1 Introduction

Application of Y-based high temperature superconducting compounds in high magnetic fields and temperatures is restricted due to strong anisotropic property, very low charge carrier density, large penetration depth and short coherence length [1, 2]. However, the materials are especially preferred in researches and applications used in industry and technology because of their intrinsic optical and electronic properties, high transition temperature, smaller power losses, their remarkable high current and magnetic field carrying capacity [3, 4]. It is well known that the slight variations in the crystalline configuration of a high temperature cuprate superconductor (HTCS) form major changes in its magnetic properties [5]. In this respect, the chemical doping and substitutions of cations or anions are generally used for the structural variations [6] as a result of the alteration of the location and occupancy of the atoms positioned in the charge reservoirs CuO or Cu–O2 consecutively stacked layers. Similarly the superconducting transition temperature (abbreviated as T c ) increases or decreases considerably [7–11]. Moreover, the structure of a HTCS can be changed by altering the vacancy of the oxygen atoms in the charge reservoirs. When the oxygen content of superconductor YBa2Cu3O7 (Y-123) is lesser than the amount of 6.4, the crystalline structure changes from orthorhombic Pmmm lattice structure to tetragonal P4/mmm lattice structure as antiferromagnetic insulator. On the other hand, the maximum Jc peak occurs in Jc versus applied magnetic field graph owing to structural defects such as intrinsic pinning of vortices at the Cu–O2 planes [12] or oxygen vacancy clusters [13, 14]. This is known as the peak effect [15, 16]. It is thought that parameter of Jc decreases with increasing temperature as a consequence of the enhanced heat activity (flux movement) in the superconducting system. Accordingly, the flux pinning is strong in low temperature. Furthermore, the Jc values rely sensitively on the impurity in superconducting matrix, and high Jc parameter is observed in the presence of large local flux pinning centers. The long and short of it is that both the Jc and Tc values of the superconductor depend strongly on the substitution and doping concentration.

The activation energy directly related to the pinning ability is defined as a measure of flux pinning strength of a superconductor material [17, 18]. The flux pinning energy value is calculated from the thermally assisted flux flow (TAFF) model described by Arrhenius equation.

In the current work, we exert the magnetoresistivity measurements of a bulk YBa2−yNdyCu3O7 samples among the applied magnetic field strengths of 0–7 T in the temperature range of 10–120 K to find any possible structural influences stemmed from the foreign nanoparticles. The crystal structures of bulk YBa2−yNdyCu3O7 materials might be affected by grain boundaries, secondary phases and structural defects. Thus, we deduce the values of critical transition temperatures, critical fields, coherence lengths and penetration depths from the magnetoresistivity curves. Moreover, the flux pinning energy values of the YBa2−yNdyCu3O7−δ samples studied are obtained with the aid of the TAFF model. As a result, both the critical temperature and activation energy values increase with the enhancement of Nd content up to y = 0.250 beyond which they decrease owing to the presence of excess magnetic ions. In other words, pair-breaking mechanism in the system plays dominant role on the degradation of the Y-123 phase formation [19].

2 Experimental details

In the previous study [20], the change of the structural, mechanical, electrical and superconducting properties with the Nd inclusions inserted in the Y-123 superconducting matrix is investigated according to bulk density, X-ray diffraction, scanning electron microscopy, dc resistivity, microhardness and transport critical current density measurements. The samples are prepared by solid-state reaction method. In the present study, we attempt to scrutinize the impact of the Nd impurities on the magnetoresistivity behavior of the Y-123 superconducting system by use of the zero field cooling (ZFC) procedure. Thus, the resistivity of all the samples decreases linearly in ZFC, and transition from superconducting to normal sample (\(T_{c}^{onset}\)) in ZFC is much clearer than \(T_{c}^{onset}\) in FC. The measurements with the four-probe method are made between 10 and 120 K by applying magnetic field in range 0–7 T and constant current of 5 mA. The external magnetic fields, which are perpendicular to the c-axis, are applied normal to the direction of the current. The critical transition temperatures (\(T_{c}^{onset}\) and \(T_{c}^{offset}\)), room temperature resistivity (ρ 300K ), residual resistivity ratio (RRR), irreversibility field (μ0Hirr), upper critical field (μ0Hc2), thermodynamic critical field (μ0Hc), activation energy (U0), penetration depth (λ), coherence length (ξ) and Ginzburg–Landau (\(\kappa\)) parameters are calculated from the experimental magnetoresistivity graphs. In this paper, we represent the undoped sample as Nd0 and Nd stoichiometries such as 0.025, 0.050, 0.100, 0.250 and 0.500 in the Ba-site Nd substituted Y-123 superconductor are called Nd1, Nd2, Nd3, Nd4 and Nd5, respectively.

