Electronic properties of doped fullerenes › 5b9a › 3fe63d... · Materials made of large organic molecules exhibit a characteristic separation of energy scales. In many organic

INSTITUTE OF PHYSICS PUBLISHING REPORTS ON PROGRESS IN PHYSICS

Rep. Prog. Phys. 64 (2001) 649–699 www.iop.org/Journals/rp PII: S0034-4885(01)65223-0

Electronic properties of doped fullerenes

Laszlo Forro1 and Laszlo Mihaly2

1 Department of Physics, Ecole Polytechnique Federale, CH-1015 Lausanne, Switzerland2 Department of Physics and Astronomy, State University of New York, Stony Brook,NY 11794-3800, USA

Received 8 May 2000, in final form 5 March 2001

Abstract

The most abundant fullerene molecule, C60, has just the right combination of size, chemicalstability, and activity to serve as the building block for a large variety of solids with fascinatingproperties. After a short overview of the related carbon compounds, first the structures of theC60 molecule and the pure crystalline C60 are summarized. Experimental and theoretical workson the vibrational and the electronic properties of the molecule and the solid are reviewed.Next, some of the relevant concepts widely used in solid state phyics, like electron–phononand electron–electron interactions, Hund’s rule and Jahn–Teller distortion are discussed in thecontext of doped fullerides. The structural and electronic instabilities of the conducting C60

compounds are reviewed in detail. Finally, a few open questions are discussed.

0034-4885/01/050649+51$90.00 © 2001 IOP Publishing Ltd Printed in the UK 649

650 L Forro and L Mihaly

Contents

Page1. Introduction 6512. New carbon structures 653

2.1. Fullerenes 6532.2. Nanotubes 6542.3. Carbon onions 654

3. C60 and its derivatives 6543.1. C60 molecule and the crystal structure of the C60 solid 6543.2. Chemical modifications, endohedral and exohedral doping 6563.3. Electronic structure of the C60 molecule and the band structure of the C60 solid 6583.4. Molecular vibrations of the C60 molecule and the phonons in solid C60 660

4. Concepts and interactions 6624.1. Electron–phonon coupling 6634.2. On-site Coulomb repulsion 6644.3. Jahn–Teller instability 6664.4. Berry’s phase and orbital ordering 6684.5. Superconductivity 669

5. Conducting C60 compounds 6715.1. Structural instabilities 6735.2. Electronic instabilities 678

6. Open questions 6897. Conclusions 693

Acknowledgments 693References 693

Electronic properties of doped fullerenes 651

1. Introduction

Materials made of large organic molecules exhibit a characteristic separation of energy scales.In many organic solids the covalent bonds within the molecules are distinctly stronger than theinter-molecular binding, and the constituents preserve their molecular character. Quite often,the theory of an organic insulator is not much different from the theory of the molecule itself.

An entirely different situation is encountered in organic conductors, where, by definition,electrons must be able to move from one molecule to the other. The internal structure andlarge size of an organic molecule may facilitate the hopping of electrons. We may still havethe separation of the energy scales between inter- and intramolecular bindings. The moleculescan be still treated, at least in a first approximation, as separate entities within the unit cellof the crystal. But the low-energy vibrational and electronic excitations within the moleculefall in the same energy range as the collective excitations of the solid, leading to a veryrich behaviour, and great challenges to the theory. Concepts like ‘small polarons’ and ‘on-siteCoulomb interaction’ acquire a new meaning in molecular solids. Understanding the molecularproperties is a (crucial) first step, but it is not the end of the road.

Many of the organic solids exhibit an interesting separation of the length scales, as well.Chemical forces determine the shape of the molecule, and there is no good reason for theresulting object to fill space properly. In van der Waals solids, maximizing the binding energyof the solid is nearly equivalent to minimizing the empty space, but the molecular ’lego’ isnever perfect. In a solid made of large organic molecules there is usually plenty of empty space.Other atoms, ions or smaller molecules can easily fill the void. One can prepare, starting fromthe same organic compound, a great variety of materials with similar backbone structures, butwith different electronic properties.

The fullerene molecule, C60, is the ultimate building block for organic conductors inboth senses. The molecule is stable, and it has rich vibrational and electronic structure. Theunsaturated molecular π orbitals are ready to form electronic bands in the solid state. Theshape of the molecule is approximately spherical, leaving plenty of empty space for otherinorganic and organic constituents. The intense effort to study the novel materials, launchedby the discovery of volume synthesis of C60 by Kratschmer et al [1], comes as no surprise.

Fullerides are members of a larger family of organic conductors. Research on the organicsynthetic metals started in the 1970s and was largely inspired by the prediction of Little [2]that in low-dimensional organic conductors an excitonic mechanism of pairing could giverise to a high-temperature superconductivity (SC). Organic charge transfer salts of chain-like structure and reasonably high conductivity were synthesized. But SC is only one of thepossible instabilities of a quasi-one-dimensional (1D) electronic band. Microscopic parameterswere such that charge density wave (CDW) or spin density wave (SDW) developed beforesuperconducting pairing [3]. Most organic conductors exhibited a metal–insulator transitionat low temperatures. SC was finally achieved in the TMTSF salts, but the critical temperature(Tc) was of the order of 1 K, much lower than expected [4] (figure 1). The low Tc was attributedto strong 1D fluctuations, destroying the SC at higher temperatures. Only the weak interchaininteractions stabilize the long-range order.

The search started for new structures, with a more three-dimensional (3D) electronic band,and quasi-two-dimensional (2D) organic conductors based on the BEDT–TTF molecule indeedyielded Tc as high as 13 K [5]. Efforts to find a truly 3D organic conductor which would exhibitan even higher Tc were unsuccessful, but two decades of research on the low-dimensional or-ganic conductors was not in vain. The milestones of this research include a better understandingof the competition between different electronic instabilities, the role of electron–phonon andelectron–electron interactions, the signs of the non-Fermi liquid behaviour, the study of the


Figure 1. Electrical resistivity of the quasi-1D organic conductors TTF–TCNQ, (TMTSF)2PF6and (TMTSF)2ClO4. They have CDW, SDW and superconducting ground states, respectively(after [4]).

Frohlich SC, cascades of magnetic-field-induced phase transitions and quantum Hall effect.SC was discovered at high temperatures in the 3D organic conductors K3C60 (Tc =

19 K) [6] and Rb3C60 (Tc = 30 K) [7] in 1991. Although this breakthrough did not followdirectly from the earlier efforts to produce excitonic superconductors, the excitonic mechanismis still one of the contenders for the explanation of the superconducting pairing in fullerides.Just as in other organic conductors, a wealth of electronic and structural ground states werefound. The discovery of C60 compounds with 1D and 2D polymeric structures makes the linkto the TMTSF– and BEDT–TTF-based synthetic metals even stronger.

In this review we focus mainly on the conducting fullerides. After a brief introduction ofthe C60 molecule and the related carbon nanostructures, the main ingredients of the fullerenephysics (like the Jahn–Teller (JT) effect, Berry phases and orbital ordering, strong electron–electron and electron–phonon interactions, SC) is discussed. We review the structure of thevarious conducting compounds, and the structural instability resulting in the conducting C60

polymer. The electronic properties, including SC and other electron correlation effects, arediscussed next. Finally, a short summary of some of the open questions is presented.

The early activities in the field have been reviewed in a collection of papers edited withcommentary by Stephens [8]. Ramirez [9] focused on the superconducting C60 compounds.The major developments of the field are reviewed by the leading researchers in the proceedingsof the ‘anniversary meeting’ edited by Andreoni [10]. The polymeric fullerenes are discussedin detail by Janossy et al [11]. Reviews by Gunnarsson [12] and by Pickett [13] discuss theelectron–phonon coupling and electron correlation effects, including SC. Optical studies havebeen reviewed by Kuzmany [14]. Pintschovius summarizes the neutron scattering results,including measurements on the intra- and intermolecular phonon modes [15]. Pennington andStenger [16] review the NMR results on the pure and superconducting compounds. A recentwork by Sundquist [17] discusses the high-pressure studies on fullerene compounds. The bookby Dresselhaus et al [18] is the most comprehensive review of fullerenes, summarizing thestate of the art as of 1996.


Figure 2. Two schematic views of the C60 molecule.

2. New carbon structures

Graphite and diamond are the two well known allotropic forms of carbon, with markedly differ-ent physical properties. Diamond is a large-gap semiconductor. The non-conducting propertyfollows from the sp3 bonding scheme of the carbon atoms in the crystal structure. Graphiteconsists of parallel sheets of graphene, an sp2 bonded honeycomb lattice of carbon atoms. Asingle graphene sheet would be a zero-gap semiconductor, but the interlayer interactions turngraphite into a semi-metal. Intercalation of graphite by alkali metals was extensively studied,and SC in graphite/alkali metal compounds was reported as early as 1965 [19].

The discovery of fullerenes by Kroto and co-workers [20] initiated a new era in carbonchemistry. A large number of stable carbon structures were synthesized, using either variationsof the original method (laser evaporation of graphite [20]), or the arc discharge technique ofKratschmer and Huffman [1]. Most of the processes in use today produce a soot consistingof a mixture of amorphous carbon and a variety of ‘interesting’ materials in the first stage.Purification of the soot is a major part of the preparation. The typical end products includethe reactive C36 [21], a range of stable fullerenes (C60, C70, and up) [22], multilayeredcarbon onions [23], single [24] and multiwalled [25] nanotubes, and nanotube bundles. Thepreparation of these materials have been discussed in several excellent reviews [18].

2.1. Fullerenes

The lowest molecular weight structure that has some stability under ambient conditions is C36,described by Zettl and co-workers [21]. It has an icosahedral symmetry, similar to that ofC60. Due to the strongly curved C–C bonds, the electron–phonon interaction is expected tobe stronger than in higher fullerenes, and superconductors with higher Tc were predicted fromdoped C36 based compounds [26]. The first measurements, however, indicate that the C36

molecule is strongly reactive and in the solid state it spontaneously forms covalently bondedstructures.

C60, shown in figure 2, will be discussed later in great detail. The most effective way ofproducing C60 is the carbon arc method or its variations [1]. This fullerene can be purifiedwith a good yield, and it is readily available on the market at a reasonable price. It is stable attemperatures well above room temperature. It is chemically active, but stable in air.

The most abundant fullerene higher than C60 is C70. Its structure can be constructed byadding a ‘belt’ of five hexagons in one of the equatorial planes of C60. The lower symmetry ofC70 breaks the orbital degeneracy and the electronic energy levels are more structured than that


of C60. In the solid state, due to the elongated shape of the C70 molecule, there is a variety ofdifferent phases depending on molecular packing and orientational order. At high temperaturesfcc and hcp crystal structures were identified. Doping studies resulted in strongly disorderedconductors but no C70-based metal or superconductor was found [18].

A low fraction of the soluble soot extract contains fullerenes heavier than C70: for example,C76, C78, C82 etc. The number of isomers increases rapidly with the increasing fullerene size,representing some difficulty for the determination of their structure (mainly done by NMRmeasurements and theoretical calculations). Although they are produced in less abundantquantities, the higher fullerenes have an important role in the study of endohedral molecules:due to their larger size they can easier incorporate in their cage atoms like La, Ce, Tm, etc,resulting in, for example, Tm@C82 [18, 27].

2.2. Nanotubes

The discovery of the C60 motivated the search for other carbon compounds made of ‘curved’graphene layers. The efforts bore fruit in 1991, when carbon nanotubes were synthesized byIijima and co-workers [25]. These multiwalled nanotubes were made of concentric cylindersof rolled-up graphene sheets, capped with semi-fullerenes. The length of a tube was in therange of a few µm, the diameter was 10–20 nm. As the size increases, these structures exhibitproperties between fullerenes and graphite.

A major breakthrough was reached when the single-walled carbon nanotubes weresynthesized using metallic catalyzer particles [24]. These structures can show 1D metallicor semiconducting properties, depending on how the graphene sheets are rolled up. Theelectronic structure is 1D. Exotic theoretical predictions, like charge-spin separation or non-Fermi liquid behaviour, could be tested on these objects. The electronic properties stronglydepend on the bending and twisting of the nanotubes, and on the interaction with the substratematerial as well.

2.3. Carbon onions

Carbon onions were first observed in electron microscope after electron irradiation of graphiticmaterials [23]. They consist of concentric spherical layers stacked one inside the other, likeRussian dolls. Carbon onions were produced in various ways: by electron irradiation ofpolyhedral carbon particles which are present in the soot of arc discharge experiments, byheat treatment of diamond nanoparticles and by high-dose carbon ion implantation into copperfoil. A perfect spherical structure can be obtained by annealing the onions above 600 K [28].The size of these onions ranges from 3 to 50 nm. These objects are larger than the typicalfullerenes, but very often smaller than the carbon nanotubes. Quantum size effects may showup in their electronic structure.

Under extended electron irradiation in transmission electron microscope sp3-like defectswere seen in carbon onions. The carbon shells contract and exert pressure on the underlyinglayers. The pressure can reach such high values that the carbon in the core of the oniontransforms into diamond [29].

3. C60 and its derivatives

3.1. C60 molecule and the crystal structure of the C60 solid

The vast majority of research on the fullerene-based solids has been done on C60. In thismolecule the carbon nuclei reside on a sphere of about 7 Å diameter, with the electronic


Figure 3. Pure fullerene crystal (left), and the illustration of the crystal structure of the solid (right).(The single crystal was grown by H Berger, EPFL, Lausanne; the structural model is by J W Lauher,SUNY, Stony Brook.)

wavefunctions extending inside and outside by about 1.5 Å (figure 2). The diameter of themolecule is approximately 10 Å, and there is a 4 Å diameter cavity inside. The atoms areactually positioned at the 60 vertices of a truncated icosahedron (or ‘soccerball’) structure,with 90 edges, 12 pentagons and 20 hexagons. The two different C–C bond lengths in C60

(1.40 and 1.46 Å), indicate that the π electrons are not delocalized evenly over all bonds [30].(For comparison, the sp3 bond in diamond is 1.544 Å the C–C distance in graphite is 1.42 Å,and a typical the double-bond distance is about 1.35 Å.) The 30 short bonds are on the edgesthat are shared by two hexagons; the bond length at the 60 edges shared by a hexagon and apentagon are longer.

