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Next: Comparison with experiment Up: Hydrogen in diamond Previous: Introduction

A new lowest energy site for hydrogen in diamond

As mentioned above, only common interstitial sites for H in diamond were investigated so far. Estreicher et. al. [77] suggested that for group IV elements, bond lengths and bond strength determine the stability of interstitial H in the crystal. The C-C bond is the strongest and shortest among group IV elements. Thus, in order to incorporate a H atom on a BC site in diamond, a larger relaxation of the C-C bond (52 % of the C-C diamond bond length, as found here), with a larger amount of energy is required, compared to other group IV crystals. It is therefore plausible that a site displaced from the C-C bond may be energetically favorable for H in diamond [125].

To search for such a site of lowest energy, we simulate an annealing process for a diamond sample, with a H atom initially located at the T$_{\rm d}$ site [126]. We start the simulated annealing by molecular dynamics (MD) calculations at 1200 K and then slowly lower the temperature down to 100 K, for 50 psec. We find that even at high temperatures, the system rapidly enters configurations where the H atom oscillates around equivalent sites (denoted ET) away from the BC and the T$_{\rm d}$ sites. Upon cooling, the amplitudes of oscillations of the atoms are reduced, however with no change in the equivalent configurations obtained at high temperature. The annealing process, as applied here, ensures that the final site of the H atom is a site of global minimum energy [126].

Next, we calculate the energy of the sample with a H positioned at such an ET site, by fully relaxing the supercell obtained after the annealing process. To ensure that the ET site is not a distorted version of the BC site, we also relax a sample initially in the ideal diamond lattice configuration, with H near the ET site. The same final configuration is obtained after relaxation, with exactly the same energy. We find this configuration to be lower in energy than that of H at the BC site by an amount of 1.4 eV. This newly discovered ET site is found here to be the lowest energy site for H in diamond.

At this stage, it is important to investigate the effect of the multiplicative factor, $\chi $, on this result. In figure (11.2), we show the cohesive energy of the sample with H at the ET site, as a function of $\chi $. It can be seen that the ET site remains the site of lowest energy for H in diamond, over the complete range of $\chi $investigated [127]. With $\chi =
0.925$ (which corresponds to an energy difference of 1.7 eV between H at BC and H at T$_{\rm d}$), the energy difference between H at the BC site and H at the ET site is still rather large, $\sim$ 1 eV. Thus, the ET site found here is not an artifact of the tight binding parametrization and does indeed correspond to the most favorable site for H in diamond.

This site is six fold degenerate with respect to the C-C bond, and located at a distance of $\sim$ 0.77 Å from an unrelaxed C-C bond, near the antibonding site. These six energetically equivalent sites are located around the C-C bond, at the corners of two equilateral triangles, which are rotated with respect to each other by 60$^{\rm o}$ around the same unrelaxed C-C bond, are perpendicular to it, and are at a distance of 1/3 of the bond length. In figure (11.3) we enumerate selected ET sites. Those numbered 1 to 6 surround one carbon-carbon bond (labeled A-B), while the sites 7 and 8 surround an adjacent bond.

In this configuration, the H atom is bonded to one C atom (say A-1 in figure (11.3)), with a bond length of 1.08 Å. The initial carbon-carbon bond (A-B in figure (11.3)) is broken, and relaxes by 43 %. These may be some of the reasons for the ET site to be favored over the BC site: it involves less C-C relaxation and the C-H bond length remains close to the average bond length in organic molecules. The outward relaxation of the C atom which is not bonded to the H atom (atom B in figure (11.3)) favors hybridization of sp2 character for the bonding to its three nearest neighbors, and the creation of one dangling bond. We calculate the distance between the H atom and the unpaired C atom (atom B in figure (11.3)) to be $\sim$ 1.8 Å.

Figure: Location of selected ET sites for H in the unrelaxed diamond lattice. Bright spheres are C atoms and red spheres are the positions of the ET sites. The triangles 1-2-3 and 4-5-6 shown in the figure are equilateral, rotated with respect to each other by 60$^{\rm o}$ around the same carbon-carbon bond (A-B), and perpendicular to it. The sites 7 and 8 surround an adjacent C-C bond.
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The presence of this dangling bond is confirmed when electronic energy levels are calculated within the TB model. The calculation of the energy difference between the highest occupied and the lowest unoccupied orbitals in pure diamond yields $\sim$ 5.8 eV, a value close to the measured energy gap in diamond of 5.48 eV [117], and $\sim$ 24.1 eV for the width of the valence-band, in excellent agreement with experiment (24.3 eV) [117]. When we compute (with 60 special k-points) the density of electronic states (DOS) for diamond with H located on the newly found stable ET site, and compare it with that of diamond (see figure (11.4)), a new state $\sim$ 0.5 eV above the middle of the energy gap emerges; it is caused by the dangling bond created by the relaxation and the breaking of the C-C bond [128].

Due to this dangling bond, there is a strong tendency for another H atom to pair with the second carbon atom (atom B in figure (11.3)). This C-H-H-C complex (one H at site 1 and another H at site 4 in figure (11.3), for example; see also figure (11.5)) is found by us to be lower in energy by 2.5 eV than the configuration proposed by P. Briddon et. al. [78], where one H is located at an antibonding site and the other at a BC site. As expected, we find that the bonding of the second H atom removes the dangling bond and eliminates the mid-gap state.

Figure 11.4: DOS of pure diamond (dashed line) and diamond with H at the ET site (solid line). The zero of the energies is taken at the bottom of the conduction band.
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Figure 11.5: Configuration of the C-H-H-C complex in the unrelaxed diamond lattice. Bright spheres are C atoms, red spheres are occupied ET sites, and white spheres are unoccupied ET sites.
\begin{figure}\centerline{\epsfysize=8.0cm \epsfbox{}}

next up previous contents
Next: Comparison with experiment Up: Hydrogen in diamond Previous: Introduction
David Saada