Depletion approximation

The depletion approximation replaces the junction's transition region by a region without mobile carriers, leaving only fixed ionized dopants. For a uniformly doped abrupt p-n junction,

ρ = - e N_{A}

on the p side,

ρ = + e N_{D}

on the n side and

ρ ≃ 0

outside. Charge neutrality gives

N_{A} x_{p} = N_{D} x_{n}

. Integrating Poisson equation gives the electric field and the electrostatic band bending.

Table of contents
- Abrupt-junction depletion width and capacitance Depletion approximation
- Charge and carrier density of an abrupt p-n junction Depletion approximation

Abrupt-junction depletion width and capacitance ( $W = 2 ε_{s} (V_{bi} - V) (N_{A} + N_{D}) / (e N_{A} N_{D})$ )

 0  0

For a planar abrupt p-n junction with ionized densities

N_{A}, N_{D}

, p-side width

w_{p}

and n-side width

w_{n}

, zero external field requires

N_{A} w_{p} = N_{D} w_{n}

. Integrating Poisson equation gives

V_{bi} - V = e (N_{A} w_{p}^{2} + N_{D} w_{n}^{2}) / (2 ε_{s})

, with positive

V

denoting forward bias. Therefore

W = w_{n} + w_{p}

has the displayed value,

w_{n} = N_{A} W / (N_{A} + N_{D})

and

w_{p} = N_{D} W / (N_{A} + N_{D})

. The positive depletion capacitance per area is

C_{A} = ε_{s} / W

. It equals

d Q_{p} / d V = - d Q_{n} / d V

when

Q_{p} = - e N_{A} w_{p}

and

Q_{n} = e N_{D} w_{n}

are signed space charges per area. Differentiating a positive charge magnitude with respect to forward voltage instead gives a negative derivative. The square-root law assumes partially depleted regions, negligible mobile charge and no substantial injected-carrier storage.

Charge and carrier density of an abrupt p-n junction

 0  0

For a zero-bias abrupt p-n junction, choose the p side at

x < 0

and the n side at

x > 0

. Ionized acceptors and donors remain fixed while mobile electrons and electron holes diffuse and establish a built-in electric field. In the depletion approximation, mobile carriers are neglected on

- x_{p} < x < x_{n}

and the net charge density is

ρ (x) = ⎩ ⎨ ⎧ - e N_{A}, + e N_{D}, 0, - x_{p} < x < 0, 0 < x < x_{n}, outside the depletion region .

(1)

Charge neutrality gives

N_{A} x_{p} = N_{D} x_{n}

. The quasi-neutral bulk can have a large mobile carrier density while its net charge density is nearly zero, because fixed dopant charge compensates it. With complete ionization and nondegenerate carriers, equilibrium also gives