Pairing energy of hole-doped C60

Pairing energy of hole-doped C60

3 April 1995 PHYSICS ELSEVIER LETTERS A Physics Letters A 199 (1995) 391-394 Pairing energy of hole-doped c(j() G.P, Zhanga,b, R.T. Fuavb, X. Sun...

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3 April 1995 PHYSICS




Physics Letters A 199 (1995) 391-394

Pairing energy of hole-doped c(j() G.P, Zhanga,b, R.T. Fuavb, X. Sunavb, K.H. Lee”, T.Y. Park” a T.D. Lee Laboratory and Department of Physics, Fudan University, Shanghai 200433. China h National Laboratory of Infrared Physics, Academia Sinica, Shanghai 200083, China c Departments of Chemistry and Physics Education, Won Kwang University, Iri 570-749. South Korea

Received 8 February 1995; accepted for publication 15 February 1995 Communicated by L.J. Sham

Abstract The pair-binding energy of hole-doped Cm is calculated by the Gutzwiller scheme. It is found that the pairing energy of hole-doped Cm is higher than that of electron-doped Ca. The implication of this difference for superconductivity is discussed.

1. Introduction The superconductivity of alkali metal doped fullerene A&60, such as Rb#& [ 1 I, K&O [ 21 has attracted the attention of many research works. Although it is still controversial, an increasing number of experiments support the phonon-mediated mechanism [ 3-51. A novel approach to the mechanism has been put forward [ 61. In Ref. [ 61 the possible relation between the transition temperature and the lattice constant has been studied and some suggestions to improve the transition temperature have been given. But in comparison to high T, oxide superconductors, the transition temperature of A3C60, such as Tc = 15 K in K3C60 [l] and T, = 30 K in Rb3C60 [2], is much lower. Up to now almost all superconductors originated from C60 are electron-doped materials since the first consideration for superconductivity is that C60 has high electron affinity [7], EA = 2.65 eV, and can take up to six electrons. However, recently it was reported that iodine-doped C60, which is a hole-doped fullerene, can have a higher transition temperature [8]. This surprising result strongly motivates us to

Elsevier Science B.V. SSDI 03759601(95)00160-3

pursue a theoretical explanation. In this paper we employ the Gutzwiller variational scheme to study the correlation effect of hole-doped C60. Although some small clusters can be well described by perturbation theory, the perturbation predictions for large clusters, such as Cm, are not reliable, and the Gutzwiller scheme has been demonstrated to be successful in predicting the bond structure in Cc0 and the MX chain [9]. For convenience, we adopted the Chakravarty, Gelfand, and Kivelson [3] (CGK) convention for the pair-binding energy, E$r = EN+I f EN-I - ZEN, where EN is the ground-state energy of an isolated C60 molecule with N additional electrons (holes). Ep”r measures the tendency toward pair formation at a doping of N electrons (holes) per fullerene. Hereafter we abbreviate pair-binding energy as pairing energy. With the Gutzwiller scheme, we find that: (i) In comparison to electron-doped C60, holedoped C60 has higher pairing energy, i.e., in the holedoped case Eitir = -0.075 eV while in electron-doped C60 qair = -0.053 eV, which implies that hole-doped C60 can have higher transition temperature.


G. P. Zhang et al. /Physics

(ii) EPR experiments disproportional reaction

can determine

whether the

C& + C& --+ C60 + C;o’


occurs in solution or not. Based on our theoretical result if ( 1) occurs, the on-site U must be smaller than U, = 3.0 eV. Together with EPR experiments for electron-doped COO,we can find the region for the onsite repulsion parameter II within 2.0 < U < 3.0 eV, This kind of measurement of on-site U by EPR may be practical and very direct. However, as we know, there is not such EPR test yet available. This paper is arranged as follows. In Section 2 we briefly describe the Gutzwiller scheme while the main results and discussions are presented in Section 3.

Letters A I99 (I 995) 391-394

Gutzwiller variational approach to C60 has been carried out by many authors. Joyes et al. [ 141 examined the stability of SDW and CDW, but ignored the electron-phonon interaction. Scherrer et al. [ 1.51 investigated the effects of electron interaction on the electron-lattice coupling in C60. Metal-insulator transitions and magnetism in C60 have been studied by Lu [ 161 and Sheng et al. [ 171, respectively. Recently electronic correlation effects have been calculated by Krivnov et al. [ 181. However, none of the studies above noticed a possible higher transition temperature in hole-doped C60. We address this problem here using the Gutzwiller variational scheme, and our variational treatment for C60 is traditional. First we construct the variational wave function as [ 91 (3)

Ifi) = e%o)3 2. Scheme Originally the Gutzwiller method [ 101 was proposed for a bulk system but, in various works by comparison to exact solutions, it has been verified that it can be extended to finite-size systems with reasonable accuracy [ 111. In C60 there are sixty 7~ electrons on the spherical surface. Band theory calculation showed that the interball electron hopping is very small, thus the band width is very narrow. Fitting ab initio results, Chen et al. [ 121 found interball hopping typically at 0.01 eV, which is much smaller than intraball hopping at 1.8 eV [ 131. Therefore the bulk properties of C60 are well characterized by on-ball features. Hence we can write down the Hamiltonian explicitly as A= - Ctii(Clreis, i.j,u

