New Research In
Physical Sciences
Social Sciences
Featured Portals
Articles by Topic
Biological Sciences
Featured Portals
Articles by Topic
 Agricultural Sciences
 Anthropology
 Applied Biological Sciences
 Biochemistry
 Biophysics and Computational Biology
 Cell Biology
 Developmental Biology
 Ecology
 Environmental Sciences
 Evolution
 Genetics
 Immunology and Inflammation
 Medical Sciences
 Microbiology
 Neuroscience
 Pharmacology
 Physiology
 Plant Biology
 Population Biology
 Psychological and Cognitive Sciences
 Sustainability Science
 Systems Biology
Unified picture of the oxygen isotope effect in cuprate superconductors

Contributed by Hokwang Mao, January 2, 2007 (received for review November 1, 2006)
Abstract
Hightemperature superconductivity in cuprates was discovered almost exactly 20 years ago, but a satisfactory theoretical explanation for this phenomenon is still lacking. The isotope effect has played an important role in establishing electron–phonon interaction as the dominant interaction in conventional superconductors. Here we present a unified picture of the oxygen isotope effect in cuprate superconductors based on a phononmediated dwave pairing model within the Bardeen–Cooper–Schrieffer theory. We show that this model accounts for the magnitude of the isotope exponent as functions of the doping level as well as the variation between different cuprate superconductors. The isotope effect on the superconducting transition is also found to resemble the effect of pressure on the transition. These results indicate that the role of phonons should not be overlooked for explaining the superconductivity in cuprates.
Understanding the hightemperature superconductivity in cuprate superconductors is at the heart of current research in solidstate physics. The isotope effect is an important experimental probe in revealing the underlying pairing mechanism of superconductivity. Early measurements (1) of the isotope effect on the superconducting transition temperature, T _{c}, provided key experimental evidence for phononmediated pairing and supported strongly the Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity in conventional materials. When hightemperature superconductivity was discovered in copper oxides, the oxygen isotope exponents, defined by α ≡ −dlnT _{c}/dlnM, with M being the isotopic mass, were promptly measured. The initial finding of the small value of α in La_{1.85}Sr_{0.15}CuO_{4} (2, 3) and near zero value in YBa_{2}Cu_{3}O_{7−δ} (δ ∼ 0) (4–6) was taken to be convincing evidence against electron–phonon interaction as the dominant mechanism in these materials. However, later experiments have revealed a much richer and more complex situation (7). There is a striking variation of α with doping level in a given cuprate material (8–11). Meanwhile, α also varies among various compounds of different cuprate families (7, 12). Interestingly, α is a remarkably monotonic function of the number of CuO_{2} layers in a way opposite to T _{c} in optimally doped compounds (12). Such elaborate isotope effects strongly suggest that one should include phonons in the theory of highT _{c} superconductivity. Studies of neutron scattering (13), angleresolved photoemission spectroscopy (14, 15), and scanning transmission electron microscopy (16) provide further evidence for phonons being an important player in the basic physics of highT _{c} superconductivity.
Compared with isotope substitution, pressure has been realized not only to be another effective way to change T _{c} for a superconductor (17) but also to yield information on the interaction causing the superconductivity. So far, the record high T _{c} of 164 K was achieved under high pressure in HgBa_{2}Ca_{2}Cu_{3}O_{8+δ} (18). Observing how the pressure influences the individual parameters of the lattice in the normal and superconducting state of a single sample allows a clean quantitative test of the validity of theoretical models. In early 1964, Olsen et al. (19) showed that there is a correlation between the pressure coefficient of the effective interaction and the isotope effect in conventional metal superconductors. For highT _{c} materials YBa_{2}Cu_{3}O_{7−δ}, Franck and Lawrie (20) found a similar doping dependence for both the copper isotopic exponent and pressure coefficient of T _{c}, dlnT _{c}/dP. These findings directly point to the equivalent importance of the pressure and isotope effects on superconductivity. Because many materials, among them 23 elements (17) and even spinladder cuprate (21), become superconductors only under sufficiently high pressure, measurements of the isotope effect in these materials must be performed under high pressure. Meanwhile, Crespi and Cohen (22) proposed that cuprates under high pressure are a hopeful candidate for attaining negative values for α, which would provide support to an anharmonic model for highT _{c} superconductivity. Until now, there is little information regarding how α changes with pressure in highT _{c} cuprates. Such investigations are highly desirable for elucidating the mechanism of highT _{c} superconductivity. The purpose of this paper is to offer an interpretation for the elaborate oxygen isotope effect in cuprate superconductors based on a phononmediated dwave BCSlike model. We find that the effective nextnearestneighbor hopping dominates the variation of α for the optimally doped cuprates. Based on the measurements of both T _{c} and Raman shift of the B _{1g} phonon mode, we predict that the pressure dependence of α for the optimally doped YBa_{2}Cu_{3}O_{7−δ} is similar to that of the pressure coefficient of T _{c}. A unified picture for the isotope effect in cuprates is given at ambient condition and under high pressure, which is supported by good agreements between theory and experiments.
