PHYSICAL REVIEW B 1 NOVEMBER 1997-IIVOLUME 56, NUMBER 18

Threading dislocation lines in two-sided flux-array decorations

M.-Carmen Miguel and Mehran KardarPhysics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

~Received 13 June 1997!

Two-sided flux decoration experiments indicate that threading dislocation lines~TDL’s!, which cross theentire film, are sometimes trapped in metastable states. We calculate the elastic energy associated with themeanderings of a TDL. The TDL behaves as ananisotropic and dispersivestring with thermal fluctuationslargely along its Burger’s vector. These fluctuations also modify the structure factor of the vortex solid. Botheffects can, in principle, be used to estimate the elastic moduli of the material.@S0163-1829~97!05842-6#

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Recent two-sided flux decoration experiments haproven an effective technique to visualize and correlatepositions of individual flux lines on the two sides oBi2Sr2CaCu2O8 ~BSCCO! thin superconductor films.1,2 Thismaterial belongs to the class of high-Tc superconductors~HTSC’s!, whose properties have drawn considerable atttion in the last few years.3 Due to disorder and thermal fluctuations, the lattice of rigid lines, representing the idealhavior of the vortices in clean conventional type-IIsuperconductors, is distorted. Flux decoration allows onequantify the wandering of the lines as they pass throughsample. The resulting decoration patterns also include difent topological defects, such as grain boundaries and dcations, which in most cases thread the entire film.

Decoration experiments are typically carried out by coing the sample in a small magnetic field. In this procevortices rearrange themselves from a liquidlike state at htemperatures, to an increasingly ordered structure, until tfreeze at a characteristic temperature.4,5 Thus, the observedpatterns do not represent equilibrium configurations of linat the low temperature where decoration is performed,metastable configurations formed at this higherfreezing tem-perature. The ordering process upon reducing temperatrequires the removal of various topological defects fromliquid state: Dislocation loops in the bulk of the sample cshrink, while threading dislocation lines~TDL’s! that crossthe film may annihilate in pairs, or glide to the edges. Hoever, the decoration images still show TDL’s in the latticeflux lines. The concentration of defects is actually quite loat the highest applied magnetic fieldsH ~around 25G!, butincreases asH is lowered~i.e., at smaller vortex densities!.Given the high-energy cost of such defects, it is most likthat they are metastable remnants of the liquid state.~Meta-stable TDL’s are also formed during the growth of somsolid films.6!

Generally, a good correspondence in the position of invidual vortices and topological defects is observed as tcross the sample. Nevertheless, differences at the scalefew lattice constants occur, which indicate the wanderingthe lines. Motivated by these observations, we calculateextra energy cost associated with the deviations of a Tfrom a straight line conformation. ThemeanderingTDL be-haves like an elastic string with a dispersive line tenswhich depends logarithmically on the wave vector of tdistortion.

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By comparing the experimental data with our resultsmean-square fluctuations of a TDL, it is in principle possibto estimate the elastic moduli of the vortex lattice. Henthis analysis is complementary to that of the hydrodynammodel of a liquid of flux lines, used so far to quantify thecoefficients.10 On the other hand, the presence of evensingle fluctuating TDL considerably modifies the denscorrelation functions measured in the decoration expments. The contribution of the fluctuating TDL to the lonwavelength structure factor is alsoanisotropicand involvesthe shear modulus, making it a good candidate for the demination of this coefficient.

In the usual experimental setup, the magnetic fieldH isoriented along to thez axis, perpendicular to the CuO planeof the superconductor. The displacements of the flux linfrom a perfect triangular lattice at a point (r,z), are describedin the continuum elastic limit, by atwo-dimensional~2D!vector field u(r,z). The corresponding elastic free-energcost is

H5E d3r

2@c66~¹u!21~c112c66!~¹•u!21c44~]zu!2#,

~1!

where¹5(]xx1]yy); andc11, c44, andc66, are the com-pression, tilt, and shear elastic moduli, respectively. Duethe small magnetic fields involved in the experiments, nolocal elasticity effects3 are expected to be weak, and will bneglected for simplicity. In addition, at the temperatures cresponding to the freezing of decoration patterns, disordinduced effects should be small, and will also be ignored

To describe a dislocation line, it is necessary to specifyposition within the material, and to indicate its charac~edge or screw! at each point. The latter is indicated by thBurger’s vectorb, which in the continuum limit is defined byrLdu52b, with L a closed circuit around the dislocation.11

For the TDL’s in our problem, the Burger’s vectors lie in thxy plane, and the line conformations are generally describy the position vectorsRd(z)5R(z)1zz ~see Fig. 1!. Unlikea vortex line, the wanderings of a TDL are highlyaniso-tropic: In an infinite system with a conserved number of fllines, fluctuations of the TDL are confined to theglide planecontaining the Burger’s vector and the magnetic field. Thopping of the TDL perpendicular to its Burger’s vect~climb! involves the creation of vacancy and interstitial line

