Heating-frequency-dependent thermal conductivity: An analytical solution from diffusive to ballistic regime and its relevance to phonon scattering measurements

Abstract

The heating frequency dependence of the apparent thermal conductivity in a semi-infinite body with periodic planar surface heating is explained by an analytical solution to the Boltzmann transport equation. This solution is obtained using a two-flux model and gray mean free time approximation, and verified numerically with a lattice Boltzmann method and numerical results from the literature. Extending the gray solution to the non-gray regime leads to an integral transform and accumulation-function representation of the phonon scattering spectrum, where the natural variable is mean free time, rather than mean free path as often used in previous work. The derivation leads to an approximate cutoff conduction similar in spirit to that of Koh and Cahill [Phys. Rev. B 76, 075207 (2007)] except that the most appropriate criterion involves the heater frequency rather than thermal diffusion length. The non-gray calculations are consistent with Koh and Cahill's experimental observation that the apparent thermal conductivity shows a stronger heater-frequency dependence in a SiGe alloy than in natural Si. Finally these results are demonstrated using a virtual experiment, which fits the phase lag between surface temperature and heat flux to obtain the apparent thermal conductivity and accumulation function.

Modeling ballistic effects in frequency-dependent transient thermal transport using diffusion equations

“…iωt , where ω is the angular frequency [15,20]. Case II: there is an oscillating heat current at the surface I Q (t) = I 0 Q e iωt [5].…”

“…These differences originate in how the angle-dependence of phonon transport is approximated; we replace the x-projected phonon velocity distribution with its angle-averaged value, v + x , while in Refs. [15,20] solutions come from taking the two lowest order moments of the BTE. Note that an isotropic medium assumption is not required, and that the input parameters v + x and λ can be extracted for any phonon dispersion and scattering time (see Eqns.…”

Heat dissipation in the quasiballistic regime studied using the Boltzmann equation in the spatial frequency domain

“…Vermeersch et al reported a truncated Lévy formalism to analyze the quasiballistic motion of thermal energy in semiconductor alloys [15,16]. Two-channel models, in which low-and highfrequency phonons are described by the Boltzmann transport equation (BTE) and heat equation, have been applied to analyze the quasiballistic thermal transport in transient grating [17] and thermoreflectance experiments [18,19]. A McKelveyShockley flux method was adapted to describe phonon transport [20].…”

Theory of the dynamical thermal conductivity of metals

“…Experimentally, it is also studied in the context of semiconductor alloys and it is found that the magnitude of the phononic thermal conductivity reduces as the frequency increases 11 . Theoretically, the electronic and the phononic dynamical thermal conductivity is discussed in the recent past by Shastry 12 and others [13][14][15][16] in different contexts such as in open systems, strongly correlated systems, semiconductor crystals, etc. In the present work, we explicitly derive the various expressions for the electronic thermal conductivity in case of metal with electron-impurity and electron-phonon interaction.…”