Thesis On Quantum Hall Effect Ahmer

FEDERAL URDU UNIVERSITY of Arts,Science and Technology Gulshan-e-Iqbal Campus

Department Of Physics

A Thesis Report On:

“INTEGRAL QUANTUM HALL EFFECT”

STUDENT NAME:

s.m.hasan.ahmer

SUPERVISOR NAME:

PROF. DR. V.E. ARKHINCHEEV

M.Sc. Final 2006

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C E RT I F I C AT E

This is to certify that s.m.hasan.ahmer has successfully completed the thesis entitled THE INTEGER QUANTUM HALL EFFECT Under my supervision and guidance.

_____________________ Prof Dr V.E. Arkhincheev (Supervisor)

___________________ Sir Rashid Tanveer (Internal Examiner)

___________________ Dr Ferooz Ahmed (External Examiner)

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ACKNOWLEDGEMENT

By the mercy of Almighty Allah, the most beneficent and merciful, I have completed my Thesis successfully. Special thanks to PROF. DR Valeriy Arkhincheev for guidance and moral support to me for the completion of my thesis. I also thankful to the Chairman of Physics Department Madam DURDANA RAZI for her cooperation with me I want to thanks and appreciate all our teachers on their continuous guidance, moral support through understanding of this research work for my thesis It is my pleasure to have a company of my classmates AZEEM, ABBAS, AKBAR, NASEEM & TAHIR for cooperating with me in my whole session. I hope that this work will give me a great opportunity in future, by the grace of Almighty Allah.

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DEDICATION

I dedicate my thesis to my humble mother, who prays for me every day and my father who was not with us, and also for my brothers and sisters

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TABLE OF CONTENTS Chapter 1: History of Hall Effect

(1-15)

1.1 Introduction to Hall Effect

1

1.1.1 Explanation of Hall Effect

2

1.2 Evolution of Resistance Concepts

3

1.2.1 The Hall Effect & the Lorentz force

4

1.2.2 Van der Pauw Technique

5

1.3 Resistivity & Hall Measurements

7

1.3.1 Sample Geometry

7

1.4 Definitions for Resistivity Measurements

8

1.4.1 Resistivity Measurements

8

1.4.2 Resistivity Calculations

9

1.5 Definitions for Hall Measurements

9

1.5.1 Hall Measurements

10

1.5.2 Hall Calculations

11

1.6 Applications related to Hall Effect:

12

1.6.1 Advantages over other methods

12

a) Split ring clamp-on sensor

13

b) Analog multiplication

13

c) Power sensing

13

d) Position and motion sensing

14

e) Automotive ignition and fuel injection

14

f) Wheel rotation sensing

14

Chapter 2: Quantum Hall Effect

(16-22)

2.1 What is it?

15

2.1.1 Background information

16

The Movement of Electrons in Magnetic Fields

16

2.2 Two-dimensional Electron Systems:

17

2.3 The Quantized Hall Effect:

18

2.3.1 Explanation of the Quantum Hall Effect

19

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2.3.2 Dirt and disorder

19

2.4 What the Quantum Hall effect requires:

20

2.5 Disappearance of Quantum Hall Effect:

20

2.6 Why is the Hall Conductance Quantized?

21

Chapter 3: Integer Quantum Hall Effect:

(23-37)

Overview of IQHE:

23

3.1 Classical theories

23

3.1.1 The Drude model

23

3.1.2 Classical electron trajectories

25

3.2 Quantum mechanical treatment

26

3.2.2 Disorder

27

3.2.3 The high field model

28

3.3 Transitions between quantum Hall states

30

3.4 Low field quantum Hall effect

33

3.5 Gauge arguments

35

3.6 The open conductor approach

36

Conclusion of thesis

38

References

39

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ABSTRACT

”The quantum Hall effect is a quantum-mechanical version of the Hall effect, observed in two-dimensional systems of electrons subjected to low temperatures and strong magnetic fields, these systems do not occur naturally, but, using advanced technology and production techniques developed within semiconductor electronics, it has become possible to produce them The quantization of the Hall conductance has the important property of being incredibly precise. Actual measurements of the Hall conductance have been found to be integer or fractional multiples of e²/h to nearly one part in a billion. This phenomenon, referred to as "exact quantization", has been shown to be a subtle manifestation of the principle of gauge invariance.”

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CHAPTER 1 History of Hall Effect:

1.1 Introduction to Hall Effect: The Hall Effect was discovered by Edwin Hall in 1879 when he was a graduate student in the Johns Hopkins University under the advisory of Professor Henry A. Rowland, after whose name this department is named now. But at that time, even the electron was not experimentally discovered. Clear understanding had to wait until quantum mechanics came into apperance. In 1930, Landau showed that for quantum electrons, unlike classical electrons, the electron's orbital motion gave a contribution to the magnetic susceptibility. He also remarked that the kinetic energy quantization gave rise to a contribution to the magnetic susceptibility which was periodic in inverse magnetic field. We can see later that Landau levels along with localization can explain the integer quantum Hall effect satisfactorily. The first inversion layer Hall conductivity measurements in strong magnetic fields were done by S.Kawaji and his colleagues in 1975. Using a somewhat different experimental arrangement which measured the Hall voltage rather than the Hall current, Klaus von Klitzing and Th. Englert had found flat Hall plateaus in 1978. However, the precise quantization of the Hall conductance in units of

was not

recognized until February of 1980. Five years later, in 1985, Klaus von Klitzing was awarded Nobel Prize in Physics for the discovery of quantum Hall effect. This was not the end of the story. In 1982 D.C.Tsui, H.L.Störmer, and A.C.Gossard discovered the existance of Hall steps with rational fractional quantum numbers, which is called fractional quantum Hall effect. R.B.Laughlin's wave functions established a very good, though not yet perfect understanding of this phenomenon. Today, the study of quasiparticles of fractional charge and fractional statistics are still active areas of research -8-

1.1.1 Explanation about Hall Effect: The Hall effect comes about due to the nature of the current flow in the conductor. Current consists of many small charge-carrying "particles" (typically electrons) which experience a force (called the Lorentz Force) due to the magnetic field. Some of these charge elements end up forced to the sides of the conductors, where they create a pool of net charge. This is only notable in larger conductors where the separation between the two sides is large enough. One very important feature of the Hall Effect is that it differentiates between positive charges moving in one direction and negative charges moving in the opposite. The Hall Effect offered the first real proof that electric currents in metals are carried by moving electrons, not by protons. Interestingly enough, the Hall effect also showed that in some substances (especially semiconductors), it is more appropriate to think of the current as positive "holes" moving rather than negative electrons. By measuring the Hall voltage across the element, one can determine the strength of the magnetic field applied. This can be expressed as

where VH is the voltage across the width of the plate, I is the current across the plate length, B is the magnetic flux density, d is the depth of the plate, e is the electron charge, and n is the bulk density of the carrier electrons.

