Phonon Scattering by Conduction Electrons and by Lattice Vacancies in Carbides of the Transition Metals

Abstract

The scattering of phonons by point defects in high concentrations and by electrons has been studied in the cubic transition metal carbides. The specimens were single crystals containing up to 24% carbon-atom vacancies: $\mathrm{Ti}{\mathrm{C}}_x$, $\mathrm{Zr}{\mathrm{C}}_x$, and $\mathrm{Nb}{\mathrm{C}}_x$, with $0.76\le x\le 0.96$. Although the lattice thermal conductivity $K_l$ is indeed low in these materials at low temperatures, the data and a Callaway analysis show that the observed suppression of $K_l$ cannot be explained solely by Rayleigh scattering from point defects. Furthermore, the dependence of $K_l$ on vacancy concentration is weak and of the wrong sense. A good fit to the data is obtained by introducing an additional term in the inverse relaxation time proportional to ${\omega }^2$. Pippard has shown that such a term represents phonon scattering by conduction electrons if the concentration of conduction electrons $n$ is sufficiently high, and the electron mean free path ${\lambda }_e$ is less than the dominant phonon wavelength. For the carbides, these conditions are met: $n\sim {10}^{21}/{\mathrm{cm}}^3$, and ${\lambda }_e$ is only a few lattice constants even at low temperatures, because of the scattering of electrons by lattice vacancies. We conclude that at low temperatures $K_l$ for the carbides is dominated by the scattering of phonons by conduction electrons. However, point defects influence the thermal conductivity of these solids both directly by scattering phonons and electrons and indirectly by altering the phonon-electron interaction.