Concept:Increasing temperature increases the intrinsic carrier concentration \( n_i \), which reduces the built-in potential barrier across the junction.
Formula:$$\frac{dV_B}{dT} \approx -2\text{ mV/}^\circ\text{C} \quad (\text{for Silicon and Germanium})$$
Solution:- The built-in potential is given by \( V_B = \frac{k_B T}{e} \ln\left(\frac{N_A N_D}{n_i^2}\right) \).
- Because \( n_i^2 \propto T^3 e^{-E_g / k_B T} \) increases rapidly with temperature, the logarithmic term decreases faster than the linear \( T \) factor increases.
- As a result, the barrier potential decreases by approximately \( 2\text{ mV} \) for each \( 1^\circ\text{C} \) increase in temperature (\( -2\text{ mV/}^\circ\text{C} \)).
Why other options are incorrect:- Option A: The barrier potential has a negative temperature coefficient; it decreases rather than increasing.
- Option C: While dopant concentration is constant, intrinsic carrier generation increases with temperature, altering the barrier potential.
- Option D: The barrier potential decreases approximately linearly over standard operating temperature ranges.
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