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Vortex matter in presence of nano-scale magnetic defects
Physica C 408–410 (2004) 466–469 www.elsevier.com/locate/physc
Vortex matter in presence of nano-scale magnetic defects q Mauro M. Doria
*
Instituto de Fısica, Universidade Federal do Rio de Janeiro, C.P. 68528, Rio de Janeiro RJ 21941-972, Brazil
Abstract A rigid ferromagnetic ground state, formed by a rectangular array of magnetic moments, is embedded in a superconductor. Insulating spheres are pinning centers with magnetic moment, and their interaction with the vortex matter is described here through a Ginzburg–Landau theory that treats superconducting and insulating regions on the same footing. For a critical value of the magnetic moment vortices undergo a deconfinement transition giving rise to a collective state of vortices and anti-vortices. Ó 2004 Elsevier B.V. All rights reserved. PACS: 74.20.De; 74.60.Ge; 74.25.Ha; 75.30.)m Keywords: Vortex; Anti-vortex; Magnetic transition; Ginzburg–Landau theory
1. Introduction The study of superconducting samples with periodic arrays of artificially made pinning centers became possible thanks to advances in nanotechnology. Arrays of pinning centers with magnetic properties are specially interesting since the coexistence of superconductivity and magnetism is conflictual by nature. Such superconducting systems have been fabricated with magnetic defects in the surface of the material [1,2]. The nucleation of a vortex is possible near a magnetic region with sufficiently strong magnetic moment. The field of a magnetic dipole, B m=jxj3 , approaches the upper critical field ðHc2 ¼ U0 =2pn2 Þ, very near the defect, namely for a distance equal to the coherence length n, in case m ¼ U0 n=2p. The vortex is confined to the same region of its onset even in case there are many of such magnetic regions, but still sparsely distributed inside the superconductor. However a subtle change occurs in case the density, or the magnetic moment, of these regions reaches a critical value. The vortex interconnects several magnetic regions, and so crosses the superconductor
q This work was supported by CNPq, Instituto do Mil^enio and FAPERJ. * Tel.: +55-2125-627-335; fax: +55-2125-627-368. E-mail address: [email protected] (M.M. Doria).
from one side to the other, thus acquiring system length size. Therefore magnetic insertions inside a superconductor can lead to a vortex deconfinement transition. To study this transition a regular three-dimensional lattice of magnetic defects is embedded in the superconductor. The defect lattice has an orthorhombic unit cell of coherence length size described by LX, LY, and LZ. Two systems are studied here, namely, (i) LX ¼ LY ¼ 11.3n, LZ ¼ 7.3n, and (i) LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n. The defects are insulating spheres of radius R ¼ n, and magnetic moment m ¼ U0 d=2p. Notice that the magnetic moment is characterized by a length, d, in terms of which the Bohr magneton becomes d ¼ re , re being the electron classical radius. The vortex deconfinement transition is observed in the range d of a few n for the above systems. In this paper the magnetic moments are rigidly oriented along the x direction forming a ferromagnetic ground state. The system is kept in zero magnetic induction, B ¼ 0, and its behavior for increasing values of d=n, is studied below, and above, the deconfinement transition at dc =n, where the vortex interconnects a row of magnetic defects along the x direction. Anti-vortices appear in the system so to keep the magnetic induction null above the transition as well. Remarkably pinned vortices and depinned anti-vortices coexist in this state. The superconducting state is described by a Ginzburg–Landau (GL) theory that treats superconducting
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.03.179
M.M. Doria / Physica C 408–410 (2004) 466–469
f ¼
Z
2 4 2 dv 2 2pi 1 sn $ A w sw þ w ; V U0 2
ð1Þ
expressed in units of the critical field energy density, Hc2 =4p. The magnetic potential is that of a dipole at the center of the defect, A ¼ m x=jxj3 , because screening currents are not being included. Thus an extreme type II superconductor ðj ! 1Þ is being considered here. For the numerical treatment a smooth function is taken, namely, sðxÞ ¼ 1 2=f1 þ exp½ðjxj=RÞK g, with the parameter K ¼ 8.
