We study the transport properties of vortex matter in a superconducting thin
film separated by a thin insulator layer from a ferromagnetic layer. We assume
an alternating stripe structure for both FM and SC layers as found in [7]. We
calculate the periodic pinning force in the stripe structure resulting from a
highly inhomogeneous distribution of the vortices and antivortices. We show
that the transport properties in FM-SC bilayer are highly anisotropic. In the
absence of random pinning it displays a finite resistance for the current
perpendicular to stripes and is superconducting for the current parallel to
stripes. The average vortex velocity, electric field due to the vortex motion,
Josephson frequency and higher harmonics of the vortex oscillatory motion are
calculated.Comment: 4 pages, 2figures, Submitted to PR

Kayali, M. Amin

Pokrovsky, Valery L.

English

arXiv

Journals published by the American Physical Society can be found at http://journals.aps.org/We study the transport properties of vortex matter in a superconducting thin film separated by a thin insulator layer from a ferromagnetic layer. We assume an alternating stripe structure for both FM and superconducting (SC) layers as found in S. Erdin [Phys. Rev. Lett. 88, 017001 (2002)]. We calculate the periodic pinning force in the stripe structure resulting from a highly inhomogeneous distribution of the vortices and antivortices. We show that the transport in SC-FM bilayer is highly anisotropic. In the absence of random pinning it displays a finite resistance for the current perpendicular to stripe and is superconducting for the current parallel to stripes. The average vortex velocity, electric field due to the vortex motion, Josephson frequency, and higher harmonics of the vortex oscillatory motion are calculated

Kayali, MA

OAKTrust Digital Repository (Texas A&M Univ)

