Vortex trapping and expulsion in thin-film YBCO strips

a r X i v :0801.2283v 1 [c o n d -m a t .s u p r -c o n ] 15 J a n 2008

APS/123-QED

Vortex trapping and expulsion in thin-?lm YBa 2Cu 3O 7?δstrips

K.H.Kuit,1J.R.Kirtley,1,2,3W.van der Veur,1C.G.Molenaar,1F.J.G.Roesthuis,1

A.G.P.Troeman,1J.R.Clem,4H.Hilgenkamp,1H.Rogalla,1and J.Flokstra 1

1

Low Temperature Division,Mesa +Institute for Nanotechnology,

University of Twente,P.O.Box 217,7500AE Enschede,The Netherlands 2

Department of Applied Physics,Stanford University,Palo Alto,CA,USA

3

Department of Microelectronics and Nanoscience,

Chalmers University of Technology,S-41296G?teborg,Sweden

4

Ames Laboratory-DOE and Department of Physics and Astronomy,Iowa State University,Ames,Iowa 50011,USA

(Dated:February 2,2008)

A scanning SQUID microscope was used to image vortex trapping as a function of the magnetic induction during cooling in thin-?lm YBa 2Cu 3O 7?δ(YBCO)strips for strip widths W from 2to 50μm.We found that vortices were excluded from the strips when the induction

B a was below a critical induction B c .We present a simple model for the vortex exclusion process which takes into account the vortex -antivortex pair production energy as well as the vortex Meissner and self-energies.This model predicts that the real density n of trapped vortices is given by n =(B a ?B K )/Φ0with B K =1.65Φ0/W 2and Φ0=h/2e the superconducting ?ux quantum.This prediction is in good agreement with our experiments on YBCO,as well as with previous experiments on thin-?lm strips of niobium.We also report on the positions of the trapped vortices.We found that at low densities the vortices were trapped in a single row near the centers of the strips,with the relative intervortex spacing distribution width decreasing as the vortex density increased,a sign of longitudinal ordering.The critical induction for two rows forming in the 35μm wide strip was (2.89+1.91?0.93)B c ,consistent with a numerical prediction.

PACS numbers:74.25.Ha,74.25.Qt,74.25.Op,74.78.Bz

I.INTRODUCTION

In principle,when a parallel magnetic ?eld is applied to an in?nitely long,defect-free superconducting cylin-der,all magnetic ?ux should be expelled as the tempera-ture T is lowered through the superconducting transition temperature T c ,provided that the applied magnetic ?eld is below either the critical ?eld H c (T )for a type-I super-conductor,or the lower critical ?eld H c 1(T )for a type-II superconductor.1In practice,real samples have ?nite size and often contain defects,which can pin magnetic ?ux.Moreover,nonellipsoidal samples,even those not contain-ing defects,naturally possess geometric energy barriers that can trap magnetic ?ux during the cooling process.Pinned or trapped vortices are nearly always observed in thin-?lm type-II superconductors,even when cooled in relatively low magnetic ?elds.In general,this can be attributed both to pinning of vortices by,for example,defects and grain boundaries,and to trapping by the geo-metric energy barriers.Understanding such pinning and trapping e?ects is important for superconducting elec-tronics applications.

The present work is motivated by applications of high-T c superconducting sensors such as SQUIDs 2and hybrid magnetometers based on high-T c ?ux concentrators.3These sensors are used in a broad ?eld of applications,such as geophysical research 4and biomagnetism.5The sensitivity of these sensors is limited by 1/f noise in an unshielded environment.The dominant source of this noise is the movement of vortices trapped in the sensor.This noise can be eliminated by dividing the high-T c body

into thin strips.2,6The strips have a certain critical in-duction below which no vortex trapping occurs,resulting in an ambient ?eld range in which these sensors can be e?ectively operated.We investigated vortex trapping in thin-?lm YBCO strips in order to incorporate the results in a hybrid magnetometer based on a YBCO ring tightly coupled to,for example,a GMR (giant magneto resis-tance)or Hall sensor.

