Superconductor Science and Technology
Supercond. Sci. Technol. 30 (2017) 105009 (7pp)

https://doi.org/10.1088/1361-6668/aa841f

Numerical modelling of iron-pnictide bulk
superconductor magnetization
Mark D Ainslie1 , Akiyasu Yamamoto2,3, Hiroyuki Fujishiro4,
Jeremy D Weiss5 and Eric E Hellstrom6
1

Bulk Superconductivity Group, Department of Engineering, University of Cambridge, Cambridge CB2
1PZ, United Kingdom
2
Department of Applied Physics, Tokyo University of Agriculture and Technology, 2-24-16 Nakacho,
Koganei, Tokyo 184-8588, Japan
3
Materials Research Center for Element Strategy, Tokyo Institute of Technology, 4259 Nagatsuta, Midori,
Yokohama, Kanagawa 226-8503, Japan
4
Department of Physical Science and Materials Engineering, Faculty of Science and Engineering, Iwate
University, Morioka 020-8551, Japan
5
Department of Physics, University of Colorado, Boulder, CO 80309, United States of America
6
Applied Superconductivity Center, National High Magnetic Field Laboratory, Florida State University,
Tallahassee, FL 32310, United States of America
E-mail: mark.ainslie@eng.cam.ac.uk
Received 19 June 2017, revised 24 July 2017
Accepted for publication 4 August 2017
Published 5 September 2017
Abstract

Iron-based superconductors exhibit a number of properties attractive for applications, including
low anisotropy, high upper critical magnetic ﬁelds (Hc2) in excess of 90 T and intrinsic criticalcurrent densities above 1 MA cm−2 (0 T, 4.2 K). It was shown recently that bulk iron-pnictide
superconducting magnets capable of trapping over 1 T (5 K) and 0.5 T (20 K) can be fabricatedwith ﬁne-grain polycrystalline Ba0.6K0.4Fe2As2 (Ba122). These Ba122 magnets were processedby a scalable, versatile and low-cost method using common industrial ceramic processing
techniques. In this paper, a standard numerical modelling technique, based on a 2D axisymmetricﬁnite-element model implementing the H-formulation, is used to investigate the magnetisation
properties of such iron-pnictide bulk superconductors. Using the measured Jc(B, T)
characteristics of a small specimen taken from a bulk Ba122 sample, experimentally measured
trapped ﬁelds are reproduced well for a single bulk, as well as a stack of bulks. Additionally, theinﬂuence of the geometric dimensions (thickness and diameter) on the trapped ﬁeld is analysed,with a view of fabricating larger samples to increase the magnetic ﬁeld available from suchtrapped ﬁeld magnets. It is shown that, with current state-of-the-art superconducting properties,
surface trapped ﬁelds >2 T could readily be achieved at 5 K (and >1 T at 20 K) with a sample ofdiameter 50 mm. Finally, an aspect ratio of between 1 and 1.5 for R/H (radius/thickness) would
be an appropriate compromise between the accessible, surface trapped ﬁeld and volume ofsuperconducting material for bulk Ba122 magnets.
Keywords: bulk superconductors, critical current density (superconductivity), ﬁnite-element
method, iron pnictides, numerical simulation, trapped ﬁeld magnets(Some ﬁgures may appear in colour only in the online journal)
1. Introduction

superconductivity in materials other than cuprates and a great
deal of research has been carried out towards practical implementation in thin ﬁlm, wire and bulk forms. Bulk superconducting materials, in particular, can be used as super-strength,

The discovery of superconductivity in iron-pnictide compounds
in 2008 [1] raised the prospect of ﬁnding high-temperature
0953-2048/17/105009+07$33.00

