 Spin and valley factors in the semiconductor mass-action constant

In a nondegenerate semiconductor, the semiconductor mass-action law is $np=N_c{N}_v{e}^{-E_g/(k_{B}T)}$. Its prefactors count states, so spin degeneracy and valley degeneracy must be included once each.

For equivalent parabolic conduction valleys with one-valley density-of-states effective mass $m_d$, $N_c=g_s{g}_v(m_d{k}_{B}T/(2\pi {\hslash }^2){)}^{3/2}$. The analogous valence expression must sum distinct relevant bands when one effective band is insufficient. If a tabulated effective mass or effective density of states already incorporates a degeneracy, multiplying by it again overcounts states. This changes the mass-action prefactor even though the exponential gap factor is unchanged.