Densities of phonon states for (GaSb${)}_{1\mathrm{-}x}$(${\mathrm{Ge}}_2$${)}_{\mathrm{x}}$

A theory of the phonon spectral densities of states is reported for the long-lived, metastable, crystalline alloys (GaSb${)}_{1\mathrm{-}x}$(${\mathrm{Ge}}_2$ ${)}_{\mathrm{x}}$, assuming both a zinc-blende–diamond phase-transition model of the metastable structure and an on-site substitutional model without a phase transition. Comparison of both models with the data strongly favors the phase-transition model, but not so strongly as to be absolutely compelling evidence for the model. The theory is evaluated with use of the recursion method and a rigid-ion approximation with first- and second-nearest-neighbor force constants. The evolution of the spectra with increasing x is predicted, and is considerably more complicated than that given by either a virtual crystal or a persistence approximation. Principal spectral features are associated with vibrations of various bonds. All of the major anomalies in the Raman data are explained: (i) The alloy dependence of the Ge-like LO Raman linewidth is related to the entropy per site. (ii) The discontinuity as a function of x in the Raman peak position of the Ge-like LO mode is due to the changing importance of vibrations associated with Ge–Ge and Ge–Ga bonds. (iii) The anomalous asymmetries of the GaSb-like and Ge-like LO Raman peaks are attributed to spectral features associated with Sb–Sb bonds and Ga–Ge bonds, respectively. (iv) The LO-TO splitting at k=0 is proportional to the order parameter of the phase transition and decreases from the GaSb value to zero as x increases to $x_c$≃0.3. The model lends strong but indirect support to the phase-transition model over the on-site model, because, as a function of x, the maximum Raman linewidth coincides with the maximum entropy in the phase-transition model, but not in the on-site model.