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 a long mean free path is diminished in a specific way. Heat current $J(x)$ and temperature gradient $\ensuremath{\nabla}T({x}^{\ensuremath{'}})$ have a nonlocal relationship, via $\ensuremath{\kappa}(x\ensuremath{-}{x}^{\ensuremath{'}})$, over a distance $|x\ensuremath{-}{x}^{\ensuremath{'}}|$ 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 $[\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{e}\ensuremath{\propto}cos(2\ensuremath{\pi}x/L)]$ to improve convergence of numerics.