All About Quantum Hall Effect

The quantum Hall effects remains one of the most important subjects to have emerged in condensed matter physics over the past 20 years. The fractional quantum Hall effect, in particular, has opened up a new paradigm

”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.”

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

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. 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 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, barshaped 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 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) 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.

History of Hall Effect:

(5)

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

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.

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 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, barshaped 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 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)

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

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. 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

(8)

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.

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 ps = 8 x 10-8 IB/[q(VC + VD + VE + VF)] if the voltage sum is positive, or (12) -8

ns = |8 x 10 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

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 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 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

Quantum Hall

extended vehicle "handling" enhancements

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 is

well

understood.

Entirely

and

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 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 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.

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 e 2 /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 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 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.

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 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.

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 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.

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

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)

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 degeneracy of 2πl2 B per Landau level:

and a

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

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

_ 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 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 e 2./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 ,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 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 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 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, 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

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. 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 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. quantum mechanics is one of the pillars of modern physics.Quantum Hall effect The quantum Hall effect is a quantum-mechanical version of the Hall effect, observed in two-dimensional electron systems 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 (ν = 2/7, 1/3, 2/5, 3/5, 5/2 etc.)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 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. The fractional quantum Hall effect continues to be influential in theories about topological order

My research at the Institute for Theoretical Physics: the quantum Hall effect Between 1995 and 1999 I worked on the quantum Hall effect at the  Institute for Theoretical Physics, University of Amsterdam. I had a much­visited webpage there,  which I have now put online again in a slightly updated version.  Content:  • • •

• •

The classical Hall effect.  The quantum Hall effect.  Why do we care?   Simple theory for the integer effect.   Simple theory for the fractional effect.   An avalanche of interesting physics.   Recent experiments.  Quantum Field Theory.  My own modest contributions. 

For   people   who   can   read   Dutch,   there   is   a  short article by Kareljan Schoutens,  published in the march 1997 issue of the faculty quasi­periodical "Afleiding". 

The classical Hall effect The Hall effect was discovered by Edwin Hall in 1879.  It   is   well   known   that   a   charged   particle   moving   in   a   magnetic   field   feels   a  `Lorentz' force perpendicular to its direction of motion and the magnetic field. As  a direct consequence of this Lorentz force, charged particles will accumulate to  one   side   of   a   wire   if   you   send   current   through   it   and   hold   it   still   in   a  (perpendicular) magnetic field. This is called the Hall effect. The voltage drop at  right angles to the current is called the Hall voltage; The current divided by the  Hall voltage is called the Hall conductance.  The Hall effect can be put to use in several ways. One application is magnetic  field strength measurement. Since the Hall voltage is proportional to the current  and the field strength, sending a known current through a medium and measuring  the   Hall   voltage   tells   you   the   field   strength.   Another nice thing is that you can reveal the nature of the mobile charges in a  current­carrying   medium.   The   Lorentz   force   will   push   a   moving   hole   (positive  charge) and a moving electron (negative charge) in exactly the same direction,  since they travel in opposite ways; from the sign of the Hall voltage you can tell if  there are more mobile holes than electrons or vice versa. 

The quantum Hall effect The  discovery of the  quantised  Hall  effect in  1980  won  von Klitzing  the  1985  Nobel prize.  Investigating  the  conductance  properties of two­dimensional  electron  gases  at  very   low   temperature   and   high   magnetic   fields,   his   group   obtained   curious  results: The Hall conductance of such a system plotted as a function of the ratio nu := electron density*h / magnetic field*e shows extremely flat plateaux at integer multiples of e²/h around integer values of  the ratio `nu'. (h is Planck's constant, e is the electron charge.) Furthermore, the  `ordinary'   conductance   plotted   as   a   function   of  nu   is   zero   everywhere   except  where the Hall conductance has a transition from one plateau to another. In other  words, there are whole intervals of nu where the voltage drop is completely at 

