A major difficulty which is not easily overcome by the *ab initio*
and empirical methods used previously is the non-periodicity displayed
in amorphous materials. For this reason it is necessary to use the
supercell method again. Also, in order to get a reasonable range of
bonding configurations a large number of atoms per supercell are
required. Due to the compute intensive nature of *ab initio*
methods coupled with the high energy plane wave cut off required for
the carbon pseudopotential, a 64 atom supercell was the largest that
could be reasonably dealt with.

The initial starting configuration of atoms in the supercell was that
of randomly packed particles constrained by a minimum separation. This
is equivalent to a random packing of hard spheres. Both the carbon and
silicon simulations were started with the identical random
configuration. In order to make a comparison to the complex structures
studied in chapters 3 and 4, the density of the amorphous carbon was
set to 3.4g/cm which, experimentally, is found to form a
distorted diamond-like structure. The density chosen for amorphous
silicon was 2.6g/cm and also the slightly lower density of
2.3g/cm which is about the same density as the amorphous silicon
made from evaporated silicon samples which, again, is thought to
predominantly formed from distorted tetrahedral units. In the *ab
initio* calculations in Chapter 3, it was found that LDA tended to
underestimate the lattice parameter of structures by about 2%. The
density of the structures are therefore overestimated. For this reason
the density chosen above are higher than experiment by this amount.

Earlier calculations performed by modelling the atomic interactions by
the empirical Tersoff potential[118, 134] or a
tight-binding approach[121, 122] were computationally fast
enough to allow quenching from the melt to be performed in order to
construct the model amorphous materials. A similar approach has been
tried with a full *ab initio* calculation[117] with a
smaller supercell of 54 atoms (at the graphitic-like density of
2g/cm ), but this required the rather unrealistic cooling rate of
the bulk material of K/s.

In the calculations given here, the atoms are simply allowed to relax
by a conjugate gradients algorithm under the influence of the
Hellmann-Feynman forces - there is no finite temperature molecular
dynamics performed for annealing due to the long simulation times
required. This will only create a single structure in the large phase
space of possible configurations at the respective densities, but it
will give a reasonable range of bonding topologies and will highlight
the differences in the nature of amorphous silicon and diamond-like
amorphous carbon. Since the calculations are fully *ab initio*
there are no *a priori* bonding characteristics required unlike
other large empirical calculations that have been performed (where
normally either or bonding is assumed depending only on
the coordination number of the respective atom, and therefore ignores
the more unusual bonding topologies such as two-fold *sp* bonds or
three-centre orbitals). This will allow any unusual types of bonding to
be found such as the two or five fold coordination[125].

A FFT grid was required for the silicon using a plane wave cutoff of 250eV. The calculation for carbon required a using a cutoff of 408eV.

Thu Oct 31 19:32:00 GMT 1996