Ferromagnetism in Defective Polymerised C60

Carbon-Based Magnetism T. Makarova and F. Palacio (Editors) © 2006 Elsevier B.V. All rights reserved.

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Ferromagnetism in Defective Polymerised €50 J.A. Chan^, B. Montanari^'^ andN.M. Harrison^ Department of Chemistry, Imperial College London, South Kensington campus, London SW7 2AZ, UK ^CCLRC Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OXll OQX, UK ^CCLRC Daresbury Laboratory, Daresbury, Warrington, WA4 4AD, UK

1 Introduction The recent observation of high temperature ferromagnetism in polymerised carbon fullerenes has fuelled a renewed interest in metal-free organic magnetic materials. Their existence not only presents a new class of potentially highly tunable materials, but also challenges the current understanding of magnetism. When cubic €50 fullerenes are subjected to pressures between ~2-8 GPa and temperatures between ^700-1150K, two-dimensional tetragonal (T) and rhombohedral (Rh) polymerised phases are formed [1]. Only samples that are synthesised in a specific temperature region near the stability limit of the fullerene cages exhibit, when quenched to room temperature, ferromagnetism with Curie temperatures above '^500K [2-4]. Polymerised phases created outside these conditions show no magnetic behaviour, suggesting that defect structures could be responsible for ferromagnetism. The unpaired electrons located at the defects could give rise to local moments, which would need to interact via a long range coupling mechanism in order to form a magnetically ordered structure. Unfortunately, the experimental characterization of the magnetic phases has proven difficult and, at present, their atomic structure is not known. In-situ X-ray diffraction measurements reveal a thermally activated process which converts the Rh-phase (Fig. 1) into the highly disordered graphite-like phase, which displays very broad Bragg peaks [5]. The detailed structure of the magnetic phase cannot be determined from this

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Fig. 1. The basal plane of pristine polymerised rhombohedral Ceodata. Transmission electron microscopy (TEM) points at an apparently well ordered crystalline structure in which the Ceo cages are largely intact and still in a Rh-Ceo - like arrangement [3]. Magnetic force microscopy (MFM) studies of the carbon phase based on polymerized fullerenes have established that approximately 30% of the material is magnetic with the magnetism occurring in well defined domains [6]. Several experimental studies have also shown that the presence of non-metallic impurities has some effect on the magnetic behaviour of carbon-based materials [7-9]. In particular, some of the studies have suggested that the presence of hydrogen may be important. For example, the observed saturation magnetisation of magnetic amorphous carbon is found to increase with the hydrogen concentration of the starting material [9], and proton irradiation of highly orientated pyrolytic graphite induces intrinsic room temperature ferromagnetic ordering, whilst a similar treatment with helium ions results in reduced magnetic signals [8]. As the characterization of the magnetic phase is problematic, theoretical calculations have an important role to play in determining possible local geometries. A number of previous theoretical studies have addressed the origin of magnetism in Rh-Ceo. Boukhvalov et al. used density functional theory (DFT) in the local spin density approximation to compute the electronic structure of pristine polymerised Rh-Ceo which indicate that it is not magnetic, in agreement with observation [10]. Other authors have studied various defective Rh- Ceo structures. In the work of Andriotis et al, for instance, a carbon atom was removed from each fullerene cage (Fig. 2(a)) [11]. Tight-binding molecular dynamics and cluster ah initio calculations were then used to analyse the magnetic properties of the resulting structure. An ionic model due to McConnell was invoked to suggest that inter-cage, through space coupling could result in long range magnetic coupling between cages linked via the interfullerene bonds of the polymerised structure [11, 12]. A re-examination of this hypothesis is reviewed here. Ribas-Arino et al (Chapter 22) used hybrid exchange DFT (B3LYP) calculations to simulate a Ceo dimer, and complete active space self consistent field calculations to compute the interactions within an isolated Ceo cage, with hydrogen atoms replacing the

