Isotope Effect in High-Tc Materials: Role of Non-Adiabaticity and Magnetic Impurities

Z. Phys. B 104, 759–763 (1997) ZEITSCHRIFT FUR¨ PHYSIK B c Springer-Verlag 1997

Isotope effect in high-Tc materials: role of non-adiabaticity and magnetic impurities

A. Bill1, V.Z. Kresin1, S.A. Wolf2

1 Lawrence Berkeley Laboratory, University of Calfornia, Berkeley, CA 94720, USA 2 Naval Research Laboratory, Washington, DC 20375-5343, USA

Abstract. Based on previous calculations of the isotope ef- oxygen isotope effect of Tc in Zn and Pr-doped (YBCZnO fect (IE) in superconductors, we present a detailed study and YPrBCO) as well as in oxygen-depleted YBa2Cu3O7 δ of the influence of non-adiabaticity and magnetic impurities (YBCO). Here we present more detailed calculations for− on the value of the isotope coefficient (IC). We focus on YPrBCO and YBCO and demonstrate that the description the combined effect of these factors and examine how their of the experimental results (see, e.g., [4–6]) for the IE of relative weight affects the IC. The isotope effect of the su- Tc requires to take into account both non-adiabaticity and perconducting critical temperature Tc, and of the penetration magnetic impurities. As for the isotope coefficient of the depth δ are discussed. It is shown that both non-adiabaticity penetration depth δ, we present calculations similar to the and magnetic impurities have to be taken into account to IC of Tc. There are, however, no experimental data available describe the oxygen isotope effect of Pr-doped and oxygen- yet for the above mentioned systems. We emphasize that the depleted YBa2Cu3O7 δ (YBCO). The calculations suggest calculated effects can also be observed in conventional su- that the effect of magnetic− impurities is stronger for Pr-doped perconductors. than for O-depleted YBCO. We also present new results for We calculate in the following the isotope coefficient of the IC of δ in O-depleted YBCO. Tc and of the penetration depth δ defined respectively by α β T M − and δ M − . Thus, α and β are determined c ∼ ∼ by α = (M/∆M)(∆Tc/Tc) and β = (M/∆M)(∆δ/δ) PACS: 74.20.Fg; 74.72.h; 74.72.Bk where ∆M− is the mass difference between− the two isotopes and ∆Tc, ∆δ are the shifts induced by isotopic substitution. In Sect. II we present the concept of the non-adiabatic IE introduced in [1] and establish a relation between the ICs of Tc and δ for London superconductors. In Sect. III I. Introduction we demonstrate that magnetic impurities affect the IE of Tc and discuss how it induces an IE of δ. In the latter case, In the theory of conventional superconductors the isotope we confine our considerations to temperatures near Tc. The effect (IE) has often been considered as a signature of the situation near T = 0 is discussed in [3]. In Sect. IV we electron-lattice interaction mediating the pairing and lead- analyze the data on the oxygen IE in YBa2Cu3O7 δ related ing to the instability of the normal phase. In several recent materials (see also [2–6]). We give special care in− separating papers [1–3], however, we have shown that a number of fac- the effect of non-adiabaticity and magnetic impurities on the tors, not related to the pairing mechanism, and some of them IC. The comparison between theory and experiment allows not related to lattice dynamics, strongly affect the value of then to determine the values of the parameters of the model the isotope coefficient (IC). Specifically, we have shown that [1–3]. non-adiabaticity, magnetic impurities and the proximity effect increase the IC of Tc. This holds both for conventional and high-Tc superconductors. II. Non-adiabatic isotope effect In the present article, we study the isotope effect of Tc and of the penetration depth δ. The general theory and its It has been shown in [1, 2] that the presence of non- application to high-temperature superconductivity has been adiabaticity in a superconductor can strongly affect the IE. presented in [1–3]. Here we focus on the different roles We consider specifically the situation