11 903 © 1997 The American Physical Society

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11 904 56M.-CARMEN MIGUEL AND MEHRAN KARDAR

as well as the potential crossing of flux lines.7,8 These defectsare very costly, making the TDL climb unlikely, except, foinstance, in the so-calledsupersolidphase, in which intersti-tials and vacancies are expected to proliferate.9 Nevertheless,for a sample of finite extent, introduction and removal of fllines from the edges may enable such motion. This seembe the case in some of the decoration experiments wherenumber of flux lines is not the same on the two sides.1 Inorder to be completely general, at this stage we allow forpossibility of transverse fluctuations inR(z), bearing in mindthat they may be absent due to the constraints.

We decompose the displacement fieldu into two parts:us@r2R(z),z#, which represents the singular displacemedue to a TDL passing through pointsRd(z) in independenttwo-dimensional planes; andur , a regular field due to thecouplings between the planes. By construction, the formethe solution for a two-dimensional problem with the circution constraint,11 while the latter minimizes the elastic energy in Eq.~1!, and is consequently the solution to

c66¹2ur1~c112c66!¹¹•ur1c44]z

2ur52c44]z2us.

After Fourier transforming this equation and substitutingformal solution in Eq.~1!, the energy cost of a fluctuatinTDL is obtained asHd5Hstr1DH, whereHstr is the stan-dard energy of a straight dislocation line. Keeping only tlowest order terms in the small deviationsR(z) of the TDLaxis from the straight line, gives the extra energy costdistortions as

DH5E dkz

2p@A'~kz!uR'~kz!u21Ai~kz!uRi~kz!u2#. ~2!

Here, R(kz) is the Fourier transform ofR(z); R' and Ristand for its components perpendicular and parallel toBurger’s vector, respectively; and

A'~kz!5b2c44

16pkz

2H lnS 11c66L

2

c44kz2 D

1F224c66

c1113S c66

c11D 2G lnS 11

c11L2

c44kz2 D J , ~3!

FIG. 1. Schematic plot of a TDL in a superconductor film. TBurger’s vectorb lies in the plane perpendicular to the magnefield H.

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lnS 11c11L

2

c44kz2 D G .

~4!

The above expressions are obtained after integrating oveq,with a long-wave-vector cutoffL at distances of the order othe flux-line lattice spacing, below which the continuutreatment is not valid. If we also take into account a showave-vector cutoffL* due to finite sample area, the depedence of the kernels onkz has different forms. For values okz!L* Amin(c66,c11)/c44, the logarithms in Eqs.~3!, ~4! re-duce to the constant value 2 ln(L/L* ), and both kernels aresimply proportional to kz

2 . In the opposite limit, ifkz@LAmax(c66,c11)/c44 all the logarithms can be approxmated by the first term of their Taylor expansion, andA' andAi are independent ofkz . In between these extremes, thform of the kernels is globally represented through Eqs.~3!,~4!. In practice, the smallest wave vectorkz that can beprobed experimentally is limited by the finite thickness of tsample, and is ultimately constrained~by the measured values ofc11, c66, andc44) to the last two regimes. From Eqs~2!–~4!, we conclude that the TDL behaves as an elasstring with a dispersive line tension (ed} lnkz), indicating anonlocal elastic energy.~A single flux line shows a similardispersive behavior, as pointed out by Brandt.12!

Equilibrium thermal fluctuations of a TDL are calculatefrom Eq.~2!, assumingthat one can associate the Boltzmaprobabilitye2DH/kBT to this metastable state. After averaginover all possible configurations ofR(kz), the mean-squaredisplacements are obtained as

^uR'~kz!u2&5kBTL

A'~kz!, and ^uRi~kz!u2&5

kBTL

Ai~kz!, ~5!

respectively, whereL is the thickness of the film. In terms othe function

^uR~kz!u2&c11[

8pkBTL

b2c44kz2

ln21S 11c11L

2

c44kz2 D , ~6!

these quantities satisfy the simple relation

^uR'u2&212^uRiu2&215^uRu2&c11

21S 12c66

c11D 2

. ~7!

Thus, even if the TDL is allowed to meander without costraints, its fluctuations areanisotropic, as A'(kz)5Ai(kz)only for c115c66. In HTSC materials,c66!c11, so thatA'(kz).Ai(kz), limiting fluctuations largely to the glideplane.

