Among the advantages of graphene as medium for spin currents is the weak spinorbit coupling in this material, in accordance with the low atomic number of carbon. This feature is beneficial wherever long spin lifetimes are relevant, as in spintronics networks. Assessing and optimizing the efficiency of graphene and other carbon nanostructures as elements of spintronic circuits requires a detailed understanding of spin-orbit coupling mechanisms. Beyond practical considerations, examination of spin-orbit interactions also provides insight into the basic architecture of graphene and nanosystems derived from it. As outlined in Chapter 4, the two Dirac points are each characterized by an electronic state of fourfold degeneracy, corresponding to two spin and two sublattice degrees of freedom. Addition of the spin-orbit effect to the description of graphene lifts the spin degeneracy and thus opens a gap at the points K and, separating two states with twofold degeneracy. Spin-orbit interaction thus modifies the fundamental electronic structure of graphene.

As specified in Section 8.1, the intrinsic spin-orbit gap in graphene is extremely small. Use of the Rashba effect, on the other hand, makes it possible to tune the gap by applying an external electric field to the graphene sheet. This device allows us to close the gap, or to widen it by several orders of magnitude. Further, a physical sheet of carbon atoms is never ideally flat, and curved segments of the graphene layer turn out to be associated with effective electric fields. This feature implies a critical dependence of the spin-orbit effect on the nanostructure geometry. It acquires major importance as one goes from graphene to carbon nanotubes. Section 8.3 is dedicated to spin-orbit coupling in SWCNTs and emphasizes the geometric origin of this effect. In contrast to graphene, this interaction is very pronounced in SWCNTs, inducing a marked energy splitting between two Kramers doublets at the Dirac point. As it is masked by competing effects, its definite experimental characterization was a relatively recent event [288].

Section 8.2 deals with orbital magnetic moments in SWCNTs and an orbital shift undergone by the Dirac cones as a consequence of magnetic interactions involving these moments. This mechanism turns out to be essential for a proper appreciation of spin-orbit processes in nanotubes, as will be outlined in Section 8.3.

Any hollow, cage-like molecule consisting entirely of carbon atoms in a state of sp2 hybridization and composed of five- and six-membered rings is referred to as a fullerene [124]. This class of carbon nanostructures thus comprises the quasi-spherical species that are popularly known as buckyballs as well as axially confined carbon nanotubes, among other systems. As every carbon atom on the fullerene surface is threefold coordinated, a fullerene may be understood as a rolled-up graphene sheet, as illustrated in Figure 6.1 by the example of a buckyball. Euler's theorem, applied to polyhedra that consist exclusively of hexagons and pentagons, confines the number of pentagons to twelve [125], while there may be an arbitary number of hexagons, excepting a single one [126]. This implies that the smallest fullerene of this type is a combination of twelve pentagons, C20. Quasi-fullerenes contain rings beyond pentagons and hexagons, such as heptagons [127] or octagons [128]. The effect of these structural irregularities is a major enhancement of fullerene reactivity.

C60

The most readily available fullerene species, and the prototypical molecular allotrope of carbon, is C60, also known as Buckminster fullerene, making fully explicit the name patron of the fullerenes, the American architect, author and visionary Richard Buckminster Fuller (1895–1983), whose name is famously associated with designs based on geodesic dome structures. The C atoms of C60 are positioned at the vertices of a regular truncated icosahedron. The point group of C60 is I h, the number of symmetry operations of the molecule, i.e. operations that map C60 into itself, is 120.

Fullerenes satisfy the Isolated Pentagon rule, stating that structures with pentagons entirely surrounded by hexagons are more stable than those that admit pentagon adjacency. The geometry of C60, where all pentagons are isolated is unique among the fullerenes C n with n ≤ 70. For topological reasons, all other species of this class involve adjacent pentagons. While pure cages containing these substructures are deemed too unstable to be fabricated in the laboratory [129], they can be stabilized by external or internal impurities (e.g. [130, 131]).

The average bond length between the atoms of C60 is 1.44 Å [132]. Each atom forms two single bonds and one double bond with adjacent atoms.

The label carbon nanostructure magnetism joins two notions that do not seem to go together easily, carbon and magnetism. The main carbon allotropes, after all, are known to be non-magnetic. This appears to be true not only about the wellknown solid phases diamond and graphite, but also with respect to the nanoscopic phases manufactured first in the eighties of the last century, and later: fullerenes and carbon nanotubes, single- and few-layer graphene. While intrinsic magnetism is the rule in the d- and f-blocks of the periodic table, it is extremely unusual in the second period, containing light elements with p electrons in their valence shells. Magnetic derivatives from carbon-based nanostructures, such as metallofullerenes with finite magnetic moments in their ground state have been known for decades, but the magnetism of these composites is inherited from elements different from carbon, such as lanthanide atoms with high spins localized in their 4f shells.

By the beginning of this century, however, sightings of intrinsic magnetism in carbon complexes became increasingly frequent and made headlines, not rarely heralded with adjectives like surprising [1], unexpected [2] or exotic [3]. While in the meantime, the initial surprise about carbon magnetism has somewhat worn off, astonishing discoveries continue to be made in this field, such as the first experimental demonstration of spin transport in graphene at the micrometer scale [4] or the first detection of strong spin-orbit coupling in carbon nanotubes [5].

