# Introduction

The 3ω, or 3 omega method, is a thermal measurement technique of most commonly bulk materials and thin layers. The material's thermal conductivity is measured in the form of temperature oscillations which are frequency dependent.

The method has been widely developed during the late 1980s by D. G. Cahill. The method characterizes the thermal conductivity of mostly solid materials(also gas and liquid characterization have been reported). This is especially useful for thin-film materials to find out their physical properties. For example, the development of thermoelectric materials, discs, and optical components benefit from the 3ω-method.

# The basic principle of the method

The basic measurement principle of the 3ω method is shown in Figure 1. As shown in the figure, the surface of the material is covered with a narrow metal line including four pads. The idea is to conduct a cross-plane thermal conductivity measurement of the sample. Between the outer pads (1.) an angular frequency current is heating the surface of the sample. The two inner pads (2.) works as a temperature sensor which determines the rise of the temperature of the surface. The metal line is evaporated on the surface before the measurements Figure 1. The basic principle of the 3ω method. In the figure, a four-point metal probe is introduced onto the surface of the material. (Figure: Bea Siuro)

I(ω) is current with angular frequency and it creates joule heating at 2ω. As shown in Formula1, the resistance of the metal line oscillates at 2ω, and combined with current at 1ω an oscillation at 3ω is produced. With the help of the formula1, a temperature change, ΔT, can be calculated from which the thermal conductivity can be calculated (these require many more formulas which can be found in the source).

$V_3ω= \frac{1}{2}I_0 R_0 \Delta T (\frac{1}{R} \frac{d R}{R dT} )$

Formula 1. Oscillation at 3ω, where $$V_3ω$$ = 3ω component of the voltage, $$I_0, R_0$$  = dc components of current and resistance, $$\Delta T$$ = amplitude of the heater's temperature rise, $$R$$ = line resistance, and $$\frac{d R}{R dT}$$ = resistance change / temperature of line.

The following example of characterization of a thin film (SiO2) shows how the calculated thermal conductivity can be utilized to find out physical properties of a material.

# Characterization of a thin film (SiO2)

3ω- method can be used to characterizes thin films and bulk materials as told in the introduction. One example is characterization of SiO2 thin films which are used as insulating materials in semiconductor devices.

These films can be produced by using different techniques, for example, thermal oxidation, chemical vapor deposition, evaporation, and sputtering. By using 3ω method (combined with infrared absorption spectroscopy and FTIR technique) the difference in the microstructure of the "same" SiO2 film (produced with different deposition methods) can be determined.

As shown in Figure 2,

• the thermal conductivity of the SiO2 film varies based on the deposit method (picture on the left)
• this affects for example the porosity of the film which is determined from the void parameter intensity (picture on the right)
• which affects on the functionality of the film Figure 2. Film thickness of SiO2 film on the left shows the difference of thermal conductivity in different deposition methods. On the right, the porosity of the film can be determined by using FTIR and 3ω method. (Note: the figures are simplified versions compared to the original ones, which can be found from the source.) (Figure: Bea Siuro)

# References

  1 2 Hahtela, O., Ruoho, M., Mykkänen, E., Ojasalo, K., Nissilä, J., Manninen, A., & Heinonen, M. Thermal-Conductivity Measurement of Thermoelectric Materials Using 3 ω Method. International Journal of Thermophysics, 2015, 36(12), 3255-3271. (  1 Cahill, D. G., & Pohl, R. O. Thermal conductivity of amorphous solids above the plateau. Physical review B, 1987, 35(8), 4067. (https://doi.org/10.1103/PhysRevB.35.4067)  1 Wang, C., Lan, T., Xu, J., Liu, H., Liu, X., & He, M. Influence of polar solvent on the thermal conductivity of ionic liquid based on the developed two-wire 3ω method. Journal of Molecular Liquids, 2022, 118674. (https://doi.org/10.1016/j.molliq.2022.118674)  1 Gauthier, S., Giani, A., & Combette, P. Gas thermal conductivity measurement using the three-omega method. Sensors and Actuators A: Physical, 195, 2013, 50-55. (https://doi.org/10.1016/j.sna.2012.12.032)  1 2 3 Huxtable, S. T. Heat transport in superlattices and nanowire arrays. 2002, University of California, Berkeley.  1 2 Yamane, T., Nagai, N., Katayama, S. I., & Todoki, M. Measurement of thermal conductivity of silicon dioxide thin films using a 3ω method. Journal of applied physics, 2002, 91(12), 9772-9776. (

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