Concept:Intrinsic carrier concentration depends exponentially on the bandgap: \( n_i \propto e^{-E_g / 2kT} \). Germanium's smaller bandgap (\( 0.67\text{ eV} \) vs \( 1.12\text{ eV} \) for Si) leads to a much higher intrinsic carrier density at room temperature.
Formula:$$n_i \approx \sqrt{N_c N_v} \exp\left(-\frac{E_g}{2 k T}\right) \implies n_i(\text{Ge}) \approx 2.5 \times 10^{13}\text{ cm}^{-3} \gg n_i(\text{Si}) \approx 1.5 \times 10^{10}\text{ cm}^{-3}$$
Solution:- At \( 300\text{ K} \), thermal energy (\( k T \approx 0.026\text{ eV} \)) more readily excites electrons across Germanium's smaller \( 0.7\text{ eV} \) bandgap.
- This produces roughly \( 1000 \) times more intrinsic electron-hole pairs in Germanium than in Silicon, resulting in higher intrinsic conductivity.
Why other options are incorrect:- Option A: Both Germanium and Silicon form covalent diamond cubic lattices.
- Option B: Both elements are in Group IV and have four valence electrons.
- Option D: Germanium is a stable element and does not undergo spontaneous fission.
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