Theoretical Studies on Electronic States of RhC 60 . Possibility of a Room-temperature Organic Ferromagnet

A possible mechanism for a ferromagnetic interaction in the rhombic (Rh) form of C60 (Rh-C60) is suggested on the basis of theoretical studies in relation to cage distortion of the C60 unit in the polymerized 2D-plane. Band structure calculations on Rh-C60 show that cage distortion leads to competition between diamagnetic and ferromagnetic states, which give rise to the possibility of thermally populating the ferromagnetic state.


Introduction
The fullerene C 60 has attracted much interest since its discovery by Kroto et al. [1] because of its symmetric structure.After the discovery of a synthetic route to fullerenes by Krätschmer et al. many interesting physical properties have been reported [2].It is well-known that C 60 is the most symmetric molecule, exhibiting icosahedral symmetry (I h ) with 120 symmetry operators.This symmetry leads to a highly degenerate set of molecular orbitals (MO), including five-fold degenerate HOMO (H g ) and three-fold degenerate LUMO (T u ).It has been reported that there are several types of lattice structures in solid states C 60 such as f.c.c.[3,4] and b.c.c., as well as a two dimensionally (2D) polymerized rhombohedral lattice [5].This rhombohedral phase (Rh-C 60 ) [Figure 1 (A)] was originally produced by Iwasa et al. under high pressure and high temperature and a number of studies on its properties have been reported [6,7].Recently, Makarova et al. reported weak ferromagnetism in Rh-C 60 [8] with a conspicuously high Curie temperature (Tc ≈ 500K).Magnetic studies on a number of other molecule-based fullerene derivatives have exhibited substantially lower magnetic ordering temperatures e.g.tetrakis(dimethylamino)ethylene-fullerene (TDAE)-C 60 becomes ferromagnetic below 16.1K [9], whilst La@C 82 is paramagnetic [10] and antiferromagnetic ordering has been observed in (NH 3 )K 3 C 60 below 45 K [11,12].In all these cases the magnetism is believed to originate from the magnetic moment on the charged fullerene after doping.Since Rh-C 60 is constructed only by carbon, there appears to be no doping mechanism, which leads us to a different mechanism from other fullerenes.
Interactions between C 60 units in 2D-Rh-C 60 may be classified into two types; inter-layer and intralayer.The first type is weak and is almost the same as interactions observed in pristine C 60 .The second one is considered to be a strong interaction between hexagon faces of neighbouring C 60 units, illustrated by gray and black circles, respectively in Figure 1 (B).Although many experimental and theoretical studies have been performed for this material, the electronic states have not still been elucidated.In this study, the electronic states of Rh-C 60 are examined using ab initio band structure calculations, focusing particularly on the effect of distortions of the C 60 cage in the polymerized 2Dplane, specifically: (i) the sp 2 -sp 3 rehybridization, (ii) the band structure and the effect of distortion of C 60 cage with variation of distortion parameter d and (iii) the relative stability of low spin and high spin state of Rh-C 60 with variation of d.

