The quantum Hall effect (or integer quantum Hall effect) is a quantummechanical version of the Hall effect, observed in twodimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall conductivity σ takes on the quantized values
where e is the elementary charge and h is Planck's constant. The prefactor ν is known as the "filling factor", and can take on either integer (ν = 1, 2, 3, .. ) or rational fraction (ν = 1/3, 2/5, 3/7, 2/3, 3/5, 1/5, 2/9, 3/13, 5/2, 12/5 ..) values. 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 integer quantum Hall effect is very well understood, and can be simply explained in terms of single particle orbitals of an electron in a magnetic field (see Landau quantization). The fractional quantum Hall effect, is more complicated, as its existence relies fundamentally on electronelectron interactions. Interestingly, it is also very well understood as an integer quantum Hall effect, not of electrons but of chargeflux composites known as composite fermions.
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Applications
The quantization of the Hall conductance has the important property of being incredibly 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", has been shown to be a subtle manifestation of the principle of gauge invariance. 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} = h/e^{2} = 25812.807557(18) Ω.^{[1]} This is named after Klaus von Klitzing, the discoverer of exact quantization. Since 1990, a fixed conventional value R_{K90} is used in resistance calibrations worldwide.^{[2]} The quantum Hall effect also provides an extremely precise independent determination of the fine structure constant, a quantity of fundamental importance in quantum electrodynamics.
History
The integer quantization of the Hall conductance was originally predicted by Ando, Matsumoto, and Uemura in 1975, on the basis of an approximate calculation which they themselves did not believe to be true. Several workers subsequently observed the effect in experiments carried out on the inversion layer of MOSFETs. It was only in 1980 that Klaus von Klitzing, working with samples developed by Michael Pepper and Gerhard Dorda, made the unexpected discovery that the Hall conductivity was exactly quantized. For this finding, von Klitzing was awarded the 1985 Nobel Prize in Physics. The link between exact quantization and gauge invariance was subsequently found by Robert Laughlin. Most integer quantum Hall experiments are now performed on gallium arsenide heterostructures, although many other semiconductor materials can be used. The integer quantum Hall effect has also been found in graphene at temperatures as high as room temperature^{[3]}, and recently has been observed in the oxide ZnOMn_{x}Zn_{1x}O^{[4]}.
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