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 σ undergoes certain quantum Hall transitions to take on the quantized values

\sigma = \frac{I_\text{channel}}{V_\text{Hall}} = \nu \; \frac{e^2}{h},

where $I_\text{channel}$ is the channel current, $V_\text{Hall}$ is the Hall voltage, 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 fractional (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 striking feature of the integer quantum Hall effect is the persistence of the quantization (i.e. the Hall plateau) as the electron density is varied. Since the latter remains constant when the Fermi level is in a clean spectral gap, it must be that the Fermi level is immersed in essential spectrum (i.e. of finite density of states), though Anderson localized. The quantization and stability of the Hall conductance in such conditions was achieved in 1994 by Bellissard, van Elst and Schulz-Baldes using the methods of Non-Commutative Geometry.^[1]

The fractional quantum Hall effect is more complicated, as its existence relies fundamentally on electron–electron interactions. Although the microscopic origins of the fractional quantum Hall effect are unknown, there are several phenomenological approaches that provide accurate approximations. For example the effect can be thought of as an integer quantum Hall effect, not of electrons but of charge-flux composites known as composite fermions. In 1988, it was proposed that there was quantum Hall effect without Landau levels. This quantum Hall effect is referred to as the quantum anomalous Hall (QAH) effect. There is also a new concept of the quantum spin Hall effect which is an analogue of the quantum Hall effect, where spin currents flow instead of charge currents.^[2]

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.^[3] 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/e² = 25812.807557(18) Ω.^[4] This is named after Klaus von Klitzing, the discoverer of exact quantization. Since 1990, a fixed conventional value R_K-90 is used in resistance calibrations worldwide.^[5] 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.^[6] It was only in 1980 that Klaus von Klitzing, working at the high magnetic field laboratory in Grenoble with silicon-based 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, who connected the quantized conductivity to the quantized charge transport in Thouless charge pump.^[7]^[8] Most integer quantum Hall experiments are now performed on gallium arsenide heterostructures, although many other semiconductor materials can be used. In 2007, the integer quantum Hall effect was reported in graphene at temperatures as high as room temperature,^[9] and in the oxide ZnO-Mg_xZn_1-xO.^[10]

Integer quantum Hall effect – Landau levels

Animated graph showing filling of landau levels as B changes and the corresponding position on a graph of hall coefficient and magnetic field 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:

\textstyle E_n = \hbar \omega_c (n+1/2),

where ω_c = 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 E_n). Specifically, for a sample of area A, in magnetic field B, the degeneracy of each Landau level is

\textstyle N = g_s BA/\phi_0,

where g_s represents a factor of 2 for spin degeneracy, and ϕ₀ = 2·10⁻¹⁵ Wb 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

Hofstadter's butterfly

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 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) are important for both the 'integer' and 'fractional' effects. In addition, Coulomb interaction is also essential in the fractional quantum Hall effect. The observed strong similarity between integer and fractional quantum Hall effects is explained by the tendency of electrons to form bound states with an even number of magnetic flux quanta, called composite fermions.

References

↑ Bellissard; van Elst; Schulz-Baldes (1994). "The noncommutative geometry of the quantum Hall Effect". J. Math. Phys. 35: 5373–5451.
↑ Ezawa, Zyun F. (2013). Quantum Hall Effects: Recent Theoretical and Experimental Developments (3rd ed.). World Scientific. ISBN 978-981-4360-75-3.
↑ Laughlin, R. (1981). "Quantized Hall conductivity in two dimensions". Physical Review B 23 (10): 5632–5633. Bibcode:1981PhRvB..23.5632L. doi:10.1103/PhysRevB.23.5632. Retrieved 8 May 2012.
↑ Tzalenchuk, Alexander; Lara-Avila, Samuel; Kalaboukhov, Alexei; Paolillo, Sara; Syväjärvi, Mikael; Yakimova, Rositza; Kazakova, Olga; Janssen, T. J. B. M.; Fal'ko, Vladimir; Kubatkin, Sergey (2010). "Towards a quantum resistance standard based on epitaxial graphene". Nature Nanotechnology 5 (3): 186–189. arXiv:0909.1220. Bibcode:2010NatNa...5..186T. doi:10.1038/nnano.2009.474. PMID 20081845.
↑ "conventional value of von Klitzing constant". NIST.
↑ Wakabayashi, J.; Kawaji, S. (1978). J. Phys. Soc. Jpn (44): 1839. Missing or empty |title= (help)
↑ Laughlin, R. B. (1981-05-15). "Quantized Hall conductivity in two dimensions". Physical Review B 23 (10): 5632–5633. doi:10.1103/PhysRevB.23.5632.
↑ Thouless, D. J. (1983-05-15). "Quantization of particle transport". Physical Review B 27 (10): 6083–6087. doi:10.1103/PhysRevB.27.6083.
↑ Novoselov, K. S.; Jiang, Z.; Zhang, Y.; Morozov, S. V.; Stormer, H. L.; Zeitler, U.; Maan, J. C.; Boebinger, G. S.; Kim, P.; Geim, A. K. (2007). "Room-Temperature Quantum Hall Effect in Graphene". Science 315 (5817): 1379. arXiv:cond-mat/0702408. Bibcode:2007Sci...315.1379N. doi:10.1126/science.1137201. PMID 17303717.
↑ Tsukazaki, A.; Ohtomo, A.; Kita, T.; Ohno, Y.; Ohno, H.; Kawasaki, M. (2007). "Quantum Hall Effect in Polar Oxide Heterostructures". Science 315 (5817): 1388–91. Bibcode:2007Sci...315.1388T. doi:10.1126/science.1137430. PMID 17255474.