Stability and transition between vortex configurations in square mesoscopic samples with antidots

19 Pages·2003·2.03 MB·English

PHYSICALREVIEWB68,174521 ~2003!
Stability and transition between vortex configurations in square mesoscopic samples with antidots
G. R. Berdiyorov, B. J. Baelus, M. V. Milosˇevic´, and F. M. Peeters*
DepartementNatuurkunde,UniversiteitAntwerpen(CampusDrieEiken),Universiteitsplein1,B-2610Antwerpen,Belgium
~Received3July2003;published17November2003!
UsingthecompletenonlinearGinzburg-Landautheory,weinvestigatedthesuperconductingstateandphase
boundaries for mesoscopic square samples containing one to four submicron antidots in the presence of a
uniformperpendicularmagneticfield.Thepropertiesofthedifferentvortexstates,possibledegeneracies,and
thetransitionsbetweenthemarestudied.Duetotheinterplayofthedifferenttypesofsymmetry,aqualitative
difference in the nucleation of the superconducting state in samples with different number or arrangement of
antidotsisfound.Thesuperconducting/normalstateH-T phaseboundaryofthesestructuresrevealsanoscil-
latory behavior caused by the formation of different stable vortex configurations in these small clusters of
pinning centers ~antidots!. We analyze the stability of these configurations and compare the superconducting
phaseboundarywithexperimentalresults.
DOI:10.1103/PhysRevB.68.174521 PACSnumber~s!: 74.20.De,74.25.Dw,74.25.Ha
I.INTRODUCTION phase boundary using the complete Ginzburg-Landau
formalism.
Recent progress in microfabrication and measurement For the superconducting structures, containing a large
techniques made it possible to study the properties of super- number of submicron holes ~antidots!, it is not a simple ex-
conductingsampleswithsizescomparabletothepenetration ercise to take into account the vortex-vortex interaction due
depth l and the coherence length j. In this case the proper- to the very large number of interacting vortices. From this
ties of a superconductor are considerably influenced by con- point of view, a microdot with an antidot cluster (232, 3
finement effects. Therefore, for such mesoscopic samples 33, etc.! with a small number of interacting vortices is a
nucleation of the superconducting state depends strongly on very promising ‘‘intermediate’’system between a single su-
the boundary conditions imposed by the sample shape,1 i.e., perconductingloopwithafinitestripwidth3andasupercon-
ducting film with a large regular array of antidots. The re-
on the topology of the system.
duced number of interacting vortices simplifies the
Recent technical and theoretical advances have led to a
calculationsandtheresultsmaybeextrapolatedtotheanaly-
revivaloftheinterestinthemagneticpropertiesofsupercon-
sis of the vortex state in substantially larger antidot arrays.
ducting networks and artificially structured superconducting
In the last decade, several experimental studies11–13 were
films. Many different topologies have been studied experi-
published, where the vortex state of superconducting (2
mentally and theoretically, which can be classified into
32)-antidot clusters made of different kinds of supercon-
simplesingleloops,1–6multiloopstructures,7–15andlargein-
ducting material were investigated. In Ref. 12 the authors
finitenetworks.16–18Thosestructuresareconsidered,e.g.,for
studied the transport properties of a superconducting Pb/Cu
singlefluxquantumlogicapplicationsinanalogywithsemi- microdot with a (232)-antidot cluster, measuring the S/N
conducting 232 quantum dot systems.19 In a superconduct-
phase boundary, the magnetoresistance, the critical currents,
ing (232)-antidot cluster, the vortices play the role of the and the V(I) characteristics. They compared their experi-
electrons in the quantum dots and the antidots play the role mental results with calculations in the London limit of the
of the quantum dots.The advantage of such a system is that Ginzburg-Landau ~GL! theory and in the framework of the
the different vortex states can be studied on a macroscopic deGennes–Alexandermodel.Itwasshownthatvorticescan
level and they can even be visualized.20,21Also, these super- be pinned by the antidots forming a cluster and that the
conducting structures have attracted attention as potential ground-state configurations of the vortices are noticeably
new components for low-temperature electronics. modified by the current sent through the structure. The au-
The theories used to explain experimental results for the thors of Ref. 13 considered a 232 aluminum antidot cluster
H(T) boundaryofthesestructuresweremainlybasedonthe and a microsquare containing two submicron holes. It was
linearized Ginzburg-Landau theory, using either the London foundthattheS/N phaseboundaryT (H) ofthesestructures
c
limit,22,23 where the modulus of the order parameter is as- shows quite different behavior in low and high magnetic
sumed to be spatially constant, or the de Gennes–Alexander fields.
formalism,7,24allowinguC(x)u tovaryalong~butnotacross! Inthiswork,weinvestigatedsystematicallythesupercon-
the strands. ducting state of mesoscopic square samples in the presence
Fomin etal.5 studied the superconducting state in a nar- of a uniform perpendicular magnetic field for six different
row mesoscopic square loop and analyzed phase boundaries topologies~Fig.1!.Itiswellknownthatdifferentgeometries
on the basis of a self-consistent solution of the Ginzburg- of mesoscopic superconductors will favor different arrange-
Landau equations. Baelus and Peeters3 considered mesos- mentsofvorticesandwillmakecertainvortexconfigurations
copic superconducting disk structures containing a circular morestablethanothers.Here,weinvestigatehowthevortex
antidot and investigated the vortex structures and the H-T configuration and the critical parameters are influenced by
0163-1829/2003/68~17!/174521~19!/$20.00 68174521-1 ©2003TheAmericanPhysicalSociety
BERDIYOROV,BAELUS,MILOSˇEVIC´,ANDPEETERS PHYSICALREVIEWB68,174521 ~2003!
