From thermodynamics we have [itex]dU=Tds-Pdv+\mu dN[/itex]. So the chemical potential is the energy change due to adding an extra particle when S and V are constant. Now consider an intrinsic semiconductor at T=0 in which the valence band is all-occupied and conduction band is empty. If we add an extra electron to the lowest point of conduction band (an specified point in the conduction band) the energy change would be [itex]E_c[/itex] and so the Fermi energy (chemical potential). Then, why the Fermi energy is in the middle of band gap and is not [itex]E_c[/itex]?
Why not E_v, as you can as well remove an electron? Besides that, this is quite an academic discussion as T=0 cannot be reached. For finite temperatures mu is well defined. Hence you can take the limit T to 0.
Let's complicate things a little bit since you chose to discuss the T=0 case. If you add an electron to the semiconductor, then this will lead to a finite configurational entropy due to the degenerate states in which you can add the electron. But this would violate the third law of thermodynamics that is S=0 at T=0. Discussing imperfections at 0K would always lead to complications. But it still fun to think about them!
DrDu, I agree with you if the minimum of the conduction band is non-degenerate, but that this is always the case, is something new to me.