Thermal conduction in artificial sapphire crystals at low temperatures I. Nearly perfect crystals

Abstract

In order to obtain a detailed verification of the theory of thermal conduction in dielectric crystals, measurements have been made on a number of artificial sapphire crystals between 2° and 100° K. In the region of the maximum there are variations in conductivity between crystals from different sources. The highest conductivities measured are about 140 W/cm deg., which suggests that estimates of several hundred watts for the maxima of ideal sapphire crystals are not unreasonable. At sufficiently low temperatures the conductivity of a very perfect, long crystal with rough surfaces is observed, in agreement with Casimir’s theory of boundary scattering, to be proportional to T 3 and to the radius; the phonon mean free path is then nearly equal to the crystal diameter. Imperfect crystals show some anomalous effects. The extension of Casimir’s theory to apply to short specimens has been verified. Perfect crystals with smooth surfaces exhibit some specular reflexion of phonons; a statistical description of the surface is proposed which leads to the observed variation of this effect with temperature and is compatible with the results of interferometric examination of the surface.

Boundary scattering of phonons: Specularity of a randomly rough surface in the small-perturbation limit

“…The importance of surface specularity was realized early on in studies of thermal conductivity of single crystal rods at low temperatures [3,4]. Subsequently, the specularity parameter, i.e., the probability for a phonon to undergo a specular reflection rather than get diffusely scattered by the surface, became ubiquitous in the analysis of boundary-limited thermal transport [5][6][7].…”

Rayleigh waves, surface disorder, and phonon localization in nanostructures

“…Existing specularity parameter models are tied to the scalar wave model, which does not support mode conversion. Indeed, Ziman adapted the concept of a specularity parameter from electromagnetic waves, which do not undergo meaningful mode conversion [19]. Our results underscore the importance of the ongoing work to extend the specularity parameter model to account for mode conversion [112] and Rayleigh waves [11].…”

“…Wave scattering from rough, random surfaces has universal features such as randomizing the direction of the outgoing energy. For that reason, early models of phonon-surface scattering were adapted from electromagnetic wave scattering [19,37], and many newer phonon-surface scattering models are based on scalar and elastic waves [25, 27, 28, 41-46, 50, 52].…”

“…As the roughness is increased, energy is drawn from the center of the ribbon to the edges, which are the most disordered parts of the system and are where the energy is least likely to propagate. We also investigate the effects of mode conversion at surfaces, the limitations of Casimir's phonon-surface scattering model [19,37], and the limitations of models for phonon-surface scattering that rely on a single scalar wave and do not include mode conversion or Rayleigh waves.…”

“…So, early work on Rayleigh wave scattering focused on their decay into bulk waves in the presence of disorder [10,12,13,16,17], but did not address phonon transport. Nakayama [14] identified Rayleigh-to-bulk mode conversion as a cause of diffuse phonon-surface scattering, which is important to many phonon transport models [18][19][20]. Maznev [11] investigated elastic wave scattering from a nearly smooth surface using a Green's-function technique and found that most energy from a normally incident longitudinal wave can be converted into Rayleigh waves, but conversion at other angles was not addressed.…”