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 quantum Hall transitions to take on the quantized values
where Ichannel is the channel current, VHall 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/, 2/, 3/, 2/, 3/, 1/, 2/, 3/, 5/, 12/,…) 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 electron density remains constant when the Fermi level is in a clean spectral gap, this situation corresponds to one where the Fermi level is an energy with a finite density of states, though these states are localized (see Anderson localization).
The fractional quantum Hall effect is more complicated, as its existence relies fundamentally on electron–electron interactions. The fractional quantum Hall effect is also understood as an integer quantum Hall effect, although 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.
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/ 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/ = 812.807557(18) Ω25. This is named after Klaus von Klitzing, the discoverer of exact quantization. Since 1990, a fixed conventional value RK-90 has been used in resistance calibrations worldwide. On 16 November 2018, the conventional value was abrogated in consequence of the decision to fix the values of h (the Planck constant) and e (the elementary charge) at the 26th meeting of the General Conference on Weights and Measures. 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 which they themselves did not believe to be true. Several researchers subsequently observed the effect in experiments carried out on the inversion layer of MOSFETs. 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. 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, and in the magnesium zinc oxide ZnO–MgxZn1−xO.
Integer quantum Hall effect – Landau levelsEdit
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:
where ωc = eB/ 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 En). Specifically, for a sample of area A, in magnetic field B, the degeneracy of each Landau level is
where gs represents a factor of 2 for spin degeneracy, and ϕ0 ≈ ×10−15 Wb2 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.
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 Bohr atom interpretation of the von Klitzing constantEdit
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 is making Kepler revolutions with angular frequency
and the induced Hall voltage as a difference between the hydrogen nucleus Coulomb potential at the electron orbital point and at infinity:
One obtains the quantization of the defined Bohr orbit Hall resistance in steps of the von Klitzing constant as
which for the Bohr atom is linear but not inverse in the integer n.
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