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 R_xy exhibits steps that take on the quantized values

${\displaystyle R_{xy}={\frac {V_{\text{Hall}}}{I_{\text{channel}}}}={\frac {h}{e^{2}\nu }},}$

where V_Hall is the Hall voltage, I_channel 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).^[1]

The fractional quantum Hall effect is more complicated and still considered an open research problem.^[2] Its existence relies fundamentally on electron–electron interactions. In 1988, it was proposed that there was a quantum Hall effect without Landau levels.^[3] 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.^[4]