Electrons In Cylindrical Quantum Wires
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PROC. 23rd INTERNATIONAL CONFERENCE ON MICROELECTRONICS (MIEL 2002), VOL 1, NIŠ, YUGOSLAVIA, 12-15 MAY, 2002
Electrons in Cylindrical Quantum Wires S.M. Stojkovi c, B.S. To si c, J.P. Setraj ci c
|
and D. Popov
Abstract The cylindrical quantum wires are analyzed in this paper. Translational symmetry holds along x-axis. The number of atoms in circular discs is several ones. For example, the Scot's model of -helix contains three molecules in discs. The cylindrical quantum wires with three and four atoms are investigated. Energies of electrons as well as currents are found. It is interesting that in both cases electron energy spectrum is degenerated. In the case of four atoms all currents are equal, while in the case of three atoms the current of non-degenerated energy level is two times higher than others. Consequently, the wire with three atoms in discs works as modulator of electric impulses.
plied Hubbard model 4,5,8] in nearest neighbor approximation:
I. Introduction
The variable represents the energy of electron localized on the lattice site, while values W are electronic hopping terms for an ideal crystal. Operators a and a are Fermi operators of electron creation and annihilation. N is number of atoms in each of discs, while NB is number of discs. Analysis will be based on the single particle wavefunction:
There has been a great deal of recent interest in ultranarrow conned semiconductor systems, called quantum wires, where the motion of electrons is essentially restricted to be one-dimensional 1-2]. The quantum size eects become signicant in these structures, which implies essentially dierent physical properties in comparison to bulk ones. The considerable interest in submicron electronic structures has been motivated by expectation of potential applications in solid-state devices, such as high-speed transistors, e cient photodetectors and lasers 3]. Our previous investigations of quantum wire models, based on Green's function method, have shown in uence of wire boundaries, as well as dierent boundary conditions to the physical properties of quantum wires 4-5]. In variance to the this previous model, with rectangular cross section, in this paper cylindrical quantum wire is analyzed. This model has been proven as fruitful platform for studying a large variety of transport and physical phenomena. The interest is mainly focused on the phenomenon called periodic conductance oscillations (the device exhibiting those oscillations are referred to as single-electron transistors) 6]. II. Electron Properties of Quantum Wires
We consider quantum wire where translational symmetry holding along x-axis. The number of atoms in circular discs is several ones. For example, the Scot's model 7] of -helix contains three molecules in discs. We start our analysis from electron Hamiltonian of simS.M. Stojkovic, B. S.Tosic and J.P. Setrajcic are with Institute of Physics, Faculty of Sciences, University of Novi Sad Trg D. Obradovica 4, 21 000 Novi Sad, Yugoslavia, E-mail: [email protected] D. Popov is with Universitatea "Politehnica Timisoara", Catedra de Fizica, Timisoara, Romania S.M. Stojkovic is with Fritz-Haber Institut der Max-Planck Gesellschaft, Berlin, Germany, E-mail: [email protected]
X aa+abaab X Wab a b aabaa b
H=
ab a b
ab
where
+
0 0
0 0
(1)
0 0
a 2 1 N ] W b b = WNb b NB b NB N 10 : b 2 2 1
2 0
1 0
8
+
X Amnamn j 0i X j Amn j = 1
=
+
2
mn
mn
(2)
where j 0i is the ground state of the electron system. In order to determine the coe cients A we shall use the standard Heisenberg's equation of motion 8]:
iha_ cd = acd H ] acd(t) = acd(0) e
i!t
(3)
and also fact that:
iha_ cd = Eacd E = h ! :
(4)
fEacd acd H ]g j i = 0 :
(5)
It leads to: Substituting Hamiltonian (1) and applying standard procedure of calculation we get: (E c )acd +
X Wcd abaab = 0 ab
X Uca (Aa d
(6)
which in nearest-neighbor approximation becomes: (E c )Acd +
a
+1
+ Aa d ) = 0 (7) 1
where Uca = Wcd a d = Wcd a d . Since along ddirection translational invariance is hold, we shall use transformation: +1
Acd =
0-7803-7235-2/02/$10.00 © 2002 IEEE
389
1
c eida0 k
(8)
where a0 is lattice constant in direction d. Substituting (8) into (7) we obtain: (E c ) c + Uca (k) a = 0 (9)
and total current is
where Uca (k) = 2Uca cos a0 k. The equation (9) will be written in nearest-neighbors approximation, taking into account that index a denotes particles located on disc's edge: (10) Uca(k) = U (k) j c a j= a0 : Electron energies of quantum wire can be found by solving the secular equation (9) (system of equations). This leads to calculation of roots of determinant: %k 1 0 0 0 0 1 1 %k 0 0 0 0 0 0 1 %k 1 0 0 0
In the case of weekly coupled electrons (degenerated electron gas) when = 6W , electron energy spectra is shown on Figure 1, where reduced electron energies E are shown. E=W
X a
D=
where:
...