right angles to the current, with Hall conductance very accurately quantised in  terms of the fundamental conductance quantum e²/h; in between these intervals,  the longitudinal conductance has a peak, while the Hall conductance goes from  one plateau value to another.  In 1982, Tsui, Gossard and Störmer, working with samples that contained less  impurities,   discovered   the   so­called   "fractional"   quantum   Hall   effect.   Here   the  conductance is quantised in fractional multiples of e²/h, like 1/3, 1/5, 2/3, 2/5 etc  etc,   always   with   odd   denominator.   Whereas   the   integer   quantisation   could  perhaps have been expected, the fractional effect came as a total surprise. Tsui  and   Störmer   were   awarded   the   1998   physics   Nobel   prize   for   the   discovery,  sharing it with Laughlin, who was the first to come with a theoretical description.  [Picture   of   the [Graph of experimental data; 11Kb] 

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setup;

 

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Why do we care? Doing   research   is   hard   work   and   condensed   matter   systems   are   particularly  opaque. So the question naturally arises: why are we working on this, are we  masochists or what? To which the answer is of course, Yes; we are theoretical  physicists,   remember?   But   apart   from   that,   there   are   many   reasons   to   be  interested in the quantum Hall effect.  •

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A very obvious first reason is that a large and growing part of the world's  information   storage   and   manipulation   depends   on   the   movement   of  electrons   through   semiconductors.   Everything   that   could   possibly   be  known about the subject should therefore be known.  For those who immediately want to know whether something has practical  applications:   The   integer   quantum   Hall   effect   is   now   used   as   the  international standard of resistance. The incredibly accurate quantisation  of the Hall resistance to approximately one part in 108 makes this possible.  The   constant  e²/h  is   proportional   to   the   `fine   structure   constant'   in  electrodynamics,   which   basically   tells   how   strongly   light   interacts   with  matter.   The   quantum   Hall   effect   provides   an   independent   way   of  accurately measuring this constant. 

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To a theoretical physicist, the fractional effect is a mouth­watering feast of  new theories, nice mathematics, exotic statistics and topology galore. I will  try to explain this below. 

Simple theory for the integer effect The discovery of the quantum Hall effect showed that the theories of electron  transport in disordered two­dimensional systems were inadequate. In the early  eighties   a   simple   explanation   was   found   for   the   integer   effect   in   terms   of  noninteracting   electrons,   i.e.   electrons   that   do   not   repel   each   other.   Such  particles `feel' each other only through Pauli's exclusion principle, which says that  no   two   fermions   can   occupy   exactly   the   same   quantum   state.   In   this  approximation you only have to figure out what the possible states are for one  electron and then combine them in a simple way to form a many electron state;  you just fill the available states with electrons, beginning with the lowest energy.  It turns out that the quantum states for an electron in a magnetic field, moving in  a two­dimensional random potential energy landscape, fall into two classes: so­ called "localised" and "extended" states. Roughly speaking, the localised states  are bound to one or more `peaks' or `chasms' in the energy landscape and have  an   energy   corresponding   to  the   `height'   were   they  are   sitting.   In   contrast,   an  extended  state  spreads  through  the whole  sample  and its energy  is  that of a  particle   that   does   not   feel   the   random   potential.   (The   energy   levels   of   an  undisturbed electron are called Landau levels.) The localised states, being bound  to   one   small   region,   cannot   contribute   to   electron   transport.   By   putting   extra  electrons into localised states it is therefore possible to change the parameter  `nu' without changing the conductance! This explains the occurrence of plateaux.  Only   when   the   new   electrons   reach   a   Landau   level   does   the   conductance  change,   because   now   the   extended   states   come   into   play.   The   fact   that   the  plateaux of the Hall conductance lie exactly at integer multiples of  e²/h  can be  explained   by   relating   the   sample   with   its   random   potential   to   a   hypothetical  situation without impurities, but I am not going to elaborate on this.  [Picture of the density of states; 3Kb]  Simple theory for the fractional effect

