The planar oxygen isotope effect coefficient measured as a function of hole doping in the Pr- and La-doped YBa2Cu3O7 (YBCO) and the Ni-doped La1.85Sr0.15CuO4 (LSCO) superconductors quantitatively and qualitatively follows the form originally proposed by Kresin and Wolf [Phys. Rev.B 49, 3652 (1994)], which was derived for polarons perpendicular to the superconducting planes. Interestingly, the inverse oxygen isotope effect coefficient at the pseudogap temperature also obeys the same formula. These findings allow the conclusion that the superconductivity in YBCO and LSCO results from polarons or rather bipolarons in the CuO2 plane. The original formula, proposed for the perpendicular direction only, is obviously more generally valid and accounts for the superconductivity in the CuO2 plane

Müller, K.

Weyeneth, S.

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J Supercond Nov Magn (2011) 24: 1235–1239DOI 10.1007/s10948-011-1151-3L E T T E ROxygen Isotope Effect in Cuprates Results from Polaron-inducedSuperconductivityS. Weyeneth · K.A. MüllerReceived: 10 February 2011 / Accepted: 14 February 2011 / Published online: 5 March 2011© Springer Science+Business Media, LLC 2011Abstract The planar oxygen isotope effect coefficient mea-sured as a function of hole doping in the Pr- and La-dopedYBa2Cu3O7 (YBCO) and the Ni-doped La1.85Sr0.15CuO4(LSCO) superconductors quantitatively and qualitativelyfollows the form originally proposed by Kresin and Wolf[Phys. Rev. B 49, 3652 (1994)], which was derived for po-larons perpendicular to the superconducting planes. Inter-estingly, the inverse oxygen isotope effect coefficient at thepseudogap temperature also obeys the same formula. Thesefindings allow the conclusion that the superconductivity inYBCO and LSCO results from polarons or rather bipolaronsin the CuO2 plane. The original formula, proposed for theperpendicular direction only, is obviously more generallyvalid and accounts for the superconductivity in the CuO2planes.Keywords Cuprates · High-temperaturesuperconductivity · Polarons · Oxygen isotope effectCopper oxides are the only compounds that show supercon-ductivity at Tc above the boiling point of nitrogen. When notdoped, these materials are antiferromagnetic insulators dueto the strong correlation splitting of the dx2−y2 -type bandinto an upper empty and a lower occupied Mott–Hubbardband. When the latter is doped and holes are present, veryhigh Tc’s are found. A two-decade-long controversy resultedfrom the question as to whether the superconducting phe-nomenon still results from the electronic correlations presentS. Weyeneth () · K.A. MüllerPhysik-Institut der Universität Zürich, Winterthurerstrasse 190,8057 Zürich, Switzerlande-mail: wstephen@physik.uzh.chor whether lattice dynamics play a role. In favor of the lat-ter, Kresin and Wolf wrote a colloquium [1] in which theyreview couplings from an extended lattice to phonons, plas-mons, exitons and polarons, both theoretically and experi-mentally. For the case of polarons, i.e., a local vibronic lat-tice deformation, they quote a formula for the oxygen iso-tope coefficient α(n), originally derived by them [1, 2] in1994 for lattice distortions along the c-direction:α(n,Tc) = γ (n) nTc(n)∂Tc(n)∂n. (1)Here n denotes the superconducting carrier density. The pa-rameter γ (n) entering (1) depends only weakly on n.Equation (1) resembles the formula proposed earlier bySchneider and Keller, derived by analysing the universalproporties of cuprates [3, 4]. Already there, an importantingredient for mapping the experimentally obtained α(n)was the detailed Tc(n) dependence. In the original report byKresin and Wolf [2], the applicability of the model is dis-cussed on the basis of the early data for α(n). The result oftheir fitting of (1) to the data yielded γ = 0.13. However, itis not really discussed how and in which manner the gradientof Tc(n) is incorporated into (1) during fitting.Experimentally, it was observed by Zech et al. [5] thatsite-selective oxygen substitution 16O → 18O in optimallydoped YBa2Cu3O7 (YBCO) results in more than 80% ofthe oxygen isotope effects being due to the planar oxygen.This is because the hole density present was shown to re-side mainly in the planar CuO2 [6]. Accordingly, the car-rier density is mainly planar n‖. Note that for the perpen-dicular direction n⊥ = 0 and therefore also α⊥ = 0. Vari-ous groups [7–10] carefully studied α||(n) for YBCO anddoped La1.85Sr0.15CuO4 (LSCO) as shown in Fig. 1, takenfrom Ref. [11]. In it, α‖(n) is compared with the resultsof a vibronic theory. The latter comprises two electronic1236 J Supercond Nov Magn (2011) 24: 1235–1239Fig. 1 (Color online) The oxygen isotope effect exponent α as a func-tion of Tc/Tc,max. The black stars refer to the calculated α when onlythe nearest-neighbor hopping integral (t1) is renormalized. The purplestars are theoretically derived, with both the second-nearest-neighborand the interplanar hopping integrals (t2, t4) renormalized. Red, blueand green data are experimental values for α for Y1−xPrxBa2Cu3O7−δ[7, 8], YBa2−xLaxCu3O7−δ [9], and La1.85Sr0.15Cu1−xNixO4 [10],respectively. The figure is taken from Ref. [11]bands present in the cuprates, a lower t–J -type d-band anda nearby, higher-lying s-band coupled by a linear vibronicterm [6, 11, 12]. Within this model, the oxygen isotopeeffects on Tc result from considering in-plane bands withfirst t1, second t2, third neighbor t3, and interplanar hop-ping integrals t4 [6, 11]. The polaronic renormalizations ofthe second-nearest-neighbor hopping integral t2 and the in-terplanar hopping term t4 yield the correct trend for α‖(n)close to optimal doping. From Fig. 1 it can be seen thatthe agreement is rather good at and below optimal doping,but clearly deviates for very low doping. Being based ona mean-field theory, this may be expected because at verylow doping individual polarons exist. A result of that theoryis that the local coupling mode is of Jahn–Teller-type t2/t4motion. From this we may infer that for a polaronic scenariothe formal α(n) [see (1)] may also be valid for the planarpolarons. Thus we identify in (1) n = n‖ and α(n) = α‖(n),as described in the next paragraph.The dependence of Tc(n) can be parameterized accordingto early work [13–15] asTc(n)Tc,max= 1 − A(nnmax− 1)2. (2)Tc,max(n) denotes the maximum transition temperature of agiven family, nmax the corresponding carrier density, and Aan empirical constant found to be ≈2–3 [13–15]. In the insetof Fig. 2 the dependence in (1) is presented. Combining (1)and (2), we obtain for the generic formula of α‖(n) in termsof z = n/nmax and t = Tc/Tc,max:α‖(z) = 2γ (n) z(z − 1)(z − 1)2 − A−1 (3)J Supercond Nov Magn (2011) 24: 1235–1239 1237Fig. 2 (Color online) The same data for α as in Fig. 1 as a function ofTc/Tc,max for Y1−xPrxBa2Cu3O7−δ [7, 8], Y1−xPrxBa2Cu3O7−δ [9],and La1.85Sr0.15Cu1−xNixO4 [10]. Clearly, (3) describes the universaldependence of α rather well. We can derive a value of γ = 0.146(8)from the fit. The inset shows the doping-dependent Tc according to (3)and (4) using the fitted A = 2.5(3)and within the underdoped region:z = 1 −√1 − tA. (4)This result can be compared with experimental data forα‖(n) from the literature. In Fig. 2, we depict data for α‖(n)as a function of Tc/Tc,max for Y1−xPrxBa2Cu3O7−δ [7, 8],for YBa2−xLaxCu3O7−δ [9], and for La1.85Sr0.15Cu1−xNixO4 [10]. Clearly, (3) describes the universal dependenceof the measured α‖(n) rather well, assuming γ (n) to be con-stant. By fitting (3) and (4) to the data, we derive a valueof A = 2.5(3), in good agreement with literature values[13–15], and γ = 0.146(8), slightly larger than the valueof γ = 0.13 reported in Ref. [2] based on early results forα(n).1 Globally, this calculation is similar to those previ-ously discussed in Refs. [2] and [3].21Note that for this analysis newer experimental data were used, ob-tained from a larger range of doping, in combination with the earlydata analyzed in Ref. [2].2Note that an obvious difference in the calculation presented inRefs. [2] and [3] and the work presented here is the functional formof Tc(n) used for calculation. The generality of this form, as presentedin (1) by the freedom of parameter A, allows an unambiguous calcula-tion of the isotope coefficient α.1238 J Supercond Nov Magn (2011) 24: 1235–1239The agreement between the measured oxygen isotope ef-fect α‖(n) as a function of doping n and the curve calcu-lated with (3) and (4), but with a constant parameter