On the origin of the sharp, low-field pinning force peaks in MgB2 superconductors

Various MgB2 thin films and single crystals were found in the literature to exhibit a sharp, narrow peak at low fields in the volume pinning force, Fp(H)-diagrams. The origin of this peak is associated with a steep drop of the current density when applying external magnetic fields and is ascribed to sample purity. We show here that bulk MgB2 prepared by spark-plasma sintering also shows the sharp, narrow peak in Fp. The peak is also seen in the volume pinning force scaling, Fp/Fp,max vs h = H/Hirr. Furthermore, polycrystalline bulk MgB2 samples prepared close to the optimum reaction temperature reveal this peak effect as well, but other samples of the series show a regular scaling behavior. The combination of magnetization data with data from electric transport measurements on the same samples demonstrates the origin of this peak effect. On increasing preparation temperature, the pinning force scaling changes from grain boundary pinning to point pinning and the grain connectivity gets worse. Hence, the sharp, low-field peak in Fp vanishes. Therefore, the occurrence of the peak effect in Fp gives important information on the grain coupling in the MgB2 samples. © 2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5133765., s


I. INTRODUCTION
The metallic superconductor with the highest transition temperature, MgB 2 , is considered as a candidate material for many applications such as trapped field magnets, tapes, and wires due to its simplicity and the simple preparation route possible, even though its Tc is much lower than most of the high-Tc superconductors. 1,2 Even in sintered, polycrystalline MgB 2 samples, it was found to be possible to establish a current flow through the entire sample perimeter as the grain boundaries (GBs) do not act as weak links, but as flux pinning sites, e.g., in Nb 3 Sn. 3,4 Together with modern cryocooling techniques, the lower Tc does not pose a serious problem. 5 On the other hand, MgB 2 has its own characteristic magnetic behavior including dendrite-like flux penetration, 6 flux jump effects, 7 and peculiar flux pinning properties, 8 which may influence possible applications. Therefore, these points must be understood properly.
For the flux pinning behavior, the analysis of the flux pinning forces Fp = jc × B (where jc denotes the critical current density, and B = μ 0 H) using the scaling approach by Dew-Hughes (DH) 9 and Kramer 10 is an important issue. The flux pinning in MgB 2 is mainly provided by GB-pinning and point-pinning. Two peculiar observations of the flux pinning analysis on MgB 2 include (i) the change of the main acting pinning mechanism with increasing temperature found in polycrystalline samples prepared with varying reaction temperatures 11 and (ii) the appearance of a sharp, narrow peak in the Fp − H diagrams at relatively low applied fields. 12-21 Such a peak was observed first in MgB 2 single crystals 22,23 and then in thin films. [24][25][26] The occurrence of this peak at relatively low fields was ascribed to the strong decrease in jc(B) due to sample purity.

ARTICLE scitation.org/journal/adv
In this contribution, we analyze magnetization data obtained on a spark-plasma sintered (SPS) MgB 2 sample with a highly dense microstructure 27 and compare the results with a series of sintered, polycrystalline bulk MgB 2 samples prepared with varying reaction temperatures. 28 On all samples, we have performed electric transport measurements, 31,32 the behavior of which enable us now to clarify the origin of the peak found in the Fp-diagrams.

II. EXPERIMENTAL PROCEDURE
The MgB 2 sample was prepared by SPS at 1200 ○ C and a uniaxial pressure of 50 MPa. The sample reached a density of 2.61 g/cm 3 (99.2%). 27 Our data are compared to sintered, polycrystalline MgB 2 samples fabricated in the solid state reaction in a pure Ar atmosphere with reaction temperatures between 750 ○ C and 950 ○ C. 28 For superconducting quantum interference device (SQUID) measurements, small samples (2 mm × 2 mm × 1.5 mm) were cut from the big pellets. For the resistance measurements, small bars (10 mm × 1 mm × 1 mm) were prepared and the contacts were fixed with silver paint, ensuring a 5 mm distance between the voltage pads. The magnetic characterization measurements were performed using SQUID magnetometry (Quantum Design MPMS3), and the magneto-resistance characteristics were recorded using an Oxford Instruments 8 T Teslatron system. The critical current densities were calculated from the magnetization loops using the extended Bean model for rectangular samples. 29 Electron backscatter diffraction (EBSD) and transmission electron microcopy (TEM) were performed on TEM-slices prepared using focused ion-beam milling. Details of these procedures are given in Ref. 30 and the supplementary material.

