QuantumPhysics.dvi

Figure 11: Schematic representation of the Hall and regular conductivity in the integer

quantum Hall effect. (from R.E. Prange inThe quantum Hall effect, Springer 1990).

A schematic rendition of the experimental observations characteristic of the (integer) Quantum
Hall Effect (or QHE) is given in Figure 11. As the magnetic fieldBis varied, it is observed that the
Hall resistivityRH=ρxydoes not follow a linear law, as would be expected from the discussion of
the classical Hall effect, but instead exhibitsplateauxon which the resistivity stays constant for a
range of variations ofB. Moreover, the values of the resistivity at the plateaux is observed to be

ρxy=

2 π ̄h

e^2

×integer (11.31)

How can this phenomenon be explained?

The key is to recall the density of carriers in a Landau level,given by

nB=

eB

2 π ̄h

(11.32)

The contribution to the conductivity of one Landau level is then obtained by including only the
density of states of one Landau leveln=nB, for which we get

(ρxy)−^1

∣∣

∣∣

1 Landau Level

=

nBe

B

=

e^2

2 π ̄h

(11.33)

Thus, the effect may be attributed to the contribution of filledLandau Levels, in which electrons
behave effectively independently. Exactly why an entire Landau Level contributes, and not say 1.5
Landau Levels would require a more detailed discussion of which states are localized and which
states are extended, and would require analysis of band structure and so on.