Models for the critical induction of thin-?lm strips have been proposed by Clem 7and Likharev.8Indirect exper-imental testing of these models was done by observing noise in high-T c SQUIDs as a function of strip width and induction.2,6,9The induction mentioned here is the magnetic induction during cooling,which is the notation throughout this paper.More direct experimental veri-?cation of these models was presented by Stan et al.10using scanning Hall probe microscopy (SHPM)on Nb strips.Both experiment and theory found that the crit-ical induction varied roughly like 1/W 2.However,the experimental 10and theoretical 7,8pre-factors multiplying this 1/W 2dependence di?ered signi?cantly.In this pa-per we propose a model for vortex trapping in narrow superconducting strips which takes into account the role of thermally generated vortex-antivortex pairs.

To test this model we performed scanning SQUID microscopy (SSM)11measurements on thin-?lm YBCO strips.We found excellent agreement between the depen-dence of critical induction on strip width and the present model for both our experiments on YBCO and for the previous work on Nb.In agreement with this previous work and as predicted by the present model,we found

2

that in YBCO the number of vortices increased for in-ductions above the critical induction linearly with the di?erence between the applied induction and the criti-cal induction.In a follow-up to the paper of Stan et al.,Bronson et al.12presented numerical simulations for the vortex distribution in narrow strips.These simula-tions showed that for inductions just above the critical induction the vortices are trapped in the centers of the strips.For higher inductions the vortices formed more complex ordered patterns,?rst in two parallel rows,then for higher inductions in larger numbers of parallel rows. We performed statistical analysis of the vortex distribu-tion in our measurements and found agreement with this model.

II.THEORY OF VORTEX TRAPPING IN A

THIN FILM STRIP

Whether or not a vortex gets trapped in a strip is de-termined by the Gibbs free energy.This energy exhibits a dip in the center of a superconducting strip for applied inductions above a certain critical value.This dip gives rise to an energy barrier for the escape of the vortex.The models proposed by Clem7and Likharev8di?er from the present model only in the minimum height of the energy barrier required to trap vortices.

A.The Gibbs free energy of a vortex in a strip Consider a long,narrow,and thin superconducting strip of width W in an applied magnetic induction B a. The vortex trapping process occurs su?ciently close to the superconducting transition temperature that the Pearl lengthΛ=2λ2/d,withλthe London penetration depth and d the?lm thickness,is larger than W.In this limit there is little shielding of an externally applied magnetic induction B a.The resultant superconducting currents in the strip can be calculated using the?uxoid quantization condition:13

B·d S+μ0λ2 J s·d s=NΦ0.(1)

In this equation the?rst integration is over a closed sur-face S within the superconductor,the second is over a closed contour surrounding S, B is the magnetic induc-tion, J s is the supercurrent density,and N is an inte-ger.SI units are used throughout this paper.If we take the strip with its long dimension in the y direction,with edges at x=0and x=W,and an applied induction perpendicular to the strip in the z direction,a square closed contour can be drawn with sides at y=±l/2and x=W/2±?x.If we assume uniform densities n v and n a of vortices and antivortices in the?lm,with n=n v?n a being the excess density of vortices over antivortices,the ?rst integral in Eq.(1)becomes2B a?xl,the second be-comes2J s l,N=2nl?x,and the supercurrent induced in response to the applied induction is:

J y=?

1

2πμ0Λ

ln αW W

?

Φ0(B a?nΦ0)

3 In these models the Gibbs free energy from Eq.(3)is

used in the limit of n→0.The critical induction model

by Likharev8states that in order to trap a vortex in a

strip the vortex should be absolutely stable.This hap-

pens when the Gibbs free energy in the middle of the

strips equals zero and leads to

B L=2Φ0

ξ ,(4)

whereαis the constant in Eq.(3).