On the other hand, magnetism in carbon nanostructures also continues to be a topic in tension. Foremost, the proposal of intrinsic carbon magnetism due to net magnetic moments at edges or vacant sites in carbon networks is charged with controvery. Proponents point not only at a large body of theoretical and computational work, predicting these effects (for an overview, see [6]), but also at numerous experiments that appear to confirm these predictions. Detractors call attention to the great difficulty of reliably separating signatures of intrinsic magnetism from artifacts due to small admixtures of magnetic impurities [3] and refer to the rather marginal, if not vanishing, net effects yielded by some recent experimental examinations of carbon magnetism [7].

In view of the ongoing debate, it would be premature to state that magnetism in carbon nanostructures is a firmly established discipline within condensed-matter physics or nanoscience.

In this chapter, as well as the following two, we present the structural protoypes of carbon nanosystems, namely graphene, carbon nanotubes and fullerenes. These represent nanoscopic allotropes of carbon in two, one and zero dimensions. With respect to the dimensionality and also to the chronology of carbon nanostructure research, Chapters 4 to 6 of this text go in reverse order. The reason for this choice is the fundamental significance of graphene for intrinsic magnetism in carbon-based materials. Besides providing general information about graphene, this chapter introduces the dimensionally reduced graphene nanoribbons. The zigzag types of these species have been shown to display magnetism in their ground state and may be understood as the basic units of carbon nanostructure magnetism.

In elemental carbon, four valence electrons occupy the 2s and 2px/py orbitals, yielding the occupation scheme 1s2, 2s2, 2p2 for the electronic shell of the carbon atom. The ground state of the carbon atom is, in spectroscopic notation, 3 P 0, involving a spin triplet in conjunction with a total orbital angular momentum L = 1, and a total angular momentum J = 0. This configuration confines carbon to two chemical bonds only, as exemplified by the carbene CH2. Carbon compounds, however, usually involve three or four bonds. This is achieved by promoting a 2s electron into a 2p orbital, which leads to the formation of hybridized molecular orbitals. We distinguish between sp, sp2 and sp3 hybridization. In these cases, the 2s and two 2p orbitals combine, giving rise to two (sp), three (sp2) or four (sp3) mixed s-p orbitals, as represented by linear combinations of atomic orbitals of s and p character. In the regime of carbon-based molecules and polymers, examples for these hybridization schemes are provided by acetylene (the sp case), polyacetylene (sp2), and methane (sp3). Among carbon allotropes, the sp2 and sp3 pattern are prototypically realized by the graphite and diamond phases of carbon, respectively.

Graphene realizes a perfectly planar form of carbon, which may be understood as a single layer of graphite. Thus, carbon atoms are arranged in a two-dimensional lattice of monatomic thickness, and with hexagonal symmetry. While several layers of graphite display largely graphitic behavior, new features emerge in a perfectly two-dimensional array of carbon atoms.

This chapter deals with magnetic composites consisting of carbon nanostructures and impurity species. The host systems included here are the prototypes fullerene, graphene and carbon nanotubes. In each case discussed, emphasis will be placed on the mechanisms by which the electronic ground state of the aggregate adopts magnetic properties. In many cases, the considered carbon allotrope provides a framework that preserves the magnetism of the guest species, as paradigmatically realized by some metallofullerenes with enclosed lanthanide atoms or clusters. In some systems, however, magnetism is not imported by the guest species but evolves as the carbon allotrope interacts with the externally added moieties. This situation is well exemplified by the much-studied compound tetrakis-dimethylamino-ethylene-C60 (TDAE-C60).

In sections 12.1 and 12.2, we refer first to magnetic metallofullerenes, arguably the most traditional among the structures included here. After all, research interest in fullerenes with enclosed metal components emerged soon after the discovery of C60 [139]. From metal atoms or metal atom clusters as guest species inside fullerene cages, we turn to single group V atoms as encapsulated components. Systems of the form A@C60, with A = N, P, have proven to be efficient in preserving the spin of the enclosed atom, which makes them interesting as physical realizations of qubits in quantum computing. The following Section (12.3) discusses a variety of magnetic phases that originate from electron transfer to fullerenes. This mechanism is operative in very diverse composites, ranging from fullerides to hybrids of fullerenes and organic molecules.

The remainder of this chapter deals with compounds involving graphene and nanotubes. Specifically, graphene is considered as substrate of two adatom types, hydrogen and fluorine, and computational as well as experimental findings on the magnetic phases of hydrogenated and fluorinated graphene are surveyed. The final section of this chapter summarizes various results on carbon nanotubes in combination with magnetic metal components, ranging from atoms to nanoparticles.

Magnetic Metallofullerenes

The magnetism of endohedral metallofullerenes with enclosed paramagnetic guest species arises from the interplay of various effects. Thus, the intrinsic magnetic moment of the enclosed metal component may affect the magnetic moment of the unit as a whole. Further, this component may be in a cationic state, as a consequence of electron transfer from the encapsulated atom or cluster to the fullerene enclosure.