Role of zirconia nanoparticles on microstructure, excess conductivity and pinning mechanism of BSCCO superconductor ceramics

Abstract

Using the solid-state reaction process, we prepared (Bi,Pb)₂Sr₂Ca₂Cu₃O_10-δ (Bi-2223 for brevity) ceramics with the addition of zirconium dioxide (ZrO₂) nanoparticles. The proportion x of ZrO₂ per the total masse of superconductor ranged from 0 wt.% to 0.2 wt.%. We report the intragrain critical current density (J_c), pinning mechanisms, and thermal fluctuations induced excess conductivity. Employing numerous characterization techniques, combining X-rays diffraction, electrical transport measurements, scanning and transmission electron microscopes (SEM and TEM), and DC magnetization hysteresis measurements, we find that the ceramic sintered with 0.1 wt.% ZrO₂ nanoparticle showed the best superconducting performance. From the analysis of excess conductivity using Aslamazov–Larkin model, we assessed the crossover temperatures between different regimes of conductivity, the penetration depth, the coherence length, as well as the upper and the lower critical magnetic fields (B_c1, B_c2) at zero kelvin for all the ceramics. Critical current density versus temperature deduced from magnetization measurement is raised by NP-ZrO₂ addition. Thereby, NP-ZrO₂ strengthened the role of δl pinning cores and enhanced the flux pinning ability.

1 Introduction

Because of their prospective applications, high-temperature superconductors (HTS) have gathered great interest [1, 2]. Among HTS materials, Bi₂Sr₂Ca_n–1Cu_nO_2n+4-δ is considered as a unique system, which possesses three compounds Bi-2201, Bi-2212, and Bi-2223 with critical transition temperature (${T}_{c}$) of 10 K, 90 K, and 110 K, respectively. Bi-2223 owns some benefits including high ${T}_{c}$, high density of critical current (${J}_{c}$) and magnetic field-carrying capability, leaving this compound the most significant in applications. However, Bi-2223 compound presents disadvantages such as brittle linkages between grains and frail capability of magnetic flux pinning (FP) that restricts their exploitation as superconductors in potential applications. Moreover, due to its reduced thermodynamic stability domain and the complex mechanism of its formation, the production of pure Bi-2223 compound is a hard job and Bi-2201 and Bi-2212 phases cannot be avoided during the growth of Bi-2223.

Lead addition to the BSCCO system is considered as a hopeful strategy for enhancing the stability and purity of the Bi-2223 compound [3, 4]. The Bi-2223 polycrystalline superconducting traits at zero as well as under an external magnetic field are closely allied with the level of crystal defects inside the product and the weak link between grains. Chemical substitution and addition processes are broadly used to remedy the limitations of ${J}_{c}$ and fulfill the requirements of better FP capacity. The convenient density of flux pinning cores (FPC) with strong pinning ability are vital factors that decide the values of critical parameters ${J}_{c}$ and H_c. Effective FPC related to crystal defects, like lattice strains, structural inhomogeneities, and non-superconductive phase inclusions, depend on the synthesis process, on the doping contents and types, and on the kind of atoms that could substitute the distinct position in Bi-2223. Massive efforts have been performed to optimize proceedings that enable the production of Bi-2223 polycrystalline materials with convenient ${T}_{c}$, ${J}_{c}$ and ${H}_{c}$ and best FP [5,6,7,8]. Some chemical inclusions lead jointly to the deterioration of intergranular properties and intensification of intragranular properties [9,10,11,12]. For example, Bi₂Te₃ nanoparticles added as a foreign phase in Bi-2223 settle in intragrain and intergrain regions to harm the crystallinity and reduce the electrical properties [9]. However, encouraging effects of many nano-oxide entities addition on J_c, B_c2, and pinning properties have been identified [13,14,15,16,17,18,19,20,21]. The inclusion of a small concentration (0.2 wt.%) of TiO₂ nanorods improves ${J}_{c}$ by 64% roughly [11]. Oztürk et al. [21] revealed that addition of 0.1 wt. % of carbon-encapsulated nano boron enhances the volume fraction of Bi-2223 to 79.87% and rises the ${J}_{c}$ at a temperature of 10 K by a factor of about three times. Jannah et al. [13] worked on Ag-sheathed Bi-2223 superconducting tapes with the addition of Co₃O₄ nanoparticles and showed that the value of ${J}_{c}$ of 0.02 wt % Co₃O₄ (30 nm) added sample is more than six times compared to the non-added one.

