The isotope effect in the doped-fullerene superconductors

Solid State Communications,

Vol. 98, No. 12, pp. 1101-l 104, 1996 Copyright 0 1996 Else&r Science Ltd Printed in Great Britain. All rights reserved 003&109X/96 $12.00 + .OO

Pergamon

0038-1098(95)00724-5

THE ISOTOPE EFFECT IN THE DOPED-FULLERENE

SUPERCONDUCTORS

I.M. Tanga*b and P. Winotaia aFaculty of Science, Mahidol University, Bangkok 10400, Thailand bInstitute of Science & Technology for Research & Development, Mahidol University, Salaya Campus, Nakorn Patom 73170, Thailand (Received 19 September 1995 by C.N.R. Rao)

Measurements [C.C. Chen and CM. Lieber, Science 259,655 (1993)] of the isotope effect in Rb3C60 superconductors exhibit a larger shift in the T, ofa Rb3(‘2C60)o.s(13C60)o.5su Perconductor than expected on the basis of existing theories. It is proposed that the transfer of intermolecular H, vibrations (vibron) from one C6Ocluster to another in a single cluster type compound Rb3( 13Co,st2Co,s)60,is different from that in the two cluster type compound, Rb3( 12C60)o,s(13C60)o.s.This difference leads to a reduction of the electron-vibron coupling constant for the latter superconductor. The T,s of mixed Rb3( 12Cso)l_c( ‘3C60)c sunerconductors are calculated for c < 0.5. Copyright 0 1996 Elsevier Science Ltd

THE DISCOVERY of superconductivity in the doped fullerenes [l] has created much excitement in the field since pair condensation in the fullerene was so unexpected. Their T, s (that of cS3c60 being 40 K [2]) are high enough to allow them to be classified as high T, superconductors, except that they do not have the layer structure which is thought to be all important to the phenomenon of high T, superconductivity. Just as was the case for the copper oxide high T, superconductors, exotic pairing mechanisms have been proposed [3,4]. The discoveries of a large isotope effect in Rb3Cso [5, 61 point to the electron-phonon interaction as being the dominant mechanism for superconductivity in the doped fullerene. It has been established, both experimentally [7-91 and theoretically [lo], that the active phonons are the intramolecular H, vibration modes (vibrons). Manini et al. [ 1l] recently studied the electron-vibron interaction in the fullerene and showed that an enhancement of the electronic pairing interaction would occur. Chen and Lieber [12] have fabricated a series of Rb3C60 superconductors in which none, 55% or all of the ‘*C ions within a fullerene cluster were replaced by 13C ions. The T,s of these superconductors, Rb3 ‘*C60, Rb3( 13Co,55‘2c0.45)60 and Rb3 ‘3C60 were 29.6 K, 29.1 K and 28.9 K with an error margin of fO.l K, respectively. These T,s yielded an isotope

effect exponent a of 0.30 f 0.05 [T, 0; rnma (where m was taken to be the average mass of the carbon ion within a cluster)]. This exponent value is close to the value 0.37 obtained earlier by Ramirez et al. [6]. For a Rb3C60 superconductor in which 50% of the fullerene clusters were 13C60and the other 50% were ‘*C6O, Chen and Lieber found a T, of 28.7K instead of 29.2 K, the value expected from an m-o.3 dependence of T,. By simultaneously fitting the McMillan formulas [ 131 for T, and a to Chen and Lieber data, values of the coupling strength in the two fullerenes, Rb3 (13c60)0.5(12c80)0.5

and

Rb3(12Co.s

13c0.5)60,

can

be

obtained. Assuming that the intramolecular vibrations varies as m-o.5, the average frequency in the double cluster type (former) superconductor would be 1373cm-’ (m = 12.48amu) while in the single cluster type (latter) superconductor, it would be 1372cm-’ (m = 12.5amu). Using the upper value of a (0.35) and the average frequency value 1372cm-’ in the McMillan formulas for T, and a, we find that the simultaneous fitting of the two formulas are achieved for the Rb3(‘3C60)o,s(12C60)o.s when ,&, v) = 0.719 and superconductor p* = 0.161 and for the Rb3(‘2Co.s r(3Co.5)60superconductor when Xel_v= 0.724 and pL*= 0.162. This anomalous change in the electron-vibron coupling

