# 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: 