Quantum Hall effect

The quantum Hall effect is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance exhibits steps that take on the quantized values
where is the Hall voltage, is the channel current, is the elementary charge and is the Planck constant. The divisor can take on either integer or fractional values. Here, is roughly but not exactly equal to the filling factor of Landau levels. The quantum Hall effect is referred to as the integer or fractional quantum Hall effect depending on whether is an integer or fraction, respectively.
The striking feature of the integer quantum Hall effect is the persistence of the quantization as the electron density is varied. Since the electron density remains constant when the Fermi level is in a clean spectral gap, this situation corresponds to one where the Fermi level is an energy with a finite density of states, though these states are localized.
The fractional quantum Hall effect is a more complicated state whose existence relies fundamentally on electron–electron interactions. In 1988, it was proposed that there was a quantum Hall effect without Landau levels. This quantum Hall effect is referred to as the quantum anomalous Hall effect. There is also a new concept of the quantum spin Hall effect which is an analogue of the quantum Hall effect, where spin currents flow instead of charge currents.

Applications

Electrical resistance standards

The quantization of the Hall conductance has the important property of being exceedingly precise. Actual measurements of the Hall conductance have been found to be integer or fractional multiples of to better than one part in a billion. It has allowed for the definition of a new practical standard for electrical resistance, based on the resistance quantum given by the von Klitzing constant. This is named after Klaus von Klitzing, the discoverer of exact quantization. The quantum Hall effect also provides an extremely precise independent determination of the fine-structure constant, a quantity of fundamental importance in quantum electrodynamics.
In 1990, a fixed conventional value was defined for use in resistance calibrations worldwide. Later, the 2019 revision of the SI fixed exact values of and, resulting in an exact

Research status

The quantization of the Hall resistance in integer and fractional quantum Hall effects is considered exact. The integer quantum Hall effect is considered a solved research problem and understood in the scope of TKNN formula and Chern–Simons Lagrangians.
The fractional quantum Hall effect arises due to strongly correlated states of electrons. These are extremely well understood in terms of the composite fermions, which are charge-flux composites that experience a significantly reduced magnetic field than electrons. The composite-fermion paradigm not only makes a multitude of nontrivial predictions, but also provides a quantitative theory. In particular, the integer quantum Hall of composite fermions produces sequences of fractions n/ where m and n are integers. The observed odd-denominator fractions are consistent with this prediction.

History

The MOSFET, invented by Mohamed Atalla and Dawon Kahng at Bell Labs in 1959, enabled physicists to study electron behavior in a nearly ideal two-dimensional gas.
In a MOSFET, conduction electrons travel in a thin surface layer, and a "gate" voltage controls the number of charge carriers in this layer. This allows researchers to explore quantum effects by operating high-purity MOSFETs at liquid helium temperatures.
The integer quantization of the Hall conductance was originally predicted by University of Tokyo researchers Tsuneya Ando, Yukio Matsumoto and Yasutada Uemura in 1975, on the basis of an approximate calculation which they themselves did not believe to be true. In 1978, the Gakushuin University researchers Jun-ichi Wakabayashi and Shinji Kawaji subsequently observed the effect in experiments carried out on the inversion layer of MOSFETs.
In 1980, Klaus von Klitzing, working at the high magnetic field laboratory in Grenoble with silicon-based MOSFET samples developed by Michael Pepper and Gerhard Dorda, made the unexpected discovery that the Hall resistance was exactly quantized. For this finding, von Klitzing was awarded the 1985 Nobel Prize in Physics. A link between exact quantization and gauge invariance was subsequently proposed by Robert Laughlin, who connected the quantized conductivity to the quantized charge transport in a Thouless charge pump. Most integer quantum Hall experiments are now performed on gallium arsenide heterostructures, although many other semiconductor materials can be used. In 2007, the integer quantum Hall effect was reported in graphene at temperatures as high as room temperature, and in the magnesium zinc oxide ZnO–Mg_xZn_1−xO.

