Intrinsic-semiconductor chemical potential (${\mu }_i=(E_c+E_v)/2+(k_{B}T/2)\ln (N_v/N_c)$)

For nondegenerate conduction bands and valence bands, charge neutrality gives $N_c{e}^{-(E_c-{\mu }_i)/(k_{B}T)}=N_v{e}^{-({\mu }_i-E_v)/(k_{B}T)}$. Taking logarithms yieldsFor single spin-degenerate parabolic extrema this becomes $(E_c+E_v)/2+(3k_{B}T/4)\ln (m_h^{\ast }/m_e^{\ast })$. Valley multiplicities or additional hole bands belong in $N_c,N_v$ and change the logarithm. The intrinsic chemical potential lies exactly at midgap only when the two effective density of states values agree.