Vortex trapping and expulsion in thin-film YBCO strips

0.35 MB·English

by  K. H. Kuit

APS/123-QED
Vortex trapping and expulsion in thin-film YBa Cu O strips
2 3 7−δ
K. H. Kuit,1 J. R. Kirtley,1,2,3 W. van der Veur,1 C. G. Molenaar,1 F. J. G. Roesthuis,1
A. G. P. Troeman,1 J. R. Clem,4 H. Hilgenkamp,1 H. Rogalla,1 and J. Flokstra1
1Low Temperature Division, Mesa+ Institute for Nanotechnology,
University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
2Department of Applied Physics, Stanford University, Palo Alto, CA, USA
3Department of Microelectronics and Nanoscience,
Chalmers University of Technology, S-41296 Gœteborg, Sweden
8 4Ames Laboratory-DOE and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
0
(Dated: February 2, 2008)
0
2 A scanning SQUIDmicroscope was used to image vortex trapping as a function of the magnetic
n inductionduringcoolinginthin-filmYBa2Cu3O7−δ (YBCO)stripsforstripwidthsW from2to50
a µm. WefoundthatvorticeswereexcludedfromthestripswhentheinductionBawasbelowacritical
J inductionBc. Wepresentasimple modelfor thevortexexclusion process which takesintoaccount
5 thevortex-antivortexpairproductionenergyaswellasthevortexMeissnerandself-energies. This
1 model predicts that the real density n of trapped vortices is given by n = (Ba −BK)/Φ0 with
BK = 1.65Φ0/W2 and Φ0 = h/2e the superconducting flux quantum. This prediction is in good

] agreementwithourexperimentsonYBCO,aswellaswithpreviousexperimentsonthin-filmstrips
n ofniobium. Wealso reportonthepositionsofthetrappedvortices. Wefoundthatatlowdensities
o thevorticesweretrappedinasinglerownearthecentersofthestrips,withtherelativeintervortex
c
spacingdistributionwidthdecreasingasthevortexdensityincreased,asignoflongitudinalordering.
r- The critical induction for two rows forming in the 35 µm wide strip was (2.89+1.91−0.93)Bc,
p consistent with a numerical prediction.
u
s PACSnumbers: 74.25.Ha,74.25.Qt,74.25.Op,74.78.Bz
.
t
a
m
I. INTRODUCTION into thin strips.2,6 The strips have a certain critical in-
- ductionbelowwhichnovortextrappingoccurs,resulting
d
in an ambient field range in which these sensors can be
n In principle, when a parallel magnetic field is applied
o effectively operated. We investigated vortex trapping in
to an infinitely long, defect-free superconducting cylin-
c thin-filmYBCOstripsinordertoincorporatetheresults
der,allmagneticflux shouldbe expelledasthe tempera-
[ inahybridmagnetometerbasedonaYBCOringtightly
tureT isloweredthroughthesuperconductingtransition
coupled to, for example, a GMR (giant magneto resis-
1 temperatureT ,providedthattheappliedmagneticfield
v c tance) or Hall sensor.
isbeloweither thecriticalfieldH (T)foratype-Isuper-
3 c
conductor,orthe lowercriticalfield H (T)fora type-II Modelsforthecriticalinductionofthin-filmstripshave
8 c1
superconductor.1Inpractice,realsampleshavefinitesize been proposed by Clem7 and Likharev.8 Indirect exper-
2
2 and often contain defects, which can pin magnetic flux. imental testing of these models was done by observing
. Moreover,nonellipsoidalsamples,eventhosenotcontain- noise in high-Tc SQUIDs as a function of strip width
1
ing defects, naturally possess geometric energy barriers and induction.2,6,9 The induction mentioned here is the
0
8 that can trap magnetic flux during the cooling process. magneticinductionduringcooling,whichis the notation
0 Pinned or trapped vortices are nearly always observed throughout this paper. More direct experimental veri-
: in thin-film type-II superconductors, even when cooled fication of these models was presented by Stan et al.10
v
in relatively low magnetic fields. In general, this can be using scanning Hall probe microscopy (SHPM) on Nb
i
X attributed both to pinning of vortices by, for example, strips. Both experiment and theory found that the crit-
r defectsandgrainboundaries,andtotrappingbythegeo- ical induction varied roughly like 1/W2. However, the
a metric energy barriers. Understanding such pinning and experimental10 andtheoretical7,8 pre-factorsmultiplying
trapping effects is important for superconducting elec- this 1/W2 dependence differed significantly. In this pa-
tronics applications. per we propose a model for vortex trapping in narrow
superconducting strips which takes into accountthe role
Thepresentworkismotivatedbyapplicationsofhigh-
of thermally generated vortex-antivortexpairs.
