The Quantum Hall Effect

THE QUANTUM HALL EFFECT

Abstract

The Quantum Hall effect (QHE) is the observation of the Hall effect in a two-dimensional electron gas system (2DEG) such as graphene and MOSFETs. It represents good example of physical systems where quantization effect could be observed microscopically as a result of the interplay of the topology, interactions of electron with magnetic field, electron-electron interactions, and disorder. The main conditions for this phenomenon to be observed are extremely low temperatures and the presence of a strong magnetic field in a 2DEG. In this study, the classical Hall effect was considered followed by a detailed review of the QHE and its underlying principles. Integer Quantum Hall Effect (IQHE) and Fractional Quantum Hall Effect (FQHE) which forms two important categorizations of the QHE were analyzed. In IQHE, it was found that the quantized resistivity assumed a plateau-like (shooting at specific magnetic field) while in the case of the FQHE, the coulombic interaction and the electron-electron correlation resulted in the measurement of certain fractional n values. Finally, the current technological applications of the phenomenon were considered. 

1.0 Introduction

The Quantum Hall effect is the observation of the Hall effect in a two-dimensional electron gas system (2DEG) such as graphene and MOSFETs etc. The two-dimensional electron gas has to do with a scientific model in which the electron gas is free to move in two dimensions, but tightly confined in the third. This implies that at least for some phases of operation of the device, the carriers are confined in a potential such that the motion is only permitted in a restricted direction thus, quantizing the motion in this direction leaving only a two-dimensional momentum or k-vector which characterizes motion in a plane normal to the confining potential. The 2DEG is mostly found as a layer of electrons found in MOSFETs. Here, the electrons occupying the gate oxide layer become confined to the semiconductor-oxide interface, and thus occupy well defined energy levels. In this case the charge carrier density in the 2DEG can be controlled by varying the gate voltage.

In 1980, Klaus von Klitzing of the University of Würzburg, Germany, was conducting experiments at the Max Planck High Magnetic Field Laboratory in Grenoble, France, using high-quality MOSFETs built by his colleagues specifically for Hall voltage measurements [1]. He observed certain anomalies in the measurement of the resistance in the presence of high magnetic field even though no one had looked for a Hall voltage. He discovered that rather than a continuous increase of the Hall voltage with changing carrier density or magnetic field, as would be expected from the classical Hall effect, he found that the Hall resistance increased in discrete steps.

2.0 The Body

2.1 The classical Hall Effect

The classical Hall Effect was first discovered by Edwin Hall in 1879 at the Johns Hopkins University [2]. This discovery came about an attempt to examine the direct effect of magnet on current flow as opposed to the general belief that wire bearing the current. The experiment was conducted using a metal (iron, silver, gold) leaf mounted on a glass plate and placed between the poles of an electromagnet. Edwin discovered that a new action of the magnet on electric currents and this later on became the Hall effect [3] [4].

The Hall effect describes what happens to current flowing through a conducting material such as a metal or a semiconductor. It relates to the motion of charge carriers in magnetic fields. It is the earliest experiments classical Hall effect that established the sign of charge carriers in conductors. More often, it has been used to measure the numerical density of charge carriers in conducting materials as well as the drift velocity when a uniform electric field is applied. It also finds important application in in magnetic field sensors that are reliable, accurate, and relatively inexpensive [2].

To analyze the Hall effect, consider a potential difference applied to an ordinary piece of metal (which is not superconducting) to overcome its electrical resistance in order to drive current through. What happens is that, the electric field flows in the direction of the current. Hall discovered that when a strong magnetic field was applied perpendicular to the flowing current, the electrons were pushed sideways by a velocity-dependent force, the Lorentz Force [3]. This phenomenon can be viewed by a strong wind that runs through moving cars and pushes that sideways along their path. The push of the electrons through to the sides is at right angles to the velocity and at right angles to the magnetic field. This results in a voltage drop called the Hall voltage (V_H) (Figure 1) that occurs at right angles to the current flow.

