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Critical currents in vicinal YBa 2Cu3O7−δ films

J. H. Durrell,∗ G. Burnell, Z. H. Barber, M. G. Blamire, and J. E. EvettsDepartment of Materials Science and Metallurgy, University of Cambridge, Pembroke Street, Cambridge, CB2 3QZ, UK.

(Dated: July 9, 2004)

Most measurements of critical current densities in YBa2Cu3O7−δ thin films to date have been performed onfilms where thec-axis is grown normal to the film surface. With such films, the analysis of the dependence ofjc

on the magnetic field angle is complex. The effects of extrinsic contributions to the angular field dependence ofjc, such as the measurement geometry and disposition of pinning centres, are convoluted with those intrinsicallydue to the anisotropy of the material. As a consequence of this, it is difficult to distinguish between proposedFLL structure models on the basis of angular critical current density measurements onc-axis films. Films grownon mis-cut (vicinal) substrates have a reduced measurementsymmetry and thus provide a greater insight intothe critical current anisotropy. In this paper previous descriptions of the magnetic field angle dependence ofjc in YBa2Cu3O7−δ are reviewed. Measurements on YBa2Cu3O7−δ thin films grown on a range of vicinalsubstrates are presented and the results interpreted in terms of the structure and dimensionality of the FLL inYBa2Cu3O7−δ. There is strong evidence for a transition in the structure of the flux line lattice depending onmagnetic field magnitude, orientation and temperature. As aconsequence, a simple scaling law can not, byitself, describe the observed critical current anisotropyin YBa2Cu3O7−δ. The experimentally obtainedjc(θ)behaviour of YBCO is successfully described in terms of a kinked vortex structure for fields applied near parallelto thea-b planes.

I. INTRODUCTION

As in all Type-II superconductors it is the structure and pin-ning behaviour of the Abrikosov lattice of flux vortices thatdetermines the critical current properties of YBa2Cu3O7−δ

(YBCO). A comprehensive understanding of this is thereforean essential prerequisite to the development of technologicallyuseful applications of this material. Low-temperature super-conductors are either isotropic or slightly anisotropic1. In con-trast the more recently discovered high temperature supercon-ductor (HTS) materials are all layered, strongly anisotropicmaterials. Superconductivity is associated with the cuprateplanes which lie in thea-b planes of these materials. Forthe case of YBCO the Ginzburg-Landau anisotropy param-eter,γ, is 5-7, whereas in Bi2Sr2CaCu2O8+x (BSCCO 2212)the value is∼ 200. This large difference has been attributed tothe cuprate chains found along theb-axis between the cuprateplanes in YBCO2,3,4 which may exhibit superconductivity.

For the more strongly anisotropic HTS materials it is rea-sonable to start by approximating them as two-dimensionalsuperconductors with purely Josephson coupling between thesuperconducting layers5. This is the two dimensional super-conductor described by Lawrence and Doniach6. This ap-proach does not suffice for the case of YBCO. Indeed, overa wide range of applied field angles it appears that the fluxlines in YBCO are the elliptical vortices which would be ex-pected from anisotropic Ginzburg-Landau theory7,8. For fieldsapplied nearly parallel to thea-b planes Blatteret al.9 havepredicted that there is a transition to a kinked vortex statewhere the contiguous vortex lines consist of alternating vor-tex string and pancakes segments parallel and perpendicularto thea-b planes. Measurements on single crystals indicatethat the vortex lines only fully ’lock-in’ to the planes whenthe field is within approximately0.2◦ of thea-b planes10. Thefully locked-in state will only be seen therefore in very perfectcrystals.

The variation of the structure of individual flux lines as the

magnetic field direction changes with respect to thea-b planesmeans that no single scaling law will describe the magneticfield angle dependence of the superconducting properties ofYBCO. The presence of anisotropic pinning centres will fur-ther complicate the observed behaviour. In order to elucidatethe several contributions to the observed dependence of criti-cal current on applied magnetic field angle it is necessary toreduce the measurement symmetry, ideally by arranging thatthe Lorentz force is not directed along a crystallographic axis.In this study we achieve this aim by employing YBCO thinfilms grown on mis-cut (vicinal) substrates. Thin films grownon single crystal substrates are a convenient experimentalsys-tem especially since the step-flow growth favoured on vicinalsubstrates can lead to a very clean microstructure.

II. THEORY

There has been extensive earlier work on the angular de-pendence of critical currents in thin film YBCO. Howeverthese studies have all employedc-axis orientated films whichgreatly complicates the interpretation of the data obtained.

The first report of the nature of the variation of critical cur-rent with applied magnetic field angle was by Roas and co-workers11. Although the angular resolution of their data waspoor they identified the two most prominent features found inthe jc(θ) behaviour of YBCO, whereθ is the angle betweenthe applied field and thea-b planes. When the field is appliedparallel to thea-b planes they observed an ’intrinsic pinning’peak which was attributed to pinning by the layered structureof the superconductor. Additionally a smaller peak was seenwhen the field was aligned parallel to thec-axis. This peakwas associated by Roaset al. with pinning by twin planes.It is important to note that in their experiment the field wasrotated in a plane perpendicular to the current so as to keepj ∧ B maximized. If the field is swept in a plane containingthe current direction the peak in the critical current observed

2

atθ = 0 is due to the ’force-free’12 effect.It is immediately useful to note that these three peaks arise

for entirely different reasons. The ’intrinsic’ pinning peak issimply due to the anisotropy of the superconductor itself andthe ’force-free’ peak arises as the Lorentz forcej ∧ B tendsto zero. It has since been shown that the primary source ofpinning in c-axis films is due to dislocations at the edge ofgrowth grains13. It is these dislocations that lead to thec-axispeak since they are most effective as pinning centres when theapplied field is aligned with them. While the ’intrinsic’ peakwill not be expected to be sample dependent, thec-axis peakwill depend on the film microstructure. This is supported bythe wide variation in the prominence of thec-axis peak seenin different samples.

