PHYSICAL REVIEW B 66,

Transcription

1 PHYSCAL REVEW B 66, Cupled messcpic supercnductrs: Ginzburg-Landau thery B. J. Baelus, S. V. Yamplskii,* and F. M. Peeters Departement Natuurkunde, Universiteit Antwerpen (UA), Universiteitsplein 1, B-610 Antwerpen, Belgium Received 7 December 001; revised manuscript received 11 March 00; published 4 July 00 The magnetic cupling between tw cncentric messcpic supercnductrs with nnzer thickness is studied using the nnlinear Ginzburg-Landau thery. We calculated the free energy, the expelled field, the ttal field prfile, the Cper-pair density, and the current density distributin. By putting a smaller supercnducting disk r ring in the center f a larger ring, the prperties change drastically. Extra grund-state transitins are fund, where the ttal vrticity stays the same, but the vrticity f the inner supercnductr changes by 1. Due t the magnetic cupling, the current in the external ring exhibits extra jumps at the transitin fields where the vrticity f the inner supercnductr changes. n this case, fr certain temperatures, re-entrant behavir and switching n and ff f the supercnducting behavir f the rings are fund as a functin f the magnetic field. A H-T phase diagram is btained fr the situatin where the inner ring has a higher critical temperature than the uter ring. An analytic expressin fr the magnetic cupling is btained fr thin rings and extreme type- supercnductrs. DO: /PhysRevB PACS numbers: 74.0.De, 74.5.Ha, 74.5.Dw. NTRODUCTON Single messcpic supercnducting rings 1 11 and disks 1 5 have attracted a lt f attentin during the last years. The supercnducting prperties f these single samples have been studied experimentally and are rather well described and understd theretically. Many f these studies were limited t disks 1,13,0 3 and rings 7,9 f zer thickness. n such a case the magnetic field induced by the supercurrents are neglected and ne assumes that the ttal field equals the external field, which is taken unifrm. n the present paper disks and rings with finite thickness are cnsidered. n this case the supercnductr tries t expel the field. Supercurrents are induced creating a lcal magnetic field ppsite t the external ne. This leads t a decrease f magnetic field inside the supercnductr and an increase near the sample edges. Fr sufficiently large disks and magnetic fields, the magnetic field penetrates the supercnductr and vrtices are created, which crrespnds t regins f lcal-field cmpressin. n small disks, nly circular symmetric vrtex states nucleate in the center, which are called giant vrtex states. Encircling the vrtex n a clsed lp, the phase change f the rder parameter equals L times, where L is the angular mmentum r vrticity. n larger disks, seperate vrtices can nucleate creating a multivrtex state and the vrticity L is just the number f vrtices. n the case f rings, the magnetic field can be expelled t the inside r t the utside f the ring r partially t bth sides. n the present paper we want t understand what will happen if tw messcpic supercnductrs are put clse t each ther but are electrically islated. When a unifrm magnetic field is applied, it will be lcally altered by each f the supercnductrs. n sme regins f space the field will be expelled frm the supercnducting disk r ring, while in ther regins it will be cmpressed int vrtices penetrating the sample r cmpressed int the inside f a supercnducting ring. This results in a strngly nnunifrm ttal field, which is the superpsitin f the applied field and the field created by the supercurrents. Anther supercnductr will interact with this nnunifrm field. n this way the supercnductrs are cupled by the magnetic field and they interact with each ther. This cupling will influence the prperties f the tw supercnductrs. t is this cupling that will be studied in the present paper. The magnetic cupling between nrmal messcpic rings is well knwn. Wang et al. 6 studied theretically the persistent currents f N nrmal rings placed peridically n the same plane. Because f the mutual inductance between tw rings, the electric current in ne ring prduces an induced flux in the ther ring, creating an extra current in this ring. They fund that the mutual inductance between nrmal rings enhances the persistent current as was the case in the experiments described in Ref. 7. Crrelatins in arrays f magnetically cupled supercnducting aluminum rings were investigated experimentally by Davidvić et al. 8 They used ultrasensitive susceptibility techniques and scanning Hall prbe micrscpy t study arrays f electrically islated supercnducting rings f micrn size. When the external flux is clse t 0 /, the magnetic mments prduced by the supercurrents in such rings are analgus t sing spins. Magnetic mments parallel t the applied magnetic field can be called spin up, while thse in the ppsite directin spin dwn. Via their diplar magnetic fields, neighbring rings can interact antiferrmagnetically and the different rings influence each ther. Recently, Mrelle et al. 9 studied experimentally the magnetic interactin between tw supercnducting messcpic aluminum rings, clse t the supercnducting/nrmal phase transitin. n their sample, a smaller ring was placed in the center f the larger ring. Using resistivity measurements the phase bundary was btained fr the tw-ring structure as well as fr the reference single ring. n bth systems, Little- Parks scillatins were bserved in the H-T phase diagram. The mdificatin f the T c (H) scillatins f the uter ring is seen in the Furier spectrum f the T c (H) line due t the /00/66/ /$ The American Physical Sciety

2 B. J. BAELUS, S. V. YAMPOLSK, AND F. M. PEETERS PHYSCAL REVEW B 66, FG. 1. Schematical utline f the cnsidered cnfiguratins; the ring-disk cnfiguratin left and the ring-ring cnfiguratin right. cupling between the uter and the inner ring. They suggested that an inner ring made frm a different supercnductr with a higher critical temperature wuld increase the magnetic cupling between the tw rings. n the present paper we present a theretical investigatin f the prperties f tw cupled messcpic supercnductrs. Our main attentin will g t the interactin between the tw supercnductrs. Hw they influence each ther? Hw d the supercnducting prperties f a single ring change when anther supercnductr is placed in its center? Therefre we cnsider tw different cnfiguratins: i a ring-disk cnfiguratin where a small disk is placed in the center f a larger ring, and ii a ring-ring cnfiguratin where a small ring is placed in the center f the larger ring as in the experiment f Ref. 9. We will als give an example f a ring-ring system where the inner ring is made frm a different supercnductr with a higher critical temperature. Our theretical analysis is based n a full self-cnsistent numerical slutin f the cupled nnlinear Ginzburg-Landau equatins. Since we cnsider sufficiently narrw rings and small disks, nly axial symmetric giant vrtex states will nucleate. 