What is intrinsic carrier concentration?

Intrinsic carrier concentration (n_i) is the number of electrons (or holes) per cm³ in a pure, undoped semiconductor at thermal equilibrium. For silicon at 300 K it is approximately 1.5 × 10¹⁰ cm⁻³. Both electron and hole concentrations equal n_i in an intrinsic material.

What is the law of mass action?

At thermal equilibrium, the product of electron concentration (n) and hole concentration (p) equals n_i²: n · p = n_i². This holds regardless of doping level, as long as the semiconductor is non-degenerate.

What are N_C and N_V?

N_C is the effective density of states in the conduction band and N_V is the effective density of states in the valence band. They represent the equivalent number of energy states available to electrons and holes and depend on the effective masses and temperature. For silicon at 300 K: N_C ≈ 2.8 × 10¹⁹ cm⁻³, N_V ≈ 1.04 × 10¹⁹ cm⁻³.

How is majority carrier concentration calculated for n-type material?

Using charge neutrality: n ≈ N_D/2 + √((N_D/2)² + n_i²). When N_D >> n_i, this simplifies to n ≈ N_D. The minority hole concentration is then p = n_i² / n.

What is a degenerate semiconductor?

When doping concentration approaches or exceeds N_C or N_V, the Fermi level enters the band and the material becomes degenerate. The simple formulas used here no longer apply. This tool warns when N_D or N_A exceeds 10% of N_C or N_V respectively.

Why does n_i increase with temperature?

Higher temperature provides more thermal energy to excite electrons across the band gap, exponentially increasing n_i. This is why semiconductors become more conductive at higher temperatures, unlike metals.

Semiconductor Carrier Concentration Calculator

Calculate intrinsic, n-type, and p-type carrier concentrations in semiconductors using effective density of states and the law of mass action.

Semiconductor Parameters

Material preset (300 K)

Band gap E_g (eV)

Eff. density of states N_C (cm⁻³)

Eff. density of states N_V (cm⁻³)

Temperature T (K)

Doping (optional)

Select a preset or enter parameters and click Calculate

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Summary

Carrier concentration determines how well a semiconductor conducts electricity. This calculator computes the intrinsic carrier concentration (n_i) from the effective density of states (N_C and N_V) and band gap, then applies the law of mass action (n·p = n_i²) to find electron and hole concentrations in doped n-type or p-type material. Built-in presets cover silicon, germanium, and gallium arsenide at 300 K.

How it works

Select a preset material or enter custom parameters (band gap, effective density of states N_C and N_V, temperature).
The intrinsic concentration is computed as n_i = √(N_C · N_V) · exp(−E_g / 2k_B T).
For n-type doping, enter donor concentration N_D; the tool solves for majority electrons using charge neutrality.
For p-type doping, enter acceptor concentration N_A; the tool solves for majority holes.
Minority carriers are found from the law of mass action: n·p = n_i².
Results are shown in cm⁻³ with scientific notation and a log-scale comparison chart.

Use cases

Determine intrinsic carrier concentration in silicon at room temperature.
Find electron and hole concentrations after introducing a known dopant density.
Verify that a doping level keeps the semiconductor non-degenerate.
Compare carrier concentrations across Si, Ge, and GaAs at 300 K.
Explore how temperature affects intrinsic carrier concentration.
Calculate minority carrier concentration for diode or BJT design.
Check whether low-doped material is still intrinsic or extrinsic.
Estimate resistivity from carrier concentration and mobility data.

Frequently Asked Questions

Last updated: 2026-06-19 · Reviewed by Nham Vu