First principles study on structural and magnetic properties of small and pure carbon clusters ( C n , n = 2 − 12 )

We have demonstrated the structural and magnetic properties of small carbo n lusters (Cn, n = 2−12) in the framework of collinear approximation using density functional theory. The calculations were performed for amorphous, linear, and ring carbon clusters using full-p otential local-orbital (FPLO) method. We have obtained stable structures, total energies, total magnetic mome nts, and HOMOLUMO energy gap of these clusters. We have found that the amorphous c arbon clusters with minimum energy are not magnetic clusters whereas their less-stable isomers with spe cial configurations are ferromagnetic objects. Two robust magnetic moments were found for C 5 carbon ring (6μB) and for pentagonal pyramid C 6 structure (4μB).


Introduction
The intriguing possibility of magnetism and structural properties of carbon clusters has triggered great research interests due to their potential applications in molecular magnetism and spin functional nanodevices. Especially, nanodiamond particles can be used as a potential candidate to increase wear resistance and microhardness, and decrease the coefficients of friction and corrosion in composite functional layers such as hard disks and magnetic heads [1]. In addition, nanoscale magnetic sensors have been developed on the bases of nanodiamonds including nitrogen-vacancy impurities [2]. It has been found that carbon atoms can be magnetized around a vacancy defect in graphene [3] and also those carbon atoms located at the edges of a graphene nanoribbon [4,5]. As a consequence, one can examine a magnetic feature for carbon clusters with some especial configurations. Therefore, it is prime important to know the magnetic properties of carbon structures and to know in which structure the carbon clusters are magnetized.
In the present work, our interest is concerned with the theoretical density functional theory (DFT) calculations to investigate the structures, electronic, and especially the magnetic properties of pure and

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neutral small carbon clusters. In this context, many experimental and theoretical studies have been conducted on the pure carbon clusters [6][7][8][9][10][11][12]. Raghavachari and Binkley, using spin-unrestricted version of Hartree-Fock method, have obtained the structures and energies of small carbon clusters (C n , n = 2 − 10) and revealed an odd-even pattern in the cluster geometries and that the odd-numbered clusters prefer linear structure and the even-numbered clusters prefer irregular cyclic structure [13]. Xu and co-workers have studied a number of annular carbon structures (C n , n = 3 − 31) and linear carbon clusters (C n , n = 2 − 12). They have found electronic properties of these carbon clusters such as total energies, energy gaps between highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), and bond lengths at the level of DFT/B3LYP/6-31G * approach [14,15]. Zhang and co-workers have optimized the structure of C n clusters (n = 2 − 30) and have calculated binding energy for these clusters using the genetic algorithm associated with simulated annealing method [16]. The electronic properties and relative energies of carbon clusters with 2n atoms (2 ≤ n ≤ 16) for stable isomers including chains, rings, cages, and graphitic (plate and bowl) structures have been found by R.O. Jones [17]. Also, López-Urías and co-workers, using single-band Hubbard model and pseudo-potential approach implemented in the SIESTA package, have studied the electronic and magnetic properties of sp 2 carbon isomers C n (n = 10,12,14) [18]. They have found a triplet state for some studied isomers in which this state strongly depends on the cluster geometry [18]. It has to be taken into account that carbon atoms are hybridized together in the form of sp, sp 2 , and sp 3 . Therefore, one can examine that there are many stable isomers for a typical pure carbon cluster including n atoms. To the best of our knowledge, there is no systematic study on magnetic and structural properties of all isomers of carbon clusters even for small structures.
In this paper we made an exhaustive search in carbon isomers to find out magnetic and stable objects with amorphous structure. In addition, the binding energies, HOMO-LUMO energy gaps and magnetic properties of sp, sp 2 , and sp 3 carbon clusters (C n , n = 2 − 12) were studied. These properties were investigated for carbon clusters in the form of amorphous wire and ring structures using full-potential local-orbital method which is developed on the basis of linear combination of atomic orbitals (LCAO) method. In Section 'Computational method,' the numerical approach is discussed. Section 'Results and discussion' presents the structural properties of different carbon isomers, calculated magnetic moments, binding energies, etc. Finally, the paper is summarized in Section 'Conclusions.'

Computational method
All magnetic calculations were performed using density functional theory [19] implemented within a self-consistent full-potential local-orbital basis band structure method, FPLO.9 package [20]. In this scheme, the scalar relativistic calculations were implemented within a spherical average on the spin-orbit interaction, using four-component Kohn-Sham-Dirac equation [21]. The generalized gradient approximation of (GGA) Perdew-Burke-Ernzerhof 96 was used for the exchange-correlation potential [22]. The valence basis set of carbon consisted of 1s, 2s, 2p, 3s, 3p, and 3d states so that for the sake of completeness, carbon core electrons and its polarization states were considered as valence electrons. For each size of studied clusters, several initial configurations were considered. In order to optimize atomic geometries, the position of all carbon atoms was fully relaxed. In the optimization process, the total energy and force converged to 0.001 meV and 0.001 eV/Å, respectively. The binding energy per atom (E b ) for all studied clusters was determined as E b = (nE s − E t )/n, where n is the size of the cluster (number of carbon atoms), E s is the energy of the single carbon atom and E t is the total energy of the relaxed cluster.

