How this instrument works
In an undoped semiconductor, every electron sitting in the conduction band left behind a hole in the valence band, so the two populations are equal — call that shared count the intrinsic carrier concentration, n_i. Thermal energy alone promotes electrons across the bandgap; no dopant is supplying free carriers. The count follows n_i² = N_c·N_v·exp(−E_g ⁄ kT), where N_c and N_v are the effective densities of states in the conduction and valence bands. Both scale as T^1.5, a result of integrating a three-dimensional free-electron density of states, so their product carries T^3 and the square root that yields n_i itself carries the T^1.5 sitting in front of the exponential above.
The exponential term does almost all of the work. Silicon's bandgap, 1.12 eV, is roughly 43 times the thermal energy kT at 300 K (about 0.0259 eV), so exp(−E_g ⁄ 2kT) is an extraordinarily steep function of temperature — near room temperature, n_i roughly doubles for every 8 to 9 kelvin of warming. That steepness is why a power diode's reverse leakage current, which scales with n_i², can climb by orders of magnitude between a cold start and a hot engine bay, and why datasheets derate maximum reverse voltage as case temperature rises.
The formula treats E_g as fixed, but a real bandgap narrows slightly as a crystal heats — silicon's shrinks from about 1.17 eV near absolute zero to 1.12 eV at 300 K, a drift usually captured by the empirical Varshni relation. Anchoring the calculation at a known 300 K value sidesteps computing N_c and N_v outright, but it keeps that small assumption baked in, so results drift from measured behavior far from room temperature: deep cryogenic cold, where dopant freeze-out takes over, or several hundred kelvin above it, where the narrowed bandgap starts to matter.
- Enter the material's Intrinsic concentration at 300K, cm⁻³ — its textbook reference value (silicon ≈ 1.5×10¹⁰, germanium ≈ 2.4×10¹³, GaAs ≈ 2.1×10⁶).
- Enter the Bandgap energy, eV for that same material (silicon ≈ 1.12, germanium ≈ 0.66, GaAs ≈ 1.42).
- Set Temperature, K to the operating condition you care about — a junction temperature, an ambient reading, or a cryostat setpoint.
- Read Intrinsic carrier concentration at T — the thermally generated carrier density, in cm⁻³, at that temperature.
Worked example — silicon exactly at the 300 K reference point
A device engineer keys in silicon's textbook figures to sanity-check the instrument before trusting it on a hot junction: Intrinsic concentration at 300K, cm⁻³ = 1.5×10¹⁰, Bandgap energy, eV = 1.12, and Temperature, K = 300 — the reference point itself. The temperature ratio (300 ⁄ 300)^1.5 collapses to exactly 1, and the exponential's bracket, (1 ⁄ 300 − 1 ⁄ 300), collapses to exactly 0, so exp(0) = 1 too. Every factor in the formula reduces to unity, and the result is n_i = 1.5×10¹⁰ cm⁻³ — the input value returned unchanged, confirming the scaling law is correctly anchored at its own calibration point.
Raise the same silicon sample to 400 K, a plausible reading just inside a running engine bay, and the picture changes sharply: the instrument returns n_i ≈ 5.19×10¹² cm⁻³, a roughly 346-fold jump from the 300 K figure for only 100 kelvin of warming. That single comparison is why thermal design matters for semiconductor junctions — the carrier population, and with it any leakage current that depends on n_i², is not a fixed material property but a function that moves by orders of magnitude across an ordinary operating range.
Questions
What makes a carrier concentration 'intrinsic'?
Intrinsic means the semiconductor is undoped — pure enough that every free electron in the conduction band is matched by a hole it left behind in the valence band, with no dopant atoms adding extra carriers of one sign. n_i is that shared, thermally generated count. Doped material instead follows the mass-action law n·p = n_i², so n_i still sets a floor even where dopants dominate the current.
Why does the concentration rise so fast with temperature?
Because the formula is dominated by an exponential, not the T^1.5 prefactor. A material's bandgap is typically tens of times the thermal energy kT, so a small rise in T shrinks the negative exponent quickly and the exponential swells fast. Near 300 K, silicon's n_i roughly doubles for every 8 to 9 kelvin of warming — also why reverse leakage in a hot diode can run orders of magnitude worse than at room temperature.
Why do silicon, germanium, and gallium arsenide give such different values?
Because the exponential term is exquisitely sensitive to bandgap energy. Germanium's narrow 0.66 eV gap lets thermal energy promote carriers easily, giving a 300 K n_i near 2.4×10¹³ cm⁻³; silicon's wider 1.12 eV gap holds it near 1.5×10¹⁰; gallium arsenide's 1.42 eV gap pushes it down near 2.1×10⁶. That is also why germanium diodes leak badly when warm and GaAs devices tolerate heat far better than silicon.
Where does this simple formula stop being accurate?
It assumes the bandgap E_g stays fixed with temperature, which is only approximately true. Real gaps narrow slightly as a crystal warms, an effect the empirical Varshni equation captures, so results drift from measured behavior far from the 300 K anchor point. At cryogenic temperatures a separate mechanism, dopant freeze-out, takes over entirely, and this formula no longer describes what is happening.
How does n_i connect to the reverse leakage current of a real diode?
A p-n junction's ideal reverse saturation current is proportional to n_i², so anything that changes n_i changes leakage current by its square. Because n_i itself grows exponentially with temperature, saturation current grows even faster — the physical reason datasheets specify a maximum junction temperature and derate reverse voltage as a device runs hotter.
Can I use this for a doped, not intrinsic, sample?
The formula describes n_i, the undoped baseline, not the carrier density of a doped sample directly. What it feeds is the mass-action law n·p = n_i², which then lets you find the minority-carrier concentration once you know the majority-carrier density set by doping — useful for estimating leakage or diffusion current in an actual doped device.