Electron Correlation and Jahn–Teller Effect in Alkali-Metal-Doped C60

Electron Correlation and Jahn– Teller Effect in Alkali-Metal-Doped C60 Shugo Suzuki, Tadahiko Chida and Kenji Nakao Institute of Materials Science, University of Tsukuba, Tsukuba 305-8573, Japan

Abstract We investigate the role of the electron correlation and the Jahn – Teller effect in the electronic structure of alkali-metal-doped C60, AxC60 where A is alkali metal. We show that the cooperation between the two effects in A2C60 and A4C60 results in the insulating ground states while the competition between the two effects in A3C60 results in the metallic ground state. We also show that the superconductivity in A3C60 is induced by the dynamic Jahn –Teller effect. Contents 1. Introduction 2. Basis of electronic structure 2.1. t1u Bands 2.2. Electron correlation and Jahn – Teller effect 3. Photoemission spectrum 3.1. A4C60 3.2. A3C60 4. Superconductivity 4.1. Dynamic Jahn– Teller effect 4.2. Order parameter 4.3. BCS theory for A3C60 5. Summary and open questions Acknowledgements References

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1. INTRODUCTION Since the discovery in 1985 [1], the C60 molecule has been widely known as a molecule consisting of only carbon atoms with the shape of a soccer ball. The study on solid C60 started after the establishment of a method for preparing a large quantity of the molecules [2]. Subsequently, superconductivity was observed in K3C60 with a critical temperature of 18 K [3]. So far, the critical temperature has been raised up to 33 K in RbCs2C60 [4]. In the last decade, not only the superconductivity but also ADVANCES IN QUANTUM CHEMISTRY, VOLUME 44 ISSN: 0065-3276 DOI 10.1016/S0065-3276(03)44035-5

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a variety of properties of alkali-metal-doped C60, AxC60 where A is alkali metal, have been investigated extensively [5,6]. As well as A3C60, A2C60 and A4C60 have been studied in detail and, as a result, it was found that A2C60 and A4C60 are nonmagnetic insulators. Figure 1 shows the results of the optical conductivity measurement [7]. It is clear that K4C60 is an insulator because there is no Drude-like behavior, which is a characteristic feature of metals, while K3C60 shows such a behavior in agreement with the fact that the material is a metal. Figure 2 shows the results of the photoemission experiment [8]. It is also clear that K4C60 is an insulator because there is no Fermi edge while K3C60 shows a clear one. Furthermore, the results of the mSR study on K4C60 [9] show that the minimum energy for the electronic excitation is about 0.3 eV; this gives the lower bound of the energy gap [10]. It is, however, possible that the actual energy gap of K4C60 is much larger than this lower bound if the excitonic effect is significant. The point to be noticed is that A2C60 and A4C60 are insulators although they are expected to be metals according to the rigid-band picture, as explained below. The lowest unoccupied molecular orbital of the C60 molecule is the t1u orbital, which is 3-fold degenerate. As shown in Fig. 3, the t1u orbitals form three conduction bands, the t1u bands, in solid C60. These bands can accommodate conduction electrons up to 6 per C60 molecule. It is thus expected that A2C60 and A4C60 are metals because the filling of the t1u bands is incomplete in these materials. Nevertheless, the experimental results show that A2C60 and A4C60 are insulators. That is, it is impossible to explain the insulating nature of A2C60 and A4C60 by the

Fig. 1. Optical conductivity spectra of AxC60 (x ¼ 0; 3, 4, and 6) [7]. K3C60 is a metal, which shows a Drude-like behavior at low energy region, while K4C60 is an insulator, which does not show such a behavior.

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Fig. 2. Photoemission spectra of KxC60 [8]. K3C60 (x ¼ 0:1; 1.3, and 2.2) is a metal, which shows a clear Fermi edge. Also, there exists a small shoulder at 1.6 eV. On the other hand, K4C60 ðx ¼ 4:2Þ is an insulator, which does not show a Fermi edge. The two phases coexist for x ¼ 2:8 and 3.7.

rigid-band picture, which is only based on the electronic transfer between the C60 molecules, T [11]. The following question then naturally occurs. Is A3C60 a simple metal although the material is between the anomalous insulator phases, A2C60 and A4C60? It seems

Fig. 3. t1u Bands. The origin of energy is taken at the bottom of the bands. They consist of three conduction bands with a width of about 0.4 eV.

