High magnetic shielding properties of an MgB₂ cup obtained by machining a spark-plasma-sintered bulk cylinder

MgB₂ commercial powder was mixed with hexagonal BN (BNh, henceforth). The mixture was loaded into a graphite die with 20 mm inner diameter and processed by spark plasma sintering at 1150 °C for a dwell time of 8 min. The maximum pressure applied on the sample during sintering was 95 MPa. More details of the fabrication process are reported in [30]. The as-prepared cylinder (25 mm in height) was fully machinable [31] and, at first, it was partially bored by using drill bits and then it was refined in the final cup-shape by means of a lathe machine. Two pictures of the final product (simply named cup, henceforth) are shown in figure 1. Its geometrical parameters are specified in the figure caption and its aspect ratio, defined as the ratio of the external height over the outer radius is 2.2.

Zoom In Zoom Out Reset image size

The structural analysis of the sample was carried out with a Bruker-AXS D8 ADVANCE diffractometer (CuK_α1 radiation, λ = 1.5406 Å). Phase concentration, crystallite size and lattice parameters of MgB₂ were extracted from Rietveld analysis. The bulk density of the full cylinder before shaping into cup was determined by the Archimedes method using toluene as weighting medium.

Magnetic measurements were performed with a physical properties measurement system (PPMS 14 T, Quantum Design, US) on a sample with dimensions 1.5 cm × 1.5 cm × 0.5 cm cut from residual parts obtained in the shaping process of the as-spark plasma sintered (as-SPSed) cylinder. Transition temperature, T_c, is defined as the onset temperature of the superconducting transition in the curve of magnetisation versus temperature, M(T), measured applying a magnetic field μ₀H_appl = 10 mT after zero-field-cooling. From magnetisation loops carried out at different temperatures the critical current density, J_c(H), curves were extracted by using the Bean model [32]. The irreversibility field, H_irr, obtained from the same measurements, was defined as the field at which J_c is 10² A cm⁻².

For the shielding experiment, the cup was placed in tight thermal contact with the second stage of a cryogen-free cryocooler. More details on the experimental setup are reported in [18]. First, the cup was cooled in zero field down to the working temperature and then a homogenous dc field was applied either parallel or perpendicular to the sample's axis with an average rate of 1 × 10^–4 T s⁻¹. The local magnetic flux density was measured by means of five cryogenic Ga–As Hall probes [33] located along the cup's axis as sketched in figure 2. The probes were always oriented to measure the component of the magnetic induction parallel to the applied field (i.e. B_z).

Zoom In Zoom Out Reset image size

Figure 3 compares the x-ray diffraction (XRD) patterns of the starting MgB₂ raw powder and corresponding spark-plasma sintered sample added with BNh. In the starting powder, besides the MgB₂ main phase, impurity phases of MgO and Mg were present. Their weight concentration is summarized in table 1, together with the average crystallite size. It is worth mentioning that the MgB₂ phase concentration in the raw powder and in the SPSed bulk was about 10% higher than in the case of the cylinder sample investigated in [18]. We also note that the ratio among secondary phases in the raw MgB₂ powder is much different in this work compared to the previous one [18]: for MgO it is 1.8 against 4.6 wt%, for MgB₄ 0 against 7.1 wt%, and for Mg 1.2 against 0.3 wt%, respectively. The lower amount of non-superconducting phases, such as MgO, in the sample from this work is expected to positively affect J_c since the presence of impurity phases reduces the supercurrent effective cross-section [34].

Zoom In Zoom Out Reset image size

The density of the final bulk sample was 2.50 g cm⁻³. Since the theoretical density of the composite is 2.62 g cm⁻³, as determined by considering the wt% of the component phases and their theoretical densities [30], it results a relative density of ∼95%, indicating a good grain packing.

