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Question
The conductivity of a pure semiconductor is roughly proportional to T3/2 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?
Solution
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
\[\sigma = T^{3/2} e -^\Delta E/2\text{kT }\]
\[ \therefore \frac{\sigma_2}{\sigma_1} = \left( \frac{T_2}{T_1} \right)^{3/2} \frac{e^{- \Delta E_2 /2k T_2}}{e^{- \Delta E_1 /2k T_1}}\]
\[ \Rightarrow \frac{\sigma_2}{\sigma_1} = \left( \frac{300}{4} \right)^{3/2} \frac{e^{- 0 . 67/(2 \times 8 . 62 \times {10}^{- 5} \times 300)}}{e^{- 0 . 74/(2 \times 8 . 62 \times {10}^{- 5} \times 4)}}\]
\[ \Rightarrow \frac{\sigma_2}{\sigma_1}={10}^{463}\]
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