shape and complexity at the atomic scale: the case of

10.1098/rsta.2004.1440

Shape and complexity at the atomic scale:the case of layered nanomaterials

By Humberto Terrones1, Mauricio Terrones

1,

Florentino L o p e z-Ur ı a s1, Julio A. Rodr ı gue z-Manzo

1

and Alan L. Mackay2

1Advanced Materials Department, Instituto Potosino de InvestigacionCientıfica y Tecnologica (IPICyT), Camino a la Presa

San Jose 2055, Lomas 4a seccion, 78216 San Luis Potosı, Mexico([email protected]; [email protected])2Birkbeck College, University of London, Malet Street,

London WC1E 7HX, UK

Published online 9 August 2004

In nature there are numerous layered compounds, some of which could be curved so asto form fascinating nanoshapes with novel properties. Graphite is at present the mainexample of a very flexible layered structure, which is able to form cylinders (nano-tubes) and cages (fullerenes), but there are others. While fullerenes possess positivecurvature due to pentagonal rings of carbon, there are other structures which couldinclude heptagonal or higher membered rings. In fact, fullerenes and nanotubes coulddisplay negative curvature, thus forming nanomaterials possessing unexpected elec-tronic and mechanical properties. The effect of curvature in other nano-architectures,such as in boron nitride and metal dichalcogenides, is also discussed in this account.Electron irradiation is a tool able to increase the structural complexity of layeredmaterials. In this context, we describe the coalescence of carbon nanotubes and C60molecules. The latter results now open up an alternative approach to producing andmanipulating novel nanomaterials in the twenty-first century.

Keywords: graphite; boron nitride; curvature; fullerenes; topology

1. Introduction

D’Arcy Thompson in his famous book ‘On growth and form’ (Thompson 1917)showed the importance of multidisciplinary research for understanding the naturalworld. Thompson hardly mentions atoms, but the interrelationships among math-ematics, geometry, mechanics (physics) and biology are emphasized. The scientifictools in D’Arcy Thompson’s time were limited to what we may now term the macro-scopic and mesoscopic scales. Nowadays we have electron microscopic and spectro-scopic techniques that enable us to work in the same topics, but at a deeper level,particularly at the atomic level. State-of-the-art characterization techniques such aselectron microscopy, scanning tunnelling microscopy (STM) and atomic force micro-scopy (AFM) now enable the identification of single atoms in complex non-crystalline

One contribution of 12 to a Theme ‘Nanotechnology of carbon and related materials’.

Phil. Trans. R. Soc. Lond. A (2004) 362, 2039–20632039

c© 2004 The Royal Society

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2040 H. Terrones and others

structures. X-ray crystal structure analysis uses the crystal as a scattering ampli-fier, neglecting the phase information in the diffraction patterns, and does not dealwith those more general structures where the surroundings of individual atoms andmolecules are not identical throughout the specimen.

Despite the differences in resolution, several themes proposed before high-resolu-tion electron microscopy (HREM) still remain as challenging research topics today.At present, we try to understand what can be done in controlling nature at a smallscale, not only at the atomic or molecular level, but also at the higher levels whichare significant to living systems, as well as for materials, and which arise as a con-sequence of the detailed microscopic structure. As a consequence, concepts such asnanoscience and nanotechnology are becoming popularly recognized characters ofa ‘second scientific/industrial revolution’ (the first was about 1968, when scientificproducts were introduced on an industrial scale, while previously science had beenapplied mainly to the improvement of already existing processes) but what do wehave today? What are the signals and scientific developments which justify multi-disciplinary research on nanosciences and nanotechnology, besides the ever-increasingdemands of the military? It is too early to say, because we are still at the beginning.Perhaps nanotechnology is at the same stage as electricity was in Michael Faraday’stime (1791–1867), when he replied to the British Chancellor of the Exchequer, MrGladstone, who asked about the practical uses of electricity, ‘One day, Sir, you maytax it’.

