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Hydrogenated Diamond Surfaces.
Carbon (100) and (111) facets
are commonly observed during the growth of diamondlike films via
chemical vapour deposition, and have for this
reason been the subject of increasing attention in recent
years. For C(100), atomically smooth surfaces can be grown, and
so the study of this surface is in some ways more attractive.
Moreover, it is unique amongst the lowindex
surfaces in that it possesses two dangling bonds per surface
carbon atom, leading to some particularly interesting
surface chemistry. Chemical vapour deposition occurs from a
hydrogen atom / hydrocarbon mixture, and hydrogenation of the
carbon surface is an intrinsic part of the process. An understanding
of the hydrogenated surface is therefore of particular importance.
A quantum mechanical description
is deemed necessary because the extended electronic states of
the carbon substrate lead to
manybody forces, not easily describable by classical
potentials. The advantage of a tightbinding (TB) formulation
is that, in exchange for a limited
amount of parameterisation, one can simulate larger samples
of the chosen system
than is at present possible with ab initio methods.
We present results for several hydrogenated C(100) surfaces
based on a new TB parameterisation.
Parameterisations.
In the simplest tightbinding formulation, the total energy
of the system is written as a band energy
term plus a sum over repulsive pair potentials.
The band energy term equals the sum of
occupied eigenvalues obtained from a minimal basis
TB Hamiltonian. Harrison suggested a universal
parameterisation for spbonded materials, with the transfer
integrals t(r) of the TB Hamiltonian varying with
the interatomic distance r as (r_0/r)**2,
and the pair potentials phi(r) as (r_0/r)**4.
Although Harrison's parameterisation
gives good results for the diamond structures of C and Si
close to equilibrium,
its transferability to other structures is poor. Consequently,
Goodwin, Skinner and Pettifor for Si, and later Goodwin for C,
proposed a modified scaling behaviour which gave improved
transferability:
s(r) = s(r_0) * (r_0 / r)^n * exp [ n ( (r / r_c)^n_c + (r_0 / r_c)^n_c ) ]
with s = t or phi. Following the GoodwinSkinnerPettifor
approach, Wang and Mak have derived CC and CH parameters
suitable for hydrocarbons in a wide range of bonding situations.
Their parameter set is, however, relatively large.
Adopting the original CC parameters of
Goodwin, we have derived new parameters for CH and HH interactions,
such that the total parameter set is limited in size
("Model 1"). We believe that many
properties of C/H systems should follow from the form of
the spbonding, and not be dependent on extensive
parameterisation.
Xu, Wang, Chan and Ho developed
a similar TB model for carbon, but replaced the pairwise form for
the repulsive energy by the embeddedatomlike form:
E_rep = \sum_i f (\sum_j \phi(r_{ij}) )
where the function f is a 4th order polynomial.
Davidson and Pickett have extended this parameterisation
to C/H systems, and given some results for hydrogenated
carbon surfaces (using 2 C atoms per layer, kpoint
sampling, and steepest descent minimisation). They include a term
E_U = U \sum_i (q_i  q_i0)^2
in the energy to reduce charge transfer. We give results also
for this model ("Model 2").
Methods.

Geometry:
Slab geometry, periodic boundary
conditions in xy plane, lower and upper surfaces
studied, 6  16 C layers with 8  16 C atoms per layer,
0  2 H atoms per surface C atom.

Zerotemperature structure:
Conjugategradient (CG)
minimisation used to locate minimum energy atomic configuration.
Energy and forces calculated exactly at each step, i.e.
TB Hamiltonian is diagonalised exactly and HellmannFeynman
forces calculated. Different initial atomic configurations
used. This method is fast (typically 3040 cycles needed),
but can get trapped in local minimum (e.g. C(2x1):1.5H case,
see below).

Finitetemperature:
TightBinding Molecular Dynamics (TBMD)
performed. Exact HellmannFeynman forces calculated at
each timestep, and configuration propagated with 4th order
Gear predictorcorrector algorithm. To date,
TBMD has been used to test whether minima found by CG are absolute
or local. When the modelling of the ground state properties
are satisfactorily concluded, the aim is to use TBMD
to study finiteT properties.
Results.
Tables IVI compare the predicted geometries from CG
minimisation for Models (1) and (2), with the ab intitio
calculations of Furthmü ller et al, Europhys. Lett.
28, 659 (1994) and Yang et al, Phys. Rev. B
48, 5261 (1993). Representative geometries are shown
in Figs. 16.
[ Tables IVI ].

For the bare C surface, and for low H coverage,
C atoms in the upper layer dimerise, given (2x1) symmetry.
For bare C surface, dimer length is similar to typical double
bond length (1.38 Å ). On hydrogenation, dimer bond length
increases.

Only minimal buckling of the C dimers
is observed (insignificant in comparison with that observed
for Si), in agreement with most other studies.

On performing a CG minimisation for
C(2x1):1.5H and Model (1), two possible structures were
found, namely (i) dimerised (Table IV and Figure 3) and
(ii) bridging (Table V and Figure 4). The bridging configuration
has been observed in previous TB studies. On running a TBMD
simulation at 600K, the bridging configuration was found
to be stable, whereas the dimers dissociated.

Because of steric effects, the HCH angle
is always less than the ideal tetrahedral angle of 109.4.
The angle decreases with increasing H coverage.
Problems and future direction

The C(100)(1x1) to C(100)(2x1) reconstruction
energy obtained from Model (1) reproduces ab initio predictions
remarkably well. The adsorption energy for H is, however, severely
overestimated for Models (1) and (2), due to a large value of
the E(CH) bond energy.

The largest excess charge predicted by Model~(1)
for the bare C surface is only 0.08e, despite the fact that
Model (1) incorporates no Coulomb terms. For the hydrogenated
surfaces, however, the excess charge on each H atom is typically
0.4e, suggesting that a Coulomb term should be included.
(Surprisingly, however, Model (2) which does include a Coulomb term
predicts comparable charge transfer).

Once the above problems are dealt with, via
a refined parameterisation and a treatment of Coulomb effects,
we intend to use TBMD to study dynamic processes at the C(100)
surface. In particular, desorption and adsorption processes
are of great interest.
m.d.winn@dl.ac.uk
Last modified: Mon Jun 30 12:21:06 BST 1997