Effect of local order on phonon density of states

Recent interest in the vibrational properties of amorphous solids has raised questions as to the importance of local structure in producing peaks in the density of states. In order to investigate this, we have studied small clusters of atoms in Si and Ge. We use the Born model with nearest-neighbor central and noncentral forces and impose an external potential upon the surface atoms. The coefficients appearing in this potenal are determined by demanding that the mean-square amplitude of vibration of the surface and interior atoms is the same at each frequency. As an example of this technique, we take five atoms in a tetrahedral arrangement, i.e., one interior and four surface atoms, and obtain a continuous density of states with a sharply defined "optic-mode" peak and a broad "acoustic-mode" peak. The crystalline density of states has similar peaks, which therefore do not result from long-range order and so are retained in the amorphous state.