Concept:Silicon's smaller atomic number (\( Z = 14 \) vs Ge \( Z = 32 \)) and smaller atomic radius create stronger covalent bonds, resulting in a larger energy band gap (\( 1.1\text{ eV} \) vs \( 0.7\text{ eV} \)) and a higher built-in potential barrier.
Formula:$$V_0 \approx \frac{E_g}{2q} + \frac{k T}{q}\ln\left(\dots\right) \implies V_0(\text{Si}) \approx 0.7\text{ V} > V_0(\text{Ge}) \approx 0.3\text{ V}$$
Solution:- Silicon's valence electrons are closer to the nucleus and more tightly bound than those in Germanium.
- This stronger bonding creates a larger forbidden gap (\( 1.12\text{ eV} \) for Si vs \( 0.67\text{ eV} \) for Ge).
- The larger bandgap reduces the intrinsic carrier concentration \( n_i \), which directly increases the contact barrier potential \( V_0 \).
Why other options are incorrect:- Option A: Both Silicon and Germanium are Group IV elements with four valence electrons.
- Option B: Silicon has a smaller atomic mass (28 vs 72.6) and a smaller lattice constant than Germanium.
- Option C: Both materials form covalent diamond cubic crystal structures.
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