Whatever the defects generated naturally or artificially inside the HTS system, they stimulate the thermal fluctuations of the superconducting order parameter above ${T}_{c}$. These fluctuations induce an excess of conductivity Δσ. Owing to elevated ${T}_{c}$, small coherence length, and large anisotropy, the behavior of fluctuation-induced conductivity (FIC) is largely amplified in the HTS. Many researchers extended the study of Bi-2223 compound by exploring the experimental data of FIC to discuss the dimensionality of conductivity and determine the crossover temperatures, the coherence length, the interlayer coupling, the inter-plane spacing, and critical physical parameters at zero Kelvin, as upper and the lower critical magnetic fields (${B}_{c1}(0)$, ${B}_{c2 }(0)$) and critical current density ${J}_{c}(0)$ [22,23,24,25,26,27,28]. Some studies were devoted to the impact of nano-oxide, and few stressed the non-oxide nano-entities. Aftabi et al. [22] examined the FlC of (Bi_1.6Pb_0.4)Sr₂Ca₂Cu₃O_10+δ superconductor added with ZnO nanoparticles and they conclude that when the ZnO NPs content increases up to 0.3 wt. % the width of the 3D regime of fluctuations is narrowed but that of 2D is augmented. ZnO NPs nanoparticles proved their ability to enhance ${B}_{c1}(0)$, ${B}_{c2 }(0)$ and ${J}_{c}(0)$ and the best are obtained for 0.2 wt. % ZnO NPs addition. The inclusion of nano-NiO up to 0.05 wt.% in (Bi,Pb)-2223 yields an increase in ${\upxi }_{\mathrm{co}}$ and d [25]. Habanjar et al. [26] analyzed the FIC of (BaSnO₃)_x/(Bi_1.6Pb_0.4)Sr₂Ca₂Cu₃O_10-δ samples and demonstrated that the parameters ${\xi }_{co}$ and d were reduced by increasing the concentration of nano-BaSnO₃ up to 0.5%, then increased with further addition. Roumié et al. [27] worked on (nano-Fe₂O₃)_x/Bi_1.8Pb_0.4Sr₂Ca₂Cu₃O_10+δ samples where x varies from 0 to 1.5 wt. %. They confirmed that nano- Fe₂O₃ increased the parameters ${\xi }_{co}$, $\mathrm{J}$ and d, and worsened ${B}_{c1}(0)$, ${B}_{c2 }(0)$ and ${J}_{c}(0)$. Loudhaief et al. [23, 28] focused on the effect of the CuS, (Cu_0.99,Eu_0.01)S and (Cu_0.99,La_0.01)S nanomaterials addition in Bi-2223. They analyzed the FIC by five fluctuations regions and reported variations in ${\xi }_{co}$ and d. The 0.2 wt% of nano-(Cu_0.99,La_0.01)S proves their capability to strengthen ${B}_{c2 }(0)$ value from 271 to 896 T, however a little increase of ${J}_{c}(0)$ was recorded.

In the present report, we aim to investigate the effect of Zirconium dioxide (ZrO₂) nanoparticles (NPs) on the superconducting properties of Pb-doped Bi-2223. The strong interest in ZrO₂ particles stems from their attractive physical properties and their promising applications. NPs of ZrO₂ have special properties such as high thermal stability, low thermal conductivity at high temperatures, quite high thermal expansion coefficient, good ionic conductivity, and high mechanical strength. These properties have led to the utilization of ZrO₂ nanoparticles in a wide range of industrial applications including oxygen sensors, fuel cells, catalysts, solid electrolytes, and the fabrication of dense ceramics [29,30,31,32]. ZrO₂-NPs is an attractive additive in BSCCO superconductors because its melting temperature of ~ 2715 °C is so high as compared to the sintering temperature of Bi-2223 (~ 835 °C). ZrO₂ is chemically inactive, and its presence cannot hamper the Bi-2223 phase formation. Hence, ZrO₂ could be desirable criterion to make superconductor composites with promising physical properties. Moreover, it has been reported that ZrO₂-NPs included in HTS systems work as effective FPC and enhance the critical current densities [33, 34]. The ${J}_{c}$ values under a low applied magnetic field of ZrO₂-added Bi-2223 were performed via magneto-resistivity measurements and it was found that the grain boundaries are heightened and the ${J}_{c}$ is upgraded with the addition of 0.1wt% ZrO₂ [35]. In this study, the dependence of x wt% NPs ZrO₂ (x = 0.0, 0.1, and 0.2) addition on the FIC of the BSCCO system prepared by a three-stage solid-state reaction route was described. It is intended to determine some important superconducting parameters at a temperature of 0 K, as ${J}_{c}(0)$, ${B}_{c1}(0)$ and ${B}_{c2 }(0)$, and survey the relationship between the FIC above T_co and the pinning mechanisms displayed in the superconducting state. Thereby, the pinning mechanisms are estimated through DC magnetic measurements at various operating temperatures from 10 to 77 K.

