  In this article, we are proposing a thorough analysis of the cross, and the
in-plane thermal conductivity of thin-film materials based on the 3$\omega$
method. The analysis accommodates a 2D mathematical heat transfer model of a
semi-infinite body and the details of the sample preparation followed by the
measurement process. The presented mathematical model for the system considers
a two-dimensional space for its solution. It enables the calculation of the
cross-plane thermal conductivity with a single frequency measurement, the
derived equation opens new opportunities for frequency-based and
penetration-depth dependent thermal conductivity analysis. The derived equation
for the in-plane thermal conductivity is dependent on the cross-plane thermal
conductivity. Both in and cross-plane thermal conductivities enable the
measurements in two steps of measurements, the resistance-temperature slope
measurement and another set of measures that extracts the third harmonic of the
voltage signal. We evaluated the methodology in two sets of samples, silicon
nitride and boron nitride, both on silicon wafers. We observed anisotropic
thermal conductivity in the cross and the in-plane direction, despite the
isotropic nature of the thin films, which we relate to the total anisotropy of
the thin film-substrate system. The technique is conducive to the thermal
analysis of next-generation nanoelectronic devices.