Experimental background
The rhombohedral form of C 60 was first reported by Iwasa et al. [5] who found that fullerene can be converted into two different structures by the application of high pressures and temperatures.Between 573 K and 673 K at 5GPa, a f.c.c.structure was produced with lattice parameter a = 13.6Å,whilst at higher temperatures (773 to 1073 K) at the same pressure, C 60 was transformed into a rhombohedral structure with hexagonal lattice parameters of a = 9.22Å, c = 24.6Åand space group R-3m.These two structures are metastable and revert to pristine C 60 upon reheating up to 573 K at ambient pressure.
Subsequent studies by Makarova et al. have investigated the physical properties of Rh-C 60 which showed some dependence on the preparative conditions [8].The stability limit of the C 60 cage is 1073 -1173 K at 6 GPa, above which there is a transition from Rh-C 60 to amorphous carbon.As a consequence, samples of ferromagnetic Rh-C 60 were synthesized close to the stability limit of the C 60 cage; Rh-C 60 prepared at 923 K and 6 GPa exhibits semiconductor-like behavior, while materials synthesized at temperatures above the polymerisation limit showed some anisotropy of their electrical properties.Five out of six samples synthesized between 1025 K and 1050 K exhibited ferromagnetic behavior.This behavior was observed in 2D polymerized Rh-C 60 but not 1D orthorhombic C 60 .
This ferromagnetic Rh-C 60 exhibits a pentagonal pinch A g (2) vibrational mode at 1407cm -1 in its Raman spectra.The mode frequencies of them correspond to those of normal Rh-C 60 and suggest only subtle structural differences between ferromagnetic Rh-C 60 and normal Rh-C 60 .Indeed X-ray diffraction patterns of ferromagnetic Rh-C 60 indicates that its cell constants (a = 9.204Å, c = 24.61Å,with space group R-3m) are quite similar to those of normal Rh-C 60 .The observed magnitude of magnetic susceptibility of ferromagnetic Rh-C 60 was 100 times larger than that for graphite and 10,000 times larger than that for pristine C 60 .The spin concentration was 5× 10 18 cm -3 .The magnetic moment was estimated as 0.4µ B , where µ B is the Bohr magneton.
Makarova et al. [13] rationalised the experimental observations of the electronic properties of Rh-C 60 as follows: (i) anisotropic properties corresponding to in-plane and out-of-plane conductivity and (ii) metallic-like behavior due to in plane conductivity at high temperature which decreased with decreasing temperature.However, in relation to this metallic behaviour, the necessary sp 2 -sp 3 rehybridization mechanism required for the [2+2] cycloaddition between facial carbons of fullerenes, cannot be possible since the sp 2 -sp 3 rehybridization process supports the magnetic behaviour of Rh-C 60 rather than its metallic properties.The facial C atoms forming sp 3 hybrid orbitals will bear electron spins.However, as mentioned above, experimental results, conductivity mitigate against this mechanism since the 'dangling bond' associated with an sp 3 hybridised facial C indicates strong localization of the electron on pristine C 60 , while the observed conductivity indicates itinerant behaviour.
Production of magnetic domains in Rh-C 60 was independently reported by another group [14].Wood et al. synthesized polymerized Rh-C 60 in the region 700~1200 K under 9 GPa pressure.They indicated that Rh-C 60 synthesized at 800 K exhibited magnetism, while the C 60 cage collapsed above 850 K in good agreement with previous observations.Therefore, near the stability limit of C 60 cage, they also found the ferromagnetic fullerene.The summary of these experimental results can be expressed in the form of a phase diagram (Figure 2)

Computational details
In this study, the LCAO tight binding approximation has been utilized with hybrid density functional theory, using the CRYSTAL98 program package.The computational method was Becke's 3 parameter functional combined with the non-local Lee-Yang-Parr correlation (B3LYP) and a 6-21G basis set of Gaussian-type orbitals.As the auxiliary basis set for fitting exchange and correlation potentials as well as the electron density, one s-type orbital, two d-type orbitals and one f-type orbital were applied.In the CRYSTAL98 program package, numerical accuracy was controlled mainly by five parameters (ITOL1-ITOL5) and the number of k points.ITOLx (x=1 -5) control truncation criteria for two-electron integrals.Because low ITOLx values lead to inaccurate density matrix, we set ITOL1-4 = 10, ITOL5 = 18.The space group and cell parameters were taken from experimental results (space group R-3m, a=9.204Å, c=24.61Å).The number of k-points in the irreducible Brillouin Zone (BZ) was 32 points.All C-C bonds within C 60 were assumed to be 1.40Å.
Several allotropes of carbon were examined using this approach, including graphite, diamond and f.c.c.C 60 in order to verify the accuracy of this approach, with respect to band gap and structure.These data are available as Supplementary Information at the end of this paper.These calculations reproduce the semi-metal character of graphite, as well as a band gap in diamond (5.5 -5.8 eV) in good agreement with experimental values (5.4 -5.6 eV [15]).For f.c.c.C 60 the calculated band gap was 2.0 eV (c.f.experimental value of 1.5 eV ).In order to examine the possibility of sp 2 -sp 3 rehybridization, that is, production of dangling bonds on the C * atom [labeled (1) and (1') in Figure 1 (B)], we have performed a comparative study of their bond population in comparison to sp 3 hybridised diamond (bond population = 0.319) and sp 2hybridised graphite (bond population = 0.440).Table1 lists the bond populations on the C * (1)-C(2) and C * (1)-C * (1') bonds.On the basis of these results, Rh-C 60 is considered to have a sp 2 hybrid orbital.Despite the fullerene cage distortion, the bond population on C * (1)-C * (1') bond was not changed substantially.Since f.c.c.fullerene also exhibits an sp 2 hybrid orbital, the possibility of sp 2 -sp 3  rehybridization is unlikely and in agreement with the metallic character of Rh-C 60 reported by Makarova et al.