-0.18 -0.20
Free Energy
and insulating regions on equal footing [3–5] through a step-like function sðxÞ, zero in the insulating and one in the superconducting regions, respectively. The free energy density is the expansion below,
467
d/ξ =17
-0.22 d/ξ =16
-0.24 -0.26 -0.28
NX=NY=18 NZ=12 NX=NY=30 NZ=20
-0.30 -0.32 10
12
14
16
18
20
d/ξ Fig. 1. The free energy densities, computed using two meshes, versus the magnetic moment are shown here for the system LX ¼ LY ¼ 12n, LZ ¼ 8n.
-0.15
2. Main results
d/ξ =19 -0.20 d/ξ =8
-0.25
Free Energy
The Ginzburg–Landau theory is numerically solved using the simulated annealing method [6]. The free energy of Eq. (1) is discretized in a gauge invariant way, and the unit cell is described by a cubic mesh of NX.NY.NZ points: LX ¼ (NX-1)n, LY ¼ (NY-1)n, and LZ ¼ (NZ-1)n. Two different meshes are used for each of the two lattices of defects discussed in this paper. They correspond to two kinds of distances between nearest neighbor mesh points, 2n=3 and 2n=5, respectively. Thus for the lattice LX ¼ LY ¼ 11.3n, LZ ¼ 7.3n, the meshes are NX ¼ NY ¼ 18, NZ ¼ 12 and NX ¼ NY ¼ 30, NZ ¼ 20, and, similarly, for the lattice LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n, they are NX ¼ 12, NY ¼ NZ ¼ 18 and NX ¼ 20, NY ¼ NZ ¼ 30. The major results of this paper are found to be mesh independent. All the three-dimensional plots shown here display a single iso-surface of constant order parameter density, jwj2 , chosen to be ðjwj2max þ 2jwj2min Þ=3. The density is equal to one for the full superconducting state, and zero for the insulating state. Thus the iso-surface is drawn for one third, fact that is approximately true for all the plots shown here. Inside the iso-surface the density drops to zero. The arrow shown there gives the direction of the magnectic moment m. Fig. 1 shows the free energy versus magnetic moment for the LX ¼ LY ¼ 11.3n LZ ¼ 7.3n system. The abrupt change of slope between d=n ¼ 16 and d=n ¼ 17 sets the deconfinement transition. The free energy shows a slight dependence on the mesh, but the transition point dc =n does not. The density jwj2 , shown for both d=n ¼ 16 (Fig. 3) and d=n ¼ 17 (Fig. 4), is obtained using the NX ¼ NY ¼ 18 NZ ¼ 12 mesh. The d=n ¼ 16 vortex state is below the transition, and Fig. 3 shows two confined vortices, the handles around the defect. The states d=n < 16 show a growth of the handles out of the
d/ξ =18
d/ξ =7
-0.30
d/ξ =15 d/ξ =14
-0.35 -0.40
NX=12 NY=NZ=18 NX=20 NY=NZ=30
-0.45 -0.50
0
5
10
15
20
d/ξ Fig. 2. The free energy densities, computed using two meshes, versus the magnetic moment are shown here for the system LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n.
Fig. 3. Iso-surface of the order parameter density for d=n ¼ 16 and system LX ¼ LY ¼ 11.3n, LZ ¼ 7.3n is shown here.
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M.M. Doria / Physica C 408–410 (2004) 466–469
Fig. 4. Iso-surface of the order parameter density for d=n ¼ 17 and system LX ¼ LY ¼ 11.3n, LZ ¼ 7.3n is shown here. Fig. 5. Iso-surface of the order parameter density for d=n ¼ 7 and system LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n is shown here.