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difficult to give a positive answer to this question. In fact, we can find several anomalous behaviors in A3C60 as well. Although K3C60 is a metal according to the photoemission spectrum shown in Fig. 2, the width of the spectrum is very broad, which is over 1 eV, in comparison to the width of the t1u bands, which is about 0.4 eV. Furthermore, of great interest is a small shoulder at 1.6 eV in the spectrum. This shoulder is found only in K3C60 but not in either K4C60 or K6C60. It is thus important to elucidate the origin of this shoulder for obtaining useful information to understand the electronic properties of A3C60. One purpose of the present study is to give an explanation for the origin of this shoulder [12]. Recently, it has been found that a striking anomaly of A3C60 is revealed by introducing the NH3 molecule. This results in the disappearance of superconductivity and, at the same time, the material is transformed into an anti-ferromagnetic insulator. The details may be found elsewhere [13,14]. Furthermore, A3C60 shows several unresolved anomalies related to the excited states. For example, although a mid-infrared absorption is observed in the optical conductivity spectrum of K3C60 shown in Fig. 1, the origin of this absorption has not been elucidated yet. We thus find several anomalous behaviors not only in A2C60 and A4C60 but also in A3C60. The key to understand the anomalous behaviors in AxC60 is the electron – electron interaction, U, and the electron – phonon interaction, S. In AxC60, the important interaction as U is the Coulomb repulsion between the t1u electrons. The importance of U has been pointed out based on the results of the photoemission experiments [15]. Also, in AxC60, the important interaction as S is the coupling of the t1u electrons to the intramolecular phonons of the C60 molecule. The importance of S has also been pointed out based on the results of the Raman experiments [16]. Nevertheless, a complete understanding of the anomalous behaviors in AxC60 has not been established yet. The reason is that the system with the orbital degree of freedom in which both U and S are important has not been known so far [17 – 20]. Furthermore, what is the mechanism of superconductivity in A3C60 which has the orbital degree of freedom with both U and S being important? Originally, U and S were pointed out to be important in relation to the mechanism of superconductivity in A3C60. In the early 1990s, there was a point of view which claims that it is exotic superconductivity due to U [21] while there was another point of view which claims that it is usual superconductivity due to S [22]. Among such suggestions, there were two notable proposals; one is the proposal of the Moskalenko –Suhl –Kondo mechanism of the superconductivity in A3C60, which is important to multi-band superconductors [23 – 25], and the other is the proposal of the importance of the dynamic Jahn –Teller effect as the origin of the attractive interaction between the t1u electrons [26,27]. A recent study has revealed a close relationship between the Moskalenko –Suhl –Kondo mechanism and the dynamic Jahn –Teller effect and showed that the orbital degree of freedom is indispensable to the superconductivity in A3C60 [28]. The purpose of the present study is to show a unified picture for understanding the electronic properties of AxC60 based on a simple model. In this model, both the electron – electron interaction U and the electron – phonon interaction S are taken

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into account as well as the 3-fold t1u bands described by the electronic transfer T. In particular, we discuss the dependence of the electronic properties on the valence, x, of the C60 molecule. We show that the insulating nature of A2C60 and A4C60 is due to the cooperation between U and S while the metallic nature of A3C60 is due to the competition between U and S. Furthermore, it is shown that the superconductivity in A3C60 is induced by the dynamic Jahn – Teller effect when S overcomes U. Moreover, a close relationship between the dynamic Jahn – Teller mechanism and the Moskalenko– Suhl – Kondo mechanism, the stabilization mechanism of the superconductivity in multi-band superconductors, is pointed out. Finally, summary and open questions are given.

2. BASIS OF ELECTRONIC STRUCTURE In this section, a model which gives the basis of the present study is introduced to investigate the electronic properties of AxC60 [17]. First, the one-electron part of the Hamiltonian, which describes the itinerant motion of the t1u electrons in terms of the electron transfer T, is given. Next, the electron –electron interaction U and the electron – phonon interaction S are examined; U represents the Coulomb repulsion between the t1u electrons and S represents the coupling of the t1u electrons to the intramolecular phonons of the C60 molecule. In particular, it is noteworthy that both U and S are almost the same or larger in magnitude than the width of the t1u bands. Finally, the importance of the dynamical aspect of S is pointed out. 2.1. t1u Bands Most of the molecular orbitals and the intramolecular phonons of the C60 molecule are highly degenerate, reflecting the high symmetry of the molecule, the icosahedral symmetry Ih. To study the electronic properties of AxC60, the 3-fold degeneracy of the t1u orbital, which is the lowest unoccupied molecular orbital of the C60 molecule, is important. It may be useful for visualizing the t1u orbital to imagine the p atomic orbital because the basis functions of the t1u orbital are transformed in the same way as those of the p atomic orbital; in the present study, three basis functions of the t1u orbital are accordingly denoted by x, y, and z. Also, one of the two intramolecular phonons to which the t1u electrons couple is called the Hg mode, as explained below in detail; the Hg mode is 5-fold degenerate. It may be useful for visualizing the Hg mode to imagine the molecular distortion similar to the d atomic orbital because the normal coordinates of the Hg mode are transformed in the same way as the basis functions of the d atomic orbital. In the solid state, the C60 molecules are crystallized into the fcc lattice and the conduction bands, the t1u bands, are formed of the t1u orbitals of the C60 molecules. Saito and Oshiyama carried out the electronic structure calculations on solid C60 and found that it is a semiconductor with a calculated band gap of 1.5 eV, preserving the molecular characteristics strongly [29]. Their results also show that the width of

