# Example 2: General Solution for Narrow Base Diode

### Overview

1. Situation where the quasi-neutral region in the solar cell is small, and therefore there is no recombination.
2. The boundary conditions for the narrow base diode particular solution are:

(1)

(2)

### 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:

At the edge of the depletion region

The excess minority carrier concentration Dn must be zero at x = W, or Dn(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 xn is the depletion width in the p-type material.

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: