Numerical study of low-frequency vibrations in amorphous silicon

Exact numerical vibrational eigenvectors and eigenvalues are studied for atomistic models of amorphous silicon $(a$-Si) with 216, 1000, and 4096 atoms in the periodic repeat unit. At the lowest frequencies, eigenvalues are sparse and eigenvectors are fairly plane-wave-like. However, some eigenvectors are “quasilocalized” or “resonant.” They are temporarily trapped in local regions of undercoordination. The present paper finds the following. (1) The “quasilocalized” modes are to a large extent artifacts of the finite size of the model systems. (2) The lower energy modes of realistic models in the harmonic approximation are broadened versions of the corresponding crystalline acoustic vibrations, with fairly well-defined wave vectors Q. The intrinsic broadening due to glassy disorder increases rapidly with Q, until at intermediate frequencies a meaningful Q can no longer be assigned. (3) The intrinsic broadening due to disorder is strong enough to suppress thermal conductivity to the level seen experimentally, with no need for special anharmonic effects or localization, except for the influence of two-level systems on the modes at very low frequencies. (4) There is no inconsistency between the broadened propagating-wave description of low-energy modes and the occurrence of “excess modes” in specific heat or in spectra. However, amorphous silicon seems to have very few such excess modes. (5) “Excess modes” and the plateau in $\kappa \mathrm{}\left(T\right)\mathrm{}$ are not closely related, since the former is absent and the latter present in both experiment and in our calculations for a-Si. Our analysis agrees closely with the recent study of amorphous ${\mathrm{SiO}}_2$ by Dell’Anna et al.