Symmetries of the C60 include the inversion symmetry, the mirror planes, the rotation by72◦ or 144◦ around axes piercing the centres of two opposing pentagons, 120◦ rotations aroundaxes through opposing hexagons and the 180◦ rotations around the axes going through themidpoints of opposing double bonds. The relationship of the 120 possible different symmetryoperations defines a symmetry group that is equivalent to the icosahedral group Ih. Thisgroup can be decomposed into irreducible representations of dimensionality d = 1, 3, 3, 4and 5 denoted by A, F1, F2, G and H , respectively. Due to the inversion symmetry, therepresentations come in pairs with symmetric and antisymmetric character, yielding the gand u labels, respectively, for each representation. (Notice that the number of elementsin a d-dimensional representation is equal to d2, and the total number of elements in thegroup is indeed 120 = 2(12 + 32 + 32 + 42 + 52). The factor of 2 stands for the u andg symmetries.) The electronic states and molecular vibrations are labelled according tothe irreducible representations, using lower- and upper-case characters, respectively. Forexample, f1u labels a threefold-degenerate asymmetric molecular electronic state, and Hg

denotes a fivefold-degenerate symmetric vibrational mode. For a more detailed discussion ofthe symmetry group and its application to C60 and other fullerenes we refer the reader to theextensive literature on the subject [18].

The molecule dissolves in various solvents. Crystallization from solution usually yieldsa structure that includes the solvent molecules [31]. However, sublimation in vacuum or inertgas atmosphere may yield pure C60 crystals of several mm or even cm size [32]. A particularlynice specimen of crystalline C60 is shown in figure 3. NMR [33], muon spin rotation [34]and x-ray diffraction [35] established that at room temperature the buckyballs are randomlyrotating between various orientations. In time average the molecules look like spheres and all


Figure 4. Two different ways of placing a fullerene molecule into a cubic environment. Each ofthe choices preserves the same number of symmetries.

of the molecular sites are equivalent. The well defined molecular centres and a nearly freemolecular rotation are not unique properties of the fullerene molecule, but are characteristicof many organic solids, where the binding is due to van der Waals forces, and the moleculehas ‘rounded’ symmetric shape [36]. More common materials, such as solid CH4 and CCl4,share this property with the C60 solid.

The centres of the C60 molecules make a fcc lattice, with a nearest-neighbour C60–C60

distance that corresponds to the diameter of the molecule. At room temperature the size of thecubic unit cell is 14.17 Å, and the nearest-neighbour distance is 10.02 Å [35].

As the temperature is lowered, the rotation of the molecules stops. The centres of themolecules remain in the same place, but the incompatibility of the icosahedral molecularsymmetry and the cubic lattice symmetry makes it impossible to describe the structure interms of the simple fcc lattice (with one C60 molecule of fixed orientation each unit cell).However, it may be still possible to view the structure in terms of a larger unit cell, and asimple cubic lattice. The freezing of the rotational motion actually happens in two stages.First, at 261 K the rotational axis of each molecule becomes constrained. The building blockof the new structure consists of four C60 molecules arranged at the vertices of a tetraeder,with each one of them spinning around a different, but well defined, axis [37]. Then, at lowertemperatures, the rotation around these axes slows and gradually stops.

Below about 90 K the molecules are entirely frozen, but they never order perfectly. In solidC60 there is always a ‘merohedral disorder’, due to the different molecular orientations [38].Figure 4 illustrates the two principal ways of orienting a fullerene molecule in a cubic lattice.

3.2. Chemical modifications, endohedral and exohedral doping

The linear size of a fullerene molecule is ten times larger than a typical atom. Even in aclose-packed structure, large empty spaces between the molecules are freely available forsmaller atoms, ions or molecules. Due to the unsaturated character of the C–C bonds on thefullerene, there are plenty of electronic states to accept electrons from appropriate donors. Thecombination of alkali and alkaline earth elements with fullerenes results in an unusual varietyof AnC60 materials, that are mostly near-stoichiometric (A can stand for Na, K, Rb, Cs Ca,Sr and Ba). Organic and inorganic molecules were also used to produce charge transfer saltswith fullerenes. The term ‘doped fullerene’ is often used to describe the product (althoughin its original sense, ‘doping’ means a small, non-stoichiometric amount of charge transfer,typically in a semiconductor).

In the close-packed fcc structure the interstitial sites have either octahedral or tetrahedralsymmetry. The tetrahedral sites are smaller, and there are twice as many of them as octahedral


Figure 5. Structure of alkali-doped C60 compounds [42], derived from the fcc structure of undopedC60. The upper and lower rows display identical structures in terms of a cubic and a tetragonal unitcell, respectively, for better view [43]. A3C60 is made with K, Rb and with a mixture of Cs andother alkali metals, so that the Cs occupies the octahedral sites. Only sodium produces the A6C60and A10C60 structures. (In A6C60 the Na4 sodium clusters in the octahedral sites are randomlyoriented [43].)

Figure 6. Alkali-intercalated C60 compounds viewed in terms of the (approximate) bcc arrangementof C60 molecules. The leftmost structure is identical to the A3C60 shown in figure 5. Doping withalkaline earth metals produces the structure labelled A′

3C60. The A4C60 and A6C60 structuresshown here are known to exist with A = K, Rb and Cs.

sites. A series of AnC60 (with n = 0, 2, 3, 4, 6, 10) materials are based on the original fccstructure of pure C60, as shown in figure 5. Binary AnC60 compounds of fcc structure ofn = 1, 2, 3 were made with K and Rb [35, 39, 40]. The tetrahedral site is too small to accepta Cs ion, and only Cs1C60 can be made [39]. On the other hand, Na is small enough for morethan one of it to fit into the octahedral site, and n = 6, and 10 (11) were made (the compoundthat was originally believed to be Na11C60 is probably Na10C60) [41, 43].

A number of other compounds are known in which the C60 abandons the close-packedstructure [44]. Two of these, the body-centred tetragonal A4C60 and the body-centred cubicA6C60 are illustrated in figure 6 [43, 45]. There is a series of compounds with alkaline earthmetals with a simple cubic A15 structure [46, 47], also shown in the figure. The prototypematerial is Ba3C60, where the centres of the C60 molecules are on a bcc sublattice with aparticular orientation of the bonds, defined by the 90◦ rotation of the twofold axis of theneighbouring molecules [46].

In contrast to the pure C60 solid, in the doped material the nearly free rotation of themolecules is often hampered by the dopant. In most structures, only the right orientation of thebulges and dents on the molecules can provide enough space for all elements. The orientationof the molecule does not follow from simple symmetry arguments as there is no ‘best’ way toput the icosahedron into a cube (see figure 4). The real structure can be explained in terms ofelaborate optimization in terms of the rotational position of the molecules. The primitive unit


Figure 7. Endohedral (left) and ‘on-site’ (right) doping of the fullerene molecule.

cell contains several C60 molecules [35, 39, 40].The cage-like structure of the C60 molecule naturally offers two other possibilities for

doping: inserting a foreign atom M inside the C60 cage and replacing one or several carbonatoms in the C60 cage with atoms having different electronic structure. In the first casethe M@C60 endohedral super-molecule is obtained (figure 7), and in the second one thecomposition is C59M .

Endohedral C60 molecules can be prepared by a ‘brute force’ method, where ions ofatoms are accelerated and implanted to the C60 cage [48]. The ions should have just enoughenergy to open up the cage and enter. The first collision should absorb and redistribute agood part of the initial kinetic energy so that the atoms does not escape the cage. Endohedralmolecules of M@C60 with M = N , P, Li, Ca, Na, K, Rb were produced this way in smallquantities. Larger yield can be achieved by co-evaporation [49] of the carbon and the metalin an arc discharge chamber (typical for fullerene production). In this process mostly higherendohedral fullerenes, like M@C82, can be extracted from the soot on a chromatographiccolumn. Wudl and co-workers [50] have demonstrated that there is a ‘soft’ chemical way toopen the C60 cage and than to reclose it; however, the insertion of an atom inside the moleculeis far from straightforward.

ESR studies and detailed quantum chemical calculations show that the endohedral dopantatom does not always transfer its charge to the cage. If the dopant remains neutral, it stays inthe centre of the cage, as in N@C60 [51]. Compounds like La@C60 are very interesting, sincecomplete charge transfer from La to the C60 cage results in a triply charged molecule, just asin Rb3C60. A crystal of La@C60 is predicted to be an air-stable superconductor [52].

On-ball doping has been achieved by replacing one carbon atom with a nitrogen atomobtaining C59N (azafullerenes) [53]. The replacement of N for C adds one extra electronto the cage, changes the structure locally, lowers the symmetry of the molecule and splitsthe degeneracy of the electronic orbitals. Furthermore, this chemical substitution renders themolecule very reactive with a high electron affinity. At ambient temperature the C59N existsonly in a dimerized phase [53]. The intercalated azafullerene K6C59N has been investigatedin detail [54]. Its structure was found to be similar to K6C60, but, in contrast to K6C60, it waspredicted to be a metal [54], although the prediction has not been tested experimentally.

3.3. Electronic structure of the C60 molecule and the band structure of the C60 solid

Before turning to the electronic properties of solids, it is instructive to take a look at theelectronic structure of an isolated C60 molecule. At first we will take the simplest possible


approach, based on the Huckel molecular orbital calculations of Haddon [55]. Correlationeffects will be discussed later.

C60 has 240 valence electrons, but each carbon atom has three sigma bonds to itsneighbours, using up a total of 180 electrons for this purpose. The energy of these electronsis well below the Fermi surface. They stabilize the structure, but do not contribute to theconduction. The remaining 60 electrons are distributed around the molecule on orbitals thatoriginate from the (much less tight) carbon–carbon π orbitals. These orbitals are somewhatsimilar to the π electron orbits of a graphene plane, with two important differences. First,the three bonds around a carbon atom in C60 (or in any other fullerene) do not make a plane.Whereas in graphene the electrons had equal probability of being ‘below’ and ‘above’ theplane, in fullerenes the π electrons tend to spend more time outside of the ball than inside.Second, in C60 the C–C bond lengths are not uniform; theπ electrons are not truly ‘delocalized’around the six-member carbon rings (like in benzene or graphene), but they are distributedover 30 ‘bulges’ of electronic orbits that stick out of the C60 molecule. A somewhat lowerelectron density belongs to the other 60 orbits connecting the carbon pairs with longer bondlengths. The overlap between these orbits on adjacent molecules will determine the propertiesof the conduction electron band of the doped solids.

A first insight into the nature of the molecular orbitals can be obtained by borrowing ideasfrom the early days of nuclear physics, when the quantum mechanics of particles confined in aspherical potential well was first considered. In this case the states are still labelled by quantumnumbers n, l and m, but the peculiar degeneracy of the different angular momentum states, thatcharacterizes the textbook treatment of the hydrogen atom, does not apply. In a steep-walledpotential well (like a nucleus or a C60 molecule) the lower l states will have lower energy.

Let us consider 60 non-interacting electrons, confined to a sphere on a high-n orbit withvarious l. The first two electrons will fill the l = 0 state. Proceeding to higher l, one has tocount the number of available states, N = 2(2l + 1) (where the first factor of 2 stands for spin).It is easy to see that, on reaching l = 4, altogether 50 electrons are consumed. The remainingten electrons will all go to the l = 5 state.

Huckel molecular orbital calculations find that the simple picture described above workswell for l = 0, 1, 2, 3, 4, but for higher energies the ‘spherical potential’ approximation is nolonger useful. In order to go any further we need to look at the energy splitting due to thetrue atomic potentials. In the real C60, l is not a good quantum number, and the electronicorbitals should be labelled according to the irreducible representations of the icosahedralsymmetry group. The orbitals available for the remaining ten electrons are, in order ofincreasing energy, the hu, the f1u and the f1g levels [55]. The degeneracy of these levels(including spin) is 10, 6 and 6 respectively. The hu level is completely filled by the tenremaining electrons, becoming the highest occupied molecular orbit (HOMO), and the f1u

level becomes the lowest unoccupied molecular orbit (LUMO). The HOMO–LUMO gap isabout 2 eV. In the spherical approximation the HOMO and the LUMO bands all belong to l = 5,and the corresponding Huckel molecular wavefunctions carry some of the l = 5 character. Forexample, a representation of the LUMO wavefunction, shown in figure 8, resembles stronglythe l = 5, m = 0 spherical harmonic [55]. The remaining six states in the l = 5 representation(labelled f2u in the icosahedral group) are pushed to such a high energy that the sixfold-degenerate f1g orbitals follow the LUMO band. This level, with an l = 6 character, becomesthe LUMO + 1 level.

In the solid state the bands originating from the HOMO and LUMO levels are the mostimportant ones, as they are in the immediate vicinity of the Fermi surface. The crystal fieldsreduce the molecular symmetry and the energy levels split. The overlap of orbitals broadensthe levels to bands. In the simplest possible cubic environment a few of the molecular levels


HUCKEL MOLECULAR ORBITALStg

t1g

t1u

t1u

t2u

t2u

gu

gu

hu

hu

hg

hg

gg

ag

gg + hg

-2

-1

0

EN

ER

GY

(

1

2

3

)

Figure 8. Huckel molecular orbitals and schematic illustration of some of the electronicwavefunctions (from [55]).

preserve their symmetry and survive without splitting (including the f1u orbitals that sharethe cubic symmetry with the t1u orbits of a cubic system). But the relative orientation of themolecules has a strong influence on the overlap of molecular orbitals [56]. In a realistic modelthe unit cell must include more than one molecule and the calculation becomes increasinglycomplex. Typical bandwidths of 0.5eV are obtained for the t1u bands (see figure 9) [57]. Atleast in a few important cases there is a residual randomness (e.g., the merohedral disorder,discussed above) to the structure. The translational invariance, suggested by the molecularcentres, is broken by the orientational disorder. Strictly speaking, the electronic wavenumberis not a good quantum number, and the band structure is not a good concept. Yet the metallicbehaviour cannot be excluded: just as in amorphous metals, the electrons can still propagatein the lattice, except that the propagation is diffusive rather than ballistic. The density of statescan be calculated and a bandwidth of about 0.5 eV is obtained for the LUMO band [56, 58].

