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Abstract

It has been known that sample dimension and lattice defects affect critical behavior of superconductivity. Particular type of porous samples called fractal bodies have well-defined non-integer dimensions dependent exclusively on the geometrical feature of structures. Differences in sample dimension and lattice defects are quantitatively described by such fractal dimensions (Ds). Powder samples of high-T_C cuprates with 2.5 £ D £ 3 were prepared, and their superconducting properties were compared as regards critical temperature T_C, critical current J_C, and critical magnetic field H_C. While the variation in H_C for different Ds was within experimental error (for the lower H_C) or too high to be measured (for the upper H_C), T_C and J_C correlated with D: they both exhibited complex patterns of D-dependence immediately below D = 3 and maximum enhancement of ∽0.7% in T_C and ∽7% in J_C were observed. The observed behavior is semi-quantitatively explained by considering fractal structures of the samples. In addition, the D-dependences of T_C and J_C are explained with the same quantitative discussion as those of Néel temperatures of fractal CoO powder samples. This suggests underlying universal physics for cooperative phenomena in fractal dimensions.

Key words

High-TC cuprate, magnetization curve, sponge structures, superconducting characteristics, critical phenomena

Introduction

YBa₂Cu₃O_7-d (YBCO) and its related copper oxides have been the centre of interest in the research field of superconductivity, mainly because of their exceptionally high critical temperatures (T_Cs) [1,2]. During the hundred years since the discovery of superconductivity, the highest T_C has increased by ~150 K, which is found in HgBa₂Ca₂Cu₃O_y [3-7]. Besides T_C, the superconducting characteristics include critical magnetic fields H_C and critical current densities J_C. They are also important properties in both terms of basic and applied research. There are extensive studies to enhance H_C and J_C. It is well known that these characteristics vary depending not only on the chemical formulae and the crystal structures, but also on the sample forms such as thin films and fibres. Similarly lattice defects also affect these superconducting properties. Generally speaking, samples with low-dimensional forms and with many defects often exhibit evidently low T_C, H_C, and J_C. Such a trend has been empirically known, yet there are few systematic or quantitative experimental studies on the relation between the dimensionalities of samples and the superconducting properties, though there are extensive theoretical studies using fractal dimensions (Ds) [8-15]. The effects of defects on the superconducting properties have been studied by irradiating ions or neutrons at the single crystals [16-23]. The important findings include enhancement of T_C or J_C after introduction of controlled defects by the particle-irradiation. The YBCO sponges are synthesized by use of a sol-gel method with appropriate additives/dopants, and the sponges exhibit improved J_Cs [24]. Although the authors do not refer to the fractal dimensions of the sponges, they apparently form fractal bodies judging from the SEM/TEM images (Figures1b-1d [24]). In the meantime, the XRD patterns clearly indicate that the sponges contain impurities such as BaCuO₂ and Y₂BaCuO₅, the net effects on J_C of the sample form (sponge) and impurities cannot be distinguished.

Alternatively, defects can be introduced simply by mixing the powder samples with alkylketene dimer (AKD), and the amount and distribution of defects can be quantitatively estimated by fractal dimensions of the mixed powder samples [25-27]. This method cannot produce geometrically controlled defects such as columnar defects in single crystals, yet it is a facile and reproducible way to produce isotropic defects in a powder sample with a finely controlled value of D (2.50 £ D £ 3.00). Using this method, the magnetic ordering in the cobalt oxide CoO having fractal dimensions has been recently studied [27]. The results show that the magnetic ordering temperatures, i.e. the Néel temperatures (T_Ns) of CoO depend on D, and rapidly enhance immediately below D = 3. The observed behaviour (T_N vs. D) is explained by a simple calculation considering the interactions between magnetic sites (i.e. spins) assumed to be distributed to give a fractal dimension D. Since this model is general and appears to be appropriately applied to other cooperative phenomena, the superconducting properties T_C, H_C and J_C are examined on the powder sample of YBCO with different Ds in this work. The detailed results and discussion are described below.

