Semi-implicit Lax-Wendroff kinetic scheme for electron-phonon coupling

TL;DR

This work addresses non-equilibrium electron–phonon heat transfer in metals at micro/nano scales where Fourier diffusion is inadequate. It introduces a semi-implicit Lax–Wendroff kinetic scheme to solve the coupled electron–phonon Boltzmann transport equations, incorporating relaxation times $\tau_e$, $\tau_p$ and a coupling constant $G$ within a unified framework. The key innovation is reconstructing interfacial distribution functions via finite differences to couple particle migration, scattering, and electron–phonon exchange within a single time step, removing restrictions imposed by $\tau$ and mean free paths $\lambda$. Numerical tests show accurate capture of ballistic to diffusive heat conduction and strong agreement with DUGKS, while exhibiting improved nanoscale performance over traditional TTM, highlighting its potential for microelectronic thermal management.

Abstract

A semi-implicit Lax-Wendroff scheme is developed for electron-phonon coupling process in metals based on the two-temperature kinetic equations. The core of this method is to integrate the evolution information of physical equations into the numerical modeling process, which leads to that the time step or cell size is not limited by the relaxation time and mean free path. Specifically, the finite difference method is used to solve the kinetic model again when reconstructing the interfacial distribution function, through which the particle migration, scattering and electron-phonon coupling processes are coupled together within a single time step. Numerical tests demonstrate that this method could efficiently capture electron-phonon coupling or heat conduction processes from the ballistic to diffusive regimes. It provides a new tool for describing electron-phonon coupling or thermal management in microelectronic devices.

Figures (3)

• Figure 1: Spatial distributions of the phonon temperatures with different thickness $L$, where $T^* = (T - T_R) / (T_L - T_R)$.
• Figure 2: Time-dependent electron and phonon temperatures at $x= L/4$ with different $L$, where $T_{ref}$= 300 K, $\Delta T$= 1 K, $T_{e}^*=\left ( T_e-{T}_{ref}\right )/\Delta T$, $T_{p}^*=\left (T_p-{T}_{ref}\right )/\Delta T$. The data predicted by the DUGKS and TTM is obtained from previous reference ZHANG2024123379.
• Figure 3: Time-dependent electron and phonon temperatures at $x$= L/4 with different $L$, where $T_{ref}$= 25 K, $\Delta T$= 1 K, $T_{e}^*=\left ( T_e-{T}_{ref}\right )/\Delta T$, $T_{p}^*=\left ( T_p-{T}_{ref}\right )/\Delta T$. The data predicted by the DUGKS and TTM is obtained from previous reference ZHANG2024123379.