Landauer formula for phonon heat conduction: relation between energy transmittance and transmission coefficient

TL;DR

This paper establishes a straightforward relation between energy transmittance and transmission coefficient for phonons in a one-dimensional harmonic chain, with implications for higher-dimensional systems.

Contribution

It provides a simple, direct demonstration of the relation between energy transmittance and transmission coefficient in phonon heat conduction, extendable to complex geometries.

Findings

Derived the relation between T(ω) and τ(ω) for 1D systems

Extended the approach to higher-dimensional geometries

Clarified the physical meaning of transmittance and transmission coefficient

Abstract

The heat current across a quantum harmonic system connected to reservoirs at different temperatures is given by the Landauer formula, in terms of an integral over phonon frequencies \omega, of the energy transmittance T(\omega). There are several different ways to derive this formula, for example using the Keldysh approach or the Langevin equation approach. The energy transmittance T({\omega}) is usually expressed in terms of nonequilibrium phonon Green's function and it is expected that it is related to the transmission coefficient {\tau}({\omega}) of plane waves across the system. In this paper, for a one-dimensional set-up of a finite harmonic chain connected to reservoirs which are also semi-infinite harmonic chains, we present a simple and direct demonstration of the relation between T({\omega}) and {\tau}({\omega}). Our approach is easily extendable to the case where both system and reservoirs are in higher dimensions and have arbitrary geometries, in which case the meaning of {\tau} and its relation to T are more non-trivial.