Intrinsic Carrier Concentration Calculator

Calculate intrinsic carrier concentration from your physical measurements using the standard formula with consistent SI units.

Quick Facts

Governing formula
ni = √(Nc·Nv) · exp(−Eg/2kT)
Carrier concentration falls off exponentially as the bandgap-to-thermal-energy ratio grows.
Effective density of states
Nc,v = 2(2πm*kT/h²)3/2
Set by each band's effective mass, temperature, and Planck's constant.
Typical values at 300 K
Si ≈ 1×10¹⁰, Ge ≈ 2×10¹³, GaAs ≈ 2×10⁶ cm⁻³
Smaller bandgaps mean far more thermally excited carriers.
Temperature sensitivity
ni roughly doubles every 8-10 °C near 300 K
This is why reverse-bias leakage current climbs quickly with heat.

Your Results

Calculated
Intrinsic Carrier Concentration
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n_i, in cm⁻³
Intrinsic Carrier Concentration (SI)
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n_i, in m⁻³
Conduction-Band DOS
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N_c, in cm⁻³
Valence-Band DOS
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N_v, in cm⁻³

Ready

Enter the bandgap energy, effective mass ratios, and temperature, then press Calculate.

About Intrinsic Carrier Concentration

In a pure (undoped) semiconductor, thermal energy constantly excites a small fraction of valence electrons across the bandgap into the conduction band, leaving behind an equal number of holes. The equilibrium density of these thermally generated electrons (equal to the density of holes) is the intrinsic carrier concentration, ni. It is given by ni = √(Nc·Nv) · exp(−Eg / 2kT), where Nc and Nv are the effective densities of states at the conduction- and valence-band edges, Eg is the bandgap energy, k is Boltzmann's constant, and T is the absolute temperature.

How the calculation works

The effective densities of states are Nc = 2(2πme*kT/h²)3/2 and Nv = 2(2πmh*kT/h²)3/2, where me* and mh* are the density-of-states effective masses of electrons and holes (entered as a ratio to the free-electron mass m0) and h is Planck's constant. Enter the bandgap energy, the two effective-mass ratios, and the temperature; the calculator computes Nc and Nv from the fundamental constants, multiplies them together and takes the square root, then applies the Boltzmann exponential exp(−Eg/2kT) to get ni. The leading factor of 2 in each density-of-states expression accounts for electron spin degeneracy.

Common mistakes

  • Using Celsius instead of kelvin: T must be absolute temperature — 300 K (about 27 °C / 80 °F) is standard "room temperature" in semiconductor physics, not 300 °C.
  • Confusing Eg with Eg/2: the exponent uses Eg divided by 2kT (not Eg/kT) because, symmetric around mid-gap, thermal excitation only needs to supply half the gap on average to create an electron-hole pair.
  • Underestimating the sensitivity of ni to Eg and T: because both sit inside an exponential, small errors in bandgap or temperature cause large swings in the result — double-check these two inputs first.

Real-world applications

  • Diode and transistor reverse-saturation (leakage) current scales with ni², so this value sets the floor for off-state leakage in every silicon p-n junction.
  • Comparing ni across Si, Ge, and GaAs explains why GaAs devices tolerate much higher operating temperatures before intrinsic conduction overwhelms doped behavior.
  • Doping levels are chosen to be many orders of magnitude above ni so extrinsic (doped) behavior dominates; this calculator shows how much headroom a given doping level has at a given temperature.
  • Solar-cell and sensor designers use ni(T) to estimate dark current and thermal-noise contributions.

Frequently Asked Questions

What is intrinsic carrier concentration?
It is the density of free electrons (equal to the density of holes) present in a pure, undoped semiconductor at thermal equilibrium, created purely by thermal excitation across the bandgap. It is denoted ni and is usually reported in cm⁻³.
Why is intrinsic carrier concentration so sensitive to temperature?
ni depends on temperature through the Boltzmann factor exp(−Eg/2kT) as well as the T3/2 dependence of Nc and Nv. Because the bandgap term sits in an exponent, ni roughly doubles for every 8-10 °C increase near room temperature — a much stronger effect than the T3/2 prefactor alone.
How does bandgap energy affect the result?
A smaller bandgap means less thermal energy is needed to excite an electron across it, so ni is larger. This is why germanium (Eg ≈ 0.66 eV, ni ≈ 2×10¹³ cm⁻³) has far more intrinsic carriers than silicon (Eg ≈ 1.12 eV, ni ≈ 1×10¹⁰ cm⁻³), which in turn has vastly more than gallium arsenide (Eg ≈ 1.42 eV, ni ≈ 2×10⁶ cm⁻³).
What is the difference between N_c, N_v, and n_i?
Nc and Nv are the effective densities of available electron states right at the conduction- and valence-band edges — they describe how many states exist, not how many are occupied. ni combines them with the Boltzmann exponential to give the actual number of those states that are thermally populated with carriers.

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