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interfullerene bonds of the Rh-C6o structure [13, 14]. In both cases, their model of single broken interfullerene bonds between the cages led to a localised spin density and, in the latter case, ferromagnetic or antiferromagnetic states depending on the position of the broken interfullerene bonds. Various other defective Rh-C6o structures have been constructed with broken interfullerene bonds within a semiempirical Hartree-Fock/AMl formalism and the resultant band structure, which contains flat bands at the Fermi edge, was used as an indication of a tendency towards magnetic localisation [15]. Periodic B3LYP calculations have also been used to show that a distortion of the Rh-Ceo structure about the interfullerene bonds, while constraining all other bond lengths in the cage to 1.40 A, can also give rise to a spin polarised ground state [16]. In all of the above studies, the possibility of local moment formation around various types of defects has been established and there are clearly a large number of local defect structures that can be created and which will result in such local moments. An explanation of the observed ferromagnetism, however, requires one to establish both that the defects are credible and that there is strong ferromagnetic coupling between the local moments. The task therefore is to narrow down the possibilities to likely defects, examine the magnetic properties of these defects, and see if strong, long range ferromagnetic coupling exists between the defects. This chapter reviews the work presented in Refs. [17, 18] where a series of defective structures was investigated by means of periodic first principles calculations based on hybrid exchange DFT in the B3LYP form: (i) the vacancy structure previously proposed by Andriotis et al, where a carbon atom is removed from each fullerene cage [11]; (ii) a similar structure where the vacancies between two adjacent C6o cages are moved closer together in pairs; (iii) a new defective structure with a fractured intrafullerene bond that is spontaneously generated by applying isotropic pressure, and (iv) a structure consisting of a vacancy defect, the displaced atom forming an additional inter-cage link, spontaneously generated by simulating the high pressure, high temperature treatment used to generate the ferromagnetic material in the laboratory. In all these cases, local spin moments around the defects arise, but no evidence of long range ferromagnetic interaction between these moments is found. First principles calculations based on DFT currently play an important role in the characterisation of many materials, and in particular in determining structure-property relationships. Mixing non-local and semi-local exchange in hybrid-exchange flinctionals, as in the now very widely used B3LYP functional [19-21], yields a good quantitative description of thermochemistry [22], and optical band gaps [23]. Moreover, it has recently been shown that hybrid exchange DFT functionals provide a significantly more accurate description of the ground state electronic structure, magnetic coupling energies and magnetic moments, and metal insulator transitions in strongly correlated systems than the more commonly used generalized gradient approximations [24-28]. In addition, each of the defective structures is here analysed using a simple model for predicting ground state spin configurations, which was developed for non defective, planar, TT -conjugated carbon and hydrocarbon systems such as organic radicals [29-32] and extended bipartite lattices [33, 34]. This model is based on the observation that, in these systems, the spins of unpaired electrons belonging to carbon atoms which share a

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covalent bond are antiparallel to each other. As a resuh, the spin polarisation of the TT electrons simply alternates between the bonded atoms. In the remainder of this chapter, this model will be therefore referred to as the "spin alternation rule". The effect of the structural curvature and defects on the validity of this model will be addressed here. Inelastic neutron scattering (INS) measurements and B3LYP calculations investigating the role played by hydrogen in this material will also be reviewed. It was found that, in the presence of hydrogen intercalation, the defective structure that occurs spontaneously under the simulated high pressure, high temperature treatment, possesses a ferromagnetic ground state. This chapter is organised as follows: the methodology is described in section 2, and is followed by an examination of the pristine structure of Rh-C6o in section 3. Defective structures of pure polymerised Rh-C6o are discussed in sections 4, 5 and 6, and a defective structure in the presence of hydrogen is described in section 7. The work is then summarised in section 8.

2 Methodology The methodology used has previously been described in some detail [17, 18] and involves a combination of reactive force-field simulations and B3LYP calculations. The force field simulations are used to suggest likely structures, the structure and stability of which are refined using the B3LYP calculations. The latter are also used to compute the magnetic coupling between the dominant spin centres. The starting geometry is the experimentally determined crystalline structure of Rh-C6o, for which the lattice parameters are a = 9.19 A, c = 24.5 A in the hexagonal unit cell [35]. As the work reviewed here focuses on the study of the covalent intercage bonds as a mechanism for long-range coupling, the system is restricted to a single layer of Rh-C6o, which is here described without any symmetry constraints. In order to generate realistic defect structures, reactive force field simulations under high pressure conditions are used to produce a number of different defective structures. For the calculations with finite temperature, reactive force field molecular dynamics (MD) simulations are performed within the NVT ensemble with a time step of 0.1 fs and run until equilibrium was reached. These simulations are carried out with the most recently developed variant of the bond order potential due to Brenner et al [36], as implemented in the GULP program [37], which allows an adequate modelling of bond breaking and formation processes. The internal coordinates of the structures generated with the static, force field calculations are relaxed within the unrestricted Hartree-Fock (UHF) method, and those of the structures generated with the MD simulations are relaxed with B3LYP calculations. All geometry optimisations were performed in the ferromagnetic configuration using an algorithm proposed by Schlegel et al. [38]. The magnetic properties of all structures are then examined within the B3LYP approximation. The periodic, B3LYP calculations are performed by using the CRYSTAL program [39], where the crystalline wavefunctions are expanded as a linear combination of atom centred Gaussian orbitals (LCAO) with s, p, ov d symmetry. In the current study all-electron calculations are performed, which make no assumptions about the

