Thermal Conductivity Measurement of Supported Thin Film Materials Using the 3$\omega$ method

TL;DR

This paper presents a comprehensive 2D mathematical analysis of the 3ω method for measuring both cross-plane and in-plane thermal conductivities of thin films, enabling more precise thermal characterization of nanoelectronic materials.

Contribution

It introduces a novel 2D heat transfer model for the 3ω method that allows simultaneous measurement of in-plane and cross-plane thermal conductivities from a single frequency measurement.

Findings

Demonstrated anisotropic thermal conductivity in silicon nitride and boron nitride films.

Validated the 3ω method's effectiveness for thin film thermal analysis.

Revealed the influence of substrate interactions on apparent anisotropy.

Abstract

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.