Intrinsic Carrier Concentration Calculator — n_i from band gap and temperature

The intrinsic carrier concentration calculator above computes how many electron-hole pairs thermal energy alone creates in a pure semiconductor. It is the quantity written n_i, and it sits underneath almost every semiconductor calculation that follows: the built-in potential of a junction, reverse leakage current, the Fermi level position in doped material, and the temperature at which a device stops working as designed.

Arb Digital publishes free engineering calculators that show the intermediate quantities rather than a single opaque output. This page prints the conduction and valence band effective density of states separately, along with the thermal energy and the temperature rise needed to double the answer, because those three numbers explain the result far better than n_i on its own does.

What This Intrinsic Carrier Concentration Calculator Does

In a perfectly pure semiconductor at absolute zero, the valence band is full and the conduction band is empty, and the material is an insulator. Raise the temperature and thermal energy starts promoting electrons across the band gap. Each electron that crosses leaves a hole behind, so in intrinsic material the electron and hole concentrations are equal, and that common value is n_i.

The calculation has two parts. The effective density of states tells you how many states are available near each band edge, and it grows as the three-halves power of temperature. The exponential Boltzmann factor tells you what fraction of those states thermal energy can actually reach across the gap. The exponential wins comfortably, which is why the answer changes by orders of magnitude over a few hundred kelvin.

The hero figure is n_i per cubic centimetre, the unit almost all semiconductor literature uses. The grid gives N_C and N_V, the thermal energy in electron-volts, and the temperature rise that would double the intrinsic concentration.

How to Use It

1. Pick a material or enter your own figures. The presets load published room-temperature values; anything you type afterwards overrides them.
2. Set the temperature. This is the input the answer is most sensitive to after the band gap, and it should be the junction temperature rather than the ambient.
3. Use the right effective masses. Density-of-states effective masses are what belong here. Conductivity effective masses, which appear in mobility calculations, are different numbers for the same material.
4. Adjust the band gap for temperature if it matters. The gap narrows as the lattice expands, so a room-temperature figure used at 500 K underestimates the carrier concentration.
5. Compare against your doping. Entering a doping level shows the margin between intrinsic and extrinsic behaviour, which is what sets the upper operating temperature of a device.

The Formula: How n_i Is Calculated

The intrinsic concentration is n_i = √(N_CN_V) × exp(−E_g ÷ 2kT). The two densities of states come from the standard parabolic band result, N_C = 2(2πm_e*kT ÷ h²)^3/2 and N_V the same with the hole effective mass. The factor of two in the exponent's denominator is there because the Fermi level in intrinsic material sits near the middle of the gap, so each carrier only has to be promoted half of it in energy terms.

The tool computes these from first principles using the exact NIST CODATA value for the Boltzmann constant of 1.380649 × 10⁻²³ J/K, together with the Planck constant and the electron rest mass, rather than using a rounded room-temperature shortcut.

Work the silicon defaults through by hand. At 300 K the thermal energy kT is 0.02585 eV. With an electron effective mass of 1.08 the conduction band density of states comes to 2.816 × 10¹⁹ cm⁻³, and with a hole effective mass of 0.81 the valence band gives 1.829 × 10¹⁹ cm⁻³. Their geometric mean is 2.270 × 10¹⁹. The exponent is −1.12 ÷ (2 × 0.02585) = −21.66, and exp(−21.66) = 3.91 × 10⁻¹⁰. Multiplying gives n_i = 8.88 × 10⁹ cm⁻³. The commonly quoted measured value for silicon is about 1.0 × 10¹⁰ cm⁻³, so this simple parabolic-band model lands within about 10 per cent, which is as much as it claims.

Why the Band Gap Dominates Everything Else

Compare three materials at the same temperature and the pattern is stark. Germanium, with a gap of 0.66 eV, has an intrinsic concentration around 10¹³ per cubic centimetre at room temperature. Silicon, at 1.12 eV, sits near 10¹⁰. Gallium arsenide, at 1.42 eV, is close to 10⁶. A doubling of the gap has moved the answer by seven orders of magnitude.

That single fact explains most of the semiconductor materials landscape. Germanium devices leak heavily and lose their doping distinction at modest temperatures, which is why they were displaced despite better carrier mobility. Wide-gap materials such as gallium nitride and silicon carbide have negligible intrinsic carriers even at several hundred degrees, which is exactly why they are chosen for high-temperature and high-power work. The effective masses matter too, but only through a three-halves power, so they shift the answer by factors rather than by decades.

