High temperature cuprate superconductors exhibit extremely local nanoscale phenomena and strong sensitivity to doping. While other experiments have looked at nanoscale interfaces between layers of different dopings, we focus on the interplay between naturally inhomogeneous nanoscale regions. Using scanning tunneling microscopy to carefully track the same region of the sample as a function of temperature, we show that regions with weak superconductivity can persist to elevated temperatures if bordered by regions of strong superconductivity. This suggests that it may be possible to increase the maximum possible transition temperature by controlling the distribution of dopants.
We have fabricated quasi-two-dimensional arrays of nano-scale Pb grains coupled by an overlayer of Ag grains. Their temperature dependent resistive transitions follow predictions for an array of mesoscopic superconductor-normal-superconductor junctions. The decrease of their transition temperatures with Ag overlayer thickness systematically deviates from the Cooper limit theory of the proximity effect as the Pb grain size decreases. The deviations occur when the estimated number of Cooper pairs per grain is less than or equal to 1 and suggest the approach to a superconductor to metal transitio…
We show that quantum vortex-loop proliferation may be one possible explanation of the super-long-range proximity effect observed in an insulating underdoped cuprate, using the dual theory of quantum vortices. As a test of this scenario, we propose that a dual proximity effect experiment can confirm the superfluid motion of the quantum vortices in the vortex-proliferated insulator and can measure the divergent correlation lengths near the superconductor-insulator transition.
The magnetization in a superconductor induced due to the inverse proximity effect is investigated in hybrid bilayers containing a superconductor and a ferromagnetic insulator or a strongly spin-polarized ferromagnetic metal. The study is performed within a quasiclassical Green function framework, wherein Usadel equations are solved with boundary conditions appropriate for strongly spin-polarized ferromagnetic materials. A comparison with recent experimental data is presented. The singlet to triplet conversion of the superconducting correlations as a result of the proximity effect with a ferrom…
Resonant proximity effect in the normal metal / insulator / diffusive ferromagnet / insulator / $s$- and d-wave superconductor (N/I/DF/I/S) junctions is studied for various regimes by solving the Usadel equation with the generalized boundary conditions. It is shown that the resonant proximity effect originating from the exchange field in DF layer strongly modifies the tunneling conductance and density of states.
We study the proximity effect in superconductor - metallic or ferromagnetic Aharonov Bohm loop hybrid structures self consistently using the Bogoliubov-deGennes formalism within the two dimensional extended Hubbard model. We calculate the local density of states, the pair amplitude, and persistent currents as a function of several modulation parameters, the position around the loop, the exchange field, the magnetic flux through the loop, the loop size, and the chemical potential in the loop atoms. We find that the parameters above can modulate the proximity effect.
Considerable evidence for proximity-induced triplet superconductivity on the ferromagnetic side of a superconductor-ferromagnet (S-F) interface now exists; however, the corresponding effect on the superconductor side has hardly been addressed. We have performed scanning tunneling spectroscopy measurements on NbN superconducting thin films proximity coupled to the half-metallic ferromagnet La2/3Ca1/3MnO3 (LCMO) as a function of magnetic field. We have found that at zero and low applied magnetic fields the tunneling spectra on NbN typically show an anomalous gap structure with suppressed coheren…
The Tomasch effect (TE) is due to quasiparticle interference (QPI) as induced by a nonuniform superconducting order parameter, which results in oscillations in the density of states (DOS) at energies above the superconducting gap. Quantum confinement in nanoscale superconductors leads to an inhomogenerous distribution of the Cooper-pair condensate, which we found triggers the manifestation of a new TE. We investigate the electronic structure of the nanoscale superconductors by solving the Bogoliubov-de Gennes (BdG) equations self-consistently and obtain the TE determined by two types of proces…
We show that in multiband superconductors even small interband proximity effect can lead to a qualitative change in the interaction potential between superconducting vortices by producing long-range intervortex attraction. This type of vortex interaction results in unusual response to low magnetic fields leading to phase separation into domains of a two-component Meissner states and vortex droplets.