IV. Transitions between these vortex states are presented in
Sec. V. In Sec. VI we show the dependence of the free en-
ergy and magnetization on the sample parameters, such as
theGinzburg-Landauparameterandthesamplethickness.In
Sec. VII we present the superconducting/normal phase
boundaryforthe(232) antidotcluster.TheH-T diagramof
this structure is compared with experimental results. Our re-
sults are summarized in Sec. VIII.
II.THEORETICALFORMALISM
Inthepresentpaper,weconsiderthinflatsuperconducting
samples of different geometry which are immersed in an in-
FIG. 1. Model configurations: full microsquare ~the reference
sulating medium in the presence of a perpendicular uniform
sample!~a!,superconductingsquareswithone~b!,two~c,d!,three
magnetic field H . To solve this problem we follow the nu-
~e!, and ~f! four antidots. W denotes the size of the side of the mericalapproach0ofSchweigertandPeeters.27,28Forthinsu-
samples, W is the side of holes, and W is the distance between
i 0 perconductingsamples(d,j,l) itisallowedtoaveragethe
antidots.kistheGinzburg-Landauparameter,andthesamplethick-
GLequations over the sample thickness for superconductors
nessequalsd50.1j.
ofarbitrarygeometry.Usingdimensionlessvariablesandthe
the sample geometry. We studied ~see Fig. 1! a filled mi- London gauge divAW50 for the vector potential AW, we write
crosquare ~a!, a square with one ~b!, two ~c,d!, three ~e! and the system of GLequations29–31 in the following form
four~f!identicalantidots,whereeachantidotcaninprinciple
contain a different number of flux quanta. ~2i„W 2AW!2C5C~12uCu2!, ~1!
2D
Our theoretical analysis is based on a full self-consistent
numerical solution of the coupled nonlinear GL equations. d
As an example we took the Ginzburg-Landau parameter k 2D AW5 d~z!Wj , ~2!
3D k2 2D
50.28, whichistypicalforAlthindisks.25Noapriorishape
or arrangement of the vortex configuration is assumed. The
where
magnetic-field profile near and in the superconductor is ob-
tained self-consistently, and therefore the full demagnetiza- 1
tion effect is included in our approach. We calculated quan- Wj 5 ~C*„W C2C„W C*!2uCu2AW, ~3!
2D 2i 2D 2D
tities such as the free energy, the magnetization, the Cooper-
pair density, the total magnetic-field profile, and the current- isthedensityofthesuperconductingcurrent.Theindices2D
density distribution. Due to the interplay of the different and 3D refer to two- and three-dimensional operators, re-
kindsofsymmetry,thereexistsaqualitativedifferenceinthe spectively. The superconducting wave function satisfies the
nucleation of the superconducting state in samples with dif- boundary conditions (2i„W 2AW)Cu 50 at the sample sur-
fereWntencuamlcbuelarteodf hthoeleHs ~-aTntsiudpoetsr!c.onducting phase boundary, face and the vector poten2tiDal is givnen by AW521H0reWf far
awayfromthesuperconductor.Herethedistanceismeasured
which shows characteristic oscillations at specific values of
in units of the coherence length j, the vector potential in
the magnetic flux coming from the limited number of pos- c\/2ej, and the magnetic field in H 5c\/2ej25kA2H .
siblevortexconfigurations.Acomparisonbetweenthecalcu- c2 c
lated H-T phase diagram and the experimental data13 con- Thethinflatsuperconductorisplacedinthe(x,y) plane,and
the external magnetic field is directed along the z axis.
firms that these effects are due to fluxoid quantization and
To solve the system of Eqs. 1~a,b!, we generalized the
vortex pinning at the antidots. Fluxoid quantization around
approachofRef.28forcirculardiskstoflatsuperconductors
each antidot leads to characteristic minima in T (H), corre-
c with an arbitrary geometry.We apply a finite-difference rep-
sponding to the transitions between different vortex states,
known as Little-Parks oscillations.26 resentation for the order parameter and the vector potential
on a uniform (x,y) Cartesian space grid, with typically 128
In small systems vortices may overlap so strongly that it
3128gridpointsfortheareaofthesuperconductor,anduse
is more favorable to form one big giant vortex. In order to
thelinkvariableapproach33andaniterationprocedurebased
see clear multivortex configurations we chose sufficiently
on the Gauss-Seidel technique to find C. The vector poten-
large values for the sizes of the samples. A square sample
[email protected]~a!#. tial is obtained with the fast Fourier-transform technique
Thedimensionsoftheholesforallstructuresweretakenthe where we set AWuxu5R ,uyu5R 5H0(x,2y)/2 at the boundary
S S
same W 52.0j, and W 51.0j is the lateral distance be- of the square simulation region with width typically four
i 0
tween the antidots. The thickness of all samples is taken d times the width of the superconductor.
