Since graphene was initially synthesized in 2004 by Novozelov et al.  , a great deal of research started due to its unusual electronic and magnetic properties. Due to the fact that it is considered a zero-gap semiconductor and presents an unusual form for conductivity due to the Dirac electrons, it is worthwhile to perform a consensus analysis for its properties. An excellent review has been presented by Castro-Neto et al.  . Graphene is well accepted as a zero-gap semiconductor and presented by Wallace  , and in addition, yields an unusual behavior when the Dirac electrons were subject to a magnetic field. This phenomenon is known as anomalous inter quantum Hall effect, first reported by Novoselov et al.  and later by Zhang et al.   .
On one hand, graphene has many industrial applications, to mention few of them such as: due to its ballistic electronic applications in the production of field-effect union type p-n and p-n-p materials  , graphene quantum dots reported by Milton-Pereira et al.  , Molecular detectors reported by Barraza-Ji- menez, et al.  and Spin injections by Hill et al.  . On the other hand, group V Transition Metal Dichalcogenides have been studied extensively   because they present very peculiar charge density waves (CDW) and superconductivity. 2H-NbSe2 presents two transition temperatures Tc at 7.4 K and other at 35 K. The later is attributed to Charge Density Waves (CDW) transition. Moreover, 2H-NbSe2 irradiated with different doses of electrons presents nanotubes of different lengths and size  .
Electronic properties were performed under Density Functional Theory, employing DMOL3  program package. For each structure, geometrical optimization was performed with an energy cut off of 2.74 × 10−4 eV and a threshold of the same value was used throughout the calculations   . For the exchange-correlation an LDA with Perdew-Burke Ernzerhof scheme was employed  . For the wave functions for each atomic species considered, DND basis set, which could be compared to 6 - 31 G, 6 - 31 G (d) and 6 - 31 G (d, p) Gaussian- type basis set was used. No spin-restrictions were considered, in such a way as to leave the multiplicity in automatic.
2.1. Structural Optimization
Figures 1-3 provide information regarding a propose unit cell for graphene, graphene with a NbSe2 cluster, graphene with vacancy defect  , and graphene with NbSe2 cluster located over the defect for Armchair, Chiral and Zig-zag configurations. On each figure, the identification could be provided as follows: carbon atoms are grey balls, Nb with light blue balls and Se with mustard balls. Each structure was properly optimized and relaxes in such a way as to reach a minimum of energy. On each Figures 1-3, three rows of figures are provided. Starting from the top left hand corner, where the graphene (pristine-original) unit cell is located, continuing toward the right corner, is graphene with NbSe2 cluster, following graphene with a defect, and last graphene with defect and NbSe2 cluster located over the defect. In the same figures from top to bottom, the middle row of figures, a charge type of distribution (HOMO-LUMO, Highest Occupied Molecular Orbital and Lowest Unoccupied Molecular Orbital) is provided for the cases mentioned above. On the same page, last row of figures, a side view for the cases already enunciated is provided. For our analysis, let us
Figure 1. Distinct optimized configurations for graphene with armchair symmetry. First column corresponds to graphene (pristine), second to graphene with NbSe2, third to graphene with defect and fourth to graphene with defect and NbSe2. Second and third row shows (at two distinct orientations) the HOMO-LUMO distributions for the respective configurations.
Figure 2. Distinct optimized configurations for graphene with chiral symmetry. First column corresponds to graphene (pristine), second to graphene with NbSe2, third to graphene with defect and fourth to graphene with defect and NbSe2. Second and third row shows (at two distinct orientations) the HOMO-LUMO distributions for the respective configurations.
Figure 3. Distinct optimized configurations for graphene with zigzag symmetry. First column corresponds to graphene (pristine), second to graphene with NbSe2, third to graphene with defect and fourth to graphene with defect and NbSe2. Second and third row shows (at two distinct orientations) the HOMO-LUMO distributions for the respective configurations.
concentrate on the last row of figures for Figure 1, the Armchair case. Notice that the graphene (pristine) presents an almost flat charge distribution, while graphene with defect and NbSe2 cluster yield indication that the cluster locates above two adjacent carbon atoms forming bonds in-between them. This kind of bonding is an uneven distribution.
Figure 2, provides information for the Chiral case, notice that the pristine configuration presents a different kind of charge distribution, when compared to the Armchair case, also for the graphene with defect and NbSe2, the cluster locates by itself over the defect, but in the middle of the defect. Moreover, the bonding formed with the carbon atoms of the defect is stronger when compared with the former case. An important observation is that there is a bump and a valley for the graphene structure produced by the force exercised by the cluster. In addition, Figure 3 provides information for a Zig-zag configuration. Notice that for graphene (pristine), charge protrudes up differently than the two former cases. Also, for graphene with defect and NbSe2 cluster, the cluster is located above the carbon atoms forming a different kind of bonding with them when compare with the former cases.
The differences behavior encountered between the three figures could be attributed to the different form of termination (either Armchair, Chiral or Zig- zag) for each structure, the brim concept reported in another investigations such as MoS2 reported by our group   in a former investigation.
2.2. Energy Bands
Due that we are interested in the electronic properties presented for the cases mentionated in the former paragraphs, energy band analyses was performed. Energy Band analysis is a powerful technique employed in order to indagate about the electronic properties (isolator, semiconductor or metal) for a material in question.
