Issue 
Eur. Phys. J. Appl. Phys.
Volume 94, Number 1, April 2021



Article Number  10401  
Number of page(s)  8  
Section  Nanomaterials and Nanotechnologies  
DOI  https://doi.org/10.1051/epjap/2021200217  
Published online  09 April 2021 
https://doi.org/10.1051/epjap/2021200217
Regular Article
An ab initio investigation of the electronic and magnetic properties of graphite and nickeldoped graphite
^{1}
ENS, Cadi Ayyad University,
Marrakech, Morocco
^{2}
Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Neel,
38000
Grenoble, France
^{3}
Angevin Laboratory of Mechanics, Processes and InnovAtion (ALMPI). ENSAM ParisTech,
Angers, France
^{4}
ENS, Cadi Ayyad University,
Marrakech, Morocco
^{*} email: selamabdo1976@hotmail.com
Received:
6
July
2020
Received in final form:
3
January
2021
Accepted:
4
March
2021
Published online: 9 April 2021
In this paper, the KKR (Korringa, Kohn, and Rostoker) is presented with coherent potential approximation methods which is used to investigate the electronic and magnetic properties of allotropic graphite forms of carbon and nickeldoped graphite. The density of states (DOS), band structure, total energy, and the magnetic moments of atoms are computed. The crystallographic structure optimization is carried out by evaluating the total energy as a function of unit lattice parameters. The DOS analysis reveals a partially metallic behavior of the compound. The magnetism vs the Nidoping content in C_{1−x}Ni_{x} is also investigated by computing moments induced on atoms; the sensitivity of the magnetism to Nidoping is also analyzed.
Note to the reader: Further to the publication of an erratum, the citation of the article was modified on 19 July 2021.
© EDP Sciences, 2021
1 Introduction
Graphite is a stable form of carbon at a room temperature and under ambient pressure. It manifests a semi metallic state that has been systematically investigated due to its potential use in the technological and to its archetypal bidimensional properties. The study of graphite intercalation compound (GIC) is closely linked to the nature of the interlayer binding. The crystal structure of the graphite is a planar hexagonal network showing layers that generally adopt an ABAB⋯ piling sequence (Bernal structure, Fig. 1b). The natural and the synthesized crystalline graphite is a mixture of Bernal and rhombohedral graphite where the layers are piled in an ABCABC pattern, with proportions of 80% Bernal, 14% rhombohedral, and 6% disordered graphite. In (GIC) structure, the layers of carbon can be piled in various ways, either similar to the Bernal structure (ABAB⋯ stacking), or with layers of carbon atoms located directly in a pile like structure (AAA⋯ stacking). These two modifications of graphite are illustrated in Figure 1. In disordered or pregraphite Carbon typically called turbostratic graphite, the Carbon atoms are in a consecutive layers directly above one another. This configuration of simple hexagonal graphite has not been observed in crystalline graphite. All previous calculations have been focused on the structural and electronic properties of the stressed graphite having the Bernal or rhombohedral structure. In this paper, an ab initio calculations were performed in order to investigate the electronic and magnetic properties. The band structure, DOS, total energy and magnetic moments are computed. For the structural optimization of volume, then the ratio (c/a), the optimization procedure is used reported in reference [1]. The procedure was repeated iteratively in order to obtain the lattice parameters of the proposed structure whith 0.001 bohr as precision. Such optimisation, based on the functional theory of density [2,3] using linearized augmented plane waves plus orbitals (LAPW + lo) [4], were implemented in the Wien2K code [5]. The calculations involved both the local density approximation (LDA) [6] and the generalized gradient approximation (GGA) [7]. The results are gathered in Tables 1 and 2.
In 1924, Bernal reported graphite as a set of parallel hexagonal sheets of carbon atoms arranged in an AB stacking pattern [8]. Later, Wyckoff [9] reviewed graphite using the lattice parameters mentioned by Baskin and Mayer, a = 0.246 nm and c = 0.667 nm [10].
