= Linearly graded p-n junction
A <linearly graded p-n junction> has net ionized doping $N_D-N_A=\alpha x$ near the metallurgical junction. In the <depletion approximation>, $\rho=e\alpha x$. With vanishing external field, neutrality places the depletion edges at $\pm w/2$ and integration of <Poisson equation> gives
$$
E_x(x)=\frac{e\alpha}{2\varepsilon_s}\left(x^2-\frac{w^2}{4}\right).
$$
The field points from n to p and has largest magnitude $e\alpha w^2/(8\varepsilon_s)$ at $x=0$. The potential is cubic inside the layer.
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