Theory of alloys. II. Embedded-cluster calculations of phonon spectra for a one-dimensional ternary alloy

The embedded-cluster method is generalized for application to ternary alloys and applied to the calculation of the frequency-distribution spectra of the random, mass-disordered, one-dimensional ternary alloy $A_x{B}_{1-x}C$. The spectra for one-dimensional models of some representative III-V and II-VI ternary alloys (diatomic mixed crystals) are calculated and compared with exact numerical spectra obtained for 50 000-atom random chains by the use of the negative eigenvalue theorem. For a cluster containing eight unit cells embedded in a coherent-potential-approximation effective medium, the embedded-cluster method reproduces all of the major features of the "exact" spectra for all alloy compositions and over a wide range of mass ratios. This reconfirms the accuracy of the method and strengthens its potential practicality for application to real semiconductor alloys.