Abstract
The dispersion relations for optical and acoustic phonons are examined in bulk Si and Ge, Si and Ge strained layers grown on (001) and (111) surfaces, and ultrathin Si/Ge superlattices at ambient pressure and under hydrostatic pressure by using a modified Keating model. This model includes four interactions, which involve up to the fifth-nearest-neighbor atom, and force constants that depend on strain. These strain-modified force constants are related to specific cubic anharmonic terms in the potential energy and also to the empirical parameters p, q, and r that have been used to describe zone-center phonon shifts and splittings arising from strain. This model is used to obtain the mode Grüneisen parameters throughout the Brillouin zone for bulk c-Si and c-Ge, and explicit analytic expressions for at the zone center and boundaries. Biaxial strain in the (001) plane is shown to affect phonon dispersion very differently than in previous work that used a much simpler model. For Si and Ge grown on a (111) substrate, the frequency shift due to the biaxial strain for the TO-phonon mode is found to be almost independent of wave vector. The pressure-induced change in the frequency of confined LO phonons in a superlattice predicted by the model agrees with the change measured previously by Raman scattering. This modified Keating model is also used to obtain the second- and third-order elastic constants for Si and Ge.
- Received 27 May 1993
DOI:https://doi.org/10.1103/PhysRevB.48.17938
©1993 American Physical Society