Ballistic Heat Conduction and Mass Disorder in One Dimension

TL;DR

This paper investigates the transition from ballistic to subballistic heat conduction in disordered one-dimensional chains, identifying a critical length scale and analyzing how mass composition influences this crossover using NEGF and analytical models.

Contribution

It introduces a critical crossover length scale for ballistic conduction in disordered chains and provides analytical expressions for transmittance and thermal conductance.

Findings

Existence of a critical length scale ($L_C$) for ballistic conduction.

Analytical expressions for phonon transmittance and thermal conductance.

Mass composition modifications can enhance ballistic heat conduction.

Abstract

It is well-known that in the disordered harmonic chain, heat conduction is subballistic and the thermal conductivity ($\kappa$) scales asymptotically as $\lim_{L\rightarrow\infty}\kappa\propto L^{0.5}$ where $L$ is the chain length. However, using the nonequilibrium Green's function (NEGF) method and analytical modeling, we show that there exists a critical crossover length scale ($L_{C}$) below which ballistic heat conduction ($\kappa\propto L$) can coexist with mass disorder. This ballistic-to-subballistic heat conduction crossover is connected to the exponential attenuation of the phonon transmittance function $\Xi$ i.e. $\Xi(\omega,L)=\exp[-L/\lambda(\omega)]$, where $\lambda$ is the frequency-dependent attenuation length. The crossover length can be determined from the minimum attenuation length which depends on the maximum transmitted frequency. We numerically determine the dependence of the transmittance on frequency and mass composition as well as derive a closed form estimate which agrees closely with the numerical results. For the length-dependent thermal conductance, we also derive a closed form expression which agrees closely with numerical results and reproduces the ballistic to subballistic thermal conduction crossover. This allows us to characterize the crossover in terms of changes in the length, mass composition and temperature dependence, and also to determine the conditions under which heat conduction enters the ballistic regime. We describe how the mass composition can be modified to increase ballistic heat conduction