The **quantum Hall effect** (or **integer quantum Hall effect**) is a quantized version of the Hall effect and which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance *R*_{xy} exhibits steps that take on the quantized values at certain level

- Applications
- History
- Integer quantum Hall effect
- Landau levels
- Density of states
- Longitudinal resistivity
- Transverse resistivity
- Photonic quantum Hall effect
- Mathematics
- The Bohr atom interpretation of the von Klitzing constant
- Relativistic analogs
- See also
- References
- Further reading

where *V*_{Hall} is the Hall voltage, *I*_{channel} is the channel current, *e* is the elementary charge and *h* is Planck's constant. The divisor *ν* can take on either integer (*ν* = 1, 2, 3,...) or fractional (*ν* = 1/3, 2/5, 3/7, 2/3, 3/5, 1/5, 2/9, 3/13, 5/2, 12/5,...) values. Here, *ν* 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 *ν* 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).^{ [1] }

The fractional quantum Hall effect is more complicated, its existence relies fundamentally on electron–electron interactions. The fractional quantum Hall effect is also understood as an integer quantum Hall effect, although not of electrons but of charge-flux composites known as composite fermions. In 1988, it was proposed that there was quantum Hall effect without Landau levels.^{ [2] } This quantum Hall effect is referred to as the quantum anomalous Hall (QAH) 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.^{ [3] }

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 *e*^{2}/*h* to nearly one part in a billion. This phenomenon, referred to as *exact quantization*, is not really understood but it has sometimes been explained as a very subtle manifestation of the principle of gauge invariance.^{ [4] } 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 *R*_{K}. 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 *R*_{K-90} = 25812.807 Ω was defined for use in resistance calibrations worldwide.^{ [5] } On 16 November 2018, the 26th meeting of the General Conference on Weights and Measures decided to fix exact values of *h* (the Planck constant) and *e* (the elementary charge),^{ [6] } superseding the 1990 value with an exact permanent value *R*_{K} = *h*/*e*^{2} = 25812.80745... Ω.^{ [7] }

The MOSFET (metal-oxide-semiconductor field-effect transistor), invented by Mohamed Atalla and Dawon Kahng at Bell Labs in 1959,^{ [8] } enabled physicists to study electron behavior in a nearly ideal two-dimensional gas.^{ [9] } 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.^{ [9] }

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.^{ [10] } 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.^{ [11] }

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.^{ [12] }^{ [9] } 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.^{ [4] }^{ [13] } 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,^{ [14] } and in the magnesium zinc oxide ZnO–Mg_{x}Zn_{1−x}O.^{ [15] }

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 tightly confined in the z direction. Then, it is applied a magnetic field along 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. The total energy becomes then, the sum of two contributions . The corresponding two equations are:

In z axis:

To simply the solution it is considered as an infinite well, thus the solutions for the z direction are the energies and the wavefunctions are sinusoidal. For the x and y directions, the solution of the Schrödinger equation is the product of a plane wave in y-direction with some unknown function of x since the vector potential does not depend on y, i.e. . 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 xy plane are given by the product of a plane wave in y and Hermite polynomials, which are the wavefuntions of an 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.

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 less 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 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 ( for free electrons) 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. 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.

It is possible to relate the filling factor to the resistivity and hence, to the conductivity of the system. When is an integer, the Fermi energy lies in between Landau levels where there are no states available for carriers, so the conductivity becomes zero (it is considered that the magnetic field is big enough so that there is no overlap between Landau levels, otherwise there would be few electrons and the conductivity would be approximately ). Consequently, the resistivity becomes zero too (At very high magnetic fields it is proven that longitudinal conductivity and resistivity are proportional).^{ [16] }

Instead, when is a half-integer, the Fermi energy is located at the peak of the density distribution of some Landau Level. This means that the conductivity will have a maximum .

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.

From the classical relation of the transverse resistivity and substituting one finds out the quantization of the transverse resistivity and conductivity:

One concludes then, that the transverse resistivity is a multiple of the inverse of the so-called conductance quantum . Nevertheless, in experiments a plateau is observed between Landau levels, which indicates that there are in fact charge carriers present. These carriers 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.

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 quantum mechanical phase, therein creates an artificial magnetic field.^{ [17] } 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.

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.

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 *.

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 a Hall current. Defining the single atom Hall current as a rate a single electron charge is making Kepler revolutions with angular frequency

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

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

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

Relativistic examples of the integer quantum Hall effect and quantum spin Hall effect arise in the context of lattice gauge theory.^{ [18] }^{ [19] }

The **quantum harmonic oscillator** is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

* Bremsstrahlung*, from

In the physical sciences, the **wavenumber** is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.

