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ISRN Condensed Matter Physics
VolumeΒ 2011Β (2011), Article IDΒ 961832, 5 pages
http://dx.doi.org/10.5402/2011/961832
Research Article

Theory of Isotope Effect in YBa2Cu3O7βˆ’πœΉ

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 YBa2Cu3O7βˆ’π›Ώ (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 𝑇𝑐, Tβˆ—, pseudogap at 𝑇𝑐, gap at 0 K, and number density of supercurrent carriers at 0 K. This paper points out that the observed inverse isotope effect of 𝑇𝑐 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-𝑇𝑐 is still controversial [1–7]. The isotope effect is an important experimental probe in revealing the underlying pairing mechanism of superconductivity. When high-𝑇𝑐 superconductivity was discovered in cuprates, the oxygen isotope exponents, defined by 𝛼𝑇𝑐 = βˆ’dlnTc/dlnM, with M being the oxygen isotopic mass, were promptly measured [8–17]. In [1, 7] the authors think that such elaborate isotope effects strongly suggest that high-𝑇𝑐 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-𝑇𝑐 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-𝑇𝑐 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, 18–20]. 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 18O and 16O as the origin of isotope effect in YBCO and make numerical calculations for the oxygen isotope effects of all quantities such as 𝑇𝑐, π‘‡βˆ—, pseudogap at 𝑇𝑐, gap at 0 K, and number density of supercurrent carriers at 0 K. 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-p𝜎 model to describe the CuO2 plane of high-𝑇𝑐 cuprates is [6]𝐻=βˆ’π‘–π›Όπ›½π‘ π‘‡π›Όπ›½π‘+𝛼𝑠𝑝𝛽𝑠+π½πΎξ“π‘–π›Όπ›½π‘ π‘ β€²Μ‚π’π‘–β‹…βƒ—πœŽπ‘ π‘ β€²π‘+𝛼𝑠𝑝𝛽𝑠′+𝐽𝑖≠𝑗̂𝐒𝑖⋅̂𝐒𝑗,(1) where the summation over 𝛼 and 𝛽 is for the oxygen sites around 𝑖th Cu++ site, 𝑝𝛼𝑠 annihilates Op𝜎 hole with spin s at site 𝛼, ̂𝐒𝑖 is the local spin operator of Cu++ at site i ,βƒ—πœŽ Pauli matrix vector, and i and j are the nearest neighbors. Expand 𝑝𝛼𝑠 in 𝐀 space. Here, 𝐀 is the wave vector in Brillouin zone of the oxygen lattice in the CuO2 plane. The second term in (1) is Kondo Hamiltonian, H𝐾, which implies that the Oπ‘πœŽ holes with 𝐀↑ and βˆ’π€β†“ can have interactions with the local spins of Cu++ at sites i and j, respectively. The third term in (1) is Heisenberg interaction, HH, between the two nearest neighbor local spins at sites i and j. π½βˆπ‘‘4, π½πΎβˆπ‘‘2, and 𝑑 is the hopping integral between site of Cu++ and Oβˆ’βˆ’ ions. The effective interaction between the two Oπ‘πœŽ holes with 𝐀↑ and βˆ’π€β†“, mediated by two nearest neighbor local spins at i and j, is called TLSMI.

Using the extended Abrikosov’s pseudo-Fermion method in [6], the expression of TLSMI π‘ˆπ€π€β€²=βˆ’π΄(𝑇)𝐹𝐀𝐀′ isπ‘ˆπ€π€β€²=βˆ’π΄(𝑇)𝐹𝐀𝐀′,(2)𝐴(𝑇)=𝐽𝐽2πΎπ‘ξ…žξ…ž/π‘ξ…žπ‘€(𝐽/𝑇)𝑇2+64𝐽𝐽2πΎβˆ‘π€π©ξ‚†π‘”(𝐀,𝐩)/1+32πœ‹2𝐽2𝐾𝑁𝐸𝐹2,β„Ž(πͺ)(3)ξ‚΅1𝑔(𝐀,𝐩)=2𝑁Cuξ‚Ά2π‘“ξ€Ίπœ–(𝐀)βˆ’πΈπΉξ€»πΉ1(𝐀)𝐹1(𝐩)𝐹5(𝐀)𝐹5(𝐩)πœ–(𝐩)βˆ’πœ–(𝐀)×𝑒𝑖(π€βˆ’π©)β‹…(π‘π‘–β€²βˆ’π‘π‘–),(4) where β„Ž(πͺ)=[cos(π‘žπ‘₯π‘Ž/2)cos(π‘žπ‘¦π‘Ž/2)]4, 𝐹𝐀𝐀′=βˆ‘4𝑖=1⋅𝐹𝑖(𝐀)𝐹𝑖(π€ξ…ž)𝐹5(𝐀)𝐹5(π€ξ…ž), 𝐹1(𝐀)=cos(π‘˜π‘₯π‘Ž)cos(π‘˜π‘¦π‘Ž), 𝐹2(𝐀)=sin(π‘˜π‘₯π‘Ž)sin(π‘˜π‘¦π‘Ž), 𝐹3(𝐀)=sin(π‘˜π‘₯π‘Ž)cos(π‘˜π‘¦π‘Ž), 𝐹4(𝐀)=cos(π‘˜π‘₯π‘Ž)sin(π‘˜π‘¦π‘Ž), 𝐹5(𝐀)=cos2(π‘˜π‘₯π‘Ž/2)cos2(π‘˜π‘¦π‘Ž/2), 𝑓(π‘₯) is Fermi distribution, πœ–(𝐀) energy of O𝐩𝜎 holes, 𝐸𝐹 Fermi energy, 𝑁(𝐸𝐹) density of states, π‘ξ…ž the number of Cu++ in a cluster with antiferromagnetic short-range order in the CuO2 plane, π‘ξ…žξ…ž the number of Cu++ in the nearest-neighbor position in the same cluster, and π‘π‘–β€²βˆ’π‘π‘– are any position vectors of two local spins in nearest neighbor. The bar represents the average on Fermi surface. 𝑀(𝐽/𝑇) is the transformation factor in the extended Abrikosov’s pseudo-Fermion method [6].

