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I was looking at the mean free time formula in the Drude model, and realized that it is proportional to 1/E. I am having trouble grasping this idea. I am aware that in practical cases the drift velocity is very small compared to the effective speed of the electrons. But if the force applied by the electric field increased appreciably, then a greater than average fraction of the electrons would only have a component only in that direction. If the electric field were to be increased until the effective speed became negligible, it is clear to see that the mean free time would increase.

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Keep in mind that that there is more going on, and that thinking about electrons as a gas is going to have limitations.

As you increase the electric field and the electrons accelerate, if the mean free path between scattering is the same, the mean velocity will be higher and the mean free time between collisions will be shorter. So that checks out.

However the material also has a band structure. In different directions depending on the type of crystal the electrons are moving through.

So the resultant mobility or conductivity would be according to the orientation of the field. So if you applied fields in different directions you could get different nobilities or conductivities, and if you wanted to you can write he conductivity as a tensor. Sometimes people working with the Hall effect do that.

But if you start turning up the electric field, you can’t do that forever. At some point you will reach a saturation velocity. Looking at it from a band structure point of view, that would depend on the band structure. Or if you have a lot of scattering for some reason perhaps limited by the mean free path.

So the Drude model is really useful conceptually but doesn’t necessarily capture all the physics.

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  • $\begingroup$ I don't understand what you mean here: "if the mean free path between scattering is the same, the mean velocity will be higher and the mean free time between collisions will be shorter. " If an electron moves with the same velocity and in the same direction as a fraction of other electrons, there would be no collision in that direction. $\endgroup$
    – Piksiki
    Jul 25, 2022 at 1:17
  • $\begingroup$ The electrons as a gas are moving in all directions all the time at high speeds. When you apply the field you are adding or subtracting a velocity. If you plot all the electrons with the axis of the plot being the velocities with out the field you would get the Fermi sphere. When you apply the field the new sphere is shifted on the plot by the drift velocity. The amount of velocity you are adding to the electron depends on the field and mean free time but that electron to could have been traveling opposite of the applied field or along it or any direction. $\endgroup$
    – UVphoton
    Jul 25, 2022 at 2:01

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