Structural, electrical and mechanical properties of the (NdFeO₃)_x/(CuTl)-1223 superconductor phase

Abstract

The impact of adding the orthoferrite NdFeO₃, at the nanoscale, on the (Cu_0.5Tl_0.5)Ba₂Ca₂Cu₃O_10–δ phase formation, microstructure, electrical and mechanical performance was studied. The nano-(NdFeO₃)_x/(CuTl)-1223 composites, with x = 0.00, 0.25, 0.50, 0.75, 1.00, and 2.00 wt.%, were prepared using the solid-state reaction technique. The X-ray diffraction (XRD) confirmed that supplementing the host (CuTl)-1223 phase with nano-NdFeO₃ did not alter the unit cell parameters (a and c) and preserved the tetragonal structure. SEM micrographs suggested that the inclusion of nano-NdFeO₃ reduced the number of voids and boosted the inter-grain connections. The energy-dispersive X-ray (EDX) spectra revealed the elemental compositions of the various superconductor composites. The superconducting transition temperature (T_c) and the critical current density (J_c) increased with the inclusion of nano-NdFeO₃ up to x = 0.50 wt.%. X-ray photoelectron spectroscopy (XPS) spectra analysis revealed the elemental composition and oxidation state of all elements. The Vickers microhardness (H_v) measurements were analyzed and compared with five different theoretical models. According to H_v measurements, the modified proportional sample resistance (MPSR) model provided the best theoretical analysis of H_v within the plateau limit regions. Moreover, various mechanical parameters were estimated as a function of the inclusion of nano-NdFeO₃. The indentation creep test showed that the composite samples with high nanoparticle concentration (x > 0.50 wt.%) have dislocation creep mechanism, while those with x ≤ 0.50 wt.% followed a dislocation climbs creep mechanism. A comparison between NdFeO₃ nanoparticles and other magnetic ferrites and nano-magnetic oxides was reported.

1 Introduction

High-temperature superconductors (HTSCs), including (Cu_1–xTl_x)Ba₂Ca_n–1Cu_nO_2n+4–δ (n = 2, 3, 4,…), are still being investigated in depth to better understand the mechanism of high-T_c superconductivity and improve critical superconducting parameters such as the superconducting transition temperature (T_c) and critical current density (J_c). (Cu_0.5Tl_0.5)Ba₂Ca₂Cu₃O_10–δ [(CuTl)-1223] represents the most appealing phase of the (Cu_1–xTl_x)Ba₂Ca_n–1Cu_nO_2n+4–y cuprates superconductors due to its distinctive features and the simplicity and affordability of its synthesis in atmospheric air [1,2,3]. Nevertheless, the performance of the bulk superconductors is limited by several energy dissipation mechanisms, one of which is the magnetic flux dynamics that adversely affect the critical superconductivity parameters. Although superconductors contain natural flux pinning centers such as inter-grain voids, micro-cracks, oxygen vacancies, and impurity phases, their ability to pin the flux motion is still limited. Therefore, minimizing structural defects, filling the pores with suitable materials, and enhancing intergranular connectivity constitute practical approaches towards improving the flux pinning ability of (CuTl)-1223 phase [4]. In addition to the superior qualities already mentioned for (CuTl)-1223 phase, this phase should be optimized to attain the necessary mechanical strength for the versatile and malleable applications in cables, coils, and tapes [4].

In this regard, the incorporation of nanoparticles as artificial pinning centers in the superconductor matrix constituted a feasible approach to efficiently enhance flux pinning and increase T_c and J_c. Not only are the superconducting properties of the (CuTl)-1223 phase improved by the addition of nanoparticles, but also its mechanical properties, which is essential to evaluate its utility in real-world applications [5]. When the size of the pinning center is on the order of the coherence length, the pinning effect is more dominant [6]. The magnetic properties of nanoparticles serve as the second prerequisite for the actual centers of pinning in bulk HTSCs [7,8,9]. Therefore, research challenges in this field revolve around the types, dimensions, densities, and distribution of the nanostructures [10,11,12,13,14,15,16]. Adding various oxide nanomaterials boosted the superconducting parameters of various HTSCs [17,18,19,20,21]. In particular, the (CuTl)-1223 superconductor's microstructure density and porosity were both improved by the addition of SnO₂ nanoparticles [22]. Khan et al. [23, 24] reported that T_c and J_c were improved upon adding CuO, CaO₂, and BaO nanoparticles to the (CuTl)-1223 superconductor. In (CuTl)-1223 superconducting bulk, the effect of nano-BaSnO₃ addition showed that the values of H_v increased until x = 0.25 wt.%, then decreased with further addition until x = 1.00 wt.% [25]. The (CuTl)-1223 main phase volume fraction increased to 85.6% at x = 0.6 wt.% due to the addition of In₂O₃ nanomaterials, then decreased to 85.3% at x = 1.0 wt.% [26]. (CuTl)-1223 superconducting properties were reported to be suppressed with the addition of Al₂O₃ nanoparticles due to the charge carrier scattering across the new nanoparticles [23, 27]. Imran et al. [28] demonstrated an enhancement in flux dynamics upon adding Co₃O₄ nanoparticles to (CuTl)-1223 superconductor.

The perovskite-structured rare-earth orthoferrites (RFeO₃) stand out among others as good candidates for a wide range of applications because of their unique structural, magnetic, electrical, and optical characteristics [29,30,31,32,33]. In specific, NdFeO₃ is one of these orthoferrites that found applications in electrochemical sensing because of its remarkable properties such as thermal and chemical stability, high catalytic activity, and oxygen content variations. The unique magnetic properties of NdFeO₃ are attributed to the strong interaction between Nd and Fe ions [33, 34]. In addition, NdFeO₃ is nontoxic, cost effective, environmentally friendly, and shows multiferroic behavior at room temperature [35].

To the best of the authors’ knowledge, the current study presents the first analysis of the factors affecting the structural, compositional, physical, electrical and mechanical properties of (CuTl)-1223 superconductor supplemented with NdFeO₃ nanoparticles. For this purpose, nano-(NdFeO₃)_x/(Cu_0.5Tl_0.5)Ba₂Ca₂Cu₃O_10–δ composites with $0.00 \le x \le 2.00 \mathrm {wt}.\%$, were synthesized through the solid-state method. The potential of nano-NdFeO₃ to enhance the physical characteristics of the (CuTl)-1223 superconductor was examined by investigating the inter-grain connectivity and mechanical characteristics of the host phase. With the intervention of nano-NdFeO₃, the pores between superconductor grains could be filled. The microstructure and phase composition of the samples were evaluated by X-ray diffraction (XRD), scanning electron microscopy (SEM), and the Energy Dispersive X-Ray (EDX). Resistivity (ρ–T) and I–V characteristics were used to investigate the electrical properties for all samples. X-ray photoelectron spectroscopy (XPS) spectra examined elemental compositions and oxidation states for all elements. Vickers microhardness measurements studied the mechanical behavior of the prepared composites.

2 Experimental techniques

2.1 Sample preparation

The co-precipitation method was used to synthesize the NdFeO₃ nanoparticles. In brief, the required quantities of NdCl₃.6H₂O and FeCl₃.6H₂O with 1:1 molar ratio were dissolved in 100 mL of deionized water, and the reaction mixture was magnetically stirred for 1 h at room temperature. The solution was titrated with 4 M NaOH to obtain pH = 12 causing the formation and precipitation of the nanoparticles. The reaction mixture was then heated for 4 h at 353 K. Suction filtration was used to collect the nanoparticles, which were then washed with deionized water until the mother liquor had a pH of 7. The samples were left overnight to dry at 373 K. The dried powders were then ground and annealed at 1223 K for 2 h.