8 Therefre the equatins can be slved fr a fixed value f the vrticity. The magnetic-field prfile utside and inside the supercnductr is btained self-cnsistently and therefre the full demagnetizatin effect is included in ur apprach. The paper is rganized as fllws: n Sec. we describe the theretical frmalism. Our results fr the ring-disk cnfiguratin are discussed in Sec.. n Sec. V the ring-ring cnfiguratin is studied. n Sec. V we calculate the H-T phase diagram fr the tw ring system, where the inner ring is made f a different material with a higher critical temperature. n Sec. V we analytically calculate the energy f tw cupled thin rings made f a different material fr high values f. Finally, in Sec. V ur results are summarized.. THEORETCAL FORMALSM We cnsider a supercnducting ring with inner radius R i, uter radius R, and thickness d immersed in an insulating medium fr example, vacuum. n the center f this ring a supercnducting disk with radius R * r anther supercnducting ring with inner radius R i * and uter radius R * is placed with the same thickness see Fig. 1. The whle sample is placed in a perpendicular unifrm magnetic field H (0,0,H 0 ). T slve this prblem, we expand ur previus apprach fr thin supercnducting disks 17 t a system f tw axial symmetric supercnductrs each made f a different material. n the present paper we slve the system f tw cupled nnlinear Ginzburg-Landau equatins which determine the distributin f bth the supercnducting rder parameter (r) and the vectr ptential A (r) inside and utside the supercnductr, 1 i ea m c i, i,, A 4 c j, 1a 1b where fr the inner supercnductr the parameters are i, i, i, i, and fr the uter ring i,, i,. The density f supercnducting current j is given by j e im * * 4e mc A, and m is the mass f the Cper pair. Since we nly cnsider circular symmetric rings and disks, we use cylindrical crdinates. Any psitin in space will be expressed by r (,,z), where is the radial distance frm the center, is the azimuthal angle, and z is the perpendicular directin. The sample lies between zd/ and zd/. Equatins 1a have t be supplemented by bundary cnditins fr (r) and A (r). The bundary cnditin fr the supercnducting cndensate at the supercnductr/ insulatr surfaces is given by n i ea c R 1 *,R *,R 1,R 0, 3 where n is the unit vectr in the radial directin. This cnditin expresses that the supercnducting current in the radial directin vanishes at the sample surface. The bundary fr the vectr ptential has t be taken far away frm the disk, A r 1 e H 0, where the field equals the applied magnetic field H (0,0,H 0 ). T rewrite Eqs. 1a in dimensinless variables, we express all distances in units f the cherence length f the uter supercnductr /m, the rder parameter in 0, /, the vectr ptential in c/e, and the magnetic field in H c, c/e H c,, where H c, 4 / is the thermdynamical critical field f the uter supercnductr and / is the Ginzburg-Landau parameter fr this supercnductr. The penetratin depth f the uter ring is given by m/(c/4e)/ 0,. Using these dimensinless variables and the Lndn gauge, div A 0, we can rewrite Eqs. 1a in the fllwing frm: i A i,, 5a i,

3 COUPLED MESOSCOPC SUPERCONDUCTORS:... A 1 i * * A, 5b where i, i, i, i in the inner supercnductr, and i,, i, in the uter ring. The first bundary cnditin becmes n i A R1 *,R *,R 1,R 0. Prvided that Eqs. 1a are fulfilled, the difference between the Gibbs free energy f the supercnducting state and the nrmal state is determined by the expressin F V 1 A A 0 j i, 4dr, where the integral is ver the ttal vlume V f the supercnducting samples and A 0 1 e H 0 is the external vectr ptential in the absence f a supercnductr. The free energy is measured in H c, V/8. The dimensinless supercurrent j is given by j(* *)/i A. We restrict urselves t samples with thickness d, which implies that we are allwed t assume that the rder parameter des nt vary in the z directin. 15,17 On the ther hand, fr the vectr ptential the variatin in the z directin is retained and fr d this is very imprtant due t the Meissner effect. 15,17 We cnsider sufficiently narrw rings and small disks, and therefre nly axial symmetric giant vrtex states will nucleate. 8,17 Cnsequently, the equatins can be slved fr a fixed value f the angular mmentum L ut in the uter ring and L in in the inner supercnductr that leads t the rder parameter, f e il in,ut, where L in,ut L ut in the uter ring and L in,ut L in in the inner supercnductr. Cnsequently, bth the vectr ptential and the supercnducting current density are directed alng the azimuthal directin e. Fr fixed angular mmenta L ut and L in, Eqs. 5a and 5b can be reduced t the fllwing frm: 1 f L in,ut A f f i, 8a 1 A A i, L in,ut z A f f, z d, 6 7 8b where (x)1 fr x1 and (x)0 fr x1, A Ae, and indicates averaging ver the disk thickness f (r) (1/d) d/ d/ f (z,)dz. Because the supercnducting cndensates f the inner and uter supercnductrs are discnnected frm each ther they cannt influence each ther directly. The cupling is entirely due t the magnetic field, r equivalently the vectr ptential, as is expressed by the secnd term in Eq. 8a. The ttal magnetic field is a sum f the applied field and the field created by the supercnducting currents f the inner ring and the uter supercnductr which is described by Eq. 8b where f () f in () f ut () and f in () f ut () is nly different frm zer in the interval R i *R * R i R. The magnetic field created by the supercurrent has a H 1/r 3 dependence fr large r. Therefre we can restrict ur calculatins t a regin with radial size R s which we tk typically five times the sample size and lngitudinal size d s typically ten times the sample thickness. The bundary cnditins fr the uter parameter can be written as f 0, R 1 *,R *,R 1,R fr the ring-ring cnfiguratin, and f R *,R 1,R 0, f 0, fr the ring-disk cnfiguratin. The last cnditin f Eq. 10 fllws frm the rtatinal symmetry f the system. The current density has t vanish in the center f the disk. The cnditin 4 fr the vectr ptential taken at infinity can be transferred t the bundaries f ur simulatin regin, Az,R s 1 H 0R s, Azd s, 1 H Fllwing the apprach f Ref. 17 we apply a finite difference representatin n the space grid n,z m t slve Eqs. 8a and 8b. Since the size f ur simulatin regin exceeds by far thse f the sample, we apply nnunifrm space grids t diminish the cmputer time. The space grid is taken unifrm inside the sample, and we increase the grid spacing expnentially with distance utside the sample. This allws us t use the same number f grid pints, typically 18, inside and utside the sample. T btain steady-state slutins fr a system f tw supercnductrs, the fllwing iteratin prcedure was used: f f n k n1/ n1/ k k f n1 f n n1/ L in,ut PHYSCAL REVEW B 66, f k k n f n1 n1/ n n n1 A f k n n f f k1 n i, f n k1 3, i, f k n 3 i, f n k1 k f n 1a