T superconductingsensorssuchasSQUIDs2 andhybrid
c
magnetometers based on high-T flux concentrators.3 To test this model we performed scanning SQUID
c
These sensors are used in a broad field of applications, microscopy (SSM)11 measurements on thin-film YBCO
such as geophysical research4 and biomagnetism.5 The strips. Wefoundexcellentagreementbetweenthedepen-
sensitivity of these sensors is limited by 1/f noise in an denceofcriticalinductiononstripwidthandthepresent
unshielded environment. The dominant source of this model for both our experiments on YBCO and for the
noise is the movement of vortices trapped in the sensor. previous work on Nb. In agreement with this previous
Thisnoisecanbeeliminatedbydividingthehigh-T body work and as predicted by the present model, we found
c
2
that in YBCO the number of vortices increased for in- in response to the applied induction is:
ductions above the critical induction linearly with the
1
difference between the applied induction and the criti- J =− (B −nΦ )(x−W/2). (2)
cal induction. In a follow-up to the paper of Stan et y µ0λ2 a 0
al., Bronson et al.12 presented numerical simulations for
The assumption of a uniform density of vortices is good
the vortex distribution in narrow strips. These simula-
at high trapping densities, andat zero density, but is in-
tions showed that for inductions just above the critical
correct at low densities, as we shall discuss later. Equa-
induction the vortices are trapped in the centers of the
tion(2)differsfromtheexpressiongiveninRef.7bythe
strips. For higher inductions the vortices formed more
term −nΦ : As vortices are nucleated in the film, they
0
complexorderedpatterns,firstintwoparallelrows,then
reduce the screening currents induced in the film by the
for higher inductions in larger numbers of parallel rows.
appliedinduction. TheequationderivedinRef.7forthe
We performed statistical analysis of the vortex distribu-
Gibbs free energy of an isolated vortex (upper sign) or
tioninourmeasurementsandfoundagreementwiththis
anantivortex(lowersign)atapositionxinsidethe strip
model.
is then slightly modified as:14,15
Φ2 αW πx
II. THEORY OF VORTEX TRAPPING IN A G(x)= 0 ln sin
THIN FILM STRIP 2πµ0Λ (cid:20) ξ (cid:16)W (cid:17)(cid:21)
Φ (B −nΦ )
0 a 0
∓ x(W −x). (3)
Whether or not a vortex gets trapped in a strip is de- µ0Λ
termined by the Gibbs free energy. This energy exhibits
The Gibbs free energy consists of two terms. The first
a dipinthe centerofa superconductingstripforapplied
term, which is independent of the applied magnetic in-
inductions above a certain critical value. This dip gives
duction B , is calculated to logarithmic accuracy, as it
a
risetoanenergybarrierfortheescapeofthevortex. The
includesonlythekineticenergyofthesupercurrents,and
modelsproposedbyClem7 andLikharev8 differfromthe
it is equal to Φ I /2, where I is the supercurrent
0 circ circ
presentmodel only in the minimum height of the energy
circulating around the vortex. This term, which has a
barrier required to trap vortices.
dome shape and decreases monotonically to zero as the
vortexreachesadistanceξ/2fromtheedgesofthestrip,
is also equal to the work that must be done to move the
A. The Gibbs free energy of a vortex in a strip
vortexfromitsinitialpositionatx=ξ/2orx=W−ξ/2
to its final position at x againstthe Lorentz forces of at-
Consider a long, narrow, and thin superconducting
traction between the vortex and an infinite set of neg-
strip of width W in an applied magnetic induction B .
a ative image vortices at −x + 2mW, m = 0,±1,±2,...
The vortex trapping process occurs sufficiently close
. Here ξ is the coherence length, which is assumed to
to the superconducting transition temperature that the
obey ξ ≪ W. We also assume that the vortex core ra-
Pearl length Λ = 2λ2/d, with λ the London penetration
dius is ξ, such that the constant α = 2/π as in Ref. 7.
