= Intrinsic-semiconductor chemical potential
{title2=$\mu_i=(E_c+E_v)/2+(k_BT/2)\ln(N_v/N_c)$}
For nondegenerate <conduction bands> and <valence bands>, <charge neutrality> gives $N_c e^{-(E_c-\mu_i)/(k_BT)}=N_v e^{-(\mu_i-E_v)/(k_BT)}$. Taking logarithms yields
$$
\mu_i=\frac{E_c+E_v}{2}+\frac{k_BT}{2}\ln\frac{N_v}{N_c}.
$$
For single spin-degenerate parabolic extrema this becomes $(E_c+E_v)/2+(3k_BT/4)\ln(m_h^*/m_e^*)$. 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.
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