Integer Quantum Hall Effect

The idea of this simple simulation is to use the density of states to explain the integer quantum Hall effect; this argument can be found in the references below. The idea is simple: for a 2D electron gas in a magnetic field, the electrons move in Landau levels. If the Fermi energy lies between the levels, the scattering that leads to the normal resistance (i.e., where the current flows in the direction of the voltage drop) goes to zero. We assume, for the sake of simplicity, that only the states at the Fermi surface can contribute to this scattering, and we thus see peaks as the Landau levels move through the Fermi energy.

The Hall resistance for the current flowing in the perpendicular direction to the voltage, which is determined by the number of electronic states occupied, gets quantized since the number of states in a Landau level is quantized (and is proportional to the magnetic field). The approximation made in the calculations shown above is the lack of distinction between localized and extended states. For details see the references.

[1] D. Yoshioka, The Quantum Hall Effect, Berlin: Springer, 2002.

[2] T. Chakraborty and P. Pietiläinen, The Quantum Hall Effects: Integral and Fractional, Berlin: Springer, 1995.

[3] K. von Klitzing, Nobel lecture, 1985.

The analysis shown here is strongly influenced by a movie by Glenton Jelbert that can be found at Wikimedia. Like many discussions of the effect, this suffers from the standard deficiency that the magnitude of the density of states (DoS) is shown as fixed, rather than growing with , as it should.