 Linearly graded junction depletion width

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.