Faculté des Sciences Appliquées

Département d’Électricité, Électronique

et Informatique

Magnetic properties of structures combining

bulk high temperature superconductors

and soft ferromagnetic alloys

Thèse présentée par

Matthieu P. PHILIPPE

en vue de l’obtention du grade de

Docteur en Sciences de l’Ingénieur

Juin 2015

3

Remerciements / Acknowledgements

Au terme de la rédaction de ma thèse, je souhaite dire merci à toutes les personnes qui m’ont

aidé, soutenu, encouragé au cours de ce long chemin. Mon merci s’adresse aussi à celles et

ceux qui vont lire ce manuscrit.

Ma gratitude va d’abord au Pr Philippe Vanderbemden, mon promoteur, qui m’a, il y a

quelques années déjà, proposé un poste d’assistant dans son service et donné l’opportunité de

faire un doctorat. Je le remercie pour les nombreuses discussions claires et précises toujours

menées dans un agréable cadre de travail où l’on trouve une solide équipe qui unit ses efforts

dans une ambiance sympathique. Cette équipe, d’ailleurs réputée, pouvant s’appuyer sur des

infrastructures robustes et du matériel de qualité. Surtout, je le remercie pour son suivi actif et

son indéfectible soutien jusqu’au bout de l’élaboration de cette thèse, je le remercie aussi

pour ses nombreuses relectures et corrections, rigoureuses et complètes dans des délais

parfois fort courts. Je lui suis, en outre, très reconnaissant pour le cours de « Mesures

Électriques », au cours duquel, grâce aux remarquables labos, j’ai découvert mon goût pour

l’électricité. J’ai ensuite eu le privilège d’enseigner cette matière et de faire partager mon

intérêt pour cette discipline.

Au Pr Benoit Vanderheyden vont aussi mes remerciements pour ses nombreux conseils, pour

les discussions sérieuses teintées d’humour, pour les multiples explications et précisions sur

les aspects mathématiques et physiques de l’électromagnétisme qui me semblent plus

compréhensibles aujourd’hui, pour ses relectures attentives de mes textes toujours

accompagnées de remarques pertinentes.

I would like to thank Prof Christophe Geuzaine, Prof Hervé Caps, Prof Kevin Berger and

Dr Xavier Granados for agreeing to be members of the jury and to read my manuscript. I also

thank all those who provided me with advice and gave me the opportunity to carry out my

experimentations using the best superconducting samples. I am grateful to Prof David

Cardwell, team leader of the Bulk Superconductivity Group at the University of Cambridge, UK,

for his collaboration and his close reading of my papers. Among Prof Cardwell’s team, I wish to

thank Prof Archie Campbell for his insight and comments, Dr Zhihan Xu and Dr Mark Ainslie for

their excellent modelling work for our common papers, Dr Yunhua Shi and Mr Anthony Dennis

for providing me with top-quality superconducting samples. My thanks also go to Dr Hidekazu

Teshima and his co-workers from Nippon Steel & Sumitomo Metal Corporation, Futtsu, Japan,

for sending us bulk superconductors with outstanding performances. Finally I wish to thank

4

Dr Devendra Kumar Namburi for his kindness, his wonderful optimism, his colourful

presentations and his great help with the PPMS.

Je sais gré de son accueil à toute l’équipe de recherche au sein de laquelle j’ai eu la chance

d’évoluer dans une ambiance de travail harmonieuse et pragmatique. Merci à Jean-François

Fagnard pour son aide tant pratique que théorique, pour les réalisations minutieuses de

bobines et autres capteurs, pour sa très bonne connaissance du laboratoire, pour son aide

dans la réalisation des mesures, pour ses nombreuses explications précises, et pour ses

relectures soigneuses de mes textes. Merci à Laurent Wéra, Kévin Hogan et Sébastien Kirsch

pour les réalisations et discussions de qualité. Merci à Pascal Harmeling et Angel Calderon pour

leur précieuse aide technique. Merci à Joseph Simon pour les rigolades, les pauses café et

l’usinage. Je tiens aussi à remercier Grégory Lousberg et Cédric Marchal pour leurs explications

et débats qui m’ont beaucoup appris. Merci aussi à Philippe Laurent pour son enseignement

des labos du cours de mesures électriques et pour les judicieux conseils techniques. Je

souhaite aussi remercier vivement Raphael Egan et Simon Debois qui ont respectivement

développé le magnétomètre et l’option rotator pour le PPMS lors de leur travail de fin d’étude.

Le doctorat offre également de belles opportunités de découverte, de partage et de

rencontres avec la communauté scientifique mondiale. J’ai eu l’occasion de présenter mes

résultats à plusieurs conférences internationales, et ce avec le soutien de l’Université de Liège

(Patrimoine, ARC 11/16-03), et du FRS-FNRS.

Toute cette aventure n’aurait pas été la même sans l’ambiance agréable et propice de

Montefiore et la présence de nombreux collègues exceptionnels. Dans le désordre : Guy,

Etienne, Gauthier, Stéphane, Anne, Maxime, Laura, Denis, Vincent pour les très bons moments

passés à Montef, de Norbert à Syntia, en passant par les boulets, la Kfet et bien d’autres

occasions heureuses.

Je tiens également à remercier tous mes amis à qui ce travail parle peu mais qui m’ont soutenu

par leur présence, leur humour, leurs idées concrétisées de vacances hors du commun qu’ils

ont voulues dynamiques et joyeuses.

Je remercie toute ma famille pour son soutien et sa patience à mon égard, mes parents de

m’avoir écouté et encouragé, ma petite sœur, mes grands-parents, avec une pensée

particulière pour les lundis midi, mon parrain et ma marraine.

Et surtout, je remercie Elodie pour son amour, son écoute, ses encouragements, son courage,

ses concessions, et son immense compréhension à mon égard.

5

Summary

The purpose of the present work is to determine experimentally and numerically how the

magnetic flux density both inside and outside a large grain, bulk high temperature

superconductor is modified when placed in the vicinity of axisymmetric ferromagnetic

components. The investigated superconductors are bulk (RE)Ba2Cu3O7 cylinders of a few cm³.

The ferromagnets are of various sizes and shapes, machined out of well characterized, soft

magnetic alloys of high permeability. Both superconductor and ferromagnet have the same

diameter and are cooled at the liquid nitrogen temperature (𝑇 = 77 K).

The properties of the superconductor / ferromagnet (SC/FM) hybrid structures are

investigated through surface (Hall probe) and volume measurements carried out in the fully

magnetized remanent state and under complete cycles of field applied either parallel or

perpendicular to the sample c-axis. The design of a suitable volume characterization method

for large samples and its validation through comparison with other measurements is part of

the present work. Numerical models based on the Brandt and finite element methods are

compared with the measurements and used to investigate configurations and physical

quantities that are not accessible experimentally.

We show that the ferromagnet acts as a magnetic short-circuit and creates a low reluctance

path that drives the flux lines directly towards the edges of the superconducting magnet. The

zone above the ferromagnet is shielded from the flux trapped inside the superconductor.

Conversely, the flux density can often be enhanced on the face opposite to the ferromagnet

(superconductor side); it is suggested that the flux density increase due to the ferromagnet is

more significant when combined to a rather thin superconducting pellet. In all the studied

cases, the presence of the ferromagnet was found to increase both the average trapped flux

inside the superconductor and the magnetic moment of the whole SC/FM structure (including

the ferromagnet).

The ferromagnet ability to divert magnetic flux lines depends on whether it is partially or fully

saturated. This saturation is found to be governed in the remanent state by the ferromagnet

thickness d, its saturation magnetization 𝑀sat and the flux produced by the superconductor

(proportional to its critical current density 𝐽c). The amount of diverted / shielded flux does not

change once the ferromagnet is fully saturated and the effect of the ferromagnet becomes

relatively less important as the generated flux increases. Hybrid structures are therefore

6

relevant even when the trapped flux density exceeds the typical saturation magnetization of

ferromagnetic materials (≈ 2 T). For a given critical current the higher the saturation

magnetization and/or the thicker the ferromagnetic material, the larger the shielding effects

on the ferromagnet side and the higher the trapped field on the superconductor face for the

investigated sample. For a ferromagnetic disc with the same diameter 𝑎 as the

superconductor, the most suitable ferromagnet thickness 𝑑∗ can be roughly estimated by a

simplified analytical expression 𝑑∗ ≈ 𝐽c0 𝑎2 6𝑀sat⁄ , below which nearly full saturation of the

FM occurs and above which weak thickness dependence is observed. The magnitude of the

supercurrent is an important parameter but its particular 𝐽c(𝐵) dependence is found not to be

a crucial parameter affecting the remanent state properties of the modelled hybrid

configurations. In addition, the time relaxation of the trapped magnetization was found not to

be influenced by the presence of the ferromagnet, within measurement uncertainties.

The ferromagnet also influences the magnetic flux density when the assembly is subjected to

an external magnetic field. The penetration of the magnetic flux inside the superconductor is

delayed in the vicinity of the ferromagnet. The hysteresis loops of the flux density averaged on

the superconductor volume show a combination of diamagnetic and ferromagnetic behaviour

for SC/FM structures. It appears to be the simple superimposition of the hysteresis cycles of

the superconductor and the ferromagnet at high applied field (i.e. exceeding the apparent

saturation of the ferromagnet). This simple “addition” rule can be used to effectively predict

the magnetic behaviour of larger or more complex hybrid structures and to modulate the

magnetic flux density. As in the remanent sate, a thicker ferromagnet has a more significant

effect on the whole hysteresis loop of the superconductor, but this effect varies less than

proportionally to the increase in the ferromagnet volume.

For a given volume of ferromagnetic material, the shape of an unsaturated ferromagnet is not

of prime importance provided that the ferromagnet covers the entire surface of the

superconductor. A succession of holes and ferromagnetic sections could be used to modulate

the flux density. A similar volume of ferromagnetic material was shown to have more influence

on the average remanent volume flux density if it is split on both faces of the superconductor

to form a FM/SC/FM structure instead of a SC/FM structure.

Two additional advantages of ferromagnets were identified in this work. First, the ferromagnet

was shown to be beneficial in improving the field gradient – and therefore the magnetic

levitation force – outside the superconductor. Second, the addition of the ferromagnetic disc

on one side of the superconductor reduces the collapse of the trapped flux density when

subjected to several cycles of magnetic field applied perpendicularly to its remanent

magnetization (parallel to the c-axis), i.e. in the so-called “crossed-field configuration”.

7

Résumé

L’objectif de ce travail est de déterminer à l’aide d’expériences et de simulations numériques

comment l’induction magnétique est modifiée dans le volume et au voisinage d’un

supraconducteur (SC) à haute température critique placé à proximité d’une pièce

ferromagnétique (FM). Les supraconducteurs étudiés sont des cylindres massifs et monograins

de quelques cm3 en (RE)Ba2Cu3O7. Les pièces ferromagnétiques sont toujours axisymétriques,

mais de différentes formes et tailles, fabriquées dans des alliages magnétiques doux à haute

perméabilité et dont les propriétés ont été mesurées préalablement. Dans chaque structure, le

supraconducteur et la pièce FM ont le même diamètre et sont refroidies à la température de

l’azote liquide (𝑇 = 77 K).

Les propriétés des structures hybrides supraconductrices / ferromagnétiques (SC/FM) sont

étudiées via des mesures en surface (mesures par sonde de Hall) et en volume, tant pour des

échantillons supraconducteurs entièrement aimantés que lors de cycles complets de champ

appliqué soit parallèlement, soit perpendiculairement à l’axe « c » de l’échantillon. Une

méthode de mesure adaptée pour de gros échantillons est développée puis validée par

comparaison avec d’autres mesures. Les résultats de modèles numériques basés sur les

méthodes de Brandt et des éléments finis sont validés par comparaison avec les mesures. Ces

modèles sont ensuite utilisés pour investiguer des configurations et des grandeurs physiques

qui ne sont pas accessibles par l’expérience.

Nous montrons que la pièce FM agit comme un court-circuit magnétique en créant un chemin

de faible réluctance qui conduit les lignes de flux directement vers les bords du

supraconducteur aimanté. La zone au-dessus de la pièce FM est protégée du flux piégé dans le

supraconducteur tandis que l’induction magnétique peut être augmentée sur la face opposée

(côté supraconducteur). De ce côté, il est suggéré que l’augmentation relative de l’induction

est plus importante lorsque que le cylindre supraconducteur est relativement plat. Dans les cas

étudiés, on a observé que la présence de la pièce FM entraine une augmentation du flux

moyen dans le supraconducteur et du moment magnétique de la structure SC/FM complète.

L’état de saturation (complet ou partiel) de la pièce FM influence la redirection des lignes

d’induction. En l’absence de champ externe, on observe que cette saturation est régie par

l’épaisseur 𝑑 de la pièce FM, par son aimantation à saturation 𝑀sat et par le flux produit par le

supraconducteur (ce flux étant proportionnel au courant critique 𝐽c). La quantité de flux

8

redirigé/blindé ne change presque plus une fois que la pièce FM est complètement saturée ;

l’effet de la pièce FM devient alors relativement moins important lorsque le flux généré

augmente. Les structures hybrides se montrent dès lors utiles même si le flux piégé dépasse la

valeur habituelle de saturation des matériaux ferromagnétiques (≈ 2 T). Pour un courant

critique donné dans le supraconducteur, plus l’aimantation à saturation est élevée et/ou plus

la pièce FM est épaisse, plus le blindage sera important du côté FM et plus l’induction en

surface sera augmentée côté SC dans les configurations qui ont été étudiées. Si la pièce FM est

un disque de même diamètre 𝑎 que le supraconducteur, l’épaisseur caractéristique

𝑑∗ ≈ 𝐽c0 𝑎2 6𝑀sat⁄ a été introduite pour estimer le régime de saturation de la pièce FM. Pour

une épaisseur inférieure à 𝑑∗, la pièce FM est presqu’entièrement saturée, tandis qu’une faible

influence de l’épaisseur est observée dans le cas contraire. L’amplitude des courants

supraconducteurs est un paramètre important mais leur dépendance vis-à-vis de l’induction

magnétique n’influence que peu le comportement magnétique des structures hybrides

simulées. Aucune influence de la pièce FM n’a été observée sur le taux de relaxation temporel

de l’aimantation piégée et ce dans la limite des incertitudes de mesures.

Lorsque la structure hybride est soumise à un champ magnétique externe, la pénétration du

flux dans le supraconducteur est retardée à proximité de la pièce FM. La courbe d’hystérèse de

l’induction volumique moyenne dans le supraconducteur est une combinaison de courbes

diamagnétique et ferromagnétique. Lorsque le champ appliqué est suffisant pour amener la

saturation apparente de l’entièreté de la pièce FM, on observe une simple addition des cycles

d’hystérèse mesurés pour le supraconducteur et la pièce FM séparément. Cette simple règle

d’addition peut être utilisée pour estimer efficacement le comportement magnétique de

structures hybrides larges et complexes et/ou pour en moduler l’induction magnétique.

Comme dans l’état rémanent, une pièce FM plus épaisse a plus d’influence et son effet sur la

courbe d’hystérèse augmente moins que proportionnellement avec son volume.

Pour un volume donné de matériau ferromagnétique, la forme de la pièce FM n’a que peu

d’importance tant que cette pièce n’est pas saturée et qu’elle couvre l’entièreté de la surface

du supraconducteur. Une succession de trous et pièces FM pourrait être utilisée pour moduler

l’induction magnétique en surface. On a observé qu’un même volume de matériau

ferromagnétique a plus d’influence sur l’induction moyenne dans le supraconducteur s’il est

réparti sur les deux faces du supraconducteur pour former une structure FM/SC/FM au lieu

d’une structure SC/FM.

Deux avantages supplémentaires ont été observés. D’une part, la pièce FM augmente le

gradient de l’induction — et donc la force de lévitation — en surface du supraconducteur.

D’autre part, l’addition d’un disque FM sur une face du supraconducteur réduit la décroissance

du flux piégé lorsque la structure est soumise à plusieurs cycles de champ magnétique

perpendiculaires au champ piégé selon l’axe « c » (« configuration en champs croisés »).

9

Contents

Remerciements / Acknowledgements .................................................................................. 3

Summary ............................................................................................................................. 5

Résumé ............................................................................................................................... 7

Contents ............................................................................................................................. 9

Chapter 1 Context and introduction ........................................................................... 13

1.1 Context ...................................................................................................................... 13

1.2 Goal ........................................................................................................................... 14

1.3 Structure of the manuscript ...................................................................................... 15

Chapter 2 Magnetic properties of bulk high temperature superconductors ................... 17

2.1 Superconductivity and superconducting materials .................................................. 17

2.2 Magnetization of trapped field magnets and Bean model ....................................... 20

2.2.1 Magnetization techniques .............................................................................. 20

2.2.2 The Bean critical state model ......................................................................... 21

2.2.3 Flux relaxation and power law model ............................................................. 23

2.3 Demagnetizing field .................................................................................................. 24

2.4 Properties and applications of bulk HTS and ferromagnetic materials .................... 25

2.5 Summary ................................................................................................................... 28

Chapter 3 Experimental methods ................................................................................. 29

3.1 Permeameter ............................................................................................................ 29

3.2 Measurement systems based on the PPMS® ............................................................ 31

3.2.1 Using PPMS temperature and field control for coil and Hall probe

measurements ............................................................................................................. 31

3.2.2 ACMS option ................................................................................................... 39

3.2.3 The rotator ...................................................................................................... 41

3.3 The magnetometer for large samples ....................................................................... 42

10 Contents

3.4 Hall probe mapping set-up ........................................................................................ 44

3.5 Summary of the measurement set-ups .................................................................... 45

Chapter 4 Modelling methods ..................................................................................... 49

4.1 The Brandt algorithm ................................................................................................ 50

4.2 The Finite Element Method ...................................................................................... 52

4.2.1 The A-formulation ........................................................................................... 53

4.2.2 The H-formulation ........................................................................................... 57

4.2.3 Campbell’s equation ....................................................................................... 58

4.3 Summary of the models and collaborations ............................................................. 59

Chapter 5 Characterization of the superconducting and magnetic materials ................. 61

5.1 Bulk YBCO superconductor (ESC2) ............................................................................ 61

5.1.1 Determination of the critical 𝑛-exponent ....................................................... 63

5.1.2 Determination of the critical current density 𝐽𝑐(𝐵) ...................................... 65

5.1.3 Probing inhomogeneities with sensing coils and Hall probes ........................ 73

5.1.4 Modelling of hysteresis curves ....................................................................... 77

5.1.5 Measurement and modelling of the magnetic flux density distribution ........ 78

5.1.6 Ageing of the superconductor ........................................................................ 82

5.2 Bulk GdBCO superconductor (ESJc) ........................................................................... 83

5.3 The ferromagnetic materials ..................................................................................... 86

5.3.1 Supra50 ........................................................................................................... 86

5.3.2 Permimphy ...................................................................................................... 89

5.3.3 C45 steel .......................................................................................................... 90

5.3.4 Ferrite .............................................................................................................. 91

5.4 Summary ................................................................................................................... 93

Chapter 6 Superconductor / ferromagnetic disc hybrid structure .................................. 97

6.1 Reference configuration (D2) .................................................................................... 97

6.1.1 Modelling of the flux density .......................................................................... 98

6.1.2 𝛥𝐵 hysteresis curves ..................................................................................... 103

6.1.3 Surface profiles of the flux density ............................................................... 108

6.2 Influence of the disc thickness ................................................................................ 111

6.2.1 Surface profiles of the magnetic flux density ............................................... 111

Contents 11

6.2.2 𝛥𝐵 Hysteresis curves .................................................................................... 117

6.3 Influence of the type of ferromagnetic material .................................................... 119

6.4 Influence of the critical current of the superconductor ......................................... 122

6.5 Summary ................................................................................................................. 126

Chapter 7 Investigation of more complex structures................................................... 131

7.1 SC/FM structures with different ferromagnet shapes ............................................ 131

7.1.1 𝛥𝐵 Hysteresis curves .................................................................................... 132

7.1.2 Flux density profiles ...................................................................................... 133

7.2 FM/SC/FM configurations ....................................................................................... 136

7.2.1 FM/SC/FM structures with ferromagnetic discs ........................................... 136

7.2.2 FM/SC/FM structures with different ferromagnet shapes ........................... 139

7.2.3 FM/SC/FM structure made with Permimphy discs ....................................... 140

7.3 SC/FM structure with the GdBCO superconducting sample ................................... 142

7.3.1 Magnetic moment ........................................................................................ 142

7.3.2 “Crossed” magnetic fields ............................................................................. 144

7.4 Summary ................................................................................................................. 149

Chapter 8 Conclusions and outlook ............................................................................ 151

Publications ..................................................................................................................... 157

Papers in international journals ........................................................................................ 157

Participation at conferences ............................................................................................. 157

References ...................................................................................................................... 159

12 Contents

13

Chapter 1

Context and introduction

1.1 Context

Bulk, high temperature superconductors are able to trap record magnetic fields and have a

significant potential for use as powerful permanent magnets in a variety of applications. The

typical trapped field in bulk superconductors is well beyond the saturation magnetization of

conventional ferromagnets. Flux densities above 3 T were demonstrated in bulk MgB2 [1, 2, 3,

4] and values exceeding 17 T were reported in bulk (RE)Ba2Cu3O7 large grain materials [5, 6],

where (RE) denotes a rare-earth element (Y, Dy, Gd, etc.). This makes these materials

extremely promising as a competing technology for traditional permanent magnets in various

engineering applications [7, 8], such as motors and generators [9, 10], magnetic levitation

systems [11, 12], magnetic bearings [13, 14], cancer therapy [15] and waste water treatment

[16]. This thesis focuses on bulk high temperature superconductors of the (RE)Ba2Cu3O7 family

which are characterized by critical superconducting transition temperatures 𝑇c in the range

90–95 K and which can therefore operate at the liquid nitrogen temperature (𝑇 = 77 K). They

are generally produced in the form of a cylindrical puck of a few cm³.

The combination of ferromagnetic and superconducting materials can enhance the

performance of the superconductor. Bulk superconductors and ferromagnetic materials

modify the path of the magnetic flux in magnetic circuits in fundamentally different ways, so

that any combination of these materials often leads to very interesting magnetic behaviour of

the composite structure. Superconductors may either repel individual, quantized flux lines

completely (i.e. perfect diamagnetism) or, in the case of type II superconductors containing

strong pinning, restrict their movement and act potentially as very strong quasi-permanent

magnets. In contrast, due to their high magnetic permeability, ferromagnetic materials tend to

channel magnetic flux by providing a low reluctance path. Superconductors can therefore be

combined with ferromagnets to modify the distribution of the magnetic flux lines and to

improve the superconducting properties of the composite structure. Typical applications

include e.g. rotating machines, where superconductors are naturally in the vicinity of

ferromagnetic frame (either in the stator or in the rotor) and in which both flux shielding and

flux trapping of superconductors can be exploited [17, 18]. By reducing the reluctance, a

ferromagnetic yoke can also be used to improve the efficiency of the magnetization process

[19, 20].

14 Chapter 1. Context and introduction

Although these studies are of extremely valuable practical interest, they do not include a

detailed study of the magnetic properties of a whole “hybrid” sample consisting of a bulk

superconductor attached to a piece of soft ferromagnetic material. In particular, such

application-related works focus on the magnetic flux density in the vicinity of the

superconductor. This magnetic flux density is probed by surface measurements outside the

bulk superconductor, e.g. using Hall sensors. In doing so, the measured signal is mainly

influenced by current loops circulating in a thin layer close to the surface and is insensitive to

“true” volume effects arising in the bulk sample. Volume measurements, on the other hand,

would be better representative of the intrinsic physical behaviour of the hybrid structure. Due

to both their large size and magnetic moments, however, there is no “off-the-shelf”

measurement device suitable for studying such large bulk samples.

1.2 Goal

The purpose of the present work is to determine experimentally and numerically how the

magnetic flux density both inside and outside a large grain, bulk high temperature

superconductor is modified when placed in the vicinity of axisymmetric ferromagnetic

components. The investigated superconductors are bulk (RE)Ba2Cu3O7 cylinders of a few cm³.

The ferromagnets are of various sizes and shapes, machined out of well characterized, soft

magnetic alloys of high permeability. In contrast to previous studies, the properties of the

ferromagnet / superconductor hybrid structures will be investigated through combined surface

and volume measurements. The design of a suitable volume characterization method and its

validation through comparison with other measurements will be part of the present work.

These different characterization methods will be used to understand in detail the influence of

the ferromagnet on the performances of the superconductor and to determine the most

relevant physical or geometrical parameters that need to be taken into account.

Without the ferromagnet, the magnetic flux distribution above a bulk superconducting magnet

is strongly non-uniform (conical profile predicted by the Bean model [21, 22]). The natural

question to be addressed is whether ferromagnets can be used either to increase the

maximum value of the flux distribution flux or to shape this magnetic induction and improve

the flux uniformity. Knowing that the saturation magnetization of ferromagnets is physically

limited to around 2 T or less, another important question is to determine whether the

ferromagnets can still be used in a regime where the trapped flux density in the

superconductor exceeds this value. An additional specificity of the work reported here is that

we will investigate the full hysteretic behaviour of the hybrid structure when it is subjected to

one complete cycle of the applied field. The field will be either swept along one given

direction, or (in the last chapter of this work) perpendicularly to the remanent magnetization.

When possible, numerical models are compared with the measurements and then used to

investigate configurations and physical quantities that are not accessible experimentally.

Chapter 1. Context and introduction 15

In summary, the questions that will be addressed are as follows.

(1) How does a ferromagnetic piece modify the magnetic behaviour of a large grain, bulk

superconductor when it is brought close to it and what are the relevant parameters that

influence this behaviour?

(2) Is it possible to deduce some general design rules to investigate the significance of

ferromagnet effects?

(3) Is it possible to shape the trapped flux profile available above the superconductor using

ferromagnets?

(4) How do the ferromagnet sections behave once they are saturated by the flux lines

produced by the superconductor?

(5) Are there other advantages of using ferromagnets in the vicinity of bulk

superconductors?

1.3 Structure of the manuscript

This manuscript describes the measurement and modelling techniques used to study the

magnetic interaction of superconductors and ferromagnetic pieces. The results are then

presented and discussed. The manuscript is organised as follows:

Chapter 2 gives a short reminder of the theory of superconductivity and describes

various applications in which they are combined with ferromagnetic materials.

Chapter 3 describes the experimental set-ups used to characterize the individual

materials and the superconductor / ferromagnet hybrid structures.

Chapter 4 describes the modelling frameworks used to model the hybrid structures and

to investigate the magnetic properties.

Chapter 5 is devoted to the determination of the magnetic properties of each material

separately. Modelling results are compared to measurement results to assess the

suitability of the models. The magnetic properties of both the ferromagnetic material

(hysteresis cycle) and the superconductors (critical current density, 𝑛-value describing

the flux relaxation) are determined from independent experiments.

In Chapter 6, we discuss the modelling and experimental results obtained on a

superconductor / ferromagnet (SC/FM) hybrid structure made with a ferromagnetic disc

and a YBCO superconducting sample. Modelling methods reproducing the observed

measurement results are used to understand in detail how the ferromagnetic

components modify the magnetic flux density above each side of the SC/FM assembly.

The models are then used to predict the magnetic behaviour in other cases. The

influence of (i) the thickness of the ferromagnet, (ii) its saturation properties, and (iii)

the superconductor properties (critical current density 𝐽c) on the magnetic flux

distribution are successively investigated.

16 Chapter 1. Context and introduction

In chapter 7, we investigate more complex structures made of ferromagnets of different

shapes and FM/SC/FM configurations. In the last part of this chapter, we investigate a

second, GdBCO superconducting sample and specifically the influence of a

ferromagnetic disc on its magnetic moment and under magnetic fields applied

perpendicularly to its magnetization.

Finally, chapter 8 gives a general conclusion of this dissertation and suggests further

prospects.

17

Chapter 2

Magnetic properties of bulk high

temperature superconductors

Superconductivity is a remarkable physical property of certain materials that can drive an

electric current without resistance and interact in a unique way with the magnetic field. This

chapter gives a brief reminder about superconductivity and focuses on the magnetic

properties of large grain, bulk high temperature superconductors. The applications of these

superconductors are then detailed, with a particular emphasis on their possible interaction

with ferromagnetic materials.

2.1 Superconductivity and superconducting materials

Superconductivity occurs when certain materials are cooled below a critical temperature 𝑇𝑐,

i.e. in the so-called superconducting state. Specifically, their electrical resistance drops to a

non-measurable value and the magnetic field is expelled from their body. Three

interdependent conditions must be met for a superconductor to be in the superconducting

state: its temperature must be lower than the critical temperature 𝑇c, the magnetic field must

be lower that the critical field 𝐻c and the current flowing in the material must be lower than

the critical current density 𝐽c. Fig. 2.1 shows a schematic representation of the 𝐽 − 𝑇 − 𝐻

phase diagram with the surface separating the material superconducting state from its normal

state. These three parameters are functions of each other. For example, the maximum current

density in a superconductor will decrease with increasing temperature and/or magnetic field

(which includes the self-field generated by this current).

Superconducting materials are divided in two groups based on the way they interact with the

magnetic field. Type I superconductors are perfectly diamagnetic materials. The magnetic field

is totally expelled from the volume of the superconductor provided that this magnetic field

remains below the critical field 𝐻c. This phenomenon, called the Meissner effect, occurs even

if the magnetic field was present before cooling the superconductor below 𝑇c. This expulsion

of the magnetic field is due to a current that flows in a thin layer at the surface of the

superconductor. The characteristic depth 𝜆 of this layer is the London penetration depth, and is

of the order of 0.1 µm for many superconducting materials. The 𝐻–𝑇 phase diagram at a fixed

temperature of type I superconducting materials is shown in Fig. 2.2 (left). Many pure metallic

18 Chapter 2. Magnetic properties of bulk high temperature superconductors

elements are Type I superconductors with a critical temperature of a few kelvins, e.g.

aluminium (𝑇c = 1.2 K) or indium (𝑇c = 3.4 K) [23].

Fig. 2.1 Schematic representation of the 𝐽 − 𝑇 − 𝐻 phase diagram of a

superconductor delimiting the superconducting state of the material.

Fig. 2.2 Phase diagram of type I (left) and type II (right) superconducting materials

in the 𝐽 = 0 plane. Typical values are 𝜇0𝐻c = 0.1 T (for elements), 𝜇0𝐻c1 = 0.01 T and

𝜇0𝐻c2 = 100 T at 0 K (for high-temperature superconductors).

Type II superconductors exhibit an intermediate magnetic state, called the mixed state,

between the Meissner and normal states, as shown in Fig. 2.2 (right). In the mixed state, the

material still has a zero electrical resistance when the current density is smaller than 𝐽c but is

penetrated by a certain amount of magnetic flux. As shown in Fig. 2.2 (right), this penetration

occurs for a magnetic field lying between two critical fields 𝐻c1 and 𝐻c2 whose typical values

at 0 K are 𝜇0𝐻c1 0.01 T and 𝜇0𝐻c2 100 T for the superconductors studied in this work.

Below 𝐻c1, the superconductor is in the Meissner state and the flux is totally expelled from the

material body. Above 𝐻c2, the magnetic material is no longer superconducting. Between 𝐻c1

and 𝐻c2, the flux penetrates inside the superconductor in the form of vortices: the number of

𝐻

𝑇c(0)

𝐽

𝑇

𝐽c(0)

𝐻c(0) Normal state

Superconducting

state

Chapter 2. Magnetic properties of bulk high temperature superconductors 19

vortices increases with increasing applied field. Each vortex can be viewed as a filament of

magnetic flux flowing through a normal core surrounded by supercurrents. The magnetic flux

associated to each vortex is quantified and equal to 𝜙0 = ℎ/2𝑒 ≈ 2.07 10-15 Wb (ℎ is the Planck

constant and 𝑒 the electron charge). The radius of the core is the coherence length which is

of the order of 0.1 µm for type I materials and a few nm for cuprates. Supercurrent loops flow

over a characteristic length 𝜆 around a vortex. In a hypothetic defect-free material, vortices

are free to move; they rearrange themselves as a hexagonal Abrikosov lattice. This is the

reversible regime and any vortex motion causes power dissipation. In practice, however,

superconducting materials usually contains a number of defects acting as pinning centres to

which the vortices are attracted. When vortices are pinned, the superconductor is said to be in

the irreversible mixed state. In some materials, a large enough magnetic field can drive the

material from the irreversible to the reversible state; this field is called the irreversibility field

𝐻irr.

When the material is subjected to both a magnetic field and an electric current, the vortex

lattice experiences a Lorentz-like force tending to unpin the vortices. If the pinning force is

sufficient, however, vortices do not move. Therefore there is no flux variation, the electric field

remains zero and a current can therefore flow through the material without losses. This

current can be either injected in the material (transport current) or induced magnetically. In

the latter case, the persistent current loops that flow in the material generate magnetic flux

density trapped inside the material and the superconductor behaves as a permanent magnet.

In practice, the defects acting as pinning centres are often introduced intentionally and their

proportion can be adjusted to improve the material critical current. Type II superconductors

include elements like vanadium (𝑇c = 5.03 K) or niobium (𝑇c = 9.26 K), metallic alloys like NbTi

(𝑇c = 10 K), copper oxide-based superconductors (𝑇c = 30–130 K), iron-based materials (𝑇c = 9–

56 K) and magnesium diboride MgB2 (𝑇c = 39 K) [23, 24, 25, 26, 27, 28].

The superconducting materials can also be classified according to their critical temperature.

While Type I materials have critical temperature of a few kelvins, several type II materials have

a 𝑇c above 30 K, these are called high-temperature superconductors (HTS). Among them,

several copper oxide-based superconductors have a critical temperature higher than the

boiling point of liquid nitrogen (77 K), which is of great interest for the cooling process. The

two materials studied in this work are among them: YBa2Cu3O7-δ (𝑇c = 93 K) and GdBa2Cu3O7-δ

(𝑇c = 95 K) [24, 29]. At 77 K, their critical fields 𝐻c1 and 𝐻c2 are approx. a few milliteslas and a

hundred of teslas, respectively. These two materials belong to the family (RE)Ba2Cu3O7-δ where

(RE) denotes a rare earth element such as Y, Dy, Gd, etc. For easier reference, these material

names are often shortened. For example, YBa2Cu3O7-δ can be written Y-123, YBaCuO, or YBCO.

Superconducting materials exist in different shapes (thin or thick films, cables, tapes, bars, bulk

pellets, etc.). This work focuses on bulk ceramic monoliths. The basic concept is to induce large

persistent current loops in them to create so-called trapped field magnets (TFM) that behave

like permanent magnets (PM). The total flux trapped in such material increases with both the

20 Chapter 2. Magnetic properties of bulk high temperature superconductors

radial size of the pellet and the current density flowing into it. In ceramic material, one has to

distinguish between the current flowing either in individual grains (intragranular current) or

across adjacent grains (intergranular current). In general, the intergranular current decreases

strongly with increasing misorientation angle between adjacent grains and can be significantly

smaller than the intragranular current, especially in non-textured (RE)BCO materials.

In this work, we will study (RE)Ba2Cu3O7-δ superconductors that can be produced as bulk

monoliths containing grains of macroscopic size (i.e. the term “large grain”) for which, in the

best case, the material consists of one “single grain”, also called “single domain”. Strictly

speaking, a single domain differs of a single crystal in that it contains nanometer-sized defects

that can act as efficient pinning centers for flux lines. The absence of grain boundaries in such

samples is crucial to achieve high current densities, and therefore high trapped flux densities.

Note also that the typical irreversibility field of (RE)BCO materials is much larger than that of

other HTS materials, which naturally makes them good candidates for trapped field magnets.

The processing technique in which one grain of macroscopic size can be achieved is called “top

seeded melt growth” (TSMG) or “melt processing”. The technique is based on the particular

phase diagram of (RE)BCO materials. It involves heating a (RE)BCO ceramic above its peritectic

decomposition temperature and subsequent cooling at a slow rate. If a small (RE’)BCO single

crystal is placed against the top face of the ceramic during this process, the slow cooling

results in epitaxial growth from this single crystal, acting as a “seed”. The rare-earth element

RE’ of the single crystal differs from that of the main superconductor and is chosen, among

others, so that the crystal does not decompose at the highest temperature used for

processing. If the rare-earth is yttrium, the resulting, large grain microstructure consists of a

superconducting YBa2Cu3O7-δ (Y-123) phase matrix containing discrete Y2BaCuO5 (Y-211)

inclusions [30, 31]. This epitaxial growth of the superconducting grain is limited by defects and

the quality of the melt-processed material (i.e. 𝐽c) tends to decrease away from the seed.

Multi-seeding techniques can be used to produce larger samples [32, 33].

2.2 Magnetization of trapped field magnets and Bean model

2.2.1 Magnetization techniques

Basically, three main activation techniques can be used to trap a certain amount of magnetic

flux in a bulk, type II high temperature superconductor: the zero field cooled process, the field

cooled process, and the pulsed field magnetization technique.

The zero-field cooling (ZFC) process consists in cooling the material below 𝑇c in the absence of

magnetic field. If a magnetic field 𝐻 < 𝐻c1 is then applied to the sample, surface currents will

first be induced to prevent the field from penetrating into the bulk. Under an increasing field

𝐻c1 < 𝐻 < 𝐻c2, vortices will penetrate into the sample, starting from the edges. If the pinning

is strong, these vortices will stay close to the sample surface. They will move towards the

Chapter 2. Magnetic properties of bulk high temperature superconductors 21

centre only if they are pushed by new vortices entering the sample under a higher external

field. Once the field is removed, the vortices close to the surface will leave the sample but a

given number of vortices will remain trapped in the superconductor as a result of the pinning

force.

In the field cooling (FC) process, the sample is first subjected to a magnetic field 𝐻 and then

cooled down below 𝑇c. If 𝐻 < 𝐻c1, the field will be expelled from the material, this is the

Meissner effect. If 𝐻c1 < 𝐻 < 𝐻c2, vortices are created inside the type II superconductor and,

in the case of strong pinning, a given number of vortices are trapped in the sample once the

external magnetic field is switched off.

