Cu(NCS

PHYSICAL REVIEW B VOLUME 50, NUMBER 21 1 DECEMBER 1994-1

Evidence for Josephson vortices in (BEDT-TTF)&Cu(NCS)q

Paul A. Mansky' and P.M. ChaikinDepartment of Physics, Princeton University, Princeton, New Jersey 08544

R.C. HaddonATS T Bell Laboratories, Murray Hill, New Jersey 07974

(Received 16 May 1994)

We present evidence for and study the properties of coreless Josephson vortices (parallel to thesuperconducting layers) in the organic superconductor (BEDT-TTF)&Cu(NCS)s, using ac suscepti-bility measurements in the mixed state. We observe (1) extremely weak-pinning restoring forces forfiux motion parallel to the layers, due to the absence of the vortex normal core; (2) single vortex(noncollective) pinning, due to weak interactions between Josephson vortices; (3) the "lock-in" ofthe vortices parallel to the layers when the dc field is applied at an arbitrary angle; and (4) highlynonlinear response to the ac Seld h in the lock-in state, above a threshold of h 0.5 G. The be-havior of tilted Bux lines is dominated by the much stronger pinning and collective interactionsof two-dimensional Abrikosov "pancake" vortices, and linear response is largely restored when thefiux lines unlock from the layers. We measure the Josephson penetration depth to be Az(T = 5

K) 20Q tom, yielding a penetration depth anisotropy of p = A~/A~~ 16Q —350.

I. INTRODUCTION

Many superconductors of recent interest have layeredcrystal structures, and are believed to consist of two-dimensional superconducting planes coupled by Joseph-son tunneling. This has important efFects on the struc-ture and behavior of magnetic vortices. For Hux linesmaking a large angle with the layers, the conventionalpicture of a vortex as a continuous, rigid fiux tube mustbe replaced by a description of a Hux line as a stackof two-dimensional "pancake" vortices in the individuallayers [Fig. 1 (top)]. These resemble a cross sectionalslice of a conventional Abrikosov vortex. The field andsupercurrent are distributed over a circular area with ra-dius A~~, the penetration depth associated with in-planesupercurrents; and the superconducting order parameteris suppressed within a normal core region of radiusthe in-plane coherence length. Pancake vortices in adja-cent layers interact only weakly, via their magnetic fieldsand the tunneling currents, leading to greatly enhancedfIuctuations of the vortex lines, and this can strongly af-fect the dc magnetization, s'4 transport, s fiux creep, s andother properties.

The interesting case of a magnetic Geld parallel ornearly parallel to the layers has been fairly well stud-ied theoretically, but has received much less experimentalattention. From the theoretical point of view, fIux linesparallel to the layers are expected to difFer &om perpen-dicular vortices in two important ways. First, all lengthscales for the vortex lattice are highly anisotropic, beingstretched out parallel to the layers. Endividual vor-tices have elliptical cross sections of area A~A~~, whereA~ ——

pA~~ is the Josephson penetration depth, associatedwith weak interlayer currents [Fig 1(bottom. )]. The pen-etration depth anisotropy p & 1 is a key parameter for

characterizing a layered superconductor. The vorticesare located on a distorted triangular lattice, with lengthscales /4o/pB in the direction normal to the layersand gp@o/B parallel to the 1 ayers, sand this has beenconfirmed by decoration experiments. ~

Second, and more importantly, a vortex line parallel tothe layers is believed not to have the usual normal coreregion. ~~'~2 This is often attributed to the fact that theinterlayer spacing s exceeds the coherence length defined

sss~assssm$$ 1

LJ = yS

FIG. 1. Top: a tilted Bux line consists of a stack oftwo-dimensional Abrikosov "pancake" vortices, each of whichcontains a normal core (dark area). The vortex makes anangle P with the layers. Bottom: structure of a Josephsonvortex, viewed along the Seld direction. The superconduct-ing order parameter maintains its full value everywhere on thelayers The Josephson ."core" (shaded area) is the region inwhich the tunneling current is comparable to the Josephsoncritical value Jo and has width LJ = ps )

0163-1829/94/50(21)/15929(16)/$06. 00 50 15 929 1994 The American Physical Society

15 930 PAUL A. MANSKY, P. M. CHAIKIN, AND R. C. HADDON 50

by g~ ——(~~ /p below some crossover temperature, approx-

imately given by (T/T, ) = [1 —((~~(0)/ps) ] ~, so thatthe normal core "fits between the layers. " More precisely,the divergence of the current and magnetic 6eld near thecenter of the vortex is cut off when the interlayer currentbecomes comparable to the Josephson critical current Jo.The in-plane currents remain small compared to the crit-ical value J, o cH, /A~~, at which the kinetic energydensity would become comparable to the condensationenergy H2/8x. (In a conventional Abrikosov vortex, thesupercurrents become comparable to J, o at r (, andthe order parameter goes to zero in order to avoid thedivergence and minimize the total energy. ) The orderparameter can thus maintain its full value everywhere onthe superconducting layers. The length scale over whichthe tunneling current is comparable to and limited by Jois the Josephson length I~ = ps ) (~~, and I~ replaces

(~~ as the length scale for the Josephson vortex core in thedirection parallel to the layers [Fig. 1 (bottom)]. Vorticesparallel to the layers are in many ways similar to vorticesparallel to a single Josephson junction and are referredto as Josephson vortices.

The absence of the normal core for Josephson vorticeshas its most dramatic effects for lux lines tilted at verysmall angles P & tan (1/p) relative to the layers (see,for example, Refs. 14—19). In this angular regime, thevortex core consists of alternating two-dimensional (2D)normal cores and Josephson cores, linked together in astaircase fashion. One well-known experimental conse-quence of this is intrinsic pinning. ' A large energybarrier opposes the motion of a Josephson vortex throughthe superconducting layers, leading to a suppression ofthe in-plane resistivity when the flux lines are nearly par-allel. Closely related to this is the magnetic lock-in ef-fect, in which the flux lines are trapped parallel to thelayers even when the external field is applied at an angle.A transition to tilted vortices occurs when the appliedperpendicular field component exceeds a threshold value.The lock-in effect was first observed experimentally in Bi-2212 by Zavaritskii and Zavaritskii, through dc magne-tization measurements. Anomalies in torque magnetiza-tion measurements have been observed in various high-T,compounds at small tilt angles, and attributed to the ef-fects of the layered structure. '2 Beyond this, there havebeen few experiments which directly probe the propertiesof Josephson vortices and relate them to the currently ac-cepted theoretical models.

In a previous publication, we presented evidence forthe lock-in effect in the layered organic superconductor(BEDT-TTF)2Cu(NCS)2, hereafter referred to as "ET."The experimental probe was low frequency ac magneticsusceptibility, which allows the measurement of the vor-tex pinning force "spring constant" k„,and the signatureof the lock-in effect was a striking nonmonotonic behav-ior of the susceptibility (see below). The quantitativeinterpretation of the data focused almost exclusively onthe angle and temperature dependence of the threshold6eld for the lock-in transition, and the details of the field,angle, and temperature dependence of the susceptibilitywere discussed only qualitatively. The data and analysispresented in this paper provide further strong evidence

for the existence of Josephson vortices in ET, and showthat the behavior of parallel flux lines is quite differentthan that of tilted flux lines in several respects. We listhere a summary of our key findings:

Josephson penetration depth and anisotropy. We mea-sure the penetration depth associated with interlayer cur-rents to be A~(5 K) 200 pm, by studying the sam-ple size dependence of the susceptibility in the Meissnerstate. Combined with A~~ 0.6—1.2 pm &om other mea-surements, this gives an anisotropy of p 160—330, inagreement with the result p & 200 &om torque magneti-zation measurements.

Anisotropy of the microscopic pinning mechanism.

The pinning force constant k„for Josephson vorticesmoving parallel to the layers is extremely weak, due tothe absence of the normal core; perpendicular vortices

are pinned much more strongly, k+/k„500. We relatethe measured force constants to the appropriate energyand length scales for the vortex core: the condensationenergy H, /8vr and coherence length (~~ for perpendicularvortices, and the Josephson coupling energy hjo/2e andJosephson length I g ——ps for Josephson vortices. Basedon our experimental results, we estimate the depinningcritical currents for both types of vortices, and obtainfair agreement with the results of remnant magnetiza-tion measurements.

Anisotropy of collective sects and the shear modulus.

Despite the weakness of Josephson vortex pinning, k„isindependent of the vortex density (field strength), show-

ing that the interactions between vortices are even weakerand do not affect the pinning. On the other hand, thepinning force decreases with field for perpendicular vor-tices, which we attribute to collective effects. This is inagreement with calculations which show that the shearmodulus of the vortex lattice is a factor of p 10smaller in parallel fields as a result of the anisotropy.

