## INTRODUCTION

**The discovery of fullerenes has disclosed** a new wide area of research in fundamental condensed matter physics,1-3’4-1 as well as in chemistry[2,5] and materials science,[6] confirming carbon as the most versatile element of nature. Fullerenes constitute a family of cage-like carbon molecules where each carbon atom is threefold coordinated and forms covalent sp2 bonds with the three nearest neighbors like in a single graphite sheet. The atoms of fullerenes lie on a closed surface topologically equivalent to a sphere. The actual shapes of fullerene clusters are polyhedra whose structural features can be largely understood from pure topological arguments. The number N of atoms of a fullerene molecule vary from a minimum of 20 to indefinite large values, presumably in the range of several thousands, although the far most stable fullerene is made of 60 atoms (C60) and takes the shape of a soccer ball, otherwise known as a truncated icosahedron. The shape of this highly symmetric polyhedron, having 60 equivalent vertices that can be generated by the 60 operations of the icosahedral point group, was well known to ancient Greek mathematicians, notably to Archimedes who described the 13 possible semiregular polyhedra. Semiregular polyhe-dra, otherwise known as the Archimedean polyhedra, are made of two different kinds of regular polygons and are obtained by vertex truncation of the five regular (Platonic) polyhedra. The first known representations of the truncated icosahedron have been given by the celebrated Renaissance artists and scholars Piero della Francesca and Leonardo da Vinci (Fig. 1).

**In this article,** the structural and topological features of graphite-like carbon are described, with special emphasis on the novel families of graphene materials that have been discovered and thoroughly investigated after the first identification of the fullerenes C60 and C70. In ”Historical Background,” an overview is given on the historical development and the scientific and technical motivations of studies on the different forms of graphite-like carbon. The topology of graphene structures is presented in ”Graphene Topology.” This section contains an elementary introduction to the general topological properties of graphenes, which allows for a classification of the graphene families in terms of topological connectivity. This is followed by four subsections describing the main graphene families: fullerenes, nanotubes,and amorphous sp2 carbon. ”Topology vs. Total Energy and Growth Processes” shows that the energetics and the growth processes of graphene structures can also be related to and, to some extent, understood from general topological arguments.

## HISTORICAL BACKGROUND

**The first idea that** graphite sheets could be bent, at the cost of replacing a number of six-membered rings with five-membered ones, so as to form graphite balloons, dates back to the mid-1960s and is due to D. E. H. Jones.a The conjecture that 60 carbon atoms could arrange themselves in a stable molecule having the shape of a truncated icosahedron was investigated in 1970 by the Japanese chemist Eiji Osawa and confirmed shortly after by the Russian chemists Bochvar and Galpern with a Hiickel calculation (1973) and by R. A. Davidson (1980) by means of a group theoretical analysis. It is now recognized that fullerenes are rather frequent in nature, especially in soot and probably in interstellar dust. However, it was in 1985 that the cluster C60 was experimentally identified by Robert F. Curl, Harold W. Kroto, and Richard E. Smalley with their graduate students J. R. Heath and S. C. O’Brien at Rice University.[1] The molecule was recognized among the various products of a cluster beam experiment where carbon clusters were generated by laser vaporization of graphite—a technique developed by E. A. Rohlfing, D. M. Cox, and A. Kaldor a year before and then adopted by Richard Smalley for his experiments at Rice. The large abundance of C60 in the cluster beam, as signaled by mass spectra, was indicative of its exceptional stability whereas the occurrence of a single line in the nuclear magnetic resonance spectrum suggested that all 60 atoms are equivalent, which led to one single possible geometry for the molecule, the truncated icosahedron. Although the number of possible fullerenic isomers rapidly increases with the cluster size and many isomers have a comparable cohesive energy per atom, the truncated icosahedron is definitely the most stable form of C60 among its 1812 possible isomers. The resemblance of these new molecules with the geodetic domes invented by the renowned architect Buckminster Fuller led to the general name of buckminsterfullerenes, subsequently abbreviated into fullerenes.

