Vortex patterns in a superconducting-ferromagnetic rod Antonio R. de C. Romagueraa, Mauro M. Doriab, Franc¸ois M. Peetersc aDepartamento de F´ısica, Universidade Federal Rural de Pernambuco, 52171-900 Recife, Pernambuco, Brazil bDepartamento de F´ısica dos S´olidos, Universidade Federal do Rio de Janeiro, 21941-972 Rio de Janeiro, Brazil cDepartement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium 0 1 0 2 Abstract n a JA superconducting rod with a magnetic moment on top develops vortices obtained here through 3D 1calculations of the Ginzburg-Landau theory. The inhomogeneity of the applied field brings new prop- 1 erties to the vortex patterns that vary according to the rod thickness. We find that for thin rods (disks) ] the vortex patterns are similar to those obtained in presence of a homogeneous magnetic field instead n because they consist of giant vortex states. For thick rods novel patterns are obtained as vortices are o ccurve lines in space that exit through the lateral surface. - r pKey words: Vortex pattern, Ginzburg-Landau, Magnetic dot uPACS: 74.20.-z, 74.20.De s . t a m 1. Introduction onset of vortices and also the nature of the vortex - patterns. d n The minimal condition for the onset of a vor- The shape of a vortex shifts from a flat coin otex in a mesoscopic superconductor depends on to a curved line in three-dimensional space, as c [geometrical parameters. A thin disk of radius R the thickness D tales the rod from a disk to a 1 (thickness D ∼ ξ, ξ is the coherence length) can tall rod. Consequently the vortex pattern con- vonly exist in the Meissner state for R ∼ ξ. But sists of top to bottom giant vortices and top to 5 1for ξ < R < 2ξ, giant vortex states are allowed side multi-vortices in these two extreme limits, 7and for R > 2ξ multivortex states become pos- namely, D ∼ ξ and D (cid:29) ξ, respectively. Rods of 1 sible, as reported in [1] and recently observed in intermediate thickness display highly non trivial . 1 Refs. [2, 3]. Mesoscopic superconductors have 0 0new and interesting properties[4], and also pro- 1vide an interesting playground to understand the : vco-existence of magnetism and superconductivity i X[5, 6]. For instance magnetic dots on top of su- rperconducting film have been investigated both a theoretically and experimentally [7–10]. Inthispaperwereportatheoreticalstudydone on superconducting rods of radius R and varying thickness D, with a magnetic dot on top, a sys- tem which displays curved vortices triggered by Figure 1: (Color online) Superconducting rod with radius the inhomogeneity of the magnetic field. We ad- R and thickness D. The magnetic dot is oriented along dress the minimal geometrical conditions for the the x-axis and positioned 2ξ above the top surface. Preprint submitted to Physica C January 11, 2010 vortex patterns showing features of both extreme by the following reduced units: the coherence limits. Our results are obtained in the context lengthξ(T) = [−(cid:126)2/2m∗α(T)]1/2 (lengths), H = c2 of the Ginzburg-Landau theory in the limit of no φ /2πξ(T)2 (magnetic field), H ξ = Φ /2πξ(T) 0 c2 0 magnetic shielding. We take a magnetic dot with (vector potential), µ = H ξ(T)3 = Φ ξ(T)/2π 0 c2 0 (cid:112) magnetic moment µ positioned 2ξ above the rod’s (magnetic moment), Ψ = −α(T)/β (order pa- 0 top surface and oriented along the rod’s axis, see rameter), and F = α(T)2/2β (free energy). The 0 Fig. 1. magnetization is calculated using the expression (cid:126) (cid:82) (cid:126) Although curve vortices also appear in presence M = (cid:126)r×J(r)dv. of a homogeneous field tilted with respect to the Intermsofthesedimensionlessunits, theGibbs surface, as vortices must always emerge perpen- free energy difference becomes, dicular to the surface [11], the inhomogeneity of (cid:90) 1 the field brings new effects. The dipolar magnetic F = 2 dV{−|Ψ|2 + |Ψ|4 + 2 field weakens with distance and may become so + |(∇(cid:126) −iA(cid:126))Ψ|2}, (1) dimatthebottomoftherodtothepointthatthis region becomes unable to sustain vortices. Conse- The integration is restricted to the volume of the quently the Meissner state is kept at the bottom rod. The boundary condition in dimensionless of the rod. This favors a reentrant vortex state, units becomes, as a top to side vortex must disappear at the top (cid:12) (cid:126)n·(∇(cid:126) −iA(cid:126))Ψ(cid:12) = 0. (2) surface,thesameoneofitsonset. ThustheMeiss- (cid:12) boundary ner phase is retrieved at a higher magnetic mo- For the magnetic dot we use field produced by a ment. This qualifies superconducting rods with a point-like dipole A(cid:126) = (µ(cid:126) ×(cid:126)r)/r3. magnetic moment on top for technological appli- cations as logical switchers, since vortices can be 3. Numerical results expelled or accepted by fine tuning the external magnetic moment dot strength. We solved Eqs. 1 and 2 using the simulated an- nealing procedure [4]. We set the following pa- 2. Theoretical approach rameters for the rod: the radius R ranges from 1ξ to 4ξ; the thickness D ranges from 2ξ to 8ξ and According to the standard Ginzburg-Landau the magnetic moment µ ranges from 0 to 100µ . 0 approach, the Gibbs free energy of the super- For the thinnest rod considered here, D = 2ξ, we conductor near the critical temperature T can c find that the inhomogeneity of the field does not be expanded in powers of the complex order pa- matter, since the free energy and the magnetiza- rameter Ψ((cid:126)r) that gives the density of Cooper- tion are similar to the homogeneous applied field pairs: |Ψ((cid:126)r)|2. Hence the Gibbs free energy differ- case. Clearly this is because the dot’s magnetic encebetweenthesuperconductingandthenormal field does not vary significantly inside the rod. states is, For R = 1ξ and any thickness we find no vor- (cid:90) (cid:110) 1 tex state, thus the Meissner state prevails up to F −F = dV α(T)|Ψ|2 + β|Ψ|4 + s n the normal state. We do not observe anti-vortices 2 (cid:126)2 (cid:16) 2π (cid:17) (cid:111) or Meissner state with opposite orientation in the + | ∇(cid:126) −i A(cid:126) Ψ|2 rods because we limit the present study to rods 2m∗ Φ 0 with a maximum radius of 4ξ. Thus only giant with the phenomenological constants α(T) ≡ vortex states (GVS) are observed. For R = 2ξ α (T − T ) < 0, β > 0, m∗ (the mass of one and for D = 2ξ, only one vortex is allowed. For 0 c Cooper-pair), and Φ = hc/2e, the fundamental larger R more than one vortex is possible and 0 unit of flux. Boundary conditions are imposed to their number increases very rapidly with the ra- theexternalsurfacesoftherod. Weexpressquan- dius. Although there is more space to accom- tities in this theory in dimensionless units defined modate extra vortices the magnetic field becomes 2 D = 2 x Fig. 3 shows the free energy for the R = 4ξ and D = 6ξ rod. 0,0 -0,2 The 3D view of the Cooper pair density, |ψ|2, shows curved top-to-side vortices, that is, vortices -0,4 0 that enter from the top surface and exit perpen- / F-0,6 F dicularly in the side surface. We call them N-fold R = 1 x -0,8 R = 2 x states, where N describes the number of vortices. R = 3 x Along the energy curve vs µ in Fig. 3 the states R = 4 x -1,0 are labeled according to this definition. The N- 0,0 fold states are not clearly seen in Fig. 3 because their free energy lines is superposed to the Meiss- -0,4 ner line from where is their onset and disappear- -0,8 ance. However these states are clearly seen to 0 / M-1,2 exist by analyzing the |ψ|2 isosurface, as shown M in Fig. 4. In the two biggest rods considered here -1,6 R = 1 x R = 2 x we observed the nucleation of N-fold states fol- R = 3 x -2,0 R = 4 x lowed by Meissner and GVS. This features are justified by the growth of a normal region closer 0 20 40 60 80 100 m / m to the the magnetic dot position. When this nor- 0 mal region occupies a significant part of the rod Figure2: (Coloronline)FreeenergyF/F