1© 2017 IOP Publishing Ltd Printed in the UK

Supercond. Sci. Technol. 30 (2017) 105009

M D Ainslie et altrapped ﬁeld magnets (TFMs) and magnetic ﬁelds greater than
17 T [2] have been achieved in large, single-grain (RE)BCO(where RE = rare earth or Y) bulk superconductors. This makes
them attractive for a number of engineering applications that relyon high magnetic ﬁelds, including compact and energy-efﬁcientelectrical machines, magnetic separation and magnetic drugdelivery systems.
The iron-based superconductors exhibit a number ofproperties attractive for applications, including low anisotropy, high upper critical magnetic ﬁelds (Hc2) in excess of
90 T and intrinsic critical current densities above 1 MA cm−2
(0 T, 4.2 K). It was shown recently in [3] that bulk ironpnictide superconducting magnets capable of trapping over
1 T (5 K) and 0.5 T (20 K) can be fabricated with ﬁne-grain
polycrystalline Ba0.6K0.4Fe2As2 (Ba122). These Ba122 magnets were processed by a scalable, versatile and lowcost method using common industrial ceramic processing
techniques.
In this paper, a standard numerical modelling technique,based on a 2D axisymmetric ﬁnite-element model implementing the H-formulation, is used to investigate the magnetisation
properties of such iron-pnictide bulk superconductors. Using themeasured Jc(B, T) characteristics of small specimens taken frombulk Ba122 samples, the experimentally measured trapped
ﬁelds in [3] are reproduced to good effect. Additionally, theinﬂuence of the geometric dimensions (thickness and diameter)on the trapped ﬁeld is analysed, with a view of fabricatinglarger samples to increase the magnetic ﬁeld available fromsuch TFMs.

Figure 1. 2D axisymmetric models for numerical simulation of ironpnictide (Ba122) bulk superconductors. (a) Single Ba122 bulk,‘Ba122-Lower,’ of diameter (D) 10 mm and thickness (H) 4.8 mm,
and (b) stack of Ba122 bulks, comprising the same ‘Ba122-Lower’bulk from (a), but polished to a thickness of 3.7 mm, and an upper
bulk, ‘Ba122-Upper,’ comprised of bulks of diameter 10 mm andtotalling thickness 14.7 mm. A 1.6 mm spacer (not modelled) exists
between the two bulks to accommodate a Hall sensor, as describedin [3].

bulk superconductors. The model shown in ﬁgure 1(b)represents experimental setup for the bulk Ba122 magnet
stack measured in [3], and ﬁgure 1(a) shows the single lower
bulk, ‘Ba122-Lower,’ in this stack before it was machined tothe thickness of 3.7 mm.
The electromagnetic properties are simulated using the2D axisymmetric H-formulation, implemented in the commercial software package COMSOL Multiphysics 5.2a andused previously by the authors to simulate bulk high-temperature superconductors [5–9], as well as bulk MgB2 [10].
Isothermal conditions are assumed because the magnetisation
process is slow; hence, no thermal model is included. The
governing equations are derived from Maxwell’s equations—
namely, Faraday’s (1) and Ampere’s (2) laws:

2. Modelling framework and veriﬁcation
2.1. Sample information

To fabricate the bulk Ba122 samples investigated in thisstudy, as described in [3], Ba, K, Fe, and As in a molar ratioof 0.6:0.42:2:2 were reacted together by a mechanochemical
reaction, followed by sintering in a hot isostatic press (HIP) at
600 °C using a procedure described elsewhere in [4]. Afterbulk synthesis and subsequent re-milling, approximately
3–5 g of Ba122 powder was pressed into 15.9 mm diameterpellets and then further densiﬁed in a cold isostatic press at
276 MPa. These were then wrapped with Ag foil and inserted
into a steel tube that was carefully machined to the diameterof the pellet and foil. Both ends of the steel tube were weldedshut under vacuum using chamfered plugs that helped thesteel tube to compress the pellet. After welding, the tubes
were swaged and cold isostatically pressed to further shapeand densify them, reducing the diameter of the samples
approximately 10%. Finally, the samples were sintered for
10 h at 600 °C in the HIP. After heat treatment, the steel tubes
were sliced with a diamond saw to reveal the pellet surfaces.

