Energy Levels of a Disordered Alloy

A study is made of the one-electron energy levels of a disordered alloy by means of perturbation theory, extending the results of Nordheim and Muto. To the accuracy of first-order perturbation theory, a disordered alloy is equivalent to a particular perfect crystal, the "virtual crystal," as was shown by Muto. For a certain rather general model, the effects of second-, third-, and fourth-order perturbation theory upon this "virtual-crystal" approximation are analyzed. The question of convergence of the perturbation approach is studied. A certain basic limitation of the perturbation approach is discussed, namely, the limitation to nonlocalized states. Accurate results obtained by Landauer and Helland for a hypothetical one-dimensional alloy are compared with results obtained by perturbation theory. It is pointed out that the same approach can be used with equal validity in discussing not only disordered alloys but also other types of imperfect crystals; e.g., imperfections resulting from dislocations. The most striking prediction of perturbation theory, i.e., the "tailing-off" of the density-of-states curve into a forbidden band, appears to have some experimental verification.