The resistance, $R_N$, of junctions between grain boundaries (GB) of bulk cuprate high temperature superconductors (HTS) affects critical current density ($J_c$) in a complex way even though GBs are known to be effective pinning centres [1], [2], [3], [4], [5]. Any factors that affect $R_N$ are therefore very important in vortex physics. $R_N$ is affected by several factors such as the orientation of the grains, the presence of impurities in the inter-granular regions, the roughness and local environment of the grain boundaries [6], [7], [8], [9]. Understanding and controlling of $R_N$ is very crucial for the development of the HTS-based devices for practical applications [10], [11].

Generally with an increase in higher superfluid density the effectiveness of resistance of GBs drops significantly resulting in increase in $J_c$ in a complex manner [12], [13], [14], [15], [16]. There is no known critical value of $j_s$, the superfluid phase stiffness (proportional to the superfluid density) below which the zero resistive or even a low resistive state can be observed. Moreover, the misalignment angle between the superconducting grains plays an important role in controlling the resistive state of GBs [17], [18]. It is very important to understand how the temperature variation of $J_s$ and $R_N$ (T) are related in bulk HTS.

We have used current–voltage (IV) characteristics and micrographs of bulk YBa₂Cu₃O_7−δ (YBCO) and DyBa₂Cu₃O_7−δ (DyBCO) superconductors to extract $R_N$. $R_N$ has been determined as a function of temperature by using (i) Ambegaokar-Baratoff (AB) and (ii) de Gennes (dG) denoted as $R_{N}^{A B}$ and $R_{N}^{d G}$, respectively [19]. Both $R_{N}^{A B}$ and $R_{N}^{d G}$ are obtained using two different energy gap functions, $Δ_{BCS}$ and $Δ_{HTS}. For $$R_{N}^{d G}$ (following the dG equation) a functional form of the correlation length ($ξ_N$) with parameter b is used. Normalised superfluid density, $ρ_s$(T)/ $ρ_s$(0) of both samples has also been extracted. How normalised superfluid density (NSD) dominates $R_{N}^{A B}$ (T) and $R_{N}^{d G}$ (T) has been investigated.

Synthesis of bulk polycrystalline YBa₂Cu₃O_7−δ (YBCO) and DyBa₂Cu₃O_7−δ (DyBCO) superconducting samples has been carried out following the standard solid state reaction method [20], [21]. We have synthesized samples by mixing pure oxides in the appropriate stoichiometric ratio and the mixed powder is pressed into pellets. We have carried out calcination at approximately 850 °C for 24 h. The samples have been sintered thrice at temperatures ranging from 930 °C to 950 °C with intermediate grindings. Using a uniform rate of flow of oxygen annealing is carried out at 450 °C for a duration of 50 h [22]. Granularity of samples are characterised by using Scanning electron microscope (SEM). The typical dimensions of the bar-shaped samples used for the transport measurements are 6.2 mm × 2.4 mm × 0.46 mm. Resistivity (ρ) as a function of temperature (T) has been measured by using the standard four-probe method with the help of a closed cycle cryogenerator (Janis, USA). Electrical contacts are well separated and the typical distance between the voltage probe is approximately 1.6 mm. A dc current in the range of 100.0 nA through 5.0 mA is used for all transport measurements [23], [24].

In Fig. 1(A) and (B) we have shown the scanning electron micrograph (SEM) of YBCO and DyBCO respectively.We have used ImageJ software to determine the average grain size.The average sizes of grains are found to be 6.01±2.11μm and 5.39±2.03μm for YBCO and DyBCO respectively. Most of the grain boundaries are found to be well defined in both samples. Grain boundaries are basically Josephson junctions below the onset critical temperatures [25], [26]. Below $T_c$, individual grains are in the superconducting state whereas junctions of varying thickness remain in the normal state. With the decrease in T below $T_c$, the junction (<1μm) resistance varies in presence of the increasing SPS in the superconducting grains (>1μm) [27].