2 Experimental details

Bi₂Sr₂Ca₂Cu₃O_10-δ bulk ceramics fortified by ZrO₂ nanoparticles with an average size of about 20 nm were synthesized via the solid-state reaction process. The heat treatment process was made in crucibles of alumina. First, stoichiometric amounts of high-purity (99.9%) raw reactants of strontium carbonate (SrCO₃), calcium carbonate (CaCO₃), and oxides of copper (CuO), filling the chemical composition Sr_1.9Ca_2.1Cu₃O₈, were crushed and calcinated in the air at 920 °C during 48 h. Next, we added to the beforehand obtained precursor bismuth oxide (Bi₂O₃) and lead oxide to fulfill the composition Bi_1.6Pb_0.4Sr_1.9Ca_2.1Cu₃O_10-d. The mixture was crushed, compacted, and then sintered at 835 °C for 144 h in air. The Bi_1.6Pb_0.4Sr_1.9Ca_2.1Cu₃O_10-d powders resulting from pellets obtained in the previous stage were reground with 5 wt.% PbO and x% of ZrO₂ nanoparticles (x = 0 wt.%, 0.1 wt.%, 0.2 wt.% per the total mass of superconductor), pressed into pellets with a diameter of 13 mm under 12 tons, sintered at 835 °C for 72 h in air and then finally cooled slowly to room temperature. The amounts of ZrO₂ were selected based on a previous report [29], which reveals that the optimal values of ${J}_{c}$ measured at 77 K are achieved for ZrO₂ concentration close to 0.1wt.%. In the rest of the text, we mention samples as B-0.0, B-0.1 and B-0.2 in accordance with the concentration x = 0 wt.%, 0.1 wt.%, 0.2 wt.% ZrO₂ respectively.

X-ray powder diffraction (Bruker D8, XRD) with CuKα radiation is used for crystal structure examination and quantitative analyses of mixtures of solid phases. The data were recorded in the range of 2θ = 2°–50° by steps width of 0.02°. A scanning electron microscope (SEM, JEOL JEM-5510) was used for the morphological observations. FEI Tecnai G2 TEM coupled with an EDX system was utilized to examine the microstructure and the chemical composition. To estimate ${J}_{c}$, DC magnetization versus magnetic field measurements were performed at temperatures varying from 10 to 77 K using a SQUID magnetometer (PPMS-7) with a maximum field of 7 T, with an applied field H_a, growing from zero to an extreme value of 60 kOe, then falling to zero, reversing in sign to an extremely negative value of 60 kOe, and finally growing again to 60 kOe. The temperature dependence of electrical resistivity (ρ(T)) was executed from ambient temperature (285 K) to 20 K using the four-probe electrical contact method.

3 Method of excess conductivity analysis

The granular character, anisotropic, low charge carrier density, and short superconducting coherence length of HTS materials predispose them to display a significant broad rounding in the transition region (TR). The deviation from the linear relation between resistivity and temperature in the TR is linked to the fluctuation of the order parameter ψ of superconducting carriers. Beyond the normal-superconductor transition, it is likely that superconducting pairs can have an existence with a finite lifetime. Because of the thermal disequilibrium superconducting pairs are simultaneously created and destroyed as the temperature approaches ${T}_{c}$. The creation of the Cooper pairs gives an increase in the excess conductivity ($\Delta \sigma$) of the carriers. $\Delta \sigma$ investigation permits also to extract some important physical critical quantities for a temperature of 0 K, such as ${J}_{c}(0)$, ${B}_{c1}(0)$ and ${B}_{c2 }(0)$ and to acquire information about the properties of flux trapping of HTS materials. The most simplified method to inspect the FIC as a function of temperature is proposed by Aslamazov-Larkin (A-L) [36]. Nearby the superconducting transition temperature, the excess conductivity is defined as:

$$\Delta \sigma \left(T\right)={{\rho }_{m}^{-1} \left(T\right)-{\rho }_{n}^{-1} \left(T\right)=A}_{F}{t}^{{\lambda }_{D}},$$

