Landauer formula for phonon heat conduction: relation between energy 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 ω, of the energy transmittance T(ω). There are several different ways to derive this formula, for example using the Keldysh approach or the Langevin equation approach. The energy transmittance T(ω) is usually expressed in terms of nonequilibrium phonon Green's function and it is expected that it is related to the transmission coefficient τ(ω) 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(ω) and τ(ω). 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 τ and its relation to T are more non-trivial.

17 pages, 1 figure

References in corpus (7)

• Heat Transport in low-dimensional systems
• Quantum thermal transport in nanostructures
• Nonequilibrium Green's Function Approach to Phonon Transport in Defective Carbon Nanotubes
• Nonequilibrium Green's function approach to mesoscopic thermal transport
• Conductance calculations for quantum wires and interfaces: mode matching and Green functions
• Heat transport in harmonic lattices
• Interfacial thermal transport in atomic junctions

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