Fractional Josephson vortices at YBa Cu O grain boundaries 2 3 7−x R. G. Mints and Ilya Papiashvili School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel (February 1, 2008) 1 ◦ We report numerical simulations of magnetic flux patterns in asymmetric 45 [001]-tilt grain 0 0 boundariesinYBa2Cu3O7−x superconductingfilms. ThegrainboundariesaretreatedasJosephson 2 junctions with the critical current density jc(x) alternating along the junctions. We demonstrate n theexistenceof Josephson vortices with fractional fluxquantafor both periodic and randomjc(x). A method is proposed to extract fractional vortices from experimental fluxpatterns. a J 8 ] n Numerousrecentstudies ofelectromagneticproperties banks of the junction. The Josephson current density o of grainboundaries in high-T superconducting films are j(x) depends on the total phase difference ϕ(x)+α(x). c c driven by necessity to probe the fundamental symmetry Assuming j(x) ∝ sin[ϕ(x) + α(x)] one can develop a - r of the order parameter and the flux quantization [1–3]. model of the electromagnetic properties of the grain p Althoughinterpretationofthe results is nontrivial,most boundaries in YBCO films [6]. Values of the phase α(x) u of the data can be understood in terms of the conven- depend on the relative orientation of the neighboring s . tional model of a strongly coupled Josephson junction facets. In the case of an asymmetric faceted 45◦ grain t a [4]. The asymmetric 45◦ [001]-tilt grain boundary in boundary we have an interchange of α = 0 and π and m YBa2Cu3O7−x films is a notable exception of this rule. j(x) = jc(x)sinϕ(x), where the alternating critical cur- - We showinthis letterthattheseboundariesshouldhave rent density j (x)∝cosα(x). d c unusualelectromagneticpropertiesunprecedentedforthe In this letter we report numerical simulations of flux n o physics of standard Josephson junctions. patternsintheasymmetric45◦[001]-tiltgrainboundaries c Experimentally the asymmetric 45◦ [001]-tilt grain inYBa2Cu3O7−xsuperconductingfilms. Theboundaries [ boundaries in YBa2Cu3O7−x films exhibit an anoma- are treated as Josephson junctions with an alternating 1 lous dependence of the critical current Ic on the ap- critical current density jc(x). We find two types of frac- v plied magnetic field H [5,6]. Contrary to the usual tional Josephson vortices for each stationary state with a 5 Fraunhoffer-type dependence with a major central peak a spontaneous flux in the grain boundaries, which exists 8 atH =0andminorsymmetricside-peaks,thesebound- for both periodic and random sequences of facets. One 0 a 1 aries demonstrate a pattern without the central major type of vortices contains the magnetic flux φ1 < φ0/2; 0 peak. Instead, two symmetric major side-peaks appear theothertypecarriesφ2 >φ0/2withacomplementarity 1 at certain fields H =±H 6=0 [5–7]. Other remarkable condition φ1 +φ2 = φ0, where φ0 is the flux quantum. a p 0 feature is the spontaneous disorderedmagnetic flux gen- We suggest a method to extract the fractional vortices / t erated at the asymmetric 45◦ [001]-tilt grain boundaries from the data on flux patterns. a m in YBa2Cu3O7−x films [8]. It is worth noting that the Thealternatingdependence jc(x)isimposedbyapar- spontaneous flux is observed only in samples exhibiting ticular sequence of facets along the boundary and there- - d the anomalous dependence Ic(Ha). forejc(x)hasthesametypicallength-scalelasthefacets. n It is convenient for the further analysis to write j (x) as Clearly, the major side-peaks reveal a specific hetero- c o geneity of the Josephson properties. Indeed, a fine scale c j =hj i[1+g(x)], (1) : faceting of grain boundaries in YBCO thin films has c c v been recorded by the transmission electron microscopy i