Abstract
An analysis of the field dependence of the pinning force in different, high density sintered samples of MgB_{2} is presented. The samples were chosen to be representative for pure MgB_{2}, MgB_{2} with additives, and partially oriented massive samples. In some cases, the curves of pinning force versus magnetic field of the selected samples present peculiar profiles and application of the typical scaling procedures fails. Based on the percolation model, we show that most features of the field dependence of the critical force that generate dissipation comply with the DewHughes scaling law predictions within the grain boundary pinning mechanism if a connecting factor related to the superconducting connection of the grains is used. The field dependence of the connecting function, which is dependent on the superconducting anisotropy, is the main factor that controls the boundary between dissipative and nondissipative current transport in high magnetic field. Experimental data indicate that the connecting function is also dependent on the particular properties (e.g., the presence of slightly nonstoichiometric phases, defects, homogeneity, and others) of each sample and it has the form of a single or double peaked function in all investigated samples.
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Introduction
Magnesium diboride, MgB_{2}, is one of the most exciting superconductors discovered in the last two decades due to a series of advantages that makes it attractive for applications. It has also a very interesting physics that brings it in the spotlight among other high temperature superconductors. One of the most important properties is the capacity to transport a high super current in an applied magnetic field. The analysis of this process showed that grain boundaries act as the main pinning structure though other mechanisms could not be neglected. The analysis of the field B and temperature T dependence of the pinning force F_{p} = J_{c} × B can provide important information on the mechanisms involved in the pinning process. For metallic, low temperature superconductors, DewHughes^{1} showed that the field dependence of F_{p} obeys the general law:
where h is the reduced magnetic field h = H/H_{c2} with H_{c2} the upper critical field and K a constant. The exponents p and q depend on the pinning mechanism and on the dimension of the pinning manifold. Table 1 shows the value of the exponents according to the DewHughes model^{1}. Moreover, the plots of reduced pinning force f_{p} = F_{p}/F_{p,max} vs. h at different temperatures, with F_{p,max} being the maximum value of F_{p}(h), would peak at \({h}_{p}=\frac{p}{p+q}\) and collapse on the same curve. However, if this scaling seems to work for some low temperature superconductors, its validity for the new classes of superconductors is unclear and the attempts to fit f_{p}(h) data using the exponents given in the Table 1 were not always successful^{2}. Many puzzling results on this topic are reported for superconducting MgB_{2}, single crystals, ceramics, and tapes^{3,4,5,6,7,8,9,10}. An analysis of the limitations of this model was presented in the Ref.^{11}. Several authors tried to circumvent this drawback using a series of the type \({f}_{p}=\sum_{i}{A}_{i}{h}^{{p}_{i}}{\left(1h\right)}^{{q}_{i}}\) with p_{i} and q_{i} from the Table 1. Besides the fact that the physics beneath such a direct summation of different mechanisms is questionable, the exponents p_{i} and q_{i} proved to be different from those predicted in Table 1^{12,13}. Ihara and Matsushita^{14} proposed a Pythagorean summation for the associated critical current density when several types of pinning contribute. In that case f_{p} is depicted as \({f}_{p}=B{\left(\sum_{i}{J}_{ci}^{2}\right)}^{1/2}\).
Considering that the pinning force is related to the critical current density, the effort was driven to find hints for the field dependence of J_{c} using different combinations of H, J_{c} and different derivatives of J_{c} leading to a linear dependence. However, these combinations seemed to work only in a limited field range, thus, introducing two or three crossover fields. If different fieldrelated regimes can be valid in superconducting cuprates, where the interplay between weak pinning, short coherence length, and long penetration depths generate different regimes of the collective pinning^{15}, it would raise difficulties regarding their interpretation in the case of MgB_{2} with a much longer coherence length and stronger pinning.
