In a pure semiconductor, thermal energy frees electrons across the band gap, leaving holes behind. Their density rises steeply with temperature – a big reason chips heat up and leak current. Silicon has about 10¹⁰ carriers per cm³ at room temperature.
How to use the Intrinsic Carrier Concentration Calculator
- Material
- Effective density of states N_c at 300 K (cm⁻³)
- Effective density of states N_v at 300 K (cm⁻³)
- Band gap at 300 K (eV)
- Temperature T
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Choose the material or enter custom parameters, then the temperature.
Inputs explained
| Field | What to enter |
|---|---|
| Material | Choose one: Silicon (Si), Germanium (Ge), Gallium arsenide (GaAs), Custom |
| Effective density of states N_c at 300 K (cm⁻³) | A number |
| Effective density of states N_v at 300 K (cm⁻³) | A number |
| Band gap at 300 K (eV) | A number |
| Temperature T | Units: K, °C, °F |
What each result means
- Band gap at T = [E3] – [a]×([Temperature T]^2÷([Temperature T] + [b]) – 90000÷(300 + [b]))
- Intrinsic carrier concentration n_i = sqrt([NcNv])×([Temperature T]÷300)^1.5×exp(-[Band gap at T]÷(2×8.617333262e-5×[Temperature T]))
Example values
How the main result changes when Effective density of states N_c at 300 K changes and everything else stays at the example values above:
| Effective density of states N_c at 300 K (cm⁻³) | Intrinsic carrier concentration n_i (cm⁻³) |
|---|---|
| 1.43e19 | 11,648,712,251.0334 |
| 2.15e19 | 11,648,712,251.0334 |
| 2.86e19 | 11,648,712,251.0334 |
| 3.58e19 | 11,648,712,251.0334 |
| 4.29e19 | 11,648,712,251.0334 |
| 5.72e19 | 11,648,712,251.0334 |
Formula
n_i = √(N_c·N_v)·exp(−E_g/(2kT)), N ∝ (T/300 K)^{3/2}, E_g(T) = E_g(300) − α·[T²/(T + β) − 300²/(300 + β)]
Worked example
Inputs — Material: Silicon (Si); Temperature T: 300 K.
Results — Band gap at T = 1.12 eV; Intrinsic carrier concentration n_i = 11,648,712,251.03336 cm⁻³.
Key terms
- Band gap
- Energy gap between the valence and conduction bands.
Frequently asked questions
What is n_i for silicon at 300 K?
About 10¹⁰ cm⁻³ (values from 1.0 to 1.5 × 10¹⁰ are quoted).
Why does n_i grow so fast with temperature?
It depends exponentially on −E_g/(2kT).
What does "intrinsic" mean?
Undoped – carriers come only from thermal generation.
Why does the band gap shrink with temperature?
Lattice expansion and vibrations (Varshni relation).