VOLUME 74, NUMBER 6 PHYSICAL REVIEW LETTERS 6 FEBRUARY 1995

Metallic Screening and Correlation Effects in Superconducting Fullerenes

P. E. I ammert

Groupe de Physico-Chimie Theorique, Ecole Superieure de Physique et de Chimie Industrielles de la Ville de Paris,F-75231 Paris Cedex 05, France

D. S. RokhsarDepartment of Physics, University of California, Berkeley, California 94720

S. Chakravarty and S. KivelsonDepartment of Physics, The University of California at Los Angeles, Los Angeles, California 90024

M. I. SalkolaTheoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

(Received 21 June 1994)

We consider the frequency depende-nt Coulomb interaction between electrons in a molecular metalin the limit in which the conduction bandwidth is much less than the plasma frequency, which inturn is much less than intramolecular excitation energies. In particular, we compute the effectiveinteractions at the Fermi energy in alkali-doped C« to second order in the screened interactions.The frequency dependence of screening substantially reduces the effects of the long-range part of theCoulomb interaction, leading to the possibility of effective attraction between electrons.

PACS numbers: 74.20.Mn, 71.27.+a, 74.70.Wz

The alkali- and alkaline-earth-doped fullerenes [1] aremolecular solids, characterized by strong intramolecu-lar electron-phonon and electron-electron interactions, butweak intermolecular hybridization. They can be metals,magnets, insulators, or even superconductors, with transi-tion temperatures as high as 33 K.

Electrons at the Fermi energy are strongly coupledto intramolecular phonons, especially the high frequencytangential modes with energy hen, = 200 meV. Severalmicroscopic calculations [2,3] estimate the BCS interac-tion due to the exchange of intramolecular phonons tobe roughly 75 meVC60. The bands at the Fermi energyare narrow (bandwidth W = 200 meV), so the electronicdensity of states at EF is correspondingly large: &(EF) =15/eV spin C6o. The dimensionless electron-phonon cou-pling constant A

—= %(EF)Vacs is then of order unity.Since the electronic bandwidth and characteristic phononenergies are comparable, the naive Born-Oppenheimer-Migdal-Eliashberg approximation is no longer justified.

Electron-electron interactions are also important in thedoped fullerenes: The typical intramolecular Coulomb en-

ergy is e /eRb = 2 eV, where Rb = 3.5 A. is the fullereneradius. Coulomb scattering in the doped fullerenes wasemphasized by Chakravarty and co-workers [3], who cal-culated the effective interaction at the Fermi energy [4] us-ing an on-site Hubbard repulsion U to model the screenedCoulomb interaction. Although the bare Hubbard interac-tion is repulsive, Chakravarty and co-workers found (us-ing both perturbation calculations [3] and exact solution ofsmall fullerene analogs [5]) that moderate Hubbard U cangenerate effective interactions at the Fermi energy whichare anomalously small or even negative (indicating at-

traction), suggesting an electronic pairing mechanism dueto intramolecular correlation effects. Much of the phe-nomenology of fullerenes can be understood in the contextof such correlation effects [3,6].

It has been argued [7,8] that the long-range part ofthe Coulomb potential generates effective interactions atthe Fermi energy that are substantially more repulsivethan those found with a simple Hubbard model. Whilethese calculations raise the important issue of long-range Coulomb effects, the extended Hubbard models ofRefs. [7] and [8] omit the crucial dynamics of metallicscreening.

In this Letter we consider frequency dependent me-tallic

screening and its effect on the energetics of pairing.Our approach relies on the separation of energy scalesfor intermolecular hopping (W), metallic screening (theDrude plasma energy hto„), and intramolecular hopping(the width D of the relevant complex of molecular states),as sketched in Fig. 1 and discussed below. Althoughcorrelation effects cannot be computed exactly, they canbe studied in a controlled manner using second-orderperturbation theory. This permits a comparison with otherperturbative calculations.