depth and d the film thickness, is larger than W. In
Other values of α, suchas 1/π as in Ref. 15, or 1/4as in
this limit there is little shielding of anexternallyapplied
Ref. 8 correspondto different assumptions regardingthe
magnetic induction B . The resultant superconducting
a core size. The second term in Eq. (3) is the interaction
currents in the strip can be calculated using the fluxoid
energy between a vortex (upper sign) [or an antivortex
quantization condition:13
(lower sign)] and the screening currents induced by the
external magnetic induction. It is the negative of the
B~ ·dS~ +µ0λ2 J~s·d~s=NΦ0. (1) work required to bring a vortex (or antivortex) in from
Z Z I
the edge against the Lorentz force due to the induced
In this equation the first integrationis over a closedsur- supercurrentgiven in Eq. (2). The upper sign in Eq. (3)
face S within the superconductor, the second is over a corresponds to the fact that J tends to drive vortices
ay
closed contour surrounding S, B~ is the magnetic induc- into the film, and the lower sign indicates that antivor-
tion, J~s is the supercurrent density, and N is an inte- tices are driven out. When Ba is sufficiently large, this
ger. SI units are used throughout this paper. If we take termmakesa minimumin G(x) inwhichvorticescanbe
the strip with its long dimension in the y direction, with trapped. For wider strips this minimum occurs at lower
edges at x = 0 and x = W, and an applied induction values of the induction.
perpendicular to the strip in the z direction, a square
closed contour can be drawn with sides at y =±l/2 and
x=W/2±∆x. Ifweassumeuniformdensitiesn andn B. Previous models for the critical induction
v a
of vorticesandantivorticesin the film, with n=n −n
v a
being the excessdensityofvorticesoverantivortices,the There are two existing models which predict the crit-
first integralin Eq. (1) becomes 2B ∆xl, the second be- ical induction for vortex trapping when the applied per-
a
comes 2J l, N = 2nl∆x, and the supercurrent induced pendicular magnetic induction is small (B ∼ Φ /W2).
s a 0
3
In these models the Gibbs free energy from Eq. (3) is a vortex-antivortexpair is given by the pairing energy:1
used in the limit of n→0. The critical induction model
by Likharev8 states that in order to trap a vortex in a E = Φ20 , (6)
pair
strip the vortex should be absolutely stable. This hap- 4πµ Λ
0
pens when the Gibbs free energy in the middle of the
the vortex-antivortex pair generation rate is given by a
strips equals zero and leads to
pre-factor times the Arrhenius factor exp(−E /k T),
pair B
where k is the Boltzmann constant. The vortex escape
2Φ αW B
0
BL = πW2 ln(cid:18) ξ (cid:19), (4) rate is given by an attempt frequency times a second
Arrheniusfactorexp(−E /k T),whereE isthediffer-
B B B
enceintheGibbsfreeenergybetweenthelocalmaximum
where α is the constant in Eq. (3).
and the minimum in the center of the strip. Since E
B
Another model for the critical induction is proposed andE havethesametemperaturedependences(recall
pair
by Clem,7 who considers a metastable condition. In that 1/Λ is proportional to T −T), the vortex genera-
c
this view vortex trapping will occur when the applied tion rate and its rate of escape will be exactly balanced
magnetic induction is just large enough to cause a mini- at all temperatures (aside from a logarithmic factor in
mum in the Gibbs free energy at the center of the strip, the ratio of the two pre-factors) when E and E are
B pair
d2G(W/2)/dx2 =0, leading to equal. This occurs at a critical magnetic induction B
K
which is the solution of the equation
πΦ
0
B = . (5)
0 4W2 max[G(x)−G(W/2)]=Epair, (7)
which leads to the condition
C. Our model for the critical induction πx 2πBa W2
max ln sin + −x(W −x)
(cid:26) h (cid:16)W (cid:17)i Φ0 (cid:20) 4 (cid:21)(cid:27)
The model proposed here is intermediate between the 1
= , (8)
models presented in Ref. 7 and 8. As the strip is cooled 2
just below the superconducting transition temperature
where the maximum value of the left-hand side of the
T , thermal fluctuations cause the generation of a high
c
equation is taken with respect to x. This equation can
density of vortex-antivortex pairs. Similar to the pro-
be solved numerically, resulting in
cesses determining the equilibrium densities of electrons
and holes in semiconductors, the equilibrium densities Φ
0
of vortices and antivortices very near Tc are determined BK =1.65W2. (9)
by a balance between the rate of generation of vortex-
antivortex pairs, the rate of their recombination, and Itappearstobeaninterestingnumericalcoincidencethat
the rates with which vortices are driven inwards and an- the solution to this equation gives a pre-factor (1.6525)
tivortices are driven outwards by the current J . Ac- that is only different from π2/6 by about 0.5%.
y
cordingly, very close to T , the densities n and n of As the temperature decreases, Λ decreases and be-
c v a
vortices and antivortices equilibrate such that their dif- comes much less than W. This means that once the
ference n = n −n is very nearly equal to B /φ and vortices are trapped in the local minimum and are clus-
v a a 0
the current J [see Eq. (2)] is practically zero. When teredaroundthemiddleofthestrip,thepotentialwellin
y
B < B < B , it is energetically unfavorable for vor- whichtheysitchangesshape. Recallthatthecalculation
0 a L
tices and antivortices to be present in the strip, and as ofbothtermsinEq.(3)assumethatΛislargerthanW.
