The minimum energy structure of C32 is a hollow cage with open windows ... to form closed (cage-like) structures in order to minimize the number of dan-.
STRUCTURAL AND THERMAL PROPERTIES OF CARBON NANOSTRUCTURES: FULLERENES AND NANOTUBES E. Hernandez, M.J. Lopez, P.A. Marcos, J.A. Alonso, and A. Rubio
Departamento de Fsica Teorica. Universidad de Valladolid. E-47011 Valladolid. Spain Abstract
In this paper we review the results of theoretical investigations on structural, dynamical and electronic properties of fullerenes and nanotubes recently conducted in our group. Such investigations have been carried out with a range of theoretical methods, including empirical potentials, tight-binding and rst-principles electronic structure simulation techniques.
1 Introduction Since their discovery in the mid-eighties, fullerenes have attracted an enormous degree of attention in both the Chemistry and Physics communities. Not only a hitherto unknown form of carbon had been found, it also raised much expectation regarding its potential applications in novel organic chemistry and
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materials science. More recently, the discovery of nanotubes has rekindled the expectations of important practical applications. Nanotubes can be either metallic or small band-gap semiconductors, depending on their chirality; their high aspect ratio (length vs. radius) makes them essentially one-dimensional microscopic systems, where quantum eects on electron conduction can be (and indeed are) investigated. Their seamless tubular structure makes them highly resistant and possibly very sti, properties that could make them very valuable as the building blocks of new materials. It is thus no wonder that fullerenes and nanotubes are being studied both experimentally and with theoretical methods by many groups throughout the world. But in spite of all the progress achieved in recent years in the science of carbon based nanostructures, signi cant gaps persist in our understanding of these materials. For example, very little is known about the growth mechanisms at play during their synthesis. Thus far direct observation of the growth process has not been possible, and those growth models that have been proposed do not seem to account correctly for all the experimental observations. Clearly, a more detailed understanding of fullerenes and fullerene-based materials is needed before the potential of these novel structures can be fully exploited. In this paper we review the results of a number of theoretical calculations which constitute our contribution towards this growing understanding. The structure of the paper is as follows: we begin (section 2) with a brief introduction to the theoretical methods employed in the calculations reported in subsequent sections. In section 3 we report results of studies of fullerene clusters, focusing in particular on the possible isomeric forms of small clusters, their thermal behaviour and the dynamics of the Stone-Wales isomerisation reaction in C60. Section 4 is devoted to the study of carbon nanotubes. We discuss results of investigations on their electronic, elastic as well as dynamical properties.
2 Theoretical methods Due to the large number of atoms involved in the formation of nanotubes and fullerenes, realistic rst-principles simulations are very dicult and costly to perform. To circumvent this problem empirical and semiempirical methods have been developed over the years, and great experience has been gained in their application to carbon based materials. The structural properties are quite well described by the Terso potential that includes, in an eective way, many-body interactions . More realistic models including electronic eects are obtained through tight-binding methods using an sp-minimum basis-set with parameters tted to reproduce ab-initio results for structural, elastic and vibrational properties of dierent carbon phases. When the size of the nanostructure decreases or atoms other than carbon are incorporated in the nanotubes, a more predictive and less empirical theory is desirable as it is the case of the ab-initio rst-principles methods based on the density-functionaltheory (DFT). In this section we give a brief description of those theoretical methods that are used to perform the simulations reported later in the paper. Plane-wave based calculations using DFT theory and the pseudopotential approximation are a well established technique in condensed matter physics . Since the seminal paper of Car and Parrinello , this technique has become extremely popular for the ab initio simulation of materials, and has been applied to many important problems in materials science. Though its main asset is its accuracy and reliability, it has the disadvantage of being extremely computationally expensive, and occasionally this makes advisable the use of semiempirical models, such as tight-binding (TB) or empirical potential methods, which, in spite of their simplicity, are often capable of providing surprisingly accurate results. We have performed the ab-initio calculations using the standard plane-wave pseudopotential total-energy scheme [2, 4, 5] in the local
