Theory of Isotope Effect in YBa2Cu3O7−𝜹

Abstract

This paper is the first to demonstrate that a pure nonphonon mechanism can quantitatively explain all isotope effect experiments in YBa2Cu3O7−𝛿 (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_𝑐, T∗, pseudogap at 𝑇_𝑐, gap at 0 K, and number density of supercurrent carriers at 0 K. This paper points out that the observed inverse isotope effect of 𝑇_𝑐 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-𝑇_𝑐 is still controversial [1–7]. The isotope effect is an important experimental probe in revealing the underlying pairing mechanism of superconductivity. When high-𝑇_𝑐 superconductivity was discovered in cuprates, the oxygen isotope exponents, defined by 𝛼_𝑇𝑐 = −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-𝑇_𝑐 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-𝑇_𝑐 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-𝑇_𝑐 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 𝑇_𝑐, 𝑇^∗, pseudogap at 𝑇_𝑐, 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-𝑇_𝑐 cuprates is [6]𝐻=−_{𝑖𝛼𝛽𝑠}𝑇_𝛼𝛽𝑝^+𝛼𝑠𝑝_𝛽𝑠+𝐽_𝐾𝑖𝛼_{𝛽𝑠𝑠′}𝐒̂_𝑖⋅𝜎⃗_𝑠𝑠′𝑝^+𝛼𝑠𝑝_𝛽𝑠′+𝐽_𝑖≠𝑗𝐒̂_𝑖⋅𝐒̂_𝑗,(1) where the summation over 𝛼 and 𝛽 is for the oxygen sites around 𝑖th Cu⁺⁺ site, 𝑝_𝛼𝑠 annihilates O_p𝜎 hole with spin s at site 𝛼, 𝐒̂_𝑖 is the local spin operator of Cu⁺⁺ at site i ,𝜎⃗ Pauli matrix vector, and i and j are the nearest neighbors. Expand 𝑝_𝛼𝑠 in 𝐤 space. Here, 𝐤 is the wave vector in Brillouin zone of the oxygen lattice in the CuO₂ plane. The second term in (1) is Kondo Hamiltonian, H_𝐾, which implies that the O_𝑝𝜎 holes with 𝐤↑ and −𝐤↓ can have interactions with the local spins of Cu⁺⁺ 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. 𝐽∝𝑡⁴, 𝐽_𝐾∝𝑡², and 𝑡 is the hopping integral between site of Cu⁺⁺ and O⁻⁻ ions. The effective interaction between the two O_𝑝𝜎 holes with 𝐤↑ and −𝐤↓, 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 𝑈_𝐤𝐤^′=−𝐴(𝑇)𝐹_𝐤𝐤^′ is𝑈_𝐤𝐤^′=−𝐴(𝑇)𝐹_𝐤𝐤^′,(2)𝐴(𝑇)=𝐽𝐽^2𝐾𝑁^/𝑁^𝑤(𝐽/𝑇)𝑇²+64𝐽𝐽^2𝐾∑_𝐤𝐩𝑔(𝐤,𝐩)/1+32𝜋²𝐽^2𝐾𝑁𝐸_𝐹²ℎ(𝐪),(3)𝑔(𝐤,𝐩)=12𝑁_Cu²𝑓𝜖(𝐤)−𝐸_𝐹𝐹₁(𝐤)𝐹₁(𝐩)𝐹₅(𝐤)𝐹₅(𝐩)×𝑒_{𝑖(𝐤−𝐩)⋅(𝐑𝑖′−𝐑𝑖)}𝜖(𝐩)−𝜖(𝐤),(4) where ℎ(𝐪)=[cos(𝑞_𝑥𝑎/2)cos(𝑞_𝑦𝑎/2)]⁴, 𝐹_𝐤𝐤^′=∑^4𝑖=1⋅𝐹_𝑖(𝐤)𝐹_𝑖(𝐤^)𝐹₅(𝐤)𝐹₅(𝐤^), 𝐹₁(𝐤)=cos(𝑘_𝑥𝑎)cos(𝑘_𝑦𝑎), 𝐹₂(𝐤)=sin(𝑘_𝑥𝑎)sin(𝑘_𝑦𝑎), 𝐹₃(𝐤)=sin(𝑘_𝑥𝑎)cos(𝑘_𝑦𝑎), 𝐹₄(𝐤)=cos(𝑘_𝑥𝑎)sin(𝑘_𝑦𝑎), 𝐹₅(𝐤)=cos²(𝑘_𝑥𝑎/2)cos²(𝑘_𝑦𝑎/2), 𝑓(𝑥) is Fermi distribution, 𝜖(𝐤) energy of O_𝐩𝜎 holes, 𝐸_𝐹 Fermi energy, 𝑁(𝐸_𝐹) density of states, 𝑁^ the number of Cu⁺⁺ in a cluster with antiferromagnetic short-range order in the CuO₂ plane, 𝑁^ the number of Cu⁺⁺ in the nearest-neighbor position in the same cluster, and 𝐑_𝑖′−𝐑_𝑖 are any position vectors of two local spins in nearest neighbor. The bar represents the average on Fermi surface. 𝑤(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 𝑡, and gave an isotope mass (M)-dependent formula of the hopping integral, which is 𝑡=𝑡(𝑎_∘)(1+3.71×ℏ×10¹⁶/(2𝑀𝜔_∘1.6²)), and 𝜔_∘ is circular frequency of optical mode, 𝜔_∘=500K. According to our estimation, ¹⁸𝑡/¹⁶𝑡=0.99952. From (3) and [22], we see that TLSMI is nearly proportional to 𝑡⁶, and thus, a small value of ¹⁸𝑡/¹⁶𝑡 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 𝑇_𝑐 is𝛼_𝑇𝑐=−ln¹⁸𝑇_𝑐−ln¹⁶𝑇_𝑐ln18−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 𝑥-dependence in Figure 1 is as follows. Both the experimental data and theoretical calculations show that the 𝑥-dependence of 𝑇_𝑐 is nonmonotonic [6, 11]. For example, 𝑇_𝑐=0 at 𝑥=0.05, 0.27, and the maximum value of 𝑇_𝑐 is at 𝑥=0.16. Because the values of 𝑇_𝑐 at 𝑥≅0.05 and ≅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 𝑇_𝑐 at 𝑥=0.16 is maximum, even the same effect of isotope substitution, as that at 𝑥≅0.05 and ≅0.27, occurs at 𝑥=0.16 will lead to the small exponent of isotope effect.

