Local Phonon Density of States in an Elastic Substrate

TL;DR

This paper calculates the local phonon density of states in an elastic substrate, revealing surface enhancements and anisotropy, with implications for understanding surface and bulk vibrational properties.

Contribution

It provides a detailed model of the local phonon DOS in a semi-infinite elastic medium, including surface effects and anisotropy, with specific application to silicon.

Findings

Surface local DOS proportional to frequency squared with enhanced constant

Surface local DOS is anisotropic and non-quadratic in frequency

Deep inside the substrate, local DOS approaches Debye behavior

Abstract

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, 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 non-quadratic in frequency. However, at large depths, the local DOS approaches the isotropic Debye value. The results are applied to a Si substrate.