Bent Strips in External Magneti Fields
and With Applied Currents
Leonardo R.E. Cabral and J.Albino Aguiar
Departamento deFsia,Universidade FederaldePernambuo,
50670-901Reife,Brasil
Reeivedon28February,2002
Current and magneti eld proles of bent strips in external magneti elds and with applied
urrentsarenumeriallyalulated. Theatstripisorretlyobtainedasthelimitingaseofbent
stripswhenthebendangleiszero. TheMeissnerresponseandtheuxpenetrationarestudied,the
latterundertheonsiderationofhighritial urrentsandsmallbends. Thedependeneofthese
prolesonthereepexponent(byamaterialrelationE/(j=j) n
)isalsostudied. Inthepresene
ofanexternalmagnetieldtheuxpenetrationisdelayedinbentstripsomparedwithatstrips.
Intheaseofanappliedurrenttheuxpenetrationfromthestripedgeisalsodelayed,butaux
frontstartingfromtheornerisalsoobserved.
I Introdution
Theeletromagnetiresponseofsuperondutorsin
ex-ternal elds has been extensively studied, and speial
attention has been given to the role of the speimen
shape. The Meissner statean bestudied byLondon
theoryandiseasilyobtainedinspeimenswith
negligi-ble demagnetization fator(longbarsand ylindersin
parallel elds). However, itturns out tobe adiÆult
taskwhenspeimenswithlargeand/ornon-uniform
de-magnetization fatorsareonsideredin the
perpendi-ulareldgeometry(inthisase,theeldisapplied
per-pendiular tothe speimen largerdimensions). Inthe
limitwhere ismuhsmallerthan thesample
dimen-sions, analytial solutions are known for a few ases,
e.g.,theMeissnerresponseinperpendiulareldinat
thin strips[1, 2℄and disks [3,4℄, longbarswith
ellip-ti[5℄andretangularross-setion[6℄.
Theunderstandingofvortexpenetrationand
evolu-tioninsideatype-IIsuperondutorinthemixedstate,
anbeomeaverydiÆultproblem,duetothe
vortex-vortex and vortex-defets interations, besidesthe
in-uene of thermal utuations [7, 8℄. At nite
tem-perature vorties an be depinned from their
prefer-ential positions by thermal utuations. This
gener-ated motion of vorties, known as ux reep, indues
aneletrield,E,relatedtotheurrentdensity, j,by
E(j)=E
exp( U(j)=k
B
T),whereU(j)isthepinning
potential[10,11℄.
Theritialstateapproahanbeusedtostudythe
wayhowmagnetiuxevolvesintothesuperondutor
withoutarryingoutthemorefundamentalLondonand
duting bars and ylinders in parallel eld geometry,
theritialstateiswellunderstood. However,few
ana-lytialresultsareknownforspeimensintheso-alled
perpendiulargeometry [1,2,9℄.
Although analytial results are important,
numer-ial methods an be employed in order to obtain the
Meissner response and the ux penetration into
su-perondutors of several shapes[12, 13℄. In this way
Brandt formulated a numerial method whih
alu-lates the urrent density, j, inside ylinders and long
bars(withniteaspetratio)inexternalmagnetield
orwith applied urrents [12, 14℄. The magneti eld
andthe magnetizationare alsoobtainedfrom the
al-ulatedj. AmaterialrelationE=E(j)alsoallowsthe
investigation of ux reep eets, whih areabsentin
theoriginalritialstateapproahandareanalytially
alulatedonlyinparallel eldgeometry.
Inthisworkwepresenttheeletromagnetiresponse
of superonduting bent strips in external magneti
elds and with applied urrents. The Meissner state
andtheuxpenetrationareobtainedapplyingBrandt's
method[12,14℄withtheappropriatehangeof
oordi-nates for bent strips. As a result points on the bent
striparedesribedbyauniqueoordinate,. The
nu-merial alulated urrent and magneti eld proles
are also dependent. The dependene of these
pro-lesandofthemagnetimomentonthebend angleis
studied. Theux frontomingfrom thestripedges is
delayed for bent strips ompared with at ones,both
inexternalmagneti elds orwith anapplied urrent.
