Method

The calculations presented here were performed using an ab initio local density functional cluster method on large H-terminated clusters (AIMPRO). A C atom was inserted into the central region of a 71 atom tetrahedral cluster Si H to investigate the structure of C . The migration energy of C was found by inserting a C atom into a trigonal 86 atom cluster Si H , oriented along 111 . A central Si atom was then replaced by C to investigate the di-carbon defect. Norm-conserving pseudo-potentials for C and Si were taken from [20].

The electronic wave functions were expanded in s- and p- Gaussian orbitals, centered on nuclei and the center of all bonds. The basis used here for the structural and local mode calculations was as following: eight Gaussian fitting functions of different exponents were used on the carbon atom(s) and inner silicon atoms, and a fixed linear combination of eight Gaussians were placed on the remaining Si atoms. A linear combination of three Gaussians were used for the terminating hydrogens. Additionally, three s- and p- Gaussian orbitals were placed at the center of every C-Si and Si-Si bond.

The electron charge density was fitted in exactly the same way as the wavefunctions. The self-consistent energy was then found, and the forces on each atom calculated. The whole structure is then relaxed using a conjugate gradient algorithm, until the equilibrium structure is determined. To explore the saddle points needed for migration energies, the relative bond lengths of the carbon atom were constrained during the relaxation as discussed previously [21]. The second derivatives of the energy with respect to atomic positions were calculated for the carbon atom(s) and their neighboring Si atoms. Energy second derivatives for the remaining atoms were taken from the Musgrave-Pople potential found previously [22], and the dynamical matrix of the cluster then constructed. From the dynamical matrix, the vibrational modes, and their normal coordinates can be calculated.

We have previously shown that the dipole moments for small molecules can be found accurately using this method [23], and this can be used to evaluate the effective charge associated with the vibrational modes of the defect. However, a larger basis is required for the accurate determination of so in this calculation the number of Gaussians used to fit the wavefunction and charge density of C and Si was increased to ten. It is the effective charge which controls the intensity ratio of the two IR absorption lines. In this work, is calculated for the two highest local vibrational modes of C , having and symmetries. These modes largely involve motion of the carbon atom along [001] and [110] respectively together with movements of the Si neighbors. is found by calculating the change in the adiabatic dipole moment of the cluster, when the atoms of the cluster are displaced by / . Here is the mass of the displaced atom.