In this paper, we investigate the field dependence of the pinning force in MgB_{2} high density samples obtained by spark plasma sintering (SPS). We selected a series of samples whose field dependence of the pinning force strongly depends on the additives and on the procedures applied to the green samples. It is an attempt to find the common features of the pinning and of the reasoning behind the dependence between the parameters p, q and the temperature.
Methods
Five bulk samples of high density magnesium diboride, pure or containing small amounts of additives, were prepared by spark plasma sintering (SPS) technique. The mass density of the samples is higher than 95% of the ideal MgB_{2}. The details of raw materials, additives, preparation conditions, as well as the structure, microstructure and physical properties of the samples are presented in the references attached to each sample. The samples selected for analysis are: (i) pure MgB_{2} ^{16}; (ii) (MgB_{2})_{0.99}(Te_{0.25}(HoO_{1.5})_{0.75})_{0.01}^{17}; (iii) (MgB_{2})_{0.99}(B_{4}C)_{0.01}^{18}; (iv) weakly oriented MgB_{2} (orientation degree ~ 21%)^{19}; (v) highly oriented MgB_{2} (orientation degree ~ 40.5%)^{20}. The partial caxis orientation was induced in the green compacts of the samples (iv) and (v) by field assisted slip casting (FASC) under a high magnetic field of 12 T. The subsequent SPS procedure enhanced the orientation.
Samples were cut from the center of the sintered disc with a diameter of 2 cm and a thickness of 0.4 cm. The size of the randomly oriented samples (i)–(iii) was 1.5 × 1.5 × 0.5 mm^{3}, while the partiallyoriented samples were 1 mm^{3} cubes. The magnetization loops at different temperatures (5–35 K) of the asprepared samples were measured by using a MPMS7 T magnetometer (Quantum Design). The field dependence of J_{c} was determined with the Bean model. For all the samples, the macroscopic irreversibility field was used as the scaling field instead of H_{c2}. The irreversibility field was obtained from the field dependence of the critical current density with the criterion J_{c}(H_{irr}) = 100 A cm^{−2}.
Results
Figure 1 shows the dependence of the reduced pinning force f_{p} on the reduced field h for all five samples at the same temperature T = 15 K. Following the suggestion of Ref.^{21}, the plots of \(\frac{dln\left({f}_{p}\right)}{dh}\) vs. h are shown in the insets. They were interpreted as consisting of three linear parts which implies two crossover fields. It is worthy to note that asobtained linearity would suggest a Gausslike hdependence of f_{p}.
A closer examination of these plots shows that the position of the peak of f_{p}(h) is dependent on the samples’ features for a given temperature in a large hrange. For example, at T = 15 K, the value of h_{p} spans from h_{p} = 0.13 for the weakly oriented sample (iv) measured in perpendicular geometry (Fig. 1d) to h_{p} = 0.26 for the highly oriented one (v) measured in parallel geometry (Fig. 1e). The only samples showing a peak at a hvalue close to the theoretical one of h_{p} = 0.20 for the grain boundary pinning are (ii) and (iii) added with Te/Ho_{2}O_{3} and B_{4}C, respectively. It is remarkable that in the sample (ii) there are no substitutions in the crystal structure of MgB_{2}, while in the sample (iii) carbon supplied from B_{4}C substitutes for boron. Another observation of interest is that in the samples doped with tellurium and rare earth oxide, (MgB_{2})_{0.99}(Te_{x}(HoO_{1.5})_{y})_{0.01}^{17}, h_{p} shifts to lower values with increasing ratio y/x. For example, at T = 5 K, h_{p} = 0.15 for the sample with the composition (MgB_{2})_{0.99}(Te_{0.25}(HoO_{1.5})_{0.75})_{0.01} but h_{p} = 0.19 for (MgB_{2})_{0.99}(Te_{0.31}(HoO_{1.5})_{0.69})_{0.01}. Such values of h_{p} smaller than the theoretically predicted ones were previously reported by other groups^{22,23}.
Other features noticeable in some samples are a shoulder and/or several inflection points (Fig. 1b,d). These details are easily visible on the graph of the derivative d(f_{p)})/dh. Shoulders and inflections were also reported by other authors^{24}.
