Thermal conductivity and localization in glasses: Numerical study of a model of amorphous silicon

Numerical calculations of thermal conductivity κ(T) are reported for realistic atomic structure models of amorphous silicon with 1000 atoms and periodic boundary conditions. Using Stillinger-Weber forces, the vibrational eigenstates are computed by exact diagonalization in harmonic approximation. Only the uppermost 3% of the states are localized. The finite size of the system prevents accurate information about low-energy vibrations, but the 98% of the modes with energies above 10 meV are densely enough represented to permit a lot of information to be extracted. Each harmonic mode has an intrinsic (harmonic) diffusivity defined by the Kubo formula, which we can accurately calculate for ω>10 meV. If the mode could be assigned a wave vector k and a velocity v=∂ω/∂k, then Boltzmann theory assigns a diffusivity ${\mathit{D}}_{\mathbf{k}}$=1/3vl, where l is the mean free path. We find that we cannot define a wave vector for the majority of the states, but the intrinsic harmonic diffusivity is still well-defined and has a numerical value similar to what one gets by using the Boltzmann result, replacing v by a sound velocity and replacing l by an interatomic distance a. This appears to justify the notion of a minimum thermal conductivity as discussed by Kittel, Slack, and others. In order to fit the experimental κ(T) it is necessary to add a Debye-like continuation from 10 meV down to 0 meV.