+ h.c.) +

where S = -iv xi niTnil and I&) is the uncorrelated Fermi sea in the Hartree-Fock space. 77 is the variational parameter, which is determined by minimizing the electronic energy, (4) where He is the electron part in the Hamiltonian (i.e. the first two terms). Actually E,cannot be calculated exactly. Using the linked-cluster technique and expanding es to the second order in v, we can clearly rewrite the electronic energy as



(v+oIHel~o)c - ;d~ol~~~He~l~o/o)c

+ ~TJ2(1/101{@Y {wf&})l~oL


(2) i,i where C,: creates a r electron at site i with spin u and nrr = C$Ci,, tij is the hopping integral between the nearest neighbor (NN) atoms at ri and rj. tij = to cu( Jri - tjl) while to is the average hopping constant. U is the on-site electronic repulsion. The third term in (2) is the lattice elastic energy. Ref. [ 131 gives for the above parameters to = 1.8 eV, CY= 3.5 eV/8, and K = 15.0 eV/A2, by fitting the optical energy gap and the two kinds of bond length.


where{A,B} =AB+BA,w=zin~rnil,andthesubscript of ( )c means that only the connected diagrams have to be taken into account. Clearly one should be very careful about the actual value of 17. The validity of the expansion depends on 17, namely q should be smaller than one. Hence the electronic energy is &=Tl+Q



where the explicit expressions for Tl ,VI,T2, V2,T3, Vs can be found in the papers on electron-doped C6a [ 91.

G.P. Zhang et al. /Physics


Letters A 199 (1995) 391-394 0.045



~ -



0.015 0.45 r 0.30

,A -0.045

0.1 5

0.00 i I




ON-SITE Fig. 1. Variational





Ee+ EL,

Fig. 2. Comparison electron-doped Cm.

7) versus II.


where EL refers to the elastic energy. By minimizing the total energy with respect to v and the bond length, the real total energy and the bond lengths can be determined.

3. Results and discussions Firstly, in order to test the reliability of our variational scheme, we show v versus U in Fig. 1. Obviously if U < 5 eV, 77 is much smaller than one. Thus our expansion is reasonable for U < 5 eV. Our aim is to determine which state is more favorable: two additional electrons (holes) go to one C60 leaving the other Crju undoped, or each C60 gets one electron (hole). Then the energy difference of these two states E+‘) = ( E2 + Par

Eo)- 2E,

-0.075 L 0.0



is the measure of pairing. Obviously, if E”p$r’becomes negative, the electrons (holes) are paired. The numerical results of E”,$’ for electron- and hole-doped C60 are plotted in Fig. 2. It is very clear that there are many similarities between electron- and hole-doped C60. (a) Pairing does appear if the on-site U is smaller than (I,, but different kinds of doping have different values of UC,namely, UC= 2.0 eV for electron-doped C60 while UC = 3.0ev for hole-doped C60.


//’ ;

’ ”


/ ,,’ J I




Therefore the total energy of the system can be expressed as E(Tota1)

/ / _,






U (eV)




-B W



z y2 -0.015


of the pairing



U (eV) energy

for hole-doped


(b) The pairing energy for both kinds of doping is reduced by increasing the on-site U.Thusthe electronelectron interaction breaks the pairing. (c) The electron-phonon interaction alone (i.e. U = 0.0eV) can induce pairing. At U = 0.0eV, the pairing energy for electron doping I&, = 0.053 eV, which is smaller than the hole doping Eitir = 0.075 eV. This difference indicates that the condensation energy wh of the hole doping case is likely to be greater than that of electron doping. Since Tca exp ( - 1/ wh) , one can expect that Tccan be higher in hole-doped C60, which is exactly what the experiments reveal. Our theory shows that the on-site interaction U alone on a ball of C6a cannot induce the so-called pureelectron attraction although a perturbative treatment predicted this attraction as possible. But EPR tests disproved that prediction. Furthermore, we propose that EPR experiments can help us to determine the on-site U.Namely if EPR experiments confirm that a disproportional reaction such as C~o+C~o -+ C6O+Ci0+ does not occur, the on-site U must lie in the region 2.0-3.0 eV. However, to the best of our knowledge, there is no EPR test available for hole-doped C60. We strongly suggest that these experiments should be performed immediately. Besides the static study of the disproportional reaction, it is suggested that the photoexcited EPR test of the disproportional reaction, which means performing EPR while the system is photoexcited, is a very powerful tool to detect the variation of on-site U in excited states, from which one may find the variation of the electron affinity in excited states, which is expected to give much information on photoinduced charge transfer in C60 as well as pho-


G.P. Zhang et al. /Physics

tosynthesis. Both theoretically and experimentally the disproportional reaction looks very intriguing. Finally we point out that this disproportional reaction is very similar to Marcus’s earlier suggestion for the isotopemarked reaction in solution [ 191.

Acknowledgement This work is supported by the National Natural Science Foundation of China, Advanced Material Committee of China and KOSEF 94 080011013.

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C 227

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