Nonlocality in microscale heat conduction

Abstract

Thermal transport at short length and time scales inherently constitutes a nonlocal relation between heat flux and temperature gradient, but this is rarely addressed explicitly. Here, we present a formalism that enables detailed characterisation of the delocalisation effects in nondiffusive heat flow regimes. A convolution kernel $\kappa^{\ast}$, which we term the nonlocal thermal conductivity, fully embodies the spatiotemporal memory of the heat flux with respect to the temperature gradient. Under the relaxation time approximation, the Boltzmann transport equation formally obeys the postulated constitutive law and yields a generic expression for $\kappa^{\ast}$ in terms of the microscopic phonon properties. Subsequent synergy with stochastic frameworks captures the essential transport physics in compact models with easy to understand parameters. A fully analytical solution for $\kappa^{\ast}(x')$ in tempered L\'evy transport with fractal dimension $\alpha$ and diffusive recovery length $x_{\text{R}}$ reveals that nonlocality is physically important over distances $\sqrt{2-\alpha} \,\,x_{\text{R}}$. This is not only relevant to quasiballistic heat conduction in semiconductor alloys but also applies to similar dynamics observed in other disciplines including hydrology and chemistry. We also discuss how the previously introduced effective thermal conductivity $\kappa_{\text{eff}}$ inferred phenomenologically by transient thermal grating and time domain thermoreflectance measurements relates to $\kappa^{\ast}$. Whereas effective conductivities depend on the experimental conditions, the nonlocal thermal conductivity forms an intrinsic material property. Experimental results indicate nonlocality lengths of 400$\,$nm in Si membranes and $\simeq 1\,\mu$m in InGaAs and SiGe, in good agreement with typical median phonon mean free paths.