1 Introduction

It has been established that the resistive transition between the normal and superconducting states in high critical temperature superconductors (HCTS) takes place in a well-defined two-step process: a first transition called the pairing transition and a second known as the coherence transition [1]. In the first, fluctuations in the amplitude of the superconducting order parameter, as predicted by the Ginzburg-Landau (GL) theory are relevant [2], with long-range order within the granular entities of the material. On the other hand, in the second stage, the significant physical quantity is the phase of the order parameter, which varies up to the long-range order range between the grains of the material, taking place through Josephson-type effects across the intergranular barriers due to the polycrystalline characteristic of this family of HCTS. In this description, when electrical resistivity is measured with decreasing temperature, the appearance of evanescent Cooper pairs is expected even in the normal state, whose density increases as the system reaches temperature values close to the volumetric critical temperature Tc. The analysis of these Gaussian fluctuations characteristic of the transition in type II superconductors is performed by means of the excess of conductivity Δσ, known as paraconductivity, which allows studying the different dimensional stages of the fluctuations in the vicinity of Tc, in accordance with the theory of Aslamov-Larkin (AL) [2]. Within its limit of validity, this mean-field model constitutes an excellent technique to determine the relevant critical superconducting parameters of the material, such as the coherence length ξ(0), the thermodynamic critical field BC(0), the penetration depth λ(0), the critical magnetic fields Bc1(0) and Bc2(0) and the critical current density Jac(0) [3]. In polycrystalline HCTS, very close to Tc, the system is dominated by genuinely critical fluctuations and the GL theory diverges, so the analysis of the excess conductivity is usually performed by implementing scaling models that facilitate the determination of the type of dynamic universality of the superconducting transition [4]. Below Tc, before reaching zero resistivity Tcs at the so-called coherence transition, a regime occurs in which microstructural defects in the polycrystalline material give rise to dissipative effects strongly dependent on the applied magnetic field intensity [5]. In this resistive region of the superconducting transition, the effects of granularity have been studied under the assumption that the intergranular barrier can be treated as a disordered matrix of Josephson junctions [6].

On the other hand, in resistivity measurements as a function of temperature in the presence of low magnetic fields, the appearance of Gaussian exponents near Tcs suggests the occurrence of a precursor behavior of the coherence transition [5]. In this scenario, close to this transition, the vortex dynamics has been analyzed through scaling models, whose critical exponents are characteristic of a vortex-glass type transition, with additional contributions to resistive dissipation before reaching the state of pure superconductivity in Tcs. Meanwhile, a relevant concern has to do with the case where resistivity measurements are not performed in constant magnetic fields applied to the material in a specific spatial direction, but under the application of different transport currents, which could give rise to local vortex effects due to grain shape and additional scattering at the granular interfaces.

With the aim of establishing the limit of the relevance of granularity versus vortex dynamics related effects in polycrystalline HTSC, in this work a careful study of the coherence transition regime in the electrical conductivity of GdBa2Cu3O7-δ under the application of different (low) transport current densities, whose respective magnetic fields are established by Maxwell’s laws [7], is presented. For these purposes, the analysis is performed by implementing the AL model in the vicinity of the coherence transition, evaluating the dynamics of the dispersive behavior in the temperature regime Tcs < T < Tc.

2 Experimental

GdBa2Cu3O7-δ material was synthesized via solid-state reaction method. Precursor oxides of GdO3, BaCO3 and CuO, of 99.99% purity, were dried and weighed in stoichiometric proportions to obtain a sample of 400 mg. After a mechanical grinding process for 3 h in an agate mortar, the powdered mixture was subjected to a pressure of 300 kg-f/cm2 in a hardened steel die to form a cylindrical pellet of 12.0 mm diameter. Subsequently, the sample was calcined at 860 °C for a time of 96 h, after which it was ground for 30 min, pressed again and sintered at 960 °C for 96 h. A final treatment at 400 °C for an additional time of 96 h in oxygen atmosphere was applied. Although the samples were initially produced in the shape of a cylinder, to facilitate resistive measurements, the pellets were cut in the shape of parallelepipeds with dimensions 11.35 × 4.55 × 1.45 mm3. The lattice parameters obtained by analysis of the experimental X-ray diffraction data were a = 3.843 Å, b = 3.905 Å and c = 11.721 Å. For electrical measurements, silver dye contacts were fixed by heat treatment at 200 °C for 1 h on the sample surface, specifying that those for the application of the transport current were located over the entire parallelepiped caps in order to ensure approximately parallel lines in the transport current path along the sample. Subsequently, copper wires were attached to the contacts for the application of the respective current and measurement of the potential difference. Electrical resistivity measurements were carried out by means of an AC resistometer at low frequency. The cryogenic environment of the setup allows measurement from liquid nitrogen temperature upwards, for which a platinum temperature sensor with an accuracy of 2 mK was used. Calibration of the system for the sample dimensions specified above allowed calculation of an equivalence of 1.135 ± 0.005 G of magnetic field intensity per 1 mA of applied transport current. Under these conditions, resistivity measurements were carried out as a function of temperature for several current densities.

