Heat conduction and phonon localization in disordered harmonic crystals

TL;DR

This paper studies heat conduction in disordered harmonic crystals, showing how phonon localization affects thermal transport and verifying Fourier's law in three dimensions through numerical simulations.

Contribution

It provides the first numerical verification of Fourier's law in three-dimensional disordered harmonic solids and analyzes phonon localization effects on heat conduction.

Findings

High-frequency phonon modes are localized, reducing heat conduction.

Fourier's law holds in 3D disordered solids under typical boundary conditions.

Localization in 2D pinned systems leads to insulating behavior.

Abstract

We investigate the steady state heat current in two and three dimensional isotopically disordered harmonic lattices. Using localization theory as well as kinetic theory we estimate the system size dependence of the current. These estimates are compared with numerical results obtained using an exact formula for the current given in terms of a phonon transmission function, as well as by direct nonequilibrium simulations. We find that heat conduction by high-frequency modes is suppressed by localization while low-frequency modes are strongly affected by boundary conditions. Our {\color{black}heuristic} arguments show that Fourier's law is valid in a three dimensional disordered solid except for special boundary conditions. We also study the pinned case relevant to localization in quantum systems and often used as a model system to study the validity of Fourier's law. Here we provide the first numerical verification of Fourier's law in three dimensions. In the two dimensional pinned case we find that localization of phonon modes leads to a heat insulator.