where ${\rho }_{m} (T)$ is the recorded resistivity during the temperature variation, ${\rho }_{n} (T)$ is the normal resistivity in which the material has a metallic behavior. The coefficient ${A}_{F}$ designate the amplitude of conduction,$t=\frac{T}{{T}_{cmf}}-1$, where ${T}_{cmf}$ denotes the mean-field temperature and ${\lambda }_{D}$ depicts the conduction dimensionality. Closer to${T}_{cmf}$, the resistive transition occurs in two stages; a pairing transition happens in small and homogeneous isolated regions at temperatures slightly above ${T}_{cmf}$ (para-conductivity zone) and a coherence transition (para-coherent zone) which takes place in the weak links between the superconducting grains at temperatures lesser than${T}_{cmf}$. Value of ${\lambda }_{D}$ and the expressions of ${A}_{F}$ for the three possible regimes of fluctuation of conductivity in the mean-field region (MFR) are regrouped in Table 1. The 3D and 2D regimes of conductivity cross at the Lawrence–Doniach [37] temperature${T}_{3D-2D}={T}_{cmf}\left[1+4({\xi }_{co}/d{)}^{2}\right]$, where ${({\xi }_{co}/d)}^{2}=\mathrm{J}$ denoted the coupling factor between adjacent planes. MFR is embraced at low temperatures by a critical (Cr) fluctuation regime with exponent ${\lambda }_{1cr}=\text{-2/3}$ for two and ${\lambda }_{2cr}=\text{-1/3}$ for three dimensions conductivity, and at high temperature by a short-wave (SW) fluctuation regime with exponent${ \lambda }_{SW} \sim -3$. At temperature${T}_{G}$, a transition from critical to 3D fluctuation regimes will take place. For 0 K, the anisotropy factor,$\gamma$, and the thermodynamic critical field,${B}_{co}$, are allied to ${T}_{G}$ (Ginsburg temperature) as$\left|({T}_{G}/{T}_{c})-1\right|=\frac{1}{2}[{k}_{B}/{{B}_{co}}^{2}{\gamma }^{2}{\xi }_{co}^{3}{]}^{2}$, where ${k}_{B}$ depicts the Boltzmann constant. The penetration depth ${\lambda }_{o}$ at zero temperature could be evaluated with the help of the expression${B}_{co}=\frac{{\Phi }_{o}}{2\sqrt{2}{ \pi {\lambda }_{o}\xi }_{co}}$, here ${\Phi }_{o}$ represent the flux quantum number.${B}_{c1}(0)$,${B}_{c2}(0)$, and ${J}_{\text{c }}(0)$ will be estimated by using these relations: ${B}_{\text{c2}} (0)=\sqrt{2}\kappa {B}_{co}$, ${B}_{\text{c1 }}(0)=\frac{{B}_{\text{co}}}{\kappa \sqrt{2}}\mathit{ln}\kappa$ and${J}_{\text{c }}(0)=\frac{{4\kappa B}_{\text{c1}}}{3\sqrt{3}{\lambda }_{o}\mathit{ln}\kappa }$, where $\kappa =\frac{{\lambda }_{o}}{{\xi }_{co}}$ designates the Ginzburg–Landau parameter.

Table 1 ${\lambda }_{D}$ and ${A}_{F}$ for the three possible conduction dimensionalities

4 Results and discussion

4.1 Structure and microstructure analyses

The collected XRD patterns of Bi-0.0, Bi-0.1, and Bi-0.2 ceramics are shown in Fig. 1. The reflection peaks denoted by ″3″ correspond to Bi-2223 phase and those denoted by ″2″ are allied with Bi-2223 phase. For the three ceramics, most peaks correspond to the Bi-2223 phase and a small number of peaks agree with the Bi-2212 phase. A minor peak intensity of the non-superconducting Ca₂PbO₄ phase is also observed in the free and ZrO₂-added ceramics. Within the sensitivity limit of the XRD instrument and the too-low concentration of additives, no ZrO₂ peaks were detected in the patterns. The XRD patterns also attest that ZrO₂ addition does not provoke any shift in the diffraction angles positions, which indicate that ZrO₂ did not enter the crystal structure. Intensities ($I$) of peaks are commonly utilized to quantify the volume fraction of Bi-2223, Bi-2212, and Ca₂PbO₄ phases as follows [14, 15]:

Fig. 1

XRD patterns of Bi-0.0, Bi-0.1, and Bi-0.2 ceramics

$${V}_{Bi-2223}=\frac{\sum {I}_{Bi-2223}}{\sum {I}_{Bi-2212}+\sum {I}_{Bi-2223}+{I}_{{\mathrm{Ca}}_{2}{\mathrm{PbO}}_{4}}} \times 100,$$

$${V}_{Bi-2212}=\frac{\sum {I}_{Bi-2212}}{\sum {I}_{Bi-2212}+\sum {I}_{Bi-2223}+{I}_{{\mathrm{Ca}}_{2}{\mathrm{PbO}}_{4}}} \times 100,$$

$${V}_{{\mathrm{Ca}}_{2}{\mathrm{PbO}}_{4}}=\frac{{I}_{{\mathrm{Ca}}_{2}{\mathrm{PbO}}_{4}}}{\sum {I}_{Bi-2212}+\sum {I}_{Bi-2223}+{I}_{{\mathrm{Ca}}_{2}{\mathrm{PbO}}_{4}}} \times 100.$$

In these relations $\sum {I}_{Bi-2223}$ and $\sum {I}_{Bi-2212}$ denote the sum of intensities of Bi-2223 and Bi-2212 phases, respectively. The volume fractions of the three existent phases are determined. Both free-added and ZrO_2-added ceramics exhibit almost identical volume fractions of the Bi-2223, Bi-2212, and Ca₂PbO₄ phases of about 86%, 13%, and 1%, respectively. A slight quantity of ZrO₂ added to Bi-based superconductor ceramics does not impress the traits of the liquid produced at the reaction temperature and does not alter the growth kinetics of Bi-2223.

