Enhanced trapped field in MgB₂ bulk magnets by tuning grain boundary pinning through milling

The MgB₂ superconductor, discovered in 2001, has the highest critical temperature T_c (∼40 K) among metallic superconductors [1]. MgB₂ exhibits unique superconducting characteristics, such as multi-gap superconductivity with Δ_π(0) = 2.2 meV and Δ_σ(0) = 7.1 meV [2, 3], a high upper critical field μ₀H_c2^//ab(0) ∼ 48 T and μ₀H_c2^//c(0) ∼ 33 T with small anisotropy ∼1.5–4.5 [4, 5], and a weak-link free transport critical current in polycrystalline forms [6]. A high critical current density J_c has been developed in polycrystalline bulks, tapes, and wires, and currently over 10⁵ A cm⁻² is realized in the temperature range of 4–30 K using various fabrication processes [6–11]. Because MgB₂ is a simple binary line-compound composed of light and abundant elements, magnesium and boron, and can be operated at cryogen-free temperatures, MgB₂ is one of the most promising candidates for cryo-magnet applications.

Superconducting bulk magnets are a direct manifestation of quantum phenomena on the macroscopic scale. The origin of the trapped field in superconducting bulk magnets is the circulating current, which can be remotely induced by magnetization below T_c. As long as the bulk magnets are kept cold, they continue to trap magnetic field and can be used as tesla-class quasi-permanent magnets. Superconducting bulk magnets should be characterized by compactness and very high field strength, principally because their equivalent current density can be higher than those of coils and spin-based permanent ferromagnets. These characteristics yield unique magnet applications for measurement, transport, and energy [12]. Rare-earth barium copper oxide (REBCO) melt-solidified, single-grain bulks have been studied as a model material for superconducting bulks, and high trapped fields of 2–3 T at 77 K and over 10 T at lower temperatures has been achieved by controlling the flux pinning and mechanical strength [13, 14]. In single-grain REBCO bulks, the predominant flux pinning centers are intra-grain precipitates, such as RE₂BaCuO_y (RE211) and artificially introduced defects [15, 16].

For polycrystalline MgB₂ bulks, the trapped field is spatially and temporally uniform because of the weak-link-free current over nano-scale grain-boundaries [17]. Recently, the record of trapped field in bulk MgB₂ has been rapidly increased by applying high pressure and densification processes [18–25]. Given the nature of polycrystalline, metallic MgB₂, strategies for enhancement of the trapped field are different than for REBCO. Multi-band transport, connectivity, and grain boundary flux pinning are known major current limiting mechanisms in polycrystalline MgB₂. First, in randomly oriented polycrystalline forms of MgB₂, the different dimensionality of the σ and π bands should affect the intergranular current transport [3, 26] but has not been well clarified yet. Second, in addition to intrinsic anisotropic multi-band transport, current percolation due to electromagnetic anisotropy [27] and structural defects [28] restricts the macroscopic current transport, which is known as the connectivity problem [29]. In this sense, the fabrication of dense MgB₂ bulks by high pressure [18–23] or infiltration growth [24, 25] techniques, which is currently the main focus, is favorable for the trapped field by increasing the connectivity and grain boundary density. Finally, impurity doping, such as with carbon-containing compounds [30–38], ball milling [39–41], and processing optimization [42–44], have been observed to be effective in enhancing the flux pinning of MgB₂ either by introducing additional pinning centers and/or strengthening the original grain boundary pinning.

In the present study, the tuning of natural pinning centers, i.e., grain boundaries, was performed to investigate the impact on the trapped field of polycrystalline bulk MgB₂. MgB₂ bulk samples with systematically varied grain size were prepared by ball milling, and their structural, transport, and magnetic properties were evaluated.