PHYSICAL REVIEW B 69, 132501 ~2004!Anisotropic transport properties of ferromagnetic-superconducting bilayersM. Amin Kayali1 and Valery L. Pokrovsky1,21Department of Physics, Texas A & M University, College Station, Texas 77843-4242, USA2Landau Institute for Theoretical Physics, Moscow, Russia~Received 12 December 2003; published 5 April 2004!We study the transport properties of vortex matter in a superconducting thin film separated by a thininsulator layer from a ferromagnetic layer. We assume an alternating stripe structure for both FM and super-conducting ~SC! layers as found in S. Erdin et al. @Phys. Rev. Lett. 88, 017001 ~2002!#. We calculate theperiodic pinning force in the stripe structure resulting from a highly inhomogeneous distribution of the vorticesand antivortices. We show that the transport in SC-FM bilayer is highly anisotropic. In the absence of randompinning it displays a finite resistance for the current perpendicular to stripe and is superconducting for thecurrent parallel to stripes. The average vortex velocity, electric field due to the vortex motion, Josephsonfrequency, and higher harmonics of the vortex oscillatory motion are calculated.DOI: 10.1103/PhysRevB.69.132501 PACS number~s!: 74.78.2w, 74.25.Ha, 74.25.Qt, 75.70.2iThe interest in heterogeneous ferromagnetic-super-conducting systems has grown rapidly in recent years. Thisinterest stems not only from their possible technological ap-plications but also from new physical phenomena arisingfrom the interaction between two order parameters. Typicallysuch a system consists of a superconductor ~SC! placed inclose proximity with a periodic ferromagnetic structure ~FS!such as an array of ferromagnetic dots or holes. The twosystems are separated by an infinitely thin layer of insulatoroxide that guarantees the suppression of proximity effects. Itwas demonstrated experimentally1–3 that the interaction be-tween the superconductor and the ferromagnet may lead toformation of superconducting vortices interacting with theFM film. Theoretical studies of such systems have been donein Refs. 4–7.Recently, Erdin et al.7 studied the equilibrium structure ofa FM-SC bilayer ~FSB!. They have proved that it representsa two-dimensional periodic stripe domain structure consist-ing of two equivalent sublattices, in which both the magne-tization mz(r) and the vortex density nv(r) alternate. Thus,they predicted spontaneous violation of the translational androtational symmetry in the bilayer. In this Brief Report westudy the transport properties of the FSB. They are associ-ated with the driving force acting on the vortex lattice froman external electric current. We show that the FSB exhibitsstrong anisotropy of the transport properties: the bilayer maybe superconducting for the current parallel to the domainwalls and resistive when the current is perpendicular to them.The force acting on a vortex from other vortices whichdetermines the value of critical current can be characterizedas the periodic pinning. An extensive development of theoryand experiment related to the pinning and its influence ontransport in superconductors was discussed in an exhaustingreview by Blatter et al.8 Our work differs from the studiesconsidered in this review by two features. First, in the pre-ceding works the magnetic field was assumed to be constantin space, whereas in our problem the average magnetic fieldis zero, it is strongly inhomogeneous in space. Therefore, inour system equal numbers of vortices and antivortices par-ticipate in the motion. Second, in these works the pinningforce was assumed to be random, whereas in our case the0163-1829/2004/69~13!/132501~4!/$22.50 69 1325dominant pinning forces are periodic and regular. Martinoliet al.9 created artificially periodic pinning barriers in super-conducting films modulating their thickness periodically. Themain difference of their modulated structure from one con-sidered in our work is that the domains in the FSB are notconfined to the crystal lattice and can move together with thevortices.Periodic pinning forces in the direction parallel to thestripes do not appear in continuously distributed vortices,their reappearance is associated with the discreteness of thevortex lattice. Therefore, we need to modify the theory7 toincorporate the discreteness effects. Let us assume that thesaturation magnetization per unit area of the FM film is mand its width is L. The energy necessary to create a singlePearl vortex10 in the superconductor is ev05e0ln(l/j) withe05f02/16p2l , where f0 is the flux quantum, l5lL2 /ds isthe effective penetration depth,11 lL5Amec2/4pnse2 is theLondon penetration depth, ds is the thickness of the super-conducting layer, and j is its coherence length. It was shownin Ref. 7 that the interaction between the superconductingvortices and the magnetization in the stripe structure renor-malizes the single-vortex energy to the value ev˜5ev02mf0 which must be negative to allow development of thestripes. The density of the superconducting vortices increaseswhen approaching the domain walls and can be expressed asnv(x)5(pm˜ /Lf0)@1/sin(px/L)#, where m˜ 5m2(ev0 /f0) isthe renormalized magnetization of the FM stripe. The vorti-ces spontaneously appear in the superconductor. We assumethat the vortices inside one stripe are arranged in parallelchains ~Fig. 1!. Each chain is periodic with the same latticeconstant b along the chain, whereas the distance between kthand (k11)th chain ak depends on k. The correspondencebetween this discrete arrangement and continuousapproximation7 is established by the requirement that the lo-cal vortex density nv(xk) calculated in Ref. 7 must be equalto (bak)21. The coordinate xk is determined in terms of akas a sum: xk5(k850k21ak8 . For definiteness we choose the ori-gin in the center of the stripe. We assume that the total num-ber of the vortex chains 2N in a stripe is large. Then some ofthem are located very close to the domain walls. Let us re-©2004 The American Physical Society01-1BRIEF REPORTS PHYSICAL REVIEW B 69, 132501 ~2004!mind that, in the continuous approximation7 nv5(pm˜ /Lf0)@1/sin(px/L)#, where L is the domain width.Considering the nearest to the domain-wall vortex chain~with the number N!, we put nv(xN)51/baN . On the otherhand xN5L2aN . Since aN /L!1, we find: b5f0 /m˜ . Thetotal number of chains in a stripe is 2N , where N5b*0L2lnv(x)dx5 12 ln(L/l). We cut the integration ~andsummation! at a distance ;l from the domain wall wherethe continuous approximation breaks. Thus, the minimumvalue of a is l . When transport current passes through thesuperconducting film, the vortices start to move. To simplifythe problem we assume that all vortices in each stripe movetogether as well as all antivortices in the neighboring stripedo. We denote their positions r15(x1 ,y1) and r25(x2 ,y2), respectively. Forces acting on a moving vortexare the Magnus force, the viscous force, and the periodicpinning force. The Magnus force is fm5pns\ds(vs2r˙)3zˆ ,where ns is the superconducting electron density, vs is thevelocity of the superconducting electron, and r˙ is the vortexvelocity. The viscous ~friction! force is ff52hr˙, where h5f0Hc2ds /rnc2 is the Bardeen-Stephen drag coefficient,12Hc2 is the upper critical magnetic field, rn is the resistivityof the superconducting sample in the normal state, and c isthe speed of light. The periodic pinning forces are due to theinteraction of the vortex with the pinning centers and thedomain walls. In the FM-SC bilayer the pinning force is dueto the interaction of the domain walls with the vortices andantivortices and the vortex-vortex interaction Uvv given byUvv512E E nv~r!V~r2r8!nv~r8!d2rd2r8, ~1!where V(r2r8) is the pair interaction between vortex lo-cated at r and another at r8. When ur2r8u@l the pair inter-action can be written as V(r2r8)5(f02/4p2)(1/ur2r8u).13The interaction energy between two parallel chains located atxl and xl8 and vertically shifted with respect to each other byan interval bs (s!1) readsFIG. 1. Schematic vortex distribution in the FM-SC bilayer. Thesign 6 refers to the vorticity of the trapped flux.13250U~xl ,xl8 ,s !5 (n ,m51N0f028p2A~xl2xl8!21~n2m2s !2b2, ~2!where N0 is the number of vortices or antivortices in a singlechain. For infinite chains (N0→‘) Eq. ~2! can be rewrittenasU~xl ,xl8 ,s !5 (k52‘‘N0f028p2A~xl2xl8!21~k2s !2b2. ~3!The sum in Eq. ~3! can be calculated using Poisson summa-tion formula.14,15 Since the force is zero in the continuousapproximation, it is possible to retain the lowest nonzeroharmonic in the Poisson summation. Thus, we arrive at thefollowing interaction energy of two chains:U~xl ,xl8 ,s !5N0f024p2bcos~2ps !x ll8 , ~4!where x ll85e22p(uxl2xl8u/b). The distance between twochains uxl2xl8u is larger or equals l hence x ll8;x5e2(2pl/b). Typical value of x ll8 is e2(dm/4p), where dm5mf0 /e05gS(nmdm /nsds), with g is Lande factor, S is theferromagnet elementary spin, nm and dm are the electronsdensity and thickness of the magnetic film, respectively.We conclude that the amplitude of the periodic potentialfor displacements parallel to the domains in units of e0 isexponentially small near the transition temperature. Relativedisplacements in perpendicular direction have energy barrier;e0 even in continuous approximation. We model therestoring forces by simple sines dependencies f x5 f’sin@(2p/a)(x12x2)#, f y5 f uusin@(2p/b)(y12y2)#, wheref’;e0 /a and f uu;(e0 /b)e2(dm/4p)! f’ .When the supercurrent is perpendicular to domains, theequations of motion for a vortex and antivortex arehy˙ 15F2Fvsx˙ 12 f uusinS 2pb ~y12y2! D , ~5!hx˙ 15Fvsy˙ 11 f’sinS 2pa ~x12x2! D , ~6!hy˙ 252F1Fvsx˙ 21 f uusinS 2pb ~y12y2! D , ~7!hx˙ 252Fvsy˙ 22 f’sinS 2pa ~x12x2! D , ~8!where F5pns\dsvs , f’5e0 /a and f uu5xe0 /b . If the cur-rent is smaller than a critical value, Eqs. ~5!–~8! accept astatic solutionx15x25Fb4phvsarcsinS Ff uu D , ~9!1-2BRIEF REPORTS PHYSICAL REVIEW B 69, 132501 ~2004!y152y25b4parcsinS Ff uu D . ~10!It is valid at F< f uu5xe0 /b . For F. f uu or equivalently, if thecurrent is larger than its critical value, vortices and antivor-tices start to move. The solution of Eqs. ~5!–~8! for F.Fcreadsx12x250, ~11!x11x25Fhvs~y12y2!, ~12!y12y25bparctanS f uuF 1A12 f uu2F2tan~v0’t !D , ~13!y11y250, ~14!where v0’52phvs2Ab2F22x2e02/b2(F21h2vs2) is the Jo-sephson frequency. Thus, the vortices and antivortices ac-quire the same velocity components v1x5v2x in the direc-tion of the current and opposite velocity components v1y52v2y in the direction perpendicular to the current. Thedomain walls do not interfere such a motion if they move inthe direction of the current with the same velocity vdw5v1x5v2x as vortices and antivortices. Such a motion is aGoldstone mode. The solution ~11!–~13! displays an oscilla-tory motion of the vortices and antivortices in the directionparallel to the domain walls ~Fig. 2!, in addition to theirmotion together with the domain walls along the direction ofthe current. Higher harmonics of the vortex motion can becalculated analytically. The distribution of vortices ~antivor-tices! is inhomogeneous in the direction perpendicular to thedomains. The local electric field E due to the vortex motionis related to its time-average velocity ^v1& as E52(qvf0 /c)nv(r)(^v1&3zˆ).8 Therefore, the local fieldproduced by vortices in the direction parallel to the domainsis equal but opposite in sign to the one produced by antivor-tices, while the local field produced by vortices and antivor-tices in the direction perpendicular to the domains has bothequal magnitude and sign. The time average of the vortex~antivortex! velocity over a period T52p/v0’ isFIG. 2. The vortex displacement as a function of time in theovercritical regime. Time is measured in units of 1/v0’ and y1 inunits of b and x51024.13250^v1y&56hAF22 f uu2S h21 F2vs2 D , ~15!