3 omega method

The 3-omega-method is a measurement method for determining the thermal conductivity of bulk material (i.e. solid or liquid substances) and thin layers. Here, a metal heater applied to the sample is periodically heated and the resulting temperature oscillations are measured. The thermal conductivity and thermal diffusivity of the sample can be determined from their frequency dependence. The process goes back to Prof. David Cahill.

method

Signals that contribute to the development of the 3-omega component.

Example of the course of a ΔT curve during a 3-omega measurement.

Heater structure used for 3-omega measurements on a glass sample.

In the 3-omega method, the metal wire placed on the sample is used both as a heater and a thermometer. The production of this wire is mostly done by microstructuring and thermal evaporation . A periodic current with the angular frequency is fed into the metal wire : ${\displaystyle \omega }$

${\displaystyle I(t)=I_{0}\cdot \cos \left(\omega t\right)}$.

The thermal power fed into the heating wire is then:

${\displaystyle P(t)=I^{2}\cdot R=I\left(t\right)^{2}\cdot R={\frac {I_{0}^{2}R}{2}}\cdot \left(1+\cos \left(2\omega t\right)\right)}$,

where is the electrical resistance of the heater. Due to the power fed in, the temperature of the heater changes with the same frequency as the power signal: ${\displaystyle R}$

${\displaystyle T(t)=T_{0}\left(t\right)+\Delta T\cdot \cos \left(2\omega t+\varphi \right)}$.

The amplitude and the phase shift in relation to the feed power depend on the thermal conductivity of the material and the frequency . is the zero position of the temperature oscillation and depends on the power and coupling of the sample to the environment. The temperature oscillation of the heating wire leads to a resistance oscillation of the same, which ${\displaystyle \Delta T}$ ${\displaystyle \varphi }$ ${\displaystyle \omega }$ ${\displaystyle T_{0}}$

${\displaystyle R(t)=R_{0}\cdot \left[1+\alpha \cdot \left(T\left(t\right)-T_{0}\left(t\right)\right)\right]=R_{0}+\Delta R\cdot \left[\cos \left(2\omega t+\varphi \right)\right]}$

and

${\displaystyle \Delta R=\alpha \cdot R_{0}\cdot \Delta T}$

given is. Here is the temperature coefficient of the electrical resistance of the heater, the resistance at temperature and the amplitude of the resistance oscillation. From this relationship and the first equation, the voltage measured across the heating wire can now be calculated: ${\displaystyle \alpha }$ ${\displaystyle R_{0}}$ ${\displaystyle T_{0}}$ ${\displaystyle \Delta R}$

${\displaystyle {\begin{aligned}U\left(t\right)&=R\left(t\right)\cdot I\left(t\right)\\&=\left[R_{0}+\Delta R\cos \left(2\omega t+\varphi \right)\right]\cdot I_{0}\cos \left(\omega t\right)\\&=R_{0}I_{0}\cos \left(\omega t\right)+\Delta RI_{0}\cdot \cos \left(2\omega t+\varphi \right)\cos \left(\omega t\right)\\&=R_{0}I_{0}\cos \left(\omega t\right)+\Delta RI_{0}\cdot {\frac {1}{4}}\left(e^{i2\omega t+i\varphi }+e^{-i2\omega t-i\varphi }\right)\cdot \left(e^{i\omega t}+e^{-i\omega t}\right)\\&=R_{0}I_{0}\cos \left(\omega t\right)+\Delta RI_{0}\cdot {\frac {1}{4}}\left(e^{i3\omega t+i\varphi }+e^{-i\omega t-i\varphi }+e^{i\omega t+i\varphi }+e^{-3i\omega t-i\varphi }\right)\\&=R_{0}I_{0}\cos \left(\omega t\right)+{\frac {\Delta RI_{0}}{2}}\cdot \left[\cos \left(3\omega t+\varphi \right)+\cos \left(\omega t+\varphi \right)\right]\end{aligned}}}$.

