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# Quantum Hall effect

The quantum Hall effect (or integer quantum Hall effect) is a quantum-mechanical version of the Hall effect, observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall conductance $\,\sigma$ takes on the quantized values

$\sigma = \nu \; \frac{e^2}{h},$

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, $\,\nu$ takes on integer values ( $\,\nu$ = 1, 2, 3, etc.). There is another type of quantum Hall effect, known as the fractional quantum Hall effect, in which $\,\nu$ can occur as a fraction ( $\,\nu$ = 2/7, 1/3, 2/5, 3/5, 5/2 etc.)

## 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: the resistance unit $\,h/{e^2}$, roughly equal to 25812.8 ohms, is referred to as the von Klitzing constant RK (after Klaus von Klitzing, the discoverer of exact quantization) and since 1990, a fixed conventional value RK-90 is used in resistance calibrations worldwide. 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. 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. Integer quantum Hall effect has also been found in graphene at temperatures as high as room temperature.

## Mathematics

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's 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. The phase diagram is fractal and has structure on all scales.

## References

• T. Ando, Y. Matsumoto, and Y. Uemura, J. Phys. Soc. Jpn. 39, 279 (1975) doi:10.1143/JPSJ.39.279
• K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980) doi:10.1103/PhysRevLett.45.494
• R.B. Laughlin, Phys. Rev. B. 23, 5632 (1981) doi:10.1103/PhysRevB.23.5632
• Integral quantum Hall effect for nonspecialists, D. R. Yennie, Rev. Mod. Phys. 59, 781 (1987) doi:10.1103/RevModPhys.59.781
• 25 years of Quantum Hall Effect, K. von Klitzing, Poincaré Seminar (Paris-2004). Postscript.
• Quantum Hall Effect Observed at Room Temperature, Magnet Lab Press Release [1]
• Room-Temperature Quantum Hall Effect in Graphene, K. S. Novoselov et al., Science 315 1379 (9 Mar 2007) doi:10.1126/science.1137201
• J. E. Avron, D. Osacdhy and R. Seiler, Physics Today, August (2003)