4 B. J. BAELUS, S. V. YAMPOLSK, AND F. M. PEETERS PHYSCAL REVEW B 66, k a A m,n n1/ n1/ k n1a m,n1 n1 n k n A m,n k na m,n z m1/ z m1/ A m1,n z m1 z m k k A m,n k n1 A m,n1 n n1 A k m,n k A m1,n z m z m1 L in,ut n k A m,n f k n a A k1 m,n, 1b where A m,n A(z m, n ), f n f ( n ), n1/ ( n1 n )/, and z m1/ (z m1 z m )/. The upper index k dentes the iteratin step. T speed up the cnvergency we intrduced the iteratin parameters f and a, and we expanded the nnlinear term ( f k n ) 3 ( f k1 n ) 3 3(f k1 n ) ( f k n f k1 n ).. RNG-DSK CONFGURATON First, we cnsider a supercnducting ring with a supercnducting disk in the center. We investigate the influence f the disk n the prperties f the ring. As an example, we take a ring with inner radius R i 1.5 and uter radius R.0 and a disk in the center with radius R *1.0. Bth supercnductrs have the same thickness, d0.15, and the Ginzburg-Landau parameter was taken t be 0.8, which is a typical value fr messcpic Al supercnductrs. 14,15 Figure a shws the free energy fr the cnsidered system as a functin f the applied magnetic field. First we cnsider the uncupled system and calculated the free energy fr the disk in the center thick dashed curve and the free energy fr the different giant vrtex states in the uter ring thick dtted curve. The results fr single rings and disks were exhaustively described in previus papers. 17,8 Ntice that, in this paper, the free energy is expressed in units f F 0 H c V/8, where V is the sum f the disk and the ring vlume. This is the reasn why the free energy f the disk and the ring are nt equal t F 0 at zer magnetic field as it was in Refs. 17 and 8. The size f the disk is such that nly the Meissner state, i.e., the L in 0 state, can nucleate. At applied magnetic fields H 0 /H c.9 the disk is in the nrmal state, which results in F0. n the single ring, n the ther hand, different giant vrtex states with vrticity L ut 0 up t L ut 10 can nucleate befre the ring becmes nrmal at H 0 /H c 6.8. Next, we intrduced the magnetic cupling between the disk and the ring and the results are given by the slid curves in Fig. a. The different axial symmetric states are determined by the vrticity f the disk L in and the ttal vrticity L ut, which is equal t the vrticity f the ring. Therefre we characterize the states by (L ut,l in ). Fr the cnsidered cnfiguratin, we find states with L in 0 and L ut 0uptL ut 5. We als find states with L in 1 and L ut 0 up t L ut 10, which equal the giant vrtex states f the single ring, because the disk is nw in the nrmal state. Ntice further that we culd als write FG.. a The free energy and b the expelled field as a functin f the applied field fr a ring with inner radius R i 1.5 and uter radius R.0 thick dtted curves and a disk in the center with radius R *1.0 thick dashed curves and fr the cupled ring-dt cnfiguratin thin slid curves. All supercnductrs have the same thickness, d0.15, and 0.8. The thin dashed curves give the sum f the free energies f the single disk and the single ring. (6,0) instead f (6,1) because fr the applied magnetic fields where the L ut 6 state in the ring exists the disk is nrmal even fr L in 0. We have chsen t write L in 1 because this expresses mre clearly that there is flux ging thrugh the disk. f bth the disk and the ring are supercnducting, the free energy f the ttal sample is different frm the sum f the free energies f the single disk and the single ring. T investigate these new states in mre detail we cnsider as an example the (,0) state. Figures 3a c shw the magnetic-field distributin, the current density, and the Cper-pair density, respectively, as a functin f the radial psitin fr five different applied magnetic fields, i.e., H 0 /H c 0.1, 0.5, 1.5,.0, and.5. Near H 0 /H c 0 the (,0) state equals the L in 0 state f the disk and the ring is in the nrmal state. The reasn is that the applied field is s lw that a lt f magnetic flux has t be attracted t create a state with L ut in the uter ring. Therefre a very high supercnducting current has t flw thrugh the uter ring which leads t the destructin f supercnductivity in this ring. The slid curves in Figs. 3a c shw that the Cper-pair density and the current density are indeed zer in the ring. The magnetic-field distributin shws the flux expulsin frm the disk. nside the disk the field decreases and at the edge there is a peak which illustrates a higher cncentratin f field because f the demagnetizatin effects. With increasing external field less flux has t be attracted and the current in the uter ring decreases. At H 0 /H c 0.17 supercnductivity is restred in the external ring see the dtted curves in Fig. 3c. The dtted curves in Fig. 3b

5 COUPLED MESOSCOPC SUPERCONDUCTORS:... PHYSCAL REVEW B 66, The abve discussin shws clearly the interplay between the supercnducting state f the disk and the ring. Next, we investigate the interactin between the disk and the ring. Therefre we added the sum f the free energies f the single disk and the single ring thin dashed curves in Fig. a and cmpare this result with the result f the ring-disk cnfiguratin slid curves in Fig. a. Ntice that there is a small difference between the tw set f curves, which is due t the cupling between the tw supercnductrs. Belw we will shw that this difference becmes mre prnunced fr thicker samples. Nw, we will determine the attractin r expulsin f the magnetic field by the cupled supercnducting system. Figure b shws the magnetic field expelled frm the sample, M, as a functin f the applied magnetic field: M HH 0, 4 FG. 3. a The magnetic-field distributin, b the current density, and c the Cper-pair density as a functin f the radial psitin fr the (L ut,l in )(,0) state f the ring-dt cnfiguratin f Fig. at H 0 /H c 0.1, 0.5, 1.5,.0, and.5. shw that the current in the uter ring flws in the ppsite directin than the current in the disk. The supercnducting currents in the disk expel the flux, while the currents in the ring are attracting flux, which is cmpressed in the regin between the disk and the ring see the dtted curve in Fig. 3a. The free energy becmes nw mre negative as cmpared t the free energy f the single disk see Fig. a. ncreasing the magnetic field further leads t less attractin f flux and hence t a higher Cper-pair density in the ring and a mre negative free energy. When the external flux becmes cmparable with the flux needed fr the L ut state, the uter part f the ring