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Tight binding studies of exohedral silicon doped C60
Composites Science and Technology 63 (2003) 1499–1505 www.elsevier.com/locate/compscitech
Tight binding studies of exohedral silicon doped C60 P.A. Marcosa, J.A. Alonsob, M.J. Lo´pezb, E. Herna´ndezc,* a Departamento de Fı´sica, Universidad de Burgos, Burgos, Spain Departamento de Fı´sica Teo´rica, Universidad de Valladolid, 47011 Valladolie, Spain c Institut de Cie`ncia de Materials de Barcelona (ICMAB-CSIC), Campus de Bellaterra, 08193 Barcelona, Spain b
Received 4 September 2002; received in revised form 25 November 2002; accepted 15 December 2002
Abstract We study the exohedral doping of C60 fullerenes by silicon atoms. The structural properties and stability of C60Sim aggregates, with m in the range 1–15, are investigated using a non-orthogonal tight binding model. For each cluster size an extensive set of possible configurations has been considered in order to increase the likelihood of determining the lowest energy structure. Each configuration was then minimised using the conjugate gradients algorithm. We find that for cluster sizes up to C60Si5 the trend is for Si atoms to aggregate, forming a monolayer on the C60 surface, while for C60Sim with m equal to 5 and beyond the trend changes towards the formation of three dimensional Si clusters attached to the fullerene surface. For m58 the additional Si atoms place themselves on the Si already present, and not directly on the buckyball surface. We have also conducted dynamical simulations of cluster fragmentation, which indicate that when the fragmentation of C60Sim takes place, it generates two homonuclear pieces, i.e., a Sim cluster and the C60 fullerene, which remains intact, in agreement with photofragmentation experiments. # 2003 Elsevier Ltd. All rights reserved. Keywords: Fullerenes; Cluster fragmentation; Silicon doping
1. Introduction Doping is a popular and fruitful method of modifying and tayloring at will the properties of many materials. From their infancy, carbon fullerenes [1] have also been a target for doping [2]. Almost any species including rare gas atoms [3], metals [2,4–6], non-metals such as B [7] and N [8,9], etc., have been successfully used in the doping of fullerenes. Three different doping configurations, i.e., endohedral, exohedral, and substitutional, can be achieved depending on the dopant species and the production technique. The direct production of fullerenes in the presence of foreign species may lead to endohedral [2] or substitutional fullerenes [7,8]. Endohedral fullerenes can also be produced by energetic collisions of foreign atoms with fullerene molecules [3,4] whereas the gentle deposition of heteroatoms on preformed carbon fullerenes would favour the exohedral type of doping [5,6]. A particularly interesting case is that of silicon doped fullerenes. On one hand, C and Si have similar valence
* Corresponding author.. E-mail address: [email protected] (E. Herna´ndez). 0266-3538/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0266-3538(03)00073-3
electronic structure, and one might expect an easy replacement of C atoms by Si in the fullerene cage. On the other hand, they exhibit quite different hibridization capabilities, which prevent silicon from forming fullerene-like structures, just as it does not form graphiticlike structures in bulk. Moreover, there is a large size mismatch [10] between carbon and silicon, which plays against an easy replacement of C by Si in the fullerenes. However, despite these size and electronic mismatches between silicon and carbon, both, substitutional and exohedral Si doped fullerenes have been produced by laser vaporization of mixed Si–C targets [11–14] and by laser vaporization of two independent, C60 and Si, targets [15,16]. Moreover, (C60)nSi+ m complexes have also been produced by the later method [16]. The silicon assisted polymerization of fullerenes in those complexes has been suggested to interpret their stability. One could then envisage the use of Si-doped fullerenes to develop new fullerides or assembled materials with novel electronic and structural properties as compared with other fullerene-based or C/Si materials. Photofragmentation experiments [14] indicate that at least up to seven C atoms can be substituted by Si in a fullerene. Ab initio calculations [17] confirm the possible substitution of up to 12 Si atoms in C60 without
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deforming the original fullerene structure too much. In contrast, only a limited (small) number of Si atoms can be directly attached to the fullerene surface [15]. In the light of photofragmentation experiments it has been suggested that silicon does not wet (cover) the fullerene surface. Instead Si atoms prefer clustering into three dimensional structures attached to the fullerene by a relatively small number of C–Si heterobonds. From the theoretical side, most studies have focused on substitutional Si-doped fullerenes [13,15,17–19] whereas exohedral doping [20] remains almost unexplored. In this paper we have investigated the minimum energy structure of exohedral silicon doped C60 fullerenes containing up to 15 Si atoms. Begining with five silicon atoms, three-dimensional Si structures develop on the fullerene surface. Our results clearly support the experimental suggestion that Si does not wet fullerenes [16]. We have also studied the thermal fragmentation of these exohedral Si-doped fullerenes. They fragment in two homoatomic pieces, i.e., a silicon cluster and the C60 fullerene which remains intact. The fragmentation pattern obtained in our simulations is in perfect agreement with the photofragmentation experiments [16] of Pellarin et al. and provides further support for the interpretation of these experiments as arising from segregated Si structures. The structure of the paper is as follows. In Section 2 we sketch the non-orthogonal tight binding methodology used throughout the paper. Section 3 is devoted to the results. First in Subsection 3.1 we introduce the computational strategy used in the search of the minimum energy configurations of the exohedral Si-doped fullerenes. Subsection 3.2 describes the minimum energy structures. The stability of the heterofullerenes is disccussed in Subsection 3.3. Finally dynamical studies of the fragmentation are presented in Subsection 3.4. We finish with a brief summary in Section 4.
