Lattice dynamics and elastic phonon scattering in silicon

We have examined the propagation of acoustic phonons in silicon on the basis of two lattice-dynamics models: a bond-charge model and an effective-force-constant model consisting of only short-range forces. Phonon propagation is governed predominantly by phonon focusing due to elastic anisotropy and elastic scattering from isotopic atoms in the medium. Both of these effects depend on phonon dispersion. We find that the dispersive effects on the isotope scattering, i.e., the deviation of the scattering rate from a ${\nu }^4$ dependence, are discernible only at frequencies higher than 1.5 THz. However, the effects of dispersion on phonon focusing are observable at frequencies as low as ∼0.4 THz. In a slotted-crystal geometry, the calculated ratio of the scattered to the total phonon intensities yields very similar values for these two lattice-dynamics models and for elasticity theory, validating our previous results on the measurement of the elastic-scattering rate of phonons in silicon.