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the t1u bands is about 0.4 eV and there are other unoccupied bands just above the t1u bands apart by about 1 eV. Accordingly, the conduction electrons donated by alkali metal atoms occupy the t1u bands and it is thus reasonable to consider only the t1u bands for understanding the electronic properties of AxC60. It should, however, be stressed again that one cannot explain the reason why A2C60 and A4C60 are nonmagnetic insulators as long as we consider only these one-electron properties. The Hamiltonian H0, which describes the itinerant motion of the t1u electrons, is given by X mn † X 0 † H0 ¼ tab ams bns ¼ jak aks aks ð1Þ m;n;a;b;s

ak s

The three curves shown in Fig. 3 are the ones calculated by using this Hamiltonian. mn Here, tab is the electronic transfer T between the t1u orbital a of the mth C60 molecule and the t1u orbital b of the nth C60 molecule, where a and b denote x, y, mn and z; tab is chosen so as to reproduce the result of the electronic structure calculations. We also use a†ms ðams Þ to denote the creation (annihilation) operator of the s-spin electron in the t1u orbital a of the mth C60 molecule. Furthermore, j0ak is the band energy of the t1u electron of the band index a (a ¼ 1; 2; and 3) and the wavenumber k; the band energies are obtained by diagonalizing the Hamiltonian H0 and we use a†ks ðaks Þ to denote the corresponding creation (annihilation) operators.

2.2. Electron correlation and Jahn –Teller effect Here, we describe the electron – electron interaction U and the electron – phonon interaction S. These two interactions were pointed out to be important by many researchers soon after the discovery of the superconductivity in A3C60. The magnitude of each interaction was estimated theoretically by using the Hartree – Fock and density functional calculations and was also estimated experimentally by using the photoemission and Raman spectroscopy techniques. Although most of early studies focused on only one of the two interactions, it has been gradually revealed that both U and S are indispensable for understanding the electronic properties of AxC60 as experimental results were accumulated. We first examine U. The most important interaction as U is the Coulomb repulsion between two t1u electrons on a C60 molecule. To estimate the order of magnitude of U, it may be useful to regard the C60 molecule as a sphere of a ˚ . The lower bound of U can be obtained by diameter D, which is about 7 A considering the Coulomb repulsion between two electrons which move on the surface of the sphere always being at the opposite position to each other separated by the distance D. We then find the lower bound of U to be e2 =e D , 0:4 eV; where e , 4 – 5 is the dielectric constant of undoped solid C60. On the other hand, the upper bound of U can be obtained by considering the Coulomb repulsion between

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two electrons which move on the surface of the sphere independently. We then find the upper bound of U to be 2e2 =e D , 0:8 eV: The actual U may be between these two values and it is likely to be about 0.6 eV. This value of U is larger than the width of the t1u bands, which is about 0.4 eV, and thus the effect of U must be taken into account. On the other hand, the exchange interaction, JC, originated in the Coulomb repulsion between the t1u electrons does not play an important role in understanding the electronic properties of AxC60. To show this, it is necessary to note that there are two kinds of repulsion as U to be precise; one is the intraorbital repulsion, Uintra, which represents the Coulomb repulsion between two t1u electrons in an equal t1u orbital and the other is the interorbital repulsion, Uinter, which represents the Coulomb repulsion between two t1u electrons in different t1u orbitals. It can be shown that the difference between Uintra and Uinter is one order of magnitude smaller than Uintra and Uinter themselves; the difference is about 10% of U. On the other hand, since the equation Uintra ¼ Uinter þ 2JC holds due to the symmetry of the t1u orbital, JC is found to be about 0.03 eV. Accordingly, the magnitude of Hund’s coupling between the t1u electrons is of this order and it is overcome by the Jahn – Teller effect due to S as shown in Section 3. We next examine S. The most important interaction as S is the coupling of the t1u electrons to the intramolecular phonons of the C60 molecule. Among a number of intramolecular phonons, the modes which couple to the t1u electrons are of two kinds, i.e., the Ag mode and the Hg mode. The former is the breathing mode, which is totally symmetric, and it affects only the center of energy of the t1u levels. On the other hand, the latter is the Jahn – Teller mode and it can split the t1u levels. The coupling constants of theses modes can be estimated from the results of the calculations on the Jahn – Teller distortion of the C2 60 molecule by Koga and Morokuma [30]. It is shown that the coupling constant of the Ag mode, SAg, and that of the Hg mode, SHg, are both about 0.4 eV. We thus find that the electron – phonon interaction S is almost the same in magnitude as the width of the t1u bands. That is, it is also indispensable to take into account S for understanding the electronic properties of AxC60. Furthermore, of great importance is the dynamic aspect, or the nonadiabatic effect, of S. Since the t1u orbital consists of the p orbitals which are perpendicular to the surface of the C60 molecule, the modes which significantly affect the distances between the carbon atoms are responsible for changing the p bonding considerably. The frequencies of such modes are, however, about 0.2 eV; this is too large to treat them by using the adiabatic approximation as usual. That is, it is appropriate to treat S in AxC60 by using the anti-adiabatic approximation instead of the adiabatic approximation. An important interaction then arises as follows; a pair of t1u electrons occupying an equal t1u orbital can be transferred into another t1u orbital without breaking the pair. This is the pair-transfer interaction, which is largely affected by the nonadiabatic effect due to S. It is this interaction that induces the dynamic Jahn – Teller effect and, consequently, the superconductivity in A3C60 as shown in Section 4.

