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Introduction

In the present calculations, we also use a TB model to describe the interactions between carbon and hydrogen (H) atoms [118]. The present model is made self-consistent by the inclusion of local charge neutrality, and has been shown to be transferable to environments outside those included in the parameter fitting, like hydrocarbon molecules, diamond surfaces and amorphous diamondlike carbons [118,119]. We have paid particular attention to the Brillouin-zone sampling for convergence of the supercell calculations [120,121,104]. We use a small supercell of 64+1 atoms and a larger one of 216+1 atoms, and different sets of special k-points. For H at high symmetry sites (Bond Center (BC) and tetrahedral (T$_{\rm d}$) sites), convergence in the energy is already achieved with 2 special k-points and the small supercell. For H at the ET site (to be discussed below), we use 10 special k-points for accuracy with the small supercell. The calculations with the supercell of 216+1 atoms and 2 special k-points yield quantitatively the same results as those with the small one, justifying the use of the latter.

We first consider, like others [77,78,122,79], the BC and T$_{\rm d}$interstitial sites for H in diamond. The other possible interstitial sites (see figure (11.1)) are sites of lower energy, or even of local maximum energy, as found by us and by others [77,78,122]. All the atoms of the samples are allowed to relax by the conjugate gradient algorithm. We initially use the tight binding parametrization of Ref. [118]. We find, as found by the others, that H at the BC site is lower in energy than at the T$_{\rm d}$ site, however, by only 0.5 eV, a value smaller than the wide range of estimates obtained by others [77,78,122,79]. Nevertheless, our results are in full agreement with those of Ref. [122], regarding the structure of the atoms in the vicinity of the H atom at the BC site. In particular, we calculate the C-H bond length to be 1.17 Å[123] and the C-C bond to increase by 52 %, in full agreement with Ref. [122]. The nearest neighbor atoms of H at a T$_{\rm d}$ site are found to relax by $\sim$ 0.1 Å, also in very good agreement with Ref. [122]. It should be noted that these lengths are slightly shorter than those found by others [77,78]. We speculate that this is probably because in Refs. [77,78], local relaxation was applied to small H-terminated clusters, while in our work and in that of Ref. [122], full relaxation was applied to supercells.


  
Figure: Location of selected interstitial sites for H in the unrelaxed diamond lattice. Bright spheres are C atoms. The BC site is in black, the T$_{\rm d}$ site in red, the hexagonal site in blue, the C site in green, and the M sites in magenta.
\begin{figure}\centerline{\epsfysize=9.0cm \epsfbox{interstis.ps}}
\end{figure}

Let us now consider the origin of the small energy difference found here between H at the BC site and H at the T$_{\rm d}$ site, and show that this has no noticeable effects on the motion and migration barriers of H in the crystal. The tight binding parameters for the C-H interaction [118] were obtained from fitting to properties of the CH4 molecule, as calculated by first principle models. In the case of H in silicon, the reference molecule is SiH4. However, to extend the tight binding model to H in crystalline silicon (c-Si), the hierarchy of energies for H at different high symmetry interstitial sites, as obtained from ab initio calculations, has to be reproduced. Boucher et. al. [124] found that this hierarchy for H in c-Si is not maintained with the tight binding parameters obtained from SiH4. To properly describe H in c-Si, the authors, therefore, suggested to decrease the strength of the Si-H repulsive potential obtained from SiH4, by using a multiplying factor $\chi \leq 1$. The optimum $\chi $ value found yielded the right hierarchy of energies for interstitial H in c-Si.

In our case, the tight binding parameters deduced from CH4 lead to the right hierarchy for interstitial H in diamond, however with different relative energies than the ab initio data (see above). We find that the agreement of the relative energies with ab initio results can be improved by reducing the C-H repulsive energy, in the same way as suggested by Boucher for the case of H in c-Si. In figure (11.2), the cohesive energy of diamond with H at the BC and T$_{\rm d}$ interstitial sites is shown, as a function of the multiplying factor $\chi $. It can be seen that the relative energy between H at BC and H at T$_{\rm d}$ can approach that obtained from other calculations (marked in the figure by arrows) by changing the value of $\chi $.


  
Figure: The cohesive energy of H in diamond at three different interstitial sites (BC, T$_{\rm d}$ and ET), as a function of the multiplying factor $\chi $ (see text). The energy difference between H at BC and H at T$_{\rm d}$, as calculated by others, are indicated. a: Ref. [77], b: Ref. [78], c: Ref. [122], and d: Ref. [79].
\begin{figure}\centerline{\epsfysize=9cm \epsfbox{fig2.ps}}
\end{figure}

In the rest of our investigation, the calculations of the energetics, the energy barrier and the motion of H in diamond are carried out with two different values for the multiplying factor: $\chi = 1$, which corresponds to the initial tight binding parametrization, and $\chi =
0.925$, which yields an energy difference of 1.7 eV between H at BC and H at T$_{\rm d}$ (see figure (11.2)), well in the range of Refs. [77,78,122,79], and close to the ab initio pseudopotential result of Ref. [78]. In both cases, the qualitative and quantitative results are similar. We thus present below only the results obtained from the initial parametrization ($\chi = 1$), unless stated otherwise.


next up previous contents
Next: A new lowest energy Up: Hydrogen in diamond Previous: Hydrogen in diamond
David Saada
2000-06-22