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 Incoming links: Linearly graded junction depletion width

For a linearly graded p-n junction, the signed p-side depletion charge per area is

Q_{p} = - e α w^{2} /8

. Differentiating with respect to positive p-side voltage

V

C_{A} = \frac{d Q _{p}}{d V} = \frac{ε _{s}}{w} = [\frac{e α ε _{s}^{2}}{12 ( V _{b} - V )}]^{1/3} .

Thus

C_{A}^{- 3}

is linear in

V

. If stored charge denotes the positive n-side magnitude

Q_{n} = - Q_{p}

V

is negative; the positive capacitance is instead

- d Q_{n} / d V = 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.

V_{b} - V_{ext} = - \int_{- w /2}^{w /2} E_{x} (x) d x = \frac{e α}{2 ε _{s}} \int_{- w /2}^{w /2} (\frac{w ^{2}}{4} - x^{2}) d x = \frac{e α w ^{3}}{12 ε _{s}} .

Consequently

w (V_{ext}) = [\frac{12 ϵ _{0} ϵ ( V _{b} - V _{ext} )}{e α}]^{1/3}, w (0) = (\frac{12 ϵ _{0} ϵ V _{b}}{e α})^{1/3} .

The linearly graded junction depletion width has this cube-root dependence because of the linearly growing space charge: the charge on one side scales as

w^{2}

w^{2}

, and its potential integral as

w^{3}

. The formula assumes

V_{b} - V_{ext} > 0

and that the depletion approximation remains applicable.