First-principles simulations of heat transport

Advances in understanding heat transport in solids were recently reported by both experiment and theory. However an efficient and predictive quantum simulation framework to investigate thermal properties of solids, with the same complexity as classical simulations, has not yet been developed. Here we present a method to compute the thermal conductivity of solids by performing ab initio molecular dynamics at close to equilibrium conditions, which only requires calculations of first-principles trajectories and atomic forces, thus avoiding direct computation of heat currents and energy densities. In addition the method requires much shorter sequential simulation times than ordinary molecular dynamics techniques, making it applicable within density functional theory. We discuss results for a representative oxide, MgO, at different temperatures and for ordered and nanostructured morphologies, showing the performance of the method in different conditions.

Figure 1

Top: Temperature profile along the heat transport direction $z$, averaged over 30 samples, for a classical molecular dynamics run performed at 1000 K for MgO. Smooth, continuous solid lines represent analytical solutions of the heat equation. Note the rapid decay of the sinusoidal profile to zero over 10 ps (blue line). Bottom: Difference in the average temperature $\left[\mathrm{\Delta }T\right(t\left)\right]$ between the hot and cold side of a periodic slab representing MgO, as a function of time, during a molecular dynamics run at constant volume and energy, carried out after the application of a sinusoidal temperature profile [Eq. (1)]. We show first-principles results (black line) obtained for a slab with 960 atoms at 500 K, and classical results (red line) for the same size slab, but averaged over 30 samples. Solid lines are the results of a fit to Eq. (4). The rate of decay of $\mathrm{\Delta }T$ is proportional to the thermal conductivity.

Figure 2

Thermal conductivity $\left(\kappa \right)$ of crystalline MgO computed at 500 (top panel) and 1000 K (lower panel), as a function of the length of the periodic slab $\left(L\right)$, using the approach to equilibrium molecular dynamics (AEMD; red curves), the sinusoidal approach to equilibrium (SAEMD; black curves), and classical potentials. We compare results obtained with the two methods using classical potentials and we show (blue curve) first-principles results obtained with the SAEMD method. Solid and dotted lines represent a fit to Eqs. (5) and (6), respectively.

Figure 3

Thermal conductivity $\left(\kappa \right)$ of a periodic slab representing nanocrystalline MgO, as a function of the slab length $L$, computed at 1000 K using a classical potential. The average radius of nanocrystalline grains is 2 nm. We compare simulation results obtained with the approach to equilibrium molecular dynamics (AEMD method; red dots), the sinusoidal approach to equilibrium molecular dynamics (SAEMD; black dots), and equilibrium molecular dynamics using a Green-Kubo (GK) formulation (blue lines). The blue lines represent the results of converged GK simulations as a function of size (see the Supplemental Material [24]). Convergence was obtained for $L\approx 8$ nm with a cubic supercell; the top and bottom blue lines represent the population standard deviation of the samples used for GK calculations (see the Supplemental Material).