Two samples of amorphous silicon have been generated at slightly different densities using the same initial random configuration. The higher density sample will be referred to as system-I, the other being system-II. After relaxation, the final configurations are found to be rather different. Given in Figures 8.1 and 8.2 are the radial distribution functions, g(r), of each sample.
Figure 8.1: Radial distribution function of amorphous silicon calculated at a density of 2.6g/cm .
Figure 8.2: Radial distribution function of amorphous silicon calculated at a density of 2.3g/cm .
The radial distribution function of system-I is in excellent agreement to that of experiment[131, 135]. For comparison, an experimental radial distribution function for an amorphous silicon sample which has a density of 2.45g/cm is given in Figure 8.3.
Figure 8.3: Experimental radial distribution function of amorphous silicon as found from neutron diffraction measurements.
System-II differs slightly in that the second neighbour peak at 3.5-4Å is too low and slightly wider than experiment. Integration under this part of the curve (taken from 2.9Å to 4.3Å) however indicates that the average number of nearest neighbours per atom is similar in each case. The lower density of system-II allows for a wider spread in second neighbour distances than system-I.
In both cases the height and width of the first peak of the radial distribution functions are in agreement with experiment. This implies that the average coordination number is correct in both cases. To find the coordination number, the maximum length of a silicon bond must be known. Unfortunately, the first minimum in the radial distribution function does not go to zero showing that there is a continuous range of neighbour distances. For this reason a maximum bond length is chosen arbitrarily to be 2.55Å. This then defines all the bonds within the structure. Some structural details of systems I and II are summarised in Table .
The coordination number of each atom can now be calculated. It is found that most atoms are four-fold coordinated (70.3% for system-I and 81.3% for system-II) while only a few atoms are either 3 or 5 fold coordinated. In system-I a single two-fold coordinated site is found. Such a feature has not been included in reverse Monte Carlo studies of amorphous silicon. In figure 8.4 is a schematic diagram of some 2, 3, and 4 fold coordinated atoms found in system-I. The 4-fold coordinated site is typical of the atomic structure of most of the atoms in the sample. It consists of a distorted tetrahedral bonding arrangement with a more distant, but unbonded, 5 neighbour, somewhat similar to that found in the BC8 structure. In the case shown in Figure 8.4, the bond angles for each site are given in Table .
Figure 8.4: Schematic diagram showing a group of atoms in the amorphous silicon structure of various coordination numbers. The full lines show covalent bonds while the dashed lines indicate unbonded close neighbouring atoms. The bonded/unbonded nature of neighbouring atoms is determined by examining 3d charge density plots. The distances are shown in Å.
The average bond angle for each coordination number is also shown in Table . As expected, the mean angle for the 4-fold coordinated sites are approximately that of the perfect tetrahedral angle. It may have been expected that the mean bond angle at a 3-fold site to be similar to that of an graphitic-like region, but instead it is found that it is less than . This leads to the implication that 3-fold sites are tending to have p-like character (at an angle of ). The bonding topology of the 3-fold site resembles a triangular pyramid with a well defined bond to the three neighbours of a central atom. A non-bonding orbital is formed at the top of the pyramid indicating that the `defect' site still retains bonding characteristics.
A typical five fold site is shown in Figure 8.5.
Figure 8.5: A five-fold coordinated silicon atom. The solid lines show the five covalent bonds from the central atom. The bond lengths are given in Å.
On examination of the valence electron charge density it is found that there are no bonds formed between atoms more that 2.6Å distant (hence the choice of when calculating the coordination number). There exist several sites with 5 atoms much closer than this (there are no 6-fold sites found) and hence covalent bonding is expected to occur. On examination of the valence electron charge density around these sited, we find that the sites tend to be fully 3 or 4 fold coordinated where the remaining atoms are relatively close and form slightly weaker bonds. When a fifth atom is found in an otherwise tetrahedral configuration it tends to weaken the longer bonds further. Such a configuration is shown in Figure 8.5 where the two more distant atoms form much weaker bonds that the three closer ones. Also, it is found that usually one of the neighbours of a five-fold coordinated atom has only three neighbours. This suggests that the electrons for the fifth bond is `donated' from the undercoordinated atom. Note that this is similar to the five-fold configuration found in the Si-BC8 surface in Chapter 5.
