Electrical and mechanical properties of (Bi,Pb)-2223 substituted by holmium

The effect of the partial substitution of Ca2+ by Ho3+ ions on the electrical and mechanical properties of the superconducting phase (Bi,Pb)-2223 was studied. Superconducting samples of the type (Bi1.8Pb0.4)Sr2Ca2.1−xHoxCu3.1O10+δ were prepared by solid-state reaction technique under ambient pressure, and characterized by means of X-ray powder diffraction (XRD) and scanning electron microscopy (SEM). The superconducting transition temperature Tc and pseudogap temperature T* were estimated from electrical resistivity measurements, while the critical current density Jc was determined from I–V curves. The electrical resistivity data were discussed according to bipolaron model in the absence of thermally excited individual polarons. The sample with x = 0.025 showed the highest phase volume fraction, Tc, and Jc. Room temperature Vickers microhardness measurements were carried out at different applied loads (0.25–5 N) in order to investigate the performance of the mechanical properties of (Bi1.8Pb0.4)Sr2Ca2.1−xHoxCu3.1O10+δ phase. It was found that all the samples exhibit normal indentation size effect (ISE). The Vickers microhardness number HV increased as x increased from 0 to 0.025. The experimental results were discussed in view of Meyer’s law, Hays–Kendall (HK) approach, elastic/plastic deformation (EPD) model, and proportional specimen resistance (PSR) model. The load independent (true) microhardness of (Bi1.8Pb0.4)Sr2Ca2.1−xHoxCu3.1O10+δ superconducting samples showed identical behavior to that of the PSR model.


Introduction
(Bi,Pb)-2223 phase is the most promising one among the bismuth strontium calcium copper oxide (BSCCO) system to synthesize tapes and wires for large scale and high current applications, owing to its high superconducting transition temperature T c and favorable characteristics such as atmospheric stability, high current and magnetic field carrying capacity, and electronic properties in view of a possible use in superconducting devices operated at liquid nitrogen temperature [1]. On the other hand, there are some  problems which limit the practical applications of (Bi,Pb)-2223; the most limiting parameters for the practical applications of this phase are critical current density J c and mechanical properties [2]. Since the discovery of high temperature superconductors (HTSCs), many substitutions have been carried out in BSCCO system in order to improve its T c and J c , and better understand the structural, physical, and mechanical properties [3][4][5][6][7][8]. The rare-earth element chemical substitution has an effective route in improving the flux pinning centers, which in turn enhance J c and other transport properties of (Bi,Pb)-2223 phase [9]. The partial substitution of Ca 2+ ions by rare-earth element R 3+ ions on the (Bi,Pb)-2223 phase has a significant effect on its normal and superconducting properties [10]. The previous studies showed that T c values are gradually decreased with the increase of R 3+ content [10][11][12][13][14][15]. The effect of partial substitution of Sm and Eu at Ca site in (Bi,Pb)-2223 system on the superconducting, microstructure, and mechanical properties was studied [12][13][14][15][16]. It was found that the volume fraction of (Bi,Pb)-2223 phase decreases with increasing Sm content. Moreover, the superconducting properties and mechanical properties degrade with increasing Sm content. Also, J c decreases by increasing Eu content and this decrease is related to the weakness of the grain connectivity and the increment of the porosity for (Bi,Pb)-2223 phase. In addition, the lattice parameter c decreases and the crystallographic structure is found to change slightly from tetragonal to orthorhombic.
The normal state properties of HTSCs, especially the metallic like behavior of electrical resistivity, can be discussed according to the bipolaron model [17]. Bipolarons could be created by the interaction between holes and optical phonons in HTSCs. Alexandrov and Mott [18] reported that for a system with large electron-phonon coupling constant, a Bose condensation of bipolarons can take place in the real space. Due to the extremely short coherence length of HTSCs, a cross-over region from BardeenCooper Schrieffer (BCS) to Bose-Einstein condensation (BEC) superfluids is considered in some theoretical studies related to their pairing mechanism. In the bipolaron model, however, BEC is only considered as paired state [19]. The bipolaron model is sensitive to both substitution concentration and sample crystal structure [20,21].
The understanding of normal state properties of HTSCs would provide important evidences about the basic mechanisms responsible for superconductivity in HTSCs. Thus, the normal state properties of HTSCs remain an area of intense research. One of the clearest differences between the BCS scenario of superconductivity and high T c cuprates is the existence of a pseudogap. According to the band theory representation, pseudogap refers to some gapped regions in the Fermi surface, while other parts retain their conducting character. The chemical substitution changes the gapped region and metallicity of the material. The pseudogap is a fundamental property of the under-doped copper oxides, where the presence of pseudogap leads to striking deviation of resistivity from simple linear behavior [22][23][24][25]. However, the temperature dependence of resistivity for under-doped HTSCs shows a distinguishable deviation from linear behavior at the pseudogap temperature T * . The recent literature indicated that, there are two scenarios for the explanation of the pseudogap temperature in HTSCs. According to the first scenario, pseudogap is related to short-range ordering fluctuations, e.g., antiferromagnetic fluctuations, spin, and charge density waves, etc. [22]. The second scenario assumes that Cooper pairs are formed at temperatures much higher than T c ; T * > T c , and further phase coherence realizes at T < T c [26,27].
The mechanical properties of small size samples can be measured using Vickers indentation method [28][29][30][31]. The effect of chemical substitution/addition on Vickers microhardness for (Bi,Pb)-2223 phase was studied by many groups [12][13][14][15][32][33][34]. It was found that the substitution of Sm at Ca site increases H V due to the enhancement in grain connectivity, and the modified proportional specimen resistance model was found to be the most successful model describing the load independent microhardness [32]. Addition of Gd 3+ ions to (Bi,Pb)-2223 degrades the mechanical properties [33,34]. This degradation is related to the increase of voids, impurity phase segregation, and the modification of the grain boundaries.
In this work, we present a detailed study of the effect of Ho 3+ substitution at Ca 2+ site in (Bi,Pb)-2223 phase. This study includes X-ray powder diffraction (XRD), scanning electron microscopy (SEM), electrical resistivity, I-V characteristics, and Vickers microhardness for (Bi 1 . The stoichiometric mixture of the starting materials was ground manually in an agate mortar and sifted using a 65 μm sieve. Then, the resulting powder was subjected to two stages of calcinations in air at 820 ℃ for 24 h for each with intermediate grinding and sieving. This is important to obtain the proper precursor phases necessary for the formation of (Bi,Pb)-2223. The phase assemblage is sensitive to the calcination conditions [35]. After that, the mixture was cooled to room temperature and then ground and sifted. Then, the powder was pelletized into a disc with diameter of 1.5 cm and thickness of about 0.3 cm at a pressure of 15 tons/cm 2 . The pellets were sintered in air (atmospheric pressure) at 845 ℃ for 96 h with a heating rate of 4 ℃/min, and they were cooled with a rate of 2 ℃/min to room temperature. After that, the sintered samples were ground, sifted, and then pressed again in the form of disc and re-sintered in air at 845 ℃ for 48 h with a heating rate of 2 ℃/min. Finally, the samples were cooled with a rate of 1 ℃/min to room temperature.
The prepared samples were characterized by XRD using a Philips X'Pert powder diffractometer with Cu Kα radiation (λ = 1.54056 Å) in the range of 4° ≤ 2θ ≤ 70°. The grain size and morphology of the samples' surface were identified using a Jeol scanning electron microscope JSM-5300, operated at 30 kV, with a resolution power of 4 nm.
The electric resistivity of the prepared samples was measured by the conventional four-probe technique from room temperature down to zero resistivity temperature T 0 via a closed cryogenic refrigeration system (Displex) employing helium gas as a working medium. The transport critical current density was measured by the conventional four-probe technique at 77 K in a flow of liquid nitrogen using the standard 1 µV/cm criterion. The temperature of the samples was monitored by Fe-Au thermocouple and stabilized with the aid of a temperature controller within ±0.1 K.