2. Model The calculations reported later in this paper have been performed using a Density Functional Tight Binding model (DFTB), due to Porezag et al. [21]. DFTB is a total-energy non-orthogonal Tight-Binding (TB) model [22] which has been extensively used for the simulation of condensed matter systems, as well as for clusters and nanostructures. In particular it has been widely used with great success in the context of carbon-based systems [30,31], including crystalline [21] and amorphous solids [23], fullerenes [24–27] and nanotubular structures [28,29]. It has also been used for silicon based systems and SiC [32,33]. A DFTB parametrisation is constructed from a series of density functional theory (DFT) calculations using a basis set consisting of confined atomic-like orbitals,
similar to the fire-ball orbitals used in other contexts [34,35]. The overlap matrix elements are tabulated as a function of the inter-atomic distance for each pair of angular momenta. For the Hamiltonian matrix elements, only two-centre contributions are considered, and the tabulation is constructed just as for the overlap matrix elements. The total energy is calculated in the standard way [22] for TB models, namely as the sum of a band structure term and a repulsive pair potential term. The band structure term is given as the sum of the occupied eigen-values resulting from the generalised eigen-value problem defined by the Hamiltonian and overlap matrices. The repulsive pair potential is designed to account for the ionic repulsion and the double counting of the electron–electron interaction implicit in the band structure contribution. It is adjusted by requiring that the total energy of the TB model matches that of the full DFT calculations performed for a number of reference systems (usually the dimer at a series of inter-atomic distances). For further details on the DFTB model and its construction, the reader should consult the paper by Porezag and coworkers [21].
3. Results 3.1. Cluster generation Generating cluster configurations that can serve as starting guess for the relaxation calculations is relatively straight forward when the number of attached Si atoms is small. For the larger clusters, however, it is necessary to adopt a systematic strategy of isomer generation, given the large number of structural possibilities. In this work we have used the following strategy: we take the equilibrium fullerene structure as given, and on this place each additional Si atom on a spherical coordinate < < < grid defined within the limits: 5< ¼ ¼ r¼ ¼ 10, 0 ¼ ¼ ¼ ¼ /2 and < < 0 ¼¼ ¼¼ 2; here r is the distance from the center of mass of the fullerene, and is measured in Angstron. The range of variation of the radial distance r is selected such that only exohedral configurations of silicon are generated, including the possibility of silicon atoms not directly attached to the fullerene surface. A sufficiently fine grid is generated using r=0.5 A˚, =3 , and =15 . For practical reasons we exclude those configuratios in which the additional silicon is closer to any of the previously added silicon atoms than a significant fraction of the silicon–silicon equilibrium bond length. In this way we generated an average of several hundreds of initial configurations for each C60Sim cluster, and each of these configurations was used as initial guess in a minimization routine using conjugate gradients [36]. The different minima (isomers) found for each cluster were classified according to their energy.
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Below we consider each cluster size and report the most stable structure found in each case. 3.2. Equilibrium structures 3.2.1. C60Si1 and C60Si2 First we concentrate on the first stages of the exohedral doping of C60 fullerenes. The first silicon atom can be attached to the fullerene surface in several structural configurations, namely, (i) on top a C atom, (ii) above the mid point of a C–C bond (edge configuration), (iii) above the centre of a pentagon, and (iv) above the center of a hexagon. All these structures lead to stable isomers (see below). Fig. 1 shows the interaction energy between the fullerene and the silicon atom approaching the fullerene in several radial directions: (i) on top a C atom, (ii) through the mid point of an edge between a pentagon and a hexagon, and (iii) through the mid point of an edge between two hexagons. Radial distances of the Si to the centre of mass of the fullerene corresponding to exohedral and endohedral Si are considered. Clearly, exohedral Si-doping of fullerenes is favoured with respect to the endohedral doping. Endohedral Si-doped fullerenes have been discarded on the base of mobility measurements performed by Fye and Jarrold [12]. The minimum energy structure corresponds to the edge configuration in which the silicon atom is above a C–C bond between two hexagons. This result is in agreement with preliminary molecular dynamics simulations [16] performed in the frame of density functional theory (DFT). Next in energy come the structure with the silicon atom above a pentagon, with an energy of 0.6 eV above the minimum structure, followed by that with
Fig. 1. Interaction energy between a fullerene and one silicon atom approaching the fullerene in several radial directions: (i) on top a C atom (dashed line), (ii) through the mid point of an edge between a pentagon and a hexagon (dotted line), and (iii) through the mid point of an edge between two hexagons (solid line), as a function of the distance of the Si to the center of mass of the fullerene.
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the Si atom above a C–C bond between a pentagon and a hexagon, 0.616 eV above the most stable one, and the structure in which the silicon atom is attached to a carbon one with an sp3 hybridization. These structures have very similar energies, so it is questionable that the simple model used here is really able to differenciate among them, but the energy difference between them and the reported ground state is sufficiently large as to give us confidence that we have really determined the ground state. Much higher in energy is the structure with the atom at the centre of a hexagon, which lies 1.13 eV above the most stable structure. The second Si atom prefers also the edge configuration and attaches above one of the hexagonal edges adjacent to the first Si. In this way the second silicon atom binds directly to the fullerene cage and to the first Si. The tendency of the second silicon atom to bind to both the fullere and the first Si is also observed in the on top isomers. Fig. 2 shows the lowest energy on top and edge structures of one- and two-Si doped heterofullerenes. 3.2.2. C60Si3 The most stable C60Si3 structure we have found consists of an equilateral triangle with the three Si atoms at its vertices; this triangle is placed on the fullerene cage such that its centre is above the centre of a hexagonal ring, with the three Si atoms located above the three hexagon–hexagon edges, as illustrated in Fig. 3. We
Fig. 2. On top and edge configurations of exohedral heterofullerenes containing one and two silicon atoms. The energies (in eV) below each structure are given with respect to the minimum energy configuration. The dark circles represent carbon and the light ones silicon.
