Dimer Formation

  Given a uniform distribution of O_i throughout the sample, statistically there is a chance that some of these atoms will be close enough that they are able to form dimers through standard O_i diffusion. When the O_i concentration is very high, as in Cz-Si, the number of dimers able to form in this way may be appreciable. Taking the dimer concentration equation from [143], and assuming no breakup of the dimers, then initially:

$$\rm [O_{2i}] = \rm 8 \pi r_c D_{O_i} [O_i]^2 t$$

$$\rm D_{O_i} = \rm 0.13 e^{-2.53 /kT}$$

Taking $\rm [O_i] = 10^{18} cm^{-3}$, and the cut-off radius, $\rm r_c$, to be 4 Å, we can now vary either the temperature or the anneal time, as shown in Table 6.5 (capture radius, $\rm r_c$, is normally taken to be the separation at which the interaction energy is equal to kT [158]).

Note that this result does not allow for dimer dissociation. It is useful to compare these figures to the equilibrium dimer concentration, using

$$\rm [O_{2i}]_{equilibrium} = \frac{[O_i]^2}{N}e^{B/kT}$$

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 [O_i] = 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 500$^\circ$C after a 1 hour anneal a zero dissociation approximation predicts a dimer concentration higher than the equilibrium value.

 

$$^\circ$$ 2c1hr anneal Equilibrium

$$^\circ$$ [O_2i] (cm^-3) [O_2i] (cm^-3) Time (mins) [O_2i] (cm^-3)

$$\times10^{10} \times10^{16} \times10^{13}$$ 300 5

$$\times10^{11} \times10^{16} \times10^{14}$$ 320 10

$$\times10^{12} \times10^{16} \times10^{14}$$ 340 15

$$\times10^{12} \times10^{16} \times10^{14}$$ 360 20

$$\times10^{13} \times10^{16} \times10^{14}$$ 380 25

$$\times10^{13} \times10^{16} \times10^{14}$$ 400 30

$$\times10^{14} \times10^{16} \times10^{14}$$ 420 35

$$\times10^{14} \times10^{15} \times10^{14}$$ 440 40

$$\times10^{15} \times10^{15} \times10^{14}$$ 450 45

$$\times10^{15} \times10^{15} \times10^{14}$$ 460 50

$$\times10^{15} \times10^{15} \times10^{15}$$ 480 55

$$\times10^{16} \times10^{15} \times10^{15}$$ 500 60

These results suggest that for a typical 450$^\circ$C 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, $x=\sqrt{\rm Dt}$ where x is the distance between BC sites, we obtain hop rates as shown in Figure 6.10.

  

Figure 6.10: Time taken for a single O_i to hop from one bond centred site to another at varying temperature. Calculated using $x=\sqrt{\rm Dt}$where the distance between sites, x, comes from AIMPRO cluster calculations and is set to 3.639Å. $\rm D= 0.13 e^{-2.53 / kT}$.

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 $1.5\times10^{15}$cm^-3 are not inconsistent with this analysis [239]. This is also particularly important in heavily carbon doped material, where C_s exerts a long range strain field in the lattice and can attract O_i 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[159].

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 [O_i] during annealing agrees with that expected for O_i-O_i interaction with the normal O_i diffusion constant [82]. It therefore seems to be the case that the dominant O_i 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 O_i followed by decomposition into two dimers).