What becomes attractive to scientists of different disciplines is that nowadays wehave the tools and methods to start manipulating atoms and molecules in order tocreate materials that nature does not form. Materials in nature have formed as aresult of the slow processes of evolution. Every structure has a history. Nowadays itis possible to conceive a structure without a history, and to proceed directly fromthe prediction/design to materialization. The interaction of inorganic and living sys-tems opens up exciting avenues in multidisciplinary research. Information is nowessentially part of the structure of materials—inorganic as well as those of biologicalorigin. Mann and Ozin (Mann & Ozin 1996; Ozin 2000) have begun to show howcertain features, such as the curved structures found in nature, can be created in thelaboratory using inorganic materials. Increasingly, crystals can be seen as being onlya special case. Moreover, Antonietti & Ozin (2004) have pointed out the importanceof mesoscale arrangements developing concepts such as mesobonding, mesoepitaxyand pattern recognition, putting them into the meso-world context and emphasizingthe importance of curvature.

In this account we demonstrate that different morphologies in layered materialslead to the fascinating properties of new materials.

2. The shape and complexity of carbon nanostructures

The discovery of buckminsterfullerene (C60) in 1985 (Kroto et al . 1985) openedup a new area of research which was extended with the discovery/identification ofcarbon nanotubes by Sumio Iijima in 1991 (Iijima 1991). Both structures are anexample of how the same type of atoms (carbon atoms) can arrange themselvesin different shapes with different electronic and mechanical properties. This mightbe called atomic combinatorics. It is worth mentioning that Oberlin et al . (1976)pointed out the existence of single-walled carbon nanotubes. Although other types

Phil. Trans. R. Soc. Lond. A (2004)

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Shape and complexity at the atomic scale 2041

(a)

(b) (c)

Figure 1. (a) High-resolution transmission electron micrograph of a capped carbon nanotube,which shows the presence of a pentagonal ring (positive curvature in region A) and a heptagonalring (negative curvature in region B). (Reproduced courtesy of Sumio Iijima.) Computer simu-lations showing a tube with (b) pentagonal and (c) heptagonal rings (darker rings) of carbon inorder to explain the situation shown in (a).

of hollow cylinders made of imogolite were recognized even before this, and are nowbeing studied in the context of nanomaterials (Bursill et al . 2000).

Carbon nanotubes can be understood as rolled planar graphitic layers, and carboncages (fullerenes) can be visualized as graphene with pentagonal carbon rings. Inthe case of C60, 12 pentagons and 20 hexagons are needed to close completelythe arrangement using sp2 hybridized carbon atoms (every atom has exactly threefirst neighbours). Diamond is another structure made of carbon atoms. However,each atom is coordinated to four carbon atoms in a tetrahedral way (sp3 hybridizedatoms). In fullerenes and nanotubes the coordination of each atom is three (seeTerrones & Terrones 2003).

More generally, the concept of curvature in graphitic nanostructures can be under-stood by introducing different sizes of carbon rings in the hexagonal framework. Forexample, graphite is composed only of hexagonal rings; fullerenes contain 12 pentag-onal rings of carbon and any number of hexagons except one. Carbon nanocylinders(open or uncapped nanotubes) are composed only of rolled hexagonal sheets. How-ever, more complex structures could embed heptagonal or octagonal rings, as hasbeen shown by (Mackay & Terrones 1991, 1993; Terrones & Mackay 1992). Thisdemonstrates that graphite is so versatile that it can exhibit all the different geome-tries present in three dimensions: Euclidean, hyperbolic and spherical (Terrones etal . 1995).

It was first confirmed experimentally by Iijima and colleagues that graphite isable to curve negatively (Iijima et al . 1992; see figure 1). These curvature changesmay result in complex carbon arrangements with different electronic, mechanical


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(a) (b)

Figure 2. Graphitic cones. (a) Atomic model of a graphitic cone and(b) an arrangement of stacked cones forming a fibre.

Figure 3. Scanning electron micrograph of fibres composed of stacked cones obtained by thepyrolysis of palladium organometallic precursors (see Terrones et al . 2000). The cones are heldtogether by van der Waals forces.

and vibrational properties when compared with graphite or diamond. It is impor-tant to point out that curvature can be associated to complexity and complexity tofunctionality. Therefore, if nanotechnology is looking for applications of graphite-likenanostructures, these should exhibit various types of curvature and functionalities.

