The integer quantum Hall effect

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At high magnetic field B, the electronic density of states becomes a set of discrete Landau levels due to the confinement produced by the field. The following diagram shows the Fermi circle in two dimensional k-space, with a series of Landau levels inside it.

When these levels are well resolved, if a voltage is applied between the ends of a sample, the voltage drop between voltage probes along the edge of a sample can go to zero in particular ranges of B, and the Hall resistance becomes extremely accurately quantised (the Quantum Hall Effect, QHE). The special values of field are those at which the ratio of the number of electrons per unit area to the number of units of flux h/e is an integer. This ratio is called the filling factor, and is denoted by the Greek letter nu.

The QHE can be described in terms of a set of states, one per Landau level, travelling along the edges. The nature of these edge states is of great interest, and much work has been done to try to examine their structure and the way in which they modify the potential of the edge.

In real space, taking a cross-section across the sample, the Landau levels follow the potential, rising up at the edges of the sample, as shown above. A typical "Hall bar" sample is sketched below, showing, with crosses, the Ohmic contacts to source, drain and voltage probes. Edge states are indicated by lines with arrows.