^v1x&5F^v1y&hvs. ~16!The time-averaged local-field components areEx52hm˜acAF22 f uu2S F2vs2 1h2D , ~17!Ey57m˜ FavscAF22 f uu2S F2vs2 1h2D . ~18!The upper sign in Eqs. ~15! and ~18! refers to the vorticesvelocity and produced field along the domain while the lowersign refers to those due to antivortices. Nonzero averageelectric field due to all vortices and antivortices in the FSBappears only in the direction perpendicular to the domains.The critical current Jc is related to Fc as Jc5(c/f0ds)Fc .Plugging Fc5e0mx/f0 into the expression for Jc and ac-cepting x51024 –1022, b51024 –1025 cm, and ns51022 cm23 we find: Jc;103 –105 A/cm2.When the current flows parallel to the stripes, the FMdomain walls stay at rest while vortices and antivorticesmove both parallel and perpendicular to the domains. Thesolution of equations of motion for vortices and antivorticesshows that they move opposite to one another both in x andy directions. Their motion along x is oscillatory with funda-mental frequency v0uu5@2phvs2Aa2F22e02/a2(F21h2vs2)# .The motion of vortices and antivortices in the parallel direc-tion proceeds until the distance between them becomes halflattice spacing b/2. Once the vertical shift between the vor-tices and antivortices reaches b/2, their motion freezes. Thecritical current in this case is Jc5nsmB/2a , the lattice spac-ing a is of the order of l;1025 –1024 cm, hence the criticalcurrent Jc is of the order 107 –108A/cm2, which is at least102 times larger than the critical current for parallel current.Therefore, the system may be superconducting for the cur-rent parallel to the stripes and exhibit finite resistance forperpendicular current. The difference in the critical currentsfor parallel and perpendicular directions is due to the expo-nential factor x which is small if b!l . The anisotropy ispronounced when dm is large which can be achieved by us-ing thicker FM layers and decreasing the density of the su-perconducting electrons. dm is temperature dependent andeventually decreases when temperature decreases startingfrom Ts . However, at the temperature of vortex disappear-ance Tv,Ts the value m˜ turns into zero and x again be-comes exponentially small. Thus, anisotropy has a minimumbetween Tv and Ts .Kopnin and Vinokur16 considered a collection of super-conducting grains with the wash-board pinning potential a1-3BRIEF REPORTS PHYSICAL REVIEW B 69, 132501 ~2004!model of random pinning. They obtained a similar result forvortex sliding in external magnetic field with a supercurrentapplied. In contrast to their work ~they considered vorticesonly!, we consider vortices and antivortices in the field ofperiodic pinning and completely neglect the random pinning.Let us discuss briefly how the magnetic field generated bysupercurrent changes our result. In Ref. 17 it was shown thatat sufficiently small critical magnetic field the domains van-ish. Therefore, in general, magnetic field suppresses both theanisotropy and periodic pinning at a critical field for whichdomains disappear. At such critical field only random pin-ning prevails. However, the total current per unit length isproportional to the thickness of the SC film and can be keptsmall.In conclusion, we studied the transport properties of theFM-SC bilayer in a state with stripe domains of alternatingmagnetization and vorticity. In the absence of a drivingforce, the vortices and antivortices are arranged in straightchains configuration. The force between two chains of vorti-ces falls off exponentially as a function of the distance sepa-rating the chains. We argued that, in the vicinity of the su-perconducting transition temperature Ts and vortexdisappearance temperature Tv , the distances between chains13250become much larger than the distances between vortices inthe same chain. We solved the equations of motion for vor-tices and antivortices for the driving current direction paral-lel and perpendicular to the domains. The calculated parallelto domains critical current is much higher than the perpen-dicular one at least in a vicinity of the two transition tem-peratures. This strong transport anisotropy is due to the factthat, for the perpendicular current the induced motion is aGoldstone mode specific for a system of mobile domains andvortices. We expect the ratio of the parallel to perpendicularcritical current to be in the range 102 –104 close to the super-conducting transition temperature Ts and to the vortex disap-pearance temperature Tv . The anisotropy decreases rapidlywhen the temperature goes from the ends of this intervalreaching its minimum somewhere inside it. The anisotropycan be destroyed by a rather weak magnetic field perpendicu-lar to the bilayer. This anisotropic transport behavior couldserve as a diagnostic tool to discover spontaneous topologi-cal structures in magnetic-superconducting systems.This work was supported by the NSF Grants Nos. DMR0103455 and DMR 0072115, U.S. DOE Grant No. DE-FG03-96ER45598, and Telecommunication and InformationTask Force at Texas A & M University.1 D.J. Morgan and J.B. Ketterson, Phys. Rev. Lett. 80, 3614 ~1998!.2 J.I. Martin, M. Velez, J. Nogues, and I.K. Schuller, Phys. Rev.Lett. 79, 1929 ~1997!.3 M.J. Van Bael, J. Bekaert, K. Temst, L. Van Look, V.V. Mosh-chalkov, Y. Bruynseraede, G.D. Howells, A.N. Grigorenko, S.J.Bending, and G. Borghs, Phys. Rev. Lett. 86, 155 ~2001!; M.Lange, M.J. Van Bael, L. Van Look, K. Temst, J. Swerts, G.Guntherodt, V.V. Moshchalkov, and Y. Bruynseraede, Europhys.Lett. 51, 110 ~2001!; S.J. Bending, G.D. Howells, A.N. Grig-orenko, M.J. Van Bael, J. Bekaert, K. Temst, L. Van Look, V.V.Moshchalkov, Y. Bruynseraede, G. Borghs, and R.G. Hum-phreys, Physica C 332, 20 ~2000!.4 I.F. Lyuksyutov and V.L. Pokrovsky, Phys. Rev. Lett. 81, 2344~1998!.5 I.F. Lyuksyutov and V.L. Pokrovsky, cond-mat/9903312 ~unpub-lished!.6 S. Erdin, M.A. Kayali, I.F. Lyuksyutov, and V.L. Pokrovsky,Phys. Rev. B 66, 014414 ~2002!.7 S. Erdin, I.F. Lyuksyutov, V.L. Pokrovsky, and V.M. Vinokur,Phys. Rev. Lett. 88, 017001 ~2002!.8 G. Blatter, M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin, andV.M. Vinokur, Rev. Mod. Phys. 66, 1125 ~1994!.9 P. Martinoli, M. Nsabimana, G.A. Racine, H. Beck, and R. Clem,Helv. Phys. Acta 56, 765 ~1983!.10 J. Pearl, J. Appl. Phys. Lett. 5, 65 ~1964!.11 A. A. Abrikosov, Introduction to the Theory of Metals ~North-Holland, Amsterdam, 1986!.12 J. Bardeen and M.J. Stephen, Phys. Rev. A 140, 1197 ~1965!.13 P. G. de Gennes, Superconductivity of Metals and Alloys~Addison-Wesley, New York, 1989!.14 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, andProducts, 5th ed. ~Academic Press, Boston, 1994!.15 A. P. Malozemoff and J.C. Slonczewski, Magnetic Domain Wallsin Bubble Materials ~Academic Press, New York, 1979!.16 N.B. Kopnin and V.M. Vinokur, Phys. Rev. Lett. 83, 4864 ~1999!.17 Ar. Abanov, V. Kalatsky, V.L. Pokrovsky, and W. Saslow, Phys.Rev. B 51, 1023 ~1995!.1-4

