- About this Journal ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

ISRN Condensed Matter Physics

VolumeΒ 2011Β (2011), Article IDΒ 961832, 5 pages

http://dx.doi.org/10.5402/2011/961832

## Theory of Isotope Effect in YBaCuO

Department of Physics, Beijing University, Beijing 100871, China

Received 31 July 2011; Accepted 24 August 2011

Academic Editor: H.Β Eisaki

Copyright Β© 2011 Fu-sui Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper is the first to demonstrate that a pure nonphonon mechanism can quantitatively explain all isotope effect experiments in YBaCuO (YBCO) and to conclude that the influence of zero-point oscillation on the two local spin-mediated interaction (TLSMI) causes the isotope effects in YBCO. This paper is the first to calculate the doping dependence of exponents
of oxygen isotope effect for all quantities of YBCO, such as , *T*, pseudogap at , gap at 0 K, and number density of supercurrent carriers at 0 K. This paper points out that the observed inverse isotope effect of comes also from zero-point oscillation.

#### 1. Introduction

Understanding the high-temperature superconductivity in cuprate superconductors is at the heart of current research in solid-state physics. However, the pairing mechanism responsible for high- is still controversial [1β7]. The isotope effect is an important experimental probe in revealing the underlying pairing mechanism of superconductivity. When high- superconductivity was discovered in cuprates, the oxygen isotope exponents, defined by = βdlnTc/dlnM, with M being the oxygen isotopic mass, were promptly measured [8β17]. In [1, 7] the authors think that such elaborate isotope effects strongly suggest that high- superconductivity should be of phonon mechanism. Although [3] proposes a t-J model including phonons, it cannot explain the isotope effect quantitatively. Therefore, until quite recently, there are still two problems which have not yet been solved. The first problem is that there is not a unified microscopic theory for both isotope effect and all other properties of the high- cuprates. The second problem is that there is not a quantitative theory of the isotope effect, which is based on a pure electron mechanism. This paper tries to solve those two problems.

The two local spin-mediated interaction (TLSMI), which is a pure electron mechanism, of high- cuprates was proposed in [18] and can explain nearly all experiments except the isotope effect [6]. An electron mechanism similar to that in [18] is also proposed in [19, 20]. Therefore, isotope effect is a key criterion for the correctness of theory in [6, 18β20]. This paper uses TLSMI to explain quantitatively isotope effects and to give a series of predictions related to the isotope effect. The TLSMI is introduced briefly in Section 2. In Section 3, we take the zero-point oscillation of and as the origin of isotope effect in YBCO and make numerical calculations for the oxygen isotope effects of all quantities such as , , pseudogap at , gap at 0βK, and number density of supercurrent carriers at 0βK. Based on the quantitative comparisons between theory and experiments, Section 4 makes conclusions and discussions.

#### 2. Two Local Spin-Mediated Interaction

The effective Hamiltonian of Hubbard-Emery d-p model to describe the plane of high- cuprates is [6]
where the summation over and is for the oxygen sites around th site, annihilates hole with spin *s* at site , is the local spin operator of at site *i * Pauli matrix vector, and *i* and *j* are the nearest neighbors. Expand in space. Here, is the wave vector in Brillouin zone of the oxygen lattice in the plane. The second term in (1) is Kondo Hamiltonian, , which implies that the holes with and can have interactions with the local spins of at sites *i* and *j*, respectively. The third term in (1) is Heisenberg interaction, H_{H}, between the two nearest neighbor local spins at sites i and j. , , and is the hopping integral between site of and ions. The effective interaction between the two holes with and , mediated by two nearest neighbor local spins at *i* and *j*, is called TLSMI.

Using the extended Abrikosovβs pseudo-Fermion method in [6], the expression of TLSMI is where , , , , , , , is Fermi distribution, energy of holes, Fermi energy, ) density of states, the number of in a cluster with antiferromagnetic short-range order in the plane, the number of in the nearest-neighbor position in the same cluster, and are any position vectors of two local spins in nearest neighbor. The bar represents the average on Fermi surface. is the transformation factor in the extended Abrikosovβs pseudo-Fermion method [6].