**Fig. 1 The truncated icosahedron according to Piero della Francesca (Libellus de Quinque Corporibus Regularibus, 1492 manuscript at Bibliotheca Vaticana) and Leonardo da Vinci (in L. Pacioli, De Divina Proportione, Bologna 1498).**

**A further breakthrough in the chemistry** and physics of fullerenes was made by Wolfgang Kratschmer and Donald R. Huffman (1990) who devised a method for the production of fullerenes in gram quantities.[7] This allowed for the growth of fullerene crystals, either pure (fullerite) or in a compound form with other elements (fullerides), and led to the discovery of superconductivity in alkali fullerides at moderately high temperatures (1991).[8] Carbon clusters can also be considered as the building blocks for a large variety of carbon-based, cluster-assembled materials and novel nanostructured carbon forms.[9] Quite often clusters have peculiar properties, originating from the low dimensionality, which are frustrated in the corresponding three-dimensional solid. The concept behind cluster assembling is that some of the interesting properties and functions that occur in the composing clusters may be preserved in a three-dimensional robust structure. Much of the materials science based on fullerenes relies on their peculiar chemistry and on various functionalization and polymerization processes that have been discovered and extensively studied starting from the early 1990s.[10] The chemistry of fullerenes also includes the synthesis of endohedral fullerenes having the formula Cn@Me, where Me stands for a metal atom inside the fullerene cage. Indeed, these studies have suggested enormous potentialities and a wide range of applications for carbon-based materials, thanks to the possibility of tailoring their physical properties and performances. It was suggested, for example, that the superconducting critical temperature of doped fullerite increases with the curvature of fullerene cages, namely with the reduction of the cluster size from C60 down to C36,[11] and perhaps C28 and C20. Moreover, an unexpected ferromagnetic behavior has been recently described in fullerenic materials.[12]

**The surprising wealth of physical and chemical** properties found in fullerenes has stimulated an extensive search for additional forms of carbon.[13] The peculiar cage-like geometry of fullerenes has provided a new ingredient for the structural analysis and growth of some widely studied forms of aggregation of sp2 carbon, such as carbon blacks, which are currently obtained by dehydrogenation of hydrocarbons and modeled as concentric shells of graphitic segments.[14] Indeed, single fullerenes may work as templates for the formation of a highly idealized form of sp2 carbon nanoparticles known as carbon onions. Carbon onions consist of concentric hollow carbon spheres that are produced with a strong electron beam irradiation of carbon nanoparticles such as those present in carbon blacks. Carbon onions have been found to act as precursors for the formation of diamond under suitable high-temperature electron beam irradiation. The low-density aggregation of sp2 clusters leads to the formation of highly porous, low-density carbon phases (carbon aerogels). Among these structures,discussed below, are of special interest from the topological point of view.

**Another important class of sp2 carbon materials is that of carbon fibers,** which can be viewed as assemblies of closely connected carbon nanotubes. A single-wall carbon nanotube (SWCN) consists of graphite sheet rolled up to form a cylinder of a given constant diameter and indefinite length. The diameter can be as small as about 0.7 nm. When a nanotube is closed (capped) at both extremities, the nanotube can be viewed as a very elongated fullerene, and is indeed topologically equivalent to a fullerene. A single-wall nanotube can be dressed by other coaxial nanotubes in the very same way as a carbon onion grows around a fullerene and forms a multiwall carbon nanotube (MWNT). Carbon nanotubes have been discovered by high-resolution transmission electron microscopy (TEM) by S. Iijima in 1991[15] and since then have stimulated an enormous interest for their great mechanical robustness combined with excellent transport properties and potential applications in field emitters, supercapacitors, nanoelec-tronics, etc. Carbon nanotubes can be grown from the decomposition of hydrocarbons on metal nanoparticles acting as catalysts. Like endohedral fullerenes, nanotubes can be filled with metal ions that modify their one-dimensional properties; for example, their electrical behavior and functions as nanowires. Indeed, nanotubes are considered as elementary components for nanocircuits, T- and Y-shaped junctions, and possibly semiconducting n-p junctions where the difference between the two sides of the junction is achieved either with two unequal nano-tube structures or through doping, e.g., with endohedral metal atoms.