Anisotropic transport properties of ferromagnetic-superconducting bilayers

Texas A&M University

PHYSICAL REVIEW B 69, 132501 ~2004!Anisotropic transport properties of ferromagnetic-superconducting bilayers
M. Amin Kayali1 and Valery L. Pokrovsky1,2
1Department of Physics, Texas A & M University, College Station, Texas 77843-4242, USA
2Landau Institute for Theoretical Physics, Moscow, Russia
~Received 12 December 2003; published 5 April 2004!
We study the transport properties of vortex matter in a superconducting thin film separated by a thin
insulator layer from a ferromagnetic layer. We assume an alternating stripe structure for both FM and super-
conducting ~SC! layers as found in S. Erdin et al. @Phys. Rev. Lett. 88, 017001 ~2002!#. We calculate the
periodic pinning force in the stripe structure resulting from a highly inhomogeneous distribution of the vortices
and antivortices. We show that the transport in SC-FM bilayer is highly anisotropic. In the absence of random
pinning it displays a finite resistance for the current perpendicular to stripe and is superconducting for the
current parallel to stripes. The average vortex velocity, electric field due to the vortex motion, Josephson
frequency, and higher harmonics of the vortex oscillatory motion are calculated.
DOI: 10.1103/PhysRevB.69.132501 PACS number~s!: 74.78.2w, 74.25.Ha, 74.25.Qt, 75.70.2iThe interest in heterogeneous ferromagnetic-super-
conducting systems has grown rapidly in recent years. This
interest stems not only from their possible technological ap-
plications but also from new physical phenomena arising
from the interaction between two order parameters. Typically
such a system consists of a superconductor ~SC! placed in
close proximity with a periodic ferromagnetic structure ~FS!
such as an array of ferromagnetic dots or holes. The two
systems are separated by an infinitely thin layer of insulator
oxide that guarantees the suppression of proximity effects. It
was demonstrated experimentally1–3 that the interaction be-
tween the superconductor and the ferromagnet may lead to
formation of superconducting vortices interacting with the
FM film. Theoretical studies of such systems have been done
in Refs. 4–7.
Recently, Erdin et al.7 studied the equilibrium structure of
a FM-SC bilayer ~FSB!. They have proved that it represents
a two-dimensional periodic stripe domain structure consist-
ing of two equivalent sublattices, in which both the magne-
tization mz(r) and the vortex density nv(r) alternate. Thus,
they predicted spontaneous violation of the translational and
rotational symmetry in the bilayer. In this Brief Report we
study the transport properties of the FSB. They are associ-
ated with the driving force acting on the vortex lattice from
an external electric current. We show that the FSB exhibits
strong anisotropy of the transport properties: the bilayer may
be superconducting for the current parallel to the domain
walls and resistive when the current is perpendicular to them.
The force acting on a vortex from other vortices which
determines the value of critical current can be characterized
as the periodic pinning. An extensive development of theory
and experiment related to the pinning and its influence on
transport in superconductors was discussed in an exhausting
review by Blatter et al.8 Our work differs from the studies
considered in this review by two features. First, in the pre-
ceding works the magnetic field was assumed to be constant
in space, whereas in our problem the average magnetic field
is zero, it is strongly inhomogeneous in space. Therefore, in
our system equal numbers of vortices and antivortices par-
ticipate in the motion. Second, in these works the pinning
force was assumed to be random, whereas in our case the0163-1829/2004/69~13!/132501~4!/$22.50 69 1325dominant pinning forces are periodic and regular. Martinoli
et al.9 created artificially periodic pinning barriers in super-
conducting films modulating their thickness periodically. The
main difference of their modulated structure from one con-
sidered in our work is that the domains in the FSB are not
confined to the crystal lattice and can move together with the
vortices.
Periodic pinning forces in the direction parallel to the
stripes do not appear in continuously distributed vortices,
their reappearance is associated with the discreteness of the
vortex lattice. Therefore, we need to modify the theory7 to
incorporate the discreteness effects. Let us assume that the
saturation magnetization per unit area of the FM film is m
and its width is L. The energy necessary to create a single
Pearl vortex10 in the superconductor is ev05e0ln(l/j) with
e05f0
2/16p2l , where f0 is the flux quantum, l5lL
2 /ds is
the effective penetration depth,11 lL5Amec2/4pnse2 is the
London penetration depth, ds is the thickness of the super-
