Altermatt Lecture:   The PV Principle


2.6:  Recombination


Recombination is when an electron drops from the conduction to the valence band.

We saw on the voltage production page, that the excited carriers recombine, say, within about 1 millisecond. This commonly happens via a state in the band gap, caused for example by an impurity atom or a crystal defect, as shown in the left figure above (but it may also happen directly without such a defect state involved, as will be discussed in the chapter "Intrinsic recombination").

Instead of looking at elecrons only, one may look at both electrons and holes by going from the top left to the top middle figure. Then, we can say that the defect state needs to capture both an electron and a hole. The probability that recombination happens is thus proportional to the density of holes times the density of electrons: ~pn.

All this implies that the recombination rate (the number of recombination events per time per volume) is limited by the less abundant carriers, i.e. by the minority carriers, see the top right figure.

Because the recombination rate is proportional to the pn product, knowing how V influences p and n helps us to know how V influences the recombination rate. In the following, let us look at how p and n depend on V.

On the voltage production page, it was said that the photo-excited carriers thermalize to the band edges. This is true for most of the carriers, but after thermalization, some carriers still have an energy larger than the band edge.

This is so even if no light shines on the cell, because the carriers are constantly activated to higher energies by thermal excitation: the higher the temperature, the more motion there is in the atoms of the Si crystal, the more energy from the atoms is spontaneously transferred to the electrons and holes. They thermalize again to the band edges, but are constantly excited again by thermal activation, so a steady-state energy distribution of the carriers sets in, which is shown in the figure below as red (for lectrons) and blue (for holes): the lighter the color, the smaller is the density of the carriers having the respective energy.

In fact, the density of carriers decreases exponentially with energy away from the band edges:

Equation for Boltzmann factor

This is called the Boltzmann factor, where T is the temperature and k is Bolzmann's constant (k = 8.61758 x 10–5 eV/K).

Band diagram

Figure 1: Band diagram when (a) V = 0 and (b) V > 0.


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