The normalised magnetisation of the sintered bulk displays a very sharp superconducting transition, as reported in the inset of figure 4. The transition temperature is 38.9 K. This high value of T_c indicates the lack of chemical reactions between MgB₂ and BNh powders [35]. Moreover, we argue that possible MgB₂ contamination with carbon from the graphite mould system during SPS processing can also be neglected since C entering on B sites strongly reduces T_c [36].

Zoom In Zoom Out Reset image size

The main frame of figure 4 shows the critical current density calculated from magnetic hysteresis loops as a function of the external field and for temperatures ranging from 5 to 35 K. The self-field J_c0 (figure 5(a)) reaches values of 4.0 × 10⁵ and 2.7 × 10⁵ A cm⁻² at T = 5 and 20 K, respectively, which are comparable or slightly higher than J_c measured on undoped MgB₂ samples used for practical applications as bulk magnets [37, 38]. Moreover, both J_c0 and in-field J_c are higher than for machinable SPSed sample from [18], demonstrating the high influence of the raw MgB₂ powder on samples quality and performance. However, these values are lower than the highest ones for non-machinable optimally-doped SPS MgB₂ [39]. Indeed, the price to pay to obtain a fully machinable material, which is crucial for applications, is to have slightly lower material's performance in terms of critical current density and correlated parameters, such as the quality product (${J}_{c0}\times {\mu }_{0}{H}_{\mathrm{irr}}$—see figure 5(b)) and the maximum pinning force, F_p,max (figure 5(c), being F_p calculated as ${\mu }_{0}{H}_{\mathrm{appl}}\times {J}_{c}$). Conversely, the machinable sample from this work retained a high irreversibility field (figure 5(d)). Considering this observation, it is of much interest to further investigate the pinning mechanism details.

Zoom In Zoom Out Reset image size

To this aim, the pinning forces versus applied field curves were normalised following the universal pinning force scaling procedure reported in [40, 41]. Then, the experimental data was fitted by the law:

$$\begin{eqnarray}&&{f}_{{\rm{p}}}=A{h}^{p}{\left(1-h\right)}^{q},\end{eqnarray} \tag{ 1 }$$

where ${f}_{{\rm{p}}}={F}_{{\rm{p}}}/{F}_{{\rm{p}},\max },$ $h={H}_{{\rm{appl}}}/{H}_{{\rm{irr}}}.$ A, p and q are fitting parameters, and in particular, p, and q values are correlated to vortex pinning mechanisms [41]. Moreover, considering percolation aspects and defining ${{h}}_{0}={h}\left({{f}}_{{\rm{p}}}=1\right),$ the pinning-force related parameter ${k}_{{\rm{n}}}=h({F}_{{\rm{p}}}/2)/{h}_{0}$ introduced by Eisterer [42] was also taken into account. Theoretical values for grain-boundary pinning (GBP) (i.e. surface pinning) are h₀ = 0.2, p = 0.5, q = 2, k_n = 0.34 and for point pinning (PP) they are h₀ = 0.33, p = 1, q = 2, k_n = 0.47 [41, 42].

The parameters h₀, and k_n are plotted as a function of temperature in figure 6(a). They show an increase with temperature and this suggests a strengthening of the point pinning at high temperatures, considering that the values k_n (as well as h₀ at high temperatures) are close to their respective theoretical values for PP mechanism (h₀ = 0.33 and k_n = 0.47). This is the typical case for MgB₂ pristine and BN added samples fabricated by SPS [39].