⎛ dB ⎞d (m 0 m r H )´E+⎜ ⎟=´E+= 0,⎝ dt ⎠dt(1 ) ´ H = J,

(2 )where μ0 is the permeability of free space, and for thesuperconducting and surrounding sub-domains, the relative
permeability is simply μr=1.
The results of the model depend strongly on the in-ﬁeld,
temperature-dependent critical current density, Jc(B, T), of thesuperconducting material [11], and the experimental data forJc(B) of a small specimen taken from the lower bulk, ‘Ba122Lower,’ measured at 4.2, 10, 20 and 25 K in ﬁelds up to 10 T
is shown in ﬁgure 2 (solid lines). Estimated Jc(B) characteristics for the upper bulk, ‘Ba122-Upper,’ are also included inﬁgure 2 (dashed lines), which are estimated from remnant
trapped ﬁeld data that suggested a self-ﬁeld Jc, i.e. average,
in-ﬁeld Jc, of around 41.1 kA cm−2 (compared to 74 kA cm−2
for the lower sample). It is assumed that the upper bulk has
the same ﬁeld dependence as the lower one, but such that Jc isalways 41.1/74 of the lower one.
The Jc(B, T) data is input into the model using a twovariable, direct interpolation, as described in [12, 13] to

2.2. Modelling framework

Figure 1 shows the 2D axisymmetric models used in thispaper to simulate the magnetisation of iron-pnictide (Ba122)2Supercond. Sci. Technol. 30 (2017) 105009

M D Ainslie et al

Figure 2. Jc(B, T) characteristics for a small specimen taken from the

Figure 3. Comparison of experimental (solid lines) and numericallower bulk, ‘Ba122-Lower,’ in ﬁgure 1(b) measured at 4.2, 10, 20
and 25 K in ﬁelds up to 10 T (solid lines). Estimated Jc(B)characteristics for the upper bulk, ‘Ba122-Upper,’ are also included(dashed lines), which are estimated from the remnant trapped
ﬁeld data.

simulation (symbols) results for the single Ba122 bulk, ‘Ba122Lower,’ shown in ﬁgure 1(a) and using the measured Jc(B) data
shown in ﬁgure 2.

‘Ba122-Lower’ (shown in ﬁgure 1(a)). For the experimental
measurement, the bulk is FC (applied ﬁeld, Bapp=6 T) toapproximately 5 K, with the ﬁeld subsequently reduced tozero, followed by increasing the temperature at a rate of
0.5 K min−1. Hall sensors located at the centre of the top andbottom surfaces of the bulk measured the magnetic ﬁeld
density, Bz.
In the numerical simulations, the trapped magnetic ﬁeld,
Bt, is calculated at t=+10 min after the application andremoval of the magnetising ﬁeld, at z=+2.9 mm, r=0 mm
and z=–2.9 mm, r=0 mm, corresponding to the top andbottom Hall sensor locations, respectively (the surfaces of thebulk are located at ±2.4 mm). The simulation results are
given for discrete points between 5 and 25 K at increments of
5 K, within the range of the input Jc(B) data detailed in theprevious section.
The simulations reproduce the experimental results very
well, validating the model assumptions, and since the results
are reproduced using only a single, small specimen takenfrom the bulk, this suggests the sample has a homogeneous
Jc distribution on a macroscopic scale, like bulk MgB2 [17]and in contrast to the inhomogeneous Jc distribution that canoccur during the growth of c-axis seeded, single-grain(RE)BCO bulk superconductors [18]. Nonetheless, it should
be noted that pulsed ﬁeld magnetisation is a more useful toolfor highlighting the inhomogeneity of the Jc distribution in abulk superconductor than FC magnetisation [19].