The   simple   explanation   for   the   integer   effect   completely   fails   to   predict   the  fractional effect. In the fractional effect the Coulomb repulsion between electrons  plays an essential role. The similarity between the experimental results for the  integer and fractional case, however, simply begs for a common description. And  indeed one exists. It is called the "composite fermion" theory, formulated by Jain  in   1989,   and   it   states   that   mumblemumble   somehow   mumblemumble   the  Coulomb repulsion has the net effect of attaching an even number of magnetic  flux quanta to every electron. Such composite objects obey the Pauli principle,  which is where the name composite fermion comes from. Since a large part of  the magnetic field has gone into defining the new composite particles, the field  that these particles feel is much smaller than the original one; in fact it exactly  mimics the integer effect's field strength. In this way the fractional quantum Hall  effect   is   explained   as   the   integer   effect   for   composite   fermions.   This   elegant  picture is widely accepted, even though the equivalence between electrons and  composite fermions is not of course 100% exact.  Before   the   composite   fermion   theory   was   formulated,   a   many­electron   wave  function was written down by Laughlin in 1983 for the fractional plateaux around  nu=1/(odd integer), in the idealised case where there are no impurities in the  sample. Even though it neglects the disorder, this wave function gives a lot of  insight.  It clarifies how the  Coulomb  interaction makes it possible  for  the  Hall  quantisation to be non­integer. It also gives a hint how flux attachment works.  And   it   shows   that   when   a   plateau   occurs,   the   electrons   form   a   so­called  "incompressible quantum liquid" whose density, as the name implies, is not easily  changed.  [Picture of the Laughlin 1/3 wave function; 56Kb]  An avalanche of interesting physics The simplest theoretical explanations for the Hall effects already generated new  ideas   like   magnetic   flux   attachment,   incompressible   quantum   fluids   and   the  importance of the `size' of wave functions (instead of only the question how many  electron states exist in a certain energy interval). It didn't stop there. By taking  these ideas a little step further, interesting predictions were made and links were  discovered with other branches of theoretical physics. 

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The   incompressible   quantum   fluid   and   composite   fermion   picture  immediately   predicts   the   existence   of   particle­like   disturbances   of   the  electron   gas   ("quasiparticles")   with   very   unusual   properties.   These  quasiparticles   can   have   a  fractional   charge  like   1/3   or   1/5   and   also  fractional   statistics.   The   term   `statistics'   refers   to   the   behaviour   of   a 

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wave function under exchange of identical particles. All known particles  are   either   fermions   (the   wave   function   acquires   a   factor   ­1   under  exchange) or bosons (factor +1). The quasiparticles in the fractional effect  can have factors like (­1)1/3. The only other physical system that we know  of today where fractional statistics may be found is liquid helium.  It turns out that interesting things happen at the edge of a quantum Hall  system.   This   can   be   roughly   understood   by   noting   that   you   can   not  cheaply change the density of an incompressible fluid, but you can change  its form; and that happens at the surface. The low­energy disturbances at  the   edge   are   described   by   bosonic   particles   moving   in   one   direction,  "chiral edge bosons". Roughly speaking, these bosons can combine to  form a quasiparticle on the edge that behaves like a composite fermion or  like a particle with fractional charge and statistics.  The theory of edge bosons (when neglecting Coulomb interactions) has  so­called conformal symmetry. I will not elaborate on this, but symmetry  always   comes   in   handy   when   you   want   to   solve   a   problem,   and   two­ dimensional   conformally   invariant   systems   have   an   infinite   number   of  symmetries.   Conformal   field   theory,   developed   at   lightning   speed   after  Belavin,   Polyakov   and   Zamolodchikov's   famous   1984   paper,   has   been  successfully   applied   to   many   problems   in   statistical   mechanics   and  condensed matter.  Particles with unusual statistics can be described by Chern­Simons field  theory. This is a topological theory, which means that it cares only about 