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opposing C atoms of the hexagonal ring, with the other two Si atoms located above the mid point of two C–C bonds, as illustrated in Fig. 3. A variation of this structure exists, which is obtained by rotating the orientation of the rhombus some 90 in the plane which contains it, such that it is now the shorter diagonal Si atoms that lie above two C atoms, the other two being displaced towards the center of two neighbouring carbon rings. This structure is ca. 0.084 eV higher in energy than the previous one, and it is likely to reconvert easily into it. Other structures we obtain for this cluster consist of the triangular motif found for C60Si3 with a fourth Si atom placed above the triangle and not directly in contact with any carbon atoms, although these structures are much higher in energy, by at least 0.8 eV. It is interesting to note that the most stable freestanding Si4 cluster is predicted by several authors [38] to have the same structure that we find in this work; the presence of the fullerene does not significantly distort the structure of this Si4 cluster.
Fig. 3. Minimun energy structures of exohedrally Si doped fullerenes containing from 3 to 15 silicon atoms. The clustering tendency of the silicon atoms is clearly apparent.
recall that in buckminsterfullerene each hexagon is surrounded by three hexagons and three pentagons, in an alternating fashion. There is very little distortion of the fullerene cage, preferentially localized at the hexagonal ring wich supports Si atoms. The common pentagonal–hexagonal edge length is increased only by 1%, and the hexagonal– hexagonal one, is increased by about 3%. The distances between Si atoms (2.6 A˚), however, are significantly stretched (about 11%) with respect to those in the bulk phase of silicon [10]. Interestingly, the Si3 free cluster has also a triangular structure, although in this case it is an isosceles triangle rather than equilateral. The two short bonds are 2.22 A˚ and they form an angle of 83 , in good agreement with the first-principles results of Binggeli and Chelikowsky [37], who obtain 2.17 A˚ and 80 . 3.2.3. C60Si4 The structure we have found to be most stable for the C60Si4 complex consists of an Si4 rhombus placed above one of the hexagonal faces of the fullerene, in such a way that the two Si atoms forming the long diagonal of the Si4 rhombus are positioned above two
3.2.4. C60Si5 In this cluster a new structural trend is started. The smaller clusters favoured structures in which all Si atoms were directly in contact with the fullerene cage. For C60Si5 we find that the most stable structure is a distorted square-base pyramid, with the fifth atom at the apex, pointing outwards of the fullerene cage. The base of the pyramid is located over a hexagonal ring, having the same overall shape and orientation as the rhombus found in the C60Si4 structure (see Fig. 3). 3.2.5. C60Si6 The most stable structure found in this case consists of a pentagonal pyramid, with its apex slightly displaced from the central point of the hexagonal ring of the fullerene over which the Si aggregate locates itself. The five Si atoms forming the pentagonal base of the pyramid are located over carbon atoms (not necessarily in a radial direction) except one of them, located over a C–C bond. A slight deformation is induced due to its unnatural placement over the hexagonal ring, as can be seen in Fig. 3. 3.2.6. C60Si7 and larger structures The C60Si7 most stable structure is obtained from the previous one by adding a new Si atom directly attached to the fullerene surface in on top configuration which also binds to one of the edges of the base of the pentagonal pyramid of Si6 (see Fig. 3). Beyond this size, the trend is to accumulate additional Si atoms onto the already existing Si structure, rather than to continue covering the surface of the fullerene. Thus in C60Si8 the structure found is the same as in C60Si7, with an extra Si
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atom bonded to the other Si atoms, and not in contact with the fullerene cage, as shown in Fig. 3. As larger C60Sim (m> 8) are considered, the emerging trend is to form structures in which the Si atoms tend to segregate from the fullerene, and build three-dimensional Si structures attached to the fullerene surface by a small number (five to six) of silicon atoms (see Fig. 3). It is clear that, in contrast to metals, silicon does not wet the fullerene surface. 3.3. Structural stability Fig. 4 shows the cohesive energy per atom of the exohedrally Si doped fullerenes, with respect to the cohesive energy per atom of the C60 fullerene, as a function of the number of Si atoms. This quantity decreases almost linearly with increasing doping dose, reflecting the lower cohesion of silicon (about 40% smaller) than that of carbon. A measure of the strength of the attachment of silicon to the fullerene surface is given by the interaction energy between the two homoatomic subsystems, calculated as Eint ¼ EðC60 Sim Þ þ EðC60 Þ þ EðSim Þ
ð1Þ
where the energies of the homoatomic subsystems, E(C60) and E(Sim), are calculated for the same structures as in the heterofullerene. Fig. 5 shows the interaction energy as a function of the number of silicon atoms. The interaction energy increases (with the exceptions of m=2 and 4) with the number of Si atoms at small doping dosis (m47) when the silicon atoms tend to form two dimensional islands covering the fullerene surface. At higher doping dosis (m510) the structural trend changes and the the additional Si does not bind directly to the fullerene but to the Si already present, forming three dimensional structures of Si. Thus the number of silicon atoms in direct contact with the fullerene remains almost constant; so does the interaction energy. The elevated plateau observed in the interaction energy for m=7–9 corresponds to the structures with the higher
Fig. 4. Cohesive energy per atom of the exohedrally Si doped fullerenes, with respect to the cohesive energy per atom of the C60 fullerene, as a function of the number of Si atoms.