The nature of the ’intrinsic’ peak is not straightforward.Similar behaviour is observed in BSCCO 2212 which can bequite satisfactorily described as a two dimensional supercon-ductor. In BSCCO 2212 the flux lines take the form of stacksof pancake vortices localized within thea-b planes14. A con-sequence of this is that any angular dependent behaviour canbe taken as depending entirely on thec-axis component ofthe applied field, the ’Kes law’15. In small fields the vor-tex structure does, however, show more complex behaviour16.The ’Kes law’ has been found to agree very well with exper-iment in BSCCO 221217. Although Jakobet al.18 reportedthat this scaling law also worked well for the case of YBCO,close examination of their data shows that while the fit forBSCCO is indeed very good, the correlation is much less sat-isfactory for YBCO. Moreover a consequence of the Kes lawis that BSCCO 2212 will not exhibit a ’force-free’ effect12 asthe local field is not aligned with the current direction, what-ever the orientation of the external field . This is experimen-tally observed as constant critical current when the appliedmagnetic field is rotated about thec-axis19 with a constantangle between the field and thea-b planes. YBCO does ex-hibit such an enhancement of critical current in the ’force-free’ geometry8,20,21. YBCO cannot, therefore, be treated asapproximating to a two dimensional superconductor.

Tachiki and Takahashi22 considered the effect of layering inYBCO by noting that forT <70 K the core size for a vortexlying along thea-b plane is less than the inter-planar spacing.They extended their model to cover orientations other thanaligned along thea-b plane by envisaging kinked vortex linesconsisting of locked-in vortex segments connected by shortlengths crossing the cuprate planes23. Although some authorsfound their data broadly fitted their predictions24,25,26 nonefound the predicted lock-in plateau27. This is in all likelihoodbecause the lock-in angle is small0.2o and is suppressed byonly slight imperfections in the crystal structure. The lock-ineffect is however seen in high quality single crystals10.

The intrinsic pinning peak does not disappear at higher tem-peratures where the vortex core is larger than the interlayerspacings. If the superconductor is simply considered as beinganisotropic, Ginzburg-Landau theory predicts that the vortexcores will become progressively more elliptical as the fieldistilted. This may be described by considering an angle depen-dent mass anisotropy parameter1,7 ǫθ. This parameter is givenby ǫ2θ = ǫ2cos2θ + sin2θ whereǫ is the mass anisotropy pa-

rameter,ǫ = 1/γ =√

mab

mc

. From this, scaled versions of

various experimental parameters may be obtained, for exam-ple Bc2(θ) = Bc2/ǫθ. For the case of a tilted vortex lineparallel to thea-b planes the vortex core size in thec directionis reduced toǫθξab, whereξab is the G-L coherence length inthea-b planes. If it is assumed that the smaller core is moreeffectively pinned, then an ’intrinsic’ pinning peak will stillbe found. The expected angular dependence forjc,

jc(B, θ) = jc(ǫθB, θ = 90o) (1)

is not straightforward as, in general,jc does not depend onBin a simple fashion. Divergence from this expected behaviourat a particular angle,θ, can however reasonably be used toinfer the existence of anisotropic pinning, as has recentlybeendiscussed by Civaleet al.28

Blatter et al.9 reconciled these various ideas by proposingthe existence of a cross-over from a lattice of conventionalrectilinear, but anisotropic, Abrikosov vortices to a regime ofkinked vortices, similar to the Tachiki and Takahashi model.In the kinked regime pancake vortices in the cuprate planesare linked by Josephson string vortices in the interlayers.Thepancake vortices have a fully developed normal core of ex-tent ξab. The vortices laying between the cuprate planes areJosephson in nature and do not exhibit full suppression of thesuperconducting order parameter.

Blatter introduced two critical angles,θ1 and θ2. Defin-ing θ = 0 to be when the applied field is aligned with thecuprate planes forθ > θ1 the flux lines are conventionalstraight Abrikosov vortices. Forθ1 > θ > θ2 the flux vor-tices are distorted towards the kinked state whilst forθ < θ2

the kinked state is fully formed. They expressθ1 as

tan(θ1) =d

ξab(t)(2)

t is the reduced temperature (T/Tc), d is the interlayer spac-ing andTc is the superconducting transition temperature. Thesecond critical angleθ2 is expressed as

tan(θ2) = ǫ (3)

Above a certain critical temperatureTcr the transition is ex-pected to be suppressed and rectilinear Abrikosov vorticesareseen for all orientations of applied magnetic field. This criti-cal temperature is defined as being that at whichξab(Tcr) =

d/√

2. In YBCO these values are29 θ1 ∼ 35◦ andθ2 ∼ 11◦

for t=0. Tcr is predicted to be about 80 K. From Eq. 3 we maydeduce thatθ2 will depend only weakly on temperature sinceǫ = ξc/ξab and bothξab andξc have a similar temperaturedependence.

