An intrinsic semiconductor has equilibrium mobile electron and electron hole densities set predominantly by thermal excitation across its band gap, rather than by semiconductor doping. In the nondegenerate regime, charge neutrality gives .
In a symmetric model with constant density of states per volume and energy in two broad bands, charge neutrality gives and . In the nondegenerate limit, . Thus the leading exponential activation law can coexist with an algebraic temperature prefactor.
For symmetric touching parabolic dispersions with constant density of states per energy band, including spin degeneracy, charge neutrality fixes . The Fermi-Dirac distribution gives and the thermal internal energy, including electrons and holes, is . Thus . Its large ratio to is compatible with the equipartition theorem because this excitation count grows with temperature and the heat capacity includes both species.
The chemical potential fixed by equal conduction-electron and valence-hole densities in an intrinsic semiconductor. In the nondegenerate parabolic-band model, .
For nondegenerate conduction bands and valence bands, charge neutrality gives . Taking logarithms yieldsFor single spin-degenerate parabolic extrema this becomes . Valley multiplicities or additional hole bands belong in and change the logarithm. The intrinsic chemical potential lies exactly at midgap only when the two effective density of states values agree.
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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An intrinsic semiconductor is a pure semiconductor material that has no significant level of impurities or dopants added to it. The electrical properties of intrinsic semiconductors are determined solely by the characteristics of the semiconductor material itself. Common examples of intrinsic semiconductors include silicon (Si) and germanium (Ge). ### Key Characteristics of Intrinsic Semiconductors: 1. **Pure Material**: Intrinsic semiconductors are chemically pure, meaning they contain no added impurities that would significantly alter their electrical properties.