We find that a long-ranged but frequency-dependentscreened interaction strikingly reproduces the results ob-tained from the Hubbard model [3]. Frequency depen-dence is a crucial ingredient in this somewhat surprisingresult, which can be understood heuristically as follows:The direct repulsive interaction of electrons at the Fermienergy [see Fig. 2(a)] occurs at a frequency below theplasma frequency and is therefore heavily screened. Thehigher-order corrections [which can be attractive in sec-

0031-9007/95/74(6)/996(4)$06. 00 1995 The American Physical Society

VOLUME 74, NUMBER 6 PH YS ICAL REVIEW LETTERS 6 FEBRUARY 1995

emptybands

plasma frequency

1/2-filled band0at Fermi energy

I ~ R

3I I I' '-

3

filledbands

FIG. 1. The electronic excitations of metallic fullerenes arecharacterized by three energy scales: the bandwidth W due tointerfullerene hopping, the plasma frequency co„, and the widthD of the ~ complex.

(a)

(b)

I&

IIIIIIs-

(c)

ond order, Figs. 2(b) —2(e)], however, involve excitationsabove the plasma energy and are therefore unscreened.The weakening of the first-order repulsion relative to thesecond-, order attraction explains the physics of our re-sults. Similar considerations may also apply to clustercompounds known as Chevrels [9].

Frequency dependent screening. —In alkali-doped C6p,the bands at the Fermi energy are derived from a set ofthree symmetry-related molecular orbitals. These orbitals

are part of a wide 2p~ complex spread over D = 15 eV.The screening response of the metal is governed [10] bythe Drude plasma energy hen„= 1 —1.5 eV.

The separation of energy scales D ) h~„) W(see Fig. 1) suggests a simple analytical model for thescreened Coulomb interaction. Since the bandwidth Wis the smallest frequency in the problem, intermolecularhopping occurs only on long time scales. We there-fore neglect intermolecular hopping at this stage of thecalculation except insofar as it generates a polarizablemetallic environment around each fullerene. The chargefluctuations of these metallic surroundings can thenbe integrated out exactly to obtain the time-dependentscreened Coulomb interaction between electrons on asingle molecule.

Armed with this screened interaction, we can perturba-tively integrate out the ~ complex to obtain the renor-malized interaction at the Fermi energy (the "pair bindingenergy" discussed below). Note that by accurately treat-ing the frequency dependence of the metallic screening weremove any potential ambiguity [11] in the proper calcu-lation of these renormalizations. It is only when this fre-quency dependence is neglected that problems can arise,as noted in Ref. [11].

The separation between the plasma frequency andtypical intramolecular excitation energies allows us tosimplify further. All high-frequency interactions (i.e., in-terband matrix elements) will be essentially unscreened,since such transitions occur too rapidly for the metal to re-spond. Low-frequency interactions (i.e., matrix elementsbetween degenerate molecular states), however, will bemaximally screened by image charges in the surroundingmetal. We therefore approximate the screened Coulombinteraction by its high- and low-frequency parts:

V3o(r) if ru ( cd&,V r, ru

Vh; (i") 1 f ru ) co&,(1)

(d) (e)

where r is the distance between a pair of ~ electrons on agiven fullerene molecule.

The high frequency inter-action Vh; (r) is simply thedielectrically screened Coulomb potential:

eVh;(r) = —. (2)6'r

FIG. 2. The first-order contribution to the effective interac-tion between two electrons in the spin singlet, orbital singletchannel is represented by (a) direct scattering by the screenedinteraction. There are five second-order terms: (b) the un-crossed ladder, (c) the crossed ladder, (d) the vertex correction,(e) screening due to on-ball particle-hole pairs, and (f) a three-body interaction. In the second-order diagrams (b) —(f), at leastone of the thick propagators must lie away from the Fermienergy.

The dielectric constant e = 2 is determined by the polar-izability of degrees of freedom that we have left out ofour model, especially the carbon 2pa- complex.

The low frequency scree-ned interaction V3, (r) betweenelectrons on a single molecule is calculated by consideringa molecule of radius Rb at the center of a spherical cavityof radius R, inside a metal (Fig. 3). Consideration of theimage charges formed in the metal gives

1 R,V.() = ———/(g2 g2)2 + g2r2 (3)

Since the distance from the center of one moleculeto the nearest carbon nucleus of a neighboring molecule

997

VOLUME 74, NUMBER 6 PHYSICAL REVIEW LETTERS 6 FEBRUARY 1995

R II

I I I R

Ill'

&I& I &II 888 I'

8l8 I i&PP&i IW

8 !B R R III \ I II I I

I I ~ i&8 8 RIIII IIRll

R I I'

&l ' ! ! '! '" "8/iI ll I 18 dl8

RRR II &I

8&PPPPII PI&II 88

Im llll, 818 8@,IN l88 8i& RIR&888Ig

8811 Pal ul I I I III I 18&i

8 & 88 8 I 8 8 I&l 8 I 8 IR R I R I I IRR" '

