Compact stochastic models for multidimensional quasiballistic thermal transport

Abstract

The Boltzmann transport equation (BTE) has proven indispensable in elucidating quasiballistic heat dynamics. Experimental observations of nondiffusive thermal transients, however, are interpreted almost exclusively through purely diffusive formalisms that merely extract 'effective' Fourier conductivities. Here, we build upon stochastic transport theory to provide a characterisation framework that blends the rich physics contained within BTE solutions with the convenience of conventional analyses. The multidimensional phonon dynamics are described in terms of an isotropic Poissonian flight process with rigorous Fourier-Laplace single pulse response P ( ξ, s) = 1/[s+ψ( ξ )]. The spatial propagator ψ( ξ ), unlike commonly reconstructed mean free path spectra κΣ(Λ), serves as a genuine thermal blueprint of the medium that can be identified in compact form directly from raw measurement signals. Practical illustrations for transient thermal grating (TTG) and time domain thermoreflectance (TDTR) experiments on respectively GaAs and InGaAs are provided.

Quasiballistic heat removal from small sources studied from first principles

“…This form is suggested by generic tendencies observed in first-principles computations and has been verified to yield accurate fits (with b = 1.06) to transient thermal grating measurements on bulk GaAs [17]. The transition length scale r CF is on the order of dominant phonon mean free paths (typically about 1 micron).…”

“…Semiconductor thermal transport at short length scales has attracted growing interest over the past decade [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. In addition to its immediate relevance to thermal management of (sub)microscale electronics [1], this field of study has opened up pathways to phonon spectroscopy techniques [2][3][4] which help illuminate the fundamentals of heat conduction in nonmetallic materials.…”

“…We will analyse thermal conduction in and around small heat sources through the RTA-BTE under the additional assumption that the thermal dynamics can be considered as isotropic. This simplifies the mathematics considerably while still maintaining a high level of physical accuracy [14,17,18], especially for cubic crystals. Under the stated conditions and at time scales exceeding phonon relaxation times, the single pulse response for the deviational thermal energy density P ≡ C v ∆T (with C v the volumetric heat capacity) is known [14,17] to take the following form in Fourier-Laplace domain ( r ↔ ξ, t ↔ s)…”

“…The propagator function ψ( ξ ) can be derived directly from wavevector-resolved phonon heat capacities C, relaxation times τ and mean free paths Λ = v τ as [17] …”

Analysis of nonlocal phonon thermal conductivity simulations showing the ballistic to diffusive crossover

“…Advances in measurement 8 include coherent xray thermal probing of strip-line arrays [9][10][11] , transient thermal gratings [12][13][14][15] , and time-domain thermal reflectance [16][17][18] . Theory has become increasingly powerful [19][20][21] and evolves togther with experiment [22][23][24] .…”