the temperature decreases and the energy scales of the It would not even be possible to magnetically image the
terms in Eq. (3) increase, the densities of both vortices vortices in the vicinity of the freeze-in temperature be-
and antivortices decrease. While vortices and antivor- cause the local field produced by each vortex is then so
ticescontinuetobethermallygenerated,theantivortices spread out. However, as the temperature decreases, the
are quickly driven out of the strip by the combination numberoftrappedvorticesperunitlengthremainsfixed
of the self-energy and field-interaction energy [note the andthe appliedfieldremainsconstant. For n>>1/W2,
lower sign in Eq. (3)]. The antivortex density thus be- the distribution of vortices (averaging over the intervor-
comesmuchsmallerthanthevortexdensityandbecomes tex spacing) takes on a dome-like shape, and vortex-free
so small that the recombination rate is negligible. The zonesappear atthe edges ofthe strip. The z component
value of n ≈ n drops below B /φ . Although when ofthe localmagneticinduction shouldthen be described
v a 0
B < B < B it is energetically unfavorable for a vor- by the equations given in Sec. 2.1 of Ref. 16. The field
0 a L
tex to be present in the strip, the vortex’s Gibbs free distribution in a single strip containing a central vortex
energy has a local minimum at the center of the strip dome in which the current density is zero is closely re-
and the vortex must overcome the energy barrier before latedtothefielddistributionofapairofparallelcoplanar
it can leave the strip. Since the energy required to form strips with a gap between them.17,18
4
D. Behavior above the critical induction

Because in the present model the screening-current
density [Eq. (2)] and the Gibbs free energy [Eq. (3)] de-
0.8
pend on n, the areal density of vortices (when no an-
tivortices are present), we can expect that for applied
inductions Ba well above the critical induction BK the 0.6
balance between the rates of vortex generation and es-
n
cape occurs when n/eff 0.4
Φ
0
B −nΦ =B =1.65 , (10)
a 0 K W2 0.2
which can be inverted to give the density n of trapped
vortices as a function of applied induction, 0.0
Ba−BK 0 2 4 6 8 10
n= . (11)
Φ0 nW2
However, for B just above B , where n ≪ B /Φ ,
a K K 0
we need to take into account the interactions between FIG. 1: Normalized effective vortex density neff/n vs nW2
vorticesmorecarefully. As shownby Kogan,19 the inter- from Eq. (14).
action energy between a vortex at (x,0) and another at
(x ,y )in a stripofwidth W >Λ=2λ2/dandthickness
i i which is plotted in Fig. 1. Since numerical calculations
d is
of U (x) as approximated above show that
int
Φ2 cosh(y¯)−cos(x¯+x¯ )
ǫint(x)= 2πµ00Λlnhcosh(y¯ii)−cos(x¯−x¯ii)i, (12) Uint ≈ neµffΛΦ20x(W −x), (15)
0
wherex¯=πx/W,x¯i =πxi/W,andy¯i =πyi/W. Toob- we can use the argument that led to Eq. (10) to state
tain the interaction energy of the vortex at (x,0) with that the balance between vortex generation and escape
all the vortices when the vortex density is very large occurs when
(n ≪1/W2), it makes sense to convert the sum over all
Φ
i to an integral, assuming a uniform density n over the B −n Φ =B =1.65 0 . (16)
a eff 0 K W2
strip width. The integral can be evaluated analytically,
and the result is exactly equal to the term associated The value of n corresponding to each value of B then
a
with nΦ0 in Eq. (3). However, when the average vortex can be obtained from Eq. (14) or Fig. 1.
density is small, we can obtain an approximation to the
interaction energy of the vortex at (x,0) with all other
vortices by again converting the sum to an integral, as- III. MEASUREMENTS ON YBCO STRIPS
suminguniformdensityofvorticesnoverthestripwidth
but only for |yi| > 1/2nW. In other words, we exclude We performed SSM11 measurements on YBCO strips.
