Intrinsic carrier concentration

The intrinsic carrier concentration $n_i$ is the number of free electrons (equivalently, of holes) per unit volume in pure, undoped semiconductor at thermal equilibrium. In an Intrinsic semiconductor electrons and holes appear in pairs, so $n=p=n_i$, and $n_i$ sets the baseline that all doped material still obeys through the Mass-action law.

The formula

$n_i=B{T}^{3/2}\exp \left(-\frac{E_g}{2kT}\right)$

where:

• $T$ is the absolute temperature in kelvin.
• $E_g$ is the Bandgap energy (for silicon, $E_g\approx 1.12\text{eV}$).
• $k$ is the Boltzmann constant, $8.617\times 10^{-5}\text{eV/K}$ (equivalently $1.381\times 10^{-23}\text{J/K}$).
• $B$ is a material-dependent constant; for silicon $B=7.3\times 10^{15}{\text{cm}}^{-3}{\text{K}}^{-3/2}$.

The structure of the expression carries the physics. The exponential Boltzmann factor $\exp (-E_g/2kT)$ is the probability that thermal energy is enough to break a bond and create a pair — it is the dominant, ferociously temperature-sensitive term. The $T^{3/2}$ prefactor comes from how many quantum states are available for the carriers to occupy and varies slowly by comparison. The exponent uses $E_g/2$ rather than $E_g$ because each generation event makes a pair, so the activation energy is effectively shared between the electron and the hole.

Worked example: silicon at room temperature

Plug in the canonical values: $T=300\text{K}$, $E_g=1.12\text{eV}$, $B=7.3\times 10^{15}{\text{cm}}^{-3}{\text{K}}^{-3/2}$.

First the thermal energy: $kT=(8.617\times 10^{-5}\text{eV/K})(300\text{K})\approx 0.02585\text{eV}$.

The exponent:

$-\frac{E_g}{2kT}=-\frac{1.12}{2(0.02585)}=-\frac{1.12}{0.0517}\approx -21.66$

so the Boltzmann factor is $\exp (-21.66)\approx 3.9\times 10^{-10}$.

The prefactor: $T^{3/2}=300^{1.5}\approx 5196$, so $B{T}^{3/2}\approx (7.3\times 10^{15})(5196)\approx 3.8\times 10^{19}{\text{cm}}^{-3}$.

Multiply:

$n_i\approx (3.8\times 10^{19})(3.9\times 10^{-10})\approx 1.5\times 10^{10}{\text{cm}}^{-3}$

which is the canonical room-temperature value for silicon.

[Background from general knowledge, not the source PDF]

How small that really is

Silicon has roughly $5\times 10^{22}$ atoms per cm³. With $n_i\approx 1.5\times 10^{10}{\text{cm}}^{-3}$, only about one atom in $3\times 10^{12}$ has donated a free carrier. That is why intrinsic silicon barely conducts — and why even a modest dose of Doping (donor or acceptor atoms at, say, $10^{16}{\text{cm}}^{-3}$) swamps the intrinsic carriers by a factor of a million and completely dominates the conductivity.

Temperature sensitivity

Because $E_g/2kT$ sits inside an exponential, $n_i$ depends enormously on temperature: warming silicon from $300\text{K}$ to about $400\text{K}$ raises $n_i$ by several orders of magnitude. This is why semiconductor devices are temperature-sensitive — leakage currents, which are carried by carriers proportional to $n_i^2$, climb steeply with heat. The product $np$ at any temperature is fixed at $n_i^2$ by the Mass-action law, and the same $kT$ scaling reappears as the Thermal voltage $V_T=kT/q$ in junction equations.

$n_i$ depends strongly on temperature through the Boltzmann factor $\exp (-E_g/2kT)$.