Vortex Congurations on Mesosopi Cylinders
with Square Cross Setion
Mauro M. Doria
InstitutodeFsia,UniversidadeFederaldoRiode Janeiro,
C.P.68528, 21945-970,RiodeJaneiroRJ,Brazil
and GilneyFigueiraZebende
Departamento deFsia,UniversidadeEstadual deFeiradeSantana,
Feirade Santana,BA,Brazil
Reeivedon28February,2002
Verylongandthinsuperondutingylinderswithsquarerosssetionarestudiedusingamodied
GinzburgLandautheorythatinorporatestheboundaryonditionsintothefreeenergyexpression.
Reent developments in the nanofabriation
teh-nologies brought new interestinto thestudy of
meso-sopi superondutors [1, 2℄. Consequently theorists
went bak to disuss many problems that ould have
beensolvedlong agoif the motivation existed before.
Theonnementofvortiesinsampleswithsizesofthe
orderoftheoherenelengthprodueinterestingeets
suh asthe onset of giant vorties and meta-stability,
properties notfoundin puresystemsonthebulk. For
this reason suh smallsale superondutorshave
be-omeofgeneralinterest [3,4,5℄.
We study here a type II superondutor with the
shapeofaverythinandlongylinderwithsquareross
setion. The applied eld is parallel to the ylinder
and the side of the square is just a few times larger
than theoherene length
0
. Beause theylinder is
verylongtheproblembeomestwo-dimensionalandit
isenoughto study aplaneperpendiularto the
ylin-der. WeassumethattheLondonpenetrationlengthis
muh larger than the side of the square, L, and
on-sequently, than
0
too. There is no shielding of the
applied eld and the magneti eld is onstant
every-where, being equal to the applied eld. To desribe
the equilibrium ongurations of this system we use
a modied Ginzburg-Landau theory whih
automati-allyinorporatestheappropriateboundaryonditions
intothetheoryandpresentsauniedviewofboth
nor-malandsuperondutingregionsindependentlyoftheir
shape [6℄. The presenttreatment of thesquare shape
rosssetionshowsthatthistheoryisalsoabletotreat
method treatssuperondutingandnormalregions on
equal footing through a step-like funtion (~x), zero
in thenormalandoneinthesuperondutingregions,
respetively, aording to the free energy density
ex-pansionbelow,
F = (1=V) Z
dvf(~x )j ~
D j 2
=2m
+ (~x)
0 [T T
℄j j
2
+j j 4
=2g; (1)
where ~
D(~=i) ~
r q
~
A =,and ~ r ~ A= ~ H.
Thewell-knownSaint-James{deGennes[7℄
bound-aryondition,
^ n
~
D =0: (2)
is automatially satised at the
normal-superondutinginterfaesbythistheory. Thesolution
in the normalregion is trivial sineEq.(1) redues to
F = (1=V) R
dvf j j 4
=2g, with the trivial solution
=0.
The solutions of the above modied
Ginzburg-Landautheoryinthesquaregeometryarefound
numer-iallyusingadisretizedversionofthetheory. Anarray
of N 2
points is introdued and onsequently the eld
(x;y) is replaed by (n
x ;n
y ),n
x
= 1;:::;N, and
n
y
=1;:::;N. Thedisretizedversiontheoryisgauge
invariant. The numerial method used to searh for
solutions is the Simulated Annealing method[8℄. The
Simulated Annealing method is a Monte Carlo
ther-malization proedure that ontainsa temperature
(0),largerthana. Wehoosetodooursimulationsfor
T = 0and this is equivalent to xing the parameters
R = 0 and 0 =a i
to onstantvalues.
Thestep-likefuntion(~x )thatdesribesthesquare
is,
(~x)=
2
1+exp[(2jxj=L) K
℄
2
1+exp[(2jyj=L) K
℄ (3)
where wetakeK=8.