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number (six) of Si atoms in direct contact with the fullerene surface. The average strength of the Si–C interaction (about 1 eV per Si atom in contact with the fullerene) in the heterofullerenes doped with more than one Si atom is much lower than either the cohesive energy (about 6.5 eV/atom) of fullerenes or the cohesive energy (about 4 eV/ atom) of small Si clusters. The small value of the heteroatomic interaction is responsible for the growth pattern, i.e., clustering, of Si deposited onto the fullerene surface and will have consequences in the high temperature behaviour of the heterofullerenes as well (see below). 3.4. Dynamical studies In order to probe the stability and dynamical behaviour of these silicon-fullerene aggregates, we have performed a series of molecular dynamics (MD) studies at different temperatures. All dynamical simulations were conducted under NVE (microcanonical) conditions. The equations of motion have been solved using the velocity Verlet algorithm [39–41] with a time step of 0.5 fs, which is sufficiently small to provide adequate energy conservation during the length of the simulation runs. Each cluster considered was taken in its lowest energy configuration, and a dynamical simulation was initiated at a temperature of 50 K. The temperature of the system was then stepwise increased in steps of 50 K between successive simulation runs. At every fixed temperature the dynamical simulation was carried out for 50 ps and the system was monitored every 0.025 ps to extract the relevant dynamical characteristics. A slower heating rate, temperature increase between runs=10–20 K, was used when required, e.g., strong reorganization of the atoms took place within the simulation time. This process of successive simulation and heating was repeated until some degree of cluster fragmentation was observed.
Fig. 5. Interaction energy between the two homoatomic pieces, i.e., the C60 fullerene and the Sim cluster, of the exohedral C60Sim heterofullerenes as a function of the number of Si atoms. (See text for the definition of ‘‘interaction energy’’.).
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Fragmentation in two pieces was detected in all the cases studied (C60Sim, m=1–10). One of the fragments is the whole silicon cluster, the other one being the hosting fullerene cage, which remains intact. This fragmentation pattern emerges from the clustering tendency of silicon deposited onto the fullerene and the small interaction energy between the two homoatomic pieces in the heterofullerenes as compared with the stability of each of those pieces separately (see above). The same fragmentation channel has been obtained in photofragmentation experiments by Pellarin et al. [16] and therefore there is good agreement between experiment and simulation. Those photofragmentation experiments also give support to the three dimensional seggregated silicon structures we have obtained as the lowest energy isomers. Fig. 6 shows the fragmentation temperature [42] as a function of the Si content in the heterofullerenes. The first thing to notice is that the fragmentation temperatures of the exohedrally doped fullerenes are much lower than the fragmentation temperature of pure C60 [43] and are even below the temperatures at which structural changes of C60 [43] were observed, within similar simulation time scales as the ones considered here. This explains why the C60 fullerene remains intact upon fragmentation of the heterofullerenes. The overall decreasing behaviour of the fragmentation temperature with the number of silicon atoms correlates with the decrease in the cohesive energy shown in Fig. 4. The pronounced down steps in the fragmentation temperature observed at m=3 and m=8 are a consequence of the new structural patterns begining at those doping dosis (see above).
4. Summary We have performed extensive molecular dynamics simulations in the framework of a non-orthogonal tight
binding methodology to investigate the structural properties, stability and fragmentation pattern of exohedral silicon doped fullerenes, C60Sim, containing m=1–15 Si atoms. A systematic strategy of cluster generation has been utilized to perform a wide search of possible structural configurations of the silicon atoms attached to the fullerene surface. The minimum energy structures, as a function of the number of Si atoms, follow a growth pattern that begings (m44) with the formation of two dimensional islands of silicon covering the fullerene. However, at an early stage of the doping process (m=5) the pattern changes towards the formation of three dimensional silicon clusters attached to the fullerene. Silicon does not wet the fullerene surface, and this will probably prevent the use of fullerenes as templates to grow two dimensional, sp2, structures of silicon. Exohedral Si doped fullerenes are stable although the strength of the interaction beteween the two homoatomic subsystems, the silicon cluster and the fullerene, is relatively small with respect to the cohesive energy within each of them. At finite temperatures (T=800–2500 K) the heterofullerenes fragment in their two homoatomic constituent parts, keeping the fullerene cage intact. The deposition of fullerene molecules to form films on different substrates has been studied experimentally (see Ref. [44] for a review) and by computer simulations [45–47]. Soft deposition can lead to well ordered films with the structure of the fullerite solid, in which fullerenes interact weakly and rotate easily. Let us speculate about the possible outcome of the soft deposition of C60Sim clusters. Since these clusters are highly reactive with other Si atoms, we could expect that strong Si–Si bonds would be established between clusters landing in nearby positions. Those bonds may produce a rather rigid structure formed by fullerenes linked through massive Si bridges. The expected rigidity also would mean that the whole deposited structure of interlinked fullerenes would be rather disordered, in contrast with the crystalline structure of the corresponding fullerite. On the other hand, for energetic cluster deposition the first effect one can expect is a fragmentation of the C60Sim clusters into C60 and Sim pieces. As Fig. 6 shows, this effect will occur more easily as the Si content of the clusters increases. At sufficiently high energies it is possible to even break the fullerenes [45,47]. What would then be the resulting structures is more difficult to guess, although interesting new composites could result of the process. Evidently, further work is necessary to advance on firm ground in this issue.
Acknowledgements Fig. 6. Fragmentation temperature of the exohedral C60Sim heterofullerenes as a function of the number of silicon atoms.
