Step 1: Solve for properties in depletion region

As in most devices, the solution for the electrostatic properties in the depletion region does not change, and so is given here.

Step 2: Solve for carrier concentration and current in quasi-neutral regions

Find U and G

We will set G equal to zero and in the n-type material (in p-type material ).

Find general solution

Using low injection recombination and constant generation gives the equation:

Note that

since (where p0 is a constant) so the derivative (and second
derivative) of Δp(x) is the same as the derivative of p(x). In addition for simplicity, we introduce a variable
change using: .

The overall differential equation now becomes:

or , which has a general solution:

For electrons (p-type material), the differential equations and solutions are:

and

Particular solution for wide base diode

We need two boundary conditions these are:

(1) At the edge of the depletion region,

(2) There is a finite surface recombination at the surface, such that

at x=0,

Rearranging gives

Plugging A back in gives:

or

The equation for electrons in p-type material, Δn(x′), can be similarly derived as:

This is plotted below for G=0.

Differentiating and plugging into equation for current gives:

Making the change from x to x′ gives

Find total current

The change in the current across the depletion region is:

Assuming that there is no generation and recombination, then ΔJn = 0 and

This case is shown in the graph below.

Note: We need to change the sign of Jn since it is an electron current, and standard definition is a hole
current.

If there is a constant generation across the depletion region, then , where xn is the depletion
width in the p-type material and xn +xp = W.

Jn at the edge of the depletion region in the p-type material is:

Jn at the edge of the depletion region in the n-type material is:

An analogous equations exists for Jp, and the total current is:

Typically, we write the equation in the form:

or

where