Experimental determination of correlations between spontaneously formed vortices in a superconductor

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by  Daniel Golubchik

Experimental determination of correlations between spontaneously formed vortices in
a superconductor
Daniel Golubchik, Emil Polturak, and Gad Koren
∗
Department of Physics, Technion - Israel Institute of Technology, Haifa 32000, Israel
Boris Ya. Shapiro and Irina Shapiro
Department of Physics, Bar Ilan University, Ramat Gan, 52900, Israel
(Dated: January 4, 2011)
1
1 We have imaged spontaneously created arrays of vortices (magnetic flux quanta), generated in a
0 superconducting film quenched through its transition temperature at rates around 109K/s. From
2 these images, we calculated the positional correlation functions for two vortices and for 3 vortices.
WecomparedourresultswithsimulationsofthetimedependentGinzburgLandauequationin2D.
n
The results are in agreement with the Kibble-Zurek scenario of spontaneous vortex creation. In
a
J addition, thecorrelation functions are insensitive to thepresence of a gauge field.

2

Vortices(magneticfluxquanta)aretopologicaldefects implemented. It assumes that minimal phase gradient
]
h of the order parameter in a superconductor. These de- will always be favorable due to minimal energy consid-
c fects are predicted to appear spontaneously, as a result eration. Under this assumption, only one vortex can be
e
of a rapid quench through the phase transition into the created at a vertex between 3 ordered regions. Hence,
m
superconducting state. During a rapid quench, close to the geodesic rule restricts the number of topological de-
-
the transition temperature (T ), the relaxation time of fects created by the system. This description is limited
t c
a the system becomes larger then the quench time. Under to relaxationofthe phase gradientsanddoes notinvolve
t
s these conditions, the system is necessarily driven out of any additional dynamics. One of the predictions of this
t. equilibrium. One model that describes the outcome of modelis strong,shortrangecorrelationbetweenvortices
a such phase transition is the Kibble-Zurek scenario, first and anti-vortices. If the size of frozen fluctuations ξˆis
m
suggestedbyKibble1,2inacosmologicalcontext. Among assumed to have a gaussian distribution, one can calcu-
- his other contributions,Zurek3 proposedterrestrialtests late the vortex-vortex correlation function4. Hence, the
d
n of this model in condensed matter systems, where the fluctuations distribution above the critical temperature
o broken symmetry is U(1), such as superfluids, BEC and leaves its mark on the emerging vortex array. By mea-
c superconductors. The behavior of these systems can be suring the positions of the vortices and calculating the
[ describedbyacomplexorderparameter. Abovethecrit- correlation function it is possible to investigate the fluc-
1 icaltemperature Tc, the orderparameterfluctuates with tuation distribution above Tc.
v acharacteristicsizeofthefluctuationbeingξ. Isthesam-
We note that the KZ model does not address the
9 pleiscooledinfinitelyslowlytowardsT ,ξ willgrowuntil
c critical coarsening process which may occur after the
0 it reaches the size of the sample. At finite cooling rates
transition5. In many systems, coarsening affects dras-
4
thefluctuations”freeze”atsomepoint,formingisolated,
0 tically the outcome of the transition due to defect-
uncorrelated domains of the ordered state. The typical
. antidefect annihilation. This may not be the case for
1 sizeofsuchadomain,ξˆ,insidewhichtheemergingorder
superconducting films. If the quench is fast enough, the
0
parameter is coherent, depends on the cooling rate. In
1 systemreacheslowtemperaturesbefore vorticescantra-
equilibrium,theorderparameterinthefinalstateshould
1 versethe distance to a nearbyantivortexandannihilate.
be uniform across the system. However, the initial mis-
: AttemperaturesmuchlowerthanT vorticesarestrongly
v match of the phase of the order parameter between dif- c
pinned,sothatanymotionispracticallyimpossible. Asa
i
X ferent regions leads to the appearance of topological de- result, vortex annihilation and coarsening is suppressed.
fects. In superconductors, these are vortices carrying a
r Superconductorsthereforeofferusanuniqueopportunity
a quantum of magnetic flux Φ ≡h/2e.
0 toinvestigatethe orderparameterfluctuationsaboveT ,
c
by measuring the vortex distribution after a quench.
IntheKZmodel,belowT uncorrelateddomainsofthe
c
orderedstateareformed. Eachofthesedomainspicksup The validity of KZ mechanism in systems with local
a random value of phase of the order parameter. Single- gauge symmetries (such as superconductors), has been
valuedness of the order parameter requires that the in- questioned by severalauthors6. For these systems an al-
tegral of the phase accumulated along the circumference ternative mechanism of flux trapping was suggested7–9.
of any loop must be an integer multiple of 2π. If the In this mechanism thermal fluctuations of the magnetic
phases of different domains are random, there is a fi- fieldarefrozeninsidethesuperconductorduringthetran-
niteprobabilitythatthephaseaccumulatedalongaloop sition. Asaconsequencevorticesareformedinclustersof
around a vertex between 3 domains will be ±2π, lead- equal sign. The resulting vortex-vortexcorrelationfunc-
ing to formation of topological defect. To calculate the tion should decay as a power law10. However, the am-
probabilityof this formation,the geodesic rule is usually plitude of trapped magnetic field fluctuations in conven-
2
tionalsuperconductingfilmsshouldbesolowthatvortex
formationshouldbebetterdescribedbytheKibbleZurek
mechanism9.