Another model for the critical induction is proposed by Clem,7who considers a metastable condition.In this view vortex trapping will occur when the applied magnetic induction is just large enough to cause a mini-mum in the Gibbs free energy at the center of the strip, d2G(W/2)/dx2=0,leading to

B0=πΦ0

4πμ0Λ

,(6)

the vortex-antivortex pair generation rate is given by a

pre-factor times the Arrhenius factor exp(?E pair/k B T),

where k B is the Boltzmann constant.The vortex escape

rate is given by an attempt frequency times a second

Arrhenius factor exp(?E B/k B T),where E B is the di?er-

ence in the Gibbs free energy between the local maximum

and the minimum in the center of the strip.Since E B

and E pair have the same temperature dependences(recall

that1/Λis proportional to T c?T),the vortex genera-

tion rate and its rate of escape will be exactly balanced

at all temperatures(aside from a logarithmic factor in

the ratio of the two pre-factors)when E B and E pair are

equal.This occurs at a critical magnetic induction B K

which is the solution of the equation

max[G(x)?G(W/2)]=E pair,(7)

which leads to the condition

max ln sin πxΦ0 W2

2

,(8)

where the maximum value of the left-hand side of the

equation is taken with respect to x.This equation can

be solved numerically,resulting in

B K=1.65

Φ0

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-

pend on n,the areal density of vortices(when no an-

tivortices are present),we can expect that for applied

inductions B a well above the critical induction B K the

balance between the rates of vortex generation and es-

cape occurs when

B a?nΦ0=B K=1.65

Φ0

Φ0

.(11)

However,for B a just above B K,where n?B K/Φ0,

we need to take into account the interactions between

vortices more carefully.As shown by Kogan,19the inter-

action energy between a vortex at(x,0)and another at

(x i,y i)in a strip of width W>Λ=2λ2/d and thickness

d is

?int(x)=Φ20

cosh(ˉy i)?cos(ˉx?ˉx i) ,(12)

whereˉx=πx/W,ˉx i=πx i/W,andˉy i=πy i/W.To ob-tain the interaction energy of the vortex at(x,0)with all the vortices when the vortex density is very large (n?1/W2),it makes sense to convert the sum over all i to an integral,assuming a uniform density n over the strip width.The integral can be evaluated analytically, and the result is exactly equal to the term associated with nΦ0in Eq.(3).However,when the average vortex 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-suming uniform density of vortices n over the strip width but only for|y i|>1/2nW.In other words,we exclude from the integral a rectangular region of height h and width W around the origin associated with one vortex, where n=1/W h.After changing variables of integra-tion,the resulting interaction energy U int(W/2)at the center of the strip can be expressed as

U int(W/2)=

4nΦ20W22

dθ φn0dφtanh?1(sinφcosθ)

n =

162

dθ φn0dφtanh?1(sinφcosθ)

μ0Λ

x(W?x),(15)

we can use the argument that led to Eq.(10)to state

that the balance between vortex generation and escape

occurs when

B a?n effΦ0=B K=1.65

Φ0

5 c)

a)b)

d)

FIG.2:Scanning SQUID microscope images of35μm wide

YBCO strips cooled in magnetic inductions of a)5μT,b)10

μT,c)20μT,and d)50μT.

In Fig.

2SSM images are displayed of35μm wide

strips for several inductions from5to50μT.The strips in

these images are darker than their surroundings because

of a change in the inductance of the SQUID sensor as it

passed over the superconducting strip.The bright dots

are trapped vortices.As the inductions increased the

vortex density also increased until it became di?cult to

distinguish one vortex from the other(Fig.2d).In Fig.

2a and Fig.2b it is clear that at low trapped vortex

densities the vortices tended to form one single row in

the center of the strip where the energy is lowest.In Fig.

2c two parallel lines have been formed,but with some

disorder.

A.Critical induction vs.strip width

The results of the measurements of the critical induc-

tion vs.strip width are displayed in Fig.3together with

the various models.The measurements were performed

on strips varying from6-35μm in width.Measurements

on strips narrower than6μm were unreliable because the

critical induction was high enough to degrade the SQUID

operation.The critical induction for40and50μm wide

strips was smaller than the uncertainty in the applied

induction.