The pulse field magnetization (PFM) technique [34] is a variant of the ZFC process. In this

technique, a large, short (≈ ms) pulse of magnetic field is applied to the zero-field cooled

superconductor. The short pulse has the advantage to reduce the heating of the magnetizing

coil, e.g. a copper coil with currents of typically a few kA can be used. The downside is that the

rapid flux penetration inside the superconductor will dissipate a non-negligible amount of

power [35, 36]. The superconductor temperature increases locally, which results in a decrease

of the critical current density and therefore a trapped field lower than what would be

expected for an isothermal sample. Higher trapped fields can be reached using a modified

multi pulse technique combined with stepwise cooling [37].

For irreversible type II superconductors with strong pinning, the motion of vortices inside the

superconductor can be described by the Bean model, or critical state model.

2.2.2 The Bean critical state model

The critical state model has been introduced by Bean [21, 22]. It describes the penetration of

the magnetic flux in type II materials which exhibit strong pinning. The following text gives a

summary of the model and more detailed explanations can be found in [38, 39].

Basically, the Bean model assumes that the macroscopic current 𝐽 can only take three values :

either zero (in flux free regions) or ± the maximum (critical) current density 𝐽c (in the regions of

the superconductor penetrated by the magnetic flux). In addition to strong pinning, the

following hypotheses are made: the sample is much larger than the London penetration depth

𝜆 and the surface barriers as well as the lower and upper critical fields are neglected (𝐻c1 = 0,

𝐻c2 → ∞).

When a small field is applied to a zero-field cooled superconductor, vortices are created at the

sample edge. These vortices are strongly pinned at the periphery of the superconductor while

the centre of the material stays free of magnetic field. The flux density 𝐵 is therefore highly

non-uniform in the material. From Ampere’s law, there must be a current flowing through the

sample, given by ∇ × 𝑩 = 𝜇0 𝑱. The Bean model states that “any electromotive force, however

small, will induce the critical current to flow locally” [22], i.e. the strong pinning condition

22 Chapter 2. Magnetic properties of bulk high temperature superconductors

requires that 𝐽 takes its maximum value, i.e. 𝐽 = 𝐽c. A current density 𝐽c will therefore flow in

the sample over a layer such that the applied field is cancelled out (or shielded) in the central

vortex-free region of the superconductor. As the applied field is progressively increased, the

magnetic flux penetrates further inside the sample and the layer in which supercurrents are

flowing increases. Once there are supercurrents in the whole material, i.e. the flux front

reaches the sample centre, the sample is said to be fully penetrated. The corresponding

applied field is called the penetration field 𝐻p. The external magnetic field can increase

without destroying the superconductivity but the amplitude of induced currents will not

increase anymore. When the external field is removed, some currents are still flowing through

the superconductor and the superconductor remains magnetized. In this approach, the

magnetic moment of the sample is due to macroscopic currents. This is in contrast with

ferromagnetic materials in which the permanent magnetic moment results from phenomena

arising at the atomic scale.

The Bean model reduces to a 1D problem in the case of an infinite cylinder of radius 𝑎 with the

applied field parallel to the cylinder axis. In this case, ∇ × 𝑩 reduces to a simple derivative in

the radial direction and 𝜇0𝐽c is the slope of the 𝐵(𝑟) curve:

d𝐵

d𝑟=

−𝜇0𝐽c

0+𝜇0𝐽c

Fig. 2.3 shows the flux penetration in this particular case with a constant critical current

density. The infinitely long superconductor shown in (a) is cooled below 𝑇c, then an external

flux density 𝐵a is applied. Fig. 2.3(b) shows the penetration of the flux as it is increased to

𝐵max. The superconductor is fully penetrated when 𝐵a = 𝐵p, the penetration flux density

(𝐵p = 𝜇0𝐻p). In Fig. 2.3(c), the applied field is decreased to zero and upon a reversal of the

supercurrents, the trapped field exhibits a conic profile. In this example, the value 𝐵max is

chosen higher than 2𝐵p, which is the condition to fully magnetize the sample. If the sample is

cooled under a magnetic flux density 𝐵0 (FC process), 𝐵0 needs only to be higher than 𝐵p to

fully magnetize the superconductor.

In the Bean model, the critical current density is not required to be a constant. In large grain,

bulk (RE)BCO superconductors, a useful model to describe the field dependence of the critical

current 𝐽c(𝐵) is the Kim model [40]:

𝐽c(𝐵) = 𝐽c1 1

(1 + 𝐵 𝐵1⁄ ) (2.1)

where 𝐽c1 and 𝐵1 are macroscopic material-dependent parameters that can be determined by

fit of an experimental 𝐽(𝐵) curve.

Chapter 2. Magnetic properties of bulk high temperature superconductors 23

Fig. 2.3 Illustration of the magnetization of a superconducting according to the

Bean model. (a) Schematic representation of an infinite type II superconducting

cylinder of radius 𝑎 with a constant critical current density 𝐽c. (b) Magnetic flux

density 𝐵z inside the superconductor for increasing values of the applied field up to

𝐵max > 2𝐵p. (c) Evolution of 𝐵z as 𝐵a is decreased from 𝐵max to zero.

2.2.3 Flux relaxation and power law model

One of the main limitations of the Bean model is that it refers to a steady state; the time

during which vortices move and rearrange themselves is assumed to be negligible. The trapped

vortices are moving only when the external magnetic field changes and the sweep rate of this

external field does not influence the flux repartition. As a consequence, the electric field is

always zero in the superconductor, except when the vortices are moving. In addition, the

vortices have also a finite probability of overcoming the pinning force and move due to their

thermal energy. This thermally-activated motion of vortices is called the flux creep. One of the

main consequences is the time relaxation of the trapped magnetization with time. This

phenomenon is dissipative and implies the existence of an electric field 𝐸(𝐽).

In this work, the relationship between the electric field and the current in the bulk

superconductors will be approximated by a power law model [41, 42, 43, 44]:

𝑬(𝑱) = 𝐸c (|𝑱|

𝐽c)

𝑛𝑱

|𝑱| (2.2)

where 𝐸𝑐 is a threshold electric field, arbitrarily chosen to 𝐸c = 10-4 V/m, 𝐽𝑐 is the critical

current density which can be field dependent and the 𝑛-exponent is directly related to flux

creep. Fig. 2.4 shows the normalized power law (2.2) for 𝑛 = 25 and 45 in green and blue,

respectively. The limit cases for 𝑛 = 1 (ohmic material) and 𝑛 → ∞ (Bean model) are shown in

violet and red. An increasing 𝑛 means stronger pinning and lower flux relaxation. This 𝑬(𝑱)

power law being analytical with no discontinuities, it can be easily incorporated in usual

numerical methods to solve Maxwell’s equations. Typical values for 𝑛 in (RE)BCO

superconductors range between 20 and 30 at low fields and 𝑇 = 77 K [45]. Note however, that

𝑛 is usually a decreasing function of temperature and can be field-dependent, as discussed by

Berger et al. [36].

24 Chapter 2. Magnetic properties of bulk high temperature superconductors

If we assume that the magnetization of the sample follows a same decay law as the current

density, the magnetic moment 𝑚 of the sample can be described by [46]:

𝑚(𝑡) = 𝑚0 (1 +𝑡

𝑡0)

11−𝑛

where 𝑚0 is the initial magnetic moment of the sample (before relaxation) and 𝑡0 is a

characteristic time usually of the order of 1 second or less.

Fig. 2.4 Relation between the electric field and the current density as described by

the power law (2.2) for 𝑛 = 1 (violet), 25 (green) and 45 (blue). The vertical red line

represents the Bean model (limit case for 𝑛 → ∞).

2.3 Demagnetizing field

In this section we describe briefly the demagnetizing field and its consequences on magnetic

measurements. When a magnetic sample of finite size is subjected to an external magnetic

field 𝑯ext, the magnetization 𝑴 induces a magnetic field in and around the sample. As this

field is opposed to the magnetization in the material, it is usually called the demagnetizing field

𝑯d [39, 47, 48, 49]. The strength of the demagnetizing field is dependent on the shape and

magnetization of the material. As a consequence, the internal magnetic field 𝑯int becomes

𝑯int = 𝑯ext + 𝑯d. In general, 𝑯d is neither uniform nor parallel to the applied field and its

relation to the magnetization is given by

𝑯d = −‖𝑫(𝑥, 𝑦, 𝑧)‖ 𝑴

where ‖𝑫(𝑥, 𝑦, 𝑧)‖ is, in general, a second order tensor. The tensor is reduced to a

proportional factor in some particular sample geometries, e.g. when the magnetization 𝑴 is

Chapter 2. Magnetic properties of bulk high temperature superconductors 25

directed along one of the main directions of an ellipsoid. For a cylindrical sample magnetized

along its symmetry axis, the demagnetizing factor is higher near the edges than along the axis.

An average demagnetizing factor must be defined, e.g. as 𝑯d,av = −𝐷𝑴av with the subscript

“av” accounts for the volume average of the quantity [49].

The contribution of the demagnetizing field opposing to the magnetization is the reason why

the applied field required to reach the full magnetization increases with decreasing aspect

ratio of the sample. The curvature of the flux lines resulting from the demagnetizing field is of

great importance in measurements of the magnetic flux. As a consequence, the demagnetizing

field must either be included in calculations of the magnetic quantities or be reduced as much

as possible in the sample, e.g. with a ferromagnetic yoke that drives the return flux lines out of

the sample.

2.4 Properties and applications of bulk HTS and ferromagnetic

materials

As introduced in the first chapter, bulk, high temperature superconductors (HTS) have a

significant potential for use as powerful permanent magnets in a variety of practical

applications. The interaction between ferromagnetic and superconducting materials may

improve the performances of various applications involving bulk type II superconductors used

as quasi-permanent magnets with a large flux density:volume ratio. Several combinations of

bulk HTS have been studied in the literature; this section highlights some important results.

Bulk HTS inserted in a ferromagnetic yoke

The trapped field generated by bulks HTS can be increased by the insertion of the HTS in a

closed magnetic circuit made of soft ferromagnetic material. This was shown by experiment

and modelling carried out by Parks et al. [50]. Because of the external yoke, the magnetic flux

profile changes from the conical profile of the Bean model to a uniform field in the

ferromagnetic material. The flux creep rate was found to remain unaltered by the presence of

the ferromagnetic yoke.

On the opposite, Smolyak et al. have found that the magnetic relaxation in a trapped field

magnet can be retarded in presence of a ferromagnet placed in vicinity of the superconductor

[51]. This property depends on the sequence of magnetization and the approach of the

superconductor to a ferromagnet. The flux relaxation is found to be fully suppressed if the

superconducting sample is magnetized first, and then is brought close to a ferromagnet.

Sandwiching a bulk HTS between two soft ferromagnetic materials can also improve the pulse

field magnetization process [19].

In applications involving magnetic fields, Genenko has studied numerically the current

distribution in a superconducting ring in the Meissner state when the ring is located between

26 Chapter 2. Magnetic properties of bulk high temperature superconductors

two coaxial cylindrical soft magnets of high permeability [52]. This unusual structure is used to

prevent the entry of magnetic flux from the edges of the ring.

Motor and generators

Bulk superconductors can be found in many applications in rotating machines (motors and

generators), where they are naturally in contact with ferromagnets. Several topologies of

superconducting rotating machines can be envisaged but the superconductor is most of the

time placed in the machine rotor [53], where either flux shielding or flux trapping properties

can be exploited. As an example, the magnetic flux generated in an iron rotor can be increased

above the saturation field by the use of an YBCO frame to block the flux within the yoke, as

investigated by Granados et al. [17]. The use of an iron yoke surrounded by a bulk-based ring

around the yoke increases the magnetization and allows large reversible variations of the

external magnetic field [9].

Hull and Strasik reviewed and discuss the concepts for using trapped-flux bulk high-

temperature superconductor in motors and generators [18]. It is emphasized that

ferromagnetic materials can reduce the reluctance of the magnetization circuit and should be

considered for in situ magnetization. To overcome the fact that the saturation magnetization

of iron (approx. 2 T) is lower than the expected field with trapped field magnets, dysprosium

(Dy) can be used. Dysprosium is paramagnetic at ambient temperature but has a much higher

saturation magnetization than conventional iron at temperatures below 80 K (i.e. the expected

operating temperature range for bulk HTS). Hence it is possible to design a trapped-flux HTS

motor with bulk HTS and a Dy core in both the rotor and stator components [18]. In this

concept, a magnetization of more than 4 T can be reached but hysteretic losses in the Dy core

might reduce these performances.

Magnetic flux shaping and levitation

Hybrid structures can be used to modulate the shape of the magnetic field produced by

superconductors and/or to increase the magnitude or gradient of the flux density used in

levitation devices. As an example, Kim et al. have studied to the spatial homogeneity of the

magnetic flux trapped by a stack of bulk GdBCO annuli [54]. The insertion of an iron ring into

the cold bore of the HTS annuli was found to reduce the maximum trapped field but improved

the homogeneity of this trapped field. These experimental results are part of the development

of a compact HTS bulk NMR magnet [55]. Del-Valle et al. presented a theoretical framework

that can describe the magnetic response of an ideal soft ferromagnet bar immersed in an

applied field and interacting with other elements such as a superconducting bar [56]. This

model shows that the shape, size, and position of the ferromagnetic bar can be optimized to

improve the levitation force above levitation tracks or modulate the magnetic flux density

above the bar.

Chapter 2. Magnetic properties of bulk high temperature superconductors 27

Fujishiro et al. investigated the spatial modulation of the magnetic field generated by trapped

field magnets by changing the geometric properties of a ferromagnetic circuit made of silicon

steel [57]. They adjusted the gap between sheets of silicon steel to improve the field

modulation and magnetic force and use it for the solution growth of organic semiconductors.

In a similar way, the assemblies of permanent magnets, bulk HTS and soft ferromagnetic core

can be used to design and improve magnetic levitation systems [58], for example by the use of

an inverse E shape ferromagnetic device core [59]. An original mixed-𝜇 magnetic suspension

has been suggested by Joyce et al. [60] for levitated ground transport. It uses the

diamagnetism of superconducting materials to stabilize the attraction between a

superconducting magnet and a ferromagnetic rail. In this structure, the diameter of the

ferromagnetic yoke must be smaller than the one of the bulk HTS to have a stable levitation

[61].

Finally, Takao et al. have investigated a magnetic levitation system that uses the magnetic

shielding effect of bulk HTS [62, 63]. A bulk HTS and a permanent magnet are used in a moving

vehicle that levitates under a fixed steel bar. Interestingly, a sufficient force is achieved by

inserting an additional ferromagnetic plate under the permanent magnet in the mobile.

Other applications involving bulk superconductors

Bulk HTS superconductors with particular geometries can be combined with ferromagnetic

materials to improve their performances. As an example, a bulk (RE)BCO monolith containing a

regular array of artificial holes can be filled with a ferromagnetic powder to increase the field

trapping properties of the composite sample [38, 64].

It is also of interest to mention studies involving HTS tubes. Such studies involve e.g. the so-

called magnetic shielding fault current limiters, in which an HTS cylinder is used as the

secondary of a transformer [65]. Since a hollow superconducting tube forms the basis of a

passive magnetic shield, it is possible to enhance the magnetic shielding properties of the sole

superconducting tube by using an additional ferromagnetic tube placed concentrically around

the HTS one [66]. The same kind of improvement is also observed in an MgB2/Fe hybrid

structures consisting of two coaxial cups subjected to a magnetic field applied parallel to their

axis [67].

Finally, the combination of ferromagnetic and superconducting materials can lead to new

applications. For example, metamaterials consisting of different arrangements of

superconducting and ferromagnetic pieces can be used for “magnetic invisibility” (or

“magnetic cloaking”) as well as concentration of static magnetic fields [68].

Tapes wires and films

Similarly to their use with bulk superconductors, ferromagnetic materials can be used as

sheaths around multifilament wires and tapes [69, 70] or as magnetic flux diverters to modify

28 Chapter 2. Magnetic properties of bulk high temperature superconductors

the flux distribution around tapes and superconducting coil magnets [71, 72, 73, 74, 75, 76, 77,

78, 79, 80, 81]; thereby improving their electrical properties (i.e. increasing the critical current

and reducing AC losses). The interplay between ferromagnetic particles and the bulk

superconducting microstructure has been studied at a small (sub-micron) scale [82, 83, 84, 85]

to investigate their impact in the distribution and pinning of individual (quantized) magnetic

flux lines, e.g. in thin film structures.

2.5 Summary

The first section of this chapter was devoted to a reminder of the basic properties of

superconductors and specifically to the description of the properties of bulk, type II,

superconductors (HTS) used as trapped field magnets. Then, the concept of demagnetizing

field was briefly presented and its implications for magnetic measurements were outlined.

Finally, applications and previous studies about the interaction between bulk HTS and

ferromagnetic materials were reviewed.

29

Chapter 3

Experimental methods

In this chapter, we describe the experimental methods and measurement systems used to

characterize the magnetic behaviour of the ferromagnetic and superconducting materials and

the hybrid ferromagnet / superconductor structures that are investigated in this work.

Basically, the quantities that can be measured are the magnetic moment 𝑚 of the whole

sample, the magnetic flux 𝜙, i.e. the flux of 𝐵 that flows through a given surface 𝑆, or the local

flux density 𝐵.

Miniature Hall probes are placed in close proximity of the sample and can be stationary or

moved against the sample surface to perform surface measurements of the magnetic flux

density. The magnetic moment 𝑚 of a trapped field magnet and the average flux density 𝐵

through the sample cross section are determined from volume measurements of the magnetic

flux. These volume measurements can be carried out using pick-up coils that can be tightly

wound around the sample (fluxmetric measurements), or of a diameter several times larger

than that of the sample (magnetometric measurements). It should be emphasized that most

commercial devices are well suited to the determination of the room-temperature magnetic

flux threading large samples (e.g. ferromagnetic rods) or of the temperature-dependent

magnetic moment of sub-centimetric size samples (e.g. single crystals). The low temperature

characterization of volume properties of bulk superconductors or superconductor /

ferromagnet hybrids as those investigated in this thesis (i.e. a few cm³) requires bespoke

experimental systems.

The measurement set-ups described in this chapter, either commercial or home-made, use

coils, Hall probes, or both simultaneously to characterize the magnetic behaviour of the

samples. For each of the experimental systems, we specify the measured physical parameters,

the corresponding theory of operation, and the practical laboratory implementation. The last

section includes a table summarizing the physical parameters that can be measured in each

experimental set-up and a list of the personal realizations achieved in this work.

3.1 Permeameter

The permeameter method is used to measure the DC magnetic properties of magnetically soft

materials at room temperature, and particularly their intrinsic 𝐵(𝐻) hysteresis loop. This

method is normalized and is described in the international standard IEC 60404-4 [86, 87]. The

30 Chapter 3. Experimental methods

system used in our laboratory is similar to that described in the standard with slight

differences that are specified here under.

Fig. 3.1 shows a schematic representation of the permeameter. The test specimen is a long rod

of a magnetically soft material. This rod is clamped between two massive C-shaped iron yokes

made of a magnetically soft material. These yokes provide a near-zero reluctance closure path

for the flux lines leaving the test specimen. In so doing, the geometry approaches that of a true

closed magnetic circuit and the demagnetizing field inside the test specimen can, in excellent

approximation, be neglected. Two magnetizing coils generate a known magnetic field 𝐻 within

the test specimen. A rapid, but monotone change in the magnetizing current will induce a

magnetic flux variation inside the test specimen. An electromotive force (emf) develops across

a pick-up coil wound as close as possible of the central part of the test specimen. This emf is

integrated over time to give the magnetic flux variation and hence the variation of the average

flux density inside the test specimen. Starting systematically from one extremity of the

hysteresis cycle, the whole hysteresis loop 𝐵(𝐻) can be determined point by point.

The test specimens are 550 mm long rods with a 10.5 mm diameter for the permeameter

available in the laboratory. This permeameter is of type A as specified in the IEC 60404-4

standard. The integration is made using a Grassot fluxmeter [88]. The main difference between

our measurement procedure and the IEC standard is that the magnetic field strength is not

directly measured by additional so-called “H-coils” but calculated as 𝐻 = 𝑁𝑖/𝐿, where 𝑁 is the

number of turns of the magnetizing coil, 𝑖 is the current in these coils and 𝐿 is the length

between the two ends of the test specimen connected to the yokes. This difference impacts

only slightly the measurement accuracy since the yoke cross-sectional area is much larger than

that of the test specimen, i.e. the magnetic field 𝐻 in the high-permeability yoke is assumed to

be zero. Another difference relates to the calibration of the fluxmeter, which is done in our

case by measuring a known flux variation generated by two calibrated concentric air coils with

high aspect ratio.

Fig. 3.1 Cross section of the permeameter. The device is used to measure the

intrinsic 𝐵(𝐻) curve of long rods of soft ferromagnetic materials at room

temperature.

Chapter 3. Experimental methods 31

3.2 Measurement systems based on the PPMS®

The PPMS® is a Physical Property Measurement System from Quantum Design [89]. It is a

versatile instrument designed to carry out physical measurements as a function of

temperature and under high magnetic fields. Fig. 3.2 shows a schematic representation of

(top) a typical PPMS dewar and (bottom) a probe assembly that fit inside the dewar. The

model available in our laboratory allows fields up to ±9 T to be applied; the temperature can

be varied from 1.9 K to 400 K. The cylindrical sample chamber is 2.5 cm in diameter. Such

dimensions limit obviously the size of the samples that can be measured but allows excellent

field homogeneity (0.01%) to be achieved within the measuring region [90]. The field is applied

by a superconducting DC magnet cooled in liquid helium. The maximum sweep rate is of

170 Oe/s (= 17 mT/s). The sample can be instrumented directly by the user but options can be

added to the PPMS to carry out specific measurements. The sections below describe the

measurement systems used in this work. First, we describe the home-made experimental set-

up developed to measure hysteresis cycles of superconductors and superconductor /

ferromagnet hybrid structures. This set-up makes advantage of the PPMS for its temperature

and field capabilities. We explain how these low-level measurements can be affected by

parasitic signals and we detail the experimental precautions taken to reduce these parasites

and their influence. Next we describe the ACMS option used for DC and AC magnetic

characterisation of small samples. In the last section we provide a description of the “rotator”

option developed in our laboratory to measure the influence of transverse magnetic fields on

the magnetic moment of magnetic samples at low temperatures.

3.2.1 Using PPMS temperature and field control for coil and Hall probe

measurements

The PPMS offers the opportunity to directly and accurately control the magnet current and the

temperature of the sample chamber. In addition, twelve electrical terminals are available on

each sample holder and wired to the external part of the measurement system accessible to

the user. Home-made measurements can thus be performed using specific sensors (in the

measurement chamber) connected to external devices (outside the measurement chamber).

In this work, a measurement system using Hall probes and coils was developed to study the

magnetic behaviour of large, single grain bulk superconductors under a time-varying magnetic

field [91, 64]. This method is particularly suitable for large samples that cannot fit inside

traditional magnetometers. This set-up was used to measure magnetic hysteresis loops, e.g.

for an applied field swept initially up to 3 T and then cycled between 3 T and -3 T at a rate of

15 mT/s. Additionally, the critical temperature and magnetic flux relaxation properties of

superconducting samples can be determined with this set-up.

32 Chapter 3. Experimental methods

Fig. 3.2 Schematic representation of (top) a typical PPMS dewar and (bottom) a

probe assembly that fits inside the dewar. Pictures from [92, 93].

Chapter 3. Experimental methods 33

The sample and its sensors are clamped inside a closed brass tube (21.5 mm inner diameter) to

avoid movement due to possibly strong magnetic forces experienced by the sample during the

measurement. This prevents any damage to the PPMS that would be caused, e.g. if a

ferromagnetic piece was unstuck from the sample holder. However, the presence of the brass

tube and its clamping system imposes a longer cooling time to make sure that the whole

sample has reached the target temperature.

The close sample holder allows a maximum of four differential signals to be measured

simultaneously, in addition to the current injection to the Hall probes. More sensors can be

used provided that the measurement procedure is carried out a second time after having

changed the sensor connections in the sample holder. In this work, we used up to two Hall

probes and five coils on a ferromagnet / superconductor hybrid structure. The four output

signals are measured by two Agilent 34420A nanovoltmeters. The PPMS and nanovoltmeters

are PC-controlled through GPIB using the NI LabVIEW development system [94]. The

measurements are analysed afterwards using Matlab [95]. Special attention is paid to the

calibration and offsets on the Hall probes and coils signals. The next paragraphs describe in

more details the experimental arrangements and the measurement precautions taken for (i)

the Hall probes and (ii) the coils.

Fig. 3.3 Left: Sample holder designed for the PPMS. Several coils are wrapped

around the superconducting sample and Hall probes are attached against its

surface. The tube shown on the left is used to enclose the sample. A cross section

of a sample equipped with two Hall probes and four coils is shown as an example in

the right panel of the figure.

Hall probes

Hall probes are semiconducting sensors based on the Lorentz force acting on the moving

charge carriers: when the probe is fed with a (known) control current, the application of a

perpendicular magnetic flux density results in a transverse voltage. It should be emphasized

that the measured physical quantity is the magnetic induction averaged on the probe active

34 Chapter 3. Experimental methods

area which can be small but remains finite. The term “local measurement”, although common,

is thus used abusively. In this experiment, two Arepoc AHP-H3Z Hall probes, each with 1 mm2

active area, are aligned on the central axis of the sample and sample holder, as shown

schematically in Fig. 3.3. Both probes are placed on each side of the sample, as close as

possible to the sample surface. The probes are fed with a 1 mA current. This current is

intentionally chosen to be lower than the maximum current rating (usually 10 to 15 mA).

Although the measurement sensitivity is decreased accordingly, this experimental precaution

is mandatory to limit the dissipated power inside the PPMS sample chamber. In the present

configuration the two Hall probes are connected in series. The Joule heating is approx.

750 Ω × (1 mA)2 = 750 μW. This dissipated power remains acceptable and can be

compensated by the PPMS temperature control. The liquid helium consumption, however, was

found to be increased.

Before each measurement cycle, the voltage offset across each probe is measured at the

measurement temperature. This offset is then subtracted from the measured Hall voltages.

The calibration curves of the Hall probes were measured at 100 K in the range [-4T; +4T]. Fig.

3.4 shows the calibration curve of the Arepoc AHP-H3Z Hall probe placed at the top of the

sample. Because of the non-linearity of the probe, measured voltages are converted into

magnetic flux density values by linear interpolation between the points of the calibration

curve. The calibration was not carried out at 77 K for technical reasons. Nevertheless, side

measurements showed that the systematic error on the probe sensitivity is smaller than 1 %

between 100 K and 77 K.

Fig. 3.4 Calibration curve of an Arepoc AHP-H3Z Hall probe at 100 K.

Chapter 3. Experimental methods 35

Coils

The pick-up coils used for probing the magnetic flux density threading the sample cross section

are wound either around the full thickness of the sample, or around a fraction of its height.

They are made of thin varnish-insulated copper wires, typically 80 µm in diameter. Fig. 3.5

shows schematically a coil wound tightly around the full height of a disk-shaped

superconductor. In this section we discuss the experimental set-up, the mathematical

operations on the measurements and the experimental precautions taken to reduce the

influence of parasitic signals on the final results. For the sake of clarity, these concepts will be

illustrated with a measurement obtained on the ESC2 sample studied in a large part of this

work.

Fig. 3.5 Schematic representation of a coil wound tightly around the full height of a

superconductor.

The basic principle of all fluxmetric techniques is to measure the voltage 𝑉 across the pick-up

coil when it embraces a time-varying magnetic flux 𝜙, as a result of the Faraday-Lenz law:

𝑉 = −d𝜙

d𝑡= −

d

d𝑡∬ 𝑩 ∙ d𝑺t

where 𝑆t is the total area of the coil, i.e. the number of turns multiplied by the area of each

turn 𝑆. This value is accurately determined through calibration. The induced voltage is then

integrated over time and divided by 𝑆t to determine the time-evolution of the average

magnetic flux density 𝐵 through the coil. Note that a time-varying magnetic flux, resulting e.g.

from a time-varying applied field, is mandatory for this measurement. In this work, the method

is used with a continuously changing magnetic field that is swept at constant (known) sweep

rate (triangular wave). Fig. 3.6 shows an example of time-varying magnetic field applied to the

sample (green) and the distorted voltage waveform (brown) measured across the coil wound

around the sample. In this example, a sweep rate of 15 mT/s is used. The amplitude of

measured voltage cycles is 60 µV, which requires a measurement accuracy of ≈0.01 µV to

detect the steep variations around zero.

36 Chapter 3. Experimental methods

Fig. 3.6 Time-varying magnetic field applied to the sample (green) and voltage

(brown) measured across the coil wound around a bulk superconducting sample.

The value of 𝑉 depends on (i) the total area 𝑆𝑡; (ii) the rate of the time-varying applied external

field, and (iii) the magnetic induction inside each turn at a given instant. Here are some

comments on these parameters:

(i) The exact value of the total surface 𝑆t is not calculated from the geometric dimensions of

the sample but rather determined indirectly through a calibration procedure at 100 K, i.e.

when the sample is not in the superconducting state. The value of 𝑆t is adjusted so that the

resulting 𝐵(𝜇0𝐻app) curve is the bisector of the first and third quadrants.

(ii) The rate of the time-varying applied field is limited by the PPMS maximum sweep rate of 17

mT/s. In this work, the typical sweep rate is 15 mT/s, which ensures a “decent” signal

amplitude (a few tens of µV) with a sufficient number of measurement points, determined

from the sampling rate of the nanovoltmeter. Several lower sweep rates were applied and the

corresponding measured data were compared. They show the influence of magnetic flux

relaxation (flux creep) inside the superconductor.

(iii) The distribution of the local magnetic flux density across the coil depends on the external

applied field and on the properties of the material. The maximal applied field is usually chosen

to be larger than twice the full-penetration field of the superconducting sample. This is

required to fully magnetize the superconductor. Moreover, it is usually useful to apply higher

fields in order to study the behaviour of the sample in its fully magnetized state, e.g. when the

field-dependant critical current 𝐽𝑐(𝐵) characteristic needs to be determined. As shown in Fig.

3.6, a maximum field of 3 T is usually chosen in this work.

Chapter 3. Experimental methods 37

Once the voltage 𝑉 is measured, it is integrated numerically over time to give the magnetic

flux. This resulting flux is divided by the total surface 𝑆t to determine the average magnetic flux

density across the sample. The resulting 𝐵(𝐻) curve, as shown in Fig. 3.7(a), is an extremely

narrow hysteresis cycle where the useful information is precisely “hidden” in slight difference

between the ascending and descending branches. Therefore, the applied magnetic flux density

is subtracted from the measurement to plot the Δ𝐵 = 𝐵𝑚𝑒𝑎𝑠 − 𝜇0𝐻𝑎𝑝𝑝 curve, as shown in Fig.

3.7(b).

Fig. 3.7 Example of (a) a 𝐵(𝐻) hysteresis curve and (b) the corresponding

Δ𝐵 = 𝐵𝑚𝑒𝑎𝑠 − 𝜇0𝐻𝑎𝑝𝑝 obtained in the framework of this work.

To obtain the hysteresis curves, low level voltages must be measured. The sensitivity and

accuracy of today nanovoltmeters are impressive, but relying on such performances is usually

not sufficient: the measured voltage must arise only from the coil around the sample. A

number of experimental precautions, therefore, should be taken in order to reduce and/or

suppress the influence of the unwanted parasitic voltages. In the present experiment, the most

significant sources of error are the thermoelectric effects and the voltages induced by

inductive or capacitive coupling. The connections between the PPMS and the nanovoltmeter

terminals are direct (no intermediate material) and tight. They are enclosed in a grounded

ferromagnetic box acting as an electric and magnetic shield. The box plays also an important

role in reducing the temperature gradient by avoiding air flow on around the connections.

Additionally, the wires are tightened near the terminals to reduce spurious mechanical

vibrations and the associated low-level induced voltages. Finally, it is worth noting that the

coils have a low output impedance. For example, a copper coil of 53 turns at 77 K around a

cylindrical sample of 16.5 mm diameter and 6 mm height is estimated to have a resistance of

1.5 Ω and an inductance of 0.12 mH (when the superconductor is in the normal state).

Therefore, its output impedance at 50 Hz is about 1.5 Ω. This means that the loop impedance

of the measurement circuit remains low, which decrease the influence of external parasitic

signals caused by capacitive coupling.

38 Chapter 3. Experimental methods

Fig. 3.8 The Physical Property Measurement System (PPMS) from Quantum Design

and, on the right, the contact box.

Even with these experimental precautions, an unavoidable DC voltage offset is present across

the coil. This voltage offset is found to vary slightly for the total duration of the measurement.

Therefore, the offset value is measured twice for each sensor: once before and once after each

measurement. The offset is assumed to have varied linearly during the measurement cycle.

The interpolated linear variation is therefore subtracted from the raw data. Sometimes a slight

manual adjustment of the linear interpolation is required in order to ensure that the measured

data do not lead to unphysical behaviour, i.e. the hysteresis curves should close on

themselves. Nevertheless, the actual variation of the offset is not linear and a slight non-

closure of the hysteresis curve may still be observed after the adjustment of the offsets, as can

be seen between 2 T and 3 T in Fig. 3.7(b). In order to illustrate the influence of an extremely

small DC offset, Fig. 3.9(a) shows the data corresponding to the hysteresis loop shown in Fig.

3.7(b) when a constant -50 nV offset is added intentionally to the raw data before numerical

integration. As can be seen, the last branch of the hysteresis loop and the virgin branch do not

merge into one curve, as would be expected physically. The accurate determination of the

total area 𝑆t embraced by the coil also has a strong influence of the results. Fig. 3.9(b) shows

the hysteresis loop calculated from experimental data shown in Fig. 3.7(b) when the total

surface is decreased by 1 %. An unphysical tilt of the curve appears clearly. This 1 % variation

of the surface is equivalent to neglect the 80 µm diameter of the coil wire wound around a

sample of 16.5 mm in diameter, as considered in a large part of this work. This result gives

evidence that the coil area must be carefully determined with accuracy better than 1%.

Because of unavoidable dimensional changes of the coil area resulting from thermal

contraction / expansion caused by the successive cooling / warming, the calibration procedure

was done multiple times at 100 K in the absence of ferromagnet. In the presence of a

Controller and current source

Superconducting coil up to 9 teslas

Contact box

Chapter 3. Experimental methods 39

ferromagnet, the value 𝑆t was corrected when needed to avoid an unphysical tilt of the

magnetization curve of the ferromagnet.

Fig. 3.9 Illustration of unwanted effects on the hysteresis loop from Fig. 3.7(b): (a) a

-50 nV offset is added to the raw data before numerical integration; (b) the total

surface area 𝑆𝑡 of the coil is decreased by 1 %.

3.2.2 ACMS option

The commercially available “AC Measurement System (ACMS)” option for the PPMS includes

both a DC magnetometer and AC susceptometer [90]. The option consists of a powerful,

automated magnetic workstation connected to the temperature and field control system of

the PPMS. The central part of the ACMS is an insert that fits directly in the PPMS sample

chamber within the uniform magnetic field region of the PPMS. This insert includes the AC

excitation coil and a set of detection (pick-up) coils used for the measurement of both AC and

DC signals. An additional thermometer and electrical connections for the ACMS system are

also provided. The measurements are performed by a detection coil set made of two sets of

copper coils connected in series-opposition and separated by several centimetres. These coils

are arranged in a first order gradiometer configuration which helps isolate the sample signal

from uniform background sources.

3.2.2.1 Extraction method (DC magnetometer)

The ACMS option allows the sample DC magnetic moment to be measured by the extraction

method. In this method, a uniform field is applied in the measurement region. The sample

holder is translated longitudinally through the detection coil set by a DC servo motor. The

rapid, smooth sample movement will induce a signal in the detection coils according to

Faraday’s Law. This signal is integrated numerically and fit to the known waveform for a dipole

using a regression algorithm. This signal analysis enables the DC dipolar magnetic moment 𝑚

of the sample to be determined.

40 Chapter 3. Experimental methods

For the ACMS option, the sample speed is approximately 100 cm/s, corresponding to a

measurement time of approximately 0.05 s. This high speed and short scan time gives an

induced voltage of relatively high amplitude and ensures that contributions of time-

dependent errors (drift, 1/f noise) remain within acceptable limits. In practice several scans are

carried out (e.g. 10) for each measurement and the data are averaged.

The two drawbacks of this commercial system are that (i) the sample holder can accommodate

only small samples (< 1cm³) and (ii) the measurable magnetic moment should be low enough

to prevent clipping of the output voltage. More precisely, the measurement range is limited to

5 emu (= 5 × 10-3 Am2) which corresponds to samples of typically a few mm3 when they are

made of HTS superconductors or ferromagnetic materials. Fig. 3.10 shows two samples

magnetically characterised with the ACMS option in the framework of this work.

Fig. 3.10 Samples suitable for the ACMS option, but that are close to its

measurement range: (a) a ferromagnetic section of Supra50 (0.56 × 1.13 × 4.92

mm); (b) sections from a bulk YBCO superconductor (1.68 × 1.81 × 0.65 mm). These

samples will be described in chapter 5.

3.2.2.2 AC susceptibility measurements

The ACMS option can also be used to measure the sample magnetic moment response to an

AC excitation field, possibly superimposed to the DC magnetic field of the PPMS magnet. Both

the in-phase and out-of-phase components of the sample response are measured. The

measured quantity is the change in magnetic moment d𝑚 (expressed in Am²) caused by a

change of the applied field d𝐻. Knowing the sample volume 𝑉, the change in the sample

magnetization d𝑀 = d𝑚 𝑉⁄ (expressed in A/m) can be determined. The (dimensionless) AC

susceptibility 𝜒𝐴𝐶 is defined as: 𝜒𝐴𝐶 = d𝑀/d𝐻 and corresponds therefore to the local slope of

the sample magnetization curve 𝑀(𝐻). The AC susceptibility 𝜒𝐴𝐶 is usually a complex number

𝜒𝐴𝐶 = 𝜒𝐴𝐶′ − 𝑗𝜒𝐴𝐶

′′ where 𝜒𝐴𝐶′ is the in-phase component that represents the field

concentration inside the sample and 𝜒𝐴𝐶′′ is the out-of phase component proportional to the

losses occurring within the sample subjected to the AC field [96, 47].