where charge-transfer played by non-adiabatic charge-transfer processes and mag- processes occur through ions that display a non-adiabatic be- netic impurities in the determination of the isotope coeffi- haviour. One example of such a system is given by high-Tc cient. We show the influence of each of these two factors cuprates where the charge-transfer occurs from a charge- on the IE and discuss how their combination affects the IC. reservoir (e.g. the CuO chains in YBCO) to a conduct- The theory was applied in [2, 3] to the description of the ing subsystem (e.g., the CuO2 planes). Another example is 760

given by manganites [(La,Ca)MnO3] [7] where the charge function of temperature. Near Tc one has ϕ 1 (T/Tc) transfer occurs between Manganese atoms via an oxygen [4], whereas ϕ 1 near T = 0. From this, one' can− deter- ion located between them. Several experiments [8, 9] as mine the isotope' coefficient β of the penetration depth as well as theoretical studies [10] have established that in both β = βph + βna (see [3]). The first term βph is the usual above-mentioned systems oxygen and copper ions display BCS contribution arising from the fact that δ depends on a non-adiabatic behaviour. This affects the charge trans- Tc through the temperature dependency ϕ(T/Tc). Tc expe- fer between the reservoir and the conducting layer, because riences the usual BCS isotope effect and induces thus an the latter process involves the motion of the non-adiabatic isotopic shift of δ. This effect is strong near Tc. There is, ions. As a consequence, the charge-carrier density in the however, an unconventional, non-adiabatic contribution to conducting layer depends on the ionic masses n = n(M) the IE (βna) arising from the relation n(M). This second, (M is the ionic mass; see [1, 2]). Qualitatively, the time non-adiabatic contribution is given by [3] spent by a charge carrier on the non-adiabatic ions before M ∂n being released as a “free” carrier in the conducting sys- βna = . (3) tem depends on the mass of the ions. For example, if the 2n ∂M charge transfer occurs via the apex oxygen, a higher iso- Comparing this expression with the definition of the param- topic mass will imply a lower charge-transfer frequency, eter γ it follows that βna = γ/2. The non-adiabatic IC of which means that the charge-carrier “spends more time” on δ is thus a constant (independent− of doping and tempera- the non-adiabatic bridging apex oxygen. This influences di- ture; see also next section). From (1) and (3) one can write rectly the charge-carrier density n in the CuO2 plane and a general relation between the IC of Tc and δ for a London consequently n = n(M). Clearly, this effect has to be un- superconductor [3] derstood as a (stationary) dynamical effect, since there is a n ∂Tc continuous particle exchange between the reservoir and the αna = 2βna . (4) conducting layer. − Tc ∂n Since Tc is a function of doping in cuprates [that is, Note that according to this expression, αna and βna have Tc = Tc(n)], one can calculate the isotope coefficient of Tc opposite sign in underdoped high-Tc superconductors. for the non-adiabatic channel (αna). It has been shown [1] Thus, non-adiabatic charge-transfer leads to a novel iso- that the IC can be written as α = αph + ana, where αph is tope effect of Tc and δ due to the dependence n(M). As will the usual (BCS) phonon contribution and the non-adiabatic be shown in Sect. IV, a complete description of Pr-doped and part is given by O-depleted YBCO requires to take into account the presence of both non-adiabaticity and magnetic impurities induced by n ∂Tc αna = γ . (1) the doping process. The next section discusses how magnetic Tc ∂n impurities affect the IC. The parameter γ = M/n(∂n/∂M) has a weak logarith- mic dependence on −M (see [1]). It follows from (1) that the isotope coefficient depends on the doping of the su- III. Isotope effect and magnetic impurities perconducting layer and on the relation Tc(n). In general this latter relation is not known. Experimentally, one