The pin junction consists of three differently doped regions. As the name suggests,
there is an intrinsic or undoped layer sandwiched between a p- and an n-doped region.
Typically this kind of junction is fabricated from amorphous silicon with a band gap of
about 1.8 eV. For the sake of simplicity, let us assume assume the same doping in the
p- and in the n-layer. Using *N*_{a} = *N*_{d} = 10^{18} cm^{-3}
and an intrinsic carrier concentration n_{i} = 10^{10} cm^{-3},
we obtain the following band diagram. The p- and n-layers are represented by the flat bands
to the left and right, respectively, and the intrinsic layer extends between 0
and 1 on the horizontal axis.

Taking the relation for the Fermi level in a doped semiconductor from the chapter on doping, we finally find:

So far the junction was in equilibrium. In operation the pin junction is contacted to an external electric circuit, and once it is illuminated we can draw a current. We would like to understand what goes on inside the cell. First of all we have to describe the properties of the charge carriers because in the end they make up the current. The following considerations treat the problem according to an analytic model for pin junctions that was proposed by Taretto [1].

We start by calculating the equilibrium charge carrier profile; from the previous sections we know that the
introduction of charge carriers by doping can change the Fermi level with respect to the band edge. This
is obviously the case for the p- and n-layers. In the intrinsic layer, however, the Fermi level
is given, and we find that the valence band edge position linearly decreases with respect to the
flat Fermi level. For the conduction band it is just the other way round. With given band edge and
Fermi level, we can calculate the charge carrier profile throughout the i-layer by solving the above
equation for the carrier density which is now unkonwn. For a position *x* between 0 an 1 we find
the equilibrium profile of the carrier density:

The situation changes once we apply an electric bias to the cell. Under a forward bias
*V* the potential drop throughout the cell decreases from the built in potential
*V*_{bi} to *V*_{bi}-*V*. In the doped regions we have
free charges which can follow the field and eventually cancel it out. Thus, the bands will
stay flat like in metals, and the Fermi levels stay at their position inside the band gaps.
In the intrinsic layer the bands still connect linearly between the doped regions, alas
with a different slope. However, there emerges a problem with the Fermi level because
it assumes different values at the opposite ends of the layer, it is longer unique. We
can continue to use the convenient concept of the Fermi level if we split it into two separate relations, one
for majority carriers and one for and minority carriers. These are called **Quasi Fermi Levels**,
and generally it is assumed that the Fermi level for the majority carriers remains flat at a level
equal to the one in the adjacent doped region.

In the two halfes of the i-layer we can calculate the profiles of the two majority carrier densities just as above. The density of the minority carriers, however, must be determined by solving the continuity equation. The following is written down for electrons between 0 and 0.5, for holes it is just the same between 0.5 and 1.

- Generation of carriers: normally this takes place by by the absorption of light
- Recombination of excess carriers: every deviation from the equilibrium distribution
*n*_{0}is likely to recombine with a lifetime τ - Diffusion of carriers from places with high density towards places with lower density. According
to the Einstein relation, the product of the diffusion constant
*D*and the lifetime yields the square of the diffusion length*L*. - Drift of carriers with a mobility μ along an electric field
*F*

Having arrived at the carrier profiles across the junction, we would like to come back to the idea of the quasi Fermi level. We assumed them to be flat for the majority carriers, but we did not know their shape for the minority carriers. Knowing the minority profile and the linear variation of the band edges, we can easily calculate them. This is shown in the figure below, again for the case of 0.4 V forward bias; we observe that also for the minorities they stay essentially flat, but with a strong drop towards the contacts.

[1] K. Taretto, U. Rau, J. H. Werner, Appl. Phys. A, 77, 865-871 (2003) and K. Taretto, U. Rau, J. H. Werner, Appl. Phys. A, 86, 151 (2007)

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