As would be expected, the Josephson string vortices local-ized within thea-b planes are more weakly pinned than thepancake vortices21. In spite of this, a vortex channelling ef-fect cannot be seen inc-axis films since there is never a com-ponent of the Lorentz force directed in the weakly pinneddirection. In the reduced symmetry found in measurementson vicinal films the expected intrinsic vortex channelling ef-fect is observed29,30. As predicted from Eq. 2 the width of

3

the vortex channelling effect has been shown to vary withchanging superconducting anisotropy in calcium doped andde-oxygenated YBCO films31.

III. EXPERIMENTAL TECHNIQUE

Thin films of YBa2Cu3O7−δ were prepared by pulsed laserdeposition. The films all had thicknesses of between 100 and200 nm and were grown on single crystal SrTiO3 substratesmis-cut by an angle,θv, towards the (001) direction. Vicinalfilms of YBCO prepared on mis-cut substrates typically ex-hibit ’step-flow’ growth which leads to a terraced structurewith the terraces perpendicular to the miscut32,33,34,35,36. Itshould be noted that the films had a low density of anti-phaseboundaries and stacking faults, this was confirmed by HREMstudies30. These types of defects disrupt thea-b planes andmake the interpretation of observed critical current in termsof the FLL structure and anisotropy impossible. Such defects,when present, do however strongly increase the observed criti-cal current37,38,39. The different growth modes of YBCO filmson vicinal substrates have been recently discussed by Mauriceet al.36

Current tracks were patterned by photo-lithography andAr-ion milling to allow four terminalIV measurements, thetracks were 200x10µm between the voltage contacts and werepatterned both perpendicular (T) and parallel (L ) to the vicinalstep direction33. Contacts were prepared by sputtering Ag/Aubi-layers. The samples were characterized using a two-axisgoniometer40 mounted in an 8 T magnet. The geometry of themeasurement, for aT track is shown in Fig. 1.

An IV curve was recorded at each set of experimental pa-rameters (B,T ,θ,φ). Critical current values were determinedfrom these curves using a criterion of 0.5µV. The choice ofa voltage criterion to determine critical current,Ic from IVcurves is arbitrary, this value was chosen as the lowest prac-tical value given the noise in the experiment. It was notedthat the overall form of all the observedjc characteristics didnot change when the savedIV characteristics were reanalyzedwith different voltage criteria.

IV. EXPERIMENTAL RESULTS

In order to provide a fuller characterization of thejc(θ)behaviour in vicinal films, and in particular the vortex chan-nelling effect first reported by Berghuiset al.29, jc(T, B, θ, φ)studies were performed on YBCO films grown on substrateswith varying mis-cut angles. The data presented in this articlewere all obtained on (T) tracks, since it is only in this geom-etry that the vortex channelling effect is observed. This oc-curs since there is no Lorentz force component in the weaklypinned direction on the vortex strings in measurements on (L )tracks.

Fig. 2 shows data taken in a 1 T field for a range of temper-atures on a10◦ vicinal film. The vicinal channelling min-imum is prominent and as expected disappears above 80K

θv

n

j

θv j

B

θ φ

c

b

a

B

c

b

a

FIG. 1: Geometry of the measurement of aT track. The upper partof the figure shows the orientation of thea-b planes with respect tothe surface of the film. The mis-cut (vicinal) angle of the substrate isgiven byθv. The lower part indicates how the tilt angleθ and rotationangleφ specify the orientation of the external magnetic field.θ=0 isdefined as when the field is parallel to the planes. Atθ = θv φ=0 thefield is aligned with the current.

0

2

4

6

-80 -60 -40 -20 0

80K70K60K50K40K30K

c

θ (degrees)

B=1 T φ=90

J c/1010

(A

/m2 )

0

2

4

6

8

10

0 20 40 60 80

J c/101

0 (A

/m2 )

B=1 T φ=0

(a)

0

2

4

6

8

10

-5 0 5 10

J c/1010

(A

/m2 )

θ (degrees)

(b)

FIG. 2: jc data taken on aT track patterned on a YBCO film grownon a 10◦ mis-cut substrate. The upperjc(θ) plot (a) was obtainedwith φ = 0◦ and in a 1 T field, the inset shows an enlargementof the channelling minimum. The lower plot (b) was obtained withφ = 90◦. The vortex channelling effect is visible in both plots, in(a) the peak arises from the force free geometry whilst in (b)thepeak which is partly suppressed by channelling arises from intrinsicpinning.

4

2

3

4

5

6

7

8

9

-20 -15 -10 -5 0 5 10 15 20

φ=90φ=80φ=70φ=60φ=50φ=40φ=30φ=20φ=10φ=0

J c/1010

(A

/m2 )

θ (degrees)

B=1 T T=40 K

FIG. 3: jc(θ) near the vortex channelling minimum, at 1 T, 40 Kand varying values ofφ, for a YBCO film grown on a 10◦ mis-cutsubstrate.

(Tcr) when the flux lines are conventional Abrikosov vorticesfor all field orientations29.

In Fig. 2a, withφ=0 the channelling minimum, correspond-ing to the field being aligned with thea-b planes, is offsetby 10◦ from a peak injc, which corresponds to the ’force-free’ orientation. This confirms that the superconducting filmindeed has itsc-axis offset by 10◦ with respect to the sur-face normal. In Fig. 2b whereθ is swept withφ=90◦ thechannelling minimum can be seen to be coincident with theintrinsic peak injc which is consequently suppressed. Thechannelling minimum is seen for allφ values, as is shown inFig. 3. The critical current of the minimum can be seen tobe higher atφ=90◦ than atφ=0◦ whenθ=0◦. Although bothθ=0◦,φ=0◦ andθ=0◦,φ=90◦ are orientations of the magneticfield that give rise to vortex channelling, in the former casetheLorentz force is purely directed along thea-b planes whereasin the latter case there is a component directed along thec-axisas well.

