Intrinsic Carrier Concentration Calculator

Intrinsic carrier concentration, n_i: The number of free electrons in a pure semiconductor per unit volume. In intrinsic material, the hole concentration is the same as n_i [1].

Semiconductor: A material that conducts electricity better than an insulator but not as easily as a metal [3].

Band gap energy, Eg: The energy difference between the valence band and conduction band. A larger gap usually means fewer carriers at the same temperature.

Effective density of states in conduction band, Nc: A model value that tells how many electron states are available near the conduction band edge.

Effective density of states in valence band, Nv: A model value that tells how many hole states are available near the valence band edge.

Silicon empirical method: A fitted equation for silicon that estimates n_i from temperature using constants A1 and A2 instead of separate Eg, Nc, and Nv values.

Natural log of n_i: The logarithm of the concentration value in cm^-3. It is useful when results span many orders of magnitude.

Mistake: Entering Celsius in Temperature (K) instead of kelvins.
Fix: Use absolute temperature in kelvins. For example, room temperature is about 300 K, not 27.

Mistake: Leaving Band gap energy, Eg (eV) , Effective density of states in conduction band, Nc (cm^-3) , or Effective density of states in valence band, Nv (cm^-3) blank while Calculation method is set to General semiconductor.
Fix: Fill in all three with positive values, or switch Calculation method to Silicon empirical.

Mistake: Typing Nc or Nv in m^-3 even though the labels say Nc (cm^-3) and Nv (cm^-3) .
Fix: Enter those inputs in cm^-3 exactly as labeled. The calculator converts the final result to Intrinsic carrier concentration, n_i (m^-3) for you.

Mistake: Editing silicon constants in Silicon empirical constant A1 (cm^-3 K^-3/2) and Silicon empirical constant A2 (K) but expecting them to affect the General semiconductor result.
Fix: Those constants are only used when Calculation method is Silicon empirical.

Mistake: Turning on Show silicon method comparison for germanium, GaAs, or Custom and expecting a comparison percent every time.
Fix: The comparison output is only shown when Material preset is silicon and both silicon methods can be computed.

Mistake: Reading Natural log of n_i as if it were another concentration value.
Fix: Use Intrinsic carrier concentration, n_i (cm^-3) or Intrinsic carrier concentration, n_i (m^-3) for the actual concentration, and use the log value only for easier comparison across huge ranges.

• This calculator is for intrinsic, undoped semiconductor behavior at thermal equilibrium. It does not model extra carriers from doping, injection, or strong external fields.
• Temperature (K) must be greater than 0. Zero or negative kelvin is a blocked input error.
• In the General semiconductor method, Band gap energy, Eg (eV), Effective density of states in conduction band, Nc (cm^-3), and Effective density of states in valence band, Nv (cm^-3) are treated as the values you supply. If real material properties change with temperature and you do not update them, the result can differ from a more detailed model.
• The Silicon empirical method is a silicon-only approximation. It should not be used for germanium, GaAs, or another custom material unless you intentionally provide fitted constants for that case.
• Intrinsic carrier concentration, n_i (cm^-3) and Intrinsic carrier concentration, n_i (m^-3) can become extremely small or extremely large. Very extreme inputs may underflow, overflow, or trigger a safe numeric-range error.
• If Display format is set to plain decimal, very large or very small results may still be shown in a more readable automatic format so the output stays usable.
• The comparison output is only meaningful for silicon when both methods are available. A large signed percent difference means model choice matters for the chosen inputs.

Formulas used

The calculator uses one of two equations, depending on Calculation method.

n_i = sqrt(Nc x Nv) x exp(-Eg / (2 x kB x T))

General semiconductor method, where kB is the Boltzmann constant in eV/K.

kB = 8.617333262145 x 10^-5 eV/K

n_i = A1 x T^(3/2) x exp(-A2 / T)

Silicon empirical method, using the user-visible constants A1 and A2.

n_i(m^-3) = n_i(cm^-3) x 10^6

ln(n_i) = natural log of n_i in cm^-3

difference (%) = ((n_i,alt - n_i,primary) / n_i,primary) x 100

How the calculator chooses inputs

If Calculation method is set to General semiconductor, the calculator reads Temperature (K), Band gap energy, Eg (eV), Effective density of states in conduction band, Nc (cm^-3), and Effective density of states in valence band, Nv (cm^-3). If the method is Silicon empirical, it reads Temperature (K), Silicon empirical constant A1 (cm^-3 K^-3/2), and Silicon empirical constant A2 (K). Hidden inputs are not used.

Worked mini-example

Using the general method for silicon at 300 K with Eg = 1.12 eV, Nc = 2.82 x 10^19 cm^-3, and Nv = 1.83 x 10^19 cm^-3:

sqrt(Nc x Nv) = sqrt(2.82 x 10^19 x 1.83 x 10^19) approx 2.272 x 10^19

Eg / (2 x kB x T) = 1.12 / (2 x 8.617333262145 x 10^-5 x 300) approx 21.66

Using the unrounded exponent, exp(-Eg / (2 x kB x T)) approx 3.912 x 10^-10

n_i approx 2.272 x 10^19 x 3.912 x 10^-10 approx 8.887 x 10^9 cm^-3

n_i(m^-3) approx 8.887 x 10^15 m^-3

This matches the expected size of the calculator result for the same inputs.

Interpretation

A higher n_i means more thermally generated electrons and holes are available in the pure semiconductor, so intrinsic conductivity tends to be higher [1]. Very strong temperature increases can raise n_i quickly [4].

Assumptions behind the result

The result assumes intrinsic material, thermal equilibrium, and consistent units. In the general method, accuracy depends on whether your chosen Eg, Nc, and Nv are appropriate for the temperature you entered. In the silicon empirical method, accuracy depends on how well the fitted constants match the silicon model you want to use.