There are two data points in Fig.3for each strip width:

The upper point indicates the lowest induction at which

vortices were observed trapped in the strip,and the lower

point indicates the highest induction at which vortices

101010

10

10

10

10

B

[

T

]

W [m]

FIG.3:Critical inductions for vortex trapping as a function

of strip width.The squares represent B c+,the lowest induc-

tions in which trapped vortices were observed,and the dots

are B c?,the highest inductions in which trapped vortices were

not observed.The dashed-dotted line is the metastable criti-

cal induction B0[Eq.(5)],the short-dashed and long-dashed

lines are B L[Eq.(4)],the absolute stability critical inductions

calculated at a depinning temperature T dp=0.98T c,with the

constantα=2/π7orα=1/4.8The solid line is B K[Eq.(9)].

were not observed.This provides an upper and lower

bound for the actual critical induction.It was apparent

from this log-log plot that the critical induction depended

on strip width as a power law.The best chi-square?t of

the experimental data to the two parameter power law

B c=aΦ0/W p yielded a=1.96+0.13?0.15and p=

1.98±0.03,taking a doubling of the best-?t chi-square

as the uncertainty criterion.If we set the exponent to

be p=2(B c=aΦ0/W2),a one parameter?t yielded

a=1.55±0.27.This is to be compared with a=1.65

for the present model[Eq.(9)],plotted as B K in Fig.

3.It should be emphasized that there were no?tting

parameters in plotting B K.

Comparison of the experiment with the models of Eqs.

(5)and(9)is straightforward,since they are only de-

pendent on the strip width.In order to evaluate Eq.(4)

one must make an estimate of the temperature at which

vortex freezout occurs because of the temperature depen-

dence ofξ.The depinning temperature T/T c=0.98used

in Fig.3for both B L curves was calculated by Maurer et

al.20for YBCO.In addition we usedξY BCO(0)=3nm,

a critical temperature of T c=93K,and the two-?uid

expression for the temperature dependence of the coher-

ence length,resulting inξ(T dp)=10.39nm.To the best

of our knowledge the depinning temperature of YBCO

has never been determined experimentally.Analysis of

Eq.4shows that a T dp closer to T c could give better

agreement between theory and experiment for some strip

widths.However the di?erence in slopes between theory

6

and experiment becomes larger for higher T dp ,making it appear unlikely that this is the correct model for our re-sults.The dependence of

the Likharev model predictions on T dp is displayed in Fig.4for

α=2/π.For lower T/T c ratios the curve moves further away from experiment.

10

1010

B

[T ]

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 B K [Eq.(9)].

We also compare results of the present model with pre-vious work on Nb strips by Stan et al.10using SHPM.This paper reported critical inductions for 3di?erent strip widths:1.6,10and 100μm.The critical inductions have been compared to the various models in Fig.5.The depinning temperature of T/T c =0.9985used in this ?gure was experimentally determined.10Using ξNb (0)=38.9nm results in the value ξNb (T =T dp )=320nm used for the B L curves in Fig.5.A reasonably good agreement exists between the measurements and the predictions of the present model.

B.

Trapped vortex density as a function of applied

induction

In Fig.6the experimentally determined density of trapped vortices as a function of induction for two strip widths is displayed.This density depends nearly lin-early on the di?erence between the induction and the

critical induction,with a slope nearly Φ?1

0,in agreement with previous work on Nb strips by Stan et al.10The 35μm strip width data can be ?t to a linear depen-dence of the vortex density n on B a with a slope of

(3.86±0.08)×1014(Tm 2)?1=(0.83±0.02)Φ?1

0,with an intercept of 3.8±1.3μT.The dashed and solid lines in Fig.6are the prediction of the present model [Eqs.(11)and (16)respectively]without any ?tting parame-ters.Reasonable agreement exists between the present model and measurements.In the case of the 6μm strips,there is an apparent saturation in the vortex density for

10

10

10

10

10

B

[T ]

FIG.5:with 025*******

125

150

n [μm

]

B [μT]

FIG.6:Plot of the number density of vortices trapped in YBCO strips 35μm and 6μm wide as a function of mag-netic induction (dots).The dashed lines are the predictions of Eq.(11),without any adjustable parameters.The solid lines indicate the predictions of Eqs.(14)and (16)

inductions higher than 130μT.This may,however be an artifact due to the ?nite resolution of our SQUID sen-sor.The direction of the applied induction was reversed for three points in the W =35μm strip data to check for an o?set in the applied induction.Such an o?set,if present,was small,as indicated by the symmetry of the data around zero induction.

