Thermal conductance of one-dimensional disordered harmonic chains

Abstract

We study heat conduction mediated by longitudinal phonons in one dimensional disordered harmonic chains. Using scaling properties of the phonon density of states and localization in disordered systems, we find nontrivial scaling of the thermal conductance with the system size. Our findings are corroborated by extensive numerical analysis. We show that a system with strong disorder, characterized by a 'heavy-tailed' probability distribution, and with large impedance mismatch between the bath and the system satisfies Fourier's law. We identify a dimensionless scaling parameter, related to the temperature scale and the localization length of the phonons, through which the thermal conductance for different models of disorder and different temperatures follows a universal behavior.Introduction-The study of heat transport via phonons in low dimensional (spatial dimension d < 3) classical and quantum mechanical systems has attracted considerable theoretical and experimental attention in recent years [1][2][3][4][5][6][7][8][9][10][11]. One of the main objectives of these studies is to understand the scaling of heat flux J which, according to Fourier's law [4], should scale with the system size L as J ∝ L −1 (L is measured along the direction of heat propagation). But extensive numerical and analytical studies in the past few decades have revealed the possible violation of Fourier's law for low dimensional systems [10,[12][13][14][15][16]. These studies show that J ∝ L (γ−1) with γ = 0 which in turn implies L-dependent thermal conductivity, κ = lim

Thermal transport in disordered harmonic chains revisited: Formulation of thermal conductivity and local temperatures

“…In numerical calculations, disorder averaging is performed over 10 4 random configurations unless stated otherwise. Note that the uniform distribution describing strong disorder is equivalent to the power-law distribution P (X) ∝ X ǫ−1 with ǫ = 1, which was employed in the previous study [17,18].…”

“…The previous studies, which generally assume spatially uncorrelated mass disorder, support anomalous transport in momentum-conserving systems. Nonetheless, κ N scaling normally with N has been found in recent years for particular classes of disorders such as correlated mass disorder [16] and uncorrelated bond disorder [17,18]. The recovery of normal conductivity despite total momentum conservation seems to disprove the prevailing conjecture, if the normality is also verified for local temperatures in the interior of the system.…”

On the definitions and simulations of vibrational heat transport in nanojunctions

“…These expressions can be readily generalized beyond onedimension to treat more complex structures. conduction in periodic and disordered one-dimensional chains [21][22][23][24][25][26][27] , (ii) recreate experimental setups to reveal transport mechanisms [28][29][30] , and (iii) provide guidelines for enhancing or suppressing the thermal conductance (oftentimes based on the harmonic force-field and using quantum scattering methods) [31][32][33][34][35] . Naturally, one wonders: What problems in this field remain unresolved?…”

Aspects of the disordered harmonic chain

“…Regarding ongoing studies of the dependence of J(N ) as function of boundary conditions and the spectral properties of the heat baths, see e.g. [32][33][34][35][36][37][38][39][40][41].…”