conducting layer, and j is its coherence length. It was shown
in Ref. 7 that the interaction between the superconducting
vortices and the magnetization in the stripe structure renor-
malizes the single-vortex energy to the value ev˜5ev0
2mf0 which must be negative to allow development of the
stripes. The density of the superconducting vortices increases
when approaching the domain walls and can be expressed as
nv(x)5(pm˜ /Lf0)@1/sin(px/L)#, where m˜ 5m2(ev0 /f0) is
the renormalized magnetization of the FM stripe. The vorti-
ces spontaneously appear in the superconductor. We assume
that the vortices inside one stripe are arranged in parallel
chains ~Fig. 1!. Each chain is periodic with the same lattice
constant b along the chain, whereas the distance between kth
and (k11)th chain ak depends on k. The correspondence
between this discrete arrangement and continuous
approximation7 is established by the requirement that the lo-
cal vortex density nv(xk) calculated in Ref. 7 must be equal
to (bak)21. The coordinate xk is determined in terms of ak
as a sum: xk5(k850
k21
ak8 . For definiteness we choose the ori-
gin in the center of the stripe. We assume that the total num-
ber of the vortex chains 2N in a stripe is large. Then some of
them are located very close to the domain walls. Let us re-©2004 The American Physical Society01-1
BRIEF REPORTS PHYSICAL REVIEW B 69, 132501 ~2004!mind that, in the continuous approximation7 nv
5(pm˜ /Lf0)@1/sin(px/L)#, where L is the domain width.
Considering the nearest to the domain-wall vortex chain
~with the number N!, we put nv(xN)51/baN . On the other
hand xN5L2aN . Since aN /L!1, we find: b5f0 /m˜ . The
total number of chains in a stripe is 2N , where N
5b*0
L2lnv(x)dx5 12 ln(L/l). We cut the integration ~and
summation! at a distance ;l from the domain wall where
the continuous approximation breaks. Thus, the minimum
value of a is l . When transport current passes through the
superconducting film, the vortices start to move. To simplify
the problem we assume that all vortices in each stripe move
together as well as all antivortices in the neighboring stripe
do. We denote their positions r15(x1 ,y1) and r2
5(x2 ,y2), respectively. Forces acting on a moving vortex
are the Magnus force, the viscous force, and the periodic
pinning force. The Magnus force is fm5pns\ds(vs2r˙)3zˆ ,
where ns is the superconducting electron density, vs is the
velocity of the superconducting electron, and r˙ is the vortex
velocity. The viscous ~friction! force is ff52hr˙, where h
5f0Hc2ds /rnc2 is the Bardeen-Stephen drag coefficient,12
Hc2 is the upper critical magnetic field, rn is the resistivity
of the superconducting sample in the normal state, and c is
the speed of light. The periodic pinning forces are due to the
interaction of the vortex with the pinning centers and the
domain walls. In the FM-SC bilayer the pinning force is due
to the interaction of the domain walls with the vortices and
antivortices and the vortex-vortex interaction Uvv given by
Uvv5
1
2E E nv~r!V~r2r8!nv~r8!d2rd2r8, ~1!
where V(r2r8) is the pair interaction between vortex lo-
cated at r and another at r8. When ur2r8u@l the pair inter-
action can be written as V(r2r8)5(f02/4p2)(1/ur2r8u).13
The interaction energy between two parallel chains located at
xl and xl8 and vertically shifted with respect to each other by
an interval bs (s!1) reads
FIG. 1. Schematic vortex distribution in the FM-SC bilayer. The
sign 6 refers to the vorticity of the trapped flux.13250U~xl ,xl8 ,s !5 (
n ,m51
N0
f0
2
8p2
A~xl2xl8!
21~n2m2s !2b2
, ~2!
where N0 is the number of vortices or antivortices in a single
chain. For infinite chains (N0→‘) Eq. ~2! can be rewritten
as
U~xl ,xl8 ,s !5 (k52‘
‘
N0f0
2
8p2
A~xl2xl8!
21~k2s !2b2
. ~3!
The sum in Eq. ~3! can be calculated using Poisson summa-
tion formula.14,15 Since the force is zero in the continuous
approximation, it is possible to retain the lowest nonzero
harmonic in the Poisson summation. Thus, we arrive at the
following interaction energy of two chains:
U~xl ,xl8 ,s !5
N0f0
2
4p2b
cos~2ps !x ll8 , ~4!
where x ll85e
22p(uxl2xl8u/b)
. The distance between two
chains uxl2xl8u is larger or equals l hence x ll8;x
5e2(2pl/b). Typical value of x ll8 is e
2(dm/4p), where dm
5mf0 /e05gS(nmdm /nsds), with g is Lande factor, S is the
ferromagnet elementary spin, nm and dm are the electrons
density and thickness of the magnetic film, respectively.
We conclude that the amplitude of the periodic potential
for displacements parallel to the domains in units of e0 is
exponentially small near the transition temperature. Relative
displacements in perpendicular direction have energy barrier
;e0 even in continuous approximation. We model the
restoring forces by simple sines dependencies f x
5 f’sin@(2p/a)(x12x2)#, f y5 f uusin@(2p/b)(y12y2)#, where
f’;e0 /a and f uu;(e0 /b)e2(dm/4p)! f’ .