7

05101520253035

P (x )

x [μm ]

05101520253035P (x )

x [μm ]

05101520253035P (x )

x [μm ]

05101520253035

P (x )

x [μm ]

05101520253035P (x )

x [μm ]05101520253035

P (x )

x [μm ]

FIG.7:Histograms of the probability of trapping as a func-tion of the lateral vortex position in a 35μm wide YBCO strip at various inductions.At low inductions the vortices trapped in a single row near the center of the ?lm,but above a induction of about 10μT they started to reorder.At a in-duction of 18μT the vortices were

trapped in two relatively well de?ned rows.

C.Vortex spatial distribution

The

local minimum in the Gibbs free energy at W/2of Eq.(3)makes it energetically favorable for vortices to be trapped in the center of the strip.However,as the vortex density increases,the vortex-vortex repulsive in-teraction makes it energetically more favorable to form an Abrikosov-like triangular pattern.Simulations on the trapped vortex position in strips was described by Bron-son et al.12In particular they predict that there should be a single line of vortices for inductions B c

We have investigated the distribution of vortices trapped in our strips at various inductions.As can be seen from the images of Fig.2,even though there was sig-ni?cant disorder in the vortex trapping positions,there

0100200300400N

y [μm ]050100150

N

y [μm ]

0510152025N

y [μm ]

0510152025

y

y

B

[μT]

FIG.8:(a-c)Histograms of the longitudinal spacing between

vortices trapped in a 35μm wide YBCO strip for selected inductions.(d)Plot of the standard deviation of the distri-bution of longitudinal spacings,divided by the mean of this distribution,as a function of induction.The relative widths of the distributions became narrower as the induction increases,indicative of ordering in a single row,until at a critical in-duction of about 10μT there was an abrupt increase in the relative width as two rows started to form.

was also some apparent correlation between the vortex positions.An example can be seen in Fig.7,where a histogram is displayed of the lateral positions of vortices trapped in the 35μm wide strip for several inductions.At low inductions,the vortex lateral position distribution peaked near the center of the ?lm because the vortices were aligned nearly in a single row.At a second critical induction of B c 2=11±1μT the distribution started to become broader.At 18μT there were two clear peaks in the distribution,corresponding to two https://www.wendangku.net/doc/0d8976906.html,ing the value of B c =3.8±1.3μT for the critical induction of the 35μm wide strips from our linear ?t of the vortex density vs.applied induction curve of Fig.6,we found B c 2=(2.89+1.91?0.93)B c .This is consistent with the prediction of B c 2=2.48B c of Bronson et al.12In the same paper the critical induction for the transition from the two-row to the three-row regime is given to be B c 3=4.94B c .This gives B c 3=18.77±6.42μT using the same value for B c .In our measurements we saw no evidence for a three row regime.It was not possible to perform analysis at higher ?elds than reported here be-cause of limitations to the spatial resolution of the SSM.We also saw evidence for longitudinal ordering.In Fig.8a-c histograms are displayed of the longitudinal dis-tances ?y between vortices in the 35μm wide strip for various inductions.As expected the inter-vortex spac-ing distributions became narrower as the inductions in-creased,since the vortex mean spacings decreased.How-

