Quasiballistic heat transfer studied using the frequency-dependent Boltzmann transport equation

Quasiballistic heat transfer occurs when there is a temperature gradient over length scales comparable to phonon mean free paths (MFPs). This regime has been of interest recently because observation of quasiballistic transport can lead to useful information about phonon MFPs, knowledge of which is essential for engineering nanoscale thermal effects. Here, we use the Boltzmann transport equation (BTE) to understand how observations of quasiballistic transport can yield information about MFPs. We solve the transient, one-dimensional, frequency-dependent BTE for a double-layer structure of a metal film on a substrate, the same geometry that is used in transient thermoreflectance experiments, using a frequency-dependent interface condition. Our results indicate that phonons with MFPs longer than the thermal penetration depth do not contribute to the measured thermal conductivity, providing a means to probe the MFP distribution. We discuss discrepancies between our simulation and experimental observations which offer opportunities for future investigation.

Figure 1

Phonon dispersions of Al (left) and Si (right). Scattering is permitted between phonons of the same polarization and frequency. High-frequency phonons lacking a corresponding state are diffusely backscattered, indicated by the cutoff mismatch in the figure.

Figure 2

(a) Lattice surface temperature (relative to a reference temperature of 300 K) versus time predicted by the BTE and the heat equation and (b) the interface conductance $G$ calculated from the BTE with relaxation times truncated to 10 ps. (c) and (d) are the same figures except calculated with unmodified relaxation times which may be arbitrarily long. The surface temperature plot (a) with truncated relaxation time agrees almost exactly with the diffusive prediction, while case (c) shows a discrepancy due to quasiballistic effects. Similarly, the truncated-relaxation-time interface conductance (b) is essentially a constant close to the specified value of $1.1\times {10}^8$ W/m${}^2$K, while the interface conductance in the unmodified case (d) changes with time.

Figure 3

Lattice surface temperature (relative to a reference temperature of 100 K) versus time predicted by the heat equation and BTE with full relaxation times at $T=100$ K. Ballistic effects are more apparent at low temperatures where relaxation times are longer.

Figure 4

(a) Surface temperature (relative to a reference temperature of 300 K) versus time predicted by the BTE (solid line), along with the solutions from Fourier's law for $k=140$ W/mK (dashed line) and $k=100$ W/mK (dotted line) at $T=300$ K. The BTE solution almost exactly matches the curve with a reduced effective thermal conductivity of 100 W/mK. (b) Thermal penetration depths and cutoff MFPs required to obtain the effective thermal conductivity from the BTE solution. The two quantities agree well, supporting the idea that phonons with MFP longer than the penetration depth do not contribute to the measured thermal conductivity.

Figure 5

Typical multipulse responses at different modulation frequencies calculated from the single-pulse response for the BTE and Fourier's law at $T=300$ K. (a) Amplitude and (b) phase responses at a modulation frequency of 15 MHz; (c) amplitude and (d) phase responses at a modulation frequency of 6 MHz. Both fits correspond to effective thermal conductivity $k_{\mathrm{eff}}=100$ W/mK and interface conductance $G=1.0\times {10}^8$ W/m${}^2$K and therefore do not not predict a modulation-frequency dependence of thermal conductivity.

Figure 6

Surface temperature (relative to a reference temperature of 300 K) versus time for silicon with a dilute-point-defect scattering rate predicted by the BTE (solid line), along with the solutions from Fourier's law for $k=18$ W/mK (dashed line) and $k=0.6$ W/mK (dotted line) at $T=300$ K. The BTE solution almost exactly matches the curve with a reduced effective thermal conductivity of 0.6 W/mK, indicating that there is no modulation-frequency dependence of the thermal conductivity.