Magnetic lock-in effect. A magnetic field applied atan arbitrary angle with the layers initially penetrates thesample only in the form of weakly pinned Josephson vor-

tices, leading to a rapid decay of ac screening. The vor-tices unlock &om the layers when the perpendicular fieldcomponent exceeds a threshold, and the screening recov-ers due to the onset of strong pinning of the pancakevortices.

Scaling with normal field component Well above t.helock-in threshold, for angles larger than about 5, theac susceptibility is determined by the perpendicular 6eldcomponent, including the collective pinning effects. Thisshows that both the pinning and interactions of tiltedflux lines are determined by the properties of the pancakevortices. We present a simple model which semiquantita-tively reproduces the observed phenomenology over thewhole range of fields and angles.

Nonlinearity in parallel fields We observe hig. hly non-linear response (strong suppression of screening) to acfields exceeding h 0.5 G in the absence of perpendic-ular flux in the sample, i.e., in zero dc 6eld or in thelock-in state. Linear behavior is largely restored whenthe flux lines unlock &om the layers. While not yet fullyunderstood, this effect further establishes that Josephson

50 EVIDENCE FOR JOSEPHSON VORTICES IN (BEDT-TT'F)2Cu(NCS) 2 15 931

vortices dier qualitatively from tilted vortices.The experimental apparatus and techniques are de-

scribed in reference Ref. 24. The insets to Figs. 6 and 10show the configuration of the experiment and the shapeof a typical crystal. The measurements described in thispaper were made on a single sample, with dimensions1.2 x 0.8 x 0.15 mm, but all of the phenomena reportedhere have been observed in several samples (althoughmeasurements were made below 4.2 K on only one sam-ple). The ac susceptibility was measured by a standardmutual inductance technique. The ac field h was parallelto the conducting layers and the longest sample dimen-sion, and perpendicular to the dc field H. The samplecould be rotated about the ac field axis with 0.0025' reso-lution, allowing the angle between H and the conductingplanes to be varied while keeping H J h. The ac Geld

amplitude and &equency were 0.1 G and 2.5 kHz, exceptwhere otherwise noted.

An overview of the properties of the organic supercon-ductor (BEDT-TTF)2Cu(NCS) 2 is given in Ref. 28. Thecritical temperature T, 10 K, and the interlayer repeatdistance is s 15 A. Other properties (upper and lowercritical fields, coherence lengths, penetration depths) willbe discussed at appropriate points in the text.

II. AC SUSCEPTIBILITY OF ANISOTROPICSUP ERCONDUCTORS

We begin with a brief discussion of the ac susceptibil-ity of superconductors, including the eKects of anisotropyand of elastic vortex motion in the mixed state. Dis-sipative vortex motion will be mentioned only brieBy.Throughout this paper, we will use H and h to denotethe applied (external) dc and ac fields, respectively, andB and b for the internal dc and ac magnetic fields. 8 willdenote the angle between H and the conducting layers,

and P is the angle between the layers and B [see, Fig.13(a)].

The ac magnetic susceptibility of a superconductor,y = M, /h, is never exactly equal to —I/4m, the valueassociated with complete field exclusion (perfect Meiss-ner efFect). The applied field decays exponentially intothe interior of the specimen, over the length scale of thepenetration depth A. If the field is applied parallel to aninfinite slab of thickness L, it is easily shown (see Ref.29, for example) that the susceptibility per unit volumeis given by

2A (Li—4m' = 1 ——tanh~

—~,L i2A&' (2.1)

$2b (92bA„+A, —b= 0.y z

(2.2)

Following the derivation of Clem and Coffey for theisotropic case (A„=A, ), we obtain the equation

which depends only on the ratio A/L. The "screeningstrength" —4m' is a conveniently normalized form of thesusceptibility, with —4xy = 1 indicating complete exclu-sion of h (A « L), and —4m' = 0 indicating complete acfield penetration (A» L).

If the sample has a finite cross section L„XL, butthe penetration depth is isotropic [Fig. 2(a)], Eq. (2.1)can still be used as long as the aspect ratio is largeand Geld penetration across the smaller sample dimen-sion (which we call L, ) dominates. For a superconduc-tor with a large penetration depth anisotropy, however,the situation shown in Fig. 2(b) can occur, in which

A„/L„&A, /L„and field penetration along the longer tJ

direction can significantly contribute to or even dominatethe permeability of the sample to the ac field. A moregeneral equation for y is therefore needed, which can beused for arbitrary values of Av/L„and A, /L, . We beginwith the anisotropic London equation for b

~[ z,

2A, ( L, l 16A„. tanh(q„L„/2)L, ~ 2A 'L„~.(2n+ I)'(k2A2+1)»2'

u(2.3)

with

2n+ 1k„=L

jr andk„'A,'+ 1

Ly

f

O

1&

Lzlk

It is easily seen that Eq. (2.3) is a function only of Az/L&and A, /L, We have check. ed both analytically and nu-merically that this equation gives physically reasonableresults. If either A; = 0, Eq. (2.3) reduces to the formof Eq. (2.1), and if either or both A; -+ oo, y = 0. Wehave numerically verified that the results of Eq. (2.3) aresymmetric under the interchange of the labels y and z.Keeping terms to n = 50 was found to be more thansufBcient summing to higher n did not lead to any

Xy~

~,':,'::,:::::;:&.:',.:,-:,-:.,.'.',:.'..:.'.~.'-,.:'.:'-.",.,''.'

~Z

FIG. 2. Field penetration (shaded areas) in isotropic andanisotropic superconductors with rectangular cross sections.(a) For an isotropic superconductor, Beld penetration fromthe edges can be ignored if L„))L, . (b) This is not the casein an anisotropic superconductor if A„/L„))A /L .


observable changes in the results of calculations usingEq. (2.3).

The above discussion holds regardless of the mecha-nism of ac field penetration. In the Meissner state, A„will be the Josephson penetration depth A~, and A,the I ondon penetration depth A~~. In the mixed state,the motion of vortices in response to the ac field pro-vides another mechanism for ac field penetration. Wewill base our discussion of this on the simple model firstproposed by Campbell and later elaborated by otherworkers. These are mean field models in the sensethat the starting point is a single-vortex equation of mo-tion. The parameters of the model are the pinning restor-ing force "spring constant" k~ and the viscosity rl (seebelow), which are assumed to be the same for all vor-tices. Recent experimental work ' has shown that thisapproach does not at all correctly describe the observedfrequency dependence of the complex impedance, whilemodels based on a scaling approach ' work much bet-ter. Nonetheless, in the limit of low &equencies and lin-ear response, the "mean field" approach seems to be anadequate framework for interpreting our results. Lim-ited studies of the frequency dependence of the complexsusceptibility in ET (Ref. 40) show a weak, roughly log-arithmic frequency dependence, similar to that observedin Ref. 37 and Ref. 41. Order of magnitude changes inthe frequency typically lead to only small changes in thesusceptibility, which we take as justification for using theapproximation of linear, elastic vortex motion.

We discuss the situation shown in Fig. 3, with theac and dc fields perpendicular to each other; this cor-responds to all of the experiments in the mixed statedescribed in this paper. The dc field H establishes thevortex lattice and determines the average vortex den-sity and orientation. The perturbing ac field h inducesa screening current near the surface, and this exerts aforce on the fiux lines of J x 4o/c per unit length, which

h „+

causes them to tilt in the direction of the ac field. Thisforce is balanced by pinning and viscous forces, accordingto the equation

1—J x 40 ——k„u+qu, (2 5)

where u is the displacement of the vortex from its equi-librium location. The pinning force constant k„and vis-cosity g are per unit length of vortex. The tilting of theAux lines leads to enhanced penetration of the ac field.In the low frequency limit of this model, ~ (( ufo

——k„/q,the viscous forces can be ignored. The vortex displace-ments and the field and current distribution must becalculated self-consistently, and several workers ' haveshown that the resulting total penetration depth is

A2 A2 A2 A2Bc'0

a,c + v +4K p

(2.6)

This is the equation which we will use throughout thispaper [together with Eq. (2.1) or (2.3)j to determine thepinning force constant kz from the susceptibility and pen-etration depth. The first term on the right hand side ofEq. (2.6) is the square of the London or Josephson pene-tration depth (depending on the geometry of the experi-ment), and the second term gives the square of the vortexmotion contribution to the penetration depth A„.

