Quantum Hall effect

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The 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 vulgar fraction ( $\,\nu$ = 2/7, 1/3, 2/5, 3/5, 5/2 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 $\,{e^2}/{h}$ 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 R_K (after Klaus von Klitzing, the discoverer of exact quantization) and since 1990, a fixed conventional value R_K-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.

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 integers that appear in the Hall effect are examples of topological quantum numbers. They are known in mathematics as the second 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 represnt the integer Hall conductances. Warm colors represent positive integers and cold colors negative integers. The phase diagram is fracal and has structure on all scales.

The fractional effect is due to completely different physics, and was experimentally 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 Robert B. 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. Although it was generally assumed that the discrete resistivity jumps found in the Tsui experiment were due to the presence of fractional charges (i.e., due to the emergence of quasiparticles with charges smaller than an electron charge), it was not until 1997 that R. de-Picciotto, et. al., indirectly observed fractional charges through measurements of quantum shot noise. Fractionally charged quasiparticles are neither Bosons nor Fermions and exhibit anyonic statistics. The fractional quantum Hall effect continues to be influential in theories about topological order. Quantum Hall effect properties have been found in graphene. This has been observed at room temperature, the only case thus far.

[edit] 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
D.C. Tsui, H.L. Stormer, and A.C. Gossard, Phys. Rev. Lett. 48, 1559 (1982) DOI:10.1103/PhysRevLett.48.1559
R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983) DOI:10.1103/PhysRevLett.50.1395
R. de-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin and D. Mahalu, Nature 389, 162-164 (1997)
K. von Klitzing, 25 years of Quantum Hall Effect, Poincaré Seminar (Paris-2004). Postscript.
Magnet Lab - Quantum Hall Effect Observed at Room Temperature
K. S. Novoselov et al, Science 315 1379 (9 Mar 2007)
J. E. Avron, D. Osacdhy and R. Seiler, Physics Today, August (2003)