0 1
0 0
0 0 0 0
1 %k 1 0 1 %k
%Sk = ESU (k) S U (k) = 2W cos a0 k :
hek : j = 3 mN d
(16)
=0
(11) (12)
On the basis of these results it is possible to nd electrical current of quantum wire from following expression 9]: h e ( r r ) : (13) j = 2m ei In this work energies of electrons as well as currents of the quantum wires with three and four atoms in discs are found. A. Quantum Wire With Three Atoms In this part quantum wire with three atoms in each of discs is considered. It is assumed that atoms are distributed on the way forming triangle of equal sides. Electron dispersion law can be found from determinant (11) which, in this case, is of 3x3 order. Energies of electrons in quantum wire with tree atoms are: E1 = E2 = + 2W cos a0 k E3 = 4W cos a0 k : (14) Electron energy spectrum is degenerated. Electrical current of each energy level is found from expression (13): hek E1 : j11 = 41 j12 = j13 = 6mN d 1 h ek (15) E2 : 4 j11 = j12 = j13 = 6mN d h ek E3 : j11 = j12 = j13 = 3mN d
Fig. 1. Electron energy spectra of quantum wire with three atoms
Current of degenerated energy level has a dierent components, while in the current of non-degenerated level all components are same. On the other hand, currents of non-degenerated level are two times higher or lower than currents of degenerated energy level. B. Quantum Wire With Four Atoms In this case is also assumed that atoms are distributed at equal distances, so they form square. Determinant (11) is of 4x4 order and its solution gives us electron energies:
E1 = E2 = E3 = + 4W cos a0 E4 = 4W cos a0 k :
(17) (18)
Electron spectrum is also degenerated as in case of wire with three atoms. It can be seen that degenerated energy level lies in the middle between other two energy levels. Electron energy spectrum in the case of weekly coupled electrons is shown on Figure 2. Electrical current is found from expression (13) and it is shown that all current's components are same and 390
However, real quantum wire has di erent kind of disorders, impurities and roughness which can easily destroy conductance quantization. In the last few years this problem has drawn considerable interest of theorist 6,12,13]. III. Conclusion
Fig. 2. Electron energy spectra of quantum wire with four atoms
equal to: hek i k = 1 2 3 4 : jik = 4mN d
(19)
Total current is than:
hek : j = 4 mN d
(20)
On the basis of these two analyzed cases one can see that energy spectrum of quantum wire is degenerated and that number of levels with di erent energies is in both cases N 1, where N is number of atoms in disc. Furthermore, from expressions (16) and (19) we can come to conclusion that total electrical current is always: h ek : (21) j = N mN d
According to Landauer-Buttiker formula 10,11] (which treats electrons in mesoscopic quasi-onedimensional systems as non-interacting entities) the conductance of narrow ballistic channels (pure metals) is a staircase like function of the Fermi energy, with steps approximately 2e2 =h height. Our results for current are in agreement with this fact (our quantum wire is of innite length and without disorders). Actually, we can say that current in innite quantum wire without any disorders is integer multiple of current quant: h ek=mNd.
