Junctions between differently doped semiconductors are the base of
every electronic device. Standard cases of p-n junctions are quite profoundly treated in
most textbooks. Here we would like to address a rather special case which
is of paramount importance for thin film silicon solar cells - the p-i-n junction.
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 Na = Nd = 1018 cm-3
and an intrinsic carrier concentration ni = 1010 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.
When all layers are in contact, the Fermi levels of the doped regions must align on
the same height. This is shown by the horizontal line with symbols. Because there are
no charges in the i-layer, the potential variation is just linear between
the values that are fixed by the doped layers. The potential differece across the
i-layer is called built in potential Vbi. It is calculated
by subtracting from the band gap the values of the Fermi levels in the doped
regions with respect to their band edges:
qVbi = Eg - EF,n - EF,p
Taking the relation for the Fermi level in a doped semiconductor from the chapter on doping,
we finally find:
This yields a built in potental of about 900 meV, and in the doped layers the Fermi levels are
located about 450 mV from the respective band edges using a typical band gap of 1.8 eV
for amorphous silicon. Note that the term QFL in the legend is the abbreviatin of "Quasi Fermi Level".
The "quasi" is not necessary here, but it will be used later.
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 .
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:
This yields an exponential increase of the electron concentration n(x) from 102
cm-3 to 1018 cm-3, while at the same time
the hole concentraton decreases exponentially from 1018 cm-3 to
102 cm-3. The situation is illustrated in the figure below, where again the legend
anticipates some considerations which will follow below.
An important observation emerges from the figure above; along the horizontal axis the hole density is
higher than the electron density between 0 and 1/2, after that it is the other way round. We conclude that
in the first part holes are the majority carriers, after that they are minority carriers.
This is just a consequence of the mass action law, the product of n and p
must equal ni2 at all places throughout the junction. The
carrier densities of this situation are referred to equilibrium densities n0
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
Vbi to Vbi-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.
The continuity equation takes care of the following processes:
The result is a linear differential equation of second order with inhomogeneity. The solution
is quite straightforward, but we must still define the boundary conditions; first, the
carrier density at x=0.5 can be easily determined by using the reduced potential drop across the
junction. Second, the drift current must equal the diffusion current. We specify this condition
at x=0 because at a later stage it permits us to introduce recombination losses due to
surface recomination. The figure below shows the result for the case of a foward bias of 0.4 V.
Here we should add a few words on the different curves that are shown: the first thing
to calculate is the majority carrier concentration between 0 and 0.5, shown by the full red triangles.
This is just the exponential drop due to the linear variation of the potential
Vbi - V across the junction. This gives us the value for the bounary
condition at x=0.5. We continute by calculating the solution of the continuity equation which
gives the profile of the minority carrier concentration shown by the full blue squares. Then, we just
mirror the relations into the second half, as shown by the open symbols. There is one more curve shown
by a black line; this is the solution for the minority carriers between 0 and 0.5, but plotted beyond its
range of validity into the range between 0.5 and 1. In fact, the curve is hardly different from the
electron majority profile in this range, indicating that the the model is not that bad.
- Generation of carriers: normally this takes place by by the absorption of light
- Recombination of excess carriers: every deviation from the equilibrium distribution
n0 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
 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)