3.4. Molecular vibrations of the C60 molecule and the phonons in solid C60

For a molecule of 60 atoms there are 3 × 60 − 6 = 174 vibrational modes (the term 6 standsfor the three translational and the three rotational modes to be subtracted from the total numberof modes). The symmetries of the molecule cause degeneracies in the vibrational frequencies.Similar to the electronic orbitals, the irreducible representations of the icosahedral group areused to label the vibrational (vibron) modes. Group theory classifies the 180 modes into twoAg modes, three F1g modes, five F2g modes, six Gg modes, eight Hg modes, and an equalnumber of antisymmetric (u) counterparts. (The number of modes belonging to a degenerategroup is equal to the dimensionality of the representation; the enumeration of the modes,2(2 × 1 + 3 × 3 + 5 × 3 + 6 × 4 + 8 × 5) = 180 accounts for all 180 degrees of freedom.) OneF1g mode corresponds to pure rotation, and one F1u mode is pure translation. This yields 46different vibrational frequencies in the ideal C60 molecule, with equal number of symmetricand asymmetric modes [59].


Figure 9. Electronic density of states from ab initio LDA–LMTO calculations, from [57]. Thelines labelled n = 1, 2, 3 indicate the positions of the Fermi energy at the corresponding dopinglevels.

The intermolecular vibrational frequencies range from ≈250 to ≈1600 cm−1 (30 to200 meV). A few of the relevant frequencies can be measured in the gas phase or in solutionusing the two most popular spectroscopic techniques, IR and Raman spectroscopy. Only thefour F1u modes have the transformation properties of vectors, a requirement for IR activity.Ten modes (the Ag and Hg modes) have tensor properties, making them Raman active. Therest of the modes were studied by inelastic electron [60] and neutron scattering [61–63] (in thesolid phase), photoluminescence [64], fluorescence [65] and phosphorescence [66] (in solidrare gas matrices). In these methods the selection rules do not restrict the detection to the IR orRaman modes only. Chemical labelling of the fullerene molecule also breaks the symmetry,and enables the optical methods to detect otherwise forbidden modes. For example, IR studieshave been performed on C60O, C61H2 [67] and on C60-tetraphenylphosphoniumiodide [68].

The semi-empirical model calculations of the fundamental frequencies are ratherstraightforward, but the results provide only a qualitative understanding of the spectrum [69–71]. With the rapid progress in computer speed and memory capacity the molecule became atesting ground for cutting edge ab initio works [72–75]. These methods yield a good agreementwith the experiments, although some details of the spectrum are still under discussion [75].

The intermolecular binding forces are much weaker than the chemical bonds within themolecule, and the C60 molecule in the solid can be considered a rigid body. The very same sixdegrees of freedom that were dropped from considerations of intramolecular vibrations leadto the elementary vibrational excitations of the solid: phonons and librons, derived from therigid translations and rotations of the molecule, respectively.

In the high-temperature fcc phase there is only one molecule in the unit cell. Thecorresponding phonon spectrum is of textbook simplicity with three acoustic modes only [15,62]. Since the molecules are rotating freely, all libron frequencies are zero. At low temperaturesthe unit cell includes several molecules, and therefore optical branches appear in the phononspectrum. At the same time the rotational motion acquires a restoring force, and the libronsmove to finite frequencies. In a good agreement between experiment and theory, the highestphonon frequencies are around 6–7 meV. The libron spectrum does not have much dispersion,and the energies are between 2 and 5 meV (figure 10) [62]. Notice that the lowest-energy


Figure 10. Measured and calculated vibrational density of states due to phonons and librons(from [62]).

intramolecular mode is about five times higher in energy than the intermolecular vibrations,lending support to the ‘rigid body’ approximation for intermolecular vibrations.

In the solid state the frequencies of the intramolecular modes do not change much. Theyremain Einstein modes with a small dispersion. Figure 11 shows the IR transmission spectrumof a thin film of microcrystalline C60, with the four characteristic absorption lines very close tothe IR frequencies of the isolated molecule [76]. The effect of crystal fields is evident in high-resolution IR and Raman measurements: the resonance lines broaden and/or split, dependingon the temperature [77, 78]. In high-sensitivity IR [79, 80] and Raman [81] measurementsmany new weak lines have been seen (in addition to the four IR and ten Raman resonancesallowed by the icosahedral symmetry group). Most of the ‘new’ lines are due to the anharmoniccharacter of the chemical bonds within the molecule, leading to weak lines at the frequenciesapproximately corresponding to the sums and differences of selected fundamental resonancemodes [82]. These lines show up in the solid only because there are so many more molecules forthe light to interact with. A few new lines appear at frequencies corresponding to fundamentalmodes. These lines are directly related to solid-state effects: in the crystal the selection rulesderived from the icosahedral symmetry are not valid any more, and most of the ‘forbidden’modes acquire weak IR and Raman activity. In a study of the temperature dependence of theIR transmission spectrum [83] a clear correlation was observed between the intensity of someIR lines and the freezing of the rotational motion of the fullerene molecules. To explain theseobservations, the concept of ‘motional diminishing’ of the IR lines was introduced [84], inanalogy to the ‘motional narrowing’ of the NMR lines.

4. Concepts and interactions

In the C60 solids the LUMO bandwidth is about 0.5 eV, significantly less than the HOMO–LUMO gap. The pure material is an insulator, with a completely filled HOMO-derived band.In the AnC60 alkali metal compounds, or in other doped materials, electrons are donated to theLUMO band. Assuming that the structure remains simple (one C60 molecule per unit cell), aband model predicts metallic behaviour for any 0 < n < 6, corresponding to partially filledbands. Many experiments contradict this prediction.

The interaction between electrons and phonons and the electron–electron interactions canmodify, or drastically change, the simple band theory results. The catch-all phrase ‘electroncorrelation effects’ is often used to describe the resulting new phenomena. Instead of tryingto deal with the full complexity of a solid, the theoretical models describing these processes


Figure 11. Infrared transmission spectrum of a thin C60 film. The four IR-active vibrational modesare clearly visible. The ripple in the baseline is due to interference effects between the front andback surfaces of the sample and the substrate [76].

are often simplified, and focus on a few important parameters. In fullerides, the separationof energy and length scales between the molecule and the solid results in a benefit in thisrespect: important parameters of the interactions can be derived from the investigation of asingle molecule, or a small cluster of molecules. Concepts and interactions can be illustratedon a molecular level.

4.1. Electron–phonon coupling

The dimensionless coupling constant of the electrons to the phonon mode ν is defined as

λν = 2

N(0)

∑q

1

ωνq

∑n,m,k

|gnk,mk+q(ν)|2δ(εnk)δ(εmk+q − εnk − ωνq) (1)

where N(0) is the electronic density of states at the Fermi energy, εnk is the energy of theelectron of wavevector k in band n, and ωνq is the energy of the phonon of wavevector q.The matrix element between the two electronic states nk and mk + q created by the phonon isgnk,mk+q(ν). A typical calculation of the coupling constant requires performing a double sumover the wavevectors, with a calculation of the matrix element at each stage. However, whenthe intermolecular vibrations of the fullerene molecule are considered, one may assume thatthe phonon modes are dispersionless [85]. In this much simplified formulation the couplingconstant is expressed in terms of the vibration frequency and the energy shift �ενα of theelectronic state α, induced by the atomic displacements belonging to the vibrational mode:

λ = N(0)∑να

�ε2να

ωνα

. (2)

The advantage of this formulation is that the effects of intermolecular electron hopping canbe incorporated into the density of states N(0), whereas the quantity λν/N(0) is determinedby the properties of a single molecule. Since electrons on the molecule have the t1u symmetry,only the Hg and Ag Raman modes can have non-zero matrix elements. This approach workswell for the Hg modes, but fails for the Ag modes. An Ag-type distortion shifts all three t1u

levels by the same energy. In a truly isolated molecule, equation (2) leads to quite significant


coupling to the symmetric Ag vibrations. In a metallic solid, where the Fermi energy is in thet1u band, the distortion will not cause a change in the electronic energy; instead a charge transferto/from other molecules will result [86]. This behaviour has been illustrated experimentallyby the Kuzmany group [87].

When phonons are coupled to electrons on the Fermi surface the coupling shortens thelifetime of the vibrations, resulting in a broadening of the phonon line. The width of thephonon is [88, 125]

γν = 2πω2νN(0)λνnν

, (3)

wherenν is the degeneracy of the phonon mode. This relationship is the key to the spectroscopicdetermination of the electron–phonon coupling parameter. Electrons residing on the t1u orbitalscouple to the Hg phonons that happen to be Raman active as well. In principle, the Ramanlinewidths could be used to derive the experimental values of λν .

Unfortunately, the ‘frozen-in’ orientational disorder typical of many fulleride compoundscomplicates the evaluation of the Raman measurements. With no disorder, the Ramanscattering would probe only the q ≈ 0 phonons, and the linewidth would be due to the lifetimebroadening described by equation (2). However, the disorder causes the phonon dispersion tocontribute to the linewidth. Although the dispersion is small relative to the phonon frequency,it may be quite significant compared with the linewidth caused by electron phonon coupling.

There is a large body of literature on the calculation of electron–phonon coupling energies,summarized in an excellent review [12] by Gunnarsson. The first step is to determine thephonon eigenvectors and frequencies. The quality of the calculation can be tested to someextent by comparing the measured and calculated vibrational frequencies. The electronicenergy shifts are quite sensitive to the phonon eigenvectors, and slight differences in the phononeigenvectors lead to quite a range of calculated values for λν/N(0) [12], yet there is no directway of measuring the atomic displacements. The electron–phonon coupling constants weredetermined by photoemission on C−

60 vapour, [89, 90], by studying the linewidth of variousmodes in Raman [91] and neutron spectroscopies. Assuming N(0) of 8 eV−1 for each spinstate, the typical values are around λ ≈ 0.5.

4.2. On-site Coulomb repulsion

How much is the energy cost of placing two electrons on the same fullerene molecule? Theanswer to this question may determine whether the material is a metal or an insulator. Consider,for example, the simple one-band Hubbard model

H =∑

tij a+iσ ajσ +

∑Un↑n↓, (4)

where a+σ i and aσi creates and annihilates electrons of spin sigma at site i, n = a+a, t is the

hopping integral and U is the on-site electron–electron interaction. With one electron peratom and U = 0 the model yields a half-filled tight binding band of bandwidth W ≈ 6t .Metallic behaviour follows: the electrons are uncorrelated and there is a 25% probabilityof finding two electrons on the same site. For large on-site repulsion, however, the doubleoccupancy of the sites is effectively prohibited. There is a gap of approximately � = U −W

between the lower energy band, that is completely full, and the upper (empty) band. There isno general solution for the problem of ‘intermediate’ U , but in a simple one-band model theratio U/W ≈ 1 separates the insulating and metallic behaviours (Mott–Hubbard transition).Gunnarsson et al [92] argued that the degeneracy of orbitals shifts this condition toU/W ≈ 2.5.

In fullerenes the competition between the Hubbard U and the bandwidth determines theinsulating or metallic character. The first estimate of U can be obtained by spreading two


Figure 12. Self-convolution of photoemission (labelled PES ⊗ PES), compared with the Augerspectrum [96].

electrons uniformly over a sphere of radius R = 3.5 Å, and yields U = 4.1 eV. This is clearlyan upper limit. For example, placing the electrons on the orthogonal t1u orbits will certainlyensure that they ‘meet’ less than in the case of uniform distribution. More appropriate, anddemanding, calculations include the rearrangement of charges on the molecule as the electronis added. The local density approximation (LDA) is a standard method for performing thesecalculations and the result is around 3.0 eV [93, 94].

An experimental estimate of the molecular electron–electron repulsion energy is obtainedas the difference between the energy cost of removing an electron from the C−

60 ion (ionizationenergy) and the energy required to place the electron to another C−

60 ion (electron affinity),U = I (C−

60) − A(C−60) = 2.7 eV [95].

In the solid state this energy is considerably decreased by the screening effects. The largeC60 molecules, closely packed around the ‘test’ molecule, are quite effective in shielding theelectric fields. Calculations taking into account this shielding yield U = 0.8–1.3 eV [93, 94].

Measuring U in the solids is far less straightforward than measuring it for molecules:one needs to compare measurements involving one electron with measurements wheretwo electrons are excited simultaneously. The combination of Auger spectroscopy andphotoemission spectroscopy was suggested for this purpose [96, 97]. In photoemission theincident light excites an electron that leaves the sample. The probability of this process isdetermined by the density of electronic states. In the Auger measurement a core electron isremoved first from the carbon K level by a high-energy photon. In the study relevant here, twoelectrons leave the valence band simultaneously: one of them exits the sample, and the otherone fills up the core level. As long as there is no interaction between the two electrons, theconvolution of the valence electron density of states with itself yields the probability of theAuger signal. Lof et al [96] compared the self-convolution of the photoemission spectrum (asa measure of density of states) to the Auger spectrum. The spectra are similar (see figure 12),but there is an energy shift of 1.6 eV. The shift is due to the interaction between the twoelectrons involved in the Auger process. As long as the interaction energy is the same for allof the electronic states involved, a U = 1.6 eV explains the experimental observation. For thet1u band U = 1.4 eV was estimated [96, 97].

The energy increase due to Coulomb repulsion depends on the orientation of the electronspins. The ‘exchange hole effect’ (i.e. the electrons’ attempt to satisfy the Pauli principle and


avoid each other) reduces the likelihood of the two electrons being at the same place [98].Consequently, electrons of parallel spins have to pay less price in terms of Coulomb energythan electrons with opposite spins. This leads to Hund’s well known (first) rule in atomicphysics, governing the occupancy of degenerate atomic orbitals. Similarly, a ferromagneticcoupling is expected between the electrons on the t1u orbitals of the C60.

