- Email: [email protected]
Triplet vortex state in magnetic superconductors – Effects of boundaries
Available online at www.sciencedirect.com
Physica C 468 (2008) 572–575 www.elsevier.com/locate/physc
Triplet vortex state in magnetic superconductors – Effects of boundaries Mauro M. Doria a,b,*, Antonio R.de C. Romaguera a,b, M.V. Milosˇevic´ a,c, Francßois M. Peeters a b
a Universiteit Antwerpen, Groenenborgerlaan 171, BE-2020 Antwerpen, Belgium Universidade Federal do Rio de Janeiro, Caixa Postal: 68.528, Rio de Janeiro – RJ, 21941-972, Brazil c Department of Physics, University of Bath, Claverton Down, BA2 7AY Bath, United Kingdom
Accepted 30 November 2007 Available online 7 March 2008
Abstract A mesoscopic superconductor with a magnetic moment in its center exhibits confined vortex loops with threefold symmetry in its center. Here, we show how the boundaries can affect this symmetry by considering a superconducting cube and sphere. Ó 2008 Elsevier B.V. All rights reserved. PACS: 74.25.Ha; 74.25.Op; 74.78.Na Keywords: Ferromagnetic superconductors; Superconducting ferromagnets, Vortex state; Ginzburg–Landau theory
Superconductivity and magnetism are mutually exclusive types of order known to coexist in the recently discovered ferromagnetic superconductors [1] and in the superconducting ferromagnets [2,3]. Nano-engineered superconductors with magnetic defects in its surface show coexistence between superconductivity and magnetism in the vortex pattern [4,5]. Recently, several nano-composites were also fabricated, such as MgB2 with embedded magnetic Fe2O3 nanoparticles [6], or Gd particles incorporated in a Nb matrix [7]. We believe that such compounds can be used to obtain the vortex pattern predicted here, which is made of confined vortex loops (CVLs) and external vortex pairs (EVPs). In mesoscopic superconductors the volume to surface area ratio is small [8], and strong lateral confinement imposes the formation of giant vortices [9,10], while the shape of the boundary directs the symmetry of the final vortex configuration [11]. Similar effects are also observed here in our vortex patterns. *
Corresponding author. Address: Universiteit Antwerpen, Groenenborgerlaan 171, BE-2020 Antwerpen, Belgium. E-mail address: [email protected] (M.M. Doria). 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.11.078
In this paper we consider the vortex pattern resulting from a static magnetic moment l in the center of a finite size superconductor. A sub-micron superconducting particle has its center taken by a point like magnetic dipole whose magnetic field leads to CVLs. These loops eventually spring to the surface and give rise to vortex pairs, because the magnetic moment’s north and south poles are the only source and sinkhole of vortices, respectively. Thus, as seen from the surface, a vortex and an anti-vortex are just the tips of a single EVP. The number of CVLs and of EVPs determines the vortex state and in the present study we restrict the magnetic moment strength to l P l0 , l0 U0 n=2p. We define the magnetic moment through a length, d, l U0 5d=8p, and thus, l=l0 ¼ 5d=4n. All the temperature dependence of the present system is in the ratio l=l0 . Recently, we showed [12] that CVLs arise in triplets from the H c2 core surrounding the magnetic moment. The growth of a CVL beyond the superconductor boundaries will eventually transform it into an EVP state. With increasing l the CVLs approach the external surface, which is expected to have some influence on the symmetry of the state. In this paper we find that the CVLs remain in a
M.M. Doria et al. / Physica C 468 (2008) 572–575
573
triplet state if far from the surface. for this purpose two different geometries are studied whose surfaces eventually affect the vortex pattern: a sphere with diameter 30 n, and a cube with side 30 n. In the latter case the magnetic moment is taken parallel to two of the cube’s faces. The vortex pattern, as a function of l, is obtained by numerically solving a gauge invariant discrete version of the Ginzburg–Landau (GL) theory on a cubic cell with side 36n. The numerical minimization of the GL free energy is done through the method of simulated annealing [12] using a mesh grid of N 3 points ðN ¼ 46Þ, containing the finite size superconductors inside. The Meissner shielding is not included, and so, Ampe´re’s law is safely ignored. The present description is restricted to a hard type II superconductor whose free energy is given by, f ¼
Z
dv 2 2pi 1 2 2 4 sn j ð$ AÞwj s j wj þ j wj ; V U0 2
ð1Þ