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shape of the ionic potential or electron charge density. Basis sets of double valence quality (6-2IG* for C and 6-3IG* for H) are used. Reciprocal space sampling with a (4x4) Monkhorst-Pack grid [40] is sufficient to converge the total energies to within 0.1 meV per C60 cage in all structures considered. The Gaussian overlap criteria which control the truncation of the Coulomb and exchange series in direct space have been described in detail elsewhere [39]. In this work they were set to 10"^ 10'^, 10'^ 10'^, 10'^ and 10"^^. In order to compare the energies of various magnetically ordered states, it is necessary to converge stable self-consistent field solutions for different electronic spin configurations. This is achieved by using a superposition of spin-polarised ionic charge and spin densities for particular atomic states to provide a suitable initial wavefiinction. In some cases, more direct control of the spin occupancy patterns has been used to control the initial conditions, such as, for instance, the total spin. We emphasise, however, that this only affected the initial wavefunctions and that all solutions presented are unconstrained.

3 Pristine Rh-C6o The electronic structure of the three-dimensional, experimentally determined pristine structure of Rh-C6o (a = 9.19 A, c = 24.5 A in the hexagonal unit cell) [35], is calculated within the B3LYP approximation, and results in an indirect fundamental band gap of 1.48 eV. The system is non-magnetic and insulating in agreement with previous studies [10, 41, 42]. When a single layer of the pristine Rh-C6o was relaxed within the reactive force-field formalism, the resulting lattice constant, a = 9.23 A, was found to be only 0.4% larger than the experimental bulk lattice constant. This result documents the ability of the reactive force field employed to reproduce the structure of polymerised C60.

4 Prototype Defective Structures: Vacancy Defects 4.1 A Central Vacancy Defect A defect structure, originally proposed by Andriotis et al [11] is created starting from the experimental geometry [35] and removing one carbon atom from each fiillerene cage, as shown in Fig. 2(a). The equilibrium lattice constant is found to be a = 9.22 A. Many different spin configurations are generated and the one with the lowest energy amongst them is identified to be the ground state. This configuration is between -0.06 and 0.44 eV per cage lower in energy compared to all other metastable configurations found. The spin density map of the ground state. Fig. 2(b), shows that the spin density localises at the dangling bonds, as expected. This configuration has a total magnetic moment of 2.0 JUB per C6o cage, and the spin polarisation pattern of the undercoordinated carbon atoms, shown in Fig. 2(b), is consistent with the spin alternation rule described in Sec. 1. In order to test for inter-cage magnetic coupling, a (2x1) supercell, containing two C60 cages, is considered. Two spin configurations are examined. In the ferromagnetic

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Fig. 2. The central vacancy structure proposed in Ref. [11] with the undercoordinated atoms, labelled as CI, C2 and C3, (a). The spin density maps of the ground state in the (b) parallel (FO), and (c) antiparallel (AF) inter-cage spin configurations. The light and dark isosurfaces correspond to values of+0.015 and-0.015 JUB, respectively. (FO) state, the spin configuration is identical to the ground state obtained for the (1x1) cell and described above. In the antiferromagnetic (AF) state, all spin orientations on one of the two €50 cages in the supercell are reversed, as shown in Fig. 2(c). The results are quantified in Table 1 and show that there is no significant difference between the energies of the FO and AF states and thus no significant coupling exists between the spin moments of neighbouring cages. Also, no significant change in the Mulliken