The exchange effects are described by the U2 term in the effective Hamiltonian

Hint = U

2n2 +

U2

6

[4(n2

1 + n22 + n2

3) − n1n2 − n2n3 − n3n1 + 3(�12 + �23 + �13)], (5)

where the i = 1, 2, 3 labels the three orbitals, ni = ∑σ c

+iσ ciσ is the number of electrons

on the ith orbital, and �ij = ∑σ c

+iσ cjσ + H.c. switches electrons between the orbitals. For

two electrons (C2−60 ) the lowest energy corresponds to a triplet state, followed by two singlet

states at 2U2 and 5U2 above the ground state. The corresponding electron configurations areanalogous to the 3P, 1D and 1S multiplets of a p2 shell (like that of the carbon atom), exceptthat the states are labelled in terms of the icosahedral group as 3T1g, 1Hg and 1A1, respectively.A similar multiplet structure arises for three electrons, in analogy to the p3 shell of nitrogen.C4−

60 is the electron–hole counterpart of C2−60 .

A singlet–triplet splitting of J = 2U2 = 0.2 ± 0.1 eV was determined from thecombination of various measurements by Lof et al [96]. Calculations [12, 13, 99] yield U2

in the 0.05 eV range. Yet, in contrast to the expected ‘high spin’ state, direct electron spinresonance (ESR) measurements on electrochemically produced ions [100] indicate that theground state of C2−

60 is singlet. There are no known doped C60 compounds in which the on-sitespin configuration follows Hund’s rule.

In summary, there is a school of thought arguing that all materials Cn−60 with integer n must

be Mott–Hubbard insulators. In this case the metallic character of the AC60 and A3C60 maybe due to non-stoichiometric composition. Even if this proves not to be the case, it is safe tosay that Coulomb correlations (U ) are large in all doped C60 compounds.

The value of the exchange coupling seems to be rather small. Other effects suppress the‘high spin’ ground state. One of these, the JT effect, is discussed next.

4.3. Jahn–Teller instability

The JT effect [101] occurs when a distortion of the lattice causes splitting of degenerateelectronic states that are only partially occupied. The perturbation caused by the latticedistortion is considered small: some electronic energies move up, while others move down(and yet others may be insensitive to the particular distortion). The electrons will occupy thelow-lying states, causing the total electronic energy to decrease. The deformation of the latticeis quadratic in terms of the deformation amplitude δx. As long as fluctuations and temperatureeffects are neglected, the electronic energy shift is first order in δx and the JT distortion isfavourable.

To illustrate this effect on a system that is similar to C60, yet substantially simpler, letus consider electrons on a fictional octahedral ‘molecule’, with threefold-degenerate, p-likeorbitals. The three orbits belonging to these states are along the x, y and z direction of thesystem of reference. Their properties are similar to the t1u orbits on the fullerene molecule.

Three types of distortions are illustrated in figure 13 causing first-order changes in theelectronic energies. A symmetric squeezing or enlargement of the octahedron will cause theenergy of all three orbits to move up or down, and does not lead to splitting of the levels.(Whether the squeezing results in increase or decrease of the energy depends on the detailsof the model, and is not important here.) A uniaxial deformation leaves two of the orbitalsdegenerate, whereas a biaxial deformation eliminates all degeneracies.


Figure 13. Distortions of an octahedron, and the corresponding change in the energies of p-likeorbitals. The direction and magnitude of the energy change depends on the actual electron–phononcoupling, but the removal of the degeneracy of the energies does not.

In considering the JT effect the first question is: how many electrons do we have on thethreefold-degenerate orbit? For zero or six electrons the size of the octahedron may change,but the shape will not be influenced. For one or two electrons a uniaxial distortion may befavourable, so that the energy of one (partially or totally filled) orbit goes down, and the energyof two (empty) orbits go up. With no on-site Coulomb repulsion and everything else beingequal, the distortion will be stronger with two electrons. A uniaxial distortion may also bepreferred for four or five electrons, but it will be opposite in direction, so the single level movesup. Finally, for three electrons, the best deal may be achieved with biaxial distortion, so thatthe energy of two electrons go down and the third one sits on the level that does not move.

The JT effect on the LUMO band of the C60 molecule resembles the case discussed here.The two highly symmetric Ag modes couple to the electronic energies, but they do not causeJT effect, just like the symmetric squeezing of the octahedron in the example above. In termsof t1u wavefunctions of x, y and z symmetry, the coupling to any of the fivefold-degenerateHg modes can be described by [102]

Hνint = gνhων

2

∑σ

(c+x,σ , c

+y,σ , c

+z,σ

) q1 − √3q4 −√

3q3 −√3q2

−√3q3 q1 +

√3q4 −√

3q5

−√3q2 −√

3q5 −2q1

(cx,σcy,σcz,σ

), (6)

where gνhων is the coupling strength between the electrons and the distortions (written interms of the dimensionless coupling gν and the vibrational frequency ων), the c are electronoperators and the qm (labelled m = 1, . . . , 5) are the amplitudes belonging to the five normal-mode distortions. Lannoo and co-workers [85] treated this problem in the quasiclassical(strong coupling) approximation in analogy with a vacancy in silicon. This Hamiltonian hasan ‘accidental’ higher symmetry, being invariant under the rotations of the 3D rotational groupSO(3) [103]. As a result of this symmetry it is possible to introduce a combination of normalmodes in such a way that the electron energies depend only on two vibron coordinates, r and z:

Hνint = hωνgν

2

[(z2 + r2

)+ z(n1 + n2 − 2n3) + r

√3(n1 − n2)

], (7)

where n1, n2 and n3 are the electron occupation numbers of the orbitals of x, y and z symmetry,


Figure 14. Distortions of a fullerene molecule insequence from (a) to (d). The arrow represents thedistortion axis and the eigenvector of the electronic stateas well. The distortion is identical at (a) and (d), yet thewavefunction acquired a phase factor ofπ . (From [112].)

respectively [104]. In the adiabatic limit the problem is further simplified by the single-modeapproximation of O’Brien [106], introducing ωeff = (∑

ωνg2ν

)/(∑

g2ν

)and g2

eff = ∑g2ν .

The JT energy gain is Eν = (hων/2) g2ν , 4Eν , 3Eν , 4Eν and Eν for electron number of

n = 1, 2, 3, 4 and 5, respectively. Auerbach et al [104, 105] calculated the contributions dueto the change in the zero-point vibrations for various electron numbers. They illustrated theunimodal and bimodal nature of the distortions for n = 1, 2, 4, 5 and 3, respectively. TheJT energies calculated from first principles [107], or estimated from the measured/calculatedelectron–phonon coupling constants [89, 108], are in the 100–150 meV range.

For the JT-active modes the connection between the JT energies and the electron–phononcoupling constant λ is quite straightforward [85]:

λ = N(0) 56

∑ν

Eν. (8)

There is a clear-cut incompatibility between JT effect and the molecular Coulombcorrelations, characterized by U and J and discussed in the previous section. For an evennumber of electrons the JT effect prefers singlet states, with two electrons per orbit, workingagainst Hund’s rule (corresponding to the calculated positive J = 2U2 in equation (5)). Sinceno ‘high spin’ configuration was observed [100], we must conclude that the JT effect wins thecompetition. On the other hand, for odd n the addition of another electron enhances the existingJT distortion, thereby creating an effective attraction between the electrons [12, 85, 104, 109].The magnitude of this interaction energy is hωνg

2ν , and it acts as a negative contribution to the

Hubbard U . There is a general agreement that this energy is not sufficiently large to overcomethe Coulomb repulsion and to create an effective on-site attraction.

Long et al [68] found indirect evidence for JT distortion in a far-infrared study of theC60 mono-anion in C60-tetraphenylphosphoniumiodide. The JT distortion has been observeddirectly in the salt of bis(triphenylphosine)iminium ion (PPN+) and C60 [110]. In [PPN+]2C60

the fullerene molecule exhibits an axial elongation with a rhombic squash. In contrast to theideal icosahedral structure, where the distances of the carbon atoms from the centre of themolecule are exactly equal, in the distorted structure there is a spread of this distance betweenof ≈0.04 Å around the mean value of 3.542 Å.

4.4. Berry’s phase and orbital ordering

What happens when the octahedral molecules make a solid, so that there is a (small) overlapbetween the electronic orbitals? For n = 0 or 6 the JT effect has no role, and the materialremains an insulator. For any other electron number the JT distortion may happen. The


competition between the JT energy and the delocalization energy of the electrons can create arich physics, which has not been fully tested on fullerenes.

One of the intriguing phenomena related to the JT instability is the appearance of a newdegree of freedom, the ‘Berry phase’, in the electronic wavefunction. In general, this phasefactor is picked up as the system is moved around a sequence of states by changing the externalparameters adiabatically, so that at the end of the process the original parameter settings areimposed again. The observable quantities do not change, but the phase of the wavefunctionmay. In our JT system the deformation of the C60 molecule represents the ‘external parameter’.Auerbach et al [104, 111] discuss the Berry phase in C60 for the unimodal and bimodaldistortions of the molecule. (The uni- and bimodal distortions of a sphere are analogousto the uni- and biaxial distortions of the octahedron discussed in the previous paragraph.)Figure 14 illustrates, for a unimodal distortion, how the electron wavefunction can pick up aphase of π , when the distortion of the molecule returns to its original form [112].

The Berry phase factor influences the zero-point vibrations of the molecule [104, 111].When more than one electrons are on the molecule, this energy shift renormalizes the electron–electron interactions, and may contribute to SC. In a solid, the interference between the wave-functions on neighbouring molecules raises the possibility of directly observing the effects ofthe Berry phase. The orbital ordering observed in some magnetic perovskites has been recentlydiscussed in these terms [113], but no corresponding experiments have been done on fullerenes.

4.5. Superconductivity

In the BCS/Eliashberg theory of phonon-mediated SC the coupling between electrons on theFermi surface and phonons of frequency ω is characterized by the function α2F(ω). Thecritical temperature is determined by the dimensionless coupling constant [114, 115]

λ = 2∫(dω/ω)α2F(ω). (9)

The BCS theory is obtained in the weak coupling limit, λ 1. The coupling is expressedin the form of λ = N(EF)V , where N(EF) is the density of states at the Fermi level, and V isthe BCS interaction energy. In the weak coupling limit the quantity ω0 exp(−1/λ) is the onlyrelevant combination of parameters, where ω0 is a properly defined average phonon frequency.The critical temperature is Tc ∝ ω0 exp(−1/λ), where the constant of proportionality dependson the actual definition of the phonon frequency. If, for example, acoustic phonons areresponsible for the SC, the Debye frequencyωD is a natural choice, and the expression becomes

kBTc ≈ hωD

1.45exp −1/λ = hωD

1.45exp −1/NV . (10)

In the weak coupling limit the ratio of the low-temperature value of the superconductingenergy gap to the critical temperature is always 2�/kBTc = 3.53.

As we will see later, in fullerenes the phonon frequency ω0 and the BCS couplingparameter V are related to intramolecular processes, and therefore they are independent ofthe interfullerene distance. On the other hand, the density of electronic states is inverselyproportional to the bandwidth, N ≈ 1/W , and the bandwidth increases when the latticespacing is decreased. Therefore, equation (10) suggests a decrease in the critical temperatureif the lattice is compressed.

In many of the real materials the coupling is intermediate or strong [116]. Solving theself-consistent equations requires numerical integration. There is no analytical solution, andthe details depend on the shape of the α2F(ω) function. To incorporate the electron–electron


Figure 15. Scaling of superconducting parameters of various materials, based on [115]. Tc isthe transition temperature, �0 is the gap at zero temperature and ωln is the characteristic phononfrequency. The value of 2�0/kBTc for Rb3C60 was taken from [172]; the Tc/ωln was calculatedassuming a single-phonon frequency corresponding to the lowest intramolecular Hg mode.

repulsion parameter µ∗, McMillan [115, 117] derived the semi-empirical relationship

kBTc = hωln

1.2exp − 1.04(1 + λ)

λ − µ∗(1 + 0.62λ), (11)

where we have replaced the phonon frequency with the one defined by Allen and Dynes [114]:

ln(ωln) = 2∫

dω ln(ω)

ωα2F(ω). (12)

A somewhat simpler expression relates �0 to the same parameters. For strong couplingthe ratio 2�/kBTc is always larger than the BCS value. The semi-empirical equation

2�/kBTc = 3.53[1 + 12.5(Tc/ωln)2 ln(ωln/2Tc)] (13)

suggested by Carbotte [115] is valid for a wide range of materials, as can be seen from figure 15.In this simplified form there are only three parameters in this theory: the phonon frequency

ω0 (or ωD or ωln), the coupling between electrons and phonons λ, and the electron repulsionparameter µ∗. Typical values of µ∗ are around 0.1. If the coupling is strong, then the measuredvalue of the Tc and �0 provides us with just enough information to determine λ and ω0.

The Eliashberg theory is derived in the limit when Migdal’s theorem [118] is valid, andthe characteristic phonon frequency is much less than the bandwidth (or Fermi energy) of theconduction energies, ω0/W 0. In regular metals, with a Debye energy of a few tens of meVand bandwidth of several eV, this condition is satisfied. Yet in the fullerene superconductorsthis condition may be violated: the bandwidth is about 500 meV, and the intramolecular phonon


Figure 16. Resistivity of doped C60 at subsequently higher doping levels. The measurementswere made on a single crystal of 3 mm length and approximately 80 µm × 40 µm cross section(from [119]).

energies range from 30 to 200 meV. The possible consequences of the breakdown of Migdal’stheorem are discussed by several authors, and a critical summary of this issue is provided byGunnarsson [12].