wherehj iistheaveragevalueofthecriticalcurrentden- X [3,9–11]. Thefacetshaveatypicallength-scaleloftheor- c sity over distances L≫l: r derof10–100nmandawidevarietyoforientations. This a grain boundary structure combined with a predominant 1 L dx2−y2 wave symmetry of the order parameter [2,12–14] hjci= L Z jc(x)dx. (2) 0 form a basis for understanding both the anomalous de- pendence Ic(Ha) and the spontaneous flux [7,8,15,16]. The dimensionless function g(x) in Eq. (1) character- Inthe caseofadx2−y2 wavesuperconductorthe phase izestheJosephsonpropertiesofthegrainboundary. This differenceoftheorderparameteracrossthegrainbound- function alternates with a typical length-scale l and has ary consists of two terms. The first term ϕ(x) is caused a zero average: hg(x)i = 0. The maximum amplitude by a magnetic flux inside the junction and the second of alternations of |g(x)| may vary from |g(x)|max >∼ 1 to termα(x) iscausedbyamisalignmentofthe anisotropic |g(x)|max ≫1dependingondetailsofthestructureofthe 1 faceted grain boundary. We assume that λ ≪ l ≪ ΛJ, vortex ψ− = ψγ, ψ+ = 2π − ψγ, the phase difference where λ is the London penetration depth and being 2π −2ψ , and thus this vortex contains the flux γ φ2 =(1−ψγ/π)φ0 >φ0/2. Thesetwofractionalvortices cφ Λ2 = o . (3) are complementary: φ1+φ2 =φ0. J 16π2λhj i c In our numerical study we solve Eq. (4) exactly. We is an effective Josephson penetration depth. With this treatthestationarystatesaswellastherelaxationtothe notation, the phase difference ϕ(x) satisfies stationary states using a time-dependent model [16] Λ2Jϕ′′−[1+g(x)]sinϕ=0. (4) ϕ¨+αϕ˙ −ϕ′′+[1+g(x)]sinϕ=0, (11) A model grain boundary with a periodic critical cur- rent density j (x) has been considered analytically by where α is a decay constant which we take from the in- c means of a two-scale perturbation theory which requires terval 0.1 < α < 1. The term αϕ˙ in Eq. (11) describes l ≪ Λ [15]. In this approximation, the phase ϕ(x) is a dissipation driving the system into one of the stable sta- J sum of a smooth part ψ(x) with a length scale Λ and a tionary states described by the solutions of Eq. (4). J rapidly oscillating part ξ(x) with a length scale l and a We begin our numerical simulations with verification small amplitude |ξ(x)|≪1: of the results obtained by means of the two-scale ap- proximation for the grain boundary with periodic j (x). c ϕ(x)=ψ(x)+ξ(x). (5) To study the fractional vortices we start the simulations from a certain initial phase ϕ (x) under the condition i We have for the phases ψ(x) and ξ(x): ϕ (L)−ϕ (0)=2πn,wheretheboundarylengthL≫Λ . i i J Λ2ψ′′−sinψ+γsinψcosψ =0, (6) In this case the numerical procedure converges well to a J final stationary state. ξ(x)=ξ (x)sinψ, (7) g InFig.1weshowastablestationarysolutionforapair where the function ξ (x) is given by offractionalvortices. We compute ϕ(x) using the model g g(x) = g0sin(2πx/l) with g0 = 100 and l = 0.1ΛJ. The Λ2ξ′′ =g(x), (8) value of γ calculated by means of Eq. (9) is given by J g and the dimensionless parameter γ >0 is defined as g2l2 γ = 0 . (12) γ =−hg(x)ξg(x)i=Λ2Jhξg′2i. (9) 8π2Λ2J Itisworthmentioningthatbothξg(x)andγ dependonly This yields γ ≈ 1.27 and ψγ ≈ 0.66; thus φ1 ≈ 0.21φ0, onthespatialdistributionofjcandthereforecharacterize φ2 ≈ 0.79φ0. As is seen in Fig. 1, the simulation gives the individual Josephson properties of a particular grain the same value of ψ . The magnified insets in Fig. 1 γ boundary. We stress that this approximation is valid if demonstrate that ϕ(x) indeed consists of a smooth part l ≪ ΛJ and |ξ(x)| ≪ |ψ(x)| (the latter condition results superimposed with a small fast oscillating term. There- in |g(x)|≪4π2Λ2J/l2). fore, the numerical simulations for single fractional vor- Intheframeworkofthe two-scaleperturbationtheory, tices confirm the qualitative and quantitative results of a single Josephson vortex is described by the solution of the approximate analytic approachdescribed above. Eq.