**Other exotic morphologies of sp2 carbon that** have been obtained with the use of different catalytic metal particles, laser-ablation processes, or simply by flash chemical vapor deposition techniques are carbon parallel-epipeds,[16] single-wall[17] and multiwall[18] ring-shaped nanotubes, graphitic nanocones,[19] carbon nanohorns,[20] and nanotrees.[21] Nanotubes as well as cones and similar morphologies can be obtained in principle by a continuous bending of a graphite sheet without breaking any bond, namely without introducing disclinations. As will be explained in the next sections, from the topological point of view, this kind of deformation keeps the Gauss curvature of the surface constant, in this case zero as for the originally flat graphite sheet. For comparison, the bending of a graphite sheet and its closure into a fullerene implies a change of the Gauss curvature from zero to a positive value and the creation of a certain number of positive disclinations associated with the conversion of some hexagonal rings into smaller polygons.

**The synthesis of new sp2 carbon forms,** such as ful-lerenes and nanotubes, and the observation of important properties, such as superconductivity in alkali metal-doped fullerenes,[22] field-emission,[23] and supercapaci-tance[24] from arrays of nanotubes, are opening fascinating perspectives for nanostructured carbon as a novel, all-purpose material. However, fullerenes and nanotubes, as well as graphite, aggregate into van der Waals three-dimensional (3-D) solids. For many technological applications, there is a need of highly connected, fully covalent sp2-bonded carbon forms, combining the valuable properties of fullerenes and nanotubes with a robust 3-D architecture. A basic question is whether a graphite sheet can be transformed into a surface characterized by a negative Gauss curvature everywhere through the creation of a sufficient number of negative disclinations where some hexagonal rings transform into larger polygons. A special case of negative Gauss curvature occurs when the mean curvature is zero everywhere, which corresponds to what is known in differential geometry as a minimal surface. The conjecture that a minimal surface is particularly stable has stimulated much theoretical work on hypothetical graphite sheets (graphenes) with the shape of a periodic minimal surface.[25-31] These structures have been called, after the name of the mathematician H. A. who investigated the differential geometry of this class of surfaces at the end of the 19th century.[32] Similar theoretical sp2 carbon structures such as polybenzenes[33] and hollow graphites[34,35] can be ascribed to the general family. In the late 1990s, possible routes to the synthesis of schwarzites through the assembling of small carbon clusters in supersonic cluster beam deposition (SCBD) experi-ments[36,37] have been theoretically investigated by means of classical molecular dynamics.[37,38] Indeed, there is a clear evidence that random schwarzites, characterized by a highly porous, fully 3-D graphite-like carbon, are formed under special conditions.[28,39] Recently, periodic structures, with a porosity on the mesoscopic scale and fascinating perspectives in optoelectronics, have been synthesized by chemical methods.[40]

**While most of the theoretical studies concerned periodic schwarzites,** experimental evidence has been recently obtained for an amorphous schwarzitic form of carbon corresponding to the random schwarzite previously envisaged by various authors.[28,41,42] This new structure, which is fully covalent in three dimensions, represents the fourth form of sp2 carbon; the first form, fullerite, is the aggregation of zero-dimensional objects (fullerenes); the second form, nanotubes bundles, is the aggregation of one-dimensional objects (nanotubes); and the third form, graphite, is the aggregation of two-dimensional objects (the graphite sheets or graphenes). Random schwarzites have been obtained by supersonic cluster beam deposition (SCBD) of carbon clusters in the presence of very small catalyst nanoparticles,[39] and look like a highly porous spongy carbon. The numerical simulation of the TEM images suggests that random schwarz-ites grow in the form of a self-affine minimal surface, with relevant and not yet fully understood topological features. The topology and structural properties of the different classes of sp2 carbon are discussed in ”Graphene Topology.”

