Calculate intrinsic carrier concentration (ni), Fermi level, and thermal voltage for semiconductors like Si, Ge, and GaAs at any temperature.
The intrinsic carrier concentration (ni) is a fundamental property of any semiconductor — it determines the number of free electrons and holes in a pure, undoped crystal at a given temperature. This quantity depends exponentially on the ratio of the band gap energy to the thermal energy, making it extremely sensitive to both temperature and material properties.
Understanding ni is essential for semiconductor device design: it governs leakage currents, junction potentials, threshold voltages, and the temperature behavior of transistors and diodes. Silicon at room temperature (300 K) has ni ≈ 1.5 × 10¹⁰ cm⁻³, while wide-bandgap materials like GaN have negligibly small intrinsic concentrations.
This Intrinsic Carrier Concentration Calculator computes ni from the band gap energy, effective densities of states (Nc, Nv), and temperature. It also reports the thermal voltage, intrinsic Fermi level position, and estimated resistivity. Preset buttons provide parameters for common semiconductors, and a temperature sweep shows how ni changes with temperature. A material comparison table rounds out the reference.
Use this calculator to connect band gap, temperature, and density-of-states parameters to intrinsic carrier density and the basic thermal behavior of a semiconductor. It is a quick way to compare materials or temperature points before you move on to a fuller device analysis. That makes it a fast material check before more detailed device modeling. It also gives you a compact way to compare how sensitive two materials are to the same temperature change.
ni = √(Nc × Nv) × exp(−Eg / 2kT) kT = Boltzmann constant × Temperature Fermi Level: Ef ≈ Eg/2 + (kT/2) × ln(Nv/Nc) Thermal Voltage: kT/q (26 meV at 300 K) Resistivity: ρ ≈ 1 / (ni × q × (µe + µh))
Result: ni = 1.08 × 10¹⁰ cm⁻³, kT = 25.9 meV
Silicon at 300 K has an intrinsic carrier concentration of about 10¹⁰ per cubic centimeter — low enough that even small doping levels dominate conductivity.
Intrinsic carrier concentration sets the baseline electron and hole population in an undoped semiconductor. It influences leakage, depletion behavior, junction properties, and how strongly temperature will shift device behavior even before intentional doping is considered.
Because the relation includes an exponential dependence on band gap over thermal energy, small temperature changes can move `ni` dramatically. That is one reason semiconductor leakage and thermal behavior matter so much in practical device design.
This calculator is useful for material comparison and first-pass device reasoning. For detailed device simulation, mobility models, band-gap narrowing, and temperature-dependent material parameters usually need a more complete semiconductor model than a single closed-form estimate.
The number of free electrons (or holes) per unit volume in a pure semiconductor at thermal equilibrium. Electrons and holes are created in equal numbers by thermal excitation.
Higher temperature provides more thermal energy to excite electrons across the band gap. The relationship is exponential, so ni increases very rapidly with temperature.
Nc and Nv represent the effective number of quantum states available for occupation near the band edges. They depend on the effective mass of carriers in the material.
A larger band gap means fewer electrons can be thermally excited, giving a much smaller ni. This is why wide-bandgap semiconductors like GaN work at high temperatures.
kT/q ≈ 26 mV at room temperature. It appears in Boltzmann statistics and sets the scale for diode and transistor voltages.
Junction leakage current is proportional to ni (or ni² in some configurations). Higher ni at elevated temperatures increases leakage, which limits device operation.