For AC susceptibility measurements using the ACMS option, the so-called “AC-drive coil set”,

wound longitudinally around the detection coil sets, is fed with a known AC current. The

Chapter 3. Experimental methods 41

resulting AC magnetic field amplitude is limited to approx. 10 Oe 800 A/m, corresponding to

𝜇0𝐻 = 1 mT (this limit increases with decreasing temperature and frequency). The frequency

range is 10 Hz – 10 kHz. To perform an AC measurement, the sample subjected to the known

AC magnetic field is successively positioned in the centre of each detection coil. Using a phase-

sensitive detection of the voltage appearing across the detection coils (wound in series-

opposition) the in-phase and out-of-phase components of the sample AC magnetic moment

can be determined.

Phase shifts due to the measurement system (and not to the sample response) are corrected

by measuring the instrument-dependent phase shift in absence of the sample for each

measurement point. This is possible thanks to the presence of a calibration coil inside each

detection coil that simulates a sample with a strict in-phase (i.e. no imaginary component)

response. As mentioned by an application note of Quantum Design [97], a spurious peak in the

temperature-dependence of the out-of-phase component of the AC susceptibility may

sometimes appear between 25 and 35 K. The peak results from a magnetic phase transition of

the Inconel material used as feedthrough at the bottom the sample chamber. This artefact is

observable at extremely low signal amplitude, i.e. for weakly magnetic materials and/or small

size samples. It was not detected in the measurements carried out in the framework of this

thesis.

3.2.3 The rotator

In the experimental methods described above, the direction of the applied field with respect

to any of the sample reference axes is kept constant. In this section we describe a rotating

sample holder to be used with the PPMS. The purpose of the rotator is to apply a magnetic

field at any angle with respect to the sample. In the framework of this thesis, this functionality

will be used to measure the influence of a magnetic field applied at some angle with respect to

the sample permanent magnetization. In practice this means that the sample must rotate in

the experimental chamber since it is obviously not possible to rotate the PPMS

superconducting DC magnet. An “off-the shelf” rotator insertion device is commercially

available as a PPMS option. This system, however, accommodates samples of maximum 3-4

mm in thickness, which is clearly below the typical size of the large grain, bulk samples (an a

fortiori, hybrid configurations) investigated in this thesis. Our research team designed a similar

device for larger samples. Fig. 3.11 shows a picture of the rotator, it was developed by Simon

Debois in the framework of his master thesis during the academic year 2013-2014 [98].

The sample holder is made from aluminium. Samples up to 9 mm diameter and 7 mm height

can fit in the central hole visible in Fig. 3.11. A Hall probe (Arepoc HPP-VU) is centred at the

bottom of the sample holder and is used to measure the magnetic flux density as close as

possible of the sample bottom surface, on its symmetry axis. Additionally, a Pt100 resistive

temperature detector (RTD) is placed in close vicinity of the sample to monitor its

temperature. In the current version of the rotator, the sample can be rotated over 90 degrees

42 Chapter 3. Experimental methods

through manual action. Note that the mechanical components of the sample holder were

designed carefully to withstand the possibly high magnetic torque resulting from the

interaction between the magnetized sample and an externally applied transverse magnetic

field. A typical measurement cycle includes (1) magnetizing the superconducting sample along

its c-axis through a field cooled or zero field cooled process; (2) removing the field and waiting

a defined amount of time for flux relaxation effects; (3) rotating the sample by 90°; (4)

applying a magnetic field which is now perpendicular to the c-axis of the sample and recording

the magnetic flux density at the surface of the sample while the transverse field is ramped to

some value 𝐻max and then cycled between +𝐻max and −𝐻max. The output voltages of the

Pt100 temperature sensor and of the Hall probe are measured by two nanovoltmeters that are

PC-controlled using a LabVIEW program.

Fig. 3.11 Photograph of the rotator developed for the PPMS. The sample (up to

9.5 mm diameter and 7 mm height) is placed in the central hole and can be rotated

by 90° around the brass axis. Picture from [98].

3.3 The magnetometer for large samples

The magnetometer described in this section is similar, in its physical theory of operation, to

the magnetometer of the PPMS (ACMS option). This device was designed to measure the

magnetic moment of samples of much larger size and/or of much larger magnetic moment

than those allowed in the PPMS. It can accommodate bulk, large samples up to 17 mm in

diameter either at room temperature or at fixed temperature in a cryogenic environment.

Importantly, the device is able to measure magnetic moments in excess of 1 Am² (1000 emu),

which is more than two orders of magnitude above the maximum magnetic moment of

commercial cryogenic magnetic measurement systems. The experimental set-up was designed

Brass rotation axis Sample location

Toward the

top of the

chamber

Puck for electrical

connections,

fitting in the

bottom of the

sample chamber

Chapter 3. Experimental methods 43

and developed by Raphael Egan in the framework of his master thesis during the academic

year 2013-2014 [99, 100]. A brief description of the system is given below.

The magnetometer is able to measure the magnetic moment 𝑚 of a magnetized sample,

typically a permanent magnet (PM) or a superconducting trapped field magnet (TFM) placed in

a liquid nitrogen bath. It is based on the extraction method: a relative motion between two

pick-up coils and the sample induces a voltage across the coils. The dipolar magnetic moment

of the sample is deduced from the integration of this induced voltage between carefully

chosen bounds. Fig. 3.12 shows the developed magnetometer. Unlike the ACMS in which the

sample is moved through the coils, the two coils wound in series-opposition move linearly over

a distance of 75 cm, while the sample (magnetized using an external magnet) is stationary and

kept at constant temperature. At this stage, is worth pointing out how this magnetometric

method differs from the fluxmetric measurements described in Sect. 3.1 (permeameter for

long ferromagnetic rods) or in Sect. 3.2.1.2 (measurement of flux threading the cross section of

a superconductor). In the latter, the pick-up coil is wrapped closely around the sample surface

and the basic measured quantity is the magnetic flux or the magnetic flux density 𝐵 averaged

over the sample cross section. In the magnetometer, however, the diameter of the pick-up coil

is purposefully chosen to be much larger than that of the sample. The consequence is that the

return flux density lines outside the sample are intercepted by the pick-up coil as well.

Although the amplitude of the induced emf is smaller, a sensing coil of large cross section is a

“sine qua non” condition for measuring the true magnetic moment [101]. In the present

device, both the sensing coil dimensions and integration bounds were optimized so that the

signal is influenced by the dipolar magnetic moment of the whole sample and that the

moments of higher order have only little influence. More precisely the measurement error due

to moments of higher order is less than 0.1 %. Note that this design also ensures that for a

given magnetic moment, the signal is weakly dependent on the (equivalent) current

distribution in the sample as well as its aspect ratio.

Since the device does not include excitation coils, the sample must be magnetized outside of

the magnetometer. There is therefore a time delay of about 600 s between the end of the

magnetizing process and the first measurement of the magnetic moment. The sensitivity and

accuracy of the system enables the time-decay of the magnetic moment of trapped field

magnets to be measured. This is done typically by carrying out measurements for at least 104 s

(≈ 3 hours) to ensure a full decade of measurements.

44 Chapter 3. Experimental methods

Fig. 3.12 Photograph of the magnetometer developed in the Electrical

Measurement laboratory [99, 100]. The coil (here located at the bottom) moves

vertically. The sample placed at the bottom of a thermally insulated sample holder

(grey tube) is immersed in liquid nitrogen (𝑇 = 77 K).

3.4 Hall probe mapping set-up

In the experimental system described in Sect. 3.2, Hall probes are used to measure the local

magnetic induction at one or a few well-defined locations above the surface of the sample. In

view of understanding physical phenomena related to the magnetization process of

superconductors, the measurement of the distribution of the magnetic flux density is also of

importance. The so-called Hall probe mapping technique [102, 103], involving moving a

miniature Hall probe above the surface sample, is an ideal technique to determine the

distribution of the magnetic flux density.

In this work, the vertical component of magnetic flux density above the magnetized sample is

mapped with a miniature Hall probe (AREPOC AXIS-3H) controlled by a XYZ micropositioning

Optical sensor for

reference position

Pick-up coils

Sample position in the

thermally insulated

sample holder

DC electric motor (in a

ferromagnetic shield) and

optical position sensors

33

cm

Coils full range

displacement

(75 cm)

Chapter 3. Experimental methods 45

stage. The probe is placed 0.5 mm above the measured surface of the whole SC/FM hybrid

structure. The bottom coating under the probe is estimated to be 0.35 mm thick, which means

that the flux density is measured approximately 0.85 mm above the highest point of the

measured surface. This small but finite gap is necessary to avoid contact, but it should be kept

in mind that details of the flux distributions may be slightly flattened out. A full mapping

corresponds to a 30 mm wide square with 1 mm long steps. The sample and the SC/FM hybrid

structures are magnetized along the superconductor’s c-axis using the field cooling (FC)

method under a uniform field of 670 mT produced by an iron-cored electromagnet, which

ensures full magnetization of the sample. The mapping procedure always starts 20 minutes

after the magnetizing field has been removed to minimize the influence of flux relaxation. The

sample is fully immersed in liquid nitrogen (77 K) during the whole experiment (magnetization

and subsequent measurement). For the hybrid SC/FM configurations, the ferromagnet was

present during the field cooled magnetization (and not added after the magnetization

process).

Fig. 3.13 Photograph experimental set-up for Hall probe mapping. The sample is a

superconductor / ferromagnet hybrid structure clamped in a PVC ring.

3.5 Summary of the measurement set-ups

In this chapter the measurements systems used throughout this work were described. Table

3.1 summarizes these set-ups together with the corresponding measured physical quantities,

their main advantages and limitations. The first described set-up, the permeameter, is used

only to measure the 𝐵(𝐻) curve and the intrinsic permeability of long rods of soft

ferromagnetic materials. Next, the Physical Property Measurement System (PPMS) was

described together with the different sensors and options added to the core system. A

particular attention was paid to the coils and Hall probes measurement set-up. The

commercial ACMS option for AC and DC magnetic measurements was briefly explained and the

rotator inset used for studying the transverse field configuration was presented. Finally, the

magnetometer and Hall probe mapping set-ups were described.

Among the experimental systems described above, the personal realizations that are part of

thesis work are the improvement and calibration of the coil and Hall probe system to be

0.5 mm gap

3-axis miniature Hall probe AREPOC AXIS-3H

3 cm

46 Chapter 3. Experimental methods

inserted in the PPMS. This includes the development of the LabVIEW program for data

acquisition, the connection box and the sample holder. As mentioned already, the rotator

system, the magnetometer and the micropositioning stage of the Hall probe mapping set-up

were designed and developed by other members of our research team. As a part of this work,

the sample holder for the magnetization, transfer and Hall probe mapping was developed and

the LabVIEW control and acquisition program was upgraded to increase speed and improve

the ease of use. Finally, several Matlab programs were developed to process measurement

data obtained by the various devices.

Table 3.1 Summary of the experimental set-ups used throughout this work.

Measurement set-up Measured quantities Advantages & limitations

Permeameter DC 𝐵(𝐻) curves and intrinsic DC permeability

Intrinsic properties, only for rods of ferromagnetic material of specific geometry (length 550 mm, diameter 10.5 mm).

PPMS depending on the specific set-up used (see below)

All PPMS measurements can be performed at temperatures down to a 1.9 K, and under magnetic fields up to 9 T, ramped at a sweep rate up to 17 mT/s.

PP

MS

spec

ific

set

-up

Home-made measurement system with coils and Hall probes

Average magnetic flux density inside the bulk samples and magnetic flux density at two locations near the sample surface.

The average magnetic flux density can be determined only under time-varying external field. Can be used with samples up to 21.5 mm in diameter.

ACMS – DC mode DC magnetic moment Samples up to ≈ 10 mm3, the maximum size depends on both 𝜇r and the applied field (maximum DC moment = 5 × 10-3 Am²).

ACMS – AC mode 𝜒′ and 𝜒′′ (AC magnetic susceptibility)

Samples up to a ≈ 10 mm3, for the same reasons as above.

Rotator Magnetic flux density near the sample surface

Used for measurements under transverse applied field and hysteresis loop measurements. Suitable for cylindrical samples up to 9 mm in diameter and 7 mm in height.

Chapter 3. Experimental methods 47

Table 3.1 (continued) Summary of the experimental set-ups used throughout this

work.

Magnetometer DC magnetic moment Can only be used for measuring the remanent magnetic moment of the sample (no magnetizing coil). Suitable for samples up to 17 mm in diameter.

Hall probe mapping Distribution of magnetic flux density above the surface of the sample

Full distribution of the magnetic flux density at 0.85 mm of the sample surface or at larger distance. Measurements carried out at 77 K (liquid nitrogen bath). Used only in the remanent state (limited space available for a magnetizing coil).

48 Chapter 3. Experimental methods

49

Chapter 4

Modelling methods

The set-ups presented in chapter 3 bring important experimental information about the

behaviour of a magnetized sample, including the flux density distribution around the sample

and the average magnetic properties. Numerical modelling of high-temperature

superconducting and ferromagnetic materials is a powerful tool to determine the physical

parameters that are not directly available through measurements, to help explain

experimental results, and to investigate the physical mechanisms that are responsible for the

observed behaviours. Additionally, modelling can be used to design superconductor /

ferromagnet structures and predict their performance as a function of the material

parameters.

Apart for the investigation of the “crossed-field” effects (cf. Sect. 7.3), the configurations

studied in this work always exhibit an axial symmetry for the sample and the applied field.

They will therefore be modelled in a 2D axisymmetric geometry. In this chapter we present the

modelling framework used in this thesis along with a reminder of the corresponding

mathematical basis and the underlying physics related to each model. The Biot-Savart

equation is solved using the Brandt algorithm to determine the field-dependent current

distribution inside the superconductor. The Maxwell equations under the magnetodynamics

approximation are solved by the finite element method using one of two formulations: the A-

formulation and the H-formulation. These models are used to study the magnetic hysteresis

curves and magnetic flux distributions generated by the superconductors and hybrid

structures. They are also used to predict the magnetic properties of the structures assuming

different constitutive laws and geometries. The particular case of the A-formulation in

magnetostatics will be presented as a simple model to predict roughly but easily the influence

of a ferromagnet on the flux distribution generated by a hypothetic fully magnetized

superconductor with constant 𝐽c. Finally, the finite element method is also used to solve

Campbell equations based on the force-displacement curves of individual flux lines in a type-II

superconductor in order to understand the dynamics of magnetic flux penetration.

Modelling results presented in this work were obtained through collaborations with other

researchers, except for the A-formulation of Maxwell’s equations which was part of this work.

The analysis of the modelling results was also part of this work. A summary of the different

methods and of the people who contributed to these models is presented at the end of this

chapter.

50 Chapter 4. Modelling methods

4.1 The Brandt algorithm

The Brandt algorithm [104] is a method used to determine the distribution of current density

inside the volume of a superconductor with a 1D or 2D symmetry subjected to an applied

magnetic field. It allows one to predict the magnetic flux penetration properties in

superconducting strips and slabs [105], plain cylinders and disks [104] as well as hollow tubes

[106, 107].

This algorithm is based on the numerical integration of the Biot-Savart and Faraday equations

where only the superconducting domain is discretized. The current density 𝑱(𝒓, 𝑡) is

determined inside the superconductor and the magnetic flux density is then deduced from the

current density distribution.

Consider the axially symmetric superconductor schematically illustrated in Fig. 4.1, subjected

to a uniform magnetic flux density 𝑩a = 𝐵a𝒆z. Both the current density and the electric field

are azimuthal, 𝑱 = 𝐽𝒆θ and 𝑬 = 𝐸𝒆θ. The Brandt algorithm consists in integrating the following

integral equation over time:

𝐽(𝑟, 𝑡) = −1

𝜇0∫ ∫ 𝑄cyl

−1(𝑟, 𝑧, 𝑟′, 𝑧′) [𝐸(𝐽(𝑟′, 𝑧′, 𝑡)) − 𝐸a(𝑟′, 𝑧′, 𝑡)] d𝑧′𝑏

0

𝑑𝑟′𝑎

0

(4.1)

where 𝑄cyl−1(𝒓, 𝒓′) is formally the inverse function of the kernel 𝑄cyl(𝒓, 𝒓′) given by

𝑄cyl(𝒓, 𝒓′) = 𝑓(𝑟, 𝑟′, 𝑧 − 𝑧′) + 𝑓(𝑟, 𝑟′, 𝑧 + 𝑧′)

with

𝑓(𝑟, 𝑟′, 𝜂) ≜1

2𝜋∫

𝑟′ cos 𝜃

(𝜂2 + 𝑟2 + 𝑟′2 − 2 𝑟 𝑟′ cos 𝜃)1 2⁄

𝜋

0

d𝜃,

i.e. 𝑄cyl(𝒓, 𝒓′) is obtained by integrating the 3D Green function of the Laplace equation

1 4𝜋|𝒓 − 𝒓′|⁄ over the angle 𝜃, while taking into account the mirror symmetry about 𝑧 = 0. In

practice, once the domain is discretized, 𝑄cyl(𝒓, 𝒓′) is evaluated on the mesh points 𝒓𝑖 and

𝑄cyl−1(𝒓, 𝒓′) is obtained from the inverse of the matrix 𝑄cyl(𝒓𝑖, 𝒓𝑗).

Chapter 4. Modelling methods 51

Fig. 4.1 Schematic representation of a cylindrical superconductor (diameter 2𝑎,

height 2𝑏) in which the current density is computed with the Brandt algorithm.

In Eq. (4.1), the constitutive equation 𝐸(𝐽) is the power law presented in Chap. 2, Eq.(2.2):

𝑬(𝑱) = 𝐸c (|𝑱|

𝐽c(𝐵))

𝑛𝑱

|𝑱| (4.2)

and 𝐸a is the electric field due to the external source.

In this work, the Brandt method was used to determine the critical current density of the

superconductor from magnetic measurements carried out on this superconductor alone (i.e.

without ferromagnet). The model was used to compute the hysteresis curve of Δ𝐵avg, i.e.

Δ𝐵 = 𝐵 − 𝜇0𝐻app averaged over the volume of the sample. This quantity can be obtained

experimentally using a pick-up coil tightly wound around the sample (cf. Chap. 3). Starting

from an “educated guess” of 𝐽c(𝐵), the hysteresis curve is computed for the same applied field

ramp as that used in experiments. The procedure consists in adjusting the parameters of the

superconductor critical current law to fit the modelled curve to the measured one.

More precisely, a 𝐽c(𝐵) dependence that follows the Kim law [40] 𝐽c(𝐵) = 𝐽c1 (1 + 𝐵/B1)−1

— as introduced in Chap.2, Eq. (2.1) — is assumed in the first instance for the superconductor.

This 𝐽c(𝐵) law is used in the Brandt algorithm to determine the current distribution inside the

superconductor for roughly estimated values of 𝐽c1 and 𝐵1. The magnetic induction inside the

sample is, in turn, determined from the modelled current distribution, and its axial component

is averaged over the sample. The resulting hysteresis curve for a full cycle of applied field is

compared to the measurements and the sample parameters 𝐽c1 and 𝐵1 are determined by

minimizing the least square error between the computed and measured curves using a trust

region algorithm. Once the 𝐽c1 and 𝐵1 parameters are known, the distribution of current

density inside the sample can be determined at any value of the applied magnetic field 𝐻.

Then the average DC magnetization 𝑀 defined as the total magnetic moment 𝑚 divided by the

volume 𝑉 of the sample is determined from the current distribution, i.e.

𝑀 =𝑚

𝑉=

1

𝑉 ∫

𝒓 × 𝑱

2d3𝒓. (4.3)

𝑎

𝑏

𝒆z

𝒆r

𝒆θ

52 Chapter 4. Modelling methods

4.2 The Finite Element Method

The finite element method (FEM) is used to easily determine the magnetic flux density

distribution around and inside the samples and to better understand the influence of the

ferromagnet on the magnetic behaviour of superconductor / ferromagnet hybrid structures.

FEM can be used for modelling materials with nonlinear characteristics, such as

superconductors and ferromagnets. It is also well suited for complex geometries.

Basically, the finite element method (FEM) is a numerical method used to find an approximate

solution to a problem described by partial differential equations (e.g. Maxwell’s equations) and

boundary conditions. The finite element method is based on the double discretization of the

geometric domain and of the function spaces of the unknown scalar and vector fields [108].

The problem to be solved is first expressed with scalar and vector potentials to help compute

the unknown fields. The resulting expressions, or formulations are equivalent to the original

model but differ by the potentials used and therefore by the independent variables solved in

the discretized problem. These so-called “strong forms” of the problem are then transformed

in “weak forms” suited to discretization by the finite element method. The unknown functions

are expressed as a linear combination of basis functions of the discrete function space

(Galerkin method). Finally, the equation of the problem is computed numerically by minimizing

the Galerkin residual of the weak formulation. The resulting weak solution is an approximation

of the exact solution of the problem.

In this work, the finite element method is used to study the penetration of the magnetic flux

density and the trapped flux inside a high temperature superconductor or a superconductor /

ferromagnet hybrid structure. The underlying physics is described by Maxwell’s equations in

the magnetodynamics approximation, i.e. where the displacement currents are neglected:

∇ × 𝑬 = −

𝜕𝑩

𝜕𝑡∇ × 𝑯 = 𝑱∇ ∙ 𝑩 = 0

The material properties are introduced in the model by the two constitutive laws 𝑩 = 𝜇 𝑯

and 𝑱 = 𝜎 𝑬, where the magnetic permeability 𝜇 for magnetic materials and the electrical

conductivity 𝜎 for superconductors are non-linear functions of the fields 𝑯 and 𝑬. Two distinct

formulations of the basis equations are used in this work: the A-formulation and the H-

formulation. These two formulations are described in the following sections. As a third

application, the finite element method is also used to resolve Campbell’s equation. This

equation gives directly the critical state in the superconductor, based on the force-

displacement curve of the flux lines.

Chapter 4. Modelling methods 53

4.2.1 The A-formulation

In this formulation, we define a vector potential 𝑨 such as ∇ × 𝑨 = 𝑩, which ensures

that ∇. 𝑩 = 0. The divergence-free magnetic flux density is therefore guaranteed by this

formulation. This property is of interest since the magnetic flux is the quantity directly

available in experiments. This formulation has been studied in details [108, 109] and has also

been used for modelling superconductors [110, 111]. This section describes the specific case of

the 2D axisymmetric problem.

The 2D axisymmetric representation of the system is shown schematically in Fig. 4.2. The

applied induction is chosen uniform and directed along 𝒆z: 𝑩a = 𝐵a𝒆z and the vector

potential 𝑨 in the azimuthal direction: 𝑨 = 𝐴θ𝒆θ.

Fig. 4.2 Left: Schematic representation of a cylindrical superconducting pellet (SC)

with a ferromagnetic disc (FM). Right: Axisymmetric geometry and mesh from the

mesh generator Gmsh [112] with the reference axes. 𝑩a is the direction of applied

magnetic flux density.

The magnetic law 𝑩 = 𝜇0 𝜇r(𝐻) 𝑯 can be expressed using the magnetic reluctivity 𝜈 = 1 𝜇⁄ :

𝑯 = 𝜈(𝐵) 𝑩.

In the superconductor, 𝜈 = 1 𝜇0⁄ and the reluctivity 𝜈 of the ferromagnet is defined using an

Akima interpolation [113] of (𝐵, 𝜈(𝐵)) pairs obtained from the intrinsic magnetization curve of

the ferromagnetic material.

In an excellent approximation, the conductivity of the ferromagnet is taken equal to zero since

the sweep rates of 𝑩 investigated in this work (i.e. |d𝐵 d𝑡⁄ | <15 mT/s, typical period of 800 s)

is too low to induce significant eddy currents in it. As with the Brandt model, the conductivity

of the superconductor is taken from the 𝑬(𝑱) law of the superconductor presented in Chap. 2,

Eq. (2.2):

“Free space”

𝒆z

𝒆r

𝒆θ

𝑩a

SC FM

Domain boundary

Symmetry axis

SC

FM

54 Chapter 4. Modelling methods

𝑬(𝑱) = 𝐸c (

|𝑱|

𝐽𝑐)

𝑛𝑱

|𝑱|

(4.4)

which can be expressed as 𝑱 = 𝜎(𝐸)𝑬, where

𝜎(𝐸) =𝐽c(𝐵)

𝐸c1 𝑛⁄

𝐸(

1−𝑛𝑛

).

A Kim model [40] approximation 𝐽c(𝐵) = 𝐽c1 (1 + 𝐵/B1)−1 is assumed for the field-

dependent critical current density.

The magnetic flux density is expressed as the sum of the applied induction and the induced

induction by the superconductor and the ferromagnet: 𝑩 = 𝑩a + 𝑩i = ∇ × 𝑨a + ∇ × 𝑨i

where 𝑨a = (𝑟 2⁄ ) 𝐵a 𝒆θ and 𝑨i = 𝐴i𝒆θ.

With these hypotheses, the azimuthal component of magnetic induction 𝑩 and the magnetic

field 𝑯 is always zero and the current 𝑱 and the electric field 𝑬 are directed along the

azimuthal direction 𝒆θ.

The Faraday law can be expressed in terms of the vector potential 𝑨:

∇ × 𝑬 = −𝜕𝑩

𝜕𝑡= −

𝜕

𝜕𝑡(∇ × 𝑨a) −

𝜕

𝜕𝑡(∇ × 𝑨i)

which will hold if

𝑬 = −𝜕𝑨a

𝜕𝑡−

𝜕𝑨i

𝜕𝑡− ∇𝜙.

Under our hypotheses, ∇𝜙 = 0 because of the axisymmetric geometry. Therefore,

𝑬 = −𝜕𝑨a

𝜕𝑡−

𝜕𝑨i

𝜕𝑡= −

𝜕𝑨

𝜕𝑡

where − 𝜕𝑨a 𝜕𝑡⁄ = (𝑟 2⁄ )(𝜕𝑩a 𝜕𝑡⁄ )𝒆θ. Using the previous expressions, we can rewrite

Ampere’s equation:

∇ × 𝑯 = 𝑱

⇔ ∇ × 𝜈 𝑩 = 𝜎 𝑬

⇔ ∇ × (𝜈 ∇ × 𝑨) + 𝜎 𝜕𝑨

𝜕𝑡= 0

This equation can be expressed as

∫ [∇ × (𝜈 ∇ × 𝑨) + 𝜎𝜕𝑨

𝜕𝑡] . 𝑨′dΩ

Ω

= 0 ∀𝑨′ (4.5)

Chapter 4. Modelling methods 55

where the integral has to be verified for any function 𝑨’. The integration by parts of (4.5) gives

∫ [(𝜈 ∇ × 𝑨) × 𝒏] ⋅ 𝑨′𝑑ΓΓ=∂Ω

+ ∫ (𝜈 ∇ × 𝑨) ⋅ (∇ × 𝑨′) 𝑑ΩΩ

+ ∫ 𝜎𝜕𝑨

𝜕𝑡⋅ 𝑨′𝑑Ω

Ω

= 0 ∀𝑨′.

(4.6)

To obtain (4.6), we used the Green identity for reducing the order of (4.5)

(𝒖, ∇ × 𝒗) − (∇ × 𝒖, 𝒗) = < 𝒖 × 𝒏, 𝒗 >

where following notations are used for the surface and volume integrals:

(𝑢, 𝑣) = ∫ 𝑢(𝒓)𝑣(𝒓)𝑑³𝒓Ω

and < 𝑢, 𝑣 >= ∫ 𝑢(𝒓)𝑣(𝒓)𝑑𝒞Γ

.

With these notations, equation (4.6) is written:

(𝜈 ∇ × 𝑨i , ∇ × 𝑨′) + (𝜈 𝑩a , ∇ × 𝑨′) + ⟨𝜈 (∇ × 𝑨i) × 𝒏 , 𝑨′⟩ + ⟨𝜈 𝑩a × 𝒏 , 𝑨′⟩

+ (𝜎 𝜕𝑨i 𝜕𝑡⁄ , 𝑨′) + (𝜎 𝜕𝑨a 𝜕𝑡⁄ , 𝑨′) = 0 ∀𝑨′. (4.7)

where the last two terms are computed inside the superconductor only since the electrical

conductivity 𝜎 is negligible in the rest of the domain and eddy currents are neglected inside

the ferromagnet (cf. above).

Boundary conditions have to be imposed in order to ensure the unicity of the solution. This will

lead to the annulation of the two surface terms. The vector potential 𝑨 is set to zero on the

axis for symmetry reasons. To ensure that the induced magnetic field 𝑩i decreases to zero at

infinity, 𝑨i is set to zero on the edges of the domain. With these conditions, the magnetic

vector potential is known on the external boundaries of the domain and, within the Galerkin

method, the test functions vanish on the boundaries of the domain. Therefore, the surface

terms in the weak formulation (4.7) are always equal to zero and can be withdrawn. A gauge

condition is usually introduced to ensure the unicity of the solution. Here, the Coulomb gauge

∇. 𝑨 = 0 is implicit since 𝑨 = 𝐴θ𝒆θ.

The solution of the problem is then computed numerically by minimizing the Galerkin residual

of the weak formulation (4.7). The unknown functions 𝑨 are expressed as a linear combination

of basis functions 𝑨𝑘 that are known

𝑨 = ∑ 𝑠𝑘𝑨𝑘

𝑀

𝑘=1

where 𝑀 is linked to the number of meshing elements and the basis function are non-zero only

near a meshing element. These 𝑨𝑘 functions will ensure the continuity of 𝐴θ between two

mesh elements.

56 Chapter 4. Modelling methods

To compute the flux distribution in the fully magnetized remanent state, we use a limited

number of time steps, following Lousberg et al. [110]: for a type II superconductor subjected to

a ramp of linearly varying field, a single time step was shown to be sufficient for obtaining the

sample magnetization provided that the values of the 𝑛-exponent were sufficiently large. For 𝑛

values similar to those investigated in this thesis, the error was estimated to stay below 3 %,

even for applied fields exceeding the penetration field. We apply this method to fully

magnetize the sample by computing its magnetic state twice: at the maximum applied field

and at the zero field (remanent state). This method allows the computation time to be

drastically decreased.

The solver used for this formulation is GetDP [114], developed by the Applied and

Computational Electromagnetics group of the University of Liège [115].

Particular case: the A-formulation in magnetostatics

In addition to the previous modelling, comparisons of results in extremely simple

configurations can be made with the QuickField FEM software. QuickField is a commercial

finite element analysis package for electromagnetic, thermal, and stress design simulation with

coupled multi-field analysis [116].

In this work, QuickField is used for simple models in magnetostatics. It will be used for the

modelling of the remanent induction in fully penetrated superconductors with a constant

critical current. In this configuration, the details of the magnetic flux penetration in the

superconducting state during the magnetization process of the sample are not taken into

account. The azimuthal current density in the permanently magnetized sample is assumed to

be constant. These simpler situations can be rapidly and easily implemented. In

magnetostatics, all time dependencies in Maxwell’s equations are neglected. The problem is

then driven by the two equations:

∇ × 𝑯 = 𝑱

∇. 𝑩 = 0

The latter can be expressed as the Poisson’s equation for the vector potential 𝑨 (as previously,

𝑨 is defined as ∇ × 𝑨 = 𝑩 since ∇. 𝑩 = 0):

∇ × (1

𝜇 ∇ × 𝑨) = 𝑱.

In the QuickField software, the models can be represented in axisymmetric configuration with

automatically made assumptions similar to the ones in the magnetodynamics models

presented above. In this work, a constant source current density 𝑱 is used to represent the full

magnetization of the superconductor. The magnetic behaviour of the ferromagnet is described

by (𝐵, 𝐻) pairs obtained from the measurements after neglecting the coercive field of the

material, i.e. without hysteresis.

Chapter 4. Modelling methods 57

4.2.2 The H-formulation

The finite element modelling framework is also used with the H-formulation in

magnetodynamics to model the remanent flux density above a superconductor / ferromagnet

hybrid structure. This modelling was carried out in the Bulk Superconductivity Group at the

University of Cambridge (UK). The numerical model developed here combines the

electromagnetic equations governing the behaviour of the superconductor based on the 2D

axisymmetric H-formulation [117, 118, 119, 120] with modifications to include magnetic

subdomains with a relative permeability 𝜇r(𝐻) as described in [121, 122, 77, 78].

Like in the A-formulation, the governing equations are derived from Maxwell’s equations —

namely, Faraday’s (4.8) and Ampere’s (4.9) laws:

∇ × 𝑬 +d𝑩

d𝑡= ∇ × 𝑬 +

d(𝜇0𝜇𝑟𝑯)

d𝑡= 0 (4.8)

∇ × 𝑯 = 𝑱 (4.9)

where 𝑯 = 𝐻r 𝒆r + 𝐻z 𝒆z represents the magnetic field, 𝑱 = 𝐽θ 𝒆θ represents the current

density and 𝑬 = 𝐸θ 𝒆θ represents the electric field. 𝜇0 is the permeability of free space, and

for the superconducting and air subdomains, the relative permeability is simply 𝜇r = 1. For the

ferromagnet subdomains, an appropriate constant value for 𝜇r is used, resulting in a linear

𝐵(𝐻) curve, until the saturation value 𝜇0𝑀sat is reached. Thus, 𝜇r is represented in the

numerical simulation by the following equation:

𝜇r =

𝜇r,max for 𝐵 < 𝜇0𝑀𝑠𝑎𝑡

(1 +𝑀sat

𝐻) for 𝐵 ≥ 𝜇0𝑀𝑠𝑎𝑡

As in the previous models, a Kim-model [40] approximation 𝐽c(𝐵) = 𝐽c1 (1 + 𝐵/B1)−1 is

assumed for the in-field behaviour of the bulk critical current, and the electrical properties are

modelled using a non-linear 𝐸(𝐽) power law given by equation (4.2)(4.4):

𝑬(𝑱) = 𝐸c (|𝑱|

𝐽c(𝐵))

𝑛𝑱

|𝑱| (4.10)

where 𝑛 will be determined experimentally.

Under the above assumptions in the 2D axisymmetric geometry, the induced current flows in

the azimuthal direction and equation (4.10) gives 𝐸θ = 𝜌 𝐽θ where 𝜌 is the resistivity of the

material. Ampere’s law (4.9) is written [119]:

𝐽θ = ∇ × 𝑯 =∂𝐻r

∂𝑧−

∂𝐻z

∂𝑟

58 Chapter 4. Modelling methods

and Faraday’s law (4.8) becomes:

[−

∂𝐸θ

∂𝑧1

𝑟

∂(𝑟𝐸θ)

∂𝑟

] = −𝜇0𝜇r [

∂𝐻r

∂𝑡∂𝐻z

∂𝑡

]. (4.11)

This model is used to study the remanent magnetization of a superconductor magnetized

under the field cooled procedure. However, isothermal conditions are assumed; hence, no

thermal model is included. In order to simulate the field cooled procedure used in the

experiment, a zero field cooled process is employed in the simulation, and based on a simple

Bean model [21, 22] approximation, as long as the externally applied field is twice the full

penetration field of the bulk sample, the trapped field will be the same as in the case of field

cooling [123]. A ramped, external magnetizing field up to 3 T is applied to the superconductor /

ferromagnet structure, then removed, over a period 𝑇 = 400 s (i.e. 𝐵a = 3 T at 𝑡 = 200 s), by

applying appropriate boundary conditions in the model [118].

The model is implemented in the commercial FEM software package COMSOL

Multiphysics 4.3a [124] using the general form partial differential equation (PDE) interface.

4.2.3 Campbell’s equation

Another set of numerical modelling studies was performed using a modelling framework

developed at the University of Cambridge (UK) [125, 126] in order to understand the dynamics

of magnetic flux penetration. This modelling framework is used to simulate various

magnetization processes of bulk superconductors by solving Campbell’s equation [125, 127]:

∇ × (∇ × 𝑨) = 𝜇0𝑱p + 𝒌 [1 − exp (− |(𝐴 − 𝐴p)𝜇0𝐽c

𝑘𝐴r|)]

where

𝒌 = 𝜇0𝑱c sign(𝐴p − 𝐴) − 𝜇0𝑱p.

Here, 𝑱c is the critical current density, 𝑨p and 𝑱p are the magnetic vector potential and the

corresponding current density of the initial state, respectively. 𝐴r is dependent on the material

and is related to the rate of the current density variation across a flux front. For obtaining the

remanent state, this equation is integrated twice. A first integration starts with 𝑨p and 𝑱p set

to zero to determine the first magnetization. A second integration starts with 𝑨p and 𝑱p

determined at the end of the first magnetization, to compute the return to the remanent

state. This equation is based on the force-displacement relation of magnetic flux lines [128,

129] and gives the distribution of flux density in critical state directly. The 𝐽c(𝐵) relation of the

bulk YBCO superconductor is described by the Kim model [40] 𝐽c(𝐵) = 𝐽c1 (1 + 𝐵/B1)−1. For

superconductor / ferromagnet structures, the properties of the soft ferromagnetic material

Chapter 4. Modelling methods 59

are incorporated by modifying the left hand side of Campbell’s equation from ∇ × (∇ × 𝑨) to

∇ × ((∇ × 𝑨) 𝜇𝑟⁄ ), where 𝜇r is estimated by the function

𝜇r = 𝜇r−max for 𝐵app = 0

𝜇r = 1 + sgn (𝐵app) 𝜇0𝑀sat

𝐵app[1 − exp (−

μr−max |𝐵app|

𝜇0𝑀sat)] for 𝐵app ≠ 0

In this equation 𝜇r−max denotes the maximum relative permeability and 𝑀sat the saturation

value of the ferromagnetic material. The 𝐵(𝐻) curve deduced from this function agrees

reasonably well with the experimental curve obtained with the permeameter, and it ensures

that the slope at 𝐵 = 0 is equal to 𝜇r−max. The phenomenon of hysteresis, however, is not

taken into account in the model since the coercive field measured is small.

