Magnetic shielding properties of a superconducting hollow cylinder containing slits: modelling and experiment

An enclosure made of superconducting material can be regarded as an excellent passive screen against magnetic fields over a large frequency range [1–3]. At low frequency in particular, the efficiency of superconductors is usually higher than that of conventional ferromagnets [1, 2]. A low magnetic field background is encountered in various domains, including SQUIDs (superconducting quantum interference devices) [4], magneto-encephalography [5], cryogenic current comparators [6] or naval military applications [7]. The present work deals with enclosures made of bulk high temperature superconductors (HTS). They can be used either at liquid nitrogen temperature (77 K)—at which magnetic shielding occurs up to a few tens of mT—or at low temperature (10 K and below) where shielding of magnetic fields in excess of 1 T was demonstrated recently [8, 9]. Several materials are candidates for efficient magnetic screens, including MgB₂ [8, 10], YBa₂Cu₃O₇ [3, 11] or Bi-based compounds [12–15]. The basic shape for studying magnetic shielding is a hollow cylinder, typically of 2–5 cm diameter and 5–20 cm high. In view of increasing size of the shielded volume, either cylinders of larger size should be synthesized or the cylinder could be cut in two halves subsequently separated in order to make a larger structure, as illustrated in figure 1(a). In the present work we model and characterize the magnetic shielding properties of a bulk hollow superconductor with two axial slits (framed illustration in figure 1(a)). The aim is to investigate whether numerical modelling is able to predict and quantify the effect of such slits on the magnetic shielding performance. The modelled data will be compared with an experiment carried out on bulk Bi₂Sr₂CaCu₂O₈ (Bi-2212) materials obtained by the melt cast process (MCP) [16–18].

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The magnetic properties of superconducting hollow cylinders have been documented extensively in the literature [19–21]. The advantage of a passive shield (i.e. a simple superconducting cavity) over an active shield (i.e. a set of coils fed with appropriate currents) is that it does not require an a priori knowledge of the orientation of the magnetic field to be screened. Basically, two simple configurations can be considered for a cylinder: (i) the axial configuration, i.e. the external magnetic field is parallel to the axis of the tube and (ii) the transverse configuration, i.e. the external magnetic field is perpendicular to the axis of the tube. The axial configuration is the simplest to consider. When the aspect ratio (height/diameter) of the cylinder is large enough (let us say  > 3), valuable information can be obtained from a one-dimensional (1D) analysis based on the Bean model [22, 23], possibly with a field-dependent critical current density J_c [21, 24]. For short cylinders, finite size effects need to be considered and a 2D model should be used in order to account for the two mechanisms of field penetration, i.e. from the lateral surface and through the opening ends [14, 25, 26]. The investigation of magnetic screening in the transverse configuration involves at least a 2D model. At low applied field (i.e. before the magnetic induction reaches the inner wall of the cylinder), the magnetic flux penetration profiles are similar to those in a bulk cylindrical wire subjected to a transverse field [27] and exhibit an 'eye-shape' structure. The flux profiles of an infinite cylinder in transverse fields can be obtained semi-analytically [28–30], using a semi-analytical model developed by Brandt [31, 32] or a finite element model [33]. In the present work dealing with a hollow cylinder containing two axial slits, only the transverse configuration will be studied, as illustrated in figure 1(b). We will take advantage of the possibilities offered by 3D finite element modelling [34–37] to investigate the flux penetration by taking into account both the finite height effects and the magnetic flux density through the slit.

Our paper is organized as follows. In section 2, we present the finite element model and the constitutive laws that are used. Section 3 describes the experimental system used to achieve precise magnetic shielding measurements. The modelling results are presented and analysed in section 4. These results will be compared to experimental data obtained with the same conditions. The conclusions are drawn in section 5.