from the integral a rectangular region of height h and A sample was prepared on a SrTiO substrate with a
3
width W around the origin associated with one vortex, pulsed laser deposited 200 nm thin film of YBCO. The
where n = 1/Wh. After changing variables of integra- sample was structured into strips varying from 2-50 µm
tion, the resulting interaction energy Uint(W/2) at the inwidth by Ar ionetching. The SSM, inwhichthe sam-
center of the strip can be expressed as ple was cooled, was placed in a liquid helium bath cryo-
stat with three layers of µ-metal shielding. The SQUID
U (W/2)= 4nΦ20W2 π2 dθ φndφtanh−1(sinφcosθ), used in the SSM had a pickup loop which was defined
int π3µ0Λ Z0 Z0 sinφ by focused ion beam milling and had an effective area of
(13) 10-15 µm2 during imaging. A magnetic induction per-
where φn =sin−1[1/cosh(π/2nW2)]. For a given vortex pendiculartothesamplewasproducedbyasolenoidcoil
density n, the interaction energy Uint(W/2) is less than which was placed around the sample and SQUID. After
the value that would be expected from the second term the desired magnetic induction was applied the sample
in Eq. (3). We may define an effective vortex density was cooled to 4.2 K and the sample was scanned. Many
neff by equating Uint(W/2) to neffΦ20W2/4µ0Λ, where differentfieldvalueshavebeenappliedtothesampledur-
ing cooling to determine the critical induction for the
neff 16 π2 φn tanh−1(sinφcosθ) variousstripwidths. The sample was warmedup to well
= dθ dφ , (14)
n π3 Z0 Z0 sinφ above Tc between different cooling cycles.
5
a) b)
-3
10
B0
BL ( = 2/ )
BL ( = 1/4)
10-4 BK
T] Bc+
B [ Bc-
-5
10
c) d)
-6
10
-6 -5 -4
10 10 10
W [m]
FIG. 3: Critical inductions for vortex trapping as a function
of strip width. The squares represent Bc+, the lowest induc-
tions in which trapped vortices were observed, and the dots
areBc−,thehighestinductionsinwhichtrappedvorticeswere
not observed. Thedashed-dottedline is themetastable criti-
FIG. 2: Scanning SQUID microscope images of 35µm wide cal induction B0 [Eq. (5)], theshort-dashed and long-dashed
YBCOstripscooled inmagnetic inductionsofa)5µT,b)10 linesareBL[Eq.(4)],theabsolutestabilitycriticalinductions
µT, c) 20 µT, and d) 50 µT. calculatedatadepinningtemperatureTdp =0.98Tc,withthe
constantα=2/π7 orα=1/4.8 ThesolidlineisBK [Eq.(9)].
In Fig. 2 SSM images are displayed of 35 µm wide
stripsforseveralinductionsfrom5to50µT.Thestripsin were not observed. This provides an upper and lower
these images are darker than their surroundings because bound for the actual critical induction. It was apparent
of a change in the inductance of the SQUID sensor as it fromthislog-logplotthatthecriticalinductiondepended
passed over the superconducting strip. The bright dots on strip width as a power law. The best chi-squarefit of
are trapped vortices. As the inductions increased the the experimental data to the two parameter power law
vortex density also increased until it became difficult to B = aΦ /Wp yielded a = 1.96+0.13−0.15 and p =
c 0
distinguish one vortex from the other (Fig. 2d). In Fig. 1.98±0.03, taking a doubling of the best-fit chi-square
2a and Fig. 2b it is clear that at low trapped vortex as the uncertainty criterion. If we set the exponent to
densities the vortices tended to form one single row in be p = 2 (B = aΦ /W2), a one parameter fit yielded
c 0
thecenterofthestripwherethe energyislowest. InFig. a = 1.55±0.27. This is to be compared with a = 1.65
2c two parallel lines have been formed, but with some for the present model [Eq. (9)], plotted as B in Fig.
K
disorder. 3. It should be emphasized that there were no fitting
parameters in plotting B .
K
ComparisonoftheexperimentwiththemodelsofEqs.