Disk-shaped MgB₂ bulk samples, with 30 mm diameters and 10 mm thicknesses, were synthesized using a modified powder-in-closed-tube technique [45]. Raw powders of magnesium (325 mesh in size, 99.9% purity) and boron (300 mesh, 99%) at a molar ratio of 1:2 were mixed and pulverized by ball milling using a tungsten carbide media for 1 h with a rotation speed of 200–600 rpm. The milling energy was estimated according to Häßler's model [39]. Here, 0 MJ kg⁻¹ of ball milling implies that the raw powder was manually pulverized with an agate mortar. The milled powder was uniaxially pressed to form a disk-shaped pellet. To prevent the magnesium from oxidizing, the pellet was heat treated in a tube furnace at 850 °C for 3 h under an argon atmosphere with ambient pressure. MgB₂ rectangular small specimens with dimensions of ∼1 × 3 × 10 mm³ were synthesized using an identical process for microstructural analysis and local electromagnetic property measurements.

The constituent phases of the samples were analyzed using powder x-ray diffraction (XRD) using Cu–K_α radiation, and microstructural observations were performed on the cross-sectional polished surfaces using scanning electron microscopy (SEM; JEOL: JSM-7001F). The mass density of the samples was determined using a micrometer caliper and a weighing balance. Transport measurements were performed using the conventional four-point probe method with a current density of ∼1 A cm⁻² using a physical property measurement system (Quantum Design: PPMS Model 6000). The normal-state electrical connectivity K was estimated using the equation K = Δρ_g/Δρ, where Δρ_g = ρ_g(300 K) − ρ_g (40 K) ≡ 6.32 μΩ cm [28] and Δρ = ρ(300 K) − ρ(40 K), which are the differences in the resistivity of ideal MgB₂ grains and that of the sample, respectively. The upper critical field (H_c2) and irreversibility field (H_irr) were determined at 90% and 10% of the ρ(T, B) transition curves, respectively. Magnetization was measured using a SQUID magnetometer (Quantum Design: MPMS-XL5s). J_c of the bulk samples with typical dimensions of 1 × 1 × 0.5 mm³ was calculated from the width of the magnetization hysteresis loops based on the extended Bean model using the equation J_c = 20ΔM/a(1 − a/3b), where ΔM = M⁺ − M⁻ is the hysteresis loop width, and a and b (a ≤ b) are the dimensions of the rectangular sample. The trapped field as a function of temperature for the MgB₂ bulks was measured at the center of the surface of the bulk samples using a Hall sensor (HGCT-3020, Lake Shore). Magnetization of MgB₂ bulks was performed using the field-cooling magnetization technique. The MgB₂ bulks were cooled by a Gifford–McMahon cryocooler (CRTHE05-CSFM, Iwatani Gas) to the measuring temperature (5, 10, and 15 K) in a 6 T field, followed by the removal of the external field.

3.1. Structural properties

First, we investigated the effects of ball milling of the raw powder on the constituent phases, lattice parameters, lattice distortion, and grain size of the MgB₂ bulk samples. The powder XRD patterns for MgB₂ samples synthesized from powder with different milling energies are presented in figure 1. Nearly single phase MgB₂ with a small amount of MgO impurity was obtained for all the samples. Among the XRD reflection peaks, the (002) peak, which originates from inter-plane diffraction, and the (110) peak, which originates from in-plane diffraction, exhibited a representative change with milling. The full-width at half-maximum (FWHM) of both the (002) and (110) peaks increased with milling, especially below ∼5 MJ kg⁻¹ (figure 2(a)). In addition to the peak broadening, a systematic shift of the (110) peak was observed (figure 1(b)). Figure 2(b) shows the evolution of the a- and c-axis lattice parameters with milling. Shortening of the a-axis was observed, whereas the c-axis lattice parameter was unchanged.

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The results for the in-plane lattice distortion indicate that unintentional carbon substitution on the boron site occurs during milling. Indeed, T_c systematically decreased from 38.4 to 33.2 K with milling (figure 7(a)). The a-axis lattice parameter dependence of T_c for the milled samples fits well with previously reported carbon-doped MgB₂ polycrystalline bulks [31] and single crystals [46] (figure 2(c)). The estimated carbon substitution level of each sample is listed in table 1. The broadening of the XRD peaks originates from the refinement of the crystal grains and/or an increase in lattice distortion. The more prominent increase in the FWHM of the (110) peak compared with that of the (002) peak indicates that the milling process mainly introduced in-plane lattice distortion rather than inter-plane distortion because of carbon substitution [31]. Therefore, we conclude that the ball-milling process induced combined structural effects of crystal size refinement and in-plane lattice distortion by carbon substitution. The contamination of carbon is considered to originate in the milling medium, i.e., tungsten carbide media and/or organic solvent.

