Localization of electrons in condensed matter

Abstract

Recent developments in the electron theory of solids and fluids are leading us more and more to the picture of condensed matter as built up out of localized electron distributions, often only weakly interacting with their environment, and significantly different objects from isolated atoms. The merits of such an approach seem to be twofold: In theories where coulombic correlations are not dominant we might hope to obtain a unified description of one-electron states in crystals, amorphous and/or disordered solids, and liquids, in terms of such localized electron distributions. This description will, of course, have to be equivalent to the Bloch-Brillouin theory of energy bands in perfectly periodic structures. Correlation effects seem to be most usefully described in terms of electron localization, and therefore a description as in (i) promises to be particularly favourable for incorporating many-body effects at a second stage. In this article we give particular attention to the nature of electron states in the presence of a local potential built up as a superposition of screened one-centre potentials. If these screened potentials are centred on the sites of a regular lattice, it is shown that, once the problem of scattering of electron waves off the one-centre potential is solved, the exact construction of the energy bands is possible, at least in principle. In practice, only for muffin tins, for which potential the density of states can be derived from the phase shifts, has it proved possible to carry through this programme exactly. But we also describe a density matrix method based on the idea of the partition function of a crystal being built up from products of one-centre partition functions, by means of which we can solve approximately for the band structure generated by overlapping screened ion potentials, once the single-centre scattering has been calculated. In this method, the one-centre scattering is defined by an effective potential U which is asymptotically related to the one-centre input potential V L. The size of the overlaps of the effective potential U on neighbouring sites determines the accuracy of the approximate theory. The way in which electron correlations can be incorporated into such a model is then considered. In the Sommerfeld model, where the positive charge is uniformly distributed, a localized orbital method is described. Further, in a lattice, the one-centre method has its counterpart in the many-body problem, though no applications have so far proved possible. Then the relation of the local potential theory to the many-body problem is described. Also included in the discussion are Wigner and Mott transitions, excitonic phases, localized moments in metallic alloys and the recent progress in the theory of strong correlations in narrow energy bands. The role that x-ray scattering measurements can play in helping to define localized electron distributions is also discussed.