Concept:At elevated temperatures, intrinsic thermal generation across the bandgap exceeds the fixed donor concentration (\( n_i \gg N_D \)), causing the semiconductor to behave intrinsically and shifting \( E_F \) back toward the middle of the bandgap.
Formula:$$E_F(T \to \text{high}) \longrightarrow E_i = \frac{E_c + E_v}{2} \quad (\text{as } n_i(T) \gg N_D)$$
Solution:- At moderate temperatures, \( n \approx N_D \) (extrinsic range) and \( E_F \) sits near the conduction band.
- At high temperatures, thermal electron-hole pair generation dominates (\( n_i(T) \gg N_D \)), so \( n \approx p \approx n_i \).
- As the material transitions to intrinsic behavior, \( E_F \) shifts downward toward the middle of the forbidden gap.
Why other options are incorrect:- Option A: \( E_F \) moves into the conduction band under heavy low-temperature doping (degeneracy), not at high temperatures.
- Option B: The Fermi level is dynamic and changes position with temperature.
- Option C: \( E_F \) shifts continuously; it does not drop discontinuously to zero.
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