has ei- The influence of magnetic impurities on the properties of a ther Tc(x) where x is the concentration of the dopant (as, superconductor has been widely studied since the first paper e.g., the Pr or Zn concentration in Y1 xPrxBa2Cu3O1 δ by Abrikosov and Gor’kov [11]. The main effect of mag- − − and YBa2(Cu1 xZnx)3O7 δ, respectively) or Tc(σ)(σ netic impurities is to reduce the superconducting condensate − − ∼ ns/m∗, where ns is the superconducting carrier density and density by breaking Cooper pairs. This affects, among other m∗ the effective mass) as obtained from µSR experiments. properties, the critical temperature and the penetration depth In the next sections we use the results obtained from µSR of a superconductor. Let us define Tc and δ (Tc0 and δ0)as measurements. In this case, one can write: the critical temperature and the penetration depth in the presence (absence) of magnetic impurities. Because of the non- n ∂Tc σ ∂Tc αna = γ = γ . (2) linear equation relating Tc (δ) and Tc0 (δ0) [11], the value of Tc ∂n Tc ∂σ the IC is also affected by magnetic impurity scattering (for Thus, the calculation of α does not rely on any model relat- details, see [2, 3]). In the case of Tc, the isotope coefficient ing n to x. As in [2, 3], we consider the parameter γ in (2) αm in the presence of magnetic impurities is given by [12, as an adjustable parameter of the model. We emphasize that 2] this is the only fitting parameter of the theory used for the α α = 0 , (5) description of the experimental results. m 1 ψ (γ +1/2)γ As mentioned in the introduction, the non-adiabatic − 0 s s charge transfer does not only affect the isotope coefficient where γs = Γs/2πTc (Γs = Γ˜snm is the spin-flip scattering of Tc. Indeed, any quantity depending on the charge-carrier amplitude; nm is the concentration of magnetic impurities concentration n will be function of the isotopic mass because and Γ˜s is a constant [11]) and ψ0 is the derivative of the of the dependence n = n(M). The penetration depth of a psi function. α0 is the IC of Tc0, that is in the absence of magnetic field is such a quantity. Indeed, both for weak and magnetic impurities. Equation (5) expresses a universal re- 2 strong-coupling London superconductors one has δ− n . lation (without any free parameter) between α and T in ∼ s m c The quantity ns is the superconducting charge-carrier den- the presence of magnetic impurities. Indeed, the experimen- sity related to n by ns = nϕ(T/Tc), where ϕ is a universal tally determined values of Tc0, Tc and α0 completely define 761

αm. The only unknown in (5) is the spin-flip scattering amplitude γs. However, this quantity can be determined in the framework of the Abrikosov-Gor’kov theory by the equation relating Tc to Tc0. This procedure for the determination of the magnetic impurity contribution to the IE allowed to describe the IC of Tc in Zn-doped YBCO (see [2]). If there is a contribution both of the non-adiabatic and the magnetic impurity channels to the IC of Tc, then α0 in (5) has to be replaced by (1). This will be used in the next section to discuss the IC of Tc for Pr-doped YBCO (see also [2]). Let us now focus on the isotope effect of the penetration depth δ in the presence of magnetic impurities. This effect has been studied in [3]. The isotope coefficient βm has to be Fig. 1. Influence of non-adiabaticity and magnetic impurities on the isotope ˜ calculated separately for temperatures near Tc and for T =0. coefficient of Tc for Y1 xPrxBa2Cu3O7 δ . Γs =0(dash-dotted line), − − Here we present only the result for T Tc and refer to [3] 60 K (dashed), and 123 K (solid). The solid line is the best fit (calculated in for its derivation as well as for the calculation∼ at T = 0. Note [2, 3]) to the experimental points (taken from [4, 5]). γ =0.16, α0 =0.025 that contrary to the non-adiabatic coefficient βna calculated above, the magnetic impurity contribution βm to the IC is doping and temperature dependent. magnetic impurities by increasing Γ˜s. The points are exper- As