implement the complex Jc(B, θ) behaviour of high-temperature
superconducting (HTS) coated conductors and as used in [8, 9]
to simulate the Jc(B, T) properties of a bulk Gd-Ba-Cu-O
superconductor. A key advantage of this modelling technique isthat it is ﬂexible to include different materials and different
superconducting properties, and it can predict trapped ﬁelds
with good accuracy, despite the fact that the upper and lower
bulks have different Jc characteristics. This variation in Jc wasobserved experimentally between the two separate batches fromwhich the samples came from, but little variation was seen
between samples within each particular batch.
The E–J power law (E α Jn) [14, 15] is used to simulate
the nonlinear electrical resistivity of the superconductor,
where n=50 and E0=1×10−4 V m−1. It should be notedthat the n value is higher than that used commonly for HTS
materials, since the observed ﬂux creep in the Ba122 bulks iscomparatively much weaker (see ﬁgure 4 in [3], for example,
where the trapped ﬁeld at 5 K decayed approximately only
3% after one day). Such a high n value also gives a good
approximation of critical state models.
To simulate the ﬁeld-cooled (FC) magnetisation process as
carried out experimentally in [3], a zero-ﬁeld-cooled (ZFC)
magnetisation process is employed in the simulations byapplying a large, slowly-ramped external magnetic ﬁeld that isseveral times larger than the full penetration ﬁeld of the bulk. By
setting appropriate boundary conditions, a uniform backgroundmagnetising ﬁeld is ramped from 0 to 5 T at 1.5 T min−1, thenramped back to 0 T at the same rate. After waiting +10 min forrelaxation of the trapped ﬁeld, the magnetic ﬂux density is thencalculated at locations corresponding to the approximate locationof the Hall sensors used to take the experimental measurements,
which will be speciﬁed in the following sections.

2.4. Stack of Ba122 bulks

Figure 4 shows a comparison of the experimental (as presented in [3]) and numerical simulation results for the stack of
Ba122 bulks. This stack comprises the same ‘Ba122-Lower’bulk analysed in section 2.3, but polished to a thickness of
3.7 mm, and an upper bulk, ‘Ba122-Upper,’ of diameter
10 mm and thickness 14.7 mm. A 1.6 mm spacer (notmodelled) exists between the two bulks to accommodate aHall sensor.

2.3. Single Ba122 bulk

Figure 3 shows a comparison of the experimental andnumerical simulation results for the single Ba122 bulk,
3Supercond. Sci. Technol. 30 (2017) 105009

M D Ainslie et al

Figure 4. Comparison of experimental [3] and numerical simulation

Figure 7. Numerical simulation results for diameter dependence of

results for the stack Ba122 of bulks shown in ﬁgure 1(b) and using
the measured and estimated Jc(B) data shown in ﬁgure 2.

the trapped ﬁeld across the top surface (z = +0.5 mm) of the Ba122
bulk with ﬁxed thickness, H, of 10 mm, for an operating temperatureof 20 K. The B(r=0 mm) values correspond to the values given inﬁgure 5.

Figure 5. Numerical simulation results for the diameter dependenceof the trapped ﬁeld at r=0 mm, z=+0.5 mm above the top
surface, Bt, of the Ba122 bulk with ﬁxed thickness, H, of 10 mm, for
operating temperatures of 5 and 20 K. The measured Jc(B, T)
characteristics of ‘Ba122-lower’ presented in section 2.2 areassumed.

Figure 8. Numerical simulation results for the thickness dependenceof the trapped ﬁeld at r=0 mm, z=+0.5 mm above the top
surface, Bt, of the Ba122 bulk with ﬁxed diameters, D, of 30 and
50 mm, for operating temperatures of 5 and 20 K. The measured
Jc(B, T) characteristics of ‘Ba122-lower’ presented in section 2.2 areassumed.

For the experimental measurement, the bulk is FC(Bapp = 8 T) to approximately 5 K, with the ﬁeld subsequently
reduced to zero, followed by increasing the temperature at a
rate of 0.2 K min−1 [3]. Hall sensors located at the centre of
the 1.6 mm spacer in between the upper and lower bulks(labelled H2) and at the centre of the bottom surface of thelower bulk (labelled H1) measured the magnetic ﬁeld density,
Bz [3].
In the numerical simulations, the trapped magnetic ﬁeld,
Bt, is calculated t = +10 min after the application andremoval of the magnetising ﬁeld, at z=−5.5 mm, r=0 mm
and z=−10.5 mm, r=0 mm, corresponding to these Hallsensor locations. As per the previous section, the simulation
results are given for discrete points between 5 and 25 K at
increments of 5 K, within the range of the input Jc(B) data.
Again, the simulations reproduce the experimental results
well, and there is of course scope for improvement of theaccuracy of the simulation by using a more exact, measured