This result applies to purely elastic and reversible vor-tex motion, and the complex susceptibility y = y' + i y"is real (y" = 0) and frequency independent. We do infact observe a small but finite dissipation y" g 0 (Fig. 6),and a weak, logarithmic frequency dependence. How-ever, the magnitude ~A~ of the complex penetration depthA = iA~e' can be determined from y' alone as long asy" is not too large. The phase angle b can have anyvalue between b = 0 (elastic dynamics) and b = vr/4 (vis-cous dynamics, pure fiux fiow resistance). The compleximpedance Z iA is thus real (resistive) for b = 7r/4 andimaginary (reactive) for b = O.ss's Figure 4 shows thereal and imaginary parts of the susceptibility as a func-tion of ~A~, for several values of b. The calculations weremade by expanding Eq. (2.1) into its real and imaginary

&) eehit ~B

(b)

0.8-C3(fl

(a 0.6

0-4-

FIG. 3. (a) Relationship between ac field penetration andflux line pinning. The total external field H+ h executes asmall amplitude tilting motion (top). Left: rigidly pinnedflux lines: B = H inside while B = H+ h outside, and so

—1/4ir. Right: if there are no forces opposing 6uxline motion, then B = H + h inside the sample and y, = 0.Center: in the general case, pinning forces lead to a vortexcontribution A„to the ac penetration depth. (b) Geometryfor discussion of angular dependence of A„.

0.2-

l 010 10 10

kl/D10

FIC. 4. Real and imaginary parts of the complex suscepti-bility y(A) vs ~A~, with A = ~Aie', for b = 0 (purely reactiveimpedance), s'/16, s/8, and s'/4 (purely resistive impedance,A =classical skin depth). X'(A) depends only weakly on b for8 ( 7r/8.

50 EVIDENCE FOR JOSEPHSON VORTICES IN {BEDT-TTF)xCu{NCS)x 15 933

parts. 2s It can be seen that for h' & vr/8 (correspondingto a small peak in y"), y' is determined to good accuracy

by ~A~ only, as claimed above.If the vortices make an angle P with the surface but

are still perpendicular to the ac field [Fig. 3(b)j, Eq. (2.6)is replaced by

B@pA, =A + sin (2.7)

III. MEASUREMENT OF THE JOSEPHSONPENETRATION DEPTH

where we have allowed for a possible angular dependenceof the pinning force constant k„.One factor of sin P arises&om the angular dependence of the Lorentz force be-tween the vortex and the surface currents, F = J x 4p/c,and the other is essentially due to the fact that the ac6eld is most effectively transmitted along the direction ofthe Hux lines, which are parallel to B. A„is zero when thevortices are parallel to the surface ac screening currents.

K T,/2. The real part of the susceptibility y'(T) wasmeasured for five ET samples of various sizes, in the ge-ometry shown in the inset to Fig. 5(b). The apparatuswas calibrated using reference samples of three conven-tional superconductors, slabs of Nb, Pb, and Ta with di-mensions sixnilar to the ET crystals. The reference sam-ples gave sharp transitions in y' at the correct T 's, andthe ratio of the step in the coil output to the samplevolume was the same for all three saxnples, to about 5%accuracy. The reference samples were thus assumed tohave a susceptibility of —I/4vr per unit volume.

Measurements were made on five ET crystals. The ab-solute values of the screening strength per unit volume

47ry'—(T) for three of the samples are plotted in Fig. 5(a).Recall that —4xy' = 1 indicates a perfect Meissner effect.The transitions are broad, the screening does not satu-rate down to T = 4.5 K T,/2, and the screening per unitvolume is weaker for smaller samples. The same data areshown again in Fig. 5(b), normalized to approximatelycoincide at T = 5 K. It can be seen that the transition issharper for larger samples. All of these features are qual-itatively consistent with the behavior expected if one of

The in-plane penetration depth A~~ of (BEDT-TTF)qCu(NCS)2 has been measured by muon spinprecession, reversible dc magnetization, and mi-crowave cavity perturbation, which give values of A~~ (0)ranging from 0.65 pm to 1.2 pm. The anisotropy p canthen be determined in a number of ways. The upper crit-ical fields at T = 0, determined &om the transition in thein-plane resistance, 4s are H,2(0) 20 T and H, 2 5 T,yielding p 4. A similar anisotropy was obtained &omspecific heat measurements of the critical 6elds close toT,. (A larger anisotropy, p 20, is obtained f'rom theresistive phase boundaries near T„due to the positivecurvature of the resistive phase boundary. 4s) However, pis related to the critical 6eld anisotropy in this way only

if H, 2 is determined by orbital effects, e.g. , the overlapof the normal cores. The critical 6eld measurements give

$~~(0) 80 A and Q = (I/p —20 A. s, suggestingthat ET is at least close to the limit of weakly coupledlayers and that soxne other mechanism might determine

H,z. We note that H,2(0) is fairly close to the Paulipair-breaking limit of (18.4 kG/K) xT, .

A xnuch larger anisotropy, p & 200, was obtained &omreversible torque magnetization experixnents, implyingthat A~ 130—240 pm. Kanoda et al. measured A~(T)of ET directly, using calibrated low &equency ac suscep-tibility xneasurements in the Meissner state, and found amacroscopic value of A~(0) 1 mm. Although this resultwas controversial, it is at least consistent with the lowerlimit &om the torque measurements. dc susceptibilityexperiments had previously provided evidence for macro-scopically large Josephson penetration depths in the or-ganic superconductor (TMTSF)zC104 (Ref. 48) and inthe intercalation coxnpound TaSz(@II)xyz.

We have used the method of Kanoda et al. for mea-suring A~. We focus on the size of the penetration depthat one temperature, not the details of the temperaturedependence, and have taken data only down to T 4.5

0.8

¹3o ¹1

8

0.2- ¹4 ooI 84O

(s)

I I I

T(K) 10

(b)

e 0.8-lk

II

0.4

0 I I

T (K)10

0.8-

Q 06-

0.4-

0.2-

(c)

00 L (mm)

Y

FIG. 5. (a) Absolute value of the screening strength —4z'g'for three samples with difFerent lengths L„.(b) Same data asin (a), normalized to coincide at T = 5 K. (c) —4z y' (5 K) vsL„for all five measurements. The fit is described in the text,and gives A~ (5 K)=200 ym.

15 934 PAUL A. MANSKY, P. M. CHAIKIN, AND R. C. HADDON

the penetration depths is comparable to the appropri-ate sample dimension. Because the in-plane penetrationdepth A~~(0) - 1 pm is much smaller than the typicalsample thickness L 100 pm, field penetration normalto the layers cannot be responsible for this behavior. Wetherefore attribute it to a macroscopically large penetra-tion depth A~ associated with interlayer currents, andassume that the situation is as shown in Fig. 2(b).

The value of A~ at T = 5 K was obtained by fittingy'(T = 5 K) from all five samples to Eq. (2.1), withthe penetration depth A~(T = 5 K) as the only fittingparameter [Fig. 5(c)]. The quality of the fit is fairly good,showing that the susceptibilities of all of the samples canbe explained by the same value of the penetration depthA~, and the result is A~(5 K) 195 6 20 ym. Thepenetration depth A~ and Josephson critical current Jqare related by Jo ——c@o/8x2sA~&, which gives Jo 400A/cm2. Given the range of reported values 0.55 ym &

A~~& 1.2 pm, using A~(0) 200 pm gives 160 & p ( 350

for the anisotropy. This is consistent with the torqueresult, p & 200. Tokumoto et al. observed similarbehavior of the dc susceptibility with a weak field parallelto the layers, and we obtain A~(T = 5 K) 200 ym Roman analysis of their data. (The two largest dimensions oftheir sample were 1.2 and 0.7 mm. We use I„=1 mmsince the orientation of the field is not specified in Ref.50.) We can only attribute the discrepency between ourresult and the value of 1 mm of Kanoda et a/. 47 to adifference in sample quality. Cracks in a sample wouldreduce the effective sample size, leading to enhanced acfield penetration and an overestimate of A~

Bbo 1600

5(9CS0)

lQ 3

o~o

FIG. 6. Real and imaginary parts of the complex suscep-tibility at T = 4.2 K, with the dc magnetic field applied (a)normal to the layers, and (b) parallel to the layers.

IV. PINNING MECHANISM ANISOTROPY

Application of a dc magnetic field 8 either parallel orperpendicular to the conducting layers (and perpendic-ular to the ac field h) causes a monotonic decrease ofthe screening strength —4m'', as shown for T = 4.2 K inFigs. 6(a) and 6(b), but the behavior of y(H) in thesetwo orientations differs in several ways: (1) The screen-

ing —4ay'(H) decreases much more rapidly for H paral-lel to the layers. This is the opposite of the anisotropyin the resistively determined critical fields H+2 2 Tand H, 2 17 T. The midpoint of the "transition" iny'(H) for H parallel to the layers occurs at H 100 G,a factor of 10 loioer than the parallel field resistivetransition at 17 T. (2) The form of y'(H) is quali-

tatively different for the two orienations, with a roughlyGaussian shape for perpendicular H, and positive curva-ture throughout, and a long tail at high fields for parallelH. (3) The peak in the imaginary part g"(H) is largerfor perpendicular H. (4) The field above which y'(H)is reversible (for H increasing and decreasing) is muchlower for the perpendicular orientation. In this section,we present data in both orientations over the tempera-ture range 2.4—9 K, and focus on the interpretation offeatures (1) and (2) in terms of the anisotropy of the mi-

croscopic pinning mechanism and of the shear modulusfor parallel and perpendicular vortices. The differences

A =A (T)+4~k~II (T)

(4.1)

with the pinning force constant kz a constant for each fit.The fit is very good at all temperatures, and the values of

k~ extracted from these data sets (and others not shown)II

are showa in the inset to Fig. 7(a). The typical value

of the restoring force constant is k„~10 dyn/cm .

in dissipation and reversibility will not be discussed indetail.