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So-called "Hall Effect sensors" are readily available from a number of different manufacturers, and may be used in various sensors such as fluid flow sensors, power sensors, and pressure sensors. In the presence of large magnetic field strength and low temperature, one can observe the quantum Hall effect, which is the quantization of the Hall resistance. In ferromagnetic materials (and paramagnetic materials in a magnetic field), the Hall resistivity includes an additional contribution, known as the Anomalous Hall Effect (or the Extraordinary Hall effect), which depends directly on the magnetization of the material, and is often much larger than the ordinary Hall effect. (Note that this effect is not due to the contribution of the magnetization to the total magnetic field.) Although a well-recognized phenomenon, there is still debate about its origins in the various materials. The anomalous Hall effect can be either an extrinsic (disorderrelated) effect due to spin-dependent scattering of the charge carriers, or an intrinsic effect which can be described in terms of the Berry phase effect in the crystal momentum space (k-space).

1.2 Evolution of Resistance Concepts: Electrical characterization of materials evolved in three levels of understanding. In the early 1800s, the resistance R and conductance G were treated as measurable physical quantities obtainable from two-terminal I-V measurements (i.e., current I, voltage V). Later, it became obvious that the resistance alone was not comprehensive enough since different sample shapes gave different resistance values. This led to the understanding (second level) that an intrinsic material property like resistivity (or conductivity) is required that is not influenced by the particular geometry of the sample. For the first time, this allowed scientists to quantify the current-carrying capability of the material and carry out meaningful comparisons between different samples. By the early 1900s, it was realized that resistivity was not a fundamental material parameter, since different materials can have the same resistivity. Also, a given material might exhibit different values of resistivity, depending upon how it was synthesized. This is especially true for semiconductors, where resistivity alone could not explain all observations. Theories of electrical conduction were constructed with varying degrees of success, but until the advent of quantum mechanics, no - 10 -

generally acceptable solution to the problem of electrical transport was developed. This led to the definitions of carrier density n and mobility µ (third level of understanding) which are capable of dealing with even the most complex electrical measurements today.

1.2.1 The Hall Effect and the Lorentz Force The basic physical principle underlying the Hall Effect is the Lorentz force. When an electron moves along a direction perpendicular to an applied magnetic field, it experiences a force acting normal to both directions and moves in response to this force and the force effected by the internal electric field.

For

an

n-type,

bar-shaped

semiconductor shown in Fig.1, the carriers is predominately electrons of bulk density n. We assume that a constant current I flow along the x-axis from left to right in the presence of a z-directed magnetic field. Electrons subject to the Lorentz force initially drift away from the current line toward the negative y-axis, resulting in an excess surface electrical charge on the side of the sample. This charge results in the Hall voltage, a potential drop across the two sides of the sample. (Note that the force on holes is toward the same side because of their opposite velocity and positive charge.) This transverse voltage is the Hall voltage VH and its magnitude is equal to IB/qnd, where I is the current, B is the magnetic field, d is the sample thickness, and q (1.602 x 10-19 C) is the elementary charge. In some cases, it is convenient to use layer or sheet density (ns = nd) instead of bulk density. One then obtains the equation ns = IB/q|VH|. (1) Thus, by measuring the Hall voltage VH and from the known values of I, B, and q, one can determine the sheet density ns of charge carriers in semiconductors. If the measurement apparatus is set up as described later in Section III, the Hall voltage is negative for n-type semiconductors and positive for p-type semiconductors. The sheet resistance RS of the semiconductor can be conveniently determined by use of the van der Pauw resistivity measurement technique. Since sheet resistance involves - 11 -

both sheet density and mobility, one can determine the Hall mobility from the equation µ = |VH|/RSIB = 1/(qnSRS). (2) If the conducting layer thickness d is known, one can determine the bulk resistivity (ρ = RSd) and the bulk density (n = nS/d).

1.2.2 The van der Pauw Technique In order to determine both the mobility µ

and the sheet density ns, a

combination of a resistivity measurement and a Hall measurement is needed. We discuss here the van der Pauw technique which, due to its convenience, is widely used in the semiconductor industry to determine the resistivity of uniform samples (References 3 and 4). As originally devised by van der Pauw, one uses an arbitrarily shaped (but simply connected, i.e., no holes or non conducting islands or inclusions), thin-plate sample containing four very small ohmic contacts placed on the periphery (preferably in the corners) of the plate. A schematic of a rectangular van der Pauw configuration is shown in Fig. 2. The objective of the resistivity measurement is to determine the sheet resistance RS. Van der Pauw demonstrated that there are actually two characteristic resistances RA and RB, associated with the corresponding terminals shown in Fig. 2. RA and RB are related to the sheet resistance RS through the van der Pauw equation exp(-πRA/RS) + exp(-πRB/RS) = 1 which can be solved numerically for RS.

(3)

The bulk electrical resistivity ρ can be calculated using ρ = RSd.

(4)

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To obtain the two characteristic resistances, one applies a dc current I into contact 1 and out of contact 2 and measures the voltage V43 from contact 4 to contact 3 as shown in Fig. 2. Next, one applies the current I into contact 2 and out of contact 3 while measuring the voltage V14 from contact 1 to contact 4. RA and RB are calculated by means of the following expressions: RA = V43/I12 and RB = V14/I23.

(5)

The objective of the Hall measurement in the van der Pauw technique is to determine the sheet carrier density ns by measuring the Hall voltage VH. The Hall voltage measurement consists of a series of voltage measurements with a constant current I and a constant magnetic field B applied perpendicular to the plane of the sample. Conveniently, the same sample, shown again in Fig. 3, can also be used for the Hall measurement. To measure the Hall voltage VH, a current I is forced through the opposing pair of contacts 1 and 3 and the Hall voltage VH (= V24) is measured across the remaining pair of contacts 2 and 4. Once the Hall voltage VH is acquired, the sheet carrier density ns can be calculated via ns = IB/q|VH| from the known values of I, B, and q. There are practical aspects which must be considered when carrying out Hall and resistivity measurements. Primary concerns are ohmic contact quality and size, sample uniformity and accurate thickness determination, thermomagnetic effects due to non uniform temperature, and photoconductive and photovoltaic effects which can be minimized by measuring in a dark environment. Also, the sample lateral dimensions must be large compared to the size of the contacts and the sample thickness. Finally, one must accurately measure sample temperature, magnetic field intensity, electrical current, and voltage

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1.3 Resistivity and Hall Measurements The following procedures for carrying out Hall measurements provide a guideline for the beginning user who wants to learn operational procedures, as well as a reference for experienced operators who wish to invent and engineer improvements in the equipment and methodology.

