There are a multitude of techniques available for studying oxygen in silicon, both experimentally and theoretically.
These include various spectroscopic techniques such as I.R. spectroscopy and Raman spectroscopy, which allow very accurate measurements to be made of the local vibrational modes of the defects. These can be used in conjunction with isotopic implantation, where different isotopes of oxygen can be implanted into the silicon. Since these have different masses their vibrational modes will shift, which helps in characterising the various vibrational modes observed, and assigning them to oxygen defects.
There are also techniques such as EPR and ENDOR, which look at the magnetic spin properties of the defect. This can in theory allow very accurate measurement of the symmetry properties of the defect.
Theoretical investigations vary greatly in size and complexity depending on the method used and the approximations made. We use density functional theory, which allows us to model the system in terms of the charge density of the electrons rather than their wavefunctions. It is an ab initio (first principles) method, i.e. in theory we need only to input the atomic masses of the atoms involved in order to obtain our results. We start with a reasonable guess at the final structure, and then calculate all the forces on the atoms. The atoms are moved according to these forces, and this step is repeated until the forces are zero, i.e. the structure is at equilibrium. A sample animation of this process is available; this is an oxygen interstitial pair inside a large hydrogen terminated silicon crystal. NB: The animation is in xmol format; if you simply get a list of atom coordinates then your viewer needs to be configured differently, there is information available on how to do this.
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