The nature of the phase diagrams of HTSC is clarified by discussing two kinds of phase diagrams, that of the host crystalline lattice, and that of the dopant glass. The latter is associated with changes in the electronic properties, while the former is much more accessible to direct experimental identification, by diffraction, of nanoscale phase separation. Careful examination of electronic properties in both the normal and superconductive states reveals that there are several electronic miscibility gaps in YBa2Cu3Ox and La2-xSrxCuO4 that have been previously overlooked. Recent experiments on …
We study heat transport in hybrid normal metal - superconductor - normal metal (NSN) structures. We find the thermal conductance of a short superconducting wire to be strongly enhanced beyond the BCS value due to inverse proximity effect. The measurements agree with a model based on the quasiclassical theory of superconductivity in the diffusive limit. We determine a crossover temperature below which quasiparticle heat conduction dominates over the electron-phonon relaxation.
Vortex matter in mesoscopic superconductors is known to be strongly affected by the geometry of the sample. Here we show that in nanoscale superconductors with coherence length comparable to the Fermi wavelength the shape resonances of the order parameter results in an additional contribution to the quantum topological confinement - leading to unconventional vortex configurations. Our Bogoliubov-de Gennes calculations in a square geometry reveal a plethora of asymmetric, giant multi-vortex, and vortex-antivortex structures, stable over a wide range of parameters and which are very different fr…
It is shown that the order parameter $Δ$ induced in the normal part of superconductor-normal-superconductor proximity system is modulated in the magnetic field differently from vortices in bulk superconductors. Whereas $Δ$ turns zero at vortex centers, the magnetic structure of these vortices differs from that of Abrikosov's.
A Green functions approach is used to study superconductivity in nanofilms and nanowires. We show that the superconducting condensate results from the multimodal entanglement, or internal Josephson coupling, of the subcondensates associated with the manifold of Fermi surface subparts resulting from size-quantisation. This entanglement is of critical importance in these systems, since without it superconductivity would be extremely weak, if not completely negligible. Further, the multimodal character of the condensate generally results in multigap superconductivity, with great quantitative cons…
We suggest a mechanism which promotes the existence of a phase soliton -- topological defect formed in the relative phase of superconducting gaps of a two-band superconductor with s+- type of pairing. This mechanism exploits the proximity effect with a conventional s-wave superconductor which favors the alignment of the phases of the two-band superconductor which, in the case of s+- pairing, are pi-shifted in the absence of proximity. In the case of a strong proximity such effect can be used to reduce soliton's energy below the energy of a soliton-free state thus making the soliton thermodynam…
We report measurements of the critical temperature of YBCO-Co doped YBCO Superconductor-Normal bilayer films. Depending on the morphology of the S-N interface, the coupling between S and N layers can be turned on to depress the critical temperature of S by tens of degrees, or turned down so the layers appear almost totally decoupled. This novel effect can be explained by the mechanism of quasiparticle transmission into an anisotropic superconductor.
The discovery of noncentrosymmetric superconductors, such as CePt$_3$Si, and chiral superconductors, such as Sr$_2$RuO$_4$, calls for experimental methods to identify the presence of spin-triplet pairing. We here demonstrate a method which accomplishes this in an appealingly simple manner: a spin-sensitive proximity effect in a ferromagnet$\mid$triplet superconductor bilayer. It is shown how the orientation of the field can be used to unambiguously distinguish between different spin-triplet states. Moreover, the proximity effect becomes long-ranged in spite of the presence of an exchange field…
We study the proximity effect in good contacts between normal metals and high Tc (d-wave) superconductors. We present theoretical results for the spatially dependent order parameter and local density of states, including effects of impurity scattering in the two sides, s-wave pairing interaction in the normal metal side (attractive or repulsive), as well as subdominant s-wave paring in the superconductor side. For the [100] orientation, a real combination d+s of the order parameters is always found. The spectral signatures of the proximity effect in the normal metal includes a suppression of t…
We solve the Bogoliubov-de Gennes equations self-consistently for d-wave superconductors with [110] surfaces. We find spontaneous breaking of time-reversal symmetry (TRS) caused by the spontaneous occurrence of a complex order parameter with an extended s-wave symmetry along the [110] surfaces. In the TRS-breaking phase, the d-wave order parameter itself becomes complex. We also show that vortex-antivortex pairs can appear along the [110] surfaces of nanoribbons. These pairs are detectable with surface sensitive probes.
