Size Effects in Thermal Conduction by Phonons

Abstract

Heat transport in nanoscale systems is both hard to measure microscopically, and hard to interpret. Ballistic and diffusive heat flow coexist, adding confusion. This paper looks at a very simple case: a nanoscale crystal repeated periodically. This is a popular model for simulation of bulk heat transport using classical molecular dynamics (MD), and is related to transient thermal grating experiments. Nanoscale effects are seen in perhaps their simplest form. The model is solved by an extension of standard quasiparticle gas theory of bulk solids. Both structure and heat flow are constrained by periodic boundary conditions. Diffusive transport is fully included, while ballistic transport by phonons of long mean free path is diminished in a specific way. Heat current J(x) and temperature gradient dT(x')/dx' have a non-local relationship, via k(x-x'), over a distance |x-x'| determined by phonon mean free paths. In MD modeling of bulk conductivity, finite computer resources limit system size. Long mean free paths, comparable to the scale of heating and cooling, cause undesired finite-size effects that have to be removed by extrapolation. The present model allows this extrapolation to be quantified. Calculations based on the Peierls-Boltzmann equation, using a generalized Debye model, show that extrapolation involves fractional powers of 1/L. It is also argued that heating and cooling should be distributed sinusoidally (de/dt~cos(2pi x/L)) to improve convergence of numerics.

9 pages (includes 5 figures). In June, 2015 a minor error in one equation was corrected by an erratum sent to Phys. Rev. B. The new arXiv version has the error corrected in the text

References in corpus (7)

• Calculation of nanowire thermal conductivity using complete phonon dispersion relations
• Direct Measurement of Room Temperature Non-diffusive Thermal Transport Over Micron Distances in a Silicon Membrane
• Heat Transport in Mesoscopic Systems
• Towards More Accurate Molecular Dynamics Calculation of Thermal Conductivity. Case Study: GaN Bulk Crystals
• Onset of non-diffusive phonon transport in transient thermal grating decay
• Thermal balance and quantum heat transport in nanostructures thermalized by local Langevin heat baths
• Landauer formula for phonon heat conduction: relation between energy transmittance and transmission coefficient

Cited by in corpus (4)

• Influence of Boundaries and Thermostatting on Nonequilibrium Molecular Dynamics Simulations of Heat Conduction in Solids
• Equivalence of the equilibrium and the nonequilibrium molecular dynamics methods for thermal conductivity calculations: From bulk to nanowire silicon
• Analysis of nonlocal phonon thermal conductivity simulations showing the ballistic to diffusive crossover
• Spatiotemporal flux memory in nondiffusive transport