Accordingly, a component with frequency 3 can be observed in the voltage signal , which is the amplitude ${\displaystyle \omega }$

${\displaystyle U_{3\omega }={\frac {\Delta RI_{0}}{2}}={\frac {\alpha \cdot R_{0}\cdot \Delta T\cdot I_{0}}{2}}}$

possesses, is proportional to the temperature amplitude of the heater and is therefore suitable for measuring the temperature amplitude. In order not to lose the phase information of the temperature oscillation, one can consider a complex quantity by multiplying the phase factor. With modern lock-in amplifiers it is possible to filter out and measure the 3-omega component of the voltage signal and the phase dependency. ${\displaystyle \Delta T}$ ${\displaystyle \Delta T}$

To determine the thermal conductivity of the sample, a relationship between the temperature oscillation and the thermal conductivity is required. For a semi-infinitely expanded volume material and an infinitely narrow heater there is the following approximation: ${\displaystyle \Delta T}$

${\displaystyle \Delta T(r)={\frac {P}{\pi l\lambda }}\left[{\frac {1}{2}}\ln \left({\frac {D}{r^{2}}}\right)-{\frac {1}{2}}\ln \left(2\omega \right)+\ln \left(2\right)-\gamma -i{\frac {\pi }{4}}\right]}$.

Here is the power fed in, the length of the heating wire, the thermal conductivity of the sample, the Euler-Mascheroni constant and the thermal diffusivity . In particular, the real part of is proportional to the logarithm of the frequency and the thermal conductivity is a proportionality factor. If the temperature oscillation is measured at different frequencies, the thermal conductivity of the sample can be determined. This approximation is valid in most cases for heater widths in the range of several micrometers. ${\displaystyle P}$ ${\displaystyle l}$ ${\displaystyle \lambda }$ ${\displaystyle \gamma }$ ${\displaystyle D}$ ${\displaystyle \Delta T}$

Use on thin films

To determine the thermal conductivity of thin layers , two measurements are carried out: one on the substrate and one with the layer to be characterized between heater and substrate. If the heater width is large compared to the layer thickness and the substrate has a high thermal conductivity compared to the layer, the heat transport can be viewed as one-dimensional through the layer. The layer is then a thermal resistor connected in series between the heater and the substrate and ensures an increase in the temperature oscillations compared to the measurement without the thin layer. ${\displaystyle \Delta T_{\text{f}}}$

From this increase, the thermal conductivity of the layer can be determined with the help of Fourier's law : ${\displaystyle \lambda _{\text{f}}}$

${\displaystyle \lambda _{\text{f}}={\frac {P\cdot d_{\text{f}}}{2\cdot b\cdot l\cdot \Delta T_{\text{f}}}}}$.

Here is the heating power, the film thickness, half the heater width and the heater length. The heater geometry should be identical for both measurements, as this has an influence on the course of the values. As an alternative to two measurements, if the substrate is known exactly, the measurement on the substrate can also be simulated, but this can lead to greater inaccuracies. ${\displaystyle P}$ ${\displaystyle d_{\text{f}}}$ ${\displaystyle b}$ ${\displaystyle l}$ ${\displaystyle \Delta T}$

literature

• Jason Randall Foley: The 3-Omega method as a nondestructive testing technique for composite material characterization. 1999.

Individual evidence

1. ^ ^{A b} David G. Cahill, RO Pohl: Thermal conductivity of amorphous solids above the plateau . In: Phys. Rev. B . tape 35 , no. 8 , 1987, pp. 4067-4073 , doi : 10.1103 / PhysRevB.35.4067 .
2. ↑ David G. Cahill, M. Katiyar, JR Abelson: Thermal conductivity of a-Si: H thin films . In: Phys. Rev. B . tape 50 , no. 9 , 1994, pp. 6077-6081 , doi : 10.1103 / PhysRevB.50.6077 .
3. ↑ T. Borca-Tasciuc, AR Kumar, G. Chen: Data reduction in 3ω method for thin-film thermal conductivity determination . In: Review of Scientific Instruments . tape 72 , no. 4 , 2001, p. 2139-2147 , doi : 10.1063 / 1.1353189 .