Results and discussion
In order to obtain stable carbon clusters, all amorphous structures were taken with most possible symmetry as initial configurations. However, it has to be pointed out that after relaxation process, most of the selected structures were unstable and deformed. The relaxed isomers and related point group symmetries of amorphous carbon clusters are illustrated in Figure 1. The relaxed carbon rings and their related point groups are shown in Figure 2. Table 1 shows the total energies of the lowest amorphous, linear, and ring structures. In Figure 3, we have indicated the calculated binding energies, related HOMO-LUMO gaps, and second difference energies for linear, ring, and amorphous structures with minimum energy.    At first, the calculations were performed for carbon dimer. The optimized bond length is 1.316 Å, and its magnetic moment was obtained 2µ B . The calculated C-C bond length at the DFT (GGA) level is larger than that in experimental value (1.243 Å) by about five percent [23]. Our calculated GGA bond length for C 2 dimer can be compared to the calculated bond length (1.315 Å) done by Congie Zhang [16].
For C 3 cluster, we considered linear and triangular structures. Linear isomer with D ∞h symmetry was found to be the most stable structure whose magnetic moment is zero. Triangular structure (D 3h ) with

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magnetic moment 2µ B lies 564 meV higher in total energy compared to its linear isomer. The calculated bond length for triangular structure is 1.377 Å, which is in agreement with the calculated bond length done by Martin and co-workers [24]. The calculated C-C bond length of the C 3 linear cluster is 1.304 Å, where it can be compared to the experimental value of 1.297 Å, determined by Hinkle et al. [25] and calculated bond length (1.305 Å) done by Hutter and co-workers [26].
We have studied square and tetrahedron structures for C 4 cluster but after relaxation due to Jahn-Teller effect, they distorted to Figure 1 (4a) and (4b) structures, respectively. The most stable structure is a rhombic structure (Figure 1 (4a)) with D 2h point group symmetry and zero magnetic moment. The Figure 1 (4b) structure with C 2v symmetry is a ferromagnetic object with magnetic moment 2µ B lying 807 meV higher in total energy compared to rhombic isomer (Figure 1 (4a)). For the C 5 cluster, we examined two different initial configurations: triangular bipyramid and squared pyramid. It is found that both isomers were deformed to the same structure after relaxation (see (5a) in Figure 1). The symmetry of C 5 structure is D 3h , and it was found to be a nonmagnetic object.
We have investigated three different geometries for C 6 . A distorted trigonal prismatic (Figure 1 (6a), C 2v ) was found to be the most stable structure and nonmagnetic cluster. The next two isomers are a pentagonal pyramid (6b, C 5v ) and a distorted octahedron (6c, C 2v ), which lie at 614 and 1041 meV higher than the structure with minimum energy whose magnetic moments are 4 and 2µ B , respectively. Our findings show that the pentagonal pyramid structure has the largest magnetic moment among the tridimensional carbon structures investigated in this work. For C 7 , we obtained two structures: hexagonal prism and plane-like structure. Our calculations have indicated that the planar C 2v isomer is the most stable structure with 4,386 meV energy below the hexagonal prism with C 6v point group symmetry. We have also studied five structures for C 8 cluster. Our calculations indicated that Figure 1 (8a) with D 4d symmetry is the most stable structure, and the planar D 2h isomer in Figure 1 (8b) with energy difference of 2,016 meV lies higher than the D 4d isomer. The next isomer with C 2v symmetry lies 2330 meV higher compare to D 4d structure. The Figure 1 (8d) structure has a cubic form with O h symmetry and energy difference of 3950 meV higher than Figure 1 (8a) structure. The most nonstable isomer (Figure 1 (8e)) has also C 2v symmetry with 5.011 eV higher with respect to the most stable structure.
For C 9 cluster, two initial structures (capped tetragonal prism and tricapped trigonal prism) were deformed to Figure 1 (9a) and (9b) structures. The square pyramid structure was also stable (Figure 1  (9c), C 4v ). The Figure 1 (9a) cluster with C 2v symmetry was found as the structure with minimum energy. The Figure 1 (9b) structure also with C 2v symmetry lies 4,489 meV higher in energy and square pyramid in Figure 1 (9c) is less stable with energy difference of 5,959 meV higher than the Figure 1 (9a) cluster. In the case of C 10 cluster, we have studied three structures, but only the square dipyramid structure Figure 1 (10c) retained its initial geometric configuration. However, this structure is the most nonstable isomer in comparison to the other studied C 10 clusters and lies 11,070 meV higher in energy with respect to the minimum energy structure (Figure 1 (10a), C s ). The (10b) carbon cluster in Figure 1 with C s symmetry and energy difference of 502 meV lies higher than Figure 1 (10a) structure. For C 11 , two stable structures were investigated. It was found that both relaxed configurations have amorphous feature with no symmetry. The most stable Figure 1 (11a) carbon structure lies below Figure 1 (11b) structure with energy difference of 715 meV. The last tridimensional cluster studied in this work is C 12 . The initial structure for C 12 was chosen as an icosahedron with I h symmetry. The total energy calculated for structure Figure 1 (12b) was −12, 384.260 eV, whereas after relaxation the initial structure was totally deformed (see (12a) structure in Figure 1). This structure has C s symmetry and its total energy is −12, 417.34 eV. However, we have found that the icosahedral C 12 cluster is a magnetic object and its calculated magnetic moment is 2µ B whereas it is not a stable structure at room temperature and ambient pressure.

