Flux saturation number of superconducting rings
D. V. Denisov, D. V. Shantsev, Y. M. Galperin, T. H. Johansen
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov Flux saturation number of superconducting rings
D. V. Denisov, D. V. Shantsev, Y. M. Galperin,
1, 2, 3 and T. H. Johansen ∗ Department of Physics and Center for Advanced Materials and Nanotechnology,University of Oslo, P. O. Box 1048 Blindern, 0316 Oslo, Norway A. F. Ioffe Physico-Technical Institute, Polytekhnicheskaya 26, St. Petersburg 194021, Russia Argonne National Laboratory, 9700 S. Cass Av., Argonne, IL 60439, USA (Dated: October 25, 2018)The distributions of electrical current and magnetic field in a thin-film superconductor ring iscalculated by solving the London equation. The maximum amount of flux trapped by the hole,the fluxoid saturation number, is obtained by limiting the current density by the depairing current.The results are compare it with similar results derived for the bulk case of a long hollow cylinder[Nordborg & Vinokur, Phys. Rev. B , 12408 (2000)]. In the limit of small holes our result reducesto the Pearl solution for an isolated vortex in a thin film. For large hole radius, the ratio betweensaturation numbers in bulk and film superconductors is proportional to the square root of the holesize. PACS numbers: 74.25.Qt, 74.25.Ha, 68.60.Dv
Use of thin film superconductors integrated in nanode-vices requires precise knowledge of the film behavior inthe presence of both external and self induced magneticfields. Recent experiments have shown that properly de-signed arrays of dots and antidots can serve as effectivetraps for magnetic flux.
It has also been shown thatpatterned superconducting films allow for the motion ofmagnetic vortices to be guided over the film area, opening up for a new field often called fluxonics.The models for flux trapping used in past years did nottake into account the precise geometry of the patternedfilm samples. Those conventional models were based onapproximations applicable either to a single vortex inan infinite film, equivalent to having a hole of zero ra-dius, i.e., a Pearl vortex , or to an infinitely long hollowcylinder. Both geometries are far from realistic forthin film devices. However several recent works consider-ing dynamics of vortices, current and field distributions inthin-film superconductor with finite hole size have beenpublished.
Yet, the flux saturation number,defined as the maximum number of flux quanta whichcan be trapped inside the hole, have so far not been cal-culated. Since the saturation number is a key conceptfor the vortex-antidot interaction and the overall effectof film patterning, we dedicate the present work to thiscalculation.In this paper, we examine the problem of trapped fluxin a thin-film superconducting ring by solving the Lon-don equations. The ring geometry allows modelling of arealistic situation with an isolated anti-dot in a thin-filmsample. We consider the case of zero external magneticfield, and show that the saturation number for a thin-filmring can differ significantly from that of a hollow cylin-der with the same radius. It is shown how the differencedepends on the ratio between the hole radius and thethickness of the ring.Consider a ring where the outer radius, r , is muchlarger than the inner radius, r , see Fig. 1. It is as-sumed that the film thickness, d , is negligible compared d B r r j d B r r j FIG. 1: The thin ring geometry. to both the hole radius and the effective penetrationlength λ eff = λ /d , where λ is the London penetrationdepth of the superconductor. We want to determine theamount of flux trapped in the ring in a remanent statewith flux present due to some magnetic prehistory. Itwill be assumed that flux pinning elsewhere is absent.In the superconductor r < r < r , and − d/ < z 2, the distributions of current density j and induction B are given by the London equation, which in terms ofthe vector potential A , where curl A = B , readscurl( λ µ j + A ) = 0 . (1)Due to the symmetry, the current density and vectorpotential have in cylindrical coordinates only azimuthalcomponents, j = (0 , j, 0) and A = (0 , A, µ λ j ( r ) + A ( r ) = Φ f πr , (2)where flux quantization implies that the constant Φ f ,the London fluxoid, is a free parameter restricted toan integer number of the flux quantum, Φ f = n Φ ,where Φ = h/ e . The second term on the left-hand-side represents the flux through the area of radius r ; R r B ( r ′ )2 πr ′ dr ′ = A ( r ) 2 πr , where B is the inductionin the film plane, and is directed perpendicular to theplane.In the absence of an applied field, the vector poten-tial is only due to the current in the ring, and can beexpressed as A = µ π Z j | r − r ′ | d r ′ . (3)For the film geometry one can neglected variations ofthe current across the the sample thickness and averageEq. (3) over z . Introducing the sheet current J ( r ) = R d/ − d/ j ( r ) dz , together with the dimensionless variables˜ J ( r ) = ( µ λ / Φ f ) J ( r ) and ˜ r = r/λ eff the Eq. (2) be-comes,˜ J (˜ r ) + 14 π Z ˜ r ˜ r Z π ˜ J ( ˜ r ′ ) cos θ p (˜ r/ ˜ r ′ ) + 1 − r/ ˜ r ′ ) cos θ dθd ˜ r ′ = 1 / π ˜ r . (4)This Fredholm integral equation of the second kind wassolved numerically by converting it into a set of linearequations corresponding to discrete values of the radialcoordinate,˜ J i + 14 π X ij Q ij ˜ J j = 12 π ˜ r i , (5) Q ij ≡ Z π cos θ p (˜ r i / ˜ r j ) + 1 − r i / ˜ r j ) cos θ dθ . (6)Results of such calculations are presented in Fig. 2. Theupper panel shows the field and current distribution forthe case where r = λ eff , r = 20 λ eff . The field is mostlyconcentrated in the hole, where it has a pronounced peakat r , while a small peak from the return field is seennear r . The shielding current flows predominantly inthe vicinity of the hole, reaching a maximum at the edge.Plotted in the lower panel is the normalized cur-rent distribution for holes of different sizes ranging from r /λ eff = 1 to 100, with all rings having r /λ eff = 1000.For comparison, the figure includes also the current dis-tribution around a single Pearl vortex in an infinite film, ˜ J Pearl (˜ r ) = [ S (˜ r/ − K (˜ r/ − /π ] / . (7)Here S is the first order Struve function and K thefirst order modified Bessel function of the second kind.For the small hole case, r = λ eff , our numerical resultis very close to the Pearl solution over the whole ringarea except near the outer edge, where the sheet currenthas an upturn. The curves for the two larger hole radiialso follow Eq. (7) for intermediate r , but are seen tohave an additional upturn at the inner edge. The cur-rent enhancement near r is found to increase with thehole size, whereas near the outer edge the behavior is r/ λ eff B (r) a nd J (r) , [ a r b . un it s ] J(r)B(r) −6 −4 −2 r/ λ eff Sh ee t c u rr e n t , ˜ J r r =100 λ eff r =25 λ eff r = λ eff FIG. 2: (Upper) Distribution of magnetic field H and current J in the superconducting ring. Inner radius of the ring r isequal to λ eff and outer radius r is equal to 20 λ eff . (Lower)Current density distribution in superconducting thin ringswith 3 different hole radii, r , and the same overall size r . Forcomparison, the graph includes also the Pearl vortex currentdistribution, plotted as the dash-dotted line. marginally influenced by the hole. Similar results for su-perconducting rings were reported previously by Brandtand Clem .Our main interest lies in finding the fluxoid saturationvalue as function of the size of the hole in a thin film. Itfollows from Eqs. (2) and (3) that the magnitude of thesheet current depends linearly on Φ f . Thus, the max-imum Φ f corresponds to having a current at the inneredge, J ( r ), equal to the maximum current supported bythe superconductor. We take this maximum value to bethe depairing current, j dp = Φ √ πµ λ eff ξ . (8)Thus, the fluxoid saturation number, n sat , satisfies, n sat Φ j dp d = Φ f J ( r ) , (9)which gives n sat = λ eff √ πξ ˜ J (˜ r ) . (10)Shown in Fig. 3 is the fluxoid saturation number as afunction of the hole size. The following material parame-ters were here used, ξ = 3 nm, λ = 150 nm, d = 100 nm,corresponding to YBa Cu O x at low temperatures. Thestepwise increase of n sat versus r seen in the main plotis essentially linear, but with an additional weak upwardcurvature. The inset shows that this behavior continuesalso as for larger holes, r > λ .For small holes we found that the current near theinner edge behaves essentially according to Eq. (7), whichin this limit is well approximated by ˜ J (˜ r ) = 1 / π ˜ r . Thus,for small holes the saturation number is given by, n sat = 2 r √ ξ , r ≪ λ eff . (11)For larger holes where r ≫ λ eff , the current upturn nearthe edge prevents us from using the Pearl solution di-rectly. Instead, we make a conjecture that the sheet cur-rent distribution is described by˜ J (˜ r ) = 1 π ˜ r p ˜ r − ˜ r . (12)Firstly, this analytical form fits excellently when com-pared to the current enhancement calculated numerically.Secondly, this form has the same asymptotic behavior asthe Pearl solution, ˜ J (˜ r ) = 1 /π ˜ r for ˜ r ≫ 1, consistentwith the close agreement seen in Fig. 2 over a wide in-termediate range of r . Finally, the Eq. (12) has the di-verging factor 1 / √ r − r , commonly present in the edgebehavior of thin superconductors in the Meissner state. As usual, the divergence is cut off at a distance λ eff fromthe edge, giving for the ring geometry a maximum sheetcurrent of ˜ J (˜ r ) = 1 / ( √ π ˜ r / ). It then follows fromEq. (10) that for large holes one has, n sat = √ r / √ ξλ / , r ≫ λ eff . (13)It is of interest to compare these results with the flux-oid saturation number associated with a hole in a bulksuperconductor. The case of a circular hole through aninfinite superconductor was discussed in Ref. 11, usingthat for this geometry with r being the hole radius, theinduction is given by B ( r ) = n Φ CK ( r/λ ) , r ≥ r , (14)where 1 /C = πr λ [2 K ( r /λ ) + ( r /λ ) K ( r /λ )], and K , K are the zeroth-order and first-order modifiedBessel functions. The current density around the holehere obtained simply by, µ j ( r ) = B ′ ( r ), and with F l ux s a t u r a ti on nu m b e r Hole size, r / λ FilmBulk Film Bulk FIG. 3: Trapped flux inside the hole of thin superconductingfilm (blue line) and inside the infinite cavity in bulk super-conductor (red line) depending on the size of the hole/cavity. j ( r ) = j dp , the maximum number of trapped flux quantabecomes, n bulk = 23 √ r ξ (cid:20) r λ K ( r /λ ) K ( r /λ ) (cid:21) . (15)The bulk saturation number is also plotted in Fig. 3,where one sees the same tendency as for the thin filmcase, namely that a larger hole size allows more flux tobe trapped. For small r the two curves are actuallyoverlapping. This is because in Eq. (2), which is validfor any cylindrical geometry, the vector potential termbecomes negligible for small r , giving µ λ j ( r )2 πr = Φ f as the asymptotic behavior independent of sample thick-ness. Thus, the linear hole size dependence in Eq. (11)holds also for the bulk case.As r increases the curves deviate, showing that thinfilm superconductors will not trap as much flux as abulk sample of the same material. For r /λ ≫ 1, theBessel functions in Eq. (15) can be simplified, and onegets asymptotically a quadratic hole size dependence, n bulk = r √ ξλ , r ≫ λ . (16)For large holes, the ratio in flux trapping capability issimply n bulk n sat = r r d . (17)For typical patterned films with hole size, r = 500 − r ≪ λ eff ,the result is approximately equal to the flux saturationnumber for bulk superconductors with a cylindric cav-ity. For large holes, r ≫ λ eff , the saturation number inthin films is less than in bulks by a factor p r / d . Thepresent results, obtained by assuming the sample size tobe much larger than the hole, should apply to supercon-ducting films patterned with a single antidot. 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