(91) Miller, R. E., 274(96) Millers, A., 165(57) Minchau, B. J., 145(25), 155(25) Moak, C. D., 233(8), 236-237(20), 250(20), 267-268(20), 273(91), 283(8), 287(8), 303(151) Morel, P., 195(100) Moriarty, J. A,, 20(47-49), 21(58-59), 22 Moritz, W., 76(146) Moruzzi, V. L., 31(81), 45 Mott, N. F., 255(49), 277(104) Munsfeldt, H., 274(93)

N Naaman, R., 273(89) Nakayama, M., 183(71)

315

AUTHOR INDEX Neelavathi, V. N., 236(20), 250-252(20), 267(20,66), 268(20) Nelson, D. R., 155(41-42), 208(108), 214(108) Neufeld, J., 235(14), 236(20), 250-252(20), 267-268(20) Newns, D. M., 187(86) Nicholson, D. M., 63(112) Niemenen, R. M., 19(44), 30(75), 278(113), 279(114), 283(124), 284-285(126) Nightingale, M. P., 151(33) Nerskov, J. K., 30(73,76,78), 32-33(78), 34,35-36(78) Northcliffe, L. C., 233(9) Nozieres, P., 165(57), 184(79), 212(122), 215( 125-126), 239-240(27), 242(34)

0 Oda, N., 269(73) Oh, D. J., 67(124) Oh, Y., 78(153) O’Hori, T., 275(99) Ohta, Y., 4(9), 30(71-72), 31(9), 70(72) Ohtsuki, Y. H., 232(4), 236(4), 245(36), 252-253(36), 264-265(4), 275(99), 289(131), 292( 141), 295( 131) Okorokov, V. V., 273(90) Outuka, A., 271(82) Overhauser, A. W., 187(89-91)

P Pandey, K. C., 86(199) Panke, H., 271(85-86) Papanicolaou, N., 44(91) Parrinello, M., 3(7), 72(139), 87(203) Pascual, P., 238(21) Pawlowska, Z., 66(119) Paxton, A. T., 4(9), 31(9) Pearson, E., 79(159), 80(c, 159), 81(159), 84, 85(185) Pelcovits, R. A , , 145(25), 155(25,42) Pendry, J. B., 295(144) Penn, D. R., 253(47), 295(145) Peo, M., 68(c) Pepper, M., 151(29) Perrot, F., 159(49), 170-171(49), 176(49) Pettifor, D. G., 4(9, l l ) , 19(11), 24(11,65),

27(68), 30(71-72), 31(9), 45(95), 48(95, 97), 50-53(95), 70(72) Phillips, J. A., 303(149), 304 Phillips, J. C., 24(66) Phillips, R. B., 52, 53(102) Pierce, T. E., 233(9) Pines, D., 165(57), 184(74-75), 215(125), 235(16), 239-240(24-25,27), 242(24, 34), 243(24), 245(25) Pinski, F. J., 63(112) Platzman, P. M., 135(15), 173(59-61), 177(59-61), 289( 133) Plesser, J., 273(89) Ponce, V. H., 269(75) Popovic, Z. D., 278-279(111) Poulsen, U. K., 45,48(98) Prange, R. E., 157(45) Puska, M. J., 30(75,78), 32-33(78), 34, 35-36(78), 278(113)

Q Quinn, J. J., 242-243(30)

Raether, H., 307(165) Raghavachari, K., 86(200) Rahman, A,, 85(181, 194), 86(194) Ramakrishnan, T. V., 219(130), 222(130) Ramesh, S., 85(188) Rasolt, M., 18(42), 20, 136-137(16), 138(20), 141(21), 153(21), 154(20), 157(16), 159(16,49), 170-171(16,49), 176(16,49), 179-180(63), 183(63), 195(99), 197(99), 200,201(106), 204( 106), 206( 106). 218( 129) Ratkowski, A., 269(74) Ray, J . R., 85(192, 194), 86(194) Read, N., 187(86) Rebonato, R., 67(128) Reimann, C. T., 78(151) Reinheimer, J., 253(48) Remillieux, J., 269(77) Rezayi, E. H., 160(52), 161(53-54), 182(64) Ribarsky, M. W., 85(188-190) Rice, T. M., 112(5), 184(78), 187(85), 210(116), 212(116) Richards, W. G., 14(32)

