LOCALIZATION OF ELECTRONS IN ORDERED AND DISORDERED SYSTEMS .2. BOUND BANDS

The question is: how should we describe the electron states in an assembly of 'atoms' with overlapping bound-state wave functions? In a regular lattice, these states are non-localized Bloch functions, forming a band of width B, say. Attention is focused on the effects of 'cellular disorder', where a statistical variation wl is imposed on the energy of the bound state of the lth atom. A very simple version of the argument of Anderson (1958) demonstrates his conclusion that the states should all become localized if wl is distributed uniformly over a range of width somewhat greater than B. The same argument applied to the 'equiconcentration binary alloy', where wl takes the values ±1/2W at random, shows that the band of propagating states is not destroyed, but splits into two narrower bands when W>>B. In an attempt to confirm these results, successive approximations for the average Green function in such a system are formulated, starting from the propagator of the Bloch states of the 'ordered' system and treating wl as a perturbation. The problem of allowing for the statistical correlations induced by repeated scatterings from the same centre is demonstrated and a simple type of self-consistent t-matrix formula - nearly the same as various other published formulae based upon the same principles - is developed. It turns out, however, that none of these 'medium propagator' methods leads to splitting of the band in the equiconcentration alloy. The technique of Matsubara et al. is then sketched out, and shown to be entirely different in principle. As in the Anderson theory, the Green function is expanded in powers of the overlap integral, which 'perturbs' the bound states of the isolated atoms. By Fourier transformation, cumulant averaging, and self-consistent approximation, Matsubara et al. obtained formulae which are rather complicated but which appear to be consistent with the Anderson criteria for localization. This is, therefore, much the best approach to the problem. The consequences of 'structural' disorder are also discussed, somewhat inconclusively.