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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?

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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 $\kappa$.

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