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

Motion of hydrogen in diamond

Next, we investigate the diffusion of H in diamond. First we qualitatively follow, for 50 psec, the trajectories that the H atom, initially placed at the T$_{\rm d}$ site, assumes at different temperatures, using MD calculations and visualization techniques [129,57] to trace the path of the H atom. Only the $\Gamma$ point is used to sample the Brillouin zone in these calculations. The equations of motion are solved by a velocity-Verlet algorithm, with a time step of 10-15 sec, and the temperature is controlled by a Gaussian thermostat. We find that up to a temperature of $\sim$ 400 K, the H atom vibrates around the initial T$_{\rm d}$ site. By $\sim$ 600 K, it already finds its way to the nearest ET site, and jumps between the three equivalent ET sites of the same triangle (sites 1,2 and 3 in figure (11.3), for example). Around 1200 K, the H atom begins to jump back and forth between the six ET sites around the same C-C bond. The trajectory of an H atom is depicted in figure (11.6) for 30 ps at 1200 K. Finally, at a temperature of $\sim$ 1700 K, it jumps to the nearest ET site of the neighboring bond (site 7 or 8 in figure (11.3), for example). At this stage, real diffusion sets in. It has to be noted that the temperatures mentioned above are not precise, but rather provide a hierarchy of the energetic barriers involved in the H motion in diamond. The present results can be compared with the experimental temperature of $\sim$450 K at which the Mu-to-Mu$^\star$ transition occurs in diamond [82].


  
Figure 11.6: Trajectory of the H atom at 1200 K, for 30 psec. The gray scale indicates the relative distance of H from ET sites (dark for short distances, bright for long distances). The C atoms are represented by spheres, in their equilibrium sites, for clarity. The lines join ET sites of the same equilateral triangle, to guide the eye.
\begin{figure}\centerline{\epsfysize=9.0cm \epsfbox{fig4.ps}}
\end{figure}

When a second H atom is added to the sample at an adjacent non-occupied ET site (as in our C-H-H-C complex mentioned above; see figure (11.5)), the two H atoms are found to jump between the ET sites of their respective equilateral triangle, even at 1700 K, and are unable to jump to another ET site of an adjacent bond, for a simulation time of 50 psec. The presence of the second neighboring H atom should therefore reduce the possibility of diffusion.

In order to quantify the migration process of H in diamond, we evaluate the potential barrier that separates two adjacent ET sites on neighboring bonds (sites labeled 6 and 7 in figure (11.3), for example), as it is the relevant barrier for actual diffusion. We notice that the jump of H from one carbon-carbon bond (A-B in figure (11.3)) to the adjacent one (associated with site 7) involves the rebonding of bond A-B and the breaking of the adjacent one. This change of bonding is due to a coordinated motion of both the H (from site 6 to 7) and the carbon (labeled B) atoms (see figure (11.7)). Therefore the diffusion of H involves also the passage of the C atom over a barrier. Exactly such a coupled-barrier diffusion process has been found by Ramamoorthy et. al. [130] for diffusion of oxygen in silicon.


  
Figure 11.7: Simultaneous displacement of the two atoms involved in the couple barrier diffusion. The bright spheres are C atoms, and the dark spheres are positions of the H atom. Bi and Bf indicate the initial and the final positions of the carbon atom, and 6 and 7 the initial and the final positions of the H atom repectively.
\begin{figure}\centerline{\epsfysize=10cm \epsfbox{couplebar.ps}}
\end{figure}

We therefore calculate the energy of the sample as a function of the positions of both atoms (H and adjacent C), in an adiabatic migration process. We move the H atom step by step from the initial relaxed site (number 6) to the final one (number 7), along the direction 6-7. Concurrently, the carbon atom (labeled B) is displaced step by step along the direction defined by its initial and final relaxed positioned. At each step, we relax the sample such that the H and the C atom are constrained to remain on the plane perpendicular to their respective directions of displacement. Four atoms are fixed on the faces of the supercell to prevent center of mass motion. This procedure yields the shape of the barrier for the coupled-barrier diffusion, shown in figure (11.8). As can be seen from the figure, the barrier is symmetric, with a maximum of 0.9 eV at the center of the paths. It is found that during the coordinated motion, the distance between H and its adjacent C atom remains $\sim$ 1.08 Å.


  
Figure 11.8: Energy barrier for the adiabatic coordinated migration of H and its adjacent C atom. The top and bottom axis indicate the displacement of these two atoms.
\begin{figure}\centerline{\epsfysize=8cm \epsfbox{fig5.ps}}
\end{figure}


next up previous contents
Next: ριλεν Up: Hydrogen in diamond Previous: Comparison with experiment
David Saada
2000-06-22