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Comparison with ab initio calculations

The Tersoff potential has been explicitly adjusted to correctly describe many physical properties of both diamond and graphite. The suitability of this potential for the accurate description of the conversion from diamond to graphite, as required for the present annealing calculations, was validated by us by comparing our results with those of a quantum mechanical calculation. The quantum mechanical case used ab initio MD [110] to describe the conversion to graphite of a diamond sample. This ab initio model is based on density functional theory, using local density approximation for the exchange-correlation interaction. Plane waves has been used in these calculatons [110] as a basis for the electronic wave functions, with a pseudopotential to describe the interaction between the valence electrons and the ionic core. The atoms are considered as classical, and the Newton's equations are solved to obtain the atomic positions and velocities as a function of the time. The sample used contains 192 atoms (see Fig. 9.1), and is heated to a temperature of 2500K.

As we will show below, this conversion process plays a central role in the system that we are studying, and the good agreement between our results and those of the ab initio study provides a powerful validation for our approach. We present a series of side-by-side comparison of pictures reproduced from Ref. [110] with results of our calculations performed for an identical case to that of Ref. [110], however with the use of the Tersoff potential. Figures 9.2(a)-9.4(a) are based on the atomic coordinates obtained from the ab initio study, and 9.2(b)-9.4(b) are the results of our MD calculations.

Our MD calculations with the Tersoff potential were performed under conditions identical to those used in the ab initio calculations [110], namely: a Nose-Hoover thermostat [111] was applied to regulate the temperature, an identical sample with 192 atoms was used, and free boundary and periodic boundary conditions were applied in the z-axis (oriented along the [111] crystallographic axes of cubic diamond) and the x-y plane respectively. The slab was initially terminated on both sides by a (2x1) reconstructed surface, which is the stable surface geometry at 0K.

Figures 9.2(b) and 9.3(b) show the progression of the graphitization throughout the slab, until in figure 9.4(b) the sample is completely graphitized, due to heating at 2500K. White and black spheres indicate threefold and fourfold coordinated atoms respectively, the coordination number being determined here by counting the number of nearest neighbors within a radius of 1.9 Å from each atom. The ab initio calculations [110] showed that graphitic planes form in diamond at 2500 K, not only by gradual layer by layer conversion of diamond surfaces to graphitic layers, but also by the conversion of <111> diamond planes to graphitic layers penetrating into the bulk diamond. Figures 9.2, 9.3 and 9.4 show the progression of the graphitization of the sample at three different stages towards its conversion to graphite (without stacking) at 2500 K. The complete graphitization is clearly visible, with all carbon atoms being sp2bonded in planes.

It has to be mentioned that the Tersoff potential, compared with the ab nitio model, do not correctly stabilize the five-fold ring of the (1x2) reconstructed surface, as can be seen from figures 9.2, 9.3 and 9.4. This, however, has no influence on the progression of the graphitization into the slab. The similarity between the two cases shown in figures 9.2, 9.3 and 9.4 is obvious, giving confidence that the Tersoff potential can properly describe the thermally driven conversion of sp3 to sp2 bonding (i.e. the graphitization of diamond).

The present study deals with the transformation that damage regions buried inside an intact diamond matrix undergo, when subjected to high temperature annealing. As will be shown below, the periphery of the damaged region exhibits "surface graphitization" which much resembles the above simulated case, giving confidence that the present computational approach is applicable also to the present study.

  
Figure 9.1: (1x2) reconstructed (111) diamond surface at 0K. This is the initial state for both calculations. White and black spheres indicate threefold and fourfold coordinated atoms respectively.
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Figure: Early stage of the graphitization through the slab at 2500K. a) Results of ab initio calculations (taken from Ref. [110]). b) Results of our classical molecular dynamics. White and black spheres indicate threefold and fourfold coordinated atoms respectively. Note that some bonds between the first and the second layers have been broken.

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Figure: Progression of the graphitization through the slab at 2500K. a) Results of ab initio calculations (taken from Ref. [110]). b) Results of our classical molecular dynamics. White and black spheres indicate threefold and fourfold coordinated atoms respectively. At this stage, bonds between inner layers, oriented in the z direction, have been broken.

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Figure: Complete graphitization of diamond at 2500 K. a) Results of ab initio calculations (taken from Ref. [110]). b) Results of our classical molecular dynamics. White and black spheres indicate threefold and fourfold coordinated atoms respectively.

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next up previous contents
Next: Generation of a damaged Up: General description and justification Previous: General description and justification
David Saada
2000-06-22