Fluctuation induced conductivity and pseudogap state studies of Bi1.6Pb0.4Sr2Ca2Cu3O10+δ superconductor added with ZnO nanoparticles

The major limitations of the Bi1.6Pb0.4Sr2Ca2Cu3O10+δ superconductor are weak flux pinning capability and weak inter-grains coupling that lead to a low critical current density and low critical magnetic field which impedes the suppleness of this material towards practical applications. The addition of nanoscales impurities can create artificial pining centers that may improve flux pinning capability and intergranular coupling. In this work, the influences of ZnO nanoparticles on the superconducting parameters and pseudogap properties of the Bi1.6Pb0.4Sr2Ca2Cu3O10+δ superconductor are investigated using fluctuation induced conductivity analyses. Results demonstrate that the ZnO nanoparticles addition improves the formation of the Bi1.6Pb0.4Sr2Ca2Cu3O10+δ phase significantly. Various superconducting parameters include coherence length along c-axis (ξc(0)), penetration depth (λpd(0)), Fermi velocity (vF), Fermi energy (EF), lower and upper critical magnetic fields (Bc1(0) and Bc2(0) respectively) and critical current density (Jc(0)), are estimated for samples with different amounts of ZnO nanoparticles. It is found that the values of the Bc1(0), Bc2(0), and Jc(0) are improved significantly in the 0.2 wt% ZnO added sample in comparison to the ZnO-free sample. The magnitude and temperature dependence of the pseudogap Δ*(T) is calculated using the local pairs model. The obtained values of Tpair, the temperature at which local pairs are transformed from strongly coupled bosons into the fluctuating Cooper pairs, increases as the added ZnO nanoparticles concentration enhances up to 0.2 wt%. Also, the estimated values for the superconducting gap at T = 0 K (Δ(0)) are decreased from about 26 meV in ZnO-free sample to about 22 meV in 0.2 wt% ZnO added sample and then increases for higher values of additive.

The superconducting pairing mechanism in high-temperature superconductor (HTS) materials remains rather controversial, more than three decades after their discovery 1,2 . It has been clear that the superconductivity mechanism in HTS materials can be understood by investigating the normal state properties of these materials [3][4][5] . The short coherence length, high anisotropy, and low density of carriers in the HTS materials lead to a progressive deviation of the resistivity ρ(T) curve from the linear metallic-like behavior in the normal state, at a representative temperature T* > T c . This deviation is followed by a considerable rounding that is observed above the transition temperature T c 6,7 . This deviation from linear behavior indicates thermal fluctuations of Cooper-pairs. These fluctuations are responsible for the finite probability of the Cooper-pairs formation above T c 8 . Thermal fluctuations, in turn, can result in an excess conductivity above T c . this is called the fluctuation induced conductivity (FIC) [8][9][10] . FIC provides an opportunity to investigate the superconducting Cooper-pairs fluctuations behavior in a broad range of temperatures above T c . The study of the FIC has attracted significant attention in the research of the HTS materials [10][11][12][13][14] . These interesting studies are keys to providing information about microscopic and superconducting parameters of the HTS materials such as Fermi velocity and Fermi energy of charge carriers, coherence length, cross over temperatures, phase relaxation time (lifetime) of fluctuating pairs, critical magnetic fields, and critical current density 6,8,15 . Moreover, theoretical conceptions of the FIC region or Cooper-pairs generation could be examined 16  www.nature.com/scientificreports/

Methods
To investigate the effects of ZnO NPs on the microscopic and superconducting parameters of the (Bi, Pb)-2223 phase, a series of composite samples of Bi 1.6 Pb 0.4 Sr 2 Ca 2 Cu 3 O 10+δ /(ZnO)x with 0.0 ≤ x ≤ 1 wt% (x = 0.0, 0.1, 0.2, 0.3, 0.5 and 1 wt%) were prepared by conventional solid-state reaction method. Appropriate stoichiometric quantities of high-purity Bi 2 O 3 , Pb 3 O 4 , SrCO 3 , CaCO 3 , and CuO powders (all of the analytical grades with minimum purities of 99.9%) were used as raw materials. As a first step, the raw materials were mixed and grounded in an agate mortar for about 1.5 h to ensure homogeneity. The mixed materials were calcined at 820 °C for 24 h in the air followed by furnace cooling to room temperature. Then calcinated materials reground into a fine powder. The calcining and grinding procedures were repeated three times. In the next step, different amounts of ZnO NPs (0.0 to 1 wt%) with a mean crystallite size of 15 ± 2 nm were added to the calcined powder and ground in agate vials for 2 h using a planetary ball mill (FRITCH P7). The mixed powder samples were pressed into blocks (15 mm × 4 mm and about 2 mm in height), under the pressure of 7 tons/cm 2 . In the final step, the composite samples were sintered at 810 °C in the air for 120 h. The phase identification of the prepared samples was carried out by powder X-ray diffraction using a Bruker D8 Advance X-ray diffractometer, with Cu Kα radiation (λ = 1.506 Å) in the diffraction angle range of 3° ≤ 2θ ≤ 80°. The concentrations of the (Bi, Pb)-2223 and (Bi, Pb)-2212 phases formation were estimated from the X-ray diffraction (XRD) peaks intensities. The microstructure and grain morphology of the various composites have been recognized by a scanning electron microscope (SEM) (Philips, XL30 model).