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the electrons' global motion, for instance how often they circle around one  another. Flux attachment is incorporated in a natural way. By arguments of  gauge invariance, Chern­Simons theory is directly related to chiral edge  bosons.  The chiral edge bosons are related to the Calogero­Sutherland model and  Luttinger liquids.  In everything mentioned above, the electron spin has been neglected. (In  a strong enough magnetic field all spins point in the same direction.) If one 

does take these spins into account, one finds interesting spin structures  called Skyrmions.  Recent experiments [This was still under construction in 1999 and therefore not so "recent" any more.]  Particles with fractional charge and statistics can at present only be probed at the  edge.   Chang   et   al.   have   done   an   experiment   where   electrons   tunnel   from   a  normal   metal   into  the   edge   of   a   nu=1/3   quantum   Hall   sample.   They   found   a  current­voltage relation for the tunneling that goes like I ~ V for low voltage and  like  I  ~  V3  for   higher   voltage,   in   accordance   with   theory.   The   temperature  dependence was also in good agreement with theory.  In an experiment by Milliken et al., the tunneling current was measured between  two edges of a nu=1/3 quantum Hall sample as a function of temperature and  gate voltage.  In an experiment by Chang et al., electrons were tunneled from an ordinary metal  into quantum Hall samples with a very sharp edge, for filling fractions between  1/4 and 1. They obtained the surprising result I ~ V1/nu. Surprising for two reasons:  First, one would naively expect the power of V (the so­called tunneling exponent)  to be quantised  when  the  conductances  are quantised.  Instead,  the tunneling  exponent   varies   continously   with   the   (non­quantised)   filling   fraction!   Second,  even   right   in   the   middle   of   conductance   plateaux   the   1/nu   result   contradicts  calculations made purely on the basis of chiral edge boson theories, except for  the   simplest   cases   nu=1   and   nu=1/3. We   have   proposed   an  explanation  for   this   experiment,   based   on   Coulomb  interactions between edge bosons and localised states in the bulk.  (Here I wanted to add a few words on shot noise, nuclear magnetic resonance,  Knight shift, Skyrmions etc, but never managed to find the time.) 

Quantum Field Theory [An apology: In spite of all my good intentions, this part is completely unreadable  for non­physicists and perhaps even for many physicists.] 

The   attentive   reader   will   have   noticed   that   in   the   `simple   theories'   the  combination of disorder and Coulomb interactions has been carefully avoided.  The reason is that there is nothing simple about this combination. It is, in fact, a  notoriously difficult problem. The Coulomb interactions prevent you from using  the single­particle wave functions with which you can attack the disorder, while  the   disorder   breaks   the   symmetry   that   would   help   you   tackle   the   interaction  problem.  The only hope left is quantum field theory. Write down an action that contains all  the ingredients: 2D electrons in a magnetic field, a random potential, Coulomb  interaction and a Chern­Simons gauge field that will generate flux attachment.  Put the action in an imaginary time path integral. Then perform the "shake and  stir" of field theory: Take the disorder average by integrating over the random  potential. Identify the massive modes and integrate them out in order to obtain an  effective   action   for   the   physically   interesting   massless   modes.   Finally,   do   a  renormalisation   group   analysis   that   tells   you   how   observables   will   depend   on  length scale.  Each   of   these   steps   introduces   its   own   problems,   which   are,   in   principle,  solvable. The solutions are sometimes quite peculiar. The presence of disorder,  for example, requires you to average the logarithm of the partition sum,  not  the  partition sum itself. This forces you to perform, on top of everything else, the so­ called replica trick, taking N identical copies of the system and then sending N to  zero. Somewhere in the derivation of the effective action, you are forced to put a  cutoff on frequency space, destroying the gauge invariance of the theory. Only by  sending this cutoff to infinity at the end of all calculations is the gauge invariance  restored. 

My own modest contributions AND WHY DO YOU THINK I DIRECTED YOU TO THE STABLES? THINK CAREFULLY, NOW. Mort  hesitated. He had been thinking carefully, in between counting wheel barrows. He'd wondered if it had  been to coordinate his hand and eye, or teach him the importance, on the human scale, of small tasks,  or   make   him   realise   that   even   great   men   must   start   on   the   bottom.   None   of   these   explanations  sounded exactly right. "I think", he began. YES? "Well, I think it was because you were up to your  knees in horseshit, to tell you the truth." Death looked at him for a long time. Mort shifted uneasily from  one   foot   to   the   other.   ABSOLUTELY   CORRECT,   snapped   Death.   CLARITY   OF   THOUGHT.  REALISTIC APPROACH. VERY IMPORTANT IN A JOB LIKE OURS. 

