Reconstruction of phonon relaxation times from systems featuring interfaces with unknown properties

Abstract

We present a method for reconstructing the phonon relaxation-time function τ ω = τ (ω) (including polarization) and associated phonon free-path distribution from thermal spectroscopy data for systems featuring interfaces with unknown properties. Our method does not rely on the effective thermal-conductivity approximation or a particular physical model of the interface behavior. The reconstruction is formulated as an optimization problem in which the relaxation times are determined as functions of frequency by minimizing the discrepancy between the experimentally measured temperature profiles and solutions of the Boltzmann transport equation for the same system. Interface properties such as transmissivities are included as unknowns in the optimization; however, because for the thermal spectroscopy problems considered here the reconstruction is not very sensitive to the interface properties, the transmissivities are only approximately reconstructed and can be considered as byproducts of the calculation whose primary objective is the accurate determination of the relaxation times. The proposed method is validated using synthetic experimental data obtained from Monte Carlo solutions of the Boltzmann transport equation. The method is shown to remain robust in the presence of uncertainty (noise) in the measurement.

Phonon relaxation time reconstruction from transient thermal grating experiments and comparison with density functional theory predictions

“…[1][2][3][4][5][6][7][8][9][10][11][12][13] One area of significant interest is the extraction of phonon relaxation times and free path distribution from thermal spectroscopy experiments. [1][2][3][4][5][6] This information is crucial in a variety of applications; examples include thermal management in nano-electronic circuits and devices, 7,8 microelectromechanical sensors, 9 and nano-structured materials for improved thermoelectric conversion efficiency. 6,10 Although density functional theory (DFT) has been widely used to predict phonon properties, 11,12 it has yet to rise to a stature level where it can replace experimental results, in part due to its failure to consistently reproduce experimental observations.…”

“…4 The reconstruction is performed using the recently proposed inverse problem framework. 2,3 Boltzmann transport equation (BTE) solutions are provided by the adjoint deviational Monte Carlo (MC) methodology; [14][15][16][17] in other words, validity of Fourier's law is never assumed. In addition to the above comparison, we evaluate the ability of the reconstruction methodology to perform on real experimental data and under conditions where measurements are available for only a portion of the length scales spanned by material free paths; the latter is typically severely limiting for effective thermal conductivity-based methodologies.…”

“…In addition to the above comparison, we evaluate the ability of the reconstruction methodology to perform on real experimental data and under conditions where measurements are available for only a portion of the length scales spanned by material free paths; the latter is typically severely limiting for effective thermal conductivity-based methodologies. 3 Performing on real experimental data can be challenging not only due to the presence of noise but also due to other spurious signals [see Fig. 1(b)]; in addition, it tests the hypothesis, implicit in this methodology, that the experimental setup can be adequately modeled using the BTE (subject to appropriate boundary conditions).…”

“…The parameter c L that provides the optimal scaling between S m and T BTE can be calculated using a convex optimization formulation that has an analytical solution (see supplementary material). As in our previous work, 2,3 the reconstruction is based on an optimization process utilizing the Nelder-Mead algorithm. 18 The description of the algorithm and typical parameter values can be found in Ref.…”

Second-order diffusion limit for the phonon transport equation-asymptotics and numerics

“…Taking D ω = 1, we have, in this case, C ω , as defined in (5) taking the form of C ω = (10ω) 2 e 10ω e 10ω −1 2 . We also choose τ (ω) = 1 10ω and v g = 10ω as suggested in [20]. The reference solution is once again computed using ∆x = ∆v = 2 −9 and α = 0.01, using the standard finite volume with upwinding fluxes.…”