3 Results and discussion

3.1 Magnetoresistivity measurement results

The magnetotransport measurement findings constitute the main part of this paper due to the crucial parameters (\(T_{c}^{onset}\), \(T_{c}^{offset}\), ρ 300K , RRR, μ0Hirr, μ0Hc2, μ0Hc, U0, λ, ξ and \(\kappa\)) evaluated from the experimental results. In this work, the effects of Nd foreign nanoparticles on the magnetoresistivity characteristics are investigated at various external fields (0 T ≤ H ≤ 7 T). Figure 1 shows the experimental magnetoresistivity curves for the pristine and Ba-site Nd substituted Y-123 superconductor. It is visible from the figure that both the offset and onset transition critical temperature values tend to increase with the increment in the Nd concentration level up to the certain value of y = 0.250 above which the value (especially \(T_{c}^{offset}\)) reduces towards to the global minimum value for the maximum content level of Nd impurity due to new induced artificial dislocations (lattice defects), bulk defects (cracks, voids and pores) and permanent disorders in orientation of consecutively stacked layers [21, 22]. On the other hand, the \(T_{c}^{offset}\) values decrease considerably as a consequence of the fluxion motions along with adjacent layers by the enhancement of the external magnetic field applied whereas the \(T_{c}^{onset}\) values are hardly affected due to the intrinsic superconducting characteristics of the Y-123 compound [23]. Based on the combination of the Nd concentration level and applied magnetic fields, it is fair to conclude that the excess Nd additions in the Y-123 superconducting system lead to not only reduction in the coupling property of the cooper-pairs, and structural inhomogeneities in the oxidation state of the inter-grains and grain boundaries but also enhancement of the weak connectivity between the superconducting grains under the external magnetic fields (Fig. 1) [24]. In numerically, the maximum \(T_{c}^{offset}\) (\(T_{c}^{onset}\)) value is observed to be about 97.25 K (98.93 K) for the Nd4 superconducting sample while the minimum value is 32.97 K (54.84 K) for the Nd5 compound at zero applied magnetic fields. What stands out clearly here that the excess Nd decorations in the Y-123 superconducting matrix lessen considerably the mobile hole concentrations [25, 26].

Fig. 1
figure 1

Change of magnetoresistivity against temperature at several applied magnetic fields in the range of 0–7 T

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As for the magnetotransport measurements, the external field strengths suppress the critical transition temperature values for the YBa2−yNdyCu3O7−δ. Abrikosov and Gorkov (AG) explained the suppression in the critical transition temperatures with the pair-breaking theory in which the interaction between the magnetic impurities of spin S and conduction electron spins in the Cooper pairs makes the change in the spin states of the electrons and hence the Cooper pairs tend to break suddenly [27]. Furthermore, the significant decrement in \(T_{c}^{offset}\) parameter is associated with the increment of de Gennes factor (G = (g − 1)2 J(J + 1)) consisting of Lande g factor and total angular momentum for the Hund’s rules in the ground state owing to the presence of the magnetic impurities in the system. In this work, the excess magnetic Nd impurities (magnetic moments) result in the hole filling (localization) in the conduction band [28–30] as a result of the hybridization (cooper-pair breaking mechanism) of Cu3d–O2p states in the Cu–O2 layers [31–34]. As for the experimental findings under the external magnetic field strengths, the \(T_{c}^{offset}\) value of pristine material decreases from 84.52 to 53.10 K with the increasement of the applied magnetic field up to the constant field of 7 T. Similarly, the value of best (Nd4) sample is felt down to 82.04 K. In fact, we could not measure the offset critical transition temperature value of the Nd5 compound after the external field strength of 0.1 T due to the destruction of the superconducting properties. It is to be mentioned here that each \(T_{c}^{offset}\) parameter is much more lowered as compared to the \(T_{c}^{onset}\) parameter with the increment of Nd foreign impurities and especially applied magnetic field. Under these circumstances, do the \(T_{c}^{onset}\) values pertaining to the superconducting materials hardly alter when the \(T_{c}^{offset}\) values decrease dramatically. Namely, the degree of broadening (variation of ∆Tc) for the virgin and best materials is observed to be about 41.59 and 16.89 K, respectively with the increment in the applied magnetic field. Shortly, the enhancement of the applied magnetic field perpendicular to the c-axis brings about an extensive broadening of the superconducting transition due to the decrement in the flux pinning ability of the superconducting samples [35] as a result of the dissipation defined by the TAFF [8]. Moreover, the crucial results of measurements performed at the applied magnetic fields confirm that the retrogression in the pinning of two-dimensional 2D pancake vortices (improvement in the recoupling linelike) stems from both the presence of the decoupling of the adjacent layers and the suppression of the interlayer Josephson coupling with the Nd substitution level in the crystal textures [36]. The decrement of \(T_{c}^{offset}\) with increment of magnetic field applied stems from the grain boundary weak-links arisen from porosity and non-superconducting phases. Magnetic field begins to enter into the grain boundary weak-links and harms the superconducting properties seriously. Besides, the Nd+3 with strong magnetic ions exhibits high value of magnetic moment in the same direction of the magnetic field applied. Thus, the magnetic impurities inserted in the crystal structure increase the interior magnetic field. This is known as the main reason of decrease in the \(T_{c}^{onset}\) and \(T_{c}^{offset}\) values.