Experimental

Materials

YBCO and AKD were synthesized by standard procedures described in Supporting Information. The value of d in the chemical formula of YBa₂Cu₃O_7-d was determined to be d = 0.01-0.04 by iodometry using nine different samples with 3-time titrations each. The powder samples of YBCO with desired fractal dimensions were prepared by the same procedures applied to the CoO powder samples having fractal dimensions [27]; AKD and YBCO were mixed and well ground in the agate mortar to be homogeneous fine powder (~for an hour). Because AKD is independent of superconducting mechanism, AKD plays a role of defects, corresponding to void spaces and holes in fractal bodies. The fractal dimension D was controlled by the mixing ratio between YBCO and AKD. Details of how we estimated the fractal dimension D are described in Appendix A. Experimental errors are ±0.05 in D and << 0.1 % in AKD content. The errors in D were estimated from the arbitrariness and statistical errors (standard deviations) in the curve-fitting analysis of neutron diffraction of the mixed powder samples with AKD [25, 27]. The arbitrariness was examined by independent curve-fitting analyses with different ranges and initial parameters. Thus obtained mixed powder samples were characterized by powder X-ray diffraction (PXD; Figure S2; one sample for each D), X-ray photoelectron spectroscopy (XPS; Figure S3; five samples for each D) and scanning electron microscopy (SEM; Figure S4; seven samples for each D) in order to check any artefact originated from the sample. All these results of characterization clearly indicated that the mixing with AKD changes none of the crystal structure and d in the chemical formula of YBCO.

Physical measurements

The electrical resistivity was measured on the compaction pellets by a four-probe method. The electrical contacts were made by gold wires (25 mmf) and gold paste (Tokuriki Chemical Research Co., Ltd. No. 8560). The equipment was home-made liquid-nitrogen-cryostat consisting of a digital voltmeter (Keithley Nanovoltmeter 2182A), a current/voltage source (Keithley SourceMeter 2400), a digital temperature controller (LakeShore, Model 331), and a diffusion/rotary pumping system (DIAVAC, DS-A412N). In order to minimize the joule heating of the sample and a resultant thermoelectric power, a constant current of 0.3-2 mA was applied for 20 ms, and the voltage drop was measured in a synchronized way with the current source. Immediately (~ 20 ms) after the first measurement, the voltage was reversed and the same measurement was carried out. Then the average was taken to cancel out the thermoelectric power. The measurement was carried out on two samples having the same D to confirm sample dependences and reproducibility. The errors in the electrical resistivity measurements were estimated to be ~±0.25 for T [K] and ~ ±0.3 for r [Wcm] based on scatterings and sample dependences of the data.

The magnetic susceptibility was measured using a SQUID (Quantum Design, MPMS-XL7minBXR3). The powder sample was formed into a compaction pellet of 1-2 mm in height and 2.0 mm in diameter (~2-18 mg), and set in a gelatin capsule. The capsule was cylindrical (EM-Japan, G7330, No.4, 5.0 mm in inner diameter, 13.9 mm in height) and diamagnetic (~ -10^-4 emu) and had some pinholes for ventilation. The capsule was set in the middle of a polystyrene straw (Quantum Design), and subjected to a DC measurement. Both of field cooled (FC) and zero-field cooled (ZFC) processes were examined. For the determination of T_C, the magnetization were measured in an FC process from 300 to 2 K with a magnetic field H = 100 G. For H_C and J_C, the magnetization curves were measured from 0 to 4 T, 4 to - 4 T, and then - 4 to 0.5 T at 2 K. The measurement was carried out on five samples having the same D to confirm sample dependences and reproducibility. The errors in the magnetic susceptibility measurements are estimated to be ~±1´10^-5 [T] for the applied magnetic field H, ~±0.1% for observed magnetization, and ~ ±0.25 [K] for the temperature control based on the specification of the equipment utilized and reproducibility observed.

Estimation of J_C

J_C was estimated by following reported methods [28,29]. For a type II superconductor with a cylindrical shape (radius = R), J_C can be estimated from the observed hysteresis in the magnetization curve using equation (1).

where C is a constant dependent on experimental conditions including the shape of the sample, and M⁺ (H,T) and M^- (H,T) are observed values of magnetization in M-H curves at a given magnetic field H and temperature T in the decreasing- and increasing-field processes, respectively (M ⁺ > M ^- ).