These modified Campbell’s equations were solved using the finite element method (FEM) in

the commercial software package FlexPDE [130].

4.3 Summary of the models and collaborations

A number of different modelling frameworks were presented in this chapter. The

corresponding physics was described together with the resolution method.

First, the Biot-Savart equation is solved with the Brandt method. This work was made by

Sébastien Kirsch (Department of Electrical Engineering and Computer Science, University of

Liege). The corresponding work is published in [131] and is used to determine the critical

current parameters of the bulk high temperature superconductor.

Then Maxwell’s equations are solved with the finite element method. The A-formulation is first

presented under the magnetodynamics approximation of Maxwell’s equations. This

formulation is resolved in GetDP and is full part of this thesis work. It allows computing

hysteresis curves of the average flux density inside the superconductor as well as obtaining the

distribution of flux lines.

The same formulation is also used under the magnetostatics approximation using the

commercial software QuickField. These modelling results are also part of this work. This simple

model is used to predict roughly but easily the influence of a ferromagnet on the flux line

distribution of a superconducting trapped field magnet with constant 𝐽c.

The H-formulation of the Maxwell’s equations under the magnetodynamics approximation is

also solved with the finite element method. This modelling work was made by Mark Ainslie

(Bulk Superconductivity Group, University of Cambridge, UK) using the commercial software

Comsol. The corresponding results are part of a paper submitted to Superconductor Science

and Technology in April 2015.

60 Chapter 4. Modelling methods

Finally, the finite element method is used to solve Campbell’s equation This work was done by

Zihan Xu (Bulk Superconductivity Group, University of Cambridge, UK) using the commercial

software FlexPDE. These results, published in [131], show the distribution of flux lines and the

hysteresis curve of the average flux density inside the modelled superconductor / ferromagnet

structures.

61

Chapter 5

Characterization of the superconducting

and magnetic materials

This chapter deals with the characterization of the superconducting samples and magnetic

materials studied in this work. Two large grain, bulk superconducting samples are

characterized in the first part of the chapter. The first is a YBa2Cu3O7 (YBCO) sample (called

“ESC2”) synthesised at the University of Cambridge, UK and with which the majority of the

results were obtained. The second is a GdBa2Cu3O7 (GdBCO) superconductor made by Nippon

Steel & Sumitomo Metal Corporation, Japan and called “ESJc” in this work. The second part of

the chapter focuses on the magnetic properties of four commercial magnetic materials. These

include three ferromagnetic metallic materials (Supra50, Permimphy, and C45 steel) and one

ferrimagnetic ceramic compound.

The purpose of this individual characterization is to know precisely the magnetic behaviour of

each material in view of understanding their mutual influence when they are combined in

superconductor / ferromagnet hybrid structures. Additionally, the physical parameters and

constitutive laws are necessary inputs to perform accurate modelling of the studied structures.

Some characteristics of the samples and materials were measured by different methods; the

corresponding results will be compared when possible. Similarly, the modelling results

obtained by different models for a given material will be compared in order to assess the

suitability of these models. Obviously, the modelling results will be compared to the

measurement results.

5.1 Bulk YBCO superconductor (ESC2)

The sample ESC2 is shown in Fig. 5.1. It is a solid, cylindrical bulk YBCO superconductor of

diameter 16.5 mm and height 6.32 mm, with its c-axis parallel to its thickness. The material

was synthesized using conventional top seeded melt growth (TSMG) at the University of

Cambridge (UK). The melt-processed, large grain microstructure consists of a superconducting

YBa2Cu3O7-δ (Y-123) phase matrix containing discrete Y2BaCuO5 (Y-211) inclusions [30, 31]. The

top and bottom faces of the as-processed grain were polished prior to characterization.

62 Chapter 5. Characterization of the superconducting and magnetic materials

Fig. 5.1 Left: Photograph of the YBCO superconducting sample ESC2 on a 5 x 5 mm

squared sheet. Centre: Photograph of the same sample with measuring coils tightly

wound around it. Right: Schematic representation of ESC2.

Referring to the superconducting constitutive laws described in Chap. 2, the electric field vs.

current density is assumed to follow an 𝑬(𝑱) power law, i.e.

𝑬(𝑱) = 𝐸𝑐 (|𝑱|

𝐽𝑐)

𝑛𝑱

|𝑱| (5.1)

where 𝐸𝑐 is the threshold electric field, arbitrarily chosen to be 𝐸𝑐 = 10-4 V/m. Both the

𝑛-exponent and the critical current density – possibly field-dependent 𝐽c(𝐵) – must be

determined through measurements. Fig. 5.2 shows the procedure followed. The 𝑛-exponent is

determined through the measurement of the relaxation of permanent superconducting

current loops in the bulk superconductor. This can be done by measuring the time-

dependence of (i) the trapped flux density at the surface of the sample (using a Hall probe) or

(ii) the trapped magnetic moment of the whole sample (using the magnetometer described in

Sect. 3.3). The critical current density is determined inductively. The usual method, i.e.

measuring a hysteresis curve of the sample magnetic moment under a varying external

magnetic field, cannot be performed on the whole bulk sample using a commercial set-up

because of its size. In the present work, we use pick-up coils wound around the sample to

record the magnetic flux density, and determine the magnetic moment through numerical

modelling using the Brandt method. This requires knowledge of the 𝑛-exponent and some

assumption on the 𝐽c(𝐵) law. A final run of measurements was performed after the

investigation of superconductor / ferromagnet hybrid structures. It involved extracting small

sub-specimens from the parent superconductor, which is necessarily destructive in nature. The

magnetic moment of such small samples could be measured using a commercial set-up (ACMS

option of the PPMS).

Chapter 5. Characterization of the superconducting and magnetic materials 63

Fig. 5.2 Schematic illustration of the experimental procedure followed to determine

both the 𝑛-exponent and the 𝐽c(𝐵) characterstic law of the superconducting

material.

5.1.1 Determination of the critical 𝑛-exponent

As presented in chapter 2, the critical exponent 𝑛 can be estimated assuming that the

remanent magnetization of the superconductor vs. time follows a power law relationship

𝑚(𝑡) = 𝑚0 (1 +𝑡

𝑡0)

1(1−𝑛)

, (5.2)

where 𝑚0 is the initial magnetic moment and 𝑡0 is a characteristic time before the beginning

of the magnetic relaxation [132]. Note that the field dependence of the critical current is

neglected in the determination of 𝑛 presented below. Physically, the flux relaxation is related

to the magnetic flux density distribution in the sample; the latter being affected by any field-

dependence of the superconducting current 𝐽c(𝐵). The practical consequence is that the

experimental value of the 𝑛-exponent determined through magnetic measurements is

affected by this field-dependence. As discussed by Vanderbemden et al. [91] and Fagnard [46],

the “apparent” measured exponent (𝑛′) can be related to the “true” exponent 𝑛 through

𝑛′ = 𝑛(1 + 𝛾), where 𝛾 is a dimensionless parameter related to the particular 𝐽c(𝐵)

dependence. This influence of 𝐽c(𝐵) on 𝑛 is neglected here.

In the first method, a Hall probe is used to measure the trapped flux density on the symmetry

axis of the sample just above its surface 𝐵z(𝑡) [103]. The home-made measuring set-up

described in section 3.2.1 is used. The sample is oriented such that its seed is placed against

the measuring probe. The sample is first cooled at 77 K in zero field. A magnetic field of 3 T is

applied with a sweep rate of 17 mT/s and set to zero at the same rate. The trapped flux density

is then measured for more than two days (2.18 × 105 s).

64 Chapter 5. Characterization of the superconducting and magnetic materials

Fig. 5.3 shows the measured time relaxation of the trapped magnetic flux density using either

a linear (left) or logarithmic (right) scale. From Eq. (5.2), the time relaxation in logarithmic scale

is expected to be a straight line of slope 1 (1 − 𝑛)⁄ for 𝑡 ≫ 𝑡0. As can be seen in Fig. 5.3(b),

experimental data follow this expected linear behaviour. Interestingly, two different regimes

are observed: the slope from 𝑡 = 104 s corresponds to 𝑛 = 44.9 while from 𝑡 = 500 to 𝑡 = 5000 s,

a smaller value (𝑛 = 36.9) is found. This value measured on a shorter time period, will be

compared to the magnetometric measurements described below. The value of 𝑛 = 45 —

obtained before running the first models — is used in modelling throughout this thesis. The

modelled magnetic quantities (e.g. the full penetration field [105]), however, are weakly

influenced by the exact value of 𝑛, as long as it is large enough.

Fig. 5.3 Relaxation of the trapped flux density in ESC2 as measured by a Hall probe

against the top face of the sample, on the symmetry axis; (Left) Linear scale; (Right)

Logarithmic scale.

In the second method, the magnetic moment 𝑚(𝑡) of the same bulk sample is measured

directly with the magnetometer [99, 100]. The following measurement results are taken from

these two references. Fig. 5.4 shows the time dependence of the remanent magnetic moment

of the ESC2 sample. For this measurement, the sample was first magnetized along its c-axis in a

1 T Halbach array permanent magnet through a field-cooled process. This procedure ensures

that the sample, whose penetration field 𝜇0𝐻p is 0.54 T — as determined from the

magnetization curve in the next section — is fully magnetized under 1 T. A 200 s time interval

is elapsed for allowing insertion in the magnetometer. The time-dependence of the magnetic

moment is then recorded for more than one hour. The initial magnetic moment can be

estimated from a fit of experimental data using Eq. (5.2).This procedure gives 𝑛 = 37.4 and 𝑚0

= 0.431 ± 0.025 Am2. The value of 𝑛 is in excellent agreement with the one of 𝑛 = 36.9

obtained above in the interval 𝑡 = 500 to 𝑡 = 5000 s. The value of 𝑚0 will be compared to the

one obtained from the measurement results detailed in the next section.

Chapter 5. Characterization of the superconducting and magnetic materials 65

Fig. 5.4 Time dependence of the magnetic moment of ESC2 measured with the

magnetometer. Figure reproduced from [100]. The superconductor was

magnetized with a 1 T Halbach array permanent magnet. The blue circles are the

measured values and the red solid line is a fit to relation (5.2).

In summary, the results obtained in this section give experimental evidence that the time

dependence of the flux density measured against the sample surface 𝐵𝑧(𝑡) behaves like the

magnetic moment of the whole sample 𝑚(𝑡). Now that the 𝑛-exponent is known, the field-

dependent critical current 𝐽c(𝐵) of the superconductor can be determined as presented in the

next section.

5.1.2 Determination of the critical current density 𝐽𝑐(𝐵)

In this section, the critical current density of the superconductor is estimated by three

different methods. The first method is based on the Bean model and assumes a constant

critical current, which is determined from the trapped magnetic moment found in the previous

section. The second method is used to determine the field dependent critical current density

𝐽c(𝐵) through measurement of the magnetic flux density over the whole volume of the sample

and on modelling with the Brandt algorithm. The last method consists in measuring the field

dependence of the magnetic moment of small superconducting samples extracted from the

bulk superconductor. This is a destructive method that must therefore be applied a posteriori.

Constant 𝐽𝑐 approximation

In a first approximation, a constant, uniform critical current is assumed. Under this hypothesis,

the trapped magnetic moment of the whole sample 𝑚0 measured by the magnetometer can

be directly related to the critical current (Bean model), as shown schematically in Fig. 5.5. If

finite size effects are neglected, one has 𝐽𝑐 = 3𝑚0 𝑎 𝑉⁄ where 𝑎 is the radius of the sample

and 𝑉 its volume. From the magnetic moment 𝑚0 = 0.431 Am2 determined in the previous

66 Chapter 5. Characterization of the superconducting and magnetic materials

section, one finds 𝐽c = 1.16 × 108 A/m2. This value is an approximated average value in the

remanent state of a fully magnetized trapped field magnet which does not take the field

dependence of 𝐽c into account.

Fig. 5.5 Magnetization profile of a bulk superconductor under the Bean model: a

constant current 𝐽𝑐 is assumed in the whole sample volume.

Determination of 𝐽𝑐(𝐵) for the whole bulk sample

The following method involves three steps: (i) measurement of the average magnetic flux

density inside the bulk superconductor, (ii) choice of a model describing the 𝐽c(𝐵) law, and (iii)

determination of the parameters of the 𝐽c(𝐵) law through modelling using the Brandt

algorithm.

As discussed in Chap. 3 (Sect. 3.2.1), the average magnetic flux density inside the bulk

superconductor is measured using a pick-up coil wrapped around the entire thickness of the

sample, (Fig. 5.6). The voltage appearing across this pick-up coil when the external field is

changed is integrated to obtain ⟨𝐵⟩, the z-component of the magnetic flux density averaged

over the whole sample. In this experiment, the applied field is cycled slowly between 3 T and –

3 T using a sweep rate of 15 mT/s. In order to obtain a “classical” hysteresis cycle, the applied

magnetic field 𝐻app multiplied by 𝜇0 is subtracted from the measured flux density; Fig. 5.7

represents (dotted blue curve) the measurement results Δ𝐵 = ⟨𝐵⟩ − 𝜇0𝐻app as a function of

𝜇0𝐻app.

Fig. 5.6 Schematic representation of a coil wound tightly around the full height of a

superconductor. In practice, the coil has 53 turns and the wire is 80 µm in

diameter.

Chapter 5. Characterization of the superconducting and magnetic materials 67

Fig. 5.7 Comparison between the measured Δ𝐵 = ⟨𝐵⟩ − 𝜇0𝐻app hysteresis curve

(dotted, blue curve), the computed Δ𝐵 hysteresis curve (solid, red curve) and the

computed volume average DC magnetization 𝑀int (solid, green curve).

A 𝐽c(𝐵) dependence following Bean-Kim law [40] 𝐽c(𝐵) = Jc1 (1 + 𝐵/B1)−1 was assumed in

the first instance. Numerical modelling was then carried out using the Brandt method

described in Chap. 4 (Sect. 4.1) in order to determine the field-dependent critical current of

the sample. An 𝑛-exponent of 45 was used in the Brandt algorithm. The values of 𝐽c1 and 𝐵1

parameters were determined as 𝐽c1 = 1.38 × 108 A/m2 and 𝐵1 = 0.987 T by minimizing the least

square error between the computed and measured curves. The modelled hysteresis curve (red

line) is compared to the experimental Δ𝐵 = ⟨𝐵⟩ − 𝜇0𝐻app curve in Fig. 5.7.

The true DC magnetization curve as defined by

𝑀 =𝑚

𝑉=

1

𝑉 ∫

𝒓 × 𝑱

2d3𝒓 , (5.3)

is then calculated using the Brandt algorithm and the previously determined values of 𝐽c1 and

𝐵1. As shown by the results shown in Fig. 5.7 (green line), this modelled DC magnetization

differs considerably from the measured and modelled Δ𝐵 hysteresis curves. The magnetization

is approximately 2.47 times greater than Δ𝐵. Similarly, a ratio of ≈ 2.40 is found between the

initial slopes of the two modelled curves. The differences between the Δ𝐵 and 𝜇0𝑀 curves

underlines the fact that, since the sample is of finite height, the experimental data obtained

with the sensing coil do not correspond to the DC magnetization of the superconductor [101],

although their general appearance is similar. Finite size effects have already been studied on

short (low aspect ratio) type-II superconductors using numerical modelling [133, 134], as well

68 Chapter 5. Characterization of the superconducting and magnetic materials

as magnetization measurements [135, 136]. However, these studies relate to the

magnetization 𝑀(𝐻app) rather than the Δ𝐵 determined experimentally in the present work.

The computed magnetization curve in Fig. 5.7 enables the full-penetration field 𝐻p to be

determined at the point where the first magnetization curve reaches the full hysteresis cycle.

In this case, 𝜇0𝐻p≈ 0.54 T. The remanent magnetization (i.e. when 𝜇0𝐻app = 0), is 𝜇0𝑀 =

0.401 T. Multiplying the magnetization by the sample volume, one obtains a magnetic moment

of 0.432 Am2. Remarkably, this value is in excellent agreement with the magnetic moment

found previously using the magnetometer (𝑚0 = 0.431 ± 0.025 Am2). This result gives

confidence on the suitability of our method to determine the critical current of the sample.

It can be concluded from the above analysis that the experimental set-up using sensing coils

wrapped around the superconductor yields magnetic hysteresis loops that differ quantitatively

from, but are closely related to, the DC magnetization. The significant advantage of this

experimental method is that it is suitable for investigating the DC volume magnetic properties

and the critical current 𝐽c of large samples. However, the method requires the 𝐽c(𝐵) variation

to be assumed and some curve fitting is necessary to determine the value of 𝐽c. Having

established its validity, therefore, this experimental method can be applied to the

superconductor alone and to the superconductor/ ferromagnet hybrid structures, which will

be done in the next chapters.

Note that this method using sensing coils could in principle be applied at 20 K to determine the

critical current at this temperature. This is of interest, for example, to study the influence of

𝐽c(𝐵) on the magnetic behaviour of superconductor / ferromagnet hybrid structures. When

the experiment was carried out at 20 K, however, the large Lorentz forces exerted on the

sample during the magnetization procedure at such a low temperature lead to the destruction

of the sample. Because of the strong flux gradient of the trapped field and the strong pinning,

large forces are transferred to the crystal lattice and create tensile stresses during the ramp

down of the magnetic field [137, 6]. They usually lead to mechanical deformation and in the

worst case to sample cracking or destruction (as was the case here). The consequence is that

several of small samples of the parent superconductor were available for magnetization

measurements with a commercial magnetometer. These are presented below.

Determining 𝐽𝑐(𝐵) on a small sample of superconductor

The field-dependent critical current 𝐽c(𝐵) of the sample can be extracted directly from the

magnetization loop [134] using commercially available devices provided the sample size is

small enough (typically < 1 cm3). Fig. 5.8 shows a small rectangular prism that was selected

from the ESC2 sample after it broke. Its rectangular basis is 1.68 × 1.81mm and its thickness is

0.65 mm. Unfortunately, it was not possible to identify from which part of the bulk pellet this

sample comes from, which might be appropriate in the analysis since superconducting

properties vary along the radius and height of the sample as discussed in Sect. 5.1.3 below.

Chapter 5. Characterization of the superconducting and magnetic materials 69

The crystalline orientation of the sample, however, could be determined easily since melt-

textured YBCO materials are prone to fracture along the ab-planes [138].

Fig. 5.8 YBCO sample selected for magnetization measurements using the ACMS

option of the PPMS. The face cleaved parallel to the ab-plane is 1.68 × 1.81mm.

The thickness (parallel to the c-axis) is 0.65 mm.

The DC magnetic moment 𝑚 of the small section of ESC2 was measured at 77 K for an applied

field parallel to the c-axis cycled between +7 T and –7 T. The volume average

magnetization 𝑀avg = 𝑚/𝑉 is shown in Fig. 5.9. The irreversibility field determined at the

point where the increasing and decreasing branches of the loop merge together is equal to

𝜇0 𝐻irr ≈ 6.9 T. This value is within the range that can be expected for near optimally doped

YBa2Cu3O7−δ material [139]. The penetration field of this particular sample is 𝜇0𝐻p ≈ 0.24 T.

The critical current density is calculated from the parts of the irreversible magnetization loop

for which the sample is in the critical state. This condition is met in the range [𝐻p; 𝐻irr − 𝐻p]

for our short sample [134, 137, 104]. In the case of an infinitely long sample with a rectangular

cross-section (2𝑎 × 2𝑏, with 𝑎 < 𝑏), the critical current is given by

𝐽𝑐(𝐵) =Δ𝑀

𝑎(1 − 𝑎 3𝑏⁄ ) (5.4)

where Δ𝑀 is the width of the 𝑀(𝐻) curve [137].

The right panel of Fig. 5.9 shows (blue curve) the critical current determined using equation

(5.4). To obtain this curve, Δ𝑀 is taken as the width of the magnetization curve in the

magnetic field range [0.3 T; 5 T]. The same procedure is applied to the [-5 T; -0.3 T] interval

and the average of the two curves is taken. A fit of the measured 𝐽c(𝐵) data to the Bean-Kim

law gives 𝐽c1 = 3.8 × 108 A/m2 and 𝐵1 = 0.38 T. The resulting curve is shown in red in the right

panel of Fig. 5.9. The zero-field value of 𝐽c is 2.75 times higher than the average zero-field 𝐽c

determined previously from Δ𝐵 hysteresis curves on the whole bulk sample (1.38 × 108 A/m2).

This confirms that the superconducting properties are not uniform throughout the volume of

the superconductor and that the extraction of one small specimen out of the sample is not

sufficient to fully characterize this sample. The smaller value of 𝐵1 indicates a stronger

magnetic-field dependence of the critical current density for this particular sample.

70 Chapter 5. Characterization of the superconducting and magnetic materials

Note also that the measured 𝐽c(𝐵) curve exhibits a slight change in concavity between

approximately 2 T and 3 T. This behaviour indicates a “fishtail” effect (or so-called “peak

effect”) that requires specific 𝐽c(𝐵) models, for example an extended Bean-Kim model [140].

The fishtail peak may result in an increase in the current density with the magnetic field at low

temperature [137], as investigated below. In the present work, the 𝐽c behaviour above 3 T will

not be investigated as the measurements on the whole bulk pellet were carried out to 3 T

maximum.

Fig. 5.9 Left: Magnetization hysteresis loop measured at 77 K on a small specimen

extracted from the ESC2 bulk sample. Right: Determination of the field-dependent

critical current at 77 K: measurement (blue) and fit (red) to a Bean-Kim law.

Fig. 5.10 shows the magnetization data of the same sample measured at 20 K (black) and 50 K

(red); the previously presented curve at 77 K is shown for reference (blue). The penetration

field 𝜇0𝐻p is 0.8 T at 50 K and 2.5 T at 20 K. The critical current density, determined using

equation (5.4), is shown in Fig. 5.11. The magnetic field intervals in which 𝐽c is determined are

restricted at low temperature since 𝐻p is higher. At 77 K, the fishtail effect is clearly visible in

the semi-logarithmic scale. At 50 K, a fishtail effect is present between 3 T and 6 T and results

in an apparently constant 𝐽c in the semi-logarithmic scale. At 20 K, no secondary peak is visible

and the fit of the Bean-Kim law to measurement data gives 𝐽c1 = 45 × 108 A/m2 and 𝐵1 = 5.3 T.

At 𝐵 = 1 T, the corresponding 𝐽c at 20 K is ≈ 38 × 108 A/m2 (against ≈ 1.04 × 108 A/m2 at 77 K).

These values constitute a good approximation of how the superconductor behaves at lower

temperature. Numerical modelling of the bulk sample with high 𝐽c will be investigated in

Chap. 6 of this thesis.

Chapter 5. Characterization of the superconducting and magnetic materials 71

Fig. 5.10 Magnetization curves measured at 50 K (red) and 20 K (black) on the YBCO

sample shown in Fig. 1.8. The curve at 77 K is shown for comparison.

Fig. 5.11 Field-dependence of the critical current density calculated at 77 K, 50 K

and 20 K from the magnetization loops shown in Fig. 5.10.

72 Chapter 5. Characterization of the superconducting and magnetic materials

The 𝐽c results obtained in this section are summarized in Table 5.1. In this table, the values of

𝐽c are shown at three chosen magnetic fields (0 T, 0.33 T and 1 T). The values of 𝐽c at 0 T

corresponds to the parameter 𝐽c1 of the Kim model, when applicable. The 1 T value is often

taken as a reference for comparison between different superconducting materials [141]. In the

present case, the “0.33 T” field was also selected for the following reason. Numerical

modelling of the flux density distribution within the bulk superconductor (see Fig. 5.18 below)

magnetized under a 3 T amplitude (ZFC process) showed that the minimum and maximum

fields on the symmetry axis were 0 T and 0.66 T. Therefore it seems reasonable to take 0.33 T

as a reference magnetic field to compare the “constant 𝐽c” and “field-dependent 𝐽c” values, as

is done in Table 5.1.

Table 5.1 Critical current density results obtained for the YBCO ESC2 sample at

77 K.

Method Model for 𝐽c

𝐵1 (T) 𝐽c(𝐵 = 0 T) (10

8 A/m

2)

𝐽c(𝐵 = 0.33 T) (10

8 A/m

2)

𝐽c(𝐵 = 1 T) (10

8 A/m

2)

Remanent magnetic moment (whole bulk sample)

Constant — (∞) 1.16 1.16 1.16

Sensing coils (whole bulk sample)

Kim 0.987 1.38 1.03 0.69

𝑀(𝐻) loop (small sample)

Kim 0.38 3.8 2.03 1.04

As can be seen, the constant approximation calculated from the remanent magnetic moment

is well within the field-dependent values determined at 0 and 1 T for the whole bulk sample.

At the reference field 𝐵 = 0.33 T, the “constant 𝐽c” estimation is slightly larger than the value

determined from the 𝐽c(𝐵) law on the whole sample. One possible reason for this small

difference lies in the different magnetizing procedures used in both cases. For the

magnetometric measurement, the sample was magnetized in a field cooled (FC) procedure in a

1 T Halbach array permanent magnet. For the fluxmetric measurement, the sample was

magnetized using a slow 3 T triangular pulse applied after a zero field cooled procedure.

Finally, we note that the 𝐽c value determined in one of the small sub-specimens is much larger

than for the whole single grain. This behaviour is likely to be due to unavoidable radial and

axial inhomogeneities in melt-textured YBCO materials as investigated in the next section.

Note that since the sweep rates of the applied field were similar in both cases, the different

induced electric fields in both cases are unable to account for the observed difference. From

Faraday’s law applied to an infinitely long cylindrical sample (radius 𝑎), the induced electric

field at the sample surface 𝐸 is given by 𝐸 = 𝜇0(𝑎 2⁄ ) d𝐻app d𝑡⁄ , where 𝐻app is the applied

magnetic field [101, 142]. The electric field induced — and hence the current density — is

therefore expected to increase with increasing sample diameter; this behaviour is in contrast

Chapter 5. Characterization of the superconducting and magnetic materials 73

with the results shown in Table 5.1. It is therefore likely that the small sample comes from a

part of the pellet with good superconducting properties.

The field dependence of the critical current described by the Bean-Kim law, with the

parameters as 𝐽c1 = 1.38 × 108 A/cm2 and 𝐵1 = 0.987 T will be used in the following part of the

work to describe the superconductor, together with an 𝑛-exponent of 45. If a constant 𝐽c

approximation is required, the value determined at 𝐵 = 0.33 T will be used, i.e. 𝐽c = 1.03 × 108

A/cm2.

5.1.3 Probing inhomogeneities with sensing coils and Hall probes

The above results definitely suggest that the magnetic properties across the sample are

inhomogeneous. It is of interest to investigate to which extent such inhomogeneities can be

probed by non-destructive measurements on the whole bulk sample. In this section we study

how the 𝐽c variations within the single domain may influence the signals given by different

sensors. We use the experimental method presented in Chap. 3 (Sect. 3.2.1) to study the

quasi-static magnetic behaviour of the sample. We first present the Δ𝐵 hysteresis curves

obtained by four coils wound around different cylindrical sections of the sample and two Hall

probes centred on the top and bottom faces of the cylindrical sample.

In this experiment, six sensors were attached to ESC2 to enable magnetic characterisation of

the bulk samples at different locations. These sensors consisted of four pick-up coils and two

Hall probes, as shown schematically in Fig. 5.12. The pick-up coils were made of 80 µm-

diameter copper wire wound tightly around the superconductor. The first coil was already

used above for the determination of the critical current. It had 53 turns and was wrapped

around the entire height of the cylinder in a single layer (i.e. with its principal axis parallel to

the thickness of the sample). The three other coils were wrapped around the top, middle and

bottom sections of the solid cylinder (as shown in Fig. 5.12). The top and bottom coils

consisted of 19 turns (one single layer) whereas the middle coil, centred on the median plane

of the cylinder, consisted of 44 turns wound in two layers. There was a 1 mm gap between two

adjacent coils. The two Hall probes (Arepoc AHP-H3Z, with a 1 mm2 active area), driven by a

current of 1 mA, were placed at the centre of top and bottom surfaces of the cylinder to probe

the axial component (i.e. normal to the surface) of the magnetic field. All measurements

involved cooling down the sample in zero field and then applying a slowly time-varying

magnetic field parallel to the c-axis of the sample. The field was swept initially up to 3 T and

then cycled between 3 T and -3 T using a sweep rate of 15 mT/s.

Fig. 5.13(a) shows the hysteresis curves of Δ𝐵 = ⟨𝐵⟩ − 𝜇0𝐻app as a function of applied field

𝜇0𝐻app measured by the four coils wound around the superconductor only as shown in Fig.

5.12. The curve measured with the coil wound around the entire height was already used to

determine the field-dependent critical current of the whole sample (cf. Fig. 5.7). The four

experimental curves presented here approximate well to each other, which indicates a

74 Chapter 5. Characterization of the superconducting and magnetic materials

relatively uniform distribution of the average flux as a function of sample height. The

agreement between the data shows also that coils are well suited to measure the average

magnetic behaviour of the superconductor, despite the relatively small signals recorded for

coils of less than 20 turns.

Fig. 5.12 (a) Photograph of the YBCO sample with the three external coils visible, (b)

dimensions of the YBCO sample and (c) sectional view of the YBCO sample and the

six sensors consisting of four coils (wound around the entire height and individually

around the bottom, middle and top sections of the YBCO sample) and two Hall

probes (centred on the top and bottom surfaces).

Fig. 5.13 (a) Δ𝐵 magnetic hysteresis curves measured with the four coils on the

superconductor only. The coil wound around the entire height of the

superconductor is shown in blue. The top, middle and bottom coils are shown in

green, orange and red, respectively. (b) Magnetic hysteresis curves measured by

the Hall probes placed against the top (green) and bottom (red) faces of the

superconductor.

Chapter 5. Characterization of the superconducting and magnetic materials 75

Fig. 5.13(b) shows the hysteresis curves measured by the Hall probes positioned at the centre

of either the top (green) and bottom (red) faces of the bulk superconductor, using the same

cycle of applied field employed for the sensing coil measurements. By analogy with the data

shown in Fig. 5.13(a), we plot 𝐵 − 𝜇0𝐻app, where B now denotes the magnetic flux density at

the Hall probe location, as a function of the applied field. Note that the seeded surface of the

sample corresponds to the bottom face of the bulk single domain in the configuration studied

here. As can be seen from Fig. 5.13(b), the remanent magnetic field measured by the top Hall

probe (on the face opposed to the seed) is approximately 25 % lower than that measured by

the bottom probe positioned against the seed. Similar discrepancies have been observed in

other bulk samples fabricated by the TSMG process [143, 144, 145, 146, 147], and are related

to the unavoidable radial and axial distribution of both 𝑇c and 𝐽c within the single domain.

In the present case, it is of interest to investigate in more detail the reason why a non-uniform

𝐽c in the bulk sample has a larger effect on the local value of 𝐵 than on the average induction

probed by the pick-up coil. First, there might be an effect of a greater 𝐽c in a small volume

around the seed. In the present case, the section (perpendicular to the c-axis of the sample) of

the seed is approximately 2 mm x 3 mm. Since the sample was polished, it is difficult to

estimate the remaining thickness of the seed inside the YBCO sample. Additionally, the critical

current measurements carried out in the previous section have shown that the critical current

is higher in the measured small section of ESC2 than on average over the whole sample.

Second, there is also a variation of 𝐽c along the z-axis, which may result on a layer with weaker

𝐽c near the face of the superconductor opposite to the seeding surface. We believe this second

effect is predominant here, and can be explained as follows. We assume an extreme situation

in which the weaker 𝐽c layer far from the seed would be a thin non-superconducting layer

(approximately 0.5 mm), the rest of the sample having a uniform 𝐽c. This configuration is

shown schematically in the left panel of Fig. 5.14 and will be compared with the fully

superconducting sample: we compute by how much either the central 𝐵 (against the surface)

or the average 𝐵 (probed by the top sensing coil) is reduced when comparing both

configurations.

The decrease of the central B against the sample surface can be calculated using the analytical

expression of the magnetic flux density along the central axis of the sample in the fully

penetrated remanent state [102, 148]:

𝐵z(𝑧) =1

2𝜇0 𝐽c (𝑧 +

𝐿

2) ln (

𝑎 + √𝑎2 + (𝑧 + 𝐿 2⁄ )2

|𝑧 + 𝐿 2⁄ |)

− (𝑧 −𝐿

2) ln (

𝑎 + √𝑎2 + (𝑧 − 𝐿 2⁄ )2

|𝑧 − 𝐿 2⁄ |)

(5.5)

76 Chapter 5. Characterization of the superconducting and magnetic materials

Using this formula, it can be shown easily that the central 𝐵 on the surface decreases by

approximately 25 % against the non-superconducting layer (Fig. 5.14) compared to the original

sample. This decrease is similar to the one observed in the experiment.

The decrease of the remanent flux embraced by the top coil was computed using the

QuickField FEM software presented in Sect. 4.2.1.1. The remanent induction in the fully

penetrated superconductors is modelled with a constant critical current. The average flux

density embraced by the top coil was computed first for the dimensions of ESC2 and found to

be 0.151 T. Then, the same calculation was carried out for the sample where a 0.5 mm layer is

assumed to be non-superconducting. Fig. 5.14 shows the results obtained by the QuickField

software. The sample is modelled in axisymmetric configuration. The black lines are the

contour lines of the vector potential multiplied by the radius; they embrace a constant

magnetic flux. The flux density, averaged on the red area which corresponds to the small coil

including the non-superconducting layer, was found to be 0.133 T, which corresponds to a

decrease of about 12 %. In spite of the quantitative disagreement with experimental data, the

result can be understood qualitatively: the sensing coil still embraces a large part of fully

superconducting material whereas the effect on the local 𝐵 is equivalent to inserting a small

air-gap between the Hall probe and the sample, leading to a large reduction of the

measurement.

Of course, we could consider a more complex 𝐽c axial (or radial) dependence and other

experimental parameters that may play a role, e.g. small misalignments between the coils, but

these are likely to have a minor impact in the analysis.

Fig. 5.14 Left: Cross section of the sample with the sensors and drawing of a

hypothetical non-superconducting layer of 0.5 mm. Right: Modelling of the

remanent state using the QuickField software for the superconductor shown on the

left; axisymmetric geometry with the symmetry axis on the left. The flux density is

averaged on the red area which corresponds to the small coil including the non-

superconducting layer.

Significantly, this study indicates that the trapped field measured at the surface of the bulk

sample is influenced predominantly by a relatively thin layer of material in the vicinity of the

sensor. In addition, Hall probes measurements are sensitive to local variations of 𝐽c, which can

Chapter 5. Characterization of the superconducting and magnetic materials 77

be higher near the seed but also lower on the face opposite to the seeding surface. In contrast,

sensing coils measure the average magnetic behaviour over the section of the superconductor

they surround, and are influenced much less by non-homogeneities in the bulk sample, as

evidenced by the results in Fig. 5.13(a).

5.1.4 Modelling of hysteresis curves

In the previous sections, the superconducting properties of ESC2 were determined

experimentally. Before using these parameters to study the behaviour of the superconductor

in the presence of surrounding ferromagnets, we want to check that our modelling methods

combined with these parameters are able to reproduce the hysteresis curve of Δ𝐵 as

measured by the coil over the entire height of the superconductor. Two modelling methods

are investigated for this purpose: the A-formulation of Maxwell’s magnetodynamics equations

in a finite element framework and the semi-analytical Brandt method used to determine the

critical current parameters, as follows : 𝑛-exponent = 45 and 𝐽c(𝐵) = Jc1 (1 + 𝐵/B1)−1 where

𝐽c1 = 1.38 × 108 A/cm2 and 𝐵1 = 0.987 T.

Fig. 5.15 shows (green) the Δ𝐵 = ⟨𝐵⟩ − 𝜇0𝐻app hysteresis curve of the average Δ𝐵 over the

entire volume of the superconductor computed with the FEM A-formulation. This curve is

compared to the corresponding Δ𝐵 hysteresis curves (blue) measured with the coil wound

over the entire height of the superconductor and (red) computed with the Brandt method.

These two latter curves were already shown in Fig. 5.7. As can be seen, the curve modelled by

FEM is extremely close to that given by the Brandt method. The FEM-computed remanent

induction is seen to be slightly underestimated (7.2 % lower than the measured value,

compared to 2.5 % using the Brandt method). In spite of these minor differences, the results

plotted in Fig. 5.15 give evidence that these models are able to represent the behaviour of the

superconductor alone. The A-formulation will be used later in this work to study the magnetic

behaviour of the superconductor.

A note of caution must be made here. For these FEM results, it was realized a posteriori that

the external edge of the domain was set somewhat too close to the studied structure either

the superconductor alone (in this chapter) or superconductor / ferromagnet hybrid structures

(in the next chapters). The main consequence is a slight decrease of the magnetic flux on the

symmetry axis. Additional modelling carried out with boundary sets at twice the distance from

the sample allowed us to estimate this error to be about 3 %. Such small difference does not

affect any of the conclusions of the work. The modelling results were in good agreement with

the measurements and the models were not run again.

78 Chapter 5. Characterization of the superconducting and magnetic materials

Fig. 5.15 Average Δ𝐵 hysteresis curves over the entire volume of the

superconductor measured with the coil wound over the entire height of the

superconductor (blue), computed with the FEM A-formulation (green) and with the

Brandt method (red).

5.1.5 Measurement and modelling of the magnetic flux density

distribution

In this section the distribution of 𝐵z, the vertical component of the magnetic flux density above

the surface of the superconductor, was investigated by Hall probe mapping. In this set-up, a

Hall probe is moved over the surface of the fully magnetized superconductor (trapped field

magnet). The effective height of measurement is approximately 0.85 mm above the surface.

Prior to these experiments, the sample ESC2 was permanently magnetized by field cooling (FC)

down to liquid nitrogen temperature (77 K) under a 670 mT uniform magnetic field. Mappings

start 20 minutes after the magnetizing field has been removed to minimize the influence of

flux relaxation on the measurement.

The left panel of Fig. 5.16 shows the measured magnetic induction 𝐵z 0.85 mm above each

face of the superconductor alone (SC). The seed is on the top face for mapping measurements.