defect body for increasing magnetic moment to the point that for d=n ¼ 16 nearest neighbor handles touch each other. The d=n ¼ 17 case is above the transition, and Fig. 4 shows two deconfined vortices that arise in one defect and end in the next one, along the x axis. Notice that the unit cell has periodic boundary conditions. Thus above the transition the two vortices have infinite length and are pinned by a row of magnetic pinning centers along the x axis. Each of the two anti-vortices, seen in the ny faces of the rectangular unit cell, are split into two halves, in the ny ¼ 1 and ny ¼ 18 faces, respectively. Indeed the electromagnetic current, plotted in Fig. 4 for the nx ¼ 18 face, of the two vortices, which are pinned by the magnetic defect, and of the two anti-vortices, have opposite circulation. Fig. 2 shows the free energy versus magnetic moment for the LX ¼ 7.3n LY ¼ LZ ¼ 11.3n lattice. Several discontinuities are observed when critical magnetic moments are reached, namely, as d=n goes from 7 to 8, from 14 to 15, and from 18 to 19. The different results obtained for the NX ¼ 12 NY ¼ NZ ¼ 18 and NX ¼ 30 NY ¼ NZ ¼ 20 meshes, specially around the 14–15 transition can be explained based on the meta estability features that exist around such transitions [7]. The density plots for this lattice are obtained using the NX ¼ 30 NY ¼ NZ ¼ 20 mesh. The deconfinement transition occurs between d=n ¼ 7 (Fig. 5) and d=n ¼ 8 (Fig. 6). Fig. 6 shows the d=n ¼ 8 state, made of a giant vortex with double magnetic flux pinned by the defect and of two anti-vortices with unit magnetic flux in the ny ¼ 1 and ny ¼ 18 unit cell faces. A first flux jump transition occurs between d=n ¼ 14 (Fig. 7) and d=n ¼ 15 (Fig. 8), where the flux trapped by the magnetic defect increases by one. Fig. 7 shows that the state d=n ¼ 14 has pinned vortex with double magnetic flux,
Fig. 6. Iso-surface of the order parameter density for d=n ¼ 8 and system LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n is shown here.
like in the d=n ¼ 8 case. However the anti-vortex is a giant state with double magnetic flux located at the four edges of the unit cell. Fig. 8 shows that for d=n ¼ 15 a jump takes place and the pinned vortex acquires triple magnetic flux since three anti-vortices are seen, one at the edges and two at the ny and nz faces. Fig. 8 also shows the electromagnetic current in the nx ¼ 20 face, which reveals, similarly to Fig. 4, anti-vortices at the faces and edges with opposite current circulation to the pinned vortex at the center. A second flux jump transition occurs between d=n ¼ 18 and d=n ¼ 19 (Fig. 9) and
M.M. Doria / Physica C 408–410 (2004) 466–469
Fig. 7. Iso-surface of the order parameter density for d=n ¼ 14 and system LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n is shown here.
469
Fig. 9. Iso-surface of the order parameter density for d=n ¼ 19 and system LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n is shown here.
3. Conclusion A ferromagnetic ground state, formed by a rectangular array of magnetic insulating defects, with coherence length size, is embedded inside a superconductor. The zero magnetic induction properties of this system is studied using a Ginzburg–Landau theory. Confined vortices live around a single defect of low magnetic moment. For a critical magnetic moment, vortices interconnect several defects and acquire system size length. Then the existing arrangements of vortices and anti-vortices undergo several transitions marked by special numbers of flux trapped inside the magnetic defect. References
Fig. 8. Iso-surface of the order parameter density for d=n ¼ 15 and system LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n is shown here.
once again the flux trapped by the magnetic defect increases by one. The d=n ¼ 18 state is basically the same configuration of the d=n ¼ 15 state (Fig. 8), and for this reason not shown here. The d=n ¼ 19 state has four magnetic flux trapped in the pinned vortex since four anti-vortices are seen at the faces. Notice the asymmetry in the vortex anti-vortex distribution in this case.
[1] J.I. Martin, M. Velez, J. Nogues, Ivan K. Schuller, Phys. Rev. Lett. 79 (1997) 1929. [2] 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. [3] M.M. Doria, S.C.B. de Andrade, Phys. Rev. B 60 (1999) 13164. [4] M.M. Doria, G.F. Zebende, J Magn. Magn. Mater. 226 (2001) 358. [5] M.M. Doria, G.F. Zebende, Phys. Rev. B 66 (2002) 64519. [6] M.M. Doria, J.E. Gubernatise, D. Rainer, Phys. Rev. B 41 (1990) 6335. [7] I.K. Marmokos, A. Matulis, F.M. Peeters, Phys. Rev. B 53 (1996) 2677.