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We give the Hamiltonian of the effective interaction between the t1u electrons, which takes into account both U and S as follows: 1X X Hint ¼ V a† c † d b ð2Þ 2 s;t m;a;b;c;d abcd ms mt mt ms Then the total Hamiltonian of the system is given by H ¼ H0 þ Hint

ð3Þ

In equation (2), Vabcd are classified into the intraorbital repulsion Vintra, the interorbital repulsion Vinter, the exchange interaction J, and the pair-transfer interaction K as schematically shown in Fig. 4. These are defined by 1 S 2 2 Ag 1 ; Uinter 2 SAg þ 2 3 J ; JC 2 SHg 8

Vintra ; Uintra 2 Vinter

1 S 2 Hg 1 S 4 Hg

ð4Þ ð5Þ ð6Þ

and K ; KC 2

3 S 8 Hg

ð7Þ

In equation (7), KC is the pair-transfer interaction originated in the Coulomb repulsion and its value is the same as JC. Although J and K have the same value, it is noteworthy that these interactions represent different physical processes. Furthermore, it should be noted that the equation Vintra ¼ Vinter þ 2K still holds due to the equation Uintra ¼ Uinter þ 2JC : Consequently, the independent parameters of the effective interaction are two. In the present study, we employ Vintra and K as two independent parameters. It is found that Vintra is about 0.2 eV, Vinter is about 0.4 eV, and both J and K are about 2 0.1 eV owing to the estimation already given. Of great importance here is the

Fig. 4. Effective interaction between the t1u electrons.

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following two points; both Vintra and Vinter are positive due to the effect of U while both J and K are negative largely affected by the nonadiabatic effect due to S.

3. PHOTOEMISSION SPECTRUM In this section, we study the characteristics of the ground states of AxC60 by examining the photoemission spectra. In particular we focus on the reason why A2C60 and A4C60 are nonmagnetic insulators while A3C60 is a metal. To study the photoemission spectra theoretically, we calculate the imaginary part of the oneparticle Green function, or the spectral function, by using the effective interaction (2) [12] and employing the dynamical mean field theory [31]. The results given below are those for A4C60 and A4C60 and A3C60; the electronic structure of A2C60 can be obtained by exchanging electrons and holes in the electronic structure of A4C60. 3.1. A4C60 We show the results of the calculations on A4C60 in Fig. 5. The spectral function of A4C60 is shown in Fig. 5(a) and that of the isolated C42 60 molecule is shown in Fig. 5(b); here, the latter is calculated analytically. The origin of energy is taken at the Fermi level. The spectral function of A4C60 clearly shows that the material is an insulator. Furthermore, a detailed analysis has shown that the ground state of A4C60 is nonmagnetic [12]. The energy gap is about 1 eV. It is reasonable to consider that the origin of this energy gap is the energy gap of about 1.5 eV which originally exists in the spectral function of the isolated C42 60 molecule shown in Fig. 5(b) although the energy gap of A4C60 is slightly smaller due to the electronic

Fig. 5. Calculated spectral functions of A4C60. (a) The spectral function of A4C60 and (b) that of the isolated C42 60 molecule. The origin of energy is taken at the Fermi level. The result shows that A4C60 is an insulator. The energy gap of A4C60 is originated in the energy gap of the isolated C42 60 molecule.

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transfer T; one can see the effect of T in Fig. 5(a) as the broadness of the peaks in the spectral function with a width of about 0.4 eV. We now study the reason why A4C60 is a nonmagnetic insulator. To this end, it is useful to consider the ground state of the isolated C42 60 molecule as schematically shown in Fig. 6. We begin with the origin of the nonmagnetic nature of the ground state. In Fig. 6, 3-fold degenerate t1u levels are split into 2-fold degenerate lower ones and a nondegenerate upper one; the former is occupied by four t1u electrons while the latter is unoccupied. The situation illustrated in Fig. 6 is unusual in a sense that the spins of electrons in degenerate levels are usually aligned parallel to each other due to Hund’s coupling. The reason why the ground state of A4C60 is nonmagnetic is that the Jahn –Teller effect overcomes Hund’s coupling in AxC60. In other words, the exchange interaction given by equation (6) is negative due to the coupling of the t1u electrons to the Hg intramolecular phonons. We next examine the origin of the energy gap. It should be noted that there is a large contribution to the energy gap not only of S but also of U as shown in Fig. 6; the latter contribution is exactly of the same kind as the origin of the energy gap in the Mott insulator. If we neglect the contribution of U, which is about 0.6 eV, the energy gap of A4C60 is underestimated by 60% smaller than its actual value. It is thus indispensable to take into account both U and S for understanding the origin of the energy gap of A4C60. Emphasizing this aspect, A4C60 is often called the Mott – Jahn – Teller insulator [18]. Consequently, by taking into account the effect due to T, the energy gap of A4C60 is given by U þ S 2 T: It is also instructive to reproduce the above result from another point of view. To this end, consider two C42 60 molecules far apart from each other as schematically shown in Fig. 7. To estimate the energy gap of A4C60, we calculate the energy necessary to transfer an electron from one molecule to the other [10]. In Fig. 7(a), which illustrates the situation before the transfer of an electron, the energy loss due to U is minimized because the charge distribution is uniform. At the same time, in this state, the energy gain due to S is maximized because there exists no unpaired electrons. On the other hand, in Fig. 7(b), which illustrates the situation after the transfer of an electron, the energy loss due to U is not minimized because the charge distribution is not uniform; the energy loss in Fig. 7(b) is larger by U than in Fig. 7(a). At the same time, in this state, the energy gain due to S is not maximized

