Intrinsic Carrier Concentration Calculator
Enter the semiconductor material and temperature to instantly calculate the intrinsic carrier concentration (ni). The calculator uses the Varshni equation to model band-gap narrowing with temperature, scales the effective densities of states, and applies the full ni = sqrt(Nc * Nv) * exp(-Eg / 2kT) formula. For silicon specifically, you can switch to the highly accurate Misiakos-Tsamakis empirical formula. Preset material constants are included for silicon, germanium, and GaAs, and a custom mode lets you enter your own values.
Formula
Worked example
For silicon at 300 K: Eg(300) = 1.17 - (4.73e-4 * 300^2)/(300 + 636) = 1.12 eV. Nc = 2.82e19, Nv = 1.83e19 cm^-3. kT = 8.617e-5 * 300 = 0.02585 eV. Exponent = -1.12 / (2 * 0.02585) = -21.66. ni = sqrt(2.82e19 * 1.83e19) * exp(-21.66) = 2.27e19 * 4.25e-10 = 9.64e9 cm^-3.
What is intrinsic carrier concentration?
In a pure (intrinsic) semiconductor, electrons and holes are generated only by thermal excitation across the band gap. For every electron promoted to the conduction band a hole is left in the valence band, so the electron concentration n and the hole concentration p are equal: n = p = ni. The value ni is called the intrinsic carrier concentration. It depends exponentially on temperature and on the band-gap energy: a smaller band gap or a higher temperature produces more thermally generated carriers. At room temperature ni is about 9.65 x 10^9 cm^-3 for silicon, 2.4 x 10^13 cm^-3 for germanium, and only about 2 x 10^6 cm^-3 for the wider-gap GaAs. In a doped semiconductor the np product still equals ni^2 (the mass-action law), which makes ni a critical quantity for calculating minority carrier concentrations in p-n junctions, bipolar transistors, and solar cells.
The formula and how it works
The analytic expression is ni = sqrt(Nc(T) * Nv(T)) * exp(-Eg(T) / (2*k*T)), where Nc and Nv are the effective densities of states in the conduction and valence bands, k is the Boltzmann constant (8.617 x 10^-5 eV/K), and T is absolute temperature. Both Nc and Nv scale as (T/300)^1.5 relative to their tabulated 300 K values. The band-gap Eg also narrows with temperature, and this calculator uses the Varshni equation Eg(T) = Eg(0) - alpha*T^2/(T + beta) with material-specific constants alpha, beta, and Eg(0). For silicon only, the toggle enables the Misiakos-Tsamakis empirical formula ni = 5.29 x 10^19 * (T/300)^2.54 * exp(-6726/T), which gives higher accuracy between 78 and 340 K because it was fitted to measured carrier densities rather than derived from independently measured band-gap and effective mass data.
Temperature dependence and practical limits
Because ni appears in an exponential, it is extremely sensitive to temperature. For silicon, ni roughly doubles for every 8-10 K rise near room temperature. Above about 400 K, ni for silicon exceeds typical dopant concentrations (which are usually 10^14 to 10^17 cm^-3), and the semiconductor becomes intrinsic again regardless of doping. This sets an upper operating limit for silicon devices at roughly 150-200 degC. Germanium has a narrower band gap so it reaches this limit closer to 70-80 degC, which is one reason silicon displaced germanium in most electronics. Wide-gap semiconductors like GaAs (and SiC, GaN) have much lower ni and can operate at higher temperatures, which is why they are favoured for high-temperature and high-power applications.
How to use the custom mode
To use a semiconductor not in the preset list, select Custom and enter the effective density-of-states values Nc(300 K) and Nv(300 K) in cm^-3, along with the band-gap energy at 300 K in eV. The calculator then applies the Varshni temperature scaling using silicon Varshni parameters as a default approximation. For a more accurate result with non-standard materials, look up the material-specific Varshni alpha and beta parameters in a semiconductor handbook (for example Sze and Ng, "Physics of Semiconductor Devices", 3rd ed.) and compare the output against published values at known temperatures to validate your constants.