— Terry Pratchett, Mort 

I wrote my master's thesis  in 1995 under the supervision of prof.dr.ir. F.A. Bais at  the University of Amsterdam. It is called "Infinite symmetries in the quantum Hall  effect". Although this work did not lead to a publication, it is a nice review of the  quantum   Hall   effect   (including   the   Chern­Simons  theory,  conformal   symmetry,  Kac­Moody algebra and W­algebra involved in its effective description) that has  helped to lure several students to this great subject.  Between 1995 and 1999 I did my PhD research on the quantum Hall effect. My  PhD   supervisor   was   Aad   Pruisken.   Other   members   of   the   condensed   matter  theory club at that time were prof. Kareljan Schoutens, Mischa Baranov, Sathya  Guruswamy, Ronald van Elburg and Eddy Ardonne.  My research in a nutshell The starting point of my research can be roughly summarised in one sentence by  saying   that   we   have   discovered   a   new   symmetry   in   Finkelstein's   theory   for  interacting electrons in  a disordered medium and that we have extended  it in  such a way that the electrons can be coupled to gauge fields. The rest of my  activities has basically consisted of capitalising on this to obtain new results for  both the bulk and edge of quantum Hall systems.  The coupling is by no means a simple procedure. Simple attempts give rise to  infinities and problems with the U(1) gauge invariance. The way we did it was by  first noting that the Finkelstein theory has a hidden symmetry (which we dubbed  "F­invariance"). In the presence of long­range interactions, the theory is invariant  under a spatially constant shift of the plasmon field. (The plasmon field is roughly  speaking defined as the Coulomb potential at a certain point in the sample due to  all   the   other   charges   elsewhere   in   the   sample).   What   is   required   for   the  invariance to hold is a very special way of treating frequency cutoffs. The shift of  the plasmon field is reminiscent of a gauge transformation of the electromagnetic  scalar potential; this fact, together with the cutoff prescription, made it possible to  include U(1) gauge fields in the theory.  Having gauge fields at your disposal is obviously a great advantage. It allows you  to   do   linear   response   calculations   and   to   perform   the   Chern­Simons   flux  attachment trick, which is exactly what we have done. Apart from that, the  F­

invariance enabled us to do renormalisation group calculations to two­loop order.  My last QHE work was on edge states. In the limit of zero bulk density of states,  our   2+1   dimensional   theory   becomes   a   1+1   dimensional   theory   of   chiral  "relativistic" edge bosons that has the same structure as phenomenological edge  models, but yields new insights.  For   more   information   I   refer   to   the   publications   listed   below   and   references  therein.   I   decided   to  highlight   one   of   our   results   here,   because   we   believe   it  settles a controversy alive among people working on quantum Hall edges.  Understanding the tunneling experiment One of the nice things about the work I've been doing is that, although it may look  like a lot of arcane formalism, it can actually be directly related to experiments. In  our   approach   to   tunneling   processes   we   found   that   there   is   an   important  difference   between   a   tunneling   experiment   and   a   measurement   of   the   Hall  conductance. The Hall conductance is a non­equilibrium property (electrons are  injected into an edge channel and don't get time to equilibrate with the localised  bulk states), while by tunneling one probes the equilibrium energy eigenstates.  Due   to   the   presence   of   localised   bulk   states   and   the   Coulomb   interactions  between all states, the many body eigenfunctions are not restricted to the edge.  A tunneling experiment therefore feels the bulk as well as the edge.  We have done a calculation that shows that  the Coulomb interactions can be  effectively taken care of by writing down a theory that lives only on the edge but  has modified constants. To be more precise, if we are sitting at filling fraction nu =  nu0 + delta, with nu0 the center of a plateau, then the noninteracting theory would  contain a constant nu0 and the interactions would modify this to nu0 + delta = nu.  Another   effect   of   the   interactions   turns   out   to   be   that   the   so­called   "neutral  modes", which are degrees of freedom that are not related to the charge of the  electrons, get strongly suppressed. As a result of all this we find a current­voltage  relation of the form I ~ V1/nu, in agreement with the experiments. 

