Question

The conductivity of a pure semiconductor is roughly proportional to T^3/2 e^−ΔE/2kT where ΔE is the band gap. The band gap for germanium is 0.74 eV at 4 K and 0.67 eV at 300 K. By what factor does the conductivity of pure germanium increase as the temperature is raised from 4 K to 300 K?

Answer 1

Here,
Conductivity at temperature T₁ = \[\sigma_1\]

Conductivity at temperature T₂ = \[\sigma_2\]

Given: \[T_1\]  = 4 K

\[T_2\]   = 300 K

Variation in conductivity with respect to the temperature and band gap of the material is given by

\[\sigma =  T^{3/2} e -^\Delta E/2\text{kT }\] 

\[ \therefore \frac{\sigma_2}{\sigma_1} =  \left( \frac{T_2}{T_1} \right)^{3/2} \frac{e^{- \Delta E_2 /2k T_2}}{e^{- \Delta E_1 /2k T_1}}\] 

\[ \Rightarrow \frac{\sigma_2}{\sigma_1} =  \left( \frac{300}{4} \right)^{3/2} \frac{e^{- 0 . 67/(2 \times 8 . 62 \times {10}^{- 5} \times 300)}}{e^{- 0 . 74/(2 \times 8 . 62 \times {10}^{- 5} \times 4)}}\] 

\[ \Rightarrow \frac{\sigma_2}{\sigma_1}={10}^{463}\]