3. Oxygen Isotope Effect in YBCO

Reference [21] estimated the effect of zero-point oscillation on the hopping integral 𝑑, and gave an isotope mass (M)-dependent formula of the hopping integral, which is 𝑑=𝑑(π‘Žβˆ˜)(1+3.71×ℏ×1016/(2π‘€πœ”βˆ˜1.62)), and πœ”βˆ˜ is circular frequency of optical mode, πœ”βˆ˜=500K. According to our estimation, 18𝑑/16𝑑=0.99952. From (3) and [22], we see that TLSMI is nearly proportional to 𝑑6, and thus, a small value of 18𝑑/16𝑑 can have obvious isotope effect. Taking TLSMI as Cooper pairing potential, considering that the Fermi surface in the CuO2 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 𝑇𝑐 is𝛼𝑇𝑐=βˆ’ln18π‘‡π‘βˆ’ln16𝑇𝑐.ln18βˆ’ln16(5) 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 π‘₯-dependence in Figure 1 is as follows. Both the experimental data and theoretical calculations show that the π‘₯-dependence of 𝑇𝑐 is nonmonotonic [6, 11]. For example, 𝑇𝑐=0 at π‘₯=0.05, 0.27, and the maximum value of 𝑇𝑐 is at π‘₯=0.16. Because the values of 𝑇𝑐 at π‘₯β‰…0.05 and β‰…0.27 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 𝑇𝑐 at π‘₯=0.16 is maximum, even the same effect of isotope substitution, as that at π‘₯β‰…0.05 and β‰…0.27, occurs at π‘₯=0.16 will lead to the small exponent of isotope effect.

961832.fig.001
Figure 1: Theoretical curve of 𝛼𝑇𝑐 versus π‘₯. π‘₯ is the number of oxygen hole per cell of Cu lattice in the CuO2 plane of YBCO. The filled circle is from [10]. The open circle is from [1, 11].

The definition of exponent of oxygen isotope effect of the pseudogap at 𝑇𝑐 is𝛼Δ(𝑇𝑐)=βˆ’ln18Ξ”ξ€·π‘‡π‘ξ€Έβˆ’ln16Δ𝑇𝑐.ln18βˆ’ln16(6) 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 YBa2Cu4O8 [12]. The result of experiment in [12] is 𝛼Δ(𝑇𝑐)<0.01. Although our theoretical results are not less than 0.01, the values of 𝛼Δ(𝑇c) is only one third of the values of 𝛼𝑇𝑐 for YBCO. The physical origin of the nonmonotonic π‘₯-dependence in Figure 2 is nearly the same as that for Figure 1.

961832.fig.002
Figure 2: Theoretical curve of 𝛼Δ(𝑇𝑐) versus π‘₯ for YBCO.

The definition of exponent of oxygen isotope effect of the gap at 0 K, Ξ”(0) is𝛼Δ(0)=βˆ’ln18Ξ”(0)βˆ’ln16Ξ”(0).ln18βˆ’ln16(7) 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 π‘₯-dependence in Figure 3 is as follows. Our numerical calculations indicate that the potential of Cooper pairs is a monotonically reducing π‘₯-dependence (e.g., from 7000 K at π‘₯=0.07 to 4500 K at π‘₯=0.25). The scale of potential of Cooper pairs determines directly the scale of the gap at 0 K. Therefore, even if the effects of isotope substitution are the same, the exponent of oxygen isotope effect of the gap at 0 K and small value of π‘₯ will be less than that at large value of π‘₯.

