Why can't you add an electron to a filled valence band in a semiconductor?

[BucketOfFish]

This is a question about the band gap. In a semiconductor, the chemical potential is in between the valence and conduction bands, so that the valence band is full and the conduction band is empty at T＝0. What happens if you try to add another electron to the system? It seems that it wouldn't have enough energy to join the conduction band, but all the states in the valence band are filled!

I realize that for a real semiconductor the valence band is filled completely, with not even a single electron left over, so this problem doesn't occur. In that case, what causes the valence band to end exactly at the energy level needed to accommodate all electrons?

 

[Simon Bridge]

New electrons can only be added to the conduction band ... consider: where did it come from?
There must be an electrical connection with some part of the semiconductor, and a potential difference.

 

[BucketOfFish]

Yes, I agree that it would only make sense for a new electron to join the conduction band, but perhaps I was not making my question clear. I actually meant to ask why an electron of energy ε_C could be added if mathematically μ<ε_C. It seems like with that amount of energy the electron would end up in limbo somewhere in the band gap.

EDIT: Wait, I misread. The electron joins the valence band? But all the states there ought to be filled, right? So by Pauli exclusion shouldn't electrons be forbidden from joining the valence band?

 

[Simon Bridge]

Actually you read it right - I just wrote it wrong.
The electron can only occupy an empty quantum state.
So you need to answer the question: where does the electron come from?