Table Of ContentSpringer Series in Solid-State Sciences 198
Teruo Matsushita
Flux Pinning in
Superconductors
Third Edition
Springer Series in Solid-State Sciences
Volume 198
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Teruo Matsushita
Flux Pinning
in Superconductors
Third Edition
123
Teruo Matsushita
Faculty of Computer Science andSystems
Engineering
Kyushu Institute of Technology
Iizuka, Japan
ISSN 0171-1873 ISSN 2197-4179 (electronic)
SpringerSeries inSolid-State Sciences
ISBN978-3-030-94638-8 ISBN978-3-030-94639-5 (eBook)
https://doi.org/10.1007/978-3-030-94639-5
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Preface to the Third Edition
Seven years have passed since publishing the second edition. The author has
experiencedvariousimportantdevelopmentsinthisperiodandthinksthatitisnow
important to enrich the content of the second edition of this book based on these
results.
One such development is the unification of the pinning theory. Up to now, the
microscopic summation theory gives the macroscopic pinning force density as a
function of the elementary pinning force and number density of pinning centers,
while the macroscopic critical state model assumes that the pinning loss power
density is given by the product of the macroscopic pinning force density and the
meanvelocityoffluxlines.Noproofhasbeenpresentedontheconsistencybetween
the two theoretical approaches. The author has calculated the pinning loss power
densityusingthetimeaverage,whichisequivalenttothestatisticalaverageusedin
thesummationtheory,andprovedthatitisequaltotheproductofthepinningforce
densityderivedbythesummationtheoryandthemeanvelocityoffluxlines.Since
thecriticalstatemodelwasalreadyprovedtheoreticallybyusingfirstprinciples,this
means that the microscopic summation theory and the macroscopic critical state
theoryareunifiedintoasinglepinningtheory.Thus,anewchapterisestablishedon
thepinningtheorywiththecriticalstatetheoryandtheprincipleofminimumenergy
dissipation,whichwere given in theAppendix in thepreviousedition.
Another point is a recent trend involving observations of very high critical
currentdensitiesinrare-earthbariumcopperoxide(REBCO)andMgB thinfilms,
2
attained by improved fabrication technology, and greater interest in the attainable
value of the critical current density in comparison with the maximum supercon-
ducting current density, i.e., the depairing current density. The author has investi-
gatedthemaximumcriticalcurrentdensitybythefluxpinningmechanismusingthe
Ginzburg-Landau theory. The obtained result exceeded the target of previous
researchers, Tinkham’s depairing currentdensity. Originally, theauthor mentioned
in the first edition that the depairing does not occur under the conditions in which
Tinkhamderivedhisresult.Thismeansthatthewell-knownvalueofthedepairing
current density was underestimated. Then, the author investigated the depairing
currentdensityusingtheGinzburg-Landautheoryandobtainedavaluejusttwiceas
v
vi PrefacetotheThirdEdition
highasTinkham’sresult.Using theseresults,theupperlimitofthecritical current
densityandthepresentachievedconditionsarediscussedinthisbookfor REBCO
and MgB thin films.
2
Oneofthemethods thattheauthor usesistheprincipleofvirtualdisplacement.
ThismethodwasusedinthederivationoftheLorentzforcetocompletethecritical
state theory and in the derivation of the force-free torque in the longitudinal
magneticfieldeffect.Itlooksasifanelectromagneticenergydisappearsorappears
during the process of virtual displacement. This does not mean that the law of
energy conservation has been broken, however. Variation to or from thermody-
namic energy occurs in these cases. The corresponding thermodynamic energy is
the pinning energy. This suggests that the pinning interaction determines the situ-
ation,includingthecriticalcurrentdensity.Thismethodisusefulforidentifyingthe
key mechanism that governs the phenomenon, and it plays a conclusive role in
disproving that flux cutting events play a part in the longitudinal magnetic field
effect.Thisisbecausethefluxcuttingisapuremagneticinteraction,andhence,the
law of energy conservation would be broken. This discussion is introduced in the
chapter on the longitudinal magnetic field effect.