This work was supported by the Spanish Ministry of Education and Culture DGESIC (Grant PB98-0345),
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Tight binding studies of exohedral silicon doped C60 P.A. Marcosa, J.A. Alonsob, M.J. Lo´pezb, E. Herna´ndezc,* a Departamento de Fı´sica, Universidad de Burgos, Burgos, Spain Departamento de Fı´sica Teo´rica, Universidad de Valladolid, 47011 Valladolie, Spain c Institut de Cie`ncia de Materials de Barcelona (ICMAB-CSIC), Campus de Bellaterra, 08193 Barcelona, Spain b
Received 4 September 2002; received in revised form 25 November 2002; accepted 15 December 2002
Abstract We study the exohedral doping of C60 fullerenes by silicon atoms. The structural properties and stability of C60Sim aggregates, with m in the range 1–15, are investigated using a non-orthogonal tight binding model. For each cluster size an extensive set of possible configurations has been considered in order to increase the likelihood of determining the lowest energy structure. Each configuration was then minimised using the conjugate gradients algorithm. We find that for cluster sizes up to C60Si5 the trend is for Si atoms to aggregate, forming a monolayer on the C60 surface, while for C60Sim with m equal to 5 and beyond the trend changes towards the formation of three dimensional Si clusters attached to the fullerene surface. For m58 the additional Si atoms place themselves on the Si already present, and not directly on the buckyball surface. We have also conducted dynamical simulations of cluster fragmentation, which indicate that when the fragmentation of C60Sim takes place, it generates two homonuclear pieces, i.e., a Sim cluster and the C60 fullerene, which remains intact, in agreement with photofragmentation experiments. # 2003 Elsevier Ltd. All rights reserved. Keywords: Fullerenes; Cluster fragmentation; Silicon doping
1. Introduction Doping is a popular and fruitful method of modifying and tayloring at will the properties of many materials. From their infancy, carbon fullerenes [1] have also been a target for doping [2]. Almost any species including rare gas atoms [3], metals [2,4–6], non-metals such as B [7] and N [8,9], etc., have been successfully used in the doping of fullerenes. Three different doping configurations, i.e., endohedral, exohedral, and substitutional, can be achieved depending on the dopant species and the production technique. The direct production of fullerenes in the presence of foreign species may lead to endohedral [2] or substitutional fullerenes [7,8]. Endohedral fullerenes can also be produced by energetic collisions of foreign atoms with fullerene molecules [3,4] whereas the gentle deposition of heteroatoms on preformed carbon fullerenes would favour the exohedral type of doping [5,6]. A particularly interesting case is that of silicon doped fullerenes. On one hand, C and Si have similar valence
* Corresponding author.. E-mail address: [email protected] (E. Herna´ndez). 0266-3538/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0266-3538(03)00073-3
electronic structure, and one might expect an easy replacement of C atoms by Si in the fullerene cage. On the other hand, they exhibit quite different hibridization capabilities, which prevent silicon from forming fullerene-like structures, just as it does not form graphiticlike structures in bulk. Moreover, there is a large size mismatch [10] between carbon and silicon, which plays against an easy replacement of C by Si in the fullerenes. However, despite these size and electronic mismatches between silicon and carbon, both, substitutional and exohedral Si doped fullerenes have been produced by laser vaporization of mixed Si–C targets [11–14] and by laser vaporization of two independent, C60 and Si, targets [15,16]. Moreover, (C60)nSi+ m complexes have also been produced by the later method [16]. The silicon assisted polymerization of fullerenes in those complexes has been suggested to interpret their stability. One could then envisage the use of Si-doped fullerenes to develop new fullerides or assembled materials with novel electronic and structural properties as compared with other fullerene-based or C/Si materials. Photofragmentation experiments [14] indicate that at least up to seven C atoms can be substituted by Si in a fullerene. Ab initio calculations [17] confirm the possible substitution of up to 12 Si atoms in C60 without
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deforming the original fullerene structure too much. In contrast, only a limited (small) number of Si atoms can be directly attached to the fullerene surface [15]. In the light of photofragmentation experiments it has been suggested that silicon does not wet (cover) the fullerene surface. Instead Si atoms prefer clustering into three dimensional structures attached to the fullerene by a relatively small number of C–Si heterobonds. From the theoretical side, most studies have focused on substitutional Si-doped fullerenes [13,15,17–19] whereas exohedral doping [20] remains almost unexplored. In this paper we have investigated the minimum energy structure of exohedral silicon doped C60 fullerenes containing up to 15 Si atoms. Begining with five silicon atoms, three-dimensional Si structures develop on the fullerene surface. Our results clearly support the experimental suggestion that Si does not wet fullerenes [16]. We have also studied the thermal fragmentation of these exohedral Si-doped fullerenes. They fragment in two homoatomic pieces, i.e., a silicon cluster and the C60 fullerene which remains intact. The fragmentation pattern obtained in our simulations is in perfect agreement with the photofragmentation experiments [16] of Pellarin et al. and provides further support for the interpretation of these experiments as arising from segregated Si structures. The structure of the paper is as follows. In Section 2 we sketch the non-orthogonal tight binding methodology used throughout the paper. Section 3 is devoted to the results. First in Subsection 3.1 we introduce the computational strategy used in the search of the minimum energy configurations of the exohedral Si-doped fullerenes. Subsection 3.2 describes the minimum energy structures. The stability of the heterofullerenes is disccussed in Subsection 3.3. Finally dynamical studies of the fragmentation are presented in Subsection 3.4. We finish with a brief summary in Section 4.