In Fig. 7(a) we show —4xy'(H) for several tempera-tures between 2.4 K and T„with H parallel to the layers.The susceptibility for H = 0 is determined by A~ (T)/L„,where A~(T) is the Josephson penetration depth (as inFig. 5), while the decay of y with increasing field isdue to vortex motion (Sec. II). The vortices are par-allel to the layers, and are driven by interlayer currentsBowing at the edges of the sample. The resulting tiltingmotion parallel to the layers leads to enhanced ac fieldpenetration aloag the y direction only, not along z [iasetto Fig. 7(a)]. With reference to Fig. 2, A, = A~~(T) is in-

dependent of field and A, /L, « 1. We may therefore en-

tirely neglect ac field penetration from the faces, and usethe simple one-dimensional equation (2.1) for y(A„/L„).The solid lines are fits to Eq. (2.1) with the field- andtemperature-dependent penetration depth given by Eq.(2.6),

50 EVIDENCE FOR JOSEPHSON VORTICES IN (BEDT-I IF)zCu(NCS)~ 15 935

0.6

0.4-

& 0.2-

I

00

%~~010 OOaaaOa a a aaaaamI I

: (b)~ ~ ~ i

I

0.1 -.-4 4 Jb

~ %4404

T 3.0, 4.0,) 5.1, 8.6, 9.1 K —

pO

~ a aaa

H (G)

f4~I I

Ly

~ ~~ T(r)s

a 4 a a a a

800

~ ~

obtained from power law fits are shown in Fig. 7(c), andhave an average value of 0.45. This is consistent withthe good quality of the fits shown in Fig. 7(a), for which

P is fixed at 0.5. The observation that A„~~H and

kz const for parallel H implies that the pinning forceII

for each vortex is independent of the total vortex density, and is therefore unaffected by interactions betweenvortices. This is the case of "single-vortex pinning, " inthe terminology of collective pinning theory. We willdiscuss the self-consistency of this finding in Sec. IV A.

Figure 8(a) shows a similar data set for the perpen-dicular field orientation. Again, the susceptibility in zerofield is determined by ac field penetration in the y di-rection, due to the large Josephson penetration depthA~(T) = A„.The vortices are now normal to the largesurfaces of the crystal, and interact with in-plane surfacecurrents; so the resulting tilting motion leads to addi-tional ac field penetration in the z direction. The fulltwo-dimensional susceptibility formula (2.3) must there-

0.6

0.5--(c)

~ ~ ~4 ~ ~

- e oa+

0.01 -. '-

2.4 K~ ~ ~ ~ ~ ~ ~

10 100H (G)

ng):1000

~ r g

0.60.5-~~1~

0A- ~g~ 0.2-".-

i 0.1-'~'

(a)

0.32 6 T(K) 8 10 1000

H (G)2000

FIG 7 (a) —4s A'(H) with H parallel to the layers, at »r-ious temperatures (points). Fits are described in text. Inset:values of k„(T)from fits. Vortices allow enhanced ac fieldpenetration along y, and A' is determined by A„/L„.(b) Vor-tex motion contribution to the ac penetration depth vs Geld,extracted from susceptibility data as described in text. Solidline shows A„~Bbehavior expected for single-vortex pin-ning. Fits to a power law Geld dependence, A ~ H~, give anaverage power P 0.45 (c).

This is extraordinarily small compared to typical valuesfor the high-T, superconductor YBCO: k„(T= 0) 10dyn/cm for single-crystal YBCO,si and k„1dyn/cmsfor vortices trapped between gruin8 in granular YBCO.We discuss the relationship of A:„to the microscopic pin-ning mechanism in Sec. IVB.

We obtain the field-dependent part of the penetrationdepth from the data by numerically inverting Eq. (2.1) toobtain A„(H,T), using L = L„=0.8 mm. (We neglecty" in determining the penetration depth, as discussedin Sec. II.) This was done using an algorithm whichchooses successively better values of A„until finding onewhich gives the correct value of y' to an accuracy of 10From Eq. (4.1), the vortex contribution A„(H) is then

A„(H)= vt„(H,T) —Ai(T~). A„(H)is plotted onalog-log scale in Fig. 7(b), and is well described by a powerlaw A„(H) Hr at all temperatures. The values of P

10, '

K 7.1K 6.2K5.1KI ~ O

(b) 8.6 K9K10E

10

10

0 0cP «.0K

3.5K$.9K~R.4K

os~ggg~

X, a B(single vortex pinning)

1000

o 4oo 4

0 ooo4

4oo 4 44J tb4 4 44 a 4 4 g~ eoa4 ggaI4 ~ a ~ o o o a aoootxi

~ ~ aoooo gQQQ

~ ~ 04 44444

~ ~~O~~

00

1010

I

H (G)

3-2-

ff=l /2 (single vortex pinning)

T (K)

FIG. 8. (a) —4s.y'(H) with H perpendicular to the layers,at various temperatures. Solid lines through the data are Gtsto power law field dependence of A„;see text. (b) A vs H,extracted from the data of figure (a) as described in text. A„increases faster than ~H, indicating collective pinning effects.(c) Temperature dependence of power P, obtained by fittinghigh-field data in (b) to a power law, A„(H) HS.


fore be used in analyzing the data. The form of y'(H)varies qualitatively as the temperature is varied, and re-sembles the parallel field data only at the lowest temper-ature studied, 2.4 K. At this temperature, the suscepti-bility can be fit using Eq. 2.3, with A„=A~(T) =constand A, (B) = Bgo/4+k„, resulting in k„(2.4 K) 0.7

dyn/cm and k /kz 500 at this temperature. Tobetter understand how this anisotropy is implied by thedata, recall that the decay of the susceptibility is deter-mined by A„/L„whenthe vortices are parallel to the lay-ers, and by A, /L, when they are perpendicular. Thus fora fixed value of the screening strength (e.g. , —47ry' = 0.3,occurring at fields B~~ 80 G and Bz 1500 G), it must

be true that the vortex contributions to A, /L, and A„/L„are equal, or

II (4.2)L2kll L2k-L

Q p z p

The pinning anisotropy is thus determined by boththe ratio of field scales for the decay of screening,

and the square of the sample aspect ratio: k„/k~II

(B&/Bl*)(Lz/L, ) = (1500/80) (0.8/0. 15) 500. We

will discuss the relationship of the measured kp to themicroscopic pinning mechanism in Sec. IV B.

At intermediate temperatures (about 3.5 —7.1 K ), adifferent type of behavior occurs in perpendicular fields,with a "bump" at low fields and a roughly Gaussian orbell shape above this. Above 7.1 K, A" (H) again has

positive curvature everywhere (as at the lowest temper-ature), but now goes to zero more abruptly, without thegradual tail which occurs at other temperatures. Thefield at which y' ~ 0 is much smaller than the resis-tively determined critical field H, 2(T) at all tempera-tures. A'(H) is reversible with increasing and decreasingdc field above the low-field bump and irreversible in thelow-field region [Fig. 6(a)]. The magnitude of the re-

versibility field is about double that of reported values

of the lower critical 6eld H+z, as determined by dc mag-

netization experiments, ' and has similar temperaturedependence (with positive curvature). We focus on the

behavior at high 6elds, and do not discuss the low-field

region further.With Az

——A~(T) determined from y'(H = 0), A„=A, (H, T) can be determined from the data by numericallyinverting Eq. (2.3), using the same type of algorithmused in analyzing the parallel field data. We again ig-nore the imaginary part y". The results of this procedureare shown in Fig. 8(b), on a log-log plot. The data forT & 7.1 K appear to be approaching power law field de-pendence A„H~at high fields, but with a temperature-dependent power P. The asymptotic value of P is plot-ted against the temperature in Fig. 8(c). A„(H) in thereversible region was fit to the form A„(B)= a+ 6H~ forT & 4.2 K, and the parameters obtained from the fitswere used to calculate the solid lines in Fig. 8(a). Notethat the largest values of A, correspond to the small-est values of —4m'', and the sensitivity is greatest when

I, . The sensitivity of the measurement is lost whenA, becomes much larger than I, = .015 cm, and this iswhy data are not available for higher fields at higher tem-

peratures.We cannot at present quantitatively account for ob-

served field and temperature dependence, but can makesome general statements. One expects that A„yBonly if k„is field independent, and this implies single-vortex pinning. Whether or not A, (H) is actually well

described by a power law, it is clear from Fig. 8 that thepenetration depth generally increases faster than ~B,and this indicates that the average pinning force con-stant k„becomes weaker as the field is increased. A likelymechanism for the softening of the pinning strength athigher temperatures is collective pinning, as first pro-posed by Larkin and Ovchinnikov. Collective pinningresults from the competition between repulsive intervor-tex interactions (elasticity), which favor long range peri-odic order, and pinning by a random potential with nolong range order. If elasticity is dominant, the flux linesare essentially randomly distributed with respect to thepinning potential minima. The efI'ective restoring forceconstant for each vortex will then vary in sign, dependingon whether the curvature of the pinning potential is pos-itive or negative at the equilibrium location of the fluxline [e.g. , for a pinning potential of the form shown in

Fig. 9(c)]. This leads to a reduction of the average pin-ning force constant when kp is averaged over the field-dependent correlation volume V, . Because the elasticconstants increase with field at low fields, the correlationvolume grows and the average pinning force constant be-comes weaker as the 6eld is increased. Collective pinningis usually used to explain the decrease of the critical cur-rent with increasing vortex density.