expressed in units of the critical field energy density, H 2c =4p. The magnetic field is B ¼ curlA, A ¼ l r=jrj3 , and we impose that in the center of the cubic cell the order parameter vanishes. Notice that (i) a non-superconducting core naturally evolves around the center and acquires radius size of the order of n; (ii) the magnetic field is continuous across the boundary; (iii) the superconducting current normal to the surface vanishes (deGennes’ boundary condition). The non-linearity of the theory is fully treated in our procedure and this determines its ground and excited states. The shape of the finite superconductor enters directly into the free energy through a step-like function sðxÞ, equal to one inside the superconductor, and zero outside, as discussed in Ref. [12]. Fig. 1 shows the free energy and the induced magnetic moment (inset) for sphere and cube. We find that the vortex state undergoes several transitions for
Fig. 1. Free energy vs. magnetic moment for the mesoscopic (continuous) sphere and (dashed) cube. The cube free energy is shifted upwards by an overall constant with respect to the sphere for clarity. The inset shows the induced magnetization and branches are labeled by their number of external vortex pairs. The vortex state for the selected points are shown in the next figures.
Fig. 2. Three-dimensional iso-density plots for selected sphere states.
increasing l. The diamagnetic response of the mesoscopic superconductor is hindered by l in the total magnetization: l=l0 þ M=M 0 . An overall normalization constant is introduced here as we assume the existence of an asymptotic Meissner phase l=l0 þ M=M 0 ¼ 0 for very small l. The induced moment weakens for large l because of the EVPs, and this renders l=l0 þ M=M 0 > 0. Notice that Fig. 1 shows that both magnetization and free energy split into distinct branch lines for both cube and sphere, and these branches are associated to the number of EVPs. There is strong metastability between distinct branches because EVPs are surface pinned. Fig. 1 is limited to three and four EVPs for the cube and sphere, respectively. However as l increases along a selected branch, although the number of EVPs remains constant, the number of CVLs does not. New CVLs arise along the way and can alter the symmetry of the final state. Thus as the CVLs approach the surface, the surface symmetry can affect the state. For the cube this means that the threefold symmetry is replaced by a fourfold symmetry. We have selected a set of vortex states to illustrate some of the features of the present system in Fig. 1, according to their d=n values: 23(Sa), 35(Sb), 45(Sc), 55(Sd), and 60(Se) for the sphere and 23(Ca), 24(Cb), 38(Cc) and 40(Cd) for the cube. 1Figs. 2 and 3 display density isosurfaces that show the spatial location of the vortices. Each isosurface corresponds to a single surface taken at 20% of the maximum density and is decomposed here in two parts. The external surface part stands near the finite superconductor external border and the internal surface part due to the 1 (For interpretation of the references in colour in this figure legend, the reader is referred to the web version of this article.)
574
M.M. Doria et al. / Physica C 468 (2008) 572–575
Fig. 3. Three-dimensional iso-density plots for selected cube states.
CVL state. The order parameter, that vanishes outside, grows inside from the border to reach maximum density within a distance of the order of n and in between reaches this external surface part. The two parts are interconnected by the EVP states and form a single isosurface. For the cases labeled as Sa, Ca, and Cb, the isosurface comprises two non-intersecting surfaces, but not for the remaining cases. This is a straightforward consequence that Sa, Ca, and Cb have no EVPs. For visualization purposes we have introduced two colors to describe different parts of a single isosurface, each one associated to a distinct concentric volume. Inside an inner cubic cell the isosurface is in red and in the remaining volume, the cubic cell minus this inner cube, is in gray. Thus the CVL states are in red, whereas the EVP states are double colored, red and gray. The external isosurface, including the regions of contact with the EVP states are represented in gray color. 2Figs. 4 and 5 show the frontal view of the density isosurfaces of Figs. 2 and 3, for the sphere and cube, respectively. They also show the phase of the order parameter in the square face that cuts the unit cell in half and has the magnetic moment pointing outward. The vortex is easily detected in the phase plot by the presence of a full hue around the vortex center, ranging from blue (0) to red (2p). The border between blue and red colored regions means a discontinuity of 2p which signals a vortex. Figs. 4 and 5 demonstrate that either the phase or the density can be used to determine the number of EVPs and also of CVLs. Notice that EVPs give a hue that fills the whole area, whereas CVLs are localized near the center. Thus from Figs. 2–5 we determine the (CVL, EVP) values for the selected moments of Fig. 1: Sa ? 2 (For interpretation of the references in colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Frontal view of the iso-density (left figures) and the corresponding phase (right figures) for the selected sphere states.