Figure 1: The classical Hall effect [5]

2.2 The Quantum Hall Effect and their Principle of Operation

The Quantum Hall Effect was discovered by the Nobel Prize winner, Klaus von Klitzing in 1980 [2], just five years after his initial prediction of the phenomenon. This “new discovery” further confirmed the fact that electrons existed and can only exist in discrete energy levels. One unique observation that Klitzing made when he observed the behavior of electrons under a strong magnetic field and extremely low temperatures, he observed that the electron flow sideways did not continue in a curved continuous path as was characteristic of the classical Hall Effect but rather observed sudden jumps. He also observed sharp peaks in the resistance to flow at specific energy levels with changing magnetic fields. That is, unlike the conventional the Hall effect where the Hall resistance (R_H) varied continuously in the case of the QHE, the Hall resistance (R_QH) took quantized values which can be expressed as R_QH= $\frac{h}{{\mathit{ve}}^2}$

, where h is Planck’s constant, e is the electron energy and the v is the filling factor (it is either an integer or fractional value). The value of this Quantized Hall resistance is 25 812.807 Ω [6].

Another important surprise from the QHE phenomenon is the fact that the longitudinal resistivity could vanish to zero and as such allows electrons to be transported along the edges of the sample without energy dissipation. This makes Quantum Hall systems to act as perfect wires which could transport electrons with little energy dissipation [7]. Transport in Quantum Hall system could be viewed with as analogous to the orbital motion of electrons around the nucleus of an atom where there is no energy loss only that in this case the motion occurs on a macroscopic scale where electrons are allowed to travel macroscopic distance with no dissipation of energy. The dissipationless quantum Hall edge states arise from the special topological properties of the band structure induced by the strong magnetic field, which protects electrons from localization or backscattering. The external field required for observing the quantum Hall effect is typically as large as several tesla [7].

The importance of the QHE is that they are good examples of physical systems where quantization effect could be observed microscopically which results from the interplay of the topology, interactions of electron with magnetic field, electron-electron interactions, and disorder. Solutions to the Schrodinger wave equation gives rise to the Landau quantization of the cyclotron orbits of charged particles in magnetic fields (Landau levels). The observation of this discretization in itself is not what makes the QHE a special one since this effect has been observed in many quantum mechanical phenomena but the interesting thing about the QHE results from the discretization that is observed when microscopic quantity is Hall conductance is measured. The Hall conductance results from measuring the voltage drop perpendicular to the applied current in the presence of the strong magnetic field. It can be expressed as,

${\sigma }_{\mathit{xy}}=\frac{e^2}{h}n$

The Hall conductance could take specific integer values (n) as in the case of Integer Quantum Hall Effect (IQHE) or fraction (n) as in the case of Fractional Quantum Hall Effect (FQHE) [8].

2.3 Integer Quantum Hall Effect (IQHE)

The first experiments exploring the quantum regime of the Hall effect were performed in 1980 by von Klitzing [9]. It occurs when the Landau levels are fully filled as such, the Hall conductance becomes equal to the number of those (Landau) levels that are completely filled [10]. The uniqueness of this phenomena is that it is geometry independent. Hall conductance remains perfectly quantized even in the presence of interactions and disorder in the sample. The disorder refers to the impurities present in the sample and Klitzing found that the QHE became more and more pronounce even in the presence of these dirt [8]. This observation can be traced to the fact that the Hall conductance is related to a certain topological invariant and therefore is not affected by the local smooth changes in the Hamiltonian. The IQHE is also among the very few physical systems that can escape Anderson localization in two dimensions.

Figure 2. The integer quantum Hall effect seen as a plateau [8].

When the resistivity of the 2DEG was measured under the various strong magnetic fields, Klitzing observed plateau-like Hall resistivity. The value plateau-like resistivity is given by the equation

${\rho }_{\mathit{xy}}=\frac{{2\pi ħ}^2}{e^2}\frac{1 }{v}\mathit{ }v$

which takes on integer values in this case

.