50.1j. Contrary to circular configurations such as disks28 for
The paper is organized as follows.The theoretical formu- nonaxiallysymmetricsystemsthereexistnoaxiallysymmet-
lationoftheproblemispresentedinSec.II.Differentvortex ricgiantvortexstatesandhencethesuperconductingstateis
configurations and their stability are studied in Secs. III and alwaysamixtureofdifferentangularharmonics.Thevortic-
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STABILITYANDTRANSITIONBETWEENVORTEX... PHYSICALREVIEWB68,174521 ~2003!
ity L of a particular superconducting sample can be calcu- lations are done for a fixed temperature. Notice that the
lated by considering the phase w of the order parameter Ginzburg-Landauparameterk5l/jistemperatureindepen-
alongaclosedloopattheboundaryofthesample,wherethe dent.
total phase difference is always Dw52pL. In nonaxially
symmetric systems three possible vortex states exist: ~i! a
III.FREEENERGY,MAGNETIZATIONANDSTABILITY
multivortex state that contains separate vortices, ~ii! a super-
OFTHEDIFFERENTVORTEXSTATES
conducting state that contains one giant vortex in the center,
and~iii!astatethatisamixtureofboth~agiantvortexinthe The free energy and magnetization of the mesoscopic su-
center which is surrounded by single vortices!. The giant perconducting samples give us plenty of information on the
vortexisnotnecessarilycircularsymmetricasinthecaseof physical processes in the superconductor, and therefore we
a circular disk, but it may be deformed due to the specific will first compare the free energy and magnetization for our
shape of the sample boundary. different superconducting samples.
To find the different vortex configurations, which include Figures 2~a!–2~f! show the free energy for the reference
themetastablestates,wesearchforthesteady-statesolutions sample ~full square! and for the square superconductor with
ofEqs.1~a,b!startingfromdifferentrandomlygeneratedini- one, two, three, and four holes, respectively as a function of
tial conditions. Then we increase/decrease slowly the mag- the applied magnetic field. The insets in all figures show an
netic field and recalculate each time the exact vortex struc- enlargement of the free energy of the states with large vor-
ture.Wedothisforeachvortexconfigurationinamagnetic- ticity. Open circles indicate continuous transitions and open
field range where the number of vortices stays the same. By squares indicate the S/N transition fields. In the reference
comparing the dimensionless Gibbs free energies of the dif- [email protected]~a!#vortexstatesuptoL511cannucleate.At
ferent vortex configurations lowermagnetic-fieldsstateswithvorticityL52, 3,4,5,and
E 6aremultivortexstates~seeRef.32!.Furtherincreaseofthe
F5V21 @2~AW2AW ! Wj 2uCu4#drW, ~4! magneticfieldleadstotheformationofthegiantvortexstate
0 (cid:149) 2d in the center of the sample.
V
The insertion of one hole in the sample changes the free
where integration is performed over the sample volume V, [email protected]~b!#.Inthiscaseall
and AW is the vector potential of the uniform magnetic field, ground-statetransitionsbetweendifferentvortexstatesoccur
0
we find the ground state, the metastable states, and the at lower magnetic fields. The L51 state is stable over a
magnetic-field range over which the different states are much larger magnetic-field range than the other states. No-
stable. The free energy will be expressed in units of tice also that the superconducting/normal transition field is
Hmec2Vas/u8rpe.oTfhtehediemxepnesllieodnlemsasgmneatgincefitiezladtiofrno,mwhthicehsiasmapdlier,ecist alanrdgeHr.0T/hHisc2fi5e3ld.2i1sfHor0/tHhec2s5up2e.r0c1onfdourctthoerrwefitehreancheolsea.mTphlee
defined as 60%increaseofthecriticalmagneticfieldiscompletelydue
to the narrow superconducting area in the upper right corner
^H&2H of the sample. It is well known that superconductivity is
M5 4p 0, ~5! enhanced34 in such corners. In both cases, only transitions
between successive L states are present ~i.e., L5n!L5n
where H is the applied magnetic field. ^H& is the magnetic 11 transitions!. The maximum vorticity that can be accom-
0
field averaged over the superconducting area of the sample, modated in the full square is L511, while for the sample
and HW5rot AW. We also calculated the magnetization averag- with a single hole, this is 19.The transitions between vortex
ingthefieldovertheW3W area~superconductorandholes!. statesaftertheL510statearecontinuousandthepositionof
the transitions at which the vorticity L changes by one unit
The temperature is indirectly included in the calculation,
through j, l, H , whose temperature dependence is given are indicated by the open dots. The continuous vortex tran-
c2
sitions are a consequence of the noncircular geometry of the
by
sample and are analogous to those found earlier in ringlike
structures with nonuniform width.3,4
j~T!5A j~0! , ~6! Figure 2~c! shows the free energy of the superconducting
u12T/T u sample with two antidots, where the antidots are located
c0
along the diagonal. In this case there exist vortex states with
l~0! vorticity up to L518 with a S/N transition field of
l~T!5Au12T/Tc0u, ~7! Han0ti/dHotc2s5am3.p2l2e.wChoinchtraisryaltmoothsteidperenvtiicoaulstotwthoactaosfetsh,ehesirnegwlee-
U U foundwithincreasingmagneticfieldDL52 transitionssuch
T as L52!L54, L56!L58, L512!L514, L514!L
Hc2~T!5Hc2~0! 12T , ~8! 516, and L516!L518. Notice that the last three transi-
c0 tions are continuous and for these transitions DL52; i.e.,
where T is the critical temperature at zero magnetic field. stateswithvorticityL513, 15,17donotbecometheground
c0
We will only explicitly insert the temperature dependence if state. The energies of all superconducting states are lower
we consider the H-T phase diagrams, while the other calcu- and the ground-state transitions occur at lower magnetic
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BERDIYOROV,BAELUS,MILOSˇEVIC´,ANDPEETERS PHYSICALREVIEWB68,174521 ~2003!
