Local phonon density of states in an elastic substrate

The local, eigenfunction-weighted acoustic phonon density of states (DOS) tensor is calculated for a model substrate consisting of a semi-infinite isotropic elastic continuum with a stress-free surface. On the surface, the local DOS is proportional to the square of the frequency v, as for the three-dimensional Debye model, but with a constant of proportionality that is considerably enhanced compared to the Debye value, a consequence of the Rayleigh surface modes. The local DOS tensor at the surface is also anisotropic, as expected. Inside the substrate the local DOS is both spatially anisotropic and nonquadratic in frequency. However, at depths that are large compared with vl /v, where vl is the bulk longitudinal sound velocity, the local DOS approaches the isotropic Debye value. The results are applied to a Si substrate.