To investigate the superconducting properties, fluctuation induced conductivity, and pseudogap properties, the temperature dependence of the resistivity ρ(T) was measured by the conventional four-point-probe technique. For fluctuation induced conductivity analysis, Aslamazov-Larkin and Lawrence-Doniach models were employed to estimate the fluctuation dimensionality, zero coherence length along c-axis (ξ c (0)), penetration depth (λ pd (0)) Fermi velocity (v F ), Fermi energy (E F ), lower and upper critical magnetic fields (B c1 (0) and B c2 (0)), and critical current density J c (0). To measure critical current density, the V-I curves of the different samples have been recorded at 77 K by the four-point-probe technique. Also, to calculate the magnitude and temperature dependence of the pseudogap, the local pairs model has been used.

Results and discussion
XRD analysis and phase formation investigation. XRD patterns of the (Bi, Pb)-2223/(ZnO NPs)x (x = 0.0, 0.1, 0.2, 0.3, 0.5 and 1 wt%) composites are displayed in Fig. 1a. The pattern of the ZnO-free sample (x = 0.0 wt%) indicates that the major peaks in this sample are related to the (Bi, Pb)-2212 and (Bi, Pb)-2223 phases and some weak peaks are observed which belong to Ca 2 PbO 4 and (Bi, Pb)-2201 as minor phases. It shows that (Bi, Pb)-2212 is the dominant phase in this sample. As can be observed, the (Bi, Pb)-2223 peaks' (2θ = 5°, 24°, 26.5°, 29°, 32°, and 34°) intensity for the ZnO-added samples enhanced significantly and (Bi, Pb)-2212 peaks' (2θ = 6°, 23.5°, 25, 27.5°, 31.5°, 35.5°, and 50.5°) intensity dropped in comparison with the ZnO-free sample. The samples with ZnO additive are composed of (Bi, Pb)-2223 with tetragonal structure as the dominant phase. The relative volume fractions of the (Bi,Pb)-2223 to (Bi,Pb)-2212 phases were estimated from the XRD peaks' intensities as described in Supplementary Note 1. The calculated relative volume fractions of (Bi, Pb)-2223 and (Bi, Pb)-2212 phases versus added ZnO NPs concentrations are demonstrated in Fig. 1b. As seen, by increasing the ZnO concentration, the (Bi, Pb)-2223 phase volume fraction increases from ~ 38 wt% in the ZnO-free sample to ~ 87 wt% in the sample with x = 0.2 wt% and then shows a decreasing behavior for higher values of added ZnO NPs until it reaches to ~ 73 wt% in the sample with x = 1 wt%. On the other hand, the volume fraction of (Bi, Pb)-2212 phase reduces from ~ 58 wt% for the ZnO-free sample to ~ 8 wt% for the sample with x = 0.2 wt% and then enhances for the samples with higher values of ZnO NPs concentrations and reaches to about ~ 19 wt% in the sample with x = 1 wt%. The XRD results indicated that the small amount of the ZnO NPs improves the formation of (Bi, Pb)-2223 phase significantly, which can be attributed to the grain connectivity improvement by ZnO NPs 15 . It has been reported that the addition of impurities decreases the partial melting point of the Bi-Sr-Ca-Cu-O system 66 . It is well established that, the optimum sintering temperature is defined just below the partial melting temperature 67 . Consequently, enhancement of (Bi, Pb)-2223 phase formation by adding the ZnO NPs can be attributed to the improvement of the sintering process in added samples. Moreover, as the added ZnO NPs concentration increases, no detectable shifts in (Bi, Pb)-2223 related XRD peaks were observed. This indicates that ZnO NPs are dispersed at the grains' boundaries and do not change the host crystal structure of the (Bi, Pb)-2223 superconductor phase.
