Generalized Debye-Peierls/Allen-Feldman model for the lattice thermal conductivity of low-dimensional and disordered materials

We present a generalized model to describe the lattice thermal conductivity of low-dimensional (low-D) and disordered systems. The model is a straightforward generalization of the Debye-Peierls and Allen-Feldman schemes to arbitrary dimensions, accounting for low-D effects such as differences in dispersion, density of states, and scattering. Similar in spirit to the Allen-Feldman approach, heat carriers are categorized according to their transporting capacity as propagons, diffusons, and locons. The results of the generalized model are compared to experimental results when available, and equilibrium molecular dynamics simulations otherwise. The results are in very good agreement with our analysis of phonon localization in disordered low-D systems, such as amorphous graphene and glassy diamond nanothreads. Several unique aspects of thermal transport in low-D and disordered systems, such as milder suppression of thermal conductivity and negligible diffuson contributions, are captured by the approach.

Figure 1

(a) The integrand of ${\kappa }_{\ell }^{\mathrm{am}}$ from Eq. (7), which converges for $d=2,3$, but diverges for $d=1$. (b) Schematic of the boundary $\sigma \left(k\right)$ demarcating propagons and diffusons. The Ioffe-Regel transition occurs around wave number $k=2\pi /\xi$.

Figure 2

Comparison of generalized model (solid purple line) to experimental results (blue dots) of $\kappa$ vs $T$ for 3D amorphous silica. The isolated contributions of propagons (short-dashed red line) and diffusons (long-broken yellow line) are also shown. The inset schematically illustrates an amorphous silica sample.

Figure 3

(a) Comparison of generalized model to available experimental and density functional theory results of $\kappa$ vs $T$ for pristine graphene. The generalized model predicts a $T^{3/2}$ scaling at low $T$ consistent with a predominant contribution from ZA modes, and a $T^{-2}$ scaling at high temperature. Individual contributions from ZA, TA, and LA modes are also shown. (b) Influence of Stone-Wales defects on $\kappa$ vs $T$ according to both equilibrium molecular dynamics, and generalized model for different degrees of defect scattering ${\mathrm{\Gamma }}_m=\gamma {\mathrm{\Gamma }}_0$, where ${\mathrm{\Gamma }}_0$ is the defect scattering for crystalline graphene (see Table 2). The statistical error bars marked on the equilibrium molecular dynamics results correspond to the first standard deviation.

Figure 4

$\kappa$ vs $T$ according to generalized model and equilibrium molecular dynamics for amorphous graphene. The dominant carriers are predicted to be out-of-plane (ZA) propagons. Unlike 3D amorphous materials, diffusons contribute negligibly to $\kappa$ over the entire temperature range. The statistical error bars marked on the equilibrium molecular dynamics results correspond to the first standard deviation.

Figure 5

(a) Phonon dispersion of a (13,13) CNT (red) and of zone-folded graphene (blue). (b) Zoomed-in plot of the dispersions near the Brillouin zone $\mathrm{\Gamma }$ point; the rolled tube LA, TW, and ${\mathrm{TA}}_1,{\mathrm{TA}}_2$ modes are marked by arrows. The inset shows the unit cells used for the (13,13) CNT and for zone-folded graphene. (c) Corresponding group velocities for both cases.

Figure 6

(a) $\kappa$ vs $T$ for (13,13) CNT according to generalized model, compared to the experimental results measured for a nanotube of similar radius. (b) The influence of Stone-Wales defects on $\kappa$ versus $T$, according to equilibrium molecular dynamics and generalized model. For the generalized model the influence of defects is accounted for by the defect scattering ${\mathrm{\Gamma }}_m=\gamma {\mathrm{\Gamma }}_0$, where ${\mathrm{\Gamma }}_0$ is the defect scattering parameter for the pristine material (see Table 2). The statistical error bars marked on the equilibrium molecular dynamics results correspond to the first standard deviation.

Figure 7

$\kappa$ vs $T$ for (a) a crystalline (3,0) hydrogenated $s{p}^3$ carbon nanotube and (b) a glassy diamond nanothread, both according to equilibrium molecular dynamics and generalized model. The generalized model predicts dominant contributions from LA modes for the pristine system; for the glassy case the contributions of propagons is dominant throughout the entire temperature range and the diffuson contribution is negligible. The statistical error bars marked on the equilibrium molecular dynamics results correspond to the first standard deviation.