Lattice Vibrations of Zincblende Structure Crystals

In this article we present expressions, based on the shell model of lattice dynamics, for all important long-wavelength properties of the lattice vibration spectrum of a zincblende structure crystal, and for the vibration frequencies having wave vectors at the Brillouin zone boundary in the [100] direction. General first and second neighbor short-range force constants are used. The formulas are presented in the distortion dipole form of Mashkevich and Tolpygo, which uses one constant, $\alpha$, to describe the electronic polarizability of each atom, rather than the two redundant constants $Y$ and $k$, the shell charge and polarization spring, respectively, used in the shell model. Some of the force constants in our expression are evaluated for GaAs, InSb, AlSb, and ZnS with the aid of available experimental data. We were unable to fit all of the data for any substance with only first neighbor force constants. When small, realistic values of second neighbor constants were used, a multitude of fits resulted. The absence of a criterion determining the "best" fit results in a spread of possible ionic charge values of about one electronic charge for the 3-5 compounds, and an inability to decide whether ZnS is very ionic or rather covalent. Our results point up the necessity of calculations of atomic force constants from fundamental quantum mechanics.