Fig. 1
Different graphitic stackings: simple hexagonal graphite (a) and natural hexagonal graphite (Bernal structure) (b) are displayed. They differ in the shift of the middle plane in the ABA stacking (all planes are directly above each other in the AAA stacking). The unit cell of the simple hexagonal crystalline structure is also shown. The interlayer distance is c and the distance between the nearest neighbors is a [18]. 
Graphite AB: spatial group 194 (P6_{3}/mmc).
Graphite AA: spatial group 191 (P6/mmm).
2 Calculation methods
2.1 The Korringa, Kohn, and Rostoker method with coherent potential approximation
The Korringa, Kohn, and Rostoker (KKR) method [11,12] was used to calculate the electronic band structure and the electronic DOS within the coherent potential approximation (CPA) [13,14].
This method was used by Akai and Dederichs for the treatment of transition metal alloys [15], with the parameterization of Vosko, Wilkand Nusair (VWN). We used the MACHIKANEYAMA2002V10 code produced by Akai [16]. The Local Density Approximation (LDA) is the most widely used approximation in the functional density theory (DFT) [17]. The Green function technique is employed to solve the Schrödinger equation, without resorting to the explicit calculation of the wave functions and eigenvalues of the Hamiltonian system. In this approach, the scattering properties of each diffusion center (atom) are described by a scattering matrix, whereas multiple scattering by all the atoms in the lattice is determined by considering that the incident wave at each center is the sum of the outgoing waves from other scattering centers.
2.2 The Green’s function method
Let H be the Hamiltonian of an electron in the system then, the Green’s operator G is defined [19,20]: (1)
where η is a positive real number and i is the imaginary unit. If k⟩ and k′⟩ represent two eigenvectors of H, then: (2)
where E_{k} is the eigenvalue associated with k⟩ and is the Kronecker symbol. Accordingly, G can be redefined as follows using projectors: (3)
The trace of operator G, denoted by Tr G, is given by: (4)
It was shown [19,20] that the imaginary part of the Green operator trace, Im Tr G, was related to the electronic states density n(E) based on the following relationship: (5)
This relationship is particularly interesting since the G trace is invariant, and the G matrix can be determined in any form of the representation.
Another useful expression of the DOS is deduced from the following elementary relation: (6)
In a representation where the matrix of G is diagonal, we have: (7)
If we express n(E) in an atomic orbital base {φ_{p}}, where the index p can include an atomic position in the crystal and a spin index s, we can then deduce that: (9)
Let’s substitute the mathematical expression of G with: (10)
Then, equation (11) becomes: (13)
where p is an atom index, we obtain: (15)
Consequently, the contribution of any state of energy E_{k} to n(E) is weighted by the orbital weight φ_{p}. Therefore, n_{p}(E) represents the contribution of φ_{p} to the total DOS n(E).
It can be seen from previous relations that the calculation of DOS the requires knowledge of the Green’s operator G. In the present work, this was done using the KKRCPA approach, taking into account multiplesites scattering within the muffintin approximation.
3 The atomic properties and graphitic structure and Nidoping on graphite
3.1 The atomic properties and graphitic structure
The configuration of the ground state of carbon atoms is 1s^{2} 2s^{2}2p^{2} [21]. The core electrons are in the first orbital 1s^{2} and the remaining four electrons take place at the valence band. The energies of the 2s and 2p levels are respectively at − 13 and − 5 eV below the vacuum level. The 1s level is well below and appears at − 35 eV [22].
