A Unifying Analytical Framework for Frequency-Domain Thermal Metrology Using Sensor Functions

Measuring the properties of multilayered structures like thin film stacks has long been an important application for thermal metrologies, and some contemporary techniques have been generalized to allow for layers with anisotropic thermal conductivity. However, the corresponding mathematical analysis has tended to be specific to a particular class of experiment. For example, for laser pump-probe systems in which both the pump and probe are Gaussian in shape, Feser, Liu, and Cahill [1] developed an analytical model that allows for arbitrarily oriented thermal conductivity tensors in every layer. However, a similarly general analytical model is lacking in other key frequency-domain techniques, such as the 3-omega technique with its rectangular heater and sensor [2][3][4]. Here, we present a more universal framework in which both the heat source and the temperature sensor have arbitrary shapes, and each may be located on a different plane of a generic multilayered structure with arbitrary thermal conductivity anisotropy in each layer. We introduce a "sensor function" that captures the heat transfer within the structure and the shape of the sensor. The framework is inspired by the scattering matrix formalism [5], which also has better numerical stability at higher modulation frequencies than the more common transfer matrix formalism [6] and also, in our experience, when the sensing plane is not the same as the heating plane. We validate this new framework by recovering various known analytical limiting cases as well as through comparison with numerical COMSOL simulations for select cases of the 3-omega method, 2-omega method[7], FDTR[8] and laser-based Angstrom's method[9].

1: Feser et al., Rev. Sci. Instrum. 85, 104903 (2014)
2: Cahill, Rev. Sci. Instrum. 61, 802–808 (1990)
3: Mishra et al., Rev. Sci. Instrum. 86, 054902 (2015)
4: Borca-Tascuic et al., Rev. Sci. Instrum. 72, 2139–2147 (2001)
5: Li and Chen, J. Appl. Phys. 132, 125103 (2022)
6: Feldman, High Temp.-High Pressures 31, 293–298 (1999)
7: Ramu and Bowers, Rev. Sci. Instrum. 83, 124903 (2012)
8: Schmidt et al., Rev. Sci. Instrum. 80, 094901 (2009)
9: Gaitonde et al., Rev. Sci. Instrum. 94, 074904 (2023)