A. Critical induction vs. strip width
(5) and (9) is straightforward, since they are only de-
pendent on the strip width. In order to evaluate Eq. (4)
The results of the measurements of the critical induc- one must make an estimate of the temperature at which
tionvs. strip width aredisplayedin Fig.3 together with vortexfreezoutoccursbecauseofthetemperaturedepen-
the various models. The measurements were performed denceofξ. ThedepinningtemperatureT/T =0.98used
c
on strips varying from 6-35 µm in width. Measurements inFig.3forbothB curveswascalculatedbyMaureret
L
onstripsnarrowerthan6µmwereunreliablebecausethe al.20 for YBCO. In addition we used ξ (0)= 3 nm,
YBCO
criticalinductionwashighenoughtodegradetheSQUID a critical temperature of T = 93 K, and the two-fluid
c
operation. The critical induction for 40 and 50 µm wide expressionfor the temperature dependence of the coher-
strips was smaller than the uncertainty in the applied ence length, resulting in ξ(T )=10.39 nm. To the best
dp
induction. of our knowledge the depinning temperature of YBCO
TherearetwodatapointsinFig.3foreachstripwidth: has never been determined experimentally. Analysis of
The upper point indicates the lowest induction at which Eq. 4 shows that a T closer to T could give better
dp c
vorticeswereobservedtrappedinthestrip,andthelower agreementbetweentheoryandexperimentforsomestrip
point indicates the highest induction at which vortices widths. However the difference in slopes between theory
6
andexperimentbecomes largerforhigher T ,making it
dp
appear unlikely that this is the correctmodel for our re-

sults. ThedependenceoftheLikharevmodelpredictions
onTdp isdisplayedinFig.4forα=2/π. ForlowerT/Tc 10-3 Nb
ratios the curve moves further away from experiment.
BK
-4 B0
10 BL ( =2/ )
T] BL ( =1/4)
BL (T/Tc=0.9999) B [c10-5
-3 BL (T/Tc=0.9995)
10 BL (T/Tc=0.9985)
-6
T] BL (T/Tc=0.98) 10
B [c BK
-4 -7
10 10
-6 -5 -4
10 10 10
W [m]
-5
10 FIG. 5: Comparison of experimental results on the critical
induction for vortex exclusion in thin film niobium strips10
-6 -5
10 10 with various theories, labeled as in Fig. 3.
W [m]
FIG. 4: Variation of the prediction of Eq. (4) (using α =
2/π) for the vortex exclusion critical induction on depinning
temperature (dashed lines). The solid line is BK [Eq. (9)].
0.030
Wealsocompareresultsofthepresentmodelwithpre-
vious work on Nb strips by Stan et al.10 using SHPM. 0.025
This paper reported critical inductions for 3 different
m
stripwidths: 1.6,10and100µm. Thecriticalinductions 0.020 (cid:181)
5
havebeencomparedtothevariousmodelsinFig.5. The 3
depinning temperature of T/Tc = 0.9985 used in this -2 (cid:181)m]0.015 W =
figure was experimentally determined.10 Using ξNb(0)= n [ m
(cid:181)
38.9nmresultsinthevalueξNb(T =Tdp)=320nmused 0.010 = 6
fortheBLcurvesinFig.5. Areasonablygoodagreement W
exists between the measurements and the predictions of
0.005
the present model.
0.000
0 25 50 75 100 125 150
B. Trapped vortex density as a function of applied B [(cid:181)T]
induction
FIG. 6: Plot of the number density of vortices trapped in
In Fig. 6 the experimentally determined density of YBCO strips 35 µm and 6 µm wide as a function of mag-
trapped vortices as a function of induction for two strip netic induction (dots). The dashed lines are the predictions
widths is displayed. This density depends nearly lin- of Eq. (11), without any adjustable parameters. The solid
early on the difference between the induction and the lines indicate the predictions of Eqs. (14) and (16)
criticalinduction, with a slope nearly Φ−1, in agreement
0
with previous work on Nb strips by Stan et al.10 The
35 µm strip width data can be fit to a linear depen-
dence of the vortex density n on B with a slope of
a
(3.86±0.08)×1014(Tm2)−1 = (0.83±0.02)Φ−1, with inductions higherthan130µT.This may,howeverbe an
0
an intercept of 3.8±1.3µT. The dashed and solid lines artifact due to the finite resolution of our SQUID sen-
in Fig. 6 are the prediction of the present model [Eqs. sor. The direction of the applied induction was reversed
(11) and (16) respectively] without any fitting parame- for three points in the W = 35 µm strip data to check
ters. Reasonable agreement exists between the present for an offset in the applied induction. Such an offset, if
modelandmeasurements. Inthecaseofthe6µmstrips, present, was small, as indicated by the symmetry of the
there is an apparent saturation in the vortex density for data around zero induction.