Morphology and microstructure. The SEM micrographs of the (Bi , Pb)-2223/(ZnO NPs)x composites with x = 0.0, 0.2, and 1 wt%, as typical ones, are represented in Fig. 2a-c respectively. These micrographs show a granular structure that is a typical structure of HTS materials. Also, plate-like grains are observed that are a sign of (Bi, Pb)-2223 phase formation. Compering of the SEM micrographs shows that the concentration and the size of the plate-like grains are enhanced in the samples with ZnO NPs additive compare to the ZnO-free sample. The added samples have coarser grains and lower porosity. These results indicate a better intergranular coupling in the added samples. In addition, the SEM micrographs display that the size and concentration of the plate-like grains are decreased in the sample with x = 1.0 wt% in comparison with the sample with x = 0.2 wt%. These effects are related to the influence of the ZnO NPs on the sintering process. The obtained results from SEM micrographs are in good agreement with the XRD results.
Electrical resistivity measurements. The temperature dependence of resistivity ρ(T), it's corresponding derivative dρ/dT, and extrapolated normal state resistivity ρ n (T), for the ZnO free sample, as a typical one, has been shown in Fig. 3a. The related curves for other samples are represented in Supplementary Fig. S1 www.nature.com/scientificreports/ seen, for all samples electrical resistivity measurements demonstrate a well-defined metal-like behavior (normal state) followed by a transition to the superconducting state (zero resistance). The normal state resistivity is characterized by the stability of the Fermi surface 2 and determined from the linear fitting of ρ(T) curve, in the range 2T c ≤ T ≤ 300 K , according to the Anderson and Zou relationship 68 : where ρ 0 is the residual resistivity that is specified by extrapolation of the linear fitting to 0 K and α is the temperature coefficient of resistivity. ρ 0 is temperature independent and indicates the defect density and homogeneities of the samples while α is a temperature-dependent intrinsic parameter. For temperatures above T*, resistivity decreases linearly with temperature by a gradient α = dρ/dT. Below T* progressive deviance of the resistivity curve from the linear behavior is observed that followed by a notable rounding that displays the appearance of fluctuation induced conductivity. Some electron pairs start to appear as the temperature is decreased below T*. These electron pairs lose the long-range coherence needed for superconductivity. By further temperature decreasing, the number of formed electron pairs improves until achieved the mean-field critical temperature T mf c , where all conducting electrons are paired and act in correlation 7 . As shown in Fig. 3a, the T mf c can be determined using the derivative of the resistivity curve dρ/dT. It is the temperature corresponding to the maximum in the plot of dρ/dT versus temperature. Also, the transition temperature width (ΔT c ) was calculated by measuring the full width at half the maximum of the dρ/dT curve. To more accurate determination of T*, the criterion (ρ n (T) − ρ 0 )/αT = 1 was used 36 . The plot of (ρ n (T) − ρ 0 )/αT against T, is shown in the inset of Fig. 3a. T* is defined as a temperature which the (ρ n (T) − ρ 0 )/αT deviates from 1.
