The band gap in germanium is  $\Delta E=0.68\text{eV}$. Assuming that the number of hole – electron pairs is proportional to  $e^{-\frac{\Delta E}{2kT}}$, find the percentage increase in the number of charge carriers in pure germanium as the temperature is increased from 300 K to 320 K

The number of charge carriers in an intrinsic semiconductor is double the number of hole – electron pairs (No of holes = No of electrons). If $N_1$ be the number of charge carriers at temperature   $T_1$and  $N_2$ at   $T_2$, we have 

$\begin{array}{l}{N}_1=N_0{e}^{-∆E/2k{T}_1}\\ \end{array}$                                                

$N_2=N_0{e}^{-∆E/2k{T}_2}$

The percentage increase as the temperature is raised from  $T_1$ to $T_2$ is

$f=\frac{N_2-N_1}{N_1}\times 100=[\frac{N_2}{N_1}-1]100$                                              

$=100[e^{\frac{\Delta E}{2k}[\frac{1}{T_1}-\frac{1}{T_2}]}-1]$

$Now; \frac{\Delta E}{2k}[\frac{1}{T_1}-\frac{1}{T_2}]$

$\begin{array}{l}=\frac{0.68\times 1.6\times {10}^{-19}}{2\times 1.38\times {10}^{-23}}[\frac{1}{300}-\frac{1}{320}]\\ =0.82\end{array}$

$Thus, f=100\times [e^{0.82}-1]\simeq 127\left(\because e^{0.82}=2.27\right)$

 $\therefore$The number of charge carriers increases by  about 127 %.