It proved impossible to grow good quality vicinal filmswith a clean microstructure for angles greater than10◦. Insuch films no channelling effect was observed. This is prob-ably due to growth no longer being perfectly epitaxial, thiswill give rise to dislocations37, stacking faults and anti-phaseboundaries39 extending through the film. Such defects willsuppress in-plane vortex channelling30.

Similar data sets were also obtained on 2◦ and 4◦ vicinalfilms. jc(θ) data obtained on 4◦ films is shown in Fig. 4 withvaryingT at 1 T and in Fig. 5 with varyingB at 60 K. Asin the case of the film grown on a 10◦ mis-cut substrate weobserve minima due to vortex channelling in both geometries.As before at higher temperatures the channelling is less pro-nounced.

The variation of thejc(θ) behaviour with field is shown inFig. 5. In theφ=0◦ geometry the effect of vortex channellingis clear and the overall form of the minimum does not changewith varying field, being very sharp. In theφ = 90◦ geometryhowever the channelling effect is not very pronounced, and

0

2

4

6

8

10

12

14

-80 -60 -40 -20 0 20

20 K30 K40 K50 K60 K70 K80 K

J c/1010

(A

/m2 )

B=1 T φ=0 (a)

0

2

4

6

8

10

-80 -60 -40 -20 0 20

20 K30 K40 K50 K60 K70 K80 K

J c/1010

(A

/m2 )

θ (degrees)

B=1 T φ=90 (b)

FIG. 4: Critical current,jc versus field tilt,θ, data recorded at 1 Tand varying temperature on an YBCO film grown on a 4◦ mis-cutsubstrate. In (a)φ was set to 0◦ and in (b) 90◦.

the minimum is less distinct. As the intrinsic pinning peakand the vicinal channelling minimum are superposed this is afunction of the relative strengths of the two effects.

Even with a small mis-cut of 2◦ it is possible to obtain vici-nal growth and consequently observe vortex channelling. Thisis shown in Fig. 6. The close angular proximity of the vortexchannelling minimum and the force-free peak mean that thecritical current increases by a factor of five betweenθ=-2◦ andθ=0. As the minimum due to vortex channelling and the forcefree effect are overlapping the point of maximumjc is shifted.This is also visible to a lesser extent in data recorded on theother vicinal angles. Measurements were not performed onsubstrates with mis-cuts less than 2◦, it is expected howeverthat the vortex channelling effect only disappears when themis-cut angle is small enough that the growth becomes thec-axis island growth typically seen in YBCO thin films.

In summarizing the experimental results it is possible tomake several general observations. The vicinal channellingeffect described in29 in films grown by sputtering has beenreproduced in films grown by pulsed laser deposition. Thistends to suggest strongly that the effect is intrinsic rather thanlinked to a particular growth technique. This channelling ef-fect will therefore be seen in any YBCO sample where thecurrent is not directed along thea-b planes and the crystalstructure is free of defects that disrupt the continuity of thea-b planes30. The channelling effect extends over a range ofmiscut angles, the upper limit being due to the difficulty of

5

0

1

2

3

4

5

-80 -60 -40 -20 0 20

1 T1.5 T2 T2.5 T3 T3.5 T

J c/1010

(A

/m2 )

T=60 K φ=0

0

0.5

1

1.5

2

2.5

-80 -60 -40 -20 0 20

4 T

2.5 T

3.5 T3 T

2 T1.5 T1 T

J c/1010

(A

/m2 )

θ (degrees)

T=60 K φ=90

(a)

(b)

FIG. 5: Critical current,jc versus field tilt,θ, data recorded at 60 Kand varying field on an YBCO film grown on a 4◦ mis-cut substrate.In (a)φ was set to 0◦ and in (b) 90◦.

0

0.5

1

1.5

2

-80 -60 -40 -20 0

40K50 K60K70K80K

J c/1011

(A

/m2 )

B=1 T φ=0

FIG. 6: Critical current,jc versus field tilt,θ, data recorded at 1 Tand varying temperature on an YBCO film grown on a 2◦ mis-cutsubstrate,φ was set to 0◦.

getting epitaxial growth on substrates with a large angle ofmis-cut. As previously observed channelling is suppressedathigh temperatures irrespective of the mis-cut angle. Moreoverthe effect is most easily seen in theφ = 0 configuration wherethe channelling minimum is off-set from the ’force-free’ peak.

3

4

5

8

9

10

-10 -5 0 5 10

Unaltered track

FIB milled track

Nor

mal

Tra

ck J

c/1010

(A

/m2)

FIB

milled T

rack Jc /10

10 (A/m

2)

θ (degrees)

FIG. 7: The vortex channelling minimum measured on a 10 micronwide track in a 6◦ vicinal film at φ=90◦ before (open squares) andafter (closed circles) thinning to≈ 2µm using a focussed ion beammicroscope.

V. TRACK WIDTH

In this article we seek to identify the extent of the chan-nelling minimum with the angular range in which there isa cross-over from rectilinear Abrikosov to kinked string-pancake vortex lines. To support this argument it is necessaryto exclude the possibility that the width of a particular currenttrack has an effect on the channelling minimum by changingthe length ofa-b plane into which a vortex line need align tolock-in. This will be more pronounced in theφ=90◦ geometrywhere the vortex line must align with the 10µm width of thecurrent track.

