1 Introduction

The yttrium-based superconductors family (YBCO) involves several superconducting phases with different zero-resistance temperatures (Tco); YBa2Cu3O7−d (Y-123) with zero-resistance temperature Tco ≈ 90K, YBa2Cu4O8 (Y-124) with Tco ≈ 80K, and Y2Ba4Cu7O15 (Y-247) with Tco ranging from 30 to 95 K, depending on the oxygen content that were synthesized. Aliabadi et al. [1] have synthesized Y3Ba5Cu8O18−x (Y-358) compound that becomes superconductor at 98 K. The discovery of superconductivity in Y-358 has attracted considerable interest, and some preparation methods have been employed and superconducting transition temperatures ranging from 80 to 98 K have been reported [2, 3]. Y-358 has a crystalline structure similar to that of Y-123 with the exception of the number of CuO chains and CuO2 planes which exceed those in Y-123. The Y-358 compound has five CuO2 planes and three CuO chains. Due to the structural resemblance and similarity of the superconducting transition temperature of Y-123 and Y-358, one can encounter some difficulties to distinguish between these compounds. Recently we have reported a comparative study of nanosized particles CoFe2O4 effects on superconducting properties of Y-123 and Y-358 [4] and we have shown that these compounds exhibit dissimilar behavior.

The broad resistivity transition caused by both external magnetic fields and transport currents has prompted intense research activity to understand the dissipation phenomenon. Various theoretical models involving thermally activated flux creep model [5], Ambegaokar–Halperin (AH) phase slip [6], Kosterlitz–Thouless (KT) transition [7], thermally activated flux flow [8], fluctuations [9], etc. have been proposed in order to interpret the origin of this behavior. Some previous studies have suggested that the problem of dissipation behavior of the layered superconductors has been achieved by applying the KT theory [7]. Ausloos et al. [10] had interpreted the electrical resistivity behavior in spray-dried bulk Bi2Sr2CaCu2O8+d in terms of a 2D KT theory. Bhalla et al. [11] have also found a reasonable agreement between the KT model and the low resistivity part under magnetic fields for bulk Bi1.6Pb0.4Sr2Ca2Cu3O x . Xu et al. [12] have shown that KT model describes very well the experimental data in granular YBa2Cu3O7−d . However, other researchers have shown that a unique model is not sufficient to explain this unusual broadening of the transition resistive [13–15]. They have applied the AH phase slip model [16–18] to a medium of Josephson weak links, and they have found a satisfactory agreement between theoretical and experimental data in bulk Bi-2223 superconductor. Additionally, they have recourse to the concept of thermally activated flux creep (TAFC) model proposed by Anderson [19] for the resistivity region near Tco and they proved that the vortex dynamics seems to dominate the cause of energy dissipation. Balaev et al. [20] have also discussed the mechanisms responsible for broadening of the resistive transition in composites YBCO + CuO under different magnetic fields, and they have observed a crossover from AH to flux creep model in the intermediate magnetic field range. Other groups [11, 15] have regained the crossover from the phase fluctuation of order parameter to vortex dynamics in bulk YBCO through which they have defined a small range of temperature ΔT ∗ where both mechanisms seem to be responsible for energy dissipation. On our knowledge, there is only one paper which focuses on the study of the flux dynamic properties in both Y-358 and Gd-358 systems by relying solely on the modified flux creep model [21]. The different regimes of energy dissipation are not widely investigated up to date. The main aim of the present study is to understand the complex dissipation behavior in Y-358 sample by using different theoretical models. So we report an analysis of the physical mechanisms responsible for the broadening of the magnetoresistivity measurements of Y3Ba5Cu8O18−x . We show that the AH phase slip model and the flux pinning mechanism induced by TAFC are accounts for the dissipation range of temperature with appearance of a small temperature range in which neither the first nor the second model are directly applicable. The KT model, on the other hand, explains very well our experimental data in the whole of the broadening of resistive transition.

