Vortex patterns in a superconducting-ferromagnetic rod Antonio R. de C. Romaguera $^{a}$, Mauro M. Doria$^{b}$, Fran¸cois M. Peetersc $^{a}$Departamento de F´ısica, Universidade Federal Rural de Pernambuco, 52171-900 Recife, Pernambuco, Brazil $^{b}$Departamento de F´ısica dos S´ olidos, Universidade Federal do Rio de Janeiro, 21941-972 Rio de Janeiro, Brazil $^{c}$Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium A superconducting rod with a magnetic moment on top develops vortices obtained here through 3D calculations of the Ginzburg-Landau theory. The inhomogeneity of the applied field brings new prop- 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 because they consist of giant vortex states. For thick rods novel patterns are obtained as vortices are curve lines in space that exit through the lateral surface. The minimal condition for the onset of a vor- tex in a mesoscopic superconductor depends on geometrical parameters. A thin disk of radius R (thickness D ∼ ξ , ξ is the coherence length) can only exist in the Meissner state for R ∼ ξ . But for ξ < R < 2 ξ , giant vortex states are allowed and for R > 2 ξ multivortex states become pos- sible, as reported in [1] and recently observed in Refs. [2, 3]. Mesoscopic superconductors have new and interesting properties[4], and also pro- vide an interesting playground to understand the co-existence of magnetism and superconductivity [5, 6]. For instance magnetic dots on top of su- perconducting film have been investigated both theoretically and experimentally [7–10]. In this paper we report a theoretical study done 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 the inhomogeneity of the magnetic field. We ad- dress the minimal geometrical conditions for the The shape of a vortex shifts from a flat coin to a curved line in three-dimensional space, as the thickness D tales the rod from a disk to a tall rod. Consequently the vortex pattern con- sists of top to bottom giant vortices and top to side multi-vortices in these two extreme limits, namely, D ∼ ξ and D ≫ ξ , respectively. Rods of intermediate thickness display highly non trivial onset of vortices and also the nature of the vortex patterns. 1. Introduction arXiv:1001.1715v1 [cond-mat.supr-con] 11 Jan 2010 Figure 1: (Color online) Superconducting rod with radius R and thickness D . The magnetic dot is oriented along the x -axis and positioned 2 ξ above the top surface. November 20, 2018 Preprint submitted to Physica C Abstract Key words: Vortex pattern, Ginzburg-Landau, Magnetic dot PACS: 74.20.-z, 74.20.De