Figures 4-6 show the band structure for the former cases enunciated formerly in the manuscript. Each figure provides Energy (eV) vs. k-points, in the reciprocal space for the extended Brillouin zone, with the Fermi level (EF) located at 0 eV. The band structure for different graphenes (Figure 4(a), Figure 5(a) and Figure 6(a)) exhibit different morphology, where is evident the distinct positions along the special k-points that occupy the Dirac cones (formed at the interaction of the π and π* bands at the Fermi surface) for each case. In Table 1 is reported the forbidden energy gap (Eg) values and the kind of behavior presented for distinct configurations under consideration. Results show that all graphene configurations, presents a semiconductor behavior and the lower Eg value correspond to Zig-zag configuration. Interestingly no matter if a vacancy defect is practiced, or a NbSe2 cluster is present or both in any graphene geometry, all configurations present a metallic behavior.
2.3. Total and Projected Density of States
In order to investigate the relative contributions of distinct atoms (C, Nb and Se)
(a) (b)(c) (d)
Figure 4. Energy Bands vs. K points for: (a) Graphene (pristine); (b) Graphene with NbSe2; (c) Graphene with defect and (d) Graphene with defect and NbSe2. Armchair configuration.
(a) (b)(c) (d)
Figure 5. Energy Bands vs K points for: (a) Graphene (pristine); (b) Graphene with NbSe2; (c) Graphene with defect and (d) Graphene with defect and NbSe2. Chiral configuration.
at the Fermi level, the partial density of states (PDOS) was computed for each distinct configuration. In Figures 7(a)-(c) vertical axis corresponds to PDOS (arbitrary units) vs information data corresponding to AI, AII, AIII and AIV
(a) (b)(c) (d)
Figure 6. Energy Bands vs. K points for: (a) Graphene (pristine); (b) Graphene with NbSe2; (c) Graphene with defect and (d) Graphene with defect and NbSe2. Zig-zag configuration.
Figure 7. Partial Density of States (PDOS) computed at the Fermi level for: (a) s-orbital; (b) p-orbital and (c) d-orbital. A, C and Z stands for Armchair, Chiral and Zig-zag graphene geometries and I, II, III and IV are associated to pristine graphene, graphene- NbSe2, graphene-defect and graphene-defect-NbSe2.
Table 1. Graphene configuration (for Armchair, Chiral and Zig-zag), energy band gap (Eg in eV) and their respective behavior.
(Armchair), CI, CII, CIII and CIV (Chiral) and ZI, ZII, ZIII and ZIV (Zig-zag) configurations, respectively. In each graph, “I” refer to graphene (pristine), “II” refer to graphene with a NbSe2 cluster, “III” refer to graphene with defect while “IV” indicates graphene with defect and with a NbSe2 cluster. In addition, contribution of distinct atoms to s, p and d orbital are shown in Figures 7(a)-(c) respectively. From our analysis we found that at the Fermi level, for distinct graphene configurations the Zig-zag configuration presented the lowest value for the orbitals (carbon) ratio C[2p]/C[2d]. On the other hand, when the contribution of distinct orbitals for the distinct configurations in Figure 7 are compared, we observe (with exception to those the configurations that involves the presence of NbSe2 molecule) that the metallic behavior is promoted by increasing the C[2p] orbital. For the cases that the NbSe2 molecule is present, the deficiency of C[2p] orbital is compensated mainly by the presence of Nb[4d] and Se[4p] orbitals. Before proceed any further, it is necessary to underline that NbSe2 crystallizes in a trigonal-prismatic configuration like MoS2 as reported by Zonnevylle et al. , hence, due to the Crystalline Electric Field (CEF) effect, Nb d-orbitals which are five-fold degenerate, brakes the degeneracy and separate into one bellow two bellow two in energy levels. It is generally accepted that Nb dz2 is the lowest in energy, while the rest are randomly arranged. On the other hand, Se[4p] orbital interact with Nb[4d] orbital of the same symmetry, producing a hybridize set of orbitals. Moreover, in the graphene honeycomb network, each Carbon atom in the hexagonal ring contributes with 4 valence electrons, from which 3 out of 4 contributes to form each ring, while one of the p-orbitals points out of the plane (pz2). This unsaturated pz2 orbital from each C atom could interact with Nb d-orbital of the same symmetry which is closer to the network. Notice from Figure 7(a) that the contributions from s-orbitals are small when compare to the p- and d-contributions from C and Nb respectively and provided in Figure 7(b) and Figure 7(c). Finally, Figure 7 shows that the participation of Nb and Se orbitals is dependent of the configuration (geometry and the presence of a defect) of a graphene sheet.
From the results obtained in this study, it is extremely important to underline the relevance for the selection of Armchair, Chiral or Zig-zag in the construction of the graphene hexagonal sheet, because that final result will be affected depending on the appropriate selection. In our case, the selection for Zig-zag graphene yielded the minimum energy of 0.13 eV for the forbidden energy gap Eg. On the other hand, we observe that the configuration of a graphene sheet affects the participation of Nb and Se orbitals.
D. H. Galvan acknowledges the Departamento de Supercomputo, Universidad Nacional Autónoma de México, Proyecto LANCAD-UNAM-DGTIC-041 for the time provided in order to perform this work.
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