Vortex matter in presence of nano-scale magnetic defects q Mauro M. Doria
*
Instituto de Fısica, Universidade Federal do Rio de Janeiro, C.P. 68528, Rio de Janeiro RJ 21941-972, Brazil
Abstract A rigid ferromagnetic ground state, formed by a rectangular array of magnetic moments, is embedded in a superconductor. Insulating spheres are pinning centers with magnetic moment, and their interaction with the vortex matter is described here through a Ginzburg–Landau theory that treats superconducting and insulating regions on the same footing. For a critical value of the magnetic moment vortices undergo a deconfinement transition giving rise to a collective state of vortices and anti-vortices. Ó 2004 Elsevier B.V. All rights reserved. PACS: 74.20.De; 74.60.Ge; 74.25.Ha; 75.30.)m Keywords: Vortex; Anti-vortex; Magnetic transition; Ginzburg–Landau theory
1. Introduction The study of superconducting samples with periodic arrays of artificially made pinning centers became possible thanks to advances in nanotechnology. Arrays of pinning centers with magnetic properties are specially interesting since the coexistence of superconductivity and magnetism is conflictual by nature. Such superconducting systems have been fabricated with magnetic defects in the surface of the material [1,2]. The nucleation of a vortex is possible near a magnetic region with sufficiently strong magnetic moment. The field of a magnetic dipole, B m=jxj3 , approaches the upper critical field ðHc2 ¼ U0 =2pn2 Þ, very near the defect, namely for a distance equal to the coherence length n, in case m ¼ U0 n=2p. The vortex is confined to the same region of its onset even in case there are many of such magnetic regions, but still sparsely distributed inside the superconductor. However a subtle change occurs in case the density, or the magnetic moment, of these regions reaches a critical value. The vortex interconnects several magnetic regions, and so crosses the superconductor
q This work was supported by CNPq, Instituto do Mil^enio and FAPERJ. * Tel.: +55-2125-627-335; fax: +55-2125-627-368. E-mail address: [email protected] (M.M. Doria).
from one side to the other, thus acquiring system length size. Therefore magnetic insertions inside a superconductor can lead to a vortex deconfinement transition. To study this transition a regular three-dimensional lattice of magnetic defects is embedded in the superconductor. The defect lattice has an orthorhombic unit cell of coherence length size described by LX, LY, and LZ. Two systems are studied here, namely, (i) LX ¼ LY ¼ 11.3n, LZ ¼ 7.3n, and (i) LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n. The defects are insulating spheres of radius R ¼ n, and magnetic moment m ¼ U0 d=2p. Notice that the magnetic moment is characterized by a length, d, in terms of which the Bohr magneton becomes d ¼ re , re being the electron classical radius. The vortex deconfinement transition is observed in the range d of a few n for the above systems. In this paper the magnetic moments are rigidly oriented along the x direction forming a ferromagnetic ground state. The system is kept in zero magnetic induction, B ¼ 0, and its behavior for increasing values of d=n, is studied below, and above, the deconfinement transition at dc =n, where the vortex interconnects a row of magnetic defects along the x direction. Anti-vortices appear in the system so to keep the magnetic induction null above the transition as well. Remarkably pinned vortices and depinned anti-vortices coexist in this state. The superconducting state is described by a Ginzburg–Landau (GL) theory that treats superconducting
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.03.179
M.M. Doria / Physica C 408–410 (2004) 466–469
f ¼
Z
2 4 2 dv 2 2pi 1 sn $ A w sw þ w ; V U0 2
ð1Þ
expressed in units of the critical field energy density, Hc2 =4p. The magnetic potential is that of a dipole at the center of the defect, A ¼ m x=jxj3 , because screening currents are not being included. Thus an extreme type II superconductor ðj ! 1Þ is being considered here. For the numerical treatment a smooth function is taken, namely, sðxÞ ¼ 1 2=f1 þ exp½ðjxj=RÞK g, with the parameter K ¼ 8.