When the supercurrent is perpendicular to domains, the
equations of motion for a vortex and antivortex are
hy˙ 15F2
F
vs
x˙ 12 f uusinS 2pb ~y12y2! D , ~5!
hx˙ 15
F
vs
y˙ 11 f’sinS 2pa ~x12x2! D , ~6!
hy˙ 252F1
F
vs
x˙ 21 f uusinS 2pb ~y12y2! D , ~7!
hx˙ 252
F
vs
y˙ 22 f’sinS 2pa ~x12x2! D , ~8!
where F5pns\dsvs , f’5e0 /a and f uu5xe0 /b . If the cur-
rent is smaller than a critical value, Eqs. ~5!–~8! accept a
static solution
x15x25
Fb
4phvs
arcsinS Ff uu D , ~9!1-2
BRIEF REPORTS PHYSICAL REVIEW B 69, 132501 ~2004!y152y25
b
4parcsinS Ff uu D . ~10!
It is valid at F< f uu5xe0 /b . For F. f uu or equivalently, if the
current is larger than its critical value, vortices and antivor-
tices start to move. The solution of Eqs. ~5!–~8! for F.Fc
reads
x12x250, ~11!
x11x25
F
hvs
~y12y2!, ~12!
y12y25
b
p
arctanS f uuF 1A12 f uu2F2tan~v0’t !D , ~13!
y11y250, ~14!
where v0
’52phvs
2Ab2F22x2e02/b2(F21h2vs2) is the Jo-
sephson frequency. Thus, the vortices and antivortices ac-
quire the same velocity components v1x5v2x in the direc-
tion of the current and opposite velocity components v1y
52v2y in the direction perpendicular to the current. The
domain walls do not interfere such a motion if they move in
the direction of the current with the same velocity vdw
5v1x5v2x as vortices and antivortices. Such a motion is a
Goldstone mode. The solution ~11!–~13! displays an oscilla-
tory motion of the vortices and antivortices in the direction
parallel to the domain walls ~Fig. 2!, in addition to their
motion together with the domain walls along the direction of
the current. Higher harmonics of the vortex motion can be
calculated analytically. The distribution of vortices ~antivor-
tices! is inhomogeneous in the direction perpendicular to the
domains. The local electric field E due to the vortex motion
is related to its time-average velocity ^v1& as E
52(qvf0 /c)nv(r)(^v1&3zˆ).8 Therefore, the local field
produced by vortices in the direction parallel to the domains
is equal but opposite in sign to the one produced by antivor-
tices, while the local field produced by vortices and antivor-
tices in the direction perpendicular to the domains has both
equal magnitude and sign. The time average of the vortex
~antivortex! velocity over a period T52p/v0
’ is
FIG. 2. The vortex displacement as a function of time in the
overcritical regime. Time is measured in units of 1/v0
’ and y1 in
units of b and x51024.13250^v1y&56
hAF22 f uu2
S h21 F2
vs
2 D , ~15!
^v1x&5
F^v1y&
hvs
. ~16!
The time-averaged local-field components are
Ex52
hm˜
ac
AF22 f uu2
S F2
vs
2 1h
2D , ~17!
Ey57
m˜ F
avsc
AF22 f uu2
S F2
vs
2 1h
2D . ~18!
The upper sign in Eqs. ~15! and ~18! refers to the vortices
velocity and produced field along the domain while the lower
sign refers to those due to antivortices. Nonzero average
electric field due to all vortices and antivortices in the FSB
appears only in the direction perpendicular to the domains.
The critical current Jc is related to Fc as Jc5(c/f0ds)Fc .
Plugging Fc5e0mx/f0 into the expression for Jc and ac-
cepting x51024 –1022, b51024 –1025 cm, and ns
51022 cm23 we find: Jc;103 –105 A/cm2.
When the current flows parallel to the stripes, the FM
domain walls stay at rest while vortices and antivortices
move both parallel and perpendicular to the domains. The
solution of equations of motion for vortices and antivortices
shows that they move opposite to one another both in x and
y directions. Their motion along x is oscillatory with funda-
mental frequency v0
uu5@2phvs
2Aa2F22e02/a2(F21h2vs2)# .
The motion of vortices and antivortices in the parallel direc-
tion proceeds until the distance between them becomes half
lattice spacing b/2. Once the vertical shift between the vor-
tices and antivortices reaches b/2, their motion freezes. The
critical current in this case is Jc5nsmB/2a , the lattice spac-
ing a is of the order of l;1025 –1024 cm, hence the critical
current Jc is of the order 107 –108A/cm2, which is at least
102 times larger than the critical current for parallel current.
Therefore, the system may be superconducting for the cur-
rent parallel to the stripes and exhibit finite resistance for
perpendicular current. The difference in the critical currents
for parallel and perpendicular directions is due to the expo-
nential factor x which is small if b!l . The anisotropy is
pronounced when dm is large which can be achieved by us-
ing thicker FM layers and decreasing the density of the su-
perconducting electrons. dm is temperature dependent and
eventually decreases when temperature decreases starting
from Ts . However, at the temperature of vortex disappear-
ance Tv,Ts the value m˜ turns into zero and x again be-
comes exponentially small. Thus, anisotropy has a minimum
between Tv and Ts .
Kopnin and Vinokur16 considered a collection of super-
conducting grains with the wash-board pinning potential a1-3