The numerical modelling results are obtained by using the finite element method (FEM), using the A–ϕ formulation [34]. The Maxwell equations are solved for two independent variables: (i) the vector potential, A and (ii) the scalar potential, ϕ, defined as:

$$\begin{eqnarray} \mathbf{B}&=&{\mathbf{B}}_{\mathrm{self}}+{B}_{\mathrm{app}}(t){\mathbf{e}}_{z}=\nabla \times {\mathbf{A}}_{\mathrm{self}}+\nabla \times {\mathbf{A}}_{\mathrm{app}}, \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} \mathbf{E}&=&-\frac{\partial {\mathbf{A}}_{\mathrm{self}}}{\partial \boldsymbol{t}}-\frac{\partial {\mathbf{A}}_{\mathrm{app}}}{\partial \boldsymbol{t}}-\nabla \phi \end{eqnarray} \tag{ 2 }$$

where B_self is the magnetic flux density induced by the HTS, B_app(t) is the uniform applied magnetic flux density; A_self and A_app are the corresponding vector potentials. These fields are approximated by edge functions (first order):

$$\begin{eqnarray} &&\mathbf{A}=\sum _{i=1}^{M}{\alpha }_{i}{\mathbf{A}}_{i} \end{eqnarray} \tag{ 3 }$$

that ensure the continuity of the tangential component of A and node functions (first order):

$$\begin{eqnarray} &&\phi =\sum _{j=1}^{N}{b}_{j}{\phi }_{j}, \end{eqnarray} \tag{ 4 }$$

that ensure the continuity of ϕ.

The superconducting properties are modelled using an E–J power law, i.e. 

$$\begin{eqnarray} &&E(J)={E}_{\mathrm{c}}\left (\frac{J}{{J}_{\mathrm{c}}}\right )^{n} \end{eqnarray} \tag{ 5 }$$

where the two material-related parameters, the critical exponent n and the critical current density J_c, are determined from magnetic measurements on a plain cylinder, as detailed in the next section. In order to limit the number of modelling parameters and simplify the interpretation of results, both n and J_c are assumed to be field-independent.

Using the above formulation, Maxwell equations are reduced to the coupled equations:

$$\begin{eqnarray} &&\nabla \times \nabla \times \mathbf{A}={\mu }_{0}\sigma (\mathbf{A},\phi )\left (-\frac{\partial {\mathbf{A}}_{\mathrm{self}}}{\partial \boldsymbol{t}}-\frac{\partial {\mathbf{A}}_{\mathrm{app}}}{\partial \boldsymbol{t}}-\nabla \phi \right ) \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} &&\nabla \cdot \left \{ \sigma (\mathbf{A},\phi )\left (-\frac{\partial {\mathbf{A}}_{\mathrm{self}}}{\partial \boldsymbol{t}}-\frac{\partial {\mathbf{A}}_{\mathrm{app}}}{\partial \boldsymbol{t}}-\nabla \phi \right )\right \} =0 \end{eqnarray} \tag{ 7 }$$

where σ is the power-law conductivity:

$$\begin{eqnarray} &&\sigma (\mathbf{E})=\frac{{J}_{\mathrm{c}}}{{E}_{\mathrm{c}}^{1/n}}\left (\vert \mathbf{E}\vert \right )^{\frac{1-n}{n}}. \end{eqnarray} \tag{ 8 }$$

The equations have to be solved with the boundary (Dirichlet) conditions at infinity:

$$\begin{eqnarray} &&\mathbf{A}=0, \end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray} &&\phi =0, \end{eqnarray} \tag{ 10 }$$

that have been implemented by using a Jacobian transformation for sending the outer surface of a spherical shell to infinity [38].

The solver used for the modelling is the open-source solver GetDP developed by the Applied and Computational Electromagnetics research unit of the University of Liège [39]. Its description and details of the code used for superconducting material modelling were described earlier [34, 39, 40].