It is another probable result obtained from Fig. 1 that the YBa2−yNdyCu3O7−δ samples studied exhibit the positive temperature derivative of the resistivity (metallic behavior resulting from the interactions between the electron and phonon in the crystal lattice or the logarithmic divergence in density of states at the Fermi level) above their own critical temperature values [37, 38]. The room temperature resistivity (ρ300K) value tends to decrease with increasing the Nd inclusions in the Y-123 crystal structure up to the concentration level of y = 0.250 after which the ρ300K value immediately increases by three times and reaches to the maximum value of 8.61 mΩ cm. The regular decrease of normal state resistivity is attributed to the increment of both the metallic interactions between the superconducting grains and optimization of the mobile hole carrier concentration in the Cu-O2 slabs [39].

The magnetoresistivity curves allow us to determine the temperature-independent residual resistivity (ρ 0 ) and residual resistivity ratio (RRR) seriously. As well known, the former parameter deduced from the extrapolation method ascribes to the defects stemming from the grain boundary weak-links or possibly additional scattering centers [40, 41]; on the other hand, the latter one obtained from the ratio of ρ300K/ρ95K is in accordance with the determinant of the sample quality [42].

The ρ 300K and RRR values deduced for the materials studied in this comprehensive work are numerically tabulated in Table 1. It is apparent from the figure that the ρ 300K values are found to decrease from 7.89 to 2.77 mΩ cm with the enhancement of the Nd decorations in the superconducting system up to the content value of y = 0.250. This is associated with the decrement of impurity scattering and random defects regarding the superconducting grain boundaries, stacking faults, planar voids, cracks and micro defects in the Cu–O2 planes [43]. As for the latter (RRR) parameters, the Nd4 material obtains value of 173 while the smallest value of 0.75 is observed for the Nd5 sample. It is pertinent to mention here that the Nd4 compound with the greatest conductivity is produced in the best quality.

Table 1 Electrical measurement results including ρ 300K , RRR, \(T_{c}^{onset}\) and \(T_{c}^{offset}\) results of the YBa2−yNdyCu3O7−δ samples under different external magnetic fields
Full size table

The grain sizes are in the nm scale and they are more uniform than pure YBCO up to y = 0.250. The mean crystallite diameter obtained from XRD analysis (not shown here) using the Scherer’s formula [44] is 93 nm. The fact that nearly uniform grain sizes have been obtained after the preparation of the sample is important for the magnetic analysis. Non-regular grains imply grain boundaries with different areas and thus less uniform Josephson junctions between superconducting grains.

