Phonon focusing in piezoelectric crystals: Quartz and lithium niobate

Abstract

The propagation of ballistic heat pulses at low temperature in $\ensuremath{\alpha}$-quartz (${\mathrm{SiO}}_{2}$) and lithium niobate (${\mathrm{LiNbO}}_{3}$) is characterized both experimentally and theoretically. The phonon-imaging method is used to determine the ballistic heat flux emanating from a point source. Both crystals exhibit large flux anisotropies due to phonon focusing. It is well known that such patterns arise from elastic anisotropy. The aim of this paper is to show how the heat flux is affected by the inherent piezoelectricity of these crystals. We have extended the usual calculation of phonon focusing to include the effect of piezoelectricity in the quasistatic limit by introducing "stiffened" elastic constants into the standard calculation of phonon flux. The resulting theoretical images agree quite well with experiment and show that while the effect of piezoelectricity on phonon focusing is small in $\ensuremath{\alpha}$-quartz, it is dramatic in lithium niobate. A theoretical study is included which shows the interesting topological changes in the slowness and wave surfaces in lithium niobate as piezoelectric stiffening is gradually (hypothetically) introduced.