1101

ISOTOPE EFFECT IN DOPED-FULLERENE

1102

strength cannot be explained within the BCS theory nor within the electronic pairing theory for superconductivity in the fullerene. Deaven and Rokhsar [14] proved that the standard el-ph coupling constant is independent of isotopic disorder. Their strong coupling calculations of T,, of the fullerene superconductors did not show any anomalous dependence on the isotopic substitution. Lammert and Rokhsar [ 151 showed that the repulsion induced electronic pairing mechanism [3] could not account for the anomalous drop in the T,. In this paper, we present a viewpoint which could account for the change in the value of the el-ph coupling strength. Chen and Lieber had wondered whether the differences in the matrix elements corresponding to hopping on (‘3C0.s ‘2Cs.5)60vs on ‘3C6fl and ‘%& clusters would give rise to significant variations in the T, s. We believe that this may indeed be the case. To understand why, we first note that the coherence length of 26A [16] makes it highly likely that the el- I/pairing is the result of joint scattering by carbon ions in separate clusters. In this scenario, an electron scatters one of the carbon ions in one cluster, causing the ion to vibrate; the vibration or vibron hops to another cluster where it causes a carbon ion in the new cluster to vibrate. This carbon ion, in turn, scatters a second electron. In the presence of 13C6a clusters in the ‘*&, host crystal, the moving excitation (the host cluster’s H, vibrational energy) or vibron [17] experiences a perturbation whenever it hops onto a randomly placed ‘3C60 cluster. This leads to a renormalization of the vibron propagator. The present author [18] showed that the one-vertex correction to the exciton propagator (the vibron being a type of exciton) in an isotopically mixed (napthalene hs/napthalene ds) crystal when averaged over random positioning of the two types of cluster is given by

c =

CA/{ 1 - (1 - c)A&(E)},

one-vertex

(1)

where c is the concentration of the guest molecules (in the present case these are the 13Ce0clusters) in the host crystal (12C6scrystal); o,(E), the bare vibron propagator and where A is the difference in the energies of the intramolecular vibration, i.e., { 1 - (M/M*)“*}hw,

(2)

with M(M*) being the mass of ‘*C(13C) and w0 being the frequency of the Hs(8) mode in the 12C6scluster (taken to be 1400cm-‘). The self energy correction, equation (2), was obtained by first calculating self

SUPERCONDUCTORS

Vol. 98, No. 12

(4

0

f’

/

/’

.*-------

A \

D(p-p’, ion-io,)

\\

\

\ \

I’

f3

\ \

/’

I

’ r3 92

gl

Fig. 1. (a) Graphical representation of the exchange of the H, intramolecular excitation vibron in a ‘*CbO cluster to another ‘*C6s cluster after the exciton has been renormalized by the scattering by randomly placed ‘3C60cluster. C is the renormalization correction of the exciton (vibron) propagator due to scattering from a 13C6,,cluster due to the difference in the excitation energy of the H, modes in the two different clusters. The factor W(E,i,, En c8)) appearing at the second vertex is given by equition (3) in the text. (b) Graphical representation of the correction to the electron propagator due to the electron-vibron interaction. D[p - p’, i(q - wm)] is the vibron propagator and G(p, iwm) is the electron propagator (a 2 x 2 matrix). consistently the weights of each one vertex graph and then summing all the graphs. The energy of the excitation differs from the H, vibrational energy by this amount. The absorption of a vibron by a 13Cion in a ‘3C60 cluster would have to take into account the off resonance energy of the vibron. This can be done by including a resonance absorption factor W(E, ER) in the definition of g2(p’ - p), the coupling constant at the second vertex in the diagram appearing in Fig. 1. The usual form of W(E, ER) is

r* WE,

EfJ

=

(E

_ ER)2

+ r2

’

(3)

where ER is the energy of the H, mode in the absorbing cluster, E is the energy of the exciton being transferred and I is the half width of the absorption peak. The full analytical expression for the electron self energy correction (Fig. 1) due to the coupling with the

Vol. 98, No. 12 ISOTOPE EFFECT IN DOPED-FULLERENE

29

I

I

0.4

Concentration 13C60

Fig. 2. Decrease in the T,s of Rb(12C60)l_c(‘3C60)c for different concentration of 13Ce0 clusters in the host ‘2C60crystal. (0) are the T, s expected from an m-o.35 dependence (result if &, is not affected by the isotopic substitution). (x) are the T,s if the electronvibron coupling is reduced by W(Evibron, En,). H, modes is given by

WIEvibronI x g, (P

E(H -

73%‘,

g ‘2’)1

P’MP

-

hJ~3gdp’

P’,

hl

-

hn)

-P)>

(4)

where D(. . .) is the vibron propagator; G(. . .), the 2 x 2 matrix electron propagator in the superconducting phase; A(. . .), the proper Coulomb vertex; E(. . .), the Coulomb dielectric constant and gi(. .) is the electron-vibron matrix element at the ith vertex. Adding the corrections due to the Coulomb repulsion and following standard methods [20], the Eliashberg equations are obtained. These equations then lead to the following strong coupling expression for T, 1.04( 1 + X&J Xe,_v ~ p*( 1 + 0.62X&

’

(5)

where the Xrlpr,is given by 3c

X&, = 2 0

dw a2F(w) ~

(6)

W

with the spectral distribution function cr’F(w) being given by

c&(w) = _!f$g , (p

-

/I’)