Nanoparticle-superconductor composites (NdFeO₃)_x/(Cu_0.5Tl_0.5)Ba₂Ca₂Cu₃O_10–δ, denoted by nano-(NdFeO₃)_x/(CuTl)-1223, were created via a single-step solid-state reaction technique. Initially, Tl₂O₃, CuO, BaO₂, and CaO powders (Sigma Aldrich, > 99.9% purity) were mixed, ground, and then sifted twice in a 60-µm sieve to ensure homogeneity. The required appropriate amount of nano-NdFeO₃ (x = 0.00, 0.25, 0.50, 1.00, and 2.00 wt.%) having a grain size of 88.96 nm was then mixed with the precursor material and ground for 2 h to get a homogeneous mixture. This uniformly ground material was then pelletized for 1 min at 10 tons/cm² pressure before being encased in silver capsules for the sintering process. To prepare the requisite (NdFeO₃)_x/(CuTl)-1223 nanoparticle-superconductor composites, the pellets were sintered at 1118 K for 4 h with a heating rate of 2 K/min and then cooled to room temperature at a cooling rate of 3.5 K/min.

2.2 Characterization and measurements

The structural characterization of nano-(NdFeO₃)_x/(CuTl)-1223 composites was accomplished using XRD measurements on a Bruker D8 focus diffractometer with Cu K_α radiation (λ = 1.5406 Å) in the ${15}^{o}\le 2\theta \le {70}^{o}$ range. The morphology and elemental composition were determined using a JEOL scanning electron microscope (JSM-IT200). The apparatus was run at a high voltage of 20 kV, with a rotary pump used to achieve low pressure. The electron source was 15 mm away from the surface of the sample. The magnification at which the photographs were taken is 5000X. The oxygen content of the samples was determined by iodometric titration using potassium iodide (KI) in a two-step redox titration. Utilizing potassium thiocyanate (KCSN), the titration's endpoint was detected. KRATOS X SAM 800 spectrometer was used for X-ray photoelectron spectroscopy (XPS) analysis, which yielded considerable quantitative and chemical state information from the material's surface. The Dc-electrical resistivity of the prepared samples was measured using the conventional four-probe approach. The temperature was recorded from room temperature (300 K) down to 77 K at which T₀ and T_c were obtained. A closed cryogenic setup using liquid nitrogen as the working medium was used to take the measurement. Current–voltage (I-V) curves allowed the determination of the transport critical current density (J_c) at temperature (77 K) using the same traditional four-probe procedure and the same cooling system used in the resistivity measurement. A homemade current source (I_max = 10 A) was used to provide current and voltage across the sample and was measured by the LEYBOLD DIDACTIC 532 13 microvoltmeter. At room temperature, a digital microhardness tester (MHVD-1000IS) was used to assess the Vickers microindentation of the examined composites in atmospheric air. The applied force varied from 0.49 to 9.80 N for a loading duration of 10–90 s. Vickers microhardness (H_v) was computed by determining the average of 5 measurements at different positions on the specimen surface.

3 Results and discussion

3.1 NdFeO₃ characterization

Figure 1a–d displays the room temperature XRD pattern, Rietveld refinement on the XRD patterns using the MAUD program [36], TEM micrograph images with the corresponding grain size histograms, and the room temperature M–H loop for orthoferrite NdFeO₃ nanoparticles, respectively. The lattice parameters of the orthorhombic phase with space group Pnma [37] were determined using MAUD software (Table 1), which provide a good index for the observed peaks with theoretical data and their refinement parameters. These were used as the numerical criteria for fitting, such as the profile residual reliability factor (R_p), weighted profile residual factor (R_wp), expected weighted profile factor (R_exp), and goodness of fitting parameters (χ²) the values of which are given in Table 1. Moreover, the average grain size (D_TEM = 88.96 nm), estimated from the histogram in Fig. 1c, was consistent with the average crystallite size (D_XRD = 87.16 nm) estimated from MAUD program. The orthoferrite NdFeO₃ weak ferromagnetic nature was demonstrated by VSM analysis as shown in Fig. 1d, with M_s value comparable to earlier studies [38, 39].

Fig. 1

NdFeO₃ nanoparticles a XRD pattern, b Rietveld refinement c particle size histograms with inset TEM micrograph, and d the M-H loop

Table 1 Phase content, reliability factors, and structural parameters for the NdFeO₃ orthoferrite

3.2 X-ray power diffraction (XRD)

Figure 2a shows the various planes in XRD patterns of the nano-(NdFeO₃)_x/(CuTl)-1223 composites that were indexed to the (CuTl)-1223 phase's tetragonal structure, with P4/mmm space group [40]. Limited and indexed low-intensity peaks in the diffractograms were attributed to the presence of secondary impurity phases such as (CuTl)-1212 phase and BaCuO₂ usually formed in the single-step solid-state reaction [41, 42]. Moreover, the patterns did not show any peaks for neodymium or its oxide or iron-based compounds suggesting that the host (CuTl)-1223 superconducting phase is dominant. The addition of nano-NdFeO₃ caused no drastic change in the position of the main characteristic (110) peak as depicted in Fig. 2b. As a result, the NdFeO₃ nanoparticles had no effect on the (CuTl)-1223 phase's structure and occupied the grain boundaries rather than the inside of the unit cell of (CuTl)-1223 matrix. This was confirmed by similarity of X-ray pattern and executing the Rietveld refinement for all composite samples, using the MAUD program [36], as shown in the graphs of Fig. 3a-3c for samples with x = 0.00, 0.50, and 2.00 wt.%, respectively. The results of the refinement analysis showed a satisfactory good match in peak fitting between the measured and calculated patterns.

Fig. 2

a XRD patterns of nano-(NdFeO₃)_x/(CuTl)-1223 composites samples, with $0.00\le x\le 2.00 \mathrm{wt}.\%$, and (b) The variation of the 2θ position of the main peak (110) for nano-(NdFeO₃)_x/(CuTl)-1223 composites

Fig. 3

Rietveld Refinements of the XRD spectra of nano-(NdFeO₃)_x/(CuTl)-1223 composites a 0.00 wt.%, b 0.50 wt.% and c 2.00 wt.%

The estimated values of the relative volume fractions for (CuTl)-1223, (CuTl)-1212, and BaCuO₂ were determined based on the sum of the measured peak intensities using the relative intensity method [43] and are presented in Table 2. Compared to x = 0.00 wt.%, the volume percentage of the (CuTl)-1223 phase increased from 87.28% to 90.20% as the amount of nano-NdFeO₃ concentration increased up to x = 0.50 wt.%. At x > 0.50 wt.%, a gradual decrease in the volume percentage of the (CuTl)-1223 phase was observed, accompanied with a boost in the (CuTl)-1212 and BaCuO₂ phase percentages. Similar findings were observed in many different nanoparticles additions to superconducting materials such as (Al₂O₃ and MgO), (CdO and Cd_0.95Mn_0.05O) and (In₂O₃ and Y₃Fe₅O₁₂) when added into (Bi,Pb)-2223, (Bi,Pb)-2212, and (CuTl)-1223 phases, respectively [26, 43,44,45,46,47]. The decline in volume percentage for the higher additions of nanoparticles was justified by the solid nanoparticles' low capabilities to prevent the formation of spatial phases. This decline could also be due to the ferromagnetic nature of orthoferrite nano-NdFeO₃, which encourages Cooper pair splitting and slow down the formation of the (CuTl)-1223 phase at high significant amounts of nanoparticles [43].

