Quantized Hall conductivity in two dimensions

Abstract

It is shown that the quantization of the Hall conductivity of two-dimensional metals which has been observed recently by Klitzing, Dorda, and Pepper and by Tsui and Gossard is a consequence of gauge invariance and the existence of a mobility gap. Edge effects are shown to have no influence on the accuracy of quantization. An estimate of the error based on thermal activation of carriers to the mobility edge is suggested. There has been considerable interest in the remarkable observation made recently by von Klitzing, Dorda, and Pepper' and by Tsui and Gossard that, under suitable conditions, the Hall conductivity of an inversion layer is quantized to better than one part in 105 to integral muitiples of e2/h. The singularity of the result lies in the apparent total absence of the usual dependence of this quantity on the density of mobile electrons, a sample-dependent parameter, As it has been proposed' to use this effect to define a new resistance standard or to refine the known value of the fine-structure constant, an important issue at present is to what accuracy the quantization is exact, particularly in the regime of high impurity density, Some light has been shed on this question by the re-normalized weak-scattering calculations of Ando, ' who has sho~n that the presence of an isolated impurity does not affect the Hall current. A similar result has been obtained recently by Prange, who has shown that an isolated 5-function impurity does not affect the Hall conductivity to lowest order in the drift velocity u =cE/H, even though it binds a localized state, because the remaining delocalized states carry exactly enough extra current to compensate for its loss. The exactness of these results and their apparent insensitivity to the type or location of the impurity suggest that the effect is due, ultimately, to a fundamental principle. In this communication, we point out that it is, in fact, due to the long-range phase rigidity characteristic of a supercurrent, and that quantization can be derived from gauge invari-ance and the existence of a mobility gap. We consider the situation illustrated in Fig. 1, of a ribbon of two-dimensional metal bent into a loop of circumference L, and pierced every~here by a magnetic field Ho normal to its surface. The density of states of this system, also illustrated in Fig. 1, consists , in the absence of disorder, of a sequence of 8 functions, one for each Landau level. These broaden, in the presence of disorder, into bands of extended states separated by tails of localized ones. We consider the disordered case with the Fermi level in a mobility gap, as shown. We wish to relate the total current I carried around the loop to the potential drop V from one edge to another. This current is equal to the adiabatic derivative of the total electronic energy U of the system with respect to the magnetic flux $ through the loop. This may be obtained by differentiating with respect to a uniform vector potential A pointing around the loop, in the manner BU ~9U 04 LOA This derivative is nonzero only by virtue of the phase coherence of the wave functions around the loop. If, for example, all the states are localized then the only effect of A is to multiply each wave function by exp(ieAx /&c), where x is the coordinate around the loop, and the energy change and current are zero. If a state is extended, on the other hand, such a gauge transformation is illegal unless A=n-hc eL In the case on noninteracting electrons, phase coherence enables a vector potential increment to FIG. 1. Left: Diagram of metallic loop. Right: Density of states without (top) and with (bottom) disorder. Regions of delocalized states are shaded. The dashed line indicates the Fermi level,