Inside the core shell no p electrons is found. Due to the orthogonalization with the core state [23], the p valence electrons did not come under the impact of repulsion. The 2p wave function is quite localized [22]. The distance to the nucleus of the orbital 2p wavefunction maximum is almost the same as the orbital 2s wavefunction maximum. In both allotropic forms which are the diamond and the graphite we can find natural carbon. The bonding between atoms is different in each form, but are mainly covalent. In diamond, the bonds arise from the sp^{3} hybridization of atomic orbitales four bonds and angles of 109.5°.
In graphite, the sp^{2} (s − p_{x} − p_{y}) hybridization of atomic orbitales (denoted σ) formed a covalently bound lattice of graphitic planes with angles of 120° between the bounds connecting adjacent atoms to neighbor atoms, They are stacked one on top of the other and weakly bounded by the residual forces that arose from the nonhybridized p_{z} orbitales (denoted as π) perpendicular to the planes. The graphitic lattice (c) and the threedimensional Brillouin zones (d) are shown in Figure 2.
In a graphitic monolayer, the first four bands (σ and π) are occupied and the last four are empty. Because of symmetrical properties, π (bonding) and π^{*} (antibonding) bands degenerated at the K point of the two dimensional (2D) Brillouin zones. As result, the (2D) graphite is a zerogap semiconductor [21]. This situation is modified by interactions between graphitic planes and semimetal is created. However, these weak interactions could be considered as perturbations of the 2D situation.
The 3D structure of graphite is not a straightforward piling of planes. Instead, every other plane is moved in the horizontal plane to create the ABABAB piling pattern as displayed in Figure 2c.
The P6_{3}∕mmc space groupcontains the crystallographic structure. Four atoms exist in the 3D unit cell: two atoms for each graphitic plane and two planes per cell (AB) displayed in Figure 2c. Also, this figures presents the cell, which is high and narrow.
At 0K, the lattice parameters are c = 0.066 nm and a = 0.024 nm [25]. From this, the distance between planes 0.033 nm and between nearest neighbours 0.014 nm [21] were estimated.
Fig. 2
(a) Crystalline structure of the hexagonal graphite [21]. The parameters of the unit cell, represented by dotteddashed lines, are a_{1}, a_{2} and a_{3} = c. a is the distance between nearest neighbours; (b) graphite Brillouin zone showing several highsymmetry points. A schematic version of the graphite electron and hole Fermi surfaces located along the H − K axes [21,24]. Each symmetric point is labeled with the usual BouckaertSmoluchowskisigner notation (Γ, A, H, K, L, M). 
3.2 Nidoping on graphite
Nickel doping has attracted attention due to its basic magnetic properties, which it is hoped to confer to graphene/graphite through doping. It is a long tradition in the graphite field and graphite intercalation compounds (GIC), to take advantage of the large interplane distances of graphene (and thus the weakest bonds/interactions) to try to introduce elements of another nature, to change its physical properties. It is reasonable today, with the available numerical computation methods (KKR, GPAW …), to start with a simulation (fast and inexpensive) rather than an experiment that requires complex synthesis and analysis equipment, not necessarily available in all laboratories. This is also quoted in article [26].
4 Results and discussion
4.1 Optimization of the graphite structure
The structural optimizations for graphite were based on the KKRCPA calculations of the total energy of the graphite system as a function of the (c∕a) ratio. The equilibrium parameters corresponded to the calculated minimum energies. Figure 3 shows that these optimized parameters indicate a minimum value for:
Fig. 3
Mesh parameters optimization E(Ry) according to (c∕a). In our KKR calculations to look for energy optimization by varying the (c∕a) ratio, we started the calculation from the experimental value (c∕a = 2.73) cited in references [4,8] and when this value decreased, we observe a minimum in (c∕a = 2.682), then we continued the calculation with a step of (c∕a = 0.020). 
4.2 Density of state according to the ratio (c∕a)
To investigate the effect of the (c∕a) parameter of graphite on DOS, the following data are used:

the purecompound is the graphite carbon with hexagonal structure.

The mesh parameters found by optimization in the previous sect. (4.1) are a = 0.246 nm and c = 0.660 nm.

To improve the quality of our calculations, we added eight empty spheres (ES) with Z = 0 at the intersites of the four carbon atoms.

The positions of the carbon graphite are: C1 (0a 0b 0c), C2 (1/3a 2/3b 0c), C3 (0a 0b 1/2c) and C4 (2/3a 1/3b 1/2c).

The positions of the eight empty spheres (ES) are: Vc1 (2/3a 1/3b 0c), Vc2 (0a 0b 1/4c), Vc3 (1/3a 2/3b 1/4c), Vc4 (2/3a 1/3b 1/4c), Vc5 (1/3a 2/3b 1/2c), Vc6 (0a 0b 3/4c), Vc7 (1/3a 2/3b 3/4c) and Vc8 (2/3a 1/3b 3/4c).

The quality of the Brillouin Zone mesh is chosen in the settings of the input file of the KKR calculation as follows: bzqlty =4.
Firstly, comparing the DOS of the graphite from our calculations to the DOS reported in reference [27], we find that there is a strong consistency between both calculations. Our investigation points out to the following results: The effect ofthe graphite parameter (c∕a) on DOS is shown in Figures 5–7 (see also Figs. A.1A.4), while retaining the same values of parameters edelt and ewidth parameters whose meanings are given in Figure 7. Moreover, an energy resonance is observed at (c∕a = 2.720) Figure 8, where the DOS decreased significantly in the vicinity of the Fermi energy. Contrarily, when the (c∕a) ratio changes, the density varies in vicinity of the Fermi energy. The symmetry of the spinup and spindown density shifted with respect to the energy axis by 0.006 Ry, depending on the value of (c∕a) Figure 6. The calculated DOS evidenced that the graphite always indicated a semimetallic character, regardless of the value of the (c∕a) ratio. In addition, spin momentum displayed in Figure 4 revealed a linear dependence on the variation of (c∕a) ratio ranging from 2.682 to 2.826.
Fig. 4
The spin moment variation versus the (c∕a) ratio. The spin moment showed a linear dependence of the change in (c∕a) ratio ranging from 2.682 to 2.826. 
Fig. 5
The DOS computed for the parameter (c∕a = 2.682) which corresponds to minimal energy in the optimization of the structure on the graphite DOS. 
Fig. 6
The DOS computed for the parameter (c∕a = 2.713). On the DOS of the graphite and which shows that the symmetry of the spinup and spindown density shifted with respect to the energy axis by 0.006Ry, as a functionof the (c∕a) value. 
Fig. 7
Both parameters edelt and ewidth displayed on the DOS curve. The edelt is imaginary part of energy in the vicinity of Fermi level. The ewidth represents width of the energy contour in Ry. 
Fig. 8
The variation of energy as a function of (c∕a) and the resonance phenomenon which corresponds to the value (c∕a = 2.720). 
4.3 The doping of graphite by nickel
4.3.1 Atomic properties and nickel structure
Nickel, belongs is the space group Fm3m(225) and its cell parameter equal to 0.352 nm. It possesses an atomic number Z = 28 and an electronic configurations [Ar]3d^{8}4s^{2}.
From calculations, we observed an asymmetry of both state densities of spin up and down with respect to the energy axis. It indicates a magnetic state for Ni. The Fermi level taken as energy reference is represented by the vertical line (E = 0 Ry).
The computed magnetic moment is equal to 0.63 μ_{B} [12]. In addition, our computed DOS displayed in Figure 9 is perfectly superposable with the DOS reported in reference [12].
Fig. 9
Density of states versus energy of Nickel. Calculations are performed using space group Fm − 3m (225) and cell parameter equal to a = 0.352 nm. 
4.3.2 Nickeldoping on graphite
In this section, we present the effect of nickeldoping on the graphite properties. Precisely, we analyzed the changes in total energy, in the DOS, and in the magnetic moment carried by atoms vs Ni content. The calculated DOS for different Ni concentrations, in C_{1−x}Ni_{x} system, are reported in Figures 10 and 11 (see also Figs. B.1B.4). The symmetry with the respect to energy axis was clearly observed between the electrons of the majority spins and those of the minority spins, which permitted the assertion that the C_{x−1}Ni_{x} was a nonmagnetic material. Comparing the DOS calculated for graphite and Nidoped graphite, we find that they are somewhat similar for the majority and minority spins. However there are unoccupied states in the C_{x−1}Ni_{x} system taking place at high energies. This unoccupied state corresponds to the addition of two Ni valence electrons. The total energy and the spin moment based on the x concentration are presented as well. Here, we underline that the NiC alloy on which the calculation were performed exists and was already synthesized and related experimental results have been previously reported by Cadeville et al. [28]. They claimed that the solubility of Carbone in Ni achieved by quenching of the liquid state at 293 K (at.