(3, 0), Sb ? (6, 1), Sc ? (6, 2), Sd ? (6, 3), Ca ? (3, 0), Cb ? (4, 0), Cc ? (5, 1), Cd ? (6, 2). The threefold symmetry found for Ca, becomes fourfold for Cb because of boundary effects. For the 1 and 2 EVP branches the states evolve under threefold symmetry, at least up to the Cc and Cd, respectively. In conclusion we found threefold symmetry also for the cube, whose surface eventually introduces fourfold symmetry to the vortex pattern. Since the threefold symmetry is to be found away from the boundaries we believe it is present in the bulk superconductor with strong magnetic inclusions ðl l0 Þ, as long as they are sufficiently apart, but, in this case, eventually the confined vortices take other arrangements by influence of neighbor inclusions. When external pairs interconnect the magnetic inclusions, the spontaneous vortex phase is obtained.
M.M. Doria et al. / Physica C 468 (2008) 572–575
575
Acknowledgements A.R. de C. Romaguera acknowledges support from CNPq (Brazil). M.M. Doria acknowledges support from CNPq (Brazil), FAPERJ (Brazil), the Instituto do Mileˆnio de Nanotecnologia (Brazil) and BOF/UA (Belgium). M.V. Milosˇevic´ and F.M. Peeters acknowledge support from the Flemish Science Foundation (FWO-Vl), the Belgian Science Policy (IUAP) and the ESF-AQDJJ network. M.V. Milosˇevic´ is currently a Marie-Curie Fellow at University of Bath, UK. References [1] H. Bluhm, S.E. Sebastian, J.W. Guikema, I.R. Fisher, K.A. Moler, Phys. Rev. B73 (2006) 14514. [2] H. Sakai, N. Osawa, K. Yoshimura, M. Fang, K. Kosuge, Phys. Rev. B 67 (2003) 184409. [3] J.D. Jorgensen, O. Chmaissem, H. Shaked, S. Short, P.W. Klamut, B. Dabrowski, J.L. Tallon, Phys. Rev. B 63 (2001). [4] J.I. Martin, M. Velez, J. Nogues, I.K. Schuller, Phys. Rev. Lett. 79 (1997) 1929. [5] M.J. Van Bael, J. Bekaert, K. Temst, L. Van Look, V.V. Moshchalkov, Y. Bruynserade, G.D. Howells, A.N. Grigorenko, S.J. Bending, G. Borghs, Phys. Rev. Lett. 86 (2001) 155. [6] A. Snezhko, T. Prozorov, R. Prozorov, Phys. Rev. B 71 (2005) 024527. [7] A. Palau, H. Parvaneh, N.A. Stelmashenko, H. Wang, J.L. Macmanus-Driscoll, M.G. Blamire, Phys. Rev. Lett. 98 (2007) 117003. [8] A.K. Geim, I.V. Grigorieva, S.V. Dubonos, J.G.S. Lok, J.C. Maan, A.E. Filippov, F.M. Peeters, Nature (London) 390 (1997) 259. [9] V.A. Schweigert, F.M. Peeters, P.S. Deo, Phys. Rev. Lett. 81 (1998) 2783. [10] A. Kanda, B.J. Baelus, F.M. Peeters, K. Kadowaki, Y. Ootuka, Phys. Rev. Lett. 93 (2004) 257002. [11] B.J. Baelus, F.M. Peeters, Phys. Rev. B 65 (2002) 104515. [12] Mauro M. Doria, Antonio R. de C. Romaguera, M.V. Milosˇevic´, F.M. Peeters, Europhys. Lett. 79 (2007) 47006.