density approximation (LDA)  to the exchange correlation potential. Abinitio norm-conserving nonlocal ionic pseudopotentials have been generated by the soft-pseudopotential method of Troullier and Martins . The LDA wave functions were expanded up to a 40-Ry cuto (see refs. [4, 5] for details of the method). The TB model used here was developed by Porezag et al. , and has been used for the simulation of both bulk phases of carbon (crystalline and amorphous) as well as for hydrocarbons. In contrast to the conventional TB models which have been so popular in computational materials science, such as those due to Goodwin et al.  or Xu et al. , which are constructed by tting directly to either experimental data or to data obtained from rst principles calculations, the model of Porezag et al. is constructed directly from LCAO DFT calculations, using a series of well de ned approximations. A minimal basis set (one s orbital per H atom, one s and three p orbitals per C atom) is used. Orbitals centred on dierent atoms are not assumed to be orthogonal, i.e. this is a non-orthogonal TB model, in contrast to the previous ones [9, 10]. The Hamiltonian and Overlap matrices are tabulated as a function of the internuclear distance, and this tabulation is obtained from DFT calculations using the same basis set (for details see ref. ). As usual in TB approximations, the Hamiltonian incorporates only two-centre contributions, while three-centre integrals are disregarded. Furthermore, selfconsistency eects in the matrix elements are not included. This is usually a good approximation in situations where there are no signi cant charge-transfer eects, as it is the case for the systems under study in the present work. The total energy consists of two terms: the TB band structure energy, given by the sum of eigenvalues associated to occupied eigenstates of the Hamiltonian weighted by their corresponding occupations, and a repulsive pair potential. The repulsive pair potential is constructed so as to give the dierence between
the exact DFT energy and the band structure energy in some chosen reference system, usually the dimer. In this way, at least for the reference system, the method is guaranteed to give the same results as the LCAO DFT calculation. One of the attractive features of this TB model is that all approximations listed above can be systematically improved upon. For example, it would be possible to include more basis functions in the treatment if a more accurate description of conduction states is found necessary. Likewise, though threecentre integrals are disregarded, they could be included in a generalisation of the model, should this be necessary. For details on how this model is constructed, the reader should consult ref. . In our study of the structural and thermal properties of small carbon clusters and fullerenes, we use an empirical potential, introduced by Terso , to mimic the covalent bond between carbon atoms in the cluster. The Terso potential is based on the \bond order" ideas introduced by Abell , i.e., the bond order (or in dierent words, the strength of each bond) in real systems decreases monotonically with increasing atomic coordination. The actual dependence of the bond order on coordination should be such as to yield the intermediate equilibrium coordinations characteristic of covalent systems. Moreover, certain bond angles, i.e., the ones corresponding to the sp2 and sp3 hybridisations of carbon, should be favored by the potential. All these elements are built directly into the functional form of the Terso potential, which depends explicitly on both coordination and bond angles. A smooth Fermi-type cuto function  is used to cut the interactions between rst and second nearest neighbors. It is fair to recognize that this potential goes beyond the three-body potentials considered in the literature for describing covalent systems. The Terso potential has been adjusted to reproduce adequately the cohesive energies of several real and hypothetical carbon polytypes (i.e., the C2
dimer, graphite, diamond, and the simple cubic, the body centered cubic, and the face centered cubic lattices) as well as the lattice constant and bulk modulus of diamond. The resulting potential is able to describe the structural features and the energetics of carbon over a wide range of con gurations (i.e., as a function of coordination). It also reproduces the elastic and defective properties of diamond and graphite. This makes this potential reliable for cluster studies.
3 Fullerene clusters The easy formation of the C60 buckminsterfullerene (BF)  (buckyball for short) under the right experimental conditions  contrasts with our lack of understanding of the growth mechanisms of small carbon clusters leading to the formation of this highly symmetric, all carbon molecule. Several growth models have been proposed in the literature, e.g. the pentagon road model, the fullerene road model, or the cycloaddition model . However none of these models ts all the experimental evidences well. Other important drawbacks of these models are that they neither explain the absence of other cluster sizes, nor lead to a preferential formation of the buckyball with respect to other fullerenelike isomers of C60 (let us remind the reader that there are 1812 topologically dierent fullerenelike isomers of C60 ). In addition, all these models require, in the last stages of BF formation, an annealing mechanism, involving the rearrangement of pentagons and hexagons, driving the fullerene isomers of C60 into the most stable BF structure (the only fullerene having Ih symmetry and no adjacent pentagons). Even this last process of annealing the fullerene structures down to BF is not well understood. The diculty in understanding the formation mechanism of C60 BF stems from the fact that its production does not take place under thermal equilibrium conditions but rather it involves a kinetics-controlled process. As a consequence, knowing
the zero temperature properties of small carbon clusters and fullerenes is not sucient for explaining the formation of C60 BF. Learning about the thermal behaviour (including the possible phases, isomerization transitions and phase changes) of small carbon clusters and fullerenes would help us to understand the processes and mechanisms leading to the formation of buckyballs, fullerenelike structures, and other carbon polytypes. We have performed extensive constant energy molecular dynamics (MD) simulations with the Terso potential to extract the structural and thermal characteristics of small carbon clusters and fullerenes. The classical Newtonian equations of motion are integrated numerically, using the velocity version of the Verlet algorithm . A time step of 0.2 fs yields conservation of the total energy within 0.01% for trajectories lasting 2:5 105 time steps.