Zoom In Zoom Out Reset image size

However, a deeper investigation in the framework of Dew–Hugues model [41] seems to indicate that different mechanisms are simultaneously active. For investigated temperatures within 5–30 K range, fits by equation (1) yielded fitting parameters that do not agree with the theory (figures 6(b), (d)). In addition, by plotting the derivative ${\rm{d}}\,\mathrm{ln}\left({f}_{{\rm{p}}}\right)/{\rm{d}}\mathrm{ln}\left(h\right)$ versus $h/\left(1-h\right)$ two linear dependencies were obtained (figure 6(c)). The line intercept with y-axis and the opposite of the slope for each linear domain are the pinning force parameters p₁, q₁ and p₂, q₂, respectively. By defining $x=h/\left(1-h\right),$ the crossover coordinate between the two regions, x_cross, corresponds to the crossover scaled field ${h}_{{\rm{cross}}}=\tfrac{{x}_{{\rm{cross}}}}{1+{x}_{{\rm{cross}}}}.$ The corresponding crossover field is ${H}_{{\rm{cross}}}={h}_{{\rm{cross}}}\times {H}_{{\rm{irr}}}.$ Parameters p₁, q₁, p₂, q₂, and μ₀H_cross are given as a function of temperature in figure 6(d). Parameters p₁, p₂, and q₂ almost do not depend on temperature. The exception is q₁ that is decreasing when temperature increases. A decrease from 2.8 to 0.8 T is also visible for μ₀H_cross. The values of p₁ (∼1.7) and q₁ (4.6–5.9) and variation of the latter parameter with temperature are approximately similar to those reported in [43] for the composite non-machinable samples with starting compositions ((MgB₂)_0.99(Te_x(HoO_1.5)_y)_0.01), x/y = 0.31/0.69, 0.25/0.75. On the other hand, the values of p₂ (∼−0.6) and q₂ (∼2.1) and the variation of these parameters with temperature is very different than for the indicated samples: the dependencies p₂(T) and q₂(T) for MgB₂ samples co-added with Ho₂O₃ and Te show a higher degree of complexity and, hence, of pinning behaviour. The physical meaning for the obtained negative values of p₂ from this work remains unclear and further research is needed.

The shielding properties of the cup were investigated in the temperature range from 20 to 35 K. We explored the magnetic mitigation performance of the manufact in both axial- and transverse-field configuration in order to attain more meaningful information on its competitiveness in practical applications where generic field orientations are likely.

The shielding efficiency was estimated using two parameters: the shielding factor (SF), calculated as the ratio between the applied magnetic field and the local magnetic induction measured by the Hall probes, μ₀H_appl/B_z, and an applied field threshold, μ₀H_appl,lim, above which the shielding factor drops below a given value.

3.3.1. Magnetic shielding in axial-field configuration

The main frame of figure 7 provides the shielding factors evaluated at T = 30 K along the cup's axis as a function of the applied field, μ₀H_appl, in axial-field configuration. Despite the aspect ratio of only 2.2, which makes the flux penetration from the cup's edge not negligible, SFs exceeding 10⁴ and 10² were found 1 mm far from the closed extremity (position z₁) and in correspondence of the cup's centre (position z₃), respectively, up to μ₀H_appl = 1.0 T. Above this field, the shielding factor sharply decreases due to a flux jump occurrence [44, 45] (figure 7, inset).

Zoom In Zoom Out Reset image size

As mentioned above, we defined a threshold field, μ₀H_{appl,lim∣∣}, as the applied field above which the shielding factor drops below a reference value, that must be different in different positions since SFs span over orders of magnitude (see figure 7). We chose SF = 10⁴ for position z₁, 10³ for position z₂, and 10² for position z₃. As expected, μ₀H_{appl,lim∣∣} depends on temperature, as summarized in figure 8. Remarkably, at 20 K, SF > 10² up to μ₀H_appl = 1.8 T were achieved in half the inner volume of the shield (figure 8(c)). In the same range of applied fields, the SF increases over 10⁴ moving towards the closed extremity (figure 8(a)). A further shift of μ₀H_{appl,lim∣∣} to higher applied field values is expected by increasing the lateral wall thickness, reducing the inner radius of the cup. In comparison, a similar SF was found at 20 K for a YBa₂Cu₃O₇ cup with comparable aspect ratio and lateral wall thickness, but only up to μ₀H_{appl,lim∣∣} ∼ 1 T [10].

Zoom In Zoom Out Reset image size

Then, in order to investigate how the J_c improvement with respect to previous samples [18] affects the shielding properties of the cup, we compared the measured SFs with those expected by assuming the same J_c(B) dependence found in [18]. To this aim, we applied the numerical modelling procedure described in [18] using a cup layout with the same sizes as that characterized experimentally in this work.