2 Experimental Details

Polycrystalline samples of Y3Ba5Cu8Oy were synthesized by solidstate reaction. Commercially powders BaCO3 (99.9 % purity), Y2O3 (99.99 % purity), and CuO (99.9 % purity) were used as starting materials. The powders were mixed according to the chemical formula of Y:Ba:Cu = 3:5:8 by hand grinding in an agate mortar with an agate pestle. The mixed materials pressed into pellets and then calcined for 12 h at 900°C. Calcination step was repeated twice with intermediate grinding. The obtained precursor was pressed into pellets and sintered at 950°C for 48 h in an oxygen atmosphere and then cooled to room temperature at a rate of 4°C/min. The heat treatment processes of sample were performed in alumina crucible.

The structure and phase identification of the powder sample ground from sintered pellets were examined by powder XRD using a Philips 1710 diffractometer with CuK α radiation. The microstructure of sample was characterized using a scanning electron microscope (JEOL JSM-6390LV). The transport properties of the sample were studied by measuring the electrical resistivity temperature ρ(T) using the four-probe technique. A magnetic field was applied along the short axis of the samples, and the excitation current was injected along the length axis of the samples. The pellets were carefully cut into bar-shaped samples. Electrical contacts were made using silver paint and the contact resistance value was approximately 0.5 Ω. A low excitation current (I << I c: the critical current) is used in order not to affect the behavior of the resistivity transition of samples.

3 Results and Discussions

Figure 1a shows the powder XRD spectrum of Y-358 sample with designated Miller indices. The analysis of the data denotes a predominately single-phase structure Y-358 with orthorhombic Pmm2 symmetry. However, small impurity peaks are observed and identified with (*) and (+) on the spectrum. SEM micrograph of the surface morphology of Y-358 sample is shown in Fig. 1b. The microstructure exhibits a plate-like grain with random orientation in all directions. Figure 2 shows the magnetoresistivity measurements of the Y-358 sample under different magnetic fields. It is clear that the applied magnetic field produces a noticeable broadening of the superconducting transition and reduces the temperature at zero resistivity Tco. There are two parts in the mixed state of ρ(T,H) transitions curves: A steep part, associating with the superconductivity onset in the intragrains and a lower temperature region characterizing the grain-boundaries network or intergranular effects, which are considered to be weak Josephson type links. Inset of Fig. 2 shows the dependence of the resistive transition width, ΔT co = T c(H = 0) − T coH as a function of the applied magnetic field. The transition width is well fitted according to a power-law scaling relation \({\Delta } T\;\text {=} \alpha H^{^{-n}}\) with n is equal to 0.29 ± 0.03 and the value of the factor α is found to be 1.16 ± 0.02. Such dependence is usually remarkable in the granular superconductors [17, 22, 23]. In the case of Y-123, the parameters n and α are found to be 0.32 ± 0.01 and 1.35 ± 0.03, respectively [24].

Fig. 1
figure 1

a X-ray powder diffraction patterns of Y-358 sample. b SEM micrographs of surface view of the fracture part of Y-358 sample

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Fig. 2
figure 2

Variations of the electrical resistivity with temperature at different applied magnetic field of Y-358 sample. Inset shows the variation of the transition width ΔTco as a function of the applied magnetic field