We study the emergent proximity effect in mesoscopic circuits that involve a conventional superconductor and an unconventional pnictide superconductor separated by a diffusive normal or ferromagnetic wire. The focus is placed on revealing signatures of the proposed s+- state of pnictides from the proximity-induced density of states and Josephson current. We find analytically a universal result for the density of states that exhibits both the Thouless gap at low energies, and peculiar features near the superconducting gap edges at higher energies. The latter may be used to discriminate between …
We investigate inverse proximity effects in a spin-triplet superconductor (TSC) interfaced with a ferromagnet (FM), assuming different types of magnetic profiles and chiral or helical pairings. The region of the coexistence of spin-triplet superconductivity and magnetism is significantly influenced by the orientation and spatial extension of the magnetization with respect to the spin configuration of the Cooper pairs, resulting into clearcut anisotropy signatures. A characteristic mark of the inverse proximity effect arises in the induced spin-polarization at the TSC interface. This is unexpec…
We investigate the effects of Andreev bound states due to the unconventional pairing on the inverse proximity effect of ferromagnet/superconductor junctions. Utilizing quasiclassical Eilenberger theory, we obtain the magnetization penetrating into the superconductor. We show that in a wide parameter range the direction of the induced magnetization is determined by two factors: whether Andreev bound states are present at the junction interface and the sign of the spin-mixing angle. In particular, when Andreev bound states appear at the interface, the direction of the induced magnetization is op…
We apply the scattering matrix approach to the triplet proximity effect in superconductor-half metal structures. We find that for junctions that do not mix different orbital modes, the zero bias Andreev conductance vanishes, while the zero bias Josephson current is nonzero. We illustrate this finding on a ballistic half-metal--superconductor (HS) and superconductor -- half-metal -- superconductor (SHS) junction with translation invariance along the interfaces, and on HS and SHS systems where transport through the half-metallic region takes place through a single conducting channel. Our calcula…
We study the perpendicular transport characteristics of small superconductor/ferromagnetic insulator/superconductor (YBa$_2$Cu$_3$O$_{7-x}$/LaMnO$_{3+δ}$/YBa$_2$Cu$_3$O$_{7-x}$) tunnel junctions. At a large bias voltage $V\sim 1$ V we observe a step-like onset of excess current that occurs below the superconducting transition temperature $T<T_c$ and is easily suppressed by a magnetic field. The phenomenon is attributed to a novel type of the superconducting proximity effect of non-equilibrium electrons injected into the conduction band of the ferromagnetic insulator via a Fowler-Nordheim tunne…
We study a two-dimensional superconductor in close proximity to a two-dimensional metallic sheet. The electrons in the superconducting sheet are coupled to those in the metallic sheet by the Coulomb interaction only. We obtain an effective phase-only action for the superconductor by integrating out all the electronic degrees of freedom in the problem. The Coulomb drag of the normal electrons in the metallic sheet is found to make the spectrum of phase-fluctuations in the superconductor, dissipative at long wavelengths. The dissipative co-efficient $η$ is shown to be simply related to the norma…
We study the superconducting proximity effect between a conventional semiconductor and a disordered s-wave superconductor. We calculate the effective momentum relaxation rate in the semiconductor due to processes involving electron tunneling into a disordered superconductor and scattering off impurities. The magnitude of the effective disorder scattering rate is important for understanding the stability of the topological (chiral p-wave) superconducting state that emerges in the semiconductor, since disorder scattering has a detrimental effect and can drive the system into a non-topological st…
Superconductor/ferromagnet proximity effect has been studied for Pb/Co$_2$Cr$_{1-x}$Fe$_x$Al bilayers. Different substrate temperatures allowed us to prepare the Heusler alloy Co$_2$Cr$_{1-x}$Fe$_x$Al films with different degree of spin polarization of conduction band of the Heusler layer. We obtain a strong correlation between the dependence of the superconducting transition temperature on the Pb-layer thickness at fixed thickness of the Heusler layer and the degree of spin polarization in the ferromagnetic layer.
We study peculiarities of proximity effect in clean superconductor -- ferromagnet structures caused by either spatial or momentum dependence of the exchange field. Even a small modulation of the exchange field along the quasiparticle trajectories is shown to provide a long range contribution to the supercurrent due to the specific interference of particle- and hole- like wave functions. The momentum dependence of the exchange field caused by the spin -- orbit interaction results in the long -- range superconducting correlations even in the absence of ferromagnetic domain structure and can expl…
We study the interactions of the vortex lattice with a periodic square array of holes in a superconducting Nb film. Using high resolution magnetic field measurements of electrical losses, extremely narrow states with a magnetic field width reaching 1% of the matching field value are found at the commensurate points. They are accompanied by pronounced harmonic generation in response to ac modulation of the magnetic field or current. We relate these sharp anomalies to a locked commensurate state of the Mott insulator type.