A third peculiarity is the anisotropy of f_{p} which is displayed by the partiallyoriented samples (iv) and (v). The crystallographic texture leads to a noticeable difference between the reduced pinning forces f_{p} and f_{p⊥}. These reduced pinning forces were obtained with the measuring field applied along and perpendicular to the caxis of MgB_{2}. The anisotropy is small in the case of weakly oriented sample (iv) (Fig. 1d) with the peak fields h_{p}= 0.13 and h_{p⊥} = 0.145. In the case of highly oriented sample (v), the anisotropy is stronger and the difference between the peak fields is significant with h_{p}= 0.26 and h_{p⊥} = 0.17 (Fig. 1e) (for measurement geometry see the inset 2 of Fig. 1e).
Finally, we mention the shift of h_{p} to higher values with increasing temperature. It was interpreted as a crossover to pinning on other manifolds. Though, the dominant pinning elements in MgB_{2} are the grain boundaries and obviously they do not disappear with the increasing temperature.
A first attempt to investigate the field dependence of f_{p}(h) for our samples was to start from the DewHughes assumption and to use the reduced form of Eq. (1). Parameters p and q were determined. Specifically, we plotted the logarithmic derivative \(\frac{d\left({\text{ln}}{f}_{p}\right)}{d\left({\text{ln}}h\right)}\) vs. x = h/(h − 1) which, if the assumption is correct, the plot would be linear providing the exponents p and q representing the slope and intercept, respectively. Examples of the indicated plot are shown in Fig. 2 for the samples (i)–(iii) measured at 15 K. The curves suggest the existence of at least two field regimes with a crossover at a certain field h_{c} where the slope changes. However, the asdetermined parameters p and q do not correspond to any known pinning regime. Thus, for h < h_{c}, q takes abnormally high values in the range 4 ≤ q ≤ 44, whereas for h > h_{c}, p is negative. For the samples plotted in Fig. 2, we obtained the following values: p = 1.67, q = 5.35 for h < h_{c} and p = − 1.59, q = 1.64 for h > h_{c} in pure MgB_{2} (sample (i)); p = 10, q = 44 for h < h_{c} and p = 4, q = 1.3 for h > h_{c} in (MgB_{2})_{0.99}(Te_{0.25}(HoO_{1.5})_{0.75})_{0.01} (sample (ii)); and p = 1.33, q = 5.1 for h < h_{c} and p = − 3.22, q = 1.12 for h > h_{c} in (MgB_{2})_{0.99}(B_{4}C)_{0.01} (sample (iii)).
These plots, as well as other combinations of field which were made in an attempt to obtain the linear representation suggest a complex field dependences of the pinning force. We remind that the pinning force is in fact the result of the field dependence of the critical current density J_{c}. Consequently, different, more or less evasive mechanisms were invoked to explain the field dependence. There were attempts to apply collective pinning models although their validity was proved to be correct in the case of the cuprate superconductors, but it is questionable for MgB_{2}. Actually, bulk superconductors, and especially MgB_{2}, have a very complex structure acquired during processing depending on technology specifics and on the nature of the ingredients.