We now focus on studying the magnetic features of the YBa2−yNdyCu3O7 and any relationship with the structural characteristics of samples. Meanwhile, YBa2−yNdyCu3O7 is a type II superconductor, some of its elementary properties can be pronounced by the field-dependence magnetization, M(H), which also provides information about the trapped fields in it. This behavior is caused by the paramagnetic characteristic of its secondary phases such as Y2BaCuO5, CuO and BaCuO2. The presence of impurities or secondary phases in samples is detected by analyzing their magnetization curves as the temperature approaches T c . The M(H) dependence for the clean and defect-free cuprate superconductors is noted to be reversible. Namely, no magnetic flux is trapped within the superconductor as disappearance of the external field. The M(H) dependence becomes highly irreversible for superconductors containing defects or impurities which interact with the penetrating flux lines. Secondary phases are commonly found after the preparation of HTCSs. The phases obtained for YBCO by the preparation technique (solid state) are Y2BaCuO5, CuO and BaCuO2 [45]. The presence of secondary phases is a possible explanation for the origin of the fishtail or peak effect [46, 47]. In the case of YBa2−yNdyCu3O7, such decrease affects the critical current density values, the pinning force and could be caused by structural defects like partial occupancy of the oxygen in the Cu–O chains in the structure. The irreversible magnetization is zero at H = H irr , in contrast to the reversible magnetization disappearing at H = H c2 . Ultimately, as the external field is reduced from H c2 to H c1 the magnetic moment becomes positive over a wide range of applied field indicating that not much magnetic field has been trapped in the YBa2−yNdyCu3O7 superconductor as in YBCO. Therefore two characteristics can be observed from the ρ-T graph, the irreversibility and remnant magnetization are caused by presence of secondary phases whereas the decrease of magnetization under external field is possibly caused by structural defects.

The critical current density of a bulk superconducting compound with randomly oriented grains is strongly limited due to the weak link behavior. Only very small currents are able to flow throughout the grain boundaries when the external magnetic field is applied. This dependence is dominated by the classical 1/H law [48] reflecting the J c (H) dependence of a percolation network of Josephson junctions [49].

The peak effect in YBCO is correlated with clustering of the oxygen vacancies. The J c is controlled by two contributions, the background contribution of inter-grain currents of a bulk sample and the intra-grain contribution of field-induced pinning (originating the peak effect) which might be caused by clusters of oxygen vacancies in the Cu–O chains. The latest is better seen when the temperature is closer to T c .

3.2 Activation energy

It is known with certainty that the copper oxide plane carries the maximum value of current while the current is the minimum value through the layers perpendicular to the Cu–O2 plane. In high temperature superconductors, coherence length is very small from penetration depth. As the lower critical magnetic field BC1 of high-temperature superconductors is very low, the upper critical magnetic field BC2 value is very high. Thus, fixing the magnetic vortex weakened and in this case the critical current Ic decreases. The size of the energy barrier for fluksoid of new oxide superconductors is smaller than conventional superconductors and it is determined that a small coherence length cause the small energy barrier [50]. Unlike almost all low-temperature superconductors are isotropic, high temperature superconductor shows a high spatial anisotropy. Anisotropy shows itself in critical fields, critical current density, the depth of penetration of the magnetic field and resistance measurement. Anisotropy occurs mainly due to the layered crystal structure that is the default for superconductivity of high temperature. Free porosity, high density, strong links between particles and homogeneous structure able to be formed for superconducting material are important as building features.

The Y-based superconductors exhibit extremely weak flux pinning characteristic due to their inherent anisotropy. Even, the presence of vortices in the system prevents to pass larger currents along the materials at rather smaller magnetic field applied. The increment in the external field applied result in decrement slowly the vortex lattice period but increase the density of the vortices owing to the change in the nature of flux-line motion.

As a result, the pinning of two-dimensional (2D) pancake vortices recrudesces owing to the decoupling of the adjacent layers, and so the sample starts to show the resistivity. Namely, the excess Nd impurities deteriorate essentially the interlayer Josephson coupling by ascending the thermal fluctuations of pancake positions. The magnetic field carrying capacity is considerably retrograded by the decrease of Josephson coupling length and the increase in the elastic moduli of the vortex lattice [36]. Theoretically, the thermal activation creates the flux lines jump to neighboring states over the energy barriers (called as the statistical distribution) [51, 52] and so extra electrical energy recommended by Arrhenius law transforms into the heat energy [53].