In real space, the width of the TDL depends on quantitsuch as^uR(L)u2&c11

[1/L*dkz/2p^uR(kz)u2&c11. Its depen-

dence on the film thicknessL follows from Eq.~6!, as

^uR~L !u2&c11;

kBT

c11H 1/d0 if L! l 1 ,

LY F2l 12lnS L

l 1D G if L@ l 1 ,

~8!

where we have definedl 1[bAc44/c11, and do is a short-distance cutoff along thez axis. For length scales belowd0 ,the layered nature of the material is important.

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56 11 905THREADING DISLOCATION LINES IN TWO-SIDED . . .

In order to estimate typical fluctuations for the TDL, wassume thatc66!c11, and approximate Eq.~4! by its leadingbehavior. In this limit, uRiu2&.2^uRu2&c66

, with ^uRu2&c66de-

fined as in Eq.~6!, after replacingc11 with c66. A behaviorsimilar to Eq.~8! is obtained for uRu2&c66

, with a correspond-

ing crossover lengthl 6[bAc44/c66. Thus, longitudinal fluc-tuations of the TDL are approximately constant for sampthinner thanl 6, and grow asL/ lnL for thicker samples. Ifallowed, transverse fluctuations follow from Eq.~7!, whoseleading behavior for smallc66 gives, ^uR'u2&;^uRu2&c11

. In

Fig. 2 we have plotteduRu2&c11and 2 uRu2&c66

as a function

of the thicknessL. Both quantities are very sensitive to thelastic coefficients. We have considered the valuesc1152.831022 G2, c4458.1 G2, and c6659.631023 G2 re-ported in Ref. 2 for a sample decorated in a field of 24which shows a single dislocation. The Burger’s vectorequal to the average lattice spacingao51 mm, and the short-distance cutoff in the plane is taken to be of the same oof magnitude. The crossover lengths introduced turn oube l 1;17 mm, and l 6;29 mm, so that the experimentasample thickness (L;20 mm! approximately falls into theconstant regime in Fig. 2. From the top curve in Fig. 2,estimate uRiu2&1/2;3 mm for T;80 K, with an uncertaintyfactor of aboutA10 due to, for instance, the uncertain valuof temperature, and both the in-plane and perpendicularoffs. If unconstrained, transverse fluctuations of the TDLsmaller, and given byuR'u2&1/2;1 mm, in the same regimeThe question mark in Fig. 2 is a reminder that once thfluctuations exceed a lattice spacing, proper care mustaken to account for constraints, and their violation by dfects or surface effects.

As discussed in Ref. 2, the values ofc11 andc66 measuredin the experiments are about three orders of magnitsmaller than the theoretical predictions from GinzbuLandau theory. If we use the latter values in our computions, the crossover lengths become much shorter and T

FIG. 2. Mean-square displacements as a function of the thnessL, both measured inmm, for a BSCCO sample decorated infield of 24 G (ao5b51 mm!.

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fluctuations are reduced by three orders of magnitude. Duthis sensitivity, analysis of transverse and longitudinal flutuations of TDL’s in two-sided decoration experimenshould provide a complementary method for determiningelastic moduli. Unfortunately, the films studied so far arethe short distance regime where details of the cutoff plasignificant role. Experiments on thicker films are neededprobe the true continuum limit.

TDL’s also produce anisotropies in the flux-line densn(r,z), and the corresponding diffraction patterns. Neutroscattering studies can, in principle, resolve the full thredimensional structure factorS(q,kz)5^un(q,kz)u2&, althoughonly a few experiments are currently available for differeHTSC materials.14,15 Two-sided decoration experiments alsprovide a quantitative characterization of thetwo-dimensionalstructure factors calculated from each surfaas well as the correlations between the two sides ofsample.

The diffraction pattern from a vortex solid has Bragpeaks at the reciprocal-lattice positions. Unbound dislotions modify the translational correlations; a finite concenttion of dislocation loops can drive the long-wavelength shmodulus to zero, while maintaining the long-range orientional order.13 The resulting hexatic phase has diffractiorings with a sixfold modulation, which disappears in the liuid phase. In all phases, the diffuse scattering close toq50is dominated by the long-wavelength density fluctuatiowhich are adequately described byn5¹•u, leading to

S~q,kz!;^uq•u~q,kz!u2&. ~9!