A MgB_{2} bulk sample is a collection of superconducting grains which also include nonsuperconducting phases like MgO and higher magnesium borides, and voids. Moreover, the superconducting grains themselves might have defects. Among them we mention vacancies (mainly of Mg), substitutions (e.g. of C for B) and inclusions, all of them being responsible for the local critical parameters. In a magnetic field, the structural anisotropy plays also an important role because the superconducting properties of each grain depend on the orientation relative to the applied field. In this landscape, the supercurrent paths are very complex and vary with temperature and field. To approach this problem, a percolation model was developed by Eisterer et al.^{25,26,27}. According to this model, the critical current density J_{c}(H) is given by^{27}:
where J_{c,M}(H) is the maximum J_{c} for the material, p_{σ}(J) is the fraction of the dissipation free material at a given J among the superconducting grains, p_{s} is the fraction of MgB_{2}, p_{c} is the percolation threshold, \({p}_{c}^{*}={p}_{c}/{p}_{s}\) and t = 1.76. Thus, the unavoidable presence of insulating phases and voids increases the effective percolation threshold to \({p}_{c}^{*}\) which can be expressed as \({p}_{\sigma }({J}_{c,M}) ={p}_{c}^{*}\). The fraction of dissipation free material p_{σ} (J) decreases with increasing J due to the variation of the local irreversibility field from grain to grain. However, there is a minimal current density J_{c,m} below which p_{σ} = 1 so that Eq. (2) can be written as
Further, we consider a polycrystalline bulk sample made of grains with both similar anisotropy γ and superconducting properties. Consequently, the irreversibility field of each grain depends on the orientation θ relative to the applied field. For the angular dependence of the irreversibility field, Matsushita et al.^{28} proposed a dependence similar to the upper critical field, i.e., \({H}_{\text{irr}}\left(\theta \right)=\frac{{H}_{irr}\left(\pi /2\right)}{\sqrt{{\gamma }^{2}{\text{cos}}^{2}\theta +{\text{sin}}^{2}\theta }}\), whereas a more complex dependence is obtained if the zeroresistivity field is considered \({H}_{\text{irr}}\left(\theta \right)=\frac{{H}_{c2}\left(\pi /2\right)}{\sqrt{\left({\gamma }^{2}{\text{cos}}^{2}\theta +{\text{sin}}^{2}\theta \right)\left[\left({\gamma }^{2}1\right){p}_{c}^{2}+1\right]}}\)^{27}. However, the former expression is more suitable for a single grain while the latter seems more appropriate for the percolative transport. In both cases, if the pinning on grain boundary is considered, J_{c,m} and J_{c,M} are given by:
In Eq. (4b), H_{irr} is field that breaks the last supercurrent carrying path, i.e., \({p}_{\sigma }({H}_{irr}) ={p}_{c}^{*}\). Consequently, H_{irr}(0) < H_{irr} < H_{irr}(π/2) even though disconnected grains displaying irreversibility still survive in the field range H_{irr} < H ≤ H_{irr}(π/2). Considering Eq. (3), the critical force F_{p} = μ_{0}HJ_{c}, which defines the dissipation onset and which will be further called the pinning force, gets the form:
where A(p_{σ}(J), p_{c}, p_{s}) is the integrand of Eq. (2), with \({p}_{\sigma }({J}_{c,M}) ={p}_{c}^{*}\). The F_{p} depends on the real pinning through the local critical current, but, macroscopically, the nondissipative transport is controlled by percolation. Because A(p_{σ}(J), p_{c}, p_{s}) is a monotonous decreasing function of p_{σ}, hence, of J, applying the mean value theorem of integration^{29} one obtains:
where \(\stackrel{\sim }{p}\) is a value between p_{c} and p_{σ,max}(H), the maximal value of p_{σ} at a given field H, i.e., the fraction of grains for which H < H_{irr}. The J_{c,M}, is related to the macroscopic irreversibility field. The p_{max}(H) might be extracted from the angular distribution of the grains G(θ, ϕ), which gives \({p}_{\sigma }\left(\theta \right)={\int }_{\theta }^{\pi /2}{\int }_{0}^{2\pi }G\left(\theta {^{\prime}},\varphi {^{\prime}}\right){\text{sin}}\theta {^{\prime}}d\theta {^{\prime}}d\varphi {^{\prime}}\), and the angle dependence of H_{irr} if the right form of both G(θ, ϕ) and of H_{irr}(θ) is known. However, an analytical form for p_{σ}(θ) can be obtained only for a constant angular distribution^{26}.