stressed in the previous section, the penetration depth imental values obtained from different techniques [4, 5]. The experiences a trivial BCS isotope effect, due to its depen- parameter γ =0.16 (the single parameter of the theory) has dency on ϕ(T/Tc) and the usual isotopic shift of Tc. Since been calculated in [2] and gives the best fit, given the exper- we are only interested in the unconventional IE resulting imentally deduced value of Γ˜s = 123 K (solid line of Fig. 1; from the presence of magnetic impurities, we calculate the see also [13]). The figure demonstrates that the non-adiabatic ˜ isotope coefficient βm of δ(T,Γs)/δ(T,0). Near Tc this co- channel alone (dash-dotted line) cannot account for the ex- efficient can be written [3] perimental data on YPrBCO. The same conclusion holds for ˜ magnetic impurities alone. Indeed, near optimal doping (that βm = Rαm , (6) is, near the 90 K plateau in the plot Tc(n)) the IE is mainly where R is a complicated function of Γs, T/Tc and α0 (the defined by the non-adiabatic channel. Magnetic impurities isotope coefficient of Tc0) and was derived in [3]. alone would lead to a slowly increasing function α(Tc) with As αm, the coefficient βm is a universal function of Tc, opposite curvature as is observed, e.g., in Zn-doped YBCO since all quantities in (6) can be determined self-consistently (see [2]). On the other hand, at higher Pr-doping (lower Tc) within the Abrikosov-Gor’kov theory. (3) and (6) have been the behaviour of the IC is mainly determined by the magnetic used to discuss recent measurements on La2 xSrxCuO4 (see − impurity channel. Thus, whereas magnetic impurities fully [3]). account for the behaviour of the IC in Zn-doped YBCO [2], and non-adiabaticity accounts for the behaviour of La-doped IV. Oxygen isotope effect in high-T materials YBCO [1], the description of α(Tc) in YPrBCO requires to c take into account both channels. Note, in passing, that the In [2] we have calculated the isotope coefficient of Tc for non-adiabatic channel does not involve the apex oxygen in Zn and Pr-doped as well as oxygen-depleted YBCO and YPrBCO since the charge transfer occurs directly between compared with experiments. Furthermore, the isotope coef- Pr and the in-plane Cu and O ions. ficient β of the penetration depth has been calculated in [3] Let us now consider oxygen-depleted YBCO. As for for Pr-doped YBCO. Here we present more detailed calcu- YPrBCO there exist experimental evidence that the effect lations for these materials and show explicitely the separate of depletion is twofold. On the one hand, it reduces the influence of non-adiabaticity and magnetic impurities on the charge-carrier concentration in the CuO2 planes and, on the isotope coefficient of Tc and δ. We also present and discuss other hand, it changes the magnetic impurity concentration. new results for the IC of δ for oxygen depleted YBCO. Both factors have thus to be taken into account in the cal- Let us first consider the isotope coefficient of Tc for culation of the IC of Tc. The results are shown in Fig. 2. Pr-doped YBCO (YPrBCO). Several experiments have es- The major difference between YBCO and YPrBCO is that tablished that Pr replaces Y which is located between the the relation Tc(n) has two plateaus in the former compound two CuO2 planes of a unit cell. The doping affects YBCO (near 90 K and 60 K; see, e.g., [6]) and only one in the latter mainly in two ways. First, it was shown that the presence of (near 90 K). As a consequence, the non-adiabatic contribu- Pr decreases the charge-carrier concentration n of the CuO2 tion to the IC of Tc also vanishes at the 60 K plateau (where planes (see, e.g., [13]). Secondly, Pr introduces magnetic im- α ∂T /∂n 0) for YBCO, leading to a sharp drop of na ∼ c ∼ purities in the system [13, 14]. These experimental foundings the total IC α = αna + αm. As in the case of YPrBCO, the imply that both the non-adiabatic and the magnctic impurity isotope coefficient of Tc is mainly determined by the non- channels have to be considered for the description of the IE adiabatic contribution near the plateau regions. Farther away, in YPrBCO [2]. the magnetic impurity contribution becomes dominant. Figure 1 shows the effect of the two channels on the The best fit to the few available experimental points IC α of T , beginning with the pure non-adiabatic chan- available [6] gives γ =0.28 and Γ˜ 15 K (the value c s ' nel (Γ˜s = 0; see (5)) and then increasing the influence of of Γ˜s could not be deduced from experiment as was the 762