1.3.1 Sample Geometry It is preferable to fabricate samples from thin plates of the semiconductor material and to adopt a suitable geometry, as illustrated in Fig. 4. The average diameters (D) of the contacts, and sample thickness (d) must be much smaller than the distance between the contacts (L). Relative errors caused by non-zero values of D are of the order of D/L. The following equipment is required: Permanent magnet, or an electromagnet (500 to 5000 gauss) Constant-current source with currents ranging from 10 µA to 100 mA (for semiinsulating GaAs, ρ ~ 107 Ω·cm, a range as low as 1 nA is needed) High input impedance voltmeter covering 1 µV to 1 V Sample temperature-measuring probe (resolution of 0.1 °C for high accuracy work)

1.4 Definitions for Resistivity Measurements Four leads are connected to the four ohmic contacts on the sample. These are labeled 1, 2, 3, and 4 counterclockwise as shown in Fig. 4a. It is important to use the same batch of wire for all four leads in order to minimize thermoelectric effects. Similarly, all four ohmic contacts should consist of the same material. - 14 -

We define the following parameters (see Fig. 2): ρ = sample resistivity (inΩ·cm) d = conducting layer thickness (in cm) I12 = positive dc current I injected into contact 1 and taken out of contact 2. Likewise for I23, I34, I41, I21, I14, I43, I32 (in amperes, A) V12 = dc voltage measured between contacts 1 and 2 (V1 - V2) without applied magnetic field (B = 0). Likewise for V23, V34, V41, V21, V14, V43, V32 (in volts, V)

1.4.1 Resistivity Measurements The data must be checked for internal consistency, for ohmic contact quality, and for sample uniformity. Set up a dc current I such that when applied to the sample the power dissipation does not exceed 5 mW (preferably 1 mW). This limit can be specified before the automatic measurement sequence is started by measuring the resistance R between any two opposing leads (1 to 3 or 2 to 4) and setting I < (200R)-0.5. Apply the current I21 and measure voltage V34

(6)

Reverse the polarity of the current (I12) and measure V43 Repeat for the remaining six values (V41, V14, V12, V21, V23, V32) Eight measurements of voltage yield the following eight values of resistance, all of which must be positive: R21,34 = V34/I21, R12,43 = V43/I12, R32,41 = V41/I32, R23,14 = V14/I23, (7) R43,12 = V12/I43, R34,21 = V21/I34, R14,23 = V23/I14, R41,32 = V32/I41. Note that with this switching arrangement the voltmeter is reading only positive voltages, so the meter must be carefully zeroed. Because the second half of this sequence of measurements is redundant, it permits important consistency checks on measurement repeatability, ohmic contact quality, and sample uniformity. Measurement consistency following current reversal requires that: R21,34 = R12,43 R32,41 = R23,14 The reciprocity theorem requires that:

R43,12 = R34,21 R14,23 = R41,32

R21,34 + R12,43 = R43,12 + R34,21, and R32,41 + R23,14 = R14,23 + R41,32.

(8)

(9) - 15 -

If any of the above fail to be true within 5 % (preferably 3 %), investigate the sources of error.

1.4.2 Resistivity Calculations The sheet resistance RS can be determined from the two characteristic resistances RA = (R21,34 + R12,43 + R43,12 + R34,21)/4 and RB = (R32,41 + R23,14 + R14,23 + R41,32)/4 (10) via the van der Pauw equation [Eq. (3)]. For numerical solution of Eq. (3), see the routine in Section IV. If the conducting layer thickness d is known, the bulk resistivity ρ = RS d can be calculated from RS.

1.5 Definitions for Hall Measurements The Hall measurement, carried out in the presence of a magnetic field, yields the sheet carrier density ns and the bulk carrier density n or p (for n-type or p-type material) if the conducting layer thickness of the sample is known. The Hall voltage for thick, heavily doped samples can be quite small (of the order of microvolts). The difficulty in obtaining accurate results is not merely the small magnitude of the Hall voltage since good quality digital voltmeters on the market today are quite adequate. The more severe problem comes from the large offset voltage caused by non symmetric contact placement, sample shape, and sometimes non uniform temperature. The most common way to control this problem is to acquire two sets of Hall measurements, one for positive and one for negative magnetic field direction. The relevant definitions are as follows (Fig. 3): I13 = dc current injected into lead 1 and taken out of lead 3. Likewise for I31, I42, I24. B = constant and uniform magnetic field intensity (to within 3 %) applied parallel to the z-axis within a few degrees (Fig .3). B is positive when pointing in the positive z direction, and negative when pointing in the negative z direction. - 16 -

V24P = Hall voltage measured between leads 2 and 4 with magnetic field positive for I13. Likewise for V42P, V13P, and V31P. Similar definitions for V24N, V42N, V13N, V31N apply when the magnetic field B is reversed.

1.5.1 Hall Measurements The procedure for the Hall measurement is: Apply a positive magnetic field B Apply

a

current

I13

to

leads

1

and

3

and

measure

V24P

Apply

a

current

I31

to

leads

3

and

1

and

measure

V42P

Likewise, measure V13P and V31P with I42 and I24, respectively Reverse the magnetic field (negative B) Likewise, measure V24N, V42N, V13N, and V31N with I13, I31, I42, and I24, respectively The above eight measurements of Hall voltages V24P, V42P, V13P, V31P, V24N, V42N, V13N, and V31N determine the sample type (n or p) and the sheet carrier density ns. The Hall mobility can be determined from the sheet density ns and the sheet resistance RS obtained in the resistivity measurement. See Eq. (2). This sequence of measurements is redundant in that for a uniform sample the average Hall voltage from each of the two diagonal sets of contacts should be the same.

1.5.2 Hall Calculations Steps for the calculation of carrier density and Hall mobility are: Calculate the following (be careful to maintain the signs of measured voltages to correct for the offset voltage): VC = V24P - V24N, VD = V42P - V42N, VE = V13P - V13N, and VF = V31P - V31N. (11) The sample type is determined from the polarity of the voltage sum VC + VD + VE + VF. If this sum is positive (negative), the sample is p-type (n-type). The sheet carrier density (in units of cm-2) is calculated from ps = 8 x 10-8 IB/[q(VC + VD + VE + VF)] if the voltage sum is positive, or (12) ns = |8 x 10-8 IB/[q(VC + VD + VE + VF)]| if the voltage sum is negative, where B is the magnetic field in gauss (G) and I is the dc current in amperes (A). - 17 -

The bulk carrier density (in units of cm-3) can be determined as follows if the conducting layer thickness d of the sample is known: n = ns/d p = ps/d (13) 2 -1 -1 The Hall mobility µ = 1/qnsRS (in units of cm V s ) is calculated from the sheet carrier density ns (or ps) and the sheet resistance RS. See Eq. (2). The procedure for this sample is now complete. Sample identification, such as ingot number, wafer number, sample geometry, sample temperature, thickness, data, and operator Values of sample current I and magnetic field B Calculated value of sheet resistance RS, and resistivity ρ if thickness d is known Calculated value of sheet carrier density ns or ps, and the bulk-carrier density n or p if d is known Calculated value of Hall mobility µ

1.6 Applications related to Hall Effect: Hall Effect devices produce a very low signal level and thus require amplification. While suitable for laboratory instruments, the vacuum tube amplifiers available in the first half of the 20th century were too expensive, power consuming, and unreliable for everyday applications. It was only with the development of the low cost integrated circuit that the Hall Effect sensor became suitable for mass application. Many devices now sold as "Hall effect sensors" are in fact a device containing both the sensor described above and a high gain integrated circuit (IC) amplifier in a single package. Reed switch electrical motors using the Hall Effect IC is another application.

