Thermal Conductivity Extraction From Frequency Domain Thermo-Reflectance Experiments Using the Phonon Boltzmann Transport Equation

Abstract

Abstract The Boltzmann Transport Equation (BTE) for phonons 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 signals was computed for a pump laser modulation frequency ranging from 20–200 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) 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, while a clear superiority of one relaxation time-scale expression over the other could not be established. 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. The BTE predictions resulted in thermal conductivity values that are slightly closer to the bulk value (smaller thermal conductivity suppression) than what the experimental data suggested.