Group theoretical prediction
Before the calculation of band structures, the splitting or degeneracy of energy band is predicted from a group theory examination.Figure 3   Under the empty lattice approximation, the Bloch function is expressed as u kn (r) = e iK ⋅r (1b) for which the energy corresponding to this wavefunction is where k and K denote wave number and reciprocal lattice vectors respectively.When the behavior at Γ-point is in the question, k set to be zero in eq.( 2).Then the following eight functions (3a-h) which satisfy E = E 0 were constructed, , (3e),(3f) ),(3h) By operating the elements that belonged to the group of Γ in Table 2, the following reducible representation was obtained and it could be resolved as Table 2. Character table for G-point of rhombohedral lattice Using the projection operator method[18], the following linear combination of Bloch functions were obtained.
Using the following relations, cos x ≈ 1+x and sin x ≈ x, these wavefunctions can be rewritten as (5g),(5h) According to the character table of I h which pristine C 60 belongs to, the three-fold degenerate LUMOs split into a singly-degenerate ψ (Γ 5 ) and two-fold degenerateψ (Γ 6 ) , whilst the five-fold degenerate HOMOs split into a singly-degenerate ψ (Γ 4 ) and two-fold degenerate ψ (Γ 1 ) and ψ (Γ 3 ).These predictions were confirmed by the following band calculations.

(A) (B)
The two lowest conduction bands (LCB) were essentially two-fold degenerate at the Γ point.According to the IR-active modes [6], we confirmed that LCB showed the interaction of C * -C * bond between C 60 units.Conversely, the highest valence band (HVB) was non-degenerate.This split of HVB from the five-fold degenerated H g came from anisotropic structure of the rhombohedral phase.Namely, LCB and HVB corresponded to (x,y) and z 2 symmetry respectively.Therefore if the LCB are )× 100 D 0 stabilized, they can get closer to the HVB without repulsion because they are mutually orthogonal.The calculated band gap was 1.23 eV, reproducing the semi-conducting nature of normal Rh-C 60 .

Rh-C 60 with distortion of C 60 cage
Since the LCB expresses the inter-fullerene interaction, it is expected that these LCB will be stabilized when the distance of inter-fullerene become shorter.A distortion parameter d was defined as ( 6) where D and D 0 denote the inter-fullerene distance with and without distortion, respectively (i.e.D 0 = 2.408Å, D < 2.408Å).By varying the parameter d, the LCB were stabilized as shown in Figure 5. Table 3 shows the variation in the Fermi energy (E f ) and band gap (E g ) as a function of this distortion parameter.An analysis of these results indicates that the greater distortion of the C 60 cage leads to a decrease in the band gap, E g .Considering the Fermi-Dirac distribution at room temperature (T=300K), Rh-C 60 is expected to become metallic in agreement with the experimentally observed semiconductor-metal phase transition [8].It is noteworthy that the dispersion of the HVB is considerably small, that is, a flat band was observed.The LCB and this HVB became closer and closer as d increased.Of particular note is the case when d =11.6%.At this point one of the LCB and the HVB form a two-fold degenerate half-filled flat bands around the Fermi level as shown in Figure 6.
According to the Mielke-Tasaki theorem [17], this band structure satisfies the necessary condition for flat band ferromagnetism.Thus if on-site Coulomb repulsion U on fullerene is not zero, the ferromagnetic electronic state may be one of the possible ground states.In order to elucidate the possibility of ferromagnetic ground state, an examination of the relative stability of this ferromagnetic state is necessary.
Figure 7 shows the relative energies of the singlet (diamagnetic) and triplet (ferromagnetic) states per unit cell.From Figure 7, we observed four distinct regions as follows: (1) For small d (0 ≤ d ≤ 9.5 %, labelled A in Figure 7), the singlet state was more stable than triplet state.Since normal Rh-C 60 was considered to be semi-conducting, this is consistent with experimental results.( 2) An energy minimum for the triplet state at d = 2%.The energy of this triplet is stabilized by 0.06 eV compared to the undistorted structure (d=0%).However the triplet is still substantially higher in energy than the singlet at this point (0.68eV) and so a contribution of the triplet state to the magnetic properties would appear negligible in region A (Figure 7).( 3) In region B (d > 9.5 %) the triplet state is now more stable than the singlet.Thus in this region, a ferromagnetic Rh-C 60 electronic state is expected arising from flat band ferromagnetism.It is, however, considered that this ferromagnetic phase is different from the experimental ferromagnetic phase since threes calculations reveal a finite band gap (semi-conductor) behaviour whereas the experimental studies indicate metallic behavior.(4) At the point of d≈9.5%, crossing of the singlet and triplet states occurs.Around the boundary between regions A and B, the triplet and singlet states are competitive.From the band structure of the singlet state at this boundary, a metallic state is predicted as the ground state, consistent with the experimental study.One possibility to explain the mechanism of ferromagnetism in Rh-C 60 could be proposed.(1) The experimental ferromagnetic Rh-C 60 phase occurs around point C, which is found at a distortion d just below the intersection of the singlet and triplet energies (Figure 7) which leads to metallic behaviour observed in the experimental ferromagnetic Rh-C 60 .(2) The singlettriplet gap at the point C is so small that some thermal population of the triplet state may occur.This thermally induced triplet Rh-C 60 can be considered as an experimentally observed magnetic domain.The population in the excited state at the various temperatures were estimated by the Boltzmann distribution as illustrated in Figure 8.The percentage of triplet is expected to be 2% at 300 K with d = 9.1%.Using this percentage of triplet, the spin concentration value (n) be estimated as follows: where V and S denote the volume of primitive cell (=601.8× 10 -24 cm 3 ) and spin magnitude (=2), respectively.The estimated value is 6.6 × 10 19 cm -3 , is comparable with the experimental value (5 × 10 18 cm -3 ) despite the simplicity of this approximation.

