# Clarification on the three omega method for measuring thermal conductivity

I've recently come across the $$3\omega$$-method for measuring thermal conductivity in materials. Now I'm told that we use a lock-in amplifier to measure the 3rd harmonic in the voltage across the metal film which gives us the conductivity. What I don't understand is why the 3rd harmonic? The voltage also has a component at frequency $$\omega$$, why do we use the $$3 \omega$$ component instead of that?

The voltage has components at all multiples of $$\omega$$, i.e. $$0\omega$$, $$1\omega$$, $$2\omega$$, $$3\omega$$, and so on (though the magnitude gets lower and lower).
However if you go through the math, you'll see that at a first approximation, the thermal conductivity $$\kappa$$ is proportional to $$V_{3\omega}$$, and not to $$V_{1\omega}$$. That's why using a lock-in amplifier tuned to measure the 3rd harmonic instead of the 1st harmonic is useful to obtain $$\kappa$$.