The data therefore suggest that at the lowest stud-ied temperature of 2.4 K (and presumably below thisas well), perpendicular fiux lines are pinned individually,while the pinning becomes collective at higher tempera-tures, presumably due to the weakening of the pinningpotential by thermal fluctuations. The field dependenceof y' was similar in all samples studied for T & 4.2 K,but only one sample was studied down to T = 2.4 K.Thus our claim of a temperature-driven transition fromsingle-vortex to collective pinning is based on one sam-

ple only, but the claims of collective pinning for T & 4.2K and single-vortex pinning in parallel 6elds at all tem-peratures are based on results from several samples. Atemperature-driven transition kom individual to collec-tive pinning was observed in YBCO single crystals forH

~~c, by Krusin-Elbaum et ol.ss They found that the

critical current 1, [deduced from the dc magnetizationM(H) using the Bean model] was field independent forT ( 45 K, and decreased as 1, 1/H at high temper-atures.

We now consider in more detail whether the above re-sults are self-consistent, and consistent with theoreticalideas about collective pinning and microscopic pinningmechanisms.

A. Collective effects:Anisotropy of the shear modulus

The observation of individual pinning of Josephsonvortices and collective pinning of perpendicular vortices

50 EVIDENCE FOR JOSEPHSON VORTICES IN (BEDT-I IF)2Cu(NCS)2 15 937

may seem contradictory at first. Single-vortex pinningoccurs when pinning forces exceed elastic forces, butJosephson vortices are 500 times more weakly pinnedthan perpendicular vortices, according to the results ofour experiments. However, the anisotropy of the elasticconstants must also be taken into account. To de-termine whether our findings are self-consistent, we canmake a crude estimate of the effective restoring force dueto the vortex elasticity in both orientations, and comparethis to the measured values of k„.We do this by consid-ering a shear wave, with displacement amplitude u andwave vector g t u. The &ee energy per vortex and perunit length isss (in cgs units)

f = BCssq (4.3)

We take the coefficient of u to be an effective restoringforce constant due to interactions, and wish to comparethis to k„.

We first consider the case of fiux lines perpendicularto the layers. Since we are interested in independent dis-placements of neighboring vortices, we take q 1/a =gB/@e, and 4.3 simplifies to Cssu . The shear modulusfor perpendicular fiux lines (for H, i « H « H,2) is thesame as for an isotropic superconductor with penetra-tion depth A~~, Css —BC'p/(8z'A~~) . Using A~~ 1 pm,we get C66 0.03 B with the magnetic field B in gauss.(Css and k~ both have units of ergs/cms =dyn/cm2).The pinning constant k 0.7 dyn/cm, and the elas-

tic energy exceeds the pinning energy for B & 30 G.Thus, far &om it being surprising that collective effectsare observed, it is puzzling that single-vortex pinning isobserved at all in kG fields, even at low temperatures.

A temperature-driven transition &om single-vortexto collective pinning was predicted by Koshelev andVinokur and by Feigel'man, Geshkenbein, and Larkin(AGL), sr who predicted that that single-vortex pinningshould occur for sufBciently small B and T. However,the crossover temperatures predicted by these models forET are on the order of T = 5 mK. This small value re-sults from the use of k+(T = 2.4 K) 1 dyn/cm2 incalculating the parameters which go into the the models:the depinning critical current j, kz (~~(0)c/@e 25A/cm2 (see next section) for the calculation in Ref. 56,and the pinning energy per 2D vortex U„k„(~~s 1mK for the calculation in Ref. 57. The reason for thisdiscrepancy is not understood. The observed crossovertemperature for YBCO in Ref. 53, 45 K. agreed fairlywell with estimates &om these models.

For vortices parallel to the layers, there is both a"hard" and an "easy" shear modulus, correspondingto shear parallel (Css) or perpendicular (Css) to thelayers. Ignoring the discreteness of the layers, i.e.,in the anisotropic Ginzburg-Landau model, Kogan andCamPbells obtained Css ——Css/P and Css PCss.We have simplified Kogan and Campbell's equation (9)using the identities for the reduced effective masses intheir model, mi ——1/p2~s and ms ——p4~s. Since the lay-ered structure only affects the structure of the core, the

AGL result for C66 should also apply to layered mate-rials. The easy shear modulus for Josephson vortices isthus a factor of p 10 weaker than the shear modu-lus for perpendicular vortices, and the anisotropy of theshear modulus is much larger than the pinning constantanisotropy, k+/kz 500.

Shear parallel to the layers should dominate the dis-tortion of the lattice. We therefore take u parallel tothe layers and q perpendicular to the layers, and setq gpB/@e to account for the anisotropy of the lat-tice structure. Using this in Eq. (4.3), the prefactor ofu2 in the expression for f is pCss 8 x 10 B dyn/cm,with B in gauss. We again identify this as an efFectiveforce constant due to vortex interactions in the presenceof shear. This does not become comparable to the mea-

sured pinning force constant k„10s dyn/cm untilB 1 kG. Furthermore, the calculation of Kogan andCampbell was in the local limit, q = 0, while vortex lat-tice elastic moduli can be highly dispersive. A numericalcalculation by Sudbo and Brandt54 showed that a very

significant further suppression of Cz~sl occurs for q nearthe Brillouin zone boundary. It is thus not at all unrea-sonable for the extremely weak measured pinning forcesto exceed the even weaker elastic forces which occur inparallel fields.

B. Microscopic pinning mechanisms and criticalcurrent: Condensation energy and coupling energy

We now relate the measured pinning force constantsto models of the microscopic pinning mechanisms forAbrikosov and Josephson vortices, and to values of thedepinning critical current obtained &om remnant mag-netization experiments.

The basic principle of normal core pinning by a defectis shown in Fig. 9(a). It is energetically favorable fora defect which suppresses superconductivity to be situ-ated in the center of a vortex normal core; separatingthe vortex &om the defect causes further suppression ofsuperconductivity by the defect and an increase in theenergy. An upper limit on the pinning energy per unitlength of a perpendicular vortex line is given by the totalcondensation energy in the core, which is approximatelyU (H, /87r)(~~~. The range of the pinning potential is

(~~, and so an upper limit on the linear restoringforce per unit length is kz U/r„H, /8vr. For ET,H, (0) 500 G (Ref. 47) and H, /8x 10 dyn/cm .The measured pinning constant for perpendicular vor-tices is about 1 dyn/cm at 2.4 K, and so the observedpinning force constant is a factor of p = 104 times weakerthan the maximum core pinning value. The coherencelength (~~(0) 80 A, and the weakness of the mea-sured pinning constant thus seems to indicate that pin-ning is due to very weak, atomic scale (d « (~~) defects.In single-crystal YBCO, strong pinning is observed(p 1), due in part to the fact that the coherence lengthf s(0) 10 A. is similar in size to the atomic scale de-fects. Values of p 10 3 are typical for conventionalsuperconductors.


4ii d weaker than average coupling, and act as pinning sites. If~f ~

&& 1, the additional energy due to f can be calculatedperturbatively:

tdefect+vortex

, ]

lk lk

vortex defect

C&

3

I I

y/ k

(c)

5

]

y/ k

In a similar spirit, we may estimate an upper limit forthe pinning constant of a Josephson vortex in a layeredmaterial, for motion parallel to the layers. We first con-sider a simpler and previously studied case, pinning of aHuxon in a single Josephson junction. ' The couplingenergy E, (per unit area) between the superconductorsdepends on the gauge-invariant phase difference Ay be-tween them, and is given by

E, = U~(1 —cos Ap), (4.4)

where U~ ——5Jp/2e gives the scale for the coupling en-

ergy, and the phase difFerence is related to the Josephsoncurrent by J = Jp sin 4y. The spatial variation of thephase for a Quxon is given by the phase soliton solu-tion, App(y) = 4 tan [exp(y/A~)] [Fig. 9(b)], and theJosephson penetration depth Ag gives the scale for boththe decay of the tunneling current at large y and the sizeof the "core" region near y = 0, in which the tunnelingcurrent is comparable to and limited by Jp .