Fig. 5. Frontal view of the iso-density (left figures) and the corresponding phase (right figures) for the selected cube states.
Physica C 468 (2008) 572–575 www.elsevier.com/locate/physc
Triplet vortex state in magnetic superconductors – Effects of boundaries Mauro M. Doria a,b,*, Antonio R.de C. Romaguera a,b, M.V. Milosˇevic´ a,c, Francßois M. Peeters a b
a Universiteit Antwerpen, Groenenborgerlaan 171, BE-2020 Antwerpen, Belgium Universidade Federal do Rio de Janeiro, Caixa Postal: 68.528, Rio de Janeiro – RJ, 21941-972, Brazil c Department of Physics, University of Bath, Claverton Down, BA2 7AY Bath, United Kingdom
Accepted 30 November 2007 Available online 7 March 2008
Abstract A mesoscopic superconductor with a magnetic moment in its center exhibits confined vortex loops with threefold symmetry in its center. Here, we show how the boundaries can affect this symmetry by considering a superconducting cube and sphere. Ó 2008 Elsevier B.V. All rights reserved. PACS: 74.25.Ha; 74.25.Op; 74.78.Na Keywords: Ferromagnetic superconductors; Superconducting ferromagnets, Vortex state; Ginzburg–Landau theory
Superconductivity and magnetism are mutually exclusive types of order known to coexist in the recently discovered ferromagnetic superconductors [1] and in the superconducting ferromagnets [2,3]. Nano-engineered superconductors with magnetic defects in its surface show coexistence between superconductivity and magnetism in the vortex pattern [4,5]. Recently, several nano-composites were also fabricated, such as MgB2 with embedded magnetic Fe2O3 nanoparticles [6], or Gd particles incorporated in a Nb matrix [7]. We believe that such compounds can be used to obtain the vortex pattern predicted here, which is made of confined vortex loops (CVLs) and external vortex pairs (EVPs). In mesoscopic superconductors the volume to surface area ratio is small [8], and strong lateral confinement imposes the formation of giant vortices [9,10], while the shape of the boundary directs the symmetry of the final vortex configuration [11]. Similar effects are also observed here in our vortex patterns. *
Corresponding author. Address: Universiteit Antwerpen, Groenenborgerlaan 171, BE-2020 Antwerpen, Belgium. E-mail address: [email protected] (M.M. Doria). 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.11.078
In this paper we consider the vortex pattern resulting from a static magnetic moment l in the center of a finite size superconductor. A sub-micron superconducting particle has its center taken by a point like magnetic dipole whose magnetic field leads to CVLs. These loops eventually spring to the surface and give rise to vortex pairs, because the magnetic moment’s north and south poles are the only source and sinkhole of vortices, respectively. Thus, as seen from the surface, a vortex and an anti-vortex are just the tips of a single EVP. The number of CVLs and of EVPs determines the vortex state and in the present study we restrict the magnetic moment strength to l P l0 , l0 U0 n=2p. We define the magnetic moment through a length, d, l U0 5d=8p, and thus, l=l0 ¼ 5d=4n. All the temperature dependence of the present system is in the ratio l=l0 . Recently, we showed [12] that CVLs arise in triplets from the H c2 core surrounding the magnetic moment. The growth of a CVL beyond the superconductor boundaries will eventually transform it into an EVP state. With increasing l the CVLs approach the external surface, which is expected to have some influence on the symmetry of the state. In this paper we find that the CVLs remain in a
M.M. Doria et al. / Physica C 468 (2008) 572–575
573
triplet state if far from the surface. for this purpose two different geometries are studied whose surfaces eventually affect the vortex pattern: a sphere with diameter 30 n, and a cube with side 30 n. In the latter case the magnetic moment is taken parallel to two of the cube’s faces. The vortex pattern, as a function of l, is obtained by numerically solving a gauge invariant discrete version of the Ginzburg–Landau (GL) theory on a cubic cell with side 36n. The numerical minimization of the GL free energy is done through the method of simulated annealing [12] using a mesh grid of N 3 points ðN ¼ 46Þ, containing the finite size superconductors inside. The Meissner shielding is not included, and so, Ampe´re’s law is safely ignored. The present description is restricted to a hard type II superconductor whose free energy is given by, f ¼
Z
dv 2 2pi 1 2 2 4 sn j ð$ AÞwj s j wj þ j wj ; V U0 2
ð1Þ
expressed in units of the critical field energy density, H 2c =4p. The magnetic field is B ¼ curlA, A ¼ l r=jrj3 , and we impose that in the center of the cubic cell the order parameter vanishes. Notice that (i) a non-superconducting core naturally evolves around the center and acquires radius size of the order of n; (ii) the magnetic field is continuous across the boundary; (iii) the superconducting current normal to the surface vanishes (deGennes’ boundary condition). The non-linearity of the theory is fully treated in our procedure and this determines its ground and excited states. The shape of the finite superconductor enters directly into the free energy through a step-like function sðxÞ, equal to one inside the superconductor, and zero outside, as discussed in Ref. [12]. Fig. 1 shows the free energy and the induced magnetic moment (inset) for sphere and cube. We find that the vortex state undergoes several transitions for
Fig. 1. Free energy vs. magnetic moment for the mesoscopic (continuous) sphere and (dashed) cube. The cube free energy is shifted upwards by an overall constant with respect to the sphere for clarity. The inset shows the induced magnetization and branches are labeled by their number of external vortex pairs. The vortex state for the selected points are shown in the next figures.