The induction is seen to follow a conic profile on both faces as predicted by the Bean model

[21, 22]. Since these results exhibit clear axisymmetric behaviour, we can examine in detail the

flux density distribution by plotting measurements radially along any diameter of the

superconductor, which is shown in the right panel of Fig. 5.16.

It is immediately apparent that the maximum flux density occurs at a slightly off-centre

position. The position corresponding to the measured maximum on the bottom face (𝑟 =

−1 mm) was selected as the reference position. This selection will remain for all mappings

Chapter 5. Characterization of the superconducting and magnetic materials 79

presented in this work. The characteristic values 𝐵z∗ = 𝐵z(𝑟 = −1 mm) are 336 mT and

250 mT on the faces near and far from the seed, respectively. The field on the face far from the

seed is approximately 25 % lower than that measured on the face containing the seed (top

face). This difference between top and bottom faces is entirely consistent with that shown in

Fig. 5.13(b) and will not be discussed further.

Fig. 5.16 Hall probe mappings of 𝐵z, the vertical component of the magnetic flux

density generated by the fully magnetized bulk high temperature superconductor

(ESC2). Top: Measurement on the face containing the seed. Bottom: Measurement

on the face far from the seed. Left: Full measured mappings. Right: Radial plots

along a diameter of the superconductor. R = 8.25 mm is the radius of the

superconductor.

The above measurements of the trapped magnetic flux density were reproduced by finite

element modelling under the magnetodynamics approximation, with both the A-formulation

and H-formulation presented in Chap. 4 (Sect. 4.2). The constitutive law parameters are those

determined in section 5.1.1 and 5.1.2 and already used in numerical modelling in section 5.1.4.

Both modelling frameworks were developed in a 2D axisymmetric configuration since

measurement results exhibit an axisymmetric behaviour. Therefore, all modelling results will

be mirrored around 𝑟 = 0 (centre of the bulk) to ease the comparison with measurements.

Additionally, the results will be identical on the top and bottom faces since the model assumes

Face near the seed

Face far from the seed

80 Chapter 5. Characterization of the superconducting and magnetic materials

uniform, macroscopic 𝐽c(𝐵) properties for the bulk superconductor. This is in contrast with

experimental results where a 𝐽c(𝑟, 𝑧) dependence needs to be considered to account for the

different flux distributions observed on the top and bottom faces.

Fig. 5.17 shows the flux density 𝐵z along a diameter obtained with both the H-formulation

(red) and the A-formulation (yellow). The scan height is 0.85 mm above the superconductor

surface. The magnetic flux density follows a conic profile. The maximum values at 𝑟 = 0 are

304 mT and 298 mT for the H-formulation and the A-formulation respectively. Fig. 5.17 shows

additionally the 𝐵z curves measured above both faces of the superconductor and already

presented in Fig. 5.16. These curves are measured 0.85 mm above the face containing the seed

(blue) and the face far from the seed (green). The general shape of the modelled flux density is

in good agreement with the measurements. The modelled distribution results are an

intermediate between those measured on the top and bottom faces. This is expected

intuitively since the 𝐽c(𝐵) data used in modelling were obtained from non-destructive average

magnetic measurements over the whole bulk pellet.

Fig. 5.17 Remanent magnetic flux density 𝐵z above the surfaces of the fully

magnetized superconductor obtained by finite element modelling both with the A-

formulation (yellow) and with the H-formulation (red). Measurements presented in

Fig. 5.16 are shown for comparison: above the face including the seed (blue) and

above the face far from the seed (green). All curves are along a diameter, 0.85 mm

above the surface. R = 8.25 mm is the radius of the superconductor.

Finally, the evolution of 𝐵z along the symmetry axis can be compared between both FEM

formulations. Fig. 5.18 shows the magnetic flux density 𝐵z as a function of the position 𝑧 on

the superconductor symmetry axis for the FEM H-formulation (red) and A-formulation

(yellow). The origin 𝑧 = 0 is the centre of the superconductor. The surface of the

Chapter 5. Characterization of the superconducting and magnetic materials 81

superconductor is shown by the vertical black line. The agreement between both 𝐵z(𝑧) curves

provides evidence that both models are equivalent to compute the magnetic behaviour of the

large grain, bulk superconductor.

These curves are then compared to the analytical expression of the magnetic flux density along

the axis of the sample in the fully penetrated remanent state given by Eq. (5.5). This expression

was used with the dimensions of the ESC2 sample and its field-independent 𝐽c value equal to

1.03 × 108 A/m2, as determined from the measurements shown in section 5.1.2 above. This

model assumes a constant critical current in the sample, unlike the FEM models which take the

field dependence of the critical current into account. The magnetic induction predicted by the

analytical model is close to that predicted by the FEM models. On the superconductor/air

interface, remarkably, the analytical curve is nearly equal to both FEM results within a 2 %

error bar. Nevertheless, the general shapes of the numerical and analytical curves are in good

agreement and the maximum observed difference (6 %) is relatively low taking into account

that the analytical formula based on the Bean model assumes that the (azimuthal) current

density in the sample is uniform and equal to 𝐽c. This is much simpler than the real current

density distribution resulting from the magnetization process and predicted by the more

advanced FEM models.

Fig. 5.18 Vertical component of the remanent magnetic induction on the symmetry

axis. Three models are compared: The FEM H-formulation (red) and A-formulation

(yellow) and the simple analytical model (blue) of Eq. (5.5). The 𝑧 = 0 position

corresponds to the centre of the superconductor, the black vertical line shows the

superconductor-air boundary.

82 Chapter 5. Characterization of the superconducting and magnetic materials

The agreement between the experimental and numerical results provides evidence that our

models are able to reproduce the magnetic properties of a superconducting disc, the

properties of it being obtained by preliminary independent experiments. This gives confidence

that these models can be used to study the flux distribution of the trapped field produced by

large, bulk high temperature superconductors.

5.1.6 Ageing of the superconductor

The measurements on ESC2 presented in this work were carried out during a relatively long

period of time starting in late spring 2012 and ending in December 2014. The superconducting

properties of YBa2Cu3O7-δ ceramics depend on the oxygen concentration, which can change

over time with the contact of air or humidity. The sample was stored in a hermetic recipient

with silica gel, which acts as a desiccant, to slow down the “ageing” of the superconductor, i.e.

the evolution of its superconducting properties.

It is nevertheless of interest to measure the evolution of the hysteresis curves over time. The

left panel of Fig. 5.19 shows the Δ𝐵 hysteresis curves measured by a coil over the entire height

of the superconductor ESC2 in June 2012 (blue) and December 2014 (red). In June 2012, the

coil was the same as presented previously (53 turns of 80 µm wire). In December 2014, a new

coil had to be wound around the superconductor; this one was made of 51 turns of 80 µm

wire. The two curves are found to be nearly superimposed, which is indicative of an excellent

reproducibility.

The right panel of Fig. 5.19 shows the Δ𝐵 hysteresis curves measured by the Hall probes. The

Hall probes used for both sets of measurements (2012 and 2014) were the same. Slight

differences can be observed between two measurement runs. The main difference is on the

bottom face (i.e. the one containing the seed) where the width of the hysteresis loop is larger

in 2014 than in 2012. This might indicate that the field dependence of the critical current might

have slightly changed during this time interval. Nevertheless, these variations are small and

mainly observable at high applied field. As can be seen in Fig. 5.19, the remanent flux density

was not altered.

As a conclusion, the surface properties have changed slightly between June 2012 and

December 2014, probably because of a change in the oxygen concentration in the YBa2Cu3O7-δ

ceramics. These variations were only measured under high applied field and were negligible on

the remanent magnetization. Additionally, the volume superconducting properties do not

show an evolution over this 30 months period.

Chapter 5. Characterization of the superconducting and magnetic materials 83

Fig. 5.19 Ageing of the superconductor. Left: Δ𝐵 hysteresis curves measured by a

coil over the entire height of the superconductor ESC2 in June 2012 (blue) and

December 2014 (red). The coil was re-wound between the measurements. Right:

Δ𝐵 hysteresis curves measured by the Hall probes centred on the top (green for

June 2012 and yellow for December 2014) and bottom (red for June 2012 and

violet for December 2014) faces of the superconductor.

5.2 Bulk GdBCO superconductor (ESJc)

The second sample studied in this work is a GdBa2Cu3O7 disk-shaped bulk superconductor from

Nippon Steel & Sumitomo Metal Corporation, hereafter called ESJc. It is 9 mm in diameter and

5.16 mm in thickness. Fig. 5.20 shows a Hall probe mapping of the trapped field of the fully

magnetized sample at 77 K. This plot was performed by Nippon Steel & Sumitomo Metal

Corporation and supplied together with the sample. The trapped field follows a conic profile

that reaches approximately 0.18 T in the centre. The conic profile shows the uniformity of the

sample properties.

This second sample will be used to measure the influence of a ferromagnetic disc on its

magnetic moment and trapped field when it is submitted to transverse applied fields (crossed

field configuration). In this section, we study the magnetic moment and hysteresis curve of the

sample alone as measured by the magnetometer and rotator option of the PPMS (at fixed

angle), respectively. We will compare the values of the 𝑛-exponent obtained by these two

measurement methods.

Fig. 5.21 shows the time evolution of the magnetic moment (blue circles) of ESJc as measured

with the magnetometer described in Chap. 3 (Sect. 3.3). The sample was magnetized by a field-

cooled process under 625 mT in a separate electromagnet. The measurement cycle starts 616 s

after having switched off the magnetizing field and lasts 104 s (approximately 3 h). The red

dashed line in Fig. 5.21 is obtained by linear regression on the data set in a logarithmic scale.

84 Chapter 5. Characterization of the superconducting and magnetic materials

Fig. 5.20 Hall probe mapping of ESJ1c at 77 K; measurement carried out by Nippon

Steel & Sumitomo Metal Corporation. The grey scale is in teslas.

Since flux creep occurs during the time interval before the first measurement, the first

measurement point does not represent the maximum trapped magnetic moment. As

presented previously, it is possible to estimate this value of magnetic moment using a power

law time dependence usually observed in flux creep phenomena. As for ESC2 above, equation

(5.2) will be used:

𝑚(𝑡) = 𝑚0 (1 +𝑡

𝑡0)

1(1−𝑛)

(5.2)

where 𝑚0 is the initial magnetic moment and 𝑡0 is the time before the beginning of the

magnetic relaxation [132]. A fit of this relation to the experimental points gives 𝑚0 ≈ 0.12 Am2

and 𝑛 = 32.9. The average critical current density in the remanent state can be approximated

from the magnetic moment as in section 5.1.2. We find 𝐽𝑐 = 3𝑚 𝑎 𝑉⁄ = 2.44 × 108 A/m2 which

is amongst the highest reported values for large grain, bulk (RE)BCO superconductors (RE =

rare earth element : Y, Gd, etc.) at 77 K [149].

Fig. 5.22 shows the hysteresis cycle measured with the Hall probe of the rotator set-up of the

PPMS. This curve was measured at fixed angle, in an applied field parallel to the c-axis of the

sample. The sample was cooled at 77 K in zero field. The magnetic field is swept initially up to

2 T and then cycled between +2 T and –2 T using a sweep rate of 3.33 mT/s. The penetration

field 𝜇0𝐻p is found to be 0.5 T, which shows that the magnetizing fields used in the following

experiments largely exceed 2𝐻p and allow full magnetization of the sample to be achieved.

0 2 4 6 8 10 12 14

0

2

4

6

8

10

12

14

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Chapter 5. Characterization of the superconducting and magnetic materials 85

Fig. 5.21 Time evolution of the magnetic moment of the ESJc sample as measured

with the magnetometer (blue circles).

Fig. 5.22 Hysteresis cycle of Δ𝐵 = 𝐵HP − 𝜇0𝐻app with 𝐵HP measured on ESJc at

T = 77 K with the Hall probe of the rotator set-up of the PPMS. The applied field is

parallel to the c-axis.

86 Chapter 5. Characterization of the superconducting and magnetic materials

The relaxation of the magnetic flux density 𝐵HP against the fully magnetized sample can be

recorded by the Hall probe of the rotator set-up, as shown in Fig. 5.23. For this measurement,

the sample is magnetized by field cooling (FC) under 1 T applied parallel to the c-axis. The field

is turned down at a sweep rate of 3.33 mT/s and the flux density is subsequently measured for

1660 s. The measurements follow a straight line in a logarithmic scale after approximately 10 s,

which is in agreement with the assumed behaviour described by the power law (5.2). The

slope of the straight line in the logarithmic scale between 102 s and 103 s corresponds to an

𝑛-exponent of 34.5 in Eq. (5.2). This value obtained from the magnetic flux density against the

sample is in good agreement with that found above from magnetic moment measurements.

Fig. 5.23 Time relaxation of the trapped magnetic flux density in ESJc. The magnetic

flux density 𝐵HP is measured above the sample, on its symmetry axis.

5.3 The ferromagnetic materials

Several magnetic materials will combined with the large grain, bulk superconductors described

in the first part of this chapter. This second part of the chapter focuses on these magnetic

materials which include three ferromagnetic metals and one ferrite ceramic. The first two

metals, Supra50 and Permimphy, are FeNi alloys which can be classified as soft ferromagnetic

materials. The third, C45 steel, is a carbon steel that has a higher coercive field than the first

two. All these materials are commercially available. Their properties are studied in the

following sections.

5.3.1 Supra50

The first ferromagnetic material is a commercial, soft ferromagnetic alloy from Aperam [150]

and is named Supra50 (note that this name has nothing to do with superconductivity!). It is a

Chapter 5. Characterization of the superconducting and magnetic materials 87

metallic alloy composed mainly of iron and nickel (51.5 wt% and 47 wt%, respectively). Its

intrinsic magnetic hysteresis 𝐵(𝐻) curve was measured independently at 300 K using the

permeameter described in Sect. 3.1. Fig. 5.24 shows the measured intrinsic 𝑀int(𝐻) =

𝐵(𝐻)/𝜇0 − 𝐻 curve (green) for Supra50. The right plot in this figure is a zoom of the left one

at low applied fields. The coercive field is found to be 520 A/m and the saturation

magnetization 𝜇0𝑀sat is 1.33 T ± 0.12 T. The maximum differential permeability (i.e. the slope

of the hysteresis curve at 𝐵 = 0) is 𝜇max′ = (d𝐵 d𝐻⁄ )𝐵=0 = 1.7 × 103𝜇0.

In order to determine the magnetic properties at 77 K (which is not possible with the

permeameter), we use the ACMS option of the PPMS (see Sect. 3.2.2). Fig. 5.25 shows a

photograph and a schematic representation of the section of Supra50 whose magnetic

moment is measured along its longest side. Fig. 5.24 shows the average volume magnetization

𝑀avg(𝐻) curve (blue) measured at 300 K. At high applied field, the sample magnetization

saturates at 𝜇0𝑀(1 T) = 1.39 T. At low field, the apparent maximum differential susceptibility

𝜒a is found to be 21.9. The coercive field was too small to be determined accurately.

Fig. 5.24 M(H) hysteresis curves of Supra50 FeNi alloy (left: full-scale, right:

stretched horizontal axis). Blue: curve measured at 300 K with the ACMS. Red:

curve measured at 77 K with the ACMS. Green: Intrinsic curve for a long rod

measured with the permeameter. On this curve, the coercive field is found to be

520 A/m and the saturation magnetization 𝜇0𝑀𝑠𝑎𝑡 is about 1.4 T. The black “error

bar” on the right panel shows the ± 9 % uncertainty on the permeameter

measurement. The maximum differential permeability is 1.7 × 103𝜇0.

The saturation magnetization measured with the ACMS (1.39 T) is in excellent agreement with

that measured using the permeameter (1.33 ± 0.12 T). The apparent differential permeability

(21.9 × 𝜇0) is much smaller than the intrinsic permeability (1.7 × 103𝜇0) because of the

demagnetizing field of the finite-size sample. This behaviour can be understood as follows. The

demagnetizing factor of a rectangular prism can be calculated accurately from [48]. The

interpolation technique introduced in [48] is used to determine the magnetometric

88 Chapter 5. Characterization of the superconducting and magnetic materials

demagnetizing factor 𝐷 along the measurement direction of the sample described in Fig. 5.25.

An intrinsic susceptibility 𝜒 = 𝜇𝑟 − 1 = 1699 is used. The computed value of 𝐷 is 0.0421. The

theoretical apparent susceptibility 𝜒a can then be determined from 𝜒a = 𝜒 (1 + 𝐷𝜒)⁄ . For our

specific sample, we find 𝜒a = 23.4, which agrees within 7% with the measured apparent

susceptibility 𝜒a = 21.9.

Fig. 5.25 Section of Supra50 whose magnetic moment was measured with the

ACMS option of the PPMS.

The magnetization was then measured at 77 K (red curve in Fig. 5.24). The saturation

magnetization (1.52 T) is found to be increased by 9.4 % compared to its value at 300 K. Fig.

5.26 shows the magnetization of the sample as a function of the temperature and under a

constant applied field of 1 T. A very weak temperature-dependence is found, with a saturation

magnetization going from 1.39 T at 300 K to 1.53 T at 20 K.

Fig. 5.26 Saturation magnetization of the Supra50 ferromagnetic material as a

function of temperature under an applied field of 1 T.

Chapter 5. Characterization of the superconducting and magnetic materials 89

Two remarks must be made here. First, the permeameter measurements were obtained at the

beginning of this work and the ACMS results were obtained much later. Since the low-

temperature results were not available at the time of modelling, the room-temperature

characteristics were thus used in all models, i.e. a saturation magnetization of 1.4 T and a

relative permeability of 1.7 × 103. The coercive field of the ferromagnet was always

neglected. Second, all the measurements were made on the “raw” as-purchased material.

Better magnetic properties are ensured by the manufacturer after thermal treatment on the

finished parts. A relative permeability of 144000 and a coercive field of 3.42 A/m are expected

after a 4-hour annealing at 1150°C under hydrogen. Since the properties were already

perfectly acceptable on the as-purchased material, further annealing was not carried out.

Supra50 is the most used ferromagnetic material in this work. Discs of different thicknesses are

machined from this material and combined with the superconductors; they are presented in

Chap. 6 (Sect. 6.1 and 6.2) and Chap. 7 (Sect. 7.3). Additional ferromagnet shapes are

introduced in Chap. 7 (Sect. 7.1).

5.3.2 Permimphy

The second ferromagnetic material, Permimphy, is another soft ferromagnetic material from

Aperam [Aperam]. It is a metallic alloy composed mainly of nickel and iron (80 wt% and 14

wt%, respectively). As for Supra50, its magnetic properties were measured independently

using the permeameter. Fig. 5.27 shows the measured intrinsic 𝑀int(𝐻) = 𝐵(𝐻)/𝜇0 − 𝐻

curve for Permimphy. On this curve, the coercive field is found to be 250 A/m and the

saturation magnetization 𝜇0𝑀sat to be around 0.8 T. The maximum differential permeability is

𝜇max′ = (d𝐵 d𝐻⁄ )𝐵=0 = 2.0 × 103𝜇0. These values will be used in modelling, except for the

coercive field that will be neglected.

Fig. 5.27 Intrinsic 𝑀int(𝐻) curve of Permimphy as measured at room temperature.

90 Chapter 5. Characterization of the superconducting and magnetic materials

As for Supra50, better magnetic properties (relative permeability of 245000 and coercive field

of 0.88 A/m) are guaranteed after a 4-hour annealing at 1170°C under hydrogen. Such

annealing was not performed.

The ferromagnetic discs machined from this material and combined with the superconductors

are presented in the following chapters (Sect. 6.3 and 7.2.3).

5.3.3 C45 steel

The third material is a ferromagnetic, high carbon steel (C45) which contains 0.45 wt% of C.

The name C45 is used in European normalization; it is equivalent to the 1045 AISI/SAE steel

grade [151]. Fig. 5.28 shows the measured 𝑀int(𝐻) = 𝐵/𝜇0 − 𝐻 curve. On this curve, the

coercive field is found to be 910 A/m and the saturation magnetization 𝜇0𝑀sat to be around

2 T. The maximum differential permeability is 𝜇max′ = (d𝐵 d𝐻⁄ )𝐵=0 = 1.0 × 103𝜇0.

Since this steel is not sold for its magnetic properties, these are not guaranteed, by the

manufacturer. Nevertheless, C45 has been studied in references [152, 153], where results are

in agreement with ours. While C45 steel has a larger coercive field than the two FeNi alloys

presented above, it is cheaper, easier to machine and easily available.

The ferromagnetic discs machined from this material and combined with the superconductors

are presented in the following chapter (Sect. 6.3).

Fig. 5.28 Intrinsic 𝑀int(𝐻) curve of C45 steel as measured at room temperature.

Chapter 5. Characterization of the superconducting and magnetic materials 91

5.3.4 Ferrite

The last presented material is a ferrimagnetic, ceramic material from Laird [154]. We bought

ferrimagnetic discs of 16.51 mm diameter and 1.27 mm height (ref: MM0650-100). Long rods

suitable for permeameter measurements, however, are not commercially available. The

material was therefore characterized with the ACMS option of the PPMS. The DC magnetic

moment and AC susceptibility of a rectangular prism (1.27 × 1.25 × 5.79 mm) were measured.

Fig. 5.29 shows the 𝑀avg(𝐻) magnetization curve vs. applied field at 300 K (blue) and 77 K

(red). The saturation magnetization nearly doubles from 300 K to 77 K, going from 𝜇0𝑀avg =

0.34 T to 𝜇0𝑀avg = 0.67 T. The apparent susceptibility is 15.2 and 14.9 at 300 K and 77 K

respectively. This set of data can be compared to those given by the manufacturer. We

measure an initial relative permeability of 650, a flux density of 0.28 T under 800 A/m (=

1 mT/𝜇0) and a coercive field strength of 32 A/m. Additional data are the Curie temperature

(above 140°C) and the electrical resistivity (103 Ωm). The measured apparent susceptibility is

limited by the demagnetizing field as for Supra50 above. For the measured sample, with a

permeability of 650, the interpolation method suggested in [48] gives a demagnetizing factor

𝐷 = 0.0635, and therefore 𝜒a= 15.37, which is extremely close to the measured apparent

susceptibilities 𝜒a and approaches the theoretical “infinite” susceptibility limit 1 𝐷⁄ = 15.74.

One interesting point is that the apparent susceptibility stays almost constant as temperature

decreases, meaning that the (unknown) intrinsic susceptibility at liquid nitrogen temperature

remains high enough such that 𝜒a ≈ 1 𝐷⁄ . Fig. 5.30 shows the temperature-dependence of the

magnetization of the sample under an applied field 𝜇0𝐻 = 1 T. Interestingly, a quasi-linear

behaviour is observed. The temperature-dependence of the ferrite saturation magnetization is

found to be much larger than for Supra50.

Fig. 5.29 𝑀avg(𝐻) hysteresis curves of a rectangular prism of the ferrite material

(MM0650-100) measured either at 300 K (blue) and at 77 K (red).

92 Chapter 5. Characterization of the superconducting and magnetic materials

Fig. 5.30 Temperature-dependence of the saturation magnetization of the ferrite

material (MM0650-100) under an applied field 𝜇0𝐻 = 1 T.

As the importance of ferrite materials resides mainly in their behaviour under alternating

fields, it is of interest to investigate the temperature-dependence of their AC magnetic

susceptibility. Fig. 5.31 shows (left) the in-phase 𝜒′ and (right) the out-of-phase 𝜒′′

components of the magnetic susceptibility under a 1mT field at 103 Hz. The in-phase

component 𝜒′ is 15.01 at 300 K and is weakly temperature-dependent. The out-of-phase 𝜒′′

component shows a steep variation as temperature decreases, with a maximum of 0.98 at

89 K.

Fig. 5.31 In-phase (left) and out-of-phase (right) components of the AC magnetic

susceptibility of the MM0650-100 ferrite material. The amplitude of the applied

field is 𝜇0𝐻 = 1 mT and the frequency is 103 Hz. The applied DC field is zero.

Chapter 5. Characterization of the superconducting and magnetic materials 93

As above, the in-phase magnetic susceptibility 𝜒′ is limited by demagnetization effects, with a

measured (apparent) value (15.01 at 300 K) extremely close to theoretically calculated limit

from the sample geometrical dimensions, i.e. 1 𝐷⁄ = 15.74. The behaviour of the out-of-phase

component 𝜒′′, on the other hand, indicates an increase of magnetic losses in the material as

temperature decreases down to 100 K. It should be emphasized that the temperature at which

the 𝜒′′ peak occurs is higher than the somewhat observed artefacts of the PPMS [97] at low

temperature.

The main advantage of the ferrite material is that eddy current losses can be avoided in

variable magnetic environments such as rotating machines or pulsed field magnetization. The

saturation magnetization of 0.67 T at 77 K is close to the 0.8 T value found for Permimphy. This

places the ferrite in a good position to provide a similar magnetic behaviour without suffering

from AC losses.

5.4 Summary

In this chapter, we have studied the magnetic properties of two large grain, bulk melt-textured

superconductors and four magnetic materials that will be combined in the next chapters to

form superconductor / ferromagnet hybrid structures.

Different measurement and modelling methods introduced in chapters 3 and 4 were used and

compared to determine and understand the properties of large, bulk superconductors. Surface

measurements were carried out with Hall probes placed against the centre of one or both

circular faces and the following parameters were recorded: (i) the time relaxation of the

magnetic flux density trapped in a fully magnetized superconductor and (ii) the Δ𝐵 = 𝐵 −

𝜇0𝐻app hysteresis curves induced by a cycle of applied field 𝐻app. The distribution of the

trapped magnetic flux density was also mapped by a Hall probe moving above the surface of

the fully magnetized sample. Volume measurements were carried out, either using the

bespoke magnetometer for pristine bulk samples or the ACMS option of the PPMS for small

specimens extracted afterwards. In addition, the Δ𝐵 hysteresis loops given by coils wound

around the superconductor under a “quasi” DC (i.e. low sweep rate) magnetic field were

studied. Four coils were considered: one coil wound around the entire height and one around

each of the top, middle and bottom sections of the superconductor. Both surface and volume

measurement results were compared with modelling results to validate the models. Volume

average Δ𝐵 hysteresis curves were computed with the Brandt algorithm and the A-formulation

in the finite element framework. The surface magnetic flux density was computed with the

two formulations (A and H) of the finite element framework.

The magnetic properties of the bulk superconductor were obtained from these measurements.

The n-exponent values determined from both Hall probe and magnetometer relaxation

measurements were in excellent agreement with each other. The critical current density was

estimated under a field-independent approximation from the magnetic moment given by the

94 Chapter 5. Characterization of the superconducting and magnetic materials

magnetometer. A more precise, field-dependent critical current was calculated by assuming a

Bean-Kim 𝐽c(𝐵) law and using the Brandt algorithm to compute Δ𝐵 hysteresis curves and

“true” magnetization from fluxmetric measurements. The agreement between modelled and

experimental results showed that sensing coils wrapped around the superconductor enable

“quasi” DC (i.e. low sweep rate) magnetic hysteresis loops to be measured on large, bulk

samples. From the Brandt method modelling results, we pointed out the quantitative

differences between the as-measured hysteresis loop and the “true” DC magnetization loop

predicted numerically. The characterization method involving sensing coils is therefore helpful

for characterizing, in a non-destructive way, the volume magnetic properties, as the field-

dependent critical current, of whole large superconducting samples whose dimensions exceed

the maximum size of classical DC magnetometers.

The Δ𝐵 hysteresis cycles measured above the surfaces with the Hall probes and in volume with

the four coils were compared. These results emphasized that superconducting properties (here

𝐽c) are not uniform over the single domain; the highest 𝐽c is found near the face containing the

seed. The coils were proved to be less sensitive than the Hall probes to these property

variations. The measured hysteresis curve using the coil wound around the entire height of the

superconductor was compared to modelling results from the Brandt and FEM A-formulation

methods.

Finally the trapped magnetic flux density mapped by Hall probe was compared to the results

obtained with the A and H-formulations in the finite element framework. The good agreement

between the experimental and numerical results gives evidence that our models are able to

reproduce the magnetic behaviour of a superconducting disc, the intrinsic properties of it

being obtained by preliminary independent experiments. This gives confidence that these

models can be used to study the flux distribution of the trapped field produced by large, bulk

high temperature superconductors.

The superconducting parameters of the bulk YBCO superconductor on which the above

characterization was carried out are as follows: 𝑛-exponent = 45 and a critical current density

𝐽c(𝐵) = 𝐽c1 (1 + 𝐵/B1)−1 where 𝐽c1 = 1.38 × 108 A/m2 and 𝐵1 = 0.987 T. Under a “constant

Jc” approximation, the average value 𝐽c = 1.03 × 108 A/m2 is found. These parameters will be

used for further numerical modelling in the following chapters.

The second superconducting sample is a solid, cylindrical bulk GdBaCuO superconductor (ESJc)

that will be used for the investigation of ferromagnetic-superconducting hybrid structures in

the so-called “crossed-field” configuration. Under a “constant 𝐽c” approximation, this material

is characterized by 𝐽c = 2.44 × 108 A/m2.

Four commercially available magnetic materials were also studied, including three metallic

ferromagnets and one ferrite ceramic. The first two metals, Supra50 and Permimphy, are FeNi

alloys which can be classified as soft ferromagnetic materials. The third one, C45 steel, is a

carbon steel that has a higher coercive field than the first two.

Chapter 5. Characterization of the superconducting and magnetic materials 95

The intrinsic 𝑀int(𝐻) curves of the three metallic materials were measured at 300 K with the

permeameter. Additional measurements of the average magnetization 𝑀avg were carried out

on a small prism of Supra50 in order to investigate the magnetic properties at cryogenic

temperatures. The comparison of these 𝑀avg(𝐻) curves with the intrinsic 𝑀int(𝐻) curve

showed the influence of the demagnetizing factor on the apparent susceptibility calculated

from magnetic moment measurements. This apparent susceptibility 𝜒a was related to the

intrinsic value 𝜒 through the demagnetizing factor 𝐷 found in literature. Nevertheless, the

ACMS measurement of the saturation magnetization was in excellent agreement with the

permeameter measurement, provided that a high enough magnetic field is applied. The

saturation magnetization was found to increase slightly from 300 K to 20 K. This allows us to

assume that the value at 300 K can be used in first approximation for applications at 77 K.

The ferrimagnetic ceramic material was not available in a shape suitable for the permeameter

but was measured using the ACMS option of the PPMS. As for the metallic materials, the

apparent susceptibility at low applied field was found to be limited by the demagnetizing

effects. Unlike Supra50, however, a significant temperature dependence of the saturation

magnetization was found.

The two large grain, bulk superconductors presented in this chapter will be combined with

ferromagnetic samples cut out of the four magnetic materials to form superconductor /

ferromagnet (SC/FM) hybrid structures. The next chapter will focus on the magnetic behaviour

of a superconducting pellet with a single ferromagnetic disc. The influence of the

superconductor and ferromagnet properties will be studied and described in details using

measurements techniques and modelling frameworks presented in the previous chapters. In

the last chapter we consider the influence of other shapes of the ferromagnet materials.

96 Chapter 5. Characterization of the superconducting and magnetic materials

97

Chapter 6

Superconductor / ferromagnetic disc

hybrid structure

In this chapter, we study the influence of a ferromagnetic disc on the magnetic behaviour of a

superconductor. In the first section, we investigate a superconductor / ferromagnet (SC/FM)

hybrid structure for a reference geometry, where the ferromagnet is a 1.90 mm thick disc of

the same diameter (16.5 mm) as the 6.32 mm thick YBCO superconductor. This structure is

characterized both experimentally and numerically. In the following sections, we successively

examine the influence of the thickness of the ferromagnet (section 6.2) and of the type of

ferromagnetic material (section 6.3). In the last section (6.4), the influence of the critical

current of the superconductor is investigated.

6.1 Reference configuration (D2)

This first section focuses on the study of the magnetic behaviour of a superconductor /

ferromagnet (SC/FM) hybrid structure in a fixed geometry that will be used as a reference

configuration, shown in Fig. 6.1. It involves a 1.90 mm thick ferromagnetic disc of the same

diameter (16.5 mm) as the 6.32 mm-thick YBCO superconductor. The reference ferromagnet is

made of Supra50 and the reference superconductor is the “ESC2” large grain melt-textured

sample which was characterized in detail in Chap. 5. This disc is attached to the circular face of

ESC2 opposite to the seed. For easier reference, this configuration is called D2, (an approx.

2 mm thick disc). The sensors used for the Δ𝐵 hysteresis loop measurements are schematically

presented on the cross section (right).

In order to facilitate the explanations and understanding of the influence of the ferromagnet

on the magnetic properties, this section starts with modelling results of the magnetic flux

density in and around the SC/FM structure shown in Fig. 6.1. Then, the measurements — Δ𝐵

hysteresis curves and profiles of the magnetic flux density — are presented. The corresponding

modelling results are compared to measurements to assess the suitability of the model to

describe the behaviour of the hybrid structure.

98 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

Fig. 6.1 Schematic representations of the D2 superconductor/ferromagnet (SC/FM)

hybrid structure made of the superconducting sample ESC2 and a 1.90 mm thick

ferromagnet made of Supra50. The sensors used for the Δ𝐵 hysteresis loop

measurements are schematically presented on the cross section (right).

6.1.1 Modelling of the flux density

In this section, we start by modelling the magnetic flux density in and around the SC/FM

structure D2.

The modelling results are obtained using the constitutive laws of ESC2 and Supra50

determined in Chap. 5. Finite element modelling is carried out using the A-formulation of

Maxwell’s magnetodynamics equations presented in Chap. 4 using the GetDP solver. This

method allows the repartition of the contour lines of the magnitude 𝐴 of the vector potential

𝑨 to be determined easily. For axisymmetric geometries, the contour lines of the 𝑟𝐴 product

— where 𝑟 denotes the radial distance — give a reasonable approximation of the flux lines of

𝐵, as described in [104].

Fig. 6.2 shows the modelled repartition of the contour lines 𝑟𝐴 when an external magnetic

field of 𝜇0𝐻app = 0.3 T is applied to (a) the superconductor only and (b) the hybrid SC/FM

structure. This value of applied field corresponds to approximately half the full-penetration

field of the superconductor. Results for the superconductor alone (Fig. 6.2(a)) display the

expected shape of contour lines in this configuration [104]. When the ferromagnetic disk is

added to the superconductor (Fig. 6.2(b)), the contour lines of 𝑟𝐴 are concentrated inside the

ferromagnet and undergo a strong change of direction at the edges of the ferromagnet: the

lines are nearly perpendicular to the interface on the air and superconductor sides. These

contours indicate that the magnetic flux distribution in the bottom part of the superconductor

remains relatively unaffected, while the penetration depth of magnetic flux is smaller in the

vicinity of the ferromagnet. However, the density of 𝑟𝐴 lines in the air above the ferromagnet

(Fig. 6.2(b)) is higher than that above the superconductor alone (Fig. 6.2(a)).

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 99

Fig. 6.2 Repartition of the contour lines of the vector potential 𝐴 multiplied by the

radial distance 𝑟 for an external applied field of 𝜇0𝐻app = 0.3 T inside (a) the

superconductor only and (b) the hybrid superconductor / ferromagnet (SC/FM)

structure. These results are obtained by finite element modelling using the

A-formulation of Maxwell’s magnetodynamics equations, as described in Chap. 4.

Fig. 6.3 (a) Contour lines of the vector potential 𝑟𝐴 in the fully magnetized

remanent state for the superconductor ESC2. These results are obtained by finite

element modelling using the A-formulation, as described in Chap. 4. (b) Same

contour lines for the hybrid ferromagnet/superconductor (SC/FM) structure D2.

𝑟 (mm)

𝑧 (mm)

8.25 0

0

6.32

𝑟 (mm)

𝑧 (mm)

8.25 0

0

6.32

8.22

(a) (b)

𝑟 (mm)

𝑧 (mm)

8.25 0

0

6.32

𝑟 (mm)

𝑧 (mm)

8.25 0

0

6.32

8.22

(a) (b)

100 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

Fig. 6.3 shows the path followed by the contour lines of 𝑟𝐴 in the fully magnetized remanent

state (trapped field) for (a) the superconductor only configuration and (b) the hybrid SC/FM

structure. Compared to Fig. 6.3(a), the results shown in Fig. 6.3(b) indicate that the contour

lines are strongly concentrated inside the ferromagnet. The contours are almost perpendicular

to the ferromagnet interface in the low permeability domain (air or superconductor). The

concentration of 𝑟𝐴 lines above the top face of the SC/FM assembly is much smaller than for

the superconductor without ferromagnet.

These results emphasize that the high permeability ferromagnet acts as a magnetic short-

circuit for magnetic flux density since it creates a low reluctance path. When the assembly is

subjected to an external magnetic field (Fig. 6.2), the penetration of the magnetic flux inside

the superconductor is delayed in the vicinity of the ferromagnet. The zone above the

ferromagnet in the trapped field configuration (Fig. 6.3) is shielded from the flux trapped

inside the superconductor. Moreover, the flux lines that were closed through the

superconductor are now closing through the ferromagnet, which leads to a decrease of the

curvature of the return flux lines inside the superconductor.

The flux lines are perpendicular to the median plane of the superconductor in the absence of

the ferromagnet whereas they are perpendicular to the superconductor / ferromagnet

interface in presence of the ferromagnet. Therefore, adding the ferromagnet can be

assimilated to doubling the height of the superconductor as expected in the limit case of the

image theorem [56, 155, 156]. The image theorem predicts that adding a semi-infinite volume

(extending to infinite radius and depth) of perfectly ferromagnetic material (𝜇r → ∞) is

equivalent to doubling the height of the superconducting cylinder. A superconducting cylinder

with a higher aspect ratio experiences a lower demagnetizing field and a reduction of the edge

effect of a finite length sample. In practice, the real effect is not as significant as the image

theorem would predict, since the ferromagnet is not semi-infinite.