Figure 6. Numerical simulation results for diameter dependence ofthe trapped ﬁeld across the top surface (z=+0.5 mm) of the Ba122
bulk with ﬁxed thickness, H, of 10 mm, for an operating temperatureof 5 K. The B(r=0 mm) values correspond to the values given inﬁgure 5.
4

Supercond. Sci. Technol. 30 (2017) 105009

M D Ainslie et al

Figure 12. Numerical simulation results for thickness dependence of

Figure 9. Numerical simulation results for thickness dependence ofthe trapped ﬁeld across the top surface (z = +0.5 mm) of the Ba122
bulk with ﬁxed diameter, D, of 50 mm, for an operating temperatureof 20 K. The B(r = 0 mm) values correspond to the values given inﬁgure 8.

the trapped ﬁeld across the top surface (z = +0.5 mm) of the Ba122
bulk with ﬁxed diameter, D, of 30 mm, for an operating temperatureof 5 K. The B(r = 0 mm) values correspond to the values given inﬁgure 8.

Jc(B, T) characteristic for the upper bulk, rather than theestimation described in section 2.2. Nevertheless, the simulations are a useful tool to estimate the trapped ﬁeld in suchbulks or stacks of bulks. In the next section, the inﬂuence ofthe geometric dimensions (thickness and diameter) on thetrapped ﬁeld is analysed, with a view of fabricating larger
samples to increase the magnetic ﬁeld, since their fabricationis scalable, versatile, low-cost and based on common industrial ceramic processing techniques.

3. Inﬂuence of geometric parameters
Using the numerical model and the same assumptions for thesuperconducting properties, the trapped ﬁeld of larger Ba122
samples can be predicted without the need for further
experiments. In this section, the measured ‘Ba122-Lower’Jc(B, T) characteristics presented in section 2.2 are assumed
and an analysis of the diameter and thickness dependence ofthe trapped ﬁeld is carried out for different sample sizes.

Figure 10. Numerical simulation results for thickness dependence ofthe trapped ﬁeld across the top surface (z = +0.5 mm) of the Ba122
bulk with ﬁxed diameter, D, of 30 mm, for an operating temperatureof 20 K. The B(r = 0 mm) values correspond to the values given inﬁgure 8.

3.1. Fixed thickness, diameter variation

Figure 5 shows the diameter dependence of the trapped ﬁeld
at r=0 mm, z=+0.5 mm above the top surface, Bt, of theBa122 bulk with a ﬁxed thickness, H, of 10 mm, for operating
temperatures of 5 and 20 K. Figures 6 and 7 show the trapped
ﬁeld across the top surface (again at z=+0.5 mm) for 5 and
20 K, respectively, where B(r=0 mm) corresponds to thevalue given in ﬁgure 5. As per section 2, the Bt value is given
for t=+10 min after the application and removal of themagnetising ﬁeld. The trapped ﬁeld increases with increasing
sample diameter and the rate of increase decreases with diameter. This occurs for two reasons: ﬁrstly, from Bean’s critical state model as applied to a thin disc-shaped bulk (3):Figure 11. Numerical simulation results for thickness dependence ofthe trapped ﬁeld across the top surface (z = +0.5 mm) of the Ba122
bulk with ﬁxed diameter, D, of 50 mm, for an operating temperatureof 5 K. The B(r = 0 mm) values correspond to the values given inﬁgure 8.