FIG. 2. The free energy as a function of the applied magnetic field for the reference sample ~a! and for the square with one ~b!, two
diagonal ~c!, two top ~d!, three ~e!, and four ~f! holes. The insets show the free energy for higher vorticity. The open circles indicate
continuoustransitionsbetweendifferentvortexstatesandopensquaresshowtheS/N statetransitionfields.
fields as compared to the previous sample. Notice also that highest of all the six considered structures. There is also a
the superconducting state with even vorticity is more stable clear enhancement of the stability for states with L54, 8,
than those with odd vorticity. 12, and 16.
In order to see the influence of the sample topology, we The maximal value of the vorticity L and the S/N
max
considered the superconductor with two antidots, when they transition fields for the five different topologies are summa-
are in the top half part of the sample @see Fig. 1~d!#. Figure rizedinTable.I.ThereisasubstantialincreaseofL when
max
2~d! shows the free energy of this structure. In this case the an antidot is introduced in the square sample. L is prac-
max
freeenergyofthestateswith2<L<6 ishigherthanthefree tically independent of the number of antidots. For the two-
energy of the sample with the two antidots located along the antidot sample L 518 is one unit smaller, which may be
max
diagonal. We also found that the transitions after the L59 the consequence of the fact that only even L can nucleate as
state are continuous. The maximal number of vortices is L a ground state in the high magnetic-field region. The S/N
519 and the S/N transition field is H /H 53.28 slightly critical field increases only by 3.4% when we go from the
0 c2
higher than in previous sample. Notice, that here DL51 be- one-tothefour-antidotsample.Thisindicatesthatthesizeof
tween the different vortex states which is a consequence of thenarrowestsuperconductingareainthesamplemainlyde-
the different symmetry of the sample as compared to that of termines the S/H transition field.
Fig. 1~c!. Figure 3 shows the magnetization of the different vortex
The free energy for the three-antidot sample is shown in states for the same samples as considered in Fig. 2, i.e., for
Fig. 2~e!. In this case we found only L5n!L5n11 tran- the full square ~a! and squares with one ~b!, two ~c,d!, three
sitions, except L50!L52 transition, and in the high
TABLEI. MaximumvorticityL andtheS/N transitionfield
magnetic-field region continuous transitions are found after max
forthefivedifferentsuperconductingstructuresdepictedinFig.1
the L511 state.
The most interesting case is the superconductor with four
L H /H
antidots.ThefreeenergyofthissampleisgiveninFig.2~f!. max c3 c2
Notice that each of the vortex states has a larger stability Fullsquare 11 2.01
region, the energies of the different superconducting states Oneantidot 19 3.21
are lower as compared to all previous cases, the transitions Twodiagonalantidots 18 3.22
betweendifferentLstatesoccuratlowermagneticfields;and Twotopantidots 19 3.28
all thermodynamic equilibrium transitions are discontinuous Threeantidots 19 3.30
withDL51. VortexstateswithvorticityuptoL519canbe Fourantidots 19 3.32
nucleated and the S/N transition field H /H 53.32 is the
0 c2
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STABILITYANDTRANSITIONBETWEENVORTEX... PHYSICALREVIEWB68,174521 ~2003!
FIG.3. Themagnetizationasafunctionoftheappliedmagneticfieldforthereferencesample~a!andforthesquarewithone~b!,two
diagonal~c!,twotop~d!,three~e!andfour~f!holes.Theverticallinesshowtheground-statetransitionsbetweendifferentvortexstates,and
opencirclesindicatecontinuoustransitionsbetweendifferentvortexstates.
