Better explanation for why electrons in filled bands don't conduct?

[VortexLattice]

My book talks about it, but doesn't give a very intuitive reason why. It essentially says that from a modified version of Liouville's Theorem, the electrons in a band stay in that band. Then it says the phase space (position-wavevector phase space) has a constant density through time, and if that's the case, when you calculate the electric and energy current densities, you sum over the Brillouin zone, and each of these currents somehow come out to zero.

I believe them well enough I guess, but I'm looking for a more intuitive explanation. Can anyone direct me to one?

 

[DrDu]

Another point is that electrons nearer to the top of a band than to the bottom are accelerated in the opposite direction than one would expect as they have a negative effective mass. The reason is that increasing an electrons crystal momentum will lead to stronger Bragg reflection from the lattice thus rather decreasing its average velocity. Hence filling up electronic state near the upper band edge will reduce current.

 

[VortexLattice]

Hmmm, but that doesn't explain why the valence ones don't conduct, right?

There must be some more intuitive reason. I trust the math but it doesn't convince me fully.

 

[DrDu]

Btw, the statement is not strictly correct. An applied field will ramp up the bands and electons can tunnel from the valence to the conduction band. At high field strength this can be observed, at lower strengths it is an exponentially small effect. Another way of viewing this is to consider time dependent fields which can be expressed in terms of a vector potential A=-E/ω sin(ωt) for ω not too large (i.e. ω should be smaller than the band gap). If this is the case, in the adiabatic approximation, the vector potential cannot induce transitions between different bands.
A mixing of the states within one band will not change the expectation value of the current operator if the band is fully occupied.

 

[OhYoungLions]

When you solve for the bloch states of an electron in a periodic potential, and periodic boundary conditions, you get states with momentum in particular directions. If you want a net current of electrons, you need a net momentum of electrons in the direction of the current. When you apply an electric field, it lowers the energy of states with a particular momentum direction. As the occupancy of such states increases, there is an unbalanced momentum distribution for the electrons, and hence a net current. BUT, there has to be unfilled states for the electrons to shift into in order for such an imbalance to develop. In filled bands, everything is filled up already, so the total momentum of electrons is always zero. There's no empty states to shift into.

If you pack a box half full of tennis balls, and turn it on an angle, they'll roll towards the lower side, but if you pack it completely full, they won't move around at all.

 

[VortexLattice]

When you solve for the bloch states of an electron in a periodic potential, and periodic boundary conditions, you get states with momentum in particular directions. If you want a net current of electrons, you need a net momentum of electrons in the direction of the current. When you apply an electric field, it lowers the energy of states with a particular momentum direction. As the occupancy of such states increases, there is an unbalanced momentum distribution for the electrons, and hence a net current. BUT, there has to be unfilled states for the electrons to shift into in order for such an imbalance to develop. In filled bands, everything is filled up already, so the total momentum of electrons is always zero. There's no empty states to shift into.

If you pack a box half full of tennis balls, and turn it on an angle, they'll roll towards the lower side, but if you pack it completely full, they won't move around at all.

Thanks, this is basically what I was looking for. I need to mull this over for a while, though...