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Vortex configurations in large superconducting mesoscopic triangles
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Physica C 468 (2008) 722–725 www.elsevier.com/locate/physc
Vortex configurations in large superconducting mesoscopic triangles Leonardo R.E. Cabral *, J. Albino Aguiar Laborato´rio de Supercondutividade, Departamento de Fı´sica, Universidade Federal de Pernambuco, 50670-901 Recife, Brazil Accepted 30 November 2007 Available online 8 March 2008
Abstract In this work we study vortex configurations in mesoscopic equilateral triangles (for triangles whose sides are much smaller than the effective penetration length) within the London limit. Therefore we are able to investigate samples whose size is much larger than the coherence length. We calculate analytically the vortex–vortex interaction by adding image vortices outside the triangular domain. The vortex configurations were found by letting the vortex system evolve in time using a Langevin dynamics simulation. This allowed us to study the dependence of the vorticity on the magnetic field. Moreover, we also investigated the role of the surface barrier on the vortex entry into the superconducting triangle. Ó 2008 Elsevier B.V. All rights reserved. PACS: 74.60.Ge; 74.25.Ha; 74.60.Ec; 74.76.2w Keywords: Vortex configurations; Vortex dynamics; Mesoscopic superconductors
1. Introduction The vortex matter in superconducting materials has been intensively studied, since the prediction of a hexagonal vortex lattice in type II superconductors by Abrikosov [1]. The experimental verification of such vortex lattice was later confirmed by ferromagnetic decoration over a PbIn alloy [2]. In macroscopic and homogeneous materials each of these vortices carry one quantum of magnetic flux, U0 ¼ h=2e. They are characterized by the coherence length, n, and the penetration depth, k. The Ginzburg–Landau pffiffiffi parameter j ¼ k=n distinguishes the type-I (j < 1= 2) pffiffiffi from the type-II (j < 1= 2) superconductors. In mesoscopic superconductors, the vortex configurations can be different from the Abrikosov lattice, giving raise to the so-called vortex molecules states [3–7]. Experimental observation of such configurations has also been made [8]. In these cases, the sample geometry greatly influences the vortex interactions, since the vortex and the exter*
Corresponding author. Tel.: +55 81 2126 8450; fax: +55 81 3271 0359. E-mail addresses: [email protected] (L.R.E. Cabral), albino@df. ufpe.br (J.A. Aguiar). 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.11.082
nal magnetic field shielding currents should obey the boundary condition of zero net current flow across the interfaces. In thin disks, for sufficiently large radius (R ¼ 50n) and large number of vortices, the vortex configurations have an Abrikosov arrangement in the center surrounded by at least two vortex shells [7]. Vortex configurations have also been studied in other geometries, such as slabs [9–11], rectangles [12], squares and triangles [13,14]. In mesoscopic superconductors, there may even appear giant vortex states or vortex–antivortex configurations (although the stability of the latter might depend on the roughness of the sample boundaries [15]). It should be pointed out that, as far as we know, previous studies of superconducting triangles focused on relatively small samples (10n) using the Ginzburg–Landau theory. This investigation, however, deals with large specimens within the London limit. 2. Theoretical formalism We considered a thin superconducting mesoscopic equilateral triangle (with thickness d and side a), placed in the z ¼ 0 plane, and submitted to a homogeneous external
L.R.E. Cabral, J.A. Aguiar / Physica C 468 (2008) 722–725
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magnetic field H ¼ H^z. Here H is in units of 4pn2 H c2 =a2 and a K ¼ k2 =d. Since we consider the system within the London limit, a ¼ 100n was used, which guarantees that the triangle is large compared to systems studied by using the Ginzburg–Landau theory. In the limit a K, the Pearl vortex–vortex interaction [16] is approximately logarithmic. The requirement of zero net current flow across the superconducting boundaries is fulfilled by an infinite set of vortex images suitably placed outside the triangular domain. This means that the superconducting triangle can be replacedpffiffiby ffi an infinite set of rectangular domains, with size 3a 3a, each one containing 6 vortices (antivortices) placed at ðxI ; y I Þ [ðxI ; y I Þ]: I ¼ 1 ! ðxi ; y i Þ; I ¼ 2 ! ð0:5ðxþ 3aÞ; h 0:5y Þ; I ¼ 3 ! ð0:5xþ ; 0:5y Þ; I ¼ 4 ! ð0:5ðx 3aÞ; h 0:5y þ Þ; I ¼ 5 ! ð0:5x ; 0:5y þ Þ;