We study clean ferromagnet-superconductor-ferromagnet (FSF) nanostructures in which the magnetization of the F layers can be parallel (P) or antiparallel (AP). We consider the case where the thickness of the S layer is of order of the coherence length, with thinner F layers. We find that reversing the direction of the magnetization in one of the F layers leads in general to drastic changes in the superconductor's state. Under a wide variety of conditions, the AP geometry favors superconductivity. Magnetization reversal in one of the F layers can lead to the superconductivity turning on and off…
We report the effect on the superconducting transition temperature (Tc) of a Nb film proximity coupled to the synthetic antiferromagnet Fe/{Cr/Fe}_(N-1). We find that there is a parity dependence of Tc on the total number of Fe layers, N; locally Tc is always a maximum when N is even, and a minimum when N is odd. The Fe electron mean free path and coherence length are indicative of dirty limit behavior; as such, we numerically model our data using the linearized Usadel equations with good correlation.
We present a general theory of the proximity effect in junctions between diffusive normal metals (DN) and unconventional superconductors in the framework of the quasiclassical Green's function formalism. Various possible symmetry classes in a superconductor are considered which are consistent with the Pauli principle: even-frequency spin-singlet even-parity (ESE) state, even-frequency spin-triplet odd-parity (ETO) state, odd-frequency spin-triplet even-parity (OTE) state and odd-frequency spin-singlet odd-parity (OSO) state. For each of the above four cases symmetry and spectral properties of …
We fabricated three-dimensional disordered Pb-Cu granular structures, with various metal compositions. The typical grain size of both metals is smaller than the superconductor and normal metal coherence lengths, thus satisfying the Cooper limit. The critical temperature of the samples was measured and compared with the critical temperature of bilayers. We show how the proximity effect theories, developed for bilayers, can be modified for random mixtures and we demonstrate that our experimental data fit well the de Gennes weak coupling limit theory in the Cooper limit. Our results indicate that…
Thin film junctions and bilayers of the doped topological insulator $\rm Bi_2Se_3$ and the s-wave superconductor NbN were found to exhibit conductance spectra with prominent zero bias and coherence peaks. Various tunneling models with different pair potentials have failed to fit our data, except for the triplet $p_x+ip_y$ pair potential, which breaks time reversal symmetry, but yielded reasonably good fits. This provides evidence for proximity induced triplet superconductivity in the $\rm Bi_2Se_3$ layer near the interface with the NbN film.
Superconductor/ferromagnet/superconductor heterostructures exhibit a so-called long-range proximity effect provided some layers of conical magnet Holmium are included in the respective interface regions. The Ho layers lead to a spin-flip process at the interface generating equal-spin spin-triplet pairing correlations in the ferromagnet. These equal-spin spin-triplet pairing correlations penetrate much further into the heterostructure compared to the spin-singlet and unequal-spin spin-triplet correlations which occur in the absence of Ho. Here we present calculations of this effect based on the…
We present an extensive theoretical investigation of the proximity effects that occur in Ferromagnet/Superconductor ($F/S$) systems. We use a numerical method to solve self consistently the Bogoliubov-de Gennes equations in the continuum. We obtain the pair amplitude and the local density of states (DOS), and use these results to extract the relevant lengths characterizing the leakage of superconductivity into the magnet and to study spin splitting into the superconductor. These phenomena are investigated as a function of parameters such as temperature, magnet polarization, interfacial scatter…
The effect of proximity of the magnetism of the Pr-based manganite (Pr0.6Sr0.4MnO3) on the superconductivity of Bi-based high-temperature superconductor (Bi1.75Pb0.25Sr2Ca2Cu3O10+d) was studied based on the results obtained from the magnetotransport and magnetization measurements. Decrease in the values of the upper critical field (HC2(0)) and an increase in the width of the superconducting transition (Delta TC) of Bi1.75Pb0.25Sr2Ca2Cu3O10+d were observed in proximity with the manganite. The combined effect of magnetic exchange interaction arising from the manganite, the leakage of Cooper-pair…