–%) is equal to 1.5. Also, they report that the variation of the mesh parameters (da∕dc)(10^{−4} nm / at.−%) equal to 7.6 [28]. In addition. The experimental results about magnetism previously reported in ([29], see Fig. 3) are in accordance with our computed Ni magnetic moment (0.63μ_{B}) as well since they point out to the measured magnetic moment of about (0, 6μ_{B}).
Fig. 10
The density of states (DOS) of the doping of graphite for a concentration equal to x = 0.01 of Nickel. 
Fig. 11
The density of states (DOS) of the doping of graphite for a concentration equal to x = 0.40 of Nickel. 
4.4 Total energy and spin momentum of C_{1−x} Ni_{x}
The total energy given in Rydberg (Ry) of C_{1−x}Ni_{x} system points out to a linear variation depending on Ni content x as displayed in Figure 12,
The linearity of the data was confirmed by a coefficient of determination R^{2} = 0.987. The negative slope of the linear model showed that C_{1−x}Ni_{x} compounds possessed a minimal energies for large x values. Figure 13 shows the evolution of the spin momentum of the C_{1−x}Ni_{x} compounds as a function of x. As other result, when Ni content x varies from 0 to 0.5, the spin moment of graphite increases, whereas the spin nickel decreases. At x greater than 0.5, we notice that the spin moment of nickel increases while the spin moment of graphite decreases.
Fig. 12
The linear variation of the computed total energy versus Ni concentration in CNi alloy. 
Fig. 13
The variation of the spin moment of graphite and Nickel versus the Ni content in CNi alloy. 
5 Conclusion
The presence of nickel atoms in the disordered interstitial sites strongly affects the DOS of graphite. The insertion of Nickel atoms in graphite introduced unoccupied energy levels. We found that graphite, even when its DOS showed more distinctive peaks similar to those of the molecular DOS, recovered its semicircular character. This was because the charge values in the various bands were comparable to those of pure graphite, and the DOS was continuous in the vicinity of the Fermi level.The magnetization of the graphite decreased when the occupation of the Ni site was almost complete. This was primarily due to the rearrangement between spinup and spindown populations when Ni increased.
Author contribution statement
Abdellah Sellam performed all the simulations and wrote and supervised the manuscript. El Kebir Hlil and Rodolphe Heyd helped analyze, interpret and discuss the results. Abdelaziz Koumina helped to improve, analyze and carefully read the document. All authors contributed to the preparation of the manuscript. Also, all authors read and approved the final manuscript.
Appendix A Density of states according to the ratio (c∕a)
Fig. A.1
(c∕a = 2.560). 
Fig. A.2
(c∕a = 2.580). 
Fig. A.3
(c∕a = 2.621). 
Fig. A.4
(c∕a = 2.662). 
Appendix B Nickeldoping on graphite
Fig. B.1
Electronic density of states of C_{95}Ni_{5}. 
Fig. B.2
Electronic density of states of C_{90}Ni_{10}. 
Fig. B.3
Electronic density of states of C_{85}Ni_{15}. 
Fig. B.4
Electronic density of states of C_{80}Ni_{20}. 
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Cite this article as: Abdellah Sellam, El Kebir Hlil, Rodolphe Heyd, Abdelaziz Koumina, An ab initio investigation of the electronic and magnetic properties of graphite and nickeldoped graphite, Eur. Phys. J. Appl. Phys. 94, 10401 (2021)
All Tables
All Figures
Fig. 1
Different graphitic stackings: simple hexagonal graphite (a) and natural hexagonal graphite (Bernal structure) (b) are displayed. They differ in the shift of the middle plane in the ABA stacking (all planes are directly above each other in the AAA stacking). The unit cell of the simple hexagonal crystalline structure is also shown. The interlayer distance is c and the distance between the nearest neighbors is a [18]. 