II Numerial method
Thenumerialmethod startsfrom theintegral
formu-lation of Ampere's law, Faraday's law and a relation
E = E(j) borrowed from mirosopi theories about
ux reep and ux ow. The external magneti eld
is applied along they-axis. The transport urrent
re-sults from an uniform external eletrield alongthe
z-axis. Theurrentdensity,j, theeletrield, E,and
the vetor potential, A, are also assumed to be
par-allelto z-axis. This assumption isreasonablefor long
ilinderswithsymmetryaxisalongz-axisandarbitrary
(uniform)ross-setioninthex yplane. Thisleadsto
thefollowingequationofmotionfortheurrentdensity,
0 t j(r)= Z d 2 r 0 Q 1 (r;r 0 ) E(j)+ t A a ; (1)
where j = jz,^ r = (x;y), A
a
is the vetor potential
of the applied eld. For applied magneti or eletri
elds, A
a
= xB
a or A a = R dtE a
(t), respetively.
The inverse kernel, Q 1
(r;r 0
) an be obtained bythe
followingidentity, Z d 2 r 00 Q 1 (r 0 ;r 00 )Q(r 00
;r)=Æ(r r 0
); (2)
whereQ(r;r 0
)isthekernelrelatingthevetorpotential
atapointrwiththeurrentdensityatallpointsinside
thesuperondutor,
A(r)=A
a + 0 Z d 2 r 0 Q(r;r 0 )j(r 0 ): (3)
For long bars (long ylinders with retangular
ross-setion)this kernelisobtainedfrom theintegrationof
the3Dkernel1=(4jr r 0
j)alongthesymmetryaxis
z,furnishing Q bar (r;r 0 ) = 1 4 ln (x x 0 ) 2
+(y y 0 ) 2 + + 1 2
lnL; (4)
where L is the bar length along z. The Meissner
re-sponse may be found by formula 1 taking E = 0 (no
vortiespresent)andintegratingintime. Thisyields
0 j M (r)= Z d 2 r 0 Q 1 (r;r 0 )A a : (5)
We onsider a thin bent strip, shown in Fig. 1, a
thin strip sharply bent in the middle by an angle ,
with thiknessd and at a a. Thus the strip
ross-setion is \V" shapep. The following hange of
oordinates,
x = os sign()sen
y = jjsen + os; (6)
maybeusedintoequation4inordertoobtainthe
ker-nel forthin bent strips. Hene, Ampere's lawan be
written as
A()=A
a + 0 Z a a d 0 Q(; 0 )J s ( 0 ); (7) whereJ s ()= R d=2 d=2
dj(;)dj()isthesheet
ur-rent(theurrentdensityintegratedoverthethikness)
andthekernelQ(; 0
)is givenby
Q(; 0 ) = 1 4 ln ( 0 ) 2 os 2
+(jj j 0 j) 2 sen 2 + + 1 2
lnL: (8)
The vetor potential of the external elds is A
a =
osB
a
,foranexternalmagnetieldB
a ,orA
a =
R
dtE
a
foranapplied eletrield. Notiethat the
dependeneontheoordinate{alongthethikness{
isdroppedout,sineweonsiderthinbentstrips. This
leadstoanequationofmotionforthesheeturrent
0 t J s ()= Z a a d 0 Q 1 (; 0 ) h E J s D + t A a i ; (9)
andtothesheet urrentfortheMeissnerstate
0 J M s () = Z d 0 Q 1 (; 0 )A a (10)
where theinversekernelis foundsimilarly to eq.4for
thebar,butintegratingover,i.e.,
Z d 00 Q 1 ( 0 ; 00 )Q( 00
;)=Æ( 0
): (11)
Bydisretizing theoordinatealong thebent strip, ,
one an numerially solve the above equation for the
Meissnerresponseandintegratetheequationofmotion
forJ
s .