Vickers microhardness measurements of the studied samples were performed in atmospheric air at room temperature using a manual microhardness tester IN-412A. A Vickers pyramidal indenter with different applied loads varied from 0.25 to 5.00 N and a loading time of 10 s was used to measure the diagonals of indentation with an accuracy of ±0.1 μm. An average of 7 readings at different locations of specimen surface was taken to obtain sensible mean value for each load. The measurements were carried out in such a way that the distance between any two indentations was more than two times the diagonal of the indentation mark to avoid surface effects due to neighboring indentation. The Vickers microhardness is defined as the ratio of applied load to the pyramidal contact area of indentation, and it is calculated as [36]: where F is the applied load (N) and d is the diagonal length of the indentation mark (μm).

1 Characterization and investigation
The XRD patterns for (Bi  Fig. 1. It is observed that most of the high-intensity peaks belong to tetragonal phase of (Bi,Pb)-2223 with a space group P4/mmm [37]. There are a few low-intensity peaks belonging to other phases, such as (Bi,Pb)-2212 and Ca 2 PbO 4 . These impurity phases are typically observed in the preparation of (Bi,Pb)-2223 by solid-state reaction technique [10,35,[37][38][39]. The notations "*" and "+" indicate the peaks of (Bi,Pb)-2212 and Ca 2 PbO 4 , respectively. The characteristic peak intensity of (Bi,Pb)-2223 phase (002) increases as x increases from 0.000 to 0.025, indicating that the low content of Ho can enhance the phase formation of  increases. This means that the high Ho content retards the growth of (Bi,Pb)-2223 phase. Furthermore, no peaks belong to Ho rich compounds observed in the XRD patterns. This means that the Ho ions enter into the crystal structure of (Bi,Pb)-2223. Similar results were reported for (Bi,Pb)-2223/Ag taps substituted by Pr ions at Ca site [40]. This indicates the high solubility limit of Ho in the crystal structure of (Bi,Pb)-2223 phase.
The relative volume fractions of (Bi,Pb)-2223, (Bi,Pb)-2212, and Ca 2 PbO 4 phases are calculated using Eqs. (2)-(4), respectively [5,10,28,37], and their values are listed in Table 1 where I represents the peak intensity of the indicated phases. A little increase in the volume fraction of (Bi,Pb)-2223 phase is observed for x = 0.025, and then it decreases for x > 0.025, while the volume fraction of low-T c phase (Bi,Pb)-2212 decreases with x = 0.025 and increases for x > 0.025. These observations indicate that the low content of holmium enhances the formation of the (Bi,Pb)-2223 phase which could be due to the ability of Ho to provide a suitable medium for the conversion of the (Bi,Pb)-2212 phase to the (Bi,Pb)-2223 phase and reduce the impure phases. While the further increase of x leads to a change in the reaction rate which retards slightly the (Bi,Pb)-2223 formation. A similar behavior was observed in many previous studies, concerning the effect of R 3+ ions on (Bi,Pb)-2223 phase [11][12][13][14].
The lattice parameters a and c were calculated using the least square method through d value and (hkl) planes for tetragonal unit cell structure. Their variations versus Ho content are also listed in Table 1. It is found that the lattice parameters a and c for the un-substituted sample are very close to those obtained by Koyama et al. [41] and Abou-Aly et al. [42]. Moreover, for all substituted samples, the lattice parameter a increases slightly with increasing Ho content while the lattice parameter c decreases significantly. Similar behavior has been observed in previous works involving substitution of rare-earth elements at Ca site in the (Bi,Pb)-2223 system [10][11][12][13]43,44]. This behavior can be explained according to two reasons. The first one is attributed to the partial substitution of larger ionic radius of Ca 2+ ion, 0.990 Å, by the smaller one of Ho 3+ ion, 0.901 Å, at the same 6-fold coordination type [45]. The second reason is that the partial substitution of Ca 2+ ions by Ho 3+ ions leads to an increase in the oxygen content of the unit cell that enhances the average oxidation state, leading to smaller Cu-O distance within the copper-oxygen sheets [46]. In addition, the lattice parameter a is controlled by the length of the in-plane Cu-O bond which may be expanded as a result of electron addition into anti-bonding orbital through the partial substitution of Ca 2+ ions by Ho 3+ ions [47].