III. RESULTS AND DISCUSSION
Figures 1(a) and 1(b) present jc as a function of the applied field, μ 0 Ha (a), and the volume pinning force, Fp, as a function of the field (b) of the SPS sample at various temperatures. In (b), the definitions of H peak denoting the position of the sharp Fp peak and Hcross marking the crossover field after the peak are given. The peak and shoulder shape of the Fp(H) curves is observable up to 30 K. The inset to Fig. 1(a) gives the jc(H, T) data in a double-log plot. Here, two regions are clearly distinguishable, as indicated by the dashed lines. The flat region at low fields can be considered to be the singlevortex pinning region, whereas the connected sloped region is where weak collective pinning is dominant. [33][34][35] The values of the exponent β in jc(H) ∝ H −β in the sloped region are around 1.0 at 5-25 K, 1.15 at 30 K, and 1.2 at 35 K, indicating that the decreasing slope of jc(H) gradually becomes steeper at temperatures above 25 K due to thermal fluctuation effects. 36 The signature of the sharp Fp-peak is clearly visible in (a) and (b), whereas only traces are seen in (c), and in (d), the Fp − H behavior is fully regular. The pinning force scaling analysis is based on the approach of DH. 9 The scaled pinning force data f = Fp/Fp,max vs h = Ha/H irr can be fitted to the functional dependence given by A being a numerical parameter, and p and q are describing the actual pinning mechanism. The position of the maximum in the Fp plot, h 0 , is given by p/(p + q). For MgB 2 , the pinning at GBs (p = 0.5 and q = 2) and the pinning at non-superconducting point pins (p = 1, q = 2) are the most important functions. Figure 2(e) gives the DH-scaling of the SPS sample. The pinning force scaling is well developed, and the fit (red line) to all data yields a peak position h 0 = 0.21, which clearly points to a dominant flux pinning provided by extended defects, i.e., GBs. The pinning force scaling of all samples studied is given in Fig. S1 of the supplementary material. Now, we turn to the origin of the peak effect in the Fp(H)curves. From the literature, we can extract the following findings concerning the sharp, narrow peak in Fp(H): (i) Single-crystals and high-quality thin films exhibit the specific Fp-curve shape. (ii) Sintered, polycrystalline samples, normal thin films, and powders do not show any peculiar Fp-curve shape. (iii) The peak shows a clear temperature dependence; therefore, matching effects are excluded. (iv) The peak is located at relatively high fields, which implies that surface effects are excluded. The peak positions, Hp, and the respective crossover fields, Hcross were extracted from the Fp-data. Table I 38,39 and so no direct observations of the flux patterns around H peak are possible.
The first finding of the present work is that sintered, polycrystalline samples prepared at reaction temperatures close to the optimum value (i.e., ∼800 ○ C 28 ) do show the sharp Fp-peak, whereas all other samples prepared at higher T do not. The SPS sample with its high density also exhibits the sharp Fp-peak.
The electric transport measurements elucidate the grain connectivity K, which plays an important role for the current flow through the sample (see Table I). K is calculated from the resistance data obtained via K = Δρg/Δρ, with Δρg = 6.31 μΩ cm for randomly oriented 3D samples, and Δρ = ρ(300 K) − ρ(40 K). 40 In our series of samples, K was found to increase steeply with increasing reaction temperature from 775 ○ C to 805 ○ C and then decreases again when increasing the temperature up to 950 ○ (see Fig. S2 of the supplementary material and Ref. 32). The SPS sample shows K = 0.96 due to its high density.
Finally, Fig. 4 gives the analysis of the scaled flux pinning curve for one selected sample (S775). To elucidate the acting flux pinning mechanism, one could combine two of the pinning mechanisms of DH and try to fit the data using a linear combination. It is straightforward to assume that the GB pinning is related to the low-field Fp-peak. This is symbolized by the red dashed line (fit 1), obtained using the fitting parameters p = 1.17 and q = 4.98. The resulting fit is perfect until Hcross is reached. Now, we subtract this fit from the experimental data and rescale them with a new Hmax. The result of this procedure is drawn with a blue line. Finally, these data are again fitted using a DH function with p = 3.04 and q = 1.76 (fit 2). All this implies that we can fit the experimental data showing the sharp, low-field peak with a combination of two DH pinning functions. The low-field peak in Fp is caused by the GB pinning, whereas the large shoulder is due to pinning at point defects.
Together with the information from the electric measurements revealing the grain coupling, we can now come to the conclusion for the origin of the Fp-peak.