Figure1. Shematipitureofabentstrip. Thethikness
ismagniedinordertobettervisualization. Theoordinate
alongthe stripis andthe diretionperpendiular to is
denotedby.
In the alulations the following symmetry
y) or eletri (along z) elds, J
s
( ) = J
s () or
J
s
( )=J
s
(), respetively. Thus theaboveintegral
rangesfrom0toaallowingforgreaterauray. Hene,
thekernel(eq. 8)beomes,
Q B
(; 0
)= 1
4 ln
h
2
2
+ os
2
+ 2
sin 2
i
; (12)
foranappliedB
a =B
a ^ y and
Q I
(; 0
)= 1
4 ln
n
2
[ 2
+ os
2
+ 2
sin 2
℄
L 4
o
: (13)
forE
a =E
a ^
z(equivalenttoapplyatransporturrent),
where
=
0
. When an external and a
trans-porturrentarepresent,thekernelgivenbyequation8
should beused.
Themagneti eld is obtainedfrom theurl ofA,
whihreads
B=B^
=rA=^B
a
os ^ A
J
; (14)
where^isaunitvetorparalleltotheoordinate. For
the aseofanexternalmagnetield (noapplied
ur-rent)themagnetimomentisfoundfromthefollowing
formula
m= 1
2 Z
d 3
rrj= y^2Ldos Z
a
0 d J
s ():
(15)
In this equation it is onsidered the ontribution to
themagnetizationfromtheU-turningurrentsnearthe
strip ends, at jzj L=2[2℄. When anapplied eletri
eld ispresent,
I =2 Z
a
0 d J
s
(): (16)
givesthetotalurrentowingthroughthebentstrip.
We hose a grid of N = 100 points to represent
half of the bent strip (i.e., 0 a). It was also
adopted d = 2:510 4
a. In this way , J
s () and
B
() beome onedimensional arrays,and thekernel
Q(; 0
)beomesasquare matrix. Hene theintegrals
shownaboveareonvertedintosumsovertheindexof
these arrays and the inverse kernel Q 1
(; 0
) an be
omputedbymatrixinversion.
III Results
ThealulatedmagnetieldlinesfortheMeissner
re-sponsearedepitedin Figs.2and 3,theformerforan
externalmagnetieldB
a
k^yandthelatterfora
trans-porturrent(whihisthesameasapplyinganeletri
eld parallel to z).^ Bend angles =0 o
;30 o
;60 o
and
90 o
are shown. The magneti eld lines are obtained
fromtheequipoteniallinesofthevetorpotenial,A.
linestendtobeuniformasthedistanefrom thestrip
inrease. Forthetransport urrent,themagnetield
linesirulatearoundthestrips,beomingmorespaed
farther from the strip. In both ases it is observed
thatthemagnetieldlinesbendsharplyattheedges.
These lines are alsonot allowed to enter the stripsas
expetedin theMeissnerstate. Thease=90 o
or-retlyresultsinatwithwidtha[halfofthewidth(2a)
foraatstrip,when=0 o
℄. Interestingly,intheases
0 o
<<90 o
themagnetiuxisshielded intheinner
wedgeofthebentstrip.
Figure2. Magnetieldlinesforbentthinstripsinan
ex-ternal eld B
a
k^y. Four bend angles are shown: = 0 o
,
orrespondingtoaatstrip,(topleft);=30 o
(topright);
=60 o
(bottomleft)and=90 o
(bottomright).
