## Tuesday, June 22, 2010

### 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 conductivity σ takes on the quantized values

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

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, 1/5, 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, however, is more complicated, and its existence relies fundamentally on electron-electron interactions.

## 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 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, based on the resistance quantum given by the von Klitzing constant RK = h/e2 = 25812.807557(18) Ω.[1] This is named after Klaus von Klitzing, the discoverer of exact quantization. Since 1990, a fixed conventional value RK-90 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]

## Integer quantum Hall effect – Landau levels

In two dimensions, when classical electrons are subjected to a magnetic field they follow circular cyclotron orbits. When the system is treated quantum mechanically, these orbits are quantized. The energy levels of these quantized orbitals take on discrete values: $E_n = \hbar \omega_c (n+1/2)$, where ωc = eB/m is the cyclotron frequency. These orbitals are known as Landau levels, and at weak magnetic fields, their existence gives rise to many interesting "quantum oscillations" such as the Shubnikov-de Haas oscillations and the de Haas-van Alphen effect (which is often used to map the Fermi surface of metals). For strong magnetic fields, each Landau level is highly degenerate (i.e. there are many single particle states which have the same energy En). Specifically, for a sample of area A, in magnetic field B, the degeneracy of each Landau level is N = gsBA / φ0 (where gs represents a factor of 2 for spin degeneracy, and φ0 is the magnetic flux quantum). For sufficiently strong B-fields, each Landau level may have so many states that all of the free electrons in the system sit in only a few Landau levels; it is in this regime where one observes the quantum Hall effect.

## 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. In the figure there is an obvious self-similarity.

Concerning physical mechanisms, impurities and/or particular states (e.g., edge currents) seem to be important for the 'integer' effect, whereas in the fractional quantum Hall effect the Coulomb interaction is considered as the main reason. Finally, concerning the observed strong similarities between integer and fractional quantum Hall effect, the apparent tendency of electrons, to form bound states of an odd number with a magnetic flux quantum, i.e. composite fermions, is considered.

## Further reading

• Ando, Tsuneya; Matsumoto, Yukio; Uemura, Yasutada (1975). "Theory of Hall Effect in a Two-Dimensional Electron System". J. Phys. Soc. Jpn. 39: 279–288. doi:10.1143/JPSJ.39.279.
• Klitzing, K. von; Dorda, G.; Pepper, M. (1980). "New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance". Phys. Rev. Lett. 45 (6): 494–497. doi:10.1103/PhysRevLett.45.494.
• Laughlin, R. B. (1981). "Quantized Hall conductivity in two dimensions". Phys. Rev. B. 23 (10): 5632–5633. doi:10.1103/PhysRevB.23.5632.
• Yennie, D. R. (1987). "Integral quantum Hall effect for nonspecialists". Rev. Mod. Phys. 59 (3): 781–824. doi:10.1103/RevModPhys.59.781.
• Hsieh, D.; et al. (2008). "A topological Dirac insulator in a quantum spin Hall phase". Nature 452 (7190): 970–974. doi:10.1038/nature06843.
• 25 years of Quantum Hall Effect, K. von Klitzing, Poincaré Seminar (Paris-2004). Postscript. Pdf.
• Quantum Hall Effect Observed at Room Temperature, Magnet Lab Press Release [1]
• J. E. Avron, D. Osacdhy and R. Seiler, Physics Today, August (2003)
• Zyun F. Ezawa: Quantum Hall Effects - Field Theoretical Approach and Related Topics. World Scientific, Singapore 2008, ISBN 978-981-270-032-2
• Sankar D. Sarma, Aron Pinczuk: Perspectives in Quantum Hall Effects. Wiley-VCH, Weinheim 2004, ISBN 978-0-471-11216-7
• A. Baumgartner et al.: Quantum Hall effect transition in scanning gate experiments, Phys. Rev. B 76, 085316 (2007), doi:10.1103/PhysRevB.76.085316