Effect of strain on phonons in Si, Ge, and Si/Ge heterostructures

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 ${\gamma }_{\mathit{i}}$ throughout the Brillouin zone for bulk c-Si and c-Ge, and explicit analytic expressions for ${\gamma }_{\mathit{i}}$ 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 ${\mathrm{Si}}_{12}$ ${\mathrm{Ge}}_4$ 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.