Models with hydrogen inclusions

To generate the extra reflections characteristic of the n-diamond structure, an alternative way to vacancy inclusion is hydrogen inclusion. Due to the small scattering cross-section relative to heavier elements, hydrogen is generally un- detectable with conventional X-ray diffraction techniques [223]. Therefore, its effect on diffraction patterns would be similar to the inclusion of vacan- cies. To this end, two models have been proposed: a hydrogen-doped cubic diamond model [218] and a carbon-hydrogen zincblende compound [66]. Hydrogen-doped cubic diamond

Wen et al. [218] proposed that hydrogen can substitute carbon in both the P0 and P1 positions. DFT calculations based on the virtual crystal approx- imation (VCA) [224] were compared with the experimental XRD pattern of Fe-catalysed n-diamond [67]. According to Wen et al. [218], at H concen- trations less than 19 at.%, H-doped diamond is mechanically stable and at

4 at.% the optimised XRD pattern matches the experimental data.

This model is characterised by substitutional disorder, therefore the same approach as in Section 4.2.2 has been employed. Using as a parent lattice the 2_×2_×2 supercell of the primitive cell of cubic diamond, an arrangement at concentration χH=0.0625 was built to maximize both the symmetry and

the distance between defects, see Figure 4.5. The model presents hydrogen substitution in the (0,0,0) position. This model has 15 carbon atoms and one hydrogen atom in the unit cell. The 4_×4_×4 supercell at χH=1/128 has

also been investigated, in which there are 127 carbon and 1 hydrogen atom. The dimension of this model makes phonon calculations computationally expensive, therefore the investigation has been limited to pressures of 0 and 40 GPa.

In the present work no phonon calculations without negative eigenvalues have been found. Figure 4.6 displays the Phonon PDOS of the 2_×2_×2 and 4_×4_×4 supercells. It is clearly shown that instability is mainly due to hydro- gen atoms. At the lower H concentration (4_×4_×4 supercell) the backbone of carbon atoms is stable; conversely at χH=0.0625 it is unstable. For the

4_×4_×4 supercell the presence of populated states related to unstable phonon modes which involve carbon atoms is minimal.

It is well-known that diamond films produced by chemical vapour depo- sition (CVD) techniques in a hydrogen-rich atmosphere have both a high hydrogen content and lattice vacancies [225]. In addition, natural Argyle diamonds have high a hydrogen content [226]. In hydrogen-doped diamond, much of the hydrogen is located at the boundaries between diamond grains, or in non-diamond carbon inclusions, although within the diamond lattice itself, hydrogen-vacancy complexes are present, whereby a hydrogen atom is bonded to one of the carbon atoms in the vicinity of the vacancy [225, 227]. It is concluded here that, if there are hydrogen inclusions in the diamond structure, dopant atoms do not occupy carbon positions.

Some words must be spent on the possible application of the Virtual Crystal Approximation (VCA) to carbon-hydrogen system as an alternative

Figure 4.5: Structure used for phonon calculations in hydrogen-doped carbon: C (gray), H (white); the model is based on a 2_×2_×2 supercell of c-diamond (SG F-43M), χH=1/16. A further model based on a 4×4×4 supercell of c-

diamond (SG F-43M) has also been employed, χH=1/128.

to the approach adopted here. The VCA allows one to handle configura- tionally disordered systems at relatively low computational cost [228]. The potentials, which represent atoms of two or more elements, are averaged into a composite atomic potential, having the advantage that a single configura- tion with a small unit cell represents the disordered system. However some properties that depend on the local environment cannot be reproduced. The VCA ignores any possible local distortion and assumes that on each poten- tially disordered site there is a virtual atom that interpolates between the behaviour of the actual components. This approach cannot be expected to reproduce the finer details of the disordered structures.

In hydrogen-substituted carbon systems, the size mismatch between car- bon and hydrogen causes local distortions that cannot be captured by the VCA approach, as an averaged potential is applied. By contrast, the insta- bility due to the local environment is evidenced by the approach applied in this work, coupling symmetry analysis and phonon calculations.

Carbon-hydrogen zincblende compound

Studying the intensity ratio of forbidden and allowed diamond reflections in XRD patterns of n-diamond produced by CVD, Cowley at. al. [66] proposed that n-diamond includes hydrogen atoms in the P1 sublattice. Experimental observations were supported by total energy calculations performed within

-1000 -500 0 500 1000 0.001 0.002 0.003 f [cm -1 ] PDOS [A.U.] -1000 -500 0 500 1000 0.001 0.002 0.003 PDOS [A.U.]

Figure 4.6: Phonon partial density of states (PDOS) of H-diamond: carbon atoms (continuous line), hydrogen atoms (short dashes): on the left, model with χH=1/16; on the right, model with χH=1/128. For both computational

models, it is clearly shown that modes with negative eigenvalues are mainly related to hydrogen. For the model at lower hydrogen concentration the carbon backbone is much more stable.

the projector augmented wave (PAW) method in the framework of DFT. This model can be considered a limiting case of the previous one where one carbon sublattice is completely substituted by hydrogen.

According to the present work, this model has phonon modes with nega- tive eigenvalues at any considered pressure, denoting structural instability.

4.2.4

Summary of c-diamond-based structures: coor-