Then, we fix the defect state at a specific position in the band gap, say ED. For each position E in the band tail we can calculate an equilibrium density of defect states that we can create out of the available number of states g(E) at this particular energy; this will cost us exactly the defect formation energy U which is given by (ED - E). Close to the band edge, we have many weak bond states available, but the defect creation energy is high. Moving further into the band gap, the defect formation energy becomes more favourable, but there are less weak bonds to start with. The total number of weak bonds is obviously obtained by integration over the whole band tail.
This integral is a hard one. There is some similarity to integrals which include a Fermi-Dirac distribution which makes it tempting to use the Sommerfeld approximation procedure. However, this approach is not valid once the accompanying function contains exponentials like in the case above. Also, we want to look at defect creation at high temperatures, not around absolute zero. Street proposes a different approximation to solve this integral (see text around eq. 26 in [Street-1989prb]) which is illustrated below: The density of band tail states is illustrated by the full black line, the "probability" to create a defect by the red line. These have to be multiplied and integrated. If we neglect the one in the denominator similar to a Boltzmann approximation; this appears reasonable for the energy range up to ED, but above this approximation explodes as illustrated by the dashed line going up. Therefore, split the integral into a part below ED where we apply the Boltzmann-like approximation, above thie energy we quite simply approximate by unity.
Illustration of the approximation in the defect integral. The Fermi-Dirac like term (red) is approximated by a Boltzmann like term up to the defect energy ED, then by unity.
We'd like to compare that to some experimental data. The figures below illustrate two choices for the set of parameters against the freeze-in equilibrium defect densities of the previous section and for a collection where the defect densities and the Urbach parameter E0 have been measured.
Applying the model to the data shown in the previous section [Street-1989prb] (upper panel) and to a collection of data on defect densities with respect to their measured Urbach tails parameters E0 [Stutzmann-1989pm] (lower panel).