Analysis of the “3-Omega” method for substrates and thick films of anisotropic thermal conductivity

Abstract

The heat equation in an anisotropic material under sinusoidal excitation is solved in cylindrical-polar coordinates for the geometry conventionally used in the measurement of thermal conductivity by the “3-Omega” method. Although the solution can be carried out in Cartesian coordinates using Fourier transforms, the inverse transforms have to be evaluated numerically. The current method allows us to make contact with, and extend, the analytical results of Cahill and Pohl for isotropic substrates. We find that the 3-Omega method experiment, if performed conventionally, measures the average thermal conductivity in the in-plane and cross-plane directions, for modest anisotropies. An alternate geometry with separate heater and thermometer lines is explored, where the temperature of the thermometer is expected on physical grounds to be selectively sensitive to the in-plane thermal conductivity. The selectivity is quantified using the analytic forms derived for the temperature profile, and the results are verified by finite element method simulations.