expels the flux t the utside, while the inner part f the ring still expels flux t the hle regin. Therefre the supercnducting current changes sign in the ring regin see the dashed curve in Fig. 3b. Since the flux is expelled in bth directins, the dashed curve in Fig. 3a shws a psitive peak at bth ring bundaries. Further increasing the external field leads t external fluxes larger than the flux needed fr the L ut state and hence the ring has t expel flux in rder t keep vrticity L ut. As a cnsequence, the current in the ring has t flw in the same directin as the current in the disk, which is shwn by the dash-dtted curve in Fig. 3b. Because f the expulsin, the field between the tw supercnductrs is lwer than the external field see the dash-dtted curve in Fig. 3 a. fwe further increase the magnetic field, the supercnducting currents in the uter ring have t increase in rder t expel mre flux and cnsequently the Cper-pair density decreases in the uter ring. At H 0 /H c.4 the supercurrent becmes t high and the ring becmes nrmal again see the dash-dtdtted curves in Figs. 3a c. At this field, the free energy equals the free energy f the single disk. where H is the magnetic field averaged ver the area R, i.e., the uter radius f the ring and H 0 is the applied field. The thick dashed and thick dtted curves are the results fr the single disk and the single ring and the slid curves fr the ttal ring-disk system. By putting a disk in the center f the ring, mre field is expelled and less field is attracted. Of curse, fr H 0 /H c.9 the disk is in the nrmal state and we recver the result fr the single ring case. V. RNG-RNG CONFGURATON n this sectin we replace the disk in the center by a secnd ring and the influence f this inner ring n the uter ring will be investigated. As an example fr this ring-ring cnfiguratin, we cnsider a supercnducting ring with inner radius R i 1.5 and uter radius R.0 and a secnd ring in the center with inner radius R i *0.6 and uter radius R *1.1. Bth rings have the same thickness, d0.15 and the same Ginzburg-Landau parameter 0.8. Figure 4 shws the free energy as a functin f the applied magnetic field fr the small single ring by thick dashed curves, fr the larger single ring by thick dtted curves, and fr the cupled ring-ring situatin by the thin slid curves, where the interactin between the tw rings is taken int accunt. n the inner ring supercnducting states can nucleate with vrticity L in 0, 1, and and the supercnducting/ nrmal transitin field is at H 0 /H c 6.4. n the uter ring states with vrticity L ut 0uptL ut 10 exist and supercnductivity is destryed at H 0 /H c The supercnducting states nucleating in the duble ring system can be characterized again by the indices (L ut,l in ). Fr L in 0, supercnducting states can nucleate with L ut 0upt4,fr L in 1 with L ut 1 up t L ut 8, and fr L in with L ut 5 up t 10. Fr L in 3 the states equal the states f the single uter ring because the inner ring will be nrmal. The indices in the figure crrespnd t the grund state f the cupled ring system. Fr the numerical example studied in the previus sectin the number and the psitin f the grund-state transitins are the same as fr the uter ring. n

6 B. J. BAELUS, S. V. YAMPOLSK, AND F. M. PEETERS PHYSCAL REVEW B 66, FG. 4. The free energy as a functin f the applied magnetic field fr a supercnducting ring with radii R i 1.5 and R.0 dtted curves, a ring with R i *0.6 and R *1.1 dashed curves and the duble ring cnfiguratin slid curves. Bth rings have the same thickness, d0.15, and 0.8. The indices indicate the grund-state vrticities (L ut,l in ). FG. 5. The grund-state free energy f the duble ring cnfiguratin f Fig. 4 slid curves and the sum f the free energies f the tw rings dashed curves fr d0.15 upper curves and fr d 1.0 lwer curves. The upper curve is shifted by 0.1 fr clarity. The insets shw sme f the crssings in mre detail. the present tw ring system this is n lnger the case and the number f transitins in the cupled system are larger than fr the single uter ring case. The inner ring induces extra transitins in the cupled system each time when the vrticity f the inner ring changes with ne unit. The first extra transitin is the transitin frm (,0) t (,1) and the secnd ne is the transitin frm (6,1) t (6,). The (,1) state is the grund state in the magnetic field regin 1.43H 0 /H c 1.63 and the (6,) state in the regin 4.7H 0 /H c 4.8. Hence, by putting a ring in the center f the larger ring, the grund state shws extra transitins. This result crrespnds t the experimental result f Mrelle et al., 9 wh saw mdificatins f the T c (H) scillatins f the uter ring in the Furier spectrum f the T c (H) line due t the cupling between the uter and the inner ring. Next, we fcus further n the interactin between the inner ring and the uter ring. Therefre we plt in Fig. 5 the grund-state free energy f the cupled rings slid curves and the sum f the free energies f the tw single rings dashed curves fr the previus cnfiguratin upper curves which are shifted by 0.1), i.e., d0.15, and fr a thicker sample with d1.0 lwer curves. Fr d0.15 the difference is mst prnunced fr the (,0), the (5,1), and the (6,1) state. Bth the value f the free energy and the transitin magnetic fields are influenced by the interactin between the tw rings. The left inset shws the (6,1) (6,) (7,) transitin in mre detail. Ntice that the interactin significantly decreases the magnetic field regin where the (6,) state is the grund state. Fr d1.0 the demagnetizatin effects becme mre imprtant and therefre the interactin between the tw rings gains imprtance. This results in a larger difference between the dashed and slid curves. The value f the free energy and the transitin fields are changing cnsiderably by the fact that bth rings are influencing each ther. The tw lwer insets shw the (,0) (,1) (3,1) and the (6,1) (7,) transitins in mre detail. The magnetic-field regin where the grund state is given by the (,1) state decreases due t the interactin. Fr the (6,1) (7,) transitin cupling between the tw rings leads t the interesting result that the (6,) state is n lnger a grund state. Figures 6a and b shw the magnetic field expelled frm the sample, M, as a functin f the applied magnetic field fr the single uter ring dashed curve and fr the FG. 6. The magnetic field expelled frm the sample, M, asa functin f the applied magnetic field fr the single uter ring dashed curve and fr the duble ring cnfiguratin slid curve. The sample is the same as in Fig. 4 with thickness d0.15 a and d1.0 b

