In an exercise I look at a square lattice and consider a model where the energy band of the electron is half filled and temperature is zero. I am then supposed to explain why a half filled band corresponds to zero chemical potential. For me the most meaningful definition of the chemical potential is the energy needed to add another electron to the lattice. Why shoul this be zero? btw, the energy levels are spin degenerate.
I also doubt that this is true in general. So ##\mu=\partial E\partial N|_{T, V}##. I suppose you have calculated some band structure and can find an explicit expression for mu.