-0.18 -0.20
Free Energy
and insulating regions on equal footing [3–5] through a step-like function sðxÞ, zero in the insulating and one in the superconducting regions, respectively. The free energy density is the expansion below,
467
d/ξ =17
-0.22 d/ξ =16
-0.24 -0.26 -0.28
NX=NY=18 NZ=12 NX=NY=30 NZ=20
-0.30 -0.32 10
12
14
16
18
20
d/ξ Fig. 1. The free energy densities, computed using two meshes, versus the magnetic moment are shown here for the system LX ¼ LY ¼ 12n, LZ ¼ 8n.
-0.15
2. Main results
d/ξ =19 -0.20 d/ξ =8
-0.25
Free Energy
The Ginzburg–Landau theory is numerically solved using the simulated annealing method [6]. The free energy of Eq. (1) is discretized in a gauge invariant way, and the unit cell is described by a cubic mesh of NX.NY.NZ points: LX ¼ (NX-1)n, LY ¼ (NY-1)n, and LZ ¼ (NZ-1)n. Two different meshes are used for each of the two lattices of defects discussed in this paper. They correspond to two kinds of distances between nearest neighbor mesh points, 2n=3 and 2n=5, respectively. Thus for the lattice LX ¼ LY ¼ 11.3n, LZ ¼ 7.3n, the meshes are NX ¼ NY ¼ 18, NZ ¼ 12 and NX ¼ NY ¼ 30, NZ ¼ 20, and, similarly, for the lattice LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n, they are NX ¼ 12, NY ¼ NZ ¼ 18 and NX ¼ 20, NY ¼ NZ ¼ 30. The major results of this paper are found to be mesh independent. All the three-dimensional plots shown here display a single iso-surface of constant order parameter density, jwj2 , chosen to be ðjwj2max þ 2jwj2min Þ=3. The density is equal to one for the full superconducting state, and zero for the insulating state. Thus the iso-surface is drawn for one third, fact that is approximately true for all the plots shown here. Inside the iso-surface the density drops to zero. The arrow shown there gives the direction of the magnectic moment m. Fig. 1 shows the free energy versus magnetic moment for the LX ¼ LY ¼ 11.3n LZ ¼ 7.3n system. The abrupt change of slope between d=n ¼ 16 and d=n ¼ 17 sets the deconfinement transition. The free energy shows a slight dependence on the mesh, but the transition point dc =n does not. The density jwj2 , shown for both d=n ¼ 16 (Fig. 3) and d=n ¼ 17 (Fig. 4), is obtained using the NX ¼ NY ¼ 18 NZ ¼ 12 mesh. The d=n ¼ 16 vortex state is below the transition, and Fig. 3 shows two confined vortices, the handles around the defect. The states d=n < 16 show a growth of the handles out of the
d/ξ =18
d/ξ =7
-0.30
d/ξ =15 d/ξ =14
-0.35 -0.40
NX=12 NY=NZ=18 NX=20 NY=NZ=30
-0.45 -0.50
0
5
10
15
20
d/ξ Fig. 2. The free energy densities, computed using two meshes, versus the magnetic moment are shown here for the system LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n.
Fig. 3. Iso-surface of the order parameter density for d=n ¼ 16 and system LX ¼ LY ¼ 11.3n, LZ ¼ 7.3n is shown here.
468
M.M. Doria / Physica C 408–410 (2004) 466–469
Fig. 4. Iso-surface of the order parameter density for d=n ¼ 17 and system LX ¼ LY ¼ 11.3n, LZ ¼ 7.3n is shown here. Fig. 5. Iso-surface of the order parameter density for d=n ¼ 7 and system LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n is shown here.
