Taking , and the cut-off radius, , to be 4 Å, we can now vary either the temperature or the anneal time, as shown in Table 6.5 (capture radius, , is normally taken to be the separation at which the interaction energy is equal to kT ).
Note that this result does not allow for dimer dissociation. It is useful to compare these figures to the equilibrium dimer concentration, using
where B is the binding energy, and N is the number of BC sites. Taking B=0.3 eV (see above), N= 1023cm-3, and [Oi] = 1018cm-3, we get the results shown in the third column of Table 6.5. These show the approximation of no dimer dissociation breaks down at higher temperatures; at 500C after a 1 hour anneal a zero dissociation approximation predicts a dimer concentration higher than the equilibrium value.
|Temp (C)||[O2i] (cm-3)||[O2i] (cm-3)||Time (mins)||[O2i] (cm-3)|
These results suggest that for a typical 450C anneal, after an hour we should expect over 1014 cm-3 dimers. This result assumes a uniform oxygen distribution throughout the material, but in practise this will probably not be the case. Even in quenched anneal experiments, at higher temperatures the oxygen is able to diffuse rapidly from one BC site to another. Treating this diffusion step using simple kinetics, where x is the distance between BC sites, we obtain hop rates as shown in Figure 6.10.
This suggests that the oxygen concentration need not be uniform. Any localised clustering will increase the dimer concentration figure calculated above, and thus experimental as-grown concentrations of cm-3 are not inconsistent with this analysis . This is also particularly important in heavily carbon doped material, where Cs exerts a long range strain field in the lattice and can attract Oi in this way. It has been observed that the oxygen concentration can vary by a factor of 1.6 over a given sample, independant of cutting direction.
This result gives us an initial pool of dimers to work with, which should be present in as-grown material in quantities similar to those observed for the defect responsible for the 1012 cm-1 mode. The rate of loss of [Oi] during annealing agrees with that expected for Oi-Oi interaction with the normal Oi diffusion constant . It therefore seems to be the case that the dominant Oi loss will occur through the formation of dimers, and no enhanced diffusion process for dimer formation needs to be invoked (such as trimer migration to Oi followed by decomposition into two dimers).