1.6.1 Advantages over other methods Hall Effect devices when appropriately packaged are immune to dust, dirt, mud, and water. These characteristics make Hall Effect devices better for position sensing than alternative means such as optical and electromechanical sensing.

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HALL EFFECT CURRENT SENSOR WITH INTERNAL INTEGRATED CIRCUIT AMPLIFIER. 8MM OPENING. ZERO CURRENT OUTPUT VOLTAGE IS MIDWAY BETWEEN THE SUPPLY VOLTAGES THAT MAINTAIN A 4 TO 8 VOLT DIFFERENTIAL. NON-ZERO CURRENT RESPONSE IS PROPORTIONAL TO THE VOLTAGE SUPPLIED AND IS LINEAR TO 60 AMPERES FOR THIS PARTICULAR (25 A) DEVICE.

When electrons flow through a conductor, a magnetic field is produced. Thus, it is possible to create a non-contacting current sensor. The device has three terminals. A sensor voltage is applied across two terminals and the third provides a voltage proportional to the current being sensed. This has several advantages; no resistance (a "shunt") need be inserted in the primary circuit. Also, the voltage present on the line to be sensed is not transmitted to the sensor, which enhances the safety of measuring equipment. The range of a given feedthrough sensor may be extended upward and downward by appropriate wiring. To extend the range to lower currents, multiple turns of the current-carrying wire may be made through the opening. To extend the range to higher currents, a current divider may be used. The divider splits the current across two wires of differing widths and the thinner wire, carrying a smaller proportion of the total current, passes through the sensor.

a) Split ring clamp-on sensor A variation on the ring sensor uses a split sensor which is clamped onto the line enabling the device to be used in temporary test equipment. If used in a permanent installation, a split sensor allows the electrical current to be tested without dismantling the existing circuit.

b) Analog multiplication - 19 -

The output is proportional to both the applied magnetic field and the applied sensor voltage. If the magnetic field is applied by a solenoid, the sensor output is proportional to product of the current through the solenoid and the sensor voltage. As most applications requiring computation are now performed by small (even tiny) digital computers, the remaining useful application is in power sensing, which combines current sensing with voltage sensing in a single Hall effect device.

c) Power sensing By sensing the current provided to a load and using the device's applied voltage as a sensor voltage it is possible to determine the power flowing through a device. This power is (for direct current devices) the product of the current and the voltage. With appropriate refinement the devices may be applied to alternating current applications where they are capable of reading the true power produced or consumed by a device.

d) Position and motion sensing Hall effect devices used in motion sensing and motion limit switches can offer enhanced reliability in extreme environments. As there are no moving parts involved within the sensor or magnet, typical life expectancy is improved compared to traditional electromechanical switches. Additionally, the sensor and magnet may be encapsulated in an appropriate protective material.

e) Automotive ignition and fuel injection If the magnetic field is provided by a rotating magnet resembling a toothed gear, an output pulse will be generated each time a tooth passes the sensor. This is used in modern automotive primary distributor ignition systems, replacing the earlier "breaker" points (which were prone to wear and required periodic adjustment and replacement). Similar sensor signals are used to control multi-port sequential fuel injection systems, where each cylinder's intake runner is fed fuel from an injector consisting of a spray valve regulated by a solenoid. The sequences are timed to match

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the intake valve openings and the duration of each sequence (controlled by a computer) determines the amount of fuel delivered.

f) Wheel rotation sensing The sensing of wheel rotation is especially useful in anti-lock brake systems. The principles of such systems have been extended and refined to offer more than anti-skid functions, now providing extended vehicle "handling" enhancements

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CHAPTER 2 Quantum Hall Effect 2.1 What is it? When an electric current passes through a metal strip there is normally no difference in potential across the strip if measured perpendicularly to the current. If however a magnetic field is applied perpendicularly to the plane of the strip, the electrons are deflected towards one edge and a potential difference is created across the strip. This phenomenon, termed the Hall Effect, was discovered more than a hundred years ago by the American physicist E.H. Hall. In common metals and semiconductors, the effect has now been thoroughly studied and is well understood. Entirely new phenomena appear when the Hall Effect is studied in two dimensional electron systems, in which the electrons are forced to move in an extremely thin surface layer between for example a metal and a semiconductor. Two-dimensional systems do not occur naturally, but, using advanced technology and production techniques developed within semiconductor electronics, it has become possible to produce them. For the last ten years there has been reason to suspect that, in two-dimensional systems, what is called Hall conductivity does not vary evenly, but changes "stepwise" when the applied magnetic field is changed. The steps should appear at conductivity values representing an integral number multiplied by a natural constant of fundamental physical importance. The conductivity is then said to be quantized It was not expected, however, that the quantization rule would apply with a high accuracy. It therefore came as a great surprise when in the spring of 1980 von Klitzing showed experimentally that the Hall conductivity exhibits step-like plateaux which follow this rule with exceptionally high accuracy, deviating from an integral - 22 -

number by less than 0.000 000 1.Von Klitzing has through his experiment shown that the quantized Hall effect has fundamental implications for physics. His discovery has opened up a new research field of great importance and relevance. Because of the extremely high precision in the quantized Hall effect, it may be used as a standard of electrical resistance. Secondly, it affords a new possibility of measuring the earliermentioned constant, which is of great importance in, for example, the fields of atomic and particle physics. These two possibilities in measurement technique are of the greatest importance, and have been studied in many laboratories all over the world during the five years since von Klitzing's experiment. Of equally great interest is that we are dealing here with a new phenomenon in quantum physics, and one whose characteristics are still only partially understood.

Conductivity changes "step-wise" when the magnetic field is changed. The conductivity is said to be quantized.

2.1.1 Background information The Movement of Electrons in Magnetic Fields Under the influence of a magnetic field an electron in a vacuum follows a spiral trajectory with the axis of the spiral in the direction of the magnetic field. In the plane perpendicular to the field, the electron moves in a circle. In a metal or a semiconductor, the electron tends to move along a more complicated closed trajectory, but with fairly strong magnetic fields and at normal temperatures this ordered movement is fragmented by collisions. At extremely low temperatures (a few degrees above absolute zero) and with extremely strong magnetic fields, the effect of - 23 -

collisions is suppressed and the electrons are again forced into ordered movement. Under these extreme conditions the classical theory does not apply: the movement becomes quantized, which means that the energy can only assume certain definite values, termed Landau levels after the Russian physicist L. Landau (Nobel prizewinner in 1962) who developed the theory of the effect as early as 1930.

