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Defect charging

Apart from the fact that the creation of the netural defect D⁰ will cost an energy equal to E_D and put a single, unpaired electron into the defect state, we did not consider any further details of the charge transfer. By definition, the Fermi level will be equal to the defect energy E_D because the electron in the defect resides in a state above the valence band, thus it represents the highest occupied state.

This is a rather passive point of view. Branz was one of the first to suggest that the defects can actually acquire charges [Branz-1989sc, Branz-1989prb]. This happens as follows: Imagine that that we have some dopants around that move the Fermi level. If the Fermi level is lowered towards the valence band, it will become more and more unlikely that the electron can stay in a state above the Fermi level; eventually it will drop down to the Fermi level, resulting in a positive defect. We can estimate the energy that is required for the formation of the charged defect; first, it takes E_D to create the defect and put the electron into it (enrgy required, thus positive value). Then, the electron drops down to the Fermi level and the energy difference becomes available (we have to count this negatively). In total, this yieds E_D - (E_D-E_F) = E_F. Likewise, if the Fermi level is raised towards the conduction band, eventually a second electron will be dropped into the defect state although it may cost a little extra energy to overcome the electrostatic repulsion. The energy of the negative defect D^- is given by E_D to create the defect, we have to add the correlation energy U it takes to put two electrons into one state (both values are positive), and we can again subtract the energy between the Fermi energy and the defect level (note that we assumed the the Fermi level to be above the defect). In total this yields E_D + U - (E_F - E_D).

Altogether we obtain the following defect formation energies:

E(D⁺) = E_F

E(D⁰) = E_D

E(D^-) = 2 × E_D + U - E_F

The figure below illustrates how much energy we have to spend for the creation of the three different defect states when the Fermi level is varied throughout the band gap.

Energy required to create the various defects with respect to the Fermi level. Red, black and blue represent the positive, neutral and negative defect, respectively.

The diagram above shows that there are situations where the creation of a charged requires less energy that the creation of a neutral defect. In fact, whenever when the Fermi level moves out of midgap position, it becomes more favourable to create a charged defect rather than a neutral one. If we put these defect cretion energies into the approximate formula that was given above, we get the following the equilibrium defect densities.

Defect densities acording to the equilibrium model using the defect creation energies of previous diagram. Red, black, and blue represent positive, neutral and negative defects, respectively. The experimental data are from [Sreet-1981prb, Jackson-1982prb, Stutzmann-1987prb]

This is an important finding here: moving the Fermi level out of the midgap position results in the creation of charged defects. We can dope to certain extent, but then we create defects that capture exactly the free carriers that we wanted to create. The doping is self limiting! Just what we observed in a preceding chapter from Spear's data of 1976. Imagine that it took 15 years to develop a model that can explain this.

There is another important conclusion from this: Imagine you deposited a complete device with p-i-n junction. The last layer in such a device will be a doped one, and once it is in place, the bands throughout the undoped layer will align in order to accomodate the different Fermi levels at its two ends. In other words, the distance between valence band and Fermi level will vary linearly across the junction. Exactly the condtion for creating all sorts of charged defects! Close to the p-layer there will be many positive defects, in the centre some neutral ones, and towards the n-layer there will be a lot of negative defects. Imagine what this means for charge transport.

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Last updated August 4th, 2010