High Field Magnetotransport-Lecture III: The Fractional Quantum Hall Effect

Abstract

The existence of the fractional quantum Hall effect (FQHE) is taken to be evidence for the formation of a new highly correlated ground state of a two dimensional electron gas. This occurs at very low temperatures, in high magnetic fields, and in systems where there is only a very small amount of disorder present. The main experimental observations are that minima are observed in the electrical resistivity component ρ_xx, at fractional Landau level occupancies _ν=nh/eB=p/q, where p is an integer and q is an odd integer [1–9]; while corresponding Hall plateaus are seen at quantized Hall resistivity values of h/_νe². To date fractional states have been reported at _ν= 1/3, 1/5, 2/5, 2/7, 3/7 and 4/9, and the equivalent ‘hole’ analogous of these states have been observed at occupancies _ν= 1-(p/q). These states occur when all of the electrons lie in the lowest spin split Landau level, but it has recently been shown that they can exist in a similar manner in the upper spin state at occupancies of the form ν= 1+(p/q). Once ν>2 the electrons occupy the second Landau level. At this point the experimental position becomes less clear, with some reports of the observation of 7/3 and 8/3 states [3,4], and some suggestions that even denominator fractions may occur [8,9]. The significance of these results is that the existence of minima in the resistivity and quantized Hall plateaus may be shown, by using the gauge invariance arguements of Laughlin [10], to result from the formation of a mobility gap in the density of states. In other words the degeneracy of the individual Landau levels for isolated electrons has been lifted by the residual Coulomb interactions, leading to the formation of an energy gap between the ground and excited states of the system.