Question

The conductivity of an intrinsic semiconductor depends on temperature as $\sigma ={\sigma }_0{e}^{-\Delta E/2KT},$ where ${\sigma }_0$ is a constant. Find the temperature at which the conductivity of an intrinsic germanium semiconductor will be double of its value at T = 300 K. Assume that the gap for germanium is 0.650 eV and remains constant as the temperature is increased.

Answer 1

Since $\sigma ={\sigma }_0{e}^{-\frac{\Delta E}{2KT}}$

Given $\Delta E=0.650eV,T=300K$

$K=8.62\times {10}^{-5}eV$

According to question,

${\sigma }_0{e}^{-\frac{\Delta E}{2KT}}=2\times {\sigma }_0{e}^{\frac{-\Delta E}{2\times K\times 300}}$

$e^{\frac{-0.6250}{2\times 8.62\times {10}^{-5}\times T}}=e^{\frac{-0.650}{2\times 8.62\times {10}^{-5}\times 300}}$

$e^{\frac{-0.650}{2\times 8.62\times {10}^{-5}\times T}}=6.96561\times {10}^{-6}$

Taking 'ln' on both sides.

We get, $\frac{-0.650}{2\times 8.62\times {10}^{-5}\times 7^{\prime }}=-11.874525$

$\Rightarrow \frac{1}{T^{\prime }}=\frac{11.874525\times 2\times 8.62\times {10}^{-5}}{0.65}$

$\Rightarrow T^{\prime }=317.51178=318K$