~e!,andfour~f!holes.Themagnetizationisameasureofthe ductorswithone~b!,two~c,d!,andthree~e!holesthelowest
expelledfluxfromthesampleandiscalculatedafteraverag- magnetization is reached for the state where L equals the
ing the field only over the superconducting region excluding number of holes, i.e., in that case the largest paramagnetic
the holes. In these figures the vertical gray lines indicate the response ~i.e., 2M,0) can be realized. Only the vortex
ground-statetransitions.Noticethatintherestofthetextwe states with vorticity less or equal to the number of antidots
definethemagnetizationas2M, i.e.,thedifferencebetween exhibit a paramagnetic response. Note that for the reference
the applied magnetic field and the averaged magnetic field. sample a paramagnetic response35 can be realized in a small
The magnetization curves are strongly influenced by the magnetic-field region of the L51, 4, 5 states. It should be
presence of the antidots. In the absence of the antidots @see stressedthatthesuperconductorisinametastablestatewhen
Fig. 3~a!# the maxima in the magnetization curve decreases such a paramagnetic response is found. Notice also the dif-
with increasing L which is not so for the other samples. For ference in the magnetization for the different two-antidot
the single-antidot sample the largest flux expulsion is samples.
reached for L51, while this is realized in the case of the Inexperimentonemeasuresthemagnetizationbyaverag-
two- ~three-! antidot sample for L52 ~3!, i.e., it equals the ing the magnetic field over some area, which is determined
number of antidots. The magnetization also depends on the by the size of the detector. In the case of samples with anti-
arrangement of antidots in the sample. For example, the dots,themagnetic-fielddistributionisextremelynonuniform
magnetization of the states with 1,L,11 for the sample inside as well as outside the sample and therefore the detec-
with two antidots along the diagonal @Fig. 3~c!# is higher tor size will have an effect on the measured magnetization.
than the magnetization for the sample with two antidots on To illustrate this we calculated the magnetization by averag-
the top row @Fig. 3~d!#, while for the other vortex states it is ing the magnetic field over the area W3W. As an example,
opposite.The sample with four antidots behaves very differ- we plotted the magnetization for the four-antidot sample in
ent and we found that 2M is maximal for L58. There is Fig.4,whichexhibitsaquitedifferentbehavior.Themagne-
also a clear bunching of the magnetization curves with in- tizationdoesnotdecreasemonotonicallybyincreasingL,but
creasing the number of antidots which is absent in our refer- itisperiodical.Nowthemaximalmagnetizationisfoundfor
encesample.Thenumberofcurvesthatarebunchedtogether the states with L50 and L54. The states are now clearly
increases with the number of antidots. The Meissner state, bunched together in groups of four. Notice also that many
i.e., L50 state becomes less stable when antidots are vortexstatesexhibitparamagneticresponse,whichclearlyis
present. The transition to the L51 state occurs at a lower very sensitive to the definition of M and that there are
magnetic field indicating that the presence of the antidot~s! ground-state transitions to the states with 2M,0!
eases the penetration of the first vortex. For the supercon- Figure 5 shows the magnetic-field range DH over which
s
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BERDIYOROV,BAELUS,MILOSˇEVIC´,ANDPEETERS PHYSICALREVIEWB68,174521 ~2003!
FIG.4. Themagnetizationofthefour-antidotsampleasafunc-
tion of the applied magnetic field when the field is averaged over
theareaW3W. FIG. 6. The magnetic-field region DH over which the state
g
withvorticityListhegroundstateasafunctionofthevorticityL,
the vortex state with vorticity L is stable ~i.e., DHs forthereferencesample~fullcircles!,forthesamplewithone~open
5H 2H ) as a function of the vorticity L. circles!, two-diagonal ~full triangles!, two-top ~open triangles!,
penetration expulsion
For the reference sample the result is shown by the full three~fullsquares!,andfour~opensquare!holes.Theresultfortwo
circles,fortheone-antidotsamplebytheopencircles,forthe diagonal, two top, three- and four-antidot samples are shifted over
two-diagonal-antidot sample by the full triangles, for the 0.2H , 0.4H , 0.6H and0.8H , respectively.
c2 c2 c2 c2
two-top-antidot sample by the open triangles, for the three-
antidot sample by the full squares and for the four-antidot
clearlyaconsequenceofthecommensurabilityofthesquare
samplebytheopensquares.Theresultsforthetwo-diagonal,
vortex lattice with the square geometry of the sample as we
two-top, three- and four-antidot samples are shifted over pointed out earlier.32 For the square with a single hole, the
0.2/H , 0.5/H , 0.6/H , and 0.8/H , respectively. The
c2 c2 c2 c2 state with L51 has the largest stability range.With increas-
stabilityofeachindividualsuperconductingstateisverysen-
ing vorticity, the stability region decreases monotonically,
sitivetothetopologyofthesample.Forthereferencesample
except for the state with L55 and 17. Notice that ~1! for L
DH (L) decreases with increasing L, except for the L54
s .2 thedifferentvortexstatesinthereferencesamplehavea
state which exhibits an enhanced stability and which is
large stability region than those for the sample with one an-
tidot and ~2! for L>11 the stability region is practically in-
dependent of L, i.e., it is the magnetic field region where all
L transitions are continuous. For the superconductors with
two-diagonalandfourholes,thevortexstateswithevenvor-
ticity are more stable than those with odd vorticity. This is
less pronounced in the case of the two top antidot sample.
Forthefour-antidotsystemvortexstatesinwhichLaremul-
tiples of 4 have the highest stability. But for the supercon-
ductor with three holes the L53 and L56, 7 states have a
substantial larger stability region as compared to the other
vortexstates.TheLstateswithenhancedstabilityareclearly
a consequence of matching phenomena.
