Intrinsic Carrier Concentration Calculator

Intrinsic Carrier Concentration

In a pure semiconductor at thermal equilibrium, electrons and holes are created in equal numbers. The intrinsic carrier concentration $n_i$ depends on temperature and band gap energy. This calculator estimates $n_i$ for common materials using effective density-of-states models.

Formula

$$n_i = \sqrt{N_c N_v}\,\exp\left(-\frac{E_g}{2kT}\right)$$

With effective density of states:

$$N_c = 2.8 \times 10^{25}\,T^{1.5}\ \text{m}^{-3}$$ $$N_v = 1.04 \times 10^{25}\,T^{1.5}\ \text{m}^{-3}$$

$E_g$ is band gap in electron-volts, $T$ is absolute temperature in kelvin, and $k$ is Boltzmann's constant in eV/K.

Practical Notes

Silicon at 300 K has a band gap near 1.12 eV. Raising temperature increases $n_i$ exponentially because more electrons gain enough energy to cross the gap. Doped semiconductors use $n_i$ as a reference in mass-action calculations.

Frequently Asked Questions

What is an intrinsic semiconductor?

It is a pure material with no intentional doping. Electron and hole concentrations are equal and set by temperature and band gap.

Why does temperature matter so much?

The exponential term exp(−E_g/(2kT)) is highly sensitive to T. Small temperature changes can shift n_i by orders of magnitude.

What band gaps do the presets use?

Silicon 1.12 eV, germanium 0.66 eV, and gallium arsenide 1.42 eV at room temperature. Choose custom to enter another value.

What are N_c and N_v?

They are effective densities of states in the conduction and valence bands. They grow with T^1.5 in this model.

Is this exact for device design?

This is a standard textbook estimate. Production device models may use more detailed band-structure data and temperature-dependent band gaps.

Report

Intrinsic Carrier Concentration Calculator