Phonon Focusing and Phonon Conduction in Elastically Anisotropic Crystals

Abstract

Heat-pulse measurements have shown striking differences (up to factors of 100) in the intensity of phonons propagating ballistically in an elastically anisotropic crystal.1,2 These results were shown to arise from phonon focusing due to elastic anisotropy.3 Phonon focusing occurs when the direction of the group velocity varies more slowly with wave vector than in an elastically isotropic solid, as for example, along cuspidal edges in the group-velocity surface.4 Subsequent measurements of the thermal conductivity of silicon and calcium fluoride in the boundary-scattering regime demonstrated anisotropies of up to 50% for silicon and 40% for calcium fluoride. The predictions of Casimir’s theory,5 end-corrected for finite thermal length,6 and generalized to allow for phonon focusing gave quantitative agreement with experimental results.7,8 Similar anisotropies in the thermal conductivity have been predicted in sufficiently defect-free superconducting lead and niobium at T/Tc«1.9 Phonon-focusing effects have also been predicted in elastically anisotropic hexagonal,10 tetragonal and orthorhombic crystals.11