Results

For determination of T_C, Figures1(a)-1(e) show the temperature dependences of magnetic susceptibility c and resistivity r, in addition to T_C (midpoint) vs. D. The D-dependences of onset T_C and offset T_C are shown in Figure S5 in Supporting Information. Figure 2 shows the observed hysteresis curves at 2 K for the sample of 2.5 £ D £ 3. For more details, each curve is shown in Supporting Information (Figures S6(a)-S6(f)). The observed hysteresis curves well agree with those previously reported [29]. In all the samples, the lower critical fields H_C1 occur at H ~ 2310 G, while upper critical fields H_C2 far exceed the highest field available (7 T) in the measurements. The minimum values of magnetization, M^- are plotted against the fractal dimensions D in Figure 3. The D-dependence well agrees with that of T_N (observed behavior) of the fractal spin systems in CoO/AKD [27].

Figure 2.  Hysteretic M-H curves at 2 K for the fractal YBCO sample of 2.5 £ D £ 3. (a) data of a whole loop of H = 0 ® 4 T ® - 4 T ® 0.5 T, and (b) Quadrant IV of the same data; note that magnetization is shown by absolute values. For clarity each curve is offset by 50 Gg^-1 with its baseline (dotted line). The YBCO content (vol%) and D of each sample are designated at the top/side of the graph.

The values of J_C derived from the hysteresis in Figure 2(a) using equation (1) are shown in Figure 4. All the magnetic susceptibility data measured on the five samples were utilized to extract J_C, which are similar to each other, and typical results are shown in Figure 4. As is well known, J_C rapidly decreases with increasing external field H [30,31]. For comparison, we shall discuss the values of J_C normalized by J_C at D = 3 (Figure 5). The errors in J_C and J_C(D)/J_C(3) are approximately ±1.5% and ±2.5% of each value, respectively. They are estimated from data scatterings in Figures 4(a) and 4(b) by taking the standard deviations. With decreasing D from 3 to 2.5, J_C decreases rapidly until it takes a minimum at D = 2.92. Then it makes a sudden upturn to take a maximum at D = 2.88, and then it exhibits a plateau or a subtle increase between D ~ 2.8 and D = 2.5. This behavior does not qualitatively depend on H, and well corresponds to a reverse behavior of that in Figure 3 [i.e., (4pM)^-1 vs. D] as well as with our previous observation of T_N vs. D in the fractal spin system in CoO [27].

Figure 4.  J_C derived from the hysteresis in Fig. 2(a) using equation (1). (a) Raw data, and (b) data normalized by < em>J_C with D = 3.00. The YBCO content (vol%) and D of each sample are designated at the top of the graph.

Discussion

Superconducting characteristics of fractal YBCO

The sintered compacts of YBCO were confirmed to be an orthorhombic phase of YBa₂Cu₃O_7-d with d = 0.02±0.02 by X-ray powder patterns and iodometry, respectively. After mixing with AKD, XPS and PXD proved that YBCO retained its original crystal and electronic structures irrespective of D. The observed differences in H_C1 among the samples with different Ds are within experimental error (Figures 2 and S6). The values of H_C2 are too high to be measured. Thus we shall discuss the D-dependence of T_C and J_C below.

Relationship between D and Superconducting Properties (J_C and T_C)

Apparently, the observed J_C deviates from an exponential field dependence (Figure 4(a)). This can be explained by assuming two different pinning sites in the fractal YBCO, i.e., the defects in the AKD part and those in the net YBCO part. Thus a curve-fitting analysis was carried out using equation (2),

