Temperature dependence of carrier concentration and mobility

In a doped semiconductor, conductivity $\sigma = e(n\mu_e + p\mu_h)$. Both factors, the carrier concentration and the drift mobility, change with temperature.

Carrier concentration vs temperature

For n-type doping $N_d$:

1. Ionisation region (low T): $n = \left(\tfrac{1}{2}N_cN_d\right)^{1/2}\exp(-\Delta E/2kT)$; $n$ rises as more donors ionise.
2. Extrinsic region: all donors ionised, $n \approx N_d$ (constant).
3. Intrinsic region (high T): $n \approx n_i \propto T^{3/2}\exp(-E_g/2kT)$; rises steeply.

 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)

Drift mobility vs temperature

Mobility $\mu = e\tau/m^*$ depends on the mean time between scattering events. Two scattering processes dominate:

1. Lattice (phonon) scattering: atomic vibrations grow with $T$, so $\tau$ falls:

$$\mu_L \propto T^{-3/2}$$

2. Ionised impurity scattering: a slow carrier is deflected more by charged dopant ions; faster (hotter) carriers are deflected less:

$$\mu_I \propto \frac{T^{3/2}}{N_I}$$

where $N_I$ is the ionised impurity concentration.

3. Matthiessen's rule combines them:

$$\frac{1}{\mu} = \frac{1}{\mu_L} + \frac{1}{\mu_I}$$

At low $T$, impurity scattering controls $\mu$ (rises with $T$); at high $T$, lattice scattering controls it (falls with $T$). So $\mu$ has a peak, and heavier doping lowers the curve and shifts the peak to higher $T$.

  log μ
    |         .-''-.          lightly doped
    |       /       `-.
    |     /    .--.    `-.
    |   /    /     `-.    `-. ~T^-3/2
    |  /   /  heavily  `-.
    | /  /    doped       `-.
    |/ / ~T^3/2
    +--------------------------- log T

Result

Conductivity follows mainly $n$: rises at low $T$, roughly flat (falling slightly with $\mu$) in the extrinsic range, then rises sharply in the intrinsic range. So both carrier concentration and mobility are strongly temperature dependent, and device behaviour must account for both.