Intrinsic Semiconductor: Carrier Concentration (ni)

Every semiconductor device — diode, BJT, MOSFET — starts from the same baseline number: how many carriers exist in the pure crystal with no doping and no applied field. That number, the intrinsic carrier concentration ni, sets the leakage floor of every junction, determines how strongly temperature affects a device's off-state current, and defines the crossover point beyond which doping actually controls conductivity rather than thermal generation. Getting ni wrong by even an order of magnitude propagates directly into wrong leakage and breakdown predictions.

What "Intrinsic" Means

An intrinsic semiconductor is a perfect crystal (silicon, germanium, GaAs) with no dopant atoms and no lattice defects contributing free carriers. At T = 0 K, the valence band is completely full and the conduction band is completely empty — no conduction, a perfect insulator. At any T > 0, thermal energy breaks a small fraction of covalent bonds, promoting electrons across the bandgap Eg into the conduction band. Each such event creates two carriers simultaneously:

• One free electron in the conduction band
• One hole (the broken bond) in the valence band

Because generation always happens in pairs, the electron concentration n and hole concentration p are exactly equal in an intrinsic crystal:

n = p = ni

ni is therefore a single number that characterizes the material's own thermal carrier population, independent of any doping.

The ni Formula

From Fermi-Dirac statistics applied to the conduction and valence bands (with the usual non-degenerate approximation), the intrinsic concentration is:

ni = √(Nc·Nv) · exp(−Eg / (2kT))

Nc, Nv — effective density of states in the conduction/valence bands (temperature-dependent, ∝ T^3/2) Eg — bandgap energy (eV) k — Boltzmann constant (8.617×10⁻⁵ eV/K) T — absolute temperature (K)

This is usually written compactly as:

ni(T) = B · T^(3/2) · exp(−Eg / (2kT))

where B lumps the density-of-states prefactors (material-specific constant). The key structural feature is the exponential: ni depends exponentially on −Eg/2kT, so a small bandgap or a small temperature rise causes a large change in carrier population.

Numbers for Silicon

At T = 300 K (room temperature), silicon's bandgap is Eg ≈ 1.12 eV, and the accepted measured value is:

ni(Si, 300 K) ≈ 1.0 × 10¹⁰ cm⁻³

(Older textbooks quote 1.5×10¹⁰; modern band-parameter fits give closer to 1.0×10¹⁰ — either is acceptable for hand analysis, but be consistent within one calculation.)

Compare this to silicon's atomic density, ≈ 5×10²² cm⁻³. The ratio:

ni / Natoms ≈ 1×10¹⁰ / 5×10²² ≈ 2×10⁻¹³

Only about 1 in 5×10¹² silicon atoms has a broken bond contributing a carrier at room temperature — intrinsic silicon is an extremely poor conductor, which is exactly why doping (adding ~10¹⁵–10¹⁹ cm⁻³ dopant atoms) can increase conductivity by many orders of magnitude.

For comparison, other common semiconductors at 300 K:

The trend is unmistakable: larger bandgap → exponentially smaller ni. This is precisely why wide-bandgap devices (SiC, GaN) have vanishingly small intrinsic leakage and can operate at much higher temperatures before thermally generated carriers swamp the doped carrier population.

Temperature Dependence — Worked Example

How much does ni change from 300 K to 400 K in silicon? Using the exponential term only (the T^3/2 prefactor is a minor correction compared to the exponential):

ni(T2)/ni(T1) ≈ (T2/T1)^(3/2) · exp[ -Eg/2k · (1/T2 − 1/T1) ]

Plug in Eg = 1.12 eV, k = 8.617×10⁻⁵ eV/K, T1 = 300 K, T2 = 400 K:

1/T2 − 1/T1 = 1/400 − 1/300 = 0.0025 − 0.003333 = −8.33×10⁻⁴ K⁻¹

Eg/2k = 1.12 / (2 × 8.617×10⁻⁵) = 1.12 / 1.7234×10⁻⁴ ≈ 6497 K

exponent = −6497 × (−8.33×10⁻⁴) ≈ 5.41

exp(5.41) ≈ 224

Prefactor: (400/300)^1.5 = (1.333)^1.5 ≈ 1.54

ni(400K)/ni(300K) ≈ 1.54 × 224 ≈ 345

So ni grows by roughly 345× for a 100 °C rise — from 1×10¹⁰ cm⁻³ to about 3.5×10¹² cm⁻³.

Check: doubling of ni per roughly every 8–10 °C near room temperature is a commonly cited rule of thumb for silicon; over 100 °C that's on the order of 2^10 ≈ 1000, consistent in order of magnitude with the 345× computed above (the rule-of-thumb doubling interval itself stretches at higher T, so exact numbers diverge — the calculation above is the reliable one).

Why This Matters in Real Devices

• Reverse-bias leakage current in a p-n junction is proportional to ni² (via the diffusion/generation current terms), so leakage roughly follows the same explosive temperature dependence — this is the dominant reason diode and MOSFET off-state leakage rises sharply with junction temperature.
• Intrinsic vs. extrinsic regime boundary: doping only "wins" (sets the majority carrier concentration reliably) as long as the dopant density Nd or Na stays well above ni at the operating temperature. If a lightly doped region heats up enough that ni approaches the doping level, the material behavior degrades toward intrinsic — carrier concentration becomes temperature-controlled rather than doping-controlled. This caps the maximum junction temperature for silicon devices (roughly 150–200 °C) and is a primary reason SiC/GaN devices tolerate much higher junction temperatures — their ni stays negligible until far higher T.
• Mass-action law: for a doped (extrinsic) semiconductor at equilibrium, n·p = ni² always holds, regardless of doping level. This single relation, combined with charge neutrality, is how majority and minority carrier concentrations are calculated in every doped device — and it only works because ni is known first.

Key Takeaways

• ni is the equal electron and hole concentration in a pure, undoped semiconductor at thermal equilibrium — generation always creates an electron-hole pair.
• ni = B·T^(3/2)·exp(−Eg/2kT): the exponential dependence on bandgap and temperature dominates, making ni extremely sensitive to both material choice and operating temperature.
• Room-temperature silicon: ni ≈ 1×10¹⁰ cm⁻³, roughly 5×10¹² times smaller than the atomic density — intrinsic silicon is a poor conductor, which is why doping is so effective.
• A 100 °C rise pushes silicon's ni up roughly 345×, directly driving the exponential rise in junction leakage current with temperature.
• Wide-bandgap materials (SiC, GaN) have ni many orders of magnitude smaller than silicon, allowing much higher safe junction temperatures.
• The mass-action law n·p = ni² links intrinsic and extrinsic carrier concentrations and underlies virtually every doped-semiconductor calculation.