As described before, one of the advantages of MgB 2 is the current flow through the entire sample perimeter even in polycrystalline samples. This implies that the currents run across various GBs, affected by the crystallographic orientation of the grains. The first point enables pinning at GBs, which is beneficiary, e.g., in Nb 3 Sn. Due to the second point, the currents and flux pinning forces are affected by the given anisotropy of the material. 41 Following the calculations of Eisterer, 42 the position of the GB pinning peak may be obtained in the range between 0.1 and 0.2, which is, indeed, the case for the sharp, low-field peaks observed here. In this range, the dominating flux pinning is provided by the GBs, and the current flow is affected by the crystal anisotropy and percolation. When a field-induced decoupling occurs, the currents will only cycle in small regions with the minimum size of an individual grain. Then, anisotropy and percolation do not play the important role anymore, and we will have a situation corresponding to polycrystalline YBa 2 Cu 3 Ox. As there is no texture of the MgB 2 grains, the differences in the local currents will average out and we obtain a peak in the Fp diagram for point pinning at h ≈ 0.33, following a simple DH pinning function with p = 0.52 and q = 0.98. 11 To observe the sharp, narrow Fp-peak, it is essential that the sample is free of additions providing pinning at small, nonsuperconducting inclusions, as this would strengthen the secondary pinning mechanism. The GB pinning may then only appear as a small shoulder on the low field side, if at all. Furthermore, the sample preparation must ensure that the MgB 2 grains are well coupled (i.e., having a high grain connectivity). Such situation is realized in the SPS sample studied here. is clean, while others are obscured by MgO and MgB 4 particles, as demonstrated in Ref. 30. The decay of the current flow will then not influence the Fp diagram that much as the peak positions are close to each other. In this sense, the previous explanation that only pure samples will exhibit the sharp, narrow Fp-peak is correct.
Table I demonstrates, further, that the peak field, H peak , scales with the sample size when comparing the thin film data with ours. Therefore, this effect will not influence the currently achieved trapped fields in the bulk MgB 2 samples (typical dimensions of 3-5 cm diameter and thickness of 10-20 mm) as the current trapped field values are still relatively low. 18,[43][44][45] However, when attempting to trap much larger fields, then the change of current flow at H peak will pose a serious problem as one would obtain not one large central trapped field peak, but several smaller ones when the current flow is reduced to smaller islands. This situation would get worse the higher the applied field would be. Hence, H peak is a limiting factor for the maximum achievable trapped field. As the grain connectivity cannot be tuned to be much better (the sharp peaks are seen in single crystals, thin films, and the SPS sample), the only way out is to improve the flux pinning inside the MgB 2 grains. Therefore, an ideal MgB 2 sample for applications needs small, nano-sized grains with embedded, strong flux pinning provided by nano-sized pinning centers. In this case, the sharp, low-field peak will be overwhelmed by the flux pinning provided by point-like defects.

IV. CONCLUSIONS
To conclude, we have performed magnetic and electric measurements on a series of sintered, polycrystalline MgB 2 samples. The samples prepared using a low reaction temperature and the highdensity, spark-plasma sintered samples exhibit the sharp, low-field peak in Fp, while the other samples do not. The electric transport measurements performed on the same samples reveal a change of the grain connectivity on increasing the preparation temperature, whereas the highly dense, spark-plasma sintered sample shows a well-developed grain connectivity. As a result, from this combination of magnetic and electric measurements, we can conclude that the change of length scale of the current flow at H peak is responsible for the Fp-peak. When the grains are well coupled together, the current flow through the entire sample perimeter is enabled. In this case, the flux pinning is mainly provided by GB pinning, and the crystal anisotropy and percolation play a role. In case the coupling of the MgB 2 grains is worse, we only observe the flux pinning provided by small, non-superconducting inclusions, and the current flow is reduced to grain clusters or grains. This finding may have important consequences for the achievable trapped fields in MgB 2 .

SUPPLEMENTARY MATERIAL
See the supplementary material for specific information about the pinning force scaling of all samples, the grain connectivity, and experimental details about the EBSD analysis.