Fig. 6. Ground state of the C42 60 molecule. The ground state is nonmagnetic because the Jahn – Teller effect overcomes Hund’s coupling. There is a large contribution to the energy gap not only of S but also of U.

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Fig. 7. Schematic diagram for (a) the uniformly charged state, 2C42 60 ; and (b) the charge 52 disproportionate state, C32 60 þ C60 : The former is favorable to both U and S while the latter is not favorable to either U or S. As a result, the uniformly charged state is stabilized by the cooperation between U and S.

because there exists unpaired electrons; the energy gain in Fig. 7(b) is smaller by S than in Fig. 7(a). The energy necessary to transfer an electron from one molecule to the other is thus given by U þ S: Taking into account the effect of T, we arrive at the same conclusion that the energy gap of A4C60 is given by U þ S 2 T: We can summarize the characteristics of the ground state of A4C60 as follows. The ground state is nonmagnetic because the Jahn – Teller effect overcomes Hund’s coupling. The material is also an insulator because U and S cooperatively open the energy gap, which is given by U þ S 2 T: Furthermore, the charge fluctuation is strongly suppressed due to the cooperation between U and S and consequently the ground state is properly described as an assembly of the C42 60 molecules. 3.2. A3C60 We show the results of the calculations on A3C60 in Fig. 8. The spectral function of A3C60 is shown in Fig. 8(a) and that of the isolated C32 60 molecule is shown in

Fig. 8. Calculated spectral functions of A3C60. (a) The spectral function of A3C60 and (b) that of the isolated C32 60 molecule. The origin of energy is taken at the Fermi level. The result shows that A3C60 is a metal. The continuous features spread over ^ 1 eV around the Fermi level in the spectral function of A3C60 are originated in the peaks shown in that of the isolated 32 C32 60 molecule. On the other hand, in the spectral function of the isolated C60 molecule, there are no counterparts of the satellites at about ^1.6 eV in the spectral function of A3C60.

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Fig. 8(b); here, the latter is calculated analytically. The origin of energy is taken at the Fermi level. The spectral function of A3C60 clearly shows that the material is a metal. The spectral function is considerably broad as shown in Fig. 8(a). Furthermore, the spectrum consists of two components; one is continuous features spread over ^ 1 eV around the Fermi level and the other is the satellites at about ^ 1.6 eV. Of great importance, when comparing two spectral functions given in Fig. 8(a) and (b), is the fact that there are no counterparts of the satellites in the spectral function of the isolated C32 60 molecule although there exist original peaks corresponding to the continuous features. This is the key to understand the electronic structure of A3C60. To find the reason why there exist the satellites in the spectral function of A3C60, it is first necessary to elucidate the reason why A3C60 remains metallic despite that both U and S are significantly strong. To this end, consider two C32 60 molecules far apart from each other as schematically shown in Fig. 9. In Fig. 9(a), which illustrates the situation before the transfer of an electron, the energy loss due to U is minimized because the charge distribution is uniform. However, in this state, the energy gain due to S is not maximized because there exists unpaired electrons. On the other hand, in Fig. 9(b), which illustrates the situation after the transfer of an electron, the energy loss due to U is not minimized because the charge distribution is not uniform; the energy loss in Fig. 9(b) is larger by U than in Fig. 9(a). However, in this state, the energy gain due to S is maximized because there are no unpaired electrons; the energy gain in Fig. 9(b) is larger by S than in Fig. 9(a). The energy necessary to transfer an electron from one molecule to the other is thus U 2 S, which is considerably small. As a result, the energy gap disappears in A3C60 due to the effect of T and the material remains metallic. Furthermore, the above consideration indicates that the charge fluctuation, 22 42 2C32 60 O C60 þ C60 ; occurs in A3C60. That is, these two states coexist in A3C60, resonating with each other, because the energy necessary for the transfer of an electron from the state shown in Fig. 9(a) to the state shown in Fig. 9(b) is considerably small. On the other hand, since the spectral function shown in Fig. 8(b) is that for the isolated C32 60 molecule, the component originated in the charge 42 disproportionate state, C22 60 þ C60 ; cannot be described only by considering a single

Fig. 9. Schematic diagram for (a) the uniformly charged state, 2C32 60 ; and (b) the charge 42 disproportionate state, C22 60 þ C60 : The former is favorable to U while not favorable to S. On the contrary, the latter is favorable to S while not favorable to U. As a result, these two states coexist in A3C60 due to the competition between U and S, resonating with each other.

