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Often this transverse voltage is measured at fixed current and the Hall resistance recorded. It can easily be seen that this Hall resistance increases linearly with magnetic field. 

In a two-dimensional metal or semiconductor the Hall effect is also observed, but at low temperatures a series of steps appear in the Hall resistance as a function of magnetic field instead of the monotonic increase. What is more, these steps occur at incredibly precise values of resistance which are the same no matter what sample is investigated. The resistance is quantised in units of h/e2 divided by an integer. This is the QUANTUM HALL EFFECT.

The figure shows the integer quantum Hall effect in a GaAs-GaAlAs heterojunction, recorded at 30mK. The QHE can be seen at liquid helium temperatures, but in the millikelvin regime the plateaux are much wider. Also included is the diagonal component of resistivity, which shows regions of zero resistance corresponding to each QHE plateau. In this figure the plateau index is, from top right, 1, 2, 3, 4, 6, 8 . Odd integers correspond to the Fermi energy being in a spin gap and even integers to an orbital LL gap. As the spin splitting is small compared to LL gaps, the odd integer plateaux are only seen at the highest magnetic fields. Important points to note are:
The value of resistance only depends on the fundamental constants of physics:
e the electric charge and h Plank's constant.
It is accurate to 1 part in 100,000,000.
The QHE can be used as primary a resistance standard, although 1 klitzing is a little large at 25,813 ohm!

Explanation of the Quantum Hall Effect

The zeros and plateaux in the two components of the resistivity tensor are intimately connected and both can be understood in terms of the Landau levels (LLs) formed in a magnetic field.

In the absence of magnetic field the density of states in 2D is constant as a function of energy, but in field the available states clump into Landau levels separated by the cyclotron energy, with regions of energy between the LLs where there are no allowed states. As the magnetic field is swept the LLs move relative to the Fermi energy.

When the Fermi energy lies in a gap between LLs electrons can not move to new states and so there is no scattering. Thus the transport is dissipationless and the resistance falls to zero.

The classical Hall resistance was just given by B/Ne. However, the number of current carrying states in each LL is eB/h, so when there are i LLs at energies below the Fermi energy completely filled with ieB/h electrons, the Hall resistance is h/ie2. At integer filling factor this is exactly the same as the classical case.

The difference in the QHE is that the Hall resistance can not change from the quantised value for the whole time the Fermi energy is in a gap, i.e between the fields (a) and (b) in the diagram, and so a plateau results. Only when case (c) is reached, with the Fermi energy in the Landau level, can the Hall voltage change and a finite value of resistance appear.

This picture has assumed a fixed Fermi energy, i.e fixed carrier density, and a changing magnetic field. The QHE can also be observed by fixing the magnetic field and varying the carrier density, for instance by sweeping a surface gate.

Dirt and disorder

Although it might be thought that a perfect crystal would give the strongest effect, the QHE actually relies on the presence of dirt in the samples. The effect of dirt and disorder can best be though of as creating a background potential landscape, with hills and valleys, in which the electrons move. At low temperature each electron trajectory can be drawn as a contour in the landscape. Most of these contours encircle hills or valleys so do not transfer an electron from one side of the sample to another, they are localised states. A few states (just one at T=0) in the middle of each LL will be extented across the sample and carry the current. At higher temperatures the electrons have more energy so more states become delocalised and the width of extended states increases.

The gap in the density of states that gives rise to QHE plateaux is the gap between extended states.

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