Table 1.  Structural parameters and normal-state transport properties of the MgB₂ samples with different milling conditions: a-axis lattice parameter, estimated actual carbon content in lattice (x_A), resistivity at 40 K, connectivity-compensated residual resistivity (ρ₀), RRR and connectivity (K). x_A was calculated using the equation x_c = 2.38 × (3.087 − a) [31], where a is the a-axis lattice parameter in Å.

Milling energy (MJ kg−1)	a (Å)	xA (%)	ρ40 K (μΩ cm)	ρ0 (μΩ cm)	RRR	K (%)
0	3.0867	0.07	15.4	2.06	3.99	13.4
0.61	3.0835	0.83	21.9	3.77	2.67	17.2
2.1	3.0814	1.3	33.6	5.71	2.11	17.0
4.9	3.0810	1.5	44.0	8.89	1.71	20.2
9.6	3.0781	2.1	59.2	9.95	1.64	16.8
17	3.0752	2.8	67.7	12.3	1.51	18.1

Secondary electron images of the MgB₂ bulk samples for different milling conditions are presented in figure 3. The high-magnification images (figures 3(a), (b), and (c)) reveal that the MgB₂ grain size systematically decreased with milling from ∼1.4 to ∼0.4 μm and to ∼0.3 μm for the sample milled with 0, 2.1, and 17 MJ kg⁻¹, respectively. The reduced grain size by milling agrees well with the XRD analysis and yields 3- and 4-fold increases in the grain-boundary density for the samples milled with 2.1 and 17 MJ kg⁻¹, respectively, compared with the un-milled sample. The lower magnification images in figures 3(d), (e), and (f) clearly demonstrate that the size and distribution of the voids changed by milling. The average size of the voids decreased from ∼20 μm (un-milled) to ∼3 μm for the sample milled with 2.1 MJ kg⁻¹ and to ∼0.6 μm for the sample milled with 17 MJ kg⁻¹. Consequently, the macroscopic distribution of voids became uniform with milling. The improved macroscopic structural uniformity is believed to result from the refinement and homogenization during milling. However, the density of the MgB₂ samples did not change by milling and was 1.4–1.5 g cm⁻³, corresponding to a relative density of ∼55%.

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3.2. Transport and magnetic properties

The transport measurements revealed that the residual resistivity (ρ_{40 K}) increased, and the relative resistance ratio (RRR) decreased with milling (table 1). The connectivity-compensated residual resistivity ρ₀ = Kρ_{40 K} [9, 28] and corresponding impurity parameter α_i for the samples milled with 0, 2.1, and 17 MJ kg⁻¹ were 2.1, 5.7, and 12.3 μΩ cm and 0.13, 0.35, and 0.76, respectively, suggesting enhanced electron scattering due to lattice distortion. However, the connectivity slightly increased with milling presumably due to improved microstructural uniformity, mainly by a reduction of large voids, which would increase the current percolation length. The temperature dependence of H_c2 is shown in figure 4. $\left| {\rm d}{{H}_{{\rm c}2}}/{\rm d}T \right|$ increased with milling, and H_c2 in the lower temperature region dramatically increased because of increased electron scattering.

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The external field dependence of J_c for the milled MgB₂ samples at 5 and 20 K is shown in figure 5. The self-field J_c improved with milling and reached a maximum at 2.1 MJ kg⁻¹ (figure 7(c)). A more prominent enhancement of J_c by milling was observed under high field. J_c under 3 T, as a representative example of in-field J_c, exhibited 2.8- and 8-fold increases at 5 and 20 K, respectively, for the sample milled with 2.1 MJ kg⁻¹. Elementally, the pinning strength of grain boundaries is enhanced by carbon substitution [31]. Thus, the observed improvement in J_c is considered to be due to combined effects of carbon substitution and grain refinement, i.e., qualitative and quantitative enhancement of grain-boundary pinning.