Furthermore, thermally activated flux flow (TAFF) is strong method to scrutinize the pinning (dissipation) mechanism and potential energy barriers of the pinning centers in the Y-123 superconducting materials [54, 55]. In this work, the magnetic field dependence of the effective flux pinning (activation) energy values is determined from Arrhenius activation law abbreviated as \(\rho = \rho_{0} \exp \left( {{\raise0.7ex\hbox{${ - U_{0} }$} \!\mathord{\left/ {\vphantom {{ - U_{0} } {k_{B} T}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${k_{B} T}$}}} \right)\). In the model, when U0 symbolizes the pinning energy value, k B is the universal Boltzmann’s constant and ρ 0 is a field-independent pre-exponential factor [56]. The flux pinning energy values of our superconducting ceramics samples are obtained from the slope of linear data in the broadening tail part of Arrhenius graphs (variation of logarithmic resistivity over the reciprocal of temperature) at different applied magnetic fields in the range of 0–7 T as given in Fig. 2. The energy dissipation is steady with the thermal activation of flux across the pinning barrier. Interestingly, the figure demonstrates that the flux pinning energy values are dependent strongly upon the Nd concentration level and external magnetic field. In more detail, all the activation energy values deduced are numerically listed in Table 2. According to the table given, the activation energies tend to decrease considerably with the increase of applied magnetic fields. To illustrate, the energy values pertaining to the Nd4 sample is computed to be about 6997 and 477 K at zero and 7 T applied magnetic field, respectively. Similarly, the value is found to be about 86 K (53 K) at zero (0.1 T) magnetic field applied for the Nd5 sample. The substantial decrement in the pinning energy values are related to the contraction of distance between the pinning centers, confirming the presence of different superconducting levels within the superconducting materials at the inter-granular regions [57, 58]. In other words, the excess Nd impurities degrade significantly the energy barriers in the adjacent layers [59].

Fig. 2
figure 2

ln(ρ/ρ0) versus 1/T graphs of the bulk Y-123 superconducting materials

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Table 2 Activation energy and β values of the pure and Ba-site Nd substituted Y-123 superconductors
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Additionally, the flux pinning energy values with regard to various applied magnetic fields are graphically given in Fig. 3. While the flux pinning energy values are decreasing sharply up to the certain magnetic field value of about 1 T, they reduce slowly and slowly above it. The main reason of this behavior stems from the penetration of the magnetic field into only the inter-granular media below the field value of 1 T [60].

Fig. 3
figure 3

Variation of flux pinning energy over different magnetic fields up to 7 T for the Nd samples

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Moreover, we survey the magnetic field dependence of the flux pining energies belonging to the YBa2−yNdyCu3O7−δ compounds with the aid of the log–log plot of activation energy versus applied magnetic field (Fig. 4). It is obvious from the figure that the relation between the variables is obtained to be linear and the external field-dependence of the flux pinning energy can thus be defined by a power law as:

$$U(\upmu_{0} H)\;\alpha \;\upmu_{0} H^{ - \beta }$$
(1)

here the parameter of β is in association with the orientation of the magnetic field in terms of the basal plane and magnetic field strength. After fitting the data obtained, we find the β values in the range from 0.486 (for the Nd0 sample) to 0.307 (for the Nd4 material) as embedded in Table 2. The reduction of β value stems from the destruction of grain boundary weak interactions in the inter-granular coupling. It is noteworthy that the Nd0 (Nd4) compound exhibits least sensitivity (most sensitivity) to the applied magnetic field. Unfortunately, we cannot determine the β value belonging to the Nd5 compound due to the lack of the magnetic characteristics under the applied field which is larger 0.1 T. It is fair to conclude that the Nd nanoparticles are ploughed to improve the flux pinning ability, electrical and superconducting properties for the Y-123 matrix. To sum up, the excess Nd individuals give rise to deteriorate the correlation between copper oxide planes. In fact the excess impurities (x > 0.250) destroy the Y-123 superconducting lattice.

Fig. 4
figure 4

External magnetic field dependence of flux pinning energy (U0) for the inorganic solid compounds

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3.3 Irreversibility and upper critical field

The high temperature superconducting samples at the magnetic field applied confirm that each superconducting material attains negative surface energy playing an important role to form the quantized fluxoids on the specimen surface, and thus superconductivity in the structure does not abolish altogether, in contrast, retrogrades gradually at even higher applied magnetic field. This fact can be verified by the presence of the H c , H c1 , H c2 and H c3 parameters. Magnetic field up to the value of H c1 is completely expulsed by the material exhibiting Meissner behavior [61] and the irreversibility fields related to H c1 are obtained by means of the following relation [62]:

$$H_{irr} \cong 200 \times H_{c1}$$
(2)