Results and Discussion
We start directly from the meanfield Hamiltonian describing dwave superconductivity:
where ε_{k} is the quasiparticle dispersion, μ is the chemical potential, and c
_{kσ}
^{†} is a quasiparticle creation operator. The gap parameter Δ_{k} = N
^{−1} Σ
_{k′}V_{kk′}
〈c
_{−k′↓}
c_{k′↑}
〉, where N is the number of sites. Considering that the interaction could be of the chargecoupled type mediated by phonons, we used dwave pairing potential V_{kk′}
= Vg(k)g(k′); ∣ε_{k} − μ∣ or ∣ε
_{k′}
− μ∣ < ω_{0}, where g(k) = cos k_{x}
− cos k_{y}
, V is the inplane pairing interaction strength, and ω_{0} is the cutoff of the phonon frequency. For phononmediated pairing, ω_{0} is assumed to vary with the isotope mass, M, as M
^{−1/2}. The dwave gap function has the typical form Δ_{k} = Δg(k), and the parameter Δ is determined by
where χ_{k} = (2E_{k}
)^{−1}tanh(βE_{k}
/2), with the quasiparticle spectrum
Differentiating Eq. 2 with respect to both T _{c} and ω_{0}, one can express α ≡ (1/2)dlnT _{c}/dlnω_{0} as Once knowing the cutoff frequency ω_{0}, dispersion ε_{k}, and interaction V, one can obtain the T _{c} and α behaviors based on Eqs. 2 and 3 . We choose a typical ω_{0} = 0.060 eV (13–16). For the dispersion, we use ε_{k} = t′ _{eff}cos k_{x} cos k_{y} + t″ _{eff}(cos 2k_{x} + cos 2k_{y} ), with t′ _{eff} and t″ _{eff} being the effective nextnearestneighbor and nextnextnearestneighbor hoppings, respectively. Such a dispersion can reproduce the bandwidth and Fermisurface shape of many cuprate superconductors (25, 26). It has been shown (27) that the choice of t″ _{eff} = 0.032 eV and V = 0.03762 eV can reproduce the experimentally observed maximum value of the superconducting transition temperature, T _{c} ^{max}, ranging from 30 to 97 K of various optimally doped monolayer cuprates when changing t′ _{eff} from −0.032 to 0.0914 eV. To keep the consistency of the theory, we use the same t″ _{eff} and V values in the following calculations. The relative t′ _{eff} can be estimated by using the experimentally observed T _{c} ^{max} for an individual material.
Fig. 1 shows the calculated T _{c} and α as a function of hole concentration n _{H} for some typical monolayer cuprates. As shown, T _{c} initially increases with increasing n _{H} to a maximum around an optimal doping level n _{H} ^{opt} and then decreases with further increasing n _{H}. This parabolic relation between T _{c} and n _{H} agrees well with experimental observations in general. The maximum in the T _{c} versus n _{H} curves is shifted toward smaller n _{H} values among these materials, in remarkable agreement with experiment. This behavior implies that the diluted optimal hole concentration is in favor of the higher value of the maximum T _{c} when a material is optimally doped. Because t′ _{eff} in dispersion is the only other parameter used in calculation, the different T _{c} behaviors of these materials should come from their electronic structure. This character provides an explicit interpretation for the role played by the “chemistry” in determining the maximum T _{c} via the effect on the electronic structure, as early suggested by Phillips (28, 29).
The calculated α exhibits many interesting features. The values of α are always positive over the whole doping range for all studied monolayer materials. There exists a minimum α at some doping level on the overdoped side, away from which one can find most large values of α. A significant increase in α occurs when the sample is highly underdoped or highly overdoped. In lowT _{c} groups, such as La, Bi, and singlelayer Tlbased cuprates, α does not vary monotonically with doping and its value is relatively large at optimal doping. For highT _{c} twolayer Tl and Hgbased materials, α is found to decrease monotonically with doping on the underdoped side, reaching a minimum on the overdoped side after passing through the optimal doping level, and then it is found to increase gradually when the material is highly overdoped. Such a difference is also believed to be a result of the electronic structure effect. Currently, La_{2−x}Sr_{x}CuO_{4} is the only monolayer system whose doping dependence of α is well known over the whole doping regime (2, 3, 9). Note that the experimentally observed α behavior in this system can be reproduced by the present dwave BCSlike model, except in the region near x = 0.125. The large α value at this doping level is related to the appearance of the stripe order (30), which is beyond the present theoretical treatment. Considering that the dispersion relations used are derived from short distance antiferromagnetic correlations, our model is in reality a mixture of both electronic and phononic contributions. Therefore, a phononinduced attraction combines with a hole dispersion modulated by antiferromagnetic correlations to contribute to pairing. Although the similar doping dependence of the isotope exponent can be reproduced within van Hove scenarios (31–33), the materialdependent dispersions considered here enable us to investigate the variation of the isotope effect among various cuprates.