SEM micrographs exposed in Fig. 2 illustrate the morphology of Bi-0.0.0, Bi-0.1, and Bi-0.2 ceramics. The three micrographs show a compacted granular structure formed practically by micrometric grains with plate-like form signs of the Bi-2223 phase [14, 15]. No perceived change in the grain size is detected as a result of low additions of ZrO₂. Micrographs of Bi-0.0 and Bi-0.1 ceramics show a softly best textural structure compared to the Bi-0.2 one. The microstructural trait of ZrO₂ nanoparticles was inspected by TEM and the findings illustrate particles having an average size of about 10 to 20 nm (Inset (a) of Fig. 3). Figure 3 illustrates low magnification TEM image photographed on Bi-2223 phase for ceramic B-0.1. Nanometric inhomogeneities with an average size of about 15 nm incorporated in the Bi-2223 phase are clearly manifested. The EDX spectrum (Inset (b) of Fig. 3) verified the occurrence of Y, Ba, Cu, and O elements besides the Zr element. These results attest that the observed inhomogeneities encompass Zirconium. The fine grain structure bound in the superconducting grains could lead to a rise in the density of grain boundaries (GB) and fine-scale disorders in the superconducting grains. These microstructural modifications alter the intragranular properties, and as a consequence, could affect the FIC behavior and or raise the sites of vortices trapping.

Fig. 2

SEM images of Bi-0.0, Bi-0.1, and Bi-0.2 ceramics

Fig. 3

TEM image of Bi-0.1 ceramic showing nano-inhomogeneities. Insets (a) micrograph showing the nanosized ZrO₂ particles used in this study and b EDXS spectrum taken from an area of Fig. 3 revealed Zr element

4.2 Excess conductivity analysis

Before moving towards the excess conductivity analysis, we first present the variations of electrical resistivity with temperature ($\rho (T))$ for the three ceramics. Figure 4 shows typical metallic behavior (dρ/dT > 0) of resistivity, with room temperature resistivity (${\rho }_{285 K}$) of about 1.78, 1.34, and 1.40 Ω-cm for Bi-0.0, Bi-0.1, and Bi-0.2 ceramics, respectively. From room temperature and going down till temperature T^s that is related to a pseudogap opening [22], the resistivity deviates from the linear behavior. Some models have been proposed to describe the mechanism of conduction in the normal state of HTS, viz., the variable range hopping (VRH), the variable range hopping with Coulomb interactions (CG) [38, 39], and the bipolaron [40]. To explore the metal-like behavior of $\rho (T)$ in the normal state, the CG and bipolaron models were tested and the results of fitting reveal that conduction characteristics are more consistent with the bipolaron regime. In copper oxide superconductors, bipolaron could be generated via an interaction between optical phonons (with energy $\hbar \omega$) and holes at temperatures below T_c [41]. This model supposes that the fraction of carriers is localized by disorder and expects a $\rho (T)$ variation as:

$$\rho \left(T\right)={\rho }_{o}\frac{\mathit{exp}(-\omega /T)+(T/{T}_{L}{)}^{2}}{\left[1+M\left(\frac{T}{{T}_{C}}\right){(1-\mathit{exp}(-{T}_{o}/T))}^\frac{1}{2}\mathit{exp}(-\left(\frac{{T}^{*}}{T}\right)\right]},$$

${\rho }_{o}$ and ${T}_{o}$ depends on the number of carriers localized by the disorders and the doping concentration and ${\text{M}}\approx 6$ is the ratio of bipolaron and polaron masses. The temperature ${T}_{o}$ is very close t ${T}_{C}$ and could be extracted from $\frac{d\rho (T)}{dT}$ plot. ${T}_{L}=(b/a{)}^{1/2}$ is temperature independent parameter, varies with optical phonon scattering (b) and itinerant carrier scattering (a) contributions, and the temperature ${T}^{*}$ is a characteristic of the material. The fitting of the bipolaron model for our samples is represented in Fig. 4 at right. The Bipolaron model matches perfectly our experimental data. Table 2 lists the parameters resulting from fitting. It is noticed that the temperature ${T}^{s}$ that points the divergence of normal state resistivity $\rho (T)$ from the linear dependence is consistent with the ${T}^{*}$ temperature obtained by fitting using the Bipolaron model. Both ZrO₂ inhomogeneities embedded inside the superconducting grains and the generated defects yield an increase in disorders. Hence, the number of localized carriers rises and superconductivity suppresses.