As can be seen in figure 7, by assuming in SF calculation ${J}_{c}\left(B\right)={J}_{c,0}\,\exp \left[-{\left(\tfrac{B}{{B}_{0}}\right)}^{\gamma }\right],$ where J_c,0 = 3.0 × 10⁴ A cm⁻², B₀ = 0.83 T and γ = 2.52, i.e. the parameter values obtained in [18] at 30 K, a good agreement between experimental and computed curves is achieved only at low applied fields, where the shielding performance mainly depends on the sample geometry. This is supported also by the evidence that in this field range the SF dependence on the distance from the closed extremity of the shield can also be described by the analytical expression calculated by Claycomb et al [46, 47] for the axial field attenuation by a superconducting tube in Meissner state closed by an end cap. Following [46, 47] the expression

$$\begin{eqnarray*}&&\mathrm{SF}\left({z}_{{\rm{i}}}\right)/\mathrm{SF}{\left({z}_{{\rm{j}}}\right)}_{\left|\mathrm{calc}\right.}=\sinh \left(3.834\,\frac{{z}_{{\rm{j}}}}{{R}_{i}}\right)/\sinh \left(3.834\,\frac{{z}_{{\rm{i}}}}{{R}_{i}}\right)\end{eqnarray*}$$

is predicted, being R_i the inner radius of the tube and z the distance from the center of the end cap. Focusing on z₂, z₃ and z₄ positions (experimental data at z₁ position are too scattered), we obtained ${\rm{SF}}\left({z}_{2}\right)/{\rm{SF}}{\left({z}_{3}\right)}_{\left|{\rm{calc}}\right.}$ = 10.0, ${\rm{SF}}\left({z}_{3}\right)/{\rm{SF}}{\left({z}_{4}\right)}_{\left|{\rm{calc}}\right.}$ = 11.7 and ${\rm{SF}}\left({z}_{2}\right)/{\rm{SF}}{\left({z}_{4}\right)}_{\left|{\rm{calc}}\right.}$ = 119, in satisfactory agreement to the corresponding experimental ratios: 9.5, 10.0 and 94, respectively.

At higher fields, where the field-dependence of J_c comes into play, this analytical approach does not fit any more and the calculated curves in figure 7 clearly drop faster than the experimental ones. Considering the SF threshold values reported in figure 8, this means that the improvement of J_c(B) raised μ₀H_{appl,lim∣∣} of more than 0.3 T.

Finally, we checked if the critical current density measured on a small sample and reported in figure 4 (open symbols) can be considered representative of the actual current flowing in the whole cup. To address this topic, we measured again the magnetic induction at T = 30 and 35 K by the Hall probes positioned along the cup's axis. To minimize the flux jump occurrence, we furtherly decreased the ramp rate of the applied field down to 5 × 10^–5 T s⁻¹. Then, we calculated J_c from the B_z versus μ₀H_appl cycle measured by the Hall probe located at position z₅ (i.e. in correspondence to the cup's open extremity) as proposed by Bartolomé et al for finite superconducting rings [48]:

$$\begin{eqnarray}&&{J}_{c}=\displaystyle \frac{{\rm{\Delta }}{B}_{c}}{{\mu }_{0}{f}\left({R}_{{\rm{o}}},{R}_{i},{d}_{i},z\right)},\end{eqnarray} \tag{ 2 }$$

where ${\rm{\Delta }}{{B}}_{{\rm{c}}}={{B}}_{c}^{+}-{{B}}_{c}^{-}$ is the induction cycle width measured at a given applied field in the axial-field configuration and the function f provides the dependency of J_c on the geometry of the system:

$$\begin{eqnarray*}&&\begin{array}{l}{f}\left({R}_{o},{R}_{i},{d}_{i},z\right)=\left(\displaystyle \frac{{d}_{i}}{2}-z\right)\mathrm{ln}\left(\displaystyle \frac{{R}_{o}+\sqrt{{R}_{o}^{2}+{\left(z-{d}_{i}/2\right)}^{2}}}{{R}_{i}+\sqrt{{R}_{i}^{2}+{\left(z-{d}_{i}/2\right)}^{2}}}\right)\\ \,\,\,\,\,\,\,\,+\left(\displaystyle \frac{{d}_{i}}{2}+z\right)\mathrm{ln}\left(\displaystyle \frac{{R}_{o}+\sqrt{{R}_{o}^{2}+{\left(z+{d}_{i}/2\right)}^{2}}}{{R}_{i}+\sqrt{{R}_{i}^{2}+{\left(z+{d}_{i}/2\right)}^{2}}}\right).\end{array}\end{eqnarray*}$$

In our case R_o and R_i are the outer and inner radii of the cup, respectively, d_i the cup internal depth and z the distance from the cup's centre (i.e. z = z₃). We used the B_z data measured by the Hall probe located at z₅, since in that position the effects of the cup closure can be disregarded [18]. Notably, focusing on high field region, i.e. above the full penetration field in the first-magnetisation curve of the cup, where the Bean model is expected to give realistic J_c values [49], the agreement is good (figure 4). Comparisons at lower temperatures could not be carried out because of the multiple occurrences of flux jumps that prevented the J_c evaluation using the Hall probes.

3.3.2. Magnetic shielding in transverse-field configuration

The main frame of figure 9 presents the SFs evaluated along the cup's axis at T = 30 K and as a function of the applied field as in figure 7, but in transverse-field configuration. The shielding ability of the cup is much lower than for the axial-field configuration even at low applied fields, as expected for this geometry [29].

Zoom In Zoom Out Reset image size

Likewise in the axial-field configuration, in the low field range the shielding performance mainly depends on the sample shape. Indeed, the SF dependence on the distance from the closed extremity is still suitably described by the analytical expression calculated in [46, 47] for the transverse field attenuation by a superconducting cup in Meissner state:

$$\begin{eqnarray*}&&\mathrm{SF}\left({y}_{{\rm{i}}}\right)/\mathrm{SF}{\left({y}_{{\rm{j}}}\right)}_{\left|\mathrm{calc}\right.}=\cosh \left(1.84\frac{{y}_{{\rm{j}}}}{{R}_{i}}\right)/\cosh \left(1.84\,\frac{{y}_{{\rm{i}}}}{{R}_{i}}\right)\end{eqnarray*}$$

being y the distance from the center of the closure cap.

Fixing on y₂, y₃ and y₄ positions, we obtained ${\rm{SF}}\left({y}_{2}\right)/{\rm{SF}}{\left({y}_{3}\right)}_{\left|{\rm{calc}}\right.}$ = 3.1, ${\rm{SF}}\left({y}_{3}\right)/{\rm{SF}}{\left({y}_{4}\right)}_{\left|{\rm{calc}}\right.}$ = 3.2 and ${\rm{SF}}\left({y}_{2}\right)/{\rm{SF}}{\left({y}_{4}\right)}_{\left|{\rm{calc}}\right.}$ = 10.0, in well agreement to the corresponding experimental ratios: 3.2, 2.9 and 9.1, respectively.

On the other hand, the large values of in-field J_c guarantee a smooth decrease or even a flat behaviour of the SF versus μ₀H_appl curves up to high applied fields.

As we did in the axial-field configuration, we defined three threshold fields, μ₀H_appl,lim⊥, as the applied field above that the shielding factor goes below 40 (for position y₁), 20 (for position y₂) and 7 (for position y₃). In figure 10, μ₀H_appl,lim⊥ values are plotted as a function of temperature. As can be seen, near to the cup's closed edge (position y₁), at 20 K SF is over 40 up to μ₀H_appl = 0.8 T, while in the same field range it is over 7 in correspondence to the cup's centre (position y₃).

Zoom In Zoom Out Reset image size