The different parameters include ρ(290 K) (room temperature resistivity), ρ 0 , T c (ρ = 0) (zero-resistivity critical temperature), ΔT c , T mf c , α and T*, were calculated from ρ(T) curves for various composites. Figure 3b shows the variation of ρ(290 K) and ρ 0 versus ZnO NPs content. As observed, both ρ(290 K) and ρ 0 decrease with the increasing of the ZnO NPs concentration up to 0.2 wt%. The values of the ρ(290 K) and ρ 0 in the sample with  Fig. 3c. As can be seen, T mf c is almost constant (about 105 K) for different x values. Moreover, T c (ρ = 0) increases from 92 K for the ZnO-free sample to 97 K in the 0.2 wt% added sample and then decreases for samples with higher ZnO NPs concentrations and reaches to 83 K in the sample with x = 1.0 wt%. Enhancement of the T c (ρ = 0) is attributed to the improvement of the inter-grain coupling by adding ZnO NPs. Figure 3c also shows that transition temperature width (ΔT c ) is decreased from about 9.5 K in the ZnO-free sample to about 5.9 K in the sample with x = 0.2 wt% and then increased for higher values of the ZnO NPs concentrations. The reduction of the ΔT c is related to the improvement of the homogeneity by adding ZnO NPs up to 0.2 wt%. The α and T* values for different samples are recorded in Table 1. As it can be seen, there is not a regular trend for α and T* values with increasing the ZnO NPs concentration. The results show that the maximum T* and α values are related to the x = 0.3 and 1.0 wt% respectively.  Fluctuation-induced conductivity. The excess conductivity (Fluctuation induced conductivity) Δσ is defined as a difference between measured conductivity σ(T) and the normal state conductivity σ n (T) extrapolated to the low T region 69 . It can be calculated as: where ρ(T) is the measured resistivity and ρ n (T) is the normal state resistivity extrapolated to the low T region. The Aslamazov-Larkin (AL) model 17 and Lawrence-Doniach (LD) model 18 were used for fluctuation induced conductivity analyses. According to the AL model 17 , the fluctuation induced conductivity region consists of three different regimes include critical, mean-field, and short-wave fluctuations. That the mean-field regime comprises of three different parts indicating three-dimensional (3D), two-dimensional (2D), and one-dimensional (1D) fluctuations regimes. According to this model, the excess conductivity is written as: is a reduced temperature, and λ is a critical exponent related to the conduction dimensionality which has values of 0.3, 0.5, 1, 1.5, and 3 respectively for critical (CR), three-dimensional (3D), two-dimensional (2D), one-dimensional (1D), and short-wave (SW) fluctuations regimes. C is the temperature-independent fluctuation amplitude which is given in 1D, 2D, and 3D fluctuations by the following equations: where e is the charge of the electron, ℏ is the reduced Planck's constant, ξ c (0) stands for zero-temperature coherence length along the c-axis, d displays the effective layer thickness of the 2D system and s presents the crosssectional area of the 1D system.
The AL theory was modified by Lawrence and Doniach (LD) for polycrystalline and layer superconductors 18 . In cuprate superconductors, superconductivity takes place principally in 2D CuO 2 planes which are coupled by Josephson tunneling. In the LD model excess conductivity is expressed as: where J = (2ξ c (0)/d) 2 represents inter-layer coupling strength. For the strong coupling (J > > 1) the above equation reduces to the 3D fluctuations condition in Eq. (5) (λ = 0.5) and for the weak coupling (J < < 1) we get the 2D fluctuation condition in Eq. (5) (λ = 1). The following expression results for cross-over temperature between 3 and 2D fluctuation regimes The excess conductivity Δσ(T) was calculated according to Eq. (2). The ln-ln plots of the Δσ versus the reduced temperature ε for different samples are presented in Supplementary Fig. S2. Plots display the existence of three regions include critical, mean-field, and short-wave fluctuation regions. The mean-field region of each  Table 1. Conductivity exponents, crossover `oscopic parameters estimated from electrical resistivity measurements and fluctuation induced conductivity analyses.
x (wt%) α (μΩ.cm/K) T* (K) λ CR λ 3D λ 2D λ SW T G (K) T 3D-2D (K) T 2D-SW (K) ΔT 3D (K) ΔT 2D (K) N G × 10 −2 κ www.nature.com/scientificreports/ sample consists of two distinct linear parts comprising 3D and 2D fluctuations regimes. The various regions of the curves were linearly fitted and the values of the conductivity exponent λ were obtained from the slopes. The different regions are separated from each other by crossover temperatures. The obtained values of the conductivity exponent λ and crossover temperatures for various samples are demonstrated in Table 1.
The short-wave region is the first sector which lies at temperatures much higher than the T mf c . In this region, the fluctuation induced conductivity reduces sharply with λ SW ≈ 3 (Table 1). Also, the characteristic wavelength of the order parameter becomes comparable to the coherence length order 7 . These behaviors are related to the variation in the density of carriers or the band structures in the Fermi surface, where both of them have a major influence on the change in the order parameter 70,71 . By decreasing the temperature, a transition from the shortwave region to the 2D fluctuation region occurs at temperature T 2D-SW . As shown in Table 1, the 2D conductivity exponent λ 2D values vary between 1.03 and 1.42. In this region, the charge carriers move along the CuO 2 planes and conductivity mainly occurs from charge carriers limited in the CuO 2 layers 52 . By further temperature reduction close to T mf c , the conductivity exponent decreases, and the 3D fluctuation region starts at T 3D-2D . For different samples, the values of the λ 3D changes between 0.60 and 0.64 (Table 1). In this region, the charge carriers cross the barrier layers to reach the conducting CuO 2 layers. They move between the planes and more influenced by thermal fluctuations compared to the 2D region. This implies that the charge carriers tend to move more freely in the whole sample before the Cooper-pairs formation 6 . As shown in Table 1, by increasing the amount of ZnO nanoparticles the width of the 2D fluctuations region ΔT 2D increases and reaches to its maximum value in the sample with 0.3 wt% ZnO NPs, whereas the width of the 3D fluctuation region ΔT 3D decreased and in the sample with x = 0.3 wt% is minimum.