Fig. 4

(Left) Resistivity dependence on the temperature and (Right) fitting the resistivity data with bipolaron model for Bi-0.0, Bi-0.1, and Bi-0.2 ceramics

Table 2 Results of the Bipolaron fitting for Bi-0.0, Bi-0.1, and Bi-0.2 samples

From room temperature to T^s the measured resistivity could be described by a linear decay ${\rho }_{n}(T)\text{ = }{\rho }_{\text{rem}}\text{ + }{\rho }_{\text{TC}}T$. ${\rho }_{\text{rem}}$ terms the remanent resistivity and is attributed to the scattering phenomena. Whereas ${\rho }_{\text{TC}}$ denotes the resistivity temperature coefficient and reflects the intrinsic electronic interaction. The extracted values of ${\rho }_{\text{rem}}$ and ${\rho }_{\text{TC}}$ for both products are registered in Table 3. The lower values of ${\rho }_{\text{rem}}$ and ${\rho }_{\text{TC}}$ registered in the Bi-0.1 ceramic imply the constructive effect of ZrO₂ on the electrical conductivity. By lowering the temperature below ${T}_{cmf}$, the coupling energy among grains overcomes the thermal energy, and hence, a superconducting order over the whole granular structure will be settled at ${T}_{co}$ (zero resistivity temperature). We note here that ${T}_{cmf}$ is regarded as being the temperature of the main peak spotted in the derivative, $\frac{d\rho }{dT}$ vs T plot. The estimated values of ${T}_{cmf}$ are listed in Table 3. ${T}_{cmf}$ is slightly weakened when ZrO₂ content increases up to 0.2 wt.%. The disorder generated by ZrO₂ addition could be one the possible causes of the reduction of ${T}_{cmf}$ in HTS compounds [42, 43].

Table 3 Parameters extracted from the resistivity measurements and excess conductivity analysis for Bi-0.0, Bi-0.1 and Bi-0.2

