Temperature effect on carrier concentration

The carrier concentration in a semiconductor depends strongly on temperature because carriers are created thermally, either by ionising impurities or by breaking covalent bonds across the band gap.

Intrinsic semiconductor

Electron-hole pairs are created when thermal energy lifts electrons across $E_g$:

$$n_i = \sqrt{N_cN_v}\,\exp\left(-\frac{E_g}{2kT}\right), \quad N_c, N_v \propto T^{3/2}$$

so $n_i \propto T^{3/2}e^{-E_g/2kT}$. The exponential dominates: for Si, $n_i$ roughly doubles for every 8 to 10 K rise near room temperature.

Extrinsic (n-type) semiconductor

A plot of $\ln n$ against $1/T$ shows three regions:

1. Low temperature (ionisation or freeze-out region): only some donors are ionised.

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

2. Medium temperature (extrinsic or saturation region): all donors are ionised, $n \approx N_d$, nearly constant. This range (about 150 K to 500 K for Si with $N_d = 10^{15}$ cm$^{-3}$) is where devices work. It begins at the saturation temperature $T_s$, where the donors become fully ionised.

3. High temperature (intrinsic region): thermal generation across $E_g$ exceeds $N_d$; $n \approx n_i$ and the material behaves as intrinsic. The changeover temperature $T_i$ is where $n_i \approx N_d$.

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

Fermi level

As $T$ rises, $E_F$ moves from near $E_d$ (low $T$) towards mid-gap (high $T$), because the extrinsic carriers become a smaller fraction of the total.

The upper limit of device operation is set by $T_i$; wide band gap materials (GaAs, SiC) have higher $T_i$ and can work at higher temperatures.