Transport in Disordered Systems

A survey is given of the main mechanisms of dc electrical conductivity in disordered systems and of how the conductivity depends on temperature. If the localization length is large, metallic behaviour will be observed at high temperatures, at lower temperature the resistance will increase with a power of T-', and at very low temperatures, when variable range hopping occurs, the resistance will increase exponentially. The conditions necessary to observe these effects in thin metallic wires are discussed. During the past ten or twelve years there has been a lot of discussion of the manner in which localized electron states produced by the strong disorder in an amorphous material would affect the transport properties. This discussion has been important for two reasons. It has shown that there is a sharp qualitative distinction between extended and localized states, and that they are not just two extremes of a continuous transition. It has also led to new and powerful ways of displaying and interpreting data on transport properties. The difference between extended and localized states is shown up most sharply by the temperature dependence of the dc electrical conductivity, particularly at low temperatures. Because of the wide variety of systems involved "low" temperatures may mean room temperature for one material and millidegrees for another. Essential information does however come from other transport measurements such as the frequency dependence of the ac conductivity, the thermopower, the Hall effect, the photoelectric effect, and so on. Very interesting experiments have been done on the behaviour of injected carriers, and these eive information which cannot readily be obtained by.working close to equilibrium conditions. In this paper particular attention is paid to the differences between localized and extended states, and to the nature of the transition between them. It is important to understand very low temperatures to be able to distinguish between extended states and large localized states. It has been suggested that localized states should exist in ordinary metallic wires if they are long enough and thin enough, and a number of experimental groups are looking for evidence of localization. If it is not found it will indicate that something important is probably missing from our analysis of disordered systems. Anderson argued that in a sufficiently disordered system electrons would be localized in a region of suitable potential energy, and that the wave function would fall off exponentially from the centre of localization at a rate which can be calculated by use of perturbation theory /I/. Even in a weakly disordered system states near the band edge should be localized, as the long wavelength states are sensitive to potential fluctuations. As the disorder is decreased or the energy is increased the localization length increases until eventually it becomes infinite and the states become extended. Mott has argued strongly that at this mobility edge the extended states have a minimum metallic conductivity /2 / . If it is accepted that the electron wavelength, the distance over which the phase of the wave function changes by 2rr, cannot be much greater than the mean free path A, the distance over which it loses phase coherence, and this condition Xk >1 is used in the stanF dard formulas of kinetic theory for the free electron model, the result u~~ = n e2'/rn = e2s2X/3n2~>e2$/3n2K ( 1 ) is obtained for the minimum metallic conductivity. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786596 c61536 JOURNAL DE PHYSIQUE This formula involves a wavelength which has to be estimated, but in two dimensions the same argument