2.2 Two-dimensional Electron Systems: Two-dimensional material systems do not occur naturally. Under special circumstances, however, certain systems can behave as if they were two-dimensional - but only within very limited energy intervals and temperature ranges. The first to demonstrate this possibility theoretically was J.R. Schrieffer (Nobel prize winner in 1972). In work appearing in 1957 he showed that in a surface layer between metal and semiconductor electrons can be made to move along the surface but not perpendicular to it. Eleven years later a research team at IBM showed that this idea could be realized experimentally. The study of two-dimensional systems developed rapidly during the years that have followed. These experiments used samples employing a specially designed transistor, a so called MOSFET

(Metal-Oxide-

Semiconductor

Field

Transistor).

Other

artificial

samples

structures

have

Effect

types –

of

hetero

subsequently

been used, in which the samples have

been

developed

using

molecular beams. It should also be mentioned that advances

in

technology

and

production methods within semiconductor electronics have played a crucial role in the study of two-dimensional electron systems, and were a precondition for the discovery of the quantized Hall Effect. - 24 -

The Quantized Hall Effect: An important step in the direction of the experimental discovery was taken in a theoretical study by the Japanese physicist T. Ando. Together with his co-workers he calculated that conductivity could at special points assume values that are integer multiples of e2 /h, where e is the electron charge and h is Planck's constant. It could scarcely be expected, however, that the theory would apply with great accuracy. During the years 1975 to 1981 many Japanese researchers published experimental papers dealing with Hall conductivity. They obtained results corresponding to Ando's at special points, but they made no attempt to determine the accuracy. Nor was their method especially suitable for achieving great accuracy. A considerably better method was developed in 1978 by Th. Englert and K. von Klitzing. Their experimental curve exhibits well defined plateaux, but the authors did not comment upon these results. The quantized Hall Effect could in fact have been discovered then, the crucial experiment was carried out by Klaus von Klitzing in the spring of 1980 at the Hochfelt-Magnet-Labor in Grenoble, and published as a joint paper with G. Dorda and M. Pepper. Dorda and Pepper had developed methods of producing the samples used in the experiment. These samples had extremely high electron mobility, which was a prerequisite for the discovery. The experiment clearly demonstrated the existence of plateaux with values that are quantized with extraordinarily great precision. One also calculated a value for the constant e2 /h which corresponds well with the value accepted earlier. This is the work that represents the discovery of the quantized Hall Effect. Following the original discovery, a large number of studies have been carried out that have elucidated different aspects of the quantized Hall Effect. The national metrological (measurement) laboratories in Germany, the USA, Canada, Australia, France, Japan and other countries have carried out very detailed investigations of the precision of the quantization, in order to be able to use the effect as a standard.

- 25 -

Explanation of the Quantum Hall Effect The zeros and plateau in the two components of the resistivity tensor are intimately connected and both can be understood in terms of the Landau levels (LLs) formed in a magnetic field. In the absence of magnetic field the density of states in 2D is constant as a function of energy, but in field the available states clump into Landau levels separated by the cyclotron energy, with regions of energy between the LLs where there are no allowed states. As the magnetic field is swept the LLs move relative to the Fermi energy. When the Fermi energy lies in a gap between LLs electrons can not move to new states and so there is no scattering. Thus the transport is dissipationless and the resistance falls to zero. The classical Hall resistance was just given by B/Ne. However, the number of current carrying states in each LL is eB/h, so when there are i LLs at energies below the Fermi energy completely filled with ieB/h electrons, the Hall resistance is h/ie2. At integer filling factor this is exactly the same as the classical case. The difference in the QHE is that the Hall resistance can not change from the quantized value for the whole time the Fermi energy is in a gap, i.e between the fields (a) and (b) in the diagram, and so a plateau results. Only when case (c) is reached, with the Fermi energy in the Landau level, can the Hall voltage change and a finite value of resistance appear. This picture has assumed a fixed Fermi energy, i.e fixed carrier density, and a changing magnetic field. The QHE can also be observed by fixing the magnetic field and varying the carrier density, for instance by sweeping a surface gate.

Dirt and disorder Although it might be thought that a perfect crystal would give the strongest effect, the QHE actually relies on the presence of dirt in the samples. The effect of dirt and disorder can best be though of as creating a background potential landscape, with hills and valleys, in which the electrons move. At low temperature each electron trajectory can be drawn as a contour in the landscape. Most of these contours encircle hills or valleys so do not transfer an electron from one side of the sample to another, - 26 -

they are localised states. A few states (just one at T=0) in the middle of each LL will be extented across the sample and carry the current. At higher temperatures the electrons have more energy so more states become delocalized and the width of extended states increases. The gap in the density of states that gives rise to QHE plateaux is the gap between extended states. Thus at lower temperatures and in dirtier samples the plateaus are wider. In the highest mobility semiconductor hetero junctions the plateaux are much narrower.

What the Quantum Hall effect requires: 1. Two-dimensional electron gas 2. Very low temperature (< 4 K) 3. Very strong magnetic field (~ 10 Tesla)

2.5 Disappearance of Quantum Hall Effect: The disappearance of integer quantum Hall effect (IQHE) at strong disorder and weak magnetic field is studied in the tight-binding lattice model.\footnote D. N. Sheng and Z. Y. Weng, Phys. Rev. Lett., to be published. We found a generic sequence by which the IQHE plateaus disappear: higher IQHE plateaus always vanish earlier than lower ones, and extended levels between those plateaus do not float up in energy but keep merging together after the destruction of plateaus. All of these features remain to be true in the weak-field limit as shown by the thermodynamiclocalization-length calculation. Topological characteriz -ation in terms of Chern integers provides a simple physical explanation and suggests a qualitative difference - 27 -

between the lattice and continuum models. A comparison of our numerical results with recent experimental measurements will be made.

2.6 Why is the Hall Conductance Quantized? The Integer Quantum Hall effect, first observed by K. von Klitzing, is used to determine the fine structure constant with precision that is comparable to the precision one gets from atomic physics. It is also used as a practical and fundamental way to define the Ohm. It is instructive to look at the experimental data. The graph that looks like a staircase function has remarkably flat plateaus. The ordinates of the plateaus correspond to integer multiple of the quantum unit of conductance, and can be measured very precisely. An intriguing aspect of this phenomenon is that a precision measurement of fundamental constants is carried on a system that is only poorly characterized: Little is actually known about the microscopic details of the system, which is artificially fabricated, and whose precise composition and shape are not known with a precision that is anywhere comparable with the precision that comes out of the experiment.There are two related but somewhat distinct theoretical frameworks that attempt to answer this question. The problem we pose has to do with their mutual relation, and the extent to which they give a satisfactory answer. One framework identifies the Hall conductance with a topological invariant: The first Chern number of a certain bundle associated with the ground state of the quantum Hamiltonian. This framework applies to a rather general class of quantum Schrodinger Hamiltonians, including multiparticle ones. It has two principal drawbacks. The first is that it requires an interesting topological structure: It applies in cases where there is a Brillouin zone, and in cases where configuration space is multiply connected. The multiple connectivity can be motivated, to some extent, by the experimental setup if one includes the leads that connect to the two dimensional electron gas in the system. This makes the Hall conductance a property of the system and not just of the two dimensional electron gas. The second drawback is that the Chern number is identified with a certain average of the Hall conductance. In some cases this average comes for free, but in general it does not.

- 28 -

A second theoretical framework identifies the Hall conductance with a Fredholm index of a certain operator. This framework is known to apply to non interacting electrons in two dimensions where the Fredholm operator is constructed from the one particle Schrodinger Hamiltonian of the system. This framework applies to a particularly popular model of the Integer quantum Hall effect: non interacting electrons in two dimensions and with random potential. Some models, like non interacting electrons in homogenous magnetic field in two dimensions, and its generalization to a periodic potential can be analyzed either framework, and the results agree. In these cases the Hall conductance can be interpreted either as a Chern number or as an Index. the two frameworks are complementary: Chern allows for electron interaction while Fredholm does not, Chern assumes an interesting topology while Fredholm does not and requires that configuration space be two dimensional; Chern comes with an averaging while Fredholm does not. The Chern framework would be a satisfactory theory if one could take the thermodynamic limit and remove the averaging. Progress in this direction has been made by Thouless and Niu who described (implicit) conditions under which this is the case. The Fredholm framework would be a satisfactory theory of the integer quantum Hall effect if one could remove the restriction of non interacting electrons.

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CHAPTER 3 THE INTEGER QUANTUM HALL EFFECT Overview of IQHE: In the following we give some simple theoretical models which reflect the properties of a two-dimensional electron gas (2DEG) in a strong magnetic field. Starting from the Drude model, we show how Landau quantization occurs in the simplest quantum mechanical model, and give a rather simplified model (the high field model) that incorporates a disorder potential and shows the occurrence of localized and extended states. The sequence of different plateaus seen in the Hall resistivity in a field sweep experiment can be described theoretically as a sequence of phase transitions between different Quantum Hall States. This result in scaling laws for the transport coefficients in the proximity of the transition points that can be verified experimentally, an open question is how the quantum Hall effect will vanish at small magnetic fields in the limit of zero temperature. We will present one possibility, the levitation of extended states. We will mention the gauge argument put forward by R. Laughlin, that explains the exact quantization of the Hall conductivity by gauge invariance. Last we will mention the open conductor approach to the quantum Hall effect by M. Büttiker, that describes electronic transport in terms of reception and transmission of charge carriers.

3.1 Classical theories As there are some limiting cases where a classical description of a disordered two-dimensional electron system is very instructive for the understanding of the quantum Hall effect, we will give the results of a classical description of an electron in a magnetic field.

3.1.1 The Drude model The basic theoretical model for electrical transport is the Drude model, which, although a very simplified model, still gives a reasonably good description of transport at high temperature and usually is a good starting point for more sophisticated models. Electrons are treated as classical particles moving under the - 30 -

influence of external fields and a friction term represented by an average scattering time

Here m is the electron mass, v the velocity vector, B and E are the magnetic and electric field vectors, respectively. Choosing B along the z-direction (B = (0; 0;B)), setting ≡ 0 (steady state condition) and using the equation

for the

current density, we get the following expression for the conductivity tensor with the mobility

_

µ = eτ/m and the cyclotron frequency we = eB/m. As experiments usually measure resistances, it is convenient to convert these results to the corresponding resistivity tensor ρ

_ The Drude model gives a magnetic field independent diagonal resistivity ρxx and a Hall (transverse) resistivity ρ xy which is linear in B.

- 31 -

Figure 3.1 Resistivity and conductivity in Drude Model

3.1.2 Classical electron trajectories To find the actual electron trajectories one has to solve the equation of motion for a classical charged particle under the influence of a magnetic and electric field, as it is done before using Halmiltonian mechanism. The results for a homogenous magnetic field along z (B = (0; 0;B)) and a homogenous electric field along x (E = (E; 0; 0)) are:

vD = -E/B is called the drift velocity. The coordinates have been separated into a slowly varying part (X(t); Y (t)), and a rapidly varying part (ε(t); η(t)), where the slow motion is a constant drift with velocity vD along y, and the rapid motion is a cyclotron motion around the center coordinates

with the frequency we.

The electron performs a cycloid motion, drifting perpendicular both to the magnetic and electric field, along an equipotential line.

Figure3.2: Electron trajectory in a classical picture

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3.2 Quantum mechanical treatment The origins of the quantum Hall effect can only be found by a quantum mechanical calculation. For this, a starting point is the Hamiltonian for an electron in a homogenous magnetic field

Choosing the direction of B along the z-axis, one can use the Landau gauge for the vector potential: A = (0;Bx;0). This gauge is appropriate for systems with translational symmetry along y. Another possible gauge is the symmetric gauge A = ½ B× r, which is a good choice for systems with axial symmetry Assuming further that V (r) = V (x; y)+Vz (z), the Schrodinger equation will separate into a part depending on z, and the remaining, now effectively two dimensional part depending on x and y. Note that Vz(z) can be zero (as assumed by Landau for the 3D case), or can be given by a confinement potential imposed e. g. by a semiconductor heterostructure, therefore creating a "real" 2D system. In any of the two cases the results for the remaining 2D problem in the (x,y)-plane are the same.

3.2.1 Landau model In the case originally considered by Landau, the external potential is assumed to vanish (V (r) = 0, no electric field). The Hamiltonian then doesn't depend on y, we get a plane wave solution in the y-direction, and in the x-direction the problem becomes equivalent to a harmonic oscillator:

^ with the center coordinate

, and the solution

W is the extension of the system in y-direction, m is an integer, and Hn are the Hermite polynomials. The states

(x; y) are delocalized (plane waves) in y- 33 -

direction, and localized (harmonic oscillator states) around X in x-direction. Note however that the shape of the wave functions depend strongly on the gauge used for A. The energy eigen values are called Landau levels

As the energy of an electron is independent of its x-position, the eigen values are infinitely degenerate, and the density of states (DOS)

is ill-defined (L is the extension of the system in x-direction). To get around this problem, one considers only states with

_ and takes the limit L

afterwards. This method, also called the Landau

counting of states, gives a DOS consisting of equidistant δ-peaks separated by and a degeneracy of 2πl2 B per Landau level:

The actual wave function is delocalized across the sample along y, and localized in an area of width

around X in x-direction.Note that using the symmetric gauge

for A, one gets the same energy Eigenvalues, but the wave functions are localized on a circle with radius p2m lB (m is a non-negative integer)

