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Say we apply an AC electric field at some angular frequency $\omega$ onto a metal with a scattering time of $\tau$.

How does the relation between $\omega$ and $\tau$ affect the conductivity of the metal?

While it is easy to see from the mathematical formulation that smaller value of $\omega\tau$ leads to larger conductivity (by examining, say, the Drude theory of metals), I would like to have a more phenomenological viewpoint.

Here is how i think about it:

I can think of two interesting cases.

Case 1 is when $\omega\gg\tau^{-1}$. In this case, the electric field will have finished many oscillations by the time the electron undergoes a collision. This results in the electrons getting pushed back and forth repeatedly, not going anywhere. Thus, the conductivity is small.

Case 2 is when $\omega\ll\tau^{-1}$. In this case, many collisions will have happened by the time the electric field finishes one oscillation. While this might be bad since there is a great amount of damping due to collision, we need to note that we are working with an AC field. Some of the collisions could possibly help the field drive the electrons. Thus, the electron responds better to the AC field, and we get larger conductivity.

Another argument that comes to mind is that fast oscillating field does not change the momentum much over time, meaning that the electrons are not accelerated well by the field, hence the small conductivity. If this is the case then, what is the point of comparing it to $\tau$? I am now confused.

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  • $\begingroup$ More specifically: see the dependence of conductivity on frequency in the answer to the linked question, which is in line with your intuition (case 1 $\omega\tau\gg1$, case 2 $\omega\tau\ll 1$.) $\endgroup$
    – Roger V.
    Commented Apr 25, 2023 at 11:31
  • $\begingroup$ Unfortunately, no. As stated in my question I would love an explanation about what actually, physically happens to the conduction electrons, not from a mathematical formula. I want to know why the conductivity is high when the applied field oscillates at a lower rate than the rate at which the collisions happen, and vice versa. While there are some explanations there, I wasn't satisfied. $\endgroup$
    – hendlim
    Commented Apr 25, 2023 at 11:31
  • $\begingroup$ I think explanation is provided in your own question... and it is a rather straightforward one - I don't see where one can have doubts about it. $\endgroup$
    – Roger V.
    Commented Apr 25, 2023 at 11:33
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    $\begingroup$ Without collisions, we would not have any current flowing - electron just oscillates with the field, and moves nowhere on average - half period it is accelerated in one direction, the other half in the other. This is still the case, if the field is very fast, i.e. $\omega\tau\gg 1$. The other limit is the static field, where electron is accelerated in one direction, and collisions limit its speed. Slow field, $\omega\tau\ll 1$ can be seen as very slowly varying field. $\endgroup$
    – Roger V.
    Commented Apr 25, 2023 at 11:39
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    $\begingroup$ The collision time in metals is on the order of 1e-14s, which are commensurate with frequencies in the infrared range. My gut feeling is that for such a system classical bulk conductivity may not be a good model anymore and that we would need an actual crystal lattice and band structure based model to study the real and imaginary parts of the metal's reflectivity in this region. Or maybe my solid state physics classes were just too long ago, already. $\endgroup$
    – FlatterMann
    Commented Apr 25, 2023 at 13:23

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