An isotope substitution will cause a slight change in the phonon frequencies, resulting in ashift in the superconducting transition temperature. The ‘isotope effect’ is characterized by theexponent α = −d ln Tc/d lnM , where M is a properly defined average mass that enters intothe calculation of phonon energies. A precise measurement of α can place some constraints onthe role of electron–phonon coupling. For example, in a simple picture based on equation (10)the isotope mass influences the Debye frequency only, and α = 0.5 is obtained. However,experiments on a wide variety of phonon-mediated superconductors showed that α = 0.5 ismore of an exception and that α < 0.5 is typical. The reason for this behaviour lies in themodification of the Coulomb interaction caused by the lattice retardation, yielding an extramass dependence in the electron–electron repulsion term: µ∗ = µ/[1+ln(EF/ωD)]. The largeris µ∗, the smaller is the isotope effect. In fact, the electron–phonon coupling may provide thedominant attractive interaction even if α = 0 within the experimental error.


5. Conducting C60 compounds

The evolution of the electronic properties of alkali-doped fullerenes is illustrated in figure 16,where the temperature dependence of the resistivity of a potassium-doped C60 crystal isplotted [119]. KnC60 with n = 1 and 3 is metallic in a wide temperature range. For n = 4and 6 the material is a semiconductor. Simple band theory arguments predict that any partialfilling between 0 and 6 electrons (empty and full t1u band, respectively) should give a metallicbehaviour. The fact that the A4C60 (and Na2C60) is non-metallic shows that correlation effectsplay an important role in the basic properties of fullerides.

Since the discovery of SC in the fullerides [6], doping and transport studies have focusedmainly on theA3C60 (A = K, Rb) compounds. Much effort has been spent on the determinationof the intrinsic resistivity and its temperature dependence both in thin-film [120,121] and single-crystal measurements [122,123]. The dc resistivity measurements, combined with reasonableestimates of carrier density n, band mass m and Fermi velocity vF, yield a good estimate of themean free path l of the carriers. The lowest resistivity values for the K3C60 system are in therange of 1 m5 cm, which gives a mean free path of 3.5 Å, much less than the nearest-neighbourC60 distance. The discrepancy is even more severe at high temperatures (T > 500 K), wherethe resistivity does not show an expected tendency for saturation [124, 125], and the deducedmean free path is in the range of the C–C distance [123]. An apparent metallic behaviour witha mean free path shorter than the lattice spacing is one of the unsolved problems in condensedmatter physics. Other systems, including cuprates, manganates and organic conductors, alsoexhibit this anomalous, ‘bad metal’ behaviour.

The estimates discussed above are made in terms of a simple Drude expression, σDC =ne2τ/m, and using l = vFτ . The quantity n/m can be derived independently from the opticalconductivity. According to a very general sum rule, the integral of the real part of the opticalconductivity (σ1(ω)) over all frequencies can be expressed in terms of the electron density n0

and mass m0, independent of the particular interactions the electrons are participating in. Thesum rule is often generalized to∫ 5

0dω σ1(ω) = πne2

2m, (14)

where 5 is restricted to low frequencies, and instead of the ‘bare’ quantities m0 and n0, theband mass m and the effective number of carriers n are used. In a simple Drude metal thecorrect carrier density n is obtained if the integration is done to frequencies well above therelaxation rate 1/τ , but below the interband transitions.

Degiorgi and co-workers performed detailed measurements of the far-IR opticalconductivity of Rb3C60 and K3C60. The behaviour was quite complex even in the low-frequency regime (below ω = 2000 cm−1): a reasonable fit to the data included a Drudepeak and a mid-IR peak centred around 400–500 cm−1. The total spectral weight of thesecontributions matches reasonably well with the total carrier density of n = 4.1 × 10−21 cm−3

and effective mass of about m = 4m0. Yet the ‘Drude’ part contains only about one-tenthof the spectral weight, with surprisingly low relaxation rates (29 and 65 cm−1 for K3C60 andRb3C60, respectively).

Similar ‘two-component’ responses have been observed in high-temperature supercon-ductors [128], in BaxBi1−xPbO3 [129] and in organic conductors [130]. The presence of thisbehaviour may be a signature of highly correlated electrons. In the absence of satisfactorymicroscopic understanding, we can only notice that a tenfold reduction of the spectral weight(n/m) corresponds to a tenfold increase of the relaxation time τ . As long as the characteristicvelocity (the Fermi velocity) remains the same, this will lead to a sufficiently long mean freepath in the Bloch–Boltzmann theory of transport.


Figure 17. Magnetic susceptibility measurements of metallic Ba6C60 and Sr6C60 from [135].The inset shows the temperature dependence of the susceptibility in the low-field (5 G) region forBa6C60 and Sr6C60 and the superconducting Ba4C60 and Sr4C60.

The temperature dependence of the resistivity of K3C60 as well as that of Rb3C60 is close toquadratic at constant pressure and it was taken as a sign of strong electron–electron interactions.A similar behaviour was observed in organic charge transfer salts. However, Cooper [131]pointed out that the raw data had to be corrected for the change in lattice spacing due tothermal expansion. It turned out that even in organic conductors, where the on-site Coulombinteraction was known to be large, the corrected resistivity is rather linear in temperature. Asimilar analysis applied to K3C60 by Vareka et al [132] arrived at the same conclusion: atconstant volume the resistivity is linear down to 100 K.

The strong residual resistivity seen in figure 16 in the case of K3C60 is also a characteristicof fulleride conductors. In simple metals, where lattice defects can cause similar behaviour, athermal annealing reduces the residual resistivity. In fullerides, however, the scattering of theelectrons is due to the inherent merohedral disorder of the C60 molecules in the cubic lattice,and cannot be annealed.

In Ba4C60 and Sr4C60 the t1g-derived LUMO + 1 starts to get filled. These materials aresuperconductors at relatively low temperatures [133,134]. In alkaline-earth-doped compoundsthe hybridization between the C60 and alkaline earth orbitals is quite strong, resulting in anincomplete charge transfer and/or overlapping bands. Ba6C60 and Sr6C60 exhibit a metallicPauli susceptibility of [135], although simple electron counting would lead to a completelyfull band (figure 17).

5.1. Structural instabilities

Fullerene-based solids exhibit a rich variety of structural instabilities. Examples includethe fcc–simple cubic transition provoked by the orientational ordering of the C60 molecule,the ordering of the side-groups attached to the C60 molecule, or the ordering of the dopantmolecules as in TDAE-C60, but there are several more. Here we treat mainly the spontaneouspolymerization of charged molecules.

The surprising ability of the C60 molecule to form covalently bonded polymers canbe realized through many routes: by photo-excitation, molecular collisions, high-pressure


and high-temperature treatments, and by ionization (charge transfer). The first indicationabout polymerization of C60 molecules was reported by Eklund and co-workers [137] forlaser-irradiated thin films of C60. A toluene-insoluble material was produced and in laserdesorption mass spectrometry measurements oligomers as large as (C60)20 were observed.The polymerization happens in a Diels–Alder-type (2 + 2) cycloaddition reaction. Thisreaction is thermally forbidden between neutral molecules. However, it is allowed betweenmolecules in the ground and excited states, and the laser irradiation provided the energynecessary for bringing some of the molecules to the excited state. It is believed that (2 + 2)cycloaddition occurs in high-pressure/high-temperature polymerization as well [138]. In thiscase the high pressure increases the width of the electronic bands and reduces the gap, andthe high temperatures help the carriers to reach the excited level. High-pressure-synthesizeddisordered polymers are very attractive for applications, since they can exhibit mechanicalproperties stronger than diamond [139].

In all of these early studies the polymers were in fact relatively short oligomers, leadingto poor quality to x-ray data. For this reason, the structural refinement of these polymerswas not very successful. The bonded structures of neutral C60 (photopolymer, high-pressure-synthesized polymers) are insulators, and they barely support doping. Apparently, the chargetransfer from the dopant to the fullerene molecule causes dissociation into monomers.

On the other hand, under the right circumstances, just mixing C60 with alkali metals maycause spontaneous polymerization between the charged C60 molecules. To date, about halfa dozen bonded configurations of the C60 are known, produced by polymerization betweencharged mono- or poly-anions of the molecules [11].

The AC60 family (A = K, Rb, Cs) is the most prominent example of the polymericcompounds, exhibiting quasi-1D structure and a very rich phase diagram. (The first strongstructural evidence for C60 polymerization was obtained on KC60 and RbC60 compounds,facilitating the acceptance of other polymerized C60 phases [39].) Another linear polymerstructure has been recently described [140] in Na2AC60. As discussed later, the polymerizationin this material competes with the SC. A 2D charged C60 polymer, based on Na4C60, will bealso discussed.

5.1.1. Polymerization and dimerization of AC60 (A =K, Rb, Cs). The first of the AC60

compounds was discovered [141] by Winter and Kuzmany in Raman measurements on KC60

at high temperatures. At the time it was believed that the composition is not stable at lowertemperatures [142] and it separates into K3C60 and C60. Chauvet et al [143] have shown thatthe RbC60 phase is in fact stable upon cooling, but there is a structural change in the 350–400 K temperature range. Later, a very pronounced first-order phase transition was found inthe optical, electrical transport, ESR and in direct thermodynamic measurements. The upperleft panel of figure 16 shows the resistivity as a function of temperature. The temperaturehysteresis of the transition is quite apparent.

The surprising feature of the low-temperature structure is that the nearest neighbour C60–C60 distance is only 9.1 Å, less than the ‘diameter’ of the buckyball. In view of the shortnearest-neighbour distance Pekker et al [144] suggested that in this direction the C60 ions arecovalently bound, and the low-temperature phase consists of a conducting linear polymer.

Structural studies on KC60 and RbC60 (and later on CsC60) have refined the internalstructure of the polymeric phase, fully supporting this suggestion [39,145]. A schematic viewof the C60 backbone is shown in figure 18. The polymerization occurs along one of the facediagonals of the high-temperature fcc lattice [39, 144].

The orientation of the planes of the four-member carbon rings on the fullerene chainsis yet another degree of freedom in the structure. The various possibilities are represented


(a)

µ

b

c

(b)

(c) (d)

Figure 18. Three representations of the polymeric (AC60)n compounds. An orthorhombic unit cellis derived from the high-temperature fcc structure by a slight compression along the a direction,corresponding to one of the face diagonals of the cube. (The lattice expands in the other twodirections, but the unit cell volume decreases [39].) The centre panel shows the double-bondedC60 chain [144]. Four possible orientations of these chains within the unit cell are illustrated onthe left panel [145]. The orientation depicted in (d) corresponds to a monoclinic structure, but theangle between the b and c sides is 90◦ within the experimental error.

in figure 18. This orientation is not set by any symmetry consideration, and had to bedetermined independently. Launois et al [145] have shown that the chain orientations varyfrom compound to compound with different dopants. The orientation of the chains in KC60 andRbC60 corresponds to parts (c) and (d) of figure 18, respectively. Notice that the structure in (c)is orthorhombic, but in (d) it is monoclinic (the space group is I2/m), and the angle between theb and c sides does not have to be 90◦. However, the deviation correlates with the non-cylindricaldistribution of atoms along the C60 chains, and it has not been detected directly in experiments.

The covalent bonding in the polymer formation is convincingly demonstrated in NMRmeasurements as well. Evidence for the sp3 electronic configuration was seen in the chemicalshift of the 13C resonance frequency. The deformation of the C60 molecule in the polymericphase was also seen in the NMR. At high temperatures (T > 380 K) the C60 molecule hasan approximate spherical symmetry, and every carbon atom is equivalent on the ball, whichmanifests itself in a single NMR line (lower plot, figure 19). Due to the covalent bonding,the ‘soccerball’ molecule distorts into a ‘rugby ball’ in the polymeric phase, resulting innine inequivalent carbon sites [146–148]. The nine lines are beautifully resolved in magicangle spinning (MAS) NMR. This technique shows clearly that the interchain interactionsare different in the KC60 on the one hand and RbC60 and CsC60 on the other hand [146], inaccordance with the different orientation of the chains discussed above.

An interesting feature of the (AC60)n polymeric fullerides is that, unlike any other alkali-metal-doped fullerene, these compounds are air stable [235]. This feature helps to isolatethe pure material from the variety of other phases created during the doping procedure. Thebest single crystals of (KC60)n were prepared with this trick [149]. The co-evaporation ofC60 and potassium metal produced multiphase crystals. The crystals were exposed to air andsubsequently soaked in toluene. In contrast to monomeric C60, the polymer did not dissolvein toluene. The polymeric material survived the treatment in the form of fibres a few hundredmicrometres long and with cross section of a few tens of µm2.

The AC60 compounds exhibit a variety of metastable structures. ESR, optical and directx-ray studies demonstrated different low-temperature phases depending on the choice of thecooling. Slowly cooling the sample through 400 K gives the polymeric phase discussed above.However, there seems to be a kinetic barrier to the polymer formation: even though the polymerbecomes energetically favourable below ≈400 K, the formation of the chains requires the closeapproach of two highly charged double bounds on the neighbouring molecules. In the various


Figure 19. MAS 13C NMR spectra on monomeric (lower plot) and polymeric (upper plot) C60 inRbC60.

AC60 compounds two distinct metastable states are known, each one of them reached by rapidcooling from the high-temperature fcc phase.

Quenching the sample from the fcc phase to 77 K prevents the polymerization andpreserves a cubic structure that is distinguished from the high-temperature fcc phase onlyby the orientational order of the molecules. As we will see below, this slight difference in thestructure gives a completely different electronic ground state. The simple cubic phase is stableup to 150 K. Above this temperature the charged C60 molecules dimerize.

The dimerized state of C60 can be reached in another way, by quenching theAC60 from 400to 273 K and by subsequent cooling. In this case the high-temperature fcc phase is preserved(although this is a ‘supercooled’ state, and the molecules relax towards the stable polymerphase in due time).

There are many configurational possibilities for making a dimer from two C60 molecules.The x-ray studies by Oszlanyi et al [150] were the first to show that the (C60)

2−2 dimers are

formed through a single C–C bond with a bonding distance of 1.54 Å. The molecules arein a trans-conformation, shown in figure 20. Note that this bonding is different from thebinding scheme of the four-member rings of the AC60 polymer (see figure 18). Because of thisstructural mismatch, there is no direct dimer-to-polymer transformation, but an intermediatemonomeric (cubic) phase is needed.

