Previous theoretical models

There have been several thermal donor models proposed in the past, notably the O₂Si_i model [105], the O_3i model [249,211], and the OBS model [250].

As the initial concentration of TDs was proportional to [O_i]⁴, initial models concentrated on defects containing four O_i atoms. Kaiser et al proposed a SiO₄ tetrahedral structure [179]. This was believed to become inactive on addition of a fifth O_i, and the model was abandoned when it was discovered that there was a whole family of TDs.

Later work concentrated on trivalent oxygen in different complexes. The strongest contender here is O_3i with the central O atom in a `y-lid' trivalent structure (see Figure 9.13). This was originally proposed in 1983 [249], and has been modelled by several groups since [93,251,211]. Since this model features O on the C<sub>2</sub> axis it cannot account for TD3 or higher (ENDOR shows no O on the C<sub>2</sub> axis), however it could be TD1 or TD2. This was shown using first principle supercell methods to be 0.7 eV lower in energy than three neighbouring BC O<sub>i</sub> atoms [211]. This structure was also proposed as a core with two further flanking O<sub>i</sub> atoms, forming a 5O species [28] (see Figure 9.11). The central O atom becomes trivalent in response to the compression from the other oxygen atoms, and is the source of the electrical activity. However ENDOR excludes this as a model for TD3+ for the same reasons as the 3O `y-lid' species.

  

Figure: Schematic diagram of the `di-oxygen square' model for the thermal donor, as proposed in [252]. This structure was found to be lower energy than the di-y-lid structure.

To avoid the problem of O on the C₂ axis, a di-y-lid model was proposed by two groups [111,252] (see Figure 9.4), and later work suggested that the `y-lid' trivalent structure discussed above could transform into this with the addition of further O_i atoms [211]. CNDO/S calculations found this to be 1.5 eV unstable with respect to another alternative 4O species, the di-oxy-square [105] (see Figure 9.1), consisting of four trivalent oxygen atoms in two neighbouring squares. This model was also discarded since the donor wavefunction is concentrated on the square structures [252] in disagreement with that deduced by ENDOR. The di-ylid consisting of two oxygen atoms in y-lid configurations without the flanking O_i atoms was modelled, but this is extremely tensile and unstable with respect to other configurations such as the BC dimer [105].

  

Figure: Schematic diagram of the OSB model for the thermal donor [250]. The diagrams show the same defect along perpendicular directions. The bottom Si atom is a lattice atom displaced a long way from its original site.

Prior to the ENDOR result that all O lies on the same C_2v plane with no O on the C₂ axis, one of the more popular models was the OSB model, shown schematically in Figure 9.2 [250]. This assumes that oxygen aggregation and compression pushes a core lattice Si off its site to become a divalent bridging Si atom, which is the source of the donor activity. The donor can then become inactive through the ejection of this atom as an interstitial. The model is compressive along $\langle$110$\rangle$ and was abandoned in light of the new ENDOR data.

The model that has received strong support in the last few years is the Deák model, shown in Figure 9.3. This consists of two tri-valent oxygen atoms pinning a Si_i between them which is divalent [253,105]. The electrical activity comes from the trivalent O atoms, which donate electrons to the empty states on the Si_i.

  

Figure: Schematic diagram of the Deák / Snyder / Corbett thermal donor model containing two trivalent O_i atoms and a Si_i [253].

There are several reasons why the Deák model cannot be correct. One problem comes through the bistability of the early TDs, TD1 and TD2, which exhibit isomeric electrically inactive structures [254]. There seems no easy way of reconstructing the Deák model such that the Si_i core atom can become fully coordinated. It is possible for several Si_i to aggregate in such a way that all of the dangling bonds combine, but not with a single atom. Alternative structures with trivalent Si_i were proposed as the inactive forms, since the CNDO calculations gave its dangling bond state in the valence band [253]. However AIMPRO calculations of trivalent Si give this state deep in the band gap [255]. In addition, the vibrational mode agreement of the model with experiment is poor. The corrected calculated CNDO modes lie at 984 and 901 cm^-1, when calculated with AIMPRO we get modes at 823.5 and 805.4 cm^-1. Experimental modes for TD3 are observed at 999 and 728 cm^-1 so the agreement is poor. There are other subtleties, for example ENDOR shows only a single Si atom on the C₂ axis whereas this model has two; this has been explained in terms of low electron density on the Si_i[256].

In conjunction with this, Si_iO_i was proposed as the structure of TD1 which would rapidly migrate and trap an additional O_i. However Newman points out that Si_iO_i has been observed and is only stable to 50 $^\circ$C [257], with a binding energy of somewhere near 0.8 eV. Thus this is ruled out as a model for TD1.

Various other models have been proposed over the years. It was suggested that an O₂ molecule in a vacancy might have double donor character if its antibonding p$\pi^*$ state crossed the gap[258], but this was disproved with self-consistent local density Greens function calculations of Kelly[259].