Phonon Resonances Associated with Interstitial Atoms in Germanium and Silicon

The resonant scattering of acoustic phonons by interstitial atoms at the (½, ½, ½) position in the Si and Ge lattices has been investigated using the Green's-function matrix technique of Lifshitz. The lattice contribution to the Green's function is calculated using a spherically symmetric phonon spectrum with transverse and longitudinally polarized phonons. The observed dispersion relation along the (1,0,0) direction is included in the calculation. The interstitial contribution to the Green's function is calculated exactly and dominates at low frequencies. The interaction matrix is constructed from nearest- and next-nearest-neighbor interactions and limited in form by symmetry considerations. The interaction matrix and interstitial mass $M$ are the independent variables in the calculation. The resonance frequency $\omega$ and width $\Delta \omega$ are calculated as functions of the force constants of the interaction matrix and interstitial mass. Results show that a single effective force constant $K$ can be defined such that $\omega \propto {\left(\frac{K}{M}\right)}^{\frac{1}{2}}$ and $\frac{\Delta \omega }{\omega }\propto {\omega }^{3}M$ over most of the transverse acoustic band. Thus, a measurement of both quantities $\omega$ and $\Delta \omega$ should allow an unambiguous determination of $K$ and $M$. Several methods are proposed for observing interstitial resonances.