The magnetic-field range over which each of the vortex
states is the ground state DH is shown in Fig. 6 for the
g
different geometries as a function of the vorticity L. The
ground-state region DH also shows similar features as the
g
stability region DH . There is one main difference, DH
s s
exhibits an overall decrease with increasing L in the larger
FIG. 5. The magnetic-field region DH over which the state magnetic-field region which is not present in DHg.
s
In all cases, the vortex states show enhanced stability for
with vorticity L is stable as a function of the vorticity L, for the
reference sample ~full circles!, for the sample with one ~open commensurate vorticity, namely, when the number of vorti-
circles!,twodiagonal~fulltriangles!,two-top~opentriangles!,three ces penetrating the sample is a multiple of the number of
~full squares! and four ~open square! holes. The result for two- holes. However, for higher magnetic field these commensu-
diagonal, two-top, three- and four-antidot samples are shifted over rability effects disappear or are less pronounced, which is
0.2H , 0.5H , 0.6H ,and0.8H , respectively. due to the finite size of the sample.
c2 c2 c2 c2
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STABILITYANDTRANSITIONBETWEENVORTEX... PHYSICALREVIEWB68,174521 ~2003!
FIG. 8. Contour plot of the Cooper-pair density in logarithmic
scale for the one-antidot sample corresponding to the L56 ~a!, 7
~b!, 8 ~c!, 9 ~d!, 10 ~e!, and 11 ~f! states at H /H 51.22, 1.37,
0 c2
1.52, 1.67, 1.82, and 1.95, respectively. Dark ~light! gray regions
correspond to high ~low! density.The scales are ~a! 331024–0.9,
~b! 331024–0.9, ~c! 831025–0.9, ~d! 131025–0.8, ~e! 1
31025–0.8, and~f!131026–0.8.
FIG.7. Thephaseoftheorderparameterforthesuperconductor
states at the given magnetic fields. In these figures phases
withoneholeforthestateswithL52 ~a!,3~b!,4~c!,5~d!,6~e!,
7~f!,8~g!,and9~h!atH /H 50.62, 0.82,0.92,1.07,1.22,1.37, nearzeroaregivenbylightgrayregionsandphasesnear2p
0 c2
by dark gray regions. In order to determine the number of
1.52, and 1.67, respectively. Phases near zero are given by light
grayregions,phasesnear2pbydarkgrayregions. vortices in some region, we should ‘‘go’’around this region.
If the vorticity in this region is L, then the phase changes L
times 2p(Dw52pL).
IV.SPATIALDISTRIBUTIONOFTHEVORTICES
In the one-antidot sample the first vortex will sit in the
In the full square sample, vortices may form multivortex hole. We found that the second vortex is located in the su-
stateswithvorticityL52, 3,4,5,and6.Byfurtherincreas- perconducting material on the same diagonal where the hole
ing the magnetic field, vortices move towards the center and [email protected]~a!#.Thethirdvortexnucleatesinthesuper-
form a giant vortex state ~see also Ref. 32!. conductingmaterialandpushesthesecondvortexawayfrom
When one antidot is placed in the upper right half of the the diagonal @Fig. 7~b!#. By increasing the magnetic field
sample,thesquaresymmetryisbrokenandtheholeactslike these vortices move towards the hole. The fourth vortex is
apinningcenter.Asaconsequencethemultivortexstatewill pushed inside the hole, where now two vortices are located
become more favorable and stay stable even up to higher @Fig. 7~c!#. The fifth vortex is again nicely located on the
fields and vorticity. For the multivortex states with higher [email protected]~d!#.Thesixthvortexislocatedinthesuper-
vorticity, the separate multivortices are not visible anymore conductingregion,wheretwovorticesarelocatedveryclose
in the contour plots of the magnetic-field distribution or [email protected]~e!#.
Cooper-pair density. The reason is that the vortices are too The latter two vortices form almost a giant vortex state, but
close to each other, the size of the Cooper-pair density is they can be resolved as two separate vortices if we plot the
very small, and the spots corresponding to high magnetic phaseofC @Fig.7~e!#orifwemakealog-scalecontourplot
fieldsareoverlapping.Thereforetovisualizethevortexcon- of the Cooper-pair density @see Fig. 8~a!#. The Cooper-pair
figurations we plot the phase of the order parameter for the density variation between these two vortices is of order 3
sample with one hole in Figs. 7~a!–7~h! for the states with 31024–731024 and, therefore one will probably not be
L5229 at H /H 50.62, 0.82, 0.92, 1.07, 1.22, 1.37, abletoresolvethemexperimentallyastwoseparatevortices.
0 c2
1.52, 1.67, respectively. All states correspond to ground Increasing the vorticity to L57 the latter two vortices move
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apart and an additional vortex is added forming a local tri-
angular arrangement @see Figs. 7~f! and 8~b!#. For the states
with L58 @Fig. 7~g!# and L59 @Fig. 7~h!#, the number of
vortices in the hole is unaltered and additional vortices are
added along the diagonal passing through the antidot @Figs.
