Generalized Fourier's law for nondiffusive thermal transport: Theory and experiment

Phonon heat conduction over length scales comparable to their mean free paths is a topic of considerable interest for basic science and thermal management technologies. However, debate exists over the appropriate constitutive law that defines thermal conductivity in the nondiffusive regime. Here, we derive a generalized Fourier's law that links the heat flux and temperature fields, valid from ballistic to diffusive regimes and for general geometries, using the Peierls-Boltzmann transport equation within the relaxation time approximation. This generalized Fourier's law predicts that thermal conductivity not only becomes nonlocal at length scales smaller than phonon mean free paths but also requires the inclusion of an inhomogeneous nonlocal source term that has been previously neglected. We provide evidence for the validity of this generalized Fourier's law through direct comparison with time-domain thermoreflectance measurements in the nondiffusive regime without adjustable parameters. Furthermore, we show that interpreting experimental data without the generalized Fourier's law can lead to inaccurate measurement of thermal transport properties.

Figure 1

Experimental (a) amplitude and (b) phase versus time (symbols) from TDTR for an Al/Si sample with a pump beam size of $15\mu m$ at $T$ = 300 K for modulation frequencies of 0.97 and 4.77 MHz. Predictions from the generalized Fourier's law (shaded regions) and original Fourier's law (curves) are also shown. The shaded regions around the PBE simulations correspond to the likelihood of the measured transmissivity possessing a particular value (darker area corresponds to more probable). Using the measured transmissivity profile from uniform heating [9], the prediction from the generalized Fourier's law agrees well with the TDTR measurements, while the Fourier results overestimate the phase and underestimate the amplitude at later times. The inset in (a) shows the schematic of the sample subject to a Gaussian beam heating. The inset in (b) shows the measured transmissivity of longitudinal phonons $T_{\text{Si}\to \text{Al}}\left(\omega \right)$ that is obtained with 1D transport in Ref. [9].

Figure 2

Measured (symbols) and predicted (shaded regions) phases versus pump size at room temperature and a fixed delay time of (a) 1.5 and (b) 6 ns for modulation frequencies of 0.97 and 4.77 MHz. At temperatures of 100, 150, 200, and 250 K, measured and predicted phases are plotted versus modulation frequency at a fixed delay time of (c) 1.5 and (d) 6 ns for a pump diameter of $15\mu \mathrm{m}$. The lines show the prediction by Fourier's law with temperature-dependent thermal conductivities from DFT and the interface conductance given by Eq. (31).

Figure 3

Spectral transmissivity profiles from BAs to Al versus phonon frequency that give an interface conductance of (a) 115 and (b) 253 ${\mathrm{MW}\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ using Eq. (31). The profiles are used to generate the synthetic TDTR data. (c) Normalized thermal conductivity versus pump diameter, obtained by fitting the synthetic TDTR data at modulation frequencies of 0.51 MHz (solid curves), 5.52 MHz (dashed curves), and 10.1 MHz (dotted curves), using the transmissivity profiles in (a) (solid symbols) and (b) (open symbols) to a diffusion model based on Fourier's law. Both three-phonon and four-phonon scatterings are included in the DFT calculations of single-crystalline BAs. The calculated bulk thermal conductivity of BAs is 1412 ${\mathrm{W}\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ [61, 62].