8

ever the distributions became narrow faster than their means as the induction was increased,indicative of lon-gitudinal ordering,until the second critical induction B c2 of approximately10μT was reached.At that induction the relative distribution widthδ(?y)/has a dis-continuous jump as a second row starts to form.A similar decrease in the relative longitudinal distribution width with increasing induction is observed in the6μm wide strip,although the spatial resolution of the SSM was not su?cient 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 energy is independent of the position along the strip and the only interaction that plays a role is the inter-action between the vortices.Di?erences in longitudinal ordering as a function of the magnetic induction could arise from local minima of the Gibbs free energy caused for example by defects in the material.For relatively low inductions vortices can easily be trapped in defects since the interaction between the vortices is small because the separation between the vortices is large.For higher mag-netic inductions the number of vortices and likewise the interaction between the vortices increases.This could mean that the vortices are more likely to trap at positions determined by the minimization of the vortex-vortex en-ergy than at positions determined by local defects.

IV.CONCLUSIONS

Experiments on vortex trapping in narrow YBCO strips using a scanning SQUID microscope,as well as pre-vious measurements on Nb,10showed a critical induction for the onset of trapping and a dependence of the vortex density on the induction which were in good agreement with a new model which takes into account the energy re-quired to generate a vortex-antivortex pair.In addition, at low inductions the vortices formed a single row,with longitudinal ordering as the inductions increased.For-mation of a second row was observed at a second critical induction consistent with numerical modeling.

V.ACKNOWLEDGMENTS

This research was?nanced by the Dutch MicroNED program and a VIDI grant(H.H)from the Dutch NWO Foundation.J.R.K was supported by the Center for 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 by the Department of Energy-Basic Energy Sciences un-der Contract No.DE-AC02-07CH11358.The SSM setup used in this research was donated to the University of Twente by the IBM T.J.Watson Research Center.

1M.Tinkham,Introduction to Superconductivity,2nd Edi-tion(McGraw-Hill,New York,1996).

2E.Dantsker,S.Tanaka,and J.Clarke,Appl.Phys.Lett.

70,2037(1997).

3F.Schmidt,S.Linzen,F.Schmidl,M.Mans,and P.Seidel, Supercond.Sci.Technol.15,488(2002).

4A.Chwala,R.IJsselsteijn,T.May,N.Oukhanski, T.Schuler,V.Schultze,R.Stolz,and H.Meyer,IEEE Trans.Appl.Supercond.13,767(2003).

5S.Lee,W.Myers,H.Grossman,H.-M.Cho,Y.Chemla, and J.Clarke,Appl.Phys.Lett.81,3094(2002).

6A.Jansman,M.Izquierdo,J.Flokstra,and H.Rogalla, IEEE Trans.Appl.Supercond.9,3290(1999).

7J.Clem,Bull.Am.Phys.Soc.43,401(1998),paper K36.06,and unpublished work.

8K.Likharev,Sov.Radiophys14,722(1972).

9E.Dantsker,S.Tanaka,P.-A.Nilsson,R.Kleiner,and J.Clarke,Appl.Phys.Lett.69,4099(1996).

10G.Stan,S.Field,and J.Martinis,Phys.Rev.Lett.92, 097003(2004).

11J.Kirtley,M.Ketchen,C.Tsuei,J.Sun,W.Gallagher,

L.S.Yu-Jahnes,A.Gupta,K.Stawiasz,and S.Wind,IBM Journal of Research and Development39,655(1995). 12E.Bronson,M.Gelfand,and S.Field,Phys.Rev.B73,

144501(2006).

13M.Tinkham,Phys.Rev129,2413(1963).

14V.Kogan,Phys.Rev.B49,15874(1994),ISSN0163-

1829.

15G.Maksimova,Phys.Solid State40,1607(1998),ISSN

1063-7834.

16A.A.B.Brojeny,Y.Mawatari,M.Benkraouda,and J.R.

Clem,Supercond.Sci.Technol.15,1454(2002).

17M.Benkraouda and J.R.Clem,Phys.Rev.B53,5716

(1996).

18M.Benkraouda and J.R.Clem,Phys.Rev.B58,15103

(1998).

19V.Kogan,Phys.Rev.B75,64514(2007),ISSN1098-

0121.

20S.Maurer,N.-C.Yeh,and T.Tombrello,Phys.Rev.B54,

15372(1996).