Concept:Poisson's equation relates spatial changes in electrostatic potential to the local space-charge density \( \rho(x) \) within the depletion layer.
Formula:$$\frac{d^2 V}{d x^2} = -\frac{\rho(x)}{\epsilon_s} = -\frac{q}{\epsilon_s} \left[ p(x) - n(x) + N_D^+(x) - N_A^-(x) \right]$$
Solution:- In the depletion layer, free carrier densities are negligible (\( n, p \approx 0 \)).
- On the N-side: \( \frac{d^2 V}{d x^2} = -\frac{q N_D}{\epsilon_s} \).
- On the P-side: \( \frac{d^2 V}{d x^2} = +\frac{q N_A}{\epsilon_s} \).
- Integrating Poisson's equation yields the linear electric field profile and quadratic potential variation across the junction.
Why other options are incorrect:- Option A: \( d^2V/dx^2 = 0 \) is Laplace's equation for charge-free regions.
- Option B: First derivative gives the electric field (\( -dV/dx = E \)), not \( \rho \epsilon_s \).
- Option D: This inverts the permittivity and charge density terms.
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