Past exam of the quantum condensed matter course of the University of Cambridge / 2021 / Section B / 6 / d / Solution

Let the neutral edges be $-x_p$ and $x_n$ and take the exterior electric field to vanish. Integrating $d{E}_x/dx=\rho /{\epsilon }_s=e\alpha x/{\epsilon }_s$ between the edges gives $x_n^2=x_p^2$, so $x_n=x_p=w/2$. This is also the condition of equal and opposite integrated depletion charges.

A second integration with $E_x(w/2)=0$ yieldsIt is negative inside the layer, pointing from n to p, and zero at both edges. Its largest magnitude occurs at the metallurgical junction:For reference, choosing $V(-w/2)=0$ gives the cubic electric potential