Energy relaxation due to two-phonon scattering of electrons: Breakdown of the energy diffusion model

TL;DR

This paper investigates how two-phonon scattering dominates energy relaxation in certain electron systems at high temperatures, challenging the traditional diffusion model and revealing new temperature-dependent behaviors.

Contribution

It demonstrates that two-phonon scattering can surpass single-phonon processes in energy relaxation, invalidating the diffusion model in specific regimes.

Findings

Two-phonon scattering dominates above the Bloch-Grüneisen temperature, with a linear temperature dependence.

Energy relaxation rate scales as T^3 below the Bloch-Grüneisen temperature for soft phonons.

An intermediate T^2 regime appears for anisotropic bands between two Bloch-Grüneisen temperatures.

Abstract

Recent THz spectroscopy of the quantum paraelectric SrTiO$_3$ (arXiv:2501.15771) and a high-$T_c$ cuprate (arXiv:2503.15646) has renewed interest in energy relaxation in correlated electron systems. We consider a situation in which single-phonon scattering is forbidden by symmetry or momentum conservation, while two-phonon scattering is allowed. Solving the Boltzmann equation, we show that above the Bloch-Gr\"uneisen temperature the energy relaxation rate from two soft transverse optical phonons exceeds the single-phonon one: while the latter scales as $1/T$, the former is linear in $T$. This dominance of two-phonon scattering invalidates the usual picture of energy diffusion due to frequent scattering by subthermal phonons; instead, energy relaxes via rare scattering events involving thermal phonons. Below the Bloch-Gr\"uneisen temperature, the energy relaxation rate scales as the single-particle rate, namely as $T^3$ for soft phonons. For anisotropic electron bands, an intermediate regime appears between two Bloch-Gr\"uneisen temperatures, in which both allowed single-phonon and two-phonon processes scale as $T^2$.