˜ Fig. 2. Influence of non-adiabaticity and magnetic impurities on the isotope Fig. 3. Isotope coefficient of the penetration depth (βm+na) for coeff. of T for YBa Cu O . Γ˜ =0(dash-dotted line), 15 K (dotted), Y1 xPrxBa2Cu3O7 δ and different values of the magnetic impurity pa- c 2 3 6+x s − ˜ − 60 K (dashed) and 123 K (solid). The best fit to the experiment is given by rameter: Γs =30K(dash-dotted line), 60 K (dotted), 90 K (dashed), and the dotted line with γ =0.28 (calculated in [2]) 123 K (solid). γ =0.16 is given by Fig. 1 (see [2]) and T/Tc =0.85. The solid line was calculated in [3] case for YPrBCO). This suggests that the contribution of magnetic impurities (non-adiabaticity) is smaller (larger) for O-depleted than Pr-doped YBCO. In other words, the depletion of oxygen seems to induces a smaller amount of magnetic impurities than doping with Pr. More experimental data are, however, necessary to give a better estimate of Γ˜s (especially away from the 60 K and 90 K plateaus where the role of magnetic impurities becomes dominant). Up to know we have presented results for the isotope coefficient of Tc. As mentioned in the previous sections (see also [3]), the penetration depth δ also displays an isotopic shift. The corresponding IC is doping and temperature de- ˜ pendent. Since we are mainly interested in the effect of non- Fig. 4. Isotope coefficient of the penetration depth (βm+na) for adiabaticity vs. magnetic impurities, we fix the temperature YBa2Cu3O6+x and different values of the magnetic impurity parameter: Γ˜s =15K(solid line), 60 K (dashed), and 123 K (dash-dotted). γ =0.28 to T/Tc =0.85. The study of the temperature dependency is given by Fig. 2 (see [2]) and T/T =0.85 ˜ c can be found in [3]. Furthermore, we consider only βm given ˜ by (6) instead of β = βna + βm (since βna is a constant). ˜ It is important to note that βm is modified through the non- for YPrBCO can be done: the isotope coefficient is negative ˜ adiabatic channel. Indeed, βm depends on αm, (5), with α0 and increases with increasing Γ˜s. Moreover, one observes ˜ given by (3). To emphasize this dependency, we write βm+na the same drop of the IC β˜ + at the 60 K plateau as for | m na| in the following. the isotope coefficient of Tc (see Fig. 2). The only difference ˜ Figure 3 shows the oxygen isotope effect of δ for is that the value of the isotope coefficient βm+na is negligi- YPrBCO. This result has several important features. First, ble near Tc = 60 K because we have considered the IC of the oxygen IC of δ is negative. It has thus opposite sign δ(T,Γ˜s)/δ(T,0) [see (6)]. Again, the non-adiabatic channel compared to the IC of Tc for underdoped materials. Sec- is dominant near the 60 K and 90 K plateaus whereas mag- ˜ ondly, comparing α with βm+na one observes that the cur- netic impurities determine the behaviour away from these ˜ | | vature of βm+na is positive for all Tc’s except near 90 K. As regions. seen on Fig.