ð1Þ
I ¼ 6 ! ðxi 3a signðxi Þ=2; y i hÞ; pffiffiffi pffiffiffi pffiffiffi Here h ¼ 3a=2, x ¼ xi þ 3y i and y ¼ 3xi þ y i . The streamline function [17], defined as J ¼ r ð^zgÞ, for a vortex core at ðxi ; y i Þ is given by, gðr; ri Þ ¼
U0 4p
1 1 X X m¼1 n¼1
X I
" pffiffiffi 2 # 2 ðx xI 3maÞ þ y y I 3na ln pffiffiffi 2 ; 2 ðx xI 3maÞ þ y þ y I 3na ð2Þ
where the index I refers to the positions of the vortices and antivortices shown in Eq. (1). A schematic representation of the contour lines of the streamline function is presented in Fig. 1. It can be noticed that the current streamlines do not cross the equilateral triangle boundaries (in fact we are interested only in the vortex configurations inside one of such triangles). The rectangle also shown contains the vortices and antivortices with positions previously described [cf. Eq. (1)]. In order to model the shielding current density, we used the vector potential, Ash , calculated in Ref. [18] for an equilateral triangle, and that KJsh ¼ Ash . The equation of motion for the ith vortex is the Bardeen–Stephen equation [19] gvi ¼ g
dri ¼ Fi þ Ci ; dt
ð3Þ
where g U0 H c2 =qn is the viscous drag coefficient, PLvi is the vortex velocity, Fi ¼ U0 Ji ^z, Ji ¼ Jsh ðri Þ þ k¼1 Jk ðri Þ and Jk ðri Þ ¼ r ½^zgðr; rk Þjr¼ri obtained from Eq. (2) for the kth vortex. The vorticity L gives the number of fluxoids (or vortex cores) inside the superconducting triangle. The additional white noise term, with zero mean and obeying the fluctuation dissipation theorem hCa;i ðtÞCb;j ðt0 Þi ¼ 2gdab dij dðt t0 Þk B T accounts for fluctuations. Here, <>
Fig. 1. Contour lines of the streamline function defined in Eq. (2). A rectangle is drawn to help visualize the periodic structure and the vortex positions given in Eq. (1). Notice that the current obeys the equilateral triangular symmetry.
means average value, k B is the Boltzman’s constant, T is the ‘temperature’, and Greek and Italic indexes refer to vector components and vortex labels. The vortex equation of motion [Eq. (3)] was integrated in time by using the Euler method. For computational purposes, we used the indexes m and n appearing in Eq. (2) running from 15 to 15. The numerical procedure is described as follows: For a given value of the external magnetic field, H, L vortices are initially distributed at random inside the triangular domain (alternatively, when H started from zero, we placed the vortices along each of the triangle edges in order to investigate the entry of vortices); The ‘temperature’ was decreased slowly to let the system find a stable configuration; After finding the configuration, H was changed by an amount DH ¼ 0:1, and the process was repeated. This procedure allowed us to obtain vortex configurations in increasing and decreasing magnetic field. 3. Results We studied vortex configurations in equilateral triangles up to L ¼ 12. Examples of stable vortex states are shown in Figs. 2 and 3. The obtained configurations present usually twofold or threefold symmetry in response to the sample triangular geometry. The majority of the configurations with threefold symmetry have one or more vortex shells. The vortices within one of these shells are at a distance r ¼ const: pffiffiffi from the center of the triangle (which is located at 2a= 3 from the corners). In Fig. 2 vector plots of the current density for configurations from L ¼ 0 till L ¼ 5 are shown. Only the configurations with three vortices have threefold symmetry. The configurations with two and four vortices are symmetric
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L.R.E. Cabral, J.A. Aguiar / Physica C 468 (2008) 722–725
Fig. 2. Vector plot of the current density for the configurations with L ¼ 0, L ¼ 1, L ¼ 2, L ¼ 3, and L ¼ 4 (clockwise direction), for H ¼ 0:1 (a), 1.2 (b), 7.0 (c), 8.0 (d), 9.0 (e), and 12.0 (f). Two different configurations with L ¼ 2, (c) and (d), are shown. The former presents a range of stability much larger than the latter.