The geometry studied in the present work is a hollow cylinder of finite height with two slits made in a vertical plane along a diameter. Because of the presence of the slits, the structure is not axisymmetric and 2D simulations cannot take account of the effect of a transverse magnetic field applied at arbitrary angle; for this reason 3D modelling will be used. Because of the limited number of nodes available to allow a full 3D calculation in a decent time, however, modelling will be carried out on a shorter sample with height L ( = 20 mm) being comparable to the average diameter D ( = 17.5 mm), as illustrated schematically in the right-hand part of figure 1(b). The thickness of the cylinder wall is 5.5 mm and the two halves are separated from each other by a 1 mm gap. This gap size was chosen to properly mesh the regions inside the slits. The resulting modelled and meshed structure involves 45 000 nodes in volume and 250 000 tetrahedra. Due to the restriction on the sample height for modelling, the experiments are carried out on both long (80 mm) and short (20 mm) samples (see next section).

The use of a 3D model requires some caution. In many problems with a specific symmetry, the current density vectors are forced to be everywhere perpendicular to the local magnetic flux density. In this case, the constitutive equations are well established and the results obtained with the FEM are trustworthy. In the opposite situation of a system with no particular symmetry, as is the case for the geometry investigated here, current densities may locally develop components which are parallel to the magnetic flux density (longitudinal currents). Hence classical constitutive equations may no longer be valid and unusual critical states should be considered [41]. This largely debated problem [42, 43] is not directly addressed in the present work. Care should thus be taken in interpreting the results obtained by 3D modelling, as the contribution of longitudinal currents in the magnetic phenomena arising in bulk superconductors is still not assessed. Besides these limitations, the modelling is expected to give relevant information about the penetration of the magnetic field inside the superconductor, the flow of the persistent currents and the way they are affected by the presence of the slits which break the symmetry of the system.

The studied sample is a hollow cylinder of Bi₂Sr₂CaCu₂O₈ (Bi-2212) produced by the melt casting process [16]. This cylinder is then cut axially into equal halves using a wire saw. The two halves are insulated with Kapton tape, resulting in a gap width of  ≈ 200 μm. The dimensions of the sample are presented in table 1. The experiments are first carried out on a long sample (80 mm length), then this sample is cut in two unequal parts (20 and 60 mm) and the same data are recorded on the shorter sample which has the same length as the modelled sample (20 mm). Magnetic shielding measurements are carried out using a home made set up designed for the characterization of relatively large hollow cylinders (up to 80 mm length) in both axial and transverse geometries. The applied magnetic field is generated by a long copper coil (length 45 cm, diameter 20 cm) fed by a HP6030A DC power supply. The applied magnetic field at the centre of the coil can reach μ₀H_app = 36 mT. The field can be ramped at a constant sweep rate between 5 μT s⁻¹ and 20 mT s⁻¹. In the present work, a fixed sweep rate of 1 mT s⁻¹ is always used. The magnetic flux density at the centre of the cylinder is measured with a high-sensitivity Hall probe (Arepoc-HHP-MP), recorded by a PCI-6281 National Instrument Data Acquisition Card controlled by a PC running Labview^®. All measurements are carried out in liquid nitrogen (77 K). Two concentric mu-metal ferromagnetic enclosures outside the exciting coil ensure protection against stray magnetic fields.

A thorough characterization of the pristine Bi-2212 sample (i.e. without slits) was first performed in axial and transverse configurations in order to determine both the dimensionless shielding factor (SF) of the cylinder, i.e. the ratio of the applied and internal magnetic inductions (SF = B_app/B_in), and its limiting field B_lim, i.e. the value of applied magnetic induction above which magnetic shielding degrades and the shielding factor SF drops below a reference level, e.g. SF<100. Experimental data can be found in [44]. From the results obtained at 77 K, two relevant points should be noticed. First, the limiting field in the transverse configuration B_lim is  ≈ 18 mT, which corresponds to the best values reported so far for superconducting magnetic shielding at the liquid nitrogen temperature [9, 11–15]. Second, the average critical current density J_c = 350 A cm⁻² corresponds to the self-field  ∼ 20 mT at the surface of the cylinder. As this is comparable to the magnetic field range investigated (B < 30 mT), we assumed that both J_c and the critical exponent of the E–J power law n = 10 are field-independent. These are considered constant in model calculations for the cut sample.

