Question

A semiconducting material has a band gap of 1 eV. Acceptor impurities are doped into it which create acceptor levels 1 meV above the valence band. Assume that the transition from one energy level to the other is almost forbidden if kT is less than 1/50 of the energy gap. Also if kT is more than twice the gap, the upper levels have maximum population. The temperature of the semiconductor is increased from 0 K. The concentration of the holes increases with temperature and after a certain temperature it becomes approximately constant. As the temperature is further increased, the hole concentration again starts increasing at a certain temperature. Find the order of the temperature range in which the hole concentration remains approximately constant.

Answer 1

Given:
Band gap = 1 eV
After doping,
Position of acceptor levels = 1 meV above the valence band
Net band gap after doping = (1 − 10⁻³) eV = 0.999 eV
According to the question,
Any transition from one energy level to the other is almost forbidden if kT is less than 1/50 of the energy gap.
$$\begin{gathered} \Rightarrow k{T}_1=\frac{0.999}{50} \\ \Rightarrow T_1=\frac{0.999}{50\times 8.62\times {10}^{-5}} \\ \Rightarrow T_1=231.78\approx 232.8\mathrm{K} \end{gathered}$$

_{$T_1$} is the temperature below which no transition is possible.
If kT is more than twice the gap, the upper levels have maximum population; that is, no more transitions are possible.
For the maximum limit,
$$\begin{gathered} K{T}_2=2\times {10}^{-3} \\ \Rightarrow T_2=\frac{2\times {10}^{-3}}{8.62\times {10}^{-5}} \\ \Rightarrow T_2=\frac{2}{8.62}\times {10}^2=23.2\mathrm{K} \end{gathered}$$
$T_2$ is the temperature above which no transition is possible.
∴ Temperature range = 23.2−231.8