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 a constant and U=0.

Find general solution

We still start out with the same equation derived from the continuity equations. However, in this case the recombination is zero, so the equation becomes:

The general solution is:

Particular solution for narrow base diode with high recombination at edges

We need boundary conditions and these are:

1. At the edge of the depletion region
   
   
   
   
2. The excess minority carrier concentration Dn must be zero at x = W, or D_n(x = W)=0.
   

The first boundary condition gives:

The second boundary conditions gives:

which simplifies to

Substituting these equations into the general solutions gives the equation for the carrier concentration:

The current is found by differentiating the carrier concentration:

Simplifying this gives:

Step 3: Find total current

The change in the current across the depletion region is given by the general equation:

If there is a constant generation across the depletion region and no recombination, then

, where x_n is the depletion width in the p-type material.

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

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

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