Heat Conduction by Phonons across a Film

TL;DR

This paper develops a non-local quasiparticle theory for phonon-mediated heat conduction across thin films, providing a formula that bridges ballistic and diffusive regimes based on mean free paths.

Contribution

It introduces a non-local relation for phonon heat transport, offers a quasiparticle-based local temperature definition, and derives an interpolating formula for film thickness dependence.

Findings

The derived formula interpolates between ballistic and diffusive heat conduction.

A variational principle bounds the heat current and determines the temperature gradient.

The approach generalizes heat conduction theory to regimes where mean free paths are comparable to sample size.

Abstract

Quasiparticle theory gives a local relation between heat current and temperature gradient, provided the quasiparticle mean free path is smaller than the scale of variation of temperature. When mean free paths are comparable to sample size, the relation becomes non-local. This non-local relation is formulated for phonon carriers; an explicit form is found in the approximation where current relaxation rates are replaced by quasiparticle relaxation rates. A quasiparticle definition of local temperature is offered. To extract the spatial variation of the temperature gradient requires inverting the non-local relation. A variational principle is constructed. The heat current is bounded above when evaluated for a trial temperature gradient. The true temperature gradient is the one which minimizes heat current. The simplest variational approximation (a constant temperature gradient) gives an approximate formula for the thickness dependence of the heat transport across a film whose thickness is comparable to mean free paths of phonons. The formula interpolates between ballistic and diffusive limits.