We investigate the effect of an inhomogeneous exchange field on the proximity effect in superconductor-ferromagnet hybrid structures within the quasi-classical theory of superconductivity. As an example we study the proximity effect in a superconductor-ferromagnet bilayer with an in-plane spiral magnetic order in the ferromagnet. This model simulates a multiple magnetic domain structure where the domain walls of the Néel type are of equal size as the domains. Triplet correlations are induced in the bilayer by the superconducting proximity effect, which are sensitive to the local quantization a…
The spread of the Cooper pairs into the ferromagnet in proximity coupled superconductor - ferromagnet (SF) structures is shown to cause a strong inverse electromagnetic phenomenon, namely, the long-range transfer of the magnetic field from the ferromagnet to the superconductor. Contrary to the previously investigated inverse proximity effect resulting from the spin polarization of superconducting surface layer, the characteristic length of the above inverse electrodynamic effect is of the order of the London penetration depth, which usually much larger than the superconducting coherence length…
Proximity-induced superconductivity in a 3D topological insulator represents a new avenue for observing zero-energy Majorana fermions inside vortex cores. Relatively small gaps and low transition temperatures of conventional s-wave superconductors put the hard constraints on these experiments. Significantly larger gaps and higher transition temperatures in cuprate superconductors might be an attractive alternative to considerably relax these constraints, but it is not clear whether the proximity effect would be effective in heterostructures involving cuprates and topological insulators. Here, …
In this paper we present scanning tunneling microscopy of Bi$_2$Se$_3$ with superconducting Nb deposited on the surface. We find that the topologically protected surface states of the Bi$_2$Se$_3$ leak into the superconducting over-layer, suggesting a dual topological proximity effect. Coupling between theses states and the Nb states leads to an effective pairing mechanism for the surface states, leading to a modified model for a topological superconductor in these systems. This model is consistent with fits between the experimental data and the theory.
The proximity effect, quasiparticle transport, and local magnetic moment in ferromagnet--d-wave superconductor junctions with {110}-oriented interface are studied by solving self-consistently the Bogoliubov-de Gennes equations within an extended Hubbard model. It is found that the proximity induced order parameter oscillates in the ferromagnetic region. The modulation period is shortened with the increased exchange field while the oscillation amplitude is depressed by the interfacial scattering. With the determined superconducting energy gap, a transfer matrix method is proposed to compute the…
We study proximity effects at ferromagnet superconductor interfaces by self-consistent numerical solution of the Bogoliubov-de Gennes equations for the continuum, without any approximations. Our procedures allow us to study systems with long superconducting coherence lengths. We obtain results for the pair potential, the pair amplitude, and the local density of states. We use these results to extract the relevant proximity lengths. We find that the superconducting correlations in the ferromagnet exhibit a damped oscillatory behavior that is reflected in both the pair amplitude and the local de…
In this communication we consider generalities of the proximity effect in a contact between a conventional $s$-wave superconductor (S) nano-island and a thin film of a topological insulator (TI). A local hybridization coupling mechanism is considered and a corresponding model is corroborated that captures not only the induced unconventional superconductivity in a TI, but also predicts the spreading of topologically protected surface states into the superconducting over-layer. This dual nature of the proximity effect leads specifically to a modified description of topological superconductivity …
We study the influence of the configuration of the majority and minority spin subbands of electron spectra on the properties of atomic-scaled superconductor-ferromagnet S-F-S and F-S-F hybrid structures. At low temperatures, the S/F/S junction is either a 0 or junction depending on the energy shift between S and F materials and the anisotropy of the Fermi surfaces. We found that the spin switch effect in F/S/F system can be reversed if the minority spin electron spectra in F metal is of the hole-like type.