| 1 the| positive curvature is associated with the One should note that contrary to YPrBCO, the contribu- magnetic impurity contribution becoming dominant in the tion of magnetic impurities to the relation Tc(x) (see also determination of α. The present result shows that magnetic discussion after (1)) is not known for O-depleted YBCO. ˜ impurities affect the IC of δ already at low impurity con- It would be interesting to measure βm+na and αm for this centrations. This qualitative difference between α and β˜ case, since it would allow to determine the value of Γ˜s. should be noticeable, since the precision of measurements| | increases with decreasing impurity concentration (increas- ˜ ing Tc). A detailed comparison between βm+na and αm can V. Conclusions be found in [3]. Another interesting property of the result displayed in Fig. 3 is that an increase of the influence of We have discussed the role of non-adiabatic charge-transfer magnetic impurities (corresponding to an increase in Γ˜s for processes and magnetic impurities in the determination of the a given concentration nm, that is, a given Tc) leads to an isotope effect. The study is based on the theory developed in increase of the IC β˜ + . Thus, the influence of magnetic [1–3]. The focus of the present paper was to single out the | m na| impurities on the IC of Tc and δ is similar. influence of each factor on the value of the IC of Tc and δ.To Finally, we present calculations of the oxygen IC of δ this aim, we have calculated the oxygen isotope coefficient for oxygen depleted YBCO in Fig. 4. The same remarks as of Tc and the penetration depth δ in Y1 xPrxBa2Cu3O7 δ − − 763

and YBa2Cu3O6+x (0 x 1) for different values of the References parameters. The results≤ demonstrate≤ that near optimal doping and near the 60 K plateau where ∂Tc/∂n is small, the 1. Kresin, V.Z., Wolf, S.A.: Phys. Rev. B 49, 3652 (1994); and in: An- IC is mainly affected by the non-adiabatic channel. Away harmonic Properties of High-Tc Cuprates, p. 18. Mihailovic, D., Ruani, from these regions, the magnetic impurity contribution to C., Kaldis, E., Muller¨ K.A. (Eds.). Singapore: World Scientific 1995 2. Kresin, V.Z., Bill, A., Wolf, S.A., Ovchinnikov, Yu.N.: Phys. Rev. the isotope effect becomes dominant. The description of the B 56, 107 (1997); Bill, A., Kresin, V.Z.: proceedings of the XIIIth experimental data requires to consider both the non-adiabatic international Symposium on “Electrons and Vibrations in Solids and and the magnetic impurity channels. It would be interesting Finite Systems”. Berlin (August 1996). Z. Phys. Chem. 201, 271 (1997) to perform experiments on the isotope shift of δ for YBCO, 3. Bill, A., Kresin, V.Z., Wolf, S.A.: Preprint 4. Franck, J.P., Jung, J., Mohamed, M.A.-K., Gygax, S., Sproule, G.-I.: since it would allow the determination of Γ˜s. This suggests to use the measure of the isotopic shift of T or δ as a Phys. Rev. B 44, 5318 (1991); in: High-Tc Superconductivity, physical c properties, microscopic theory and mechanisms, p. 411. Ashkenazi, A. tool to probe the presence of magnetic impurities or non- et al. (eds.) New-York: Plenum Press 1991 adiabaticity and to determine the values of the non-adiabatic 5. Soerensen, G., Gygax, S.: Phys. Rev. B 51, 11 848 (1995) parameter γ (4) and of the spin-flip scattering amplitude Γs 6. Zech, D., Conder, K., Keller, H., Kaldis, E., Muller,¨ K.A.: Physica B (5). 219&220, 136 (1996) 7. Kresin, V.Z., Wolf, S.A.: Phil. Mag. B 76, 241 (1997) 8. Mustre de Leon, J. et al.: Phys. Rev. Lett. 64, 2575 (1990); Gasparov, A.B. thanks the Office of Naval Research and the Swiss National Science L. et al.: J. Supercond. 8, 27 (1995); Ruani, G. et al.: Solid State foundation for the financial support. He also wishes to thank the organizers Commun. 96, 653 (1995); Jesowski, A. et al.: Phys. Rev. B 52, 7030 A. Bussmann-Holder and G. Benedek for making the participation to this (1995) workshop possible. The work of V.Z.K. is supported by the U.S. Office of 9. Sharma, R.P. et al.: Phys. Rev. Lett. 77, 4624 (1997); D. Haskel et al.: Naval Research under contract No. N00014-96-F-0006. Phys. Rev. B 56, 521 (1997) 10. Johnston, K.H., Clougherty, D.P., McHenry, M.E.: in: Novel superconductivity, p. 563. Wolf, S., Kresin, V.Z. (Eds.). New York: Plenum Press 1986 11. Abrikosov, A., Gor’kov, L.: Sov. Phys. JETP 125, 1243 (1961) 12. Carbotte, J. et al.: Phys. Rev. Lett. 66, 1789 (1991); Singh, S.P. et al.: J. Supercond. 9, 269 (1996) 13. Maple, B. et al.: J. Supercond. 7, 97 (1994) 14. Zagoulev, S. et al.: Phys. Rev. B 52, 10 474 (1995); Physica C 259, 271 (1996)