In Fig. 2, we have shown resistivity, (ρ) as a function of T for YBCO and DyBCO. Normal states of the both samples exhibit the metallic behaviour down to the onset transition temperature [28], [29]. The ρ (300 K) of YBCO and DyBCO are 1.7 mΩ-cm and 2.1 mΩ-cm respectively. In Fig. 3(A) and (B), the variation of $d \rho / d T$ with T is shown. The onset transition temperature, ($T_c$) for each sample has been determined by identifying the temperature at which the abrupt change in $d \rho / d T$ occurs from the normal state. $T_c$s of YBCO and DyBCO are found to be 91.0 K and 79.5 K, respectively and are shown in Fig. 3(A) and (B).

Fig. 4(A) displays the IV characteristics of YBCO at several temperatures. Corresponding current density (J) and electric field (E) are also shown in addition to IV. The IV curves in Fig. 4(A) for temperatures ranging from 75.0 K to 92.0 K are shown together with a representative IV curve at 76.5 K. We found that the IV at the lowest measurable temperature is linear for YBCO. Fig. 4(B) presents IV curves for the DyBCO within T = 67.0 K to 80.0 K. In the inset of Fig. 4(B) we have shown IV at the lowest measured temperature of DyBCO. $J_c$ (T) has been extracted by taking into account the criterion of electric field, E = 0.01 mV/cm [30], [31]. In Fig. 5 $J_c$ (T) are shown. For the YBCO sample, the highest $J_c$ is approximately 226.16 mA/cm², while for the DyBCO, the maximum $J_c$ is 227.42 mA/cm². In the presence of the grain boundary networks, the decrease in transport $J_c$ in comparison to that of the single crystal (of YBCO) has been observed [3], [32]. Impact of GBs on vortex dynamics and hence on $J_c$ is a complex phenomenon. We have used $J_c$ in understanding several $R_N$ of both bulk YBCO and DyBCO.

Even though the determination of $R_N$ is really a challenging research, several theoretical models have been used to extract and understand it. We have made an attempt to determine $R_N$ by using two equations for intergranular junctions as follows [33], [34], [19].

Firstly, the Ambegaokar-Baratoff (AB) equation, as given in Eq. (1), is utilized to calculate the junction resistance, $R_{N}^{A B}$. Following Eq. (1), the determination of $R_{N}^{A B}$ involves $I_c$ and the energy gap, Δ. For $R_{N}^{A B}$, we have used two distinct energy gap functions, namely $Δ_{BCS}$ and $Δ_{HTS}$ [35], [36]. According to the BCS theory, Δ is represented by an equation, $\Delta_{\mathrm{BCS}}(T)=\Delta_{\mathrm{BCS}}(0)\left(1-T / T_{c}\right)^{0.5}$ and $\Delta_{\mathrm{BCS}}(0)$ is taken to be $1.74 k_{B} T_{c}$. In Fig. 6, we have displayed $R_{N}^{A B}$ with T. $R_{N}^{A B}$ (T) has a peak. For the YBCO sample, the maximum of $R_{N}^{A B}$ is found to be 6.92 Ω at 82.5 K. At 72.5 K, the maximum of $R_{N}^{A B}$ is 8.51 Ω in DyBCO. Furthermore, we have also used $Δ_{HTS}$ for the gap to calculate $R_{N}^{A B}$, which is expressed as $\Delta_{\mathrm{HTS}}(T)=\Delta_{\mathrm{HTS}}(0) \tanh \left[\frac{\pi}{\delta_{\mathrm{HTS}}(0)} \sqrt{\left.\alpha\left(\frac{T_{c}}{T}-1\right)\right]}\right.$. The value of α is taken 4/3 and $δ_{HTS}$(0) is taken as ΔHTS(0)/$k_BT_c$. In high-temperature superconductors exhibiting d-wave symmetry, the value of ΔHTS(0) is considered to be2.14kBTc. Fig. 7 displays $R_{N}^{A B}$ with T for both samples. Peaks in $R_{N}^{A B}$ (T) are observed at 82.5 K and 72.5 K for YBCO and DyBCO respectively. Peaks are 25.46 Ω and 31.62 Ω for YBCO and DyBCO respectively.