3.1 Isomer hierarchy It is well known that small clusters may exhibit a large number of dierent isomeric forms and that this number increases tremendously fast with cluster size, e.g., it has been estimated that a 55-atom cluster possesses 8:3 1011 isomers . For this reason we do not intend to obtain all the possible isomers of a given cluster but rather to generate a suciently large number of isomers for extracting the main structural features of the clusters and relating them to the thermal behaviour and, possibly, the growth mechanisms of carbon clusters and fullerenes. We use the thermal quenching procedure for obtaining the possible isomeric forms of small carbon clusters. In this procedure, a high energy con guration of the cluster is cooled down into the rigid structure corresponding to a local minimum of the potential energy surface. A relevant set of isomers is obtained by quenching a number of con gurations (500) generated along high energy trajectories originating on dierent wells of the potential energy surface corresponding to planar, bowl-type, hollow cage, and
diamond structures, respectively. There is a big controversy nowadays about what is the minimum energy structure of small carbon clusters . The theoretical predictions depend strongly on the level of the theory beeing used for studying the clusters. Thus, for intance, the structure of C20 has been extensively studied in the literature. C20 is the smallest topologically possible fullerene, having 12 pentagonal faces and none hexagonal face (see Fig. 1L). This structure has been found to be the ground state of C20 in electronic structure calculations at the level of MP2 (second order Moller-Plesset perturbation), as well as in CCSD(T) calculations  (coupled-cluster calculations with all single and double excitations and a perturbational estimate of some triple excitations), or within the local density approximation (LDA) of DFT. Hartree-Fock calculations as well as DFT with generalized gradient corrections (GGA) give, however, the monocycle ring as the minimum energy structure of C20 . Ab initio calculations using higher level exchange-correlation functionals  such as B3LYP (Becke three-parameter exchange with Lee-Yang-Parr correlation) or BPW91 (Becke 1988 exchange with Perdew-Wang 1991 correlation), as well as quantum Monte Carlo approaches  predict the bowl structure (a portion of the C60 fullerene formed by a pentagonal face surrounded of ve hexagons, (Fig. 1G)) as the most stable isomer of C20 . Empirical and semiempirical potentials (as the one used here) are not suitable for solving this controversy. However, the use of semiempirical potentials allows one to perform an extensive search of isomers on the potential energy surface and therefore to extract the underlying rules which determine the cluster structures. We nd the bowl structure as the most stable isomer of C20 in agreement with the most sophisticated ab initio calculations [19, 21]. However, as we just warned, one should not overemphasize this agreement. Besides the three geometries of C20 mentioned above, we nd also many more isomers,
Figure 1: Minimum energy structures and some selected isomers of C13 , C20 , and C32 . The energies of the isomers (in eV) are measured from the corresponding ground states of the clusters
some of which are shown in Fig. 1. Defect bowl-type structures, obtained by substituting some of the hexagonal rings of the minimum energy structure by pentagons and heptagons, appear at energies of 2.3 eV or higher above the ground state. The bowl isomers lower slightly their energies (except the ground state that, of course, increases its energy) by a window opening mechanism in which a carbon-carbon bond breaks producing an 8- or a 9-membered ring (Fig. 1I). Within 1 eV of the minimum energy structure of C20 we nd several perfect (i.e., formed exclusively by hexagons) and defective (i.e., one hexagon is substituted by one heptagon) graphitic-like isomers (Fig. 1H). We have been able to identify also some cage structures (e.g. Fig. 1J), other than the fullerene. These cages release, partially, the strain energy of the fullerene, i.e., lower their energies, by opening one or several windows in the dodecahedral structure (the window opening mechanism is the same we have reported for bowl structures). The minimum energy structure of C32 is a hollow cage with open windows (OW; see Fig. 1O). The C32 fullerene (Fig. 1P) is also quite low in energy, which means that the strain energy associated with this closed structure is small. The bowl geometry (a portion of the C60 fullerene) is higher in energy (Fig. 1Q). As we have mentioned at the begining of this section, there are 1812 topologically dierent fullerene isomers (hollow cages formed by 12 pentagons and 20 hexagons) of C60 . The most stable buckminsterfullerene (BF) structure has Ih symmetry and is the only C60 fullerene which satis es the isolated pentagon rule, i.e., there are no adjacent pentagons in the structure. Besides the fullerene isomers, many other isomers are formed by opening one or several windows in the fullerene cage. In summary we can say that there are two competing eects which determine the geometries of small carbon clusters: on one hand there is a tendency
to form closed (cage-like) structures in order to minimize the number of dangling bonds, and on the other hand the strain energy of structures with high curvature opposes the closing of small clusters. These two trends combine in small carbon clusters giving rise to a window opening mechanism which appears to help releasing the strain energy of the closed geometries. A window mechanism has been proposed for explaining the formation of endohedral fullerenes. Our simulations show that the OW structures are metastable states of the clusters with a nite lifetime that provides further support for that mechanism.