The final region is the dynamic critical region, where the fluctuations of the order parameter become comparable to the magnitude of the order parameter itself 7 . The Ginzburg-Landau theory breaks down and interaction between Cooper-pairs is assumed 52 . Crossing between the 3D region and the critical region occurs at T G . The values of λ CR vary between 0.25 and 0.33 (Table 1). These values are in good agreement with the theoretical prediction of the 3D-xy universality class, with dynamics given by the representative E-model 72,73 . Despite of numerous investigations of superconducting fluctuations in the HTS materials, the conclusions about the effects of inhomogeneities which occur at varying length scales, are contradictory. Indeed, some studies have shown that inhomogeneities crucially influence the width of the critical and the mean-field (2D and 3D) regions of superconducting fluctuation [74][75][76][77] . It has been observed 74 that by adding nanoparticles to the (Cu 0.5 Tl 0.5 )Ba 2 Ca 2 Cu 3 O 10-δ the critical region is disappeared in some nanoparticle concentrations. Also, it has been reported 77 that the width of the 3D region was reduced by the addition of nanoparticles in polycrystalline (Bi, Pb)-2223 superconducting matrix, which was explained based on the scattering of mobile carriers across the insulating nanoparticles present at the grain-boundaries.
By the values of the T 3D-2D and the LD model (Eq. (6)) the ξ c (0) and J were estimated for different samples. Variation of the ξ c (0) and J as a function of the ZnO NPs concentration are shown in Fig. 4a. The obtained values for these parameters have a good agreement with reported values for (Bi, Pb)-2223 phase 9,49,52 . The obtained short coherence length (few Å) for prepared samples is generally a trait of the HTS materials and is caused by the presence of the overlapping energy bands 15 . As seen, the ξ c (0) and J values decrease as the concentration of ZnO NPs enhances from 0.0 wt% to 0.2 wt%, and then increase for further enhancement of the ZnO NPs concentrations. The minimum value of the ξ c (0) for x = 0.2wt% displays the suppression in the density of charge carriers in conducting planes 6,74 .
Moreover, when the ξ c (0) value is obtained, the Fermi velocity v F and Fermi energy E F of the charge carriers can be estimated using the following expressions where K≈ 0.1242 is a proportionality constant and m* = 10m 0 is the electron effective mass 14,15 . As displayed in Fig. 4a, it is clear that the Fermi velocity and Fermi energy of the charge carriers are suppressed with increasing in the ZnO NPs concentration from 0.0 to 0.2 wt%. The estimated values of v F are less than that of the free electron (v F = 10 8 cm/s) 15 . The Fermi velocity depends on the density of carriers in the CuO 2 planes 78 . As observed, the addition of ZnO NPs from 0 to 0.2wt% reduces the interlayer coupling strength J that suppresses the charge transfer mechanism to the conducting planes, and led to suppressing the density of carriers in the CuO 2 planes. The obtained Fermi velocity and energy values are comparable with the results obtained by other groups 15,79 .