7 COUPLED MESOSCOPC SUPERCONDUCTORS:... PHYSCAL REVEW B 66, FG. 7. The magnetic-field range H 0 ver which the (L ut,l in ) state is the grund state, as a functin f L ut fr the single uter ring the dashed curve and pen squares and fr the duble ring system the slid curve and clsed circles. The sample is the same as in Fig. 4 with thickness d0.15 a and d1.0 b. duble ring cnfiguratin slid curve fr thickness d 0.15 and d1.0, respectively. Fr d0.15, the tw extra transitins, resulting frm the influence f the inner ring, are clearly visible by the jumps at H 0 /H c 1.43 and 4.8. Ntice that, depending n the directin f the current, a single ring expels r attracts field at a given applied magnetic field and vrticity. Expulsin attractin leads t a lwer higher magnetic-field density in the center f the ring and t a higher lwer density near the utside. Therefre, depending n the applied field, the field expulsin r attractin in the cupled ring cnfiguratin can either increase if the tw currents are in the same directin, r decrease if the directins are ppsite. This can be clearly seen frm Figs. 6a and b where the field expulsin r attractin becmes mre prnunced by putting the inner ring in the center f the uter ring. Fr d1.0 ne extra transitin results frm the influence f the inner ring, i.e., at H 0 /H c The difference between the transitin fields f the single ring and the cupled ring system becmes larger see als the lwer curve in Fig. 5 and the difference in the expulsin becmes mre prnunced with increasing the sample thickness, which indicates again that the interactin between the tw rings increases with increasing d. n Figs. 7a and b the magnetic-field range H 0 ver which the (L ut,l in ) state is the grund state is pltted as a functin f L ut fr thickness d0.15 and d1.0, respectively. This magnetic-field range crrespnds t the distance between tw cnsecutive jumps in the expelled field see Fig. 6. The results fr the single uter ring are given by the pen squares and fr the duble ring system by the clsed circles. The curves are guides t the eye. Fr d0.15 the FG. 8. The averaged current density fr the grund state in the inner ring a and the uter ring b as a functin f the applied magnetic field fr the same duble ring cnfiguratin as in Fig. 4. The results fr the single rings are given by dashed curves, thse fr the duble ring cnfiguratin by slid curves. extra transitins are clearly visible at L ut and 6 and fr d1.0 at L ut where fr the same L ut tw jumps ccur due t a transitin f the inner ring. Als fr the ther vrticities L ut, there is a difference between H 0 fr the single ring and the duble ring. The reasn is that the grund-state transitin fields are influenced by the interactin between the tw rings. This was als visible in Fig. 5. f the free energy f the duble ring was just the sum f the free energies f the tw single rings, H 0 wuld be the same fr the single uter ring and the duble ring, except fr L ut and 6, where extra transitins ccur because L in changes with ne unit. Ntice further that the difference between the results fr the single uter ring and the duble ring enhances with increasing sample thickness. Next, we investigate the effect f the interactin between the tw rings n the supercnducting current density in the tw rings fr d0.15. Figures 8a and b shw the averaged current density fr the grund state in the inner ring and the uter ring, respectively, as a functin f the applied magnetic field. The results fr the single ring are given by dashed curves, these fr the duble ring cnfiguratin by slid curves. First, we describe what happens if there is n interactin between the tw rings. n this case we can cnsider them as tw single rings. At lw fields, the grund state f a single ring is given by the L0 state r Meissner state and the ring expels the field t the utside f the sample. With increasing external field, mre flux has t be expelled frm the ring which leads t a higher current density. After the first transitin the grund state is given by the L1 state and initially flux will be trapped in the ring and the flux ging thrugh the ring is larger than the flux f the external field. T cmpress this extra magnetic field, the supercnducting

8 B. J. BAELUS, S. V. YAMPOLSK, AND F. M. PEETERS PHYSCAL REVEW B 66, current in the ring has t flw in the ppsite directin. At the transitin, the current shws a jump frm a negative t a psitive value, i.e., frm expulsin t cmpressin. With increasing external field, less flux has t be cmpressed t achieve vrticity 1 and the current density in the uter ring decreases. Further increasing the field, the external flux becmes larger than the flux needed fr L1 and flux has t be expelled. Therefre the current in the ring changes sign. Withut interactin between the tw rings, the current density in ne ring exhibits nly jumps when the vrticity f the grund state f this ring changes see dashed curves in Figs. 8a and b. n the cupled tw rings situatin j shws small jumps n tp f the previusly described expulsin cmpressin jumps. At lw fields, the grund state is given by the (0,0) state r Meissner state. Bth rings expel the field t the utside f the sample, which means that the current flws in the same directin in each ring. Since sme flux is already expelled by the uter ring, the inner ring has t expel less and therefre the current is less negative. After the first transitin the grund state changes int the (1,0) state. Nw, the uter ring cmpresses the field t achieve vrticity 1, and, as a cnsequence, the field in the hle f the uter ring is larger than the external field. This means that the inner ring has t expel mre field and the current density jumps t a value mre negative than its value withut interactin. The ther transitins can be explained analgusly. Frm Figs. 8a and b it is clear that the tw rings are influencing each ther and that the interactin between the tw rings results in extra jumps in the current density in ne ring when the vrticity f the ther ring changes. These jumps are smaller than the jumps when the vrticity f the cnsidered ring increases, but they are nt negligible. Up t nw, we cnsidered rather small samples. Fr the single ring it is knwn that by increasing the sample size i mre L states are pssible and ii the magnetic-field range ver which the state with vrticity L is the grund state decreases. 