8~c,d!#. Increasing the field further to L510 @see Fig. 8~e!
for H /H 51.82] an additional vortex enters the antidot
0 c2
where now three vortices are located. The vortex arrange-
ment for the L511 state with H /H 51.95 is shown in
0 c2
Fig.8~f!.FortheL516statefourvorticesareinsidethehole
which is the maximum number of vortices that can be con-
tained in the hole. As is apparent from Figs. 7 and 8 the
vorticity of particular vortices in the one-hole sample may
vary, but the distribution of vortices in the superconducting
regionisalwayssymmetricwithrespecttothediagonalpass-
ing through the hole. From Fig. 8 we clearly see that the
variation in the Cooper-pair density between the vortices is
very small and as a consequence we expect that experimen-
tally the multivortex area will be observed rather as a trian-
gularlike shaped giant vortex state.
The sample with two antidots @see Fig. 1~c,d!# is more
symmetric, which will have an influence on the position of
the vortices. We show in Figs. 9~a!–9~h! the phase of the
order parameter for the states with vorticity L53–10 at
H /H 50.76, 0.87, 1.02, 1.1, 1.38, 1.51, 1.67, 1.81, re-
0 c2
spectively.Atthesemagneticfieldsthevortexstatesinques-
tion correspond to the ground state. The L51 and L52
vortexconfigurationsaretrivial.ForL51 thereisonevortex
inoneoftheantidotsandthegroundstateisdegeneratewith
respect to which antidot the vortex is located. This degen-
eracy is a new aspect which was not present in previous
sample. When L52 each antidot contains a single vortex.
For L53 the extra vortex is located in one of the antidots
@Fig.9~a!#andthesuperconductingstateisagaindegenerate. FIG.9. Thephaseoftheorderparameterforthesuperconductor
withtwo-diagonalholesforthestateswithL53 ~a!,4~b!,5~c!,6
When L54 two vortices nucleate in each of the antidots
~d!,7~e!,8~f!,9~g!,and10~h!atH /H 50.76, 0.87,1.02,1.1,
@Fig. 9~b!#. The fifth and the sixth vortex are situated in the 0 c2
1.38,1.51,1.67,and1.81,respectively.Phasesnearzeroaregiven
superconducting region along the diagonal @Figs. 9~c,d!#. bylightgrayregions,phasesnear2pbydarkgrayregions.
When the seventh vortex enters the superconductor, its posi-
tion is close to the center of the sample @Fig. 9~e!# and the are placed and the rest of the sample is practically in the
state is degenerate with a similar vortex configuration in normal state.
which the additional vortex is to the right side of the diago- Inordertoshowtheinfluenceofthesymmetryweplotted
nalpassingthroughthetwoantidots.Thisnonsymmetricvor- the phase of the order parameter for the sample with two
texarrangementwithrespecttothediagonalpassingthrough antidots, where the antidots are at the upper half of the
the antidots is energetically preferred because of the narrow sample @Figs. 10~a!–10~h!#. The first and the second vortex
superconducting region between the two antidots where su- are, as expected, in the holes. The third vortex is located in
perconductivity is enhanced.34 Figure 9~f! shows that for L the superconducting region @Fig. 10~a!#. For the L54 state
58 therearetwovorticesineachholeandtheothervortices two vortices in the superconducting region are located sym-
arealongthediagonalinthesuperconductingregionforming [email protected]~b!#andthissymmetryisbrokenwhenthe
two clusters each consisting of two closely spaced vortices. fi[email protected]~c!#.UptotheL510state
The ninth vortex is stabilized in the center of the sample @Figs. 10~d!–10~g!# two vortices are in each hole and the
@Fig.9~g!#.Whenthetenthvortexentersthesuperconductor, othervorticesareinthesuperconductingregion,formingdif-
it initially forms a giant vortex in the center with vorticity ferent vortex configurations.At the lower magnetic-field re-
L52, but we find that this giant vortex is not stable and gionfortheL510state,eachholecontainstwovorticesand
eventually the giant vortex decays in two separate vortices, the other two vortices are situated close to the holes @Fig.
each of them moves to each of the holes @Fig. 9~h!#.Till the 10~h!#. By further increasing the field these two vortices
L516stateallothervorticesaresituatedalongthediagonal. eventually enter the holes.
Increasing L further we found that the number of vortices in The symmetry is reduced when we add a third antidot
each hole increases with one unit. Superconductivity is then @Fig. 1~e!#. In Figs. 11~a!–11~h! the phase of the order pa-
only preserved in the corners of the sample where the holes rameter for the superconductor with three holes is given for
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FIG. 10. The phase of the order parameter for the supercon- FIG. 11. The phase of the order parameter for the supercon-
ductorwithtwotopholesforthestateswithL53 ~a!,4~b!,5~c!, ductorwiththreeholesforthevortexstateswithL54 ~a!,5~b!,6
6~d!,7~e!,8~f!,9~g!,and10~h!atH /H 50.67,0.9,0.97,1.07, ~c!, 7 ~d!, 8 ~e!, 9 ~f!, 10 ~g!, and 11 ~h! at H /H 50.87, 0.97,
0 c2 0 c2
1.27,1.47,1.65,and1.77,respectively.Phasesnearzeroaregiven 1.02, 1.195, 1.47, 1.65, 1.77, and 1.92, respectively. Phases near
bylightgrayregions,phasesnear2pbydarkgrayregions. zero are given by light gray regions, phases near 2pby dark gray
regions.
