The problems incurred with first principles calculations on self-interstitials are similar to that of the vacancy. There have been many reports on the interstitial site and bonding configurations (see, for example references [96, 98, 99, 101, 102, 103]). The results vary depending on the method used (supercell vs. Green's function techniques), and with which basis set is used. The Green's function methods allow for only local atomic relaxation around the defect (up to approximately 3-4Å) with the further out regions treated as a macroscopically averaged region. In the supercell method, relatively small cells have also been used and are therefore not able to take longer range relaxation into account, for example Bar-Yam el al describe various self interstitial configurations with a supercell of only 8 and 16 atoms. Results were later obtained by Chadi for defects in GaAs and Si using supercells of 32 atoms.
Most calculations have been on interstitials which are located at points of high symmetry, although their are many more possible interstitial sites within the diamond structure. This is explicitly shown in the following calculations. An interstitial was introduced into an otherwise perfect crystal of diamond silicon. A supercell of 65 atoms was used (a unit cell plus one interstitial atom). The additional atom was placed at the hexagonal symmetry point as shown in Figure 7.4 and the structure allowed to relax under the forces calculated by the Hellmann-Feynman theorem. This configuration will be referred to as . Since the initial configuration of the defect was highly symmetric, the molecular dynamics method did not break this symmetry and the final configuration was that of the hexagonal interstitial. An similar simulation was also done where the starting point of the interstitial atom was offset slightly from the hexagonal symmetry point in order to investigate whether a breaking of this symmetry would result in the interstitial moving to give a different defect configuration (referred to as system ).
Figure 7.4: Schematic diagram of a 9 atom cell containing a hexagonal interstitial ( ) showing the initial and final configurations of the simulation.
It is found that on relaxation, the two similar starting configurations result in quite different bonding topologies, but have similar defect formation energies of 2.25eV for and 2.12eV for . As expected, remained in the highly symmetric hexagonal site but was found to have a lower formation energy than previously reported calculations[96, 95]. This is due to the converged plane wave basis set, which had an energy cutoff of 250eV, and the larger supercell size used here, whereas other calculations have not allowed for longer range atomic relaxation. A schematic representation of the final bonding configuration of the defect is given in Figure 7.4. The first shell of atoms surrounding the interstitial atom relax outwards leaving all atoms four fold coordinated. This creates two four-fold and two seven-fold bonded rings of atoms around the additional atom in the structure. This severely buckles the hexagon in which interstitial is located in order to sustain the four-fold coordination of all atoms. The bond lengths to the four atoms surrounding the interstitial are 2.33Å, 2.35Å, 2.57Åand 2.60Å. The next shell of atoms consist of the two atoms of the hexagon that the interstitial did not bond to. It is found that they relax outwards slightly to 2.8Å. This is slightly larger than previously reported results which did not include relaxation of more distant atoms. Here, the next shell of atoms are found at a distance of 3.3Å to 4.0Å from the defect (compared to the perfect second neighbour distance in diamond silicon of 3.84Å).
On consideration of the electronic structure (see Figure 7.5), similar to that of the vacancy, the tails of the valence and conduction bands extend slightly into the band gap closing it by 0.05eV. There are two electronic states found in the gap - one is very close to the top valence band at an energy of only +0.007eV, the other slightly further in to the gap at +0.03eV.
Figure 7.5: Schematic diagram of the electronic states near the Fermi level for Si-diamond containing a hexagonal interstitial.
The interstitial configuration is found to have a relaxed structure quite different from that of . A three dimensional `ball and stick' representation of the configuration is given in figure 7.6
Figure 7.6: The final relaxed atomic configuration of the self interstitial . In this figure the bonds are created by choosing the four nearest neighbours of each atom. It is found that this does not make any unreciprocated neighbouring atoms.
The most significant feature of this defect is that it unusually forms a three-fold ring of silicon atoms. There are no four or five-fold rings formed although all atoms remain fully four-fold coordinated. In order to investigate the bonding topology of this feature, the valence electron charge density in the plane of the ring is plotted in Figure 7.7.
Figure 7.7: The electronic charge density of the plane of the three-fold ring of silicon atoms of the interstitial configuration.
This is the plane of the supercell. The figure shows that, rather than the configuration being a three-fold ring of covalent bonds, which is generally accepted to be unstable in silicon structures, a three centre bonding orbital is formed. On consideration of the defect formation energy in comparison to other calculations[95, 96, 104, 105, 106, 107] on neutral self interstitials, this three centre orbital is found to be more stable that the highly symmetric hexagonal or tetrahedral interstitials where the configuration is found to have a defect formation energy of 2.12eV. This is a similar result to the empirical calculations of Maroudas and Brown where they also find that a low symmetry configuration, the extended self interstitial, is also a low energy defect.
The low formation energy of the interstitial is also reflected in the band structure. As with the other point defects considered here, the band gap becomes smaller as the conduction and valence bands tail off in to the gap. In this case it is found decrease by half to 0.21eV as illustrated in Figure 7.8.
Figure 7.8: Electronic structure in the region of the band gap in Si-diamond containing a low symmetry interstitial.
The highest point state loses the 3-fold degeneracy to form three states of energy , -0.046eV and -0.061eV. These states are higher in energy than the 3-fold degenerate state found in the perfect structure, hence the band gap is reduced. This is because the defect introduces a lower energy doubly degenerate state at -0.603eV, repelling the higher energy states into the gap. Note that the defect adds new states to the electronic structure which are of a lower energy than the perfect triply degenerate point state. This low energy state is in contrast to the vacancy and configurations which are found to have defect states within the band gap.
The low energy of defects such as this also imply that they could be typical of local configurations found in more complex forms of Group IV elements such as amorphous silicon. This will be discussed in more detail in Chapter 7 where the supercell method is used to model the amorphous structure of Si and C.