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isolated C32 60 molecule. This is the reason why there are no counterparts of the satellites in Fig. 8(b). In other words, it is necessary to consider not only the uniformly charged state shown in Fig. 9(a) but also the charge disproportionate state shown in Fig. 9(b) as an initial state of the photoemission process. 42 We now explain that the origin of the satellites is the initial state C22 60 þ C60 : The two initial states shown in Fig. 10(a) and (b) give rise to the two final states shown in Fig. 10(c) and (d). The important point is that the final state (d) can arise only from the initial state (b) although the final state (c) can arise from both the initial states (a) and (b). That is, the origin of the satellite is the process in which an electron is 2 emitted from the C22 60 molecule in the initial state (b) and the C60 molecule is left in the final state (d). To understand this, we estimate the energy difference between the two final states; since the final state (c) gives rise to the emission at the Fermi level, the energy difference between (c) and (d) gives the position of the satellite relative to the Fermi level. First, the energy loss due to U in the final state (d) is larger by 2U than in the final state (c) because the charge imbalance is significant in the final state (d). Next, we can find by detailed calculations that the energy gain due to S in the final state (d) is smaller by S than in the final state (c). Consequently, the satellite appears at 2U þ S, which is about 1.6 eV, from the Fermi level. The satellite obtained by our calculations most likely corresponds to the small shoulder at 1.6 eV observed in the photoemission spectrum of K3C60 shown in Fig. 2. In fact, the shoulder is observed only in K3C60 but not in either K4C60 or K6C60. Furthermore, this is also the case in the photoemission spectra of RbxC60. These experimental results strongly suggest that the shoulder is intrinsic for A3C60. Finally, it is noteworthy that the continuous features around the Fermi level are originated in the excited states of the C22 60 molecule in the final state (c) shown in Fig. 10; the details are found elsewhere [12]. We can summarize the characteristics of the ground state of A3C60 as follows. The reason why A3C60 remains metallic despite that both U and S are significantly strong is that the two interactions compete in this material. Furthermore, this competition

Fig. 10. Schematic diagram for the photoemission process in A3C60. The left part shows two initial states, i.e., (a) the uniformly charged state and (b) the charge disproportionate state, and the right part shows two final states, i.e., (c) the one, which results from both the initial states and (d) the one, which results only from the charge disproportionate initial state. The origin of the satellite is the process ðbÞ ! ðdÞ:

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22 42 induces the charge fluctuation 2C32 60 O C60 þ C60 : As a result, the charge 22 42 disproportionate state C60 þ C60 gives rise to the satellite in the spectral function of A3C60. The shoulder observed in the photoemission spectrum of A3C60 most likely corresponds to the satellite obtained in the present study.

4. SUPERCONDUCTIVITY As explained in the previous section, charge fluctuation is enhanced in A3C60 while suppressed in A4C60. This means that the interaction between the t1u electrons can be attractive in the former while repulsive in the latter. Here, we investigate the superconductivity which occurs when the interaction between the t1u electrons is attractive [28]. We first show that the attractive interaction is originated in the pairtransfer interaction which is negative due to the nonadiabatic effect of S. It is then shown that this can be interpreted as a manifestation of the dynamic Jahn –Teller effect. Next, after examining possible types of the order parameter of superconductivity, we construct the BCS theory for A3C60 and study the characteristics of the superconductivity. In particular, we point out that the origin of the superconductivity is the dynamic Jahn –Teller effect when we consider it in the real space and, at the same time, the origin is the Moskalenko –Suhi –Kondo mechanism when we consider it in the wavenumber space.

4.1. Dynamic Jahn –Teller effect The dynamic Jahn – Teller effect describes the situation where the nuclei undergoes the tunneling motion between the equivalent distorted configurations of the molecule, resulting in the stabilization of the vibronic ground state of the coupled electron – phonon system. For the one-electron Jahn –Teller problem, the degeneracy of the vibronic ground state remains the same as that of the original electronic ground state. On the other hand, in the many-electron Jahn – Teller problem, the high degeneracy of the original electronic ground state is lifted by the dynamic Jahn – Teller effect, resulting in the multiplet terms. In particular, as shown below, a nondegenerate and totally symmetric vibronic ground state is realized for the C22 60 and C42 60 molecules produced by the charge fluctuation in A3C60; this type of vibronic state may be referred to as the orbital-singlet state. For simplicity, we examine the interaction between two t1u electrons, 22 comparing the energy difference between 2C2 60 and C60 þ C60 in the vanishing T limit. First, there is no interaction energy due to Hint in 2C2 60 because the two t1u electrons reside on different molecules apart from each other. On the other hand, there exists an interaction energy due to Hint in C60 þ C22 60 because the two t1u electrons reside in an equal molecule simultaneously. For example, if the two-electron state of the C22 60 molecule is lx " x # l; the interaction energy is

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given by Vintra. Since this is positive, there is only repulsive interaction between the t1u electrons as long as we consider such a state. It is, however, important to notice that the state lx " x # l is not an eigenfunction of Hint. In fact, when applying Hint to lx " x # l; there arise ly " y # l and lz " z # l due to 22 the pair-transfer interaction K. The ground state of C60 molecule is thus the most stable state among the linear combinations of these three states. This is given by 1 pffiffi ðlx " x #i þ ly " y #i þ lz " z #iÞ 3