Translational symmetry in the y direction is broken byrandom variations of the coupling Jp, e.g. , of the formJp(y) = [1 + f (y)]Jp. Regions with f & 0 correspond to

FIG. 9. (s) Schematic of core pinning mechanism. Sepa-rating the vortex snd defect by s distance u ) ( raises theenergy per unit length by st most (H, /8m)( (b) Spat.ialvariation of the phase and tunneling current of a Huxon ina single Josephson junction. Az is the only length scale. Ins layered superconductor, J Jp for ~y~ Lz = ps, whilefor large ~y~ the current decays as exp(ky/A~). (c) Pinningpotential for s perturbation 8Jp/ Jp = nh'(y), for s Huxpn in ssingle junction. The depth of the well is ~ 4a.U& A z and therange of the potential is Ag .

UJ y cos Q(ppdy. (4.5)

One then expects the total pinning energy to be on theorder of (UgAg) (f ),per unit length. The upper limit1/2

for the pinning energy is thus rp ——Up Ag, and that forthe restoring force constant is eg/A& ——U~/A~. This isin agreement with more detailed calculations. For a linedefect, with f (y) = ab(y) and o. « 1, the pinning energyis 4o e~. Figure 9(c) shows the pinning potential for thiscase. If f (y) is random, with correlation length l, thenthe scale for the pinning energy iss@ eg[(l/A~) (f2)]i~2.

The structure of a Josephson vortex in a layered mate-rial differs &om that of a Quxon in that there are taolength scales parallel to the layers: the penetrationdepth A~, which governs the exponential decay of thefield and current at large distances, and the Josephsonlength Lg ——ps, which governs the behavior of the coreregion in which J Jp . The core region should accountfor most of the pinning energy, due to the large currentsand small length scale. Considering only the contributionof the central junction (z = ns = 0) to the pinning, theupper limit on the pinning force constant is

Lq ps 2e 16vrsp q sA~ )(4.6)

where we have used 4p ——hc/2e and Jp ——c@p/8vr sA&.

Using p 250, s = 15 A, and A~ 200 pm gives k„&3 x 10 2 dyn/cm2, and this is only a factor of 30 higherthan the typical values ( 10 dyn/cm2) measured inET at low temperatures.

The pinning force constant can be approximately re-lated to the depinning critical current J, if one assumesthat the force is linear over most of the range r„ofthe po-tential. The critical current is then defined by the forceequation for a single vortex, I" „=J,@p/c = k„r„,which gives

CJ

kyar~

p(4.7)

For perpendicular vortices, with k„(2.4 K) 0.7dyn/cm and r„(~~(0) 80 A, this gives J,(2.4 K)25 A/cm . For parallel vortices, using k„10dyn/cm and r„Lg= ps 5000 A. results in J 2.5A/cm . The critical currents were estimated by Mots etal. f'rom remnant magnetization measurements, with

the results J, (4 K) 250 A/cm2 and J, (0) 3 A/cmz,in fair agreement with our estimates. The nature andconcentration of the pinning sites will significantly affectk„and J, and so considerable variation with the condi-tions of sample growth and purity is to be expected.

Just as the anisotropy of the collective effects initiallyseems to contradict the pinning force anisotropy (k„&)kr ), it also seems surprising that the onset of reversible

50 EVIDENCE FOR JOSEPHSON VORTICES IN (BEDT-'ITF}iCu(NCS}2 15 939

(nonhysteretic) behavior in y'(H) occurs at much lowerfields in the perpendicular orientation (Fig. 6). Further-more, the peak in the imaginary part y"(H), which is re-lated to dissipation, is also larger in perpendicular fields.These results taken together seem to indicate that Hux

creep occurs much more easily in perpendicular 6elds,despite the much weaker linear (small amplitude) restor-ing force for Josephson vortices. This is in agreementwith the measurements of the creep rate —c}ln M/o} ln t ofMota et al.zs They found that the creep rate was roughlya factor of three smaller for parallel fields at all temper-atures. There is thus a complicated interplay betweenthe pinning force constant, the length scale r„,collectiveefFects, and thermal efFects in determining the magneticbehavior.

V. LOCK-IN EFFECTAND ANGULAR DEPENDENCE

Figures 10(a) and 10(c) show the screening strength—4m y'(H) for H applied at an angle 8 with the layers, fora series of angles from (}= 0 (H parallel to the layers) to8 = 64', and for temperatures of 2.4 and 4.2 K. A strikingnonmonotonic behavior is observed, as reported in Ref.24. Regardless of the angle, the screening strength ini-tially decays in exactly the same way as for parallel fields((} = 0). At an angle-dependent threshold field Hti, (8),the screening at 6nite angles begins to deviate &om theparallel field value, reaches a minimum, and then recov-ers (to nearly the zero field value for larger angles). Thisis extremely unusual in that increasing the field strengthleads to the enhancement of screening, a property charac-teristic of superconductivity. A maximum of the screen-ing occurs at a field roughly double that of the screeningminimum, and y' decays monotonically with H abovethis.

We repeat here the interpretation of these features interms of the lock-in efFect, 6rst given in Ref. 24, withreference to Fig. 11. A magnetic 6eld applied at an an-gle initially enters the sample only in the parallel direc-tion, and the vortex lattice consists only of weakly pinnedJosephson vortices. The susceptibility therefore decaysrapidly with field, just as for 8 = 0 [Fig. 11(a)], due to alarge vortex penetration depth A„&om the edges of thesample. At the threshold 6eld Hqh, the Qux lines unlock&om the layers. Although the vortex density continues toincrease, the rapid increase in the pinning strength dueto the proliferation of strongly pinned 2D normal corescauses the reduction of A„and the recovery of screening[Fig. 11(b)]. Finally, well above the lock-in threshold,B

~~H [Fig. 11(c)]. Both penetration depths A„and

increase with 6eld as the vortex density continues toincrease, leading to the monotonic decrease of the screen-ing.

In Figs. 10(b) and 10(d), the data of Figs. 10(a) and10(c) are plotted against the perpendicular field compo-nent H~ ——H sin0. The scaling of the lock-in thresholdwith the perpendicular 6eld component is immediatelyobvious when the data are plotted in this way. Anotherfeature of the data, not described in Ref. 24, is also ap-parent: The susceptibility is determined only by H~ for

4 640

3

2

00 I

H(G)

gp 4EJ

~ 3fg

2-

1I

00

~64

32'16'

40 (3}0»

H sine (G)

(b)

H.~e

1200

e=0, 0.5, 1, 2, 4, 8, 16, 32' (c)

3-L

2-

1-I

00 40'00

H(G')8000 12000

CJ

cg

2-I

00 800

H sine(G)1600

FIG. 10. y'(H) ~s data sets at temperatures of 2.4 K (a),(b),and 5 K (c),(d), plotted against both the total field and theperpendicular Seld component. Insets show the experimentalgeometry.

field strengths well above the lock-in threshold and forsuKciently large angles, 8 & 5 —10 . That is, the screen-ing is the same as would occur for a 6eld of total strengthH~ applied normal to the layers, (} = 90'. (The failure ofthis scaling at small angles is obvious in the 4.2 K data;this is not seen in the 2.4 K data because the smallestangle studied was (} = 2', and' the maximum value ofH~ was 2000 sin2 = 70 G.) The scaling behavior in-


AC Field

(A)

(B)

ffl (C)

-4z X'

e~o

FIG. 11. Schematic representation of the dc field configu-ration (left) and the ac field distribution (right), in the lock-instate (A), just above the lock-in transition (B), and well abovethe transition (C), corresponding to the labeled portions ofthe curves at bottom.

k~(P) kll + k„sin (5.1)

eludes the changes in the form of the field dependencewhich were associated with a temperature-driven transi-tion from single-vortex to collective pinning (Sec. IV).

Scaling of the in-plane resistivity and criticalcurrent with H~ has previously been observed in thecuprates, and was initially taken as evidence for a purelytwo-dimensional model, in which H~ leads to the creationof perpendicular vortices and the parallel field compo-nent Hi~ passes freely between the layers, without creat-ing vortices. A purely 2D model is clearly inappropriateto describe our experiments, since we observe the effectsof Josephson vortex motion in parallel fields. The sus-ceptibility is determined by both penetration depths A,and A„,and the breakdown of scaling at small angles isdue to the easy motion of slightly tilted vortices and theresulting large penetration depth A„.We therefore inter-pret the scaling with H~ as being due to scaling of thevortex penetration depth A for tilted vortices, togetherwith a negligible contribution of A„to ac field penetra-tion in the range of fields and angles for which scaling isobserved.

