Analytical integration of the heater and sensor 3ω signals of anisotropic bulk materials and thin films

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We derive and analytically integrate the models for the heater and sensor 3ω signals of the temperature field of anisotropic bulk materials and thin films. This integration is done by using the Fourier transform and expressing the frequency dependence of temperature in terms of the modified Bessel and Struve functions, which are well-implemented in major computation software. The effects of the radiative losses and interface thermal resistance are also evaluated for different frequency regimes. Further, by fitting the 3ω model integrated over the heater and sensor widths to experimental data recorded up to 31 kHz, the thermal conductivity and thermal diffusivity of a quartz glass wafer are determined for temperatures ranging from 300 to 800 K. The obtained results show that the usual log-linear approximation can induce an uncertainty of about 5% on the thermal conductivity values. The exact integrated models are thus expected to facilitate the accurate determination of the thermal conductivity and thermal diffusivity of anisotropic materials through a wide spectrum of modulation frequencies and without time-consuming numerical integration.