n-type carrier concentration versus temperature

In an n-type semiconductor (donor concentration $N_d$, donor level $\Delta E = E_c - E_d$ below the conduction band), the electron concentration changes with temperature through three distinct regions. A plot of $\ln n$ against $1/T$ shows them clearly.

 ln n
   |\
   | \  intrinsic
   |  \  slope -Eg/2k
   |   \
   |    \__________________  n = Nd
   |         extrinsic      \
   |                         \  ionisation
   |                          \  slope -ΔE/2k
   +----|------------------|------------ 1/T
      1/Ti               1/Ts
   (high T at left, low T at right)

  n
    |                          /
    |                         /  intrinsic
    |        ________________/
 Nd |      /   extrinsic (n = Nd)
    |     /
    |    / ionisation
    |___/________________________ T
       ~100K   Ts         Ti (~500K for Si)

1. Ionisation (freeze-out) region: low T

At very low $T$, electrons are bound to donors. As $T$ rises, donors ionise:

$$n = \left(\frac{1}{2}N_cN_d\right)^{1/2}\exp\left(-\frac{\Delta E}{2kT}\right)$$

Slope of $\ln n$ vs $1/T$ is $-\Delta E/2k$. $E_F$ lies between $E_d$ and $E_c$.

2. Extrinsic (saturation) region: medium T

Above $T_s$ almost all donors are ionised, and intrinsic generation is still small:

$$n \approx N_d$$

The concentration stays nearly constant. $E_F = E_c - kT\ln(N_c/N_d)$ moves slowly towards mid-gap. Devices are designed to operate here (about 150 K to 450 K for Si with $N_d = 10^{15}$ cm$^{-3}$).

3. Intrinsic region: high T

Above $T_i$, electrons excited across the band gap outnumber those from donors:

$$n \approx n_i = \sqrt{N_cN_v}\exp\left(-\frac{E_g}{2kT}\right)$$

Slope is $-E_g/2k$, much steeper. $E_F$ approaches $E_{Fi}$ (mid-gap) and $n \approx p$; the material loses its n-type character.

Fermi level vs temperature

  E
 Ec |-------------------------
    | EF near Ed
    |   \_
    |     `-._
 Ei |---------`-.__________ (EF -> Ei)
    |
 Ev |-------------------------  T

Summary: $n$ rises exponentially, flattens at $N_d$, then rises again exponentially at high $T$. $T_i$ increases with doping and with band gap.