Attenuation of phase excitations in charge-density wave systems

The attenuation of collective excitations (phasons) corresponding to phase modulation of a charge-density wave (CDW) caused by electron-phason interaction is studied. Phason attenuation is a nonlocal effect and must be treated microscopically because the pertinent length scale is determined by the CDW wave vector $\overrightarrow{\mathrm{Q}}$ rather than the phason wave vector $\overrightarrow{\mathrm{q}}$. In three-dimensional jellium, phasons with $\overrightarrow{\mathrm{q}}$ parallel to $\overrightarrow{\mathrm{Q}}$ are predominantly attenuated by scattering electrons (in $\overrightarrow{\mathrm{k}}$ space) near the CDW energy gaps, and the attenuation rate is independent of temperature. The phason attenuation rate is $\gamma \equiv \left(\frac{1}{E}\right)\frac{\mathrm{dE}}{\mathrm{dt}}$, for a phason with energy $E$. We find $\gamma \sim q{\cos }^2\theta$, where $\theta$ is the angle between $\overrightarrow{\mathrm{q}}$ and $\overrightarrow{\mathrm{Q}}$. For $\overrightarrow{\mathrm{q}}$ parallel to $\overrightarrow{\mathrm{Q}}$, $\gamma$ is approximately 0.3 times the phason frequency; i.e., phasons are underdamped. Phasons with $\overrightarrow{\mathrm{q}}$ perpendicular to $\overrightarrow{\mathrm{Q}}$ are not attenuated. If heterodyne gaps (caused by potentials of periodicity $\overrightarrow{\mathrm{Q}}\pm 2\pi \overrightarrow{\mathrm{G}}$) cut the Fermi surface, the attenuation is increased by a factor of 3.