Intrinsic Carrier Concentration Calculator
Calculate intrinsic carrier concentration (ni) for Si, Ge, GaAs, and custom semiconductors. Uses band gap, temperature, and effective density of states.
About This Calculator
What is Intrinsic Carrier Concentration?
The intrinsic carrier concentration (ni) is the equilibrium density of electrons and holes in a pure (undoped) semiconductor. It is a fundamental material parameter that governs the electrical behaviour of all semiconductor devices, from simple diodes to advanced CMOS integrated circuits.
The calculation follows semiconductor physics:
- Nc = 2(2pi me* kT / h^2)3/2 -- effective density of states in the conduction band
- Nv = 2(2pi mh* kT / h^2)3/2 -- effective density of states in the valence band
- ni = sqrt(NcNv) x exp(-Eg / 2kT) -- intrinsic carrier concentration
Constants used: Planck's constant (h = 6.62607015 × 10⁻³⁴ J·s), Boltzmann constant (k = 1.380649 × 10⁻²³ J/K), and electron rest mass (m0 = 9.1093837 × 10⁻³¹ kg).
Preset Materials
| Material | Eg (eV) | me*/m0 | mh*/m0 | ni @ 300K (cm⁻³) |
|---|---|---|---|---|
| Silicon (Si) | 1.12 | 1.08 | 0.56 | ~1.5 × 10¹⁰ |
| Germanium (Ge) | 0.67 | 0.55 | 0.36 | ~2.4 × 10¹³ |
| GaAs | 1.424 | 0.067 | 0.45 | ~2.1 × 10⁶ |
The presets use density-of-states effective masses that account for band degeneracy in each material. Use Custom mode for compound semiconductors or to explore how parameter variations affect ni.
Frequently Asked Questions
What is intrinsic carrier concentration?
Intrinsic carrier concentration (ni) is the equilibrium concentration of electrons and holes in a pure, undoped semiconductor. It depends exponentially on temperature and band gap energy according to ni = sqrt(NcNv) x exp(-Eg/2kT), where Nc and Nv are the effective density of states in the conduction and valence bands respectively. For silicon at 300 K, ni ≈ 1.5 x 10¹⁰ cm⁻³.
How does temperature affect intrinsic carrier concentration?
Intrinsic carrier concentration increases exponentially with temperature because more electrons gain thermal energy to cross the band gap. The ni vs T curve follows a steep exponential rise governed by the Boltzmann factor exp(-Eg/2kT). This strong temperature dependence explains why semiconductor leakage currents increase dramatically at high temperatures, a critical consideration in IC design, power electronics, and solar cells.
What are Nc and Nv in the formula?
Nc is the effective density of states in the conduction band, and Nv is the effective density of states in the valence band. They depend on temperature and the density-of-states effective masses of electrons and holes. For silicon at 300 K, Nc ≈ 2.8 x 10¹⁹ cm⁻³ and Nv ≈ 1.0 x 10¹⁹ cm⁻³. These values determine how many states are available for carriers to occupy.
What is the difference between intrinsic and extrinsic semiconductors?
Intrinsic (undoped) semiconductors have equal concentrations of electrons and holes (n = p = ni) determined purely by the material properties and temperature. Extrinsic (doped) semiconductors have impurities added to create excess electrons (n-type) or holes (p-type), making n ≠ p. Doping shifts the Fermi level and controls conductivity, which is the basis for all semiconductor devices from diodes to transistors.
What materials can I calculate with this tool?
The calculator includes presets for Silicon (Si), Germanium (Ge), and Gallium Arsenide (GaAs) with their standard band gaps and effective masses. A Custom mode lets you enter any band gap energy, electron effective mass ratio, and hole effective mass ratio, supporting compound semiconductors like InP, GaN, SiC, and emerging 2D materials.
How accurate is the intrinsic carrier concentration formula?
The formula ni = sqrt(NcNv) x exp(-Eg/2kT) is derived from Fermi-Dirac statistics using the parabolic band approximation. It is very accurate for most elemental and III-V semiconductors near room temperature. Discrepancies typically arise from temperature-dependent band gap narrowing, non-parabolic bands, or multi-valley effects, which the Custom mode can accommodate by adjusting effective masses.
Why is intrinsic carrier concentration important in semiconductor devices?
ni determines the pn-junction reverse saturation current (I0), threshold voltage temperature coefficient in MOSFETs, and the intrinsic temperature limit above which doping becomes ineffective. It is fundamental to understanding solar cell dark current, diode rectification, BJT leakage, and the performance limits of high-temperature electronics used in aerospace and automotive applications.