Quantum Hall effect

The quantum Hall effect (or integer 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

: <math> R_{xy} = \frac{V_\text{Hall{I_\text{channel = \frac{h}{e^2\nu} , </math>

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 (i.e. the Hall plateau) 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 (see Anderson localization).

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.

Applications

Electrical resistance standards

The quantization of the Hall conductance (<math> G_{xy}= 1/R_{xy} </math>) 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 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/(2mn 1) where m and n are integers. The observed odd-denominator fractions are consistent with this prediction.

History

The MOSFET (metal–oxide–semiconductor field-effect transistor), 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.

With the conductivity <math>\sigma=\rho^{-1} </math> one finds

: <math>\sigma=

\frac{1}{\det \rho}

\begin{pmatrix}

\rho_{yy}&-\rho_{xy}\\

-\rho_{yx}&\rho_{xx}

\end{pmatrix} \; .</math>

If the longitudinal resistivity is zero and transversal is finite, then <math> \det \rho \neq 0 </math>. Thus both the longitudinal conductivity and resistivity become zero.

Instead, when <math>\nu</math> is a half-integer, the Fermi energy is located at the peak of the density distribution of some Landau Level. This means that the resistivity will have a maximum due to increased scattering.

This distribution of minimums and maximums corresponds to ¨quantum oscillations¨ called Shubnikov–de Haas oscillations which become more relevant as the magnetic field increases. Obviously, the height of the peaks are larger as the magnetic field increases since the density of states increases with the field, so there are more carriers which contribute to the resistivity. It is interesting to notice that if the magnetic field is very small, the longitudinal resistivity is a constant which means that the classical result is reached.

alt=|thumb|263x263px|Longitudinal and transverse (Hall) resistivity, <math>\rho_{xx}</math> and <math>\rho_{xy}</math>, of a two-dimensional electron gas as a function of magnetic field. Both vertical axes were divided by the quantum unit of conductance <math>e^2/h</math> (units are misleading). The filling factor <math>\nu</math> is displayed for the last 4 plateaus.

Transverse resistivity

From the classical relation of the transverse resistivity <math display="inline">\rho_{xy}=\frac{B}{en_{\rm 2D</math> and substituting <math display="inline">n_{\rm 2D}=\nu \frac{eB}{h}</math> one finds out the quantization of the transverse resistivity and conductivity:

: <math>\rho_{xy}=\frac{h}{\nu e^2}\Rightarrow \sigma=\nu \frac{e^2}{h}</math>

One concludes then, that the transverse resistivity is a multiple of the inverse of the so-called conductance quantum <math>e^2/h</math> if the filling factor is an integer. In experiments, however, plateaus are observed for whole plateaus of filling values <math>\nu</math>, which indicates that there are in fact electron states between the Landau levels. These states are localized in, for example, impurities of the material where they are trapped in orbits so they can not contribute to the conductivity. That is why the resistivity remains constant in between Landau levels. Again if the magnetic field decreases, one gets the classical result in which the resistivity is proportional to the magnetic field.

Photonic quantum Hall effect

The quantum Hall effect, in addition to being observed in two-dimensional electron systems, can be observed in photons. Photons do not possess inherent electric charge, but through the manipulation of discrete optical resonators and coupling phases or on-site phases, an artificial magnetic field can be created. This process can be expressed through a metaphor of photons bouncing between multiple mirrors. By shooting the light across multiple mirrors, the photons are routed and gain additional phase proportional to their angular momentum. This creates an effect like they are in a magnetic field.

Topological classification

thumb|[[Hofstadter's butterfly]]

The integers that appear in the Hall effect are examples of topological quantum numbers. They are known in mathematics as the first Chern numbers and are closely related to Berry's phase. A striking model of much interest in this context is the Azbel–Harper–Hofstadter model whose quantum phase diagram is the Hofstadter butterfly shown in the figure. The vertical axis is the strength of the magnetic field and the horizontal axis is the chemical potential, which fixes the electron density. The colors represent the integer Hall conductances. Warm colors represent positive integers and cold colors negative integers. Note, however, that the density of states in these regions of quantized Hall conductance is zero; hence, they cannot produce the plateaus observed in the experiments. The phase diagram is fractal and has structure on all scales. In the figure there is an obvious self-similarity. In the presence of disorder, which is the source of the plateaus seen in the experiments, this diagram is very different and the fractal structure is mostly washed away. Also, the experiments control the filling factor and not the Fermi energy. If this diagram is plotted as a function of filling factor, all the features are completely washed away, hence, it has very little to do with the actual Hall physics.

Concerning physical mechanisms, impurities and/or particular states (e.g., edge currents) are important for both the 'integer' and 'fractional' effects. In addition, Coulomb interaction is also essential in the fractional quantum Hall effect. The observed strong similarity between integer and fractional quantum Hall effects is explained by the tendency of electrons to form bound states with an even number of magnetic flux quanta, called composite fermions.

Bohr atom interpretation of the von Klitzing constant

The value of the von Klitzing constant may be obtained already on the level of a single atom within the Bohr model while looking at it as a single-electron Hall effect. While during the cyclotron motion on a circular orbit the centrifugal force is balanced by the Lorentz force responsible for the transverse induced voltage and the Hall effect, one may look at the Coulomb potential difference in the Bohr atom as the induced single atom Hall voltage and the periodic electron motion on a circle as a Hall current. Defining the single atom Hall current as a rate a single electron charge <math>e</math> is making Kepler revolutions with angular frequency <math>\omega</math>

: <math>I = \frac{\omega e}{2\pi},</math>

and the induced Hall voltage as a difference between the hydrogen nucleus Coulomb potential at the electron orbital point and at infinity:

: <math>U=V_\text{C}(\infty) - V_\text{C}(r) = 0 - V_\text{C}(r) = \frac{e}{4\pi\epsilon_0 r}</math>

One obtains the quantization of the defined Bohr orbit Hall resistance in steps of the von Klitzing constant as

: <math>R_\text{Bohr}(n) = \frac{U}{I} = n\frac{h}{e^2}</math>

which for the Bohr atom is linear but not inverse in the integer n.

Relativistic analogs

Relativistic examples of the integer quantum Hall effect and quantum spin Hall effect arise in the context of lattice gauge theory.