In this paper electron energy spectrum and electrical current of cylindrical quantum wire was investigated. Particularly attention was paid to the wire with three and four atoms in discs. It is interesting that in the both cases electron energy spectrum is degenerated. In the case of four atoms all currents are equal, while in the case of three atoms the current of non-degenerated energy levels is two times higher or lower than others. Consequently, the wire with three atoms in discs works as modulator of electric impulses what could have wide practical interest. The described model and given microtheoretical approach open a possibility to study presence of di erent kind of disorders as well as wire of nite length. This corresponds to more realistic case when periodic conductance oscillations, characteristic for single electron transistor, can appear. The model presented here is suitable to serve as a basis of such research. These problems will be object of our further investigations. References
1] J.Singh, Physics of Semiconductors and Their Heterostructures, McGraw-Hill, New York, 1993, p.228. 2] C.W.J.Beenakker, H. Van Houten, Quantum Transport in Semiconductor Nanostructures, Solid State Physics vol. 44: Semiconductor Heterostructures and Nanostructures, Academic, Boston, 1989. 3] C.Weisbuch, B.Winter, Quantum Semiconductor Structures, Academic, San Diego, 1991 (Chapter 6). 4] D.Lj.Mirjanic, J.P.Setrajcic, S.M.Stojkovic, D.Sijacic, I.D.Vragovic, IEEE Proc. 22nd MIEL 1 (2000) 173. 5] S.M.Stojkovic, D.Lj.Mirjanic, J.P.Setrajcic, D.D.Sijacic, I.K.Junger, Surface Science 477 (2001) 235. 6] K.Nikolic, A.MacKinnon, Conductance uctuations of narrow disordered quantum wires, http://www.arxiv.org/list/cond-mat9405055. 7] B.Tosic, Lj.Maskovic, M.Skrinjar, D.Kapor, G.Knezevic, J.Phys 3 (1991) 7619. 8] W.Jones and N.H.March, Theoretical Solid State Physics, Dover, New York 1985. 9] A.S.Davydov, Teoriya tverdogo tela, Nauka, Moskva 1976. 10] R.Landauer, Philos.Mag. 21 (1970) 863. 11] M.Buttiker, Phys.Rev.Lett. 57 (1986) 1761. 12] J.Cserti, G.Szalka, G.Vattay, Conductance in a Periodically Doped Quantum Wires, http:// www.arxiv.org/list/condmat 9710265. 13] Y.-L.Lu, Conductance of Finite Quantum Wire Connected to Reservoirs, http:// www.arxiv.org/list/cond-mat9706081
391
Electrons in Cylindrical Quantum Wires S.M. Stojkovi c, B.S. To si c, J.P. Setraj ci c
|
and D. Popov
Abstract The cylindrical quantum wires are analyzed in this paper. Translational symmetry holds along x-axis. The number of atoms in circular discs is several ones. For example, the Scot's model of -helix contains three molecules in discs. The cylindrical quantum wires with three and four atoms are investigated. Energies of electrons as well as currents are found. It is interesting that in both cases electron energy spectrum is degenerated. In the case of four atoms all currents are equal, while in the case of three atoms the current of non-degenerated energy level is two times higher than others. Consequently, the wire with three atoms in discs works as modulator of electric impulses.
plied Hubbard model 4,5,8] in nearest neighbor approximation:
I. Introduction
The variable represents the energy of electron localized on the lattice site, while values W are electronic hopping terms for an ideal crystal. Operators a and a are Fermi operators of electron creation and annihilation. N is number of atoms in each of discs, while NB is number of discs. Analysis will be based on the single particle wavefunction:
There has been a great deal of recent interest in ultranarrow conned semiconductor systems, called quantum wires, where the motion of electrons is essentially restricted to be one-dimensional 1-2]. The quantum size eects become signicant in these structures, which implies essentially dierent physical properties in comparison to bulk ones. The considerable interest in submicron electronic structures has been motivated by expectation of potential applications in solid-state devices, such as high-speed transistors, e cient photodetectors and lasers 3]. Our previous investigations of quantum wire models, based on Green's function method, have shown in uence of wire boundaries, as well as dierent boundary conditions to the physical properties of quantum wires 4-5]. In variance to the this previous model, with rectangular cross section, in this paper cylindrical quantum wire is analyzed. This model has been proven as fruitful platform for studying a large variety of transport and physical phenomena. The interest is mainly focused on the phenomenon called periodic conductance oscillations (the device exhibiting those oscillations are referred to as single-electron transistors) 6]. II. Electron Properties of Quantum Wires