Bt = km 0 Jc R,(3 )where R=D/2 is the sample radius and k is a correction
factor to the simple Bean (slab) approximation due to the5Supercond. Sci. Technol. 30 (2017) 105009

M D Ainslie et alﬁnite thickness, H, of a disc-shaped bulk superconductor
sample, given byk=⎛RHln ⎜⎜ +
2R ⎝ H⎛ R ⎞2 ⎞1 + ⎜ ⎟ ⎟⎟.
⎝H ⎠ ⎠

readily be achieved at 5 K (and >1 T at 20 K) with a sample
of diameter 50 mm. Finally, an aspect ratio of between 1 and
1.5 for R/H (radius/thickness) would be an appropriatecompromise between the accessible, surface trapped ﬁeld andvolume of superconducting material for bulk Ba122 magnets.

(4 )It can be seen that the correction factor k decreases withincreasing aspect ratio, which reduces the trapped ﬁeld, but Jc
also reduces with increasing diameter because of the larger
magnetic ﬁeld generated from the supercurrent ﬂowing over alarger area. In contrast with MgB2 bulks, whose Jc(B) characteristics follow an exponential law that can decrease rapidly
in ﬁeld [10], rapid saturation of Bt is not observed for the
Ba122 bulks with increasing diameter. Enhancing the pinningto improve the Jc(B, T) characteristics is another obvious
route to increasing the trapped ﬁeld; however; the simulation
results suggest that, based on current superconducting properties, surface trapped ﬁelds >2 T could readily be achieved
at 5 K (and >1 T at 20 K) with a sample of diameter 50 mm.
In the following subsection, we investigate the thickness
dependence of the trapped ﬁeld for ﬁxed diameters, D, of 30
and 50 mm, which are typical sizes of practical bulk hightemperature superconductors.

Acknowledgments
Mark Ainslie would like to acknowledge ﬁnancial support
from a Royal Academy of Engineering Research Fellowship.
Hiroyuki Fujishiro would like to acknowledge ﬁnancial support from JSPS KAKENHI Grant No. 15K04646. The workat NHMFL was supported NSFDMR-1306785, by the
National High Magnetic Field Laboratory, which is supportedby the National Science Foundation under NSFDMR1157490, and by the State of Florida. The work at TUAT was
supported by MEXT Elements Strategy Initiative to Form
Core Research Center and by JSPS KAKENHI Grant No.
JP15H05519.

ORCID iDs

3.2. Fixed diameter, thickness variation

Mark D Ainslie

Figure 8 shows the thickness dependence of the trapped ﬁeld
at r=0 mm, z=+0.5 mm above the top surface, Bt, of theBa122 bulk with ﬁxed diameters, D, of 30 and 50 mm, for
operating temperatures of 5 and 20 K. Figures 9–12 show thetrapped ﬁeld across the top surface (again at z=+0.5 mm)
for 5 and 20 K for D=30 mm (ﬁgures 9 and 10) andD=50 mm (ﬁgures 11 and 12). The trapped ﬁeld saturates as
the thickness, H, of the bulk approaches D, which can be
deduced from (3) and (4), and is observed for bulk (RE)BCO
superconductors [20], as well as bulk MgB2 [10]. From theseresults, the geometry of the bulk can be optimised, and as
found in [21] for bulk (RE)BCO superconductors, an aspectratio of between 1 and 1.5 for R/H (radius/thickness) would
also be an appropriate compromise between the accessible,
surface trapped ﬁeld and volume of superconducting material
for bulk Ba122 magnets.

https://orcid.org/0000-0003-0466-3680

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4. Conclusion
In this paper, a 2D axisymmetric ﬁnite-element model
implementing the H-formulation is used to investigate themagnetisation properties of iron-pnictide (Ba122) bulk
superconductors. Using the measured Jc(B, T) characteristics
of a small specimen taken from a bulk Ba122 sample,
experimentally measured trapped ﬁelds are reproduced well
for a single bulk, as well as a stack of bulks. The inﬂuence ofthe geometric dimensions (thickness and diameter) on thetrapped ﬁeld was then analysed, with a view of fabricatinglarger samples to increase the magnetic ﬁeld available fromsuch TFMs. It is shown that, with current state-of-the-art
superconducting properties, surface trapped ﬁelds >2 T could6Supercond. Sci. Technol. 30 (2017) 105009

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