The amorphous carbon structure is generated by the same method used above for silicon where the same initial random configuration is used. This allows a direct comparison between the amorphous silicon and carbon structures. Figures 8.9 and 8.10 show the radial and angle distribution functions of the final atomic configuration of the sample.
Figure 8.9: Radial distribution function of amorphous carbon calculated at a density of 3.4g/cm .
Figure 8.10: Bond angle distribution function of amorphous carbon.
It should be noticed that, unlike both silicon simulations, the radial distribution function drops to zero after the first neighbour peak. This leads to an unambiguous method of locating bonded pairs of atoms: 1.85Å. Examination of the electronic charge density indicates that this definition of a bond is correct. Using this the coordination numbers and ring statistics can be found. These are summarised in Table 8.1.
Table 8.1: Structural data for the amorphous carbon simulation.
As can be seen from the coordination numbers there exists no atom to which a fifth nearest neighbour is bonded. On consideration of the case of BC8 carbon this is expected. It was found that carbon is unable to form highly distorted tetrahedral bonding, favouring instead multiple bonding to a single atom. In order for silicon to form a 5-fold coordinated atom it is necessary to form a wide range of bond angles (from about to was found in the two silicon samples for ). Although the chemistry of carbon allows it to form many bonding configurations, this one is unstable with respect to multiply covalent bonds.
Although the radial distribution function is a very useful quantity in determining averages for shells of neighbouring atoms it is not unambiguously related to the spatial distribution of the carbon atoms. The bond angle distribution function is also necessary to determine the types of bonding. The bond angle distribution function contains several interesting features. Other experimental and theoretical results indicate that amorphous carbon contains mainly four-fold coordinated bonded atoms. This is also evident here in the large peak at about . Averaging the bond angles subtended by all four-fold coordinated atoms gives an angle of . Also of note is the shoulder at indicating planar graphitic-like bonding is also present, although in a smaller amount. Averaging the bond angles of three-fold coordinated atoms gives which is slightly less than the expected for perfect -like bonding although the statistics are rather limited since only six sites are found. There is also an indication that the amorphous carbon may be forming some p-like bonding due to the peak in the distribution function at . A small peak also appears at . Such a small bond angle indicates the possibility of 3-fold rings exist in the sample. A ring counting calculation in fact confirms that there are two such 3-fold rings (Table 8.1).
There have been several studies on the structure of amorphous silicon and carbon using the method of reverse Monte Carlo simulations which fits trial atomic configurations to the experimental radial distribution function. This may not fully describe all the atomic bonding environments in view of the large number of possible bonding topologies evident in the bond angle distribution function. Unfortunately obtaining this three-body function experimentally proves to be extremely difficult.
The amorphous structure is found to contain only three and four-fold coordinated atoms. Example of their bonding topologies are illustrated in Figure 8.11.
Figure 8.11: Schematic diagram illustrating a typical bonding topology of a 3-fold and 4-fold coordinated carbon atom. The dashed lines indicate close, but unbonded neighbours. The 4-fold coordinated atom forms a slightly distorted tetrahedral structure. It is this type of structure which dominates the amorphous carbons structure at this density.
There is a large number of 4-fold coordinated sites (90.6%) formed from slightly distorted bonding. All of the remaining sites are found to be 3-fold coordinated, but are not all necessary bonded. Of particular interest is the 3-fold rings that are found in the structure (see Figure 8.12).
Figure 8.12: A 3-fold ring of carbon atoms found in the amorphous structure. On examination of the charge density it was found that the electronic structure within the ring is best described as a 3-centre orbital, rather than three simple covalent bonds. Also shown is a four fold ring of carbon atoms is formed from a ring of covalent bonds unlike the 3-fold ring.
Like the silicon interstitial configuration found in Chapter 6 which formed a three fold ring, the charge density for this configuration in amorphous carbon formed a three-centre bonding orbital. Such a feature will not be found on examination of a radial distribution function alone since the inter-atomic distances are close to the C-C bond length. To find the electronic band(s) which are associated with the three-centre orbital, the electronic charge density was constructed from each individual band. This unusual feature is found to be very stable with its eigenvalue lying 24eV below the highest occupied band. However, another localised bonding orbital was found be be associated with it whose eigenvalue showed that it occupied the most energetic band.