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Table 1. Total magnetic moments per C6o cage (S), MuUiken spin populations of the undercoordinated atoms labelled as in Fig. 2, and total energy differences with respect to the FO configuration, for the FO and AF configurations of the central vacancy structure. Configuration s {JUB) Atoms Spin {\e\) AE^^^ (eV/Ceo) FO lO r22 0.000 CI 1.22 C2 1.26 C3 -0.83 CI -1.20 AF 2.0 0.000 C2 -1.24 C3 0.84 charge or spin overlap populations of the interfullerene bonds between FO and AF spin configurations is seen, in agreement with the lack of interaction between the cages. If this vacancy defect structure could be realised experimentally, these calculations predict that it would only exhibit a paramagnetic behaviour. The two-dimensional band structure of this paramagnetic state has low lying, flat bands at the Fermi edge separated by a spin-flip, indirect band gap of 1.66 eV. In this case, however, the flat bands are simply a result of the localised spin orbitals with no significant inter-cage interaction and are thus not indicative of a magnetically ordered ground state. 4.2 A Migrated Vacancy Defect The high pressure high temperature treatment is likely to drive defect migration within the samples. It is therefore of interest to investigate whether the prototype defective structure examined in Sec. 4.1 would exhibit inter-fiillerene ferromagnetic coupling if the distance between the defects was reduced. A modification of the central vacancy structure is thus examined, where the vacancy defects on adjacent fullerene cages are positioned closer together in pairs. Fig. 3(a). The same lattice parameters as in the central vacancy defect structure are used in order to facilitate a meaningfiil comparison of their total energies and determine whether such defect migration is likely. The ground state configuration of this migrated vacancy defect is found to be 0.291 eV per cage higher in energy than the central vacancy structure, indicating a strong repulsion between vacancies. Figure 3(b) shows the spin density map of the spin configuration with the lowest energy amongst all the configurations found and referred to as AFl. The intra-cage geometric and electronic arrangement of the three undercoordinated C atoms around each vacancy is similar to that of the central vacancy structure. The majority of the spin polarisation is described by the spin alternation rule, with exceptions on the undercoordinated atoms C4 and C5. If the spin density on these atoms were described by the spin alternation rule, the spin density on atom C5 would be parallel to that on C6, and antiparallel to that on C4, contrary to what is observed. Fig 3(b). This suggests that the dominant spin interactions around this defects are through space, rather than through

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*n\ 1

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Fig. 3. A migrated vacancy structure with the undercoordinated atoms labelled as C4, C5 and C6, (a). The spin density maps in the (b) antiparallel (AFl), and (c), parallel (AF2) inter-cage spin configurations. The light and dark isosurfaces correspond to values of +0.015 and -0.015 JUB, respectively. bond, and therefore favour an antiparallel spin alignment, between the atoms C5 and C6, and between C4 and C6. The polarisation of the spin density also decays more slowly away from the defects and results in a cancellation of the local moments and a zero total magnetic moment per Ceo cage. Even in the absence of a net magnetic moment per cage, it is still interesting to determine the nature and extent of the magnetic interaction between the defects. The energies of the AFl state just described is therefore compared to that of a new state, called AF2, Fig. 3(c), where all the spins on one of the two Ceo cages in a (2x1) supercell are flipped. The results are reported in Table 2. An energy difference of

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16 meV per cage between the AFl and AF2 states confirms an increased spin coupling compared to the central vacancy defect. The coupling, however, is antiferromagnetic, in agreement with the spin alternation rule. The results thus indicate that this migrated vacancy structure is an unlikely candidate for the ferromagnetic phase.

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Fig. 4. A defective structure generated by applying isotropic pressure, with the undercoordinated atoms labelled as CI, C2, C3 and C4, (a). The spin density maps in the (b) AFl, and (c) AF2 configurations. The light and dark isosurfaces correspond to values of+0.010 and -0.010 JUB, respectively.