8 Therefre, fr a larger radius f the duble ring cnfiguratin, we expect many mre grund-state transitins. Figure 9 shws the grund-state free energy fr a single inner ring with radii R *.0 and R i *1.5, fr a single uter ring with radii R 3.0 and R i.6 and fr the cupled ring-ring cnfiguratin. The sample thickness is d 0.15 and the Ginzburg-Landau parameter 0.8. Fr the single inner ring, the grund state changes frm vrticity L in 0uptL in 10 and the supercnducting/nrmal transitin field is at H 0 /H c Fr the single uter ring the grund state changes frm vrticity L ut 0uptL ut 3 and supercnductivity is destryed at H 0 /H c By cmparing the free energy f the duble ring with the ne f the uter ring, we ntice that there are many mre grundstate transitins as a cnsequence f the transitins in the inner ring. Fr the single ring the minimum in the free energy f the L1 state is always less negative than the ne f the L state. Due t the interactin between the tw rings, this is n lnger always the case fr the duble ring cnfiguratin. At H 0 /H c 6.73 the free energy f the duble ring cnfiguratin equals the ne f the uter ring since the inner ring is in the nrmal state. FG. 9. The grund-state free energy fr a single inner ring with radii R *.0 and R i *1.5, fr a single uter ring with radii R 3.0 and R i.6 and fr the duble ring cnfiguratin, i.e., the cmbinatin f these tw rings. The sample thickness is d 0.15 and the Ginzburg-Landau parameter 0.8. Figures 10a and b shw the field expelled frm the regin R * f the inner ring with radii R *.0 and R i *1.5 and the regin R f the uter ring with radii R 3.0 and R i.6, respectively. The results fr single rings are given by the dtted curves and fr the duble ring cnfiguratin by the slid curves. At lw fields, the single inner ring is in the Meissner state and expels the magnetic field, i.e., M0. With increasing external field, mre field FG. 10. The field expelled frm the regin R * f the inner ring with radii R *.0 and R i *1.5 a, and the regin R f the uter ring with radii R 3.0 and R i.6 b. The results fr single rings are given by the dtted curves and fr the duble ring cnfiguratin by the slid curves (d0.15 and 0.8)

9 COUPLED MESOSCOPC SUPERCONDUCTORS:... PHYSCAL REVEW B 66, FG. 11. The H-T phase diagram fr the inner ring dashed curves, the uter ring dtted curves, and the duble ring cnfiguratin slid curves. The material parameters and the sizes f bth rings are different and are given in the figure. is expelled. At H 0 /H c 0.33 the grund state changes frm L in 0tL in 1 and flux has t be cmpressed int the hle t achieve vrticity 1. Therefre the magnetic field inside the hle will be larger than the external ne and M jumps t negative values, i.e., field cmpressin. With increasing field, less flux has t be attracted and M becmes less negative. Further increasing the external flux becmes larger than the ne needed fr L in 1 and therefre the field has t be expelled again. This means that M becmes psitive. Further increasing the field, mre flux has t be expelled and M becmes mre psitive. At H 0 /H c 0.99 the vrticity changes frm L in 1 t L in, which means that M jumps t negative values, and s frth. At H 0 /H c 6.73 supercnductivity is destryed and the field becmes equal t the external ne, and as a cnsequence, M0. The descriptin fr the single uter ring is cmpletely analgus. Placing a larger ring arund the inner ring influences the expelled field drastically see the slid curves in Fig. 10a. Due t the expulsin f the uter ring at lw fields, the magnetic field inside the hle f this ring will be smaller than the external ne. Nw, the expulsin by the inner ring results in a smaller lcal field than fr the case f the single ring and, as a cnsequence, the expelled field increases at lw fields. At H 0 /H c 0.13 the grund state f the uter ring changes frm vrticity L ut 0tL ut 1, which means that suddenly the uter ring has t attract flux t achieve vrticity 1. Therefre the field inside the hle f the uter supercnductr becmes larger than the external ne and the expulsin by the inner ring will be less prnunced than fr the case f the single ring. As a cnsequence, M jumps frm a value abve the ne fr the single ring case t a value belw this value. This interplay between the tw rings leads t a higher expulsin attractin frm the regin R * when the uter ring expels attracts flux and t a lwer expulsin when the uter ring attracts expels flux. Fr the uter ring FG. 1. The free energy as a functin f the applied magnetic field fr the inner ring dashed curves, the uter ring dtted curves, and the duble ring cnfiguratin slid curves fr the system f Fig. 11 at T0.98T c,. The inset shws an enlargement f the phase diagram Fig. 11 in the T/T c, 1 regin. an analgus explanatin can be given see the slid curves in Fig. 10b. V. TWO COUPLED RNGS OF DFFERENT MATERALS Until nw, we cnsidered always tw supercnductrs made f the same material. This means that bth supercnductrs have the same cherence length, penetratin depth, and critical temperature, i.e., i, i, and T c,i T c,. Since bth rings have the same width and the radius f the inner ring is smaller than the ne f the uter ring, the inner ring becmes nrmal at a smaller field than the uter ring. As a cnsequence, n effect f the magnetic cupling can be bserved in the H-T phase diagram. T circumvent this prblem the T c f the uter ring was artificially lwered in the experiment f Ref. 9 by applying a sufficiently large external current thrugh the uter ring. An alternative apprach will be fllwed in the present sectin where we take the inner ring f a different material such that it has a higher critical temperature than the uter ring, and als a different cherence length and penetratin depth, which leads t a different Ginzburg-Landau parameter. As an example, we take fr the uter ring the values used by Geim et al. 14 fr Al, i.e., (T0)50 nm, (T 0)70 nm, and thus 0.8, resulting in a critical temperature T c, (H0)1.3 K. Fr the inner ring, we assume a higher critical temperature T c,i 1.T c, 1.56 K, and i (T0)160 nm, i (T0)80 nm, and thus i 0.5. Fr the radii f the uter ring we take as an example R i 375 nm, R 500 nm, and fr the inner ring R i * 15 nm and R *50 nm. The H-T phase diagram is shwn in Fig. 11 fr the uncupled situatin fr the inner ring thick dashed curves and fr the uter ring thick dtted curves and the cupled duble ring situatin slid curve. At T0 the uter ring has a much higher critical field