where H, J_C,0, A_i, and t_i ( i = 1,2) indicate magnetic field strength [G], offset values of J_C [A cm^-2], amplitudes [A cm^-2], and relaxation constant [G^-1], respectively. Among the parameters of J_C,0, A_i, and t_i, only t_i (i = 1,2) has a physical meaning intrinsic to the sample and independent of the experimental conditions.  The value of J_C,0 originates from residual magnetization and depends on the amount and distribution of defects in samples and the experimental conditions such as magnetic fields. The absolute values of A_i depend on the system sizes (the volumes of the AKD and YBCO parts). The relative values of A_i, i.e., A₁/A₂ or A₂/A₁, have a physical meaning intrinsic to the fractal dimension D. The relative values of A_i describe the ratio of the number of the flux lines penetrating the AKD parts to that penetrating the YBCO parts. Because AKD (a diamagnetic insulator) and YBCO should have clearly different strengths of pinning magnetic flux lines and thus clearly different values of t_i, the curve-fitting analysis above can distinguish one from the other. The fitting results are shown in Supporting Information (Figures S7(a)-7(g) and Table S3) and the obtained parameters are summarized in Figure 6. The errors in t_i^-1, A_i, and D are approximately ±2.5 %, ±2.5 %, and ±0.01, respectively, which was estimated from the arbitrariness and statistical errors (standard deviations) in the curve-fitting analysis of J_c using equation (2). The arbitrariness was examined by independent curve-fitting analyses with different ranges and initial parameters. The values of t_i^-1 (i = 1,2) respectively remain constant through 2.5 £ D £ 3.0 (Figure 6(a)), which supports the validity of our assumption that the respective term in equation (2) is related to either of the chemical components, the AKD or (defects in) the YBCO parts. The t₁^-1 [= (1.8 ± 0.4) ´ 10^-5 G^-1] is smaller than the other [t₂^-1 = (1.9 ± 0.2) ´ 10^-4 G^-1] by one order of magnitude. Thus the magnetic flux lines penetrating the parts characterized by t₁ are more tightly pinned than the other and less sensitive to increase in magnetic field H. Therefore t₁ and t₂ should correspond to the AKD and the YBCO parts, respectively. The values of A₁/A₂ exhibit a steep rise below D = 3.00, take a sharp maximum at D = 2.98 and another local maximum at D = 2.92 (Figure 6(b)). We shall discuss the origin of this characteristic behavior later. Such behavior semi-quantitatively agrees our previous observation about T_N vs. D of the CoO/AKD fractal samples (Figure 3, inset) [27]. Although superconductivity appears to be a quite different phenomenon from magnetic ordering, the D-dependence of T_C and J_C can be explained by the same discussion as the D-dependence of T_N [26, 27] as shown in the next section. By considering an infinite cubic lattice, the D-dependence of critical (Curie or Néel) temperature T_C is described by equation (3),

where z, a, J₁, S, k_B and G(D) are the number of nearest neighbor site, a spatial decay constant of spin-spin interaction (a < 1), the exchange integral between the spin at the origin and one of the spins at the nearest neighbor sites, the spin momentum, the Boltzmann constant, and the Gamma function. The model described by equation (3) is based on a mean field approximation and a Heisenberg model. The calculated T_N of CoO within this framework is quantitatively consistent with the observed value [32]. In this model, the exchange energy between the spin at the origin and all other spins is considered to evaluate the magnetic ordering temperature T_C. The interaction between and is assumed to decay in proportional to , where b is a constant depending only on the distance between and and thus b is lattice-dependent. The Gamma function describes the number of spins with a distribution having fractal dimension D. The parameter a is introduced to approximately describe the exchange integral J_0j between   and a spin at the distance j in a progressive manner using the exchange integral J₁ between and one of its nearest neighbor spins.

The first term of the exponent (~1.37j) of a is dependent on the geometry of the lattice selected for the model, yet any lattice should give a finite number of a similar order. Thus the term

is considered to be qualitatively similar in general cases.

In a finite system we should take the surface effect into consideration by introducing a correction factor as equation (4) [27].

, where R_S/V is the ratio of surface area to volume of a given fractal body. Note that the word “surface effect” includes the effect from grain boundaries. The D-dependence of T_C described by equation (4) is shown in the inset of Figure 3 (the black line).

Fractal model in this work

In our previous work, a combination of a Heisenberg model and a fractal spin distribution was applied to explain the observed variation in the Néel temperatures of CoO/AKD mixed powder samples. Here it should be made clear how our previous model can be applied to superconductivity. Firstly, the D-dependence of a cooperative phenomenon, if any, is based on the number of sites around a particular site located at the origin. Because of the self-similarity of fractal bodies, any site can be this “particular site at the origin”. The number of sites is dependent only on the geometrical feature of the site distribution. This distribution is quantitatively and generally described by its unique fractal dimension D. Since both superconductivity and magnetic ordering belong to cooperative phenomena and both are based on interactions between sites, it is considered that their D-dependences should be qualitatively common, i.e. T_C should follow the same type of formula like equation (3).

