Intrinsic Carrier Concentration Calculator - n_i for Si, Ge, and GaAs at any temperature

An intrinsic carrier concentration calculator is a solid-state physics tool that evaluates n_i = sqrt(N_c * N_v) * exp(-E_g / (2 kB T)) for a pure, undoped semiconductor at absolute temperature T, with temperature-corrected effective densities of states and a Varshni-corrected band gap.

• • Solid-state physics homework and exam problems: Compute the equilibrium electron and hole density in pure silicon, germanium, or GaAs at any temperature and check it against the tabulated value.
• • Semiconductor device design sanity checks: Estimate the intrinsic background carrier density for a diode, MOSFET, or solar cell so the doping level required to dominate conduction can be planned around.
• • Material comparison and temperature sensitivity: Compare n_i for silicon, germanium, and GaAs at the same T and sweep T to see the negative temperature coefficient of an intrinsic semiconductor.

The intrinsic carrier concentration is the equilibrium density of conduction-band electrons in a perfect, undoped sample, and by charge neutrality it equals the valence-band hole density. At 300 K in silicon the value is about 8.8e9 /cm^3, many orders of magnitude smaller than the silicon atom density of about 5e22 /cm^3, which is why pure silicon is a poor conductor at room temperature.

The same exp(-E / (2 kB T)) suppression that drives the intrinsic carrier concentration is the engine of the related two-level population calculation, and our Boltzmann Factor Calculator returns the dimensionless ratio E / (kB T) and the two-level population ratio for any energy and temperature pair.

The calculator rescales the band gap to T with the Varshni relation, rescales N_c and N_v with the parabolic-band factor (T/300)^(3/2), and evaluates n_i = sqrt(N_c * N_v) * exp(-E_g / (2 kB T)) using the exact SI Boltzmann constant.

• N_c: Effective density of states in the conduction band at 300 K. Silicon 2.82e19, germanium 1.02e19, GaAs 4.35e17 per /cm^3. Scaled with T as (T/300)^(3/2).
• N_v: Effective density of states in the valence band at 300 K. Silicon 1.83e19, germanium 5.65e18, GaAs 7.57e18 per /cm^3. Scaled the same way.
• E_g: Band-gap energy at 300 K. Silicon 1.12 eV, germanium 0.66 eV, GaAs 1.424 eV. The form applies the Varshni relation to rescale.
• T: Absolute temperature in Kelvin. The form refuses T at or below 0 K.
• kB: Boltzmann constant, fixed at 1.380649e-23 J/K = 8.617333262e-5 eV/K since the 2019 redefinition.

The dimensionless exponent E_g / (2 kB T) is the only quantity that controls the size of the carrier density, and the form returns it explicitly.

According to the Ioffe NSM Archive: Silicon band structure and carrier concentration, silicon follows Eg = 1.17 − 4.73×10⁻⁴ T²/(T + 636) eV with Nc = 6.2×10¹⁵ T³/² /cm³, and the germanium page gives the 400 K worked example n_i ≈ 1.38×10¹⁵ /cm³.

According to NIST CODATA 2018, the Boltzmann constant is exactly 1.380649e-23 J/K = 8.617333262e-5 eV/K since the 2019 SI redefinition, with no measurement uncertainty, so the exponent E_g / (2 kB T) uses an exact, fixed value rather than a rounded alternative.

The same exp(-E / (kB T)) exponential appears in the Arrhenius rate equation for a chemical reaction with activation energy Ea, and our Arrhenius Equation Calculator accepts Ea in kJ/mol and returns the rate constant k that n_i generalises for an energy gap between bands rather than over a saddle point.

Four ideas cover the full n_i formula and the two temperature corrections.

The exponent E_g / (2 kB T) is the dimensionless ratio that drives the result.

The Bohr model gives the discrete energy levels that turn into the conduction and valence bands in a crystal, and our Bohr Model Calculator returns the orbital radii, photon wavelengths, and spectral series for the same NIST CODATA constants used here.

