Alloy Statistics and Phase Diagrams

Abstract

This chapter is devoted to the study of phase diagrams of semiconductors and their alloys, and the underlying statistics and free energies. It starts by treating the relationship between phase diagrams and free energies. The concepts of common tangent line and its relationship to quality of chemical potentials and activity coefficients are deduced. Then several approximate but analytical free-energy models frequently encountered in alloy phase diagram studies are introduced. These results provide the necessary background for a comprehensive review of statistical models and thermal data that have worked for semiconductors. The phase diagrams studied here include the liquidus curves of binary melts (such as Ga_{1– x} As _x ) in equilibrium with the stoichiometric semiconductor compounds (such as GaAs), the miscibility gap of solid solutions, and the liquidus and solidus curves of ternary alloys. To study the detailed statistical properties of pseudobinary alloys, the quasi-chemical approximation (QCA) is generalized from pairs to clusters of arbitrary sizes. A transformation is devised that partitions the alloys’ excess energies naturally into two parts: a strain-dominated part that depends only on the concentration, and a smaller “chemical” energy that controls the temperature dependence of the cluster populations. Although this generalized QCA (GQCA) calculation produces free energies and phase diagrams that are similar to those from previous empirical models, the physics is quite different. In previous models, a tendency for phase separation in alloys always implies there is a repulsive energy between the different alloying species. The present theory shows that, while the larger long-range strain energy drives phase separation, in most zinc blende alloys the chemical energies are attractive, so they favor local correlations resembling ordered compounds.