The analysis of excess conductivity $\Delta \sigma ={A}_{F}{t}^{{\lambda }_{D}}$ is done by considering logarithm form on both sides: $\mathit{ln}\Delta \sigma ={lnA}_{F}+{\lambda }_{D}lnt$. The straight-line relationship presents an easy method to determine ${\lambda }_{D}$ and ${A}_{F}$. Indeed, the slope of the curve is ${\lambda }_{D}$ and the intercept of the straight-line extrapolation with the ln axis helps us to extract the value of the amplitude ${A}_{F}$. Figure 5 shows the plots of $\mathit{ln}\Delta \sigma$ vs $\mathrm{ln}t$ of Bi-0.0, Bi-0.1, and Bi-0.2 ceramics. The adjustment of the experimental ln-ln data with the theoretical predictions is represented by solid straight lines with different colors for MFR. Each plot could be fragmented into five straight lines corresponding to the slopes ${\lambda }_{2cr}=\text{-1/3}$ for critical fluctuation region, $\lambda_{SW} \sim - 3$ for short-wave fluctuation regime and ${\lambda }_{D}=$-0.5, -1, and -1.5 for fluctuation of conductivity in 3D, 2D, and 1D. These behaviors of fragmentation are often observed in the FIC of various HTS. In references [22, 25,26,27], FIC was analyzed by four fluctuation regions, viz., critical, short-wave, 3D, and 2D fluctuation regimes. In references [23, 28], the authors reveal that the 1D fluctuation regime participates also along with the four cited regimes. Transition and crossover temperatures as T_co, ${T}_{\mathrm{cmf}}$, T_G, T_LD, T_2Dx1D, and T_1DxSW for the three ceramics are listed in Table 3. Attainment of higher temperatures in the Bi-0.1 ceramic notifies a better superconducting performance as compared to others. When the temperature is below ${T}_{G}$ and comes closer to ${T}_{cmf}$, the fluctuation in the amplitude of order parameter ψ grows up to become approximately of the same order as itself. Therefore, a large fluctuation governs the system, and the 3D-XY model previously established for the superfluid transition in liquid ⁴He, describes well this critical fluctuation regime [44, 45]. Whereas, far from ${T}_{cmf}$ and near ${T}_{c}^{on}$ at the knee of $\rho (T)$ curve (superconducting onset temperature), the fluctuation of the order parameter is analogous to the coherence length. In this region, the thermal energy is as high to hold steady the formed cooper pairs as long as possible, and as a consequence, SW fluctuations will take place. In the mean-field region, all curves advertise 3D, 2D, and 1D conductivity fluctuations. The last regime is ascribed to the survival of conducting charge strips [46]. The width of this regime of fluctuation is shortened with ZrO₂ addition. Microscopic and mesoscopic inhomogeneities could destabilize the conducting charge strips and hence shrink the 1D regime width [23]. We notice that the 2D fluctuation regime is wider compared to the 3D one. This is clear evidence that the superconductivity mainly occurs in the CuO₂ planes. BSCCO is an anisotropic system, its layered structure is formed by alternating low-conductivity layers enfolded by high-conductive CuO₂ planes, in which the charge carriers have the tendency to flow easily before the superconducting order extending over a long range. By further reducing the temperature, the charge carriers can jump between the CuO₂ planes in 3D, and thus a motion in the entire system will be established before a global superconducting state settles beyond T_co. Noting that following the ZrO₂ addition, the 2D regime width increases, and that of the 3D regime somewhat spreads over a broader region (Fig. 6a). This behavior may be attributed to the superconducting grains coupling. The higher the GB density is, the broader the 2D regime of fluctuation and the smaller thickness of effective layers of the 2D system, and therefore more 3D regime appears earlier. Indeed, our ceramics are formed by superconducting crystallites randomly oriented, and thus the microscopic parameters as d and ${\xi }_{co}$ are regarded as effectual parameters of polycrystalline medium [23, 47]. Based on the crossover temperature T_G and the amplitude expressions for 2D and 3D fluctuation regimes and given that the factor $\gamma$ frequently taken about 10, d and ${\xi }_{co}$ and the thermodynamic critical magnetic field ${B}_{co}$ and λ(Å) are extracted for zero absolute temperature. The lower value of d particularly for the Bi-0.1 ceramic proves an enhancement of the core density and a better link between grains (Fig. 6b). Our estimated values of ${\xi }_{co}$ are comparable to those stated by other groups [23, 48]. Due to the polycrystalline nature of these ceramics, d and ${\xi }_{co}$ values were higher than those reported in thin film and monocrystal [49]. Besides, the λ values were less than those reported in Ref [23], hence reflecting the best shielding of flux lines particularly in Bi-0.1 ceramic. Once the penetration depth is estimated with the help of the ${B}_{co}$ expression, values of ${B}_{\text{c1}}(0)$, ${B}_{\text{c2}}(0)$, and ${J}_{\text{c}}(0)$ were estimated and depicted in Fig. 6c and Fig. 6d. As observed these three critical parameters are enhanced with the addition of ZrO₂, and the Bi-0.1 ceramic reveals the best values. The strengthening of these critical parameters is chiefly attributed to the damping in the motion of vortices through improving the flux trapping sites inside the ceramic artificially introduced by the addition of ZrO₂ nanoparticles. The Twin boundaries, the grain boundaries and the superconducting/nanophase interface, and the local strained regions are stated as strong flux pinning sites (SFPC) [50, 51]. Noticeably the ${J}_{\text{c }}(0)$ values for our samples are of an order of magnitude higher than those reported in La‑Substituted Ca in (Bi, Pb):2223 [52].

Fig. 5

Plots of ln(Δσ) versus ln(ε) for Bi-0.0, Bi-0.1, and Bi-0.2 ceramics. The fitting of the experimental ln-ln curves is represented by solid color straight lines. ${\lambda }_{2cr}$, ${\lambda }_{3D}$, ${\lambda }_{2D}$, ${\lambda }_{1D}$ and ${\lambda }_{sw}$ refer to critical, three, two one, and short-wave dimensions of conduction

Fig. 6

Variations of (a) 2D and 3D conductivity fluctuation regions, b microscopic physical characteristics ${\xi }_{co}$ and ${\lambda }_{o}$ and c critical physical parameters ${B}_{\text{c1}}(0)$, ${B}_{\text{c2}} (0)$ and d ${J}_{\text{c }}(0)$ at 0 K as a function of the ZrO₂ nano content