γ , isreally remarkable. From optimal doping nmax down to near-vanishing superconductivity, where the measured α‖(n) ex-hibits a characteristic upturn, the data follow the curve ob-tained with (1) very well. This allows the conclusion thatthis simple expression is more generally valid than onlyfor polarons axed along the crystallographic c-direction, forwhich it was originally obtained [1, 2]. It quantitativelyyields the correct behavior for polarons sited in the CuO2plane. Now the vibronic theory of Bussmann-Holder andKeller [6, 11] reproduces α‖(n) quite well near optimumdoping, see Fig. 1. In their work, the authors deduced thatthe local polaronic lattice deformation is of t2/t4, i.e., Jahn–Teller type. From this fact, we may assume that this is alsothe case in the entire range shown in Fig. 2.3 In other words,the effects of doping-dependent oxygen isotope and the the-ories shown in Figs. 1 and 2 are evidence of the planar pola-ronic origin of the high-temperature superconductivity.The above conclusion agrees with the ones reached in theviewpoint published by the second author [16] four yearsago. In this viewpoint, the possibility for superconductiv-ity to be due to magnetic interactions was also addressed.At that time, no electronic theories were known to yield fi-nite Tc’s. Furthermore, experiments were discussed whichindicated that magnetic interactions could not be respon-sible for the high-temperature superconductivity. However,the group of Keimer at the MPI in Stuttgart and othersinvestigated the antiferromagnetic (AFM) peaks near theM-point of the Brillouin zone with inelastic neutron scat-tering [17] for a number of compounds. These peaks arevery weak, and the group was able to enhance the signalby first exciting electrons from the 2p level of oxygen withX-rays. The signal obtained in this way quantitatively fol-lows the temperature-dependent superconducting gap. Fromthis, a magnetic mechanism was deduced by assuming an in-trinsic homogeneous state of the material. However, this isnot the case! This has been pointed out recently [16] andalso earlier [18]. For example, X-ray [19] and early TmNMR [20] investigations yielded near-insulating AFM re-gions and metallic-like regions forming clusters. Close to asuperconducting cluster or stripe, weak AFM signals existat the border, owing to the continuous electronic part of thewavefunction [21]. Indeed, weak spin ordering has been de-tected by nuclear quadrupole resonance [22]. It is thus pos-sible that this weak AFM signal is coherent by interactionwith the coherent superconducting part, and is responsiblefor the AFM signal.3This behavior is similar to those in critical phenomena, where the lo-cal conformations near structural phase transitions that occur in themean field regime subsist symmetry-wise in the critical regime.The largest oxygen isotope effect at Tc shown in Fig. 1,occurring at low doping, is about α = 1.0, i.e., much lowerthan those observed at the pseudogap temperature T ∗ re-ported for LSCO [23] and HoBa2Cu4O8 [24]. For the lat-ter, α∗ = −2.2. It is sign-inverted, as expected from (1),which also applies to the pseudogap isotope effect [1]. Withthe nearly linear T ∗(n), we arrive at γ ∗ = 1.7 from thesedata. The existence of an even greater α∗ for copper inYBCO in terms of local lattice conformations [24] has beencommented on in [6, 16]. In the latter paper, the occur-rence of T ∗ has been assigned to the formation of inter-site Jahn–Teller bipolarons [16]. That these giant isotope ef-fects on T ∗, which were observed and confirmed a decadeago [23, 24], were ignored by the proponents of pure elec-tronic mechanisms for the occurrence of superconductivityin the cuprates is not acceptable from a scientific point ofview.The isotope effect with α = 0.5 in the classical super-conductors was substantial for the acceptance of the BCStheory. The carrier-dependent α‖(n) at Tc(n) quantitativelyfollows the Kresin–Wolf formula [see (1)] derived for po-larons along the c-direction over the entire doping range[see Fig. 2]. We regard this as a strong clue that, for thematerials shown, superconductivity is induced by polarons,or rather bipolarons, with spin S = 0, but that they occur inthe CuO2 plane [16]. The latter