Itwassuggestedbyseveralauthorsthattopologicalde-
fects in first order phase transitions11–14 and in sustems
showing spinodal decomposition15 may form due to dy-
namics. In this approach, the geodesic rule does not
hold anymore. Pairs of vortices and anti-vortices are
formedduringcollisionsbetweendomainwalls. Although
most of the simulations were done for first order phase
transitions, this mechanism is claimed to be generally
applicable16. The correlation function for this mecha-
nism was never calculated, but it should have two char-
acteristiclengths: theseparationbetweennearbyvortices
producedbydomainwallcollisions,andthedomainsize.
Vortex pairs of opposite sign could also arise from a
Kosterlitz-Thouless type of transition17. In this theory,
unbound vortex pairs appear above T . If the system
KT
is cooled through T , these vortices effectively annihi-
KT
late. However,ifthequenchisfast,someoftheunbound
vortex pairs can survive the quench, become pinned at
low temperatures and observed. In this theory, the den-
sity of unbound vortex pairs above T increases. Con-
KT
sequently, the observed vortex density should depend
on the temperature from which the system is quenched.
Within our resolution, we found no such dependence in
the experiment described here.
The Kibble-Zurek (KZ) model has been tested in liq-
uid helium18,19, liquid crystals20, in superconductors21,
Josephsonjunctions22,23 andsuperconductingloops24,25.
Spontaneously generated topological defects were de-
tected in several of the aforementioned experiments.
More sensitive testing of the model involves the deter-
mination of correlations between these defects. There
are only two such experiments performed to date. In FIG. 1. Spontaneously created vortices in a superconduc-
one experiment20, done with liquid crystals, an array of tor. a) A typical image of magnetic field created after a
topological defects was imaged. However,the amount of quench. The intensity is proportional to the local magnetic
data was insufficient to detect correlations beyond near- field. Bright and dark spots are vortices and anti-vortices
est neighbors. In our experiment26, we imaged sponta- respectively. The scale bar represents 10 µm. b) A typical
result of the simulation of TDGL equations. The scale bar
neously formed arrays of vortices in a superconductor.
represents 10 ξ0.
The amount of data gathered was large, allowing a pre-
cise determination of the two point correlations. In this
work, we extend this study to look at correlations be-
tween 3 defects. important, fast cooling to low temperatures (far below
T ) traps the vortices on pinning centers, preventing an-
Our experimental technique is described in detail in c
previous publications26,27. Briefly, we image the mag- nihilation ofvortices andanti-vortices. Using shortlaser
pulses26, we achievedcooling ratesas highas 2·109K/s.
netic field on the surface of a superconducting film us-
ing highresolutionMagneto-optics. The superconductor Fig.1a shows a typical image of spontaneously gener-
sampleconsistsofa200nmthickNiobiumfilmwithT of ated vortices. The average asymmetry between the den-
c
8.9K. The filmis patternedintosmallsquaresof200µm sityofpositiveandnegativevorticesislessthan1%. The
across. OntopoftheNbfilm,wedepositeda40nmlayer averagedensity of vortices was 6·105cm−2 for a cooling
of EuSe which serves as the Magneto-Optic sensor. To rate of 4·108K/s and 1.3·106cm−2 for a cooling rate of
minimize the effect of stray magnetic fields, our appara- 2·109K/s. The scaling of the density with the cooling
tus was carefully shielded using µ-metal. The residual rateisconsistentwiththeKZmodelat2D(proportional
field was less than 10 7T). to the square root of the cooling rate).
−
From our previous experiments21, we know that ex- Our experimental results are compared with numer-
tremely high cooling rates are essential for spontaneous ical simulations of the 2D time dependent Ginzburg-
generation of a measurable amount of vortices. No less Landau equations. The simulations are described in de-
3
)
u. 0.00
a.
(
n
o
ati -0.02
el
r
r
o
c
al -0.04
di
a
R
-0.06 R/Rav
0.0 0.5 1.0 1.5 FIG. 3. Schematic description of spontaneous formation of
vortices on intersections between ordered domains. a) Ran-
dom domain array. b) Close packed domain structure. Vor-
FIG. 2. The vortex-vortex correlation function G(r). Solid tices and anti-vortices are marked by + and - respectively.
red triangles represent the correlation function calculated
fromexperimentaldata. Thestatisticalerrorbarsaresmaller
than the point size. Open blue diamonds represent correla-
tionscalculated from theresultsofsimulation. Thesolid line
is a fit to theory of Liu and Mazenko4. The negative peak function has a minimum. This is consistent with the
at short distance reflects vortex-antivortex correlations pre- KZ model which predicts that nearest neighbor vortices
dicted by KZ model. shouldhaveoppositepolarities. Thecorrelationfunction
calculated from the simulations in the Fig.2 was aver-
aged over 200 realizations. Since the same scaling was
tail in28. Briefly, we used a square 200×200 grid, with used, the experimental and simulated correlation func-
the parameters κ = 1, Γ = 1. The initial conditions, tions can be compared without additional parameters.