According to the Ginzburg-Landau (GL) theory 80 , the thermodynamic magnetic critical field B c (0) can be calculated form Ginzburg Number (N G ), which is determined by the equation where T G represents the crossover temperature from critical to 3D fluctuation regime, k B stands for the Boltzmann's constant, and γ = ξ ab (0)/ξ c (0) is the anisotropy parameter with an approximate value around 35 for (Bi, Pb)-2223 system 16 , where ξ ab (0) stands for coherence length within the CuO 2 planes (in the ab plane). The penetration depth λ pd (0), lower critical magnetic field B c1 (0), upper critical magnetic field B c2 (0), and critical current density J c (0) are estimated, after determination of the B c (0), using the following GL Eqs. 6,81 : www.nature.com/scientificreports/ where � 0 = h/2e is the flux-quantum number and κ = λ pd /ξ is the Ginzburg-Landau parameter. As displayed in Fig. 4a, the value of λ pd (0) are decreased by the enhancement of ZnO NPs concentration from 0.0 to 0.2 wt% and then increased for higher values of ZnO NPs concentration. The N G parameter are presented in Table 1. The variation of N G with ZnO NPs concentration shows the same trend as λ pd (0). The estimated superconducting critical parameters include B c (0), B c1 (0), B c2 (0), and J c (0) are plotted in Fig. 4b. As observed, the critical superconducting parameters B c (0), B c1 (0), B c2 (0), and J c (0) are improved significantly with increasing of the ZnO NPs content to 0.2 wt% and then diminish with further increases in ZnO NPs concentration. Comparing of the obtained critical superconducting parameters for 0.2 wt% ZnO NPs added sample with free-sample shows that, B c (0), B c1 (0), B c2 (0), and J c (0) have been improved by about 78, 100, 58, and 150% respectively. The improvement in these critical parameters is mostly due to the reduction in the magnetic vortices' motion through improving the flux pining ability inside the composite, revealing the existence of strong pinning sources. Introduced nanoparticles to superconducting matrix plays the role of artificial pinning centers that can improve flux pining capability 6 . The obtained results indicate that the inclusion of ZnO NPs is a promising candidate to reduce the vortices' motion, and improving the critical superconducting parameters in the (Bi, Pb)-2223 phase.
Critical current density measurements. The measured V-J curves, at 77 K, for different (Bi, Pb)-2223/ (ZnO NPs)x composites are demonstrated in Supplementary Fig. S3. Critical current densities of the different composites were determined from the V-J curves, using the criterion of 2 μV/cm. Figure 5 illustrates the measured critical current density as a function of the ZnO NPs added concentration. As observed, it increases from www.nature.com/scientificreports/ about 114 A/cm 2 for x = 0.0 wt% to 249 A/cm 2 for x = 0.2 wt%. For higher values of ZnO NPs, the critical current density decreases and reaches 45 A/cm 2 for x = 1.0 wt%. As can be seen, the obtained behavior for measured critical current density at 77 K has a good agreement with the estimated one at 0 K, from fluctuation induced conductivity analyses. The improvement of the critical current density by adding the ZnO NPs up to 0.2 wt% is attributed to the improvement of the inter-grain coupling and flux pining capability.
Pseudogap temperature dependence. As noted above, the pseudogap is a special state of materials that is defined by a reduced electron density of states at the Fermi level. In HTS materials, there is a reduction in the quasi-particle density of states at T < T* (the reasons for this are not entirely discovered), which provides the required conditions for the pseudogap formation 82 . The experimental observation of the pseudogap state in HTS materials has become possible for the first time, primarily due to the development of the angle-resolved photoemission spectroscopy (ARPES) technique 83 . Accordingly, the local pairs model allows us to acquire information about the pseudogap temperature dependence by analyzing the fluctuation induced conductivity 2,11 . In the local pairs model, it is considered that the deviation of the ρ(T) from linearity in the normal state is owing to the opening of the pseudogap at T* > T c , leading to the appearance of the excess conductivity, as a consequence of local pairs (strongly coupled bosons) formation 11,36 . These local pairs are subjected to the Bose-Einstein condensate (BEC) theory. When the HTS reaches characteristic temperature T = T*, strongly coupled bosons (SCBs), which do not interact with each other, are starting to form. As the temperature decreases, the concentration of SCBs increases, and some part of them transform into fluctuating Cooper-pairs (FCPs) obey the BCS theory. With more temperature decreasing, the concentration of SCB drops, and the concentration of FCP raises as a result of the transformation of SCB into FCP. At T = T pair all of the SCBs are transferred into the FCPs and in the temperature range, Tc < T ≤ T pair all pairs exist prevailingly in the FCP form 33,48,84,85 . As noted, the excess