Table 2 Fluctuation of crystal structure parameters and average grains size of nano-(NdFeO₃)_x/(CuTl)-1223 composites, with $0.00\le x\le 2.00 \mathrm{wt}.\%$

To investigate the fluctuation in lattice parameters (a and c) caused by the addition of various nano-NdFeO₃ concentrations, Rietveld refinement was used as shown in the plots of Fig. 3a–c for samples with x = 0.00, 0.50 and 2.00 wt.%, respectively, and their values are presented in Table 2. The computed lattice parameters a and c for the pure sample were 3.8517 Å and 15.845 Å, respectively, with volume unit cell of V = 235.066 ų, which are comparable to the values reported in literature [48]. After nano-NdFeO₃ addition, there were no discernible changes in the lattice parameters and the c/a ratio was unaltered. These findings confirmed that the nano-NdFeO₃ resided between the grains boundaries rather than entering the crystal structure, as further supported by the SEM micrographs discussed below. Such observations corroborated many studies analyzing the effect of adding nanoparticles to superconducting materials [26, 43, 49].

The following equation [22] was used to determine the composite samples' porosity percentage (P%):

$$P\% = \left( {1 - \frac{{\rho_{{{\text{exp}}}} }}{{\rho_{{{\text{theo}}}} }}} \right) \times 100,$$

(1)

where ρ_theo and ρ_exp represent the theoretical and experimental densities for the prepared samples, respectively. The fluctuation of P% with x is shown in Fig. 4, where the samples had P% values ranging from 23.14 to 39.55%, which is reasonable for ceramic superconductor materials [50]. Additionally, as the nano-NdFeO₃ concentration increased up to x = 0.50 wt.%, the porosity decreased by 4.90%, and the experimental density increased linearly. This indicated that the addition of a small amount of nano-NdFeO₃ improved the microstructure of these samples due to its ability to penetrate within the pores and voids of the composite samples [50, 51]. For x > 0.50 wt.%, on the other hand, a notable increase in porosity was observed, leading to an increase in the formation of voids and pores that reduce the links between the grains of the samples.

Fig. 4

Fluctuation of porosity with x for nano-(NdFeO₃)_x/(CuTl)-1223 composites

3.3 Size, surface morphology and elemental composition

Figures 5a-f depict the SEM micrograph images with the correlating grain size histograms of nano-(NdFeO₃)_x/(CuTl)-1223 composites with $0.00\le x\le 2.00$ wt.%, respectively. The presence of the rectangular-shaped plates in all samples verified the formation of (CuTl)-1223 phase [52]. Besides, the appearance of some grains with irregular shapes was designated for (CuTl)-1212 and BaCuO₂ present as impurity phases with different percentages in all micrographs of composite samples. Additionally, the small spherical grains might be nano-NdFeO₃ in the (CuTl)-1223 phase even after sintering at high temperatures [25]. The size of the grains was estimated by ImageJ program (Table 2), and the size distribution is presented in the histograms of Fig. 5. It was noted that the average grain size increased from 4.07 to 4.66 as the concentration of nano-NdFeO₃ increased up to x = 0.50 wt.%, and then decreased upon further addition up to x = 2.00 wt.%. As a result, the addition of a small amount of nano-NdFeO₃ improved the grain’s growth and enhanced the inter-grain connectivity due to the filling of voids and cracks, which reduced the porosity of these composite samples [47]. For the composite samples with x > 0.50 wt.%, the voids between particles increased significantly, the grain size started to degrade, and the rectangular-shaped grains became irregular at higher NdFeO₃ concentrations (x = 2.00 wt.%). This is due to the fact that at high nanoparticle additions, the nano-NdFeO₃ acted as hindrances to grain growth with weak links between them [25].

Fig. 5

SEM micrographs with particle size histograms of nano-(NdFeO₃)_x/(CuTl)-1223 composites, $0.00\le x\le 2.00 wt.\%$

Figures 6a–c depicts EDX spectra for nano-(NdFeO₃)_x/(CuTl)-1223 composites, with x = 0.00, 0.50, and 2.00 wt.%, respectively. The EDX analysis revealed the presence of Nd, Fe, and O in the composite samples. The absence of any characteristic peaks for Nd and Fe in the XRD patterns (Fig. 2a) confirmed that the nanoparticles did not get into the (CuTl)-1223 phase structure, but occupied gaps between the grains. The nearly constant unit cell parameters further supported this assertion. Mohammed et al. [22] noted similar observations for the (CuTl)-1223 phase supplemented by nano-SnO₂. Moreover, the addition of nano-NdFeO₃ to the (CuTl)-1223 superconducting phase is confirmed by the rise in the weight percentage of Fe and O originating from the NdFeO₃ nanoparticles as x increases. The mass and atomic percentages of Tl, Ba, Ca, Cu, Nd, Fe and O determined from EDX analysis are listed in Table 3.

Fig. 6

Typical EDX spectra of (NdFeO₃)_x/(CuTl)-1223 superconducting phase a x = 0.00 wt.%, b x = 0.50 wt.%, and c x = 2.00 wt.%

Table 3 The mass and atomic percentage of the elemental composition analysis by EDX of nano-(NdFeO₃)_x/(CuTl)-1223 composites, with x = 0.00, 0.50, and 2.00 wt.%

3.4 Electrical resistivity measurements

Figure 7 depicts the electrical resistivity dependence on temperature of nano-(NdFeO₃)_x/(CuTl)-1223 composites with 0.00 ≤ x ≤ 2.00 wt.%. The curves exhibit linear temperature dependence from room temperature down to the onset temperature (${T}_{\mathrm{c}}^{\mathrm{onset}})$. As the superconducting transition temperature (T_c) is reached, a fast decrease in the resistivity curves was recorded. A small curvature was observed at T > T_c, indicating fluctuations of the Cooper pairs responsible for the formation of superconductivity [53]. The data of electrical resistivity of the prepared samples in the normal state are well fitted according to Matthiessen’s rule, $\rho =\alpha T+{\rho }_{0}$ [54]. The values of the fitting parameters for the residual resistivity (ρ₀), the resistivity temperature coefficient (α), and the room-temperature resistivity (ρ_300K) are presented in Table 4. These values declined as x increased up to 0.50 wt.%, after which they increased with higher nano-NdFeO₃ additives up to x = 2.00 wt.%. This confirmed the results obtained from XRD and SEM. The ρ₀ and ρ_300K values highlighted the sample's purity and the number of scattering centers that form as a result of oxygen vacancies and the cation disorder [55]. Moreover, α showed small fluctuation with x, suggesting that the nano-NdFeO₃ addition had no extensive effect on charge carrier concentrations because they did not enter the crystal structure of the (CuTl)-1223 phase [56]. Similar results were obtained by Abdeen et al. [57] and Jabbar et al. [58] who studied the effect of silver (Ag) and gold (Au) nanoparticles, respectively, on (CuTl)-1223 phase.