The effect offlux creep is significant in high-temperature superconductors with
largeanisotropy.ThistrendisgenerallyarguedusingtheGinzburgnumber.Onthe
otherhand,inthepreviousedition,theauthorshowedthatthenumberoffluxlines
inthefluxbundleplaysthedominantroleinthefluxcreepanddidnotmentionthe
associationbetweenthefluxlinenumberandtheGinzburgnumber.Inthisedition,
the relationship between the two numbers is argued in the Appendix, and it is
shownthatthereisasimilaritybetweentheminthedependenceontheanisotropy,
criticaltemperature,thermodynamiccriticalfield,andcoherencelength.Inaddition,
the flux line number also depends on other factors such as temperature, magnetic
field, electric field, and specimen size, which are not included in the Ginzburg
number. For this reason, the theoretical method based on the flux line number is
more useful in the analysis offlux creep phenomena.
Finally,theauthor hopes these revisions areuseful for researchers who workin
the field of applied superconductivity. He would like to thank Dr. T.M. Silver at
Wollongong University for correcting the English in this book.
Iizuka, Japan Teruo Matsushita
Preface to the Second Edition
ThecriticalcurrentdensityofsuperconductingRE-123coatedconductorshasbeen
improved significantly. Especially addition of artificial pinning centers has been
examined,andtheartificialpinningcentersarenowintroducedtosomecommercial
coated conductors. The introduction of pinning centers provides not only
strengtheningthepinningforcebutalsoenhancementoftheuppercriticalfielddue
totheelectronscatteringbythepinningcenters.Thelatterfactgreatlyimprovesthe
high-field performance of the superconductor through the enhancement of the
irreversibility field. Such information is newly added in Chap. 8.
In these several years the critical current density of MgB has been improved
2
appreciably, and this superconductor is now used for practical devices such as
superconducting magnets for MRI. The effect of packing factor of MgB on the
2
superconducting properties such as the critical current density and irreversibility
fieldcanalsobequantitativelydescribedusingthepercolationtheory.Inaddition,it
wasfoundthatthefluxpinningstrengthofgrainboundariesinMgB isstrongerthan
2
that in Nb Sn at 4.2 K. These factors are needed to understand the critical current
3
properties inMgB .Forthis reason thecontentof Chap. 9 on MgB isupdated.
2 2
Fifty years have passed since the appearance of the first paper by C.P. Bean on
the critical state model that is indispensable to describing the electromagnetic
phenomena in the superconductor. Recently the force-balance equation, on which
the critical state model is based, was theoretically derived from the first principle
and then, generalized to irreversible cases with the aid of the summation theory.
That is, the critical state model is no longer a phenomenological model, but a
rigoroustheoryonthecriticalstate.Sincetheirreversibilitycanbederivedfromthe
level of the first principle, this theory states that, if energy dissipation could not
occur,itwouldbecontradictorytothefirstlawofthermodynamics.Additionofthis
section in Appendix is also one of important revisions of this book.
Finally the author hopes that the new edition is useful especially for young
researchers who are involved in applications of superconductors.
Iizuka, Japan Teruo Matsushita
vii
Preface to the First Edition
Superconductivity is now a considerable focus of attention as one of the tech-
nologies which can prevent environmental destruction by allowing energy to be
used with high efficiency. The possibility of practical applications of supercon-
ductivity depends on the maximum current density which superconductors can
carry,thevalueoflosseswhichsuperconductorsconsume,themaximummagnetic
fieldstrengthinwhichsuperconductorscanbeused,etc. Thesefactors aredirectly
relatedtothefluxpinningofquantizedmagneticfluxlinesinsuperconductors.This
book extensively describes related subjects, from the fundamental physics offlux
pinning to electromagnetic phenomena caused by flux pinning events, which will
be useful for anyone who wants to understand applied superconductivity.
TheJapaneseeditionwaspublishedforthispurposein1994.Sincethen,therehas
been significant progress in the research and development of high-temperature
superconductors. In particular, the new superconductor MgB was discovered in
2
2001,followedbysteadyimprovementsinthesuperconductingpropertiesnecessary
for applications. On the other hand, there are no essential differences in the flux
pinningphenomenabetweenthesenewsuperconductorsandmetallicsuperconduc-
tors. Hence, the framework of the previous Japanese edition was kept unchanged,
whilenewdescriptionwasaddedonthesenewsuperconductorsintheEnglishedition.