316

AUTHOR INDEX

Ritchie, R. H., 233(7), 235(14, 19), 236-237(20), 242(34), 246(39-40), 248(39-40), 250(20,43), 251-252(20, 44), 259(40, 44),262(40, 54), 264(44), 267(20,40,65,68-69), 268(72), 269(74, 79), 271(83), 272(87), 273(92), 275( loo), 276( 102), 278( 102), 279( 114). 284-285(126), 295(7), 303(148), 307(161-163, 165-168), 308(170-171) Rivacoba, A., 267-268(70) Roberto, J . B., 69(f) Robinson, J. E., 283(121), 287(121) Romanov, Y. A , , 275(97) Ross, C. G., 303(157) Rossbach, J., 274(93) Rossmainth, F . , 274(93) Roth, L. M., 130(13), 201(105), 202, 207( 105) Roulet, B., 212(121) Rumyantsev, V. V., 275(97) Rutherford, E . , 231( 1)

S Sachdev, S., 214(124), 216, 217-218(124), 222( 124) Saile, B., 68(c) Sakai, A,, 32(83) Sakai, T., 79(160) Samson, J. H., 49(99-100) Sanchez, J. M., 62(108) Sarasola, C., 271(83) Saso, T., 187(88) Sato, A. K., 78(152) Scharft, M., 264(62) Scherzer, B. M. V., 303(153) Schiff, L. I., 238(21) Schilling, W., 12(27), 69(d) Schlottmann, P., 188(94) Schliiter, M., 86(201), 197(102) Schmit, J. N., 72(141), 73(b) Schneider, M., 85(181) Schober, H. R., 4(10) Schrieffer, J. R., 208(107), 244(35) Schuller, I. K., 85(181) Schultz, F., 283(120, 123), 287(120, 123) Schulz, H., 209(109) Schumacher, D., 12(27), 69(d) Seeger, A , , 12(27), 68(c-d), 69(d)

Segre, E., 232(4), 236(4), 264-265(4) Seitz, F., 3(6), 4(12), 23(12), 27(12), 56(6), 112(5), 116(8), 130(10) Sellin, I. A., 307( 169) Serene, J . W., 187(85) Shaefer, H. E., 68(c) Sham, L. J., 32(82), 115(6), 183(69-71), 115(6), 279( 116) Shapiro, M. H., 78(151) Sharff, M., 308(174) Shevelko, V. P., 289(135), 292-293(135), 295(135), 297(135), 303(135) Shibata, H., 271(82) Shirane, G., 187(83) Shuttleworth, R., 71(129), 75(144) Sigmund, P., 277(105, 107) Silbert, M., 78(155) Sinclair, J. M., 37(85), 39(85), 67(85), 68(a) Singwi, K. S., 278(112) Sitenko, A. G., 235(14) Sizmann, R., 303, 155 Skriver, H. L., 48 Sofield, C. J., 306(160) Sols, F., 289(132, 137), 291(140), 295(132), 296(140, 146), 297(146), 303(146) Solyom, J., 210(111-115, 118), 211, 212(111-115, 118). 214 Som, D. K., 9(25), 46(25) Sorella, S., 222(132-133) Sorensen, J. E., 75(b) Souloulis, C. M., 85-87( 195) Spenner-Harper, M., 306( 160) Spindler, E., 271(86) Srolovitz, D. J., 71-72(133), 75(133) Stachowiak, H., 259(50) Stehling, W., 271(85-86) Stensgaard, I., 75(b) Stern, F., 133(14), 135(14) Sterne, P., 63(112) Stever, M. F., 269(78) Stillinger, F. H., 79(158), 80(a, 169), 84, 85( 169) Stocks, G. M., 63(112-113) Stoloff, N. S., 63(112) Stoneham, A. M., 4(10) Stormer, H. L., 151(30) Stott, M., 30(74), 34(74), 78(155) Stott, M. J., 278-279(111) Su, J. J., 274(95) Sudakov, V. V . , 212(119)

317

AUTHOR INDEX Sung, C. C., 233(7), 295(7), 307(163), 308( 170) Sutton, A. P., 4(9), 30(71-72), 31(9), Suzuki, K.,275(99) Swaroop, A,, 14(35)