*!R I . IR

~~=R-' '--'RmR IRIR I!888 88

R ~RR8R

2Rb

C

FIG. 3. We model the low-frequency effects of metallicscreening by considering a single C«molecule inside aspherical cavity of radius R, in a metal.

is 6.5 A, and the carbon 2p, orbitals extend roughly1 A from the nucleus, we estimate the effective cavityradius to be R, = 5 —6 A.. While the geometric radiusof an individual fullerene molecule is 3.5 A, the 2pvrorbitals are squeezed to the outside of the molecule by itscurvature. The effective radius Rb at which the electronsmove is therefore somewhat larger than the geometricradius of the molecule, perhaps by as much as half anangstrom.

The on-site interaction is an atomic property of thecarbon 2p orbitals and must be treated separately. For acarbon 2p orbital, the unscreened Coulomb integral Vh;(0)is roughly 7—10 eV. Metallic screening of this atomicCoulomb integral should be negligible, so for simplicitywe present here results for VP, (0) = Vh;(0)

—= V(0), whereV(0) corresponds roughly to the "Hubbard U." Equa-tions (2), (3), and the on-site interaction complete the

specification of our model for frequency dependence inthe screened Coulomb interaction of metallic fullerenes.

Calculation of pairing interaction —To cal.culate theeffective interaction arising from Eqs. (1)—(3), we startwith a simple tight-binding model for the ~ molecular or-bitals of C6p. The matrix elements t and t' correspondingto nearest-neighbor intrapentagon and interpentagon hop-ping are 2RO and 2.6 eV, respectively.

The three partially occupied molecular orbitals in thealkali-doped fullerenes transform as the t]„representationof the icosahedral group. The (unnormalized) pairedstates of C6p" favored by both the intramolecular phononand the electronic pairing mechanisms are the unique spinand orbital singlets [4]

~Ag, 5 = 0) IX [c tc,t + c~tcYt + c,tc,)]" '10&, (4)where ~0) represents the neutral state of C6o and x, y, andg label the three orthogonal t]„states.

We calculate the "pair binding energy" [3] of thesestates, which is equal [4] to the effective singlet two-bodyinteraction energy plus a small contribution from three-body terms [Fig. 2(f)]. Our calculation is equivalent tointegrating out all electronic states away from the Fermienergy to second order in the Coulomb interaction. Theresulting effective interactions at FF (as well as phonon-

500—8

~RRRI~

A =2.t A

F 300—

200— ]t00—

(/j ~0

-100—

-2004 6 8

V(R = 0)/eVfo

FIG. 4. Second-order calculations of the effective singletinteraction at the Fermi energy. The two solid lines are theresults of the present work, corresponding to cavity radii R, = 5and 6 A. The dotted line is the Hubbard model calculation ofRef. [3]. The dashed lines are the results of Goff and Phillips[8]. The horizontal axis is the on-site Coulomb repulsion, V(0),or equivalently the Hubbard U. For comparison, the phononmediated attraction VBcs is roughly 75 meV.

mediated interactions) can then be introduced in a modelwhich includes intermolecular hopping to describe thelow-energy behavior of the metal and superconductor [4].

Explicitly, we compute the Feynman diagrams shownin Fig. 2. The double-dashed lines represent the screenedCoulomb interaction of Eqs. (1)—(3); the incoming andoutgoing thin lines are propagators for the singlet pairstate Eq. (4). We sum over all intermediate states with atleast one particle or hole in orbitals away from the Fermienergy. Such propagators are represented by thick linesin Fig. 2. There are over 60000 distinct terms in thissum. In this manner, all virtual particle-hole fluctuationsinvolving the higher-lying states are included to secondorder in the interaction. Our calculation includes (a) thefirst-order scattering at the Fermi energy and (b) the usualsecond-order ladder, as well as (c) the crossed ladder,(d) screening from an intramolecular particle-hole bubble(i.e., the 6 eV 2pvr intramolecular plasmon), (e) theleading vertex correction, and (f) a small three-bodycorrection.

Figure 4 shows the second-order effective interactioncalculated using the screened Coulomb interaction dis-cussed above. The results are plotted versus the on-siteinteraction V(0). Results for cavity radii R, = 5 and6 A. are shown. For comparison, we show the second-order calculations of Chakravarty and co-workers for theHubbard model and of Goff and Phillips for the exponen-tially screened interaction V(r) = Ue ~'. In these lattertwo models the frequency dependence of screening effectsis neglected.