In theφ=90◦ geometry the channelling minimum was mea-sured on a6◦ vicinal film on a 10µm wide track. The trackwas subsequently thinned in a focussed ion beam microscopeto a width of 2µm. The thinned track was subsequently re-measured. As gallium ions, used in the milling process, canhave a deleterious effect on YBCO theTc of the sample waschecked and seen to have remained the same. The milling wascarried out in such a way as to minimise the amount of galliumspread over the surface of the sample.

The results from this investigation are shown in Fig. 7. Itcan be seen that the width of the channelling minimum is un-affected. The variation in the absolute value of the criticalcurrent may be partly due to uncertainty in the width of thesuperconducting channel, the region around a cut made with agallium beam will also have suppressed superconductivity dueto gallium implantation. This measurement clearly demon-strates therefore that the width of the measurement track doesnot affect the width of thejc minimum associated with vortexchannelling.

VI. DISCUSSION

The existence of a vortex channelling effect in measure-ments on vicinal YBCO films is clear from the series of ex-periments presented in this article. In this section the magneticfield orientation at which cross-over from Abrikosov vortices

6

0

0.5

1

1.5

2

2.5

3

20 K30 K40 K50 K60 K70 K80 K

J c/Jc(φ

=10

)

(a)

0.6

0.8

1

1.2

1.4

1.6

-20 -15 -10 -5 0 5 10 15 20

20 K30 K40 K50 K60 K70 K

θ (degrees)

J c/Jc(θ

=10

)

(b)

FIG. 8: Data from Fig. 4 plotted againstjc/jc(θ = 10◦). (θv = 4◦)

to the kinked configuration occurs will be determined. Theform of the channelling minimum will modelled by consid-ering the Lorentz forces and pinning on the separate vortexelements.

As has been discussed previously the scaling law describedin Eq. 1 is not expected to work particulary well over the en-tire angular range where the vortices are conventional in na-ture. This is primarily due to enhanced pinning atθ = 0◦

from twins and dislocations. However, away from both thisenhanced extrinsic pinning and the cross-over into the kinkedvortex state the primary contribution tojc(θ) is not unreason-ably expected to arise primarily from the anisotropy of thematerial. In Fig. 8 thejc(θ) characteristic with varying tem-perature from Fig. 4 is shown plotted asjc(θ)/jc(θ = 10◦).If the scaling law applies temperature should not change theform of the rescaled curves.

From the figure this does indeed seem to be the case outsidethe range where the behaviour crosses over into the vicinalchannelling regime. In Fig. 8 the region over which thejc(θ)plots diverge has a range of about20◦, this is consistent withthe prediction of Eq. 3 sinceθ2 depends only on the ratiobetween thea-b plane andc-axis coherence lengths. In Fig.8b all the peaks follow the same form with the position of thecrossover into the channelling minimum appearing to dependon temperature.

Given that thejc(θ) characteristic cannot be described interms of the simple G-L scaling equation given in Eq. 1 it isimportant to determine if the behaviour nearθ = 0◦ is con-sistent with kinked vortex lines. If the vortex lines are indeedkinked it should be possible to predict thejc(θ) behaviour byconsidering the Lorentz forces and pinning on the two types

vortex segment, vortex strings and pancakes.In theφ = 0◦ case the external field direction lies parallel to

the current flow forθ = θv and is tilted toward thec-axis asθis varied. For an individual rectilinear vortex line the Lorentzforce is given byjc × Φ0 = fL wherefL is the Lorentz forceper unit length on a vortex line. At the critical currentfL issimply equal tofp, the pinning force per unit length. Thisanalysis assumes the pinning forces arise from strong pinningcentres. In this limit the collective pinning volume is thatofa single vortex and it is reasonable to consider pinning forcesacting on individual vortex lines. In the case of a kinked vor-tex line the Lorentz forces on and the pinning of the separatesegments will be different. For a kinked vortex line spanningtwo pointsl apart and at an angleθ to thea-b planes there willbe a lengthlp = l sin θ of pancake elements andls = l cos θof string elements. Where the vicinal angle isθv the Lorentzforce on the entire line per unit length will be (the Lorentzforces in this geometry being in the same direction on bothelements):

fL = jφ0(cos θ sin θv − sin θ cos θv) (4)

Immediately it can be seen that even a kinked vortex line willexperience a force free effect since atθ = θv fL will bezero. This is indeed what is experimentally observed, howeverexperimentally the critical current does not become infinite.There will therefore be two regimes for angle dependence ofthe critical current in the kinked vortex regime. One will ap-ply where dissipation arises from the vortices depinning, thesecond will arise in the regime near the force free orientationwhere a different dissipative mechanism must apply.

The pinning force available on the vortex lines maybe cal-culated by considering pinning on individual string and pan-cake elements. The pinning force per unit length of vortexopposing the motion of the vortices will be given by:

fp = fp,str cos θ + fp,pc sin θ (5)

where the subscript ’pc’ refers to forces on vortex pancakesand ’str’ to those on vortex strings. Atj = jc fp = fL so wemay write the following equation forjc(θ) combining Eqs. 4and 5:

jc =fp,str cos θ + fp,pc sin θ

φ0(cos θ sin θv − sin θ cos θv)(6)

The fit of the model to data on a 4◦ miscut samples at 40Kand 1 T is shown in Fig. 9, herefp,pc is 8.1x10−5 Nm−1

andfp,str is 4.9x10−6 Nm−1. The model, with simply twoparameters, reproduces the main features of the experimen-tally observed behaviour, however it predicts an infinite criti-cal current when the current is aligned with the field since, asdiscussed above, at this point the Lorentz forces on the stringand pancake segments are opposite and balance out. Threeregions may be identified; in the first the flux lines are rec-tilinear but anisotropic, in the second the vortices are kinkedbut depin conventionally and in the third regime around the’force-free’ geometry the critical current must arise fromanalternative mechanism.