T Tabet, E., 222(132-133) Takai, T., 79(159-160), 80(c, 159), 81(159), 84,85(180, 197) Tamm, I., 263(58), 272(58) Tape, J. E., 269(77) Taylor, P. A , , 85(191) Taylor, R., 4(13, 16), 18(42), 19(13), 20, 60(106) Tejada, J., 272(87) Teller, E., 81(172), 234(10), 275 Temmerman, W. M., 63(112) Ter Matriosian, K. A,, 212(119) Terreault, B., 303(157) Tersoff, J., 12, 13(29), 32(83), 88(29, 204-205), 89(29,209), 90(29,204) TesanoviC, Z., 189(96), 190, 192-193(96), 200-201(106), 204(106), 206(106), 218(129), 223 Thetford, R., 78(155) Thiaville, A,, 208(108), 214(108) Thomas, L., 287( 129) Thouless, D. J., 151(33), 155(38-40) Tichy, G., 35,38,41-42,67-68(120), 69(b), 71(120), 72(b, 120), 73(a), 75(b), 76 Tiller, W. A., 79(159-160), 80(c, 159), 81(159), 84, 85(180, 184-185, 197), 86( 184) Ting, C. S., 210(117), 212(117) Tinkham, M., 95(1) Tolchenkov, D. L., 273(90) Tolk, N. H., 289(134), 295(134) Tomanek, D., 75(145), 86(201) Tombrello, T. A , , 78(151) Tomlinson, S. M., 5(18) Tong, S. Y., 295(144) Tonks, L., 235(17) Torrens, I. M., 4(14) Torres, V. T. B., 4(10) Tosatti, E., 72(139) Travis, D. N., 14(32)

Tsui, D. C., 151(30) Tully, J. C., 289(134), 295(134) Turchi, P., 45(95), 48(95), 50-53(95) Turchi, P. E. A., 63(112) Turnbull, D., 3(6), 4(12), 23(12), 27(12), 56(6), 79(164), 112(5), 116(8), 130(10), 242-243 (29), 245(29)

U Ugalde, J., 271(83), 272(88) Upadhyaya, J. C., 20(53-54)

v Vager, Z., 262(55-57), 272(56-57), 273(89) Vanderbilt, D., 138(20), 141(21), 153(21), 154(20) Van Hove, L., 240(28) Van Hove, M. A,, 76(147), 295(144) van Schilfgaarde, M., 5(22) Varelas, C., 283( 122), 287( 122) Varma, C. M., 210(116), 212(116) Vashishta, P., 85(178) Verbeek, M., 303(153) Vinter, B., 183(69-70) Vitek, V., 26(67), 35,38,41-42, 67-68(120), 69(b), 71(120), 72(b, 120), 76, 78(153) Vlasov, A. A., 235(17) von Barth, U., 278-279(111) Vosko, S. H., 128(9), 259(51) Voss Tiweiland, G. A,, 274(93) Voter, A. F., 71-72(133), 75(133)

W Walecka, J. D . , 98(2), 155(2) Watson, R. E., 24(64) Weaire, D., 3(6), 19(6), 45(95), 48(95), 50-53(95), 56(6) Weast, R. C., 12(28), 15(28) Weber, T. A., 79(158), 80(a, 169). 84, 85(169) Wegner, H. E., 269(77) Weinert, M., 77( 149) Weiss, J . J., 159(47-48)

318

AUTHOR INDEX

Weizsacker, Z . , 234(12) Welch, D. O., 67(128) Weller, M., 68(d) Wenzl, H., 69(d) White, C. W., 289(134), 295(134) Wilets, L., 308(176) Wilkins, J. W., 187(85) Willeke, F., 274(93) Williams, A. R., 31(81), 45, 63(114) Williams, E. J., 234(12) Wills, J. M., 21(56) Wilson, P. B., 274(95) Wimmer, E., 77(149) Winograd, IS.,78(151) Winter, H., 63(113) Winther, A., 276(101) Wolf, D., 76(146) Woods, C. J., 306(160) Wragg, A., 44(91)

Y Yamazaki, Y . , 268(71), 269(73)

Yarlagadda, B. S . , 283(121), 287(121) Yaruyama, Y., 303(158), 305 Yasilove, S. M., 75(d) Yin, M. T., 86(198) Yoshioka, D., 136(17-19), 137(18), 151(34), 174- 175(62), 188(93)

Z Zabransky, B. J . , 262(56), 269(78), 272(56), 273(89) Zajfman, D., 273(89) Zak, J., 130(10) Zallen, R., 87(202) Zangwill, A,, 71(132), 73(b) Zaremba, E., 30(74), 34(74), 307(164) Zhang, F. C., 160(50-51) Zhang, X.-G., 63(112) Zhukova, Y. N., 273(90) Ziegler, J . F., 232(4), 236(4), 264-265(4), 307( 162) Ziman, J . M., 2(4), 24(63), 195(101) Zunger, A., 79(164)

