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 Rxy exhibits steps that take on the quantized values
where VHall is the Hall voltage, Ichannel is the channel current, e is the elementary charge and h is the Planck 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 and still considered an open research problem.[2] Its existence relies fundamentally on electron–electron interactions. In 1988, it was proposed that there was a quantum Hall effect without Landau levels.[3] 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.[4]
Applications[edit]
The quantization of the Hall conductance () has the important property of being exceedingly precise.[5] 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. 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 RK. 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 RK-90 = 25812.807 Ω was defined for use in resistance calibrations worldwide.[6] 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),[7] superseding the 1990 value with an exact permanent value RK = h/e2 = 25812.80745... Ω.[8]
Research status[edit]
The fractional quantum hall is considered part of exact quantization.[9] Exact quantization in full generality is not completely understood but it has been explained as a very subtle manifestation of the combination of the principle of gauge invariance together with another symmetry (see Anomalies). The integer quantum hall instead is considered a solved research problem[10][11] and understood in the scope of TKNN formula and Chern–Simons Lagrangians.
The fractional quantum Hall effect is still considered an open research problem.[2] The fractional quantum Hall effect can be also understood as an integer quantum Hall effect, although not of electrons but of charge–flux composites known as composite fermions.[12] Other models to explain the Fractional Quantum Hall Effect also exists.[13]
Currently it is considered an open research problem because no single, confirmed and agreed list of fractional quantum numbers exists, neither a single agreed model to explain all of them, although there are such claims in the scope of composite fermions and Non Abelian Chern–Simons Lagrangians.
History[edit]
The MOSFET (metal–oxide–semiconductor field-effect transistor), invented by Mohamed Atalla and Dawon Kahng at Bell Labs in 1959,[14] enabled physicists to study electron behavior in a nearly ideal two-dimensional gas.[15] 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.[15]
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.[16] 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.[17]
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.[18][15] 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.[11][19] 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,[20] and in the magnesium zinc oxide ZnO–MgxZn1−xO.[21]
Photonic quantum Hall effect[edit]
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.[23][24][25][26][27] 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.
Relativistic analogs[edit]
Relativistic examples of the integer quantum Hall effect and quantum spin Hall effect arise in the context of lattice gauge theory.[28][29]