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Table 2. Total magnetic moments per Ceo cage (S), Mulliken spin populations of the undercoodinated atoms as labelled in Fig. 3, and total energy differences of the A F l and AF2 configurations of the migrated vacancy structure. The energies are relative to the FO configuration, Table 1, of the central vacancy defect described in Sec. 4.2. Configuration s (JUB) Atoms Spin (\e\) M^^^ (eV/C6o) AFl

OO

AF2

0.0

C4 C5 C6 C4 C5 C6

-0.89 -0.67 1.18 0.90 0.68 -1.16

+0.291

+0.307

5 Spontaneous Formation of Defects from Pressure In order to search for more realistic defects, reactive force field static simulations are performed in which isotropic pressure is applied by gradually reducing the a and b lattice parameters in the (1x1) cell and relaxing all internal coordinates at each step. Defective structures appear spontaneously for a 5 - 6% reduction in the lattice constants. A number of different structures are generated in this way as the total strain and its rate of application are varied. In a commonly generated structure, bond fracture occurs for the m^ra-fullerene bond located between the two inter-fullerene bonds, as shown in Fig. 4(a). This is in contrast with the usual assumption that the inter-fallevQnQ bond would be the most likely to break [13-15]. However, the single crystal X-ray diffraction studies performed by Chen et al. [43] identified this intra-flillerene bond as the longest in the structure, thus suggesting that it may be the weakest bond in the polymerised structure. Here the computed distance between the two atoms involved in the bond increases by 62%, from 1.594 A to 2.583 A as the bond breaks. The resulting defective structure exhibits local moments on the undercoordinated atoms and is, once again, characterised by flat, spin polarised bands at the Fermi edge. Two spin configurations are analysed in a (1x1) cell, containing one fuUerene cage, in order to investigate the intm-csigQ magnetic coupling. In the FO configuration, the spin moments on the two undercoordinated atoms in the cage are parallel to each other, and in the A F l configuration they are antiparallel (Fig. 4(b)). In both cases, the spin density map shows that the spin density is localised in;?-orbitals that are directed toward each other. The AFl state tums out to be the most stable state in the (1x1) cell, as illustrated in Table 3. The favoured intra-csigQ magnetic coupling is therefore antiferromagnetic, in agreement with the spin altemation rule, and the magnitude of the coupling is sizeable. Because of its antiferromagnetic nature, however, this structure has no net magnetic moment. In order to investigate the inter-cagQ magnetic coupling, the energy of the A F l state is compared with that of the AF2 state, obtained starting from the A F l state represented in a (2x1) supercell and flipping the spins on one of the two fiillerene cages, (Fig. 4(c)). The difference between the total energies of the A F l and AF2 configurations amounts

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Table 3. Total magnetic moments per C6o cage (S), Mulliken spin populations of the undercoordinated atoms as labelled in Fig. 4, and total energy differences with respect to the FO configuration, for the FO, AFl and AF2 configurations of the structure generated by isotropic pressure. Configuration s {/UB) Atoms Spin {\e\) AE^^^ (eV/Cgo) FO lO CI 0.74 0J4 0.000 CI C2 0.66 0.74 C3 C4 0.66 CI AFl 0.0 CI -0.28 -0.735 C2 0.25 C3 -0.28 C4 0.25 -0.29 CI AF2 0.0 CI -0.29 -0.736 C2 0.25 C3 0.29 C4 -0.25

to a mere 0.5 meV, thus indicating that no significant magnetic inter-cage interactions are present. Therefore this structure can also be discarded as a possible explanation of the ferromagnetic character of this material.

6 Spontaneous Formation of Defects from Pressure and Temperature This section reviews the only defective structure to date produced by simulating the high pressure, high temperature treatment used to generate the ferromagnetic samples in the laboratory. This is achieved by employing MD simulations, the detail of which is described in Sec. 2, using a mixture of isotropic and anisotropic pressures, and temperatures similar to those used to form the magnetic phases experimentally. By using this procedure, a common defect structure naturally arises under a reduction of ~ 2 ^ % and ^4-6% in the a and b lattice constants, respectively, and at temperatures between 900-1 COOK. In this structure, shown in Fig. 5, an atom breaks away from the cage and two of its neighbours (labelled as A and B) to form an inter-cage linkage at position C. This bridge atom forms an angle of 132.1° with its neighbouring atoms. This defect is reminiscent of the vacancy-adatom pair, which is often seen in irradiated graphite [44], and the structure retains essentially intact Ceo cages within the rhombohedral symmetry, in agreement with TEM observations [3]. This structure may also be thought of as a natural precursor to the decomposition of the polymerised fixllerenes into the graphitic phase and is therefore consistent with the observation of magnetism near this phase boundary.

