Question

The conductivity of an intrinsic semiconductor depends on temperature as σ = σ₀ e^−ΔE/2kT, where σ₀ 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

Let the conductivity at temperature T₁ be ${\sigma }_1$ and the conductivity at temperature T be ${\sigma }_2$.
Given:
$T_1=300\mathrm{K}$
Band gap, E = 0.650 eV
Now,
According to the question,
$\sigma ={\sigma }_0\mathrm{e}{-}^{\frac{\Delta E}{2KT}}$
${\sigma }_2=2{\sigma }_1$
$$\begin{gathered} \Rightarrow {\sigma }_0{\mathrm{e}}^{\frac{-\mathrm{\Delta E}}{2kT}}=2\times {\sigma }_0{\mathrm{e}}^{\frac{-\mathrm{\Delta }E}{2\times k\times T_1}} \\ \Rightarrow {\sigma }_0{\mathrm{e}}^{\frac{-\mathrm{\Delta E}}{2kT}}=2\times {\sigma }_0{\mathrm{e}}^{\frac{-\mathrm{\Delta }E}{2\times k\times 300}} \\ \Rightarrow {\mathrm{e}}^{\frac{-0.650}{2\times 8.62\times {10}^{-5}\times T}}=2\times {\mathrm{e}}^{\frac{-0.650}{2\times 8.62\times {10}^{-5}\times 300}} \\ \Rightarrow {\mathrm{e}}^{\frac{-0.650}{2\times 8.62\times {10}^{-5}\times T}}=6.96561\times {10}^{-6} \end{gathered}$$

On taking natural natural log on both sides, we get

$$\begin{gathered} \frac{-0.650}{2\times 8.62\times {10}^{-5}\times T}=-11.874525 \\ \Rightarrow \frac{1}{T}=\frac{11.874525\times 2\times 8.62\times {10}^{-5}}{0.65} \\ \Rightarrow T=317.51178\approx 318\mathrm{K} \end{gathered}$$