The SEM micrographs of the fractured surface of the prepared samples with x = 0.000, 0.025, 0.075, and 0.200 are shown in Figs. 2(a)-2(d), respectively. The granular morphology for the un-substituted sample x = 0, consists primly of flaky layers of large platelet-like structure with random alignment distribution as shown in Fig. 2(a), which is a signature of (Bi,Pb)-2223 phase formation [48,49]. The platelet grains are well linked as x increases to 0.025, in which the platelet-like grains still have a flaky structure. Additionally, the sample has more uniform surface appearance with larger average grain size and lower porosity as shown in Fig.  2(b). The average grain size increases from 1.99 to 2.27 µm (calculated from the SEM) as Ho content increases from x = 0.000 to x = 0.025 and then decreases to 1.06 µm for x = 0.200. This indicates that the low Ho content increases the growth of (Bi,Pb)-2223 phase,  consistent with the enhancement of volume fraction of (Bi,Pb)-2223 phase at low concentration. As the Ho content increases, the platelet shape is not changed, whereas the grain dimensions are changed and oriented randomly.  Fig. 3. All the prepared samples have a metallic-like behavior at high temperatures followed by a superconducting transition as the temperature is lowered. The normal metallic-like behavior above T c can be interpreted by the liquid model and it reflects the spin charge separation in CuO 2 planes that results in the longitudinal transport relaxation rate 1/τ  T [51]. It is noticed that the electrical resistivity behavior does not show the existence of a secondary phase at higher Ho contents, implying that the samples are mainly single phase. This result is consistent with that obtained through the X-ray investigation. A small curvature in electrical resistivity above T c is observed for all  [52] or the opening of spin-gap that appears in the HTSCs due to magnetic impurity substitutions [53]. These superconducting thermodynamic fluctuations occur at finite temperatures just above T c due to the appearance of Cooper pairs even above this temperature.
The superconducting transition temperature T c , which displays the superconducting transition within the grains, is determined as the temperature corresponding to the crest in dρ/dT versus T curve [54]. The inset in Fig. 3 shows the variation of T c with x for (Bi 1.8 Pb 0.4 )Sr 2 Ca 2.1−x Ho x Cu 3.1 O 10+δ . It is clear that T c enhances as x increases from 0.000 to 0.025 and then it decreases with the further increase in x. The parabolic dependence of T c on x is considered to be a typical character of the substituted high temperature superconductors [11]. This relation indicates that the un-substituted sample lies in the under-doped region [12]. The enhancement of T c can be explained according to the increase in the formation of (Bi,Pb)-2223 and well linked grains for 0.000 < x ≤ 0.025. It is also probably due to a change of the average oxidation state of copper as reported by Laximi Narasaiah et al. [55] substituting Y 3+ ions in (Bi,Pb)-2223. They found an increase in T c at low Y content. There are two different explanations for the strong depression of T c for x > 0.025. The first one due to Abrikosov-Gor'kov theory [56] is related to a Cooper pair breaking mechanism as a result of the magnetic ion substitution, and leads to a disorder in the internal magnetic state. The second one is due to the scattering between the Cooper pair and magnetic ions according to spin-flip process [57]. This process is characterized by total spin conservation in the scattering event, so the spin of Ho atom must flip when the Cooper pairs are broken. The location of carriers can play important role for enhancement or decrement of T c , which will discuss later by bipolaron model.
To compare our results with different R 3+ substitution [58][59][60], the variation of the reduced transition temperature (T c (x)/T c (0)) with rare-earth elements is plotted in Fig. 4. It is clear that, the critical concentration of Ho 3+ ions required to destroy the superconductivity of (Bi,Pb)-2223 samples (x > 0.200) is higher than that reported for Sm 3+ (x = 0.2), Pr 3+ (x = 0.1), and Ce 3+ or 4+ (x = 0.1) ions [58][59][60]. This indicates that Ho 3+ ions have a higher solubility limit in the (Bi,Pb)-2223 phase and are less detrimental to superconductivity. The solubility limit of rare-earth elements R 3+ in the (Bi,Pb)-2223 phase can be understood by a comparison of their ionic radii of 0.901, 0.958, 0.990, and 1.034 Å for Ho 3+ , Sm 3+ , Pr 3+ , and Ce 3+ , respectively, with respect to 0.990 Å of Ca 2+ [45]. This means that the solubility limit may increase as the difference of ionic radii increases [10].