andmagnetiza- 0 the remaining superconducting part behaves as a tion M/M versus magnetic moment µ/µ for mesoscopic 0 0 thin rod, i.e., it does not exhibits N-fold states. rods with thickness D = 2ξ and several radii. Giant vor- tices with increasing maximum vorticity are observed by So, we retrieve the Meissner state. This feature is increasing the radius. in Table 1 and correspond to the second Meissner state. weaker at distances away from the center. Conse- quently the average magnetic magnetic field does not grow with the radius. This effect is equivalent R = 4 x a n d D = 8 x tothatstudiedbyOvchinnikovsometimeago[12]. 0,0 A sawtooth behavior in the magnetization with bigger Hc2 is clearly established for bigger radius. -0,2 This behavior is observed for R = 3ξ and D = 2ξ. In Fig. 2 we show the vortex states obtained for -0,4 different thin rods, i.e., disks. Their magnetiza- 0 / F tion curves are similar to those obtained in pres- F-0,6 M e is s n e r 3 -fo ld ence of homogeneous field. In table 1 we give the 4 -fo ld -0,8 5 -fo ld complete description of the states through the set 6 -fo ld of parameters (R,D,µ/µ ). M e is s n e r 0 -1,0 0 20 40 60 80 100 The inhomogeneity of the field becomes impor- m / m 0 tant for the thicker rods D = 6ξ and 8ξ, as shown here. Even for small radii, the rod already dis- Figure 3: (Color online) Free energy F/F vs µ/µ for a 0 0 plays behavior that differs significantly from the rod with R = 4ξ and D = 8ξ. For increasing magnetic moment the vortex pattern evolves continuously through homogeneous field case. We find smooth free en- the N-fold configurations, as labeled in the inset. See, ergy lines, with no first order transition occa- Fig. 4 for the corresponding |ψ|2 isosurface views of these sioned by the entrance of quantized vortex lines. N-fold states. 3 Table 1: The sequence of ground states for the of rods considered here. The magnetic dot is positioned 2ξ away from the rod’s top surface and oriented along the z-axis. The question mark means the normal state was not reach. R/ξ 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 D/ξ 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 State µ/µ0 µ/µ0 µ/µ0 µ/µ0 µ/µ0 µ/µ0 µ/µ0 µ/µ0 µ/µ0 µ/µ0 µ/µ0 µ/µ0 µ/µ0 µ/µ0 µ/µ0 µ/µ0 Meissner 0-36 <100 <100 <100 0-18 0-76 <100 <100 0-14 0-46 <100 <100 0-14 0-20 0-22 0-22 1GVS - - - - 20-30 78-100 - - 16-24 48-76 - - 16-22 22-100 - - 2GVS - - - - 32 - - - 26-30 78-100 - - 24-30 - - - 3GVS - - - - - - - - 32-38 - - - 32-36 - - - 4GVS - - - - - - - - 40-50 - - - 38-44 - - - 5GVS - - - - - - - - - - - - 46-50 - - - 3-fold - - - - - - - - - - - - - 34-38 24-34 24-32 4-fold - - - - - - - - - - - - - 40-44 36-42 34-40 5-fold - - - - - - - - - - - - - 46-52 44-48 42-48 6-fold - - - - - - - - - - - - - 54-56 50-54 50-58 7-fold - - - - - - - - - - - - - - 56-62 - Meissner - - - - - - - - - - - - - - 64-74 60-100 1GVS - - - - - - - - - - - - - - 76-100 - Normal >36 ? ? ? >32 ? ? ? >50 ? ? ? ? ? ? ? 4. Conclusions Using Simulated Annealing, a truly three- dimensional approach, we obtain the vortex pat- terns of mesoscopic rods with a point-like mag- netic moment on top. We conclude that rods with thickness smaller than 4ξ can be considered as thin disks since only top to bottom giant vortex states are obtained. For rods with thickness big- ger than 4ξ, we observed the appearance of N-fold vortex states, namely N top-to-side vortices. The onset and disappearance of these states is from a single GVS or Meissner state line, as contin- uous transitions from it. Consequently, for suffi- cientlargemagneticmomentsonecanretrievethe Meissner state before reaching the normal state, and this, leads to a reentrant behavior. Figure 4: (Color online) Isosurfaces of the Cooper pair densityforamesoscopicrodwithR=4ξ andD =6ξ. Ev- ery image corresponds to one of the states N-folder states 5. Acknowledgment listed in Table 1. A. R. de C. 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