The complex behaviour of the system can be interpreted in terms of the hindered rotationof the C60 molecules. The polymeric phase is the thermodynamically stable phase, but it canbe achieved only by passing the energy barrier or orienting the molecules in a direction thatwould be unfavourable in the fcc phase. At 273 K the kinetic barrier is large enough to reducethe possibility of polymerization. Cooling the fcc phase from this temperature results in thedimer formation.

Similar dimer structures occur in C59N, where the extra electron coming from the nitrogenatom is delocalized over the ball, and renders the molecule very reactive [53]. It is not surprising


Figure 20. Schematic view of the dimerized (C60)2 in the AC60 and(C59N)2 compounds.

Figure 21. Molecular view of the single-bonded linear fulleride chain of the Na2AC60 compound.

that the C59N· dimerizes into (C59N·)2, with the same structure as the dimer in the alkali-dopedC60 compounds. It is interesting to note that a C–C single bond occurs on carbon atoms whichare next to the nitrogen. One of the signatures of the dimerized state is the vanishing spinsusceptibility: the spins couple into a singlet, and most of the spin susceptibility is lost.

5.1.2. Na2AC60 compounds. The high-temperature phase of these compounds is isostructuralto the A3C60 fullerides. Around room temperature they undergo a fcc–sc structural transition(Ts = 299 K for Na2CsC60) [151]. At low temperatures the structure of these materials is sc,space group Pa3),

Various thermal and pressure treatments can produce polymeric phases as well [140,152].In the case of Na2KC60 and Na2 RbC60 the polymerization takes place spontaneously at lowtemperatures, while for Na2CsC60 a moderate pressure is needed [153]. The polymerizedNa2AC60 compounds form a linear chain with single bonding between the C60 molecules, assketched in figure 21.

When cooled slowly, polymerization is complete in Na2KC60 but is only partial inNa2RbC60 where the amount of polymeric phase depends on sample homogeneity [154] andcan be suppressed by rapid cooling.

The polymerization has a spectacular signature in the ESR signal: the broad linecharacteristic of the cubic phase gradually disappears, and a narrow line originating from thepolymeric part of the sample grows in intensity as the temperature is lowered. The integratedESR intensity reveals that the linear, singly bonded polymer is a good metal with a Paulisusceptibility. This is true for the other two members of the Na2AC60 family as well [154].

5.1.3. Polymerization of A4C60. 2D polymers are especially attractive, because with theirsynthesis one can complete the study of the effect of the dimensionality (1D, 2D, 3D) ondifferent properties. One could imagine very interesting systems ranging from 2D frustratedmagnets to 2D superconductors. The first 2D polymer was the high-pressure-synthesizedrhombohedral C60 [138,150]. In this polymer the bonding between C60 molecules is achievedthrough (2 + 2) cycloaddition as in the AC60 polymers. The only problem is that this polymeris neutral. All attempts to dope it have been unsuccessful up to now, as the charge transferprompts dissociation of the polymer.

After the first high-pressure synthesis, it seemed almost ‘natural’ that more polymerscould be best produced under extreme conditions, i.e. high pressures and high temperatures(such as 6–8 GPa, 600 K). It came as a surprise that with the same ‘soft’ chemistry used toprepare the known fullerides (e.g., K3C60, Na2C60, CsC60, etc), one can synthesize a doped 2D


Figure 22. The 2D polymer structure of Na4AC60. In the left panel the position of the constituentsis depicted. The deviation from the cubic structure is exaggerated for better view. The polymericsheets, shown in the right panel, form in one of the (110) planes of the bcc parent structure.

polymer, the Na4C60 compound [155] (see figure 22). In this polymer the same type of singleC–C bonding appears between C60 molecules as in the dimers or in the Na2AC60 polymer(figure 21).

The 2D sheets of the C60 polymers are charged due to the charge transfer from the Naatoms. This conducting 2D network is stable up to 500 K, classifying it as the ‘most stable’fulleride polymer. As far as the electronic properties are concerned, this compound seems tobe rather a semi-metal than a good metal (low and temperature-dependent spin susceptibility).Future high-pressure studies will show whether this system can have a superconducting, orother correlated ground states.

5.2. Electronic instabilities

As discussed in section 4.2, most doped fullerenes are close to the Mott localization: theratio of the on-site Coulomb energy U and the bandwidth W is close to the critical value. U isbelieved to be in the region of 1.5 eV, independent of the charge state of the C60 molecule, sinceit is determined mainly by on-site (molecular) interactions. The one-electron bandwidth is inthe 0.5–0.8 eV range. This parameter, however, is sensitive to the packing of the molecules,and it may be further narrowed by electron–phonon (polaronic) effects.

In the metallic state, electron correlations may influence the dc conductivity, as mentionedin the context of figure 16. However, resonance techniques such as ESR and nuclear magneticresonance (NMR) are better suited for the study of electron correlations on a microscopic level.The metallic behaviour is indicated by a temperature-independent Pauli susceptibility in ESRand by a Korringa relaxation rate in NMR. Correlation effects will alter these simple signaturesof metallicity. For example, in the case of the A1C60 compounds, the nuclear magneticrelaxation time T1 in K1C60 shows a simple temperature-independent (T1T )

−1 (Korringa)behaviour. Rb1C60 and Cs1C60 are also good conductors, but the subtle difference in theorientation of the polymeric chains results in a quasi-1D electronic structure. This leads tostrong antiferromagnetic fluctuations, giving rise to a non-metallic nuclear spin relaxationrate [136].

For narrow bands, localization is expected to occur. A prime example of Mott localizationis found in Cs3C60. The lattice is so much dilated by the large Cs ions that, instead of beinga metal with an expected superconducting Tc higher than that of RbCs2C60 (Tc = 32 K),the system becomes a semiconductor. Nevertheless, the U/W ratio can be shifted to themetallic side by applied pressure, and the superconducting transition appears in the 40 Krange [156].


In the localized state the exchange interaction between the nearest-neighbour spins candrive the system into an antiferromagnetic (e.g. NH3K3C60) or into a ferromagnetic (e.g.TDAE-C60) state.

SC is the most thoroughly investigated property of the doped fullerenes. Interestingly, aslong as localization does not occur, the narrower bandwidth leads to higher density of statesat the Fermi level, resulting in a higher superconducting transition temperature. This modelexplains the increase of the Tc with increasing lattice spacing, discussed later to some detail.

The metallic state may be unstable if the electron bands are strongly anisotropic. Low-dimensional electronic instabilities, like SDW and spin-Peierls transitions, are expected inbonded fullerene structures, for example in the quasi-1D RbC60 and CsC60 compounds.

5.2.1. Superconductivity in A3C60. The symmetry of the order parameter and the magnitudeof the energy gap are two of the most fundamental properties of superconductors. Sincethe original discovery in 1991, [6] great progress has been made in understanding SC inA3C60 compounds. There is a general agreement about the s-wave nature of the orderparameter [157]. It is also generally believed that the BCS formalism, with appropriateelectron–phonon coupling, describes the SC. The relatively high value of the superconductingtransition temperature is due to the role of high-frequency intramolecular vibrational modes.

A very successful line of studies was carried out on the relationship between thesuperconducting transition temperature and the lattice spacing by two different methods:pressure dependence and substitutional doping [158–160]. Modest pressures in the rangeof 10–20 kbar can cause relatively large changes in the lattice parameters. According tothe experiments these changes are accompanied by a significant variation of the transitiontemperature. On the other hand, using different alkali metals results in different latticeparameters. The combination of these two methods allowed for a continuous coverage ofthe lattice parameter range from 13.9 to 14.5 Å (figure 23) [160]. The observed drop ofthe transition temperature at smaller lattice spacing is generally attributed to the increase ofbandwidth [158] which, in turn, leads to lower density of states at the Fermi level. Thebehaviour can be understood, at least qualitatively, in terms of equation (10). The densityof states in the range of 20–30 eV−1 per molecule, derived from the pressure-dependentmeasurements [158, 161] is in a qualitative agreement with the calculations [162] and directmeasurements of the DOS by ESR [163].

A review of the literature reveals some disparity between the various measurements of thesuperconducting energy gap in K3C60 and Rb3C60. First, we consider spectroscopic methods,in which the characteristic gap energy is measured directly. Optical spectroscopy, tunnellingand photoemission belong to this category.

Most of the measurements were done on Rb3C60. The first optical study by Rotteret al [164] resulted in values of η ≡ 2�/kBTc ≈ 3–4. The error bar in this measurementwas large enough include any values from weak to intermediate couplings. FitzGeraldet al [165] obtained 3.5 from transmission measurements on thin-film samples. In more recentreflectance measurements on polycrystalline [126]and single-crystal [127] samples, Degiorgiet al [166–169] obtained η = 2.98 and 3.45, respectively. The Harvard group reportedtunnelling measurements yielding a value greater than η = 5, whereas Jess and co-workersmeasured values between η = 2 and 4 in a STM study [170]. Photoemission experiments bythe Argonne–University of Illinois collaboration [171] led to η = 4.1 ± 0.4. The main resultof that study is shown in figure 24 [171]. K3C60 and Rb2CsC60 were studied to a lesser extent.

Koller et al [172] investigated the gap in Rb3C60 by IR transmission spectroscopy and bybreak junction tunnelling (figure 25). A� = 5.2 meV energy gap was deduced, correspondingto η = 4.1. The spectroscopic results are summarized in table 1.


Figure 23. Dependence of superconducting transition temperature on the lattice parameter of alkali-metal-doped C60 compounds. In the Fm3m structure the molecules are more or less randomlyoriented between two preferred orientations. In Pa3 there are four C60 molecules in a cubicunit cell with well defined orientations. Solid and dashed curves are fits to the McMillan formula,equation (11), with the parameters indicated on the figure. Two fits are also depicted: the electronicdensity of states at the Fermi level is assumed to follow a power law (solid curve) and a lineardependence (dashed curve) on the lattice spacing (from [160]).

Table 1. Energy gap measurements in A3C60 compounds, reported in terms of the ratio 2�/kBTc.

Material IRa PEb TJc NMRd µSRe

K3C60 2–5 [165]3.6 [126] 4.2 [173] 3 [174]3.44 [127]3–5 [164]2.98 [126] 2–4 [170]

Rb3C60 3.45 [127] 4.1 [171] 5.3 [166] 4.1 ± 0.4 [174] 3.6 [175]4.2 ± 0.2 [172] 4.2 [172]

Rb2CsC60 <4 [176]

a IR: infrared spectroscopy.b PE: photoemission.c TJ: tunnelling (STM or break junction).d NMR: nuclear magnetic resonance.e µSR: muon spin rotation.

The gap can be also derived from the activated temperature dependence of the physicalproperties that are related to the single-electron excitations in the superconducting state.For example, the electronic contribution to the specific heat or the asymptotic temperaturedependence of the magnetic penetration depth were successfully used to extract the gap energyin ‘traditional’ superconductors. A more microscopic method, applied to fullerenes by severalauthors, is the study of the NMR spin-lattice relaxation rate 1/T1, or the shift of resonanceposition (Knight shift). The endohedral cavity of the C60 molecule offers a unique opportunityof trapping a muonium atom (µ+ e−) inside the C60 [177]. In a muon spin rotation measurementthe T1 relaxation of polarized and trapped muons is measured, and (similar to the NMR T1)the activation energy of the electron spin susceptibility is derived.


Figure 24. Photoemission results in the normal and superconducting state of Rb3C60 (upperand lower panels, respectively), compared with a Pt reference sample. The energy gap � wasdetermined by fitting a BCS-like density gap function [171].

An indirect measure of the electron–phonon coupling follows from the behaviour of theNMR T1 at temperature just below the critical temperature. According to the BCS theory,the relaxation rate should increase just below Tc (Hebel–Slichter peak). Suppression of thisincrease is expected as the coupling gets stronger [178].

Tycko et al [174] measured 1/T1 of the 13C nuclei in K3C60 and Rb3C60, and obtainedη = 3 and 4.1, respectively. They did not see a Hebel–Slichter peak, and concluded that theweak coupling limit is not applicable. Similar observations were made by Holczer et al [179]Stenger et al performed a detailed study of the Knight shift on 87Rb, 133Cs and 13C in Rb2CsC60.They found a weak Hebel–Slichter peak at low magnetic fields. The results were consistentwith an η � 4. In the muon spin rotation measurements of Kiefl et al on Rb3C60 the Hebel–Slichter peak was observed, and the activation energy of the relaxation rate yielded η = 3.6.These results are also listed in table 1, together with the data from other studies.

An inspection of table 1 leads to the conclusion that η = 4.1 is consistent with the mostreliable direct measurements. Assuming that the parameters satisfy the ‘universal’ relationshipplotted in figure 15, this number can be used to derive a characteristic phonon frequency,ωln = 210 cm−1. This value is close to the lowest-lying molecular Hg mode frequency. Usingequation (11), the electron–phonon coupling constant deduced from these values is λ = 0.89,if the electron–electron interaction parameter µ∗ = 0. For a more realistic µ∗ = 0.1 oneobtains λ = 1.16.

The dependence of the superconducting transition temperature on isotopic mass may beused to sort out the value of µ∗. Experiments on the Rb isotope effect [180, 181] led to the


Figure 25. Measured and calculated tunnelling conductance of Rb3C60. The dots represent themeasured conductance of a break junction. The solid curve was calculated by fitting a BCS densityof states; the dashed curve was obtained by modelling the non-ideal behaviour of the junction. Theenergy gap obtained from the two fits is the same, � = 5.2 mV [172].

Table 2. Carbon isotope effect in A3C60 compounds. Method: MP, magnetic measurement onpowder; RS, resistivity on single crystal.

Material Method 13C (%) α Ref.