According to the de Gennes formulation, Eq. (2), the junction resistance $R_{N}^{d G}$ in an SNS junction is obtained. In Eq. (2) d represents the thickness of the normal region and $ξ_N$ denotes the correlation length of the normal region [37], [38]. Below $T_c$, the correlation length is defined by $\xi_{N} \approx a_{0} \exp \left(\frac{1}{x(T)}\right)$ involving the diameter of vortex core, a0 and x(T), in which we have $x(T) \approx\left[2 b^{-1}\left(1-\frac{T}{T_{c}}\right)^{\frac{1}{2}}\right]$ near $T_c$. Here, b depends on $\frac{E_{c}^{c}}{k_{B} T_{c}}$ and $E_c$ represents the vortex energy [39], [40], [41]. For our analysis, we have taken b = 10 to obtain $ξ_N$ to estimate $R_{N}^{d G}$. Fig. 8 displays $R_{N}^{d G}$ as a function of T for both YBCO and DyBCO using $Δ_{BCS}$. RNdG (T) s have maxima of 1.72 mΩ at 77.5 K for YBCO and 1.7 mΩ at 67.75 K for DyBCO. We have also used another energy gap function, ΔHTS to determine $R_{N}^{d G}$, which is shown in Fig. 9. Using $Δ_{HTS}$, we observed that the maximum $R_{N}^{d G}$ for the YBCO is 6.5 mΩ at 77.5 K whereas for the DyBCO, the maximum $R_{N}^{d G}$ is 6.48 mΩ at 67.75 K. $R_{N}^{d G}$ is relatively lower compared to $R_{N}^{A B}$ for both energy gap functions. $R_N$ ranging from 10–50 Ω for grain boundary junctions are also reported [42]. In addition, for various thicknesses of the junction, $R_N$ values in the mΩ to μΩ range have been obtained [43], [44].

In Figs. 6, 7, 8 and 9, we have illustrated the normalized superfluid density (NSD), $ρ_s$(T)/ $ρ_s$(0) with T. For the NSD of the d wave symmetry, we have employed Eq. (3) as follows [45].

In Figs. 6, 7, 8 and 9 we have plotted the corresponding normalized $ρ_s$(T)/ $ρ_s$(0) with T. The NSD increases with the decrease in T whereas $R_{N}^{A B}$ (and $R_{N}^{d G}$) increases with T up to a certain T below which it decreases. Clearly granular NSD dominates $R_{N}^{A B}$ (and $R_{N}^{d G}$) and a metallic nature of $R_N$ (T) becomes visible. This observation is independent of the model we have used. However, we don’t observe any constant critical NSD above which $R_N$ (T) starts decreasing in decreasing T. In Table 1, we have shown NSD corresponding to the peak in $R_N$ (T) for both samples following (i) AB and (ii) dG equations. We have observed that for the BCS gap, the NSD = 0.26–0.31 whereas for the HTS gap function, the NSD is 0.40–0.44. It indicates that $R_N$ is mostly affected by the weak scattering [46], [47].

The tranport critical current density, $J_c$ extracted by using current–voltage (IV) characteristics below the critical temperature (Tc) has been used to extract RN. The grain boundary junction resistance in bulk YBCO and DyBCO has been extracted using two different equations, (i) AB and (ii) dG, with two different energy gaps Δ_BCS and Δ_HTS. $R_{N}^{d G}$is lower than the $R_{N}^{A B}$ for both nature of used gaps. The normalized superfluid density (NSD) is obtained using both energy gaps as a function of T. The NSD increased as the temperature decreased and the presence of increased superfluid density dominates the $R_N$ (T) at low temperature (T). Below a certain NSD for each case (combination of AB (or dG) and BCS (or HTS) gaps, the junctions exhibit a significant decrease in resistance, displaying a metallic behavior. The NSD corresponding to the maximum $R_N$ is found to be in the range of 0.26–0.44. It reveals that the weak scattering limit is responsible for the junction resistence in presence of the finite NSD.

Doyel Rakshit:Data curation, Formal analysis, Investigation, Writing. Sourav Das: Investigation, Data curation. Ajay Kumar Ghosh: Conceptualization, Formal analysis, Investigation, Methodology, Project administration, Resources, Supervision, Writing - original draft, Writing - review and editing.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.