3.2.2 Disorder In real semiconductor samples some kind of disorder potential, caused for example by lattice defects or ionized donors is always present. The exact calculation of the effect of a random potential onto the energy spectrum of the problem is not possible in a straightforward way, on one hand because it is by far not clear what shape the disorder potential should have (one can think of the whole range from an unscreened 1/r Coulomb potential to a completely screened δ-potential), and on the other hand - 34 -

the mathematical effort even for the simplest situation of a random arrangement of δpotentials is considerable. It is clear however, that the degeneracy of the Landau levels will be lifted by an additional potential, and the delta-peaks in the density of states transform into structures with a finite width. A prominent approach to calculate the shape of the disorder-broadened Landau levels is the self consistent Born approximation (SCBA), where only single scattering events are taken into account. The SCBA gives an elliptic function as shape for the broadened Landau levels, models including multiple scattering events give a Gaussian shape

where Γn is a Landau level dependent width. In addition to the broadening of the Landau levels, a disorder potential will change the nature of most of the electronic states in the Landau level. Except the states in the middle of the level, which will be extended over the sample, all electronic states will localize. This can be shown easily with the help of the semi-phenomenological high field model

3.2.3 The high field model Using the separation of the coordinates introduced in chapter 3.1.2, the Hamiltonian for an electron in a magnetic field and a disorder V (x; y) looks as follows:

The x- and y- coordinates do not commute

Taking the limit B

, one can neglect ζ and η in the argument of V , as their

expectation values are of the order of

The Hamiltonian then separates, and

the first part is equivalent to the Landau level energies

- 35 -

As the commutator of [X; Y ] is proportional to 1=B, X and Y can be treated as classical variables for

, and the problem can be calculated classically,

resulting in the following equations of motion

_ This implies that dV/dt vanishes, so the potential energy of the electron is

Figure 3.3: Disorder potential with closed orbits (localized states) and open orbits (extended states) constant.

We can say that, in the limit of high B, the electron is delocalized on an area of approxmately

and moves on the equipotential lines of the disorder potential. If V is

symmetric around V = 0, then electron orbits for E≠ħωe/2 (lowest Landau level), will circle around valleys or peaks of the disorder potential and will therefore be localized as shown in figure , and only for E=ħωe/2 the trajectory will traverse the sample and give a delocalized state. The electronic density of states for the Landau model with and without disorder is sketched in figure Depending on the value of EF with respect to ħωe there will be either localized states in the vicinity of the Fermi energy and the

- 36 -

system

will

be

Figure 3.4: Schematic density of states for the disordered Landau model. The grey regions represent localized states.

insulating, or extended states, resulting in the sample to show a metallic-like behavior. Changing the ratio of EF to ħωe, either by changing the carrier density or by sweeping the magnetic field will cause a series of transitions between metallic and insulating states. Note that because V will rise strongly at the boundaries of a sample in x and y-direction (as only in this case the wave function will vanish outside the sample), there will always be an extended state for all ratios of EF to ħωe propagating at the edge of the sample. This edge state can carry a current, even if all other states around EF are localized.

3.3 Transitions between Quantum Hall States At low temperatures the DOS of a 2DEG will decay into areas of extended states (in the vicinity of the Landau level centers) and areas of localized states,that surround the former (in the Landau level tails). We can identify two extremal transport regimes: the plateau region, when the Fermi energy is situated in a range of localized states, and the transition region between two plateaus, when the Fermi energy lies in an area of extended states. - 37 -

Figure 3.5: electron trajectories for the plateau region

Electron trajectories for the plateau regime are shown in figure 3.5. There is no net current flowing in the bulk of the sample, and transport takes place only in the edge states of the sample. As there are no extended states in the vicinity of EF , the longitudinal conductivity σxx vanishes. The Hall conductivity is determined by the number n of occupied Landau levels below EF , and can be shown to be equal to n e2./ħ In the transition regime, when EF lies in a region of extended states, electron transport in the bulk of the sample is possible, and therefore dissipative currents will flow in the sample giving a nonzero longitudinal conductivity and a Hall conductivity that lies between two quantized values. Typical electron trajectories for the transition region are shown in figure 3.6. An interesting question is, how the crossover between these two regimes will look like. According to the high field model (chapter 3.2.2), electron trajectories in the plateau region are closed, with the diameter of the closed loops increasing as the Fermi energy approaches an area of extended states. For a real world (finite size) sample, the system should enter the transition regime as soon as the average diameter of the electron trajectories exceeds the sample size L. Note that for finite temperatures, L has to be plateaus replaced by an effective sample size

Figure 3.6: electron trajectories for the transition region between two - 38 -

,which corresponds to the phase coherence length of the charge carriers. This length, which is usually given by LФ or Lin, depends on temperature with a powerlaw 2 ) Theoretically the transition between two quantum Hall states is being described Ф as a continuous quantum phase transition, order parameter being the localization length ξ which corresponds to the mean diameter of a closed electron trajectory. At the transition point, when different localized trajectories come close to each other, electrons are able to tunnel between different localized states close to a saddle point. In this picture, the transition between the two regimes is a quantumpercolation transition. The order parameter ξ has been predicted to diverge with a power law at the critical energy of the transition:

The most prominent

model for the calculation of the critical exponent _ is the Chalker-Coddington model, which calculates the percolation exponent for a regular lattice of saddle points. The result for an analytic solution is ν = 7/3, a value which has been verified numerically by lattice models for different disorder potentials. The critical conductivity σxx(Ec) was found to be e2/2h. In a typical quantum Hall experiment one therefore sees a series of phase transitions between different plateau states, with a values of σxx = 0 in the two neighboring plateau regions, reaching a value of σxx = 1/2 at the transition field Bc. Bc corresponds to the critical energy Ec = ħωe. As an electronic state has to be considered extended as soon as its localization length is larger than the effective sample size (ξ> Lin), the width of the area of extended states around the critical energy Ec will shrink with decreasing temperature. As Lin increases with a powerlaw for decreasing temperature, the area of extended states should shrink to zero width for T

0. The transition region between two Quantum Hall States should therefore

become more and more narrow for decreasing temperature. As it was shown by Pruisken the transport coefficients in the transition region should be determined by a regular function that only depends on a singe scaling variable:

This makes it possible to observe the product of the localization length exponent ν and the exponent of the inelastic scattering length p for example in the half width of the peak in ρxx, or the slope of ρxy at Bc:

- 39 -

_ Theoretical calculations predict a value of μ = 0:43.

Figure 3.7: Sharpening of the transition between two quantum Hall plateaus for decreasing temperature.

The critical field Bc usually corresponds to a magnetic field value where the .Fermi energy EF coincides with the center of a Landau level. However, there exists an exception to this rule.