defect body for increasing magnetic moment to the point that for d=n ¼ 16 nearest neighbor handles touch each other. The d=n ¼ 17 case is above the transition, and Fig. 4 shows two deconfined vortices that arise in one defect and end in the next one, along the x axis. Notice that the unit cell has periodic boundary conditions. Thus above the transition the two vortices have infinite length and are pinned by a row of magnetic pinning centers along the x axis. Each of the two anti-vortices, seen in the ny faces of the rectangular unit cell, are split into two halves, in the ny ¼ 1 and ny ¼ 18 faces, respectively. Indeed the electromagnetic current, plotted in Fig. 4 for the nx ¼ 18 face, of the two vortices, which are pinned by the magnetic defect, and of the two anti-vortices, have opposite circulation. Fig. 2 shows the free energy versus magnetic moment for the LX ¼ 7.3n LY ¼ LZ ¼ 11.3n lattice. Several discontinuities are observed when critical magnetic moments are reached, namely, as d=n goes from 7 to 8, from 14 to 15, and from 18 to 19. The different results obtained for the NX ¼ 12 NY ¼ NZ ¼ 18 and NX ¼ 30 NY ¼ NZ ¼ 20 meshes, specially around the 14–15 transition can be explained based on the meta estability features that exist around such transitions [7]. The density plots for this lattice are obtained using the NX ¼ 30 NY ¼ NZ ¼ 20 mesh. The deconfinement transition occurs between d=n ¼ 7 (Fig. 5) and d=n ¼ 8 (Fig. 6). Fig. 6 shows the d=n ¼ 8 state, made of a giant vortex with double magnetic flux pinned by the defect and of two anti-vortices with unit magnetic flux in the ny ¼ 1 and ny ¼ 18 unit cell faces. A first flux jump transition occurs between d=n ¼ 14 (Fig. 7) and d=n ¼ 15 (Fig. 8), where the flux trapped by the magnetic defect increases by one. Fig. 7 shows that the state d=n ¼ 14 has pinned vortex with double magnetic flux,
Fig. 6. Iso-surface of the order parameter density for d=n ¼ 8 and system LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n is shown here.
like in the d=n ¼ 8 case. However the anti-vortex is a giant state with double magnetic flux located at the four edges of the unit cell. Fig. 8 shows that for d=n ¼ 15 a jump takes place and the pinned vortex acquires triple magnetic flux since three anti-vortices are seen, one at the edges and two at the ny and nz faces. Fig. 8 also shows the electromagnetic current in the nx ¼ 20 face, which reveals, similarly to Fig. 4, anti-vortices at the faces and edges with opposite current circulation to the pinned vortex at the center. A second flux jump transition occurs between d=n ¼ 18 and d=n ¼ 19 (Fig. 9) and
M.M. Doria / Physica C 408–410 (2004) 466–469
Fig. 7. Iso-surface of the order parameter density for d=n ¼ 14 and system LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n is shown here.
469
Fig. 9. Iso-surface of the order parameter density for d=n ¼ 19 and system LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n is shown here.
3. Conclusion A ferromagnetic ground state, formed by a rectangular array of magnetic insulating defects, with coherence length size, is embedded inside a superconductor. The zero magnetic induction properties of this system is studied using a Ginzburg–Landau theory. Confined vortices live around a single defect of low magnetic moment. For a critical magnetic moment, vortices interconnect several defects and acquire system size length. Then the existing arrangements of vortices and anti-vortices undergo several transitions marked by special numbers of flux trapped inside the magnetic defect. References
Fig. 8. Iso-surface of the order parameter density for d=n ¼ 15 and system LX ¼ 7.3n, LY ¼ LZ ¼ 11.3n is shown here.
once again the flux trapped by the magnetic defect increases by one. The d=n ¼ 18 state is basically the same configuration of the d=n ¼ 15 state (Fig. 8), and for this reason not shown here. The d=n ¼ 19 state has four magnetic flux trapped in the pinned vortex since four anti-vortices are seen at the faces. Notice the asymmetry in the vortex anti-vortex distribution in this case.
[1] J.I. Martin, M. Velez, J. Nogues, Ivan K. Schuller, Phys. Rev. Lett. 79 (1997) 1929. [2] 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. [3] M.M. Doria, S.C.B. de Andrade, Phys. Rev. B 60 (1999) 13164. [4] M.M. Doria, G.F. Zebende, J Magn. Magn. Mater. 226 (2001) 358. [5] M.M. Doria, G.F. Zebende, Phys. Rev. B 66 (2002) 64519. [6] M.M. Doria, J.E. Gubernatise, D. Rainer, Phys. Rev. B 41 (1990) 6335. [7] I.K. Marmokos, A. Matulis, F.M. Peeters, Phys. Rev. B 53 (1996) 2677.