Integer quantum Hall effect

Landau levels

In two dimensions, when classical electrons are subjected to a magnetic field they follow circular cyclotron orbits. When the system is treated quantum mechanically, these orbits are quantized. To determine the values of the energy levels the Schrödinger equation must be solved.
Since the system is subjected to a magnetic field, it has to be introduced as an electromagnetic vector potential in the Schrödinger equation. The system considered is an electron gas that is free to move in the x and y directions, but is tightly confined in the z direction. Then, a magnetic field is applied in the z direction and according to the Landau gauge the electromagnetic vector potential is and the scalar potential is. Thus the Schrödinger equation for a particle of charge and effective mass in this system is:
where is the canonical momentum, which is replaced by the operator and is the total energy.
To solve this equation it is possible to separate it into two equations since the magnetic field just affects the movement along x and y axes. The total energy becomes then, the sum of two contributions. The corresponding equations in z axis is:
To simplify things, the solution is considered as an infinite well. Thus the solutions for the z direction are the energies, and the wavefunctions are sinusoidal. For the and directions, the solution of the Schrödinger equation can be chosen to be the product of a plane wave in -direction with some unknown function of, i.e.,. This is because the vector potential does not depend on and the momentum operator therefore commutes with the Hamiltonian. By substituting this Ansatz into the Schrödinger equation one gets the one-dimensional harmonic oscillator equation centered at.
where is defined as the cyclotron frequency and the magnetic length. The energies are:
And the wavefunctions for the motion in the plane are given by the product of a plane wave in and Hermite polynomials attenuated by the gaussian function in, which are the wavefunctions of a harmonic oscillator.
From the expression for the Landau levels one notices that the energy depends only on, not on. States with the same but different are degenerate.

Density of states

At zero field, the density of states per unit surface for the two-dimensional electron gas taking into account degeneration due to spin is independent of the energy
As the field is turned on, the density of states collapses from the constant to a Dirac comb, a series of Dirac functions, corresponding to the Landau levels separated. At finite temperature, however, the Landau levels acquire a width being the time between scattering events. Commonly it is assumed that the precise shape of Landau levels is a Gaussian or Lorentzian profile.
Another feature is that the wave functions form parallel strips in the -direction spaced equally along the -axis, along the lines of. Since there is nothing special about any direction in the -plane if the vector potential was differently chosen one should find circular symmetry.
Given a sample of dimensions and applying the periodic boundary conditions in the -direction being an integer, one gets that each parabolic potential is placed at a value.
The number of states for each Landau Level and can be calculated from the ratio between the total magnetic flux that passes through the sample and the magnetic flux corresponding to a state.
Thus the density of states per unit surface is
Note the dependency of the density of states with the magnetic field. The larger the magnetic field is, the more states are in each Landau level. As a consequence, there is more confinement in the system since fewer energy levels are occupied.
Rewriting the last expression as it is clear that each Landau level contains as many states as in a 2DEG in a.
Given the fact that electrons are fermions, for each state available in the Landau levels it corresponds to two electrons, one electron with each value for the spin. However, if a large magnetic field is applied, the energies split into two levels due to the magnetic moment associated with the alignment of the spin with the magnetic field. The difference in the energies is being a factor which depends on the material and the Bohr magneton. The sign is taken when the spin is parallel to the field and when it is antiparallel. This fact called spin splitting implies that the density of states for each level is reduced by a half. Note that is proportional to the magnetic field so, the larger the magnetic field is, the more relevant is the split.
In order to get the number of occupied Landau levels, one defines the so-called filling factor as the ratio between the density of states in a 2DEG and the density of states in the Landau levels.
In general the filling factor is not an integer. It happens to be an integer when there is an exact number of filled Landau levels. Instead, it becomes a non-integer when the top level is not fully occupied. In actual experiments, one varies the magnetic field and fixes electron density or varies the electron density and fixes the magnetic field. Both cases correspond to a continuous variation of the filling factor and one cannot expect to be an integer. Since, by increasing the magnetic field, the Landau levels move up in energy and the number of states in each level grow, so fewer electrons occupy the top level until it becomes empty. If the magnetic field keeps increasing, eventually, all electrons will be in the lowest Landau level and this is called the magnetic quantum limit.