Why Can't Electrons Move Inside the Valence Band?

[aaaa202]

Why can't electrons move inside the valence band? Is that the Pauli exclusion principle - and is it true that the electrons can't move even when the valence band is only partially filled?

 

[ZapperZ]

Why can't electrons move inside the valence band? Is that the Pauli exclusion principle

Yes.

- and is it true that the electrons can't move even when the valence band is only partially filled?

No, it is not true. When it is partially filled, then you have vacancies. These vacancies can move. We call them "holes". We simplify the treatment and description of the valence band by "renormalizing" the background electron charges and consider these holes to be positively-charged particles, and the move upon application of a field.

That's why you have p-doped semiconductors. These are semiconductors which holes as the majority charge carrier. These holes reside in the valence band.

Zz.

 

[aaaa202]

Okay I thought it might be the Pauli exlusion principle. So each band has only a finite number of energy and when all these are filled the electrons cannot move unless the next band is close.
But I must admit that there is one thing I really don't understand about all this. We are using the eigenstates to determine which states an electron can occupy. But this is quantum mechanics! Superposition of eigenstates are possible state to leading to almost infinite combinations of possible states. Why are these not relevant for solid state theory? I think I am misunderstanding something completely.

 

[DrDu]

Looking at it from another angle, electrons in the valence band can always move. However, electrons near the top of the valence band move in the opposite direction to electrons near the bottom of the valence band, so that in a full band no net current arises.

 

[DrDu]

Okay I thought it might be the Pauli exlusion principle. So each band has only a finite number of energy and when all these are filled the electrons cannot move unless the next band is close.
But I must admit that there is one thing I really don't understand about all this. We are using the eigenstates to determine which states an electron can occupy. But this is quantum mechanics! Superposition of eigenstates are possible state to leading to almost infinite combinations of possible states. Why are these not relevant for solid state theory? I think I am misunderstanding something completely.

You have to distinguish between eigenstates of a one particle hamiltonian and eigenstates of a many particle hamiltonian. If you consider but a single electron, all these superpositions of Bloch states are relevant. However in solid state theory, we are interested in systems containing many electrons.
There we use the Bloch one electron states only to build up a convenient basis for the many particle hamiltonian, e.g. via antisymmetrized Slater determinants.

 

[aaaa202]

Looking at it from another angle, electrons in the valence band can always move. However, electrons near the top of the valence band move in the opposite direction to electrons near the bottom of the valence band, so that in a full band no net current arises.

How does this symmetry arise? Why most the electrons move so as to cancel out any net current?

You have to distinguish between eigenstates of a one particle hamiltonian and eigenstates of a many particle hamiltonian. If you consider but a single electron, all these superpositions of Bloch states are relevant. However in solid state theory, we are interested in systems containing many electrons.
There we use the Bloch one electron states only to build up a convenient basis for the many particle hamiltonian, e.g. via antisymmetrized Slater determinants.

The problem is not that I don't know how to distinguish between them. The problem is the whole statistical approach where it only seems that the eigenstates are relevant for the statistics when in reality there exists an infinite amount of superpositions of eigenstates for the many particle Hamiltonian, which the many-particle system could be in. https://www.physicsforums.com/showthread.php?t=715106
I have tried to address the problem here but so far the answers have not been very convincing.

 

[DrDu]

How does this symmetry arise? Why most the electrons move so as to cancel out any net current?

If you look at Bloch functions, $\exp(iqx) u_q(x)$, the cell periodic part $u_q$ is a solution of a Schroedinger equation with $H=(p+q)^2 +V(x)$. If you apply an electric field E it can be included via a vector potential $A=iEt$ (don't ask me about signs and prefactors), so the effect of the field is just that of a q with a linear time dependence.
Hence all the $u_q$ will wander parallel to each other through the Brillouin zone. This motion is periodic as $u_(q+K)=u_q$. In a full band, the states for every possible q is filled and this won't change all states change q at equal path.