## 3ω measurement technique

The 3ω method, originally designed for bulk material thermal conductivity measurements, has evolved to become a widely adopted technique for characterizing thin films, even those just a few nanometers thick. This method is especially prevalent for assessing the in-plane and cross-plane thermal conductivity of anisotropic films and freestanding membranes.

In the 3ω method a thin metallic strip, which is in thermal contact with the sample, acts as both a heater and a temperature sensor. For the measurement an AC current,

with angular modulation frequency and amplitude passing through the strip, generates a heating source with power

where R_{h} is the resistance of the strip under the experimental conditions, and causes a rise in temperature in the form of

and consequentially an oscillation in the resistance of the stripe

By measuring the voltage drop across the heater, an amplitude-modulated signal with a small component at the third harmonic \( 3\omega \) is obtained, extractable using a lock-in amplifier. The calculation of the sample’s thermal conductivity and specific heat capacity involves solving the corresponding heat diffusion equation, which is dependent on the experimental setup.

For a typical cross-plane thermal conductivity approach with a thin film on a bulk substrate, a differential approach is employed with two measurements: one on the blank substrate and the other including the layer of interest. The thin film acts as a thermal resistor connected in series between the heater and the substrate, leading to increased temperature oscillations. The thermal conductivity (\( \lambda \)) can be determined using Fourier’s law:

where \( w \) and \( l \) are the width and length of the heater, and \( \Delta P \) and \( \Delta T \) are the power and temperature differences between the two measurements.

Another experimental setup to determine in-plane thermal conductivity and specific heat capacity involves a heater aligned to the middle of a membrane or suspended substrate. The thermal behavior of the suspended part can be evaluated using the thermal time constant (\( G = 2\lambda dlb^{-1} \)), where \( b \) is the width and \( l \) is the length of the membrane, and \( D \) is the thermal diffusivity.

With G=2λdlb^(-1), is the thermal time constant, b the width and l the length of the membrane and D is the thermal diffusivity.

An integrated measurement chip is commonly used for in-plane thermal conductivity measurements on thin films using the 3ω method.

### Which properties will be determined?

The 3ω measurement technique is an electrothermal method utilized to ascertain the thermal conductivity, thermal diffusivity, and specific heat capacity of bulk materials (solid or liquid) as well as thin layers. In this technique, an alternating current-driven metal strip serves as a heater applied to the sample. The periodic heating of the metal strip induces temperature oscillations, which are subsequently measured. The frequency dependence of these oscillations allows for the determination of the sample’s thermal conductivity and thermal diffusivity.