7

0

5

10

15

-20 -15 -10 -5 0 5 10 15 20

Experimental DataModel

J c/1010

(A

/m2 )

θ (degrees)

Kinked vortex depinning

AnisotropicAbrikosovVortices

Kinked vortex cutting

FIG. 9: Comparison of the model treating the pinning on individualvortex segments to experimental data in theφ=0◦ orientation. Theexperimental data was taken on a 4◦ vicinal film at 1 T and 40K.The dotted lines indicateθ2 and the dashed line the point at whichthe direction of the Lorentz force on the string segments reversesdirection.

In this distinct regime, which extends betweenθ = 0◦ andslightly beyond the ’force free’ orientation whereθ = θv, adifferent mechanism must account for the finite critical cur-rent. In this range the Lorentz forces on the vortex string andpancakes segments are in opposite directions. The vortex pan-cakes are strongly pinned and subject to only a small forceper unit length. It is likely, therefore, that dissipation occursthrough cutting and joining with adjacent vortices when eachstring segment is subject to a large enough force. Flux cuttingprocesses have been described by several authors, it is impor-tant to note that at no time in the phase slip process does a’free’ vortex end appear12,41,42,43,44.

If a single vortex string segment is considered it will besubject to certain Lorentz force, which will be opposed by theelastic tension in the vortex string as it is deformed and by thepinning force on the vortex string. When the force on the vor-tex string exceeds a certain value,fcut the vortex string cutsand joins. This force represents therefore the critical currentcondition. A similar mechanism has been identified in grainboundaries in YBCO, which is another system in which vortexchannelling is found45.

Taking a single vortex string in theφ = 0◦ geometry theLorentz force on the vortex string segment will be given byfL,seg = jcφ0d sin θv cos θ/ sin θ. The length of an indi-vidual vortex segment for a particular magnetic field tilt isgiven by d/ sin θ, whered is the spacing between cuprateplanes. In YBCOd is one third of thec-axis lattice param-eter≈0.39 nm. The available pinning is given byfp,seg =fcut + (fp,strd/ sin θ) wherefcut is the force required on asingle segment for cutting and cross-joining to occur. Com-bining these we may write the following equation for theθdependence of the critical current in the vortex cutting region.

jc =1

φ0

[

fcut tan θ

d sin θv

+fp,str

sin θv cos θ

]

(7)

2

4

6

8

10

12

-20 -15 -10 -5 0 5 10 15 20

Experimental DataModel

J c/1010

(A

/m2 )

θ (degrees)

FIG. 10: The figure shows the behaviour predicted by extending themodel shown in Fig. 9 to take account of vortex cutting when theLorentz forces on the string and pancake segments are in oppositedirections. The experimental data was taken on a 4◦ vicinal film at 1T and 40K.

The actually observed critical current behaviour for akinked vortex will be the lower of that predicted by Eqs. 6 and7 for any particular value ofθ. This simply because flux flowoccurs if it requires a smaller Lorentz force than that requiredfor the cutting process. If the simple assumption is made thatthe cutting force required per segment is independent of itslength choosing a cutting force of3.6x10−14 N for the 40 Kand 1 T data shown in Fig. 9 results in a reasonable fit to theexperimental data. This is shown in Fig. 10.

It is clear therefore that there is a region betweenθ = 0◦

andθ ≈ θv where the flux flows by cutting and cross join-ing rather than by depinning entire vortex lines. Treating theflux cutting force per segment as being independent of lengthappears to be a reasonable first approximation.

A similar approach may be used to model the behaviour oftheθ dependence of the critical current in theφ = 90◦ orien-tation by separately considering the forces on the string andpancake vortex segments. Here the field is tilted so that theflux lines are always perpendicular to the current. The forceon the vortex strings will be directed normal to the plane ofthe film while the forces on the pancake segments is directedalong thea-b planes. The strong intrinsic pinning means thatthe component of the force on the vortex strings directed outout of thea-b planes is always opposed by a an equal pinningforce and the forces need only be resolved within the plane.A section of a tilted vortex line is shown in Fig. 11. Thecomponent of the force on the strings within thea-b plane perunit length is given byjΦ0 cos θ sin θv. The force per unitlength on the vortex pancakes is given byjΦ0 sin θ cos θv, thecos θv factor arising from the fact the the pancake segmentsare not perpendicular to the current. The resulting force perunit length on the vortex is then:

fL = jcφ0

√

(cos θ sin θv)2 + (sin θ cos θv)2 (8)

.The pinning force will act in the opposite direction to

this force with the pancake pinning per unit length given by

8

fL,pc=jcφ0 sinθ cosθv

ϑ

2,

2, pcLstrLL fff +=

fL,str=jcφ0 cosθ sinθv

FIG. 11: Schematic of the forces on a segment of kinked vortexline,thec-axis is perpendicular to the page. The figure shows, schemati-cally, a vortex string connecting two vortex pancakes. One pancakewill lie in the cuprate plane above the string the other in thecuprateplane below it. The angleϑ describes the rotation of the resolvedforce on the vortex string within thea-b planes. Forces directedoutside thea-b plane may be neglected as strong intrinsic pinningrestricts the motion of vortex segments in thec-axis direction.