2. Model The calculations reported later in this paper have been performed using a Density Functional Tight Binding model (DFTB), due to Porezag et al. [21]. DFTB is a total-energy non-orthogonal Tight-Binding (TB) model [22] which has been extensively used for the simulation of condensed matter systems, as well as for clusters and nanostructures. In particular it has been widely used with great success in the context of carbon-based systems [30,31], including crystalline [21] and amorphous solids [23], fullerenes [24–27] and nanotubular structures [28,29]. It has also been used for silicon based systems and SiC [32,33]. A DFTB parametrisation is constructed from a series of density functional theory (DFT) calculations using a basis set consisting of confined atomic-like orbitals,
similar to the fire-ball orbitals used in other contexts [34,35]. The overlap matrix elements are tabulated as a function of the inter-atomic distance for each pair of angular momenta. For the Hamiltonian matrix elements, only two-centre contributions are considered, and the tabulation is constructed just as for the overlap matrix elements. The total energy is calculated in the standard way [22] for TB models, namely as the sum of a band structure term and a repulsive pair potential term. The band structure term is given as the sum of the occupied eigen-values resulting from the generalised eigen-value problem defined by the Hamiltonian and overlap matrices. The repulsive pair potential is designed to account for the ionic repulsion and the double counting of the electron–electron interaction implicit in the band structure contribution. It is adjusted by requiring that the total energy of the TB model matches that of the full DFT calculations performed for a number of reference systems (usually the dimer at a series of inter-atomic distances). For further details on the DFTB model and its construction, the reader should consult the paper by Porezag and coworkers [21].
3. Results 3.1. Cluster generation Generating cluster configurations that can serve as starting guess for the relaxation calculations is relatively straight forward when the number of attached Si atoms is small. For the larger clusters, however, it is necessary to adopt a systematic strategy of isomer generation, given the large number of structural possibilities. In this work we have used the following strategy: we take the equilibrium fullerene structure as given, and on this place each additional Si atom on a spherical coordinate < < < grid defined within the limits: 5< ¼ ¼ r¼ ¼ 10, 0 ¼ ¼ ¼ ¼ /2 and < < 0 ¼¼ ¼¼ 2; here r is the distance from the center of mass of the fullerene, and is measured in Angstron. The range of variation of the radial distance r is selected such that only exohedral configurations of silicon are generated, including the possibility of silicon atoms not directly attached to the fullerene surface. A sufficiently fine grid is generated using r=0.5 A˚, =3 , and =15 . For practical reasons we exclude those configuratios in which the additional silicon is closer to any of the previously added silicon atoms than a significant fraction of the silicon–silicon equilibrium bond length. In this way we generated an average of several hundreds of initial configurations for each C60Sim cluster, and each of these configurations was used as initial guess in a minimization routine using conjugate gradients [36]. The different minima (isomers) found for each cluster were classified according to their energy.
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Below we consider each cluster size and report the most stable structure found in each case. 3.2. Equilibrium structures 3.2.1. C60Si1 and C60Si2 First we concentrate on the first stages of the exohedral doping of C60 fullerenes. The first silicon atom can be attached to the fullerene surface in several structural configurations, namely, (i) on top a C atom, (ii) above the mid point of a C–C bond (edge configuration), (iii) above the centre of a pentagon, and (iv) above the center of a hexagon. All these structures lead to stable isomers (see below). Fig. 1 shows the interaction energy between the fullerene and the silicon atom approaching the fullerene in several radial directions: (i) on top a C atom, (ii) through the mid point of an edge between a pentagon and a hexagon, and (iii) through the mid point of an edge between two hexagons. Radial distances of the Si to the centre of mass of the fullerene corresponding to exohedral and endohedral Si are considered. Clearly, exohedral Si-doping of fullerenes is favoured with respect to the endohedral doping. Endohedral Si-doped fullerenes have been discarded on the base of mobility measurements performed by Fye and Jarrold [12]. The minimum energy structure corresponds to the edge configuration in which the silicon atom is above a C–C bond between two hexagons. This result is in agreement with preliminary molecular dynamics simulations [16] performed in the frame of density functional theory (DFT). Next in energy come the structure with the silicon atom above a pentagon, with an energy of 0.6 eV above the minimum structure, followed by that with
Fig. 1. Interaction energy between a fullerene and one silicon atom approaching the fullerene in several radial directions: (i) on top a C atom (dashed line), (ii) through the mid point of an edge between a pentagon and a hexagon (dotted line), and (iii) through the mid point of an edge between two hexagons (solid line), as a function of the distance of the Si to the center of mass of the fullerene.
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the Si atom above a C–C bond between a pentagon and a hexagon, 0.616 eV above the most stable one, and the structure in which the silicon atom is attached to a carbon one with an sp3 hybridization. These structures have very similar energies, so it is questionable that the simple model used here is really able to differenciate among them, but the energy difference between them and the reported ground state is sufficiently large as to give us confidence that we have really determined the ground state. Much higher in energy is the structure with the atom at the centre of a hexagon, which lies 1.13 eV above the most stable structure. The second Si atom prefers also the edge configuration and attaches above one of the hexagonal edges adjacent to the first Si. In this way the second silicon atom binds directly to the fullerene cage and to the first Si. The tendency of the second silicon atom to bind to both the fullere and the first Si is also observed in the on top isomers. Fig. 2 shows the lowest energy on top and edge structures of one- and two-Si doped heterofullerenes. 3.2.2. C60Si3 The most stable C60Si3 structure we have found consists of an equilateral triangle with the three Si atoms at its vertices; this triangle is placed on the fullerene cage such that its centre is above the centre of a hexagonal ring, with the three Si atoms located above the three hexagon–hexagon edges, as illustrated in Fig. 3. We
Fig. 2. On top and edge configurations of exohedral heterofullerenes containing one and two silicon atoms. The energies (in eV) below each structure are given with respect to the minimum energy configuration. The dark circles represent carbon and the light ones silicon.
