Anisotropic failure of Fourier theory in time-domain thermoreflectance experiments

Abstract

The applicability of Fourier's law to heat transfer problems relies on the assumption that heat carriers have mean free paths smaller than important length scales of the temperature profile. This assumption is not generally valid in nanoscale thermal transport problems where spacing between boundaries is small (o1 mm), and temperature gradients vary rapidly in space. Here we study the limits to Fourier theory for analysing three-dimensional heat transfer problems in systems with an interface. We characterize the relationship between the failure of Fourier theory, phonon mean free paths, important length scales of the temperature profile and interfacial-phonon scattering by time-domain thermoreflectance experiments on Si, Si 0.99 Ge 0.01 , boron-doped Si and MgO crystals. The failure of Fourier theory causes anisotropic thermal transport. In situations where Fourier theory fails, a simple radiative boundary condition on the heat diffusion equation cannot adequately describe interfacial thermal transport.

Steady state and modulated heat conduction in layered systems predicted by the analytical solution of the phonon Boltzmann transport equation

“…Even though the Fourier's law of heat conduction is not strictly valid at this length and time scales, its predictions are often compared with experimental data to determine effective thermal properties of materials. [17][18][19] Theoretical models based on the solution of the phonon BTE, aimed at describing the modulated temperature field measured through the thermoreflectance technique, are just starting to emerge. By numerically solving an integral equation for the temperature, which is derived from the phonon BTE, Koh et al 20 developed a nonlocal theory for the temperature of a semi-infinite solid excited with a periodic heating and modulation frequencies up to 100 MHz.…”

Length-dependent thermal transport and ballistic thermal conduction

“…Wilson and Cahill recently pointed out the discrepancy of previous results originated from anisotropic thermal transport in-plane and cross-plane. 11 Yet deducing the l's is not at all straightforward, as experimental complexities associated with laser spot sizes, laser modulation frequencies, temperature profiles, and interface phonon scatterings are quite involved. 11 Moreover, theoretical interpretations of the optical measurement results remain controversial.…”

“…11 Yet deducing the l's is not at all straightforward, as experimental complexities associated with laser spot sizes, laser modulation frequencies, temperature profiles, and interface phonon scatterings are quite involved. 11 Moreover, theoretical interpretations of the optical measurement results remain controversial. While it is commonly assumed that the observed deviation to be due to ballistic phonons, ab-initio calculations suggest that agreements on Si are better if simply considering harmonic and anharmonic phonon channels but without directly incorporating ballistic phonons.…”

“…However, it introduces another problem of thermal contact resistances in these optical measurements. 11 The thermal transport at the interface is difficult to model, but it certainly plays an important role in enhancing the anisotropy in an otherwise isotropic material. 11 The interpretation of optical measurement results without considering the role of interface may also lead to inconsistent conclusions.…”

“…11 The thermal transport at the interface is difficult to model, but it certainly plays an important role in enhancing the anisotropy in an otherwise isotropic material. 11 The interpretation of optical measurement results without considering the role of interface may also lead to inconsistent conclusions. In TDTR or FDTR, changes of κ's were observed by changing spot sizes or the modulation frequency of the pump laser.…”