**Supersonic cluster beam deposition experiments** have also shown that by increasing the energy per atom of the beam clusters, an increasing sp3-to-sp2 bond ratio is observed in the SCBD carbon. A natural question was whether carbon is able to form other fully covalent, fully three-dimensional sp3 solids besides diamond and lonsda-leite, which can be obtained from the coalescence of ful-lerenic cages. Much theoretical work and speculations have been made about the possible geometries and structure of carbon clathrates originating from the coalescence of small fullerenes,[43—46] but no experimental synthesis of these interesting materials has been obtained so far. Similar structures, characterized by only a partial saturation of bonds (mixed sp3/sp2 forms), have been recently synthesized by linking together through covalent bonds C36 clusters[11] or C20 clusters.[47] The topology of clath-rates is discussed in general terms in Refs. [48,49] while some topological algorithms that generate complex and even fractal clathrate structures can be found in Refs. [46,50] together with some topological algorithms that generate complex and even fractal clathrate structures.

## GRAPHENE TOPOLOGY

**From the topological point of view,** fullerenes as well as graphite sheets, nanotubes, and schwarzites [which we shall group under the general family name of graphenes (Fig. 2)] are here described as polygonal tilings of surfaces with only hexagons, pentagons, and heptagons, where each vertex corresponds to a carbon atom, each edge corresponds to a covalent bond, and each polygon corresponds to a carbon ring. The generalization to structures with larger and/or smaller polygons is straightforward. Each atom has a threefold coordination. The surface, covered by a polygonal tiling of carbon rings, defines the shape of a graphene and is characterized by its connectivity or order of connection k. According to Hilbert and Cohn-Vossen,[51] the order of connection is the number plus one of the close cuts that can be made on the given surface without breaking it apart in two pieces. The surface topology may be alternatively characterized by the Euler-Poincare characteristic w = 3—k or the genus g=(k —1)/2. For example, a simple (one-hole) torus can be cut along two closed lines without splitting it in two pieces, and therefore k =3 or w=0, g = 1. For a sphere, k =1 (g=0, w=2), whereas for an n-hole torus, k =1+2n, W=2(1 — n), and g=n. Thus the genus represents the number of ”holes” (or ”handles”) of a generalized torus.

**Fig. 2 The four allotropic forms of graphite-like carbon, which are characterized by sp2 bonds and threefold coordination and are grouped under the general name of graphenes: (a) Two lattice planes of graphite crystal, where each plane represents an ideal graphene with only hexagonal rings. (b) The fullerene C60, formed by 12 pentagons and 20 hexagons. (c) A nanotube, having a cylindrical shape and an indefinite length. (d) A three-periodic, D-type schwarzite, characterized by a lattice with the diamond structure. While the crystalline forms of fullerenes, nanotubes, and graphite are held together by van der Waals forces in three, two, and one space directions, respectively, schwarzite is entirely covalent in three dimensions.**

**While fullerenes are represented** by a closed surface topologically equivalent to a sphere (k =1), uncapped nanotubes, graphite sheets, and schwarzites are open surfaces with an infinite extension in one, two, or three dimensions, respectively. However, graphenes characterized by a periodic atomic structure can be conveniently reduced to a closed surface by applying cyclic boundary conditions. In this way, uncapped nanotubes and graphite sheets become topologically equivalent to an ordinary (one-hole) torus (k=3). On the other hand, the connectivity of an infinite periodic surface is infinite. However, if cyclic boundary conditions are applied on a finite portion of the periodic surface, the connectivity is finite, although dependent on the actual number of unit cells of the periodic structure. Thus it is convenient to define the connectivity per unit cell. This is obtained by closing the portion of surface contained in the unit cell on itself as implied by the cyclic boundary conditions, and the number gcell of handles generated by the closure operation gives the order of connection per unit cell as