4.1. Distribution of shielding currents

First we investigate the distribution of currents in a short superconducting hollow cylinder containing two axial slits (figure 1(b)), modelled numerically using the FEM method described in section 2. The simulations were carried out for several angles θ between the applied magnetic field and the normal to the plane of the slits, as shown in figure 1(b). Figure 2 shows the distribution of the axial component of the current density, J_z, in the median plane of the hollow cylinder for several orientations 0° ≤ θ ≤ 90° of the transverse field. The amplitude of the applied field is fixed at 30 mT, i.e. above the limiting field B_lim of the pristine cylinder. This ensures that the magnetic field penetrated completely through the wall of the cylinder.

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For θ = 0° (i.e. H⊥slits), the distribution of J_z exhibits an eye-shape structure which is similar to that found in an infinite cylinder without any slit [15, 27]. This is an anticipated result since the shielding currents in the median plane are always perpendicular to the applied field, i.e. parallel to the slits. Therefore the slits do not affect the current flow in this configuration. Differently stated, two orthogonal symmetry planes appear naturally, one in the direction parallel to the applied field (odd symmetry), and the other one in the perpendicular direction (even symmetry). When the magnetic field is tilted with respect to the normal of the cut, i.e. θ > 0°, the distribution of currents is modified as shown in figure 2(a). The reason is that the screening currents tend to flow along closed loops in planes perpendicular to the applied field, but are interrupted by the slits. This results in a symmetry break and the appearance of a net discontinuity (currents of opposite signs) on both sides of the slit. This configuration can be visualized more clearly in figure 2(b) where the directions of screening currents are schematically illustrated for the particular case of θ = 45°. In this case the shielding currents are forced to follow two separate shielding current loops, and this effect reduces the efficiency of magnetic shielding accordingly (see section 4.2). Finally, the situation where the magnetic field is aligned with the plane of the slits (θ = 90°) allows us to recover two symmetry planes, odd in the direction of the applied magnetic field and even in the perpendicular direction.

4.2. Penetration of magnetic flux density

Figure 3 shows the distribution of magnetic flux density in the hollow cylinder subjected to a 7.5 mT transverse field directed at θ = 45°. Two planes are considered: (a) the median plane (Z = 0) and (b) the vertical plane perpendicular to the cut (X = 0). The figures demonstrate three penetration routes for the magnetic flux density:

• (i)  
  through the walls of the cylinder, as illustrated by the black arrows in figures 3(a) and (b);
• (ii)  
  through the open ends at the top and bottom of the cylinder, as illustrated by the grey arrows in figure 3(b);
• (iii)  
  through two slits, as illustrated by the white arrows in figure 3(a).

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As can be seen in figure 3(a), the magnetic flux density at the centre of the cylinder (B_in) is clearly deflected with respect to the applied flux density (B_app). This misalignment can be considered as a result of a 'competition' between the symmetry, which tends to be imposed by the applied field, and the presence of the cut, which allows direct magnetic flux penetration. In order to quantify the relative contribution of magnetic flux density entering through the cut, we define φ as the angle between B_in at the centre of the cylinder and the external magnetic field B_app. The variation of φ with respect to the angle and amplitude of the applied field will be studied in section 4.3.