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3.3. Field-trapping properties

The temperature dependence of the trapped field (B_T) for the magnetized MgB₂ bulks is shown in figure 6. B_T near T_c decreased with milling because of the reduction of T_c by carbon substitution; however, the slope of B_T, $\left| {\rm d}{{B}_{{\rm T}}}/{\rm d}T \right|,$ increased with milling. The B_T of the bulk magnets milled with 0.61, 2.1, and 4.9 MJ kg⁻¹ showed crossover at 33.8, 31.4, and 26.3 K, respectively, with that of 0 MJ kg⁻¹, and these bulk magnets exhibited higher B_T at low temperatures. The trapped field of the sample with 2.1 MJ kg⁻¹ was the highest and reached 3.72 and 2.32 T at 5 and 20 K, respectively, at the center of the bulk surface (figure 6). The milling-energy dependence of B_T at 20 K is shown in figure 7(d). The improvement of the trapped field with lower energy milling is believed to originate from the enhancement of the in-field J_c due to enhanced grain boundary pinning (figure 7(c)). As demonstrated in figures 7(a) and (b), T_c and the irreversibility field exhibit opposite trends with milling. Consequently, the optimum J_c was determined by the balance between T_c reduction and pinning-strength enhancement by increased electron scattering. However, the decrease in the trapped field by higher energy milling is considered to be due to the reduction of T_c.

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The above results indicate that simultaneous size reduction and introduction of disorder on MgB₂ grains contributed to increase the trapped field through tuning of the grain boundary pinning by enhancing pinning strength and number of grain boundaries. The trapped field of 3.72 T obtained in the present study is the highest among the reported MgB₂ bulks synthesized under ambient pressure. Indeed, the Bean model indicates that the total magnetic flux of the bulk, ${\Phi}=\iint \frac{x}{r}g{{B}_{{\rm T}}}{\rm d}\theta {\rm d}x$ (r is radius of a bulk and g is geometrical factor and ∼1.35 [17] for the present case and ∼1 for 'bulk pair' geometry), is estimated to be 740 μWb at 20 K, which is higher than 400–500 μWb of those prepared by the high-pressure technique reported in [18, 19, 21] because of the high B_T and large volume. This result corresponds to ∼4 × 10¹¹ fluxons trapped in the bulk with Abrikosov triangular lattice spacing of ∼27 nm at the center of the bulk. The average grain size of the milled sample, ∼300–400 nm, is an order of magnitude larger than the flux line spacing, which indicates that the flux lines are collectively pinned rather than individually pinned by strong pinning centers. Given that 1–2 orders of magnitude higher J_c is realized in wires and films [7] with finer grain size (<∼100 nm) and higher H_c2, further improvement of the trapped field is expected with grain size refinement and tuning of the inter- and intra-band electron scattering rates without losing the high T_c. Because polycrystalline bulk MgB₂ can be fabricated without texturing and a high density of natural pinning centers can be obtained using rather simple techniques, i.e., ball milling and pressureless heating, a high trapped field can be attained in a reproducible and scalable manner, making MgB₂ bulk magnets advantageous for novel and unique cryo-magnet applications.

To investigate the effect of pinning centers for the trapped field of MgB₂ bulk magnets, the grain size of MgB₂ samples was controlled by ball milling of the raw powder. Grain size refinement and in-plane lattice distortion by carbon substitution proceeded with milling. The grain refinement and carbon substitution induced J_c improvement, and the highest J_c among the MgB₂ sample with milling represented a several-fold increase under self- and high-field compared with the MgB₂ sample without milling. The highest trapped field of the MgB₂ bulk magnets in this study was 3.72 T at 5 K, which was 30% higher than that of the bulk magnet without milling. This result suggests that tuning the natural pinning center of MgB₂ is an effective method for enhancement of the trapped field.

The authors gratefully acknowledge A Ishihara, T Akasaka and M Tomita (Railway Technical Research Institute) for helpful discussions and S Ohtsuka and T Moroyama (The University of Tokyo) for their assistance with the SEM analysis. This work was supported by Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (Grant Nos. 22860019, 23246110 and 24656368) and by the Japan Science and Technology Agency, PRESTO. The microstructural analysis was conducted at the Center for Nano Lithography & Analysis, The University of Tokyo, which is supported by the Ministry of Education, Sports, Culture, Science, and Technology, Japan.