We consider quantum wire where translational symmetry holding along x-axis. The number of atoms in circular discs is several ones. For example, the Scot's model 7] of -helix contains three molecules in discs. We start our analysis from electron Hamiltonian of simS.M. Stojkovic, B. S.Tosic and J.P. Setrajcic are with Institute of Physics, Faculty of Sciences, University of Novi Sad Trg D. Obradovica 4, 21 000 Novi Sad, Yugoslavia, E-mail: [email protected] D. Popov is with Universitatea "Politehnica Timisoara", Catedra de Fizica, Timisoara, Romania S.M. Stojkovic is with Fritz-Haber Institut der Max-Planck Gesellschaft, Berlin, Germany, E-mail: [email protected]
X aa+abaab X Wab a b aabaa b
H=
ab a b
ab
where
+
0 0
0 0
(1)
0 0
a 2 1 N ] W b b = WNb b NB b NB N 10 : b 2 2 1
2 0
1 0
8
+
X Amnamn j 0i X j Amn j = 1
=
+
2
mn
mn
(2)
where j 0i is the ground state of the electron system. In order to determine the coe cients A we shall use the standard Heisenberg's equation of motion 8]:
iha_ cd = acd H ] acd(t) = acd(0) e
i!t
(3)
and also fact that:
iha_ cd = Eacd E = h ! :
(4)
fEacd acd H ]g j i = 0 :
(5)
It leads to: Substituting Hamiltonian (1) and applying standard procedure of calculation we get: (E c )acd +
X Wcd abaab = 0 ab
X Uca (Aa d
(6)
which in nearest-neighbor approximation becomes: (E c )Acd +
a
+1
+ Aa d ) = 0 (7) 1
where Uca = Wcd a d = Wcd a d . Since along ddirection translational invariance is hold, we shall use transformation: +1
Acd =
0-7803-7235-2/02/$10.00 © 2002 IEEE
389
1
c eida0 k
(8)
where a0 is lattice constant in direction d. Substituting (8) into (7) we obtain: (E c ) c + Uca (k) a = 0 (9)
and total current is
where Uca (k) = 2Uca cos a0 k. The equation (9) will be written in nearest-neighbors approximation, taking into account that index a denotes particles located on disc's edge: (10) Uca(k) = U (k) j c a j= a0 : Electron energies of quantum wire can be found by solving the secular equation (9) (system of equations). This leads to calculation of roots of determinant: %k 1 0 0 0 0 1 1 %k 0 0 0 0 0 0 1 %k 1 0 0 0
In the case of weekly coupled electrons (degenerated electron gas) when = 6W , electron energy spectra is shown on Figure 1, where reduced electron energies E are shown. E=W
X a
D=
where:
...
0 1
0 0
0 0 0 0
1 %k 1 0 1 %k
%Sk = ESU (k) S U (k) = 2W cos a0 k :
hek : j = 3 mN d
(16)
=0
(11) (12)
On the basis of these results it is possible to nd electrical current of quantum wire from following expression 9]: h e ( r r ) : (13) j = 2m ei In this work energies of electrons as well as currents of the quantum wires with three and four atoms in discs are found. A. Quantum Wire With Three Atoms In this part quantum wire with three atoms in each of discs is considered. It is assumed that atoms are distributed on the way forming triangle of equal sides. Electron dispersion law can be found from determinant (11) which, in this case, is of 3x3 order. Energies of electrons in quantum wire with tree atoms are: E1 = E2 = + 2W cos a0 k E3 = 4W cos a0 k : (14) Electron energy spectrum is degenerated. Electrical current of each energy level is found from expression (13): hek E1 : j11 = 41 j12 = j13 = 6mN d 1 h ek (15) E2 : 4 j11 = j12 = j13 = 6mN d h ek E3 : j11 = j12 = j13 = 3mN d
Fig. 1. Electron energy spectra of quantum wire with three atoms
Current of degenerated energy level has a dierent components, while in the current of non-degenerated level all components are same. On the other hand, currents of non-degenerated level are two times higher or lower than currents of degenerated energy level. B. Quantum Wire With Four Atoms In this case is also assumed that atoms are distributed at equal distances, so they form square. Determinant (11) is of 4x4 order and its solution gives us electron energies:
E1 = E2 = E3 = + 4W cos a0 E4 = 4W cos a0 k :
(17) (18)
Electron spectrum is also degenerated as in case of wire with three atoms. It can be seen that degenerated energy level lies in the middle between other two energy levels. Electron energy spectrum in the case of weekly coupled electrons is shown on Figure 2. Electrical current is found from expression (13) and it is shown that all current's components are same and 390
However, real quantum wire has di erent kind of disorders, impurities and roughness which can easily destroy conductance quantization. In the last few years this problem has drawn considerable interest of theorist 6,12,13]. III. Conclusion
Fig. 2. Electron energy spectra of quantum wire with four atoms
equal to: hek i k = 1 2 3 4 : jik = 4mN d
(19)
Total current is than:
hek : j = 4 mN d
(20)
On the basis of these two analyzed cases one can see that energy spectrum of quantum wire is degenerated and that number of levels with di erent energies is in both cases N 1, where N is number of atoms in disc. Furthermore, from expressions (16) and (19) we can come to conclusion that total electrical current is always: h ek : (21) j = N mN d
According to Landauer-Buttiker formula 10,11] (which treats electrons in mesoscopic quasi-onedimensional systems as non-interacting entities) the conductance of narrow ballistic channels (pure metals) is a staircase like function of the Fermi energy, with steps approximately 2e2 =h height. Our results for current are in agreement with this fact (our quantum wire is of innite length and without disorders). Actually, we can say that current in innite quantum wire without any disorders is integer multiple of current quant: h ek=mNd.
In this paper electron energy spectrum and electrical current of cylindrical quantum wire was investigated. Particularly attention was paid to the wire with three and four atoms in discs. It is interesting that in the both cases electron energy spectrum is degenerated. In the case of four atoms all currents are equal, while in the case of three atoms the current of non-degenerated energy levels is two times higher or lower than others. Consequently, the wire with three atoms in discs works as modulator of electric impulses what could have wide practical interest. The described model and given microtheoretical approach open a possibility to study presence of di erent kind of disorders as well as wire of nite length. This corresponds to more realistic case when periodic conductance oscillations, characteristic for single electron transistor, can appear. The model presented here is suitable to serve as a basis of such research. These problems will be object of our further investigations. References
1] J.Singh, Physics of Semiconductors and Their Heterostructures, McGraw-Hill, New York, 1993, p.228. 2] C.W.J.Beenakker, H. Van Houten, Quantum Transport in Semiconductor Nanostructures, Solid State Physics vol. 44: Semiconductor Heterostructures and Nanostructures, Academic, Boston, 1989. 3] C.Weisbuch, B.Winter, Quantum Semiconductor Structures, Academic, San Diego, 1991 (Chapter 6). 4] D.Lj.Mirjanic, J.P.Setrajcic, S.M.Stojkovic, D.Sijacic, I.D.Vragovic, IEEE Proc. 22nd MIEL 1 (2000) 173. 5] S.M.Stojkovic, D.Lj.Mirjanic, J.P.Setrajcic, D.D.Sijacic, I.K.Junger, Surface Science 477 (2001) 235. 6] K.Nikolic, A.MacKinnon, Conductance uctuations of narrow disordered quantum wires, http://www.arxiv.org/list/cond-mat9405055. 7] B.Tosic, Lj.Maskovic, M.Skrinjar, D.Kapor, G.Knezevic, J.Phys 3 (1991) 7619. 8] W.Jones and N.H.March, Theoretical Solid State Physics, Dover, New York 1985. 9] A.S.Davydov, Teoriya tverdogo tela, Nauka, Moskva 1976. 10] R.Landauer, Philos.Mag. 21 (1970) 863. 11] M.Buttiker, Phys.Rev.Lett. 57 (1986) 1761. 12] J.Cserti, G.Szalka, G.Vattay, Conductance in a Periodically Doped Quantum Wires, http:// www.arxiv.org/list/condmat 9710265. 13] Y.-L.Lu, Conductance of Finite Quantum Wire Connected to Reservoirs, http:// www.arxiv.org/list/cond-mat9706081
391
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