Rb3C60 MP 33 1.4 ± 0.5 [182]Rb3C60 MP 76 0.32 ± 0.05 [183]Rb3C60 MP 60 2.1 ± 0.35 [184]Rb3C60 MP 82 1.45 ± 0.3 [185]Rb3C60 MP 99 0.3 ± 0.05 [186]Rb3(12C0.45

13C0.55)60 MP 55 0.3 [186]Rb3(12C60)0.5(13C60)0.5 MP 50 0.8 [186]Rb3C60 RS 99 0.21 ± 0.012 [188]K3C60 MP 60 1.3 ± 0.3 [184]K3C60 MP 99 0.3 ± 0.06 [187]

expected conclusion: the isotope effect is close to zero, since the vibrations of the Rb do notcouple to the conduction electrons. The carbon isotope effect has been extensively studied inA3C60, and α values ranging from 0.3 to 2.1 were reported. The results are summarized intable 2.

Fuhrer and co-workers [188] evaluated the isotope effect results in terms of the Eliashbergequations by numerical integration over the intramolecular phonon modes. They obtain theelectron–phonon coupling (defined in equation (9)) λ = 0.9+0.15

−0.1 and the renormalized electronrepulsion parameter µ∗ = 0.22+0.03

−0.02, with an average phonon frequency of ωln = 1390 K =970 cm−1.

In terms of BCS parameters, SC in the A3C60 compounds is characterized by anintermediate coupling λ∗ = 1 (to yield a 2�/kBTc > 3.5), a moderately strong electron–


electron repulsion parameter µ∗ = 0.2 and a characteristic phonon frequency in the lower partof the molecular vibrational spectrum, ωln = 200–1000 cm−1. The remarkably high criticaltemperature is therefore due to the fact that, in contrast to a simple metal, in this molecularsolid the electrons can couple strongly to high-frequency molecular phonon modes.

5.2.2. Superconductivity in Na2AC60. The Na2AC60 (A = K, Rb) compounds also have threeelectrons per C60 and, forA = Rb and Cs, are known to be superconductors. Although in manyrespects the SC in these materials and in A3C60 are similar, there is one notable difference: asfigure 23 illustrates, the pressure (lattice constant) dependence of the superconducting Tc ismuch steeper than in the other A3C60 fullerides [12].

The widely accepted description accounts for Tc in the A3C60 compounds by consideringthe same magnitude of electron–phonon and electron–electron interaction parameters. Thedifferences between various systems are attributed mostly to the lattice spacing (a) dependenceof the density of states at the Fermi level, N(EF). Thus, to explain the steeper dependenceof Tc on a in Na2AC60 fullerides, a stronger dependence of N(EF) on a was proposed [160],justified by the different crystal structures. Yet band calculations give a similar a dependenceof N(EF) for Pa3 and Fm3m [189]. NMR measurements found that N(EF) is notreduced sufficiently to explain the reduction of Tc in Na2RbC60 and Na2KC60 [190], andthe pressure dependence remains unexplained. Another related feature of the Na2AC60

compounds is the different variation of Tc with respect to physical as opposed to chemicalpressure [160, 191, 192].

A direct comparison of the electronic properties of Na2AC60 andA3C60 systems is possibleabove room temperature, in the fcc phase. Na2CsC60 exhibits an increase in the cubic latticeconstant at the structural transition [193], but the lattice is still contracted relative to otherA3C60

compounds. (In Na2CsC60, a = 14.1819 Å at T = 425 K [193], whereas a = 14.240 Åat T = 300 K for K3C60 [35], which has the smallest lattice constant among the A3C60

fullerides.) Therefore. the overlap of electronic orbitals between adjacent C60 balls must belarger in Na2AC60 than in the A3C60 compounds, and the Na2AC60 systems are expected tobe even better metals than the A3C60 fullerides. In sharp contrast to this expectation, the ESRresults shown in figure 26 demonstrate that the high-temperature fcc phase of Na2CsC60 is agapped insulator. A similar conclusion was drawn from IR reflectivity measurements [226].It appears that the mobility of the Na ions have a profound effect on the electronic properties.One cannot exclude that the Tc versus a anomaly in Na2AC60, discussed above, is also relatedto this mobility.

Interestingly, although the polymeric phase of Na2AC60 is a metal, it does not exhibitSC. The reason for this might be that the bonding, and consequently the deformation of themolecule, modifies the phonon spectrum, especially those modes which mediate the couplingbetween the electrons in the Cooper pair.

5.2.3. Superconductivity in field-doped pure C60. SC can be achieved in C60 even withoutchemical doping, as was demonstrated recently by Batlogg and collaborators [194]. A largeelectric field, applied perpendicular to the surface, can pull enough electrons to the surface of thecrystal to produce SC. The experiment was done in a field effect transistor (FET) configuration(see figure 27). The quasi-2D electron gas, formed at the interface of the C60 and the Al2O3

insulating layer, becomes conducting if the crystal and the interface is of high purity.The critical temperature of the superconducting transition depends on the polarity and

magnitude of the electric field. Figure 27 shows the results for negative gate voltage, when acritical temperature as high as 52 K was achieved. As a negative gate voltage results in hole


Figure 26. Electron spin susceptibility and ESR linewidth of Na2AC60. The inset shows the low-temperature behaviour, where the onset of SC reduces the signal. Notice the pronounced changein the temperature dependence at the sc–fcc structural transition. For comparison, the normalizedvalues for the metallic K3C60 are shown by the dashed curve. The continuous curve represents anantiferromagnet of 1 µB/C60, and Neel temperature of 200 K.

doping of the surface, this is the first example of seeing SC in hole-doped C60. Upon furtherincrease of the gate voltage, the transition temperature drops to zero.

The gate voltage can be converted to carrier concentration by assuming that all the chargeis in the first molecular layer of the surface. The highest critical temperature is achieved ata doping level of 3 holes per C60 in the HOMO band. For positive gate voltages (electrondoping) [195] the results are in excellent agreement with the SC in alkali-doped C60: themaximum critical temperature is reached at half-filled LUMO band (three electrons per C60),and the value of the critical temperature (around 11K) fits reasonably well to the data shownin figure 23, taking into account the lattice spacing of pure C60. In fact it is surprising howlittle the SC is influenced by the 2D character of the surface states.

The HOMO band is constructed from the fivefold-degenerate hu molecular orbitals. Thisband can have a maximum of ten holes, and the optimum doping level (3/C60) correspondsto less than half-filling. Having larger degeneracy reduces the effective on-site Coulombrepulsion, resulting in a system farther away from Mott localization. The larger transitiontemperature for hole doping correlates with the larger normal-state resistivity, indicating a


Figure 27. Channel resistance of the FET made with single-crystal C60. The device is shown in theinset. The negative gate voltage creates a hole-doped layer on the surface of the C60 (from [194]).

larger electron–phonon coupling constant in the HOMO band. If the increase of Tc withincreased lattice spacing proves to be the case for hole doping as well as electron doping,then critical temperatures approaching 100 K can be anticipated [194] for suitably expandedfullerene crystals.

5.2.4. Electronic transitions in AC60. Electrons in AC60 materials exhibit an unusually richbehaviour, illustrated in figure 28 for CsC60. At high temperatures the spin susceptibilityis of Curie type, corresponding to approximately 1 Bohr magneton per C60 (figure 28(a)).How did the electrons in the partially filled t1u band get localized? As discussed earlier, theelectrons residing in this band are close to the Mott transition. For CsC60 we may also considerthe possibility of random potentials acting on the electron wavefunction, causing Andersonlocalization.

In the high-temperature phase, when the fullerene molecules have random rotations andorientations, random potentials develop even in the absence of impurities: for localizedelectrons the JT effect lifts the degeneracy of the t1u level, and then the rotation of thedistorted molecule introduces a random overlap between the wavefunctions of neighbouringC60 molecules. Although there is no direct experimental evidence to support this view, it islikely that in CsC60 the Mott and the Anderson mechanisms work in tandem to localize theelectrons.

A dramatic demonstration of the proximity of the AC60 systems to the critical value ofU/W is provided by the cubic phase at low temperatures (see figure 28(b)). This phase isobtained by rapidly quenching the system with liquid nitrogen. In this low-temperature cubicphase the C60 molecules are orientationally ordered, and the random potential is absent. The


Figure 28. Magnetic susceptibility (determined from ESR) of CsC60 exposed to various heattreatments.

lattice contraction [196] increases the bandwidth with respect to the high-temperature phase.It seems that the absence of randomness and the slight increase of W is sufficient to shift thesystem into the metallic regime. Detailed NMR measurements show that this low-temperaturephase is not a simple metal: below 40 K, a spin gap opens in the electronic structure, mostlikely due to the formation of spin singlets stabilized by the JT distortion of the C60 [197].

The sc phase is unstable upon heating. Above 140 K the C60 molecules irreversiblydimerize. The dimers persist up to 240 K, where they break up into monomers (see section 5.1).The electrons on the dimer form spin singlets, and the material is a diamagnetic insulator (seefigure 28(b)).

Another metallic state is obtained when the molecules polymerize upon slow cooling(figure 28(c)). Below the polymerization temperature, down to 50 K, the spin susceptibilityhas a Pauli-like metallic behaviour with a weak temperature dependence. In the temperaturerange 30–50 K the the spin susceptibility decreases [198]. This is interpreted as the signatureof fluctuations of incommensurate short-range ordered SDWs. Detailed ESR measurementsin a wide frequency range have shown that a fully ordered, long-range SDW state sets inaround 30 K [198]. In a closely related compound, the RbC60, Mehring and co-workers [199]determined the spin-flop field of 0.34 T in a high-field ESR.

There is a strong similarity between the fulleride polymers and the quasi-1D organiccharge transfer salts such as (TMTSF)2PF6 in which the 1D-to-3D transition of SDWs has beenextensively studied [200]. In (TMTSF)2PF6 the SDW ground state has been suppressed under


hydrostatic pressure and SC has been observed above 10 kbar. Based on this analogy, one mayargue that the phase transition in CsC60 is also a consequence of the one-dimensionality of theelectronic structure, and should exhibit similar pressure dependence. Indeed, the SDW phasein the AC60 compounds also disappears under high pressure. However, the zero-resistancestate, the sign of SC, did not appear for pressures up to 15 kbar [201].

Simovic et al [202] have found that between the ambient-pressure SDW phase and thehigh-pressure metallic phase there is another ground state in a narrow pressure window closeto 5 kbar. Their detailed NMR study of CsC60 as a function of pressure shows the formation ofa long-range spin singlet ground state, possibly of a spin-Peierls nature, below 20 K. Furtherincrease of the pressure stabilizes a metallic state in the whole temperature range. Althoughit is natural to suppose that the material at high pressures has higher dimensionality than atambient pressure, little is known about the properties of the metallic state above 5 kbar.

To what extent is the electronic structure of the polymeric AC60 compounds quasi-1D?Chains of fullerene molecules with unfilled conduction bands, running parallel to each other,suggest a 1D character. However, the polymeric structure is achieved by sp3 bonding of C60

molecules, and the correspond electronic states lie well below the Fermi surface. In otherwords, the electronic transport along the chains via the closest carbon atoms is prohibited.Therefore, the transport along the chain is controlled by the overlap of the wavefunctions onthe carbon atoms next to the sp3-bonded pairs. Inspection of the structure reveals that thisintrachain overlap might be comparable to the interchain overlaps. Hence the dimensionalityof the electronic bands is very susceptible to the respective orientations of the neighbouringchains. Strong interchain overlap gives 3D electronic structure, while weak overlap results ina quasi-1D electronic structure.

MAS NMR measurements clearly show that the interchain interactions are different forthe KC60 on the one hand and RbC60 and CsC60 on the other [146]. This is corroborated bythe x-ray data of Launois et al [145]. The differences in the interchain overlaps makes theKC60 an anisotropic 3D conductor [203], while the RbC60 and CsC60 may well be quasi-1Dconductors [204]. The different electronic structures of KC60 and RbC60 are further illustratedby the temperature dependence of the spin susceptibilities in figure 29. In the polymericphase the KC60 has a temperature-independent Pauli susceptibility, while RbC60 has a weaklytemperature-dependent susceptibility showing the same type of SDW phase transition at 50 K asin CsC60. Recent 133Cs–13C double-resonance NMR data on CsC60 suggest that the conductionelectron density is concentrated on the ‘equator’ of C60 in the polymeric phase [205].

5.2.5. Ferromagnetism in TDAE-C60. The charge transfer salt tetrakis(diethylamino)ethylene-fullerene, TDAE+C−

60, was first synthesized by Wudl and co-workers [206]. The original mo-tivation of combining C60 with a strong organic donor was to make a novel fullerene-basedconductor. Instead, a ferromagnetic insulator was obtained. The ferromagnetism is basedon electron correlations between the fullerenes, and the Curie temperature is unusually high(Tc = 16 K) for an organic magnet.

The material is made in an oxygen-free atmosphere by reacting either powdered C60 withliquid TDAE at room temperature or C60 in toluene solution of TDAE. The lattice structure ofpowder samples was first determined to be monoclinic C2/m with one formula unit per unitcell [207]. Single-crystal measurements [208] resulted, at room temperature, in a monoclinicstructure, with unit cell dimensions a = 15.858(2) Å, b = 12.998(2) Å, c = 19.987(2) Å,θ = 93.37◦ and four formula units per unit cell. The space group was found to be C2/c.At room temperature the shortest C60–C60 distance is 9.99 Å, which decreases to 9.87 Å at80 K. 13C NMR measurements [209] indicated that the free rotation of the fullerene molecules,characteristic of so many C60-based solids, freezes in the temperature range below 100–150 K.


Figure 29. Spin contribution to the magnetic susceptibility of metallic RbC60 and KC60. Thedownturn around 50 K for RbC60 is accompanied by a rapid broadening of the ESR curve, indicativeof the development of some kind of magnetic order.

A very slight modification of the preparation conditions (keeping the temperature ofthe toluene solvent below room temperature) results in a non-ferromagnetic modification,α′TDAE-C60 [210,211]. The electronic ground state of this system is spin singlet [212]. Thismaterial is metastable, and it irreversibly transforms into the (ferromagnetic)αTDAE-C60 uponannealing at temperatures slightly above room temperature [213]. Although the existence ofthe two modifications has been established unambiguously, no systematic structural analysishas yet been performed on α′TDAE-C60. The different magnetic behaviour seems to be relatedto the different orientational order of the fullerene molecules at low temperatures [213–215].