A = 0,Ψ = 0,θ = 0.7, mimic a quench from tempera- The dashed line is a fit to the theoretical predictions by
tures far above Tc to T = 0.7Tc. The implicit Crank- Liu and Mazenko4 with ξˆ= 0.35rav. In their work, Liu
Nicholsonschemewasemployedonastaggeredgridwith andMazenkoassumedaGaussiandistributionofthefluc-
step in time ∆τ = 0.01 and in space ∆l = 0.2. The tuations. Even through the simulation parameters differ
time of each run was 20τGL. After this time the vor- fromourexperimentalconditions(infinitevs. finitecool-
tices are well defined and assumed to be pinned. A typ- ing rate, different initial temperatures, different interac-
ical result is presented in fig.1b. Bright and dark spots tions between vortices), the resulting correlation func-
mark the positions of vortices and anti-vortices. Several tion is the same. Qualitatively similar correlation func-
oftheparametersofthesimulationaredifferentfromthe tionwasalsocalculatedfromsimulationsofquenched2D
experimental parameters. First, the cooling rate in the XY model30. The agreement with the correlation func-
simulation is infinite, in contrastto the finite rate in the tioncalculatedby Liu andMazenko4 suggeststhat inall
experiment. Finite cooling rates in the simulation scale cases,theshapeofthecorrelationfunctionisdictatedby
the correlation length but do not affect the distribution. Gaussian fluctuations.
Second,thesimulationisfullytwodimensional,whilethe
InadditiontothetwopointcorrelationfunctionG(r−
sample used in the experiment is a thin film. The inter-
r ), we used our data to calculate an angular ( 3 point)
action between vortices in thin films is much stronger, ′
correlation function. This function involves 3 vortices.
and mediated through the magnetic field of the vortex
For each vortex in the array, we look for its two near-
outside of the film29.
est neighbors. Nearest neighbors are defined as vortices
We used images like those shown in Fig.1 to deter-
located within a distance of ∼ ξˆ and have an opposite
mine the correlations between the vortices. The two
polarity (see fig.3). If such neighbors are found, we cal-
particle vortex-vortex correlation function is defined as
G(r −r ) =< n(r)n(r ) > , with n(r) = 1 at the loca- culate the angle between the two lines connecting the
′ ′
tion of a positive vortex, −1 at the location of negative vortex to its neighbors. In the limit where the domains
are mono-dispersed and close packed (fig.3b), the angle
vortex and 0 elsewhere. The correlation function cal-
between nearest neighbors will be 120o. For randomly
culated from our data is shown in Fig.2. The distance
distributed domains (fig.3a)the angleswill have a broad
in the figure was scaled by the mean vortex separation,
r =<ρ> 1/2 as proposed by20. r is related to ξˆby distribution. Even through the connection between the
av − av fluctuation distribution and the angle distribution was
r = ξˆ , where p is the average number of vortices per
av √p never calculated, the first obviously determines the sec-
domain. For our high cooling rate of 2·109K/s, r is ond. Therefore the comparison between the angle dis-
av
8.2µm. The correlation function was averaged over 260 tribution calculated out of experimental data, and from
images, with 50,000 vortices in total. At the distance the results of a simulation can be used as a validity test
corresponding to the nearest neighbors the correlation for a model. The angle distributions calculated from ex-
4
the error bars, these two distributions are consistent.
In conclusion,we calculatedthe vortex-vortexcorrela-
0.15 A( ) tionfunctionandthe angulardistributionfunctionusing
experimentaldataandcomparedthemwiththeresultsof
TDGL simulations. Wefoundanagreementbetweenthe
experimental and simulated results. This implies that
0.10
withinexperimentalaccuracy,the TDGLequationin2D
describesthevortexformationinoursystem. Oneconse-
quenceisthatthe couplingtothe gaugefieldoutsidethe
0.05 sample does not affect the vortex distribution, at least
for the parameters used in the experiment. The correla-
tion function calculated using a quenched XY model30
gave qualitatively similar results, which suggests that
0.00 gaugefieldshavenoeffectonthesystemevolutionduring
quench.
0 30 60 90 120 150 180
FIG. 4. The angular distribution function. Solid black
squaresrepresenttheangulardistributioncalculatedfromex- ACKNOWLEDGMENTS
perimentaldata. Redpentagonsareangulardistributioncal-
culated from the results of the simulation.
WethankS.LipsonandE.Buksfortheircontribution
to this experiment. We thank S. Hoida,L. Iomin andO.
perimental data and from simulation are shown in Fig. Shtempluk for technical assistance. This work was sup-
4. Both distributions have a minimum at θ =0o and in- ported in part by the Israel Science Foundation (Grant
crease until θ ≈100o were they reach a plateau. Within No. 499/07)and by the Minerva and DIP projects.
∗ [email protected] 3824, (1997).
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