Intrinsic Carrier Concentration Calculator

An intrinsic semiconductor is an ideal, chemically pure semiconductor whose mobile electrons and holes are created by thermal excitation rather than intentional dopants. At thermal equilibrium, every electron promoted into the conduction band leaves one hole in the valence band, so the electron concentration equals the hole concentration. That common value is called the intrinsic carrier concentration, usually written ni. It provides a fundamental scale for understanding conductivity, junction behavior, leakage current, and the limits of semiconductor devices. This calculator evaluates the standard nondegenerate-semiconductor relation ni = sqrt(Nc × Nv) × exp(-Eg / (2kT)). Eg is the energy band gap in electronvolts, T is absolute temperature in kelvin, and k is the Boltzmann constant expressed as 8.617333262 × 10^-5 electronvolts per kelvin. Nc and Nv are the effective densities of available quantum states in the conduction and valence bands. Because the result uses the same volume unit as those density inputs, entering both in cm^-3 produces ni in cm^-3. The exponential term makes carrier concentration extremely sensitive to both band gap and temperature. A narrow-gap material such as germanium has a much larger intrinsic concentration than silicon at the same temperature. A wide-gap material such as silicon carbide can maintain a much smaller intrinsic population at elevated temperature. Increasing temperature supplies more thermal energy, greatly increasing the probability that electrons cross the forbidden band gap. Effective densities of states also change with temperature, commonly in proportion to T raised to the three-halves power, so accurate comparisons should use Nc and Nv evaluated at the selected temperature. The equation assumes Maxwell-Boltzmann statistics, thermal equilibrium, a spatially uniform material, and carrier populations far below the available state densities. These assumptions work well for many ordinary semiconductor calculations but can break down in heavily doped or degenerate material, under strong optical excitation, or far from equilibrium. The simple expression also does not model band-gap narrowing, traps, defects, strain, or quantum confinement. Material references may quote different effective masses and therefore slightly different density-of-states values. Use this result as a physically meaningful estimate and keep all units consistent. For device-grade work, source band-gap and effective-mass data for the exact material composition and temperature. Intrinsic concentration connects directly to the mass-action law, where equilibrium electron concentration multiplied by hole concentration equals ni squared. It therefore helps engineers estimate minority carriers, reverse saturation current, resistivity trends, and the temperature range where intrinsic conduction begins to dominate doped behavior.