3. Carbon nanotubes

It is well known that graphite conducts electricity and is a semi-metal, while graphene(a single sheet of carbon made of hexagonal rings) is a zero-gap semiconductor.However, carbon nanotubes (CNTs) can be better conductors than graphite andgraphene, depending on their diameter and helical arrangement (the way the hexag-onal carbon rings are oriented along the tubule axis) (Hamada et al . 1992). Dekker’s


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(a) (b)

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Figure 4. High-resolution transmission electron micrographs ofdifferent conical fibres with different degrees of crystallinity.

(a) (b) (c)

Figure 5. Schwarzites. (a) Four cubic cells of the schwarzite type P (primitive) with octagonal andhexagonal rings of carbon. (b) One cubic cell of the schwarzite type G (gyroid) with octagonaland hexagonal rings of carbon. (c) One cubic cell of the gyroid triply periodic minimal surface.

and Lieber’s groups have confirmed experimentally that the electronic transport insingle-walled carbon nanotubes (SWCNTs) depends on their chirality and diameter(Wildoer et al . 1998; Odom et al . 1998).

Perfect carbon nanotubes could possess a very high rigidity with a Young’s modu-lus of up to 1.2 TPa; this makes CNTs the strongest fibres so far synthesized (Treacyet al . 1992). Although important efforts have been made to scale-up this strength,in order to build an ultra-high resistance material, it has not been possible to havemacroscopic composites with comparable Young’s modulus (see Terrones & Terrones2003). Recently, it has been reported that high electron irradiation is able to poly-merize bundles of SWCNTs with bending moduli of 70% of the value for isolatedSWCNTs (Kis et al . 2004). The authors produced strong local interactions (bonds)among the SWCNTs present within the bundles, in order to overcome the weak


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−10 −5 10energy (eV)

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Figure 6. Icosahedral fullerenes. (a) Icosahedral (Ih) fullerene with 1500 atoms. (b) Electronicdensity of states (LDOS) for different icosahedral (Ih) fullerenes with only 12 pentagonal rings.Note that as the fullerene becomes bigger the density of states approaches that of graphite, anddiminishes the states at the Fermi level (located at zero).

van der Waals forces that make individual tubes to slide with respect to each other.Another challenge is to embed efficiently CNTs in polymer matrices, thus producinga strong chemical bond between the tube surface and the matrix. We believe theseproblems will be solved in the near future by doping CNTs and by making theirsurfaces more reactive (see Terrones & Terrones 2003).

Several papers in this issue cover specifically the electronic and the mechanicalproperties of carbon nanotubes. However, there are more complex arrangements,some of them still hypothetical, which need to be explained in order to witness thebeautiful shapes, complexities and properties related to carbon nanomaterials.

4. Carbon nanocones

In 1994, H. Terrones proposed the possibility of generating graphitic cones (seefigure 2a) using isometric transformations (where distances are preserved) on agraphene sheet (Terrones 1994). Three years later, Ebbesen and co-workers reportedthe existence of carbon nanocones and nanodiscs produced by the pyrolysis of heavyoils (Krishnan et al . 1997). Stacked cones forming fibres (see figures 2b, 3 and 4) canalso be obtained by thermolysing organometallic palladium precursors as catalysts(Terrones et al . 2001). Interestingly, not only carbon can form cones: boron nitride


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Figure 7. Quasi-spherical fullerenes. (a) Quasi-spherical fullerene composed of heptagons, hexa-gons and additional pentagons in order to keep sphericity. (b) Electronic density of states ofdifferent quasi-spherical fullerenes.

nanocones have also been reported (Bourgeois et al . 2000). Charlier & Rignanese(2001) have calculated the electronic properties of graphitic cones with differenttypes and geometry, finding that these are very sensitive to the pentagonal rings atthe tip. Recently, Lammert & Crespi (2004) studied the electronic properties as afunction of the distances from the cone apex to the base; they also discussed theintrinsic Aharonov–Bohm effect in nanocones (see figures 2–4).

5. Schwarzites

A. L. Mackay, H. Terrones and M. Terrones have shown that novel geometries andshapes may emerge if different types of curvature are introduced via carbon rings ofvarious sizes (Mackay & Terrones 1991, 1993; Terrones & Mackay 1992; Terrones etal . 1995; Terrones & Terrones 1996, 1997a, b). Other research groups have followedthese ideas, confirming that the electronic and mechanical properties of these curvedarrangements are attributed to non-hexagonal sp2-like carbon rings (Lenosky et al .1992; Vanderbilt & Tersoff 1992; O’Keeffe et al . 1992).

