Application of the inverse method in the 3 ω method for measuring the thermal conductivity of thin films

Abstract

Planar nano-materials generally refer to samples that are extremely thin in the direction of thickness, such as nanofilms, graphene, and superlattice. They are of great importance in various fields, including transistors, thermoelectrics, and optoelectronics, etc. In practice, materials’ thermal transport characteristics often play a crucial role in applications. More importantly, due to the confinement in the direction of thickness, the mechanisms of thermal transport process in planar nano-materials become very different from those at the macroscale. For example, the effective lattice thermal conductivities of nanofilms are always highly dependent on their thickness, since phonon coherence, ballistic transport, and boundary scattering effects could occur in this case. Therefore, probing thermal transport properties of planar materials has been a research hotspot in both the research and industrial communities. The 3 ω method has been proven to be a powerful tool that can measure the thermal conductivities of planar materials effectively. Particularly for a thin film on a substrate, the differential 3 ω method was developed. In this scheme, it is necessary to determine the thermal properties of substrate materials in the first step. Thus, a reference sample should be measured in addition to the target sample, which not only increases the complexity of measurement but also reduces the measurement accuracy of the thin film’s thermal transport properties. In the present work, we used the numerical simulations to demonstrate that in the 3 ω method the combination of finite element method (FEM) and inverse problem method can derive the thermal properties of thin film and substrate simultaneously. It was found that the thermal conductivities could be determined accurately despite the cutoff frequency of measurement, whereas the thermal diffusivities’ accuracy could be improved with increasing cutoff frequency of measurement. Furthermore, as a test, the thermal conductivities of silicon dioxide (SiO2) thin film on the silicon (Si) substrate were measured using the method above. In our experiments, the thickness of Si substrate was 450 μm, and the thickness of SiO2 thin film was 950 nm. The frequency of driving current was set from 100 to 5000 Hz (the cutoff frequency). The trust region reflective method was adopted in the fitting process. Moreover, it was found that the measurement uncertainty of temperature rise was less than 1% and thus its influence on the uncertainties of measured thermal properties could be ignored. As a result, the major uncertainties of thermal conductivity measurement came from measuring the thickness of SiO2 thin film. The thermal conductivity of Si substrate was 142.1 W/(m K) with a relative error equal to approximately 2%, and the thermal conductivity of SiO2 thin film was 1.01 W/(m K) with a relative error equal to approximately 2%, which could agree well with the reported value in literature. In summary, the present work demonstrated that the combination of FEM and inverse method could be employed for thermal properties measurement, and thus analytical models are not always necessary. For future work, the methodology presented in this paper can be extended to deal with the measurement of thermal transport properties of irregular-shape thin flakes on a substrate.