Concept:The intrinsic carrier concentration \( n_i \) depends on the effective density of states in the conduction and valence bands (\( N_c, N_v \)) and varies exponentially with the bandgap divided by \( 2kT \).
Formula:$$n_i(T) = \sqrt{N_c N_v} \exp\left(-\frac{E_g}{2 k T}\right) = A \cdot T^{3/2} \exp\left(-\frac{E_g}{2 k T}\right)$$
Solution:- Setting \( n = p = n_i \) and applying Fermi-Dirac statistics yields \( n_i = \sqrt{N_c N_v} e^{-E_g / (2kT)} \).
- This shows that intrinsic carrier concentration increases exponentially with temperature and decreases exponentially with wider bandgaps.
Why other options are incorrect:- Option A: \( n_i^2 \propto T^3 e^{-E_g / kT} \) gives \( n_i^2 \), not \( n_i \).
- Option C: The temperature dependence is exponential, not inversely proportional.
- Option D: Adding densities of states to an energy term is dimensionally invalid.
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