Six short steps cover the material picker, the three 300 K material parameters, the temperature, and the Varshni and density-of-states corrections that the form applies automatically.

1. 1 Pick a material: Choose Silicon, Germanium, GaAs, or Custom. The form auto-fills N_c, N_v, and E_g with the 300 K values from the material table.
2. 2 Override the 300 K parameters if needed: For Custom or a literature value, type N_c, N_v, and E_g at 300 K into the three numeric fields.
3. 3 Set the absolute temperature T: Enter T in Kelvin. 77 K for liquid-nitrogen-cooled devices, 300 K for room temperature, 400 K for a hot die.
4. 4 Read the Varshni-corrected band gap: The result panel shows E_g(T), the band gap the form actually used. At 300 K it equals the input E_g; at 400 K in germanium it drops to about 0.62 eV from the input 0.66 eV.
5. 5 Read the intrinsic carrier concentration: The primary output is n_i in /cm^3, with n_0 and p_0 reported alongside it. For an intrinsic sample n_0 = p_0 = n_i, which the form returns as three equal numbers.
6. 6 Sanity-check the exponent: The result panel shows the dimensionless exponent E_g / (2 kB T). At 300 K in silicon it is about 21.7.

To reproduce the silicon n_i at 300 K, pick Silicon, leave the defaults, set T = 300 K, and read n_i = 8.80e9 /cm^3. To reproduce the Ioffe germanium 400 K example, switch to Germanium, set T to 400 K, and read n_i = 1.38e15 /cm^3 with the Varshni-corrected E_g = 0.62 eV.

A purpose-built intrinsic carrier concentration calculator handles the Varshni band-gap correction, the (T/300)^(3/2) density-of-states scaling, and the exponent in one form.

• • Three built-in materials with auto-filled parameters: Silicon, germanium, and GaAs load N_c, N_v, and E_g automatically.
• • Varshni band-gap correction applied automatically: For silicon, germanium, and GaAs the form rescales E_g from 300 K to T using the standard alpha, beta parameters.
• • Parabolic-band density-of-states scaling: N_c and N_v are rescaled with (T/300)^(3/2) automatically.
• • Custom material mode: The Custom option lets you type in N_c, N_v, and E_g for any other semiconductor (InP, InAs, 4H-SiC, diamond).
• • n_0, p_0, and exponent surfaced alongside n_i: The form returns n_0, p_0, and the dimensionless exponent E_g / (2 kB T) alongside n_i.

Because the result is a carrier density in /cm^3, every output field carries a fixed unit.

The conductivity of a doped semiconductor σ = q (n μ_n + p μ_p) is set by the carrier densities (n, p) and the mobilities, so once the dopant level dominates the device current, n_i only sets the leakage floor. Our Conductivity to Resistivity Calculator handles σ and ρ with temperature correction for common metal presets, so the resistivity paired with this n_i estimate stays on the same absolute-temperature scale.

Three variables drive n_i the most, and two limitations tell you when the parabolic-band / Varshni model is not the right tool.

• • The parabolic-band density-of-states model assumes parabolic E-k dispersion in both bands, which is fine for silicon and germanium at low doping and moderate T but breaks down for narrow-gap materials like InSb.
• • Very small T or very large E_g make the exponent larger than about 700, which underflows in floating point. The form clamps n_i, n_0, and p_0 to 0 and adds an 'underflow' status.

The same exponential suppression drives the dark current of a photodiode and the off-state leakage of a MOSFET at high temperature.

According to OpenStax University Physics Vol 3, §9.6 Semiconductors and Doping, in an undoped semiconductor every electron that transitions into the conduction band leaves a hole in the valence band, so the conduction-electron density and hole density are equal (n = p), which is the charge-neutrality condition n_0 = p_0 = n_i that the form returns.

The dimensionless ratio kB T / q also drives the Nernst equation, and our Nernst Equation Calculator uses the same exact SI kB in RT / (nF) to return E from E0, T, n, and Q.