NANO LETTERS
Magnetism in Corrugated Carbon Nanotori: The Importance of Symmetry, Defects, and Negative Curvature
2004 Vol. 4, No. 11 2179-2183
J. A. Rodrı´guez-Manzo, F. Lo´pez-Urı´as, M. Terrones,* and H. Terrones* AdVanced Materials Department, Instituto Potosino de InVestigacio´ n Cientı´fica y Tecnolo´ gica, A.C. (IPICyT), Camino a la presa San Jose´ 2055, Col. Lomas 4a seccio´ n, 78216 San Luis Potosı´, S.L.P., Me´ xico Received August 11, 2004; Revised Manuscript Received September 4, 2004
ABSTRACT We demonstrate that carbon nanotori constructed by either coalescing C60 molecules along the 5-fold axis (incorporating pentagons and octagons) or by joining the ends of Haeckelite tubes (containing heptagons, hexagons, and pentagons) exhibit large magnetic moments when an external magnetic field is applied. In particular, we have used a π-orbital nearest-neighbor tight-binding Hamiltonian with the London approximation in order to study the influence of uniform external magnetic fields on various types of torous-like carbon nanostructures with negative and positive Gaussian curvature. We have calculated the ring currents and the induced magnetic moments on these structures. The results reveal the presence of an unexpected magnetic response, which is due to the presence of nonhexagonal carbon rings. Our results could also explain the existence of ferromagnetic nanocarbons. We envisage that coalesced peapods may exhibit unusual magnetic properties.
Over the past few years, ferro- and paramagnetism in carbon has resulted in a large number of experimental and theoretical investigations.1-12 The small magnetic susceptibility observed in bulk diamond and the large diamagnetic response of graphite are explained by the occurrence of sp3 and sp2 bonding respectively.5 The presence of sp2 bonding in graphite originates ring currents produced by the delocalized π-electrons under an external magnetic field. For example, in the case of benzene the London theory13 predicts that the π-electron ring currents are diamagnetic with the magnetic field normal to the plane of the six-membered ring. The discovery of novel forms of carbon such as C60, nanotubes, nanonions, and nanocones has recently attracted the interest on the magnetic response of nanocarbon systems. Haddon et al. found that C60 exhibits a smaller magnetic susceptibility when compared to that of benzene χb (χC60 ≈ -0.2χb).14 This observation is due to a ring current cancellation effect caused by the Van Vleck paramagnetic term.15 Experimentally, Ramirez et al.4 demonstrated that the magnetic susceptibility in multiwalled carbon nanotubes (MWNTs) is significantly enhanced when compared to that of highly oriented pyrolytic graphite (HOPG); these authors suggest that the induction of ring currents on the waist of the tubes by the applied magnetic field are responsible for this enhancement. After the experimental observation of carbon nanotori,16 special attention has been focused on the magnetic response * Corresponding authors. E-mail:
[email protected]; hterrones@ ipicyt.edu.mx. 10.1021/nl0486968 CCC: $27.50 Published on Web 09/29/2004
© 2004 American Chemical Society
of these type of geometries. Haddon17demonstrated theoretically that graphitic nanotori, which were first proposed by Dunlap18 and Itoh et al.,19 reveal an extremely large diamagnetism (negative susceptibility) relative to benzene (χC576 ≈ 657χb) for a C576 tori constructed using hexagons, heptagons, and pentagons.17 More recently, Liu et al.6 found, using a Hu¨ckel tight-binding model with the London approximation, that some metallic nanotubes exhibit a colossal paramagnetic moment for tori with specific “magic radii”. From the experimental point of view, it has been shown that C60 molecules are able to fill the inner core of single-walled carbon nanotubes (SWCNTs) and boron nitride multiwalled nanotubes.20,21 The structures consisting of C60@SWCNT are termed nanopeapods. Using electron irradiation treatments and thermal annealing of C60@SWCNT, the C60 molecules start to coalesce, producing nanocapsules inside the tubes, which should reveal unusual electronic properties depending on the molecular arrangement of the fullerene molecules.22 For this reason the study of the magnetic properties of coalesced C60 molecules becomes an interesting issue from the experimental and theoretical standpoint. In this letter we report, for the first time, the magnetic properties of various carbon nanotori created by coalescing either C60 molecules along three axes of symmetry (2-, 3-, and 5-fold) or Haeckelite tubules, using a π-orbital nearestneighbor tight-binding Hamiltonian with the London approximation. We find that the combination of negative and the positive Gaussian curvature (caused by the presence of
Figure 1. Models of corrugated carbon nanotori constructed by coalescing, covalently, C60 molecules along the different axes of symmetry: (a) 2-fold; (b) 3-fold, and (c) 5-fold. In (b) the necks possess heptagonal rings, whereas the structures shown in (a) and (c) contain octagonal rings within the necks.
