Anderson transition

For disordered systems, then, a quite different form of metal-insulator transition occurs—the Anderson transition. In these systems a range of energies exists in which the electron states are localized, and if at zero temperature the Fermi energy lies in this range then the material will not conduct, even though the density of states is not zero. The Anderson transition can be discussed in terms of non-interacting electrons, though in real systems electron-electron interaction plays an important part. 

In Chapter 5 we return to the Anderson transition resulting from disorder, and describe the work done in the last decade on the major effects caused by... 

Disordered systems localization, the Anderson transition and the mobility edge... 

Perhaps the earliest system to be described as an Anderson transition was cerium sulphide (Cutler and Mott (1969), on the basis of observations by Cutler and Leavy (1964)). The material in question can be written Ce3 xvxS4, where v is a cerium vacancy, the vacancies being distributed at random. The field near a cerium vacancy repels electrons, because they are negatively charged. Variation of x, then, changes the number of electrons and the number of scatterers. Figure 1.25 shows some results on the conductivity. At that time the present author believed... 

Since oc in (81) is the reciprocal of the localization length , measurements of T0 near to the (Anderson) metal-insulator transition can determine how varies with nc—n. Thus Castner and co-workers (Shafarman and Castner 1986, Shafarman et al 1986) have made measurements in Si P and found, as expected, that for systems just below the Anderson transition varies as (nc— n)-v, with v l. 

Anderson type (though affected of course by long-range interaction). Until recently it was supposed by the present author that the former is the case. We must now favour, however, the latter assumption for many-valley materials (e.g. Si and Ge), the Hubbard gap opening up only for a value of the concentration n below nc. The first piece of evidence comes from a calculation of Bhatt and Rice (1981), who found that for many-valley materials this must be so. The second comes from the observations of Hirsch and Holcomb (1987) that compensation in Si P leads to localization for a smaller value of nc than in its absence. As pointed out by Mott (1988), a Mott transition occurs when B = U (B is the bandwidth, U the Hubbard intra-atomic interaction), while an Anderson transition should be found when B 2 V, where V is some disorder parameter. Since U e2/jcuH, where aH is the hydrogen radius, and K e2/jca, and since at the transition a 4aH, if the transition were of Mott type then it should be the other way round. 

In NaxW03-yFy Doumerc (1978) observed a transition that has all the characteristics of an Anderson transition similar phenomena are observed in NaxTayW3 y03. The results are shown in Fig. 7.14. It is unlikely that this transition is generated by the overlap of two Hubbard bands with tails (Chapter 1, Section 4) this could only occur if it took place in an uncompensated alkali-metal impurity band, which seems inconsistent with the comparatively small electron mass. We think rather that in the tungsten (or tungsten-tantalum) 5d-band an Anderson transition caused by the random positions of Na (and F or Ta) atoms occurs. The apparent occurrence of amiD must, as explained elsewhere, indicate that a at the temperature of the experiments. Work below 100 K, to look for quantum interference effects, does not seem to have been carried out. 

Fig. 10.8 Expected behaviour of Knight shift K in (a) a non-crystalline solid and (b) a liquid when EF crosses Ec, giving an Anderson transition.

The main effect of both types of electron localization, of course, is a crossover from metallic to nonmetalhc behavior (a M-NM transition). Nevertheless, it would be very beneficial to have a method of experimentally distinguishing between the effects of electron-electron Coulomb repulsion and disorder. In cases where only one or the other type of localization is present this task is relatively simpler. The Anderson transition, for example, is predicted to be continuous. That is, the zero-temperature electrical conductivity should drop to zero continuously as the impurity concentration is increased. By contrast, Mott predicted that electron-correlation effects lead to a first order, or discontinuous transition. The conductivity should show a discontinuous drop to zero with increasing impurity concentration. Unfortunately, experimental verification of a true first order Mott transition remains elusive. 

In this chapter, the focus has been on the Mott (Mott-Hubbard) and Anderson transitions. When charge ordering is present, other types of transitions are also possible. A classic example is the mixed-valence spinel Fe203. There are two types of cation sites in Fc203, denoted as A and B. The A sites are tetrahedrally coordinated Fe... 

Figure 8.07 (top) One-electron tight-binding picture for Anderson transition, (a) Band width (left) and potential well structure in the absenee of disorder, (b) Variation of band disorder in site energies. The horizontal marks are the energies EjS (schematic). When the width W of the disorder exceeds the overlap bandwidth B, disorder-induced localization takes place, (bottom) Schematic density-of-states diagram for a crystalline and an amorphous semiconductor, in the vicinity of the highest occupied and lowest empty states. 

This implies for example that when a(t) has the Mott minimum value (e ifafit), the dc or large length scale (L — ) conductivity is zero. Thus, the conductivity goes continuously to zero at the Anderson transition, due to interference effects not envisaged in the Mott approximation. The exponent with which a vanishes is the same as that with which the localization length diverges. 

There is another type of transition between a localized and a delocalized, band-type wave function. We like to think of it as an Anderson transition. 

One can elaborate on the simple ideas we used to describe the Mott and Anderson transitions. For example, we can extend the models to a lattice of identical atoms, each with two valence electrons, and so forth. The general... 

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