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 accomodate all electrons?