(6)undertheboundaryconditionsψ′(±∞)=0. The latter can be written as sinψ±(1−γcosψ±)=0, where M ψ± =ψ(±∞). Itisconvenientforthefurtheranalysisto (cid:21)S(cid:3)(cid:14)(cid:3)\(cid:3)J assume that ψ− <ψ+. 6 Inthe caseofγ <1,there is only onesingle vortexso- (cid:21)S(cid:3)(cid:16)(cid:3)\(cid:3)J (cid:11)F(cid:12) lution, for whichthe phase ψ(x) increases monotonically (cid:25)(cid:17)(cid:27) 4 from ψ− = 0 to ψ+ = 2π. This solution describes the (cid:25)(cid:17)(cid:25) Josephson vortex with one flux quantum φ0. In the case of γ > 1, the spatial distribution of the smooth phase (cid:24)(cid:17)(cid:25)(cid:24) (cid:11)E(cid:12) 2 (cid:25)(cid:17)(cid:23) ψ describes two fractional vortices. For the first frac- tional vortex the phase ψ(x) increases from ψ− = −ψγ \(cid:3)J (cid:24)(cid:17)(cid:24)(cid:24)(cid:19) (cid:20) (cid:21) (cid:21) (cid:20)(cid:24) (cid:20)(cid:26) to ψ+ =ψγ, where 0 -10 0 10 20 \// (cid:3) . ψγ =arccos(1/γ). (10) FIG. 1. The phase distribution ϕ(x) computed using g(x) = g0sin(2πx/l) and γ ≈ 1.27. Two fractional vortices Thedifferenceψ+−ψ− =2ψγ andthusthisvortexcarries with φ1 ≈ 0.21φ0 and φ2 ≈0.79φ0 are clearly seen, the fine the flux φ1 = ψγφ0/π < φ0/2. For the second fractional structureof ϕ(x) is demonstrated in themagnified insets. 2 Consider now a dilute chain of fractionalvortices. Let both ϕ (x) and α due to the flux pinning induced by the i a vortex with the flux φ1 be situated somewhere in the non-uniformity of the critical current density. chain. Thephaseψ ofthisvortexchangesfrom2πn−ψγ A special role in the description of Josephson bound- to 2πn+ψγ with an integer n. Therefore, one expects aries with random alternating jc(x) belongs to the sta- the phase of an adjacent vortex to start with the value tionary state ϕ (x) which corresponds to the zero total s 2πn+ψγ and to end up with 2π(n+1)−ψγ, the total spontaneous flux and to the absolute minimum of the phase accumulation of these two vortices being 2π. In Josephson energy E{ϕ(x)} (defined in a standard way other words, the chain consists of a sequence of pairs of [17]). Our simulations show that ϕ (x) is unique for a s vortices with fluxes φ1 and φ2. This qualitative picture given boundary, stable, and independent either of initial is confirmed by numerically solving Eq. (4). Figure 2(a) guesses ϕ (x) or of the damping constant α. Therefore, i shows the result of such a calculation for which we took the phase ϕ (x) can serve as a signature of each indi- s g(x)=150 sin(20πx/ΛJ), that corresponds to γ ≈2.85, vidual boundary. It is convenient to represent ϕs(x) as ψγ ≈1.21,andthe fluxesφ1 ≈0.39φ0,φ2 ≈0.61φ0. The ϕs(x) = ψγ +ξs(x) with ψγ = const and the variable finalstationarystateofournumericalproceduresimulat- part ξ (x) having zero average, hξ (x)i = 0, and a typi- s s ing the relaxation process, depends on the choice of the cal amplitude |ξ (x)|<π/2. s initial phase ϕ (x). By taking a proper non-monotonic i Anexampleofacomputedϕ (x)isshowninFig.3(a). s dependence ϕ (x), we may end up with a stationary so- i For this simulation we took ϕ (x) = const+ξ (x) with i i lution shown in Fig. 2(b). A remarkable feature of this an arbitrary small ξ (x). As stated above, the resulting i solution is the existence of fractional vortex-antivortex phase ϕ (x) is independent of ϕ (x). It is worth men- s i pairs clearly seen in the simulation of Fig. 2(b), the pair tioning that the spontaneous self-generatedflux φ (x)= s with the fluxes ±φ1 is followed by the pair with ∓φ2. φ0ξs(x)/2π has awide