The Temperature That Ends Extrinsic Behaviour

A doped semiconductor works because the deliberately introduced impurities supply far more carriers than thermal generation does. Silicon doped at 10¹⁵ per cubic centimetre has roughly a hundred thousand times as many carriers from doping as from the intrinsic process at room temperature, so the doping completely determines its behaviour.

Raise the temperature and that margin erodes fast. When n_i approaches the doping level, thermally generated pairs start to rival the intentional carriers, junctions stop rectifying properly, leakage rises steeply and the device drifts out of specification. For lightly doped silicon this happens somewhere above about 200 °C, which is a large part of why ordinary silicon devices carry the junction temperature ratings they do.

The doubling temperature in the grid makes this concrete. Near room temperature silicon's intrinsic concentration doubles for roughly every 9 K of temperature rise. That comes from combining the three-halves power law in the prefactor with the exponential, and it is why reverse leakage in a silicon junction roughly doubles every ten degrees — a rule of thumb many engineers use without knowing where it comes from.

What This Model Leaves Out

The parabolic band approximation assumes the energy bands are simple quadratic functions of momentum near their edges, and that the Fermi level sits far enough from either band edge for Boltzmann statistics to replace the full Fermi-Dirac distribution. Both assumptions are reasonable for lightly doped material at moderate temperatures and both fail as doping rises.

Heavily doped material shows band gap narrowing, where the impurity states merge into the band edges and the effective gap shrinks by tens of millielectron-volts. That raises the effective intrinsic concentration above what this formula gives, and semiconductor device modelling handles it with a separate empirical correction. Degenerate doping, where the Fermi level enters a band, breaks the Boltzmann approximation outright.

The band gap's own temperature dependence is the other omission. Silicon's gap falls from about 1.17 eV at absolute zero to 1.12 eV at room temperature and continues downward, following the empirical Varshni relation. Because the gap sits inside an exponential, ignoring that change matters more the further you move from the temperature your band gap figure was quoted at. The OpenStax University Physics chapter on semiconductors and doping covers the underlying band picture and sets out the same relation with worked numbers.

Where This Sits Next to the Other Tools

This page gives the carrier concentration in pure material. The Fermi level calculator takes the next step and locates the Fermi energy once doping is added, which is the quantity that actually determines junction behaviour. For the energy units, the electron volt calculator converts between accelerating voltage, electron-volts and joules for individual particles, while the energy converter handles bulk energy units. The temperature converter is useful because band gap data is published in kelvin while device ratings are usually in degrees Celsius, and the Ohm's law calculator covers the circuit end once carriers become current. MIT OpenCourseWare's 6.012 Microelectronic Devices and Circuits works through the full derivation.

Common Mistakes to Avoid

• Using conductivity effective masses — the density-of-states effective mass is a different number for the same material, and substituting one for the other shifts the answer noticeably.
• Forgetting the factor of two in the exponent — the intrinsic Fermi level sits near mid-gap, so the exponent is the gap divided by twice kT, not by kT.
• Using a room-temperature band gap at high temperature — the gap narrows as the lattice expands, and because it sits inside an exponential the error compounds quickly.
• Applying this to heavily doped material — band gap narrowing and degenerate statistics both break the assumptions, and device simulators use an empirical correction instead.
• Mixing per-cubic-centimetre and per-cubic-metre figures — the two differ by a factor of a million, and semiconductor literature uses the centimetre convention almost universally.

Related Free Tools From Arb Digital

Continue with the Fermi level calculator to place the Fermi energy in doped material, and use the electron volt calculator when moving between electron-volts and joules at the single-particle scale. The energy converter covers bulk energy units, the temperature converter handles kelvin against Celsius, and the Ohm's law calculator and voltage divider calculator take you into the circuit that the semiconductor ends up inside. Everything Arb Digital publishes is on the free online tools hub.

Frequently Asked Questions

This tool is provided for educational use. It applies the standard parabolic-band, non-degenerate Boltzmann approximation with a fixed band gap, and does not model band gap narrowing, degenerate statistics, non-parabolicity or the temperature dependence of the gap itself. Published material data and device simulation govern any real design.