10 B. J. BAELUS, S. V. YAMPOLSK, AND F. M. PEETERS PHYSCAL REVEW B 66, (H nuc /H c, 6.74) than the inner ring (H nuc /H c, 4.34). Therefre the supercnducting/nrmal transitin f the duble ring cnfiguratin equals the ne f the uter ring fr lw temperatures. With increasing temperature, the nucleatin field f the uter ring, i.e., the ne f the duble ring system, changes mre quickly than the ne f the inner ring. The scillatins are the well-knwn Little-Parks scillatins. At T/T c, 0.91 bth single rings have the same transitin field H nuc /H c, At higher temperatures, the supercnducting/nrmal transitin is determined by the inner ring. The situatin where the critical field f the uter ring is larger than the ne f the inner ring is exhaustively described in the previus sectins. n Fig. 1 we shw the free energy fr the cnfiguratin f Fig. 11 at T0.98T c, where the supercnductivity f the inner ring exists at larger fields than the ne f the uter ring. The free energy f the supercnducting states f the inner ring are given by the dashed curves, the states f the uter ring by the dtted curves, and the duble ring cnfiguratin by the slid curves. Bth in the inner and the uter ring, supercnducting states with vrticity L0 and L1 exist. At T0.98T c,, the critical fields f the inner and the uter ring are H 0 /H c, 1.77 and 0.69, respectively. Ntice that in bth rings the free energies f the L0 state and the L1 state d nt crss, which means that with increasing field the grund state changes frm the Meissner state int the nrmal state and, with further increasing the field, int the L1 state and back int the nrmal state. The reasn is that near T c the supercnductivity f the ring has decreased. This means that nly rather small currents can be induced and thus nly a small flux can be attracted r expelled by the ring. n the regin between the existence f the L0 state and the L1 state the currents, which have t be induced t expel r attract the necessary flux t achieve vrticity L0 and L1, are t high. With increasing temperature, the L1 state cannt nucleate anymre and the supercnducting/nrmal transitin jumps t the field where the L0 state is destryed. The crrespnding scillatins in the H-T phase diagram Fig. 11 are the Little- Parks scillatins. Fr the duble ring cnfiguratin the (0,0), the (1,0), and the (1,1) state can nucleate. The (1,1) state is split int tw parts crrespnding t the L1 states in the tw single rings with an intermediate magnetic-field regin in which bth supercnductrs are nrmal. The grund state changes frm the Meissner state (0,0) int the (1,0) state at H 0 /H c, 0.53, which equals the (1,1) state at H 0 /H c, Further increasing the field the grund state changes int the nrmal state at H 0 /H c, 0.76, then back int the (1,1) state at H 0 /H c, 0.86 and further back int the nrmal state at H 0 /H c, Cmpared t the uncupled inner ring and the uter ring situatin, extra grundstate transitins ccur fr the duble ring case with interesting re-entrant supercnducting behavir and a switching n and ff f the supercnducting state in the inner and uter ring. V. TWO COUPLED THN RNGS N THE LMT š1 Here we will shw that in the limit f tw cupled thin rings it is pssible t btain analytical results fr the cupling energy between the tw rings. This als crrespnds t the case f 1 and allws us t slve the prblem analytically with the small parameter d/ 1. Frm the numerical calculatins f previus sectins it fllws that the radial dependence f the rder parameter in bth inner and uter rings is slw and smth. Therefre we assume that the rder parameter in bth inner and uter rings, f in and f ut, respectively, are cnstant. n the limit f the thin rings we neglect in the first apprximatin the z dependence f the vectr ptential. With this assumptin and because f the cylindrical symmetry f the prblem the vectr ptential has nly the azimuthal cmpnent A() and the magnetic field has nly the nrmal cmpnent H()A()/. The distributin f the vectr ptential due t the supercurrents inside the inner and uter ring are described by the fllwing equatin see als, fr example, Eqs. and 3 in Ref. 18: 1 in(ut) A d L in(ut) i() and utside the rings by 1 A 0. A in(ut) f in(ut), 13a 13b The crrespnding bundary cnditins are the cntinuity f bth A() and H() n the radial sides f the rings. Fr d 0 it gives everywhere A(d0)H 0 / and H(d0) H 0. T a first apprximatin the slutin f Eq. 13a becmes, fr small d/ i(), A in(ut) H 0 D in(ut) C in(ut) d 4 i() H 0 /4, f in(ut) L in(ut) ln1 14a which is valid inside the supercnductrs, i.e., in the range R i *R * and R i R. The vectr ptential utside the rings is then btained frm a slutin f Eq. 13b and is equal t 1 /, 0R i *, AD D 3 /C 3 /, R *R i, H 0 /C ext /, R. The crrespnding distributin f the magnetic field is 14b

11 COUPLED MESOSCOPC SUPERCONDUCTORS:... PHYSCAL REVEW B 66, , 0Ri*, H 0 D in f in HD1 d/ i L in ln H 0 /4, R i *R *, D 3, R *R i, H 0 D ut f ut d/ L ut ln H 0 /4, R i R, H 0, R. 15 Frm the abve bundary cnditins the integratin cnstants D and C in Eqs. 14a 15 are C in d/4 i f in R i * L in H 0 R i * /4, C 3 d/4 i f in R * R i * L in H 0 R * R i * /4, C ut C 3 d/4 f ut R i L ut H 0 R i /4, C ext C 3 d/4 f ut R R i L ut H 0 R R i /4, D ut d/ f ut L ut ln R H 0 R /4, D 3 H 0 d/ f ut L ut lnr /R i H 0 S ut /4, D in d/ i f in L in ln R *H 0 R * /4 d/ f ut L ut lnr /R i H 0 S ut /4, D 1 H 0 d/ i f in L in lnr */R i * H 0 S in /4d/ f ut L ut lnr /R i H 0 S ut /4. Ntice that within this apprximatin the magnetic field is cnstant in the inner hle and in the space between the tw rings, and has a small as the ring s thickness is small radial dependence inside the rings. nserting the expressins fr the rder parameters, the magnetic field, and the vectr ptential int the expressin fr the energy nte that in the equivalent expressin 7 the Ginzburg-Landau equatins have already been used, F V dv 1 i, i, 4 F S i f in 1 i f in 4 f in (0) in d/ i f in in f ut f ut ut 1 f 4 ut S ut (0) d/ f ut S in (1) d/ f ut () (1) ut d/ i f in (), 17 where S in (R * R i * ), S ut (R R i ), SS in S ut, and (0) ut(in) L ut(in) lnr ( * ) /R ( i * ) H 0 L ut(in) S ut(in) H 0 S ut(in) R ( * ) R i ( * ) /8, () S in L in H 0 R i * R * /4L ut lnr /R i H 0 S ut /4, (1) ut(in) L ut(in) ln R ( * ) H 0 R ( * ) (1) /4J ut(in) (1) J ut(in) R ( i * ) L ut(in) H 0 R ( i * ) () /4J ut(in) (3) J ut(in), S ut(in) L utin H 0 R ( * ) R ( i * ) /4, () J ut(in) L ut(in) lnr ( * ) /R ( i * ) H 0 S ut(in) /4, (3) J ut(in) L ut(in) ln 1L ut(in) H 0 1 ln/8h 0 4 R ( /48 * ). Ri ( * ) 18 After the minimizatin f the free energy 17 with respect t f in(ut) we btain tw equatins, i A h r H With a O(d / 4 ) accuracy we find the difference between the Gibbs free energy f the supercnducting state and the nrmal state, f in(ut) d i() i() f in(ut) i() (1) in(ut) f in(ut) d d i (0) S in(ut) in(ut) f ut(in) ()0,