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Quantum Hall Effect -- from Eric Weisstein's World of Physics Both a fractional and integer quantum effect exist. See also: Hall Effect http://scienceworld.wolfram.com/physics/QuantumHallEffect.html - 13k Hall Effect -- from Eric Weisstein's World of Physics When electrons (or holes) move in a conducting plate that is immersed in a magnetic field, they experience a Lorentz force \mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B}) (in MKS), where q is the charge, E is the electric field, v is the velocity, and B is the magnetic field. Note that the seco http://scienceworld.wolfram.com/physics/HallEffect.html - 17k SI -- from Eric Weisstein's World of Physics "SI" stands for "System International" and is the set of physical units agreed upon by international convention. The SI units are sometimes also known as MKS units, where MKS stands for "meter, kilogram, and second." In 1939, the CCE recommended the adoption of a system of units based on the meter, kilo http://scienceworld.wolfram.com/physics/SI.html - 26k - 2001-02-23 Fine Structure Constant -- from Eric Weisstein's World of Physics A dimensionless number that appears in the analysis of quantum electrodynamical Feynman diagrams. It is not currently known if it can be derived from first principals in terms of mathematical constants, but it can be determined as a conglomeration of the electron charge e, \hbar (h-bar), and the speed of http://scienceworld.wolfram.com/physics/FineStructureConstant.html - 22k - 2002-07-09 Blank Entries from Eric Weisstein's World of Physics Blank Entries from Eric Weisstein's World of Physics http://scienceworld.wolfram.com/physics/contribute/entry.html - 58k Blank Entries from Eric Weisstein's World of Physics

Quantum Hall Effect In 1985 Klaus von Klitzing won the Nobel Prize for discovery of the quantised Hall effect. The previous Nobel prize awarded in the area of semiconductor physics was to Bardeen, Shockley and Brattain for invention of the transistor. Everyone knows how important transistors are in all walks of life, but why is a quantised Hall effect significant? Over 100 years ago E.H. Hall discovered that when a magnetic field is applied perpendicular to the direction of a current flowing through a metal a voltage is developed in the third perpendicular direction. This is well understood and is due to the charge carriers within the current being deflected towards the edge of the sample by the magentic field. Equilibrium is achieved when the magnetic force is balanced by the electrostatic force from the build up of charge at the edge. This happens when Ey = vxBz .The Hall coefficient is defined as RH = Ey /Bzjx and since the current density is jx = vxNq , RH =1/Nq in the case of a single species of charge carrier. RH can thus be measured to find N the density of carriers in the material. Often this transverse voltage is measured at fixed current and the Hall resistance recorded. It can easily be seen that this Hall resistance increases linearly with magnetic field. In a two-dimensional metal or semiconductor the Hall effect is also observed, but at low temperatures a series of steps appear in the Hall resistance as a function of magnetic field instead of the monotonic increase. What is more, these steps occur at incredibly precise values of resistance which are the same no matter what sample is investigated. The resistance is quantised in units of h/e2 divided by an integer. This is the QUANTUM HALL EFFECT.

The figure shows the integer

quantum Hall effect in a GaAs-GaAlAs heterojunction, recorded at 30mK. The QHE can be seen at liquid helium temperatures, but in the millikelvin regime the plateaux are much wider. Also included is the diagonal component of resistivity, which shows regions of zero resistance corresponding to each QHE plateau. In this figure the plateau index is, from top right, 1, 2, 3, 4, 6, 8.... Odd integers correspond to the Fermi energy being in a spin gap and even integers to an orbital LL gap. As the spin splitting is small compared to LL gaps, the odd integer plateaux are only seen at the highest magnetic fields. Important points to note are: • • •

The value of resistance only depends on the fundamental constants of physics: e the electric charge and h Plank's constant. It is accurate to 1 part in 100,000,000. The QHE can be used as primary a resistance standard, although 1 klitzing is a little large at 25,813 ohm!

