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. Final2006

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CERTIFICATE

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 (disorder-

related) 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

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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

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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 ( = 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 (3)

which can be solved numerically for RS.

The bulk electrical resistivity can be calculated using

= 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 semi-

insulating 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.

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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. (6)

Apply the current I21 and measure voltage V34

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 R43,12 = R34,21

R32,41 = R23,14 R14,23 = R41,32 (8)

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The reciprocity theorem requires that:

R21,34 + R12,43 = R43,12 + R34,21, and

R32,41 + R23,14 = R14,23 + R41,32. (9)

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.

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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.

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

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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).

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)

The Hall mobility µ   = 1/qnsRS (in units of cm2V-1s-1) 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- 18 -

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.

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

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permanent installation, a split sensor allows the electrical current to be tested without

dismantling the existing circuit.

b) Analog multiplication

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

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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

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 "step-

wise" 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

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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 earlier-mentioned

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

collisions is suppressed and the electrons are again forced into ordered movement.

Under these extreme conditions the classical theory does not apply: the movement

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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 Effect

Transistor). Other types of

artificial samples – hetero

structures have 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.

The Quantized Hall Effect:

- 24 -

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.

Explanation of the Quantum Hall Effect

- 25 -

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,

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

- 26 -

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 thermodynamic-

localization-length calculation. Topological characteriz -ation in terms of Chern

integers provides a simple physical explanation and suggests a qualitative difference

between the lattice and continuum models. A comparison of our numerical results

with recent experimental measurements will be made.

- 27 -

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.

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 - 28 -

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.

- 29 -

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

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

- 32 -

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-

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)

- 33 -

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

the mathematical effort even for the simplest situation of a random arrangement of δ-

potentials 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

- 34 -

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

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

_

- 35 -

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

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

- 36 -

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.

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

- 37 -

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

,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 quantum-

percolation 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

- 38 -

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:

_

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

- 39 -

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

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

- 40 -

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 0, according to the floating up

scenario, are shown in figure 3.8.

3.5 Gauge arguments

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,

- 41 -

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 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

- 42 -

Figure 3.10: Classical representation for perfectly transmitting edge channels

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 (gxy = n.e2/h). An

illustration of the classically calculated electron orbits in the quantum Hall plateau

regime is shown in figure 3.10.

- 43 -

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

- 44 -

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.

- 45 -

REFERENCES:

[ l] A.B. Fowler, F.F. Fang, W.E. Howard and P.J. Stiles Phys. Rev. Letter 16, 901

( 1966)

[ 2] For a review see: Proceedings of the Int. Conf. on Electronic Properties of Two-

Dimensional Systems, Surf. Sci. 58, (1976), 73 (1978), 98 (1980, 113 (1982), 142

( 1984)

[ 3] F. Stern and W.E. Howard, Phys. Rev. 163, 816 (1967)

[ 4] T. Ando, J. Phys. Soc., Jpn. 51, 3893 (1982)

[ 51 J.F. Koch, Festkörperprobleme (Advances in Solid State Physics), H.J. Queisser,

Ed. (Pergamon-Vieweg, Braunschweig, 1975) Vol. XV, p. 79

[ 6] T. Mimura, Surf. Science 113, 454 (1982)

[ 7] R.B. Laughlin, Surface Science 113, 22 (1982)

[ 8] R. Kubo, S.J. Miyake and N. Hashitsume, in Solid State Physics, Vol. 17, 269

(1965). F. Seitz and D. Turnball, Eds., (Academic Press, New York, 1965)

[ 9] T. Ando, J. Phys. Soc. Jpn. 37, 1233 (1974)

[10] R. B. Laughlin in Springer Series in Solid State Sciences 53, p. 272, G. Bauer,

F.Kuchar and H. Heinrich, Eds. (Springer Verlag, 1984)

[11] R.E. Prange, Phys. Rev. B 23, 4802 (1981)

[12] H. Aoki and T. Ando, Solid State Commun. 38, 1079 (1981)

[13] J. T. Chalker, J. Phys. C 16, 4297 (1983)

[14] W. Brenig, Z. Phys. .50B, 305 (1983)

[15] A. Mac Kinnon, L. Schweitzer and B. Kramer, Surf. Sci. 142, 189 (1984)

[16] T. Ando, J. Phys. Soc. Jpn. 52, 1740 (1983)

[17] L. Schweitzer, B. Kramer and A. Mac Kinnon, J. Phys. C 17, 4111 (1984)

[18] H. Aoki and T. Ando, Phys. Rev. Letters 54, 831 (1985)

[19] E. Abrahams, P.W. Anderson, D.C. Licciardello and T.V. Ramakrishnan,

Phys.Rev. Letters 42, 673 (1979)

[20] G.A. Baraff and D.C. Tsui, Phys. Rev. B 24, 2274 (1981)

[21] T. Toyoda, V. Gudmundsson and Y. Takahashi, Phys. Letters 102A, 130 (1984)

[22] S. Kawaji and J, Wakabayashi, Surf. Schi. 58, 238 (1976)

[23] S. Kawaji, T. Igarashi and J. Wakabayashi, Progr. in Theoretical Physics 57, 176

(1975)

- 46 -

[24] T. Englert and K. v. Klitzing, Surf. Sci. 73, 70 (1978)

[25] K. v. Klitzing, G. Dorda and M. Pepper, Phys. Rev. Letters 45, 494 (1980)

[26] K. v. Klitzing, H. Obloh, G. Ebert, J, Knecht and K. Ploog, Prec. Measurement

and Fundamental Constants II, B.N. Taylor and W.D. Phillips, Eds., Natl. Burl.

Stand. (U.S.), Spec. Publ. 617, (1984) p. 526

[27] R.W. Rendell and S.M. Girvin, Prec. Measurement and Fundamental Constants

II, B.N. Taylor and W.D. Phillips. Eds. Natl. Bur. Stand. (U.S.), Spec. Publ. 617,

(1984) p. 557

[28] K. v. Klitzing, Festkörperprobleme (Advances in Solid State Physics), XXI, 1

(1981), J. Treusch, Ed., (Vieweg, Braunschweig)

[29] G. Ebert, K. v. Klitzing, C. Probst and K. Ploog, Solid State Commun. 44, 95

(1982)

[30] E. Stahl, D. Weiss, G. Weimann, K. v. Klitzing and K. Ploog, J. Phys. C 18, L

783 (1985)

[31] T.P. Smith, B.B. Goldberg, P.J. Stiles and M. Heiblum, Phys. Rev. B 32, 2696

(1985)

[32] V. Mosser, D. Weiss, K. v. Klitzing, K. Ploog and G. Weimann to be published

in Solid State Communication.

[33] E. Gornik, R. Lassnig, G. Strasser, H.L. Störmer, A.C. Gossard and W.

Wiegmann, Phys. Rev. Lett. 54, 1820 (1985)

[34] J.P. Eisenstein, H.L. Störmer, V. Navayanamurti, A.Y. Cho and A.C. Gossard,

Yamada Conf. XIII on Electronic Properties of Two-Dimensional Systems, p.

292(1985)

- 47 -