the states with L54-11 at H /H 50.87, 0.97, 1.02, 1.195,
0 c2
1.47, 1.65, 1.77, 1.92, respectively. In this case the first vor- of the antidots contains three vortices and one giant vortex
tex is located in one of the two antidots, situated along the withL52 nucleatesinthesuperconductingregion.Untilthe
diagonal of the sample. The second vortex is then placed in L515stateeachantidotcontainsthreevorticesandtheother
the opposite antidot along the diagonal. For L53 each anti- vortices are located in the superconducting region. Increas-
dot contains a single vortex and the vortex configuration is ingthefieldchangesthevortexarrangementinthesupercon-
very stable ~see Figs. 5 and 6!.The fourth vortex enters into ducting region. For example, for L512 three vortices are
the superconducting region @Fig. 11~a!#. The next vortex locatedalongthediagonalandincreasingthefieldrearranges
[email protected]~b!#.For them such that they form a triangle lattice. For L515 the
L56 the two antidots along the same diagonal contain each fourth vortex enters the lower left hole. For the L516 state
two vortices @Fig. 11~c!#. In the L57 state there are two eachholecontainsfourvorticesanduntilthesuperconductor
vortices in each of the holes @Fig. 11~d!#. The eighth vortex transforms to the normal state no more vortices are added to
[email protected]~e!#andforms the antidots.
a tight cluster of two vortices which is practically a L52 For the sample with four antidots @Fig. 1~f!# the vortices
giant vortex. For the vortex state with L59, each antidot are, up to high magnetic fields, located in the antidots.As a
contains two vortices and three separate vortices are located consequencethephaseofthesuperconductingcondensateC
in the superconducting region @Fig. 11~f!# in a triangle ar- contains less information and therefore we show in Figs.
rangement.Whenthenextvortexentersthesampleitgoesto 12~a!–12~h! the magnetic-field distribution for the sample
the upper hole and one of the vortices from the supercon- with four holes for the states with vorticity L51–4, 12, 13,
ducting region moves to the bottom right hole @Fig. 11~g!#. 14, and 17 at H /H 50.25, 0.33, 0.46, 0.67, 1.2, 2.36,
0 c2
When there are 11 vortices in the sample @Fig. 11~h!#, each 2.48, and 2.97, respectively.At these values of the magnetic
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FIG.12. Themagnetic-fielddistributionforthesuperconductor FIG.13. Themagnetic-fielddistributionforthesuperconductor
withfourholesforthestateswithvorticityL51 ~a!,2~b!,3~c!,4 with two holes for the states with vorticity L50 at H /H 50.02
0 c2
~d!,12~e!,13~f!,14~g!,and17~h!atH /H 50.25, 0.33,0.46, ~a!and0.47~b!,withvorticityL51 atH /H 50.03~c!,0.13~d!
0 c2 0 c2
0.67, 1.2, 2.36, 2.48, and 2.97, respectively. Higher magnetic field and0.53~e!andwithvorticityL52atH /H 50.08~f!,0.28~g!,
0 c2
isgivenbydarkgrayregionsandlowerbylightgrayregions. and0.92~h!.Darkgrayregionscorrespondtohighmagneticfield.
fields,theconsideredstatescorrespondtothegroundstateof [email protected]~f!#,whichisduetothesmallsizesofthe
the superconductor. The applied magnetic field is always holes that prevent them from capturing more vortices at
given by the same gray color.An increase ~decrease! of the those fields and the fact that if the extra vortex would go to
local field with respect to the applied field is indicated by a one of the antidots a very asymmetric configurations would
darker ~lighter! color. Up to the first penetration field, the beobtained,whichisenergeticallyunfavored.Whenthe14th
magnetic field is expelled from the superconductor and we vortex appears in the superconductor, the symmetry can be
see an increased magnetic field near the boundary of the restored by moving two vortices to the holes @Fig. 12~g!#.
superconductor.Buteveninthiscasethereissomeincreased StartingfromL517, thesuperconductivityinthecentralre-
magneticfieldinallantidots.AtH /H 50.56thefirstvor- gion of the sample is destroyed and vortices move to the
0 c2
tex enters the superconductor. One can expect that the posi- center. After the 19th vortex the sample transforms into the
[email protected]~a!# normal state.
~for a much smaller size of the superconductor this vortex The magnetic-field distribution in the superconductor and
canbelocatedinthecenterofthesuperconductor!.Thesec- inside the antidots is also of interest. As an example, we
ond vortex is situated in the hole opposite to the hole con- consider the magnetic-field distribution ~Fig. 13! in the su-
taining the first vortex @Fig. 12~b!#. The probability for the perconductor with two antidots for vorticity L50, 1, and 2,
third vortex to be located in one of the remaining two holes where the value of the magnetic field is taken such that it
is equal. In the depicted configuration the third vortex is in corresponds to the ground state. The dark regions in the fig-
thetoprighthole,asshowninFig.12~c!.FortheL54 state urescorrespondtohighmagneticfields.Itisclearfromthese
all antidots contain a single vortex @Fig. 12~d!#. This rule of figures that the magnetic field is nonuniform in and around
filling the antidots with vortices is repeated up to the 12th the sample and inside the antidots. For low magnetic fields
vortex @Fig. 12~e!#. The 13th vortex appears in the center of the superconductor expels the magnetic-field and the mag-
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