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.

The harmonic diffusivity becomes a Rayleigh ${\mathrm{\omega }}^{\mathrm{-}4}$ law and gives a divergent κ(T) as T→0. To eliminate this we make the standard assumption of resonant-plus-relaxational absorption from two-level systems (this is an anharmonic effect which would lie outside our model even if it did contain two-level systems implicitly). A reasonable fit and explanation then results for the behavior of κ(T) in all temperature regimes. We also study the effect of increasing the harmonic disorder by substitutional mass defects (modeling amorphous Si/Ge alloys). The additional disorder increases the fraction of localized states, but delocalized states still dominate. However, the diffusivity of the delocalized states is diminished, weakening our faith in any literal interpretation of the minimum conductivity idea.

• Received 6 July 1993

DOI:https://doi.org/10.1103/PhysRevB.48.12589