conductivity is assumed to appear in the temperature range of T c < T ≤ T* due to the local pairs' formation and pseudogap opening. This in turn means that the excess conductivity Δσ(T) is a consequence of such processes should enclose information about the magnitude and temperature dependence of the pseudogap 36 . To obtain such information, an equation is required that describes the experimental behaviors of the Δσ(T) over the entire temperature range from T* to T c and would contain the pseudogap parameter explicitly. The local pairs model describes excess conductivity by 11,28,36 where A 4 is a numerical coefficient that has the same meaning as the C factor in Eq. (3) and ε * co is a theoretical parameter that specifies the shape of theoretical curves for T > T 2D-SW 28,86 . Δ* displays pseudogap and assumed that Δ* = Δ*(T G ). Besides, (1 − T/T*) determines the number of local pairs formed at T ≤ T* and exp( Δ*/T) determines the number of local pairs destroyed by thermal fluctuations as T approaches T c . Solving Eq. (14) for Δ*(T) one can obtain www.nature.com/scientificreports/ where Δσ(T) is the experimentally obtained excess conductivity 36 . The mentioned parameters in Eqs. (14) and (15) include ε * co , A4, and Δ* are also directly determined from the experiment within the local pairs model as described in Supplementary Note S2. It is assumed that Δ* = Δ*(T G ) = Δ(0), where Δ(0) is the superconducting gap at T = 0 K 2,87 . The estimated values for prepared samples are recorded in Table 2. The Δ*(T G ) values for samples are varied between 255 K (22 meV) to 351 K (30 meV). The estimated Δ*(T G ) values are in good agreement with the reported values of the superconducting gap Δ (0), obtained from the Andreev spectra 49,82,88 . The value of Δ*(T G ) is dropped from 310 K (26 meV) in the ZnO-free sample to 255 K (22 meV) in the sample with 0.2 wt% additives and then increases to 351 K (30 meV) in the sample with 1.0 wt% ZnO NPs. These behaviors and Δ*(T G ) values are also in good agreement with experimentally obtained Δ* magnitude at T G (Fig. 6a-f). The magnitude of Δ*(T G ) was used to determine BCS ratio 2Δ(0)/k B T c = 2Δ*(T G )/k B T c in different samples. As displayed in Table 2, the BCS ratio is about 5.9 in ZnO NPs free sample and drops to 4.8 in the sample with 0.2 wt% ZnO NPs. For higher values of additive, the BCS ratio is increased. The optimal approximation of 2Δ*(0)/ k B T c for the Bismuth based cuprates is attained at values 5-7 49,89 .
The temperature dependence and magnitude of the pseudogap parameter Δ*(T) have been constructed by Eq. (15) using the obtained values for A 4 , ε * co , ξ c (0), and T* parameters. The Δ*(T) curves for different samples are demonstrated in Fig. 6a-f. All the curves show the shape typical for HTS materials. As can be seen, by temperature decreasing the pseudogap value first increases and then reduces after passing through a maximum at T = T pair . This reduction is due to the variation of the SCBs into the FCPs. With the further temperature decreasing, Δ*(T) usually shows a minimum at T min ~ T 2D-SW , and then it increases slightly and reaches a maximum at T max ~ T 3D-2D followed by a minimum always at T G . Below T G , there is an abrupt jump in Δ*(T) at T → T c . The obtained values of T pair and Δ*(T pair ) for different samples are shown in Table 2. As seen, T pair is increased from 137 to 147 K by increasing the ZnO NPs concentration from 0.0 to 0.3 wt% and then is decreased for the samples contain more ZnO NPs. Also, the maximum value of the pseudogap Δ*(T pair ) is varied between 411 K (35 meV) for the sample contains 0.1 wt% ZnO NPs to 543 K (46 meV) for the sample with 0.3 wt% additives. These obtained values have a good agreement with reported values for Bi-(2223) phase 49,82 .
These results show that fluctuation induced conductivity analysis is a sufficiently instructive and effective method to study the pseudogap state properties of HTS materials. Different mechanisms can affect the superconducting properties of the (Bi, Pb)-2223 phase by the addition of ZnO NPs. These mechanisms include intergranular coupling, flux pining capability, and intragranular properties. As observed the addition of ZnO nanoparticles up to 0.2 wt% improves the intergranular coupling by decreasing the undesirable (Bi, Pb)-2212 phase. Besides, the added ZnO nanoparticles can play the role of artificial pinning centers and led to the enhancement in the flux pining capability. Therefore, the improvement in critical current density and critical magnetic fields is attributed to the improvement of the flux pinning capability and inter-grain coupling. On the other hand, as observed, the macroscopic superconducting parameters of the samples (ξ c (0), E F , v F , λ pd , and J) were changed by the addition of the different ZnO NPs concentration. It shows that the added ZnO NPs affected the intragranular properties of the samples. Accordingly, the variation of the pseudogap properties can be related to the competition between intergranular coupling, intragranular properties, and flux pining capability.