3.2 Thermal behaviour To investigate the thermal behaviour, i.e., phases and phase transitions, of the clusters we stepwise increase their temperature (T 50 K) starting from the zero temperature con gurations corresponding to the ground state and a few selected isomers, until cluster fragmentation or evaporation occurs (T 5000 K). The temperature of the clusters is increased by scaling up the velocities of all the atoms in the cluster between successive MD runs of 20 to 50 ps each (since the time step is 0.2 fs this means a number of time steps between 1 105 and 2:5 105 per trajectory). In this way we have generated the caloric curves (time-averaged kinetic energy vs total energy) of C13 , C20 , and C32 clusters (see Fig. 2). For a given cluster size, dierent branches of the caloric curve are obtained by considering dierent isomers as the zero temperature starting point for the heating up process. The horizontal separation between branches, at zero kinetic energy, corresponds to the energy dierence between the corresponding isomers. At low energies, the dierent branches of the caloric curve for a given cluster size are well resolved and run almost parallel. This behaviour is characteristic of solid-like clusters. The atoms perform oscillations around
0.8 /at (eV)
C13 0.6 0.4 A
0.2 0.0 C32
C20 0.6 0.4 0.2
-5.5 -5.0 ETot/at (eV)
-5.5 -5.0 ETot/at (eV)
Figure 2: Caloric curves of C13 , C20 , and C32 clusters. A, G, L, etc. indicate the isomer (see Fig. 1) which generates the corresponding branch of the caloric curve. their equilibrium positions but the overall shape (structure) of the cluster is preserved. The changes in slope of the caloric curves (see Fig. 2) occurring at higher energies are associated with structural changes of the clusters, i.e., isomerization and phase transitions. At high energies all the branches of the caloric curve of a given cluster size tend to merge together in a single curve. This behaviour is characteristic of the solid-to-liquid phase transition. However, the caloric curves end up before reaching a common regime for all the branches. This re ects that small carbon clusters start to fragment or to evaporate atoms or C2 or C3 units before fully developing a liquid-like phase. Some changes in the slope of the caloric curves may be observed at energies lower than the energy range corresponding to the solid-to-liquid phase transition. These changes in slope re ect isomerization transitions occurring at a slow rate and involving only a few isomers. The pronounced jumps up exhibited by several branches of the caloric curves originating in isomers other than the minimum energy structure correspond to isomerization transitions which
take the cluster from higher to lower potential energy regions of the con guration space with the corresponding increase in the cluster kinetic energy. Despite the lack of a fully developed liquid-like phase we nd, as we progress in the energy range corresponding to the solid-to-liquid phase transition, more and more frequent isomerization transitions involving more and more isomers, what eventually results in the collapse of all the branches of the caloric curve of a given cluster size. The structural changes of the clusters which take place as the cluster energy is increased are also re ected, as almost discontinuous jumps up in the relative root mean square bond length uctuation (Fig. 3). Small (< 10%) values of correspond to solid-like clusters with well de ned structures, large (> 20%) values of correspond to clusters in which the atoms perform diusive type of motions, and a sharp increase (over a nite range of temperatures) of the magnitude of between those limiting values is the signature of a phase transition. 0.4 C13
0.3 0.2 0.1 0.0
0.3 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6 0.8 kT (eV) kT (eV) kT (eV)
Figure 3: Relative root mean square bond length uctuation of C13 , C20 , and C32 clusters. The coding of the graphs is the same as in Fig. 2. To better understand the high temperature behaviour of small carbon clus-
ters, in particular the solid-to-liquid phase transition, we have performed a thermal quenching analysis of some branches of the caloric curves of C20 and C32 . Through this analysis we are able to identify which are the isomers \visited" by the cluster depending on the branch of the caloric curve considered and the cluster energy. Our study reveals that some isomers are thermally isolated from each other (at least within the time scale of our simulation runs), i.e., there is not a thermally activated pathway of isomerization transitions connecting them. For instance the bowl-type and the graphitic like structures of C20 never converted into a fullerene, a cage, a diamond-like, nor a tubelike isomers; the fullerene and the cage structures of C20 never converted into a diamond-like nor a tube-like isomers and vice versa. Similarly the bowltype structures of C32 never converted into a fullerene nor a cage isomers. At the highest temperatures considered here the clusters tend to form quite open structures before fragmenting. However, in some cases we have observed cluster reconstruction forming either bowl-type or graphitic like structures.
