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Quantum Hall effect
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Quantum Hall effect
The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance Rxy exhibits steps that take on the quantized values
where VHall is the Hall voltage, Ichannel is the channel current, e is the elementary charge and h is the Planck constant. The divisor ν 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. Here, ν is roughly but not exactly equal to the filling factor of Landau levels. 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 and still considered an open research problem. Its existence relies fundamentally on electron–electron interactions. In 1988, it was proposed that there was a 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/h to better than one part in a billion. 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. 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 RK-90 = 25812.807 Ω was defined for use in resistance calibrations worldwide. Later, the 2019 revision of the SI fixed exact values of h and e, resulting in an exact RK = h/e2 = 25812.80745... Ω.
The fractional quantum Hall effect is considered part of exact quantization. Exact quantization in full generality is not completely understood but it has been explained as a very subtle manifestation of the combination of the principle of gauge invariance together with another symmetry (see Anomalies). The integer quantum Hall effect instead is considered a solved research problem and understood in the scope of TKNN formula and Chern–Simons Lagrangians.
The fractional quantum Hall effect is still considered an open research problem. The fractional quantum Hall effect can be also understood as an integer quantum Hall effect, although not of electrons but of charge–flux composites known as composite fermions. Other models to explain the fractional quantum Hall effect also exists. Currently it is considered an open research problem because no single, confirmed and agreed list of fractional quantum numbers exists, neither a single agreed model to explain all of them, although there are such claims in the scope of composite fermions and Non Abelian Chern–Simons Lagrangians.
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Quantum Hall effect
The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance Rxy exhibits steps that take on the quantized values
where VHall is the Hall voltage, Ichannel is the channel current, e is the elementary charge and h is the Planck constant. The divisor ν 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. Here, ν is roughly but not exactly equal to the filling factor of Landau levels. 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 and still considered an open research problem. Its existence relies fundamentally on electron–electron interactions. In 1988, it was proposed that there was a 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/h to better than one part in a billion. 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. 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 RK-90 = 25812.807 Ω was defined for use in resistance calibrations worldwide. Later, the 2019 revision of the SI fixed exact values of h and e, resulting in an exact RK = h/e2 = 25812.80745... Ω.
The fractional quantum Hall effect is considered part of exact quantization. Exact quantization in full generality is not completely understood but it has been explained as a very subtle manifestation of the combination of the principle of gauge invariance together with another symmetry (see Anomalies). The integer quantum Hall effect instead is considered a solved research problem and understood in the scope of TKNN formula and Chern–Simons Lagrangians.
The fractional quantum Hall effect is still considered an open research problem. The fractional quantum Hall effect can be also understood as an integer quantum Hall effect, although not of electrons but of charge–flux composites known as composite fermions. Other models to explain the fractional quantum Hall effect also exists. Currently it is considered an open research problem because no single, confirmed and agreed list of fractional quantum numbers exists, neither a single agreed model to explain all of them, although there are such claims in the scope of composite fermions and Non Abelian Chern–Simons Lagrangians.