Phonon frequencies for Si-Ge strained-layer superlattices calculated in a three-dimensional model.

Abstract

We present the results of full three-dimensional calculations of phonon frequencies in silicon-germanium strained-layer superlattices (SLS's) and compare with experimental data. We use a modified version of the six-parameter valence-force-potential model, incorporating the strained nature of the layers. We confirm that the dispersion relations for Si/Ge SLS's fall into bands closely associated with the bulk dispersion relations of silicon and germanium, as expected. Also, we find that our model precisely reproduces the predicted strain-induced frequency shifts to within one-tenth of 1% for epitaxial layers grown in the [001] direction, if longitudinal and transverse polarized phonons are considered separately. We also present analytical expressions for these frequency shifts. We find that the degree of penetration of confined LO phonons into adjacent layers is qualitatively different for growth in the [001] and [110] directions. Confined phonons in [110] superlattices have nearly zero penetration depth into adjacent layers, while the penetration depth for phonons in [001] superlattices is quite significant. This large penetration depth for [001] superlattices allows for coupling of transverse interface modes across thin layers, giving rise to a bifurcation of these otherwise degenerate modes. We also show that only transversely polarized interface modes are confined to the interface region, while longitudinally polarized modes with similar energies actually extend across the silicon layers and thus depend strongly on the width of the silicon layers. We compare our calculated results with measured Raman spectra of two typical samples and find that our model gives the correct position of the Raman peaks to within 1 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$, after correcting for temperature.