3.3 The Stone-Wales annealing mechanism The Stone-Wales (SW) transformation has been proposed in the literature  as a basic process for ring rearrangement in a fullerene cage and, consequently, for annealing fullerene structures down to its ground state. This transformation has been shown to have quite high activation barriers that could prevent it from being an ecient annealing mechanism for fullerenes. We have found  (through dynamical simulations) a thermally activated mechanism for the SW transformation of C60 . This mechanism involves isomerization transitions through intermediate isomers which connect the BF structure with the C2v fullerene isomer obtained via SW transformation of BF (that isomer, containing two pairs of adjacent pentagons, is the necessary last step in the annealing of fullerenes before reaching BF). A multi-step process, as the dynamically
driven mechanism presented here for the SW transformation, was completely unexpected on the grounds of the structural optimizations performed by other authors[27, 28, 29]. On the other hand, this new mechanism may help to understand the kinetics of the annealing process. Starting from the minimum energy structure (BF) of C60 at T = 0K we heat up the C60 fullerene until the break up of the cage occurs (at T 5000 K). The BF structure remains intact (the carbon atoms just oscillate around the equilibrium positions) up to a temperature of about 2500 K. At higher temperatures C60 experiences structural changes, i.e., isomerization transitions between BF and fullerene-like isomers having one or several open windows in the cage. At a temperature of about 3900 K we were able to identify the Stone-Wales transformation of BF. A detailed analysis of the trajectory reveals that the SW transformation involves a new multi-step isomerization transition process, in contrast with the commonly assumed single-step mechanism. The isomers involved in the SW transformation were identi ed using the thermal quenching procedure. The mechanism of the SW transformation found in our simulations proceeds through the following isomers (see Fig. 4): 1) a window opens between a pentagon and an hexagon; 2) a second window opens next to the rst one; these two windows share a C2 handle which lies about 10% outwards of the fullerene cage; 3) the C2 handle rotates and re-closes one of the windows, changing the relative positions of the pentagon and the hexagon which formed that window; 4) the second window re-closes changing also the relative positions of the corresponding pentagon and hexagon and producing, consequently, the SW isomer. We have generated a continuous low energy path on the potential energy surface connecting the BF and SW structures of C60 through those isomers (see Fig. 4). Between every two successive isomers, I and F, with coordinates XI and XF, respectively, the reaction coordinate is de ned
0 BF X
X X Reaction Coordinate
Figure 4: Low energy path for the SW transformation of BF. The reaction coordinate between every two successive isomers, I and F, with coordinates XI and XF, respectively, is de ned as X = (1 ) XI + XF ( = 0 1). Between isomers III and IV an intermediate con guration (indicated by the arrow in the gure) has been considered. This con guration, suggested by the dynamical process, is obtained by a 20 degrees rotation of the C2 handle in isomer III. The geometries of the isomers involved in the multi-step isomerization transition are also shown. as X = (1 ) XI + XF (with ranging from 0 to 1). The highest barrier along this path is found between isomers III and IV and lies 6.89 eV above the ground state of C60 . This barrier height can be signi cantly reduced, down to 5.58 eV above the ground state of C60 , by considering an intermediate con guration between isomers III and IV obtained by a 20 degrees rotation of the C2 handle in isomer III. This intermediate con guration is suggested by the dynamical pathway of the SW transformation and is indicated by the arrow in Fig. 4. The value of the barrier height between the BF and the SW struc-
tures obtained here is somewhat lower than the activation energies (6-7 eV) found by other theoretical methods for a direct (one-step) SW transformation. What is probably more noticeable about the multi-step mechanism proposed here is that the step-to-step isomerization transition barriers are substantially reduced (none of them exceeds 2.6 eV) with respect to the one for the global process. As a consequence the intermediate meta-stable isomers connecting two fullerene structures may play an important role in de ning the kinetics of the annealing process without relying on an auto-catalysis mechanism. Our results on the structural and thermal properties of small carbon clusters and fullerenes in conjunction with the proposed multi-step-isomerizationannealing process, opens a new road for theoretical investigations on the formation mechanisms of dierent carbon structures including fullerenes, onions or nanotubes from a hot carbon-plasma or by cluster coalescence.
4 Carbon nanotubes Nanotubes have recently attracted a great deal of attention, from theoreticians and experimentalists alike [30, 31]. This is mostly due to their novel structure and the associated properties, such as electronic conductivity, exibility, stiness, etc. These properties make nanotubes good candidates for many important technological applications. Carbon nanotubes were rst observed by Iijima  in 1991. Initially, only multi-wall nanotubes were observed, i.e. nanotubes consisting of several concentric shells, each one of them obtained by rolling up a graphene sheet in a seamless fashion. More recently, however, single-wall nanotubes have also been synthesised. The atoms can arrange themselves on each shell with dierent degrees of helicity, corresponding to the dierent ways in which it is possible to roll-up a graphene sheet to form a tube. Nanotubes are usually capped by structures containing pentagons which are reminiscent of fullerene
fragments. Multi-wall nanotubes have diameters usually in the range 1-25 nm and many microns in length; their aspect ratio varies from 100 to 1000. These peculiarities have made them the focus of attention as potentially useful materials in nanotechnology. Due in part to this, a great deal of progress has been achieved in the production and puri cation of both single and multi-wall carbon tubules. Also there has been a considerable eort in the investigation of their structural, electronic, optical, magnetic and transport properties [34, 35]. As for single-wall nanotubes, it has been recently found that they tend to appear in the form of \ropes" (or bundles), which usually consist of up to a hundred mono-disperse nanotubes packed in a perfect triangular lattice with a lattice parameter of 17 A. These nanotubes, obtained by dual pulsedlaser  or electric arc discharge techniques , appear to have a diameter of approximately 1.4 nm, which is close to that of the (10,10) nanotube. This opens the perspective of investigating the process of ion intercalation in ropes, with the possibility of conductivity enhancement  and charge transfer  such as that found in fullerenes. Ab-initio calculations of metal lling of nanotubes  have been reported showing that capillarity eects of alkali atoms are favoured for small diameter tubes due to a charge transfer from the metal atoms to the tube wall, whereas for noble metal atoms hybridisation eects do not allow for a complete charge transfer to the tubule wall, making the incorporation of atoms inside tubes of small diameter rather unlikely. For small carbon tubes we have also proposed halogen doping  as a way of getting high superconducting Tc because of the high density of states for one hole injection. We extended these studies to nanotube ropes and we have found that the interstitial channels between tubules have a unique capillarity behaviour, being easily able to accommodate foreign alkali metal atoms, noble metal atoms and halogens . Energetically, the intercalation is favoured
in these interstitial regions rather than inside the nanotubes (in contrast to isolated tubes). These results are in agreement with recent experiments on K and Br doped ropes [38, 39]. Perhaps one of the most intriguing questions in fullerene-, and in particular nanotube science concerns their growth mechanism, which is as yet a little understood phenomenon. It is known that in order to obtain single wall nanotubes it is necessary to add to the graphitic carbon used in the arch discharge a certain amount of a transition metal catalyst . Otherwise, only multiwall nanotubes are synthesised. What the role played by the catalyst might be is as yet unknown, though obviously it must somehow favour the growth of single wall nanotubes over the formation of multi-wall ones. In what follows we will focus on single-wall nanotubes, and in particular in their electronic and structural properties, as well as on the question of their growth mechanism.