In the following the content of each chapter is briefly introduced.
InChap.1variousfundamentalsuperconductingpropertieswhichdeterminethe
flux pinning and electromagnetic phenomena in type II superconductors are
described,basedontheGinzburg–Landautheory.Inparticular,itisshownthatthe
center of a quantized flux line must be in the normal state so that the Josephson
current does not diverge due to the singularity in the gradient of the phase of the
superconducting order parameter there. This causes a loss due to the motion of
normal electrons in the core that is driven by the electric field, which is induced
when flux lines are forced to move by the Lorentz force. At the same time such a
structure of the core contributes to the flux pinning event. The role of the kinetic
energy indetermination of theupper critical field isalso shown. This will help the
readers to understand the kinetic energy pinning mechanism for the artificial Nb
pinning centers introduced into Nb-Ti, which is discussed in Chap. 6.
ix
x PrefacetotheFirstEdition
InChap.2thecriticalstatemodel,whichisneededtounderstandtheirreversible
electromagnetic phenomena in superconductors, is described. The mechanism
of the irreversibility is introduced on the basis of the ohmic electric resistivity,
whichisinducedwhenafluxlineisdrivenbytheLorentzforce.Ontheotherhand,
the losses in superconductors are non-ohmic ones with a hysteretic nature. The
reason for this will also be discussed. The critical state model provides the rela-
tionship between the current density and the electric field strength, and the elec-
tromagneticphenomenainsuperconductorsaredescribedbytheMaxwellequations
coupledwiththisrelationship.Itisshownthatthecritical statemodelcandescribe
irreversible magnetizations and AC losses in superconductors. The effect of
superconductor diamagnetism will also be an important topic.
Various electromagnetic phenomena are introduced in Chap. 3. These include
geometricaleffectsanddynamicphenomenawhichwerenottreatedinChap.2.The
rectifying effect in the DC current-voltage characteristics in a superposed AC
magneticfield,fluxjumps,surfaceirreversibility,andDCsusceptibilityinavarying
temperaturearealsoincluded.Inaddition,itisshownthatanabnormalreductionin
lossesoccurs,deviatingfromthepredictionofthecriticalstatemodelwhenanAC
magnetic field is applied to a superconductor smaller than the pinning correlation
length called Campbell’s AC penetration depth. This is attributed to the reversible
motion offlux lines limited within pinning potential wells, being in contrast with
thehysteresislosswhichresultsfromthefluxmotioninvolvedindroppingintoand
jumpingoutofthepinningpotentialwells.Inhigh-temperaturesuperconductorsthe
superconductingcurrentsustainedbyfluxpinningappreciablydecayswithtimedue
tothethermal agitationoffluxlines.Thisphenomenon,whichiscalledfluxcreep,
is also discussed. In extreme cases the critical current density is reduced to zero at
somemagneticfieldcalledtheirreversibilityfield.Theprinciplesusedtodetermine
the irreversibility field are described, and the result is applied to high-temperature
superconductors in Chap. 8.
InChap.4variousphenomenawhichareobservedwhenthetransportcurrentis
applied to a long superconducting cylinder or tape in a longitudinal magnetic field
are introduced, and the force-free model, which assumes a current flow parallel to
the flux lines, is explained. Although this model insists that the force-free state is
intrinsically stable, the observed critical current density in a longitudinal magnetic
field depends on the flux pinning strength, similarly to the case in a transverse
magnetic field, indicating that the force-free state is unstable without the pinning
effect. From the energy increase caused by introducing a distortion due to the
parallelcurrenttothefluxlinelatticetherestoringtorqueisderived,andthecritical
currentdensityispredictedtobedeterminedbythebalancebetweenthistorqueand
themomentofpinningforces.Theresultantrotationalmotionoffluxlinesexplains
theobservedbreakinJosephson’sformulaontheinducedelectricfield.Apeculiar
helicalstructureoftheelectricfieldwithanegativeregionintheresistivestatecan
also be explained by the flux motion induced by the restoring torque.
Thecriticalcurrentdensityisakeyparameterwhichdeterminestheapplicability
of superconductors to various fields, and hence the measurement of this parameter
isveryimportant.InChap.5variousmeasurementmethodsarereviewed,including