There is a fundamental relationship, known as the Gauss-Bonnet theorem, which links the topological properties of a closed surface, expressed by the Euler-Poincare characteristic, to its differential geometrical properties, expressed by the principal curvature radii R1 and R2:

**There is a fundamental relationship,** known as the Gauss-Bonnet theorem, which links the topological properties of a closed surface, expressed by the Euler-Poincare characteristic, to its differential geometrical properties, expressed by the principal curvature radii R1 and R2:

where the integral extends over the whole surface S. For a periodic surface, the theorem applies to any portion of it, notably to a single unit cell, after closure through the periodic boundary conditions. The two principal radii are usually replaced by two more convenient variables, the mean curvature H and the Gauss curvature K, defined by

respectively, with the convention that radii have a positive or a negative sign according to whether they are oriented on one side of the surface or the other. Thus according to the Gauss-Bonnet theorem, the Euler-Poincare characteristic is the average Gauss curvature times S/2p.

According to Euler’s theorem, the tiling of a closed surface with a Euler-Poincare characteristic w fulfils the equation

where v is the number of vertices (atoms), e is the number of edges (bonds) and f is the number of polygonal faces (atom rings). We shall indicate by fj the number of j-membered rings present in the tiling, so that

By inserting Eqs. 5 and 6 into Eq. 4, one obtains

For a given connectivity (genus) of the surface, this equation provides a condition among the numbers of polygons. It appears that this condition is independent of the number of hexagons, which is therefore arbitrary.

Special cases are the Platonic tilings, with a single kind of polygons, and the Archimedean tilings, with two different kinds of polygons, one of which is, in the present discussion, the hexagon. In both cases

for 6, whereas the number of hexagons is zero for Platonic tilings or any other number > 1 for Archimedean tilings. If, besides hexagons, the tiling also includes pentagons and heptagons, then

**From this equation,** it appears that pentagon-heptagon pairs may be added to (or cancelled from) a tiling within the same topology of the surface. Thus if a hexagon is replaced by a pentagon in some thermally induced or stress-induced defect formation process, a heptagon must be formed as well in some other place of the graph-ene network.

**To replace a hexagon in a flat graphite layer with** a pentagon with no bond stretching, a whole sector 60°-wide of the surrounding hexagonal lattice has to be removed. This transforms the original flat lattice into a cone truncated by the pentagonal face, where the continuous surface encompassing the atoms acquires a positive Gaussian curvature: The perimeter of a circle of radius R lying on the surface and centered at the pentagon center is shorter than 2pR. This defect is an example of a positive disclination. In a similar way, a hexagon can be replaced by a heptagon with no bond stretching by inserting a 60°-wide sector, which transform the original flat lattice into a saddle around the heptagonal face, with a negative Gaussian curvature: The perimeter of a circle of radius R lying on the surface and centered at the heptagon center is longer than 2pR. This defect is an example of a negative disclination.

**The creation of pentagon-heptagon defect pairs occurs,** for example, when a cylindrical nanotube is bent. An ideally infinite cylindrical nanotube is a perfect hexagonal tiling. An elbow bending produces a positive Gauss curvature on the external side and a negative Gauss curvature on the internal side of the elbow, and this is accommodated by the creation of pentagon and heptagon defects on the two sides, respectively. These arguments are easily extended to the case where octagons are considered instead of, or in addition to, heptagons.