BRIEF REPORTS PHYSICAL REVIEW B 69, 132501 ~2004!model of random pinning. They obtained a similar result for
vortex sliding in external magnetic field with a supercurrent
applied. In contrast to their work ~they considered vortices
only!, we consider vortices and antivortices in the field of
periodic pinning and completely neglect the random pinning.
Let us discuss briefly how the magnetic field generated by
supercurrent changes our result. In Ref. 17 it was shown that
at sufficiently small critical magnetic field the domains van-
ish. Therefore, in general, magnetic field suppresses both the
anisotropy and periodic pinning at a critical field for which
domains disappear. At such critical field only random pin-
ning prevails. However, the total current per unit length is
proportional to the thickness of the SC film and can be kept
small.
In conclusion, we studied the transport properties of the
FM-SC bilayer in a state with stripe domains of alternating
magnetization and vorticity. In the absence of a driving
force, the vortices and antivortices are arranged in straight
chains configuration. The force between two chains of vorti-
ces falls off exponentially as a function of the distance sepa-
rating the chains. We argued that, in the vicinity of the su-
perconducting transition temperature Ts and vortex
disappearance temperature Tv , the distances between chains13250become much larger than the distances between vortices in
the same chain. We solved the equations of motion for vor-
tices and antivortices for the driving current direction paral-
lel and perpendicular to the domains. The calculated parallel
to domains critical current is much higher than the perpen-
dicular one at least in a vicinity of the two transition tem-
peratures. This strong transport anisotropy is due to the fact
that, for the perpendicular current the induced motion is a
Goldstone mode specific for a system of mobile domains and
vortices. We expect the ratio of the parallel to perpendicular
critical current to be in the range 102 –104 close to the super-
conducting transition temperature Ts and to the vortex disap-
pearance temperature Tv . The anisotropy decreases rapidly
when the temperature goes from the ends of this interval
reaching its minimum somewhere inside it. The anisotropy
can be destroyed by a rather weak magnetic field perpendicu-
lar to the bilayer. This anisotropic transport behavior could
serve as a diagnostic tool to discover spontaneous topologi-
cal structures in magnetic-superconducting systems.
This work was supported by the NSF Grants Nos. DMR
0103455 and DMR 0072115, U.S. DOE Grant No. DE-
FG03-96ER45598, and Telecommunication and Information
Task Force at Texas A & M University.1 D.J. Morgan and J.B. Ketterson, Phys. Rev. Lett. 80, 3614 ~1998!.
2 J.I. Martin, M. Velez, J. Nogues, and I.K. Schuller, Phys. Rev.
Lett. 79, 1929 ~1997!.
3 M.J. Van Bael, J. Bekaert, K. Temst, L. Van Look, V.V. Mosh-
chalkov, Y. Bruynseraede, G.D. Howells, A.N. Grigorenko, S.J.
Bending, and G. Borghs, Phys. Rev. Lett. 86, 155 ~2001!; M.
Lange, M.J. Van Bael, L. Van Look, K. Temst, J. Swerts, G.
Guntherodt, V.V. Moshchalkov, and Y. Bruynseraede, Europhys.
Lett. 51, 110 ~2001!; S.J. Bending, G.D. Howells, A.N. Grig-
orenko, M.J. Van Bael, J. Bekaert, K. Temst, L. Van Look, V.V.
Moshchalkov, Y. Bruynseraede, G. Borghs, and R.G. Hum-
phreys, Physica C 332, 20 ~2000!.
4 I.F. Lyuksyutov and V.L. Pokrovsky, Phys. Rev. Lett. 81, 2344
~1998!.
5 I.F. Lyuksyutov and V.L. Pokrovsky, cond-mat/9903312 ~unpub-
lished!.
6 S. Erdin, M.A. Kayali, I.F. Lyuksyutov, and V.L. Pokrovsky,
Phys. Rev. B 66, 014414 ~2002!.
7 S. Erdin, I.F. Lyuksyutov, V.L. Pokrovsky, and V.M. Vinokur,Phys. Rev. Lett. 88, 017001 ~2002!.
8 G. Blatter, M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin, and
V.M. Vinokur, Rev. Mod. Phys. 66, 1125 ~1994!.
9 P. Martinoli, M. Nsabimana, G.A. Racine, H. Beck, and R. Clem,
Helv. Phys. Acta 56, 765 ~1983!.
10 J. Pearl, J. Appl. Phys. Lett. 5, 65 ~1964!.
11 A. A. Abrikosov, Introduction to the Theory of Metals ~North-
Holland, Amsterdam, 1986!.
12 J. Bardeen and M.J. Stephen, Phys. Rev. A 140, 1197 ~1965!.
13 P. G. de Gennes, Superconductivity of Metals and Alloys
~Addison-Wesley, New York, 1989!.
14 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and
Products, 5th ed. ~Academic Press, Boston, 1994!.
15 A. P. Malozemoff and J.C. Slonczewski, Magnetic Domain Walls
in Bubble Materials ~Academic Press, New York, 1979!.
16 N.B. Kopnin and V.M. Vinokur, Phys. Rev. Lett. 83, 4864 ~1999!.
17 Ar. Abanov, V. Kalatsky, V.L. Pokrovsky, and W. Saslow, Phys.
Rev. B 51, 1023 ~1995!.1-4


Anisotropic Transport Properties of Ferromagnetic-Superconducting Bilayers

Abstract

Similar works

Full text

Available Versions

OAKTrust Digital Repository (Texas A&M Univ)

Texas A&M University