fp,pc sin θ. The string pinning force must take into accountthat the vortex string is only pinned for movement perpen-dicular to its length21 and will be given byfp,str cos θ sin ϑ.From Fig. 11 we can see that orientation of the Lorentz withinthe a-b plane,ϑ will be given bytan ϑ = fL,pc/fL,str. Wemay therefore write the following equation forjc(θ):

jc =1

φ0

(

fp,pc sin θ√

(cos θ sin θv)2 + (sin θ cos θv)2+

fp,str cos2 θ sin θv

(cos θ sin θv)2 + (sin θ cos θv)2

)

(9)

Using Fig. 9 it is possible to compare this model for theθdependence of the critical current to experimental data. Acomparison to data obtained on a 4◦ vicinal film at 40 K and1 T is shown in Fig. 12. Within the region for which themodel is expected to be valid,|θ| < θ2 a good fit to the ex-perimental data is obtained withfp,pc = 8.1x10−5 Nm−1 andfp,str = 6.2x10−6 Nm−1. This is the same pancake pinningforce as theφ = 0◦ geometry but the required string pinningforce is marginally higher. One possible source of this dis-crepancy may be that only the force balance in thea-b planehas been considered. The vortex strings are also subject to aLorentz force which tends to push them in thec-axis direction,this may increase the pinning force against movement in thea-b plane.

VII. CONCLUSION

In the high Tc superconductor YBa2Cu3O7−δ there isa temperature and magnetic field orientation dependentcrossover from a lattice of rectilinear Abrikosov flux linesto alattice of ’kinked’ vortices. These kinked vortex lines consistof vortex pancakes crossing the cuprate planes and Josephsonstrings aligned in the charge reservoir layers.

As a consequence of this, ’intrinsic’ vortex channellingis observed providing the superconducting transport currentdoes not flow parallel to thea-b planes. The minima in thecritical current versus magnetic field angle characteristic oc-curs not because the vortex lines are ’locked-in’ to thea-bplanes over a wide angular range but because the transitionto the kinked vortex state changes the angular dependence of

3x1010

3.5x1010

4x1010

4.5x1010

5x1010

5.5x1010

-20 -15 -10 -5 0 5 10 15 20

Model

Experimental Data

J c (A

/m2 )

θ (degrees)

FIG. 12: Comparison of experimental data with the model predic-tion for an external magnetic field in theφ = 90◦ geometry.Theexperimental data was taken on a 4◦ vicinal film at 1 T and 40K.

the magnitude of the avaliable pinning force. At the anglewhere the vortex lines are expected to align with the planesimperfections in the films and thermal activation probably en-sure that the vortex lines are never entirely string like. Wehave shown that the angular dependence of the critical currentin the ’kinked’ range may, to a first approximation, be con-sidered by summing the Lorentz force on the separate vortexelements and comparing this to the pinning force available tothese elements. The vortex channelling effect arises becausealthough vortex strings are strongly pinned for motion parallelto thec-axis they are comparatively weakly pinned for motionin thea-b planes.

In the regime where the forces on vortex pancakes and

9

strings are opposed it is apparent that a different mechanismserves to limit the critical current behaviour. We argue that it iscutting and cross joining of individual vortex string segmentsthat give rise to flux flow. The vortex pancakes themselves donot flow in this region.

For magnetic fields well away from parallel to thea-bplanes, the properties of the rectilinear vortices may be treatedby using anisotropic Ginzburg-Landau theory to describe howthese elliptical vortices interact with pinning centres. Thejc(θ) behaviour in thin films does not, however, scale usingthe simple scaling law obtained by Blatteret al. since thestrong pinning found in such films is highly anisotropic.

The complexity of the angular variation of the vortex struc-

ture in YBa2Cu3O7−δ is due to this material’s relativly lowsuperconducting anisotropy as compared to most other highTc superconductors. The consequent cross over transitions inthe vortex lattice make it impossible to apply a single scalinglaw to the angular dependence of the critical current.

Acknowledgments

This work was supported by the Engineering and PhysicalSciences Research council.

∗ Electronic address: jhd25@cam.ac.uk1 V. G. Kogan and J. R. Clem, inConcise Encylopedia of Magnetic

and Superconducting Materials, edited by J. E. Evetts (Pergamon,Oxford, 1992).

2 R. J. Cava, A. W. Hewat, E. A. Hewat, B. Batlogg, M. Marezio,K. M. Rabe, J. J. Krajewski, W. F. Peck, and L. W. Rupp, PhysicaC 165, 419 (1990).

3 L. F. Hou, J. Deak, P. Metcalf, and M. McElfresh, Physical Re-view B 50, 7226 (1994).

4 J. L. Tallon, C. Bernhard, U. Binninger, A. Hofer, G. V. M.Williams, E. J. Ansaldo, J. I. Budnick, and C. Niedermayer, Phys-ical Review Letters74, 1008 (1995).

5 J. R. Clem, Superconductor Science and Technology11, 909(1998).

6 W. E. Lawrence and S. Doniach, inProceedings of the 12th In-ternational Confernece on Low Temperature Physics, edited byE. Kanda (Keigaku, Tokyo, Kyoto, 1970), p. 361.