follows from the agreementof the calculated planar isotope effects at Tc(n) with the vi-bronic theory near optimum doping, see Fig. 1. Comparingthe observed isotope effects with the vibronic theory indi-cates that the local polaronic conformations are of the t2/t4Jahn–Teller type. This theory also yields the large invertedoxygen isotope effect of α∗ = −2.2 observed at the pseudo-gap temperature T ∗ and other behaviors [11]. Furthermore,already at very low doping, intersite Jahn–Teller bipolaronsare formed, which cluster in probably ramified, conductingentities, such as stripes [25].4Acknowledgements The authors would like to thank H. Keller forvarious helpful discussions and the hospitality at the Physics Instituteof the University of Zurich, and one of us (K.A.M.) thanks A. Bishopfor correspondence and M. Mali and B. Keimer for discussions. Thiswork was partly supported by the Swiss National Science Foundation.References1. Kresin, V.Z., Wolf, S.A.: Rev. Mod. Phys. 81, 481 (2009)4Note that inelastic neutron scattering and photoemission spectroscopyintegrate over the volume of the sample used. For the former, this hasalready been alluded to in the text. For the latter, this means that be-cause of the intrinsic heterogeneity at a particular instant, emissionsfrom AFM near insulating parts of the sample, close to the M-point ofthe Brillouin zone, as well as from other conducting Fermi-like ones,near X-point directions, are detected, i.e., the observed excitations re-sult from different parts of the sample. Assuming homogeneity leads toquestionable conclusions as in [26].J Supercond Nov Magn (2011) 24: 1235–1239 12392. Kresin, V.Z., Wolf, S.A.: Phys. Rev. B 49, 3652(R) (1994)3. Schneider, T., Keller, H.: Phys. Rev. Lett. 69, 3374 (1992)4. Schneider, T., Keller, H.: Phys. Rev. Lett. 86, 4899 (2001)5. Zech, D., Keller, H., Conder, K., Kaldis, E., Liarokapis, E.,Poulakis, N., Müller, K.A.: Nature (London) 371, 681 (1994)6. Bussmann-Holder, A., Keller, H.: Eur. Phys. J. B 44, 487 (2005)7. Franck, J.P., Jung, J., Mohamed, M.A.-K., Gygax, S., Sproule,G.I.: Phys. Rev. B 44, 5318 (1991)8. Khasanov, R., Shengelaya, A., Morenzoni, E., Conder, K., Savic,I.M., Keller, H.: J. Phys., Condens. Matter 16, S4439 (2004)9. Bornemann, H.J., Morris, D.E.: Phys. Rev. B 44, 5322 (1991)10. Babushkina, N., Inyushkin, A., Ozhogin, V., Taldenkov, A., Ko-brin, I., Vorobeva, T., Molchanova, L., Damyanets, L., Uvarova,T., Kuzakov, A.: Physica C 185, 901 (1991)11. Keller, H., Bussmann-Holder, A., Müller, K.A.: Mater. Today11(9), 38 (2008)12. Bussmann-Holder, A., Keller, H., Bishop, A.R., Simon, A., Mic-nas, R., Müller, K.A.: Europhys. Lett. 72, 423 (2005)13. Presland, M.R., Tallon, J.L., Buckley, R.G., Liu, R.S., Flower,N.E.: Physica C 176, 95 (1991)14. Tallon, J.L., Bernhard, C., Shaked, H., Hitlerman, R.L., Jorgenen,J.D.: Phys. Rev. B 51, 12911(R) (1995)15. Schneider, T.: In: Bennemann, K., Ketterson, J.B. (eds.) ThePhysics of Superconductors. Springer, Berlin (2004)16. Müller, K.A.: J. Phys., Condens. Matter 19, 251002 (2007)17. He, H., Bourges, P., Sidis, Y., Ulrich, C., Regnault, L.P., Paihes, S.,Berzigiarova, N.S., Kolesnikov, N.N., Keimer, B.: Science 295,1045 (2002)18. Mihailovic, D., Müller, K.A.: In: Kaldis, E., Liarokapis, E.,Müller, K.A. (eds.) High-Tc Superconductivity 1996: Ten YearsAfter the Discovery. NATO ASI Ser. E, vol. 343, p. 243. Kluwer,Dordrecht (1997)19. Bianconi, A., Saini, N.L., Lanzara, A., Missori, M., Rosetti, T.,Oyanagi, H., Yamaguchi, H., Oka, K., Ito, T.: Phys. Rev. Lett. 76,3412 (1996)20. Teplov, M.A., et al.: In: Kaldis, E., Liarokapis, E., Müller, K.A.(eds.) High-Tc Superconductivity 1996: Ten Years After the Dis-covery. NATO ASI. Ser. E, vol. 343. Kluwer, Dordrecht (1997)21. Martin, I., Kaneshita, E., Bishop, A.R., McQueeney, R.J., Yu,Z.G.: Phys. Rev. B 70, 224514 (2004)22. Raffa, F., Ohno, T., Mali, M., Roos, J., Brinkmann, D., Conder,K., Eremin, M.: Phys. 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Oxygen Isotope Effect in Cuprates Results from Polaron-induced Superconductivity

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Oxygen Isotope Effect in Cuprates Results from Polaron-induced Superconductivity

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