https://doi.org/10.1140/epjb/e2012-30640-x
Regular Article
Landauer formula for phonon heat conduction: relation between energy transmittance and transmission coefficient
Raman Research Institute,
560080
Bangalore,
India
a e-mail: meetsumand@gmail.com
Received:
18
July
2012
Published online:
19
November
2012
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 . There are several different ways to derive this
formula, for example using the Keldysh approach or the Langevin equation approach. The
energy transmittance
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
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
are more non-trivial.
Key words: Mesoscopic and Nanoscale Systems
© EDP Sciences, Società Italiana di Fisica and Springer-Verlag, 2012