Moreover, the field strength between H c1 and H c2 values penetrates partially in the form of quantized vortices within fluxoids and the materials enter in a special state referred as mixed state or Schubnikov phase [63]. At the applied critical field of H c2 , the average field in the interior equals to the external magnetic field applied and the second-order phase transition occurs in the system; that is, the bulk superconductivity disappears completely. At fairly greater field applied (H c3  > H > H c2 ), the superconductivity in a thin surface layer appears only. In other words, the surface sheath of the superconducting compound persists to rather greater critical field that is named as H c3  ≈ 1.69 H c2 [34]. Additionally, in type-II superconductors there is another crucial field known as the thermodynamic critical field (H c ) that is in association with the measure of the extent to which the superconducting state is favored over normal state in the lack of magnetic field and can be extracted from the formula given below:

$$H_{c1} = \frac{{H_{c}^{2} }}{{H_{c2} }}$$
(3)

All the critical field parameters given above can be determined by several experimental methods including the resistive transition ρ(μ 0 H, T), ac susceptibility and dc magnetization techniques to obtain the information about the vortex dynamics of the systems [64, 65]. In the present work, the resistive transition ρ(μ 0 H, T) measurements give an advantage that the systematic survey of the applied magnetic field on the resistive transitions simplifies to discuss the change in the vortex dynamics of new superconducting system [66, 67].

Temperature-dependent resistivity at the constant of the external field is obtainable as the following relations [68, 69],

$$\rho \left( {\upmu_{0} {\text{H}}_{\text{irr}} ,T} \right) = 0.1\rho_{n} \;{\text{for}}\;{\text{the}}\;\upmu_{0} {\text{H}}_{\text{irr}} \left( T \right)$$
(4)

and

$$\rho \left( {\upmu_{0} {\text{H}}_{{{\text{c}}2}} ,T} \right) = 0.9\rho_{n} \;{\text{for}}\;{\text{the}}\;\upmu_{0} {\text{H}}_{{{\text{c}}2}} \left( T \right)$$
(5)

where ρ n is related to the normal state resistivity values concerning to the inorganic solid compounds at the temperature value of 95 K. The intercepts of extrapolation to zero point on the temperature axis for the values of μ0Hirr (T) and μ0Hc2 (T) ascribe to the direct values of μ0Hirr (0) and μ0Hc2 (0), respectively [70].

It is obvious from the Fig. 5 that the μ0Hirr and μ0Hc2 curves shift to far lower temperature values with the increment in the Nd impurity content level after y = 0.250 as a result of the reduction in the flux pinning force [71]. In the case of the content level of y = 0.500, we cannot evaluate the full set values of the μ0Hirr and μ0Hc2 parameters. This is in accordance with the fact that the excess Nd content level in the Y-123 superconducting matrix demolishes the pinning ability completely due to the shrink of distance between the pinning centers.

Fig. 5
figure 5

Temperature dependences of a Irreversibility fields (μ 0 H irr ) and b Upper critical fields (μ 0 H c2 )

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At the same time, the theoretical results found are numerically listed in Table 3 in detail. It is visible from the table that the H irr parameters vary from 764.50 T (for the pure sample) to 76.39 T (for the Nd4 material). This result is another clue of the degradation of the flux pinning ability with the increase of the Nd concentration level in the Y-123 superconducting system owing to the existence of the Cooper pair-breaking mechanism.

Table 3 Temperature-independent critical field parameters (μ0Hc1, μ0Hc, μ0Hirr, μ0Hc2 and μ0Hc3) and extracted λ, ξ and κ values belonging to the superconducting samples at absolute zero temperature
Full size table

As for the H c3 and H c values calculated for the materials, there seems to be similar trend in the change of the H c3 and H c parameters in the presence of the Nd nanoparticles in the Y-123 crystal structure. The smallest values ascribe to the sample prepared by the maximum Nd concentration inserted in the Y-123 lattice. In other words, the excess Nd impurity makes rapidly the phase translation from the superconducting state to normal state of the Y-123 structure, confirming strongly both the decrease of Josephson coupling length and increase in the elastic moduli of the vortex lattice.

3.4 Coherence length, penetration depth and Ginzburg–Landau parameter

One of the important properties of coherence length is impurity of samples. If there is impurity, coherence length increases. Secondary phase in samples are effective for critical current. These phases behave like flux trap center and increase coherence length. When the coherence length is greater than 2 nm, critical current density increases.