Fig. 7

Resistivity versus temperature plots of (NdFeO₃)_x/(CuTl)-1223 superconducting phase

Table 4 Different superconducting parameters for nano-(NdFeO₃)_x/(CuTl)-1223 composites with x = 0.00, 0.25, 0.50, 0.75, 1.00, and 2.00 wt.%

The superconducting transition temperature (T_c) for the nano-(NdFeO₃)_x/(CuTl)-1223 composites with 0.00 ≤ x ≤ 2.00 wt.% was determined using the maximum point of the Gaussian fitting of (dρ/dT) curves [59, 60], as presented in the inset of Fig. 7 for x = 0.50 wt.%. When x varied between 0.00 and 0.50 wt.%, T_c increased from 114.72 K to 124.37 K, and then declined to 96.15 K for x = 2.00 wt.%. T₀ followed the same trend as T_c and its values boosted from 104.80 to 113.80 K when x increased from 0.00 to 0.50 wt.% and then it decreased to 80.00 K for x = 2.00 wt.%. Several factors contribute to the variation of T_c with the nano-NdFeO₃ addition. Firstly, T_c is influenced by the enhancement or decrement in the percentage of the volume fraction of the (CuTl)-1223 superconducting phase, which in turn depends on many factors such as the density of voids and pores, changes in grain size, intergrain coupling, microdefects, dislocations, and the optimization of oxygen content which can occupy the oxygen vacancies in the bulk (CuTl)-1223 phase [58, 61]. Additionally, inter-granular properties that are strongly correlated to grain structure rely on intrinsic characteristics linked to the Ginzburg–Landau order parameter ($\psi )$ or density of Cooper pairs (n_s) in the bulk properties of HTSCs [62]. Describing the intricate and dynamic interactions between superconducting grains as well as grain-boundary systems, inter-plane coupling, optimum carriers' density (n_s), and homogeneous distribution of carriers in CuO₂ planes is therefore crucial to understand the variation in T_c values for HTSCs. The increase in volume fraction with increasing the addition of nano-NdFeO₃ up to x = 0.50 wt.% is consistent with the findings obtained from the SEM analysis which supported enhancement in intergrain coupling, an increase in the average grain size, as well as rise in the microstructure density thus reducing porosity among the grains. These factors, which directly contribute to increase in the volume fraction, led to an increase in T_c as a result of increasing the rate of Cooper pair formation (Cooper pairs density (n_s)) upon the transition from normal to superconducting states. This was also associated with less variation in the Ginzburg–Landau order parameter, which reached its bulk value of ${|\psi |}^{2} = {|{\psi }_{\infty }|}^{2}= {n}^{*}$ (The density of Cooper pairs within the superconductor material, far from any gradients in (ψ), is the point at which the superconducting state is stable and has a minimum Gibbs free energy (G_s(0, T, n_s)) [63]. Furthermore, the Josephson coupling between the superconducting planes becomes significant. It provides phase coherence to superconducting condensates and leads to superconductivity of the entire sample if this phase coherence is long-range, allowing carriers to transport across inter-crystallite sites [64]. On the other hand, the decrease in T_c may be caused by the reduced volume fraction as revealed by the XRD data due to agglomeration and segregation of these nanoparticles at the grain boundaries in the bulk CuTl-1223 matrix, mobile hole trapping, or other reasons related to oxygen vacancy disorder mechanisms [24]. In general, the uniform distribution of nano-NdFeO₃ can strengthen weak links between superconducting grains by filling voids and pores, and assist charge carriers to cross grain boundaries. However, this effect is prominent only at an optimal level of nanoparticle inclusion (x = 0.50 wt.%), after which a noticeable drop in the volume fraction of superconducting grains occurred, leading to suppression of the superconductivity parameters. Similar outcomes were reported in the literature when adding various nanoparticles to the (CuTl)-1223 phase, such as Au [58], BaSnO₃ [25], SnO₂ [26], and Cu [65]. In samples with long-range superconducting order, T₀ indicates the onset of superconductivity (where all grains become superconducting, i.e., intragrain as well as grain boundaries become superconducting) [66]. The variation of T₀ with nano-NdFeO₃ addition can also be explained on the basis of the variation of the volume fraction, as well as the carriers’ density in the CuO₂ planes [67]. The carriers supplied by the charge reservoir layer to the CuO₂ planes depend upon the oxygen contents in the charge reservoir layer [68]. T₀ rises with the addition of nano-NdFeO₃ because the carriers' density in the CuO₂ planes has been optimized. The oxygen content in the charge reservoir layer evidently depends on the amount of NdFeO₃ nanoparticles present in the (CuTl)-1223 matrix as will further be discussed below. Additional factors that can significantly contribute to T_c variation include the effect of hole concentrations, oxygen content, and oxygen-deficient parameters as will be detailed below.

As (CuTl)-1223 is a holes-doped superconductor, it was crucial to determine the hole concentration (p), which affects the superconducting parameters of (CuTl)-1223 superconducting phase [41]. The hole concentration in nano-(NdFeO₃)_x/(CuTl)-1223 composites has a parabolic relationship with T_c, as given by Eq. 2 [69]:

$$T_{{\text{c }}} /T{ }_{{\text{c}}}^{{{\text{max}}}} = 1 - 82.6(p - 0.16)^{2} ,$$

(2)

where ${T}_{\mathrm{c}}^{\mathrm{max}}$ is the highest possible value of T_c for (CuTl)-1223 superconductor (${T}_{\mathrm{c}}^{\mathrm{max}}$ = 132 K) [4]. The computed values of hole concentration are also presented in Table 4. The plots of T_c and p against x, which are displayed in Fig. 8, exhibited opposite trends with the lowest hole concentration in the Cu–O planes associated with the optimum T_c achieved at x = 0.5 wt.% [41]. HTSCs in over-doped regions may microscopically separate into superconducting and normal-state regions, which is thought to be an intrinsic bulk property [70]. When the superconducting volume fraction in the over-doped region decreases or increases, it strongly suggests the occurrence of separation into superconducting and normal-state regions [71]. The decrease in superconducting condensation energy in the over-doped region may be the cause of the phase separation. With increasing the hole concentration towards the Fermi-liquid state, the superconducting condensation energy tends to decrease [72]. A decrease in n_s in the superconducting region is one of many causes for the decline in T_c values with increasing hole concentration. The screening of the Coulomb interaction between holes in the Fermi-liquid state is thought to be stronger than that in the normal state with lower hole concentration per Cu. Accordingly, it is possible that holes tend to gather into the normal Fermi-liquid state region out of the superconducting region, causing a reduction in n_s in the superconducting region. Such phenomenon occurs if the loss of the Coulomb energy in the normal-state region exceeds the superconducting condensation energy in the superconducting region [73]. Therefore, decreasing the hole concentration, despite the increase in oxygen content, resulted in an enhancement in T_c up to x = 0.50 wt.% due to the increase in the superconducting volume fraction and n_s.

Fig. 8

The variation of T_c (K), hole concentration (p), and δ versus x in nano-(NdFeO₃)_x/(CuTl)-1223 composites

Another factor that can affect T_c values is the proximity effect. When the microscopic phase separation into superconducting and normal-state regions occurs [74, 75], normal quasi-particles in normal-state regions can cross the boundary into superconducting regions because of the proximity effect. T_c falls because of the suppression of the development of the superconducting order parameter in each superconducting region as its size decreases.

The oxygen content (y = 10–δ) in the nano-(NdFeO₃)_x/(CuTl)-1223 composites was obtained using iodometric titration [76], and their values are summarized in Table 4. The oxygen content increased gradually as x increased, indicating that nano-NdFeO₃ supplied (CuTl)-1223 with more oxygen. The deficiency of oxygen in the processed samples may result in the production of residual secondary phases. The importance of oxygen in the construction and behavior of cuprates HTSCs is widely known. This is clear from the fact that all HTSCs have square-planar CuO₂ planes, from which superconductivity arises [77]. Therefore, the addition of excess nanoparticles may have prevented the growth of the (CuTl)-1223 matrix, thus resulting in the deterioration of the superconducting characteristics of the samples. Furthermore, changes in the oxygen content upon nano-NdFeO₃ addition and the inhomogeneous distribution and/or agglomeration of the nanoparticles at the grain boundaries may also be responsible for this non-monotonic change in the superconducting parameters [78]. Moreover, the appearance of macroscopic inhomogeneity and impurities phases may be explained by the high values of the transition width, ΔT = T_c—T₀, particularly for x > 0.50 wt.% as presented in Table 4.

