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The supercell method has been applied to several point defects of the diamond structure. The lower defect formation energies found in the ab initio calculations relative to previously published results imply that large cells are necessary for this technique. This is also shown by the large relaxation undergone by the second shell of atoms surrounding the defect. Surprisingly, a three centre bonding orbital is found in a low symmetry interstitial configuration which is also associated with a rather low formation energy. Direct calculation of the energy eigenstates of the electrons finds that this bonding configuration introduces a state below the top valence band causing the band gap to become smaller. This is in contrast to the vacancy and the hexagonal interstitial where the defects were found to introduce new states in the band gap.

The ab initio calculations on a supercell of that size are compute intensive and therefore only allowed a simple atomic relaxation under the Hellmann-Feynman forces. In order to investigate the effects of temperature on point defects it was necessary to use the empirical potential described in Chapter 4. This allowed a complete analysis of the harmonic behaviour of the defects, although the unusual three-centre bonding orbital could not be examined in this manner due to the nature of the potential.

The supercell method has allowed the calculation of phonon densities of states and elastic constants for various point defect concentrations and compare the results by applying the same method to the perfect crystal. Knowledge of the complete densities of states also allowed for the determination of the change in thermodynamic properties of the crystal caused by the defects and for the comparison to the configurational properties. The supercell method has also allowed us to calculate the small change in defect formation energies with respect to temperature. This result could not be obtained with comparable computational effort by running a molecular dynamics simulation and collecting thermal averages due to the short timestep required in the simulation compared to the lowest frequency vibrations. The molecular dynamics results were only accurate enough to show that the formation energies were not significantly changed.

It was found that the covalent bond charge potential gives a reasonable description of many properties of point defects in silicon, being able to predict the relative stability of the defects and their configuration, although it is unable to correctly describe three-centre bonding orbitals or the Maroudas-Brown configuration. It inaccurately describes the values of some elastic coefficients because of the parameterisation, but does find that defect stiffening occurs in this tetrahedrally bonded model.

Analysis of the normal modes shows that some high frequency modes are highly localized at the defect, whereas the perfect crystal has no modes closely associated with any particular atom.

The free energy of defects is dominated by the internal energy contribution, and to a first approximation the entropic and vibrational effects can be ignored. Moreover the entropy is dominated by the harmonic contribution, and again to first order anharmonic effects can be ignored. Thus is was not possible to measure any anharmonic entropy effect in the free energy of formation of point defects.

Defects such as these are expected to be good models for possible local bonding configurations in amorphous silicon and therefore should be included in any such model. The surprising result that silicon forms a three centre orbital has not been included in any model so far. This type of defect would go unnoticed in experimental measurements of the radial distribution function of amorphous silicon since the atomic separations in this bond is similar to that of the normal Si-Si covalent distance.

This work has been made possible by applying standard lattice dynamics techniques on a large supercell using a massively parallel CM200 computer as described in Chapter 4. The diagonalizations take approximately 12 minutes and the matrix inversions 7 minutes on 2 tex2html_wrap_inline7242 processors. Only memory requirements prohibit larger systems to be considered here since the size of the dynamical matrices are of order (3N) tex2html_wrap_inline6878 . A possible method of increasing the size of system and hence the complexity of the defects that could be considered is to use a large cluster of atoms without periodic boundary conditions, so that the dynamical matrices become band matrices although this would add the extra complication of surface modes, it would reduce the memory requirements of the problem, scaling as N with a large prefactor, and allow use of an algorithm for diagonalising sparse matrices which is an order of N faster than the Jacobi method we used to diagonalize dense matrices. This method could also be of use by using a short-ranged potential where interactions larger than a given distance were assumed to be zero.

next up previous
Next: Amorphous Carbon and Silicon Up: Point Defects In Silicon Previous: Effect on Elasticity

Stewart Clark
Thu Oct 31 19:32:00 GMT 1996