Analytical Green's Function of Multidimensional Boltzmann Transport Equation for Modeling Hydrodynamic Second Sound

TL;DR

This paper develops an analytical Green's function approach to solve the multidimensional Boltzmann transport equation, enabling efficient modeling of phonon hydrodynamics and second sound phenomena across various regimes.

Contribution

It introduces a new analytical formalism using Callaway's approximation for multidimensional BTE, capturing hydrodynamic phonon transport without heavy computational costs.

Findings

Successfully models transition from ballistic to diffusive regimes.

Accurately predicts temperature oscillations in ultrafast experiments.

Avoids large matrix inversions typical of full BTE solutions.

Abstract

Hydrodynamic second sound can be generated by heat pulses when the phonon-phonon interaction is dominantly momentum conserving, and the propagation of the temperature field becomes wavelike rather than diffusive. While the Boltzmann transport equation (BTE) has been widely applied to study phonon dynamics and thermal transport at the nanoscale, modeling the hydrodynamic transport regime remains challenging. The widely used relaxation time approximation (RTA) treats all phonon interactions as resistive without considering momentum conservation, resulting in the absence of phonon hydrodynamics. Rigorously solving BTE by inverting the full scattering matrix, however, is extremely computationally demanding and has been only applied to model one-dimensional temperature variations. Here, we present an analytical Green's function formalism for solving multidimensional Boltzmann transport equation (BTE) using phonon properties from first-principles calculations. This formalism involves Callaway's scattering approximation with separate relaxation times for momentum-conserving and momentum-destroying scattering events. The Green's function captures phonon dynamics in a wide range of temperature, spatial, and temporal scales, and successfully reproduces the transition from ballistic, hydrodynamic, to diffusive transport regimes. Our method avoids the computationally demanding inversion of large scattering matrices and shows good accuracies in predicting the temperature oscillation in ultrafast pump-probe characterizations with different geometries of thermal excitation.