Simulating the process of steady-state thermoreflectance for measuring the thermal conductivity of materials

Abstract

The measurement of thermal conductivity by the method of steady-state thermoreflectance is based on the effect of heating the sample by laser radiation. The power of reflected radiation from the sample is measured using an additional probe laser. The change in the reflection coefficient of the studied material due to the heating is proportional to the change of the sample temperature. The radiation power absorbed by the material acts as a volumetric heat source. The value of thermal conductivity is calculated according to the Fourier’s law of thermal conductivity. A steady-state thermoreflectance setup requires calibration, i.e., thermal conductivity measurement of reference samples. The value of the calibration coefficient, in turn, depends on the characteristics of the lasers used in the setup. In this work, a numerical model of sample heating is proposed which allows considering the shape and diameter of the incident radiation beam, the distribution of radiation power over the beam area, and the absorption and reflection coefficients of the material. A methodology for samples reflection and absorption coefficients determination, including measurements and calculations considering the Fabry-Perot effect, is proposed. The study was performed for the samples of germanium, silicon, gallium arsenide, and sitall. The irradiation was performed with a continuous wave single-mode diode laser with a wavelength of 980 nm. The incident radiation power distribution over the beam area was approximated according to the Gaussian function in the OriginPro software. The radiation power that passed through the samples and reflected from them was measured. The temperature of the samples during irradiation was measured with a thermal imager. The reflection and absorption coefficients of the samples were determined from the results of radiation power measurements using a mathematical model of the interaction of plane-polarized TE electromagnetic radiation with the material. Comparison of the calculation results with the literature data for germanium, silicon, and sitall samples showed their correspondence. For gallium arsenide there was a discrepancy between the calculation results and the literature data. For gallium arsenide samples, a model considering the Fabry-Perot effect was used, and the optical properties were determined numerically by searching for the minimum modules of the transfer functions of the transmitted and reflected radiation in the Matlab program. The model of electromagnetic heating of the investigated samples was implemented in the COMSOL Multiphysics software. A method for the determination of the reflection and absorption coefficients of materials investigated by the steady-state thermorflectance is proposed. The proposed model allows considering beam shape, width and distribution of the radiation power as well as the value of the absorbed radiation power for each sample. The difference between the calculated values of the sample temperature and the measurement results does not exceed 9 %. The model can be applied to measure the thermal conductivity of bulk and thin-film materials with unknown properties.

Keywords

• steady-state thermoreflectance
• thermal conductivity coefficient
• optical methods for thermal conductivity measurement
• electromagnetic heating modelling