the configurations with 5, 6, 7, 9, 10 and 12 vortices. The configurations with L ¼ 5 and L ¼ 6 have different symmetries, although neither of them has a vortex in the center of the triangle. For L ¼ 6, the vortices are at the same distance to the center of the triangle. In Fig. 3c–e, the configurations have one vortex in the center, with vortex shells around it. However, while for L ¼ 7 and L ¼ 10 the vortices are distributed in concentric circular shells (or a deformed shell) around the center of the triangle, for L ¼ 9 there is a mirror reflection symmetry with respect to the triangle middle axis. For the configuration with L ¼ 12, which shows a threefold symmetry, we observe three (nine) vortices in an inner (outer) shell. It should be pointed out that, although not shown here, configurations with different vorticities or with the same vorticity, but different distribution of vortices (for instance, a shell of five vortices with one in the center), have also been obtained. Fig. 4 depicts the distance of the vortices from the center of the triangle for the configurations L ¼ 1 ! 7 in decreasing magnetic field. As expected the distance increases with the decrement of H. Below certain values of H ¼ H ex (expulsion field), the curves present discontinuities. This is associated with the expulsion of one or more vortices from the sample (the vortex expulsion, as well as the vortex penetration, always occurs through the sides of the triangular domain), when the configuration ceases to be stable and the remaining vortices rearrange themselves. The values of the expulsion field H ex;L for the configuration with L vortices are: H ex;1 ¼ 1:0, H ex;2 ¼ 6:3, H ex;3 ¼ 7:6, H ex;4 ¼ 11:0, H ex;5 ¼ 15:4, H ex;6 ¼ 13:3, H ex;7 ¼ 14:1, H ex;8 ¼ 23:7, H ex;9 ¼ 25:5, H ex;10 ¼ 28:7, H ex;11 ¼ 31:0, and H ex;12 ¼ 33:1. Notice that H ex;2 refers to the configuration shown in Fig. 4b. The expulsion field for the one depicted in Fig. 4h (cf. Fig. 2d) is H ex;2 ¼ 6:9. Moreover, the configurations with L ¼ 6 and L ¼ 7 remain stable for H < H ex;5 . This is because the symmetry of the L ¼ 5 configuration is not suitable to fit inside a triangular geometry, which is not the case for the L ¼ 6 and L ¼ 7 configurations.
Fig. 3. Streamlines of the current density of the vortices for the configurations with L ¼ 5, L ¼ 6, L ¼ 7, L ¼ 9, L ¼ 10 and L ¼ 12 (clockwise direction), for H ¼ 16:0 (a), 15.0 (b), 17.0 (c), 26.0 (d), 32.0 (e), and 35.0 (f).
with respect to the triangle middle axis. Also, two different configurations with two vortices are shown in Fig. 2c and d. The former is stable for a wide range of magnetic field, while the latter appears after a transition in decreasing field from a higher vorticity configuration (e.g., when L ¼ 4 ! L ¼ 2, see Fig. 4). The current streamlines of the vortices (not including the magnetic field shielding currents) are depicted in Fig. 3, for
Fig. 4. Distance of vortices from the triangle center for L ¼ 1 ! 7. The respective configurations (with same colors) are depicted beside.