Subject Index

A

Bonding mechanisms, 4 Bond-orbital methods, 5 Born approximation of collision theory, 232 Born-Oppenheimer approximation, 2 Bound ion-electron composite, 246-248 one-electron ions, 246 with an electron gas, 246-248 Bound ion-electron composite interactions, 248-250 capture process, 249 excitation in the electron gas, 249 localized state, 249 loss process, 249 orthogonalized plane wave (OPW) functions, 259 scattering process, 249 Bow waves, 256 Bragg maximum, 333 Broken bonds, 23-42 effective interatomic potentials, 36-42 back bond, 40 bulk bond, 40 dimer molecule bond, 40 effective pair potentials, 38, 40-41 empirical pair potential, 38 local environment, 39 reference environment concept, 36-42 triplet potentials, 40-41 uniform reference environment, 39 embedded-atom analysis, 30-36 broken bond problems, 36 constant-volume potentials, 36 d-d interaction terms, 36 embedding function, 33-36 empirical curves, 35 local density approximation, 32 Pauli kinetic energy, 34 valence electrons, 34 tight-binding analysis, 23-30 band shape, 26 bandwidth, 24,29

Abelian, 112 A b initio schemes, 64-66 Abrikosov lattice theory, 201,205, 207-208 Alignment-ring patterns, 272-273 polyatomic ions, 273 Amorphous Si, 86-87 Angular forces, 4 cluster potentials, 79 semiconductors, 78 Si, 78-88 Amatz, 59 Ansiotropic Heisenberg Hamiltonian, 145, 182 Antiferromagnetic alignment in isospin-spin space, 105-109 density profile, 107 spin current density, 108-109 spin-density profile, 107 spin-density wave (SDW), 109 totally orthogonal space, 107 valley-density wave (VDW), 108 Arbitrary polarization, 177 Atomistic simulations, 3, 5, 31 Auger cascades, 231 Auger processes, 288-294 binding energies, 290-291 momentum transfer, 294 Axilrod-Teller triple fluctuating dipole interaction, 81

B Band structure symmetry, 95-98 single-particle states, 96 BCS type models, 2 Bogoliubov transformation, 154 Bohr radius, 237 Bonding energetics, 3 319

320

SUBJECT INDEX

Broken bonds-(conrd. ) tight-binding analysis-(contd. ) Pauli exclusion principle, 29 semiconductors, 24 square valence-band width, 26 total electronic band energy, 25 transition metals, 23-27 transitional-metal cohesive energies, 27 Broken symmetry states, 132

C Capture cross sections, 303 Cartesian geometry, 160 Charge density wave (CDW), 188-189 CDW state 156-157 Charge states, of moving ions, 303-306 cross sections, 304-306 function of velocity, 304-306 Cherenkov radiation, 152, 234 Circular response, 175 Cluster functionals, 8-9, 11,42-53, 88-90 angular forces, 9 Baskes scheme, 88-89 configurational energy. 8 constant volume, 53-57 densities of state, 44-46 bcc, 46 canonical d-band, 44-45 diamond and fcc for Si, 45-46 fcc and shcp, 44-46 higher moment, 43 low-order, 48 maximum entropy, 44,49 second moment, 43 Tersoff scheme, 88-90 Cluster potentials, 7-8, 79-87 angles, 7 Axilrod-Teller triple fluctuating dipole interaction, 81 BH, SW, and PTHT potentials, 80-87 diamond structure, 80-82 higher-order terms, 7-8 structural moments, 81 three-body terms, 7-8 Coherent resonant interaction, 289 Coherent resonant processes, 292-295 capture process, 294 crystal structure, 292

electrostatic potential, 292 momentum transfer, 294 moving potential, 292 pseudopotential, 295 reciprocal momenta, 294 Cohesive energy, 64,67 measuring, 64 Colinear isospin components, 217 Concentration fluctuation variables, 62 Condensed matter theory, 1 bare interactions, 2 BCS-type models, 2 Heisenberg-type models, 2 quantum mechanics, 2 Configurational energies, 7 Constant-concentration effective reactions, 63-64 analogy to constant volume, 64 matching schemes, 63 mean-field multiple scattering schemes, 63 pair interaction, 64 Constant-volume interatomic potentials, 17-23 average volume, 19 d-band metals, 20 Friedel or Ruderman-Kittel oscillations, 19 induced charge contributions, 18 interatomic d-d couplings, 21 local volume, 19 pseudopotential, 17 s-d coupling V,,, 20 three-body terms, 21 valence-based structure, 17 Constant-volume potentials, 58-59 cluster functionals, 53-57 bond-breaking cluster potentials, 56 bond-breaking series, 55 change of basis, 57 crystal structure energy differences, 53 pseudopotential, 56 single-band tight-binding model, 54 Coulomb field, 230 Coulomb hole (ch), 245 Cross sections Eikonal approximation, 297 H, He, B, C, N , Q, 296-303 Shell, Augur and resonant, 298-303 Crystal structure energy differences, 11

