= Semiconductor mass-action law
{title2=$np=n_i^2$}
= Mass-action law of nondegenerate semiconductor carriers
{synonym}
In thermal equilibrium, nondegenerate conduction <electrons> and valence <electron holes> share one <chemical potential>. Multiplying $n=N_c e^{-(E_c-\mu)/(k_BT)}$ by $p=N_v e^{-(\mu-E_v)/(k_BT)}$ cancels that <chemical potential>:
$$
np=N_cN_v e^{-E_g/(k_BT)}=n_i^2.
$$
This relation remains valid with <semiconductor doping> under the stated assumptions. Equality of the two individual densities, $n=p=n_i$, additionally requires the intrinsic neutral case. Separate electron and hole quasi-Fermi levels in nonequilibrium need not obey this equilibrium product.
With one parabolic valley and two spin states at each band edge, the prefactor multiplying $(k_BT)^3$ in the product is $4(m_cm_h)^{3/2}/(2\pi\hbar^2)^3$. Both populations must be nondegenerate; the <Pauli reduction of the pinned-edge mass-action product> quantifies a failure of the classical approximation.
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