Enhancement of the phonon heat capacity in metals at low temperature

Abstract

Landau damping of each phonon mode causes its spectral density to acquire a low-frequency tail proportional to ${\ensuremath{\omega}}^{2}$. Consequently the ${T}^{3}$ low-temperature heat capacity arising from acoustic modes, with a frequency spectrum proportional to ${\ensuremath{\nu}}^{2}$, is supplemented by the spectral-density tails from all modes in the Brillouin zone. This extra term is proportional to ${T}^{3}\mathrm{ln}(\frac{\ensuremath{\Theta}}{T})$ and is a few percent of the Debye term at 1 K. In metals with large electron-phonon interaction---e.g., superconductors---the extra term will be larger, and will be present only in the normal state. A lower ${T}^{3}$ coefficient should occur in the superconducting state, a phenomenon reported experimentally for indium 20 years ago, since the spectral-density tail falls to zero for frequencies below the energy gap.