4.3 Critical current density and flux pinning mechanism

In the rest of the paper, we assess the critical current density versus temperature ${J}_{\text{c}}(T)$. This later is determined with the help of isothermal magnetization measurements and using the beam model [53]. Based on the above-cited results we consider ${J}_{\text{c}}(T)$ of free and 0.1 wt.% added ceramics. The values of ${J}_{\text{c}}$ of Bi-0.1 was compared to those of Bi-0.0 for different temperatures and the plot of the ratio $R=\frac{{J}_{\text{c, }Bi-0.1 \, }(T)}{{J}_{\text{c, }Bi-0.0 \, }(T)}$ is shown in the inset of Fig. 7. The Bi-0.1 ceramic was found to exhibit the highest values of ${J}_{\text{c}}$ over the whole range of measurement temperature, suggesting the best flux pinning property in this ceramic. Almost analogous values of $R$ evaluated at 10 K are reported in the literature for polycrystalline (Bi,Pb)2223 superconductors doped with TiO₂-nanorod [54] and NP SiC [55]. However, the obtained values of self-field J_c measured at 77 K are larger compared to the best data stated in (Bi,Pb)2223 tape/CoFe₂O₄ [20], (Bi, Pb)2223 tape/NiF₂ [56], (Bi,Pb)2223 tape/Co₃O₄ [13] and (Bi, Pb):2223 tape/ Bi₂O₃ [57]. Cores that govern the pinning of flux lines are (1) spatial variations of the mean free path of charge carrier owing to the scattering by normal phase noted δl, (2) spatial variations of the κ parameter connected with a random distribution of the critical temperature noted δT_c and (3) stress/strain field noted δε. Analysis of the ${J}_{\text{c }}(T)$ was performed using the collective flux pinning model [58], where ${J}_{\text{c}}(T)$ is expressed against the reduced temperature (t = T/T_co) as: ${J}_{c}={J}_{co}(1-t{)}^{\alpha }(1+t{)}^{\beta }$, and for δT_c, δl, and δε trapping sites ${J}_{c}={J}_{co}(1-t{)}^{7/6}(1+t{)}^{5/6}$ [59, 60]${J}_{c}={J}_{co}(1-t{)}^{5/2}(1+t{)}^{-1/2}$, and ${J}_{c}={J}_{co}(1-t{)}^{7/6}(1+t{)}^{-11/6}$ [61], respectively. Analysis of ${J}_{c}$ versus t by means of δT_c, δl, and δε mechanisms was done for the self-magnetic field (Fig. 7). As seen the ${J}_{c}(t)$ trends can be ascribed by the contribution of δT_c core for both ceramics at low temperature and at high temperature, mainly δl core validate ${J}_{c}(t)$ behavior over a larger range of the considered temperature. The above results sign that the ZrO₂ addition betters the flux pinning mainly by normal phase at elevated temperature. Normal phases dispersed inside the superconducting matrix are frequently proven to be dominant in the FP mechanism. When the normal phase is larger than the coherence length, the pinning mostly arises from the interfaces among the superconducting grain and the normal phases and/or from the generated defects. Normal phases may be generated naturally as Ca₂PbO₄ phase or artificially by adding a suitable concentration of fine nanoparticles such as Zr-rich inhomogeneities in our case. Hence, the former phase seems to be responsible for spatial variation of the mean free path of charge carriers (δl core) for Bi-0.0 ceramic, and for Bi-0.1 ceramic both Ca₂PbO₄ and Zr-rich inhomogeneities compete in δl pinning core to produce a positive competitive effect in reinforcing pinning properties.

Fig. 7

Reduced temperature dependence of the J_c for Bi-0.0 and Bi-0.1 ceramics. The red, blue, and green lines are theoretical curves according to δT_c, δl, and δε pinning cores. Inset shows the ratio $R$ between ${J}_{\text{c }}(T)$ of Bi-0.1 ceramic and Bi-0.0

5 Conclusion

Using the solid-state reaction process, we fabricated Bi-2223 ceramics with the addition of zirconia (ZrO₂) nanoparticles of about 20 nm in size. The proportion x of ZrO₂ per the total mass of superconductor ranged from 0 wt.% to 0.2 wt.%. We report the intragrain critical current density (J_c), pinning mechanisms, and thermal fluctuations induced excess conductivity. The main Bi-2223 phase along with Bi-2212 and insignificant Ca₂PbO₄ phases are identified in all ceramics. The Bi-2223 and Bi-2212 phase volume fractions are not impacted by the ZrO₂ addition, and they are about 86% and 13%, respectively. Plots of resistivity against temperature evidenced that the critical temperature for the transition to the superconductor state, T_co, slightly decreases from 101.4 to 99.7 K when the NP-ZrO₂ content is increasing from 0 wt.% to 0.2 wt.%. With the help of the Aslamazov–Larkin model, we assessed, thorough excess conductivity curves, the crossover temperatures between different regimes of conductivity, the coherence length, the penetration depth as well as ${B}_{c1}(0)$ and ${B}_{c2 }(0)$ values for all the ceramics. It was established that NP-ZrO₂ upgrades ${B}_{c1}(0)$ and ${B}_{c2}(0)$ and the elevated values are obtained for ceramic added with an amount of 0.1 wt.% NP-ZrO₂. We also found that the ${J}_{c}(T)$ is raised by NP-ZrO₂ addition and thereby the strengthening of δl pinning cores resulted in the improvement of the flux pinning ability.