4.1 Electronic and structural properties of nanotubes. Both the tube structure and its metal/non-metal character are speci ed by an integer pair (n,m). If a1 and a2 are the translational unit vectors of a graphene sheet, then the pair (n,m) speci es the circumference vector c = na1 + ma2 which connects two equivalent sites of the sheet and de nes the diameter of the tube. The non-chiral structures of type (n,0) and (n,n) present respectively a zig-zag or armchair pattern in the direction parallel to the tube axis. From zone-folding symmetry considerations , a tube will be metallic, i.e. it will have a non-vanishing density of states at the Fermi level, if the values of n and m obey the relation 2n + m = 3q where q is an integer. Thus all (n,n) armchair tubes are expected to be metallic. Furthermore, this symmetry property is always satis ed in (n,n) tubes independent of external stretching or elongation of the tubes, at dierence with other metallic tubes where these
elastic deformations open a gap in the nanotube electronic structure. This can be easily understood by noticing that lines of allowed ~k-points always include the to K line of the graphene sheet (along the circumference of the tube). Therefore, to open a gap this symmetry has to be broken as it is the case when the tube is twisted. This type of ionic excitation is important to understand the behaviour of resistivity at low temperatures. Defects are expected to be present in nanotubes in several forms: doping and topological defects, as well as rehybridisation and incomplete bonding (due to dislocations). The presence of such defects could alter substantially both the electronic and elastic properties of nanotubes. For example, topological defects such as a pentagon-heptagon complex pair induce a change of curvature in the structure, and its eect on the electronic structure can be regarded as a pure carbon switch, creating a metal/metal, metal/semiconductor or semiconductor/semiconductor junction between two perfect semi-in nite tubes of dierent indices . Such defects involve a global change in the tube structure, and therefore cannot be incorporated into an already existing tube. We seek a way of modifying the band gap that can be achieved by introducing defects that can be incorporated a posteriori into an existing tube. We have thus studied the eect of topological defects that introduce no net disclination in the tube. In particular we have simulated an adjacent pentagon-heptagon complex pair obtained by a =2 rotation of one bond in a four-hexagon complex, i.e., similar to the Stone-Wales transformation discussed above in the context of fullerenes. It should be noticed that these defects are dicult to observe in transmission electron microscopy experiments, given that they conserve the curvature of the tube (producing no net disclination) . Coordinates of defective tubes were generated by rotating one, two or three bonds in an otherwise perfect tube. The resulting con guration was then relaxed with tight-binding molecular dynamics (TBMD) in a periodic simulation
super-cell con guration, allowing the axial length of the super-cell to vary. The role of the curvature-induced hybridisation and of the periodic boundary conditions was clari ed by unrolling the tubes and performing similar calculations for the resulting attened sheets of defective graphene. We indicate that when two defects are present in the super-cell they are evenly spaced in opposite sides of the tube. When there are three defects they distribute uniformly. These results are summarised in Figure 5. Depending on the tube symmetry, either metalisation or band gap opening can result. In the case of (n,n) tubes a line of allowed k -values runs from to K in the graphite extended zone. Moving the Fermi level along this line from K to upon introduction of rotated-bond defects should not open a gap. Instead, an increase of approximately 25 % in the density of states at the Fermi level should be expected, as the band dispersion decreases. This is illustrated in Fig. 6 for the case of the (5,5) tube (similar results are obtained for the (10,10) tube). We have checked that the results are quite insensitive to the location and alignment of defects in the tube wall. Ab-initio calculations for the smaller tubes have been performed to check the reliability of the TB-results for the electronic structure close to the Fermi level, and we have obtained good agreement between the two calculations, that gives support to the TB-results. However, motion of the graphite Fermi point can also have the result of opening a gap. For example, in the case of the (n; n + 3q) where q is an integer, there is a small hybridisation-induced band gap, in spite of the fact that zone-folding arguments predict those systems to be metallic. In these tubes the defect-induced motion of the Fermi surface from K to can pull the Fermi surface away from the adjacent line of allowed k -values, thereby opening the gap. These results are relevant for transport properties of nanotube ropes metalised by defects and doped with alkali or halogen atoms as in graphitic intercalation compounds. One open question, presently under study, is how
