Adiabatic bond charge model for the phonons in diamond, Si, Ge, andα−Sn

An adiabatic bond charge model (BCM) for the lattice dynamics of diamond-type crystals is developed. Our BCM unites elements of earlier models by Phillips and Martin, Keating, and Cochran. Four types of interactions are used: (a) central ion-ion forces, (b) Coulomb interactions of the ions and bond charges (BC's), (c) central ion-BC forces, and (d) bond-bending forces. These interactions represent the metal-like (a) and covalent (b)-(d) part of the crystal bonding. The phonon dispersion curves for Si, Ge, and $\alpha -\mathrm{Sn}$ are calculated using only four disposable parameters; for diamond, five parameters are employed. For all crystals, very good agreement with experiment is obtained. In particular, the typical flattening of the transverse acoustic phonons in the semiconducting materials is understood as a consequence of the adiabatic motion of the BC's, when the effective ion-BC coupling (b)+(c) is weak compared to the bond-bending forces (d). In an alternative representation of the BCM, the interactions (b) and (c) are replaced by central and noncentral ion-BC-ion potentials along one bond. The remaining long-range part of the Coulomb forces is unimportant; therefore, all essential interactions of the BCM are of very short range. Furthermore, the interaction parameters follow clear trends from diamond to $\alpha -\mathrm{Sn}$: type (a) increases, whereas types (b)-(d) decrease, especially the ion-BC coupling tends to vanish toward $\alpha -\mathrm{Sn}$.