Coherence length of superconductor in perfect impurity is abbreviated ξ0 and is also intrinsic characteristic properties of superconductor. ξ0 value is formulated by Pippard as following;

$$\xi_{0} \approx \left( {{{\hbar \text{v}_{\text{f}} } \mathord{\left/ {\vphantom {{\hbar \text{v}_{\text{f}} } {\uppi\Delta \left( 0 \right)}}} \right. \kern-0pt} {\uppi\Delta \left( 0 \right)}}} \right) \approx 0.18\left( {{{\hbar {\text{v}}_{\text{f}} } \mathord{\left/ {\vphantom {{\hbar {\text{v}}_{\text{f}} } {{\text{k}}_{\text{B}} {\text{T}}_{\text{c}} }}} \right. \kern-0pt} {{\text{k}}_{\text{B}} {\text{T}}_{\text{c}} }}} \right)$$
(6)

where kB is Boltzmann constant, Δ(0) is change in energy of sample that is superconductor at absolute 0 K and vf is velocity of electrons at Fermi energy level.

On the other hand, the coherence length is attributed to a cylindrical core of a radius ξ whereas the penetration depth is in related to a current circulating around the core out to a distance λ [72]. Hence, theoretical calculations including the upper critical magnetic field and irreversibility fields allow us to find the ξ(0) and λ(0) values at the absolute zero temperature as the following relations:

$$\xi (T) = \left[ {\frac{{\phi_{0} }}{{2\pi \mu_{0} H_{c2} (T)}}} \right]^{{\frac{1}{2}}} \;{\text{and}}\;\lambda (T) = \left[ {\frac{{\phi_{0} }}{{2\pi \mu_{0} H_{irr} (T)}}} \right]^{{\frac{1}{2}}}$$
(7)

where the vale of \(\phi_{0}\) is equal \(2.07 \times 10^{ - 15} Tm^{2}\). The parameter values of ξ(0) and λ(0) are shown in Table 3. It is said from the given table that both the ξ(0) and λ(0) parameters have a tendency to continuously fall with the enrichment of the Nd content level up to y = 0.250 in the Y-123 superconducting lattice and actually reach the lowest value ξ(0) of 16.47 Ǻ and λ(0) of 20.77 Ǻ for the Nd4 sample. This is correlated to the fact that the Nd inclusion in the superconducting structure is favorable for the velocity of the Y-123 phase formation and it is reasonable to conclude that the superconducting quantities pertaining to the Y-123 ceramics develop with the rise of the Nd nanoparticles. It is another possible result that the considerable increment in the penetration depth (relative to the coherence length) is the result of the reduced mean free path with the excess Nd concentration level (after y = 0.250). The change of the ratio between the coherence length and penetration depth is defined as Ginzburg–Landau parameter abbreviated as \(\kappa\) at only vicinity of the absolute zero temperature. It is pertinent to mention here that for the type II superconducting materials the value of \(\kappa\) is almost constant along the whole temperature range study. According to the results, all the compounds produced exhibit London type superconductors in electrodynamics locality as a result of the higher \(\kappa\) value of \(\frac{1}{\sqrt 2 }\) and also the regular enhancement in the \(\kappa\) value with the excess Nd content level suppresses the superconductivity as shown in the Nd5 specimen.

4 Conclusion

Although some similarities between YBa2−yNdyCu3O7 and YBCO structures do exist, there are also incredible differences in their magnetic behaviour. Below the T c , the magnetic flux can generally enter hard the YBa2−yNdyCu3O7 compounds into its vortex region (H c1  < H < H c2 ) and greater amount of fields can be trapped than in pure YBCO.

When we viewed the variable magnetic field and temperature graphs, decline of \(T_{c}^{offset}\) became more pronounced by increasing the applied magnetic field. This situation arises due to be non-superconducting phase in the superconducting particles of samples and weak links caused by the presence of porosity between the particles. Magnetic field begins to enter into the weak links boundary of samples and causes the negative effect on superconductivity. Also, if Nd impurity is considered to have great magnetic moments and to be a strong magnetic ion, magnetization or interior field is increased by these magnetic moments that try to get themselves through the direction of increasing field applied. As a result, the superconductivity will be affected negatively. This is known as the main reason for the decline in \(T_{c}^{offset}\).

