Intrinsic carrier concentration and Fermi level calculator
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KEY POINTS
The intrinsic carrier concentration is ni = √(NcNv)·exp(−Eg/2kT), about 1×10¹⁰ cm⁻³ for silicon at 300 K. In a doped semiconductor the majority carrier concentration is roughly |ND − NA|, and the Fermi level moves about 60 meV closer to the band edge for each tenfold increase in concentration at room temperature. SiC and GaN, with their wide band gaps, keep ni small at high temperature and stay extrinsic above 300 °C.
Inputs
Band diagram (Fermi level position)
Ei is the intrinsic Fermi level. EF moves toward Ec in n-type material and toward Ev in p-type material.
Change with temperature
| Temperature | ni [cm⁻³] | Majority carriers [cm⁻³] | Regime |
|---|---|---|---|
| 250 K (-23 °C) | 7.90×10⁷ | 1.00×10¹⁶ | Extrinsic |
| 300 K (27 °C) | 1.07×10¹⁰ | 1.00×10¹⁶ | Extrinsic |
| 400 K (127 °C) | 5.63×10¹² | 1.00×10¹⁶ | Extrinsic |
| 500 K (227 °C) | 2.72×10¹⁴ | 1.00×10¹⁶ | Extrinsic |
| 600 K (327 °C) | 3.89×10¹⁵ | 1.13×10¹⁶ | Transition |
| 800 K (527 °C) | 1.23×10¹⁷ | 1.29×10¹⁷ | Intrinsic (doping no longer controls) |
Once ni approaches the intended doping level, the conductivity type set by doping can no longer be maintained. The slope of this graph and the difference in band gap explain why silicon devices are limited to around 150 °C while SiC and GaN can operate above 300 °C.
Understanding ni and the Fermi level
The intrinsic carrier concentration ni is the concentration of electrons (equal to holes) created by thermal excitation in an undoped semiconductor. It depends on the band gap Eg through exp(−Eg/2kT), so a 1 eV difference in Eg changes ni at room temperature by more than eight orders of magnitude. Silicon has about 1×10¹⁰ cm⁻³ at 300 K, while 4H-SiC has only around 10⁻⁸ cm⁻³.
In a doped semiconductor, charge neutrality sets the majority carrier concentration at roughly |ND − NA|, and the minority carriers follow from the mass action law np = ni². The Fermi level expresses carrier concentration on a log scale: each tenfold increase moves it about 60 meV closer to the band edge at room temperature.
FAQ
Why can SiC devices run hotter than silicon?
ni depends exponentially on band gap. With a band gap about three times that of silicon, 4H-SiC keeps ni far below its doping level even above 300 °C, so it stays extrinsic.
When does this calculation lose accuracy?
When the Fermi level comes within 3kT of a band edge (degenerate doping), and for wide band gap materials whose deep dopants are not fully ionized at room temperature.
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Last updated: October 23, 2026