Phase diagram
The investigation of the relationship among superconductivity, diamagnetic and ferromagnetic phases seemed to be attractive work.The phase diagram of Rh-C 60 with band gap (E g ) against d was illustrated in Figure 9.As the parameter d increases, diamagnetic and ferromagnetic phases disappeared and appeared, respectively.For superconducting materials, the existence of a singlet magnetic phase (singlet superconducting or spin density wave etc.) is expected between diamagnetic and ferromagnetic phases.However, in case of Rh-C 60 , this singlet magnetic phase was not observed experimentally [8].It is considered that the occurrence of the ferromagnetic phase prevents the appearance of a singlet magnetic phase because of the relative stability of ferromagnetic phase.Conversely the triplet surperconductivity might be realized in the Rh-C 60 system.

Conclusions
In this paper, one possible mechanism to account for ferromagnetic Rh-C 60 was suggested on the basis of theoretical studies.Calculated band structures indicate the presence of a semiconductingmetallic phase transition with increase in the distortion parameter d.As d increases, the ferromagnetic state becomes stabilized in comparison with singlet state and finally becomes the ground state.This mechanism was considered to be associated with flat band ferromagnetism.However, the triplet state is expected to show semiconductor nature contrary to the experimental metallic character, whilst singlet state shows metallic results.Near the crossing of singlet and triplet states, the singlet-triplet gap is sufficiently small for some thermal population of the ferromagnetic state.This ratio was estimated as at most 2% at 300K by Boltzmann distribution, yielding a spin concentration of 6.6× 10 19 cm -3 , in agreement with the experimental one.In conclusion, the cage distortion of C 60 in polymerized 2Dplane Rh-C 60 leads to competition between diamagnetic and ferromagnetic states, with the ferromagnetic state populated by thermal excitation.Recently, several experimental and theoretical studies on Rh-C 60 polymers and related carbon systems have been reported [19][20][21][22][23][24][25][26][27].Some of them support previous results by Makarova et al.The present results provide one possible explanation of these results.Further experimental and theoretical studies are necessary towards achieving the goal of a room-temperature organic magnet.

Figure1.
Figure1.The crystal structure of Rh-C 60 .The cell parameters were obtained by Ref. 8. (A) 2D slab of Rh-C 60 on x-y plane; (B) Covalently bonding carbon (black circles) and the other carbons (gray circles); d is the distortion parameter.Only C* (black circles) were distorted.

Figure 2 .
Figure 2. The summary of experimental results.Experimentally observed phases are listed.
illustrates the B.Z. of Rh-C 60 crystal.The symmetry of this B.Z. is D 3d , which is a subgroup of I h .

Figure 3 .
Figure 3.The reciprocal space around the first B.Z.The cross points ( × ) denote reciprocal lattice points.

Figure 4 (
Figure 4 (A) The band structure of normal Rh-C 60 (d = 0%); (B) The first B.Z. with special points and line.The detail information is given in the text.

Figure 7 .
Figure 7. S-T gap of Rh-C 60 with variation of C 60 cage distortion.The information of region and point are in the text.

Figure 9 .
Figure 9.The band gap of Rh-C 60 with increase of distortion parameter d.In case of d = 0~9.5%,diamagnetic phase was obtained, while d > 9.5%, ferromagnetic phase was obtained.

Figure S1 .
Figure S1.(A) The calculated model of graphite.(B) The calculated B.Z. (C) The obtained band structure.The detail information is given in the text.

Figure S2 (A)Figures
Figure S2 (A) The calculated model of diamond; (B) The calculated B.Z; (C) The obtained band structure.The detail information is given in the text.

Figure S3 .( 2 )
Figure S3.(A) The calculated model of f.c.c.fullerene (B) The calculated B.Z. (C)The obtained band structure.The detail information is given in the text.

Table 1
Mulliken bond populations of several C-C bonds calculated by the B3LYP method.