**The topological features of graphenes are** relevant to their cohesive energy and electronic structure. Intuitively, the number of topologically different close electron orbits increases with the order of connection of the graphene surface. The topological structure of the electronic eigen-functions in graphenes, notably in schwarzites, has been theoretically investigated in some recent works by Aoki et al.[52] The electronic energy levels and their degeneracy depend also, and more specifically, on the local point symmetries. The subtle links between topological and point-symmetry properties have been elucidated by Fowler, Ceulemans, and coworkers in a series of seminal works.[53-56]

### Fullerenes

For fullerenes (k =1), Eq. 8 givesf5=12 orf4=6 orf3=4. By excluding energetically unfavorable three- and four-membered rings, it turns out that Archimedean fullerenes must always contain 12 five-membered rings, and the number of atoms grows with the number of six-membered rings as

**Therefore the most common fullerenes C60 and C70** have 20 and 25 six-membered rings, respectively, whereas the smallest fullerene is the Platonic dodecahedron C20. In general, fullerenes with abutting pentagons are less stable. To avoid abutting pentagons, each pentagon must be surrounded by five hexagons, and each hexagon cannot have more than three adjacent pentagons. Thus the smallest fullerene with no abutting pentagons must have 12×5/3=20 hexagons, and this explains the high stability of C60 in its isomeric form with the 12 pentagons regularly spaced at the vertices of an icosahedron. On the other hand, the carbon atom at the trihedral vertex of three abutting pentagons has three bonds forming angles of 108° and can therefore easily go in an approximate sp3 configuration as long as its outward dangling bond is saturated, say, by a hydrogen atom, which leads to rather stable forms of hydrogenated fullerenes. Thus C20H20, C28H4, etc. are particularly stable molecules, unlike the corresponding pure fullerenes that have a very low stability. As regards the possible isomeric forms, even restricting the fullerene isomers to those exclusively made of five- and six-membered rings, their number rapidly increases with v. As shown by Fowler et al.[57,58] in some basic works, the enumeration of isomers and the classification of their electronic structures can be achieved through the so-called spiral algorithm and leapfrog rules. Thus for C28, there are only two isomers; for C40 this number is 40, whereas for C60 there are 1812 spectrally distinct isomers. Fortunately, because of the geometrical conditions discussed above, there is, in general, at least for not too large fullerenes, only a small number of particularly stable isomers that are largely favored with respect to the others. They are the isomers with no abutting pentagons: only one isomer for C60, but five for C78, seven for C80, etc. The most stable isomers approximate as much as possible the spherical shape to minimize the distortion of the bonds from the ideal sp2 configuration. For approximately spherical fullerenes, the average diameter dv can be evaluated from the equation

where S5 ffi 1.720r52 and S6 ffi 2.598r62 are the areas of pentagons and hexagons of average edge length r5 and r6, respectively, and Sp = 12S5+f6S6 is the surface area of the fullerene polyhedron. For r5 and r6 equal to the bond length in graphite r0 = 0.142 nm, the diameter of C60 estimated from this equation is d60 ffi 0.694 nm, whereas the experimental value is 0.704 nm. This small discrepancy is fully accounted for by using the actual average bond lengths r5 = 0.145 nm and r6 = 0.142 nm.

**Giant fullerenes,** with v in the range 102-103, are believed to approach an icosahedral shape with the 12 five-membered rings at the vertices and the 20 triangular faces made by graphite sheets (Fig. 3). In this limit, the fullerene may be viewed as obtained by folding a single graphite sheet through the insertion of 12 positive disclinations. However, capped nanotubes may also be viewed as giant fullerenes where the 12 five-membered rings are concentrated in the caps, six on each side.

**Fig. 3 A giant fullerene of 1620 carbon atoms. Rather than spherical, the structure is approximately icosahedral as a result of the positive disclinations induced by the 12 pentagonal rings requested by Euler’s theorem and located at the vertices.**

**Fig. 4 (a) A single-wall carbon nanotube can be obtained by folding a graphite sheet so as to bring the point (L,M) to coincide with the origin (0,0). The vector T, orthogonal to the circumference C, defines the periodicity in the axial direction. In this example (L,M) = (4,1). (b) An achiral perpendicular nanotube of indices (5,5). (c) An achiral parallel nanotube of indices (9,0). (d) A chiral (7,3) nanotube.**