Besides diamond and graphite, carbon may produce periodic hypothetical archi-tectures with negative Gaussian curvature, called schwarzites (in honour of H. A.Schwarz, a German mathematician who, in the 1870s, contributed greatly to thestudy of periodic minimal surfaces), which exhibit channels very similar to thosepresented in zeolites (see figure 5). These nanostructures possess negative curvature


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(a) (b)

Figure 8. Haeckelites. (a) In the background, a haeckelite planar sheet with heptagons and pen-tagons only. In the front, a haeckelite tube connected to a conventional zigzag carbon nanotube.(b) A corrugated haeckelite tube with hexagons, heptagons and pentagons of carbon.

(saddle-shaped surfaces) due to carbon rings with more than six atoms (heptagonsand/or octagons). In fact, the negative Gaussian curvature provides complexity tothe structure; it can be shown that the genus of the structure increases if the numberof heptagons or octagons increases. The genus of a sphere is zero, and the genus ofa torus is one. A discussion on the topology of graphitic structures can be found inTerrones & Mackay (1992), Mackay & Terrones (1993), Terrones et al . (1995) andTerrones & Terrones (2003).

Theoretical work has demonstrated that schwarzites cover a wide spectrum ofelectronic behaviours ranging from semiconductors to metallic conductors (Huanget al . 1993; Valencia et al . 2003). Recently, the synthesis of periodic graphite foamssimilar to schwarzites has been reported. However, the phases have not been fullyidentified (Benedek et al . 2003). What makes schwarzites more interesting is thepresence of channels, which could behave as molecular sieves, and interact stronglywith different types of molecules (figure 5). It is worth mentioning that carboniza-tion experiments using carbon-impregnated zeolites as templates resulted in periodiccarbon arrangements that could be associated with structures similar to schwarzites(Ma et al . 2001). The production of periodic and crystalline schwarzites will also beof the greatest importance in photonics.


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Figure 9. High-genus icosahedral fullerenes. (a) A genus-11 holey icosahedral fullerene withheptagonal rings at the necks of the holes; vertices of two types can be distinguished accordingto their surrounding faces (6, 6, 6) and (6, 6, 7). (b) Computed surface of a vesicle with genus 11,similar in topology to the holey fullerene shown in (a). (c) Local electronic density of states ofthe holey fullerene shown in (a). Note that in the sites (6, 6, 6) the density of states is moregraphitic (fewer states at the Fermi level, which is shown with the dotted line). At the necks(holes) the electronic properties change, increasing the number of states, indicating a differentelectronic behaviour in these regions.

6. Graphitic onions

Graphitic onions, concentric giant fullerenes, which were created under high elec-tron irradiation by Ugarte (1992), have also attracted the attention of numerousresearchers (see Terrones & Terrones 2003) (figure 6). It is interesting to observethat these carbon onions synthesized by Ugarte were highly spherical, possibly dueto the presence of heptagons and additional pentagons embedded in the cage molecule(figure 7) (Terrones & Terrones 1997a, 2003). At a different scale, molecular modelsof graphitic onions resemble the skeletons of the radiolaria drawn by Ernst Haeckel(Haeckel 1887; Thompson 1917). The electronic properties of large carbon shells,classical icosahedral giant fullerenes made of hexagons and only 12 pentagons, wouldbe similar to those of a graphitic sheet. As the cage grows, it approaches the elec-tronic properties of a graphene sheet (figure 6b) (Lopez-Urıas et al . 2003). For quasi-spherical cages, in which heptagons and additional pentagons are present, the density


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(a) (b)

(c) (d)

Figure 10. Toroidal graphite. (a) Torus obtained by bending a zigzag (8, 0) carbon nanotube.This tube has just hexagonal rings of carbon. (b) A torus formed with heptagons (interior), pen-tagons (exterior) and hexagons. (c) Hemitoroidal nanotube cap with heptagons and pentagonsproduced by arcing graphite electrodes. (d) Molecular model of the hemitoroidal cap shownin (c).

of electronic states changes significantly and becomes very different from that foundin ‘graphitic’ classical icosahedral giant fullerenes (figure 7b). First-principle calcu-lations performed on quasi-spherical fullerenes have shown that these systems lie indeep local minima on the potential energy surface (Heggie et al . 1998).