The integrand in Eq. (5), hence, \(A\left(\stackrel{\sim }{p}, {p}_{c}, {p}_{s}\right)\) is a decreasing function of H no matter the angle distribution, number of phases or percolation thresholds. In fact, Eq. (5) is helpful to determine the high field (decreasing) part of F_{p}(H). The low field dependence raises more problems than it could suggest the simple form which appears as the second term in the brackets of Eq. (6). Dew Hughes^{1} proposed a local decrease of the shear modulus at grain boundaries that would lead to an alignment of the vortices along the boundaries. Possible plastic deformations, if appear, might lead to dissipation only if percolative channels develop^{30}. However, as the elastic moduli of the vortex lattice are also dependent on the orientation of vortices relative to the crystalline axes and anisotropy, the saturation of the synchronization is reached at different fields for different orientation and depends on the grain distribution and the presence of different superconducting phases. In the absence of a model that should describe such a complex process we propose to use a field dependent factor, similar to the efficiency factor proposed by Dew Hughes^{1}, that can be experimentally determined. In addition, the distribution of the irreversibility fields is required in real samples because the irreversibility is dependent on grain size ^{31}.
A general form for the reduced pinning force f_{p} = F_{p}/F_{p,max} in terms of reduced field h = H/H_{irr} can be obtained from the Eqs. (5) and (6) interpolated to the low field factor and averaged on grain size. In addition to the form proposed by Dew Hughes, it contains a field dependent coupling factor g(h,T) in polycrystalline samples that arise from the anisotropy of the samples and can be determined from the experimental data:
This equation has the advantage to preserve the same exponents p and q, hence, the pinning nature in the almost entire temperature range where H_{irr} (T) > 0. The function g(h,T) can account for the shift of the peak, the increase of the width, and for other peculiarities of f_{p}: these effects emerge as the consequence of the percolative nature of the supercurrent transport.
The attempts to fit f_{p}(h) experimental curves with Eq. (7) showed that g(h,T) is either a single or a double peaked function which depends on the sample composition and fabrication technique. These functions have the characteristics of a distribution function either Gaussian or lognormal. The reason for such a dependence is not clear and further investigations are required. Below, we present the data on f_{p}(h,T) (symbols) and their fits with Eq. (7) (continuous lines) above in the temperature range 5–30 K for all samples discussed above.
Figure 3a shows data for the sample (i) made of pure MgB_{2}. In this case, g(h) is a double peaked Gaussian function, \(g\left(h\right)={g}_{0}+\frac{{A}_{1}}{{\sigma }_{1}\sqrt{2\pi }}{\text{exp}}\left[\frac{1}{2}{\left(\frac{h{h}_{p1}}{{\sigma }_{1}}\right)}^{2}\right]+\frac{{A}_{2}}{{\sigma }_{2}\sqrt{2\pi }}{\text{exp}}\left[\frac{1}{2}{\left(\frac{h{h}_{p2}}{{\sigma }_{2}}\right)}^{2}\right]\) with slightly different amplitudes, A_{1} and A_{2}, and standard deviation, σ_{1} and σ_{2}, for each peak (See the inset to Fig. 3a for T = 15 K). This type of a double peaked Gaussian was also found for the more complex compositions corresponding to sample (ii) (Fig. 3b) and to the weakly oriented sample (iv) (Fig. 3d). The two samples have a different weight of each peak (see the insets to both figures). In the case of the sample (iii) doped with B_{4}C, g(h) is a single peaked Gaussian (inset to Fig. 3c).
More interesting is the case of the strongly oriented sample (v) (Fig. 4) for which g_{}(h) is a Gaussian and g_{⊥}(h) is a lognormal function \(g\left(h\right)={g}_{0}+\frac{A}{\sigma h\sqrt{2\pi }}{\text{exp}}\left\{{\left[\frac{{\text{ln}}\left(h/{h}_{p}\right)}{\sigma \sqrt{2}}\right]}^{2}\right\}\) (see the inset to Fig. 4). In the case of the (MgB_{2})_{0.99}(B_{4}C)_{0.01} sample (iii) (Fig. 3c) and of the strongly textured sample (v) (Fig. 4), the use of only a single peaked distribution function can be roughly understood as a result of the grains orientation. The need of a double peaked g(h) in the case of the samples (i), (ii), and (iv) might indicate the presence of two types of MgB_{2} grains with slightly different intrinsic properties (anisotropy, local irreversibility field). For example, such phases can result from gradual spatial distribution of carbon (intended or accidental doping) due to its diffusion from the grain boundaries to the core of the MgB_{2} grains. The fitting parameters for all samples as determined at 15 K are given in Table 2.