ð8Þ

because the pair-transfer interaction K is negative. We refer to this type of state as the orbital-singlet state and this can be interpreted as a manifestation of the dynamic Jahn – Teller effect. The reason is that each of lx " x # l; ly " y # l; and lz " z # l is accompanied by the different Jahn –Teller distortion in the static case. The energy of the state (8) is given by Vintra þ 2K and this is lower than that of the state lx " x # l by 2lKl: Thus, if we define g ; 2ðVintra þ 2KÞ

ð9Þ

2 C60 þ C22 60 is more stable than 2C60 when Vintra þ 2K , 0; i.e.,

g.0

ð10Þ

This is the condition of the attractive interaction between the t1u electrons originated in the dynamic Jahn – Teller effect and, for this condition to be satisfied, S must overcome U as we can see from equations (4) and (7). Extending the above consideration to the actual superconductor A3C60, we can find the condition of the attractive interaction between the t1u electrons by 22 42 comparing the energy difference between 2C32 60 and C60 þ C60 : As a result, we obtain exactly the same condition g . 0 as the condition (10). In AxC60, since Vintra is about 0.2 eV and K is about 2 0.1 eV, the attractive interaction between the t1u electrons can arise although it is rather subtle. Thus, in our model, the negative pairtransfer interaction K stabilizes the orbital-singlet state and, consequently, the attractive interaction between the t1u electrons due to the dynamic Jahn –Teller effect arises when the pair-transfer interaction K dominates the intraorbital repulsion Vintra.

4.2. Order parameter We now study the characteristics of the order parameter of the superconductivity induced by the attractive interaction between the t1u electrons originated in the dynamic Jahn – Teller effect. To this end, we estimate the energy gain per C60 molecule due to the superconductivity by using a simple consideration of Hint. It is reasonable to expect that there can be three types of order parameter, i.e., those of the Ag, Hg, and T1g symmetries, because t1u £ t1u is reduced to the sum of these three representations. We examine the three possibilities below.

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We first study the order parameter of the Ag symmetry. This type of order parameter, PAg, is given by PAg ; kx†" x†# l ¼ ky†" y†# l ¼ kz†" z†# l

ð11Þ

This is of the spin-singlet type. In this case, the energy gain DEAg is given by

DEAg ¼ 3ðVintra þ 2KÞP2Ag ¼ 23gP2Ag

ð12Þ

The superconductivity of the Ag symmetry then gives rise to the energy gain when g . 0: This condition is exactly the same as the condition (10) and is possible to be satisfied in A3C60. We next consider the cases of the order parameters of the Hg and T1g symmetries. The order parameter of the Hg symmetry is, for example, given by 1 PHg ; kx†" x†# l ¼ ky†" y†# l ¼ 2 kz†" z†# l 2

ð13Þ

This is also of the spin-singlet type as in the case of the Ag symmetry. The energy gain due to this type of superconductivity is then given by DEHg ¼ 6ðVintra 2 KÞP2Hg

ð14Þ

The superconductivity of the Hg symmetry thus can arise if Vintra 2 K , 0: It is, however, impossible for this condition to be satisfied in A3C60 because Vintra is positive and K is negative. Also, the order parameter of the T1g symmetry is, for example, given by PT1g ; kx†" y†" l

ð15Þ

This is of the spin-triplet type. The energy gain due to this type of superconductivity is then given by DET1g ¼ ðVintra 2 3KÞP2T1g

ð16Þ

The superconductivity of the T1g symmetry thus can arise if Vintra 2 3K , 0: It is, however, also impossible for this condition to be satisfied in A3C60. 4.3. BCS theory for A3C60 As expected from the above consideration, it is most likely that the order parameter of the superconductivity in A3C60 is of the Ag symmetry and spin-singlet. We confirm this by constructing the BCS wavefunction of the Ag type, lC l; as follows: Y lC l ¼ ðuak þ vak a†k" a†2k# Þ l0l ð17Þ ak

Here, uak and vak satisfy the conditions uaRk ¼ uak

ð18Þ

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and vaRk ¼ vak

ð19Þ

for lC l to be of the Ag type, where R is any symmetry operation of the system. We next calculate the expectation value of Hint by using lC l: The expectation value consists of two parts; one is the contribution of the Hartree –Fock type and the other is the contribution of the BCS type. The Hartree –Fock contribution can be taken into account by replacing the bare one-electron energy j0ak with the Hartree – Fock one-electron energy jak : The BCS contribution is given by 1 X X kl2k2l V kCla†ks a†2k2s lC l kClb2l2s bls lC l ð20Þ kClHint lC lBCS ¼ 2 s ak;bl abab kl2k2l Here, Vabab is the coupling constant of the effective interaction expressed with the band index and the wave number. The results of the calculations show that the ground state energy is given by X ð21Þ kClH0 þ Hint lC l ¼ 2 jak v2ak 2 3NgP2Ag

ak

where PA g ¼

1 X u v 3N ak ak ak

ð22Þ

with N being the total number of the C60 molecules in the system. Also, g ¼ 2ðVintra þ 2KÞ is exactly the same as the one defined by equation (9). By minimizing the expectation value (21), we obtain the gap equation and find that the condition of the occurrence of superconductivity is g . 0: This again reproduces the same result as the condition (10). The solution of the gap equation is finally given by   1 D ¼ gPAg . 2~vc exp 2 gNF

