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Junctions

Junctions between differently doped semiconductors are the base of every microelectronic 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 pin 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. The band diagram of this kind of junction is shown below, the p- and n-layers are represented by the flat bands to the left and right, respectively. 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 V_bi. 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:

qV_bi = E_g - E_F,n - E_F,p

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

$V_{bi}= 2 \frac{kT}{q}\ln \frac{N_d}{n_i}$ In the shown diagram we assumed the same doping in the p- and in the n-layer for the sake of simplicity, using N_a = N_d = 10¹⁸ cm^-3 and an intrinsic carrier concentration n_i = 10¹⁰ cm^-3. 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 [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:

$n_0(x)=\frac{n_i^2}{N_d}e^{-\frac{qV_{bi}}{2kT}x}$ 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 n_i² at all places throughout the junction. The carrier densities of this situation are referred to equilibrium densities n₀ and p₀

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.

$G-R(x)+D\frac{d^2n}{dx^2}+\mu F \frac{dn}{dx}$ The continuity equation takes care of the following processes:

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 $R(x)=\frac{n(x)-n_0(x)}{\tau}$
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

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 V_bi - 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.

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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