T_C= (constant A) × f(D) (3)’

where “constant A” is independent of D (but can be a function of any other parameters related to the system and phenomenon), and f(D) is a function of D only.

It should be noted here again that our previous model, equation (3), is an application of a Heisenberg model to a spin system having fractal distribution, and that the model is mathematically a product of two terms corresponding the two independent origins as shown by equation (3)’. For example, in the case of the Heisenberg CoO/AKD system, f(D) can be formulated as follows.

In our generalized model described by equation (3)’, all specific conditions are included in the constant A, such as Heisenberg spin system, superconducting mechanism, and so on. Thus the actual values of constant A differ between magnetic ordering and superconducting transitions. However, the Gamma function in equation (3) directly and exclusively originates from the fractal spin distribution, and a spatial decay of inter-site interaction can be generally expressed by an appropriate a (a < 1). Here inter-site interaction is meant for intermolecular or interatomic interaction that forms the electronic structure including electron-phonon interaction. Accordingly, the dependence on D of the T_Cs of different phenomena should be qualitatively or approximately described by f(D), if the assumption above is valid. In this work we are not interested in the precise value or analytical formula of constant A, but interested in how T_C depends on D, i.e. whether T_C can be connected with f(D) or not. Additionally J_C might have a similar dependence on D.

Thus a tentative discussion will be made here based on the abovementioned analogy. One can obtain the same D-dependence of T_C simply by interpreting that the Gamma function describes the number of superconducting sites instead of the number of magnetic sites (spins). Here the superconducting site means the minimum YBCO part having the dimension of the coherent lengths and exhibiting bulk superconductivity. This is why Figure 1(e) (T_C vs. D) resembles the inset (the red curve) of Figure 3 (T_N vs. D).

Figure 1.  Temperature dependences of magnetic susceptibility c and resistivity r. Fractal dimensions D are designated in each graph. (a) overall magnetic behavior below 100 K (cooled process, H = 100 G); note that 4pc is dimensionless, (b) close view around T_C, (c) overall electrical behavior, (d) close view around T_C, and (e) T_C as midpoint vs. D. In (e) red curve is a guide for eye.

The D-dependences of 4pM (Figure 3), J_C(D)/J_C(3) (Figure 5) and A₁/A₂ (Figure 6(b)) share the following features;

(A) Complicated dependence on D at ~2.8 £ D £ 3.0

(B) A sharp maximum (in Figures 3 and 6(b)) or minimum (in Figures 5 and 6(b)) at D = 2.95±0.03

(C) A broad maximum (in Figures 3 and 6(b)) or minimum (in Figure 5) at D ~ 2.8

These features also resemble the D-dependence of T_C (Figure 1(e)), as mentioned above. Thus there can be an intrinsic origin of the observed D-dependence common to these superconducting properties. The minimum values of magnetization 4pM occur at the lower critical fields H_C1, where magnetic flux lines begin to penetrate the YBCO parts in the AKD/YBCO mixed powder samples. With regard to the superconductors, the energy is lower when the magnetic flux lines penetrate the AKD parts than otherwise. However, with regard to the magnetic flux lines, the energy is lower when the magnetic flux lines keep themselves as away from each other as possible due to the repulsion between themselves. This situation generally makes them arrange in a regular geometry like a lattice, and also makes them go as straight as possible. As a result, some of the magnetic flux lines penetrate the YBCO parts. Thus the number of the magnetic flux lines penetrating the AKD parts should depend on the amount and distribution of the AKD parts in the mixed powder samples, i.e., the fractal dimension D. This is why the minimum values of magnetization 4pM (Figure 3) and A₁/A₂ (Figure 6(b)) vary with D. Because the critical current density J_C is also directly dependent on how many magnetic flux lines penetrate the YBCO parts, J_C(D)/J_C(3) (Figure 5) also vary with D.