7 G. V. Blatter, G. and A. Larkin, Phys. Rev. Lett.68, 875 (1992).8 R. Herzog, Ph.D. thesis, University of Cambridge (1997).9 G. Blatter, M. V. Feigelman, V. B. Geshkenbein, A. I. Larkin,and

V. M. Vinokur, Reviews Of Modern Physics66, 1125 (1994).10 A. A. Zhukov, H. Kupfer, P. A. J. de Groot, and T. Wolf, Journal

of Low Temperature Physics117, 1513 (1999).11 B. Roas, L. Schultz, and G. Saemannishenko, Phys. Rev. Lett.64,

479 (1990).12 A. M. Campbell and J. E. Evetts, Advances in Physics50, 1249

(1972).13 B. Dam, J. M. Huijbregtse, F. C. Klaassen, R. C. F. van der Geest,

G. Doornbos, J. H. Rector, A. M. Testa, S. Freisem, J. C. Martinez,B. Stauble-Pumpin, et al., Nature399, 439 (1999).

14 J. R. Clem, Physical Review B43, 7837 (1991).15 P. Kes, J. Aarts, V. Vinokur, and C. van der Beek, Phys. Rev. Lett.

64, 1063 (1990).16 A. E. Koshelev, Physical Review Letters83, 187 (1999).17 P. Schmitt, P. Kummeth, L. Schultz, and G. Saemannischenko,

Physical Review Letters67, 267 (1991).18 G. Jakob, M. Schmitt, T. Kluge, C. Tomerosa, P. Wagner, T. Hahn,

and H. Adrian, Physical Review B47, 12099 (1993).19 J. H. Durrell, G. Gibson, Z. H. Barber, J. E. Evetts, R. Rossler,

J. Pedarnig, and D. Bauerle, Applied Physics Letters77 (2000).20 G. Ravi Kumar, M. R. Koblischka, J. C. Martinez, R. Griessen,

B. Dam, and J. Rector, Physica C235-240, 3053 (1994).21 J. H. Durrell, R. Herzog, P. Berghuis, A. P. Bramley, E. J. Tarte,

Z. H. Barber, and J. E. Evetts, Superconductor Science and Tech-nology12, 1090 (1999).

22 M. Tachiki and S. Takahashi, Solid State Communications70, 291(1989).

23 M. Tachiki and S. Takahashi, Solid State Communications72,1083 (1989).

24 T. Nishizaki, T. Aomine, I. Fujii, K. Yamamoto, S. Yoshii,T. Terashima, and Y. Bando, Physica C185, 2259 (1991).

25 T. Nishizaki, T. Aomine, I. Fujii, K. Yamamoto, S. Yoshii,T. Terashima, and Y. Bando, Physica C181, 223 (1991).

26 T. Aomine, T. Nishizaki, F. Ichikawa, T. Fukami, T. Terashima,and Y. Bando, Physica B194, 1613 (1994).

27 D. Feinberg and C. Villard, Physical Review Letters65, 919(1990).

28 L. Civale, B. Maiorov, A. Serquis, J. O. Willis, J. Y. Coulter,H. Wang, Q. X. Jia, P. N. Arendt, J. L. MacManus-Driscoll, M. P.Maley, et al., Applied Physics Letters84, 2121 (2004).

29 P. Berghuis, E. DiBartolomeo, G. A. Wagner, and J. E. Evetts,Physical Review Letters79, 2332 (1997).

30 J. H. Durrell, S. H. Mennema, C. Jooss, G. Gibson, Z. H. Barber,H. W. Zandbergen, and J. E. Evetts, Journal of Applied Physics93, 9869 (2003).

31 J. H. Durrell, J. E. Evetts, R. Rossler, M. P. Delamare, J. D.Pedarnig, and D. Bauerle, Applied Physics Letters83, 4999(2003).

32 W. F. Hu, X. J. Zhao, W. Peng, T. S. Wang, W. Liu, L. Li, C. Lei,and Y. F. Chen, Journal of Crystal Growth231, 493 (2001).

33 L. Mechin, P. Berghuis, and J. E. Evetts, Physica C302, 102(1998).

34 U. Poppe, Y. Y. Divin, M. I. Faley, J. S. Wu, C. L. Jia, P. Shadrin,and K. Urban, IEEE Transactions on Applied Superconductivity11, 3768 (2001).

35 F. Wellhofer, P. Woodall, D. J. Norris, S. Johnson, D. Vassiloyan-nis, M. Aindow, M. Slaski, and C. M. Muirhead, Applied SurfaceScience129, 525 (1998).

36 J. L. Maurice, J. Briatico, D. G. Crete, J. P. Contour, and O. Du-rand, Physical Review B68, 115429 (2003).

37 D. H. Lowndes, D. K. Christen, C. E. Klabunde, Z. L. Wang,D. M. Kroeger, J. D. Budai, S. Zhu, and D. P. Norton, PhysicalReview Letters74, 2355 (1995).

38 C. Jooss, R. Warthmann, H. Kronmuller, T. Haage, H. U. Haber-meier, and J. Zegenhagen, Physical Review Letters82, 632(1999).

39 C. Jooss, R. Warthmann, and H. Kronmuller, Physical Review B61, 12433 (2000).

40 R. Herzog and J. E. Evetts, Review of Scientific Instruments65,3574 (1994).

10

41 M. G. Blamire and J. E. Evetts, Applied Physics Letters46, 1181(1985).

42 C. J. O. Reichhardt and M. B. Hastings, Physical Review Letters92, 157002 (2004).

43 C. Carraro and D. S. Fisher, Physical Review B51, 534 (1995).44 S. A. Grigera, T. S. Grigera, E. F. Righi, G. Nieva, and F. de la

Cruz, Physica C371, 237 (2002).45 J. H. Durrell, M. J. Hogg, F. Kahlmann, Z. H. Barber, M. G.

Blamire, and J. E. Evetts, Physical Review Letters90, 247006(2003).