Quantum Hall effect

The quantization of the Hall conductance has the important property of being exceedingly 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. This is named after Klaus von Klitzing, the discoverer of exact quantization. The quantum Hall effect also provides an extremely precise independent determination of the fine-structure constant, a quantity of fundamental importance in quantum electrodynamics.

In 1990, a fixed conventional value R_K-90 = 25812.807 Ω was defined for use in resistance calibrations worldwide.[4] On 16 November 2018, the 26th meeting of the General Conference on Weights and Measures decided to fix exact values of h (the Planck constant) and e (the elementary charge),[5] superseding the 1990 value with an exact permanent value R_K = h/e² = 25812.80745... Ω.[6]

The MOSFET (metal-oxide-semiconductor field-effect transistor), invented by Mohamed Atalla and Dawon Kahng at Bell Labs in 1959,[7] enabled physicists to study electron behavior in a nearly ideal two-dimensional gas.[8] In a MOSFET, conduction electrons travel in a thin surface layer, and a "gate" voltage controls the number of charge carriers in this layer. This allows researchers to explore quantum effects by operating high-purity MOSFETs at liquid helium temperatures.[8]

The integer quantization of the Hall conductance was originally predicted by University of Tokyo researchers Tsuneya Ando, Yukio Matsumoto and Yasutada Uemura in 1975, on the basis of an approximate calculation which they themselves did not believe to be true.[9] In 1978, the Gakushuin University researchers Jun-ichi Wakabayashi and Shinji Kawaji subsequently observed the effect in experiments carried out on the inversion layer of MOSFETs.[10]

In 1980, Klaus von Klitzing, working at the high magnetic field laboratory in Grenoble with silicon-based MOSFET samples developed by Michael Pepper and Gerhard Dorda, made the unexpected discovery that the Hall conductivity was exactly quantized.[11][8] 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.[3][12] 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,[13] and in the magnesium zinc oxide ZnO–Mg_xZn_1−xO.[14]

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:

${\displaystyle E_{n}=\hbar \omega _{\text{c}}\left(n+{\tfrac {1}{2}}\right),}$

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 "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

${\displaystyle N=g_{\text{s}}{\frac {BA}{\phi _{0}}},}$

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 magnetic 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.

Hofstadter's butterfly

The Quantum Hall effect, in addition to being observed in two-dimensional electron systems, can be observed in photons. Photons do not possess inherent electric charge, but through the manipulation of discrete optical resonators and quantum mechanical phase, therein creates an artificial magnetic field.[15] This process can be expressed through a metaphor of photons bouncing between multiple mirrors. By shooting the light across multiple mirrors, the photons are routed and gain additional phase proportional to their angular momentum. This creates an effect like they are in a magnetic field.

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. Note, however, that the density of states in these regions of quantized Hall conductance is zero; hence, they cannot produce the plateaus observed in the experiments. The phase diagram is fractal and has structure on all scales. In the figure there is an obvious self-similarity. In the presence of disorder, which is the source of the plateaus seen in the experiments, this diagram is very different and the fractal structure is mostly washed away.

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.

The value of the von Klitzing constant may be obtained already on the level of a single atom within the Bohr model while looking at it as a single-electron Hall effect. While during the cyclotron motion on a circular orbit the centrifugal force is balanced by the Lorentz force responsible for the transverse induced voltage and the Hall effect one may look at the Coulomb potential difference in the Bohr atom as the induced single atom Hall voltage and the periodic electron motion on a circle a Hall current. Defining the single atom Hall current as a rate a single electron charge ${\displaystyle e}$ is making Kepler revolutions with angular frequency ${\displaystyle \omega }$

${\displaystyle I={\frac {\omega e}{2\pi }},}$

and the induced Hall voltage as a difference between the hydrogen nucleus Coulomb potential at the electron orbital point and at infinity:

${\displaystyle U=V_{\text{C}}(\infty )-V_{\text{C}}(r)=0-V_{\text{C}}(r)={\frac {e}{4\pi \epsilon _{0}r}}}$

One obtains the quantization of the defined Bohr orbit Hall resistance in steps of the von Klitzing constant as

${\displaystyle R_{\text{Bohr}}(n)={\frac {U}{I}}=n{\frac {h}{e^{2}}}}$

which for the Bohr atom is linear but not inverse in the integer n.

Relativistic examples of the integer quantum Hall effect and quantum spin Hall effect arise in the context of lattice gauge theory.[16][17]

• Quantum Hall transitions
• Fractional quantum Hall effect
• Quantum anomalous Hall effect
• Quantum cellular automata
• Composite fermions
• Conductance Quantum
• Hall effect
• Hall probe
• Graphene
• Quantum spin Hall effect
• Coulomb potential between two current loops embedded in a magnetic field

• D. R. Yennie (1987). "Integral quantum Hall effect for nonspecialists". Rev. Mod. Phys. 59 (3): 781–824. Bibcode:1987RvMP...59..781Y. doi:10.1103/RevModPhys.59.781.
• D. Hsieh; D. Qian; L. Wray; Y. Xia; Y. S. Hor; R. J. Cava; M. Z. Hasan (2008). "A topological Dirac insulator in a quantum spin Hall phase". Nature. 452 (7190): 970–974. arXiv:0902.1356. Bibcode:2008Natur.452..970H. doi:10.1038/nature06843. PMID 18432240.
• 25 years of Quantum Hall Effect, K. von Klitzing, Poincaré Seminar (Paris-2004). Postscript. Pdf.
• Magnet Lab Press Release Quantum Hall Effect Observed at Room Temperature
• Avron, Joseph E.; Osadchy, Daniel; Seiler, Ruedi (2003). "A Topological Look at the Quantum Hall Effect". Physics Today. 56 (8): 38. Bibcode:2003PhT....56h..38A. doi:10.1063/1.1611351.
• 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; T. Ihn; K. Ensslin; K. Maranowski; A. Gossard (2007). "Quantum Hall effect transition in scanning gate experiments". Phys. Rev. B. 76 (8): 085316. Bibcode:2007PhRvB..76h5316B. doi:10.1103/PhysRevB.76.085316.
• E. I. Rashba and V. B. Timofeev, Quantum Hall Effect, Sov. Phys. - Semiconductors v. 20, pp. 617–647 (1986).