Another surprise that his experiment revealed was, the longitudinal resistivity ${\rho }_{\mathit{xx}}$

  vanishes to zero when the ${\rho }_{\mathit{xy}}$

(resistivity due to the magnetic presence) sits on the plateau but spikes whenever the ${\rho }_{\mathit{xy}}$

suddenly jumps to the next plateau. The $\frac{{2\mathit{\pi ħ}}^2}{e^2}$

referred to as the quantum of resistivity is currently being employed as the standard for measuring the resistivity of a material. IQHE was first discovered in an inversion layer of a metal-oxide-semiconductor field-effect transistor (MOSFET) and later discovered in 2D graphene sheets.

2.4 Fractional Quantum Hall Effect (FQHE).

Initially, research had taught that the Quantized Hall conductance could only take specific integer values, but later research found that this measured Hall conductance could assume specific fractional values that obey certain forms (most of them have odd denominator). This is as a result of the fact that sometimes the Landau levels becomes partially filled resulting in degenerate energy levels [11]. This can be viewed as filled Landau Levels of particles in a fictitious magnetic field, within a partially-full Landau Level of the true magnetic field. The most prominent fractions experimentally determined are $n=1/3$

and $n=2/5$

[8]. In the FQHE, there are a lot of energetically equivalent ground states and any small interaction is expected to lift the degeneracy and pick up a single ground state 

Research found that as the disorder of the 2DEG system is decreased, the integer Hall plateau become less pronounce but there is the emergence of many plateau at specific fractional values Figure 3).  FQHE phenomenon comes into play because of strong coulombic interactions and correlations among electrons  rather than just interactions with the applied magnetic field as in the case of the IQHE [12]. This complex and striking electron behavior observed in the FQHE has made the QHE a continuous source for exploring novel phenomena with most of them relying on fact that the mathematics of topology has a consequence on quantum physics [12]. FQHE states greatly enrich our understanding of quantum phases and quantum phase transitions [13]. Important examples of these application are in topological insulators, topological order and topological quantum computing. FQHE was first discovered in GaAs-GaAsAl heterostructure and later discovered in 2D graphene relativistic electrons.

Figure 3. The Fractional quantum Hall effect [8].

2.5 The Landau Levels

The Landau level form due to cyclotron motion quantization in a 2D system [12]. In the presence of a magnetic field, the energy levels of a particle become equally spaced, with the gap between each level proportional to the magnetic field B. These energy levels which are degenerate in nature are what is referred to as Landau levels.

3.0 Application of Quantum Hall Effect

3.1 Metrology

Because of the extreme precision of the Quantum Hall resistivity measurement, QHE finds significant application in metrology currently serving as the standard definition of ohms. It has allowed for the definition of a new practical standard for electrical resistance, based on the Quantum Hall resistance which is given by the von Klitzing constant RK = 25812.807557Ω [6]. This is named after Klaus von Klitzing, the discoverer of exact quantization. The Klitzing constant, RK, can be reproduced within a relative uncertainty of one part in 10⁹.

3.2 Topological Quantum computers

Another potential application of QHE is in topological quantum computations [14]. Topological quantum computers are basically theoretical computers that base their operation on two important quantum mechanical phenomena which are superposition and entanglement. Their operations employ two-dimensional quasiparticles called anyons, whose world lines pass around one another to form braids in a three-dimensional spacetime [14]. Here Quantum information encoded in the charges of the anyons is highly resistant to decoherence, and can be reliably processed by carrying one anyons around another [15].

4.0 Conclusion

The Quantum Hall Effect and the underlying principle of its operation has been carefully explored in this paper. QHE has been found to prove the discrete properties of electrons. IQHE which was initially discovered revealed that quantized Hall conductance could only take on integer values however, later research showed that this conductance could also assume certain fractional values under conditions where electron-electron correlation and coulombic interaction become prominent. This does not only lead to a complex description of the electron behavior in the 2DEG  system but also it offers rich our insights into quantum phases.

Works Cited