4.3. Angle between the applied field and the internal field

We now investigate how the angle φ between internal and applied field varies as a function of the angle θ between the applied field and the normal of the cut. The results of modelling are shown in figure 4(a) for three amplitudes of the applied field. The φ versus θ curves are found to exhibit a pronounced maximum, which can be better understood using schematic illustrations shown in figure 4(b). For θ = 0°, the applied field is perpendicular to the cut which therefore has no influence on the shielding currents (φ = 0°). For θ = 90°, the applied field is directed along the cut plane which offers a preferential channel for flux penetration; by symmetry the internal field B_in is thus aligned with the applied field B_app and one has again φ = 0°. In the intermediate situation (0° < θ <90°), the competing mechanisms discussed above lead to a misalignment between B_in and B_app, yielding a finite value of φ and a φ versus θ curve displaying a maximum, as shown in figure 4(a). Interestingly, the modelling predicts non-symmetrical curves, with the maximum deflection occurring for θ between 50° and 55° in the magnetic field range investigated. Figure 4(c) shows the experimental data measured on a Bi-2212 cylinder having the same dimensions as that used for modelling. As can be seen, the measured curves largely agree with the modelled ones. Small differences are observed. First, a deflection of φ ≈ 2.5° is measured when the field is either normal or parallel to the cut. This can be attributed to the small signal-to-noise ratio of the perpendicular component (theoretically zero) of the measured magnetic induction for these angles and will not be discussed further. Second, the experimental curves are somewhat flattened compared to the modelled curves: the maximum value of φ at 7.5 mT is 29° for the modelling against 21° for the experiment. This is likely due to the finite active area of the Hall sensor (100 μm × 100 μm), which results in a smearing of the measured flux density, and from the differences in modelled and actual gap sizes, which may affect the relative importance of a penetration occurring through the slits. Apart from these minor differences, the overall agreement is very good. It should be recalled here that the modelling was carried out without adjustment of free parameters: the J_c and n values (both supposed field-independent) being taken directly from experimental data measured on the pristine sample.

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It is of interest to investigate how the amplitude of the applied magnetic field affects the value of φ. The φ versus B_app curve modelled for the applied field directed at θ = 45° is plotted in figure 5(a). As can be seen, the modelling predicts a maximum value of φ, albeit relatively flat. This indicates that the magnetic flux channelling through the two slits dominates in some range of applied fields. This can be interpreted as follows. At low applied field B_app, the internal field B_in is very small, but the main contribution arises from flux entering from the top and bottom of the cylinder, through large open ends. The relative orientation of the cut and the applied field does not affect much the internal field, and hence the misalignment φ is small. For large applied fields B_app, the magnetic flux enters mainly through the walls of the superconductor, as is the case for a sample without a cut, and the angle φ is expected to approach 0 at high fields. In between, the angle φ exhibits a well-defined maximum, as shown in figure 5(a). The corresponding experimental data are plotted in figure 5(b). Due to the signal averaging over the active area of the Hall probe, the measured values of φ are found to be smaller than those obtained by simulation, as is the case in figure 4. Remarkably, however, the maximum predicted by modelling can be clearly seen in the experiment.

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4.4. Magnetic shielding curves

In this last section, we study the magnetic shielding curves of the cut cylinder, i.e. the magnetic flux density in the centre of cylinder B_in as a function of the applied field B_app. Because of the angle between B_in and B_app, we focus here on the component of B_in parallel to the applied field, hereafter denoted 'B_∥'. The modelled results are shown in figure 6(a) for various orientations 0° ≤ θ ≤ 90° of the transverse field. Figures 6(b) and (c) show the corresponding experimental data measured on a bulk Bi-2212 sample with the same geometry as the modelled structure (figure 6(b)), or the same radius but four times the height of the modelled structure (figure 6(c)).

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We first discuss the modelled data (figure 6(a)). When θ = 90° (magnetic field parallel to the slits) the magnetic shielding is very weak, i.e. B_∥ ∼ 0.6B_app. On lowering θ, a more pronounced shielding effect appears, yielding for the best orientation (θ = 0°) B_∥ ∼ 0.28B_app at 10 mT. Although the presence of the slits has no effect on the magnetic flux penetration at this particular angle, the magnetic shielding is relatively poor because the magnetic field penetrates the short cylinder from the top and the bottom.

As can be seen from figure 6(b), the experimental results obtained with the same geometry remarkably agree with the modelling. For θ = 0°,B_∥ ∼ 0.27B_app at 10 mT. This excellent agreement makes us confident in the fact that all penetration routes for the magnetic field are properly taken into account by our modelling tool.