Κ (n,n) remains metallic
(n,m) gap closes
(n,m+3i) gap opens
Figure 5: Schematic diagram showing the evolution of the electronic structure of a graphitic sheet upon introduction of bond rotation defects. The dashed lines show allowed k values for large-gap, small-gap and metallic tubes. ecient is the annealing of defects in the nanotubes as a function of defect concentration. The calculations described above were performed assuming in nitely long tubes. However, it is known that individual tubes behave as molecular wires and exhibit quantum eects at low temperatures, such as single electron transport through the quantised energy levels of the tube, which has been observed experimentally by Bockrath et al. . A rough estimate of the importance of quantum eects can be made within the simple \particle-in-a-box" model, for which the energy separation between levels is E = hF =2L, where F is the Fermi velocity, which can be estimated from ab-initio band structure calculations to be 8:1 105 m/s and L is the tube length. According to this model, a 3 micron long tube will have a considerable energy splitting of 0:6 meV. Within the TB approach we have a mean level spacing of E = RL6 eV A2 , where R is the tube radius. Once more, for normal diameters and tube lengths this splitting is of the order of meVs, which is a considerable energy for trans-
1 0 -1 X
Figure 6: Evolution of the band structure E(k) and density of states of (5,5) tubes upon increasing the concentration of bond rotation defects. Two rotated bonds within a 60-bond (40-atom) unit cell yield a at defect band. port and conductivity measurements. Using the TB model, we have studied nite-size eects in carbon nanotubes as a function of the length L and type of the end structure used to terminate the tube. Several structures were considered, ranging from tori to open tubes terminated with hydrogen atoms and a range of caps formed from hemispheric pieces of fullerenes. Four types of band-gap behaviour were observed  : d the gap diminished towards the in nite tube value with a 3-period oscillation and an amplitude damping as 1=L; d' the gap diminished monotonically as 1=L towards the in nite length tube value; e the gap approached exponentially fast a constant value dierent from that of the in nite tube and e' the gap amplitude is constant. The dierent band-gap behaviour correlates with the structure of the cap and the character of the frontier orbitals (these levels correspond to the top of the valence band and the bottom of the conduction band) in the following ways: delocalised or bulk-like orbitals, i.e with electron densities more or less uniform through
Figure 7: Gap versus tubule length L for dierent carbon nanotubes corresponding to the various band-gap behaviours and dierent geometric boundary conditions discussed in the text . Here the tube length L stands for the number of unit cells in the axial direction that the nite tube contains, that is, for (n; n) and (n; 0) tubules the length is 2:45L A and 4:26L A, respectively. The inset shows the C5v -symmetry cap from the icosahedral C80 fullerene. the length of the tube gave rise to behaviours d and d' , while end-localised orbitals gave rise to behaviours e' (completely end-localised) and e (loosely end-localised) . These eects should be observable in conductivity and transport measurements in single-wall nanotubes as well as in doped composite nanotubes . The presence of cap-localised states close to the Fermi level makes these nanotubes ideal candidates for scanning tunneling microscopy tips and electron emission materials. In Fig. 7 cases d , e , and e' are shown for two classes of nanotubes: gap-less tubes (curves d0, e0 and e00 ), and non-zero gap tubes (curves d6=0, e6=0 and e06=0). The d' case, though not illustrated in the gure does occur, for example for the open-ended (8,0) tube.