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opposing C atoms of the hexagonal ring, with the other two Si atoms located above the mid point of two C–C bonds, as illustrated in Fig. 3. A variation of this structure exists, which is obtained by rotating the orientation of the rhombus some 90 in the plane which contains it, such that it is now the shorter diagonal Si atoms that lie above two C atoms, the other two being displaced towards the center of two neighbouring carbon rings. This structure is ca. 0.084 eV higher in energy than the previous one, and it is likely to reconvert easily into it. Other structures we obtain for this cluster consist of the triangular motif found for C60Si3 with a fourth Si atom placed above the triangle and not directly in contact with any carbon atoms, although these structures are much higher in energy, by at least 0.8 eV. It is interesting to note that the most stable freestanding Si4 cluster is predicted by several authors [38] to have the same structure that we find in this work; the presence of the fullerene does not significantly distort the structure of this Si4 cluster.
Fig. 3. Minimun energy structures of exohedrally Si doped fullerenes containing from 3 to 15 silicon atoms. The clustering tendency of the silicon atoms is clearly apparent.
recall that in buckminsterfullerene each hexagon is surrounded by three hexagons and three pentagons, in an alternating fashion. There is very little distortion of the fullerene cage, preferentially localized at the hexagonal ring wich supports Si atoms. The common pentagonal–hexagonal edge length is increased only by 1%, and the hexagonal– hexagonal one, is increased by about 3%. The distances between Si atoms (2.6 A˚), however, are significantly stretched (about 11%) with respect to those in the bulk phase of silicon [10]. Interestingly, the Si3 free cluster has also a triangular structure, although in this case it is an isosceles triangle rather than equilateral. The two short bonds are 2.22 A˚ and they form an angle of 83 , in good agreement with the first-principles results of Binggeli and Chelikowsky [37], who obtain 2.17 A˚ and 80 . 3.2.3. C60Si4 The structure we have found to be most stable for the C60Si4 complex consists of an Si4 rhombus placed above one of the hexagonal faces of the fullerene, in such a way that the two Si atoms forming the long diagonal of the Si4 rhombus are positioned above two
3.2.4. C60Si5 In this cluster a new structural trend is started. The smaller clusters favoured structures in which all Si atoms were directly in contact with the fullerene cage. For C60Si5 we find that the most stable structure is a distorted square-base pyramid, with the fifth atom at the apex, pointing outwards of the fullerene cage. The base of the pyramid is located over a hexagonal ring, having the same overall shape and orientation as the rhombus found in the C60Si4 structure (see Fig. 3). 3.2.5. C60Si6 The most stable structure found in this case consists of a pentagonal pyramid, with its apex slightly displaced from the central point of the hexagonal ring of the fullerene over which the Si aggregate locates itself. The five Si atoms forming the pentagonal base of the pyramid are located over carbon atoms (not necessarily in a radial direction) except one of them, located over a C–C bond. A slight deformation is induced due to its unnatural placement over the hexagonal ring, as can be seen in Fig. 3. 3.2.6. C60Si7 and larger structures The C60Si7 most stable structure is obtained from the previous one by adding a new Si atom directly attached to the fullerene surface in on top configuration which also binds to one of the edges of the base of the pentagonal pyramid of Si6 (see Fig. 3). Beyond this size, the trend is to accumulate additional Si atoms onto the already existing Si structure, rather than to continue covering the surface of the fullerene. Thus in C60Si8 the structure found is the same as in C60Si7, with an extra Si
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atom bonded to the other Si atoms, and not in contact with the fullerene cage, as shown in Fig. 3. As larger C60Sim (m> 8) are considered, the emerging trend is to form structures in which the Si atoms tend to segregate from the fullerene, and build three-dimensional Si structures attached to the fullerene surface by a small number (five to six) of silicon atoms (see Fig. 3). It is clear that, in contrast to metals, silicon does not wet the fullerene surface. 3.3. Structural stability Fig. 4 shows the cohesive energy per atom of the exohedrally Si doped fullerenes, with respect to the cohesive energy per atom of the C60 fullerene, as a function of the number of Si atoms. This quantity decreases almost linearly with increasing doping dose, reflecting the lower cohesion of silicon (about 40% smaller) than that of carbon. A measure of the strength of the attachment of silicon to the fullerene surface is given by the interaction energy between the two homoatomic subsystems, calculated as Eint ¼ EðC60 Sim Þ þ EðC60 Þ þ EðSim Þ
ð1Þ
where the energies of the homoatomic subsystems, E(C60) and E(Sim), are calculated for the same structures as in the heterofullerene. Fig. 5 shows the interaction energy as a function of the number of silicon atoms. The interaction energy increases (with the exceptions of m=2 and 4) with the number of Si atoms at small doping dosis (m47) when the silicon atoms tend to form two dimensional islands covering the fullerene surface. At higher doping dosis (m510) the structural trend changes and the the additional Si does not bind directly to the fullerene but to the Si already present, forming three dimensional structures of Si. Thus the number of silicon atoms in direct contact with the fullerene remains almost constant; so does the interaction energy. The elevated plateau observed in the interaction energy for m=7–9 corresponds to the structures with the higher
Fig. 4. Cohesive energy per atom of the exohedrally Si doped fullerenes, with respect to the cohesive energy per atom of the C60 fullerene, as a function of the number of Si atoms.