The shielding performance can be improved significantly by increasing the aspect ratio of the structure. Figure 6(c) shows the B_∥ versus B_app curves measured on a sample with larger aspect ratio (length 80 mm, average diameter 21 mm), which is more representative of a practical case. With such dimensions, notable shielding (B_∥ < 0.01B_app) is observed for B_app < 18 mT and θ = 0°. For increasing θ, magnetic shielding degrades, as is the case for the short sample. This feature appears more clearly in figure 7, showing the shielding factor SF as a function of the applied magnetic field. The shielding factor SF is defined as B_app/B_in, where B_in denotes the component (B_∥) parallel to the applied field B_app. Experimentally, the shielding factor is found to exhibit remarkably high values (SF > 100) for θ ≤ 15° but the magnetic shielding efficiency deteriorates when the angle between the applied transverse magnetic field and the normal to the slit plane exceeds 15°.

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From the data displayed in figures 5 and 6, it is worth noting that the modelling is able to reproduce experimental data although Meissner currents are totally neglected. There are likely two reasons for this. The first is that the superconductor is modelled by a E–J power law and not through a critical state analysis (Bean model); the latter would possibly require amending the model by considering that vortices do not penetrate below the lower critical field H_c1. In the present case, the only constitutive law (and hence, superconducting parameters) involve J_c and n. Since these parameters are considered to be field-independent, it seems that the value of H_c1 compared to H_app can be neglected as well. Second, the data are measured at liquid nitrogen temperature (77 K), i.e. close to T_c ( ∼ 90 K). Hence the lower critical field is quite small and flux creep effects due to thermally activated vortex motion are found to play a major role. As a consequence, the threshold field of the pristine structure is found to depend on the sweep rate of the applied magnetic field (which would not be the case for Meissner currents). This feature is reflected by the relatively small n value (n ∼ 10) used in our modelling.

We have investigated the magnetic flux penetration in a bulk hollow superconducting cylinder of finite height containing two axial slits. The cylinder was subjected to a transverse field directed at several angles with respect to the cut plane. This study was performed both numerically and experimentally.

The FEM model, based on the open-source solver GetDP, was used with a 3D geometry in order to account for non-symmetrical arrangements of the magnetic field and the superconductor. The results show how the slits block the shielding current loops and act as an 'entrance channel' for the magnetic flux. Three penetration routes of the magnetic flux are identified: (i) the walls of the cylinder, (ii) the opening ends at the top and bottom and (iii) the two slits. The finite contribution of the flux entering through the slits to the total flux density in the cylinder gives rise to a misalignment between the applied field B_app and the internal field B_in. The angle φ between the two fields is found to be a non-monotonic function of both the applied field direction θ and amplitude B_app.

The modelled flux penetration was compared with the experimental results on a bulk Bi-2212 cylinder measured with a high-sensitivity Hall probe. Although the modelled angle φ is found to be overestimated with respect to experimental data, the overall agreement is very good and the qualitative features observed in the experiment can be reproduced by the model. In particular, the field amplitude dependence of the angle φ exhibits a relatively flat profile with a well-defined maximum, representative of the competition between the penetration routes for the magnetic flux as the applied field changes.

In this study we intentionally minimized the set of parameters used in the model, since the use of additional material-dependent data increases uncertainty and complicates significantly the understanding of their influence on the final results. The results obtained in the present work are based on a critical current density J_c and a power-law exponent n, assumed to be field-independent and determined from independent preliminary measurements on the superconducting cylinder before cutting. The data show clearly that the principal features can be reproduced without sophisticated constitutive laws. The excellent agreement between simulation and experiment also validates the used simulation tools. These will be a valuable help for the design of larger shielding devices, especially with respect to an optimized choice of geometry and position of the interfaces, if more than one bulk part is to be used. It is worth emphasizing that the model can be combined with more complex parameters, including e.g. field-dependent superconducting properties or any time-dependent waveform of the applied magnetic field.

We thank the FRS-FNRS, the Communauté Française de Belgique for cryofluid, travel and equipment grants, under references CR.CH.09-10-1.5.212.10 and ARC 11/16-03. Fruitful and enlightening discussions with M O Rikel (Nexans Superconductors) are also gratefully acknowledged.