#### 3. Oxygen Isotope Effect in YBCO

Reference [21] estimated the effect of zero-point oscillation on the hopping integral , and gave an isotope mass (M)-dependent formula of the hopping integral, which is , and is circular frequency of optical mode, K. According to our estimation, . From (3) and [22], we see that TLSMI is nearly proportional to , and thus, a small value of can have obvious isotope effect. Taking TLSMI as Cooper pairing potential, considering that the Fermi surface in the CuO_{2} plane has nesting structure [6], and using the values of parameters given in [6], we obtain the following theoretical doping evolutions of the isotope effects for all quantities of YBCO.

The definition of exponent of oxygen isotope effect of is The result of numerical calculations for YBCO is given in Figure 1. The theoretical results fit the data well. The physical origin of the nonmonotonic -dependence in Figure 1 is as follows. Both the experimental data and theoretical calculations show that the -dependence of is nonmonotonic [6, 11]. For example, at , 0.27, and the maximum value of is at . Because the values of at and are very small, even a very small effect of isotope substitution will lead to large exponent of isotope effect. On the contrary, because the value of at is maximum, even the same effect of isotope substitution, as that at and , occurs at will lead to the small exponent of isotope effect.

The definition of exponent of oxygen isotope effect of the pseudogap at is The result of numerical calculations for YBCO is given in Figure 2. There are no data in Figure 2. For reference, we give the data of [12]. The result of experiment in [12] is . Although our theoretical results are not less than 0.01, the values of is only one third of the values of for YBCO. The physical origin of the nonmonotonic -dependence in Figure 2 is nearly the same as that for Figure 1.

The definition of exponent of oxygen isotope effect of the gap at 0βK, is The result of numerical calculations for YBCO is given in Figure 3. There are no data in Figure 3. Therefore, Figure 3 is a prediction for YBCO. The physical origin of the monotonically increasing -dependence in Figure 3 is as follows. Our numerical calculations indicate that the potential of Cooper pairs is a monotonically reducing -dependence (e.g., from 7000βK at to 4500βK at ). The scale of potential of Cooper pairs determines directly the scale of the gap at 0βK. Therefore, even if the effects of isotope substitution are the same, the exponent of oxygen isotope effect of the gap at 0βK and small value of will be less than that at large value of .

The definition of exponent of oxygen isotope effect of the number density of supercurrent carriers at 0βK is The result of numerical calculations for YBCO is given in Figure 4. The data are from [10].

As is well known, the number density of supercurrent carriers at 0βK determines directly the value of . Therefore, the physical origin of the nonmonotonic -dependence in Figure 4 is the same as that for Figure 1.

The definition of exponent of oxygen isotope effect of the temperature , at which the pseudogap begins to open, is The result of numerical calculations for YBCO is given in Figure 5. The data are for [13]. Therefore, Figure 5 is a prediction for YBCO. is in Figure 5. However, is a theoretical result in [3]. The physical origin of the oscillatory and increasing -dependence in Figure 5 comes from many factors. Both the experimental data and our numerical calculations show that the value of is monotonically reducing from small value of (K) to large value of (K). Note that the range of variation of the values of is very high. If we just consider the value of , then the exponent of oxygen isotope effect of the temperature will tend to be monotonic increasing. However, the scale of is determined by the potential of Cooper pairs. This potential is given by (3). From (3), we see that the potential is very sensitive to temperature through the factor in denominator and the factor in the numerator. The variation of means the variation of . The -dependence of the potential in (3) is more complicated, because the , , , and so on in (3) are -dependent. Many factors determine the oscillatory and increasing -dependence.

The definition of exponent of oxygen isotope effect of effective mass in the CuO_{2} plane, , is
The experiment in [10] is , which means that . The explanation for this result is as follows. The carriers in YBCO are polaronic oxygen holes. According to [4], The mass of polaronic oxygen holes are oxygen isotope mass (M) dependent. , and . Therefore, .

