Conservation of Energy with non-allowed states

[gyroscopeq]

(This though came up while learning about band structure, so that is how I am going to explain it, but I think it applies equally well to a square well, for example).

Say you have an electron at the Fermi level of an insulator. Then, you apply an electric field. No current flows, because all of the states of higher momentum (i.e. kinetic energy) require giving the electron a bunch of energy to get to, due to the band gap. What happens to the energy that should be given to every electron, qV? It seems like applying this field needs to be giving the electrons energy. Is it the case that they are essentially just given potential energy? In that case, is it also accurate to say that there is no classical analog for the "force" that would need to be holding them in place to keep them from moving in an applied potential, given that it is just something that falls out of the SWE and not a classical potential?

Thank you!

 

[mfb]

They gain potential energy (or lose - the zero-point is arbitrary!), but so do all other electrons and all states around them. The energy states that allow conduction are still far away.

There is no classical analog for band structure in general.

 

[gyroscopeq]

They gain potential energy (or lose - the zero-point is arbitrary!), but so do all other electrons and all states around them. The energy states that allow conduction are still far away.

By this argument, why would it not be that the existence of *any* gap, regardless of size, prevents conduction? (And more to the point, there is always a small gap between levels, the continuum limit is just an approximation.)

Also, in the case of an electric field, V is by definition *not* constant, right? So why wouldn't some electrons gain more potential energy than others? (I guess at a given point the potential gain would be the same and perhaps that's what's relevant).

Sorry for so many follow ups. This just seems like something that is *very* important to understand completely.

 

[mfb]

By this argument, why would it not be that the existence of *any* gap, regardless of size, prevents conduction?

Thermal energy can bring electrons to the conduction band, for smaller band gaps that happens more often.

(And more to the point, there is always a small gap between levels, the continuum limit is just an approximation.)

Between levels, yes, but not between bands. And the differences between levels are so tiny you cannot note them at all for macroscopic objects.

Also, in the case of an electric field, V is by definition *not* constant, right?

Sure.

So why wouldn't some electrons gain more potential energy than others?

All electrons at the same place gain the same energy.

(I guess at a given point the potential gain would be the same and perhaps that's what's relevant).

Right.