It is of interest to check that the contour lines of 𝑟𝐴 give information about the direction of the

magnetic flux density, as suggested in [104]. The difficulty arises when one needs to visualize

3D flux lines in axial symmetry using a 2D plot. In axisymmetric geometry with an azimuthal

vector potential 𝑨 = 𝐴𝜃𝒆𝜃, the vector potential 𝐴𝜃 multiplied by the radial position 𝑟 is a

quantity that is directly proportional to the magnetic flux 𝜙 crossing a circle of radius 𝑟 at a

fixed height 𝑧. In axisymmetric geometry, the axial flux 𝜙 embraced by such a circle is given by

𝜙(𝑟, 𝑧) = ∬ 𝑩 ∙ 𝒏

𝑆

d𝑆 = ∬(∇ × 𝑨) ∙ 𝒏

𝑆

d𝑆 = ∮ 𝑨 ∙ 𝐝𝒍

𝒞

= 2𝜋 𝑟𝐴θ(𝑟, 𝑧)

We could consider a situation where the circle is moved along 𝑧 in such a way that 𝜙 remains

constant. Since the flux through the circle is fixed, its border must always end on lines of 𝐵.

Therefore contour lines of 𝑟𝐴 are locally parallel to the field lines of 𝐵. However, the density of

contour lines does not represent the real flux density. Fig. 6.4 shows an enlargement of the

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 101

ferromagnet in Fig. 6.3(b) with the contour lines of 𝑟𝐴 and an arrow plot of 𝑩. Each arrow

represent the direction of the local magnetic flux density, and its size is proportional to the

magnitude of 𝑩. The arrows of 𝑩 are tangeant to the contour lines which shows the suitability

of plotting the contour lines of 𝑟𝐴, which are, at first glance, easier to understand than the

arrow plots that might give a “cluttered” impression.

Fig. 6.4 Comparison of the contour lines of 𝑟𝐴 and the vector field of the magnetic

induction 𝑩. Result obtained with the A-formulation in the finite element

framework.

Results similar to those plotted in Fig. 6.3 were obtained with both other finite elements

models: the model based on solving Campbell’s equation and the one based on the H-

formulation (cf. Chap 4). Results with both modelling methods were obtained at the University

of Cambridge. These results are compared here to assess the suitability of these models to

represent the magnetic behaviour of the SC/FM structures. We first investigate the Campbell

method. Fig. 6.5 compares the contours lines of 𝐴 obtained for the SC/FM structure D2 using

the FEM A-formulation (left) and using the Campbell method (right). The results are obtained

in the fully magnetized remanent state. Note that for technical reasons, the quantity plotted in

Fig. 6.5 is the contour lines of 𝐴 in both cases. The influence of the ferromagnet on the shape

and direction of the contour lines modelled using the Campbell method is found to be in

excellent agreement with that obtained using the A-formulation.

We now turn to the magnetic flux density in and around the fully magnetized SC/FM structure

D2 obtained with the H-formulation of Maxwell’s equations as presented in Sect. 4.2.2. The

corresponding results are presented in Fig. 6.6. The grey-scale background shows the local

intensity of the magnetic flux density 𝐵. The field lines are found to behave as expected from

Fig. 6.3 (b). They are almost perpendicular to the ferromagnet interface in the low

permeability domain (air or superconductor) and concentrated inside the ferromagnet. The

grey map of the flux density clearly shows that the magnetic flux density is higher in the

102 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

ferromagnet and in the centre of the superconductor, near the ferromagnet. The additional

information is that in this case, the ferromagnet saturation is not reached (the maximum value

of the grey scale is 1.082 T while the Supra50 saturation is at 1.4 T). This will be discussed later

in this chapter.

Fig. 6.5 Contours lines of 𝐴 obtained for the D2 configuration in the fully

magnetized remanent state, either with the finite element A-formulation (left) or

using the Campbell method (right).

Fig. 6.6 Plot of the magnitude of 𝐵 (grey scale) obtained with the H-formulation in

the finite element framework. The top and bottom rectangles are for meshing

purposes and correspond to free space. The lines are plotted in the COMSOL

environment and are a guide to the eye to show the direction of the local flux

density.

SC

FM

Free space

Free space

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 103

The above results show the excellent agreement between the three models and give evidence

that these modelling methods can be used interchangeably to investigate the magnetic

properties of various SC/FM structures. These three models are used in the following part of

the text. Obviously, for a given configuration the results given by only one model will be

presented, unless differences can be found. The reason why one particular model is used is

mainly due to the different collaboration opportunities that have taken place at the successive

stages of the research work. In the configuration presented in this section, all modelling results

lead to the same conclusion: due to its high permeability when it is not saturated, the

ferromagnet acts as a magnetic short-circuit for the magnetic flux density.

6.1.2 𝛥𝐵 hysteresis curves

6.1.2.1 Measurements

This section is focused on the Δ𝐵 hysteresis curves measured on the SC/FM structure D2

shown in Fig. 6.1. The experimental method is the same as on ESC2 alone (Sect. 5.1.4); the

same arrangement of pick-up coils and Hall probes was used for the SC/FM hybrid, except for

the top Hall probe that was placed at the top of the whole structure. In this case a coil was not

wound around the ferromagnet. All measurements involved cooling down the sample in zero

field and then applying a slowly time-varying magnetic field parallel to the c-axis of the sample.

The field was swept initially up to 3 T and then cycled between 3 T and -3 T using a sweep rate

of 15 mT/s.

Fig. 6.7 shows the Δ𝐵 magnetic hysteresis loop measurements of the volume average flux

density carried out on the hybrid SC/FM configuration D2. Fig. 6.7(a) compares the hysteresis

loops recorded by the four pick-up coils. The hysteresis loop measured with the coil wound

around the superconductor only (i.e. when no ferromagnet is present) is also shown for

comparison (light blue dots). The hysteresis loops of the SC/FM hybrid structure show a

combination of diamagnetic (negative slope at the origin) and ferromagnetic behaviour (the

average 𝐵 exceeds 𝜇0𝐻app at medium and high positive applied fields). The behaviour at high

applied field (typically for 𝜇0𝐻app > 1 T) highlights some differences between the four coils:

for a given applied field, the magnetic induction measured by the top coil (green triangles) in

the vicinity of the ferromagnet is higher than that measured by the bottom coil (red triangles),

located more distant from the ferromagnet. The coils wound around the middle (orange

squares) and entire height (navy blue circles) of the sample show an intermediate behaviour

between the top and bottom coils. Compared to the superconductor alone (Fig. 5.13), the

remanent induction (measured at 𝜇0𝐻app = 0) is increased by 8.8 %, 15 % and 29 % for the

bottom, middle and top sections, respectively. The magnetic flux probed by the coil wound

around the entire height of the superconductor shows an increase of 16 % of the remanent

magnetic induction.

104 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

Fig. 6.7 (a) Magnetic hysteresis curves measured with the four coils on the SC/FM

configuration D2. The navy blue circles represent the data for the coil wound

around the entire sample height. The green triangles, orange squares and red

triangles are for the top, middle and bottom coils, respectively (i.e. the colours of

the coils refer to those shown schematically in the inset). The curve for the coil

wound around the entire height of the superconductor only is shown for

comparison (light blue dots). (b) Magnetic hysteresis curves measured by the Hall

probes centred against the SC (green) and FM (red) circular faces of the assembly.

The hysteresis curves shown in Fig. 6.7(a) provide experimental evidence of a strongly non-

uniform distribution of the magnetic induction through the thickness of the superconductor. In

addition, when the external field is modified, the evolution of Δ𝐵 depends on the proximity of

each section within the sample to the ferromagnet. The variations of the applied field have

clearly more effect on the magnetic induction in the sections near the ferromagnet. At zero

applied field, the fact that the average remanent induction 𝐵z probed by the pick-up coil

around the superconductor is larger in the presence of the ferromagnet, is entirely consistent

with the modelling results presented in the previous section: the measured 𝐵z increases since

the flux lines in the superconductor are perpendicular to the SC/FM interface and are then

driven toward the outside of the superconductor by the ferromagnet.

Fig. 6.7(b) shows surface measurements by the Hall probes on the hybrid SC/FM structure for

the configuration D2. The top probe is above the ferromagnet and the bottom probe is against

the superconductor (see inset). As is the case for the superconductor only (see Fig. 5.13), the

superconductor is oriented with the seed on the bottom of the configuration. The shape of the

magnetic hysteresis curve is roughly similar to that of the superconductor only for the Hall

probe located at the bottom (red curve), but appears somewhat tilted anticlockwise with

remanent magnetic induction increased slightly by 4%. The full penetration field does not

differ significantly from that measured for the superconductor alone. The shape of the

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 105

hysteresis curve measured by the Hall probe positioned on the top of the ferromagnetic disc

(green curve), however, is modified strongly, and appears more like the hysteresis curve of a

ferromagnetic material than that of a diamagnetic material. The initial slope of the first

magnetization curve is still negative but above ≈ 0.5 T, the quantity ⟨𝐵⟩ − 𝜇0𝐻app becomes

positive. Significantly, the remanent magnetic induction is strongly reduced compared to the

configuration without the ferromagnet.

The above measurements are consistent with the modelling results discussed in the previous

section. There is little change of the field trapped by the ferromagnet on the bottom face, as

sensed by the Hall probe. The results of Fig. 6.7 support the observation that flux lines are

concentrated into the low reluctance ferromagnetic material. Indeed, the flux lines are driven

toward the Hall probe located on the ferromagnet side during the initial magnetization of the

sample (i.e. during the field cycle 0 → 𝐻max), whereas this probe is shielded from the trapped

flux by the ferromagnet in the remanent state. Such a magnetic shielding effect leads to a

marked reduction of the magnetic field in the air above the ferromagnet. This observation,

although at first sight surprising, yields the following practical conclusion: if the

superconducting permanent magnet is used to produce magnetic field in the air, a

ferromagnet should not be added on the side where the magnetic field is produced. However,

if the relevant quantity is the average flux density trapped in the volume of the

superconductor, a ferromagnet is helpful in increasing this trapped flux.

6.1.2.2 Modelling

The Δ𝐵 hysteresis curve of the flux density averaged over the whole volume of the

superconductor, i.e. without including the ferromagnet in the average, was computed within

the finite element framework. Fig. 6.8 shows the curves computed with the Campbell method

and A-formulation of Maxwell’s magnetodynamics equations in orange and red, respectively.

The curve measured with the coil wound around the full height of the sample is shown in blue

for comparison. The general shape of the computed curves is similar to that of the measured

curve. Around the remanent state, the agreement to the measured curve is better on the

Campbell method curve than on the A-formulation result. At higher field, the agreement is

better with the A-formulation method, especially above 1 T in the first increasing branch of the

cycle 0 → 𝐻max. On reversal of the applied field, the A-formulation method predicts a lower

Δ𝐵 than actually measured. The general agreement of the measured and modelled curves

gives confidence on their suitability to represent the magnetic influence of the ferromagnet

near the superconductor in the remanent state. However, the A-formulation of Maxwell’s

magnetodynamics should be preferred to predict accurate behaviour under high applied field.

The difference between both methods seems to be related to the 𝐵(𝐻) constitutive law of the

ferromagnetic material. In the A-formulation, the ferromagnet permeability is defined using an

Akima interpolation of (𝐵, 1/𝜇(𝐵)) pairs obtained from the measured 𝐵(𝐻) curve while a

continuous function is used in the Campbell method. As a further work, it might be of interest

to refine the model of the saturation of the ferromagnet 𝐵(𝐻) curve.

106 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

Fig. 6.8 Δ𝐵 hysteresis curves computed from the magnetic flux density 𝐵z averaged

over the volume of ESC2 in the SC/FM configuration D2. These curves were

computed with the 𝑨-formulation of the Maxwell equations (red) and from

Campbell’s equations (orange) using the finite element method. The curve

measured with the coil wound around the entire height of the superconductor is

shown (blue) for comparison.

6.1.2.3 Addition of the hysteresis curves

In addition to the previous observations, it is of interest to compare the average flux density of

the D2 hybrid structure – as measured by the coil wound around the entire height – to that

measured (i) for the superconductor only and (ii) for this hybrid structure D2 when the

superconductor is in an unmagnetized state (i.e. above its critical transition temperature). Fig.

6.9(a) shows the hysteresis curve measured at 77 K using the coil wound around the entire

sample thickness for the superconductor only. Fig. 6.9(b) shows the hysteresis curve of the

ferromagnet measured using the same coil arrangement for the bulk sample (there are no

turns around the ferromagnet) measured at 100 K. Fig. 6.9(c) compares the data resulting from

the simple numerical addition of the two previous measured curves (red data points) and the

true experimental data for the hybrid structure at 77 K (blue data points). Similar results were

obtained with the three other coils (bottom, middle and top coils) and with the Hall probe on

the face opposite to the ferromagnet. Fig. 6.10 shows the modelled hysteresis curves

corresponding to those of Fig. 6.9(c), obtained by solving Campbell’s equation. Additionally, it

is worth noting that similar results were obtained – for the bottom Hall probe and the coil over

the entire height – on the measured SC/FM configurations presented later in this chapter and

in the next chapter.

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 107

Fig. 6.9 Magnetic hysteresis loops measured with the coil wound around the entire

height of the superconductor for (a) the superconductor only at 77 K, (b) the small

ferromagnetic disc only at 100 K (𝑇 > 𝑇c), (c) the hybrid structure involving this

small disc at 77 K (blue). The red curve represents the simple numerical addition of

loops (a) and (b). The insets show a sectional view of each configuration.

Fig. 6.10 Modelled magnetic hysteresis loops obtained by solving the Campbell’s

equation for the hybrid structure D2 (blue). The red curve was computed in the

same way as the red curve in figure Fig. 6.9(c), i.e. as the simple addition of the

modelled curves for the superconductor only and the ferromagnet only.

108 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

Clearly, the hysteresis data in Fig. 6.9 show that the magnetic flux density measured for the

hybrid structure approximates very well the simple superposition (i.e. addition) of the

individual hysteresis curves of the ferromagnet and the superconductor. The curves are in a

particularly good agreement for applied magnetic fields exceeding the apparent saturation

field of the ferromagnetic component (𝜇0𝐻app ≈ 1.4 T). It is worth noting that this apparent

saturation field may differ substantially from the intrinsic saturation field of the ferromagnet

for two reasons: (i) the measuring coil probing the magnetic flux (Fig. 6.9(b)) is wound below

the ferromagnet and (ii) the low aspect ratio of the ferromagnetic disk gives rise to a large

demagnetizing field. When the ferromagnet is saturated totally by the external field, the flux

densities due to the ferromagnet and the superconductor simply add together. Below this

saturation limit, however, the non-uniformity of the field may lead to saturation of only some

regions of the ferromagnet, and the attraction of the return flux lines to the non-saturated

regions of the ferromagnet explains the increase of the remanent induction, as described

previously. Remarkably, the results obtained by numerical modelling (Fig. 6.10) display the two

main qualitative features observed in the experiment, i.e. the additive behaviour at large fields

and the larger trapped field for the hybrid configuration compared to the simple numerical

addition. The quantitative agreement between Fig. 6.9(c) and Fig. 6.10 is also excellent and

underlines the suitability of the modelling method to describe the SC/FM assemblies as well as

the importance of using accurate data on the two materials (ferromagnet and superconductor)

through preliminary experimental characterization.

6.1.3 Surface profiles of the flux density

6.1.3.1 Measurements

In this section, we investigate the influence of the ferromagnet on the profile of the magnetic

flux density on both sides of the SC/FM assembly D2 through Hall probe mappings and

modelling.

Fig. 6.11 shows the Hall probe mappings 0.85 mm above each face for the D2 hybrid

configuration (a 1.90 mm thick ferromagnetic disc made of Supra50 stuck against the

superconductor). The left panel shows the whole measured mapping, while the right panel

focuses on the flux density profile along a diameter (red). The corresponding measurements

above the superconductor alone – already presented in Fig 5.18 – are shown for comparison

(blue). Please note that whole SC/FM structure is upside down: the ferromagnet is presented

on the bottom but it is still stuck against the face of the superconductor opposite to the seed.

As in Sect. 5.1.8., it is immediately apparent that the maximum flux density occurs at a slightly

off-centre position. The position corresponding to the measured maximum on the bottom face

in the absence of ferromagnet (r = -1 mm) was selected as the reference position. On the SC

side (i.e. opposite to the ferromagnet), the characteristic value 𝐵z∗ = 𝐵z(𝑟 = −1 mm)

increases from 336 mT to 358 mT when the ferromagnet is added (+6.5 %). On the

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 109

ferromagnet side, an important change in the shape of the flux density occurs: the flux density

distribution exhibits now a plate-like shape and its value on the symmetry axis is strongly

reduced. On the ferromagnet side, 𝐵z∗ is decreased from 250 mT to 42.5 mT (-83 %). There are

two reasons for this: (i) the measurement is made 1.90 mm further from the superconductor

since the ferromagnet in inserted in between, (ii) the ferromagnet influences this

measurement by shielding the flux produced by the trapped field magnet. The latter explains

that the distribution of 𝐵z changes from a conical to a plate-like shape, which is entirely

consistent with the observations made at the beginning of this chapter.

Fig. 6.11 Hall probe mappings of 𝐵z, the vertical component of the magnetic flux

density generated by the fully magnetized bulk high temperature superconductor

(ESC2) in the SC/FM hybrid configuration D2 made with a 1.90 mm thick disc. Top:

Measurement 0.85 mm above the face containing the seed and opposite to the

ferromagnet. Bottom: Measurement at 0.85 mm of the ferromagnet. Left: Full

measured mappings. Right: Radial plots along a diameter of the superconductor

(red). R = 8.25 mm is the radius of the superconductor and ferromagnet. The curves

measured without any ferromagnet are shown for comparison (blue).

6.1.3.2 Modelling

The H-formulation of Maxwell’s magnetodynamics equations in the finite element modelling

framework presented in Chap. 4 is used, together with the parameters introduced in Chap. 5,

Superconductor side (containing the seed)

Ferromagnet side

110 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

to model the magnetic flux distribution of the SC/FM configuration D2 investigated

experimentally. Fig. 6.12 compares the measurement shown in Fig. 6.11 and the modelling

results of the distribution of 𝐵z along a diameter, 0.85 mm above each surface. The flux

density 𝐵zc = 𝐵z(𝑟 = 0) on modelling reaches 304 mT and 327 mT (+7.6 %) on the

superconductor side for the SC alone and D2 configuration, respectively. On the other side, 𝐵zc

drops from 304 mT to 42.4 mT (-86 %). Relative variations are given for comparison purpose

since the values for the SC alone are not equal.

There is an excellent qualitative agreement between the measurements and modelling results.

On the superconductor side, i.e. opposite to the ferromagnetic section, the increase of the

trapped field is visible. Quantitatively, the relative variations of 𝐵z are reproduced: the model

predicts an increase of +7.6 % and the measured one was +6.5 %. On the ferromagnet side, the

variations are of -83.0 % and -86.1 % in the experimental and modelling results, respectively.

(a) Experiment (b) Modelling

Sup

erc

on

du

cto

r si

de

Ferr

om

agn

et s

ide

Fig. 6.12 (a) Magnetic flux density Bz above the top and bottom surfaces of the SC

alone (blue) and the SC/FM hybrid structures D2 (red). All measurements are

performed along a diameter 0.85 mm above the external surface of the whole

hybrid structure. R = 8.25 mm is the radius of the SC/FM assembly. (b) Modelling

results for the same SC/FM structures (same colours).

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 111

The nice agreement between the experimental and numerical results was also obtained with

other shapes of the ferromagnet; they will be presented in the next chapter. All these results

provide strong evidence that the model is able to reproduce the magnetic properties of a

superconducting pellet placed against a ferromagnetic disc, the properties of both being

obtained by preliminary independent experiments. This gives confidence that this model can

be used to study the flux redistribution and predict the influence of magnetic materials of

other properties and/or shapes on the trapped field produced by large, bulk high temperature

superconductors.

6.2 Influence of the disc thickness

In this section, we focus on the influence of the thickness of the ferromagnetic disc on the flux

density in and around the SC/FM hybrid structure. First, the distribution of 𝐵z is studied

through experiment and modelling. Second, the Δ𝐵 hysteresis curves of the average flux

density over the whole volume of the superconductor are investigated. The whole study is

relative to the YBCO superconducting sample “ESC2” and ferromagnetic discs made of Supra50

material. The corresponding parameters determined in Chap. 5 are used in the models.

6.2.1 Surface profiles of the magnetic flux density

Fig. 6.13 shows, in green, the measured distribution of 𝐵z on both sides of the SC/FM hybrid

structure D1, consisting of the superconducting sample ESC2 placed against a 0.99 mm

ferromagnetic disc made of Supra50. Both have the same diameter. The results for SC alone

and the D2 configuration are shown in blue and red for comparison. The flux distribution

measurements on D1 are consistent with the results obtained on D2. In comparison to the

results obtained for the structure D2 in Fig. 6.12, the thinner ferromagnetic disc in D1 leads to

a smaller attenuation in magnetic flux above the ferromagnet and to a smaller increase on the

face opposite to the ferromagnet. On the SC side, 𝐵z∗ increases from 336 mT to 355 mT

(+5.7 %) for D1 and to 358 mT (+6.5 %) for D2. On the FM side, the measured 𝐵z∗ is reduced

from 250 mT to 49.7 mT (-80.1 %) for D1 and to 42.5 mT (-83.0 %) for D2. The observed

behaviour is between the D2 configuration and the superconductor alone but the relative flux

increase due to the presence of the ferromagnetic disc does not seem to be proportional to its

thickness (or volume). Additional results are needed to draw conclusions; they will be provided

by modelling.

Fig. 6.14 shows the modelled flux density profiles obtained on each face of the SC/FM

structures using Supra50 ferromagnetic discs of several thicknesses. In addition to the D1 and

D2 configurations characterized experimentally above, three new configurations are

investigated: D035, D070, and D3, corresponding to disc thicknesses of 0.35 mm, 0.70 mm, and

2.90 mm, respectively. All discs have the same diameter as the superconductor. To make the

comparisons easier, the cross sections of these configurations are shown on the right-hand

side part of Fig. 6.14. The SC-alone configuration is also shown for reference.

112 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

Fig. 6.13 Magnetic flux density Bz above the top and bottom surfaces of the SC

alone (blue) and the SC/FM hybrid structures D1 (green) and D2 (red). All

measurements are performed along a diameter 0.85 mm above the external

surface of the whole hybrid structure. R = 8.25 mm is the radius of the SC/FM

assembly. Superconductor side (left) and ferromagnet side (right).

Fig. 6.14 Modelled influence of FM disc thickness on 𝐵z, modelling 0.85 mm (a)

above and (b) under the SC/FM structures sketched on the right. On the sketches,

the FM discs and their thicknesses are in red. The thickness of the superconductor

is 6.32 mm.

On the superconductor face, a thicker disc on the opposite side gives a higher flux density

compared to the reference sample. The profiles obtained for 0.70 mm and thicker discs are

similar to each other while the 0.35 mm thick disc is experimentally found to be much less

efficient. A similar redistribution for the different discs can be observed on the ferromagnet

side: the plate-like shape of the flux density is obtained for the 0.70 mm and thicker discs,

while the 0.35 mm thick disc exhibits a completely different behaviour. In the latter, the flux

(a) Superconductor side (b) Ferromagnet side

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 113

distribution is intermediate between that obtained for the superconductor alone and for the

thicker discs and displays a centred peculiar “camel-bump”. Actually, a closer look at the D070

curve shows the emergence of this bump as well (see inset of Fig. 6.14). For thicker discs, the

flux density at a constant distance (0.85 mm) from the ferromagnet is a decreasing function of

the ferromagnet thickness.

The relative increase of 𝐵zc = 𝐵z(𝑟 = 0) on the top face as a function of the thickness 𝑑 of the

ferromagnet is shown in Fig. 6.15 for a scan height of 0.85 mm. Similar curves are observed

with scan heights of 0.1 mm and 1.5 mm. The curve in Fig. 6.15 displays a “kink” occurring at

some disc thickness 𝑑∗ within the range of about 0.7 mm to 1.0 mm. The increase of 𝐵zc is

strongly thickness-dependent for 𝑑 < 𝑑∗ and then flattens for 𝑑 > 𝑑∗. The origin of this

behaviour is discussed below.

Fig. 6.15 Modelled relative increase of 𝐵zc = 𝐵z(𝑟 = 0) on the top face of the

superconductor, as a function of the thickness of the ferromagnetic disc attached

to the opposite face.

In order to obtain additional information about the particular D035 and D1 configurations (i.e.

on both sides of the kink), it is of interest to examine the distribution of magnetic flux lines

generated by the magnetized superconductor. These are shown in Fig. 6.16. The 0.99 mm disc

(D1) is found to drive a very large proportion of the flux lines towards the edges of the

superconductor, as observed for the D2 configurations modelled at the beginning of this

chapter. This disc is not fully saturated, as evidenced from the colour scale which does not

reach the saturation magnetization of 1.4 T. This behaviour is also observed on the D2 (Sect.

6.1) and D3 configurations (figure not shown). The 0.35 mm disc (D035), however, is nearly

fully saturated. The saturation magnetization is reached in a majority of the ferromagnet

volume; only small regions near the symmetry axis and near the outer face of the ferromagnet

are not saturated.

0%

2%

4%

6%

8%

10%

0 0.5 1 1.5 2 2.5 3

Rel

ativ

e in

crea

se o

f B

zc

Thickness of the ferromagnetic disc (mm)

114 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

Fig. 6.16 Magnetic flux density plots of the modulus of 𝐵 (colour scale) for the (a)

D035 and (b) D1 configurations. The contour lines show the direction of the

magnetic induction 𝑩. The ferromagnetic discs are made from Supra50 material,

characterized by a saturation magnetization 𝜇0𝑀sat = 1.4 T.

The remarkable feature of the above results is that a large volume of the ferromagnetic disc is

brought close to saturation (𝐵 ≈ 1.4 T), although the maximum flux density that would exist

against the superconductor without ferromagnet is ≈ 0.3 T, i.e. one fourth of the saturation

magnetization of the ferromagnet. The saturation is responsible for the much different

behaviour of the 0.35 mm disc, compared to thicker ones. The kink occurring in Fig. 6.15 is

thus related to the “threshold” thickness 𝑑∗ under which this saturation occurs. It is obviously

linked to (i) the superconductor generated flux – and therefore to its size and critical current –

and to (ii) the saturation magnetization of the ferromagnetic material. In order to extend the

generality of these results to other sizes and ranges of physical properties, it is tempting to

roughly estimate this “threshold” thickness 𝑑∗ from a simplified analysis based of conservation

of magnetic flux and to compare it to the interval of about 0.7 to 1.0 mm found in Fig. 6.15.

First, we assume that the superconducting disc (radius 𝑎) is characterized by a field-

independent 𝐽c, i.e. the true 𝐽c(𝐵) is replaced by a uniform 𝐽c producing almost the same flux

distribution above the superconductor alone. As estimated in chapter 5, the corresponding

field-independent 𝐽c in the present case is 𝐽c2 = 1.03 × 108 A/m2. As a first approximation, the

radial dependence of flux density in the median plane (𝑧 = 0) is given by 𝐵(𝑟) = 𝜇0 𝐽c0 (𝑎 − 𝑟).

If we further assume that there are no stray field lines through the lateral surface of the

superconductor, the total flux 𝜙 generated by the superconductor is given by radial integration

of 𝐵(𝑟), i.e. 𝜙 = 𝜇0 𝐽c0 𝜋 𝑎3 3⁄ . In the case of a perfect (infinitely permeable) ferromagnet,

axial magnetic flux lines exiting the superconductor at the SC/FM interface would be

channelled by the ferromagnet along the radial direction, i.e. they would exit the ferromagnet

through its lateral surface. For a ferromagnetic disc of thickness 𝑑, conservation of magnetic

flux imposes therefore that the magnetic flux density in the ferromagnet 𝐵FM is such that

(a) D035 (b) D1

FM

SC

FM

SC

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 115

𝐵FM (2 𝑎 𝑑) = 𝜇0 𝐽𝑐0 ( 𝑎3 3⁄ ).

A rough approximation of the flux density in the ferromagnet is therefore given by

𝐵FM = 𝜇0 𝐽c0 (𝑎2 6𝑑⁄ ).

Since this hypothetical, perfect (infinitely permeable) behaviour occurs only when the magnet

is not saturated, the minimum ferromagnet thickness required (denoted 𝑑∗) is such that

𝐵FM = 𝜇0 𝐽c0 (𝑎2 6𝑑⁄ ) < μ0𝑀sat

𝑑 > (𝐽c0 𝑎2 6 𝑀sat⁄ ) = 𝑑∗ (6.1)

Using the actual parameters (𝐽c0 ≈ 108 A/m2, 𝜇0 𝑀sat = 1.4 T, 𝑎 = 8.25 10-3 m), we obtain a

theoretical threshold thickness given by 𝑑∗ ≈ 1 mm. This value, estimated with rather “crude”

assumptions, is in fair agreement with the interval found using finite element modelling (0.7 to

1.0 mm). As a consequence, formula (6.1) can be used as a simple design rule to determine the

minimum required thickness of ferromagnetic material. Note that for some given

superconductor properties, much thicker ferromagnets will not impact significantly on the

surrounding field.

More precisely, we can derive an analytical upper bound for the relative increase of 𝐵zc on the

superconductor side when a ferromagnet of the same radius is attached to the opposite face.

This upper bound is the expected horizontal asymptote of the graph shown in Fig. 6.15 when

the ferromagnet is a perfect ferromagnetic material occupying the semi-infinite volume 𝑧 < 0.

The expected increase for an infinitely thick ferromagnetic “disc” is likely to be bounded by the

increase that would occur for a semi-infinite ferromagnet. Since the latter configuration is

equivalent to doubling the height of the superconducting disc, we can roughly estimate the

effect of the ferromagnet from the analytical formula giving the magnetic flux density along

the axis of a fully magnetized superconducting cylinder (radius 𝑎, height 𝐿, critical current

density 𝐽c) [102, 148, 8]. This formula is already presented in Chap. 5 as Eq. (5.5):

𝐵z(𝑧) =1

2𝜇0 𝐽c (𝑧 +

𝐿

2) ln (

𝑎 + √𝑎2 + (𝑧 + 𝐿 2⁄ )2

|𝑧 + 𝐿 2⁄ |)

− (𝑧 −𝐿

2) ln (

𝑎 + √𝑎2 + (𝑧 − 𝐿 2⁄ )2

|𝑧 − 𝐿 2⁄ |)

where 𝑧 is the elevation from the centre of the superconductor. At the centre of one face of

the superconductor (𝑧 = 𝐿 2⁄ ), the flux density 𝐵CF is equal to

𝐵CF = 𝐵z (𝐿

2) =

1

2 𝜇0 𝐽c 𝐿 ln [

𝑎 + √𝑎2 + 𝐿2

𝐿]

116 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

which can be rewritten as a function of the dimensionless (thickness / radius) ratio 𝜁 = 𝐿 𝑎⁄ ,

i.e.

𝐵CF =1

2 𝜇0 𝐽c 𝑎 𝑓(𝜁),

where the function 𝑓(𝜁) is introduced, given by

𝑓(𝜁) = 𝜁 ln [1

𝜁+ √1 +

1

𝜁2].

The theoretical increase that would be due to a semi-infinite ferromagnet is therefore given by

[𝑓(2𝜁) − 𝑓(𝜁)] 𝑓(𝜁)⁄ For the geometric parameters of the superconductor investigated in

this work (𝐿 = 6.32 mm, 𝑎 = 8.25 mm, 𝜁= 0.766), one obtains ≈ 13 %. This rough upper bound is

indeed a few percent above the relative increase modelled for the thickest ferromagnet

investigated (8 %). The conclusion to be drawn from this analysis is that a ferromagnet is

extremely helpful in increasing the flux density of a superconductor, but that a thickness much

larger than the threshold value 𝑑∗ defined above is not needed.

Since many applications of bulk superconductors are linked to levitation systems [157, 158,

159], it is also of interest to investigate whether a ferromagnetic disc placed at the bottom of

the superconducting pellet is beneficial to the vertical levitation force above the

superconductor top face. Along the axis of the superconductor, this levitation force is

proportional to the gradient of flux density d𝐵z d𝑧⁄ [58, 160]. Fig. 6.17 shows the modelled

magnetic induction 𝐵z along the symmetry axis (i.e. for 𝑟 = 0) for the D035 and D1

configurations. The 𝑧 derivative of these curves can be used to study the influence of the

ferromagnet on the levitation force. The modelling results show that the addition of the

ferromagnetic disc on one side increases d𝐵z d𝑧⁄ at 𝑟 = 0 on the opposite side by 3.1 % and

5.2 % with the 0.35 mm and 0.99 mm discs, respectively (values taken 0.85 mm above the

superconductor). Although the relative increase of d𝐵z d𝑧⁄ is not as high as the relative

increase in 𝐵z, it can be concluded that the presence of the ferromagnet yields an increase of

the levitation force above the superconducting pellet.

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 117

Fig. 6.17 Magnetic induction 𝐵z along the symmetry axis (𝑟 = 0) of the D035 and D1

configurations. The superconductor alone (SC) is shown for reference. The vertical

lines show the limits of the superconductor. The D035 and D1 abscissae show the

limit of the ferromagnet in the D035 and D1 configurations, respectively.

6.2.2 𝛥𝐵 Hysteresis curves

The comparison of the hybrid configurations made with magnetic discs of different thicknesses

can be carried out when recording Δ𝐵 hysteresis curves measured with the coil wound around

the entire height of the superconductor. Similar results were obtained for the three other coils

(top, middle, and bottom).

Fig. 6.18 shows the hysteresis curves measured with the coil wound around the entire height

of the superconductor for the different SC/FM hybrid structures D1, D2, D3 made with a

Supra50 disc of 0.99, 1.90, and 2.90 mm respectively. The curve for the superconductor only is

also shown (blue dots) for comparison. Table 6.1 summarizes the respective volumes of the

ferromagnetic discs and the corresponding remanent inductions of the curves in Fig. 6.18.

These increases are reported in Fig. 6.19. Each hysteresis loop exhibits a combination of

diamagnetic and ferromagnetic behaviour, as is the case for the configuration with the disc of

intermediate thickness (D2). The slope at the origin of the axes remains unchanged by the

presence of the ferromagnet, although its presence leads to a change of sign of Δ𝐵 as the

applied field increases. It can be seen further that the maximum of the curve appearing around

𝐻 ≈ 0 (strictly speaking, at a slightly negative applied field for the superconductor only) is

shifted to the right and appears now for a positive applied field for all configurations of the

hybrid structure. The remanent induction 𝐵rem (measured at zero applied field) is also

increased by the presence of the ferromagnetic component, as summarized in Table 6.1.

Top Superconductor Bottom

D035

FM

118 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

Fig. 6.18 Δ𝐵 hysteresis curves measured with the coil wound around the entire

height of the superconductor for SC/FM hybrid structures made with ESC2 and FM

discs made of Supra50 of thickness 0.99 mm (yellow, D1), 1.90 mm (green,

presented previously (D2)), and 2.90 mm (red, D3). The superconductor alone

curve is shown for comparison (blue)

Table 6.1 Evolution of the measured average remanent flux density for the

configurations D1, D2, and D3 compared with the volume of the ferromagnet (FM).

The superconductor (SC) has a volume of 1351 mm3.

SC/FM hybrid Volume: SC+FM (mm3) 𝐵rem (T) Increase of 𝐵rem (%)

S only 1351 0.168 --

D1 1351+213 0.191 14

D2 1351+406 0.196 16

D3 1351+623 0.198 18

Fig. 6.19 Relative increase of the measured 𝐵rem = Δ𝐵(𝐻app = 0) as a function of

the thickness of the ferromagnetic disc.

0%

5%

10%

15%

20%

0 0.5 1 1.5 2 2.5 3

Rel

ativ

e in

crea

se o

f B

rem

Thickness of the ferromagnetic disc (mm)

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 119

Interestingly, the plot of the measured increase of the average flux density 𝐵rem as a function

of the disc thickness in Fig. 6.19 is similar to the increase of the central flux density 𝐵zc that

were predicted by numerical modelling in Fig. 6.15. A 1 mm-thick Supra50 leads to an increase

of 14 % of 𝐵rem. The contribution of the next layers only lead to a limited increase of 𝐵rem;

from D1 to D2, the volume of ferromagnet nearly doubles (+91 %) while the remanent

induction increase only by 2 percentage points (from 14 % to 16 %). A similar increment is

measured for the last disc thickness (D3). As in Fig. 6.15, a kink is expected in the curve of Fig.

6.19. However, more results are needed to narrow the interval where the kink occurs.

Nevertheless, it is clear that it must occur below a 1 mm thickness, which is in agreement with

our previous estimation.

The position of the maximum magnetization for negative applied fields in the case of a pure

superconductor has been associated with a minimization of the magnetic flux inside the

material (i.e. consisting of the sum of the applied field and the trapped field), which leads to an

increase in the field-dependent critical current [134]. The observed shift of the position of

these maxima to positive applied fields for the hybrid SC/FM structure suggests that the

increase of the remanent field is related to the closing of the return flux lines outside of the

superconductor. A similar shift has been measured on the magnetization cycle of structures

made by an iron yoke surrounded by a bulk-based superconducting ring [9] or on (RE)BCO

materials where (RE) is a rare earth paramagnetic ion, e.g. DyBCO melt textured single

domains [161].

6.3 Influence of the type of ferromagnetic material

The value of the characteristic ferromagnet thickness 𝑑∗ = (𝐽c0 𝑎2 6 𝑀sat⁄ ) under which the

ferromagnet is mainly saturated derived in Eq. (6.1) depends on the saturation magnetization

of the ferromagnetic material. The influence of this parameter is studied in this section by (i)

modelling of magnetic flux density profiles and (ii) measurement of the Δ𝐵 hysteresis curves

on structures made with ferromagnetic discs with different saturation magnetizations.

Flux profiles are computed with the H-formulation of Maxwell’s equations for the D035 SC/FM

hybrid structure made of ESC2 and a 0.35 mm thick disc of the same diameter as the

superconductor (16.5 mm). Three ferromagnetic materials are investigated. Their permeability

and saturation magnetization are summarized in Table 6.2. The two first are Supra50 and

Permimphy which were measured in Chap. 5. A third material, “Vacoflux” from

Vacuumschmelze [162], is added here. It has not been characterized experimentally; its

magnetic properties were obtained through the manufacturer specifications.

