*Ab Initio Study of the Electronic and Energy Properties of Diamond Carbon DOI: http://dx.doi.org/10.5772/intechopen.111435*

The convergence criteria used imply a total energy change of 0.0005 eV/electron, a maximum integral of absolute density change of 0.0001 electrons, and a maximum integral of absolute eigenstate change of 4.10<sup>8</sup> (eV)2 . We also used an 8 8 8 Monkhorst-Pack grid and 512 k-points in the irreducible part of the Brillouin zone. In addition, the following parameters were considered:


After initialization and convergence of the graphene structure, we obtained the following results (see **Table 4**):

**Figure 4** plots the density of state obtained from this calculation. The calculated electronic density of state shows that diamond is a semi-metal with a non-zero gap Eg equal to 5.5 eV, which is in agreement with other calculations [19]. The electronic spectrum near the Fermi energy has a linear form. This behavior allows diamond to


#### **Table 4.**

*Results obtained after initialization and convergence of the graphene structure.*

**Figure 4.** *Density of state (DOS) of diamond.*

offer many applications in biosensors [20] and in devices for power electronics [21], because electrons have a high mobility in diamond at Dirac point of 4500 cm<sup>2</sup> V�<sup>1</sup> s �1 . Diamond is a wide-band-gap isolator. It is intrinsically insulating but can become semiconductor, then metallic and superconductor if it is suitably doped. This doping, which can be of type n or p, is obtained by the substitution of a carbon atom by a donor (n) or acceptor (p) atom, respectively.

### **3.4 PBE0 and PBE calculations for diamond**

PBE is a GGA function introduced by Perdew, Burke, and Ernzerhof, in which all parameters other than those of its local spin density component are fundamental constants [22, 23]. Adamo and Barone [24] realized a new hybrid Hartree–Fock /DFT model derived from the PBE GGA, called PBE0, in which the HF exchange contribution is also fixed *a priori* [24]. The PBE0 approach is obtained by casting the PBE function in a hybrid scheme, in which the HF ratio is fixed a priori. The integer function can be expressed as follows:

$$\mathbf{E\_{xc}^{\rm PBEO}} = \mathbf{E\_{xc}^{\rm PBE}} + \frac{\mathbf{1}}{\mathbf{4}} \left( \mathbf{E\_x^{\rm HF}} - \mathbf{E\_x^{\rm PBE}} \right) \tag{1}$$

Where EPBE xc and EPBE <sup>x</sup> are the correlation and GGA exchange contributions, respectively, and EHF <sup>x</sup> is the HF exchange [25]. It can be interesting to examine the two approaches of PBE and PBE0 to determine the lattice constant of diamond as well as its gap value.

### *3.4.1 Diamond lattice constant as a function of the number of k-points*

In this section, we perform a non-self-consistent PBE0 calculation based on a selfconsistent PBE to represent the variation of the diamond lattice constant as a function of k-point number.

We use a Monkhorst-Pack k-point grid [26], which is essentially a uniformly spaced grid in the Brillouin zone. Another less commonly used scheme is the Chadi-Cohen k-point grid [27, 28]. Monkhorst-Pack grids are specified as n1 � n2 � n3 grids, *Ab Initio Study of the Electronic and Energy Properties of Diamond Carbon DOI: http://dx.doi.org/10.5772/intechopen.111435*

#### **Figure 5.** *Calculation of the diamond lattice constant in the case of PBE0.*

**Figure 6.**

*Calculation of the diamond lattice constant in the case of PBE.*

and the total number of k-points is n1-n2-n3. The computational cost is linear with respect to the total number of k-points, so a computation on a 4 4 4 grid will be about 8 times more expensive than that on a 2 2 2 grid. Therefore, we again seek to strike a balance between convergence and ease of computation [28]. Below we examine the k-point convergence of diamond carbon. **Figure 5** presents the results of the PBE0 calculation, **Figure 6** presents the results of the PBE calculation, and **Figure 7** presents the results of the PBE0 and PBE calculation.

To plot the three figures, we need a grid of at least 8 8 8 k-points to achieve a convergence level of at least 0.002 Å for PBE0 and 0.0005 Å for PBE. We find a lattice constant between 3.560 and 3.562 Å in the case of PBE0 and between 3.5815 and 3.5820 Å in the case of PBE. PBE and PBE0 do not converge to the same lattice constant, so they do not give the same results. PBE0 was found to perform better than the PBE function in reproducing the experimentally found geometric features (a = 3.567 Å) [29].

The convergence of the k-points is not always monotonic as in this example, and sometimes very dense grids (e.g., up to 20 20 20) are required for highly
