Transverse interplanar forces and phonon spectra of strained Si, Ge, and Si/Ge superlattices.

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We present a first-principles pseudopotential study of the effects of the (001) biaxial strain on the transverse phonon spectra of Si and Ge along the $\ensuremath{\Delta}$ direction, and of the transverse phonon modes of strained Si/Ge (001) superlattices (SL's). The supercell method is used to calculate the [001] transverse interplanar force constants (IPFC's) of Si and Ge at three strain configurations corresponding to coherent growth on ${\mathrm{Si}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}$ (001) (with $x=0, 0.5, \mathrm{and} 1$) substrates. IPFC's up to the eighth-nearest-neighboring planes were included. It has been found that (i) the strain-induced frequency shift of the transverse optical (TO) branch is in excellent agreement with the available experimental data and has a weak wave vector dependence. (ii) Both the Si- and Ge-like TO confined modes in strained Si/Ge (001) SL's are well described relative to the bulk dispersions by an effective confinement length ${d}_{\mathrm{eff}}=(n+3)\frac{{a}_{\ensuremath{\perp}}}{4}$, where $n$ is the number of monolayers of Si or Ge, and $\frac{{a}_{\ensuremath{\perp}}}{4}$ is the corresponding interplanar distance. (iii) The frequencies of the interface modes are about 400 ${\mathrm{cm}}^{\ensuremath{-}1}$ and have a strong dependence on the interfaces coupling in short period SL's.