Conclusion
The impacts of the ZnO nanoparticles addition on the microscopic, superconducting, and pseudogap properties of the Bi 1.6 Pb 0.4 Sr 2 Ca 2 Cu 3 O 10+δ superconductor are investigated using fluctuation induced conductivity analyses. The XRD results show that the inclusion of small amounts of the ZnO NPs improves the (Bi, Pb)-2223 phase formation significantly. The volume fraction of (Bi, Pb)-2223 phase increases from ~ 38 wt% in the pure sample to ~ 87 wt% in the sample with 0.2 wt% additive. As the ZnO NPs concentrations increases, no detectable shifts on the XRD peaks of the (Bi, Pb)-2223 phase are observed that indicates the ZnO NPs dispersed at the grain boundaries by filling the spaces between (Bi, Pb)-2223 grains and then does not change the crystal structure. The SEM micrographs also displayed enhancement of plate-like grains which is a sign of the (Bi, Pb)-2223 phase formation and detects the improvement of the intergranular coupling in ZnO-added samples. Electrical resistivity measurements confirm the improvement of the transport properties and reduction in the room temperature and residual resistivity by increasing the ZnO NPs concentrations up to 0.2 wt%. Also, the zero-resistance critical temperature T c (ρ = 0) is enhanced by introducing ZnO NPs and reached its maximum values for the sample with 0.2 wt% ZnO NPs.
The fluctuation induced conductivity analyses were carried out using the Aslamazov-Larkin and Lawrence-Doniach models. The analyses on the prepared composites indicate the existence of four distinct fluctuation regimes include critical, 3D, 2D, and short-wave fluctuations. The dimensionality of the fluctuation regions www.nature.com/scientificreports/ and the microscopic parameters such as coherence length along the c-axis ξ c (0), inter-layer coupling strength J, Ginsberg number N G , penetration depth λ pd (0), Fermi velocity v F , and Fermi energy E F of the charge carriers were estimated. The results show that the width of the 3D fluctuations region is suppressed while the width of the 2D fluctuations region is enhanced by increasing the ZnO NPs concentration up to 0.3 wt%. It is observed that as the added ZnO NPs concentration increases up to 0.2 wt%, the obtained values for ξ c (0), J, N G , λ pd (0), v F and E F decreases and reaches their minimum values and for higher amounts of ZnO NPs they grow up. The superconducting critical parameters include thermodynamic magnetic field B c (0), lower and upper critical magnetic fields (B c1 (0), and B c2 (0)), and critical current density J c (0) were calculated using the Ginzburg-Landau theory.
The results demonstrate a significant improvement in these important superconducting parameters. The values of B c (0), B c1 (0), B c2 (0) and J c (0) increase about 78, 100, 58, and 150% respectively in the sample with 0.2 wt% additive in comparison with the ZnO NPs free sample. The improvement in these critical parameters is ascribed to the reduction of magnetic vortices motion and improvement of intergranular coupling by the appropriate small content of ZnO NPs inclusion. Since the addition of ZnO NPs develops artificial pinning centers inside the (Bi, Pb)-2223 matrices, hence the magnetic vortices motion is reduced in samples with a small content of ZnO NPs. www.nature.com/scientificreports/ Finally, the magnitude and temperature dependence of the pseudogap Δ*(T) was calculated using the local pairs model. The obtained Δ*(T) curves show the shape characteristic for high-temperature superconductors with a maximum at T pair (the temperature at which local pairs are transformed from strongly coupled bosons (SCBs) into the fluctuating Cooper pairs (FCPs)) and a minimum at T G (crossover temperature between critical and 3D fluctuation regions). The results indicate that the value of T pair increases from about 137 K for the ZnO NPs free sample to about 147 K in the 0.3 wt% ZnO NPs added sample. The value of Δ*(T G ) is dropped from 26 to 22 meV as the added ZnO NPs concentration increases from 0.0 to 0.2 wt% and then increases for higher values of additive and reaches to 30 meV in the sample contains 1.0 wt% ZnO NPs. The Δ*(T G ) is equal to the superconducting gap at T = 0 K, Δ(0). The obtained values for Δ*(T G ) was used to determine the BCS ratio of 2Δ(0)/k B T c in different samples. The BCS ratio is decreased from 5.9 in the ZnO NPs free sample to 4.8 in the sample with 0.2 wt% ZnO NPs and then enhanced by more increase in the additive concentration.
In conclusion, It is found that the addition of 0.2 wt% ZnO NPs into the (Bi, Pb)-2223 superconducting matrix improves the (Bi, Pb)-2223 phase formation, inter-grain coupling, and flux pinning capability which leads to a significant enhancement in the critical current density and critical magnetic fields.