W(&ibronE&

1103

(4) is the center of the exciton band. Equation (6) is the same as McMillan’s expression except for some redefinitions, the major one being that the excitonvibron coupling strength includes the resonant absorption factor W(. . .). As mentioned in the beginning, a simultaneous fit of the observed T, s and isotope effect exponent [lo] to the McMillan’s formula for Tc and a yields for the Rb3 ( ‘3C60)o,s( 12C60)o,s superconductor, the values Xrl_ph= 0.719 and p* = 0.161 and for the Rb3(‘2Co.s 13Co,5)60fullerene superconductor, the values Xelpph= 0.724 and p* = 0.162. cl* can also be calculated from

TE

0.2

SUPERCONDUCTORS

p* = uJ{l

f Uoln(EF/%ib&),

(8)

with U, being equal to N(E,)V,, (V,, being the screened Coulomb potential). The value of Xel_v (0.724) is in the range of values (0.3-0.9) calculated in [lo], while the value (0.162) of 1_1* is close to the one obtained by Ramirez et al. [6] (0.15 obtained from a fitting the McMillan formulas for T, and a to the data for the Rb3Ch0 superconductor). Recently, Ramirez et al. [21] showed that the set of values (obtained in [6]) were in the range of values required to account for the specific heat jump at T, of the A3Cb0 superconductors. The drop in the value of Xel_v of 0.7% in the double cluster type superconductor Rb3(‘3C60)o.s (‘2C60)o.s in our approach is due the non-resonant absorption of the vibron, i.e., the fact that (Evibron- ER)2 term in W(E, E,), equation (3), is not equal to zero. The difference E - ER is given by equation (l), the one vertex correction. We shall, however, assume that E - ER = CA, the result if only the first graph in the series of one vertex graphs is kept. Using this (with c = 0.5) along with the value of the drop, in equation (3) I’ can be determined (the value being 3.3 x lo* cm-‘). We have calculated the T,s of Rb(‘2C60)l_c(13C60)c expected from T, m-o.35 (consequence of a constant Xel_J and the T,s expected when the electron-vibron coupling strength is reduced by the resonant absorption factor W(&ibron, EH,). The results are shown in Fig. 2. As is seen. the T,s obtained on the basis of a changing Xel_v deviates more from the T,s predicted by an m-o.35 dependence as the concentration of the guest 13Cb20 clusters becomes larger. It would be interesting to have the necessary experiments done and see whether this happens.

)gz

REFERENCES

x (P’- PMP - P’,w),

(7)

1.

where B(q, w) is the spectral weight function of the vibron. The hw appearing in the prefactor of equation

2.

A.F. Hebard et al., Nature 350, 600 (1991); K. Holczer et al., Science 252, 1154 (1991). T.T.M. Palstra et al., Solid State Commun. 93, 327 (1995).

ISOTOPE EFFECT IN DOPED-FULLERENE

1104 3. 4.

S. Chakravarty, M.G. Gelfand & S. Kivelson, Science 254, 970 (1991). G. Baskaran & E. Tosatti, C’urr. Sci. 61, 33 (1991). T.W. Ebbesen et al., Nature 355, 620 (1992). A.P. Ramirez et al., Phys. Rev. Lett. 68, 1058 (1992). K. Prassides et al., Nature 354, 462 (1991). P. Zhou et al., Phys. Rev. B45, 10,838 (1992). T.W. Ebbesen et al., Physica C203, 163 (1992). CM. Varma, J. Zaanen & K. Raghavacchari, Science 254, 989 (1991). N. Manini, E. Tosatti &A. Auerbach, Phys. Rev. B49, 13,008 (1994); A. Auerbach, N. Manini & E. Tosatta, Phys. Rev. B49, 12,998 (1994). CC. Chen & C.M. Lieber, Science 259, 655

2: 7. 8. 9. 10. 11. 12.

(1993).

13. 14. 15. 16. 17.

SUPERCONDUCTORS

Vol. 98, No. 12

W.L. McMillan, Phys. Rev. 167,331 (1968). D.M. Deaven & D.S. Rokhsar, Phys. Rev. JW3, 4114 (1993). P.E. Lammert & D.S. Rokhsar, Phys. Rev. B48, 1310 (1993). K. Holczer et al., Phys. Rev. Lett. 67,271 (1991). P.N. Prasad & R. Kopelman, J. Chem. Phys. 58, 126 (1973).

18. 19. 20.

I.M. Tang, A. Sirianunpiboon & A. Vittaya, Mol. Cryst. Liq. Cryst. 101, 185 (1983). See for instance: D.J. Scalapino, in Superconductivity (Edited by R.D. Parks), Chapter 10. M. Dekker, New York (1969). A.P. Ramirez et al., Phys. Rev. Lett. 69, 1687 (1992).