3.4 Low field quantum Hall effect In the limit T

0 the single parameter localization theory predicts all two

dimensional systems to be localized at B = 0, there can be no extended states at zero field. For the quantum Hall effect in high magnetic fields however, extended states are essential, and their existence is well established. The question is what will happen to the extended states that are connected with the Landau level centers, as the magnetic field is decreased. Theoretically the possibility that these states just dissappear is difficult to establish. It were R. Laughlin and D. Khmelnitzkii who suggested that these extended states will oat up in energy as the magnetic fields approaches zero. The values of the magnetic field, where the extended state associated with the n-th Landau level will cross the Fermi energy when floating up was taken to be the value where the Drude Hall conductivity corresponds to the quantum value (n + 1/2)e2/h. This floating up scenario therefore predicts quantum Hall phases to exist even at low magnetic field

As will be shown in chapter

3.2.2, a necessary condition for the observability of a Hall plateau is a value of σxx - 40 -

1. As the only available microscopic mechanism, that could lead to a decrease of σxx in low magnetic fields is weak localization, which gives much smaller corrections than strong localization that occurs in high fields, the condition σxx

1 is usually not

fulfilled at experimentally accessible temperatures, and the quantum Hall effect at low magnetic fields cannot be observed The only experimental observations

Figure 3.8: Left: Magneto conductance for a quantum Hall system according to the floating up scenario, in the limit of very high temperature (Drude) and zero temperature. Right: Extended states in the floating up scenario. Dashed lines represent the conventional Landau levels.

Any time an extended state crosses the Fermi level, there will be a quantum Hall transition visible in the transport data of a quantum Hall transition at low magnetic fields were made in strongly disordered systems, that only show a single quantum Hall phase, and where a clear transition from the low field insulating state to the corresponding quantum Hall plateau at σxy = 1 exists. Transitions between higher quantum Hall states have only been observed in the high field regime (ωt < 1) up to now. The transport coefficients for a system in the limit T scenario, are shown in figure 3.8.

3.5 Gauge arguments - 41 -

0, according to the floating up

In one of the first theoretical papers dealing with the quantum Hall effect, R. Laughlin proposed an explanation for the exact quantization of the Hall conductance that was based on gauge considerations. An extension of his paper was published by B. Halperlin later. Both authors consider a two-dimensional system in a continuous but multiply connected geometry like a cylinder or ring geometry, e.g. as shown in figure.9. The 2D electron gas is assumed to be subject to a magnetic field B perpendicular

to

its

surface,

and

it

is

assumed

that

there

Figure.3.9: Geometry considered by R. B. Laughlin in his gauge argument for the exact quantization of the Hall conductance.

is an additional magnetic flux Φo that can be varied freely without changing the value of B, passing through the hole of the system. The system then should be gauge invariant under a flux change ΔФo by an integral multiple of the flux quantum h/e. An adiabatic change of Φ0 by a single flux quantum should therefore leave the system unchanged. Assuming a DOS as shown in the previous chapter, the effect of the flux change ΔΦo onto the electronic wave functions will depend on the nature of the states at the Fermi energy. Localized states will just acquire an additional phase factor, they won't be affected otherwise Extended states however will suffer an electromotive force, and will be pushed to the exterior of the sample. After Laughlin, gauge invariance requires an integer number of electrons to be transferred across the sample under a flux change ΔΦo = h/e, which in turn requires the Hall conductivity to be - 42 -

quantized. It should be noted that some authors claim the gauge argument presented to be incomplete. After Laughlin's gauge argument has been superceded of what is nowadays called the topological approach to the quantum Hall effect. In this theoretical approach the Hall conductivity is identified with the Chern number, which is a topological invariant

3.6 The open conductor approach A theory treating the QHE from a totally different point of view has been worked out by M. Buttiker Based on a theory of Landauer viewing conductances in terms of transmission of electrons, this theory inherently includes the presence of contacts, a fact which had been neglected in the previously mentioned theories. Associating each contact or probe of the system with an electrochemical potential Vi, the resistance of a four probe conductor is given by the two current contacts are labeled by k and l

the voltage probes m and n. the conductance coefficients are defined by

The main point of Buttiker's theory is the relation of the conductance coefficients gmn to the transmission probabilities of an electron incident at contact m with the

transmission probabilities

of an electron incident at point n in quantum state β

leaving the conductor at probe m in state α. The main problem in this approach is the calculation of the coefficients Tmn, which is

Figure 3.10: Classical representation for perfectly transmitting edge channels

- 43 -

and localized, non-current carrying states in the Buttiker picture.

simplified a little bit in the case of the quantum Hall effect. In the case of the plateau regime (EF located in a region of localized states) the only current carrying states are the previously mentioned edge states. As these edge states are moreover sufficiently isolated from all other current carrying states (e. g. on the opposite side of the sample), they are perfectly transmitting (Tmn =1), as there are no states an electron could scatter to. As a consequence of this absence of backscattering the longitudinal conductance of the sample vanishes (gxx = 0), and the Hall conductance corresponds to e2/h times the number of occupied edge states or channels (g xy = n.e2/h). An illustration of the classically calculated electron orbits in the quantum Hall plateau regime is shown in figure 3.10.

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Conclusion of Thesis: After the completion of this thesis I conclude the following facts that: The quantum Hall effect is a quantum-mechanical version of the Hall effect, observed in two-dimensional systems of electrons subjected to low temperatures and strong magnetic fields, in which the Hall conductance σ takes on the quantized values

where e is the elementary charge and h is Planck's constant. In the "ordinary" quantum Hall effect, known as the integer quantum Hall effect, ν takes on integer values (ν = 1, 2, 3, etc.). There is another type of quantum Hall effect, known as the fractional quantum Hall effect, in which ν can occur as a vulgar fraction with an odd denominator (ν = 2/7, 1/3, 2/5, 3/5, etc.) The integral quantum Hall effect can be explained solely by the filling of the Landau levels. Each Landau level has a certain capacity to accept electrons, which depends on the magnetic field B. By changing the magnetic field, we change the ability of each Landau level to accommodate electrons. When there is a match between the capacity of the Landau levels and the number of electrons in the sample, an integer number of Landau levels are exactly filled, and the integral quantum Hall effect is produced

Quantization of Hall Conductance The quantization of the Hall conductance has the important property of being incredibly precise. Actual measurements of the Hall conductance have been found to be integer or fractional multiples of e²/h to nearly one part in a billion. This phenomenon, referred to as "exact quantization", has been shown to be a subtle manifestation of the principle of gauge invariance. It has allowed for the definition of a new practical standard for electrical resistance: the resistance unit h/e², roughly equal to 25 812.8 ohms, is referred to as the von Klitzing constant RK (after Klaus von Klitzing, the discoverer of exact quantization) and since 1990, a fixed conventional - 45 -

value RK-90 is used in resistance calibrations worldwide. The quantum Hall effect also provides an extremely precise independent determination of the fine structure constant, a quantity of fundamental importance in quantum electrodynamics.

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