The ferromagnetic resonance of thin sputtered Ni80Fe20 films grown on Nb is measured. By varying the temperature and thickness of the Nb the role of the superconductivity on the whole ferromagnetic layer in these heterostructures is explored. The change in the spin transport properties below the superconducting transition of the Nb is found to manifest itself in the Ni80Fe20 layer by a sharpening in the resonance of the ferromagnet, or a decrease in the effective Gilbert damping co-efficient. This dynamic proximity effect is in contrast to low frequency studies in these systems, where the effe…
Recently, it was experimentally realized that 2D superconducting transition metal dichalcogenides (TMD) such as gated MoS$_2$ and monolayer NbSe$_2$ have in-plane upper critical magnetic fields much higher than the Pauli limit. This is due to the so-called Ising spin-orbit coupling (SOC) of TMD which pins the electron spins along the out-of-plane directions and protects the Cooper pairs from in-plane magnetic fields. However, many TMD materials with extremely large Ising SOC, in the order of a few hundred meV, are not superconducting. In this work, we show that TMD materials can induce strong …
The inverse proximity effect, i.e. the induction of a magnetic moment in the superconductor in superconductor-ferromagnet (S/F) junctions is studied theoretically. We present a microscopic approach which combines a model Hamiltonian with elements of the well established quasiclassical theory. With its help we study systems with arbitrary degree of disorder, interface transparency and thickness of the layers. In the diffusive limit we recover the result of previous works: the direction of the induced magnetization $M$ is opposite to the one of the F layer. However, we show that in the ballistic…
The pinning of quantized flux lines, or vortices, in the mixed state is used to quantify the effect of impurities in iron-based superconductors (IBS). Disorder at two length scales is relevant in these materials. Strong flux pinning resulting from nm-scale heterogeneity of the superconducting properties leads to the very disordered vortex ensembles observed in the IBS, and to the pronounced maximum in the critical current density jc at low magnetic fields. Disorder at the atomic scale, most likely induced by the dopant atoms, leads to "weak collective pinning" and a magnetic field-independent …
We have investigated the differential resistance of hybrid planar Al-(Cu/Fe)-Al submicron bridges at low temperatures and in weak magnetic fields. The structure consists of Cu/Fe-bilayer forming a bridge between two superconducting Al-electrodes. In superconducting state of Al-electrodes, we have observed a double-peak peculiarity in differential resistance of the S-(N/F)-S structures at a bias voltage corresponding to the minigap. We claim that this effect (the doubling of the minigap) is due to an electron spin polarization in the normal metal which is induced by the ferromagnet. We have dem…
We consider superconductivity nucleation in multiply connected mesoscopic samples such as thin-walled cylinders or rings placed in electrical contact with a ferromagnet. The superconducting critical temperature and order parameter structure are studied on the basis of linearized Usadel equations. We suggest a mechanism of switching between the superconducting states with different vorticities caused by the exchange field and associated with the oscillatory behavior of the Cooper pair wave function in a ferromagnet.
Topologically nontrivial superconducting phases have been engineered in topological materials by the proximity effect in contact with conventional superconductors. In this paper, by using the method of the Kronig-Penney model, we study the superconducting proximity effect in the bulk electronic states of Weyl semimetals by considering a multilayer structure consisting of Weyl-semimetal and superconductor layers. Due to the proximity effect, two Weyl nodes are decoupled into four nodes of Majorana fermions resulting in Weyl-superconductor phases or three-dimensional extension of topological-sup…
We used the proximity effect as a tool to achieve an ideal (barrier - free) NS boundary for quantitative evaluation of transport phenomena that accompany converting dissipative current into supercurrent in NS systems with unconventional superconductors, single - crystal chalcogenide FeSe and granulated pnictide LaO(F)FeAs. Using features (limitations) of Andreev reflection in the NS systems with dispersion of the electron spin subbands, we revealed direct evidence for spin - polarized nature of transport and the absence of residual magnetization in iron - based superconductors in the normal st…
We study theoretically the influence of ferromagnetic metals on a superconducting film in the clean limit. Using a self-consistent solution of the Bogoliubov--de Gennes equation for a ferromagnet-superconductor-ferromagnet double junction we calculate the pair potential and conductance spectra as a function of the superconducting layer thickness $d$ for different strengths of ferromagnets and interface transparencies. We find that the pair potential and the critical temperature are weakly perturbed by the exchange interaction and do not drop to zero for any finite $d$. On the other hand, for t…
Iron-based superconductors have great potential for high-power applications due to their prominent high-field properties. One of the central issues in enhancing the critical current density of iron-based superconducting wires is to reveal the roles and limitations of grain boundaries in supercurrent transport. Here, we finely tuned the electronic properties of grain boundaries by doping Ba1-xKxFe2As2 superconductors in a wide range (0.25<x<0.598). It is found that the intra-grain Jcintra peaks near x~0.287, while the inter-grain Jcinter has a maximum at about x~0.458. Remarkably, the grain bou…