pentagonal, heptagonal, or octagonal carbon rings) plays a crucial role in their magnetic behavior. In both cases we observe nanostructures that possess a strong magnetic moment induced by large ring currents. Our results indicate that it should be possible to find a magnetic response in pure graphitic sp2 arrangements possessing negative and positive Gaussian curvature. The structures were generated using parametrized force fields that have proved accurate for diamond, graphite, and various types of fullerene-like structures (e.g., C60).23,24 Figure 1 shows three tori generated by coalescing C60 molecules along the 2-, 3-, and 5-fold symmetry axes, respectively. To describe the effect of an external magnetic field on carbon nanostructures, we consider a π-electron tight binding Hamiltonian and the London approximation.5,13-15,17 London theory modifies the resonance or transfer integral βij between pairs of adjacent carbon atoms by a field-dependent phase factor. Thus, the current between the atom Ri to the nearest neighbor atom Rj is given by Jij ) [
∑n (cni )*(cnj )] exp
{
ie
2pc
}
[A(Ri) - A(Rj)]·(Ri + Rj) βij f (n) (1)
where cni is the eigenvector corresponding to the eigenenergy n, A(R) is the vector potential, f (n) corresponds to the Fermi function and βij is the electronic hopping at zero magnetic field. The total current Iij on the bond is found by adding Jij + Jji. We derived the magnetic moment of the nanotorus using µ ) Iσ, where I is the ring current in the torus and σ is the area enclosed by the torus. By introducing the Fermi distribution to the Iij current expression, it is possible to study the temperature dependence of the magnetic moment. Our calculations confirmed the results reported by Liu et al.,6 revealing that for tori formed by joining the extreme ends of conventional carbon nanotubes made of only hexagons, the magnitude of the ring currents and the magnetic moments depend on the size (“magic radii”); 2180
however, when having pentagons and heptagons (or octagons) in the graphitic mesh, the magnetic behavior changes dramatically. Before studying the magnetic response of our structures, the electronic density of states has been calculated and for all cases we found degenerate states around the Fermi level.25 Generally, it is not expected to find a magnetic response if a gap around the Fermi level exists. Regarding the coalescence of C60 molecules forming tori, we find three different magnetic responses when an external magnetic field is applied. Our calculations were all performed for applied magnetic fields of 0.1 T. However, we have carried out a systematic study up to 2.0 T, finding no important variations in the magnetic response. The coalesced C60 molecules along the 2-fold axis exhibit ring currents 10 times higher than that of benzene Jb (Imax ) 11.37 Jb; see ij Table 1). Due to the symmetry induced by the octagons, the total ring current is canceled, thus possessing a zero magnetic moment (see Figure 2a). Interestingly, the ring currents increase with size or diameter of the tori. When the C60 molecules are covalently connected along the 3-fold axis, the ring currents are 3 orders of magnitude ) 5530 larger than that of benzene for the smaller tori (Imax ij Jb; see Table 1); these currents are also higher than those shown in the nanotori created by connecting the ends of perfect SWNTs (only hexagons). In some cases, and depending on the size of the tori, the ring currents drop to almost zero, producing an extremely small magnetic moment. In other cases, the large ring currents are mainly localized on the heptagonal rings and are, surprisingly, parallel to the magnetic field and not perpendicular (see Figure 2b). At the same time, there are also vanishing ring currents on the hexagonal rings causing a disruption of the total current along the toroidal circuit producing a zero total magnetic moment. Tori formed by joining C60 molecules along the 5-fold axis reveal large ring currents, 3 orders of magnitude higher than that of benzene (see Table 1). For this case, the ring currents flow (clockwise) in the same direction as shown in Figure 2d. In consequence, a large magnetic moment is found for all the 5-fold symmetry nanotori. We also observed that there are specific peaks that correspond to an enhanced flow Nano Lett., Vol. 4, No. 11, 2004
Table 1. Maximum Current Iij in Units of Benzene Current Jb for Corrugated Carbon Nanotori Constructed by Coalescing C60 Molecules along the 2-, 3-, and 5-Fold Axes of Symmetry with Different Number of Atomsa 2-fold
I max ij
3-fold
I max ij
5-fold
I max ij
616 672 728 784 840 896 952 1008 1064 1120 1176 1232 1288 1344 1400 1456 1512 1568 1624 1680
11.37 11.12 10.98 10.94 10.98 11.12 11.32 11.60 11.97 12.43 13.01 13.73 14.63 15.74 17.11 18.92 21.39 24.87 30.16 39.23
840 900 960 1020 1080 1140 1200 1260 1320 1380 1440 1500 1560 1620 1680
5530 0.68 4333 0.70 4128 0.73 2510 0.73 1895 0.74 1469 0.74 1577 0.75 1377
700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300
4380 3485 3586 3148 2622 2660 2565 2194 2028 2007 1921 1770 1678 1650 1541 1430 1390
a The external magnetic field B ) 0.1 T is applied perpendicular to the nanostructure.