7. Haeckelites

Alternative hypothetical shapes for sp2 hybridized carbon arrangements consist oflayered crystals formed using different symmetries and carbon ring statistics, so thatthe nanostructure comprises heptagonal, pentagonal and hexagonal rings. In orderto compensate for the zero Gaussian curvature, the number of pentagons must equal


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(a) (b)

(c)

Figure 11. Tori and coalesced C60 molecules. (a) A torus obtained by coalescing C60 moleculesalong the 2-fold axis (octagons are formed at the necks). (b) A torus obtained by coalescing C60

molecules along the 3-fold axis (heptagons are formed at the necks). (c) A torus obtained bycoalescing C60 molecules along the 5-fold axis (octagons are formed at the necks).

that of heptagons. These structures have been called haeckelites in honour of ErnstHaeckel (Terrones et al . 2000; see figure 8.). Despite being planar and having sp2

character, haeckelites are intrinsically metallic (Crespi et al . 1996; Terrones et al .2000). When haeckelite sheets are rolled up, CNTs are formed and some of these pos-sess high electronic conductance (Rocquefelte et al . 2004). We believe that graphiticnanostructures irradiated with a TEM might exhibit, after considerable irradiationtimes, heptagonal and pentagonal defects in the graphitic lattice, thus producing thecharacteristic rugosity of haeckelite structures (see figure 8b).

8. Finite zeolites or holey fullerenes

High-genus graphitic closed arrangements with channels, similar to those found inzeolites and expected in schwarzites, have been proposed by Terrones & Terrones(1997b) (see figure 9). These finite zeolites can be envisaged as concentric fullerenesinterconnected by necks. It is in these necks that the negative Gaussian curvatureis concentrated due to the presence of heptagonal or octagonal rings of carbon.Interestingly, these sp2-hybridized structures do not close with the aid of pentagons,


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Figure 12. Electronic density of states for tori formed by the coalescence of C60 calculatedusing a Huckel approach. Note that in all cases there are states at the Fermi level.

and contain only hexagons and heptagons or octagons. The electronic properties ofthese systems, calculated using tight-binding Hamiltonians, revealed that the localelectronic density of states changes dramatically in highly curved regions (Ricardo-Chavez et al . 1997). Here, the electronic behaviour on the channels (necks), where theheptagons are located, is different from that on the hexagons (figure 9c). Therefore,it is possible to find active sites in the channels or cavities in the same arrangement.Note that these small-diameter channels or cavities are smaller than those found inthe hollow cores of SWCNTs, and are stable with little strain in holey fullerenes. Wehave to remember that carbon nanotubes smaller than 10 A in diameter are not verystable; however, channels of 4 A in diameter can be stabilized if there is negativecurvature around, which relaxes the strain in the structure.

9. Toroidal and helicoidal graphite

During early experiments with carbon fibres, Motojima et al . (1991) found coiledcarbon structures up to several micrometres in length. Three years later, Amelinckxet al . (1994) showed that helical carbon nanotubes or corkscrew nanotubes could besynthesized using thermolytic methods in the presence of metal catalysts (see Ter-rones & Terrones 2003). Amelinckx and co-workers also proposed a possible growthmechanism based on the concept of spatial velocity hodograph. Subsequently, the firstreports showing experimental evidence of toroidal carbon nanostructures appeared(Liu et al . 1997; Martel et al . 1999). In fact, the stability of these tori was stud-ied first theoretically by Itoh et al . (1993) and Ihara et al . (1993). Geometrically,


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(a) (b)

Figure 13. Tori generated from haeckelite sheets. (a) Hexagonal haeckelite (containing pentagons,hexagons and heptagons) rolled into a torus. (b) Rectangular haeckelite (containing pentagonsand heptagons only) rolled into a torus.