321

SUBJECT INDEX

Damping effects, 259 Mermin dielectric response function, 259 Delta function interaction, 209-210 Densities of states v4, 44 bcc structure, 45 fcc structure, 45 hcp structure, 45 Density-based theories, 5 Density functional approach, 278-279 induced electron density, 279 linear theory, 279 real metallic densities, 278-279 Density-functional theory (DFT), 115 Diamond-structure semiconductors, 15 dimer molecule, 16 fcc structure cohesive energy, 15 Dielectric function, 234-235 frequency, 235 random phase approximation, 235 wave vector, 235 Dipole approximation, 232 Dipole sum rule of atomic physics, 232 Dissipation in isospin space, 151-156 excitation spectrum, 151 fractional quantum Hall effect, 151 integer quantum Hall effect, 151 via the valley wave, 152-154 Bogoliubov transformation, 154 carrier density, 153 Cherenkov radiation, 152-154 impurity potential, 153-154 matrix elements, 153-154 Dynamic screening, 271-272 transition energies, 271

E Effective Hamilton frequency-dependent corrections, 120 higher order corrections from Gc, 120 isospin invariant, 118 Linhard dielectric function, 120 RPA Feynman graphs, 116 valence electron response, 120 valley electrons, 116-120

Effective interatomic potentials (see also Broken bonds), 11 Effective-mass Hamilton, 122- 125 isospin exchange contribution, 125 isospin rotation, 125 quartz interactions, 122 symmetry-breaking terms, 124 symmetry relations, 124 Effective pair potentials, 3 Eikonal approximation, 297 Eigenvectors, 239 Elastic constants, 67, 78 Cauchy discrepancy, 78 Electromagnetic radiation, 234 Electron-electron scattering, 195 Electron-hole scattering, 190 Electronic densities of state, 28 Electronic exchange processes, 288-289 Augur process, 288-289 coherent resonant interaction, 288-289 low velocities, 288 shell process, 289 Electrons, 230, 244 bound, 262 convoy, 231 gas, 244 secondary, 231 valence, 230 v << v", 231-232 Embedded-atom analysis, see Broken bonds Energy functionals, 5-10 broken bond energies, 10 cluster functionals, 8 cluster potentials, 7 crystal structure energy differences, 10 pair functionals, 8 pair potentials, 6 relaxation geometries, 10 Shrodinger equation, 9 Equation of state potential, 60 Equilibrium lattice constants, 67 Experiment (ab inifio)calculations, 11 External magnetic field, effect of, 130-132 kinetic energy, 131

F Fermi-Dirac occupation function, 192 Fermi golden rule, 247

322

SUBJECT INDEX

Ferromagnetic alignment of the isospin ground state wave function, 105 spin ground state, 102-105 unitary transformation, 102-105 Fitted functionals, 64-66 Fitted potentials, 64-66 fitted schemes, 65-66 Fock cyclic condition, 160-161 Four-body potential, 52-53 Fractional quantum Hall effect, 151 Fractional quantum Hall effect (FQI ground state in isospin space, 156- 165 CDW state, 156 interparticle interaction, 157 isosopin degree of freedom, 157 Laughlin state, 156-157 symmetry considerations, 160-165 eigenstate, 164 Fock cyclic condition, 160-161, 164 Haldane’s truncated pseudopotential method, 161 Laguere polynomials, 163 Landaux level index, 162 multilayer heterojunction, 165 permutation symmetric Hamiltonian, 164 Young symmetry, 161 tuncated pseudopotential, 157 variational considerations, 157-160 antisymmetric wave functions, 158 Cartesian geometry, 160 cylindrical gauge, 157 hyperrated chain technique, 159 Monte Carlo methods, 159 three triangular sublattice ground-state structure, 157 Friedel or Ruderman-Kittel oscillations, 19 Friedel sum rule, 278 Fukuyama approximation, 192

G Gaussian functions, 197 General polarization, 177 g factor, 132 Global interatomic potentials, see Semiconductors and transition metals

H Haldane’s truncated pseudopotential method, 161 Hartrees measurement, 237 Heisenberg-type models, 2 He nuclei in electron gas, 279-283 linear theory, 279-280 v/v,, 281 Hubbard approximation, 184-186 Hydrodynamic approximation, 255 Hyperrated chain technique, 159