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Generally, the resistivity transition in the polycrystalline HTS cannot be explained by a unique model [15, 20]. The AH phase slip model could be applicable in high temperature near Tc where Tc is the peak temperature of the d ρ/dT versus T curve defined as the critical temperature of the superconducting Josephson junction forming. The resistivity ρ(T), given by this approach in the limit of low current, I << I c(T), is written as ρ(T) = ρ 0[I 0(γ/2)]−2 [17] where I 0 is the zero-order modified Bessel function [11, 24], ρ 0 is the average normal resistance of the junction, and γ is the normalized barrier height for thermal phase slippage defined as γ(H,T) = C(H)(1-t)q where t = T/T p is the reduced temperature and C(H) is a parameter depending on the magnetic field. Note that TP is the branching point of the curve. The resistivity curves under magnetic field fitted by AH model for Y-358 sample are represented in Fig. 3. In the fitting process of the resistivity ρ(T,H), we have excluded the regions near the Tc and Tg temperatures (Tg is the glass transition temperature). We started our fitting procedure by considering the experimental values of Tp and ρ p while the other two parameters CH and q are used as free. It follows that the parameters ρ p and Tp are substantially unchanged for different magnetic fields. Furthermore the values of the exponent q are around 1 and the parameter CH decreases on increasing the magnetic field (Table 1). For all curves, it should be noted that the AH model describes a fairly large zone of the resistive transition, but it deviates from the experimental data points at temperature Tu. Many authors [13, 15] have reported that the AH model cannot explain adequately all the part of broadened resistivity curves of ρ(T, H), and they suggested that the flux creep model is the preferable regime in the tail part with the exponential relation [5, 26] as: ρ(T, H) = ρ p exp(−U(T, H)/k BT), where ρ p is the pre-exponential factor independent of the applied magnetic field, k B is the Boltzmann constant, and U(H,T) is the pinning energy that depends on temperature and magnetic field which expressed as U(H,T) ≈ U(0)(1-t)mH−n, where t = T/Tc is the reduced temperature, U(0) is the effective unperturbed pinning potential at T = 0 K, m and n are taken as fitting parameters. Previously, Anderson et al [27] have demonstrated the field dependence of Tg in HTS: H = H o(T co − T g(H)/T gH)1/n. Such relation allows to derive the magnetic field dependence U(H) which can be expressed as U(H) = (Tco/Tg(H)−1)−1 and the temperature dependence U(T) which is given by U(T) = −[k BT(T co−T g(H))/T g(H)]ln(ρ/ρ p). Typical result of U(T) for Y-358 sample is presented in Fig. 4. It is clear that the form of U(T) = U(0)(1−T/T P)m explains correctly the experimental data for a temperature below TL.m and Tp parameters were determined from fitting experimental curves. Generally, the temperature exponent, m, is usually chosen to be m = 2 [28], 3/2 [29], and 1 [30]. Within the framework of this study, the parameter fitting m is found to be 2 which corresponds to 2D vortex anisotropy state. A same value of m has been reported in the Y-123 embedded with nanoparticles of Y-deficient Y-123, generated by the planetary ball milling [15]. In order to estimate the pinning energy, we have plotted ln(ρ/ρ p) versus (1−T/TP)2/T at various magnetic fields. Typical curves are presented in Fig. 5a. From the linear behavior in the tail part below the temperature TL as indicated by solid line, we have determined the slope of each line which yields the activation energy U(H). Figure 5b shows the variations of UH with applied magnetic field of Y-358 sample. The flux pinning energy decreases with increasing the applied magnetic field, and it is of the order of magnitude of other approaches [31]. We note that the activation energy in Y-358 sample is higher than reported in Y-123 sample [15]. UH can be fitted by a power-law relation: U(H) ≈ H −n. In our case, the value of the exponent n is 0.57; this is in accordance with reports published earlier [15, 31].

Fig. 3
figure 3

AH phase slip fit (lines) for resistivity versus temperature data (symbols) under different applied magnetic fields of Y-358 sample

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Fig. 4
figure 4

Temperature dependence of the U(T) at various magnetic fields for Y-358 sample

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Fig. 5
figure 5

a Plots of the low temperature ln(ρ/ρ p) versus (1−T/Tp)m/T at various magnetic fields for Y-358 sample. b Variations of UH with applied magnetic field for Y-358 sample

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Table 1 Values of ρ p , Tp , CH and q at different magnetic fields for Y-358 sample
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Based on our data, we can conclude that TAFC model describes a restricted zone ΔT = TL−Tg, i.e., just the lower part. Figure 6 shows the experimental and calculated resistivity ρ(T) for various applied magnetic field of Y-358 sample. Solid and dashed lines correspond to the theoretical curves of AH and TAFC approaches, respectively. It follows that there exists a small range of temperature which is not calculated by these models. The width of the small region ΔT∗ = Tu−TL versus applied magnetic field ranging from 0 to 200 mT is indicated in inset of Fig. 6. It is clear that the width extends due to the increase in the applied magnetic field.