As discussed in Sec. II, the vortex penetration depthin the z direction A should have the angular dependenceA„Bsin P/k„(P). The observed scaling therefore sug-gests that the pinning constant for tilted vortices in ETobeys k„(P) sing for large P. This is exactly whatone would expect for a vortex consisting of a tilted stackof coupled pancake vortices, since the line density of 2Dnormal cores is proportional to sing, and kz is the pin-ning force per unit length. The Josephson core segmentsonly exist for P ( tan i(1/p) 0.3, and simply pro-vide the limiting pinning force as P -+ 0. We thereforepropose the simple expression

for tilted vortices moving parallel to the layers and trans-verse to the vortex axis. The second term dominates aslong as P ) k„/k„0.1', leading to the observed scal-ing of A, . The observation of collective pinning effects inthe scaled susceptibility at 4.2 K indicates that the inter-actions between pancake vortices affect their pinning ina way which depends only on the two-dimensional den-sity of pancake vortices, not on the tilt angle. In otherwords, the coefficient k in Eq. (5.1) depends only Bz.Besides being consistent with the observed scaling of A,Eq. (5.1) can be used to construct a simple phenomeno-logical model which qualitatively reproduces many of thefeatures of our data, including the nonmonotonic behav-ior associated with the lock-in effect; we describe thismodel in the Appendix.

The lock-in effect has a simple theoreticalexplanation. Keeping the flux lines locked to thelayers causes the distortion of the magnetic Geld aroundthe sample and raises energy of interaction between thesample and the external field. On the other hand, al-

lowing the flux lines to unlock leads to the introductionof pancake vortices or vortex "kinks" wherever a fluxline crosses a superconducting layer, and this raises theinternal energy of the superconductor. When these en-

ergies are balanced against each other, it is found thatthe lock-in state is favorable as long as the perpendicu-lar field component H~ ——Hsing is below a thresholdvalue,

~ 1n(ps/(Ii)

"1n(AII/(ii)(5.2)

(as long as ps ( Aii, which seems to be true for ET). H, iis the perpendicular lower critical field. The lock-in ef-

fect does not occur in the anisotropic Ginzburg-Landaumodel because the normal core is present for all orien-tations. The core contribution to the internal energy is

quadratic ( gP) in the AGL model, while for Josephsoncoupled layers it is linear ( ~P~) and there is a cusp inthe energy at P = 0. The lock-in effect and the largeanisotropy of k„thus have a common origin, in the ab-sence of the normal core for Josephson vortices.

Depending on the values of p and Aii used in Eq. (5.2),and with (Ii(0) = 80 A. , Eq. (5.2) gives Hg (0.7-

0.95)H, i for ET. We showed in Ref. 24 [Fig. 4(b)] thatour Hg(T) has a similar magnitude and temperature de-

pendence to the demagnetization-corrected lower criticalfield H+z(T) measured by Mota et al. When properlycorrected for the demagnetization factor of our sample,our H~(T) actually exceeds the lower critical field of Motaet al. by about 20%. However, the definition of thethreshold field used in Ref. 24 corresponds roughly to theminimum of screening, and is somewhat arbitrary. Defin-

ing H, i, by the first measurable deviation of y'(H) ~s from

y'(H) ~s o would give a value nearly a factor of 2 smaller.We can at most say that our HJ(T) and the H i(T) ofMota et al. agree to better than a factor of 2. We alsonote that H+~ (0) 300 G in Ref. 26, while the Ginzburg-Landau expression H, i = (4'o/4vrAII) [ln(XII/(Ii) + 0.5]

gives 0 ~ 12 G for ET. This, and the anomalous pos-itive curvature in the temperature dependence, leads us

50 EVIDENCE FOR JOSEPHSON VORTICES IN (BEDT-T'IF)pCu(NCS)p 15 941

to the conclusion that both our Hg and the H, &mea-

surements of Mota et al. are signi6cantly enhanced atlow temperatures due to pinning.

VI. NONLINEARITY IN PARALLEL FIELDS

0.6

OP

o 04-VI

Cg

g 02-I

0

I I I

h = 0.3, 0.1, 0.01,d 0003 Gauss

iphs ip

h = 4.7 Gauss pi pk JDtkk ajg~LLl 4 i

iigQI I

T(K)

10

10

~ 10'-II

E

1010

0.5

10I I

1 0-1 100h. (G)

I I

10

h = 4.7 Gauss04- T = 3 3 K ~ "'""~-

~ ~ 6=10'

0.3—fg

Q. l—'L 6=5'

00 200 400 600 800 1000 1200 1400

(c)

H (G)

FIG. 12. Nonlinear efFects. (a) X'(T) with JI = 0, for vari-ous h (ac field amplitude). Geometry is as shown in Fig. 5(b).(b) The induced ac magnetic inoment m appears to saturateat h 0.6 G. X' is taken from (a) with T = 4 K. (c) Thenonlinearity persists in the lock-in state, but the penetrationof the perpendicular Geld component restores linear response.

We have studied the linearity of the real and imaginaryparts of the susceptibility by varying the ac 6eld ampli-tude h, with f = 2.5 kHz. In the presence of a magneticfield perpendicular to the layers, we find that both y' andy" are linear for a factor of 10 increase in h. When h isreduced by a factor of 10, both y' and y" appear to belinear, but the poor signal to noise ratio did not permitan unambiguous result for y".

A dramatic nonlinearity is observed. in the absence of adc 6eld or whenever the internal field B is parallel to thelayers. In Fig. 12(a), we show —4m''(T) with H = 0, for

diferent values of h. The screening response is linear forh & 0.3 G, but the susceptibility is strongly suppressedfor h = 1 G and 4.7 G. In fact, in the temperature rangeT 4—6 K, increasing h &om 1 to 4.7 G suppresses thesusceptibility by about a factor of 4.7, indicating that theinduced ac magnetic moment has saturated [Fig. 12(b),which shows m = yh vs h]. Figure 12(c) shows the dcBeld dependence of y' for a drive amplitude of h = 4.7G at T = 3.3 K, for angles of 8 = 0 and 8 = 10 .The points are data, and the solid line is a 6t of Eqs.(2.1) and (4.1) to the 0 = 0 data. The fitting parameters

are A~ ——910 pm and k„=1.7 x 10 4 dyn/cm2. Thusthe 6eld-dependent part of the penetration depth obeysA ~B in both the linear and nonlinear regimes, forB parallel to the layers. When the perpendicular fieldcomponent penetrates above the lock-in threshold, theabsolute value of the screening —4xy' recovers to values

0.5, typical of the linear (low h) regime. In otherwords, the strong nonlinearity only occurs in the absenceof perpendicular flux.

Two explanations immediately come to xnind. First,the screening response should be limited by the interlayercritical current Jo, and become nonlinear for ac 6eldslarger than that given by V' x h hNL/A~ = 4' Jp/c&or hN1, = 4it'JpA~/c. This explanation is appealing be-cause of the observed saturation of the ac magnetizationabove the threshold. However, the penetration of theperpendicular 6eld component largely restores the linearresponse, and this would imply that the interlayer super-current can exceed Jo in the presence of perpendicularflux. Also, using the relationship Jo ——c@o/8ir2sA~& for alayered material, hNL = @p/2vrsA~ 10 G, about factorof 20 larger than the observed threshold. These observa-tions argue against this being the correct explanation forthe nonlinearity.

The reduction of the nonlinearity by perpendicular fluxsuggests another possibility: that the ac field exceeds thelower critical 6eld parallel to the layers, leading to the in-troduction of weakly pinned Josephson vortices and an-tivortices in alternate parts of the ac field cycle. Themotion of these in response to the ac field then leadsto the reduction in screening, while the introduction ofperpendicular flux either damps their motion or preventsac-6eld-induced nucleation of Josephson vortices from oc-curring. Measured values of the parallel lower critical

Beld, &om dc magnetization experiments, are H y: 0 6II

G at T = 5 K (Ref. 50) and an upper limit of H, i & 0.5G, while the value calculated &om the formula of Clemet al. is 0.06 G. Thus it is quite plausible that an acfield of amplitude 6 & 1 G would lead to the introductionof Josephson vortices.