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Figure 14. Electronic density of states for tori constructed from haeckelite sheets. (a) Hexago-nal haeckelite (hexagons, heptagons and pentagons); (b) rectangular haeckelite (heptagons andpentagons).

toroidal graphite can be constructed by joining the extremes of a carbon nanotube(see figure 10a); in this case the structure is constructed only from hexagonal ringsexhibiting genus 1 (a similar genus to that of a doughnut or of a coffee cup with onehandle). The electronic properties of these tori are similar to those of long SCWNTs.For example, if we generate a torus with a semiconducting tube, the torus will besemiconducting; likewise for metallic SWCNTs (Lopez-Urıas et al . 2003). Liu et al .(2002) published a study on the magnetic properties of graphitic tori, in which,depending on the size and diameter of the torus, the structure could reveal colos-sal paramagnetic moments. Theoretical work on the magnetic properties of carbonnanostructures has also been reported (Haddon & Pasquarello 1994; Haddon 1995).


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These studies are relevant in the field, since magnetism in pure carbon structureshas been reported experimentally (Makarova et al . 2001).

Another geometrical route for constructing tori is by the introduction of heptagonsand pentagons proportionally in a closed graphitic framework, so that the curvatureis compensated (see figure 10b). In this case, the heptagonal rings exhibit a deficit inelectronic charge, whereas the pentagonal rings reveal an excess of charge (Johnsonet al . 1994). Haddon (1997) reported that this kind of architecture might possessnovel magnetic responses. Hemitoroidal carbon nanotube tips with negative Gaussiancurvature have also been found using the arc-discharge method (see figure 10c, d).

We have calculated, using a Huckel approach, the electronic properties of vari-ous types of tori generated by coalescing C60 molecules. Here, the density of statesstrongly depends on the way the C60 cages join: along the 2-fold axis, along the3-fold axis or along the 5-fold axis (see figures 11 and 12). These results agree withthose reported by Hernandez et al . (2003), using tight binding calculations in corru-gated nanotubes obtained by coalescing C60 cages. A more complex decoration of atorus can be obtained if we use haeckelite tubes (see figure 13). According to Huckelapproach calculations, all haeckelite tori should be metallic (see figure 14).

10. Coalescence as a method of increasing complexityin carbon nanostructures

We have observed that the shape and symmetry of carbon nanostructures dictatethe electronic properties. Experimentalists are now able to produce bulk amountsof fullerenes and carbon nanotubes. However, other nanostructures with novel mor-phologies have been found unintentionally in different types of by-products. Thequestion that arises now is how, in already made carbon nanostructures, to inducechanges that increase their complexity and therefore functionality. In this respect, animportant step forward has been taken using electron irradiation in a TEM. Ugarte(1992, 1993) carried out the first experiments in the graphitic systems. He showedthat, by irradiating amorphous carbon and polyhedral graphitic particles, it waspossible to obtain quasi-spherical graphitic onions.

Another unexpected irradiation experiment, using a TEM equipped with a heatingstage, was performed by Banhart & Ajayan (1996). The authors irradiated carbononions and produced nano-diamond in the cores of the particles. Similar irradiationexperiments at high temperatures showed that SWCNTs could coalesce to formtubes of double the diameter, and produce ‘X’, ‘Y’ and ‘T’ junctions (Terrones etal . 2000a, 2002) (see figures 15 and 16). Therefore, SWCNTs could be manipulatedand used as building blocks for higher hierarchical structures. For example, two-and three-dimensional architectures made of SWCNTs are possible (see figure 17).In SWCNTs, junctions of heptagons or higher member rings should be present inorder to preserve the sp2 bonding. The rectification properties of carbon nanotubejunctions have been calculated using tight binding and other approaches (Andriotiset al . 2001).

The introduction of C60 molecules inside SWCNTs has attracted experimental-ists to study these molecules in a confined environment (Smith et al . 1998). In situcoalescence of C60 fullerenes inside a SWCNT has been observed under the TEM(Smith & Luzzi 2000). The electronic properties of different types of coalesced C60nanostructures have been calculated. We find that, when joining the molecules along


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2 nm

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Figure 15. High-resolution transmission electron micrograph showing the spontaneouscoalescence of two single-walled carbon nanotubes within a bundle of 14–15 tubes.

the 3-fold axis, the tube is semiconducting and, when joining the molecules alongthe 2-fold or the 5-fold axis, the tubes conduct electrons (Hernandez et al . 2003).Recently, Bandow et al . (2004) have studied the Raman signal from carbon nano-structures obtained by the coalescence of C60 inside a SWCNT.