The experiments in [14, 16] pointed out that the inverse isotope effect of was observed in some magnetic superconductors. The theory in Section 3 can have inverse isotope effect in principle. The reason is as follows. It is possible that [21]. In this case, the larger the is, the larger the hopping integral is. From (2) and (3), we see that the larger the is, the larger the TLSMI is, and, thus, the larger the is.

#### 4. Conclusions and Discussions

From [6] and this paper, we know clearly that the mechanism in Section 2 for the high- cuprates, that is, the TLSMI between two polaronic oxygen holes that causes the high- superconductivity in YBCO, can explain many experimental findings including oxygen isotope effects of YBCO quantitatively. Right now, the theoretical curves in Figures 2, 3, and 5 are pure predictions for YBCO. If all the predictions can be verified by future experiments, then the mechanism in [6] for the high- YBCO might be a correct unified microscopic theory for both isotope effect and all other properties of the high- cuprates.

Although the theory in this paper cannot yet explain the following two experiments, we know the reasons. (1) Upon oxygen isotope substitution ( versus ), and are shifted from 79βK and 170βK to 78.5βK and 220βK for , respectively, by means of inelastic neutron scattering [15]. However, this huge isotope shift of is absent in NMR and NQR experiments for the same material [15]. We guess that the formula of neutron cross-section, based on which [15] analysis experiment data, is wrong. Because [23, 24] pointed out that the Fermi golden rule, which is used to derive the formula in [15], is not correct. (2) The huge oxygen isotope effect of in [17] cannot be explained by the theory in this paper. We guess that this huge oxygen isotope effect comes from inhomogeneous distribution of carriers in its stripe phase [4, 25].