There are only a few models concerning the linear temperature dependence of the electrical resistivity of the HTSCs. One may exclude boson-boson and boson-phonon scatterings which are responsible for linear resistivity behavior at high and low temperatures, respectively [61]. Despite the polycrystalline nature of our samples, the data of electrical resistivity are analyzed using bipolaron model. In the first attempt of bipolaron model, normal state resistivity data are compared with a condition in which the contributions of the thermally excited individual polarons have not been taken into account [62]. This model is based on the localization of carriers by disorders and it predicts a linear behavior for the normal state resistivity. Considering the residual resistivity is zero, the temperature dependence of resistivity is given by Eq. (5): where m is the in-plane boson mass; C is a constant proportional to the deformation potential; V is the volume of the elementary cell; L N is the number of localized states per unit cell; and L bN T is the density of unoccupied potential wells.  The experimental electrical resistivity data and those fitted according to Eq. (5) are plotted in Fig. 5. In this figure, ρ(T) deviates downwards from linear behavior at temperature T * corresponding to the pseudogap opening [63]. The fitting parameters for different substituted samples are listed in Table 2. The parameter m 2 C, which corresponds to the rate of resistivity change, decreases slightly as Ho content x increases to 0.025 and then it increases with further increase in x.   value till x = 0.025. A reverse trend is observed for x > 0.025.
The T * value is also determined by analyzing the first derivative dρ/dT with respect to temperature at finite dρ/dT [64] as shown in Fig. 6. From the figure we can see that (dρ/dT)/(dρ/dT) T=160K reduces as the temperature increases and then converges to a constant value at which T * can be determined. Table 3 displays the variation of T * values with x. It is clear that T * values decrease by increasing Ho content up to x = 0.100 and then increase for x ≥ 0.150. This enhancement can be explained by anti-ferromagnetic correlation in the CuO 2 planes according to t-J model or extended t-J model [65], due to Ho substitution. The effective interplane coupling can be strongly enhanced by strong spin anti-ferromagnetic correlation, and extend over a coherence length of several lattice spacing. The enhanced coupling also leads to enhanced pairing between fermions on adjacent planes, thus is responsible for the formation of the spin gap [23]. In addition, it is observed that the values of pseudogap temperature estimated from the deviation of ρ(T) from the linearity and those from the first derivative dρ/dT with respect to dρ/dT at finite temperature are nearly close to each others.    Fig. 7. The explanation of these curves can be divided into four parts as follows: (1) The system is resistanceless, V(I) = 0, when the current is not large enough to cause vortex pair unbinding [66]. (2)  x > 0.025 and could be residual at the grain boundaries. This decreases the intergranular Josephson coupling energy and hence decreases of the pinning of intergranular vortices [70]. Thus, one can contribute the decrease in T c and J c with increasing Ho content to the suppression of the relative volume fraction for (Bi,Pb)-2223 phase and the growth of the impurity content, as well as the increase in porosity accompanied with the increase in grain boundary weak links. This result is in consistent with the XRD data.

2 Mechanical measurements
The   [14] in Bi 1.6 Pb 0.4 Sr 2 Ca 2−x Sm x Cu 3 O y phase. They attributed this behavior to the increase of void density and the resistance to crack propagation. The H V -F curve behavior shows that the H V decreases non-linearly as the applied load increases up to 3 N, beyond which the curves tend to attain saturation (nearly plateau) region. This non-linear behavior is also observed for (Bi,Pb)-2223 phase [12][13][14][15][30][31][32][33][34] and it is known as indentation size effect (ISE). The ISE behavior can be explained qualitatively on the basis of penetration depth of the indenter. Since the indenter penetrates only surface layers at small applied loads, the surface effect is more pronounced at these loads. However, as the depth of penetration increases, the effect of inner layers becomes more and more prominent, and ultimately there is no change in the values of hardness with applied load [71]. The substantial variation of microhardness is observed with increasing load between 0.25 and 3.00 N. This behavior is due to the contribution of weak grain boundaries and the process of crack formation along grain boundaries [14]. The H V -F results are analyzed according to different models as follows.