An ideal **Fermi gas** is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density, temperature, and the set of available energy states. The model is named after the Italian physicist Enrico Fermi.

In physics, a **coupling constant** or **gauge coupling parameter**, is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two static bodies to the "charges" of the bodies divided by the distance squared, , between the bodies; thus: G in for Newton's gravity and in for electrostatic. This description remains valid in modern physics for linear theories with static bodies and massless force carriers.

In quantum mechanics and quantum field theory, the **propagator** is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram. These may also be viewed as the inverse of the wave operator appropriate to the particle, and are, therefore, often called *(causal) Green's functions*.

In condensed matter physics, the **Fermi surface** is the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature. The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline lattice and from the occupation of electronic energy bands. The existence of a Fermi surface is a direct consequence of the Pauli exclusion principle, which allows a maximum of one electron per quantum state.

In physics, the **Lamb shift**, named after Willis Lamb, is a difference in energy between two energy levels ^{2}*S*_{1/2} and ^{2}*P*_{1/2} of the hydrogen atom which was not predicted by the Dirac equation, according to which these states should have the same energy.

In atomic physics, the **spin quantum number** is a quantum number that describes the intrinsic angular momentum of a given particle. The spin quantum number is designated by the letter s, and is the fourth of a set of quantum numbers, which completely describe the quantum state of an electron. The name comes from a physical spinning of the electron about an axis that was proposed by Uhlenbeck and Goudsmit. However this simplistic picture was quickly realized to be physically impossible, and replaced by a more abstract quantum-mechanical description.

The **old quantum theory** is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory is now understood as the semi-classical approximation to modern quantum mechanics.

In quantum physics, the **spin–orbit interaction** is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is one cause of magnetocrystalline anisotropy and the spin Hall effect.

In condensed matter physics, **Hofstadter's butterfly** describes the spectral properties of non-interacting two-dimensional electrons in a magnetic field in a lattice. The fractal, self-similar nature of the spectrum was discovered in the 1976 Ph.D. work of Douglas Hofstadter and is one of the early examples of computer graphics. The name reflects the visual resemblance of the figure on the right to a swarm of butterflies flying to infinity.

In quantum mechanics, **Landau quantization** refers to the quantization of the cyclotron orbits of charged particles in a uniform magnetic field. As a result, the charged particles can only occupy orbits with discrete, equidistant energy values, called Landau levels. These levels are degenerate, with the number of electrons per level directly proportional to the strength of the applied magnetic field. It is named after the Soviet physicist Lev Landau.

In quantum mechanics, a **raising** or **lowering operator** is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.

In condensed matter physics, the **Laughlin wavefunction** is an ansatz, proposed by Robert Laughlin for the ground state of a two-dimensional electron gas placed in a uniform background magnetic field in the presence of a uniform jellium background when the filling factor of the lowest Landau level is where is an odd positive integer. It was constructed to explain the observation of the fractional quantum Hall effect, and predicted the existence of additional states as well as quasiparticle excitations with fractional electric charge , both of which were later experimentally observed. Laughlin received one third of the Nobel Prize in Physics in 1998 for this discovery. Being a trial wavefunction, it is not exact, but qualitatively, it reproduces many features of the exact solution and quantitatively, it has very high overlaps with the exact ground state for small systems.

The **Planck constant**, or **Planck's constant**, is a fundamental physical constant denoted , and is of fundamental importance in quantum mechanics. A photon's energy is equal to its frequency multiplied by the Planck constant. Due to mass–energy equivalence, the Planck constant also relates mass to frequency.

A **composite fermion** is the topological bound state of an electron and an even number of quantized vortices, sometimes visually pictured as the bound state of an electron and, attached, an even number of magnetic flux quanta. Composite fermions were originally envisioned in the context of the fractional quantum Hall effect, but subsequently took on a life of their own, exhibiting many other consequences and phenomena.

The **quantization of the electromagnetic field**, means that an electromagnetic field consists of discrete energy parcels, photons. Photons are massless particles of definite energy, definite momentum, and definite spin.

**Symmetries in quantum mechanics** describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems.

**Electric dipole spin resonance** (**EDSR**) is a method to control the magnetic moments inside a material using quantum mechanical effects like the spin–orbit interaction. Mainly, EDSR allows to flip the orientation of the magnetic moments through the use of electromagnetic radiation at resonant frequencies. EDSR was first proposed by Emmanuel Rashba.

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