L.R.E. Cabral, J.A. Aguiar / Physica C 468 (2008) 722–725
For a ¼ 100n, we obtain that the magnetic field for vortex entry H en is much larger than H ex . For example, only at H en ¼ 18:3 are the first one or two vortices allowed to penetrate the sample. In increasing field, the configuration with L ¼ 3 is observed for H en 22:8, and a similar value is obtained for L ¼ 4. H en usually increases with L, but the exact value for H en seems to depend on the vortex arrangement of the previous configuration. 4. Conclusions We modeled a system of vortices inside a triangular mesoscopic superconductor within the London limit. The configurations obtained have twofold or threefold symmetry as expected from the sample geometry. We observe vortex configurations composed of shell of vortices. Regarding the barrier for vortex entry and exit, for the ratio a=n ¼ 100 there is a large difference between the entry and expulsion fields. This suggests a high surface barrier which confines a given vortex configuration. Nonetheless, further investigation is necessary. The results will be published elsewhere. Acknowledgements We would like to thank C.C. de Souza Silva for helpful discussions. This work was supported by the Brazilian Science Agencies CNPq and FACEPE. References [1] A.A. Abrikosov, Soviet Phys. JETP 5 (1957) 1174; A.A. Abrikosov, Phys. Chem. Solids 2 (1957) 199.
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[2] U. Essman, H. Trauble, Phys. Lett. A 24 (1967) 256; U. Essman, H. Trauble, Sci. Am. 224 (1971) 75. [3] A.I. Buzdin, J.P. Brison, Phys. Lett. A 196 (1994) 267. [4] J.J. Palacios, Phys. Rev. B 58 (1998) R5948. [5] B.J. Baelus, F.M. Peeters, V.A. Schweigert, Phys. Rev. B. 63 (2001) 144517. [6] B.J. Baelus, L.R.E. Cabral, F.M. Peeters, Phys. Rev. B 69 (2004) 064506. [7] L.R.E. Cabral, B.J. Baelus, F.M. Peeters, Phys. Rev. B 70 (2004) 144523. [8] I.V. Grigorieva, W. Escoffier, J. Richardson, L.Y. Vinnikov, S. Dubonos, V. Oboznov, Phys. Rev. Lett. 96 (2006) 077005. [9] G. Carneiro, Phys. Rev. B 57 (1998) 6077. [10] S.H. Brongersma, E. Verweij, N.J. Koeman, D.G. de Groot, R. Griessen, B.I. Ivlev, Phys. Rev. Lett. 71 (1993) 2319. [11] C.C. de Souza Silva, L.R.E. Cabral, J. Albino Aguiar, Phys. Rev. B 63 (2001) 134526. [12] E. Sardella, M.M. Doria, P.R.S. Netto, Phys. Rev. B 60 (2000) 13158. [13] B.J. Baelus, F.M. Peeters, Phys. Rev. B 65 (2002) 104515; G.R. Berdiyorov, L.R.E. Cabral, F.M. Peeters, J. Math. Phys. 46 (2005) 095105. [14] L.F. Chibotaru, A. Ceulemans, V. Bruyndoncx, V. Moshchalkov, Nature (London) 408 (2000) 833; L.F. Chibotaru, A. Ceulemans, V. Bruyndoncx, V. Moshchalkov, Phys. Rev. Lett. 86 (2001) 1323; V.R. Misko, V.M. Fomin, J.T. Devreese, V.V. Moshchalkov, Phys. Rev. Lett. 90 (2003) 147003. [15] A.S. Mel’nikov, I.M. Nefedov, D.A. Ryzhov, I.A. Shereshevskii, V.M. Vinokur, P.P. Vysheslavtsev, Phys. Rev. B 65 (2002) 140503R. [16] J. Pearl, Appl. Phys. Lett. 5 (1964) 65. [17] E.H. Brandt, Rep. Prog. Phys. 58 (1995) 1465. [18] L.F. Chibotaru, A. Ceulemans, M. Morelle, G. Teniers, C. Carballeira, V.V. Moshchalkov, J. Math. Phys. 46 (2005) 095108. [19] J. Bardeen, M.J. Stephen, Phys. Rev. 140 (1965) 1197A.
