Variational approach to extracting the phonon mean free path distribution from the spectral Boltzmann transport equation

Abstract

The phonon Boltzmann transport equation (BTE) is a powerful tool for studying nondiffusive thermal transport. Here, we develop a new universal variational approach to solving the BTE that enables extraction of phonon mean free path (MFP) distributions from experiments exploring nondiffusive transport. By utilizing the known Fourier heat conduction solution as a trial function, we present a direct approach to calculating the effective thermal conductivity from the BTE. We demonstrate this technique on the transient thermal grating experiment, which is a useful tool for studying nondiffusive thermal transport and probing the MFP distribution of materials. We obtain a closed form expression for a suppression function that is materials dependent, successfully addressing the nonuniversality of the suppression function used in the past, while providing a general approach to studying thermal properties in the nondiffusive regime. DOI: 10.1103/PhysRevB.93.155201 The Boltzmann transport equation (BTE) is widely used in analyzing heat transfer at length scales and time scales for which Fourier's law breaks down. In particular, there has been a growing interest recently in developing numerical and analytical solutions to the BTE to model thermal transport in phonon spectroscopy experiments [1-12] to extract phonon mean free path (MFP) distribution. The thermal conductivity accumulation function has been utilized as an elegant metric for understanding which MFP phonons contribute predominantly to thermal transport in a material [13][14][15]. Various experimental tools such as time-domain thermoreflectance (TDTR) [4,6,8,12,16,17], frequency-domain thermoreflectance (FDTR) [18,19], and transient thermal grating (TTG) [3,5,7,20] techniques have been extensively utilized recently in order to probe and observe nondiffusive transport by using ultrafast time scales or ultrashort length scales and gain key insight into the material's MFP spectrum.When the length scales in a system become comparable to the MFPs in a material, the effective thermal conductivity is reduced compared to its bulk, diffusive limit value [21,22]. A suppression function S ω is used to quantify this reduction or suppression of thermal conductivity, defined asThe variables C ω ,v ω , ω are the volumetric spectral heat capacity, the group velocity, and the MFP, respectively. The suppression function provides the ability to extend the notion of thermal conductivity beyond the diffusive regime in which it is defined from Fourier's law [21,23]. By utilizing the suppression function for a given experimental geometry, one can obtain the material's phonon MFP distribution from the experimentally measured thermal conductivity [5,6,12,23]. To obtain the effective thermal conductivity, the thermal signal from the experiment is fitted to the results of the Fourier law. The suppression function is calculated through modeling of the given experimental geometry with the BTE. However, one key assumption in this method is the universality of the suppression function, i.e....

Unifying first-principles theoretical predictions and experimental measurements of size effects in thermal transport in SiGe alloys

“…the case of very long penetration depth, the solid angle integrals vanish as ψ n (η ω , Kn ω → 0) ∝ Kn ω , and we recover the one-dimensional TTG limit as in this case the substrate essentially starts off at a uniform temperature, and we recover the previously derived effective thermal conductivity [28]. Note that information concerning the spectral contribution to heat capacity in needed, unlike prior work [30], in the equation for effective thermal conductivity.…”

“…This function is then used as a kernel in the effective thermal conductivity integral. This picture has also turned out to be an oversimplification [28,44]. Here we show that a fully spectral solution to the BTE is required to characterize the effective conductivity.…”

“…This progression from penetration depth to gray suppression to fully spectral interpretations is shown in Figure 6. from [28] to estimate the effective thermal conductivity at 1.00 um grating period.…”

“…Figure 3 demonstrates that the variational technique predicts that transport has a weak dependence on the optical penetration depth, a consequence of the kernels' weak dependence on optical penetration depth. As such, the one-dimensional limit of the TTG [28] geometry is sufficient to characterize the dependence of effective thermal conductivity on grating period.…”

“…To solve for the variational parameter, we can utilize mathematical optimization methods such as least squares on the error residual of the temperature equation [28], or impose a physical condition that we wish the trial function to satisfy. Here, we impose that the trial function must satisfy energy conservation taken over the control volume of the semiinfinite substrate over all time, analogous to the condition utilized for the thin film TTG geometry [29].…”