rangeoflength-scalesimposedby therandomj (x). Notealsothatrandomnessofj (x)re- c c sults in a considerably higher amplitude of the flux vari- (cid:11)E(cid:12) ation φ (x) as compared to a periodic j (x); this is seen s c (cid:21)S(cid:11)R(cid:14)(cid:22)(cid:12)(cid:14)\ J from comparison of Fig. 3(a) with the insets in Fig. 2. (cid:21)S(cid:11)R(cid:14)(cid:22)(cid:12)(cid:16)\ J (cid:11)F(cid:12) M (cid:21)S(cid:11)R(cid:14)(cid:21)(cid:12)(cid:14)\ S(cid:18)(cid:21) J (cid:19) (cid:21)SR(cid:14)\ J (cid:11)F(cid:12) (cid:16)S(cid:18)(cid:21) (cid:21)SR(cid:16)\ J M \ S(cid:18)(cid:21) FIG.2. Twochainsoffractionalvorticesinagrainbound- (cid:11)E(cid:12) ary with a periodically alternating critical current density: (cid:19) (a) an “ideal” chain, (b) a chain with vortex-antivortex “de- - 30 - 20 - 10 0 10 20 \(cid:19)/. fects”. Empty triangles mark the positions of the fractional FIG.3. Two stationary solutions ϕ(x) developed in a zero vortices with the fluxes φ1 < φ0/2, full triangles correspond magnetic field for a stepwise randomly alternating g(x) for to φ2 > φ0/2. The up-down orientation of triangles indicate twoinitialconditions: (a)preventing,(b)stimulatingcreation the field direction of vortices. For this particular calculation of vortices (g0 = 200, l = 0.1ΛJ, σl ≈ 0.06l, ψγ ≈ 1.515, we use g(x) = g0sin(2πx/l), g0 = 150, l = 0.1ΛJ, which φ1≈0.48φ0, φ2 ≈0.52φ0). result in γ ≈2.85, ψγ ≈1.21, and φ1 ≈0.39φ0, φ2 ≈0.61φ0. It follows from Eq. (4) that the stationary solution Next,westudyfluxpatternsforamorerealisticcaseof ϕ (x) generates two series of solutions having the same s a grain boundary with 2N facets and a non-periodic al- Josephson energy as ϕ (x): ϕ+(x) = 2πn+ϕ (x) and s n s ternatingcriticalcurrentdensityj (x). Wetreatthiscase ϕ−(x) = 2πn−ϕ (x), where n is an integer. The aver- c n s numericallyanduseastepwiseg(x)definedas: g(x)=g0 age values hϕ+n(x)i=2πn+ψγ and hϕ−n(x)i=2πn−ψγ if ai < x < bi, and g(x) = −g0 if bi < x < ai+1 interchangebeingseparatedbyhϕ+n(x)i−hϕ−n(x)i=2ψγ (i = 1,...,N). It is convenient to introduce the ran- or by hϕ− (x)i−hϕ+(x)i = 2π−2ψ , as is shown in n+1 n γ dom distances a˜ and b˜ with ha˜i=hb˜i=0 and a stan- Fig.2foraperiodicj (x). Thegapsbetweentheaverage i i i i c dard deviation σ such that b − a = 0.5(l + a˜) and values of the stationary phases ϕ+(x) and ϕ−(x) allow l i i i n n ai+1−bi =0.5(l+b˜i). The simulationsstartwithanini- for fractional vortices with the fluxes φ1 = φ0ψγ/π and tial phase ϕi(x) matching the condition ϕi(L)−ϕi(0)= φ2 =φ0−φ1 as solutions of Eq. (4) varying with a typ- 2πn. Then the numerical procedure of solving Eq. (11) ical length-scale of Λ . An example of a computed ϕ(x) J converges well to a stationary state which depends on withtwoclearlypronouncedfractionalvortex-antivortex 3 pairs and small varying part ξ(x) is shown in Fig. 3(b). one of these lines everywhere, except a few relatively sharpjumps from one line to the next; in particular, the Considernowatypicalflux(phase)patternforachain “signature” ϕ (x) is nested at ψ : ϕ (x) = ψ +ξ (x). ofvorticesforarandomlyalternatingj (x). Assumethat s γ s γ s c the chain starts with a domain with ϕ(x) = ϕ (x), i.e., Thejumps (orvortices)shouldbecenteredatϕ(x)=πn s [where according to Eq. (4) ϕ′′(x) = 0]. Then we take the phase varies slightly (|ξ (x)| < π/2) around its av- s a domain situated between lines πn and π(n +1) and erage value ψ . Therefore, ψ is the value of hϕ(x)i at γ γ form ϕ (x) = ϕ(x)∓ξ (x), choosing the minuses if the the “tail” of the neighboring vortex or antivortex. If the v s random parts of ϕ(x) and ϕ (x) are identical, and the neighbor carries the flux φ2, the average phase should s plus otherwise. The curve ϕ (x) shown by a thick curve increase from ψ to 2π −ψ in the neighbor’s domain. v γ γ at Fig. 4(b) is remarkably smooth; clearly it represents Alternatively the neighbor may carry the flux −φ1 gen- the fractional vortices described above within the model erating a decreaseof hϕ(x)i from ψ to −ψ . In general, γ γ with periodic j (x). this flux pattern is similar to the one arising for a pe- c riodic j (x). However, the relatively large amplitudes In conclusion, we have shown by numerical simula- c of the spontaneous flux and the absence of periodicity tions that two types of fractional vortices with fluxes ◦ within the chain may mask the fractional vortices. φ1 <φ0/2 and φ2 =φ0−φ1 >π/2 exist at 45 [001]-tilt grainboundariesinYBa2Cu3O7−x filmsexhibitingspon- taneous flux in zero field cooled samples. Faceted grain M boundariesaretreatedasJosephsonjunctionswithalter- (cid:23)S nating critical current density. We show how to extract (cid:22)S I(cid:3) (cid:3)(cid:21)I(cid:19) fractionalvorticesfromthedataonspontaneousfluxdis- (cid:21)S tribution. S I(cid:3) (cid:3)(cid:19) (cid:11)E(cid:12) Oneofus(RGM)isgratefultoJ.R.Clem,V.G.Kogan, and J. Mannhart for useful and stimulating discussions. (cid:19) (cid:23)S (cid:21)\J ThisresearchissupportedinpartbygrantNo.96-00048 fromthe United States – IsraelBinationalScience Foun- (cid:22)S (cid:21)S(cid:3)(cid:16)(cid:3)(cid:21)\J dation (BSF), Jerusalem, Israel. (cid:21)S S (cid:11)F(cid:12) (cid:19) (cid:16)(cid:22)(cid:19) (cid:16)(cid:21)(cid:19) (cid:16)(cid:20)(cid:19) (cid:19) (cid:20)(cid:19) (cid:21)(cid:19) (cid:22)(cid:19)\/L . FIG. 4. (a) The “signature” ϕs(x) and the phase ϕ(x) with the total flux φ = 2φ0 at the boundary calculated for [1] P. Chaudhari and S.Y. Lin, Phys. Rev. Lett. 72, 1084 g0 =90, l=0.1ΛJ, σl ≈0.015l, which gives ψγ ≈1.21, and (1994). φ1 ≈0.39φ0,φ2 ≈0.61φ0. (b)Thethinlinedepictsϕ(x),the [2] C.C. Tsuei et al., Phys.Rev.Lett. 73, 593 (1994). thicklinedepictsthephaseϕv(x)generatedbythefractional [3] J.H. Miller et al.,Phys. Rev.Lett. 74, 2347 (1995). vortices and extracted from the phaseϕ(x). [4] R. Gross et al.,Phys. Rev.Lett. 64, 228 (1990). [5] C.A. Copetti et al., Physica C 253, 63 (1995). [6] H. Hilgenkamp, J. Mannhart, and B. Mayer, Phys. Rev. At the top panel of Fig. 3 we show ϕ(x) obtained nu- B 53, 14586 (1996). merically for g0 = 90, l = 0.1ΛJ, and σl = 0.015l, the [7] R.G. Mints and V.G. Kogan, Phys. Rev. B 55, R8682 calculated value of ψ is 1.21. The bottom curve is the γ (1997). “signature” ϕs(x) corresponding to the zero total flux, [8] J. Mannhart et al.,Phys. Rev.Lett.77, 2782 (1996). φ = 0. The upper curve is calculated with the same set [9] C.L. Jia et al.,Physica C 196, 211 (1992). of input parameters for φ = 2φ0. Even a brief examina- [10] S.J. Rosner, K. Char, and G. Zaharchuk, Appl. Phys. tion of the curves shows a striking correlation between Lett. 60, 1010 (1992). the two: there are domains (e.g., −10 < x/ΛJ < 10) in [11] C. Træholt et al., Physica C 230, 425 (1994). which the “noise” patterns are nearly identical, whereas [12] D.A. Wollman et al., Phys.Rev.Lett. 71, 2134 (1993). inothers(e.g.,10<x/Λ <30)thepatternsrepeateach [13] D.A. Brawner and H.R. Ott, Phys. Rev. B 50, 6530 J other being flipped. This suggests that one can extract (1994). [14] D.J. Van Harlingen, Rev.Mod. Phys. 67, 515 (1995). the smooth part of the upper curve ϕ(x) by properly [15] R.G. Mints, Phys.Rev. B 55, R8682 (1997). subtracting the “signature” ϕ (x). s [16] R.G.MintsandIlyaPapiashvili,Phys.Rev.B62,15214 The subtraction is done as follows. First, we draw the (2000). straightlines2πn±ψγ atthegraphofϕ(x),seeFig.4(b). [17] A. Barone and G. Paterno, Physics and Applications of We see that “random” variations of ϕ(x) are nested on the Josephson Effect (Wiley, New York,1982). 4