Figure 1 

Theoretical curve of 𝛼𝑇𝑐 versus 𝑥. 𝑥 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 𝑇_𝑐 is𝛼_Δ(𝑇𝑐)=−ln¹⁸Δ𝑇_𝑐−ln¹⁶Δ𝑇_𝑐ln18−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 𝛼_Δ(𝑇𝑐)<0.01. Although our theoretical results are not less than 0.01, the values of 𝛼_Δ(𝑇c) is only one third of the values of 𝛼_𝑇𝑐 for YBCO. The physical origin of the nonmonotonic 𝑥-dependence in Figure 2 is nearly the same as that for Figure 1.

Figure 2 

Theoretical curve of 𝛼Δ(𝑇𝑐) versus 𝑥 for YBCO.

The definition of exponent of oxygen isotope effect of the gap at 0 K, Δ(0) is𝛼_Δ(0)=−ln¹⁸Δ(0)−ln¹⁶Δ(0)ln18−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 𝑥-dependence in Figure 3 is as follows. Our numerical calculations indicate that the potential of Cooper pairs is a monotonically reducing 𝑥-dependence (e.g., from 7000 K at 𝑥=0.07 to 4500 K at 𝑥=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 𝑥 will be less than that at large value of 𝑥.

Figure 3 

Theoretical curve of 𝛼Δ(0) versus 𝑥 for YBCO.

