Semiconductor mass-action law ( $n p = n_{i}^{2}$ )

In thermal equilibrium, nondegenerate conduction electrons and valence electron holes share one chemical potential. Multiplying

n = N_{c} e^{- (E_{c} - μ) / (k_{B} T)}

p = N_{v} e^{- (μ - E_{v}) / (k_{B} T)}

cancels that chemical potential:

n p = N_{c} N_{v} e^{- E_{g} / (k_{B} T)} = n_{i}^{2} .

(1)

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_{B} T)^{3}

in the product is

4 (m_{c} m_{h})^{3/2} / (2 π ℏ^{2})^{3}

. Both populations must be nondegenerate; the Pauli reduction of the pinned-edge mass-action product quantifies a failure of the classical approximation.

Table of contents
- Pauli reduction of the pinned-edge mass-action product Semiconductor mass-action law
- Spin and valley factors in the semiconductor mass-action constant Semiconductor mass-action law

Pauli reduction of the pinned-edge mass-action product

 0  0

For chemical potential at the conduction edge, divide the exact product by the classical prediction. The ratio is

F_{1/2} (0) e^{x} F_{1/2} (- x)

, where

x = E_{g} / (k_{B} T)

. It is less than one, approaches

F_{1/2} (0)

T \to 0

, and approaches its square in the formal high-temperature parabolic model.

Spin and valley factors in the semiconductor mass-action constant

 0  0

In a nondegenerate semiconductor, the semiconductor mass-action law is

n p = 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 π ℏ^{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.