Intrinsic concentration and its temperature dependence

Intrinsic concentration $n_i$ is the number of electrons per unit volume in the conduction band (equal to the number of holes in the valence band) of a pure semiconductor at temperature $T$.

Electron concentration in the conduction band

The number of electrons is the density of states $g_c(E)$ times the probability of occupancy $f(E)$, integrated over the band:

$$n = \int_{E_c}^{\infty} g_c(E)\, f(E)\, dE, \qquad g_c(E) = \frac{8\sqrt{2}\,\pi\, m_e^{*3/2}}{h^3}\sqrt{E-E_c}$$

For $E - E_F \gg kT$ (non-degenerate case), $f(E) \approx \exp[-(E-E_F)/kT]$ (Boltzmann approximation). Integrating gives

$$n = N_c \exp\left[-\frac{E_c - E_F}{kT}\right], \qquad N_c = 2\left(\frac{2\pi m_e^* kT}{h^2}\right)^{3/2}$$

where $N_c$ is the effective density of states at the conduction band edge.

Hole concentration in the valence band

In the same way, using $1 - f(E)$ as the probability that a state is empty:

$$p = N_v \exp\left[-\frac{E_F - E_v}{kT}\right], \qquad N_v = 2\left(\frac{2\pi m_h^* kT}{h^2}\right)^{3/2}$$

Intrinsic concentration (mass action law)

Multiplying $n$ and $p$, the Fermi level cancels:

$$\begin{aligned} np &= N_c N_v \exp\left[-\frac{E_c - E_v}{kT}\right] = N_c N_v \exp\left(-\frac{E_g}{kT}\right) \\ \text{For intrinsic: } n = p = n_i \;&\Rightarrow\; n_i^2 = N_c N_v e^{-E_g/kT} \\ n_i &= \sqrt{N_c N_v}\, \exp\left(-\frac{E_g}{2kT}\right) \end{aligned}$$

Since $np = n_i^2$ does not depend on $E_F$, it holds for doped semiconductors too (mass action law). Setting $n = p$ also gives the intrinsic Fermi level $E_{Fi} = \frac{E_c+E_v}{2} + \frac{3}{4}kT\ln\left(\frac{m_h^*}{m_e^*}\right)$, which is close to mid-gap.

Effect of temperature

As $N_c, N_v \propto T^{3/2}$,

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

• The exponential term dominates, so $n_i$ rises very rapidly with $T$. For Si ($E_g$ = 1.1 eV), $n_i$ is about $1.45\times10^{10}$ cm⁻³ at 300 K and rises by roughly 2 times for every 8–10 K rise near room temperature.
• A plot of $\ln n_i$ against $1/T$ is nearly a straight line with slope $-E_g/2k$; this is used to measure $E_g$.
• Larger band gap means smaller $n_i$ (Ge > Si > GaAs at the same $T$).
• Because conductivity $\sigma = n_i e(\mu_e + \mu_h)$, an intrinsic semiconductor's resistance falls sharply with temperature (negative temperature coefficient, used in thermistors).

 ln(ni)
   |\
   | \   slope = -Eg/2k
   |  \
   |   \     Ge
   |    \     \
   |     \ Si  \
   +------------------> 1/T

High $n_i$ at high temperature is also why doped devices stop working properly above a certain temperature: intrinsic carriers swamp the dopant carriers.