Heat Transport in Harmonic Lattices

Abstract

We work out the non-equilibrium steady state properties of a harmonic lattice which is connected to heat reservoirs at different temperatures. The heat reservoirs are themselves modeled as harmonic systems. Our approach is to write quantum Langevin equations for the system and solve these to obtain steady state properties such as currents and other second moments involving the position and momentum operators. The resulting expressions will be seen to be similar in form to results obtained for electronic transport using the non-equilibrium Green's function formalism. As an application of the formalism we discuss heat conduction in a harmonic chain connected to self-consistent reservoirs. We obtain a temperature dependent thermal conductivity which, in the high-temperature classical limit, reproduces the exact result on this model obtained recently by Bonetto, Lebowitz and Lukkarinen.

Crossover from ballistic to diffusive thermal transport in quantum Langevin dynamics study of a harmonic chain connected to self-consistent reservoirs

“…It is also shown in [7] that in the limit N → ∞, the steady state is a local equilibrium state with a temperature profile satisfying Fourier's law with a temperature independent, finite thermal conductivity for the classical model. The quantum version of this model is studied in [8] where a finite, temperature dependent thermal conductivity is found in the quantum regime.The quantum Langevin equations of the chain sites are,where η l is the noise generating from the lth reservoir. The correlations of noises, are such that the distributions of normal modes in isolated reservoirs follow BoseEinstein statistics.…”

“…Also it is well established that heat transport in one-dimensional momentum conserving systems (absence of external potentials) does not follow Fourier's law [4,5]. Heat conduction in a long harmonic chain connected to self-consistent reservoirs at every site shows diffusive behaviour qualifying system size independent thermal conductivity [6,7,8]. The transition from ballistic to diffusive dynamics in thermal and electrical transport has recently received a lot of attention.…”

“…It is also shown in [7] that in the limit N → ∞, the steady state is a local equilibrium state with a temperature profile satisfying Fourier's law with a temperature independent, finite thermal conductivity for the classical model. The quantum version of this model is studied in [8] where a finite, temperature dependent thermal conductivity is found in the quantum regime.…”

“…So if we want to use the scaling form of the temperature which is not so good at the boundaries for smaller wire sizes, it is better to evalute current in the middle bond of the wire. Current through (l, l + 1) spring of the chain is given by [8] …”

Linear-response formula for finite-frequency thermal conductance of open systems

“…We will obtain the solution of the equations of motion in the time-dependent steady state by using Fourier transforms in the time domain. The approach is similar to that used in the derivation of the Landauer-type formula for steady state heat current in harmonic systems, where the current is expressed in terms of phonon Green's functions [17]. Let us introduce the transforms:…”