12 B. J. BAELUS, S. V. YAMPOLSK, AND F. M. PEETERS PHYSCAL REVEW B 66, When ne f the rings is in the nrmal state the results becme mre simple. Namely, when the uter ring is in the nrmal state and the inner ring is supercnducting case r vice versa case, we have f ut 0, f in (0) in i S in i (1) 1 d in i S in, and f in 0, f ut 1 (0) S ut 1 d (1) 1 S ut, 3 FG. 13. The grund-state free energy f the duble ring cnfiguratin with the same parameters as in Fig. 5 btained frm ur analytical expressins which are valid fr type- supercnducting rings. Als the grund-state energies frm Fig. 5 are shwn. frm which we btain the equilibrium values f the rder parameters fr the three pssible situatins. When bth the inner and uter rings are in the supercnducting state case, we btain with a O(d / 4 ) accuracy f in(ut) i() i() d i 1 (0) in(ut) i d S in(ut)1 (i) i() i() (0) S ut(in) (). S in(ut) (1) S in(ut) 0 nserting these expressins int Eq. 17 we btain F F in, where F ut F int F in S 4 in S i 4 (0) S in i in (1) i d in i S in O d 4, i 1a is the self energy f the inner ring, and F ut S ut 1 (0) S S ut 1 d (1) ut S ut O d 4, 1b is the self energy f the uter ring, while F int d i 1 i (0) S in i in 1 (0) S ut () S 4 O d, is the interactin energy between the tw rings. 1c respectively. The crrespnding energies are F S in S in i F S ut S and in the d/ i() (0) S in 1 (0) S ut (1) 1 i d in i S in, 4 1 d (1) 1 S ut, 5 1 limit they cincide with F in and F ut, respectively. One can see that an interactin between the tw rings i.e., the cupling exists nly when bth rings are supercnducting. The energy f the ring-ring cupling in the cnsidered limit is prprtinal t the ring s thickness. Due t the interactin between the rings the Cper-pair density in each ring has a small prprtinal t d/ ) cntributin frm the neighbring ring. n Fig. 13 the calculated magnetic-field dependence f the grund-state energy f the cupled thin rings f the same material i.e., with i and i ) with the same radial sizes as in Fig. 5 is shwn fr d/ 0.05 slid curve. Als fr cmparisn the curves frm Fig. 5 are shwn by dashed and dash-dtted lines which crrespnd t d/ 1.98 and 1.76, respectively. One can see that all curves have the same qualitative behavir and with increasing d/ nly a small decrease f the grund-state energy takes place. Als the values f the transitin fields between the different L states are very nicely reprduced. This is quite surprising and, t make the physics f this result mre clear, we cnsider the limit f narrw rings. Let us intrduce the average radius f the tw rings, in (R *R i *)/, ut (R R i )/, respectively, and their crrespnding widths, w in R *R i *,w ut R R i. Next we expand Eqs. 18 and 1 with respect t the small parameters w in(ut) / in(ut) 1 with an O(w in(ut) ) accuracy. Within this apprximatin the energies 1 are

13 COUPLED MESOSCOPC SUPERCONDUCTORS:... F in(ut) S in(ut) (i) S i (i) L in(ut) in(ut) i 1 d i() O d F int S in d S i i() (i) i w in(ut) w in(ut) (i) i L in(ut) in(ut) in(ut) w ut ut 1 i in(ut) in(ut) L in(ut) in(ut) Ow in(ut), 6a L in inl ut ut L in in i 1 L ut ut in Ow in w ut,w ut, ut 6b where in(ut)h 0 in(ut) / is the average magnetic flux thrugh the crrespnding ring, measured in units f 0 hc/e. Frm these expressins ne can see that the dminant thickness dependent terms in F in(ut), as well as the ring s interactin energy F int, are f rder O(d/ i() )w in(ut) nt f rder O(d/ i() ), as ne culd think naively. This additinal smallness shws that an increase f the ring thickness influences the ring s self-energies and the interactin energy very slightly and fr rings which have a nt t large width as in ur case the abve results becme valid fr d/ i() 1. V. CONCLUSONS We investigated the magnetic cupling between tw cncentric messcpic supercnductrs with nnzer thickness. When a secnd supercnductr is placed in the center f a supercnducting ring, it feels a nnunifrm field, which is the superpsitin f the unifrm applied field and the field expelled frm the uter ring. Als the first ring will be influenced by the magnetic field expelled frm the supercnductr in the center. S, bth supercnductrs are cupled PHYSCAL REVEW B 66, magnetically. This results in substantial changes f the supercnducting prperties. Frm the study f the free energy we learned that extra grund-state transitins ccur in cmparisn with the single ring case. These are transitins where the ttal vrticity stays the same, but the vrticity f the inner supercnductr changes by ne unit. We als fund that the free energy f the duble ring system is nt exactly the same as the sum f the free energies f the tw uncupled single rings which is anther signature f the magnetic cupling f bth rings. This interactin enhances with increasing sample thickness. We als calculated the expelled field fr the ring-ring cnfiguratin which shwed that as cmpared with a single ring mre, r less, field can be expelled r attracted depending n the vrticities f bth supercnductrs. The behavir f the Cper-pair density, the magneticfield prfile, and the current density was calculated. Since an extra supercnductr is placed in the center, the magnetic field will be expelled frm this supercnductr r will be cmpressed in the center f it, which results in a higher r a lwer magnetic-field density between the tw supercnductrs. The current in bth rings exhibits extra jumps at the transitin fields where the vrticity f the ther ring increases r decreases by 1. The reasn is that at these applied fields the ttal magnetic field in the regin between the tw supercnductrs changes. We investigated what happens if the inner ring is made f a different material with a higher critical temperature. The H-T phase diagram shwed that the nucleatin field f the duble ring equals the ne f the uter ring at lw temperatures and the ne f the inner ring at higher temperatures. Analytical expressins are btained fr the magnetic-field distributin and the energy f tw cupled thin type- supercnducting rings. These analytical results are fund t give excellent results when d/ i() 1 and, mrever, give als gd agreement with ur exact numerical results fr sufficiently narrw rings when d/ i() 1. ACKNOWLEDGMENTS This wrk was supprted by the Flemish Science Fundatin FWO-Vl, the Onderzeksraad van de Universiteit Antwerpen GOA, the nteruniversity Ples f Attractin Prgram Belgian State, Prime Minister s Office Federal Office fr Scientific, Technical and Cultural Affairs, and the ESF VORTEX prgram. Discussins with V. V. Mshchalkv and M. Mrelle are gratefully acknwledged. *Permanent address: Dnetsk Physical & Technical nstitute, Natinal Academy f Sciences f Ukraine, Dnetsk 83114, Ukraine. Electrnic mail: francis.peeters@ua.ac.be 1 Fr a review, see, fr example, Jrge Berger and Jacb Rubinstein, Cnnectivity and Supercnductivity Springer, Berlin, 000. H. Vleberghs, V.V. Mshchalkv, C. Van Haesendnck, R. Jnckheere, and Y. Bruynseraede, Phys. Rev. Lett. 69, J. Berger and J. Rubinstein, Phys. Rev. Lett. 75, ; Phys. Rev. B 56, ; 59, V.V. Mshchalkv, L. Gielen, C. Strunk, R. Jnckheere, X. Qiu, C. van Haesendnck, and Y. Bruynseraede, Nature Lndn 373, E.M. Hrane, J.J. Castr, G.C. Buscaglia, and A. Lpez, Phys. Rev. B 53, V.M. Fmin, V.R. Misk, J.T. Devreese, and V.V. Mshchalkv, Slid State Cmmun. 101, ; Phys. Rev. B 58, V. Bruyndncx, L. Van Lk, M. Verschuere, and V.V. Mshchalkv, Phys. Rev. B 60, ; V. Bruyndncx, L. Van Lk, and V.V. Mshchalkv, Physica C 33, B.J. Baelus, F.M. Peeters, and V.A. Schweigert, Phys. Rev. B 61,