Fig. 6.20 shows the flux distribution modelled above the D035 configuration for three

ferromagnetic materials of different saturation magnetizations presented in Table 6.2: Supra50

(violet dash-dot) as shown in the previous section, Permimphy (red dash), and Vacoflux (green

line). The magnetic flux density obtained for the superconductor alone is shown for reference

(blue line). On the ferromagnet side, the three 𝐵z profiles are found to exhibit several bumps.

120 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

When increasing the saturation magnetization 𝑀sat of the ferromagnetic material, the general

trend is that 𝐵z decreases in the central region and increases near the edges of the sample. At

the highest saturation magnetization investigated, the shape tends to the plate-like shape

observed for thicker ferromagnetic discs of lower saturation magnetization. On the

superconductor side, the maximum flux density is an increasing function of the saturation

magnetization 𝑀sat of the ferromagnet. The corresponding flux density increase, as a function

of 𝑀sat, is shown in Fig. 6.21. A quasi-linear behaviour is observed in the studied range.

Table 6.2 Characteristics of the ferromagnetic materials: saturation magnetization

𝜇0 𝑀sat, maximum differential permeability 𝜇r,max and coercive field 𝐻c (neglected

in modelling).

Material 𝜇0 𝑀sat (T) 𝜇r,max 𝐻c (A/m)

Supra50 1.4 1.7 × 103 520

Permimphy 0.8 2.0 × 103 250

Vacoflux 2.2 1.7 × 103 —

C45 steel 2.0 1.0 × 103 910

Ferrite 0.67 — —

These results are similar to those obtained for several thin ferromagnetic discs of a given

saturation magnetization in the previous section (Sect. 6.2). We can therefore reasonably

assume that the explanation is similar: when the ferromagnet is saturated, only a small

fraction of the flux lines can be channelled radially towards the lateral surface of the

ferromagnet. According to Eq. (6.1), the threshold thickness below which saturation occurs –

and the “bumps” in 𝐵z appear – is inversely proportional to the saturation magnetization 𝑀sat.

This means that for 𝜇0𝑀sat = 2.2 T, this threshold thickness would be approximately 64 % of

that for the Supra50 studied earlier. Qualitatively, increasing the saturation magnetization of

the soft ferromagnet has the same effect globally as increasing its thickness. The conclusion to

be drawn is that ferromagnets with the highest saturation magnetization possible are always

preferred, but that the intrinsic (physical) limitation of the saturation magnetization of classical

ferromagnets can be overcome by using thicker ferromagnetic discs.

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 121

Fig. 6.20 Modelled magnetic flux density 𝐵z above the top and bottom surfaces for

the D035 configuration (0.35 mm thick disc) for three different saturation

magnetizations. Three materials are compared: Supra50 (𝜇0𝑀sat = 1.4 T; violet

dash-dot) as presented before, Permimphy (𝜇0𝑀sat = 0.8 T; red dash), and Vacoflux

(𝜇0𝑀sat = 2.2 T; green line). The superconductor alone is shown for reference (blue

line).

Fig. 6.21 Relative increase of the modelled flux density at the centre of the top face

of the superconductor 𝐵zc as a function of the saturation magnetization of the

ferromagnetic material placed on the bottom face.

The Δ𝐵 hysteresis curves were also measured with the coil wound around the entire height of

the superconductor on SC/FM hybrid structures involving different magnetic materials

characterized in Chap. 5 and summarized in Table 6.2. The discs thickness is approx. 1.90 mm

for each configuration except for the ferrite for which it is 1.27 mm (thickness available

commercially). Fig. 6.22 shows a schematic representation of these configurations and the

corresponding measured Δ𝐵 hysteresis curves. The general behaviour of the hysteresis curves

is as expected and results from a combination of a ferromagnetic and a superconducting

behaviour. The remanent flux density is nearly similar for all these curves; only the value

0%

1%

2%

3%

4%

5%

6%

0 0.5 1 1.5 2 2.5Rel

ativ

e in

crea

se o

f B

zc

µ0Msat of the ferromagnet (T)

(a) Superconductor side (b) Ferromagnet side

122 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

obtained for the ferrite curve is slightly lower than the three other SC/FM structures. The

influence of the material properties is however visible at higher applied field. A higher

saturation magnetization of the FM leads to a higher flux density increase at high applied field.

These results are consistent with the observations made in Fig. 6.9: at high applied field, the

magnetic flux density measured for the hybrid structure approximates very well to the simple

superposition (i.e. addition) of the individual hysteresis curves of the ferromagnet and the

superconductor. In the remanent state, it can be seen that the saturation magnetization does

not influence the result, which indicates that the ferromagnetic materials are not saturated.

The slightly lower remanent magnetization in the ferrite structure is likely to be related to the

thickness of the ferrimagnetic disc. In addition, the coercive field does not appear to influence

significantly the remanent magnetization in the investigated configurations.

Fig. 6.22 Δ𝐵 hysteresis curves measured using the pick-up coil wound around the

entire height of the superconductor for the SC/FM configurations shown on the

right. Several ferromagnetic materials are used: Supra50 (𝜇0𝑀sat = 1.4 T),

Permimphy (𝜇0𝑀sat = 0.8 T), C45 steel (𝜇0𝑀sat = 2.0 T) and the ferrimagnetic

material (𝜇0𝑀sat = 0.67 T). The curve measured on the superconductor alone is

shown for comparison.

6.4 Influence of the critical current of the superconductor

Two finite element numerical models are used in this section to study the influence of the

critical current density of the superconductor on the above results. The H-formulation is used

to compute the magnetic flux density profile above the SC/FM structures while the A-

formulation is used for the Δ𝐵 hysteresis curves. We first compare the magnetic flux profiles

obtained either with a field-dependent 𝐽c(𝐵) or with a constant 𝐽c. Next, we investigate the

properties of a large-𝐽c hybrid structure generating a field much higher than the saturation

magnetization of the ferromagnet.

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 123

Fig. 6.23 shows the flux distribution above the different hybrid SC/FM structures where the

superconductor would be characterized by a constant critical current 𝐽c2 = 1.03 108 A/m2. This

particular “average” uniform 𝐽c is used since the 𝐵z profile is nearly the same as that obtained

with original 𝐽c(𝐵) law for the SC configuration (superconductor alone). Remarkably, the

constant and field-dependent critical currents lead to similar distributions of 𝐵z on both faces

of the SC/FM assemblies. The practical conclusion is the following: although the

superconducting material is characterized by some field-dependent 𝐽c(𝐵), this 𝐽c(𝐵)

dependence is not so crucial for investigating the flux profiles at some distance for the hybrid

SC/FM structures. Note that the regime investigated here is one for which the flux density in

the superconductor is on average 50–60 % of the 𝐵1 parameter of the 𝐽c(𝐵) law, which means

that field-induced variations of the critical current density can indeed be found in the

superconducting volume.

Fig. 6.23 Comparison of modelling results of 𝐵z above each surface of the SC/FM

structure with a constant critical current density 𝐽c2 (dashed line) and the original

𝐽c(𝐵) critical current density (continuous line). The SC, D035, and D1 configurations

are shown in blue, violet, and red, respectively.

It is of interest to investigate what happens when the magnetic field generated by the

superconductor is much higher than the saturation magnetization of the ferromagnetic

material, and to know if it is still useful to use ferromagnetic materials in such high field

environments. The model is therefore run with a higher (constant) critical current ten times

larger than that used in the previous sections, i.e. 𝐽𝑐2′ = 10 × 𝐽𝑐2 = 10.3 108 A/m2. Note that the

practical way of reaching such higher critical current density would be to simply decrease the

operating temperature. Fig. 6.24 shows the modelling results obtained at 𝐽𝑐 = 𝐽𝑐2′ when the

superconductor is combined with ferromagnetic discs of thicknesses 0.35 mm, 0.99 mm, and

1.90 mm (the D035, D1, and D2 configurations, respectively); these are all made of Supra50

material (saturation magnetization = 1.4 T). To achieve full penetration of the superconductor,

a magnetizing induction of 15 T is applied.

(a) Superconductor side (b) Ferromagnet side

124 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

Fig. 6.24 Modelling results of 𝐵z above each surface of the SC/FM structure with a

constant critical current density 𝐽𝑐2′ = 10 × 𝐽𝑐2 = 10.3 10

8 A/m

2. The SC, D035, D1

and D2 configurations are shown in blue, violet, red, and green, respectively.

As can be seen, the remanent magnetic flux density is approximately ten times higher, with

maximum values on the order of 3 teslas. Nevertheless, the general behaviour is similar to the

previously obtained results. The trapped field on the top face increases with increasing

thickness of ferromagnet. The increase, however, is relatively less significant than found

previously, e.g. for D2, the increase of 𝐵zc is 3.0%, instead of 9.6% for the lower constant

critical current density 𝐽𝑐2. On the bottom face, the distribution of 𝐵z is still conical, but

attenuated at its centre and widened out at the edges.

The interesting feature of the results shown in Fig. 6.24 is that the previous conclusions are still

valid. A thicker ferromagnet leads to a larger decrease above the ferromagnet and a larger

increase above the superconductor. We also see that a higher flux needs a thicker ferromagnet

of a given material to achieve an equivalent shielding effect. According to the rough estimation

given by Eq. (6.1), the minimum thickness of ferromagnet above which it is no longer saturated

(𝑑∗) is directly proportional to the superconductor critical current density 𝐽𝑐. In the present

case this would correspond to approximately ten times the value found for the initial 𝐽𝑐, i.e.

𝑑∗ ≈ 10 mm. Since the practical thickness range investigated for Fig. 6.24 (0.35 to 2 mm) is less

than 𝑑∗, the ferromagnet is probably driven to saturation, at least partially. The similarity

between the gradual evolution of the flux profiles in Fig. 6.24(b) and in Fig. 6.14 (b) (i.e. for thin

discs of various thicknesses with the original 𝐽c) gives additional evidence that the ferromagnet

is mainly saturated.

The important conclusion to be drawn is that the ferromagnet still plays a role in increasing the

field produced by the superconductor, even though the latter is twice the intrinsic saturation

magnetization of the ferromagnetic material. Interestingly, the regime investigated is one for

which magnetic flux density measured at some distance from the ferromagnet is extremely

(a) Superconductor side (b) Ferromagnet side

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 125

sensitive to the thickness of ferromagnet. Although this is out of the scope of the present

study, this strong sensitivity to thickness could be usefully exploited to form the basis of a

contactless thickness sensor of ferromagnetic materials.

The influence of a higher critical current density can also be studied on the Δ𝐵 hysteresis

curves computed with the A-formulation. Fig. 6.25 shows the corresponding modelling results

where the 𝐽c(𝐵) Bean-Kim law is used with 𝐽c1 = 1.38 108 A/m2 and 𝐵1 = 0.987 T for the

superconductor alone (blue) and D2 configuration (red). The same curves are then computed

with the same 𝐵1 but with 10 × 𝐽c1 for the superconductor alone (purple) and the D2

configuration (green). A maximum magnetizing flux density of 6 T was chosen to fully

magnetize the superconductor, since a higher critical current implies a higher penetration

field. For comparison, the two intermediate cycles with an applied induction up to 3 T are

shown as dashed lines. The Δ𝐵 hysteresis curves computed with the original values of the

critical current were already presented in Fig. 6.8. The remanent induction increases from

0.156 T to 0.186 T (+19 % or +30 mT) in presence of the ferromagnet. At 3 T of applied flux

density, the average induction goes from -44 mT to 49 mT (+ 93 mT). The Δ𝐵 hysteresis curves

computed with the higher critical current shows that the magnetic response of the

superconductor is increased. They show an increased curvature with an increasing applied

field compared to those with the original critical current. The remanent flux density increases

from 0.932 T to 1.05 T (+13 % or +118 mT) in presence of the ferromagnet. At 6 T of applied

flux density, Δ𝐵 goes from -261 mT to -165 mT (+ 96 mT).

The increased curvature of the hysteresis curves computed with the higher critical current is a

consequence of the Bean-Kim law used with the same value of 𝐵1 in both cases. Under a local

magnetic flux density equal to 𝐵1, the critical current is reduced by a factor two. Therefore, the

higher critical current will undergo a higher absolute decrease for a same applied field. These

results show that the ferromagnet still has an influence on the average magnetic flux inside the

superconductor with large 𝐽c values. However, the general shape of the hysteresis curve is

relatively less influenced by the presence of the ferromagnet than for the lower 𝐽𝑐. This is

coherent with the addition of the effects of the ferromagnet to the magnetization of the

sample. Interestingly, the model predicts an increase of Δ𝐵 of 93 mT at 3 T with the original

current and an increase of 96 mT at 6 T with the higher 𝐽c. This emphasizes the finite

contribution of the ferromagnet once it is fully saturated under the applied field. The additive

contribution is therefore relatively lower for a higher magnetization of the superconductor.

In the remanent state, 𝐵rem is increased by 118 mT (+13 %) when the ferromagnet is added to

the superconductor. In the present case where the investigated 𝐽c is 10 times larger than for

the original material, the increase of 𝐵rem suggests that the ferromagnet is fully saturated

since its absolute contribution is limited to a value similar to the one obtained under an

important applied field. The relative contribution of the ferromagnet to the average trapped

field is therefore reduced in the remanent state, as it is at high applied fields. Note that this

126 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

behaviour is also observed on the narrower cycles computed with the higher critical current,

i.e. where the superconductor is not fully magnetized.

Fig. 6.25 Δ𝐵 hysteresis curves computed with the A-formulation of Maxwell’s

equation for the superconductor alone and D2 configuration for two critical current

densities. The superconductor critical current density 𝐽c(𝐵) is described with the

Bean-Kim law with 𝐽c1 = 1.38 × 108 A/m

2 and 𝐵1 = 0.987 T for the superconductor

alone (blue) and D2 configuration (red). The same curves are then computed with

the same 𝐵1 but with 10 × 𝐽c1 for the superconductor alone (purple) and the D2

configuration (green). In this later case, intermediate cycles with an applied field up

to 3 T are computed (dashed).

6.5 Summary

In this chapter, the magnetic properties of hybrid structures made of soft ferromagnetic discs

attached to a bulk, large grain superconductor (SC/FM) were characterized both

experimentally and numerically.

The experimental set-ups presented in Chap. 3 are used. Δ𝐵 hysteresis curves of (i) the

average volume flux density and (ii) the surface flux density were measured. Volume

measurements were carried out using four coils wound around the entire height and around

the top, middle and bottom sections of the superconductor. The surface (Hall probe)

measurements hysteresis curves were probed by a fixed Hall probe stuck at the centre of both

circular faces. Additionally, the profiles of the magnetic flux density above both faces of the

SC/FM structures were measured by Hall probe mapping.

Numerical modelling was carried out to draw important conclusions about the magnetic flux

distribution in and around the assemblies. The three 2D axisymmetric modelling method

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 127

developed within the finite element framework, and presented in Chap. 4, were used:

Campbell’s equation, A-formulation and H-formulation of Maxwell’s magnetodynamics

equations. These models use the physical parameters of the SC and FM determined from

independent characterization measurements carried out in Chap. 5. The modelling results

within the A-formulation and the Campbell method are in excellent qualitative and

quantitative agreement with the measured Δ𝐵 hysteresis curves of the average volume flux

density. The H-formulation was validated and used for further investigations into the profile of

the magnetic flux density above both faces of the SC/FM structures. The representation of the

flux distribution and direction around a SC/FM hybrid structure was discussed for the three

methods. The agreement of the models between them and with measurement validates the

models and suggests they can be used interchangeably for further investigations of the

magnetic properties of SC/FM structures.

First, we studied the influence of the FM on the flux distribution around the SC through

modelling of the flux lines. Then, the influence of the 1.90 mm thick disc made of Supra50 (D2

configuration) is studied with the three types of measurements and with modelling. Modelling

is then used to study the magnetic flux distribution and predict the behaviour for other

constitutive laws and geometries. We have investigated successively the influence of

ferromagnet thickness, its saturation magnetization, and the critical current density of the

superconductor, including the 𝐽c(𝐵) dependence. In all studied configurations, a ferromagnet

disc is attached to the large grain superconducting pellet.

All the results studied in this chapter are consistent with the following understanding of the

influence of the ferromagnetic material. In the trapped field configuration, the zone above the

ferromagnet is shielded from the flux trapped inside the superconductor. The ferromagnet

acts as a magnetic short-circuit and creates a low reluctance path which drives the flux lines

directly towards the edges of the superconductor. Conversely, the flux density is enhanced on

the face opposite to the ferromagnet (superconductor side). To a first approximation, this

increase can be understood through the image theorem. Adding the ferromagnet is equivalent

to doubling the height of the superconducting cylinder which experiences a lower

demagnetizing field. In practice, the real effect is not as significant as the image theorem

would predict, since the ferromagnet is not semi-infinite. Additionally, the presence of the

ferromagnet increases the average remanent induction inside the superconductor in all the

configurations studied in this investigation.

The ferromagnet also influences the magnetic response of the assembly under an external

magnetic field. The penetration of the magnetic flux inside the superconductor is delayed in

the vicinity of the ferromagnet. The Δ𝐵 hysteresis loops from sensing coils wound around the

superconductor show a combination of diamagnetic and ferromagnetic behaviour, with the

ferromagnet having a larger effect on the sections of the superconductor in closer proximity to

its position. The relation of the hysteresis cycle of the hybrid SC/FM structure to that of its two

constituent materials characterized separately has been investigated experimentally. The

128 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

hysteresis effects of the superconductor and the ferromagnet simply superimpose at high

applied field (i.e. exceeding the apparent saturation of the ferromagnet). Remarkably, this

behaviour is observed above the ferromagnetic saturation limit of the ferromagnet in all the

investigated configurations. Below this level of applied field, the ferromagnet contains

saturated and non-saturated regions.

The results demonstrate that the thickness and saturation magnetization of the ferromagnetic

material are important and play somewhat similar roles. The higher the saturation

magnetization and/or the thicker the ferromagnetic material, the higher the trapped field on

the superconductor face and the larger the shielding effects on the ferromagnet side. A

ferromagnet of given thickness below its full saturation will drive the majority of flux lines

radially towards its sides, therefore shielding a large part of the magnetic flux density 𝐵z. If the

trapped field is increased, the amount of diverted/shielded flux will not change after the full

saturation of the ferromagnet. Therefore, the effect of the ferromagnet will become relatively

less important as the generated flux increases.

Based on these considerations, the results show that the increase of magnetic flux vs.

thickness of ferromagnetic disc exhibits a “kink” corresponding to full saturation of the

ferromagnet. We have derived a simplified analytical expression based on magnetic flux

conservation to roughly estimate the most suitable ferromagnet thickness 𝑑∗, below which

saturation occurs and above which weak thickness-dependence is observed. A remarkable

result of this analysis is that the rough estimation of 𝑑∗ ≈ 𝐽c0 𝑎2 6𝑀sat⁄ agrees reasonably well

with that obtained with accurate 2D finite element modelling of the full assembly. An increase

of the ferromagnet thickness well beyond 𝑑∗ is of low interest for practical purposes.

Investigations for constant and field-dependent critical current densities lead to nearly similar

results. This shows that the particular 𝐽c(𝐵) dependence is not a crucial parameter in

investigating the behaviour of SC/FM structures. The field produced by the superconductor is

shown to be enhanced by the ferromagnet, even though the generated field outside the

assembly is much larger than the saturation magnetization of the ferromagnet. This occurs, for

example, when the temperature of the superconductor is lowered from 77 K to 20 K.

Conversely, thin ferromagnetic discs can be driven to full saturation even though the outer

magnetic field is smaller than the saturation magnetization of the ferromagnet.

These results help significantly in understanding of the contribution of ferromagnetic sections

to the trapped field produced by bulk high temperature superconductors, which is

fundamental to realizing practical applications of these technologically important materials.

The following “rules of thumb” were deduced from the results.

The fact that the ferromagnet acts as a shield emphasizes that such a ferromagnetic element

should not be placed in a zone where a large magnetic field in the air is desired. As far as the

volume magnetic flux through the superconductor is concerned, however, the effect of the

ferromagnet is beneficial to the trapped flux.

Chapter 6. Superconductor / ferromagnetic disc hybrid structure 129

A larger ferromagnet has a more significant effect on the whole hysteresis loop of the

superconductor, but this effect varies less than proportionally to the increase in the

ferromagnet volume. The 𝑑∗ value can be used to investigate the significance of the

ferromagnet effects. Ferromagnets with the highest saturation magnetization possible are

always preferred, but the intrinsic (physical) limitation of the saturation magnetization of

classical ferromagnets can be overcome by using thicker ferromagnetic discs.

The simple “additive” behaviour of the superconductor and ferromagnet individual magnetic

curves can be used effectively as a “rule of thumb” for predicting the magnetic behaviour of

larger or more complex hybrid structures involving ferromagnets and bulk superconductors.

130 Chapter 6. Superconductor / ferromagnetic disc hybrid structure

131

Chapter 7

Investigation of more complex structures

In this chapter, we study measurement results obtained on more complex SC/FM hybrid

assemblies than those studied in the previous chapter. First, we investigate the properties of

structures assembled with a YBCO superconductor (ESC2) attached to ferromagnetic sections

of various shapes. Then, FM/SC/FM hybrid structures are characterized experimentally. These

structures consist of the same YBCO superconductor sandwiched between two disc-shaped

ferromagnets, one against each circular face. Finally, SC/FM structures with a GdBCO

superconducting sample (ESJc) and a ferromagnetic disc are investigated. The influence of the

ferromagnetic disc on the magnetic moment of the whole assembly and in the so-called

“crossed field” configuration is measured. All these results are discussed in the framework of

the conclusions drawn in the previous chapter.

7.1 SC/FM structures with different ferromagnet shapes

In this section we investigate the properties of ferromagnetic pieces machined into a ring and

an inverted cone, the latter being shaped like an “inverted Bean profile”. These sections are

combined with ESC2 to study their influence on the flux profile above the ferromagnet. The

purpose of studying such particular ferromagnet shapes is to investigate whether the presence

of a high permeability material at the edge of the superconductor is helpful in improving (i.e.

flattening) the flux distribution and/or in increasing the trapped flux density. The flux profile

measurements will be compared to FEM modelling results computed with the H-formulation.

Before the examination of flux profiles, the Δ𝐵 hysteresis loops of the average flux density

inside the superconductor are investigated.

The inverted cone (IC) and ring (RG) were machined with the same diameter as the

superconductor (16.5 mm). The cross sections of the SC/FM structures made with these FM

section are shown schematically in the right panel of Fig. 7.1. The ring is 2.88 mm thick and has

an internal diameter of 9.52 mm. The inverted cone height is 2.88 mm at the edge and

0.09 mm at the centre. Significantly, these ferromagnetic sections have a volume similar to

that of the ferromagnetic disc D2 (≈ 412 mm3 within an accuracy of 3%, as summarized in

Table 7.1).

132 Chapter 7 Investigation of more complex structures

7.1.1 𝛥𝐵 Hysteresis curves

In this section, we examine how the shape of the ferromagnet placed against the

superconductor modifies the results obtained previously with ferromagnetic discs. Fig. 7.1

shows the hysteresis curves measured with the coil wound around the entire height of the

superconductor for the different SC/FM hybrid structures described on the right hand side of

the figure. The curve for the superconductor only is also shown (blue dots) for comparison.

Table 7.1 summarizes the respective volumes of the ferromagnetic components and the

corresponding remanent inductions of the curves in Fig. 7.1. Each hysteresis loop exhibits a

combination of diamagnetic and ferromagnetic behaviour, as is the case for the configuration

with the discs. The slope at the origin of the axes remains unchanged by the presence of the

ferromagnet, although its presence leads to a change of sign of Δ𝐵 as the applied field

increases. The remanent induction (measured at zero applied field) is also modified by the

presence of the ferromagnetic component, as listed in Table 7.1.

The shape effect can be appreciated by comparing the results for the disc (D2), the inverted

cone (IC) and the ring (RG), all of them having (almost) the same volume. This comparison is

based here on the measurements with the coil wound around the entire height of the

superconductor and similar results were obtained for the three other coils (top, middle, and

bottom — see Chap. 6). As can be seen from Fig. 7.1 and Table 7.1, the ring produces the

smallest increase of the remanent induction (12%), while the disc and the inverted cone both

lead to an increase of 16%. Such behaviour can be understood qualitatively as follows.

Whereas they both have the same volume, the ring is thicker than the disc (2.88 mm

compared to 1.90 mm) and does not cover the centre of the superconductor. The most distant

part of the ferromagnet has a small effect on the induction inside the superconductor, as

observed from the measurements on the discs of different thicknesses (Chap. 6). These

observations suggest that the presence of ferromagnetic material against the whole surface of

the superconductor is important. It is therefore preferable to locate the ferromagnet as close

as possible to the superconductor.

The experimental results show further that the inverted cone geometry and the 1.90 mm thick

disc produce a similar increase of the remanent induction in the hybrid structure, despite the

inverted cone being slightly larger (3.2%) than the disc. The present measurements can be

interpreted with the help of the observations made in Chap. 6. Above the fully magnetized

superconductor, the ferromagnet is subjected to a highly non-uniform magnetic field and

contains saturated and non-saturated regions. Magnetic flux density plots showed that the

zone located in the FM, around the symmetry axis and opposite to the SC/FM interface is the

last to reach saturation. The above results with the inverted cone confirm that this central part

might be removed, e.g. for weight considerations.

Chapter 7 Investigation of more complex structures 133

Fig. 7.1 Magnetic hysteresis loops measured with the coil wound around the entire

height of the superconductor for the superconductor only (blue dots) and the

SC/FM configurations whose cross sections are shown on the right with the

ferromagnets in red: D2 (disc), IC (inverted cone) and RG (ring). The three pieces

are characterized by the same volume of ferromagnetic material (within 3%).

Table 7.1 Evolution of the average remanent induction for the configurations D2, IC

and RG (see Fig. 7.1). The volumes of the superconductor (SC) and the

ferromagnets (FM) are presented.

SC/FM hybrid Volume: SC+FM (mm3) 𝐵rem (T) Increase of 𝐵rem (%)

SC only 1351 0.168 --

D2 1351+405 0.196 16

IC 1351+418 0.196 16

RG 1351+414 0.189 12

Additionally, it is worth mentioning that the “addition rule” presented in Chap. 6 – the fact

that the effects of the superconductor and ferromagnet are superimposed above the apparent

saturation of the ferromagnet – was verified for the coil wound over the entire height of the

superconductor in all the configurations presented here, as well as for the FM/SC/FM

structures presented in the next section.

7.1.2 Flux density profiles

Fig. 7.2(a) shows the radial measurements of 𝐵z for the measured configurations presented in

Fig. 7.1. As in the previous chapters, the position corresponding to the measured maximum on

the SC side in the absence of ferromagnet (r = –1 mm) was selected as the reference position

134 Chapter 7 Investigation of more complex structures

to take into account that the maximum flux density occurs at a slightly off-centre position. The

characteristic values 𝐵z∗ = 𝐵z(𝑟 = −1 mm) are summarized in Table 7.2, together with their

relative variation compared to the SC alone. On the SC side (i.e. opposite to the ferromagnet),

Fig. 7.2(a) shows an increase of 𝐵z for each hybrid SC/FM configuration. This increase ranges

between 2.1 % and 8.0 % depending on the shape of the ferromagnet. On the FM side, the

distribution of 𝐵z changes from a conical to a plate-like shape when the ferromagnet covers

the entire surface (D2 and IC). The flux density is the largest for a ring aligned with the edges of

the ferromagnet. The ferromagnetic ring (RG) does not cover the entire surface of the

superconductor and the measured flux density exhibits three maxima including a central

“bump”. It is of interest to note that a larger increase in 𝐵z on the SC side corresponds to a

larger decrease on the bottom face in all measured configurations.

(a) Experiment (b) Modelling

Sup

erc

on

du

cto

r si

de

Ferr

om

agn

et s

ide

Fig. 7.2(a) Profiles of the magnetic flux density 𝐵z above both surfaces of the

SC/FM hybrid structures shown in Fig. 7.1. All measurements are performed along

a diameter 0.85 mm above the external surface of the whole hybrid structure. R =

8.25 mm is the radius of the SC/FM assembly. (b) Modelling results with the

H-formulation method for the same SC/FM structures.

Chapter 7 Investigation of more complex structures 135

Table 7.2. Numerical values of the remanent magnetic flux density 𝐵z shown in Fig.

7.2. (a) 𝐵z∗ = 𝐵z(𝑟 = −1 mm) for the measurement results and (b) 𝐵z

c at the centre

for modelling. All values are taken 0.85 mm above the external surface of the

whole hybrid structure. The variations relative to the superconductor alone (SC)

configuration are presented for both measurement and modelling.

(a) Measurement (b) Modelling Superconductor side

Configuration 𝐵z∗ (mT) Relative variation 𝐵z

c (mT) Relative variation

SC 336 -- 304 -- D2 358 6.5% 327 7.6% IC 363 8.0% 328 7.9% RG 343 2.1% 314 3.3%

Ferromagnet side

Configuration 𝐵z∗ (mT) Relative variation 𝐵z

c (mT) Relative variation

SC 250 -- 304 -- D2 42.5 -83.0% 42.4 -86.1% IC 29.7 -88.1% 29.0 -90.5% RG 73.6 -70.6% 92.0 -69.7%

These flux distribution measurements are consistent with the results obtained in Chap. 6. The

zone above the ferromagnet is shielded from the flux trapped by the superconductor and on

the SC side, the magnetic flux is increased. For a given volume of ferromagnet, the shape also

plays an important role since the inverted cone (IC) enables a larger flux modification than the

disc (D2). The flux distribution on the ferromagnet side is strongly shape-dependent: when the

ferromagnetic material does not cover the centre of the surface, the magnetic flux density

arising from the supercurrent located under the uncovered zone is not shielded, leading to a

succession of minima and maxima. This result shows the possibility to modulate the induction

using a succession of holes and ferromagnetic sections. Such a “nearly sinusoidal” flux

distribution is desired in some applications, e.g. strong field undulators or wigglers to be used

in future synchrotron light sources and e-e linear colliders [163].

As in the previous chapters, the H-formulation of Maxwell’s magnetodynamics equations (see

Sect. 4.2.2) was used to model the magnetic flux distribution of the configurations investigated

experimentally. Fig. 7.2(b) shows the modelling results corresponding to the measurements

shown in Fig. 7.2(a), i.e. the profile of 𝐵z along a diameter, 0.85 mm above each surface. Table

7.2(b) gives the numerical values at 𝑟 = 0 and their variations relative to the superconductor

alone (SC) configuration. As presented in Chap. 5 and 6, the modelled curves shown in Fig.

7.2(b) exhibit mirror symmetry at r = 0 (centre of the bulk) since they are obtained from a 2D

axisymmetric model. In the absence of any ferromagnet (SC configuration), the results are

identical on both faces since the model assumes uniform, macroscopic 𝐽c(𝐵) properties for the

bulk superconductor and the modelled results are an intermediate between those measured

on each face, as expected intuitively.

136 Chapter 7 Investigation of more complex structures

As can be seen in Fig. 7.2, there is an excellent qualitative agreement between the

measurements and modelling results. Remarkably, the exact characteristics of the

experimental flux profiles can be reproduced accurately with the model. On the SC side, i.e.

opposite to the ferromagnetic section, the sequence of all modelled plots is the same as for

the experiment. Quantitatively, the relative variations of 𝐵z at the reference position, as

summarized in Table 7.2, are similar. These relative variations differ for a maximum of 1.2

percentage points (pp) on the top face and 3.1 pp on the bottom face. This agreement was

already verified for the disc shaped ferromagnets, the suitability of the model to predict the

influence of the ferromagnet on the flux surrounding the SC/FM structure is confirmed by

these new results.

7.2 FM/SC/FM configurations

This section is devoted to the study of FM/SC/FM structures in which the superconductor ESC2

is sandwiched between two ferromagnetic sections, one on each circular face. These

structures are assembled from one of the previously presented SC/FM structures on which a

ferromagnetic disc of approx. 1 mm or 2 mm is added.

7.2.1 FM/SC/FM structures with ferromagnetic discs

Two FM/SC/FM structures were assembled with discs machined in Supra50. The D1D1

structure is made of ESC2 enclosed between two discs with a thickness of approx. 1 mm

(0.99 mm and 0.97 mm precisely). The D2D2 structure is similar, but both discs have a

thickness of approx. 2 mm (1.90 mm and 1.92 mm). The cross sections of these structures are

shown in the right panel of Fig. 7.3.

Fig. 7.3 shows the Δ𝐵 hysteresis curves measured with the coil wound around the entire

height of the superconductor on the D1D1 and D2D2 structures. The same curves around the

SC alone and the D1 and D2 structures are shown for comparison. The respective volumes of

the ferromagnetic components and the corresponding remanent inductions of the curves are

summarized in Table 7.3. Each hysteresis loop exhibits features similar to those observed

previously. Each curve is a combination of diamagnetic and ferromagnetic behaviour, as is the

case for the configurations with only one ferromagnetic disc attached (D1 and D2). The slope

at the origin of the axes remains unchanged by the presence of the ferromagnet, although its

presence leads to a change of sign of Δ𝐵 as the applied field increases. The remanent

induction (measured at zero applied field) and Δ𝐵 under 3 T of applied flux density are also

increased by the presence of the ferromagnetic components, as listed in Table 7.3.

Chapter 7 Investigation of more complex structures 137

Fig. 7.3 Δ𝐵 hysteresis loops measured with the coil wound around the entire height

of the superconductor for the FM/SC/FM structures made with discs and presented

on the right hand side. The FM material is Supra50.

Table 7.3 Evolution of the average flux density 𝐵rem in the remanent state and Δ𝐵

under 3 T of applied flux density for the configurations D1, D2, D1D1 and D2D2 (see

Fig. 7.3). The volumes of the superconductor (SC) and the ferromagnets (FM) are

presented.

F/S hybrid Volume: SC+FM

(mm3)

𝐵rem (T) Increase

of 𝐵rem (%) Δ𝐵(3 T)

Increase of

Δ𝐵(3 T) (T)

SC only 1351 0.168 – -0.064 –

D1 1351+208 0.191 14 0.010 0.074

D2 1351+405 0.196 16 0.062 0.126

D1D1 1351+421 0.225 34 0.083 0.147

D2D2 1351+815 0.235 40 0.183 0.247

The results plotted in Fig. 7.3 show the influence of the volume of the ferromagnet. Under high

applied field (𝜇0𝐻app > 1 T), adding on the SC side a ferromagnetic disc similar to the one on

the other side increases Δ𝐵. For example, Δ𝐵(3 T) is increased by 0.126 T in the D2

configuration compared to the SC alone and it is increased by 0.247 T in the D2D2

configuration. This increase is approximatively the double for a total volume of ferromagnetic

material approximatively twice as big (815 mm3 against 405 mm3). A similar relation is

observed between the D1 and D1D1 configurations. Additionally, splitting a given volume of

ferromagnetic material in two equal parts with one on each side of the SC leads to a limited

improvement of Δ𝐵(3 T) in comparison with a structure where the FM is attached on one side

only. In the D2 configuration, Δ𝐵(3 T) is increased by 0.126 T using a FM volume of 405 mm3,

while in D1D1 Δ𝐵(3 T) is increased by 0.147 T by a FM volume of 421 mm3. At these high

138 Chapter 7 Investigation of more complex structures

applied fields, it was verified that the ferromagnet is saturated by the external applied field

and that the “addition rule” is applicable. Therefore, when the ferromagnet is saturated by the

external applied field, it appears that the volume of ferromagnetic material is important on the

volume average of Δ𝐵 inside the superconductor but that its repartition between both sides of

the superconductor is of reduced influence.

On the contrary, in the remanent state, a given volume of ferromagnetic material has more

effect on the trapped flux when it is split on both sides to form an FM/SC/FM structure rather

than concentrated on a single side in a SC/FM structure. The effect on 𝐵rem of a given volume

of ferromagnet is approx. twice better when split on both sides of the superconductor: D1D1

leads to an increase of 34 % of 𝐵rem with 421 mm3 of ferromagnetic material Supra50 split in

two discs while D2 only leads to an increase of 16 % with 405 mm3. These results are in

agreement with the observations made previously, the increase of 𝐵rem is less than

proportional to the volume of ferromagnet for a given configuration, but splitting this

ferromagnet in two discs will strongly increase 𝐵rem.

Fig. 7.4 shows the profile of the magnetic flux density measured over the D2D2 configuration.

The measurements over the D1, D2 and SC alone configurations are shown for comparison.

The profiles above both faces of D2D2 are plate-like shaped, as above the ferromagnet in the

D1 and D2 configurations. In Fig. 7.4 (right), 𝐵𝑧∗ = 48.2 mT is slightly higher for D2D2 than for

D2 (𝐵𝑧∗ = 42.5 mT). The shape and magnitude of the flux profiles above both faces of the D2D2

structure are similar: On D2D2, 𝐵𝑧∗ is 48.7 mT and 48.2 mT in the left and right panels of Fig.

7.4, respectively. In comparison, 𝐵𝑧∗ in the left panel is 34 % higher than in the right panel in

the absence of the ferromagnetic discs (SC configuration). Therefore, the asymmetry between

𝐵z∗ on each face of the superconductor is reduced by presence of the ferromagnetic discs.

Fig. 7.4 Profiles of the magnetic flux density 𝐵z above both surfaces of the SC/FM

hybrid structure D2D2 shown in Fig. 7.3. All measurements are performed along a

diameter 0.85 mm above the external surface of the whole hybrid structure. R =

8.25 mm is the radius of the SC/FM assembly. Mappings above the D1, D2 and SC

alone configurations are shown for comparison.

Chapter 7 Investigation of more complex structures 139

7.2.2 FM/SC/FM structures with different ferromagnet shapes

Additional FM/SC/FM structures were assembled with the non-disc shaped ferromagnetic

sections and an approximatively 1 mm thick ferromagnetic disc placed on the opposite face of

the structure. These are the RGD1 and ICD1 structures shown in the right hand side of Fig. 7.5.