Fig. 2. Three-dimensional iso-density plots for selected sphere states.
increasing l. The diamagnetic response of the mesoscopic superconductor is hindered by l in the total magnetization: l=l0 þ M=M 0 . An overall normalization constant is introduced here as we assume the existence of an asymptotic Meissner phase l=l0 þ M=M 0 ¼ 0 for very small l. The induced moment weakens for large l because of the EVPs, and this renders l=l0 þ M=M 0 > 0. Notice that Fig. 1 shows that both magnetization and free energy split into distinct branch lines for both cube and sphere, and these branches are associated to the number of EVPs. There is strong metastability between distinct branches because EVPs are surface pinned. Fig. 1 is limited to three and four EVPs for the cube and sphere, respectively. However as l increases along a selected branch, although the number of EVPs remains constant, the number of CVLs does not. New CVLs arise along the way and can alter the symmetry of the final state. Thus as the CVLs approach the surface, the surface symmetry can affect the state. For the cube this means that the threefold symmetry is replaced by a fourfold symmetry. We have selected a set of vortex states to illustrate some of the features of the present system in Fig. 1, according to their d=n values: 23(Sa), 35(Sb), 45(Sc), 55(Sd), and 60(Se) for the sphere and 23(Ca), 24(Cb), 38(Cc) and 40(Cd) for the cube. 1Figs. 2 and 3 display density isosurfaces that show the spatial location of the vortices. Each isosurface corresponds to a single surface taken at 20% of the maximum density and is decomposed here in two parts. The external surface part stands near the finite superconductor external border and the internal surface part due to the 1 (For interpretation of the references in colour in this figure legend, the reader is referred to the web version of this article.)
574
M.M. Doria et al. / Physica C 468 (2008) 572–575
Fig. 3. Three-dimensional iso-density plots for selected cube states.
CVL state. The order parameter, that vanishes outside, grows inside from the border to reach maximum density within a distance of the order of n and in between reaches this external surface part. The two parts are interconnected by the EVP states and form a single isosurface. For the cases labeled as Sa, Ca, and Cb, the isosurface comprises two non-intersecting surfaces, but not for the remaining cases. This is a straightforward consequence that Sa, Ca, and Cb have no EVPs. For visualization purposes we have introduced two colors to describe different parts of a single isosurface, each one associated to a distinct concentric volume. Inside an inner cubic cell the isosurface is in red and in the remaining volume, the cubic cell minus this inner cube, is in gray. Thus the CVL states are in red, whereas the EVP states are double colored, red and gray. The external isosurface, including the regions of contact with the EVP states are represented in gray color. 2Figs. 4 and 5 show the frontal view of the density isosurfaces of Figs. 2 and 3, for the sphere and cube, respectively. They also show the phase of the order parameter in the square face that cuts the unit cell in half and has the magnetic moment pointing outward. The vortex is easily detected in the phase plot by the presence of a full hue around the vortex center, ranging from blue (0) to red (2p). The border between blue and red colored regions means a discontinuity of 2p which signals a vortex. Figs. 4 and 5 demonstrate that either the phase or the density can be used to determine the number of EVPs and also of CVLs. Notice that EVPs give a hue that fills the whole area, whereas CVLs are localized near the center. Thus from Figs. 2–5 we determine the (CVL, EVP) values for the selected moments of Fig. 1: Sa ? 2 (For interpretation of the references in colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Frontal view of the iso-density (left figures) and the corresponding phase (right figures) for the selected sphere states.