I Impurity density, 147 Induced density fluctuation, 306 Induced screen change contributions, 245 Inelastic collision time (mean), 238 Inelastic scattering, 238 Integer quantum Hall effect (QHE), 151, 156 ground state, 156 isospin space, 132-156 Slater determinant, 156 Interatomic pair potential, 10 Interatomic potentials, 3-4, 16-17, 53, 57-61 ansatz, 59 constant-volume potentials, 16-17, 58-60 density matrix, 60 equation of state potential, 60 explicit linear transformation, 17 king parameters, 17 maximum entropy principle, 59 probability distribution, P([R}), 58 Interparticle screening, 132 Ions charge states of, 287, 303-306 energy loss of, 234,264 heavy, 262 one-electron, 246 polyatomic, 273 speed, 234 velocity, and, 275 with Z, >2, 283-286 Ising configuration, 141, 145 king ground state, 147

SUBJECT INDEX Ising-type models, 5 Isospin BS, 95 charge density wave (CDW), 188-189 critical temperature, 189, 194 electron-hole scattering, 190 external magnetic field, 187-194 graphite, 194 instabilities, 187- 190 kinetic energy, 187 Matsubara frequencies, 191 VDW, 188-189 Wigner crystal, 188 Zeeman splitting, 188-190 Isospin degree of freedom, 132, 175 Isospin flip response function, 148 Isospin fluctuations, 183 Fermi liquid state, 214-222 colinear isospin components, 217 diagonal vertex functions, 218 diffusion ladders, 219 diffusion propagators, 221 disorder, 219-221 impurity scattering, 220,222 integrands, 221 isospin degrees of freedom, 222 parameters, 217-218 quasiparticle interaction, 216 spin antisymmetric, 216 spin symmetric, 216 time reversal symmetry, 215 two-spin component, 215 Isospin space effective-mass Hamiltonian, 95, 98-114 antiferromagnetic alignment, 105-109 band-structure symmetry, 95 broken symmetries, 109-113 coherence, 111 ferromagnetic alignment, 102-105 orthogonal space, 107 Pauli matrices, 101 periodicity, 111 polarization of valleys, 109-1 10 SU,,,, continuous group, 101 symmetry in uniform magnetic field, 113-114 two-valley system, 101 fractional quantum hall effect, 156-182 arbritary polarization, 177 circular response, 175 density-density mode, 173, 175-177

323

density operator, 169 fully isospin polarized FQHE, 175-177 general polarization, 177 interparticle interaction, 178 inversion symmetry, 171 isospin degree of freedom, 175 isospin unpolarized FQHE, 175-177 kinetic energy, 168-169 longitudinal isospin response, 175-177 magnetic reciprocal lattice vector, 178 magneto roton modes, 173 QHE in two-layer heterojunction, 178-182 single mode approximation, 165-168 spin current operator, 172 spin excitation mode, 175 symmetry, 160-165 variations, 157-160 Integer quantum Hall effect, 136-156 anisotropic Heisenberg Hamiltonian, 145 antiferromagnetic isospin state, 137 a posteori, 138 BS ferromagnetic isospin ground state, 137 degeneracy, 138 disorder and isospin polarization, 142-148 dissipation in isospin space, 151-156 ground state energy configuration, 136 Hartree-Fock (HF) calculation, 136 impurity density, 147 king BS, 145-147 Landau gauge, 136 magnetic analog, 144 multilayer heterojunction, 145 normal state, 136 polarization, 135 random field, 144 silicon inversion layer, 133-138 single impurity, 143 Slater determinant, 135, 138 symmetry-breaking terms, 138-141 triangular isospin sublattices, 137 valley waves, 148 Isospin-spin rotations, 187 Isospin superconducting condensate, 199-209 Abikosov lattive, 201, 207 theory, 205,207

324

SUBJECT INDEX

Isospin superconducting condensate-(contd. ) copper pair, 207-208 delta formation, 207 field limits, 200 free energy, 207 Landau Ginsburg free energy density expansion, 199,204-205 London relation, 201 magnetic field, 201 Meissner effect, 201 relation, 204 noninteracting single-particle Green’s function, 200-201,203 order parameter, 201,204,207-208 quantum fluctuations, 208 superconducting tubes, 207-208 two-particle kernel coupling, 205 Wigner lattice, 208 Isospin in three dimensions, 182-187 absence of external magnetic field, 183 correlation energy, 183 dynamic dielectric function, 184 electron pockets, 187 Hubbard approximation, 184-186 impurity energy, 186 isospin fluctuations, 183 isospin polarization, 186 isospin-spin rotations, 187 particle-hole rescattering, 184 Raman cross-sections for n-doped Si, 183 Isospin vortices, 147-148

K Kinetic energy, 132 Kosterlitz-Thouless transition, 155 Kramers-Kronig relations, 242