961832.fig.003
Figure 3: Theoretical curve of 𝛼Δ(0) versus π‘₯ for YBCO.

The definition of exponent of oxygen isotope effect of the number density of supercurrent carriers at 0 K is𝛼𝑛𝑠(0)=βˆ’ln18𝑛𝑠(0)βˆ’ln16𝑛𝑠(0).ln18βˆ’ln16(8) The result of numerical calculations for YBCO is given in Figure 4. The data are from [10].

961832.fig.004
Figure 4: Theoretical curve of 𝑛𝑠(0) versus π‘₯ for YBCO. The data are from [10].

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

The definition of exponent of oxygen isotope effect of the temperature π‘‡βˆ—, at which the pseudogap begins to open, isπ›Όπ‘‡βˆ—=βˆ’ln18π‘‡βˆ—βˆ’ln16π‘‡βˆ—.ln18βˆ’ln16(9) The result of numerical calculations for YBCO is given in Figure 5. The data are for YBa2Cu4O8 [13]. Therefore, Figure 5 is a prediction for YBCO. π›Όπ‘‡βˆ—>0 is in Figure 5. However, π›Όπ‘‡βˆ—<0 is a theoretical result in [3]. The physical origin of the oscillatory and increasing π‘₯-dependence in Figure 5 comes from many factors. Both the experimental data and our numerical calculations show that the value of π‘‡βˆ— is monotonically reducing from small value of π‘₯ (π‘₯=0.05π‘‡βˆ—=350K) to large value of π‘₯ (π‘₯=0.27π‘‡βˆ—=20K). Note that the range of variation of the values of π‘‡βˆ— is very high. If we just consider the value of π‘‡βˆ—, then the exponent of oxygen isotope effect of the temperature π‘‡βˆ— will tend to be monotonic increasing. However, the scale of π‘‡βˆ— 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 𝑇 through the 𝑇2 factor in denominator and the 𝑀(𝐽/𝑇) factor in the numerator. The variation of π‘‡βˆ— means the variation of 𝑇. The π‘₯-dependence of the potential in (3) is more complicated, because the 𝐽, 𝐽𝐾, 𝐸𝐹, and so on in (3) are π‘₯-dependent. Many factors determine the oscillatory and increasing π‘₯-dependence.

961832.fig.005
Figure 5: Theoretical curve of π›Όπ‘‡βˆ— versus π‘₯ for YBCO. The data are for YBa2Cu4O8 [13].

The definition of exponent of oxygen isotope effect of effective mass in the CuO2 plane, π‘šβˆ—π‘Žπ‘, isπ›Όπ‘šβˆ—π‘Žπ‘=βˆ’ln18π‘šβˆ—π‘Žπ‘βˆ’ln16π‘šβˆ—π‘Žπ‘.ln18βˆ’ln16(10) The experiment in [10] is π›Όπ‘šβˆ—π‘Žπ‘=βˆ’0.061, which means that 18π‘šβˆ—π‘Žπ‘>16π‘šβˆ—π‘Žπ‘. 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. π‘šβˆ—π‘Žπ‘βˆšβˆexp(π‘Žπ‘€), and π‘Ž>0. Therefore, 18π‘šβˆ—π‘Žπ‘>16π‘šβˆ—π‘Žπ‘.

The experiments in [14, 16] pointed out that the inverse isotope effect of 𝛼𝑇𝑐 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 𝑑=𝑑(π‘Žβˆ˜)(1βˆ’3.71×ℏ×1016/(2π‘€πœ”βˆ˜1.62)) [21]. In this case, the larger the 𝑀 is, the larger the hopping integral 𝑑 is. From (2) and (3), we see that the larger the 𝑑 is, the larger the TLSMI is, and, thus, the larger the 𝑇𝑐 is.

4. Conclusions and Discussions

From [6] and this paper, we know clearly that the mechanism in Section 2 for the high-𝑇𝑐 cuprates, that is, the TLSMI between two polaronic oxygen holes that causes the high-𝑇𝑐 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-𝑇𝑐 YBCO might be a correct unified microscopic theory for both isotope effect and all other properties of the high-𝑇𝑐 cuprates.

Although the theory in this paper cannot yet explain the following two experiments, we know the reasons. (1) Upon oxygen isotope substitution (16O versus 18O), 𝑇𝑐 and π‘‡βˆ— are shifted from 79 K and 170 K to 78.5 K and 220 K for HoBa2Cu4O8, respectively, by means of inelastic neutron scattering [15]. However, this huge isotope shift of π‘‡βˆ— 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 𝑇𝑐 in La2βˆ’π‘₯Srπ‘₯CuO4 [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].

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