In the text 
Fig. 2
(a) Crystalline structure of the hexagonal graphite [21]. The parameters of the unit cell, represented by dotteddashed lines, are a_{1}, a_{2} and a_{3} = c. a is the distance between nearest neighbours; (b) graphite Brillouin zone showing several highsymmetry points. A schematic version of the graphite electron and hole Fermi surfaces located along the H − K axes [21,24]. Each symmetric point is labeled with the usual BouckaertSmoluchowskisigner notation (Γ, A, H, K, L, M). 

In the text 
Fig. 3
Mesh parameters optimization E(Ry) according to (c∕a). In our KKR calculations to look for energy optimization by varying the (c∕a) ratio, we started the calculation from the experimental value (c∕a = 2.73) cited in references [4,8] and when this value decreased, we observe a minimum in (c∕a = 2.682), then we continued the calculation with a step of (c∕a = 0.020). 

In the text 
Fig. 4
The spin moment variation versus the (c∕a) ratio. The spin moment showed a linear dependence of the change in (c∕a) ratio ranging from 2.682 to 2.826. 

In the text 
Fig. 5
The DOS computed for the parameter (c∕a = 2.682) which corresponds to minimal energy in the optimization of the structure on the graphite DOS. 

In the text 
Fig. 6
The DOS computed for the parameter (c∕a = 2.713). On the DOS of the graphite and which shows that the symmetry of the spinup and spindown density shifted with respect to the energy axis by 0.006Ry, as a functionof the (c∕a) value. 

In the text 
Fig. 7
Both parameters edelt and ewidth displayed on the DOS curve. The edelt is imaginary part of energy in the vicinity of Fermi level. The ewidth represents width of the energy contour in Ry. 

In the text 
Fig. 8
The variation of energy as a function of (c∕a) and the resonance phenomenon which corresponds to the value (c∕a = 2.720). 

In the text 
Fig. 9
Density of states versus energy of Nickel. Calculations are performed using space group Fm − 3m (225) and cell parameter equal to a = 0.352 nm. 

In the text 
Fig. 10
The density of states (DOS) of the doping of graphite for a concentration equal to x = 0.01 of Nickel. 

In the text 
Fig. 11
The density of states (DOS) of the doping of graphite for a concentration equal to x = 0.40 of Nickel. 

In the text 
Fig. 12
The linear variation of the computed total energy versus Ni concentration in CNi alloy. 

In the text 
Fig. 13
The variation of the spin moment of graphite and Nickel versus the Ni content in CNi alloy. 

In the text 
Fig. A.1
(c∕a = 2.560). 

In the text 
Fig. A.2
(c∕a = 2.580). 

In the text 
Fig. A.3
(c∕a = 2.621). 

In the text 
Fig. A.4
(c∕a = 2.662). 

In the text 
Fig. B.1
Electronic density of states of C_{95}Ni_{5}. 

In the text 
Fig. B.2
Electronic density of states of C_{90}Ni_{10}. 

In the text 
Fig. B.3
Electronic density of states of C_{85}Ni_{15}. 

In the text 
Fig. B.4
Electronic density of states of C_{80}Ni_{20}. 

In the text 
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