Beyond this, the details of the mechanism for the non-linearity are unclear at present. From Fig. 7, it can beseen that a parallel dc field of several hundred gauss isneeded to produce a suppression of —4vry' equivalent tothat achieved by an ac field of 4.7 G, implying that avery high net density of vortices must be present for theabove explanation to work. This is presumably deter-mined by the balance between vortex nucleation at theedges of the sample, and vortex-antivortex annihilation


in the bulk. The picture is further complicated by thefact that a Geld of only a few tenths of a gauss is sufficientto induce interlayer currents comparable to the depinning

critical current, J, 3 A/cm2 (Sec. IVB), so that thisshould play a role as well. The frequency dependencein the nonlinear (large h) regime is still weak, however,comparable to that observed in the linear regime, andthis is inconsistent with viscosity-limited motion of com-pletely depinned vortices. Thus while the nonlinearityprovides another striking demonstration of the qualita-tively difFerent behavior observed when the vortices areparallel to the layers, this effect is not well understood atpresent.

VII. CONCLUSIONS

We have described ac susceptibility measurements on asingle crystal of (BEDT-TTF)2Cu(NCS)2 which providestrong evidence for the existence of Josephson vorticesin this material, and clearly show that their propertiesare difFerent than those of tilted or perpendicular vor-tices. The lock-in efFect and the anomalously weak pin-ning force for Josephson vortices are direct consequencesof the absence of the usual normal core region. The pin-

ning force anisotropy is on the order of k„/k~ 500,and the force constants could be related to the appropri-ate energy and length scales for Abrikosov and Joseph-son vortex cores, and to the observed depinning criticalcurrents. k„is field independent up to H = 3 kG atII

~

all temperatures, while k+ decreases with field (exceptat T = 2.4 K). We have attributed these observationsto single-vortex pinning and collective pinning, respec-tively. This is consistent with theoretical calculationswhich show that the parallel Geld shear modulus C66 is afactor of p 10 weaker than for perpendicular Gelds,due to the anisotropic structure of the vortices and thevortex lattice. The susceptibility scales with the perpen-dicular Geld component H~ for sufficiently large fieldsand angles, showing that the dynamics of tilted vorticesare determined by the pinning and interactions of the 2Dpancake vortices. We have also observed a highly non-linear response to the ac field in the lock-in state, witha threshold of 6 0.5 G, but linear response is largelyrestored when the Bux lines unlock from the layers. Fi-nally, we have measured the Josephson penetration depthto be A~{T = 5 K) 200 pm, consistent with the result

p ) 200 from torque measurements.Our experiments show that motion of Josephson vor-

tices in the direction parallel to the layers can have anenormous effect on the electromagnetic properties of alayered superconductor, and that it is the interactionof Josephson vortices with interlayer currents which ismost significant. Several studies of the angular de-pendence of in-plane transport properties have shownanomalous enhancements of the resistance near the par-allel orientation, ' and we suggest that these may bedue in part to interlayer currents which occur near cracksor twin boundaries. There have been proposals to uti-lize intrinsic pinning in applications of high T super-conductivity, e.g. , by winding an oriented polycrystalline

ACKNOWLEDGMENTS

The authors thank L. Bulaevskii, M. CofFey, D. Huse,V. Kogan, M. Ledvij, and N.P. Ong for helpful discus-sions at various points in this work. This research wassupported by NSF Grant No. DMR 92-04581.

APPENDIX:A SIMPLE MODEL POR THE SUSCEPTIBILITY

Beyond simply providing an explanation for the scalingof A, (H~) for tilted vortices, Eq. (5.1) can also be usedto calculate A&. By combining Eq. (5.1) with Eq. (2.3)for y(Av/Iz, A, /I ), and a simple relationship betweenB and 0 which at least qualitatively describes the lock-inefFect, we can construct a simple phenomenological modelwhich captures most of the essential features of the data.With reference to Eqs. (2.7) and (5.1), and to Figs. 2(b),3(b), and 13(a), the penetration depths from the edgesand surfaces of the sample (A„and A„respectively) are

B@0cos+A~

47r (k~~' + k~ sin &j))

(A1)

tape into a solenoid with the copper oxide planes tan-gent to the surface, so that the field will be parallel tothe layers. This orientation gives the minimum resis-tance even without intrinsic pinning. The efFectiveness ofsuch an application might be limited by the occurrenceof stray interlayer currents and the resulting Josephsonvortex motion, and a detailed understanding of the na-ture of Josephson vortex pinning and the type of defectsresponsible is therefore potentially of some practical in-terest.

It is hoped that this work will stimulate further re-search on the properties and behavior of Josephson vor-tices. Besides the absence of the normal core and theweak and highly anisotropic elastic forces, vortex motionand fluctuations are essentially restricted to one dimen-sion by intrinsic pinning, and entanglement is impossiblein the absence of thermally activated kinks. All of thesefactors should have a significant efFect on the statisticalmechanics of Josephson vortices, and the consequencesfor the controversial issue of the vortex lattice melting orvortex glass phase transition have recently begun to beexplored experimentally. sr

(BEDT-TTF)2Cu(NCS)2 is an excellent candidate forsuch experiments, and in general for further work on vor-tices in layered superconductors. Not only is L ( sover virtually the entire temperature range [for T/T, (gl —((~~/ps)2 = 0.9996], but the entire (H, T) phase di-

agram is experimentally accessible, since= H, 2(0) = 20T. Large single crystals are easily grown, and the pinningis extremely weak, allowing the efFects we report here tobe observed so readily by ac susceptibility. An importantissue for future work on this and other organic supercon-ductors is the characterization of the defects responsiblefor pinning.

50 EVIDENCE FOR JR JOSEPHSON VORTICES IN (BEDT-I IF)PM is 943

0.6: 64'

I~ ~

I~

I~ ' I~ ~

(a)

gXB-

Ly

~ ~ ~ ~ I ~ ~ ~ i ~ I ~ ~ ~ ~ ~ I ~ ~ ~ ~1 2 I~ I \ ~ I I

I

0.5--

0 40ill

cg 0.3--

p. 1--

00

~ i ~ ~

1d00 H(Q) 2Ci00

0.8--

0.6-- 0.6 I I~ ~ ~~ ~

/~ ~

I III T ~ I ~~ ~ I

). 0.4--

"~ 0.2--

0-90

~ I~

I ~~

~ ~ ~ I ~~

~ ~

-60 -30( 0) 30 60 90

FIG. 13. (s) Geometry for the model calculation of the sus-

cepts z y. n'b lit . (b) Angular dependence of the peue'

penetration depthsand A, with B fixed. Vertical scale is arbitrary, ur but is the

same for both penetration depths. Dashehed lines show theeffect of the cutoff on A„(P);see the Appendix.

0.5--

0.2-

0.1-

40

0 I ~ ~ I

I

16'

660

—32', 64'

I ~ . ~ I

H sine(G) 1200

and

BC'o sin

II+

II~

4~(k~ + k~ sing)(A2)

The angular dependence of A& annd A &with B=const,A = 0 for simplicity) is shown in

Fig. 13(b). The penetration depth anisotropy in e

A /A = k+/ki /500 = 22, is much

larger than the typical sample aspect ratio I,„/I,, 5.We do not show experimental data for the angular de-pendence o y a ef ' t fixed field in this paper, but Fig.

~ ~ ~ ~

3 in Ref. 24 shows that there is a deep minimum in—4vry'(8)~~ at 8 = 0', due to the peak in A„(P),andthe half width 8th of the minimum depends on H as

data also scale with H~ for H~ )) Hg (not s own .In order to model the lock-in efFect, we need to know

the relations p e weenh l t hi between B and H near the transition.Although this has recently been studied in detai y

H~ will probably penetrate as in the Bean critical statemodel. In the absence of a detailed model for this, andin the interest of simplicity, we use the approximation

II IIh t B = H for allH while B~ ——0 for H~ ( Hg

and B~ ——H~ —HJ for H~ & HJ. This is certainly notaccurate, but at least qualitatively models the behaviornear the lock-in threshold and gives the correct limiting

FIG. 14. Results of model calculation of susceptibdzty,plotted against (s) the applied field strength H snd (b) the

snd (b), respectively.

behavior (B = H) for H~ )) H~.Equation (2.3), when combined with Eq. (5.1) for

k (P), Eqs. (Al) and (A2) for A„and A„and theabove approximation for B(H, g'ives the results shownin Fig. 14(a). This is meant to simulate t e =2.4Kdata of Fig. 10(a). The parameters used are Hg = 50 G,

of the model are plotted against H~ in Fig. . ise model that' t t th most serious deficiency o t e m

8=32'the scaling behavior is not observed until aboutfor smaller angles, the scaled screening strengt —aery' isless than the value for perpendicular fields. Becauseclearly scales with H~ for large P in the model, this mustoccur because the model predicts much larger values of A„

Multiplying the vortex term in Eq. yform exp( —~P/Po[), with Po a few degrees [dashed linein Fi . 13(b)] leads to improved scaling of the model re-sults, but this procedure is arbitrary. The reason for thisdiscrepancy is no un ers oot d stood. A less serious de6ciencyof the model is that the recovery of screening above t e

relationship etween anh b t B nd H used in the model nearthe transition.


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