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Fig. 5. A defective structure generated from MD simulations at high pressure and temperature. Atoms A, B and C are undercoordinated, and the atoms shown in white are involved in the two competing inter-cage coupling routes. Analysis of the ground state spin density reveals local moments at the dangling bonds on atoms A, B and C (Fig. 5). The orientation of the spin moments in this structure was found to be consistent with the spin alternation rule, producing a total magnetic moment of 4 JUB per Ceo- The two-dimensional electronic structure has an indirect fundamental gap of 1.31 eV, slightly smaller than that of the central vacancy case, which suggests that the states relative to the unpaired electrons are more delocalised and that there is greater interaction between them. Even so, the energy differences between the stable electronic configurations found for this structure turn out to be negligible. Analysis of this system with the spin alternation rule shows that there are two competing pathways for the inter-cage spin coupling, which are highlighted in Fig. 5. The pathways lead to competing inter-cage magnetic interactions, with path 1 generating an antiferromagnetic coupling and path 2 a ferromagnetic coupling. The coupling energies of these pathways are of similar magnitude, leading to a net cancellation of inter-cage coupling and thus to an overall ground state that has no net magnetisation.

7 A Ferromagnetic Ground State in the presence of Hydrogen As described in Sec. 1, several experiments involving magnetic carbon-based materials show that hydrogen may play an important role in the appearance of the magnetic behaviour. INS measurements were therefore performed in order to determine the hydrogen concentration in a sample of ferromagnetic Rh-C6o (see Ref. [17] for details). The resulting INS spectrum reveals a broad C-C stretch at 170 ± 7 meV,

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0.30

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0.10

0.00 200

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Energy (meV)

Fig. 6. The INS intensity, S(E), obtained with an incident energy of 600 meV. The sample was 99.8% purity Ceo converted to the magnetic rhombohedral phase by quenching from 9 GPa and 800K. together with a clear C-H stretch at 368 ± 1 meV, Fig. 6. The latter peak reveals the presence of a significant amount of hydrogen covalently bonded to carbon in the ferromagnetic samples. The H:C6o ratio is determined to be approximately 17%. This amount is of the same order of magnitude of the percentage (30%) of magnetic phase detected in the samples by MFM [6]. In order to investigate the effect that H intercalation may have, in the calculations is is assumed therefore that one H atom per C60 is present in the ferromagnetic phase. Possible bonding sites for hydrogen intercalation in the defect structure described in Sec. 6 are examined using B3LYP calculations. The choice of the sites considered is guided by the consideration that the most likely bonding sites are those where the H atom quenches dangling bonds, i.e. the three undercoordinated atoms labelled as A, B, and C in Fig. 5. The binding energies at these sites, calculated with respect to H2 gas and with fully relaxed internal coordinates, are found to be 3.09 eV for site A, 2.91 eV for site B, and 2.45 eV for site C. The geometry with the H atom bonded to atom A is therefore preferred. Table 4. Total magnetic moments per C60 cage (S), Mulliken spin populations of the undercoordinated atoms as labelled in Fig. 5, and total energy differences with respect to the FO configuration, for the FO and AF configurations of the structure generated by M P simulations at high pressure and temperature and with a H atom bonded at site A. Configuration S (//^) Atoms Spin {\e\) AE"^^^ (eV/C^^) FO

3.0

AF

0.0

B C B C

0.99 0.96 -0.98 0.94

0.000 +0.003

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Fig. 7. The spin density maps of the (a) FM and (b) AF configurations of the defective structure generated from pressure and temperature with H bonded to atom A. The Ught and dark isosurfaces correspond to values of+0.008 JUB and -0.008 /z^, respectively. The electronic ground state of this system is found to have a total magnetic moment of 3.0 JUB per C6o, Table 4. The spin density map (Fig. 7) shows that the spin density is localised predominantly on the B and C sites while that at the A site is suppressed by the H atom. The bonding of hydrogen to atom A therefore quenches the main spin centre in path 1, leaving path 2 as the dominant inter-cage coupling mechanism. This results in di ferromagnetic ground state for this system. The average magnetisation of the experimental samples was measured to be 0.34 |UB per cage [3]. For this to be the case in the theoretical structure, - 1 1 % of the C6o cages would be magnetic and the H:C6o ratio would thus be - 1 1 % , in reasonable agreement with the 17%) of H:C6o ratio obtained from the INS data reported above. The two dimensional electronic band structure for this system displays an indirect, spin-flip band gap of 1.44 eV, whilst the smallest direct band gap between states of same spin is 1.74 eV. The computed energy difference, however, between the ferromagnetic and antiferromagnetic states is only 3 meV per cage, Table 4. Although this energy difference is too small to explain room temperature ferromagnetism in Rh-C6o, the intercage linkage found does provide a mechanism for ferromagnetic coupling. It is possible

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that structures which allow for greater spin delocalisation or multiple linkages including interlayer linkages, will give rise to significantly larger inter-cage couplings.