Common semiconductor parameters at 300 K
| Material | Eg at 300 K (eV) | Nc (cm^-3) | Nv (cm^-3) | ni at 300 K (cm^-3) |
|---|---|---|---|---|
| Silicon (Si) | 1.12 | 2.82 x 10^19 | 1.83 x 10^19 | ~9.65 x 10^9 |
| Germanium (Ge) | 0.66 | 1.02 x 10^19 | 5.65 x 10^18 | ~2.4 x 10^13 |
| GaAs | 1.424 | 4.35 x 10^17 | 7.57 x 10^18 | ~2.0 x 10^6 |
Standard material constants used in semiconductor physics. Eg narrows with temperature; ni is the intrinsic carrier concentration at 300 K.
Frequently asked questions
What is the intrinsic carrier concentration of silicon at room temperature?
The best-accepted value for silicon at 300 K is approximately 9.65 x 10^9 cm^-3, based on the work of Altermatt et al. (2003). Older textbooks quote values in the range 1.0 to 1.5 x 10^10 cm^-3 due to earlier measurements and slightly different band-gap values. Room temperature is conventionally taken as 300 K (about 27 degC) in semiconductor physics.
What is the Varshni equation and why does band-gap change with temperature?
The Varshni equation, Eg(T) = Eg(0) - alpha*T^2/(T + beta), describes how a semiconductor's band-gap energy decreases as temperature rises. The narrowing arises because lattice vibrations (phonons) at higher temperatures expand the crystal lattice and increase electron-phonon interactions, which lower the energies of the band edges. The constants alpha and beta are fitted to measured data for each material. For silicon, alpha = 4.73 x 10^-4 eV/K, beta = 636 K, and Eg(0) = 1.17 eV.
What is the Misiakos-Tsamakis formula and when should I use it?
The Misiakos-Tsamakis formula, ni = 5.29 x 10^19 * (T/300)^2.54 * exp(-6726/T), is an empirical fit to measured intrinsic carrier concentrations in silicon from 78 to 340 K (Misiakos and Tsamakis, Appl. Phys. Lett. 64, 1994). It is generally more accurate than the analytic formula in that temperature range because it was fitted to direct measurements rather than computed from separately measured material parameters. Use it when you need the best possible accuracy for silicon in near-room-temperature or low-temperature applications such as photovoltaics or cryogenic sensor design.
What are Nc and Nv, the effective densities of states?
Nc and Nv are the effective densities of quantum states available to electrons in the conduction band and holes in the valence band, respectively. They are not the actual number of carriers, but rather scale factors that determine how many carriers would occupy those bands at a given temperature if they were fully populated. Both scale as T^1.5, reflecting the fact that the available energy states spread out as temperature rises. For silicon at 300 K, Nc = 2.82 x 10^19 cm^-3 and Nv = 1.83 x 10^19 cm^-3.
Why does GaAs have a lower ni than silicon despite being used at higher temperatures?
GaAs has a wider band gap (1.42 eV vs 1.12 eV for silicon) so far fewer electrons can be thermally excited across it at a given temperature. At 300 K ni for GaAs is about 2 x 10^6 cm^-3, versus about 10^10 for silicon. The smaller ni means the intrinsic carrier density stays below practical dopant concentrations up to higher temperatures, extending the useful operating range. GaAs also has high electron mobility, making it attractive for high-frequency and optoelectronic devices.
How does ni relate to minority carrier concentration in a doped semiconductor?
In a doped semiconductor the mass-action law states that the product of electron and hole concentrations equals ni^2 at thermal equilibrium: n * p = ni^2. In an n-type semiconductor with donor concentration Nd >> ni, the majority carrier concentration n is approximately Nd, so the minority hole concentration is p = ni^2 / Nd. For silicon at 300 K with Nd = 10^16 cm^-3, p = (9.65 x 10^9)^2 / 10^16 = about 9.3 x 10^3 cm^-3, many orders of magnitude below Nd.
Sources
- Misiakos, K. and Tsamakis, D. (1994). Accurate measurements of the silicon intrinsic carrier density from 78 to 340 K. Applied Physics Letters, 64(15), 2007-2009.
- Altermatt, P. P., Schenk, A., Geelhaar, F., and Heiser, G. (2003). Reassessment of the intrinsic carrier density in crystalline silicon in view of band-gap narrowing. Journal of Applied Physics, 93(3), 1598-1604.