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We have also studied the magnetic properties of ring and linear carbon clusters in which they are isoelectronics to the considered amorphous structures. It has to be pointed out that our calculated magnetic moments for even-numbered linear clusters are 2µ B and for odd-numbered linear structures have vanished. Li and co-workers have also shown the same magnetic pattern for finite-sized carbon wires [27]. It can be seen in Figure 2 that the magnetic properties of ring structures are completely dependent on the size and structure of the clusters. It has to be pointed out that C 3 (D 3h ) with triangular structure, C 5 ring with pentagonal structure (D 5h ), and C 7 , C 9 , and C 12 with C 2v symmetry are magnetic rings. Among the studied carbon clusters with n ≤ 12, the C 5 ring has the largest magnetic moment (6µ B ). According to the data in Table 1, it can be argued that the carbon rings of C 6 , C 10 , C 11 , and C 12 are more stable than their linear isomers with 362 meV, 2682 meV, 1011 meV and 547 meV lower in total energy, respectively. Martin and co-workers have found that the carbon linear structures with n ≤10 are more stable than their ring isomers [28,29]. Their findings are in contradiction with our results, since we have found that the ring structures of C 6 and C 10 are more stable than their linear structures. We have also found that the C 5 with D 3h symmetry and C 7 with C 2v symmetry are more stable than the isoelectronic ring structures (see Table 1).
The behavior of binding energies (E b ), second difference energies ∆ 2 E(n), and HOMO-LUMO gaps for the all studied carbon structures are depicted in Figure 3. It can be seen that the binding energies per carbon atom are increased with the increasing size of the cluster. For the amorphous carbon structures, it can be seen that there is a gradual increase in the binding energy in the range of n = 3 − 6 and n = 7 − 12 for tridimensional (amorphous) structure with minimum energy (see Figure 3). It has to be pointed out that also with increasing the number of carbon atoms in the clusters, the binding energies are merged to 6.5 eV. This value is very close to the binding energy of carbon macromolecules [16]. A better way to show the relative stability of a cluster is the second differences in energy ∆ 2 E(n) defined by ∆ 2 E(n) = E(n + 1) + E(n − 1) − 2E(n), where E(n) is the total energy of C n clusters. It was shown in the middle panel of Figure 3 that the C 3 , C 7 , and C 11 amorphous structures are more stable than their neighbors. The same argument can be concluded for C 6 and C 10 ring structures (see the middle panel of Figure 3). The ∆ 2 E(n) for linear structures shows a systematic behavior whereas for amorphous and rings structures it behaves very irregular. It is obvious from ∆ 2 E(n) in Figure 3 that the odd-numbered linear carbon structures are more stable than the even-numbered linear carbon structures. We have also investigated the behavior of HOMO-LUMO energy gaps (∆ H-L ) for the studied structures (see Figure 3). It can be pointed out that the HOMO-LUMO gap can give a criterion for chemical reactivity of a typical cluster, in which a larger HOMO-LUMO gap means less reactivity. In Figure 3 it can be seen that HOMO-LUMO gaps of the odd-numbered linear structures are larger than the even-numbered linear clusters. On the other hand, the behavior of HOMO-LUMO gaps for amorphous and rings clusters is completely irregular compared to their linear isomers. We have found that the C 6 carbon cluster with C 5v point group symmetry is the less reactive cluster compared to the other studied amorphous carbon clusters. We have also clarified that C 10 carbon ring is less reactive object compared to the other studied carbon rings. It can be seen in Figure 3 that the ∆ 2 E(n) and ∆ H-L have the same pattern for linear carbon structures, whereas this behavior cannot be seen for amorphous and linear structures.

Conclusions
In conclusion, we have studied the structures and magnetic properties of small carbon clusters (C n , n = 2 − 12) using density functional theory method with generalized gradient approximation. The stable geometrical carbon structures are obtained after relaxation calculations. Then we have investigated the most stable structures, magnetic moments, binding energies, the second differences of energy, and the related HOMO-LUMO energy gaps. Furthermore, we have found that the magnetizations of amorphous and ring structures are completely dependent on the cluster size and configuration of the carbon atoms in the cluster. We have also shown that the lowest amorphous carbon structures with minimum energy when formed in a tridimensional structure have no net magnetic moments. According to our results for small www.SID.ir and amorphous carbon clusters, we may expect a nonmagnetic feature for larger carbon clusters formed at room temperature and ambient pressure. Finally, we have shown that some less stable carbon isomers are magnetic in which one can imagine that they are initial building blocks for magnetic nanodiamonds at high pressure.