ð23Þ

where ~vc is the cut-off energy and NF is the density of states at the Fermi level per spin per orbital. It is noteworthy that the BCS contribution (20) contains the energy gain due to the interband pair-transfer process of the Cooper pairs, which is a remarkable feature of multi-band superconductors, as well as the energy gain due to the usual intraband pair-scattering process of the Cooper pairs. That is, the energy gain in a single-band superconductor is originated only in the pair-scattering process in the single band while the energy gain in a multi-band superconductor is originated not only in this process but also in the pair-transfer process between the different bands; this further stabilizes the superconductivity. This stabilization mechanism due to the interband pair-transfer process in multi-band superconductors is known as the Moskalenko –Suhl –Kondo mechanism [32 – 34].

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It is thus reasonable to conclude that the mechanism of the superconductivity in A3C60 is the dynamic Jahn –Teller effect when we consider it in the real space while it is the Moskalenko– Suhl – Kondo mechanism when we consider it in the wavenumber space, as schematically shown in Fig 11. That is, for A3C60 the superconductivity induced by the dynamic Jahn – Teller effect is equivalent to that induced by the Moskalenko – Suhl –Kondo mechanism. Soon after the discovery of the superconductivity in A3C60, Rice et al. [23], Asai and Kawaguchi [24], and ¨ rd [25] pointed out the importance of the Moskalenko– Suhl – Kristoffel and O Kondo mechanism in the superconductivity of A3C60. This is a remarkable suggestion although, at that time, it was too early to understand a variety of behaviors of AxC60 in a unified way. Furthermore, the considerations given above indicate that the high symmetry in the vicinity of the C60 molecule is indispensable to the superconductivity in A3C60. Otherwise, it is most likely that the formation of the orbital-singlet state is suppressed because the pair-transfer interaction, which is the origin of the attractive interaction, cannot be used efficiently. This may be justified by considering the fact that the structure of most of superconducting A3C60 is fcc and also the fact that the structure of Cs3C60, which is not a superconductor under usual conditions, is not fcc. Moreover, it is also likely that the disappearance of superconductivity and the appearance of anti-ferromagnetic state in NH3A3C60 is a result of the reduction of the symmetry of the structure from fcc to a lower one.

Fig. 11. Schematic diagram for (a) the dynamic Jahn– Teller effect and (b) the Moskalenko – Suhl – Kondo mechanism. In the former, the distorted C22 60 molecule undergoes the tunneling motion between three equivalent configurations. This results in the formation of the orbitalsinglet state. In the latter, as shown by the black arrows, the Cooper pairs are transferred from one Fermi sphere to another, which is the pair-transfer process, a remarkable feature of multiband superconductors, and stabilizes the superconductivity. Also, as shown by the white arrows, the Cooper pairs are scattered coherently within each Fermi sphere, which is the pair-scattering process in usual superconductors.

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5. SUMMARY AND OPEN QUESTIONS We have investigated the reason why A2C60 and A4C60 are nonmagnetic insulators while A3C60 is a metal and even a superconductor. These are the behaviors revealed for the first time, which arise when both U and S are important in a system with the orbital degree of freedom. One of the important points is that the cooperation between U and S results in the insulating ground states of A2C60 and A4C60 while the competition between them results in the metallic ground state of A3C60 although both U and S are significantly strong. Another important point is that the superconductivity in A3C60 arises when S dominates U and the origin is the dynamic Jahn – Teller effect when we consider it in the real space while it is the Moskalenko– Suhl – Kondo mechanism when we consider it in the wave-number space. In the future study, it is necessary to resolve the following problems. First, there remains a question why A3C60 is the only superconductor among AxC60 [35]; that is, although there exists a fcc non-superconducting metal ABa2C60, which can be regarded as A5C60, the reason why this material is not a superconductor is not revealed yet [36]. Also, the study of NH3A3C60 is important to the elucidation of the reason of the disappearance of superconductivity and the appearance of antiferromagnetic state in this material [13,14]. Furthermore, there remain other problems of the electronic properties of A3C60: the extremely high electrical resistivity of about 2 mV cm [37], which is retained even at low temperatures in the normal state, the mid-infrared absorption in the optical conductivity spectra [7], the extraordinary decrease of the nuclear relaxation time with increasing temperature [38], etc. The resolution of these problems will provide us with further understanding of the electronic properties of the materials with strong electron – electron and electron – phonon interactions.

ACKNOWLEDGEMENTS We would like to thank Y. Iwasa, M. Kosaka, K. Tanigaki, H. Tou, S. Hino, and Y. Maniwa for valuable discussions. We are also grateful for the permission to reproduce Figs 1 and 2.

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