In Fig. 2, the t′ _{eff} dependence of both the maximum T _{c} and α is plotted for optimally doped materials in the range of interest of t′ _{eff}. Clearly, t′ _{eff} dominates the maximum T _{c} behavior among various compounds, confirming previous theoretical interpretations (27, 34). The most significant observation is that α is not negligible but rather varies with t′ _{eff} as well. For small t′ _{eff} values, the material also has a small value of the maximum T _{c} but a large value of α. As t′ _{eff} increases (leading to the increase of the maximum T _{c}), α first sharply drops and then gradually decreases, reaching a minimum at around t′ _{eff} = 0.048 eV; further increase in t′ _{eff} leads to slightly increasing α. The theoretical calculations yield α of 0.26 for the optimally doped La_{2−x}Sr_{x}CuO_{4}, which is consistent with the experiments (2, 3). Therefore, this model predicts that both the maximum T _{c} and α in these optimally doped cuprates are in reality related to the modification of the electronic structure through the change of t′ _{eff}. The crucial role of t′ _{eff} has been proposed (35) to be associated to the JahnTeller active Q _{2}type mode. Our results imply that the coupling of the electronic degrees of freedom to the JahnTeller Q _{2}type mode is responsible for the material dependence of the isotope effect. According to our theoretical analysis, the optimally doped Tl_{2}Ba_{2}CuO_{6+δ} should have the smallest α value compared with other monolayer compounds. Thus, experiments are desirable to answer whether the systematic evolution of α among cuprates is correctly predicted here.
For YBa_{2}Cu_{3}O_{7−δ}, assuming that the α behavior is affected only negligibly by interlayer coupling, we can estimate the value of t′ _{eff} from Eq. 2 when taking T _{c} ^{max} = 92 K at the optimal doping. Thus, we are able to calculate the changes in T _{c} and α for this material by using Eqs. 2 and 3 . Calculation results are shown in Fig. 3 together with experimental data for comparisons. As can be seen, all of the α values are positive over the whole doping regime. The optimally doped material has the corresponding minimum α of 3.5 × 10^{−3}. An early study on YBa_{2}Cu_{3}O_{7} gives a value of α of 0 ± 0.02 (4). Another careful examination of five pairs of YBa_{2}Cu_{3}O_{7} samples (6) gives an oxygen isotope effect of α = 0.017 ± 0.006. The comparison between theory and experiments is excellent at the optimal doping level. As the material goes away from the optimal doping level, α increases with decreasing T _{c}. By changing the oxygen content, Zech et al. (8) observed that α remains rather small at T _{c} ≃ 61 K and then increases for a further decrease of T _{c} down to ≈53 K in YBa_{2}Cu_{3}O_{7−δ} (0.47 > δ > 0.01). The gradual increase of α with the dramatic reduction of T _{c} was reported in YBa_{2}(Cu_{1−x}Zn_{x})_{3}O_{7−δ} (7). As is clearly seen in Fig. 3, the theoretical curves cover most of these experimental data points (4–8), except the points at ≈50 K, where a maximum of α is believed to be associated with the chain oxygen ordering (20). It is apparent that the present theoretical model accounts for the observed isotope effects in this extensively studied highT _{c} material as well. Therefore, it is natural to expect a nearly zero oxygen isotope effect in the optimally doped YBa_{2}Cu_{3}O_{7−δ} for a BCStype electronphonon interaction pairing. It is worth noting that the doping dependence of α in YBa_{2}Cu_{3}O_{7−δ} resembles that of dlnT _{c}/dP (36, 37), signaling a nice correlation between the isotope and pressure effects.
We now show how the present dwave BCSlike model can predict the highpressure behavior of α. Just like the case at ambient conditions, the change of α with pressure is assumed to be through the changes of both T _{c} and ω_{0} and is then expressed by α(P) = α[T _{c}(P), ω_{0}(P)]. For the cutoff phonon frequency ω_{0}(P), we take the pressure dependence of the frequency ω_{B1g} of the B _{1g} Cu–O bondbuckling phonon, which has been ascribed as a bosonic mode influencing highT _{c} superconductivity (14). The information on ω_{B1g}(P) can be obtained by highpressure Raman scattering measurements. We thus have ω_{0}(P) = [ω_{0}(0)/ω_{B1g}(0)]ω_{B1g}(P). The α(P) behavior can be derived from Eq. 3 based on the T _{c}(P) and ω_{B1g}(P) determinations. We choose optimally doped YBa_{2}Cu_{3}O_{7−δ} as an example. Raman spectra in the y(xx)ȳ (A _{1g} + B _{1g}) geometry were taken at 25 K and at different pressures up to 22.3 GPa. The most pronounced spectral change under pressure is a significant enhancement of the B _{1g} phonon mode near 300 cm^{−1}. Fig. 4 shows the measured T _{c}(P) and Raman shift ω_{B1g}(P) of B _{1g} phonon mode obtained by fitting the Raman data in terms of a Green's function approach (38). Note that the measured T _{c}(P) behavior of this material is consistent with other independent measurements (39, 40).
