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, 3P0, 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.