8 Conclusions Various defective structures were examined with the aim of investigating the origin of the ferromagnetism observed in metal-free Rh-Ceo samples at room temperature. Reactive force field simulations and periodic hybrid density functional B3LYP calculations were combined to identify Rh-Ceo defective structures and determine their magnetic ground states. A recently proposed vacancy structure was used as a prototype defect. Its ground state was found to possess a net magnetic moment per cage but the coupling between the spin moments on adjacent cages was negligible. The possibility of the vacancy defects migrating closer together was also investigated. In this case, the net magnetic moment per cage was found to be zero. Significant magnetic coupling between the cages was present but antiferromagnetic in nature. The application of isotropic pressure to a layer of Rh-C6o resulted in the fracture of the intra-cage bond between the two inter-cage bonds. The unpaired electrons, localised on each cage, coupled antiferromagnetically, resulting in no net magnetic moment per cage, and no significant coupling between the defects on adjacent cages was found. Classical molecular dynamics calculations based on a reactive force field were used to simulate the high pressure, high temperature treatment used for the synthesis of the ferromagnetic samples. This simulation produced a defective structure where a carbon atom gets displaced and forms a new bond with a neighbouring fullerene cage. The undercoorinated bridge atom between the cages presented a possible new route, other than the interfullerene bonds created by the 2+2 cycloadditon, through which the local moments could couple. It was found, however, that this structure displays no net magnetisation. The electronic band structures of these defective layers are all spin-polarised, with flat bands at the Fermi edge, and with insulating band gaps. In the systems presented, however, these features in the band structures are not a reliable indication of a tendency to form a ferromagnetic ground state as they are generated by the presence of a localised spin density with no significant inter-cage coupling. Even so, the structures with central vacancies, bridge atoms, or similar defect structures that possess a net magnetic moments per cage, could be responsible for the large paramagnetic signal observed in the samples [3]. The non-metallic character of the band structures of all the systems presented here suggests that the most likely mechanism for magnetic coupling is either direct exchange or superexchange mediated by the delocalised ;;r-system. This is consisent with this work's findings, where the spin alternation rule is found to retain its validity in nearly all systems examined. In only one portion of a defective structure examined here, the spin alternation rule breaks down due to a dominant through-space, rather than through bond, spin polarisation effect. It is also noteworthy that the spin polarisation of the open-cage fullerene structures and nanotube segments investigated by Kim et al [48] is

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also consistent with the spin alternation rule. The presence of curvature and sp^ atoms, however, shortens the range of the polarisation effect, and hence the magnitude of the magnetic interactions, compared to the non-defective, planar 5;p^-hybridised systems for which the spin alternation rule was initially formulated. In fact, in planar systems, the spin polarisation can be seen to propagate across fifteen or more bonds in planar graphitic ribbons [45], and long range effects have also been observed in planar graphite [46] and graphitic ribbons [47]. This simple model is thus a very valuable guiding tool for anticipating which defect structures might generate ferromagnetic ground states in carbon-based materials. The presence of intercalated H in a concentration amounting to ~17% in the ferromagnetic samples was detected by inelastic neutron scattering experiments. In the calculations, the addition of H in the most stable site of the structure formed under pressure and temperature resulted in a ferromagnetic ground state. The inter-cage coupling, however, is insufficient to explain the very high Curie temperature observed. This structure is, to the authors' knowledge, the only proposed structure so far that has been obtained by simulating the experimental treatment leading to the ferromagnetic phase, and that is consistent with the ferromagnetic ground state. Acknowledgements The authors would like to thank S.M. Bennington, J.W. Taylor, and J.D. Gale for their contributions to the work reviewed here and the EPSRC for provision of computer time under the Materials Chemistry Consortium project, GR/S13422/01.

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