The contribution of equilibrium density fluctuations~fromlongitudinal phonons! to Eq. ~9! has the form10

So~q,kz!5kBTLAq2

c11q21c44kz

2, ~10!

whereA is the sample area. This contribution is clearly istropic, and independent of the shear modulus in the sphase.~The anisotropies of the solid and hexatic phasesmanifested at higher orders inq.! For a sample of finitethickness, the phonon contribution in rather general sittions including surface and disorder effects, was obtaineRef. 10, asSo(q,L)5S2D

o (q)R(q,L), whereS2Do (q) is the 2D

structure factor of each surface, whileR(q,L) measures thecorrelations between patterns at the two sides of the film

The above results were used in Ref. 2 to determineelastic modulic11 and c44 of the vortex array, at differenmagnetic fields. However, the decoration images usedthis purpose have the appearance of a solid structure wfinite number of topological defects. We shall demonstrhere that the presence of a single trapped TDL modifiesisotropic behavior in Eq.~10!. We henceforth decompose thdisplacement fieldu(r) into a regular phonon partuo, and acontribution ud5us1ur from the meandering TDL described byR(z). The overall elastic energy also decomposinto independent contributionsH@uo,R#5Ho@uo#1Hd@R#.To calculate the average of any quantity, we integrate osmoothly varying displacementsuo, and over distinct con-

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11 906 56M.-CARMEN MIGUEL AND MEHRAN KARDAR

figurations of the dislocation lineR(z). Thus the structurefactor in Eq.~9! becomes a sum ofphononanddislocationparts.

The results of Eq.~5! can be used to calculate the contbution from a fluctuating TDL, which has the form

Sd~q,kz!58pLb2c66

2 q'2

c112 q4

d~kz!1kBTLb2

~c11q21c44kz

2!2

3S 4c662 qi

2q'2

Ai~kz!1

~c44kz212c66q'

2 !2

A'~kz!D , ~11!

qi[q•b/b, andq' the component perpendicular to the Burer’s vector. The first term on the right-hand side~rhs! of Eq.~11! corresponds to the straight TDL, and vanishes inliquid state withc6650. The next two terms on the rhs resufrom the longitudinal and transverse fluctuations of the TDrespectively.~The latter is absent if the TDL is constrainedits glide plane.! The dislocationpart is clearly anisotropicand the anisotropy involves the shear modulusc66. Thusafter inverting thekz transform in Eq.~11!, the TDL contri-bution to the structure factors calculated from the two-siddecoration experiments can also be exploited to obtain inmation about the elastic moduli.

In conclusion, we have calculated the energy cost of manderings of a TDL in the flux lattice of a HTSC film such

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BSCCO. Flux decoration experiments indicate that sumetastable TDL’s are indeed frequently trapped in thin filin the process of field cooling. We have estimated the thmal fluctuations of a TDL in crossing the sample, as wellits contribution to the structure factor. Both effects can,principle, be used to estimate the elastic moduli of the vorsolid. However, there are strong interactions between sdefects, which need to be considered when a finite numbeTDL’s of different Burger’s vectors are present. The genalization of the approach presented here to more thanTDL may provide a better description of the experimensituation. From the experimental perspective, it shouldpossible to find samples with a single trapped TDL, proving a direct test of the theory. Other realizations of TDLcan be found in grown films,6 and may also occur in smectiliquid crystals. It would be interesting to elucidate the simlarities and distinctions between the defects in these syste

M.C.M. acknowledges financial support from the DreccioGeneral de Recerca~Generalitat de Catalunya!. M.K.is supported by the NSF Grant No. DMR-93-03667. Wegrateful to D.R. Nelson for emphasizing to us the constraon transverse motions of TDL’s. We have benefited froconversations with M.V. Marchevsky, and also thank Z. Yfor providing us with the raw decoration images from Refsand 2.

1Z. Yao et al., Nature~London! 371, 777 ~1994!.2S. Yoonet al., Science270, 270 ~1995!.3G. Blatteret al., Rev. Mod. Phys.66, 1125~1994!.4D. Grier et al., Phys. Rev. Lett.66, 2270~1991!.5M. V. Marchevsky, Ph.D. thesis, Leiden University, 1997.6E. A. Fitzgerald, Mater. Sci. Rep.7, 87 ~1991!.7R. Labusch, Phys. Lett.22, 9 ~1966!.8F. R. N. Nabarro and A. T. Quintanilha, inDislocations in Solids,

edited by F. R. N. Nabarro~North-Holland, Amsterdam, 1980!,Vol. 5.

9E. Frey, D. R. Nelson, and D. S. Fisher, Phys. Rev. B49, 9723~1994!.

10M. C. Marchetti and D. R. Nelson, Phys. Rev. B47, 12 214~1993!; ibid. 52, 7720~1995!.

11L. D. Landau and E. M. Lifschitz,Theory of Elasticity~Perga-mon, New York, 1970!.

12E. H. Brandt, J. Low Temp. Phys.26, 735 ~1977!.13M. C. Marchetti and D. R. Nelson, Phys. Rev. B41, 1910~1990!.14R. Cubbitet al., Nature~London! 365, 407 ~1993!.15U. Yaronet al., Phys. Rev. Lett.73, 2748~1994!.