L Laguere polynomials, 163 Landau Ginsburg free energy density expansion, 199,204-205 Landau level, 132 Lattice constant, 69 Laughlin state, 156-158 Linear response theory, 275-276 Fermi-Teller result, 275

Friedel oscillations, 275 stopping power, 276 Linear theory of condensed matter, 250-267 London relation (see also Meissner effect) 20 1 Longitudinal isospin response, 175

M Magnetic field external, 130-132 isospin broken-symmetries, 183-194 isospin-spin symmetry, 113-1 14 spin and isospin space, 132-133 superconductivity, 194-209 with E,,, 138-142 Magneto roton modes (see also Density-density mode), 177 Matsubara frequencies, 191 Maximum entropy principle, 59 Meissner effect, 201 relation, 204 Meissner expulsion, 199 Mermim dielectric response function, 259 Monte Carlo methods, 159 Multilayer heterojunction, 165 Multiplicative renormalization group, 212 interparticle interactions, 213 Multivalley systems, 133

N Nearest neighbor interaction, 71 Noninteracting single-particle Green’s function, 200-201,203 Nonlinear wake, 262 charge densities, 262 scattered Coulomb wave function, 262

0 Okorokov effect, 306

P Pair functionals, 11, 48, 50, 66-78 bcc and fcc transitional metals, 66

SUBJECT INDEX lattice constant, 69-70 pair potential, 68-69 relaxation volume, 67,69-70 vacancy properties, 67 Pair potentials, 5-6 Paraisospin-spin state, 100 Pauli exclusion principle, 29 Pauli repulsion, 30 Pauli term, 132 Peierls distortion, 50-51, 130-131 Permutated symmetric Hamiltonian, 164 Perturbation theory, 231,239 Phase shift analysis, 277-278 Fermi level, 277-278 Feynman diagram methods, 277 Frieda1 sum rule, 278 Pauli principle, 277 Photon excitation of atoms theory, 232 Plasmon dispersion, 252-253 hydrodynamic dielectric function, 252 polarization density, 253 singularities, 253 Plasmon pole approximation, 268 Poisson’s equation for scalar potential, 237 Polyatomic ions, 273 Probability distribution P ({R,}),58 Protons in electron gas, 279-283 linear theory, 279-280 v/vf,281 Pseudopotential, electron-ion, 4

325

superconductivity, 210 three-dimensional nature, 209 Umklapp processes, 210 VDW, 210 Quantum theory of electron gas, 235 collective oscillations, 235 equation of motion, 235 fourier space, 235 Quantum theory of wake. 29-262 quanta1 function, 259 unfolding technique, 262 swift positronium atom, 262

R Random field effect on dissipation, 154-156 critical velocity v,, 155 dispersions, 154 domains, 155 drift velocity ud, 155 finite temperature, 155 impurities, 154-155 king BS, 155 Kosterlitz-Thouless transition, 155 magnetic length, 155 Related energy functionals, 3-4 Resonant coherent excitation, 273 He+ ions, 273 Rydberg states, 231

5 Quanta1 self energy external electron, 242-246 exchange processes, 242-244 second order energy, 243 moving charge, 239-242 Hamiltonian equations, 239 projectile, 236-239 self energy, 236 Poisson’s equation, 237 Quantum fluctuations effect, 209-214 CDW, 210 delta function interaction, 209-210 one-dimensional forms, 209 scattering processes, 210 SDW, 210

Second neighbor distance, 69 interaction, 71 Second-order perturbation theory, 146 Self wake effect, 268 Semiclassical approximation, 307 Semiconductors bond-orbital methods, 5 decompositions, 2 diamond structure, 4 energy functionals, 4 global interatomic potentials, 11-16 Cauchy relations, 12 covalent bonding, 13 defects curve, 14 dimer molecule, 12 equation of state, 14

326

SUBJECT INDEX

Semiconductor-(contd. ) global interatomic potentials-(contd. ) lattice relaxation effects, 13 metallic bonding, 13 molecule potentials, 14 radical pair potential models, 13 interactions, 2 pseudopotentials, 20 Shell processes, 289,295-296 ion velocity, 295 Silicon (Si) 110 inversion layer, 145 clusters, 86 excitation spectrum gap, 151 integer quantum Hall effect, 133 inversion layer, 145 random field in isospin space, 145 surface reconstructions, 85 Single mode approximation (SMA), 165- 168 circular spin response functions, 166 collective mode, 168 density-density response function, 168 longitudinal isospin response function, 168 Single-particle effects, plasmon pole approximation, 253-256 bow waves, 256 hydrodynamic approximation, 255 incident beam, 255 wake potential, 255 Single-particle Green’s function G,( w ) . 149 Slater determinant, 156 ground state, 138 Solenoidal focusing, 274 Solid solution analogy, 61-64 binary solid solutions, 61-62 configurational energetics, 61 effective king parameters, 62 Spherical geometry, see Cartesian geometry Spin density wave (SDW), 109 Spin space, see Isospin space Static lattice potential, 272 Stationary charges, 252 positive and negative, 252 Stopping power, straggling, 263-264 dampling, 263 Mermin response function, 263 straggling parameter, 264