Figure3. Thesame as Fig. 2,butfor atransporturrent
owingalongz.^
TheuxpenetrationispresentedintheFigs4and5
foranexternalmagnetield andfor atransport
de-ofthegures. Beauseof thesymmetryof theproles
(the magneti omponent, B
, and the sheet urrent,
J
s
,are,respetively,evenandoddfuntionsof foran
external magneti eld,while B
and J
s
are oddand
evenfuntionsof forthetransporturrentase),the
guresshow the >0branh only. TheBeanritial
state assumption (j
= onstant) [15℄ was onsidered
in allasestreatedin thisartile. Intheexternaleld
asetheinreasinganddereasingbranhesareshown
intheleftandrightoftheFig.4forn=101. The
on-tinuous (dashed) line presentsthe proles taken from
the=0 0
(=30 0
)ase.
0
0.5
1
1.2
−1
0
1
ζ
/ a
J
s
−2
0
2
B
η
0
0.5
1
1.2
0
1
ζ
/ a
0
1
2
Figure4. Sheeturrent,J
s
,(bottom)andtheperpendiular
omponent ofthe indution,B
,(top) dependeneonthe
oordinateparalleltothebentstrip,,forbentstripsin
ex-ternalmagnetields.Left: inreasingbranh.Right:
de-reasingbranh.Thethinanddashedorrespondto=0 o
and30 o
,respetively. Eahgroupoftwoproles,indiated
by thearrows,are obtained atthe sameexternaleld. Js
andB=0areinunitsofjD,andweusedd=2;510 4
a.
NotiethatJsandBare,respetively,oddandeven
fun-tionson.
Theprolesfor thesetwobendangles were alulated
for the same external elds, where the arrows
indi-atethediretion inwhihtheexternaleld inreases.
The divergene of themagneti eld at the stripedge
=a,whihhadalreadybeenobtainedforaatstrip
(=0 o
)[1,2,9℄,isalsoobservedwhen>0 o
. Asthe
externaleldinreases,magnetiuxpenetratesdeeper
into the strip, and the shielded region beomes
nar-rower. Thedierenebetweenthe=0 o
and=30 o
asesappearasadelayedpenetrationinthelaterase.
Infat,thepenetrationismoredelayedasthebend
an-gleinreases[16℄. Intheregiontheuxhaspenetrated,
the sheet urrent is almost onstant, as expeted for
the Bean ritial state (the small deviation from the
onstantvalueoursduetothenitereepexponent,
n=101). J
s
dereasestowardszero,intheshielded
re-gion,asonewalkstothemiddleof thestrip, although
the deays for = 0 o
and = 30 o
present dierent
behaviors. Foreah externaleld,there isarossover
separatingaregionwhereJ
s (0
o
)>J
s (30
o
)fromthe
re-gionwhereJ
s (0
o
)<J
s (30
o
). Inthedereasingbranh
thesamefeaturesobservedininreasingbranhare
ob-startstoleavethestripthroughtheedges =aand
aregionwheremagnetiuxistrappedappears.
Inthe transport urrentase the proleswere
ob-tained at inreasing time, keeping an uniform eletri
eld in the z^ onstant (Fig. 5). This makes the
to-tal urrent (whih is the applied urrent) owing in
the strip inrease with time. The thin dashed,
dash-dotted and thik lines represent the bend angles =
0 o
;15 o
;30 o
and 45 o
, respetively. Eah groupof four
proles, indiated by the arrows, are obtained at the
sametime(or, forthesametransport urrent). Inthe
left(right),n=101(n=11)wasadopted,
orrespond-ingto weak(strong)ux reep. Theinueneofreep
isevideniatedbythesmootherprolesandthelarger
ux diusion into the stripfor n=11 omparedwith
n = 101. For both ases the ux penetration from
the strip edges = a is more delayed as the bend
angle inreases. However, a new feature is observed:
ux penetrates from the middle of the bent strip, at
= 0, where the strip bends. This does not happen
foraatstripandoursin bentstripsbeauseofthe
nonzeroperpendiularmagnetieldomponentatthe
bend(seeFig.3). Thispenetrationismorepronouned
forlargervalues. Aftertheritialurrentisattained
thesheeturrentbeomesuniformthroughthestrip[2℄,
whihorrespondsto thefullux penetration. Thisis
depited in the last prole in the right of Fig. 5
(al-though it is not shown in the left of this gure, this
stageisalsopresentforn=101).