Many factors affect the values of T_c in cuprates superconductors, one of which is the oxygen deficiency (δ = 10–y). In the current study, δ exhibited an abnormal behavior with T_c. δ decreased with increasing the concentration of nano-NdFeO₃ up to x = 2.00 wt.%. This may be due to the enhancement in the oxygen content (oxygen occupancy) upon the addition of NdFeO₃ nanoparticles acting as the source of oxygen, as revealed from the EDX data and will be discussed in the XPS analysis. While lower δ values is normally associated with higher increment of T_c [79], the sample with x = 2.00 wt.% in the current study had the lowest δ and T_c values as depicted in Fig. 8. This behavior could be explained by a variety of factors, such as the possibility that the length of the Cu apical oxygen (Cu–O_apical) bond does not directly affect T_c due to the absence of discernible changes in the c-lattice parameter. Therefore, its relevance is in relation to the concentration of holes that impact T_c [80], which increased for x > 0.50 wt.% and resulted in a decrease in T_c as discussed above. It is worth mentioning that T_c is difficult to control and comprehend due to the volatile nature of thallium and its propensity to change oxidation states (Tl¹⁺ or Tl³⁺) as will be discussed in XPS analysis, as well as different elemental defects.

Measurements of the transport critical current density (J_c) are used to assess the samples' capacity to carry current. Figure 9 shows the fluctuation in the E-J curves as a function of the nano-NdFeO₃ addition at 77 K following Eq. 3 [81]. These curves were used to calculate J_c values for all samples at a criterion of 1 μVcm⁻¹ [82]. J_c values were plotted against x as presented in the inset of Fig. 9. Obviously, J_c exhibited the same trend as T_c upon NdFeO₃ addition. J_c improved from 171.8 A/cm² for the pure sample to 262.8 A/cm² for x = 0.50 wt.%, then declined upon further addition of nanoparticles as presented in Table 5. This enhancement could be attributed to the improvement in phase formations and the enhancement in inter-grain connectivity as confirmed from XRD and SEM results, respectively. This is due to the diffusion of nano-NdFeO₃ over the twin boundaries, pore surfaces, and grain boundaries. In contrast, J_c values dropped as the concentration of nano-NdFeO₃ increased (x > 0.50 wt.%). This is caused by a variety of factors, including an increase in secondary phases, the presence of voids between the grains, disorder in grain alignment, a decrease in inter-grain connectivity caused by an excessive addition of nanoparticles, and a significant agglomeration of nanoparticles between the grains, which increased grain boundary resistance [83]. Similar findings were observed in previous studies [23, 24, 57].

$$E = \beta J^{n} ,$$

(3)

where β and n are experimental constants.

Fig. 9

Variation of the electric field as a function of the critical current density for (NdFeO₃)_x/(CuTl)-1223 composites. The insets represent the logarithmic plots of (E) versus (J) and the variation of J_c versus the NdFeO₃ content

Table 5 The variation of J_c, n and U versus x for (NdFeO₃)_x/(CuTl)-1223 composites

The homogeneity of a superconductor is described by the n-values or the indexes of transition. These values were calculated from the log(E)–log(J) curves and are depicted in Table 5. It should be noted that the higher the n values, the more homogeneous are the prepared samples. n is related to the pinning potential (U) according to Eq. 4 [81]:

$$n = \frac{U}{{TK_{{\text{B}}} }}.$$

(4)

It is evident that as x increased up to 0.50 wt.%, the values of n and U increased, supporting the beneficial impact of the inclusion of nano-NdFeO₃ on the superconducting properties of (CuTl)-1223 phase at that level of addition. On the other hand, for x > 0.50 wt.%, both n and U decrease as x increases, implying that high-level addition degraded this phase for the reasons mentioned above.

3.5 X-ray photoelectron spectroscopy

The XPS spectra of the samples with x = 0.00 and 0.75 wt.% are shown in the graphs of Fig. 10. The survey spectrum of the pure sample (x = 0.00 wt.%) showed that thallium, copper, calcium, barium, oxygen, and traces of carbon were present. In addition, the survey spectrum of the sample at x = 0.75 wt.% revealed the presence of iron and neodymium in addition to the elemental components of the bulk superconductor. The sample's contact with outside air prior to the XPS measurement is thought to be the cause of carbon presence in the spectra [84].

Fig. 10

XPS spectra of the samples x = 0.00 and 0.75 wt.%

Additionally, the high-resolution XPS spectra (x = 0.00 and 0.75 wt.%) of the main spectral lines for Tl, Cu, Nd, Fe and O were deconvoluted and displayed in Fig. 11 to further investigate their oxidation states. According to Figs. 11a, b, the high-resolution XPS spectra of Tl (Tl4f) exhibited four main peaks centered at 118, 122.29, 123.01, and 119.57 eV for x = 0.00 wt.% and three main peaks centered at 121.7, 125.33, and 118.8 eV for x = 0.75 wt.%.

Fig. 11

The main spectral lines with deconvolution plots of Tl4d, Cu2p, Nd3d, Fe2p, and O1s for x = 0.00 and 0.75 wt.%

The binding energy of the Tl4f_7/2 signal has previously been reported to be 117.4 eV for Tl³⁺ in Tl₂O₃ and between 118.7 and 119.4 eV for Tl¹⁺ depending on the ligands [85]. This tendency is consistent with the values of 118 and 118.8 eV for Tl₂O₃ and Tl₂O, respectively, in this work. The presence of NdFeO₃ nanoparticles caused a monotonic shift in the signal's direction towards higher binding energy. The sample had a 0.5–0.6 eV shift compared to the fractured one. This overall shift in the Tl4f signal was greater than the 0.25 eV chemical potential shift observed in valence band spectra. Additionally, it outperformed Tl4f signal shifts in single- and double-layer pure Tl-O systems [85,86,87]. These findings suggest that a decrease in the valency of the Tl4f signal may have contributed to its significant shift.

In the interface studies of oxide superconductors, the impact of metal atoms on Cu–O bonds has been a main topic of discussion [88]. The oxidation state of the element Cu is shown in the spectra of Fig. 11c, d, which demonstrated the presence of copper (Cu2p). The observed binding energy has been adjusted in accordance with the reference peak of C1s (285.92 eV). The strong peaks in Fig. 11c, d are characteristic for Cu²⁺ 2p_3/2 and Cu²⁺ 2p_1/2, and are located at 933.38 eV and 955.22 eV, respectively [89]. Figure 11d clearly shows that the CuO spectrum reported earlier is consistent with the Cu2p_3/2 and Cu2p_1/2 peaks, which are separated by nearly 20 eV [88]. According to XPS analysis, the excess holes in CuO ceramic directed the majority of the conduction in CuO to be caused by the hopping of holes between Cu²⁺ and Cu³⁺, which leads to the formation of grain boundaries [90].

Figure 11e shows the Nd3d XPS spectrum, and demonstrates that NdFeO₃ has been added successfully [91]. The two asymmetric peaks at 989.35 eV and 994.39 eV represent Nd3d_5/2 and Nd3d_3/2, respectively [32]. This validated that Nd exists in the + 3 oxidation state, which corresponds to Nd in Nd₂O₃.

A 2p doublet peak corresponding to the + 3 oxidation state is seen from the Fe spectrum in Fig. 11f. The binding energy of the 2p_3/2 peak for Fe, which corresponds to Fe³⁺ in Fe₂O₃, is 711.99 eV. As a result, Fe ions in the + 3 state are only found in the octahedral environment of NdFeO₃, resulting in equivalent crystallographic sites [32, 92].

