For nondegenerate bands separated by , the intrinsic carrier concentration is , using the effective density of states . For fixed parabolic band effective masses and fixed gap, .
In a nondegenerate parabolic-band semiconductor with fixed density-of-states mass, , where . ThusA single measured value of does not determine its temperature dependence. A fixed-gap intrinsic model instead has , with the two effective density of states factors in . Identifying a crossover by gives an order-of-magnitude dopant-density estimate once that activation law is specified. At exact charge neutrality with , the total mobile-electron density is , so a crossover is not a sharp equality of every density definition.
For fixed band effective masses and band gap , the intrinsic carrier concentration obeys . ThusA base-ten plot against instead has slope after the prefactor correction. The intrinsic branch must be used, rather than the extrinsic plateau or dopant freeze-out branch. A changing gap or effective mass makes this an activation-energy estimate, and a small gap uncertainty can produce a much larger multiplicative uncertainty in a room-temperature minority density.
In thermal equilibrium, nondegenerate conduction electrons and valence electron holes share one chemical potential. Multiplying by cancels that chemical potential:This relation remains valid with semiconductor doping under the stated assumptions. Equality of the two individual densities, , 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 in the product is . Both populations must be nondegenerate; the Pauli reduction of the pinned-edge mass-action product quantifies a failure of the classical approximation.
For chemical potential at the conduction edge, divide the exact product by the classical prediction. The ratio is , where . It is less than one, approaches as , and approaches its square in the formal high-temperature parabolic model.
In a nondegenerate semiconductor, the semiconductor mass-action law is . 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 , . 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.
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