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 in the limit of small mole fractions of all minority constituents of the alloy. Subject to certain approximations, the effects of all orders of perturbation theory upon the energy are determined. After introducing the pseudo quantum number k, it is shown that the expectation value of velocity and the effective mass are related to the energy in exactly the same fashion as is the case in the theory of energy bands of a perfect crystal. This pseudo quantum number k is in certain respects analogous to the ordinary k vector of band theory. It is shown that alloying has a tendency to reduce effective masses and to reduce the anisotropy of surfaces of constant energy in k space. As an example, a semiquantitative calculation is made of the dependence of the effective electron masses in germanium-silicon alloy upon alloy composition. In case there are localized bound impurity levels associated with any of the minority-constituent atoms in the host crystal, then the general analysis breaks down. It is shown that this results from the discontinuous nature of the energy (considered as a function of $f$, the mole fraction of the minority constituent) at $f=0\mathrm{}$, whenever a bound level is associated with this constituent. It is pointed out that the analysis of this paper corresponds to the determination of coherent scattering correctly to all orders of perturbation theory but does not include the effects of incoherent scattering.