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number (six) of Si atoms in direct contact with the fullerene surface. The average strength of the Si–C interaction (about 1 eV per Si atom in contact with the fullerene) in the heterofullerenes doped with more than one Si atom is much lower than either the cohesive energy (about 6.5 eV/atom) of fullerenes or the cohesive energy (about 4 eV/ atom) of small Si clusters. The small value of the heteroatomic interaction is responsible for the growth pattern, i.e., clustering, of Si deposited onto the fullerene surface and will have consequences in the high temperature behaviour of the heterofullerenes as well (see below). 3.4. Dynamical studies In order to probe the stability and dynamical behaviour of these silicon-fullerene aggregates, we have performed a series of molecular dynamics (MD) studies at different temperatures. All dynamical simulations were conducted under NVE (microcanonical) conditions. The equations of motion have been solved using the velocity Verlet algorithm [39–41] with a time step of 0.5 fs, which is sufficiently small to provide adequate energy conservation during the length of the simulation runs. Each cluster considered was taken in its lowest energy configuration, and a dynamical simulation was initiated at a temperature of 50 K. The temperature of the system was then stepwise increased in steps of 50 K between successive simulation runs. At every fixed temperature the dynamical simulation was carried out for 50 ps and the system was monitored every 0.025 ps to extract the relevant dynamical characteristics. A slower heating rate, temperature increase between runs=10–20 K, was used when required, e.g., strong reorganization of the atoms took place within the simulation time. This process of successive simulation and heating was repeated until some degree of cluster fragmentation was observed.
Fig. 5. Interaction energy between the two homoatomic pieces, i.e., the C60 fullerene and the Sim cluster, of the exohedral C60Sim heterofullerenes as a function of the number of Si atoms. (See text for the definition of ‘‘interaction energy’’.).
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Fragmentation in two pieces was detected in all the cases studied (C60Sim, m=1–10). One of the fragments is the whole silicon cluster, the other one being the hosting fullerene cage, which remains intact. This fragmentation pattern emerges from the clustering tendency of silicon deposited onto the fullerene and the small interaction energy between the two homoatomic pieces in the heterofullerenes as compared with the stability of each of those pieces separately (see above). The same fragmentation channel has been obtained in photofragmentation experiments by Pellarin et al. [16] and therefore there is good agreement between experiment and simulation. Those photofragmentation experiments also give support to the three dimensional seggregated silicon structures we have obtained as the lowest energy isomers. Fig. 6 shows the fragmentation temperature [42] as a function of the Si content in the heterofullerenes. The first thing to notice is that the fragmentation temperatures of the exohedrally doped fullerenes are much lower than the fragmentation temperature of pure C60 [43] and are even below the temperatures at which structural changes of C60 [43] were observed, within similar simulation time scales as the ones considered here. This explains why the C60 fullerene remains intact upon fragmentation of the heterofullerenes. The overall decreasing behaviour of the fragmentation temperature with the number of silicon atoms correlates with the decrease in the cohesive energy shown in Fig. 4. The pronounced down steps in the fragmentation temperature observed at m=3 and m=8 are a consequence of the new structural patterns begining at those doping dosis (see above).
4. Summary We have performed extensive molecular dynamics simulations in the framework of a non-orthogonal tight
binding methodology to investigate the structural properties, stability and fragmentation pattern of exohedral silicon doped fullerenes, C60Sim, containing m=1–15 Si atoms. A systematic strategy of cluster generation has been utilized to perform a wide search of possible structural configurations of the silicon atoms attached to the fullerene surface. The minimum energy structures, as a function of the number of Si atoms, follow a growth pattern that begings (m44) with the formation of two dimensional islands of silicon covering the fullerene. However, at an early stage of the doping process (m=5) the pattern changes towards the formation of three dimensional silicon clusters attached to the fullerene. Silicon does not wet the fullerene surface, and this will probably prevent the use of fullerenes as templates to grow two dimensional, sp2, structures of silicon. Exohedral Si doped fullerenes are stable although the strength of the interaction beteween the two homoatomic subsystems, the silicon cluster and the fullerene, is relatively small with respect to the cohesive energy within each of them. At finite temperatures (T=800–2500 K) the heterofullerenes fragment in their two homoatomic constituent parts, keeping the fullerene cage intact. The deposition of fullerene molecules to form films on different substrates has been studied experimentally (see Ref. [44] for a review) and by computer simulations [45–47]. Soft deposition can lead to well ordered films with the structure of the fullerite solid, in which fullerenes interact weakly and rotate easily. Let us speculate about the possible outcome of the soft deposition of C60Sim clusters. Since these clusters are highly reactive with other Si atoms, we could expect that strong Si–Si bonds would be established between clusters landing in nearby positions. Those bonds may produce a rather rigid structure formed by fullerenes linked through massive Si bridges. The expected rigidity also would mean that the whole deposited structure of interlinked fullerenes would be rather disordered, in contrast with the crystalline structure of the corresponding fullerite. On the other hand, for energetic cluster deposition the first effect one can expect is a fragmentation of the C60Sim clusters into C60 and Sim pieces. As Fig. 6 shows, this effect will occur more easily as the Si content of the clusters increases. At sufficiently high energies it is possible to even break the fullerenes [45,47]. What would then be the resulting structures is more difficult to guess, although interesting new composites could result of the process. Evidently, further work is necessary to advance on firm ground in this issue.
Acknowledgements Fig. 6. Fragmentation temperature of the exohedral C60Sim heterofullerenes as a function of the number of silicon atoms.
This work was supported by the Spanish Ministry of Education and Culture DGESIC (Grant PB98-0345),
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