The average coordination number for system-I is 4.03 while the slightly lower density structure of system-II is found to have a coordination number of 3.97. Most models of amorphous silicon, such as random networks, assume from experimental measurements that the structure is fully four-fold coordinated. The fact that both simulations have found a structure which is very close to those found in other calculations from an initial random packing indicates that a full annealing treatment may not be necessary. There have been previous ab initio calculations on amorphous silicon and germanium (for example, see [135, 130]) which have rapidly cooled the melt in order to form models of their amorphous structure which reduces the percentage of `wrongly' coordinated sites, but giving an average coordination number similar to that found here. This suggests that cooling from the melt followed by annealing may not be the most efficient method of obtaining a reasonable model of amorphous silicon since there results do not differ significantly from those given here.
The ring statistics are also given in Table for both systems. This can be compared to the BC8 and ST12 structures which contain a range of small ring sizes from 5 to 7 fold rings. There are no three fold rings (and therefore no three-centre bonding orbitals) found in either sample which is somewhat unexpected considering the low formation energy of the interstitial configuration found in Chapter 6.
Since the bond lengths found in amorphous silicon (and also in the BC8 and ST12 structures) are similar to that found in the diamond structure it is generally assumed that the energy associated with straining the bond angle away from the perfect tetrahedral value gives the main proportion of excess energy of the amorphous structure relative to that of diamond Si. This distortion of angles away from is illustrated in Figures 8.6 and 8.7 which show the bond angle distribution functions for systems I and II respectively.
Figure 8.6: Bond angle distribution function of amorphous silicon system-I.
Figure 8.7: Bond angle distribution function of amorphous silicon system-II.
As can be seen, a relatively small change in the size of the unit cell of the 64 atom simulation (a change of 3.5%) makes a rather large change to both the radial and bond angle distribution functions despite the same random starting configurations. The main features however seem to remain similar in both cases. There is a very large spread in bond angles centred about the maximum of . This spread is much more pronounced in system-II. This can also be seen in the radial distribution function for system-II where the second neighbour peak is lower and wider than the corresponding peak in system-I. A smaller peak exists in both cases at about indicating the possibility of a small number of planar -like bonding configurations. In fact, a search through all bonding topologies containing an angle of about shows that they belong to distorted tetrahedrally bonded structures. No such planar threefold atomic configurations were found.
Also, as found above, there is also a tendency for the bond angles to be smaller than which show p-like character. This is shown by a shoulder/peak at about in both cases.
Finally, a comparison to the BC8 and ST12 structures in silicon is given in Figure 8.8.
Figure 8.8: A comparison between the first five neighbour distances in BC8, ST12 and amorphous silicon. The points on the BC8 and ST12 graphs show the neighbour distances at several different compressions.
The average first five neighbour distances for the two amorphous silicon simulations are shown along side a similar plot silicon in the BC8 and ST12 structures. Also shown is a plot of these distances for a third simulation of amorphous silicon as a much reduced volume to emphasize this trend. The neighbour distances for BC8 and ST12 are shown over a wide range of pressures. Firstly, it should be noticed that the distance to the first four bonded neighbours remains relatively unchanged with respect to the (generally unbonded) fifth neighbour distance. It can be seen that this is increasingly true in the trend of BC8 ST12 amorphous as the structure becomes more disordered. In the similar plot for a highly compressed amorphous silicon simulation the trend in neighbour distances becomes linear. It should be noted, however, that this third simulation is done only to show this reduction in the trend of reducing the distance in the extreme case where experimental verification of this linear trend in neighbour distances may be infeasible.