Fig. 7.5 shows the Δ𝐵 hysteresis curves measured with the coil wound around the entire

height of the superconductor in the RGD1 and ICD1 configurations. The curves measured on

the SC alone, D1, and D1D1 structures are shown for reference and comparison. Table 7.4

shows the corresponding numerical values of 𝐵rem. These curves are entirely consistent with

those relative to the configurations presented previously. An increased volume of

ferromagnetic material means an increased Δ𝐵 under 3 T of applied field and in the remanent

state. However, the RGD1 combination shows a slightly lower effect on the curve than the

ICD1 configuration, even though they nearly have the same volume (622 and 626 mm3,

respectively). This is even more visible on the remanent flux density. 𝐵rem is increased by 27 %

in the RGD1 configuration and by 38 % in the ICD1 configuration. Interestingly, the D1D1

configuration reaches an increase of 34 % of 𝐵rem with a volume of ferromagnet of (only)

421 mm3 (32 % lower than in RGD1). In comparison with the D1, IC, and RG configurations, the

addition of the 1 mm disc on the SC side of the SC/FM structures has an enormous influence

on the remanent magnetization 𝐵rem. It should be emphasized that the addition rule was

observed for these configurations.

These results are fully compatible with the observations made above. The volume of the

ferromagnet plays an important role in increasing the measured Δ𝐵, especially at high applied

field. For a given ferromagnet volume, covering the entire surface of the superconductor

always leads to a larger increase of flux density. If the average remanent flux density 𝐵rem is

the relevant parameter, a ferromagnetic disc split into two thinner discs placed on both sides

yields a larger increase of 𝐵rem.

140 Chapter 7 Investigation of more complex structures

Fig. 7.5 Δ𝐵 hysteresis curves, measured with the coil wound around the entire

height of the SC. Configurations are shown on the right. SC, D1, and D1D1 are

shown for comparison.

Table 7.4 Evolution of the average remanent induction for the configurations D1,

IC, RG, D1D1, ICD1, and RGD1 (see Fig. 7.5). The volumes of the superconductor

(SC) and the ferromagnets (FM) are presented.

F/S hybrid Volume: SC+FM (mm3) 𝐵rem (T) Increase of 𝐵rem (%)

SC only 1351 0.168 --

D1 1351+208 0.191 14

IC 1351+418 0.196 16

RG 1351+414 0.189 12

D1D1 1351+421 0.225 34

ICD1 1351+626 0.232 38

RGD1 1351+622 0.214 27

7.2.3 FM/SC/FM structure made with Permimphy discs

The D2D2 structure was studied a second time with the bulk, large grain YBCO

superconducting pellet in between two approx. 1.90 mm thick ferromagnetic discs made of

Permimphy (𝜇0𝑀sat = 0.8 T) instead of Supra50 (𝜇0𝑀sat = 1.4 T). Fig. 7.6 shows the Δ𝐵

hysteresis curves measured with the coil wound around the middle section of the

superconductor on the D2D2 assembly with Permimphy discs (black). The curves measured on

the D1D1 and D2D2 made with Supra50 are shown in green and red respectively. Note that for

practical reasons, the coil wound around the middle section (and not that wound around the

Chapter 7 Investigation of more complex structures 141

whole thickness) of the superconductor is used here. The consistency between these curves

and those measured on Supra50 is found. Table 7.5 summarizes the values of 𝐵rem as well as

the volumes of ferromagnetic material in each configuration.

Fig. 7.6 Δ𝐵 hysteresis curves measured with the coil wound around the middle

section of the Superconductor for the D2D2 structure with Permimphy (black) and

Supra50 (red) and the D1D1 structure with Supra50 (green). The hysteresis loop for

SC alone is shown for reference. Cross sections are shown on the right.

Table 7.5 Evolution of the average remanent induction for the configurations D1D1

with Supra50, and D2D2 with Supra50 and Permimphy (see Fig. 7.5). 𝐵rem values

might be different from those shown above since these ones are measured with

the coil wound around the middle section of the superconductor. The volumes of

the superconductor (SC) and the ferromagnets (FM) are presented.

F/S hybrid Volume: SC+FM (mm3) 𝐵rem (T) Increase of 𝐵rem (%)

SC only 1351 0.168 --

D1D1 (Supra50) 1351+421 0.219 30

D2D2 (Supra50) 1351+815 0.231 38

D2D2 (Permimphy) 1351+809 0.221 32

Under high applied field, the curve measured on the D2D2 structure made with Permimphy

(thick discs, low 𝑀sat) is close to the one measured on the D1D1 structure made with Supra50

(thin discs, high 𝑀sat); and the D2D2 structure with Supra50 (thick discs, high 𝑀sat) shows a

larger increase of Δ𝐵. This proximity of curves measured on structures made with different

saturation magnetizations and different volumes of ferromagnetic materials confirms that a

lower saturation magnetization can be compensated with an increased thickness.

142 Chapter 7 Investigation of more complex structures

In the remanent state, however, the curve measured on the D2D2 structure with Permimphy

(thick discs, low 𝑀sat) is closer to the curve measured on D2D2 with Supra50 (thick discs, high

𝑀sat) than to that measured on D1D1 (thin discs, high 𝑀sat). Therefore, in the remanent state,

the most relevant parameter is the volume of ferromagnetic material and not its saturation

magnetization, at least for the present experimental configurations where the ferromagnetic

material is not fully saturated.

7.3 SC/FM structure with the GdBCO superconducting sample

This section is dedicated to the measurement results obtained with the second bulk, large

grain superconducting sample i.e. a GdBa2Cu3O7 (GdBCO) disc-shaped bulk superconductor

from Nippon Steel & Sumitomo Metal Corporation, hereafter called ESJc (see Fig. 7.7). This

particular sample was characterized independently in Chap. 5 (Sect. 5.2). We approximated

the critical current density at 𝐽𝑐 = 2.44 × 108 A/m2 at 77 K.

Fig. 7.7 Schematic representation of the SC/FM structure E1 made of ESJc with a

0.95 mm thick ferromagnetic disc attached to its bottom.

The GdBCO sample was measured in the magnetometer and in the rotator set-up of the PPMS,

both presented in Chap. 3. As previously, all the following characterizations were performed at

77 K. In the following experiments, the superconductor ESJc is combined with a FM disc

attached to it. The disc is 9 mm in diameter and 0.95 mm thick and forms with ESJc a SC/FM

structure hereafter called E1 and shown in Fig. 7.7.

7.3.1 Magnetic moment

The magnetometer described in Chap. 3 (Sect. 3.3) is used to measure the time-dependence of

the magnetic moment of the fully magnetized SC/FM “E1” assembly. The result is compared to

that obtained on the superconductor ESJc alone (cf. Chap. 5).

Fig. 7.8 shows the time evolution of the magnetic moment (green circles) of the E1 hybrid

structure as measured with the magnetometer. The structure was magnetized by a field-

cooled process under 625 mT in a separate electromagnet. The measurement cycle starts 428 s

after having switched off the magnetizing field and lasts 104 s (approximately 3 h). The same

curve measured on ESJc alone, and already presented in Fig. 5.23, is shown for comparison

Chapter 7 Investigation of more complex structures 143

(blue circles). The red dashed lines in Fig. 7.8 are obtained by linear regression on the data sets

in a logarithmic scale.

Since flux creep occurs during the time interval before the first measurement, the first

measurement point does not represent the maximum trapped magnetic moment. We use

equation 5.2 to determine the initial magnetic moment 𝑚0, i.e.:

𝑚(𝑡) = 𝑚0 (1 +𝑡

𝑡0)

1(1−𝑛)

(5.2)

where 𝑡0 is a characteristic time before the beginning of the magnetic relaxation [132]. A fit of

this relation to the experimental points gives 𝑚0 = 0.13 Am2 and 𝑛 = 32.1 for the SC/FM

structure E1. The values obtained for ESJc alone were 𝑚0 = 0.12 Am2 and 𝑛 = 32.9. The

presence of the ferromagnet is found to have little influence on the time relaxation of the

trapped flux and to increase the initial magnetic moment by 8 %. These results fully agree with

the previous observations obtained from the average flux density in SC/FM hybrid structures

made with a ferromagnetic disc.

Fig. 7.8 Time evolution of the magnetic moment 𝑚(𝑡) measured on the SC/FM

structure E1 with the magnetometer (green circles). The measurement on ESJc

alone is shown for comparison (blue circles). The red dashed lines are obtained by

linear regression on the data sets in a logarithmic scale.

144 Chapter 7 Investigation of more complex structures

7.3.2 “Crossed” magnetic fields

The E1 hybrid structure was also characterized using the rotator set-up of the PPMS presented

in Chap. 3. The Hall probe of this set-up is used first to measure the hysteresis cycle and flux

relaxation.

Fig. 7.9 shows the hysteresis cycle measured with the Hall probe of the rotator set-up on the

E1 structure (green). The same curve measured with ESJc alone is shown for comparison

(blue). These curves were measured at fixed angle, in an applied field parallel to the c-axis of

the superconducting sample. The structure was cooled at 77 K in zero field. The magnetic field

is swept initially up to 2 T and then cycled between +2 T and –2 T using a sweep rate of

3.33 mT/s. The penetration field 𝜇0𝐻p is found to be 0.5 T and is not significantly changed by

the presence of the ferromagnet. The Δ𝐵 hysteresis curve is tilted anticlockwise by the

addition of the ferromagnet. It shows the combination of a superconducting and

ferromagnetic behaviour. For an applied flux density above 0.5 T, the measured Δ𝐵 is higher in

presence of the ferromagnet. This trend is however inverted below 0.5 T of applied flux

density. The remanent flux density is 0.22 T for ESJc alone and 0.20 T for the SC/FM structure

(–9 %). It is worth noting that these measurements were carried out twice, and gave similar

results.

From these hysteresis curves, the magnetizing field of 1 T used in the following field-cooled

experiments with and without ferromagnet largely exceeds 𝐻p and allows full magnetization of

the sample to be achieved.

The anticlockwise tilt of the curve in presence of the ferromagnet was already observed with

the superconducting sample ESC2. At high applied field, a simple “addition rule” of the

ferromagnet and superconductor contributions is therefore likely but the addition was not

checked in this case. However, at low field, and specifically for the remanent magnetization,

the trend is opposite to the behaviour observed with the YBCO sample (ESC2) in Chap. 6. The

remanent flux density measured by the Hall probe on the SC side of the structure is decreased

by the presence of the FM on the opposite side. At the same time, the magnetometer

measured an increase of the trapped magnetic flux in the SC/FM structure. We believe that

this behaviour might result from the aspect ratio and superconducting properties of the

superconductor. While ESC2 has an aspect ratio of height over diameter of 6.32/16.5 = 0.38,

ESJc has a higher aspect ratio of 5.16/9.0 = 0.57. It seems from magnetic moment

measurements that in both cases the magnetic flux inside the assembly is increased by the

presence of the ferromagnet, and probably on the ferromagnet side. Due to the larger aspect

ratio of the EsJc sample, however, the flux lines on the SC side of the structure experience a

higher curvature and are likely to be moved away from the symmetry axis at a higher rate in

presence of the ferromagnet. On this symmetry axis, the gradient of 𝐵 is higher on the SC side

in presence of the ferromagnet than on the other side. This gradient is such that the flux

density is lowered at the Hall probe position. This phenomenon is in opposition with the idea

Chapter 7 Investigation of more complex structures 145

that an effect similar to the image theorem leads to an increase of the equivalent aspect ratio

of the superconductor when the ferromagnet is present. However, the flux profile above a

bulk superconductor of finite height with a high aspect ratio is usually flatter than the conical

profile predicted by the Bean model in an infinite sample. This is for example the case when

bulk superconductors are stacked on top of each other [164]. Additionally, flux lines are also

forced to be perpendicular to the ferromagnet in the air, which probably acts on the flux on

the SC side of the structure and, in turn, impacts the measured flux by the Hall probe. Several

effects are therefore combined. More investigations on large, bulk superconductors with

dissimilar aspect ratios are needed to refine the interpretation. In particular, a coil wound

around the entire height of the sample could confirm the average increase of 𝐵rem inside the

volume of the superconductor. Additional modelling of 𝐵(𝑧) on the symmetry axis could bring

useful information on this point. As these results were obtained last, these additional

investigations are suggested for further work.

Fig. 7.9 Δ𝐵 = 𝐵HP − 𝜇0𝐻app hysteresis curve as a function of the applied flux

density 𝜇0𝐻app parallel to the superconductor c-axis. 𝐵HP is the flux density

measured by the Hall probe of the rotator set-up in axial field. Measurements for

the superconductor ESJc alone (blue) and for the SC/FM structure E1 (green).

The relaxation of the magnetic flux density can be recorded by the Hall probe of the rotator

set-up, as shown in Fig. 7.10. For this measurement, the sample or SC/FM structure is

magnetized by field cooling (FC) under 1 T applied parallel to the c-axis. The field is turned

down at a sweep rate of 3.33 mT/s and the flux density is subsequently measured for 1530 s.

The measurements follow a straight line in a logarithmic scale after approximately 10 s, which

is in agreement with the assumed behaviour described by the power law (5.2). The slope of

the straight line in the logarithmic scale between 102 s and 103 s corresponds to an 𝑛 exponent

of 34.5 and 32.1 in equation (5.2) for the ESJc alone and the SC/FM structure E1, respectively.

146 Chapter 7 Investigation of more complex structures

These values obtained from the magnetic flux density against the sample are in good

agreement with the one found above from magnetic moment measurements. Several

additional flux creep measurements were carried out. From these results, it is not possible to

detect an influence of the ferromagnet on the flux relaxation in the bulk superconducting

sample.

The measurement results in Fig. 7.10 confirm that the trapped flux density measured by the

Hall probe is lower in presence of the ferromagnet. At the beginning of the measurements of

the flux relaxation, it is 0.221 T for the SC alone, and 0.215 T for the SC/FM structure. At the

end of the measurement run (i.e. at t=1530 s) it is 0.184 T for the SC, and 0.178 T for the

SC/FM structure.

During the measurement of the flux creep, the sample (or hybrid structure) was rotated by 90°

such that its c-axis is perpendicular to the field applied by the PPMS, this forms the so-called

“crossed” field configuration. Then, ten cycles of magnetic field are applied following a

triangular waveform at a sweep rate of 3.3 mT/s. Two experiments were carried out, one with

50 mT amplitude and another one with 100 mT, each on the SC alone and on the E1 structure.

Fig. 7.10 Measurement of the relaxation of the flux density trapped in the

superconductor ESJc in without (blue) and with (green) the ferromagnet. The flux is

measured by the rotator Hall probe on the superconductor face opposite to the

ferromagnet.

Fig. 7.11 shows a schematic representation of the rotated E1 configuration in an applied

magnetic flux density 𝐵app. 𝑀trapped is the direction of the initially trapped magnetization into

the superconductor. 𝐵app is the applied uniform magnetic flux density. In this position, the Hall

probe is used to measure the component 𝐵HP of the flux density that is in the same direction

as 𝑀trapped.

Chapter 7 Investigation of more complex structures 147

Fig. 7.11 Schematic representation of the section of the SC/FM hybrid structure E1.

𝑀trapped is the direction of the initially trapped magnetization into the

superconductor. 𝐵app is the applied uniform magnetic flux density. In this position,

the Hall probe measure the component 𝐵HP of the flux density that is in the same

direction as 𝑀trapped.

Fig. 7.12 (left) shows the flux density 𝐵HP measured by the Hall probe during 10 cycles of

crossed field with 50 mT amplitude for the SC alone (blue). The trapped flux inside the

superconductor is decreased at each cycle of field applied transversally. The first cycle has an

important effect on the trapped field (-11 %). The next one however is relatively less

detrimental to the remaining trapped field (-3.6 % over the cycle, -14 % since the original

trapped field). The same trend is observed during the following cycles. In Fig. 7.12, the first

crossed cycle applied to the E1 structure is shown in green. As measured above, the flux

density measured by the Hall probe is lower in presence of the ferromagnet than without

before applying any transverse field. Remarkably, the first cycle of crossed field only leads to a

decrease of the trapped magnetization by 3.8 %, the resulting 𝐵HP being higher than after the

first cycle applied to the SC alone.

Fig. 7.12 (left) Flux density 𝐵HP measured by the Hall probe in the configuration

shown in Fig. 7.12 for 10 cycles of applied tranverse field of amplitude 50 mT for

the SC alone (blue). The first cycle for the E1 structure is shown (green). (Right)

Normalized curves, 10 cycles are shown for the SC alone (blue) and the E1 structure

(green).

148 Chapter 7 Investigation of more complex structures

This behaviour is confirmed during the next crossed cycles. It is shown in Fig. 7.12 (right)

where the measurements are normalized over the starting trapped field to focus on the

observation on the relative variation resulting of transverse fields. The presence of the FM on

one side of the SC significantly reduces the attenuation of the available flux density due to

transverse applied fields. After ten cycles, the trapped field had collapsed by 22 % when the SC

is alone and by only 16 % when the ferromagnet is present on the other side.

Fig. 7.13 shows the evolution of 𝐵HP(𝐻app = 0) as a function of the number of full cycles of

transverse field that are applied to either the SC alone, or the E1 structure. These values are

normalized over the trapped field measured before the first cycle. The measurements cycles

with an amplitude of 50 mT are shown for the SC alone (blue) and the SC/FM structure (green).

The same experiment was carried out with an amplitude of 100 mT for the transverse cycles.

As previously, the measured trapped field is lower in presence of the ferromagnet before any

applied crossed field. The detrimental influence of transverse fields is however decreased

thanks to the presence of the ferromagnet. After the first cycle of crossed field, the measured

𝐵HP is higher with the ferromagnet than without. The relative decrease of 𝐵HP as a function of

the number of 100 mT transverse cycles is shown in Fig. 7.13 for the the SC alone (red) and the

SC/FM structure (yellow). The same trend as with the 50 mT cycles is observed, but the

reduction of 𝐵HP is higher with the 100 mT cycles.

Fig. 7.13 Flux density 𝐵HP measured by the Hall probe as a function of the number

of full transverse applied field cycles. Under cycles with an amplitude of 50 mT, the

measurements are in blue and green for the SC alone and E1 structure, respectively

and in red and yellow under an amplitude of 100 mT.

Chapter 7 Investigation of more complex structures 149

These results show that the trapped field is decreased by the applied crossed fields. The

influence of fields perpendicular to magnetization has already been observed on wires [165,

166], bulk materials [167, 91, 168, 169, 170], and stacked tapes [171]. Models have been

suggested to describe the influence of transverse fields [172, 173, 174]. In our case, the

heating of the superconductor due to the time-varying field can be neglected since the sweep

rate is slow [170] and the decay results mainly from the modification of the current

distribution inside the superconductor [91]. Additionally, the first cycle of transverse field has

been shown to have a relatively larger effect on the magnetization decay than the following

cycles [170], which is verified in our experiment. The presence of the ferromagnet slightly

affects the trapped field available in surface of the investigated superconductor. However, it

will strongly reduce the “magnetization collapse” caused by successive cycles of transverse

applied field.

7.4 Summary

In this chapter, we have studied the influence of the shape of the ferromagnetic sections used

in SC/FM hybrid structures. The involved ferromagnetic sections were machined into discs, a

ring and an inverted cone. Ferromagnetic sandwiches FM/SC/FM made of the superconductor

pellet ESC2 with a ferromagnetic section on each of its circular faces are then studied. These

investigations involved the measurement of (i) the Δ𝐵 hysteresis curves of the volume average

flux density given by the coil wound around the entire height of the superconductor and (ii) of

the flux density profiles measured with the Hall probe mapping experimental set-up.

The Δ𝐵 hysteresis curves of the volume average flux density measured of the new

configurations made with ESC2 show a combination of a diamagnetic and ferromagnetic

behaviour, independently of the shape of the ferromagnet. The influence of the ferromagnet

increases with its volume and better results are obtained when the ferromagnet covers the

entire surface of the superconductor for a given volume of ferromagnetic material.

Additionally, a similar volume of ferromagnetic material was shown to have more influence on

the average remanent flux density 𝐵rem if it is split on both faces of the superconductor to

form a FM/SC/FM structure instead of a SC/FM structure.

The additive behaviour of the superconductor and ferromagnet shown in Chap. 6 was verified

for the investigated configurations when the ferromagnet is fully saturated by the applied

field. In this case, the influence of the ferromagnet depends on its saturation magnetization

𝑀sat, on its volume and on its arrangement (SC/FM or FM/SC/FM structure). In the remanent

state, the increase of 𝐵rem inside the SC is related to the volume and arrangement of the

ferromagnet, the saturation magnetization having no measurable influence in the investigated

configurations. It is nevertheless likely that this saturation magnetization would have an

influence if the ferromagnet was fully saturated by the trapped field inside the

superconductor.

150 Chapter 7 Investigation of more complex structures

The influence of a ferromagnetic disc on the magnetic moment of the superconducting sample

ESJc was then studied with the magnetometer. The influence of transverse fields was also

investigated on the trapped magnetization of this sample, with and without the presence of

the ferromagnetic disc. The latter experiments were carried out with the rotator set-up for the

PPMS. The embedded Hall probe in this system was also used to measure Δ𝐵 hysteresis cycles

and time relaxation of the flux in surface of the SC and SC/FM configurations.

The measurements on ESJc showed that the magnetic moment of the investigated SC/FM

structure is increased compared to the SC alone configuration. However, the hysteresis and

flux relaxation measurements carried out with the Hall probe showed a decrease of the

trapped flux density available at the centre of the superconductor surface when a ferromagnet

is added on the other side. This observation is likely to be caused by the higher aspect ratio of

the ESJc sample. It is important here to note that this was observed on the Δ𝐵 hysteresis

curves measured in surface by the Hall probe and not over the volume of the superconductor

by a coil wound around its height.

Both magnetic moment and Hall probe measurements showed that the 𝑛-exponent

characteristic of the magnetic flux relaxation is not modified significantly by the presence of

the ferromagnet. The addition of the ferromagnetic disc on one side of the superconductor

reduced the collapse of the trapped flux density measured against the other face of this

superconductor when submitted to several cycles of magnetic field applied perpendicularly to

its c-axis. Even if the measured trapped field 𝐵HP was lower in the investigated SC/FM

structure before any transverse cycle was applied, the beneficial effect of the ferromagnet was

such that the SC/FM structure shows a higher 𝐵HP after the first applied full transverse cycle.

151

Chapter 8

Conclusions and outlook

Large grain, bulk superconductors are promising materials to replace conventional permanent

magnets in applications such as levitation and rotating machines. In these applications,

superconductors are, or might, be surrounded by ferromagnetic materials. The aim of this

thesis was to determine experimentally how the magnetic flux density of a large cylindrical

bulk superconductor is modified when placed in the vicinity of ferromagnetic components of

various sizes and shapes, machined out of well characterized, soft magnetic alloys of high

permeability. In all the investigated structures, the ferromagnets had the same diameter as

that of the superconductor. As large, bulk sample are usually too large to fit in commercial

magnetometers, several measurement and modelling methods were used and compared to

determine the properties of the materials individually and understand their magnetic

interaction during cycles of magnetic field. There are two main elements in which the present

work differs from previous literature studies on related topics: (i) the full magnetic hysteresis

cycle (i.e. not only the remanent state) is considered and (ii) experimental techniques that

probe both surface and volume DC magnetic properties are combined on the same large, bulk

superconductor.

Surface measurements were carried out using Hall probes that were either (a) stationary and

placed against the centre of the circular faces, or (b) moved above the surface of the fully

magnetized sample to scan the flux distribution. Volume measurements were carried out using

either the bespoke magnetometer for large bulk samples or one or several coils wound around

the sample. We have shown that sensing coils wrapped around the superconductor enable

“quasi” DC (i.e. low sweep rate) magnetic loops to be measured on large, bulk samples. We

have pointed out the quantitative differences between the measured hysteresis loop and the

‘‘true’’ DC magnetization loop predicted numerically or measured with the magnetometer for

large samples. The characterization method involving sensing coils was validated for

characterizing, in a non-destructive way, the volume magnetic properties of whole large

superconducting samples whose dimensions exceeds the maximum size of classical DC

magnetometers.

Numerical modelling was carried out to draw important conclusions about the magnetic flux

distribution in and around the assemblies and predict the behaviour for other constitutive laws

and geometries. Three 2D axisymmetric modelling method developed within the finite

element framework were used: Campbell’s equation, A-formulation and H-formulation of

152 Chapter 8. Conclusions and outlook

Maxwell’s magnetodynamics equations. The Brandt algorithm was used to compare the Δ𝐵

and “true” magnetization hysteresis curves of a superconductor. The good agreement

between the experimental and numerical results gives evidence that our models are able to

reproduce the magnetic properties of a superconducting disc.

The results obtained in this work allows one to give the following answers to the questions

raised in the introduction of this thesis.

(1) How does a ferromagnetic piece modify the magnetic behaviour of a large grain, bulk

superconductor when it is brought close to it and what are the relevant parameters

that influence this behaviour?

We first focus on the trapped field configuration. The ferromagnet acts as a magnetic short-

circuit and creates a low reluctance path that drives the flux lines directly towards the edges of

the superconducting magnet. The zone above the ferromagnet, therefore, is shielded from the

flux trapped inside the superconductor and such a ferromagnetic element should not be

placed in a zone where a large magnetic field in the air is desired.

Conversely, the flux density can often be enhanced on the face opposite to the ferromagnet

(superconductor side), which is clear advantage for practical applications. To a first

approximation, this increase can be understood through the image theorem. Adding the

ferromagnet is equivalent to doubling the height of the superconducting cylinder which

experiences a lower demagnetizing field. In practice, the real effect is not as significant as the

image theorem would predict, since the ferromagnet is not semi-infinite. This simple picture

allows one to understand that the aspect ratio of the superconductor is likely to be an

important parameter that influences the beneficial effect of the ferromagnet. In a

superconducting sample with a high aspect ratio, the presence of the ferromagnet was

observed to lead to a decrease of the trapped field on the superconductor side. It is therefore

suggested that a ferromagnet is valuable when combined with a thin superconducting pellet. In

all the studied cases, the increase of flux density on the face opposite to the ferromagnet was

found to be related to an increase of both the average trapped flux inside the superconductor

and of the magnetic moment of the whole SC/FM structure (including the ferromagnet).

The ferromagnet ability to divert magnetic flux lines depends on whether it is partially or fully

saturated. Three parameters have been identified that govern the ferromagnet to saturation in

the fully magnetized remanent sate: the ferromagnet thickness (d), its saturation

magnetization 𝑀sat and the flux produced by the superconductor; the latter being

proportional to the critical current density of this superconductor 𝐽c. For a given

superconductor, the thickness and saturation magnetization of the ferromagnetic material are

important and play somewhat similar roles. The higher the saturation magnetization and/or

the thicker the ferromagnetic material, the larger the shielding effects on the ferromagnet side

and the trapped field on the superconductor face for the investigated sample.

Chapter 8. Conclusions and outlook 153

For a ferromagnetic disc with the same diameter 𝑎 as the superconductor, the saturation

regime can be roughly estimated from a simple equation. We have derived a simplified

analytical expression based on magnetic flux conservation to roughly estimate the most

suitable ferromagnet thickness 𝑑∗ ≈ 𝐽c0 𝑎2 6𝑀sat⁄ , below which nearly full saturation occurs

and above which weak thickness-dependence is observed. An increase of the ferromagnet

thickness well beyond 𝑑∗ is of low interest for practical purposes.

Although the magnitude of the current is obviously an important parameter in investigating

the behaviour of SC/FM structures, its particular 𝐽c(𝐵) dependence, however, was found not

to be a crucial parameter affecting the remanent state properties of the modelled hybrid

configurations. In addition, the time relaxation of the trapped magnetization was found not to

be influenced by the presence of the ferromagnet, within measurement uncertainties.

The ferromagnet also influences the magnetic flux density when the assembly is subjected to

an external magnetic field. The penetration of the magnetic flux inside the superconductor is

delayed in the vicinity of the ferromagnet. The Δ𝐵 hysteresis loops from sensing coils wound

around the superconductor show a combination of diamagnetic and ferromagnetic behaviours,

with the ferromagnet having a larger effect on the sections of the superconductor in closer

proximity to its position. This behaviour is independent of the shape of the ferromagnet. The

hysteresis cycles of the superconductor and the ferromagnet simply superimpose at high

applied field (i.e. exceeding the apparent saturation of the ferromagnet). Below this level of

applied field, the ferromagnet is not fully saturated and its interaction with the

superconductor is closer to the one observed in the remanent state.

(2) Is it possible to deduce some general design rules to investigate the significance of

ferromagnet effects?

As far as the volume magnetic flux through the superconductor is concerned, the effect of the

ferromagnet is found to be beneficial to the whole trapped flux. At the surface of the

superconductor, however, either an increase or a decrease of the available trapped flux has

been observed when a ferromagnet of the same diameter as the superconductor is placed on

the opposite face of this superconductor. This strongly suggests that the added value of the

ferromagnet is more significant when combined to thin superconducting disks. In view of

future work, a ferromagnet larger than the superconductor should be investigated as it gets

closer to the image theorem hypotheses. Similarly, it would be worth studying

superconductors with different aspect ratios. Some interesting conclusions will probably

emerge concerning the aspect ratio below which the combination with ferromagnetic material

proves to be of definite interest.

For a given volume of ferromagnetic material, the shape of an unsaturated ferromagnet is not

of prime importance provided that the ferromagnet covers the entire surface of the

superconductor. A similar volume of ferromagnetic material was shown to have more

154 Chapter 8. Conclusions and outlook

influence on the average remanent volume flux density 𝐵rem if it is split on both faces of the

superconductor to form a FM/SC/FM structure instead of a SC/FM structure.

If the volume of ferromagnetic material can be varied, a thicker ferromagnet has a more

significant effect on the whole hysteresis loop of the superconductor, but this effect varies less

than proportionally to the increase in the ferromagnet volume. The 𝑑∗ value introduced in this

work can be used to easily estimate the significance of the ferromagnet effects during a design

process. Ferromagnets with the highest saturation magnetization possible are always

preferred, but the intrinsic (physical) limitation of the saturation magnetization of classical

ferromagnets can be overcome by using thicker ferromagnetic discs.

If the ferromagnet is brought to saturation, the simple “additive” behaviour of the

superconductor and ferromagnet individual magnetic curves can be used effectively for

predicting the magnetic behaviour of larger or more complex hybrid structures involving

ferromagnets and bulk superconductors. This behaviour can be used to modulate the magnetic

field by the addition of ferromagnets.

(3) Is it possible to shape the trapped flux profile available above the superconductor

using ferromagnets?

By investigating the flux density above ferromagnetic pieces of various shapes, it was shown

that a succession of holes and ferromagnetic sections could be used to modulate the flux

density. Any increase of flux density at the edge of the superconductor, not surprisingly, occurs

at the expense of reducing significantly the central flux density. A “nearly sinusoidal” flux

distribution is desired in some applications, e.g. strong field undulators or wigglers to be used

in future synchrotron light sources and e-e linear colliders. Going further, one can imagine

designing a flat profile of the magnetic induction in a particular direction and a high gradient in

the perpendicular direction, which is of great interest for levitation tracks.

(4) How do the ferromagnet sections behave once they are saturated by the flux lines

produced by the superconductor?

The answer to this question is perhaps the most relevant result of this work. A ferromagnet of

given thickness below its full saturation will drive the majority of flux lines radially towards its

sides, therefore shielding a large part of the magnetic flux density 𝐵z. If the trapped field is

increased, the amount of diverted/shielded flux will not change after the full saturation of the

ferromagnet. Therefore, the effect of the ferromagnet will become relatively less important as

the generated flux increases. The field produced by the superconductor, however, is shown to

be enhanced by the ferromagnet, even though the generated field outside the assembly is

much larger (twice in the present investigation) than the saturation magnetization of the

ferromagnet. This occurs, for example, when the temperature of the superconductor is

lowered from 77 K to 20 K. Conversely, thin ferromagnetic discs can be driven to full saturation

Chapter 8. Conclusions and outlook 155

even though the outer magnetic flux density is lower than the saturation magnetization of the

ferromagnet. Hybrid structures are therefore relevant even when the trapped flux density in

the superconductor exceeds ≈ 2T, i.e. the typical saturation magnetization of ferromagnetic

materials.

(5) Are there other advantages of using ferromagnets in the vicinity of bulk

superconductors?

Two additional advantages of ferromagnets were identified in this work. First, the ferromagnet

was shown to be beneficial in improving the field gradient – and therefore the magnetic

levitation force – outside the superconductor. Second, the addition of the ferromagnetic disc

on one side of the superconductor reduces the collapse of the trapped flux density when

submitted to several cycles of magnetic field applied perpendicularly to its remanent

magnetization (parallel to the c-axis), i.e. in the so-called “crossed-field configuration”. Even if

the measured trapped flux density was lower in the investigated SC/FM structure before any

transverse cycle was applied, the beneficial effect of the ferromagnet was such that the SC/FM

structure exhibited a higher flux density after the first applied transverse cycle.

The reduced collapse of the trapped field in presence of crossed fields is of great interest for

rotating machines where the trapped field magnets are subjected to alternating fields. The

addition of a ferromagnetic disc on one side of a superconducting pellet has therefore a

double impact: the flux density is either increased or, in worst cases, slightly reduced but

under the application of transverse field cycles the overall effect of the ferromagnetic section

is beneficial to the trapped flux. This phenomenon was investigated in the present work when

a small number of large amplitudes cycles are applied. In a real machine, however, the

superconductor / ferromagnet structure is likely to be subjected to a rather high number of

cycles (during the machine operation) of relatively small amplitude, i.e. much smaller than the

full-penetration field. It should be of interest to investigate the corresponding decrease of

magnetization — and the beneficial effect of the ferromagnet — in such conditions. The

challenge is both experimental (measurement of relative decays that are comparable to the

usual magnetic flux relaxation effects) and numerical (3D modelling would be required).

For rotating machine applications, the idea of encircling the cylindrical superconductor with a

ferromagnetic ring arises. A ferromagnetic sheath around the superconductor could possibly

be detrimental to the magnetization of the trapped field magnet but might be useful to reduce

its collapse under transverse fields. These ferromagnetic sheaths would also act as a

mechanical reinforcement ring. Traditional ferromagnetic steels can be used since the coercive

field has not be proven detrimental under 1000 A/m. A variant of this configuration is the

superconductor set in a “cup”, i.e. with ferromagnetic material on all its faces except the one

where high magnetic flux density is required.

156 Chapter 8. Conclusions and outlook

In summary, the results presented in this thesis are believed to help in designing applications

where large grain, bulk superconductors can replace efficiently permanent magnets. These

results aid significantly the understanding of the contribution of ferromagnetic sections to the

trapped field produced by bulk high temperature superconductors, which is fundamental to

realizing practical applications of these technologically important materials.

157

Publications

Papers in international journals

M. P. Philippe, J.-F. Fagnard, S. Kirsch, Z. Xu, A. Dennis, Y.-H. Shi, D. Cardwell,

B. Vanderheyden, and P. Vanderbemden, “Magnetic characterisation of large grain,

bulk Y-Ba-Cu-O superconductor-soft ferromagnetic alloy hybrid structures,” Physica C,

vol. 502, pp. 20–30, 2014.

R. Egan, M. P. Philippe, L. Wera, J. F. Fagnard, B. Vanderheyden, A. Dennis, Y. Shi,

D. A. Cardwell, and P. Vanderbemden, “A flux extraction device to measure the

magnetic moment of large samples; application to bulk superconductors,” Rev. Sci.

Instrum., vol. 86, no. 2, p. 025107, 2015.

M. Haghgoo, A. A. Yousefi, M. J. Z. Mehr, A. F. Léonard, M. P. Philippe, P. Compère,

A. Léonard, and N. Job, “Correlation between morphology and electrical conductivity

of dried and carbonized multi-walled carbon nanotube/resorcinol–formaldehyde

xerogel composites,” J. Mater. Sci., 2015, online first, DOI:10.1007/s10853-015-9148-0.

M. P. Philippe, M. D. Ainslie, L. Wéra, J.-F. Fagnard, A. R. Dennis, Y.-H. Shi,

D. A. Cardwell, B. Vanderheyden and P. Vanderbemden, “Influence of soft

ferromagnetic sections on the magnetic flux density profile of a large grain, bulk

Y-Ba-Cu-O superconductor,” submitted to Superconductor Science and Technology.

Participation at conferences

M. P. Philippe, J.-F. Fagnard, A. Dennis, Y.-H. Shi, D. A. Cardwell, B. Vanderheyden, and

P. Vanderbemden, “Magnetic behaviour of soft ferromagnetic alloys attached to bulk

(RE)BCO superconductors,” ASC 2012: Applied Superconductivity Conference, Portland,

Oregon, October 7–12, 2012; Poster presented by M. P. Philippe.

M. P. Philippe, J.-F. Fagnard, A. R. Dennis, Y.-H. Shi, D. A. Cardwell, B. Vanderheyden,

and P. Vanderbemden, “Magnetic hysteresis cycle and remnant field distribution of

bulk high temperature superconductor / ferromagnet hybrids,” EUCAS 2013: European

Conference on Applied Superconductivity, Genova, Italy, September 15–19, 2013.

Poster presented by M. P. Philippe.

158

M. P. Philippe, J.-F. Fagnard, L. Wéra, A. Dennis, Y. Shi, D. A. Cardwell,

B. Vanderheyden, and P. Vanderbemden, “Measurement of the magnetic hysteresis

cycle and remanent field distribution of bulk high temperature superconductor /

ferromagnet hybrids,” 2013 BEMEKO workshop on measurement, Liège, Belgium,

November 7, 2013. Oral presentation by M. P. Philippe.

M. P. Philippe, J.-F. Fagnard, L.Wéra, A. Dennis, Y.-H. Shi, D. A. Cardwell,

B. Vanderheyden, and P. Vanderbemden, “Magnetic flux distribution and hysteresis

properties of bulk high temperature ferromagnet / superconductor / ferromagnet

hybrid structures,” ASC 2014: Applied Superconductivity Conference, Charlotte, North

Carolina, August 10–15, 2014; Poster presented by M. P. Philippe.

159

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