11. Shapes and complexity in boron nitride and BCN nanostructures

Hexagonal boron nitride (h-BN), consists of another layered structure, which maybe curved and bent, thus forming different nano-morphologies. h-BN is constructedfrom hexagonal layers made of alternating atoms of boron and nitrogen, in which allhexagons from different layers eclipse one another (see figure 18). BN nanotubes andonion-like arrangements have been successfully synthesized in the laboratory usingdifferent methods (Chopra et al . 1995; Loiseau et al . 1996; Terrones et al . 1996a).


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5 nm(a)

(b)

Figure 16. High-resolution transmission electron micrographs depicting the coalescence of sin-gle-walled carbon nanotubes forming an ‘X’ junction. Drawings and molecular models are alsoshown.

Figure 17. A nanotube fabric. A three-dimensional network made by joiningsingle-walled carbon nanotubes. The junctions require heptagonal rings.

From the theoretical point of view, BN could be curved (similar to graphene sheets),by adding non-hexagonal BN rings, but the structure will be more stable if the boronand nitrogen atoms alternate. In other words, pentagons and heptagons will resultin the creation of B–B or N–N bonds that are less favourable in energy. However,squares and octagons will be energetically more suitable. Nevertheless, there are


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(a)

(b)

5 nm

Figure 18. Boron nitride structures. (a) Atomic structure of hexagonal boron nitride (BN).(b) HRTEM image of a multi-walled BN nanotube produced by an electric arc-discharge onBN–Ta electrodes (Terrones et al . 1996b).

calculations that show that an excess of nitrogen atoms could produce reasonablystable BxNy nanostructures (Fowler et al . 1999) (see figure 19).

In BN, the positive curvature will be due to squares, and octagons will cause thenegative curvature. BN cages could be closed with three squares; in fact this canbe seen in the tips of BN nanotubes showing right angles, close to 90◦, observed inHRTEM studies (see figure 18b). Theoretical calculations indicate that BN nanotubesare insulators in the same way as bulk hexagonal BN (Egap ≈ 5 eV). It is not expectedto find conductance in BN curved nanostructures, but they could play in importantrole as reinforcements in composites and high-temperature ceramics.

With the above information, different complex arrangements could be imaginedfor BN schwarzite-like structures, BN haeckelites, toroidal BN, etc. (see figure 20).However, there has been more interest in hybrid BxC, CNx and BxCyNz nanomater-ials, since carbon nanotubes can be doped with boron and/or nitrogen (Terrones etal . 1996a, b, 1999). Usually boron- and nitrogen-doped CNTs exhibit an enhanced


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(a)

(b)

(c)

(d)

Figure 19. Molecular models of BN nanotube tips—different views and cross-sections. B atomsare denoted by open circles, N atoms by dark circles. (a) A zigzag tubule with three squaredefects included in a BN hexagonal framework; (b) an armchair tubule with four squares andone octagonal ring; (c) a zigzag tube containing three pentagon-pair defects with an excess ofnitrogen; (d) an armchair tube with three pentagon-pairs. In (c) and (d) there is an excess ofonly two nitrogen atoms per pentalene group. It is noteworthy that the rotation around the tubeaxis demonstrates that the flat profile of the tip is preserved in all cases, which is consistentwith experimental observations. However, the two tubes containing pentagon-pair defects havemore rounded corners, due to the release of energetic strain.

electrical conductivity when compared with conventional (un-doped) CNTs; in thecase of boron doping, the conductivity is dominated by holes, whereas for nitrogendoping, the transport is ruled by donors (electrons) (Czerw et al . 2001). Anotherinteresting feature is that boron favours zigzag chiralities (Blase et al . 1999) andimproves the electron emission of carbon nanotubes (Charlier et al . 2002). Besides,CNx nanotubes are highly conductive and are able to store gaseous N2 in their hollowcores (Terrones et al . 2000b).

Note that CNx nanotubes are highly reactive and could strongly interact with inor-ganic, organic molecules and even toxic gases (Jiang et al . 2003, 2004; Villalpando-Paez et al . 2004).