Figure 3.  Minimum values of magnetization (at H ~ 2310 G) vs. D at 2 K. Broken curve is a guide for eye. Inset shows our previous observation of T_N vs. D in the fractal spin system in CoO [27].

Figure 5. Normalized J_C vs. D. Broken curves are guides for eye. Red broken curve; based on the data at H = 15100 G, and black broken curve; based on the data at H = 402.5 G.

Figure 6(a)-(b). D-dependences of best-fitting parameters in eq. (2). (a) t_i^-1, and (b) A_i. Estimated errors in t_i^-1, A_i, and D are approximately ±2.5 %, ±2.5 %, and ±0.01, respectively.

 Based on the semi-quantitative explanation by equation (4), firstly we shall discuss the observed D-dependence of J_C and T_C in a qualitative way. When the AKD content is lower, i.e., D is close to 3, some magnetic flux lines should take roundabout routes when they penetrate AKD. This situation has both thermodynamic advantage and disadvantage as discussed above. If there are not AKD parts around a flux line, it would go straight and penetrate YBCO parts. Thus it depends on the detail of distribution of AKD in the sample whether each magnetic flux line actually penetrates AKD or not. The distribution, as well as the content, of AKD is described by D. In particular, when the AKD content is comparable to the defect content in YBCO, it will sensitively depend on D whether each magnetic flux line actually penetrates AKD or YBCO in order to lower the total energy. A series of chemical analysis data shown in the Supporting Information indicated that all the YBCO powder samples used in this study contained only a small amount of defects. Therefore, the AKD content becomes comparable to the defect content in YBCO when D is particularly close to 3. This qualitatively explains the complicated D-dependence of T_C and J_C in the range of ~2.8 £ D £ 3.0.

On the other hand, when the AKD content is higher, i.e., D is clearly lower than 3, most of the magnetic flux lines can penetrate the AKD parts in the samples. This situation would not change substantially when the AKD content even more increases, i.e., when D even more decreases. This means small D-dependence at lower D.

It requires more quantitative examination of fractal bodies to explain the anomaly at D = 2.95±0.03, which is now under way in our laboratory and will be reported in due course. However, in fact, the observed T_C and J_C both exhibited anomalies at D ~ 2.9, which can be semi-quantitatively explained by the general D-dependence of cooperative phenomena described by equation (4).

Conclusion

The critical current density J_C and critical temperature T_C of YBa₂Cu₃O_7-d (d = 0.02±0.02) depend on the fractal dimension D. In particular, both J_C and T_C sensitively vary in the ranges of ~2.8 £ D £ 3.0 and ~2.9 £ D £ 3.0, respectively, and take the maximum values at D » 2.88 for J_C and D » 2.92-2.98 for T_C. When the applied magnetic field H is rather low (£ 400 G), J_C(2.88) is greater than J_C(3) by ca. 7%. Similarly, T_C(2.92-2.98) is greater than T_C(3) by ca. 0.7%. Such behavior is semi-quantitatively explained by considering the fractal structures of the samples. In addition, the D-dependences of T_C and J_C are explained with the same quantitative discussion as those of Néel temperatures of fractal CoO-AKD mixed powder samples.

Authorship and contributorship

T. Naito planed the study, collected the XPS and a part of the magnetic susceptibility data, analyzed the whole set of data, and wrote the paper. H. Yamamoto prepared and characterized the sapmles, collected conductivity and a part of magnetic susceptibility data. He derived the values of J_C by analysis of the magnetic susceptibility data. K. Konishi, K. Kubo, and T. Nakamura helped T.N. in the magnetic susceptibility measurements. H. Mayama carried out the neutron scattering experiments and analyzed the data to derive the calibration curve for estimation of fractal dimension of each sample.

Competing interest

The authors declare that they have no competing interests.

Acknowledgements

This work is sponsored in part by Grant-in-Aids for Exploratory Research (No. 19655044) from the Ministry of Education, Culture, Sports, Science and Technology, Japan (MEXT), Grants-in-Aid for Scientific Research, Scientific Research (C) (No. 23540432) from Japan Society for the Promotion of Science (JSPS), a financial support from Network Joint Research Center for Advanced Materials and Devices, and a research grant from Ehime University.

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