The CuO matrix's lattice oxygen O²⁻ generates O1s peak binding energy, which is measured at 530.96 eV (Fig. 11g). The obtained binding energy values are in good agreement with those reported in literature [93]. The increase in atomic percentage of oxygen from 69.18% at x = 0.00 wt.% to 82.58% at x = 2.00 wt.% confirmed the improvement in oxygen content after NdFeO₃ nanoparticles addition. Furthermore, the addition of NdFeO₃ nanoparticles did not significantly alter the XPS spectra of Ca and Ba.

3.6 Vickers microhardness

The Vickers microhardness (H_v) measurements were made to determine the impact of NdFeO₃ nanoparticles addition on the mechanical characteristics of the (CuTl)-1223 superconductor, using Eq. 5 [94]:

$$H_{{\text{V}}} = 1854.4\left( {\frac{F}{{d^{2} }}} \right)\left( {{\text{GPa}}} \right),$$

(5)

where d is the average diagonal length of the indentation in micrometers and F is the applied force in Newton. The fluctuation of H_v as a function of the test load F at dwell time (t = 20 s) for the nano-(NdFeO₃)_x/(CuTl)-1223 composites, with 0.00 ≤ x ≤ 2.00 wt.%, are presented in Fig. 12. The fluctuation is split into two parts: the first is a load dependent region that arises at F < 2.94 N, in which the H_v values constantly decline until F = 2.94 N, while the second is the independent load region (plateau region). However, H_v values practically remained unchanged at applied force above 2.94 N, indicating that the saturation limit area is 2.94 N. This suggested that the samples displayed the characteristic ISE (Indentation Size Effect) nature, concurrent elastic/plastic deformations in the composite samples owing to the recovering of the system, as asserted by the indenter penetration depth [95, 96]. Moreover, it is evident from Fig. 12 that nano-NdFeO₃ inclusion caused a significant variation in H_v values. The inclusion of nano-NdFeO₃ until x = 0.50 wt.% increased the H_v values by 5.6%, followed by significant decrease with further addition of nanoparticles up to x = 2.00 wt.%. H_v values were boosted upon addition of small amounts of nano-NdFeO₃, which displayed noticeable influence on repairing the microcracks and filling the voids between the grains, thus improving the mechanical properties of these composite samples. These observations corroborated the analysis of the SEM images, which revealed an increase in grain connectivity due to porosity reduction and an increase in specimen resistance to crack propagation among grains due to the addition of nano-NdFeO₃ until x = 0.50 wt.%. On the other hand, the decrement in the H_v values at higher levels of nanoparticles addition was due to weak grain boundaries, impurity phases, and an irregular grain orientation distribution as asserted from the results for the porosity calculation and SEM micrograph analysis of the composite samples [97].

Fig. 12

H_v as a function of F of nano-(NdFeO₃)_x/(CuTl)-1223 composites,$0.00\le x\le 2.00 \text{wt}.\%$

3.7 Mechanical modelling and analysis

3.7.1 Meyer’s law

Meyer’s law is valuable for analyzing the foundational mechanical properties of materials subject to impact loads. The samples behave in one of two ways: first, there is the indentation size effect (ISE), where H_v values drop as indentation test loads are applied. However, the reverse indentation size effect (RISE) reveals an unexpected rise in the H_v values with the imposed loads. Meyer's law is given as [98]:

$$F = Ad^{n} ,$$

(6)

where n denotes the Meyer index and A denotes the hardness constant. The fitting parameters n and A were derived from the experimental data fitting curves of ln F versus ln d, as shown in Fig. 13. It is believed that the value of n is the most important parameter in predicting if a material has ISE or RISE trends. ISE nature will be evident when n < 2; however, RISE behavior is implied for n > 2. Table 6 presents the values of A and n and demonstrated that the n values were less than 2 for all prepared samples, indicating that plastic and elastic deformations occur concurrently in samples. Moreover, n values ranged between 1.00 and 1.60 for the samples with nanoparticle concentration between x = 0.00 wt.% and x = 0.50 wt.%, and therefore are categorized as hard materials. However, samples with x ≥ 0.75 wt.% are categorized as soft materials, confirming the low values of their H_v [99, 100].

Fig. 13

ln F vs ln d for nano-(NdFeO₃)_x/(CuTl)-1223 composites, $0.00\le x\le 2.00 \text{wt}.\%$

Table 6 Fitting parameters obtained from Meyer, HK, EPD, PSR, and MPSR with x content

3.7.2 Hays and Kendall (HK) model

The HK model [78] states that plastic deformation is only discernible in the H_v values at levels of applied load greater than a predefined threshold value (W), to initiates the notable indentation in the sample surface’s sample [101]. This method uses an effective indentation load, F_eff = F – W, which is coupled to the indentation size in place of the applied test load, supplying the load relationship between the indentation size as given by Eq. 7 [102]:

$$F = A_{1} d^{2} + W,$$

(7)

where A₁ denotes the load-independent microhardness constant. The microhardness experimental data for nano-(NdFeO₃)_x/(CuTl)-1223 composites with $0.00\le x\le 2.00\mathrm{ wt}.\mathrm{\%}$ were fitted using the HK model and are shown in Fig. 14. The values of the parameters that fit (A₂ and W) are shown in Table 6. Obviously, the positive values of W for all samples indicated that the imposed load has the potential to cause both reversible elastic deformation and irreversible plastic deformation [53]. Additionally, the values of A₁ exhibited similar trends as the variation of H_v as a function of F when NdFeO₃ nanoparticles were added, thus supporting the NISE behavior.

Fig. 14

F versus d² according to HK model of nano-(NdFeO₃)_x/(CuTl)-1223 composites

3.7.3 Elastic/plastic deformation (EPD) model

The EPD model is believed to be useful for predicting the behavior of materials (ISE / RISE) in close-proximity indentation imprint. The indentation is thought to have an elastic component that appears close to the indentation imprint after the indenter is taken out [103]. The elastic component d_o is consequently added to the measured plastic component (d) as given in Eq. 8:

$$F = A_{2} \left( {d + d_{{\text{o}}} } \right)^{2} ,$$

(8)

where $A_{2}$ is a constant. The fluctuation of F ^0.5 versus d for the nano-(NdFeO₃)_x/(CuTl)-1223 composites ($0.00\le x\le 2.00\mathrm{ wt}.\mathrm{\%}$), and the values of estimates parameter (A₂ and d_o) based on the fitting of experimental results to Eq. 8 are shown in Fig. 15. The coexistence of both elastic and plastic deformations in this range of applied test load for all samples is confirmed by the positive values of d_o [104]. When adjusting for recovering elasticity in the observed plastic indentation, the values of d_o are too high to be considered. Consequently, it is feasible to deduce that there are other factors contributing to the ISE behavior of the prepared composites in addition to the indentation size inaccuracy brought on by impeding the tip of the indenter and the recovery of elasticity after indentation [43].

