The quantum Hall effect is a quantum mechanical version of the Hall effect, observed in two-dimensional systems of electrons subjected to low temperatures and strong magnetic fields, in which the Hall conductance σ takes on the quantized values

where e is the elementary charge and h is Planck's constant. In the "ordinary" quantum Hall effect, known as the integer quantum Hall effect, ν takes on integer values (ν = 1, 2, 3, etc.). There is another type of quantum Hall effect, known as the fractional quantum Hall effect, in which ν can occur as a fraction with an odd denominator (ν = 2/7, 1/3, 2/5, 3/5, etc.)

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 e²/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: since 1990, one "klitzing" (after Klaus von Klitzing, the discoverer of exact quantization) has been defined as the Hall resistance at ν = 4 (i.e., h/4e²). In addition, the resistance unit h/e², roughly equal to 25,812.8 Ohms, is sometimes referred to as the von Klitzing constant. When coupled with measurements of the Josephson constant, it provides an extremely precise independent determination of the fine structure constant, an extremely important quantity in quantum electrodynamics.

The integer quantization of the Hall conductance was originally predicted by Ando, Matsumoto, and Uemura in 1975, on the basis of an approximate calculation. Several workers subsequently observed the effect in experiments carried out on the inversion layer of MOSFETs. It was only in 1980 that von Klitzing, working with samples developed by Michael Pepper and Gerhard Dorda, made the totally 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.

The fractional effect was discovered in 1982 by Daniel Tsui and Horst Störmer, in experiments performed on gallium arsenide heterostructures developed by Arthur Gossard. The effect was explained by Laughlin in 1983, using a novel quantum liquid phase that accounts for the effects of interactions between electrons. Tsui, Störmer, and Laughlin were awarded the 1998 Nobel Prize for their work.

References

• T. Ando, Y. Matsumoto, and Y. Uemura, J. Phys. Soc. Jpn. 39, 279 (1975)
• K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980)
• R.B. Laughlin, Phys. Rev. B. 23, 5632 (1981).
• D.C. Tsui, H.L. Stormer, and A.C. Gossard, Phys. Rev. Lett. 48, 1559 (1982)
• R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).