current around the torus, resulting in high magnetic moments (see Figure 3). The existence of these “magic structures” should occur when the HOMO-LUMO gap closes. However, further calculations in order to explain this issue are currently underway. Nevertheless, for temperatures as low as 10 K, an important reduction of the magnetic moment is observed. For example,we find that µ goes from 65.1 µB at zero temperature to 1.9 µB at 10 K for the 2300-atom structure (see inset Figure 3). We believe that these structures are the best candidates for showing magnetic behavior in peapod coalescence experiments. In this context, we suggest to measure the magnetic properties of fullerene peapods at different temperatures. According to the Stoner criterion,26 we envisage that these structures may also be ferromagnetic. This is because these structures are indeed metallic (according to ab initio DOS calculations and electronic conductance results22) and are also paramagnetic. The two types of the Haeckelite tori analyzed in this study are the R57 (rectangular type containing heptagons and pentagons only) and the H567 (hexagonal type with pentagons, hexagons, and heptagons).27 We find that these structures exhibit large ring currents for specific sizes and do not follow a paramagnetic behavior pattern, as in the case of coalesced C60 nanotori along the 5-fold axis. For example, some of them possess large magnetic moments and others lack of them (see Figure 2e). According to our results a possible explanation of the observed ferromagnetism in the rombohedral C60 phase might be explained by the presence of ring currents due to C60 sp2 polymerization through heptagonal and octagonal rings of carbon, so circuits (nanorings or torus-like) consisting of Nano Lett., Vol. 4, No. 11, 2004
coalesced C60 molecules are formed, generating a total magnetic moment. In addition, other experimental observations have demonstrated that high electron irradiation of highly oriented pyrolytic graphite (HOPG) is ferromagnetic at different temperatures.2 According to our findings, these results might be due to the formation of heptagonal or pentagonal rings within the hexagonal lattice. More recently, the existence of ferromagnetic carbon nanofoams has been proved experimentally.28 In this context, and using the idea of introducing heptagonal and octagonal carbon rings, we have constructed interconnected concentric fullerenes (see Figure 2g) termed “holey balls”,29,30 which could be a simplification of entangled graphene surfaces, thus resembling a carbon nanofoam. It is important to emphasize that the presence of negative curvature through heptagons or octagons makes holey balls and carbon nanofoams locally similar. Our calculations demonstrate the presence of a high paramagnetic response in holey balls. From the above experiments, one may conclude that ferromagnetism could be encountered in carbon nanomaterials containing hexagonal and nonhexagonal rings arranged symmetrically in a sp2-like fashion. However, we should strongly emphasize that sp2-like carbon nanostructures containing nonhexagonal rings (heptagons or octagons) arranged in a symmetric fashion, could only be ferromagnetic provided these structures are defect free, symmetric, and exhibit a permanent magnetic moment or a diverging susceptibility at small fields. Therefore, ferromagnetic carbons produced experimentally should be carefully examined and characterized in order to neglect the presence of ferromagnetic impurities. When generalized Stone-Wales defects are introduced within our nanotori (bond rotations to keep the connectivity of the network but changing the pentagon, hexagon, heptagon statistics), the ring currents in all the structures decrease drastically, thus poisoning the presence of a total magnetic moment. Therefore, it appears that magnetism is very sensitive to the presence of defects, which break the symmetry of the systems. However, if a magnetic field is applied with an inclination angle with respect to the toroid, it may be possible to re-activate the paramagnetism. Further calculations along this line should be performed but are beyond the scope of this communication. In this context, it is noteworthy that if one or more atoms are removed (creation of vacancies) in paramagnetic nanotori systems, the magnetic moment is completely destroyed and the structure is no longer paramagnetic. Therefore, if defects are present in crystalline carbon systems containing heptagons, hexagons, pentagons and/or octagons, the chances to find paramagnetism are very low, because it is highly likely to encounter structural defects, experimentally, in synthesized structures. We have also noted that for some tori, the magnetic moment can be enlarged when adding an electron or a hole. For example, by adding four holes (or electrons) to the 840atom 3-fold symmetry tori, a magnetic moment of µ ) 13.26 µB (or µ ) 8.16 µB) is obtained. Also, for 1120-atom nanotori (coalesced C60 along the 2-fold axis), it is observed that for 2181