0
0.5
1
1.2
0
1
ζ
/ a
0
1
0
0.5
1
1.2
0
1
ζ
/ a
J
s
0
1
B
η
Figure5.Sheeturrent,J
s
,(bottom)andtheperpendiular
omponent of theindution, B
, (top)dependeneonthe
oordinateparalleltothebentstrip,,forbentstripswith
anuniform eletrield. Left: n=101. Right: n =11.
The thin, dashed, dash-dottedand thik lines orrespond
to = 0 o
;15 o
;30 o
and 45 o
, respetively. Eah groupof
four proles, indiated by the arrows, are obtained at the
sametime. J
s andB
=
0
areinunitsofj
D,andweused
d=2;510 4
a. Notiethat Js and B are, respetively,
evenandoddfuntionson.
IV Conlusions
superondut-or eletrields (thelaterorrespondingtoanapplied
urrent). A hange of oordinates is used in order to
transformatwodimensionalprobleminanone
dimen-sional problem. The sheet urrent J
s
dependene on
the oordinate along the strip, , wasalulated
em-ployinganumerialproedure[12,14℄. IntheMeissner
state the resulting magneti eld lines orretly show
the magnetishielding bentstrips should present. We
alsoomparedtheuxpenetrationforbentstripswith
thatforaatstrip. Forboththeexternalmagnetield
andtransporturrentasesthepenetrationofmagneti
ux from thestrips edges, =a, is delayedin bent
strips. However,ux penetrationfrom thestrip bend,
=0,isobservedwhenatransporturrentisapplied,
beingmorepronounedasinreases.
Aknowledgments
WewouldliketothankCleioC.deSouzaSilvafor
helpful disussions. Thisworkwaspartiallysupported
byBrazilianSieneAgeniesCNPqandFACEPE.
Referenes
[1℄ W.T.Norris,J.Phys.D3,489(1970).
[2℄ E.H.BrandtandM.Indenbom,Phys.Rev.B48,12893
(1993).
[3℄ P. N. Mikheenkoand Kuzovlev, PhysiaC 204, 229
(1993).
[4℄ J.Zhu,J.Mester,J.Lokhart,andJ.Turneaure,
Phys-iaC212,216(1993).
[5℄ J.R.Clem,R. P.Huebener, andD. E.Gallus,J. Low
Temp.Phys.12,449(1973).
[6℄ E.H. Brandtand G.P. Miktik,Phys.Rev.Lett. 85,
4164(2000).
[7℄ C.C.S.Silva,L.R.E.Cabral,andJ.A.Aguiar,Phys.
Rev.B63,4526 (2001).
[8℄ C.C.S.Silva,L.R.E.Cabral,andJ.A.Aguiar,Phys.
StatusSolidiA187,209(2001).
[9℄ E.Zeldov,J.R.Clem,M. MElfresh,andM. Darwin,
Phys.Rev.B49,9802(1994).
[10℄ G.Blatter,M.V.Feigel'man,V.B.Geshkenbein,A.I.
Larkin,andV.M.Vinokur,Rev.Mod.Phys.66,1125
(1994).
[11℄ E.H.Brandt,Rep.Prog.Phys. 58,1465(1995).
[12℄ E.H.Brandt,Phys.Rev.B54,4246(1996).
[13℄ R. Labush and T. B. Doyle, Physia C 290, 143
(1997).
[14℄ E.H.Brandt,Phys.Rev.B59,3369(1999).
[15℄ C.P.Bean,Phys.Rev.Lett.8,250(1962);idem,Rev.
Mod.Phys.36, 31(1964).
[16℄ L. R. E. Cabral and J. A. Aguiar, Physia C 341
