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 givesThe field points from n to p and has largest magnitude $e\alpha w^2/(8{\epsilon }_s)$ at $x=0$. The potential is cubic inside the layer.

For a linearly graded p-n junction, the signed p-side depletion charge per area is $Q_p=-e\alpha w^2/8$. Differentiating with respect to positive p-side voltage $V$, using the linearly graded junction depletion width, gives the positive differential junction capacitance per areaThus $C_A^{-3}$ is linear in $V$. If stored charge denotes the positive n-side magnitude $Q_n=-Q_p$, its derivative with respect to $V$ is negative; the positive capacitance is instead $-d{Q}_n/dV=d{Q}_n/d{V}_R$ for reverse-voltage magnitude $V_R=-V$. Neither definition uses the zero net charge of the whole depletion layer.

Integrating the electric field across a linearly graded p-n junction gives $V_b-V=e\alpha w^3/(12{\epsilon }_s)$, where positive $V$ means forward bias. Thus the depletion region width depends on the cube root of the remaining barrier voltage. Both the space-charge density and integration range grow with width; this differs from the square-root width law of an abrupt uniformly doped junction.