Superconductivity BS ground state, 195 Coulomb repulsion, 196-197 critical temperature, 196-197 donor-doped silicon, 197 e-e interaction, 198 e-e repulsion, 196-197 electron-electron scattering amplitude, 195 electron-phonon interaction, 195 instability above T,, 194 isospin magnetic freezeout, 198 Meissner expulsion, 199 orbital pair breaking, 195, 199 screened electron-ion potential, 196 Zeeman splitting, 194 Surface plasmon mode, 307 Surface properties, 71-78 energy, 71 reconstructions, 72,76-77 relaxations, 72, 74-75 stress, 71-74 Surface vicinage effect, 274 conducting foils, 274 solenoidal focusing, 274 Surface wake, 274-275 vicinage effect, 275 SU(2)-symmetry Lie group, 94 Swift ions, 232,236,265 Symmetry-breaking energy (EsB) with external magnetic field, 141-142 change of states, 142 static imperfections, 142 thermal functions, 142 valley coupling, 141 without external magnetic field, 138-141 Bloch functions, 138 pseudopotential representation, 141 Symmetry-breaking terms correlation ground state, 127-129 decoupling, 127 effective-mass Hamiltonian, 114-132 density-functional theory (DFT), 115 Feynman graphical respresentation, 115 Hartree contribution, 115 Slater determinants, 127-128 SU(2) terms, 125-126 xy terms, 126-127

327

SUBJECT INDEX

T Thomas-Fermi statistical model, 233 Three-body interactions, 83 Three-triangular sublattice ground-state structure, 157 Tight-binding analysis, see Broken bonds Tight-binding approach, 67 Tight-binding models, 11, 48 canonical, 48 Total energy schemes, 64-66 ab initio schemes, 64-66 fitted schemes, 65-66 Transitional metals angular forces, 4 atomistic simulations, 5 bcc-fcc energy diffeence, 48, 50, 155 bonding energy, 3-4 density of stales, 34 energy functionals, 5-11 interatomic potentials, see Semiconductors tight-binding analysis (see also Broken bonds), 23-30 Transition-metal cohesive energies, 27 Transverse response, see Circular response Transverse susceptibility, 146 Triplet potentials, 15, 84

U Umklapp process, 210

V Vacancy-vacancy interactions, 71 unrelaxed, 71 Vacancy formation energy, 64, 67 Vacancy relaxations, 70 anisotropic part 70 Valence electron gas, 234 Valley-density wave (VDW), 108, 188-189 Valley waves 110 inversion layer of Si, 148-151 dispersion, 148, 151 gap influence, 151-152 impurity concentration, 151 magnetic analog, 151 random field, 151

isospin flip response function, 148 Laguere polynomials, 151 modified Bessel function, 151 quantum Hall effect, 148 single-particle Green’s function, G,(w), 149 vertex function, 150 Van der Waals fluctuating dipole interaction, 5 Velocities, 287-289 Vicinage effect modified Coulomb cluster, 269 valence electrons. 271

W Wake binding, 267-269 capture and loss, 268 oscillatory wake, 267 plasmon-pole approximation, 267 positive and negative ions, 268 self-wake effect, 268 wake bound electrons, 267 concept, 250-252 Bessel function, 251 induced density fluctuation, 250 oscillatory potential, 251 stationary charge, 252 energy fluctuations, 264-267 dewaking of an electron, 267 energy loss of ion, 264 quantal energy content, 267 scalar electric potential, 265 RPA plasmon threshold, 259 quantal dielectric response function, 256 wake potential, 256-259 Wake-field accelerators, 273-274 Wigner crystal, 188

X xy symmetry

isospin polarized ground state, 181 anisotropic Heisenberg Hamiltonian, 182

328 xy symmetry-(contd.)

isospin polarized ground state-(contd. ) anisotropy, 181 dispersion, 182 dissipation, 182 isospin invariant, 181

SUBJECT INDEX

Y Young symmetry, 161

Z Zeeman splitting, 188-190, 195,208