Ab initiopseudopotential calculations of the transverse and longitudinal phonon spectra of Ge, α-Sn, and α-Sn/Ge superlattices

Abstract

A combination of a first-principles pseudopotential plane wave and the interplanar force constants (IPFC's) methods is used to calculate the transverse (T) and longitudinal (L) phonon spectra along [001] of Ge and biaxially strained \ensuremath{\alpha}-Sn, which correspond to a coherent growth on \ensuremath{\alpha}-Sn, $\ensuremath{\alpha}\ensuremath{-}{\mathrm{Sn}}_{0.5}{\mathrm{Ge}}_{0.5},$ and Ge(001) substrates. The calculated sets of IPFC's are also used to investigate the phonon spectra of \ensuremath{\alpha}-Sn/Ge superlattices (SL's) pseudomorphically grown on a Ge substrate. Our results for both the T and L phonon spectra of unstrained \ensuremath{\alpha}-Sn are in good agreement with the experimental data, and their variations with respect to the (001)-biaxial strain are predicted. Despite the large lattice mismatch between Ge and \ensuremath{\alpha}-Sn, the calculated phonon spectra of \ensuremath{\alpha}-Sn show a linear variation with respect to the strain state. The T and L confined and interface phonon modes in the \ensuremath{\alpha}-Sn/Ge SL's are discussed, in comparison with the corresponding phonon spectra of bulk Ge and properly strained \ensuremath{\alpha}-Sn. In particular, the Ge-like L- and T-optical confined modes in $\ensuremath{\alpha}\ensuremath{-}{\mathrm{Sn}}_{m}/{\mathrm{Ge}}_{n}$ SL's are well described relative to the bulk dispersions of Ge by effective confinement lengths equal to ${(n+1)a}_{0}(\mathrm{Ge})/4$ and ${(n+3)a}_{0}(\mathrm{Ge})/4,$ respectively.