Classical and Quantum Hall Effects
The classical Hall effect was discovered by Edwin Hall in 1879. The basic idea is that a current of electrons in a thin conducting strip, idealized as a two-dimensional plane, is subjected to a constant magnetic field in the normal direction. The Lorentz force perpendicular to the current causes a buildup of charge on the edge of the strip. This induces a voltage across the width of the strip, in addition to the voltage along the length, which is responsible for the current. One can define the Hall resistance , in addition to the usual DC resistance . It is shown in the Details section that the Hall resistance is given by , where is the density of the charge carriers and is the thickness of the strip. In semiconductors, the charge carriers can be both electrons and positive holes. If the holes dominate, the voltage across the strip can be in the opposite direction.[more]
The graphic for the "classical" Hall effect shows the dependence of and on the magnetic field for selected values of and τ, a scattering parameter in the Drude model for electrical conductivity, of the order of 1 to sec.
The quantum Hall effect was discovered by von Klitzing in 1980. It requires liquid helium temperatures (4.2 K or lower) and magnetic fields of the order of 5 or 10 T. The electrons are idealized as moving in two dimensions, which is well approximated by using a silicon metal oxide semiconductor field effect transistor (MOSFET). The observed Hall resistance exhibits a series of plateaus with quantized values , . (More recently, a fractional quantized Hall effect has been discovered, with , and so on, but this Demonstration is only concerned with the integer quantum Hall effect.) As discussed in the Details section, the integer values can be correlated with the number of filled Landau levels below the Fermi level of the semiconductor. Remarkably, the observed Hall resistance, in contrast to the ordinary resistance, is fairly insensitive to the purity or other details of the semiconductor. The results have been precise to a few parts in and have now been adopted for defining the international standard of resistance.
The graphic for the "quantum" Hall effect shows the variation of as well as the DC resistance with . The dashed line shows the corresponding classical result. The plateaus of are associated with reduction of to zero, evidently exhibiting an onset of superconductivity.[less]
The classical Hall effect can be accounted for by the Drude model for the motion of an electron in crossed electric and magnetic fields subjected to a Lorentz force plus a damping term modeling the scattering of the electron from lattice ions:
where is the velocity of the electron and is the average time between collisions. For steady-state solutions, with , the equation reduces to
which has the form of the generalized Ohm's law
Using the relations and , where is the electron number density and is the cyclotron frequency, we obtain a tensor relation for the conductivity
where is the direct current conductivity in the absence of a magnetic field. The corresponding resistivities are then given by
The quantum Hall effect arises from the energy bands comprising the Landau levels of electrons confined to a two-dimensional domain. The energy of each electron has the form
with . The second term represents the free motion of the electron in the direction and is limited by the width of the two-dimensional domain. The number density of electrons in a filled Landau level is given by
where is the thickness of the conductor. For sufficiently high magnetic field , the energy width is smaller than the separation of Landau levels, thus resulting in a series of equally spaced bands. From the Drude model, when filled Landau levels lie completely below the Fermi level of the electron system, we find
 D. Tong, "Lectures on the Quantum Hall Effect." arxiv.org/abs/1606.06687.
 S. Meng. "Integer Quantum Hall Effect." (May 20, 2020) ethz.ch/content/dam/ethz/special-interest/phys/theoretical-physics/itp-dam/documents/gaberdiel/proseminar_fs2018/07_Meng.pdf.