4.2 Elastic and dynamical properties of nanotubes. The inherent strength of the carbon-carbon bond indicates that the tensile strength of carbon nanotubes might exceed that of other known bers. Calculations of the elastic properties of nanotubes based on ab-initio  and TB  models indicate that they are extremely rigid in the axial direction, having Young modulus in the Terapascal range. Consequently they are more likely to distort in a direction perpendicular to the tube axis. Simulations also seem to indicate that nanotubes subject to large deformations switch into dierent con gurations which correspond to an abrupt release of stress energy . The bending is fully reversible up to very large bending angles, despite the occurrence of kinks and highly strained tubule regions. Therefore, nanotubes are able to sustain extreme strains with no signs of brittleness, plasticity or atomic rearrangement and have large in-plane rigidity and strength. This exibility property stems from the the ability of the sp2 network to rehybridise when deformed out of plane, the degree of sp2 sp3 rehybridisation being proportional to the curvature. We have also performed TB molecular dynamics calculations of nite single wall nanotubes with the aim of investigating dierent aspects of the growth mechanism. An isolated carbon (10,0) (zig-zag) nanotube consisting of 120 C atoms has been simulated. One end of the nanotube has been completed with 10 H atoms, which have the role of saturating the dangling bonds of the edge C atoms. These H atoms have been kept xed at their relaxed positions throughout the simulation. The edge C atoms at the other end of the tube were left unsaturated. The system at its starting con guration is illustrated in Figure 8. Canonical (constant NVT) MD simulations were started at a temperature of 1000 K. The NVT algorithm used was that due to Nose  and Hoover , and was implemented using a modi ed velocity-Verlet scheme described by
Figure 8: Initial con guration of the simulated tube. The grey atoms are carbons, and the white ones hydrogens. Frenkel and Smit . The time step was xed at 0.5 fs, a value small enough to provide excellent conservation of the total energy of the extended (nanotube plus thermostat) system. At this temperature a simulation of 10 ps was performed. The system was also simulated at 2000 K for 10 ps and at 2500 K for 20 ps. These simulations were performed sequentially, and the starting con gurations at the higher temperatures were obtained by subjecting the system to heating rates of 250 K/ps until the desired temperature was obtained. At the temperature of 1000 K, the structure of the nanotube remains essentially unaltered. The vibrations of the atoms around their equilibrium
8.5 ps 2.5 ps
Figure 9: Closure of the (10,0) carbon nanotube at a temperature of 2500 K. positions are the only noticeable feature at this temperature. However, as we increase the temperature to 2000 K, structural changes begin to be noticeable at the open edge of the nanotube. We rst observe the formation of heptagon-pentagon defects formed from adjacent hexagons. Eventually, heptagons tend to break, forming arms (or sticks in the terminology of Murry et al ) containing usually two C atoms. These arms oscillate widely, and sometimes recombine, changing the local structure at the edge. Finally, at the temperature of 2500 K the changes in the structure at the open end become more dramatic. Eventually the C atoms in the arms referred to above manage to bridge across the tube, forming structures similar to \handles" , and thus begin the process of tube closure. This process is illustrated in the se-
Figure 10: Closure of the (10,0) nanotube at 2500 K (continued). quence of snapshots in Figures 9 and 10. The process ends with the complete closure of the nanotube. Once closed, the nanotube remains unaltered over the time-scale spanned by the simulation at this temperature. It is also noticeable that in the closed structure all C atoms adopt sp2 bonding. Representative structures observed at times spanning the length of the simulation at 2500 K are illustrated in Figures 9 and 10. Before the closure of the tube occurs, we also observe the evaporation of C dimers from the tube edge. An illustrative example of this process is shown in Figure 11, from where it can be seen that this process bears some resemblance to the fragmentation of C60 to C58 by emission of a dimer, a process studied by Murry et al. . In total only 4 such evaporation events were observed.
t = 0 fs
t = 25 fs
t = 50 fs
Figure 11: An example of a C dimer evaporation from the open end of the nanotube, at a temperature of 2500 K. The TB MD simulations reported here indicate that single-wall nanotubes have a tendency to close their open ends at temperatures close to those at which synthesis takes place. The closed structures are expected to be rather unreactive, and thus it is unlikely that single-wall nanotubes can grow by adsorbing carbon clusters onto these closed ends. This could explain why, in the absence of a transition metal catalyst, single wall nanotubes are not obtained. Our ndings agree with the results of recent rst principles calculations , a fact that gives us con dence in the model used in our calculations. The fact that TB simulations are several orders of magnitude cheaper than rst principles calculations indicates that a more systematic study of the processes that come into play in nanotube growth can be carried out using this TB model. Such a study will probably be impractical with rst principles calculations due to their high computational cost. We are currently performing a set of simulations identical to the ones reported here, starting from a range of dierent initial conditions, so as to improve the statistics of our results. Similar calculations have also been undertaken for BN nanotubes, and in the near future we will extend this treatment to other composite systems.
5 Conclusions and perspectives We have shown here that theoretical simulations constitute a useful tool in fullerene and nanotube research. They are able to provide new insight into the thermodynamic, elastic and electronic properties of carbon-based and other composite materials, and in conjunction with experiments may allow us to understand the growth mechanisms of these materials. Much of the work we have presented here is still in progress; this is a lively eld of research, and many important developments can be expected in the near future, such as potential applications in nanotribology, coating of materials and optoelectronics, just to mention a few. In this review we have focused our attention on carbon-based materials, but we are already applying the techniques described here to the study of BxCy Nz materials. In particular we are presently studying the dynamical and elastic properties of BN and BCN nanotubes, and our results on this topic will be presented in due course. Our work has so far dealt with isolated carbon clusters and nanotubes. However, it is important to remember that many of the potential applications of these materials stem from their ability to form crystalline structures or in some cases polymeric forms. Thus we are also beginning the study of aggregates of fullerenes and nanotubes, and we hope to present results from these studies in the near future.
Work supported by DGES (Grant PB95-0720-C02-01), Junta de Castilla y Leon (Grant VA72/96), Fundacion Domingo Martnez, European Community TMR contract ERBFMRX-CT96-0067 (DG12-MIHT), and Centre de Computacio y Comunicaciones de Catalunya. M.J.L. acknowledges a contract granted by MEC, program I+D, associated to the project PB95-0720-C02-01.
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