Thermal conductance of one-dimensional disordered harmonic chains

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, suprisingly, the thermal conductance of a system with strong disorder, characterized by a “heavy-tailed” probability distribution, and with large impedance mismatch between the bath and the system, scales normally with the system size, i.e., in a manner consistent with 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.

Figure 1

Schematic illustration of a one-dimensional disordered harmonic chain. All particles and springs in the heat baths are identical, with mass $m$ and spring constant $k$. Particles in the system all have mass $M$, and particles $i$ and $(i+1)$ are connected by a spring with stiffness $K_i$. The masses at the two ends of the disordered chain, $i=1$ and $i=N$, are connected to heat baths through a spring of strength $k$.

Figure 2

Density of states, $\rho \left(\omega \right)$, as a function of phonon frequency $\omega$ for different disorder strengths for a one-dimensional harmonic chain with (a) uniform disorder and (b) power-law disorder and $N=2000$. (a) For $W=0,\rho \left(\omega \right)$ diverges when $\omega \to 2$ and vanishes for $\omega >2$. The presence of disorder $(W>0)$ smears out the divergence at $\omega =2$ and gives rise to finite $\rho \left(\omega \right)$ for $\omega >2$. For $\omega \to 0,\rho \left(\omega \right)$ approaches a constant for all disorder strengths. (b) For power-law disorder, $\rho \left(\omega \right)$ diverges at $\omega \to 0$ in the strong disorder regime and approaches a constant in the intermediate and weak regimes (inset). (c) The transmission coefficient $\tau \left(\omega \right)$ for various coupling strengths $\left(k\right)$ and a given realization of disorder (uniform disorder with $W=0.5$ and $N=11)$. For $k=1.0$, transmission is roughly constant up to a cutoff frequency ${\omega }_L$. For $k\ll 1$ or $k\gg 1,\tau$ is sharply peaked around the eigenfrequencies of the disordered chain.

Figure 3

The dependence of localization length, $\xi \left(\omega \right)$, on the frequency $\omega$ of the phonons for different strengths of disorder $(y$ axis in logarithmic scale) for (a) uniform disorder and (b) power-law disorder. For small frequencies $(\omega \to 0),\xi \left(\omega \right)$ diverges as ${\omega }^{-\alpha }$ where $\alpha$ depends on disorder strength. (a) For uniform disorder, $\alpha =2$ in weak as well as strong disorder regimes. (b) For power-law disorder, $\alpha \approx 2$ for weak disorder $(\epsilon >2)$ while for intermediate disorder $(1<\epsilon \le 2),\alpha \approx \epsilon$ and in the strong disorder regime $(\epsilon \le 1),\alpha \approx \frac{2\epsilon }{1+\epsilon }$ (inset).

Figure 4

Dependence of the thermal conductance $G_{qm}\left(L\right)$ on the length $L$ of the disordered chain for uniform (disorder strength $W)$ and power-law (disorder strength $\epsilon )$ disorder in a log-log plot with (a) $k=1.0$ and (b) $k=0.01$. The mean temperature is fixed at $T=0.10$. For $k=1.0,G_{qm}\left(L\right)\approx 1$ for very small $L$, while $G_{qm}\left(L\right)\propto L^{-\beta }$ for large $L$. $\beta \ne 1$ for $k=1.0$ except for $\epsilon =1$. For $k=0.01,\beta =1.0$ in the strong disorder regime $(\epsilon \le 1.0)$ while in all other cases $G_{qm}\left(L\right)$ scales anomalously. Unified description of thermal conductance for one-dimensional disordered harmonic chains with different models of disorder in terms of the variable (c) ${\omega }_L/{\omega }_T$ (for $k=1.0)$ and (d) ${\omega }_L^{s+1}/{\omega }_T$ (for $k=0.01)$. Here, ${\tilde{G}}_{qm}=\frac{(s+1)G_{qm}}{D}$ (see text for details).