Explanation of the Quantum Hall Effect The zeros and plateaux 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 quantised 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 electrons have more energy so more states become delocalised 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 heterojunctions the plateaux are much narrower.

Some Interesting Variants on the QHE In very high mobility samples extra plateaux appear between the regular quantum Hall plateaux, at resistances given by h/e2 divided by a rational fraction p/q instead of an integer. This is the fractional quantum Hall effect (FQHE). Early observations found that q was always an odd number and that certain fractions gave rise to much stronger features than others. The FQHE is much more difficult to explain since it originates from many electron correlations, but for this reason has been of great interst to theoreticians and experimentallists alike. In some materials there are more than one species of charge carrier. These may be elecrons in different conduction band minima, different spatially confined subbands or electrons and holes simultaneously present. The numbers and mobilities of all the species have to be considered to find the transport coefficients. If there are electrons and holes the total filling factor is the difference between the filling factors for electrons and holes. At certain fields this can be zero, at which point the Hall resistance itself becomes zero! Introduction The resistivity measurements of semiconductors can not reveal whether one or two types of carriers are present; nor distinguish between them. However, this information can be obtained from Hall Coefficient measurements, which are also basic tools for the determination of carrier density and mobilities in conjuction with resistivity measurement. Theory As you are undoubtedly aware, a static magnetic field has no effect on charges unless they are in motion. When the charges flow, a magnetic field directed perpendicular to the direction of flow produces a mutually perpendicular force on the charges. When this happens, electrons and holes will be separated by opposite forces. They will in turn produce an electric field ( h) which depends on the cross product of the magnetic intensity, , and the current density, J. h

=R x

Where R is called the Hall coefficient.

Now, let us consider a bar of semiconductor, having dimension, x, y and z. Let is directed along X and along Z then h will be along Y, as in Fig. 2. Then we could write

Where Vh is the Hall voltage appearing between the two surfaces perpendicular to y and I = yz Hall Effect experiment consists of the following: 1. (a) Hall Probe (Ge Crystal); (b) Hall Probe (InAs) 2. Hall Effect Set-up (Digital), DHE-21 3. Electromagnet, EMU-75 or EMU-50V 4. Constant Current Power Supply, DPS-175 or DPS-50 5. Digital Gaussmeter, DGM-102 Hall Probe (a) Hall Probe (Ge Crystal) Ge single crystal with four spring-type pressure contacts is mounted on a sunmica-decorated bakelite strip. Four leads are provided for connections with measuring devices. Technical details Material: Ge single crystal n or p-type as desire Resistivity: 8-10 Ω.cm Contacts: Spring type (solid silver) Zero-field potential: <1mV (adjustable) Hall Voltage: 25-35mV/10mA/KG It is designed to give a clear idea to the students about Hall Probe and is recommended for class room experiment. A minor drawback of this probe is that it may require zero adjustment. Hall Effect Setup (Digital), DHE-21

(b) Hall Probe (InAs) Indium Arsenide crystal with 4 soldered contacts is mounted on a PCB strip and covered with a protective layer. The Hall Element is mounted in a pen-type case and a 4-core cable is provided for connections with the measuring device and current source. Technical details Contacts: Soldered Rated Control Current: 4mA Zero Field Potential: <4mV Linearity (0-20KG): ±0.5% or better Hall Voltage: 60-70mV/4mA/KG The crystal alongwith its four contacts is visible through the protective layer. This is mainly used as a transducer for the measurement of magnetic field.

DHE-21 is a high performance instrument of outstanding flexibility. The set-up consists of a digital millivoltmeter and a constant current power supply. The Hall voltage and probe current can be read on the same digital panel meter through a selector switch. (i) Digital Millivoltmeter

(ii) Constant Current Power Supply

Intersil 3½ digit single chip A/D Converter ICL 7107 have been used. It has high accuracy like, auto zero to less than 10µV, zero drift of less than 1µV/°C, input bias current of 10pA max. and roll over error of less than one count. Since the use of internal reference causes the degradation in performance due to internal heating, an external reference has been used. Digital voltmeter is much more convenient to use in Hall experiment, because the input voltage of either polarity can be measured.