The definition of exponent of oxygen isotope effect of the number density of supercurrent carriers at 0 K is𝛼_𝑛𝑠(0)=−ln¹⁸𝑛_𝑠(0)−ln¹⁶𝑛_𝑠(0)ln18−ln16.(8) The result of numerical calculations for YBCO is given in Figure 4. The data are from [10].

Figure 4 

Theoretical curve of 𝑛𝑠(0) versus 𝑥 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 𝑇_𝑐. Therefore, the physical origin of the nonmonotonic 𝑥-dependence in Figure 4 is the same as that for Figure 1.

The definition of exponent of oxygen isotope effect of the temperature 𝑇^∗, at which the pseudogap begins to open, is𝛼_𝑇^∗=−ln¹⁸𝑇^∗−ln¹⁶𝑇^∗ln18−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. 𝛼_𝑇^∗>0 is in Figure 5. However, 𝛼_𝑇^∗<0 is a theoretical result in [3]. The physical origin of the oscillatory and increasing 𝑥-dependence in Figure 5 comes from many factors. Both the experimental data and our numerical calculations show that the value of 𝑇^∗ is monotonically reducing from small value of 𝑥 (𝑥=0.05𝑇^∗=350K) to large value of 𝑥 (𝑥=0.27𝑇^∗=20K). Note that the range of variation of the values of 𝑇^∗ is very high. If we just consider the value of 𝑇^∗, then the exponent of oxygen isotope effect of the temperature 𝑇^∗ will tend to be monotonic increasing. However, the scale of 𝑇^∗ 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 𝑤(𝐽/𝑇) factor in the numerator. The variation of 𝑇^∗ means the variation of 𝑇. The 𝑥-dependence of the potential in (3) is more complicated, because the 𝐽, 𝐽_𝐾, 𝐸_𝐹, and so on in (3) are 𝑥-dependent. Many factors determine the oscillatory and increasing 𝑥-dependence.

Figure 5 

Theoretical curve of 𝛼𝑇∗ versus 𝑥 for YBCO. The data are for YBa2Cu4O8 [13].

The definition of exponent of oxygen isotope effect of effective mass in the CuO₂ plane, 𝑚^∗𝑎𝑏, is𝛼_𝑚^∗𝑎𝑏=−ln¹⁸𝑚^∗𝑎𝑏−ln¹⁶𝑚^∗𝑎𝑏ln18−ln16.(10) The experiment in [10] is 𝛼_𝑚^∗𝑎𝑏=−0.061, which means that ¹⁸𝑚^∗𝑎𝑏>¹⁶𝑚^∗𝑎𝑏. 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. 𝑚^∗𝑎𝑏∝exp(𝑎√𝑀), and 𝑎>0. Therefore, ¹⁸𝑚^∗𝑎𝑏>¹⁶𝑚^∗𝑎𝑏.

The experiments in [14, 16] pointed out that the inverse isotope effect of 𝛼_𝑇𝑐 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 𝑡=𝑡(𝑎_∘)(1−3.71×ℏ×10¹⁶/(2𝑀𝜔_∘1.6²)) [21]. In this case, the larger the 𝑀 is, the larger the hopping integral 𝑡 is. From (2) and (3), we see that the larger the 𝑡 is, the larger the TLSMI is, and, thus, the larger the 𝑇_𝑐 is.

4. Conclusions and Discussions

From [6] and this paper, we know clearly that the mechanism in Section 2 for the high-𝑇_𝑐 cuprates, that is, the TLSMI between two polaronic oxygen holes that causes the high-𝑇_𝑐 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-𝑇_𝑐 YBCO might be a correct unified microscopic theory for both isotope effect and all other properties of the high-𝑇_𝑐 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), 𝑇_𝑐 and 𝑇^∗ 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 𝑇^∗ 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 𝑇_𝑐 in La_2−𝑥Sr_𝑥CuO₄ [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].