Ballistic phonon imaging in germanium

The anisotropic propagation of acoustic phonons in a single crystal of Ge at low temperatures is examined both experimentally and theoretically. We have devised a general heat-pulse imaging method which reveals the angular distribution of energy flux emitted from an incoherent point source of ballistic phonons. The resulting images contain remarkably complex two-dimensional variations in the phonon flux. The larger features of the flux patterns agree with previous numerical calculations of phonon focusing. Using only continuum elasticity theory and the known elastic constants for Ge, we find that the sharp features in the ballistic phonon image are explained by mathematical infinites in the phonon flux. Useful physical insights into the origin of these flux patterns are gained by graphically plotting (a) the locus of singularities on the constant-frequency $\overrightarrow{\mathrm{k}}$-space surfaces and (b) the corresponding group-velocity surfaces. For a cubic crystal only two parameters (the ratio of elastic constants $C_{11}:C_{12}:C_{44}$) completely determine the flux pattern.