In Fig. 5 we present the calculated α as a function of pressure up to 25 GPa in the optimally doped YBa_{2}Cu_{3}O_{7−δ} together with dlnT _{c}/dP for comparison. As pressure is increased, α slightly decreases in a manner similar to dlnT _{c}/dP, as expected from an anharmonic model (22), but α remains small and positive within the pressure regime studied. Our calculations allow the prediction of isotope effect changes under pressure in cuprate superconductors. The similarity of the isotope effect and the pressure effect both at ambient condition and under high pressure indicates that both effects are playing an equivalent role in shedding important light on highT _{c} superconductivity in cuprates. There has been no report of isotope measurements under high pressure in cuprates so far. The prediction made here for α(P) awaits further experimental studies.
We should emphasize that the present theoretical investigation is based on a dwave model. However, we find that the dwave character is not critical to the results obtained and conclusions drawn. An swave model can also give the similar general trend of α as functions of doping and pressure. We also would like to mention that the strong correlation effect is not directly included in our theoretical treatment, rather entering our model through the hole dispersion. Although the physics in the lowdoping regime would be dominated by antiferromagnetic correlations, electron–phonon coupling is expected to be more important in the optimal and overdoped regime. The present theoretical model bears nicely the experimentally observed generic phase diagram between the T _{c} and doping level. Therefore, we believe that all our results and conclusions are physically valid over a wide range of doping regimes.
Conclusions
We have obtained a clear picture of the oxygen isotope effects in hightemperature superconductors based on a phenomenological phononmediated BCSlike model. We demonstrate that the variation of α among various optimally doped cuprates is controlled by the effective nextnearestneighbor hopping. The obtained oxygen isotope effect resembles the pressure effect both at ambient conditions and under high pressure. The good agreement between theory and experiment strongly suggests that the role of phonons should be taken into account for explaining the hightemperature superconducting properties in cuprates.
Materials and Methods
Samples of single crystal YBa_{2}Cu_{3}O_{7−δ} were grown by a selfflux technique. We performed the measurements of T _{c} under high pressures by using a highly sensitive magnetic susceptibility technique with diamond anvil cells (41). The technique is based on the quenching of the superconductivity and suppression of the Meissner effect in the superconducting sample by an external magnetic field. The magnetic susceptibility of the metallic parts of the highpressure cells is essentially independent of the external field. Therefore, the magnetic field applied to the sample inserted in the diamond cell mainly affects the change of the signal coming from the sample. Our high pressure Raman system and experiments are detailed in ref. 38. We have used synthetic ultrapure anvils to reduce diamond fluorescence. Samples were loaded into diamond anvil cells with helium as a pressure medium, and pressure was determined by using the ruby fluorescence technique.
Acknowledgments
We thank O. K. Andersen, V. H. Crespi, T. Cuk, A. F. Goncharov, A. Lanzara, J. C. Phillips, J. S. Schilling, S. Y. Zhou, and J. X. Zhu for stimulating discussions. This work was supported by the Office of Basic Energy Science and National Nuclear Security Administration of the U.S. Department of Energy. H.Q.L. was supported by the Hong Kong Research Grants Council.
Footnotes
 ^{§}To whom correspondence should be addressed. Email: xjchen{at}ciw.edu

Author contributions: X.J.C., V.V.S., Z.W., H.Q.L., R.J.H., and H.k.M. performed research.

The authors declare no conflict of interest.
 Abbreviation:
 BCS,
 Bardeen–Cooper–Schrieffer.
 © 2007 by The National Academy of Sciences of the USA
References

↵
 Maxwell E
 ↵
 ↵
 ↵
 ↵
 ↵

↵
 Franck JP
 Ginzberg DM

↵
 Zech D ,
 Conder K ,
 Keller H ,
 Kaldis E ,
 Müller KA
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵

↵
 Schilling JS
 Schrieffer JR
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵

↵
 Tissen VG ,
 Nefedova MV
 ↵
 ↵
Citation Manager Formats
Sign up for Article Alerts
Jump to section
You May Also be Interested in
More Articles of This Classification
Physical Sciences
Related Content
 No related articles found.
Cited by...
 No citing articles found.