Equation (7) explains in a consistent way the peculiarities of the hdependence of the derivative \(\frac{d\left({\text{ln}}{f}_{p}\right)}{dh}\) and the shape of \(\frac{d\left({\text{ln}}{f}_{p}\right)}{d\left({\text{ln}}h\right)}\) vs. x = h/(h − 1) curves as were shown in the Insets to Fig. 1 and in Fig. 2, respectively. Thus, Fig. 5a and b show the plots of \(\frac{d\left({\text{ln}}{f}_{p}\right)}{d\left({\text{ln}}h\right)}\) vs. x = h/(h − 1) for the nonoriented samples as obtained with Eq. (7).
We mention that our procedure encounters difficulties around h ~ 1, i.e., for applied fields in the vicinity of H_{irr} where the data are scattered and the result is uncertain. Additional phenomena also must be taken into account close to H_{irr} where creep is strongly emphasized and proliferation of nonsuperconducting areas occurs.
In literature, the use of a distribution function was proposed to represent the voltagecurrent characteristics of high temperature superconductors. Namely, in Refs.^{32} and^{33,34} the distribution functions to describe the local critical current density were of a Gaussian or Weibull type, respectively.
Conclusion
We have shown that the reduced pinning force f_{p} dependence on the reduced field h can be described in the case of polycrystalline bulk samples by the model of pinning on grain boundaries. A connecting function is associated and it arises from the peculiar structure of each sample.
At high fields, this function is the result of the percolation processes that are characteristic for the samples with intrinsic anisotropy and distribution of the orientation of the grains. It also mirrors the local properties of the grains as they result from their size, stress, doping, and inclusions. At lower fields, the manifestation of polycrystallinity was included in a field dependent factor similar to the efficiency factor used to illustrate the pinning in isotropic materials.
These properties are typical for sintered MgB_{2} samples, but the model might be suitable and applied also to other superconductors. The proposed model preserves the framework of the grain boundary pinning. It also removes the putative crossovers inferred from the behavior of different combinations of the field, current, and/or of their derivatives as well as the need for the models consisting of the summation of different pinning mechanisms.
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Acknowledgements
This work was supported by UEFISCDI Romania through Core Program PN1903 (contract no. 21 N/08.02.2019) and the project POC 37_697 no. 28/01.09.2016 REBMAT. M.A.G. also acknowledges the support from the Operational Programme Human Capital of the Ministry of European Funds through the Financial Agreement 51668/09.07.2019, SMIS code 124705. M.B. thanks also for financial support provided through the Project PNIIIP11.1PD20190651 SUPRASHAPE.
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Conceptualization V. S.; methodology, V. S., A. M. I., G. A., M. A. G., M. B., P. B.; validation, P.B., G.A.; formal analysis, V. S., A. M. I., G. A., M. A. G., M. B., P. B.; investigation, V. S., A. M. I., G. A., M. A. G., M. B., P. B.; resources, P.B.; writing—original draft preparation, V. S.; writing—review and editing, V. S., P. B., and G. A.; visualization, V. S. and M.A.G.; supervision, P.B.; project administration, P.B.; funding acquisition, P.B.
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Sandu, V., Ionescu, A.M., Aldica, G. et al. On the pinning force in high density MgB_{2} samples. Sci Rep 11, 5951 (2021). https://doi.org/10.1038/s41598021852092
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DOI: https://doi.org/10.1038/s41598021852092
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