# Basic Equations

## Poisson's Equation

$\frac{dE}{dx}=\frac{\rho }{\epsilon }=\frac{q}{\epsilon }\left(p\left(x\right)-n\left(x\right)-{N}_{A}^{-}+{N}_{D}^{+}\right)$

where E is the electric field, ρ is the charge density and ε is the material permittivity. This equation gives the basic relationship between charge and electric field strength. In semiconductors we divide the charge up into four components: hole density, p, electron density, n, acceptor atom density, NA and donor atom density, ND.

For the ideal diode derivation NA is assumed constant in the p-region and zero in the n-region. Similarly, ND is assumed constant in the n-region and zero in the p-region.

## Transport Equations

where Jn is the electron current density, μn is the electron mobility and Dn is the electron diffusivity. Similarly, Jp is the hole current density, μp is the hole mobility and Dp is the hole diffusivity. q is the electronic charge and E is the electric field. Note that in this section we use E, and interchangeably for the electric field. Its a mistake and we should be consistent and stick to E.

The transport equations describe how carriers move, i.e. the flow of carriers or current. Its easier to use the current density, J, in A/cm2 rather than the absolute current, I, in A since we are not concerned with the area of the device. I = J × Area and for a 1 cm2 device J and I are equal.

The first term in each equation is for drift and the second term is for diffusion.

### Continuity Equations

The continuity equation keeps track of all the carriers in terms of movement, generation and recombination. They are sometimes termed 'book keeping' equations since they make sure that every carrier is accounted for.

General Conditions

$$\frac{d n}{d t}=\frac{1}{q} \frac{d J_{n}}{d x}-(U-G)$$

$$\frac{d p}{d t}=-\frac{1}{q} \frac{d J_{p}}{d x}-(U-G)$$

where U is the recombinaton rate and G is the generation rate.

Solar cells operate in steady state and we are not concerned with transients or switching times. Under thermal equilibrium and steady state conditions the carrier concentrations do not change with time so that:

$\frac{dn}{dt}=\frac{dp}{dt}=0$

Rearranging the equations above leads to:

$$\frac{1}{q} \frac{d J_{n}}{d x}=U-G$$

$$\frac{1}{q} \frac{d J_{p}}{d x}=-(U-G)$$

### Summary

We now have the five basic equations to solve:

## Poisson's Equation

$\frac{dE}{dx}=\frac{\rho }{\epsilon }=\frac{q}{\epsilon }\left(p\left(x\right)-n\left(x\right)-{N}_{A}^{-}+{N}_{D}^{+}\right)$

## Transport Equations

$$\frac{1}{q} \frac{d J_{n}}{d x}=U-G \quad \frac{1}{q} \frac{d J_{p}}{d x}=-(U-G)$$

The equations are readily solved using numerical approaches1, and there are many device simulators that perform this task. By making a few approximations it is also possible to solve the equations closed form as outlined on the following pages.