3 Analysis Method

Figure 1a shows that the application of different currents affects the part of the resistivity curve close to the value at which it becomes zero. As the current density increases, this transition regime shows more resistive dissipation and the critical temperature Tcs undergoes a shift towards the lower temperature region. This regime associated with the coherence transition is characteristic of polycrystalline HTCS [8] through the appearance of a shoulder to the left of the maximum peak of dρ/dT, which, with increasing applied current, widens dramatically in temperature, dramatically decreasing the value of Tcs. Meanwhile, the maximum in dρ/dT (identified as TP = 94.9 K in Fig. 1b) is not affected by the variation in the transport current.

Fig. 1
figure 1

Resistivity as a function of temperature (a) and derivative with respect to temperature (b) for GdBa2Cu3O7-δ under application of several transport currents

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According to AL theory [2], the behavior of the paraconductivity ∆σ as a function of temperature during the superconducting transition has the form of a potential function of the type. The data analysis is conveniently performed by applying the Kouvel-Fisher method [9, 10], which consists of linearizing the potential function, applying the concept of the logarithmic derivative

$${\chi }_{\sigma }^{-1}=-\frac{d}{dT}\left(Ln\Delta \sigma \right)=-\frac{d}{dT}\left[Ln\left(A{\varepsilon }^{-\lambda }\right)\right]=\frac{T-{T}_{c}}{\lambda }$$
(1)

The data analysis was performed as was detailed in previous work [3].

4 Discussion

In Fig. 2, two temperature regimes can be clearly identified considering the minimum value reached by each current curve with the treatment of the related data in Eq. (1). Above this minimum point, the behavior is similar for the seven currents, in this zone the paring transition occurs, there, the genuinely critical regime to determine the critical temperature Tc [10, 11], and the Gaussian fluctuations to determine the critical parameters characteristic of the sample will be studied. Below Tc, the coherence transition will be the regime studied. Here, the fluctuations in resistivity are different for each current; however, Fig. 2 suggests a scaling factor related to the magnitude of the applied electric current to model its behavior. This scaling model was proposed in a previous work [5] where different magnitudes of magnetic field were applied instead of electric currents.

Fig. 2
figure 2

χσ−1 inverse of the logarithmic derivative of the paraconductivity applied to the experimental data

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4.1 Fluctuations in the Normal State and Coherence Length Determination

In the pairing transition, the critical exponent λ value associated to the critical genuinely regime is λcr = 0.33 [2], these are the resistivity fluctuations nearest to the critical temperature Tc. In Fig. 2, the critical genuinely regime was found in the temperature range 95.40 K < T < 95.93 K, with a λcr = 0.32 on average for all currents. Extrapolating the behavior modeled by Eq. (1), at χσ−1 = 0, the critical temperature of the sample was determined as Tc = 95.07, this value has a difference 0.15 K with the maximum value of the resistivity derivative, TP = 94.92 K, see Fig. 1. The critical exponent regime for the 3D Gaussian fluctuations according to the AL is λ3D = 0.50 [2], in the experimental data, the average for the applied current values was λ3D = 0.49 in a temperature range of 95.94 K < T < 96.70 K. The procedure detailed in a previous work [3] was followed to determine the critical parameters: coherence length ξ(0) and the critical magnetic field. In the 3D Gaussian Fluctuation temperature range, it is possible to determine ξ(0) through the relation between the reduced temperature and the paraconductivity given by Eq. (2) [2],

$$\Delta\sigma={\frac{\text{e}^2}{32\hslash\xi\left(0\right)}\varepsilon^{-\lambda_{3D}}}.$$
(2)

Figure 3 shows the relation of the Eq. 2 for each current, where the slope is ξ(0).

Fig. 3
figure 3

Paraconductivity Δσ and reduced temperature ε relation, the slopes ξ(0) for each current are: 2.36, 2.38, 2.64, 2.66, 2.68, 2.72, 2.56 nm for the currents 2, 10, 25, 50, 200, 400, 600 mA respectively

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The average for all the currents is ξ(0) = 2.57 nm, this value is taken as the coherence length of the GdBa2Cu3O7-δ sample which is similar to ξ(0) = 1.7 nm reported in the literature [12]. Knowing ξ(0), it is possible to determine parameters as the critical magnetic field Bc due to the Ginzburg and Landau theory can be applied to 3D Gaussian Fluctuation, thus on average is Bc(0) = 0.35 T.