The αTDAE-C60 is a very soft ferromagnet; the saturation field is less than 50 Œ, and thehysteresis loop of the magnetization is at most a few oersted wide [216]. In contrast to earlymeasurements, where a much smaller magnetization was found, the saturation magnetizationcorresponds to one spin per formula unit [215]. The material has very small anisotropyfield (29 G) with the easy axis in the c direction, where the interfullerene distance is thesmallest [217].

In most of the known ferromagnets the microscopic coupling is due to conductionelectrons. TDAE-60 (α or α′) is an exception: optical, microwave and dc conductivitymeasurements have clearly established its semiconducting character [218–220]. The smalloverlap of electronic orbitals, resulting in a narrow band, means that the electrons aresusceptible to each one of the correlation effects discussed in sections 4.2–4.4. The mostlikely candidate is the Mott–Hubbard localization, due to the Coulomb repulsion between theelectrons. But the Mott insulators are typically antiferromagnets. How is it possible to have aferromagnetic coupling in αTDAE-C60?

The answer to this question must include the degeneracy of the t1u orbitals. In acombination of spin and orbital ordering the spin order can be ferromagnetic, whereas theorbital order is ‘antiferromagnetic’, i.e. alternating. In this configuration each electron can jumpto the empty orbit on a nearest-neighbour molecule, experience the ferromagnetic exchangecoupling (Hund’s rule) and jump back. As discussed in section 4.2, calculations of the on-site exchange energy, and (indirect) measurements, yield an on-site ferromagnetic coupling ofabout J = 2U2 = 0.1–0.2 eV (see equation (5)). The hopping of the electron renormalizesthis coupling by a t2 term. In this qualitative description, a ferromagnetic intermolecular J


Figure 30. Left: x-ray diffraction photographs of a well annealed single crystal of TDAE-60(from [213]). The arrows indicate approximate crystallographic directions. Right: an ordered stateof the fullerene molecules favouring ferromagnetic coupling (from [224]).

of the order of magnitude of a few meV seems to be reasonable. The intimate relationshipbetween the orbital order and the magnetic coupling (for example, a ‘ferromagnetic’ orbitalorder would lead to an antiferromagnetic ground state) also explains the observed extremesensitivity to the orientational ordering of fullerene molecules.

An ordered JT state must yield weak superlattice peaks in the x-ray diffraction. In a recentwork, Kambe et al found superlattice reflections developing at 150 K, most likely due to theorientational ordering of the distorted fullerenes (see figure 30 [213]). A possible ordered stateof the fullerene molecules, also shown in the figure, was suggested earlier by Kawamoto [224].Further exploration of the connection between the ordered structure and the ferromagnetismis certainly warranted.

5.2.6. Antiferromagnetism in NH3K3C60. NH3K3C60 is derived from the superconductingcompound K3C60 by intercalating the ammonia molecule into the fcc [221] lattice. The NH3

molecules do not change the charge state of the C60 molecules, but they do change the crystalsymmetry into a face centred orthorhombic one.

It has been suggested that the smaller transfer integrals of the expanded latticesuppresses SC and favours Mott–Hubbard transitions. The slightly temperature-dependentspin susceptibility of the K3C60 acquires a negative temperature slope. Nevertheless, thetemperature dependence does not correspond to a Curie–Weiss behaviour [222]. At 40 K along-range antiferromagnetic ground state develops with a low value of the spin-flop field. Thedevelopment of the magnetic state is demonstrated in the clearest way by the ESR linewidth:it strongly increases below 40 K at 9 GHz (see figure 31 [223]). At higher frequencies thelinewidth narrows. This particular frequency dependence of the linewidth is the characteristicsign of antiferromagnetism (AF) [225].

Like almost all the electronic instabilities in the fullerides, the AF state is very sensitiveto the lattice structure, and to disorder. The ordering of the ammonia molecules at 170 K orthe quenching of the ordering changes the character of the AF state [223].

6. Open questions

Although many of the doped fullerides are have been extensively studied, a number of ratherfundamental questions are still debated. Here we review a few of these, starting with the AC60

polymer.


Figure 31. Temperature dependence of the ESR linewidth of NH3K3C60 at various frequencies(9 GHz: squares, 35 GHz: circles 75 GHz: triangles). The inset shows the linewidth (in mT) above40 K [223].

The chain-like structure of the fullerenes in the polymeric form suggests a 1D electronicband. Yet, due to the σ character of the chemical bond in the AC60 polymer, the carbon atomsparticipating in the binding of two C60 molecules have all electrons in full bands well belowthe Fermi energy. The conduction band that can delocalize the electrons along the C60 chainis formed from the π–π overlap of the carbon atoms that are next to the binding sites. Butthis overlap may be comparable to interchain overlaps, rendering the system 3D. Accordingly,band structure calculations typically yield 3D character.

The interchain overlap of orbitals depends strongly on the mutual orientation of the poly-merized chains. Indeed, the different electronic properties of KC60 and RbC60 (KC60 is metallicdown to low temperatures, while RbC60 and CsC60 show a phase transition to a SDW state)seem to correlate with subtle structural differences: according to single-crystal x-ray diffrac-tion, the two compounds belong to different space groups due to different chain orientationin the unit cell (figure 18). Mehring and co-workers [205] determined that in the polymericCsC60 the conduction electron density is enhanced along the ‘equator’ of C60. This paper wouldsuggest that the 1D electron gas may be situated perpendicularly to the polymeric backbone.

High-pressure NMR study has shown that pressure can suppress the magnetic ground statein CsC60 [202]. However, before stabilizing a metallic state, a ‘spin-Peierls’-like diamagneticstate develops. Whether this is a true spin-Peierls state or a JT bipolaron distortion, orsomething else, is not known yet. Further studies, especially high-resolution neutron diffractionmeasurements under pressure, are needed.

The nature of metallic state at pressures above 5 kbar is also unknown and may beinteresting. The proximity to a quantum critical transition (the SDW state) may very well leadto non-Fermi liquid behaviour. Measurements of the power-law dependence of the resistivityon single crystals at high pressures could give an answer. How about SC in this metal? Zettland co-workers searched, but did not detect a zero-resistance state in the polymerized phaseof KC60, RbC60 and CsC60 up to pressures of 15 kbar.

KC60 (and RbC60 and CsC60 under pressure), together with the polymeric form ofNa2AC60, have bonded fullerene structures, and they all support metallic conduction. Althoughthe monomeric form of Na2AC60 is superconducting, no SC has been seen in any of thepolymers. Will SC appear at much higher pressures, or does the bonded structure somehowexclude SC altogether? For example, is it possible that the phonon mode that mediates theCooper pair formation is modified by the polymerization, resulting in the disappearance of SC?


Figure 32. 87Rb NMR spectrum of Rb2CsC60 at several temperatures. At high temperatures themotional narrowing results in a single resonance signal [229].

The monomeric Na2AC60 offers interesting questions of its own. Although the charge stateof the fullerene molecules is the same as in the A3C60 fullerides, these materials have severalinsulating ground states, depending on the orientational ordering of the fullerene molecules.When superconducting, the pressure dependence of the superconducting transition temperatureis steeper (see figure 23) than in the fcc A3C60 materials. It is natural to assign the uniquefeatures of these compounds to the mobility of the small Na ion [226]. Yet there is noclear structural evidence for correlating this mobility to the various electronic states of thecompound.

At closer inspection the structure of the A3C60 fullerides is different from the ‘ideal’structure shown in figure 5. Well established NMR results, starting from the early work ofWalstedt et al [227–229], indicate the existence of two different tetragonal sites. A particularlyclear illustration of this feature is shown in figure 32 for Rb2CsC60, where the Rb occupiesthe tetragonal site only. The surprising fact is the existence of two separate signals at lowtemperatures. (The observation of a single signal at higher temperatures is due to the rapidchange of the Rb environment leading to motional narrowing.) If exact stoichiometry and aspherical shape for the fullerene molecules are assumed, then all tetragonal sites are equivalent,and only one signal is expected. In the search of interactions between the alkali metal andthe carbon atoms on the C60 molecule, Pennington performed a series of ‘spin-echo doubleresonance’ measurements [230]. A number of possible explanations, relating to the particularorientation of the molecules, has been excluded.

Vacancies on these sites has been also suggested as an explanation by a number ofauthors [228, 231, 232]. X-ray studies [231, 233] resulted in about 2% cation deficiency inthe A3C60 structure. A reliable determination of the site occupancy at this level of accuracy isby no means a simple matter. A careful comparison toA4C60 [234], where the off-stoichiometryhas not been seen, lends support to the reliability of the x-ray work. Yet Pennington’s groupargued, based on a careful investigation of the thermal activation time scales leading to themotional narrowing of the NMR signal, that the vacancies, if present, cannot explain theexistence of two NMR signals.

It remains to be seen whether the final answer to this question is somehow related tovacancies on the alkali sites. In view of the strongly correlated nature of the electronic states inthe fullerides (large U ) the role of vacancies may be very important. Although the electronic


Figure 33. Resistivity of Rb3C60 and K3C60 single crystals over a wide temperature range(from [119]).

band is conveniently treated as half-filled by most authors, it is still possible that in the finalanalysis the electron correlations are so strong that the stoichiometric A3C60 is in fact aninsulator. Controlling the vacancy concentration, and relating it to the properties in the metallicand/or superconducting state, has been unsuccessful up to this point. On the other hand, theresults on the FETs made with pure C60 [194, 195] lend support to the argument that off-stoichiometry is not an important factor in the SC. Although the conversion of gate voltageto doping level can be debated, it looks likely that the critical temperature peaks around thedoping level of three electrons per C60. Further work in this direction will attract particularattention.

Another interesting issue related to the metallic fullerides is the mean free path of theelectrons. Ballistically propagating electronic states can only exist if the mean free path is largerthan the lattice spacing, l > a. The Bloch–Boltzmann theory fails and a resistivity saturationis expected as l becomes comparable to a [124], and localization is expected when l < a.

The normal-state conduction in A3C60 is puzzling in this respect. At room temperaturethe resistivity exhibits a metallic slope (increasing with temperature) and its absolute value is1.5 m 5 cm (K3C60), 2.5 m 5 cm (Rb3C60) [123]. Although the absolute value measurementsare particularly sensitive to sample quality, these numbers have been independently confirmed.The typical mean free path derived from these values is in the range of a few angstroms,much less than the lattice spacing or the centre-to-centre distance of the fullerene molecules(although larger than the distance between the carbon atoms within the molecule). Therefore,the existence of metallic conduction in the A3C60 compounds is somewhat unexpected. Whenexamined over a wide temperature range, as presented in figure 33, the temperature dependenceof the resistivity is indeed different from that of a simple metal [119].

The A4C60 (A = K, Rb, Cs) fullerides [45] should have a partially filled t1u conductionband and yet they are all semiconductors. The ground state is non-magnetic. Due to the inter-grain resistance contributions typical of powder samples, attempts to determine the band gapfrom dc conductivity measurements were inconclusive. The best estimate of the gap followsfrom magnetic resonance measurements, indicating a spin gap of 0.15 eV to the lowest triplet


state. The gap closes at a relatively moderate pressure of 10 kbar. Chemical pressure, achievedby the small size of Na, also closes the gap [235].

The insulating phase can be caused by the strong crystal field splitting of the t1u levelarising from the bct crystal structure, or from static JT distortion of the C60 molecules. A verythorough x-ray study has been performed in search of the JT distortion [234], and an upper limitof 0.04 Å was found for the atomic displacements. It is intriguing to observe that the knownexample of molecular JT effect [110] results in a small distortion of about this magnitude.

There is also a possibility that strong electron–electron repulsion localizes the electronicstate. The mechanism leading to semiconducting behaviour in A4C60 may be responsible forthe non-conducting character of Na2C60 [236] as well.

7. Conclusions

More than a decade after the discovery of the Kratschmer–Huffman method [1], research onfullerenes is still blooming. The high-temperature SC, the proximity to a metal–insulatortransition, and the unusual magnetic correlations have attracted a strong interest in thesecompounds. With the steady increase of computing power available for ab initio electronicstructure calculations, the 60-atom fullerene molecule will soon become an ‘exactly solvable’problem. Yet, even when the single molecule becomes better understood than today, the largechoice of complex structures realized in the solid state will provide new surprises.

Much new physics follows from the rotational freedom of the fullerene molecules. Therequirement for the molecules to line up in a particular fashion to have certain electronicproperties is an interesting new aspect of fullerene-based solids.

The bonded fullerene structures have provided several surprises, and are expected tocontinue to do so. Many of these materials are metastable. But for practical purposesthese modifications are stable enough, just as is another metastable carbon allotrope, thediamond.

Fullerene-based conductors exhibit an unprecedented flexibility: the HOMO, the LUMOand the LUMO + 1 bands can each carry SC. The A3C60 or Na2AC60 superconductors (likeK3C60 or Na2CsC60) have n = 3 electrons per ball which gives a half-filled t1u LUMO band.In divalent alkaline-earth-doped samples (like in the orthorhombic Ba4C60 and Sr4C60) thestronger hybridization of the molecular and atomic orbitals results in band overlaps, but it islikely that electrons in the t1g-derived LUMO + 1 band contribute to the SC observed in the4–6 K range (see figure 17, and [133, 134]). The electron doping achieved at the C60–Al2O3

interface in FETs [194, 195] established SC in the hu HOMO band. With this breakthrough,critical temperatures up to 100 K seem to be reasonable, if the expansion of the C60 latticeworks the same way as it did for the hole conduction. Using these FETs, a whole new rangeof experiments can be envisaged, and application in electronic devices becomes a seriouspossibility.

Acknowledgments

The authors are indebted to the Swiss National Science Foundation and the National ScienceFoundation (USA) for support of this collaboration. LM thanks Professor Libero Zuppiroli forhospitality during his stay in Lausanne. Discussions with A Janossy, P W Stephens, P B Allen,A Tutis, F Simon and M C Martin are gratefully acknowledged.


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Electronic properties of doped fullerenes › 5b9a › 3fe63d... · Materials made of large organic molecules exhibit a characteristic separation of energy scales. In many organic