Fig. 15

Linear plot of F^0.5 versus d for nano-(NdFeO₃)_x/(CuTl)-1223 composites $, 0.00\le x\le 2.00 wt.\%$

3.7.4 Proportional sample resistance (PSR) model

The PSR model's foundation is the energy dispersion linked to the appearance of voids, persistent fissures, and faults on the sample surface. It is considered a modified version of the HK model in which the term αd takes the place of W. Additionally, this model is used to describe the mechanical identification of the material showing the ISE or RISE character [105]. The following equation can be used to define the PSR model:

$$F = \alpha d + \beta d^{2} ,$$

(9)

where α and β are the surface energy and constant parameter, respectively. Additionally, there are two parts to α constant: the elastic resistance of the test specimen and the friction resistance determined by the indenter [105]. Moreover, the theoretical value of microhardness is determined by the β constant. Figure 16 displays the fluctuation of F/d versus d for (NdFeO₃)_x/(CuTl)-1223 composites with $0.00\le \mathrm{x}\le 2.00\mathrm{ wt}.\mathrm{\%}$. α and β values were estimated from the plots of Fig. 16 and displayed in Table 6. The instantaneous emergence of both elastic and plastic deformations is confirmed by the positive values of α. Additionally, the observed increase in α as x increased up to 0.50 wt.%, was attributed to dissipation of energy at the interfaces due to fractures [106].

Fig. 16

F/d versus d for nano-(NdFeO₃)_x/(CuTl)-1223 composites, with $0.00\le x\le 2.00 wt.\%$

3.7.5 Modified proportional sample resistance (MPSR) model

Another theoretical model for assessing the (CuTl)-1223 phase's mechanical characteristics is the MPSR model. It was found that the MPSR model, in contrast to PSR, includes a second constant, denoted as α₁, for the placement of the indenter's diagonal length on a specimen's surface. The MPSR model is described as given in Eq. 10 [107]:

$$F = \alpha_{1} + \alpha_{2} d + \alpha_{3} d^{2} ,$$

(10)

where α₁ denotes the minimal applied load required to cause an indentation. Surface residual stresses brought on by machining and surface polishing are linked to α₁. However, α₂ and α₃ parameters have the same physical significance as they do in the PSR model (α and β) [108]. Figure 17 displays the variation of F against d and the derived fitting parameters (α₁, α₂, and α₃) are presented in Table 6. The positive α₃ values reinforced that the samples exhibited ISE behavior that tends to expand progressively by inclusion of nano-NdFeO₃ up to x = 0.50 wt.%.

Fig. 17

Variations of F against d for nano-(NdFeO₃)_x/(CuTl)-1223 composites, x = 0.00, 0.25, 0.50, 0.75, 1.00 and 2.00 wt.%

It is possible to estimate the true microhardness or microhardness in the saturation region using the models previously mentioned, namely HK, EPD, PSR, and MPSR in accordance with the relationships shown below [104, 109]:

$$H_{{{\text{HK}}}} = 1854.4 \times A_{1}$$

(11)

$$H_{{{\text{EPD}}}} = 1854.4 \times A_{2}$$

(12)

$$H_{{{\text{PSR}}}} = 1854.4 \times \beta$$

(13)

$$H_{{{\text{MPSR}}}} = 1854.4 \times \alpha_{4}$$

(14)

Figure 18a compares the experimental H_v values in the saturation region for the nano-(NdFeO₃)_x/(CuTl)-1223 composites with H_v values computed theoretically using Eqs. (11–14), and their values are listed in Table 7. Obviously, when comparing MPSR model with the other theoretical models in this work, H_MPSR values were substantially closer to the experimental H_v values in the saturation region with deviations less than 2% and 8% for x ≤ 0.50 wt.% and x ≥ 0.75 wt.%, respectively. The fragments of evidence obtained by this method suggested that the MPSR is a feasible method for predicting the mechanical properties of nano-NdFeO₃ added to (CuTl)-1223 superconductor. A similar model was used to analyze the experimental results obtained by Barakat et al. [64] for the (CuTl)-1223 phase.

Fig. 18

a Variation of experimental and theoretical H_v in plateau region according to four models with x, b variation of E and Y for (NdFeO₃)_x/(CuTl)-1223 phase with respect to x and c variation of K and B for (NdFeO₃)_x/(CuTl)-1223 phase with respect to x

Table 7 Variation of the experimental microhardness at the plateau region and the calculated microhardness according to HK, EPD, PSR, and MPSR models versus x-content

Additional valuable mechanical parameters determined in the current study included the elastic modulus (E), yield strength (Y), fracture toughness (K), and brittleness index (B). These properties are crucial for the application of the (CuTl)-1223 phase in numerous fields. The formulas below [43] are used to calculate the values of load-dependent parameters in the plateau region:

$$E = 81.96 \times H_{{\text{v}}}$$

(15)

$$Y = \frac{{H_{{\text{v}}} }}{3}$$

(16)

$$K = \sqrt {2E\alpha }$$

(17)

$$B = \frac{{H_{{\text{v}}} }}{K}$$

(18)

The fluctuation of E, Y, K, and B with x is presented in Fig. 18b, c, respectively. The curves showed that the values of these parameters rise consistently as x increases up to 0.50 wt.% and then decrease up to 2.00 wt.%, except for B whose values decrease with increasing x up to 0.50 wt.% and then increase up to x = 2.00 wt.%. These enhancements could be attributed to nano-NdFeO₃ settlement into the voids and inter-grain spaces, which decreased the porosity of the (CuTl)-1223 as determined by the porosity calculation [97]. Furthermore, the addition of nano-NdFeO₃ prompted continued improvements in ductility and better tendency to withstand indentation fractures. Moreover, higher E values may be attributed to the bond strength between atoms or molecules in a substance, as reported by Zewnet et al. [110] who investigated the relationship between elastic modulus and the size of the grain to determine at what grain size the microcracking begins because of the anisotropic thermal stresses that arise throughout treatment. The increment in the material's fracture toughness K (the ability to sustain fracture) is linked to the increase in surface energy. The brittleness index, which assesses a material's ability to break when deformed, is also an important practical characteristic.

3.7.6 Indentation creep analysis (time-dependent Vickers microhardness)

Time-dependent Vickers microhardness addresses the time-dependent motion of a rigid indenter into a solid under a constant applied test load. A fast, precise, non-destructive method of determining how well a material performs mechanically is by measuring its indentation creep [111]. H_v is plotted with dwell time from 10 to 90 s at three applied static loads (0.98, 2.94, and 9.8 N) as shown in Fig. 19, in order to examine the indentation creep of nano-(NdFeO₃)_x/(CuTl)-1223 composites with $0.00\le x\le 2.00$ wt.%. The observed rapid decrease in H_v values with dwell time is characteristic of the indentation creep deformation behavior [112]. This inverse relationship confirmed the penetration depth of the indenter and reflected the acceptable behavior of H_v and dwell time [111]. Using the Sargent–Ashby model [112], the indentation creep behavior can be investigated in terms of the following relationship:

$$H\left( t \right) = \frac{{\varepsilon_{0} }}{{\left( {\eta c\varepsilon_{0} t} \right)^{1/\eta } }},$$

(19)

where ε₀ is known as the strain component at reference stress r₀, η is the stress component, and c is constant.

Fig. 19

Fluctuation of Vickers microhardness (H_v) as a function of dwell time (t) for nano-(NdFeO₃)_x/(CuTl)-1223 phase

The stress exponent (ƞ) can be calculated from the slope (-1/ƞ) of the ln H_v versus ln t plots using Eq. 19 as shown in Fig. 20a–c, with applied test loads F = 2.96 N, 4.9 N, and 9.80 N, respectively. ƞ values are useful in determining the processes that control deformation. According to literature [78], if the values of ƞ are close to one, the sample suffers from diffusion creep. Within the sample, the sliding of grain boundary will occur for values near two, and dislocation creep will prevail for values between 3 and 10. Furthermore, the formation and development of new cavities are influenced by two critical factors: the indentation generated by the applied force and the concentration of the additives. Table 8 shows that at low concentrations of nanoparticles (x ≤ 0.50 wt.%), the values of ƞ for the three applied loads decrease. However, higher concentrations of nanoparticles (x > 0.50 wt.%) was associated with increase in ƞ values. This behavior is supported by the porosity calculations that showed a concurrent increase in the formation of new cavities at higher additions of nanoparticles. As a result, the contribution of the cavities is anticipated to increase with the applied stress, and the contribution to the creep rate and the stress component will rise [113].