Fig. 6
figure 6

Dependence of the resistivity as a function of the applied magnetic field for Y-358 sample. Solid and dashed lines indicated AH and TAFC fits, respectively. Inset shows the dependence of the transition region \({\Delta } \mathrm {T}^{\ast }= \mathrm {T}_{\mathrm {u}}-\mathrm {T}_{\mathrm {L}} \)as a function of the applied magnetic field

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In the following of this study, we have attempted toconsider a model that describes the observed dissipation in the small range which is not described by neither AH nor flux creep models. The Kosterlitz–Thouless transition of vortex–antivortex unbinding because of thermal fluctuation has been proposed and has successfully been applied to analyse experimental results in both bulk and single crystal samples of HTS [32–35]. This model could be applied directly above a temperature called Kosterlitz–Thouless phase transition temperature (TKT) and supposed remains valid until a few degrees below the mean-field (Ginzburg–Landau) transition temperature (TGL). In most reports, TGL fixed as the temperature at which d ρ/d T showed a peak value [38]. Below TKT, there are no free vortices induced by thermal excitation and all of the vortices are frozen into vortex–antivortex pairs. Nevertheless, when the temperature is increased, some parts of the vortex–antivortex pairs will be thermally dissociated into the unbinding vortices, leading to dissipation in the superconductors. In the proximity of TGL, fluctuations of the amplitude of the order parameter are relevant favoring an ohmic behavior of the paraconductivity [37]. According to Halperin and Nelson [38], the resistivity, in terms of the mean free path ℓ and the zero-temperature coherence length ξ 0 follows exponential square root dependence in the dirty limit namely, \(\frac {\rho } {\rho _{\mathrm {N}}}= A\frac {\ell } {\xi _{0}} \exp [-\beta (\frac {\mathrm {T}_{\text {GL}}-\mathrm {T}}{\mathrm {T}-\mathrm {T}_{\text {KT}}})^{0.5}]\), where ρ n is the normal state resistivity and A and β are nonuniversal constants of the order of unity.

Figure 7 shows KT fits of the resistivity under different magnetic field in semi-logarithmic scale. In the initial fitting procedure of the resistivity ρ(T,H), we take into account the experimental values of TGL and TKT ∼ Tg, while \(\alpha = \mathrm {A}\frac {\ell } {\xi _{0}} \)and β were left free. The value of TGL is practically independent of the magnetic field and equals to 92.9 K while TKT shifts to a lower value on increasing the applied magnetic field (inset of Fig. 7). For each curve, the free parameters α and β have been determined from the fit. In our case, α = 1 and β = 1.1. These values are similar to those found by Bhalla et al. [11, 13]. One can note that KT theory describes the phenomena of dissipation in Y-358 sample in the whole of the temperature range TKT < T < TGL, the experimental data deviate from the KT model close to TKT and TGL temperatures. Deviation from the linear behavior near TGL may be due to fluctuations in the amplitude of the order parameter. For lower temperatures, the dissipation mechanisms seem to be related to the vortex dynamics.

Fig. 7
figure 7

KT fit (lines) for resistivity versus temperature data (symbols) of Y-358 sample under different applied magnetic fields. Inset shows the variation of the applied magnetic field versus the Kosterlitz–Thouless transition temperature TKT

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4 Conclusion

We have analyzed the broadening of the resistive transition in Y-358 compound through the resistivity ρ(T) measurements under applied magnetic fields ranging from 0 to 200 mT. Our results show that the resistivity broadening are described by three dissipation mechanism models, i.e., the Ambegaokar–Halperin model dominated by a phase slip process of the order parameter, thermally activated flux creep model, and Kosterlitz–Thouless theory. The AH model is found to explain the dissipation in the temperature range TL < T < Tp. At low temperatures below Tu, the mechanism of vortex dynamics seems to be predominant. AH and TAFC models. Moreover, TAFC and AH models cannot explain the variation of ρ(T,H) in the temperature range ΔT∗ = Tu−TL. While the KT model explains adequately, our experimental data in the whole of the broadening of resistive transition.