This power supply, specially designed for Hall Probe, provides 100% protection against crystal burn-out due to excessive current. The supply is a highly

Specifications Range: 0-200mV (100mV minimum) Accuracy: ±0.1% of reading ±1 digit

Fractional Quantum Hall Effect In high mobility semiconductor heterojunctions the integer quantum Hall effect (IQHE) plateaux are much narrower than for lower mobility samples. Between these narrow IQHE more plateaux are seen at fractional filling factors, especially 1/3 and 2/3. This is the fractional quantum Hall effect (FQHE) whose discovery in 1982 was completely unexpected. In 1998 the Nobel Prize in Physics was awarded to Dan Tsui and Horst Stormer, the experimentalists who first observed the FQHE, jointly with Robert Laughlin who suceeded in explaing the result in terms of new quantum states of matter. The figure shows the fractional quantum Hall effect in a GaAs-GaAlAs heterojunction, recorded at 30mK. Also included is the diagonal component of resistivity, which shows regions of zero resistance corresponding to each FQHE plateau. The principle series of fractions that have been seen are listed below. They generally get weaker going from left to right and down the page:

o o o o o o o

1/3, 2/5, 3/7, 4/9, 5/11, 6/13, 7/15... 2/3, 3/5, 4/7, 5/9, 6/11, 7/13... 5/3, 8/5, 11/7, 14/9... 4/3, 7/5, 10/7, 13/9... 1/5, 2/9, 3/13... 2/7, 3/11... 1/7....

(The fractional quantum Hall effect (FQHE) is concerned centrally with filling factor. This is usually writen as the greek letter nu, or v due to the limitations of HTML.)

Explanation of the Fractional Quantum Hall Effect Just as in the IQHE, FQHE plateaux are formed when the Fermi energy lies in a gap of the density of states. The difference is the origin of the energy gaps. While in the integer effect gaps are due to magnetic quantisation of the single particle motion, in the fractional effect the gaps arise from collective motion of all the electrons in the system. For the state at filling factor 1/3 Laughlin found a many body wavefunction with a lower energy than the single particle energy. This can also be adopted at any fraction v=1/(2m+1), but the energy difference is smaller at higher m and hence the fractions become weaker along the series 1/3, 1/5, 1/7.... All tests of Laughlin's wavefunction have shown it to be correct. The difficulty that arises is in accounting for all the other fractions at v=p/q where p>1 and simple wavefunctions can not be written down. It is also necessary to explain why q is always odd. The original explanation, developed by Haldane and Halperin, used a hierarchical model. Quasi-electrons or quasi-holes excited out of the Laughlin ground state would condense into higher order fractions, known as daughter states e.g. starting from the 1/3 parent state addition of quasi-electrons leads to 2/5 and quasi-holes leads to 2/7. Then quasi-particles are excited out of these daughter states which condense again into still more daughter states..... and so on down the hierarchy. There are several problems or unsatisfactory features within the hierarchical model: • • • •

it does not explain which daughter state (quasi-electron or -hole) should be the stronger after a few layers of the hierarchy there will be more quasi-particles than there were electrons in the original system between fractions the system is not well defined the quasi-particles carry fractional charge

More recently a model of composite fermions (CFs) has been introduced. A composite fermion consists of an electron (or hole) bound to an even number of magnetic flux quanta. Formation of these CFs accounts for all the many body interactions, so only single particle effects remain. The model exploits the similarities observed in measurements of the IQHE and FQHE to map the latter onto the former. Thus the fractional QHE of electrons in an external magnetic field now becomes the integer QHE of the new composite fermions in an effective magnetic field. The CFs have integer charge, just like electrons, but because they move in an effective magnetic field they appear to have a fractional topological charge. The composite fermion picture correctly predicts all the observed fractions including their relative intensities and the order they appear in as sample quality increases or temperature decreases. It also shows v=1/2, where the effective field for the CFs is zero, to be a special state with metallic characteristics.