#### References

- X. J. Chen, V. V. Struzhkin, Z. Wu, H. Q. Lin, R. J. Hemley, and H. K. Mao, βUnified picture of the oxygen isotope effect in cuprate superconductors,β
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 104, no. 10, pp. 3732β3735, 2007. View at Publisher Β· View at Google Scholar Β· View at PubMed - G. M. Zhao, V. Kirtikar, and D. E. Morris, βIsotope effects and possible pairing mechanism in optimally doped cuprate superconductors,β
*Physical Review B*, vol. 63, no. 22, Article ID 220506, pp. 2205061β2205064, 2001. - R. Zeyher and A. Gred, βLarge isotope effect on
*T*in cuprates despite a small electron-phonon coupling,β_{c}*Physical Review B*, vol. 80, no. 6, Article ID 064519, 2009. View at Publisher Β· View at Google Scholar - A. S. Alexandrov,
*Theory of Superconductivity from Weak to Strong Coupling*, Institute of Physics Publishing, Bristol, UK, 2003. - T. Ohno, βCu(2) NQR study of isotope effect in high-
*T*superconductor,β_{c}*Journal of the Physical Society of Japan*, vol. 69, pp. 131β137, 2000. - F.-S. Liu and Y. Hou,
*General Theory of Superconductivity*, chapter 3, Nova, New York, NY, USA, 2008. - V. Z. Kresin and S. A. Wolf, βColloquium: electron-lattice interaction and its impact on high
*T*superconductivity,β_{c}*Reviews of Modern Physics*, vol. 81, no. 2, pp. 481β501, 2009. View at Publisher Β· View at Google Scholar - D. Zech, K. Conder, H. Keller, E. Kaldis, and K. A. Müller, βDoping dependence of the oxygen isotope effect in YBa
_{2}Cu_{3}O_{x},β*Physica B*, vol. 219-220, no. 1-4, pp. 136β138, 1996. View at Publisher Β· View at Google Scholar - H. Katayama-Yoshida, T. Hirooka, A. J. Mascarenhas et al., βIsotope effect in superconducting YBa${}_{2}$Cu${}_{3}$O${}_{7-\delta}$,β
*Japanese Journal of Applied Physics*, vol. 26, no. 12, pp. 2085β2086, 1987. - G.-M. Zhao and D. E. Morris, βObservation of a possible oxygen isotope effect on the effective mass of carriers in YBa
_{2}Cu_{3}O_{6.94},β*Physical Review B*, vol. 51, no. 22, pp. 16487β16490, 1995. View at Publisher Β· View at Google Scholar - J. P. Frank,
*Physical Properties of High Temperature Superconductors*, vol. 4, World Scientific, Singapore, 1994. - G. V. M. Williams, J. L. Tallon, J. W. Quilty, H. J. Trodahl, and N. E. Flower, βAbsence of an isotope effect in the Pseudogap in YBa
_{2}Cu_{4}O_{8}as determined by high-resolution89Y NMR,β*Physical Review Letters*, vol. 80, no. 2, pp. 377β380, 1998. - F. Raffa, T. Ohno, M. Mali et al., βIsotope dependence of the spin gap in YBa
_{2}Cu_{4}O_{8}as determined by Cu NQR relaxation,β*Physical Review Letters*, vol. 81, no. 26, pp. 5912β5915, 1998. - D. R. Penn, M. L. Cohen, and V. H. Crespi, βInverse isotope effects and models for high-
*T*superconductivity,β_{c}*Physical Review B*, vol. 47, no. 9, pp. 5528β5530, 1993. View at Publisher Β· View at Google Scholar - D. R. Temprano, J. Mesot, S. Janssen et al., βLarge isotope effect on the pseudogap in the high-temperature superconductor HoBa
_{2}Cu_{4}O_{8},β*Physical Review Letters*, vol. 84, no. 9, pp. 1990β1993, 2000. - P. M. Shirage, K. Kihou, K. Miyazawa et al., βInverse iron isotope effect on the transition temperature of the (Ba,K)Fe
_{2}As_{2}superconductor,β*Physical Review Letters*, vol. 103, no. 25, Article ID 257003, 2009. View at Publisher Β· View at Google Scholar - G. M. Zhao, K. Conder, H. Keller, and K. A. Müller, βOxygen isotope effects in La
_{2-x}Sr_{x}CuO_{4}: evidence for polaronic charge carriers and their condensation,β*Journal of Physics Condensed Matter*, vol. 10, no. 40, pp. 9055β9066, 1998. - F.-S. Liu, βHigh-
*T*superconductivity enhanced by antiferromagnetism,β_{c}*Chinese Physics Letters*, vol. 6, no. 10, pp. 473β476, 1989. - F. J. Ohkawa, βAuxiliary-particle theory of strongly correlated systems,β
*Journal of the Physical Society of Japan*, vol. 58, no. 11, pp. 4156β4167, 1989. - Fusayoshi J. Ohkawa, βAnisotropic cooper pairs in high-
*T*superconductors,β_{c}*Japanese Journal of Applied Physics, Part 1*, vol. 26, no. 5, pp. L652βL654, 1987. - D. S. Fisher, A. J. Millis, B. Shraiman, and R. N. Bhatt, βZero-point motion and the isotope effect in oxide superconductors,β
*Physical Review Letters*, vol. 61, no. 4, p. 482, 1987. View at Publisher Β· View at Google Scholar - H. Matsukawa and H. Fukuyama, βEffective Hamiltonian for high-
*T*Cu oxides,β_{c}*Journal of the Physical Society of Japan*, vol. 58, no. 8, pp. 2845β2866, 1989. - F. S. Liu, K. D. Peng, and W. F. Chen, βDepartures from the Fermi golden rule,β
*International Journal of Theoretical Physics*, vol. 40, no. 11, pp. 2037β2043, 2001. - F.-S. Liu and W.-F. Chen, βNecessity of exact calculation for transition probability,β
*Communications in Theoretical Physics*, vol. 39, no. 2, pp. 209β211, 2003. - C.-P. Chou and T.-K. Lee, βMechanism of formation of half-doped stripes in underdoped cuprates,β
*Physical Review B*, vol. 81, no. 6, Article ID 060503, 2010. View at Publisher Β· View at Google Scholar