Figure 2. Ring current paths on different carbon nanostructures. Arrows represent bond currents. Clockwise currents are paramagnetic and counterclockwise currents are diamagnetic. The length of the arrows are proportional to current intensity. The maximum current (largest arrow) is different for each case. (a) Segment of a carbon toroid built by coalescing C60 molecules along the 2-fold axis of symmetry (616 atoms). Here the currents at the necks flow in opposite directions and with equal intensity, thus the total current is zero. (b,c) Portions of tori constructed by coalescing C60 fullerenes along the 3-fold axis of symmetry. Interestingly, the structure in (b), which contains 840 carbon atoms, reveals the presence of isolated current circuits, whereas in (c) the 900-atom structure does not show this effect. (d) Segment of a nanotoroid built by joining covalently C60 molecules along the 5-fold axis of symmetry (900 atoms). Here the currents at the C60 interconnections are of the same magnitude with all current intensities pointing along the same direction. (e) Carbon nanotoroid constructed by joining the ends of a rectangular Haeckelite nanotube (832 atoms) containing only heptagons and pentagons. In this case, it is clearly observed the presence of a zero magnetic moment. (f) Close up of (e). Interestingly, the currents are parallel to the applied magnetic field, instead of perpendicular. (f) Molecular model of a “holey-ball” (interconnected concentric fullerenes joined by necks) containing 1320 atoms, which exhibits a maximum paramagnetic current in the heptagonal rings. (h) Close up of (g) revealing the heptagonal rings and the currents induced. The structures shown in (g) and (h) do not contain pentagonal rings and consist of hexagons and heptagons only. 2182
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36365-E, 37589-U, J36909-E, W-8001 (Millennium Initiative), the UC-MEXUS through grant PS/CN 02-114, and the CIAM initiative (grants CO2-41464 and CO2-42428). References
Figure 3. Magnetic moment µ in Bohr’s magnetons (µB) of the torus obtained by coalescing C60 molecules along the 5-fold axis as function of the size of the system (number of atoms). Inset reveals the temperature dependence of the magnetic moment for this case with tori containing 2300 and 1000 atoms.
four holes (electrons) µ ) 9.26 µB (or µ ) 6.03 µB). These results open the possibility of finding a magnetic response of carbon nanostructures when doping with either boron or nitrogen atoms. In conclusion, it has been demonstrated that a paramagnetic response of corrugated nanotori containing hexagons, heptagons, pentagons, and octagons could be obtained for perfect arrangements of nonmagnetic atoms such as carbon. Besides the complexity and the symmetry of these systems, the negative and the positive Gaussian curvatures introduced by adding heptagons (or octagons) and pentagons, respectively, play a crucial role in determining the electromagnetic behavior of toroidal architectures not studied before. We envisage these systems also to be ferromagnetic at low temperatures if the structure are defect free. Toma´nek et al.7 using first principles confirms that magnetism due to spin polarization could be found in graphitic foams exhibiting negative curvature. Our models also resemble graphitic foams, but in one dimension. In the future, it may be possible to fabricate nanotechnological magnetic devices from nonmagnetic atoms31 for specific systems with perfect symmetries and arrangements. Unfortunately, ferromagnetism and paramagnetism in carbon nanostructures is very sensitive to the introduction of defects and the rupture of symmetry. Experimentally, small concentrations of defects or the presence foreign atoms (magnetic or nonmagnetic) may result in a magnetic response. Therefore, ferromagnetic carbons should be carefully examined and characterized in order to neglect the latter possibility. Acknowledgment. This work was done under the auspices of CONACyT (Mexico) through grants G-25851-E,
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