7

0.4 Ba=4(cid:181)T 0.4 Ba=5(cid:181)T 15 Ba = 4 (cid:181)T 30 Ba = 5 (cid:181)T
Mean = 106 (cid:181)m Mean = 53.4 (cid:181)m
0.3 0.3 Std dev = 91.3 (cid:181)m Std dev = 20.1 (cid:181)m
10 20
P(x) 0.2 P(x) 0.2 N N
5 10
0.1 0.1
0 0
0 100 200 300 400 0 50 100 150
0.0 0.0
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 y [(cid:181)m] y [(cid:181)m]
x [(cid:181) m] x [(cid:181) m] 1.0
0.4 Ba=10(cid:181)T 0.4 Ba=12(cid:181)T 80 BMae =a n1 0= (cid:181)1T2.9 (cid:181)m 0.8
60 Std dev = 3.29 (cid:181)m
0.3 0.3 y0.6
N
P(x) 0.2 P(x) 0.2 40 y0.4
20 0.2
0.1 0.1
0 0.0
0 5 10 15 20 25 0 5 10 15 20 25
0.00 5 10 15 20 25 30 35 0.00 5 10 15 20 25 30 35 y [(cid:181)m] Ba [(cid:181)T]
x [(cid:181) m] x [(cid:181) m]
0.4 Ba=18(cid:181)T 0.4 Ba=20(cid:181)T FIG.8: (a-c)Histogramsofthelongitudinalspacingbetween
vortices trapped in a 35 µm wide YBCO strip for selected
0.3 0.3
inductions. (d) Plot of the standard deviation of the distri-
P(x) 0.2 P(x) 0.2 bution of longitudinal spacings, divided by the mean of this
distribution,asafunctionofinduction. Therelativewidthsof
0.1 0.1 thedistributionsbecamenarrowerastheinductionincreases,
indicative of ordering in a single row, until at a critical in-
0.0 0.0 duction of about 10 µT there was an abrupt increase in the
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
x [(cid:181)m] x [(cid:181)m] relative width as two rows started toform.
FIG. 7: Histograms of the probability of trapping as a func-
was also some apparent correlation between the vortex
tion of the lateral vortex position in a 35 µm wide YBCO
positions. An example can be seen in Fig. 7, where a
strip at various inductions. At low inductions the vortices
histogramis displayed of the lateralpositions of vortices
trappedinasinglerownearthecenterofthefilm,butabove
trapped in the 35 µm wide strip for several inductions.
a induction of about 10 µT they started to reorder. At a in-
Atlowinductions,thevortexlateralpositiondistribution
duction of 18 µT the vortices were trapped in two relatively
well definedrows. peaked near the center of the film because the vortices
were aligned nearly in a single row. At a second critical
induction of B = 11±1µT the distribution started to
c2
become broader. At18 µT there weretwo clearpeaks in
C. Vortex spatial distribution
the distribution, corresponding to two rows. Using the
value of B = 3.8±1.3µT for the critical induction of
c
The local minimum in the Gibbs free energy at W/2 the 35 µm wide strips from our linear fit of the vortex
of Eq. (3) makes it energeticallyfavorablefor vortices to density vs. applied induction curve of Fig. 6, we found
be trapped in the center of the strip. However, as the B = (2.89+1.91−0.93)B . This is consistent with
c2 c
vortex density increases, the vortex-vortex repulsive in- the prediction of B = 2.48B of Bronson et al.12 In
c2 c
teraction makes it energetically more favorable to form the same paper the critical induction for the transition
anAbrikosov-liketriangularpattern. Simulations onthe from the two-row to the three-row regime is given to be
trapped vortexposition in strips was describedby Bron- B = 4.94B . This gives B = 18.77±6.42µT using
c3 c c3
sonetal.12Inparticulartheypredictthatthereshouldbe the same value for B . In our measurements we saw no
c
asinglelineofvorticesforinductionsBc <Ba <2.48Bc. evidence for a three row regime. It was not possible to
Above this induction range a second line of vortices is perform analysis at higher fields than reported here be-
predicted to form. As the induction is increased further cause of limitations to the spatial resolutionof the SSM.
additional lines of vortices are predicted to form into a We also saw evidence for longitudinal ordering. In
nearly triangular lattice. Fig.8a-chistogramsaredisplayedofthelongitudinaldis-
We have investigated the distribution of vortices tances ∆y between vortices in the 35 µm wide strip for
trapped in our strips at various inductions. As can be various inductions. As expected the inter-vortex spac-
seenfromtheimagesofFig.2,eventhoughtherewassig- ing distributions became narrower as the inductions in-
nificant disorder in the vortex trapping positions, there creased,since thevortexmeanspacingsdecreased. How-
8
ever the distributions became narrow faster than their viousmeasurementsonNb,10 showedacriticalinduction
means as the induction was increased, indicative of lon- for the onset oftrapping and a dependence ofthe vortex
gitudinalordering,untilthesecondcriticalinductionB density on the induction which were in good agreement
c2
of approximately 10 µT was reached. At that induction withanewmodelwhichtakesintoaccounttheenergyre-
the relative distribution width δ(∆y)/<∆y> has a dis- quired to generate a vortex-antivortexpair. In addition,
continuousjumpasasecondrowstartstoform. Asimilar at low inductions the vortices formed a single row, with
decrease in the relative longitudinal distribution width longitudinal ordering as the inductions increased. For-
with increasing induction is observed in the 6 µm wide mationof a secondrowwas observedat a secondcritical
strip,althoughthespatialresolutionoftheSSMwasnot induction consistent with numerical modeling.
sufficient to resolve vortices at the second critical induc-
tion for this width.