4.2 Josephson Vortices and Vortex-Glass Transition (Coherence Transition)

Figure 4 exemplifies the behavior of the inverse of the logarithmic derivative of the excess of conductivity. Before reaching the zero resistance, all the currents describe a similar slope whose value can be associated to a critical exponent λ = 4.8 ± 0.2 related in Eq. (1). According to the literature [13], λ = ν(2 + z-d + η), where ν = 4/3 is the contribution due to the coherence length, η ≅ 0 is related with the order parameter correlation and the dimensionality is d = 3. This leads to a dynamic critical exponent z = 4.5, which is similar to the value reported for the transition vortex glass/fluid model, z = 4.4 [13].

Fig. 4
figure 4

χ−1 as temperature function. In average for the currents for currents of 600, 400, 200, 50, 25 mA, the critical exponent, in the genuinely critical region, has the value of λ = 4.8 ± 0.2, and λ' = 0.52 ± 0.02 for 3D Gaussian fluctuations

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Through the extrapolation to χ−1 = 0 for each current, it has gotten the paracoherent critical temperature, in Table 1 these values are reported for each current applied to the sample.

Table 1 Critical temperature for each electrical current applied
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By the extrapolation to χ−1 = 0 for each current, it has gotten the paracoherent critical temperature, in Table 2 these values are reported for each current applied to the sample. Analyzing the region between 90 K and 93 K, for currents 25, 50, 200, 400, 600 mA, in average, λ = 0.52 ± 0.02 were obtained, which is as the critical exponent associated to the dimensionality 3, λ = ½ from Eq. (2). When the same analysis was done at the pairing transition for the Gaussian fluctuations, a factor multiplying the coherence length was found as it is shown in Eq. (2). This term has been named Josephson coherence length because the AL analysis takes place in the coherence region where the paraconductivity can be associated with granular effects.

Table 2 Critical exponent λ' and Coherence length ξj(0) in the coherence transition regime
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For currents 2 and 10 mA, the magnetic field generated is not strong enough to show the resistivity fluctuation necessary to determine the critical exponent λ = 1/2. In Table 2, the values founded for λ', ξj(0) and the temperature region associated to the critical exponent for each current are shown. Figure 5 shows current as a function of Josephson coherence length ξj(0). A saturation of the Josephson coherence length is observed for current values equivalent to magnetic fields greater than Bc1, indicating the formation regime of vortex lines with glass-like disorder.

Fig. 5
figure 5

Josephson Coherence length associated with applied current. The fit curve is described by the function ξj(I) = a0 + a1exp(I/τ), where a0 = (5.7 ± 0.2) × 10−12 m, a1 = (4.2 ± 0.1) × 10−11 m, τ = (67 ± 3) mA

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According to Ampere law, the magnetic field induced H es proportional to the applied current i, H ∼ i [5]. It is possible to apply the scaling model previously done [5] in the current terms,

$$-\chi \left[T-{T}_{cs}\right]\sim \left[\frac{T-{T}_{cs}\left(i\right)}{{T}_{c}-{T}_{cs}\left(i\right)}\right]$$
(3)

Assuming [14, 15],

$$\Delta \sim {i}^{-1/2(2+z\pm d)}{S}_{\pm }\left[\frac{T-{T}_{cs}\left(i\right)}{{T}_{c}-{T}_{cs}\left(i\right)}\right]$$
(4)

Figure 6 shows the scaling behavior of the resistivity fluctuation in the coherence transition region for the current applied values.

Fig. 6
figure 6

Scaling behavior of Eq. (3) of the coherence zone for each current applied

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The curve associated to the 2 mA current does not match with group because its fluctuation is not significantly in the coherence regime. For the applied current values, the scaling of the curves in Fig. 6 suggests that the system is in the equivalent superconducting glass regime at low fields, slightly above the critical field Bc1(T).

5 Conclusion

The coherence transition of the material GdBa2Cu3O7-δ shows scalable behavior under the application of different magnitudes of applied currents. In addition, the vortex glass/fluid transition was identified in the genuinely critical regime, characterizing the contribution of the granular character of the sample to the fluctuations in the conductivity at the phase transition. Finally, the analysis of the Josephson effect has been proposed through the AL theory, applying the paraconductivity model as a function of temperature for 3D fluctuations in the coherence transition regime.