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12. Shape and complexity in metal dichalcogenides

Metal dichalcogenides (MDs) such as molybdenum sulphide (MoS2), tungsten sul-phide (WS2), niobium sulphide (NbS2) and rhenium sulphide (ReS2) are layeredstructures in which the metal atoms are sandwiched by two sulphur atoms (see fig-ure 21). Reshef Tenne’s group demonstrated that MoS2 and WS2 can form nanocagesand nanotubes (Tenne et al . 1992, 1993; Margulis et al . 1993) (see figure 21).Although the synthesis process for forming curved nanostructures is different fromthose used for graphite, geometrically, tubes are envisioned by rolling up individuallayers in the same way as graphite, and thus they acquire curvature. The creationof cages of MD materials is more complicated because the positive curvature isachieved by square-like defects and the negative curvature by octagon-like defects(see figure 22).

Density functional tight binding calculations show that both armchair and zigzagMoS2 and WS2 are semiconductors with a decreasing gap as the diameter decreases.Flat and bulk MoS2 and WS2 layers show a semiconducting behaviour (ca. 2.1 eV)with a larger gap than nanotubes (Seifert et al . 2000a, b). Interestingly, MoS2 andWS2 armchair nanotubes exhibit both an indirect and direct band gap (similar tothe bulk materials), whereas zigzag tubes only display a direct gap. Nevertheless, allMoS2 and WS2 nanotubes will be semiconducting with much narrow gaps (ca. 0.2–1.5 eV). MoS2 and WS2 cages are also exceptional solid lubricants, which could beused to diminish wear in engines, etc. (Rapoport et al . 1997, 2003a, b).

Hsu et al . have reported the synthesis of doped MD with carbon producing C–MoS2, C–WS2 and WxMoyCzS2 nanotubes (Hsu et al . 2000, 2001). Similarly, WS2have been doped with Nb atoms (Zhu et al . 2001). Whitby et al . (2001) have beenable to cover carbon nanotubes with an individual layer of WS2. Therefore, furtherexperimental and theoretical research on the electronic and mechanical properties ofthese layered systems is needed.

Rao’s group reported the existence of NbS2 nanotubes (Nath & Rao 2001) soonafter their theoretical prediction by Seifert et al . (2000c). Density functional tight-binding calculations on these nanotubes reveal that all NbS2 nanotubes are metallic(Seifert et al . 2000c). The metallic character of these tubes comes from the fact thatNb has one ‘d’ electron less than Mo or W in the valence shell. In fact, the flat layerof NbS2 is metallic and becomes a superconductor at ca. 6 K. It remains a challengeto measure the electronic transport of these low-temperature superconductor nano-tubes. At present, it is necessary to produce bulk amounts of nanotubes made of newlayered materials, in order that their potential applications are exploited, and noveltechnologies arise.

13. Conclusion

It has been demonstrated that complex forms and shapes may be obtained fromlayered materials if curvature is introduced. The electronic and mechanical propertiesof curved nanostructures are expected to be very different from those found in thebulk material, so if new applications are envisaged, then complex layered materialsshould be considered as good candidates in developing nanotechnology. We believethat the different shapes observed in nature at high levels of complexity shouldinspire new morphologies in the nanocosmos. Thus, besides carbon nanotubes and


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(a) (b)

Figure 20. Molecular models of (a) a BN fullerene (cage) and(b) a genus-5 BN holey fullerene-like structure.

(a)

(b)

Figure 21. (a) A molecular model for a zigzag MoS2 nanotube. Side view shows the triple-layerarrangement with Mo between sulphur atoms. (b) A model of an Mo100S200 cage closed with sixsquare-like defects.


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(a) (b)

(c) (d)

Figure 22. Closed tips of (a) zigzag and (b) armchair MoS2. (c) A square-like defect producesthe positive curvature in MoS2 nanotubes. (d) Octagonal defects are responsible for the negativecurvature in these metal disulphide nanotubes.

fullerenes, there is a wide range of curved nanostructures awaiting to be discovered—not only those made of carbon, but those with metal dichalcogenides and otherlayered compounds.

We thank Dr Emilio Munoz-Sandoval, Daniel Ramırez-Gonzalez and Lisette Noyola-Cherpitelfor fruitful discussions and valuable assistance in some of the work shown here. We are also grate-ful to CONACYT-Mexico (Grants 36365-E, 37589-U and J36909-E), the UC-MEXUS throughGrant no. PS/CN 02-114 and the CIAM initiatives (Grants C02-41464 and C02-42428).

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