Extraction of thermal conductivity using phonon Boltzmann Transport Equation based simulation of frequency domain thermo-reflectance experiments

Abstract

The multidimensional phonon Boltzmann Transport Equation (BTE) was solved numerically in cylindrical coordinates and in time domain to simulate a Frequency Domain Thermo-Reflectance (FDTR) experimental setup. The phase lag between the pump and probe laser signals were computed for a pump laser modulation frequency ranging from 20 to 200 MHz. Results were obtained both with and without the inclusion of optical phonons, as well as with two different relaxation time-scale expressions (Holland versus Broido) for scattering, obtained from the literature for silicon. It was found that inclusion of optical phonons significantly improved the agreement between the measured and the computed phase lag. Subsequently, the thermal conductivity was extracted by fitting the Fourier heat conduction equation results—also solved numerically in time domain—to the measured and computed (using BTE) phase lag values. While both relaxation time-scales exhibited thermal conductivity suppression, a clear superiority of one time-scale expression over the other could not be established. With the Broido time-scale, the thermal conductivity extracted from BTE calculations overpredicted the value extracted from experiments regardless of whether optical phonons were included in the calculations. In contrast, with the Holland time-scale, the extracted thermal conductivity value overpredicted the value extracted from experiments when optical phonons were included, but underpredicted the value when optical phonons were excluded.