# Hydrogenated Diamond Surfaces.

Carbon (100) and (111) facets are commonly observed during the growth of diamond-like 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 low-index 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 many-body forces, not easily describable by classical potentials. The advantage of a tight-binding (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 tight-binding 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 sp-bonded 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 Goodwin-Skinner-Pettifor approach, Wang and Mak have derived C-C and C-H parameters suitable for hydrocarbons in a wide range of bonding situations. Their parameter set is, however, relatively large. Adopting the original C-C parameters of Goodwin, we have derived new parameters for C-H and H-H 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 sp-bonding, 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 embedded-atom-like 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, k-point 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 x-y 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.

• Zero-temperature structure: Conjugate-gradient (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 Hellmann-Feynman forces calculated. Different initial atomic configurations used. This method is fast (typically 30-40 cycles needed), but can get trapped in local minimum (e.g. C(2x1):1.5H case, see below).

• Finite-temperature: Tight-Binding Molecular Dynamics (TBMD) performed. Exact Hellmann-Feynman forces calculated at each time-step, and configuration propagated with 4th order Gear predictor-corrector 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 finite-T properties.

# Results.

Tables I-VI 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. 1-6.

[ Tables I-VI ].

• 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 H-C-H 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(C-H) 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