In theory there should be longitudinal ordering inde-
pendent of the magnetic induction. After all, the Gibbs
free energyis independent ofthe positionalongthe strip
and the only interaction that plays a role is the inter-
action between the vortices. Differences in longitudinal
ordering as a function of the magnetic induction could
arise from local minima of the Gibbs free energy caused
forexamplebydefectsinthematerial. Forrelativelylow
inductions vorticescaneasily be trappedin defects since
V. ACKNOWLEDGMENTS
the interactionbetween the vortices is small because the
separationbetweenthevorticesislarge. Forhighermag-
netic inductions the number of vortices and likewise the
interaction between the vortices increases. This could This research was financed by the Dutch MicroNED
meanthatthevorticesaremorelikelytotrapatpositions program and a VIDI grant (H.H) from the Dutch NWO
determined by the minimization of the vortex-vortexen- Foundation. J.R.K was supported by the Center for
ergy than at positions determined by local defects. Probing the Nanoscale, an NSF NSEC, NSF Grant No.
PHY-0425897, and by the Dutch NWO Foundation.
J.R.C’s work at the Ames Laboratory was supported
IV. CONCLUSIONS by the Department of Energy-BasicEnergy Sciences un-
derContractNo.DE-AC02-07CH11358. The SSMsetup
Experiments on vortex trapping in narrow YBCO used in this research was donated to the University of
stripsusingascanningSQUIDmicroscope,aswellaspre- Twente by the IBM T.J. Watson Research Center.
1 M. Tinkham, Introduction to Superconductivity, 2nd Edi- L.S.Yu-Jahnes,A.Gupta,K.Stawiasz,andS.Wind,IBM
tion (McGraw-Hill, NewYork,1996). Journal of Research and Development 39, 655 (1995).
2 E. Dantsker, S. Tanaka, and J. Clarke, Appl. Phys. Lett. 12 E. Bronson, M. Gelfand, and S. Field, Phys. Rev. B 73,
70, 2037 (1997). 144501 (2006).
3 F.Schmidt,S.Linzen,F.Schmidl,M.Mans,andP.Seidel, 13 M. Tinkham, Phys. Rev 129, 2413 (1963).
Supercond.Sci. Technol. 15, 488 (2002). 14 V. Kogan, Phys. Rev. B 49, 15874 (1994), ISSN 0163-
4 A. Chwala, R. IJsselsteijn, T. May, N. Oukhanski, 1829.
T. Schuler, V. Schultze, R. Stolz, and H. Meyer, IEEE 15 G. Maksimova, Phys. Solid State 40, 1607 (1998), ISSN
Trans. Appl.Supercond.13, 767 (2003). 1063-7834.
5 S. Lee, W. Myers, H. Grossman, H.-M. Cho, Y. Chemla, 16 A.A.B.Brojeny,Y.Mawatari, M.Benkraouda,andJ.R.
and J. Clarke, Appl.Phys.Lett. 81, 3094 (2002). Clem, Supercond.Sci. Technol. 15, 1454 (2002).
6 A. Jansman, M. Izquierdo, J. Flokstra, and H. Rogalla, 17 M. Benkraouda and J. R. Clem, Phys. Rev. B 53, 5716
IEEE Trans. Appl.Supercond.9, 3290 (1999). (1996).
7 J. Clem, Bull. Am. Phys. Soc. 43, 401 (1998), paper 18 M. Benkraouda and J. R. Clem, Phys. Rev. B 58, 15103
K36.06, and unpublishedwork. (1998).
8 K. Likharev,Sov. Radiophys 14, 722 (1972). 19 V. Kogan, Phys. Rev. B 75, 64514 (2007), ISSN 1098-
9 E. Dantsker, S. Tanaka, P.-A. Nilsson, R. Kleiner, and 0121.
J. Clarke, Appl. Phys.Lett. 69, 4099 (1996). 20 S.Maurer,N.-C.Yeh,andT.Tombrello,Phys.Rev.B54,
10 G. Stan, S. Field, and J. Martinis, Phys. Rev. Lett. 92, 15372 (1996).
097003 (2004).
11 J. Kirtley, M. Ketchen, C. Tsuei, J. Sun, W. Gallagher,