2. 1 Meyer's law
Meyer's law is the most basic method used in describing the ISE behavior [72,73]. It gives a relation between the applied load F and the indentation diagonal length d: where A is a constant, representing the load needed to initiate unit indentation. The exponent n is called Meyer's index and describes the ISE. The value of n is less than 2 for normal ISE, while for reverse ISE, the value of n is greater than 2. When n = 2, H V is independent of the applied load. The typical plots of the dependence of lnF on lnd for (Bi 1.  Table 5. It is obvious from Table 5 that all prepared samples have n < 2, indicating a normal ISE behavior. Here the values of A are very small to be acceptable for the ceramic materials which are characterized by their high microhardness. Thus, Meyer's law can explain the ISE behavior of the data qualitatively well, but it cannot determine the true microhardness [34].

2. 2 Hays-Kendall approach
Hays-Kendall (HK) approach displays that there exists a minimum load W (specimen resistance) to initiate plastic deformation below which only elastic deformation is occurred [74]. It is found that indentation size is proportional to an effective load where 1 A is the load independent microhardness constant. The values of W and 1 A are calculated from the linear plotting of F versus d 2 and are also listed in Table 5. Figure 9 shows the variation of F versus d 2 for (Bi 1 The values of W are positive for all the samples, meaning that the applied load is sufficient to create both the elastic and the plastic deformation. The calculated values of W are found to be larger than the acceptable values [75]. Similar results were observed by Kölemen et al. [32], and they reported that the values of W (0.128-0.163 N) seem to be large to be accepted when studied the effect of partial substitution of Ca 2+ ions by Sm 3+ ions on the superconducting properties of Bi 1.6 Pb 0.4 Sr 2 Ca 2−x Sm x Cu 3 O y . Similarly, there is reason to believe that W values (0.188-0172 N) obtained in this study seem to be large. Also, according to Ionescu et al. [76], Vickers nanoindentation tests were conducted on the similar BSCCO materials and plastic indentation impressions were observed at the load as low as 10 mN. The constant A 1 is suggested to be a measure of the load independent microhardness H HKin , and their values increase as x increases to 0.025 and then decrease for x > 0.025, confirming the experimental variation of H V with x.

2. 3 Elastic/plastic deformation model
Bull et al. [77] proposed that the dependence of indentation size on the applied load can be obtained by way of the equation: where 2 A is a constant and 0 d is related to d plastic deformation. The values of 2 A and 0 d are calculated from F 1/2 versus d curves for all the samples and are listed in Table 5. Figure 10 shows the variation of As seen in Table 5, the value of 0 d is positive for all the samples. This means that, for this range of applied loads, the elastic deformation is observed along with plastic deformation and elastic relaxation is present. The presence of elastic deformation along with plastic deformation is the reason of ISE behavior for our samples. Also, it is found that 0 d changes randomly, and similar behavior was reported by previous studies on BSCCO system [32,33,78] In order to determine the consistency of the above mentioned models with our samples, the experimental and calculated microhardness at the plateau region as well as the load independent microhardness values according to each model are listed in Table 6. It is obvious from Table 6 that the true microhardness value according to the HK model (0.751 GPa) of the sample with x = 0.000 is lower than the hardness value in the plateau region (H V = 0.802 -0.775 GPa). This behavior is also observed in all the other samples. Although the HK model has been applied to BSCCO system in Refs. [34,81,82] for example, the consistency is not very good in our case; it is found that W seems to be large to be accepted, and the values of the true hardness calculated by this model are not in the plateau region. Therefore, one can conclude that HK approach is not suitable for describing our experimental data. Similar behavior of failure of HK model in BSCCO system was observed in many previous works [14,32,33,75,76,78]. While, for EPD model, it is found that 0 d changes randomly, and the values of the true hardness