Classical model for sub-Planckian thermal diffusivity in complex crystals

Measurements of thermal diffusivity in several insulators have been shown to reach a Planckian bound on thermal transport that can be thought of as the limit of validity of semiclassical phonon scattering. Beyond this regime, the heat transport must be understood in terms of incoherent motion of the atoms under strongly anharmonic interactions. In this work, we propose a model for heat transport in a strongly anharmonic system where the thermal diffusivity can be lower than the Planckian thermal diffusivity bound. Similar to the materials that exhibit thermal diffusivity close to this bound, our scenario involves complex unit cells with incoherent intra-cell dynamics. We derive a general formalism to compute thermal conductivity in such cases with anharmonic intra-cell dynamics coupled to nearly harmonic inter-cell coupling. Through direct numerical simulation of the nonlinear unit-cell motion, we explicitly show that our model allows sub-Planckian thermal diffusivity. We find that the propagator of the acoustic phonons becomes incoherent throughout most of the Brillouin zone in this limit. We expect these features to apply to more realistic models of complex insulators showing sub-Planckian thermal diffusivity, suggesting a multispecies generalization of the thermal diffusivity bound that is similar to the viscosity bound in fluids.

Figure 1

(a) Schematics of our cubic lattice. The orange lines represent nonlinear springs. Each unit cell is composed of free balls (green circles) of mass $m$ moving within a finite mass $M$ spherical shell of radius $R$, as illustrated in (b).

Figure 2

(a) The velocity autocorrelation function for $N=40$ from simulation with $5\times {10}^6$ collisions (red open circles). The blue dashed line is the inverse Fourier transform of the fit in the momentum space, as shown in (b). (b) Velocity-velocity power spectrum (red open circles) and rational fit to the data (blue dashed lines).

Figure 3

Real and imaginary parts of the response function $\chi \left(\omega \right)$ of the shell coordinate to an external force for $N=40$ balls.

Figure 4

(a) Logarithm of the imaginary part (color) of the phonon Green's function in the frequency-momentum space for $\vec{q}\parallel (1,1,1)$, where a coherent mode is visible. (b) The vertical cut of (a) along $q=0.8$. (c), (d) Blue solid lines: Real and imaginary part of the poles ${\omega }_s(q,q,q)$ along $q\parallel (1,1,1)$ that corresponds to the peaks in (a). Orange dashed lines: The corresponding analytic form from Eqs. (19) and (20).

Figure 5

Velocity probability distribution of the shell coordinate in different directions for $N=40$ obtained from simulation with $5\times {10}^6$ collisions. The distribution functions match with each other and exhibit Gaussian shapes.