The effects of the long-range Coulomb interaction aresubstantially reduced by including frequency dependence,

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VOLUME 74, NUMBER 6 PH YS ICAL REVIEW LETTERS 6 FEBRUARY 1995

as seen by comparing our results with Goff and Phillips.In second-order perturbation theory, we find that theeffective interaction changes sign near V(0) of order7—8 eV, which is close to the value obtained for theHubbard model. The effective Coulomb interactions forthe Hubbard model and the present frequency-dependentmodel are quite similar except at small U, where thepresent model includes longer-range Coulomb interactionsthat yield a repulsive effective interaction even when thebare on-site interaction vanishes.

In conclusion, in a conventional metal the screenedCoulomb interaction is essentially instantaneous (i.e., fre-quency independent) over the bands of interest. Thusin the traditional ladder sum, all terms involve thescreened Coulomb interaction [11]. In molecular metalslike the doped fullerenes, however, second- and higher-order terms are not screened, since they involve virtuallyexcited states well above the plasma frequency. Thesehigher-order terms are therefore stronger (i.e., more at-tractive) than one might naively expect. Screening doesreduce the (repulsive) first-order term, but not the (attrac-tive) second-order terms.

We stress that strong intramolecular correlation effectsare necessarily present in a molecular metal such as C6o.We have shown that a combination of intermolecularscreening and intramolecular correlations dramatically re-duces the Coulomb repulsion between electrons. Such areduction is necessary for a consistent theory of phonon-mediated superconductivity in the fullerenes in the pres-ence of strong electron-electron repulsion. To the extentthat second-order perturbation theory is valid, frequency-dependent screening may even result in an effective pairbinding arising purely from electronic correlation effects.

This work was supported by the NSF under PYIGrant No. DMR-91-57414 (P.E.L. and D.S.R.), Grant No.DMR-92-20416 (S.C.), and Grant No. DMR-93-12606

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[4]

[5]

[6]

[8]

[9]

[10]

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For a review see A. F. Hebard, Phys. Today 45, 26 (1992).C. M. Varma, J. Zaanen, and K. Raghavachari, Science254, 989 (1991);K. H. Johnson, M. E. McHenry, and D. P.Clougherty, Physica (Amsterdam) 183C, 319 (1991);M. Schluter et al. , Phys. Rev. Lett. 6S, 526 (1992);R. A. Jishi and M. S. Dresselhaus, Phys. Rev. B 45, 2597(1992); D. M. Deaven and D. S. Rokhsar, Phys. Rev. B4$, 4114 (1993).S. Chakravarty and S. Kivelson, Europhys. Lett. 8, 751(1991); S. Chakravarty, M. P. Gelfand, and S. Kivelson,Science 254, 970 (1991).P. E. Lammert and D. S. Rokhsar, Phys. Rev. B 48, 4103(1993).S.R. White, S. Chakravarty, M. P. Gelfand, and S.A.Kivelson, Phys. Rev. B 45, 5062 (1992). These smallanalogs, however, can have orbitally degenerate pairground states, unlike that expected for C6o (see Ref. [4]).M. Salkola, S. Chakravarty, and S.A. Kivelson, Int. J.Mod. Phys. B 7, 2859 (1993); M. Salkola, Phys. Rev. B49, 4407 (1994); S. Chakravarty, S. Kivelson, M. Salkola,and S. Tewari, Science 256, 1306 (1992).G. Murthy and A. Auerbach, Phys. Rev. B 46, 331 (1992).W. Goff and P. Phillips, Phys. Rev. B 46, 603 (1992); 4S,3491 (1993).Superconductivity in Ternary Compounds I, edited byO. Fischer and M. B. Maple (Springer-Verlag, New York,1982).Intramolecular plasmonlike excitations at 6 eV of the2pm orbitals will be explicitly taken into account below[Fig. 2(e)]. Screening due to 2p o. orbitals can beabsorbed into a frequency independ-ent dielectric constantbecause the characteristic energy for such excitations islarger than D.O. Gunnarsson and G. Zwicknagel, Phys. Rev. Lett. 69,957 (1992).

(S.K.). M.I.S. was supported in part by the U.S. Depart-ment of Energy.