In this comprehensive study, we struggle to examine the variation of the flux pinning ability, electrical and superconducting characteristics of the Y-123 superconducting compounds with respect to the Nd concentration level (0 ≤ x ≤ 0.500) in the crystal structure via the magnetotransport measurements exerted in the external field range from 0 to 7 T. The main aim of the study is to obtain the regular consecutively stacked layers in order to develop both the pinning of 2D pancake vortices and the interlayer Josephson coupling by the partial replacement of Ba by Nd impurities in the system. Hereafter, the thermal fluctuation and interplane interaction energy of pancake vortices reduce noticeably and the end, the magnetic field carrying capacity is meaningfully improved by the increment of Josephson coupling length. According to the magnetoresistivity measurement, the room temperature resistivities, residual resistivity ratios, critical transition temperatures, penetration depths and coherence lengths, irreversibility fields, upper critical fields, thermodynamic critical fields, activation energies approve the regular enhancement in the flux pinning abilities of the Y-123 superconducting materials with the addition in the Nd concentration level up to y = 0.250 as a result of the regular distribution of the artificial pinning centers. Furthermore, the noteworthy conclusions from the current work are as the followings:

  • The residual resistivity values tend to decrease systematically with arising the Nd nanoparticles up to y = 0.250 as a consequence of the decrement in the superconducting grain boundaries, voids, stacking faults, planar and micro defects in the Cu–O2 planes. After y = 0.250, the specimen quality diminishes regularly owing to the lessening of the conductivity between the superconducting layers, that is localization problems in the conduction band, being even confirmed by the change in the normal state resistivity values belonging to the bulk Y-123 materials.

  • Likewise, both the \(T_{c}^{onset}\) and \(T_{c}^{offset}\) values fall fast with the excess Nd inclusions due to both the distortion between Y-123 superconducting layers, and the formation of the porosities and weak-links between the superconducting grains. Reduction of the structural inhomogeneities in the oxidation state of the grains and grain boundaries is another feasible evidence for the decrement in \(T_{c}^{onset}\) and \(T_{c}^{offset}\) parameters with the Nd decorations. Similarly, the applied magnetic field damages melodramatically the critical transition temperatures and in fact we cannot measure the temperature values for the Nd5 (exposed to excess Nd impurities) sample at higher external magnetic field value such as 0.1 T. This is in association with the Cooper pair-breaking mechanism described as the hybridization of Nd5 states with the Cu3d–O2p states in the Cu–O2 layer, being favored by the degree of the broadening. Numerically, the \(T_{c}^{onset}\) values decrease from 94.69 K towards 54.84 K whereas the \(T_{c}^{offset}\) values change in the range of 84.52–32.97 K with increasing the Nd concentration level at zero applied magnetic field. However, the \(T_{c}^{offset}\) value of the best (Nd4) sample reduces to 82.04 K at the applied filed of 7 T while we cannot measure the values for the Nd5 material at higher applied magnetic field then 0.1 T.

  • The activation energy changes are similar in the change of the critical temperatures. To be precise, the flux pinning energy values comprehended from the TAFF model tend to decrease from 6997 to 477 K with external magnetic field applied for the best sample (Nd5) due to the increment in the thermal fluctuations. This is correlated to the extensive regression of the pinning of 2D pancake vortices with the randomly distributed Nd inclusions in the Y-123 superconducting lattice. In other words, the recoupling linelike feature among the Cu–O2 slabs improves quickly and the adjacent layers desire decoupling hastily. Therefore, both the interlayer Josephson coupling length and elasticity of the vortex lattices suppress in the system.

  • Additionally, variation of the activation energies against the applied magnetic field shows that the sensitivity of the materials to the external field increases recurrently as a result of the demolition of the weak interactions in the inter-granular couplings.

  • As for the other theoretical findings obtained, the critical field parameters as regards μ0Hc, μ0Hc1, μ0Hirr, μ0Hc2 and μ0Hc3 and the extracted λ, ξ and \(\kappa\) values each present the degradation of the flux pinning ability with the Nd concentration owing to the presence of the random distribution of the artificial pinning centers and Cooper pair-breaking mechanism.

All in all, it is not wrong to assert that the Nd inclusions up to x = 0.250 are ploughed to improve the flux pinning ability, electrical and superconducting characteristics of the Y-123 superconducting texture.