Fig. 20

Fluctuation of ln H_v versus ln t at a F = 2.94 N, b F = 4.9 N and c F = 9.8 N of nano-(NdFeO₃)_x/(CuTl)-1223 phase

Table 8 Stress exponent ƞ versus F for (NdFeO₃)_x/(CuTl)-1223 composites

3.8 Comparative studies between different magnetic nanoparticles

Literature reports investigating the effect of adding different nanoparticles to the (CuTl)-1223 superconductor presented contrasting results. While some nanoparticles showed enhancement in various calculated parameters, others showed suppression. It has been reported that the increase in ferromagnetic behavior and saturation magnetization of the nanoparticle samples as their addition concentration increases can cause the destruction of Cooper pair, and suppression of T_c and J_c [43]. However, the weak ferromagnetic behavior of orthoferrite NdFeO₃ prepared in this work enhanced T_c, J_c, and H_v of the (CuTl)-1223 superconductor. The percentage enhancement rate (%E) is listed in Table 9. Soft ferromagnetic spinel ferrite (Mg_0.8Zn_0.2Fe₂O₄) (M_s = 5.29 emu/g, H_c = 89.96 Oe) prepared by Qasim et al. [114] and the typical ferrimagnetic NiFe₂O₄ (M_s = 33 emu/g, H_c = 400 Oe) reported by Nadeem et al. [5] prepared using sol–gel method revealed no enhancement in T_c values of the superconductor. Besides, nano-Co₃O₄ (71 nm) reported by Imran et al. [28] and Mumtaz et al. [42] which were prepared by the sol–gel method as well showed no enhancement in the T_c values. In addition, CuO and MnFe₂O₄ nanoparticles developed by Mumtaz et al. [27] and Naveed et al. [115], respectively, showed no enhancement in T_c values. Hematite can also play a role in enhancing the T_c values. For instance, Fe₂O₃ prepared via sol–gel method by Mumtaz et al. [4] resulted in a rise in T_c %E which is small compared to %E achieved by NdFeO₃ (Table 9). Besides, Fe₂O₃ used by Mohammed et al. [116] improved the T_c %E, however, it is still less than NdFeO₃ as reported in Table 9. Moreover, Table 9 showed that Co_0.5Zn_0.5Fe₂O₄ nanoparticles synthesized by Barakat et al. [64] revealed an improvement in T_c. Srour et al. [25] reported an enhancement in T_c upon BaSnO₃ nanoparticles addition, which is also less than NdFeO₃ as noted in Table 9. However, the addition of hard ferromagnetic Y₃Fe₅O₁₂ by Rekaby et al. [43] resulted in the best enhancement rate for T_c among all results mentioned before. The enhancement rate of J_c obtained by Srour et al. [25] was found to be 52.69%, which is slightly less than the rate obtained in this work (52.95%).

Table 9 The effect of different nanoparticles addition on (CuTl)-1223 superconducting phase

The experimental H_v values were obtained from plateau region of the curves of H_v versus F. Table 9 lists the enhancement rate for the mechanical properties of the (CuTl)-1223 superconductor supplemented with different nanoparticles. In this work, NdFeO₃ at x = 0.50 wt.% offered the high enhancement rate (H_v = 0.487 GPa, and E = 39.266 GPa). According to Barakat et al. [64], the addition of Co_0.5Zn_0.5Fe₂O₄ nanoparticles (H_v = 1.115 GPa, and E = 91.39 GPa) also enhanced the H_v and E. These rates are close to the ones obtained in this work. Besides, BaSnO₃ nanoparticles (H_v = 0.332 GPa, and E = 27.185 GPa) synthesized by Srour et al. [117] revealed an enhancement in H_v and E greater than the rate achieved by nano-NdFeO₃ in the current study as shown in Table 9. The highest enhancement rates among the nanoparticles discussed in this section were attained upon the addition of Y₃Fe₅O₁₂ (H_v = 0.59 GPa, and E = 47.27 GPa) prepared by Rekaby et al. [43]. These improvements can be attributed to the small grain sizes of Co_0.5Zn_0.5Fe₂O₄, BaSnO₃, and Y₃Fe₅O₁₂ nanoparticles which are found in the coherence length range [64] that facilitate the penetration of the nanoparticles into the grain boundaries and close pores and voids.

4 Conclusion

Nano-(NdFeO₃)_x/(CuTl)-1223 composites with x = 0.00, 0.25, 0.50, 0.75, 1.00, and 2.00 wt.% were successfully prepared, and the effect of nanoparticle addition on the superconductivity properties and mechanical characteristics was investigated. The XRD findings showed that the crystal structure of the host (CuTl)-1223 superconducting matrix was unaffected by the addition of NdFeO₃ nanoparticles that occupied intergranular sites rather than breaking or substituting any lattice site in the unit cell. SEM images revealed a boost in the grain connectivity as a consequence of nano-NdFeO₃ addition until x = 0.50 wt.%, associated with a decrease in the porosity percentage (P%) from 23.14% to 18.24%. The presence of Nd and Fe elements in the bulk samples was validated by EDX, confirming that nano-NdFeO₃ did not enter the phase crystal structure, but rather settled at the grain boundaries.

Inclusion of nano-NdFeO₃ up to x = 0.50 wt.% induced an enhancement in the superconducting transition temperature (T_c), the critical current density (J_c) and the flux pinning potential of the (CuTl)-1223 superconductor as revealed from measurements of Dc-electrical resistivity and I-V characteristics. Higher concentrations of the nanoparticles (x > 0.50 wt.%) deteriorated the same parameters. These observations were compatible with the XRD data, oxygen content, and hole concentration. Furthermore, XPS studies indicated that the superconductivity of (CuTl)-1223 is governed by the mechanism of electron transfer between Cu and Tl ions, as well as with variations in oxygen content brought on by NdFeO₃ nanoparticles addition. The impact of nano-NdFeO₃ addition on the mechanical properties of (CuTl)-1223 superconductor was examined by assessing various parameters such as H_v, E, Y, K, and B. The results showed that adding nano-NdFeO₃ improved the mechanical properties of (CuTl)-1223 composites up to x = 0.50 wt.%.

Five theoretical models were utilized for the mechanical analysis of the measured H_v values, and provided a better understanding of the effects of nano-NdFeO₃ inclusion on the mechanical properties and sustainability of the (CuTl)-1223 phase. When the models' load-independent microhardness data were compared, the MPSR model produced significantly closer results. According to the results of indentation creep, the composites samples under investigation showed grain boundary sliding across the operational creep mechanism. The enhancement rate in the superconducting properties (T_c and J_c) of (CuTl)-1223 superconductor achieved by NdFeO₃ orthoferrite nanoparticles was greater than Fe₂O₃, Co_0.5Zn_0.5Fe₂O₄, and BaSnO₃ nanoparticles, yet lower than that of Y₃Fe₅O₁₂. In addition, %E in mechanical characteristics (H_v and E) of (CuTl)-1223 superconductor upon addition of NdFeO₃ nanoparticles was comparable (e.g. Co_0.5Zn_0.5Fe₂O₄) or lower (e.g. BaSnO₃ and Y₃Fe₅O₁₂) to what was reported for other magnetic nanoparticles. This can be attributed to various factors such as grain size and the magnetic nature of materials.