Althoughthepresenttheoryhasnoparameterit
is possibleto simulate the limit! 1using the
fol-lowing trik. The magnetization is obtained through
theexpression,
~
M =(1=V) Z
dvf~x ~
J=2g; (4)
where ~
J = (q=2m)[ ~
D +:℄ is the
eletromag-netiurrent. Theaboveexpressiongivesthe
magneti-zation just from the spatial distribution of the order
parameter. However Ampere's law is not being
im-posed here and the Meissner regime does not exist in
thepresentstudy. It happensthat theabove
magneti-zationisonlyproportionaltothetruemagnetizationof
the limit!1. Theproportionality onstant, C,is
determined imposing thatB =H + C4M =0for
verysmallH values.
Let us disuss the results of our numerial
mini-mizationofthefreeenergy,desribedbyEq.(1),whose
resultsareshownin thegures.
Figure1showstwourvesonasingleplot,namely
thefreeenergyversustheappliedeldandthe
magne-tization versusthe applied eld forasquare withside
L =2
0
. Inthesmall applied eldregime the
magne-tization is approximatelylinear and this is assoiated
to theMeissnerphase. For inreasingeldthe
magne-tization reahes amaximumafter the Meissner region
tionsareobservedinthisase,eahouringatspeial
valuesoftheappliedeld. FromzeroeldtoH
A there
are no vortiesinside the system. H
B
marks the
en-traneoftherstvortex,H
C
oftheseond,H
D ofthe
third,andH
E
ofthefourth. ForeldslargerthanH
E
morevortiesenter thesquare, although amore
au-rateurvethanthepresentoneisneessaryto
hara-terizethesetransitions. Wehaveobservedthat
individ-ualvortiestendto ollapse into asinglegiant vortex
asthe eld inreases, until anew vortex enters when
theyagainareindividuallyseen. Forinstane,theeld
valueimediately before H
E
orresponds to just a
sin-glegiantvortexwithvortiitythree. Nextweshowthe
normalizedorderparameterforeahpointofourmesh,
taken with N = 60 and lattie parameter a =
0 =5.
Figs.4,5,6,7,8orrespondtotheappliedeldvalues
H A ,H B ,H C ,andH
D
,respetively. Eahdepressionof
theorder parameterorrespondto avortexinside the
system.
In onlusion we have studied here a long
super-onduitngylinderwithasquarerosssetionusinga
modiedGinzburg-Landautheorythatnaturally
inor-poratestheappropriateboundaryonditionsof
normal-superondutinginterfaes. Wenumeriallysolvea
dis-reteversionofthistheoryusingtheSimulated
Anneal-ingmethod. Weshowedthatthismethod givesagood
desription of the present square geometry. The
en-traneofvortiesprodue jumps inthe magnetization
showingthat theyindue rst ordertransitions in the
system, as shown for the ases of squares with sides
L=2
0
,L=3
0
,and L=8
0
studied here.
Aknowledgments
This work was supported in part by CNPq and
Figure1. Thefreeenergyandthemagnetizationversusthe appliedeld areshownherefor theaseof asquarewithside
L=20. Themagnetizationhasnodisontinuitiesindiatingthatnovortiespenetrateinthesystem.
Figure2. Thefreeenergyandthemagnetizationversusthe appliedeld areshownherefor theaseof asquarewithside
Figure 3. Thefreeenergyandthemagnetization versustheappliedeldare shownherefor thease ofasquarewithside
L=80. Thejumpsinthemagnetizationharaterizeentraneeldsforvorties. FromnoelduntilHA thereisnovortex
insidethesystemandatHB avortexentersthesystem. Theothereldsareassoiatedwithnewnumberofvortiesinside
thesystem,namelyHC twovorties,HD three,andHEfour.
Figure 4. The normalized orderparameterj(nx;nyj 2
is
shownhereforL=80 andeldvalueHA.
Figure 5. Thenormalized orderparameter j(nx;nyj 2
is
Figure 6. The normalized order parameterj(nx;nyj 2
is
shownhereforL=80 andeldvalueHC.
Figure 7. The normalized order parameterj(nx;nyj 2
is
shownhereforL=80 andeldvalueHD.
Figure 8. The normalized order parameterj(nx;nyj 2
is
shownhereforL=80 andeldvalueHE.
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