(3, 0), Sb ? (6, 1), Sc ? (6, 2), Sd ? (6, 3), Ca ? (3, 0), Cb ? (4, 0), Cc ? (5, 1), Cd ? (6, 2). The threefold symmetry found for Ca, becomes fourfold for Cb because of boundary effects. For the 1 and 2 EVP branches the states evolve under threefold symmetry, at least up to the Cc and Cd, respectively. In conclusion we found threefold symmetry also for the cube, whose surface eventually introduces fourfold symmetry to the vortex pattern. Since the threefold symmetry is to be found away from the boundaries we believe it is present in the bulk superconductor with strong magnetic inclusions ðl l0 Þ, as long as they are sufficiently apart, but, in this case, eventually the confined vortices take other arrangements by influence of neighbor inclusions. When external pairs interconnect the magnetic inclusions, the spontaneous vortex phase is obtained.
M.M. Doria et al. / Physica C 468 (2008) 572–575
575
Acknowledgements A.R. de C. Romaguera acknowledges support from CNPq (Brazil). M.M. Doria acknowledges support from CNPq (Brazil), FAPERJ (Brazil), the Instituto do Mileˆnio de Nanotecnologia (Brazil) and BOF/UA (Belgium). M.V. Milosˇevic´ and F.M. Peeters acknowledge support from the Flemish Science Foundation (FWO-Vl), the Belgian Science Policy (IUAP) and the ESF-AQDJJ network. M.V. Milosˇevic´ is currently a Marie-Curie Fellow at University of Bath, UK. References [1] H. Bluhm, S.E. Sebastian, J.W. Guikema, I.R. Fisher, K.A. Moler, Phys. Rev. B73 (2006) 14514. [2] H. Sakai, N. Osawa, K. Yoshimura, M. Fang, K. Kosuge, Phys. Rev. B 67 (2003) 184409. [3] J.D. Jorgensen, O. Chmaissem, H. Shaked, S. Short, P.W. Klamut, B. Dabrowski, J.L. Tallon, Phys. Rev. B 63 (2001). [4] J.I. Martin, M. Velez, J. Nogues, I.K. Schuller, Phys. Rev. Lett. 79 (1997) 1929. [5] M.J. Van Bael, J. Bekaert, K. Temst, L. Van Look, V.V. Moshchalkov, Y. Bruynserade, G.D. Howells, A.N. Grigorenko, S.J. Bending, G. Borghs, Phys. Rev. Lett. 86 (2001) 155. [6] A. Snezhko, T. Prozorov, R. Prozorov, Phys. Rev. B 71 (2005) 024527. [7] A. Palau, H. Parvaneh, N.A. Stelmashenko, H. Wang, J.L. Macmanus-Driscoll, M.G. Blamire, Phys. Rev. Lett. 98 (2007) 117003. [8] A.K. Geim, I.V. Grigorieva, S.V. Dubonos, J.G.S. Lok, J.C. Maan, A.E. Filippov, F.M. Peeters, Nature (London) 390 (1997) 259. [9] V.A. Schweigert, F.M. Peeters, P.S. Deo, Phys. Rev. Lett. 81 (1998) 2783. [10] A. Kanda, B.J. Baelus, F.M. Peeters, K. Kadowaki, Y. Ootuka, Phys. Rev. Lett. 93 (2004) 257002. [11] B.J. Baelus, F.M. Peeters, Phys. Rev. B 65 (2002) 104515. [12] Mauro M. Doria, Antonio R. de C. Romaguera, M.V. Milosˇevic´, F.M. Peeters, Europhys. Lett. 79 (2007) 47006.
Fig. 5. Frontal view of the iso-density (left figures) and the corresponding phase (right figures) for the selected cube states.