Physica C 468 (2008) 722–725 www.elsevier.com/locate/physc
Vortex configurations in large superconducting mesoscopic triangles Leonardo R.E. Cabral *, J. Albino Aguiar Laborato´rio de Supercondutividade, Departamento de Fı´sica, Universidade Federal de Pernambuco, 50670-901 Recife, Brazil Accepted 30 November 2007 Available online 8 March 2008
Abstract In this work we study vortex configurations in mesoscopic equilateral triangles (for triangles whose sides are much smaller than the effective penetration length) within the London limit. Therefore we are able to investigate samples whose size is much larger than the coherence length. We calculate analytically the vortex–vortex interaction by adding image vortices outside the triangular domain. The vortex configurations were found by letting the vortex system evolve in time using a Langevin dynamics simulation. This allowed us to study the dependence of the vorticity on the magnetic field. Moreover, we also investigated the role of the surface barrier on the vortex entry into the superconducting triangle. Ó 2008 Elsevier B.V. All rights reserved. PACS: 74.60.Ge; 74.25.Ha; 74.60.Ec; 74.76.2w Keywords: Vortex configurations; Vortex dynamics; Mesoscopic superconductors
1. Introduction The vortex matter in superconducting materials has been intensively studied, since the prediction of a hexagonal vortex lattice in type II superconductors by Abrikosov [1]. The experimental verification of such vortex lattice was later confirmed by ferromagnetic decoration over a PbIn alloy [2]. In macroscopic and homogeneous materials each of these vortices carry one quantum of magnetic flux, U0 ¼ h=2e. They are characterized by the coherence length, n, and the penetration depth, k. The Ginzburg–Landau pffiffiffi parameter j ¼ k=n distinguishes the type-I (j < 1= 2) pffiffiffi from the type-II (j < 1= 2) superconductors. In mesoscopic superconductors, the vortex configurations can be different from the Abrikosov lattice, giving raise to the so-called vortex molecules states [3–7]. Experimental observation of such configurations has also been made [8]. In these cases, the sample geometry greatly influences the vortex interactions, since the vortex and the exter*
Corresponding author. Tel.: +55 81 2126 8450; fax: +55 81 3271 0359. E-mail addresses: [email protected] (L.R.E. Cabral), albino@df. ufpe.br (J.A. Aguiar). 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.11.082
nal magnetic field shielding currents should obey the boundary condition of zero net current flow across the interfaces. In thin disks, for sufficiently large radius (R ¼ 50n) and large number of vortices, the vortex configurations have an Abrikosov arrangement in the center surrounded by at least two vortex shells [7]. Vortex configurations have also been studied in other geometries, such as slabs [9–11], rectangles [12], squares and triangles [13,14]. In mesoscopic superconductors, there may even appear giant vortex states or vortex–antivortex configurations (although the stability of the latter might depend on the roughness of the sample boundaries [15]). It should be pointed out that, as far as we know, previous studies of superconducting triangles focused on relatively small samples (10n) using the Ginzburg–Landau theory. This investigation, however, deals with large specimens within the London limit. 2. Theoretical formalism We considered a thin superconducting mesoscopic equilateral triangle (with thickness d and side a), placed in the z ¼ 0 plane, and submitted to a homogeneous external
L.R.E. Cabral, J.A. Aguiar / Physica C 468 (2008) 722–725
723
magnetic field H ¼ H^z. Here H is in units of 4pn2 H c2 =a2 and a K ¼ k2 =d. Since we consider the system within the London limit, a ¼ 100n was used, which guarantees that the triangle is large compared to systems studied by using the Ginzburg–Landau theory. In the limit a K, the Pearl vortex–vortex interaction [16] is approximately logarithmic. The requirement of zero net current flow across the superconducting boundaries is fulfilled by an infinite set of vortex images suitably placed outside the triangular domain. This means that the superconducting triangle can be replacedpffiffiby ffi an infinite set of rectangular domains, with size 3a 3a, each one containing 6 vortices (antivortices) placed at ðxI ; y I Þ [ðxI ; y I Þ]: I ¼ 1 ! ðxi ; y i Þ; I ¼ 2 ! ð0:5ðxþ 3aÞ; h 0:5y Þ; I ¼ 3 ! ð0:5xþ ; 0:5y Þ; I ¼ 4 ! ð0:5ðx 3aÞ; h 0:5y þ Þ; I ¼ 5 ! ð0:5x ; 0:5y þ Þ;
ð1Þ
I ¼ 6 ! ðxi 3a signðxi Þ=2; y i hÞ; pffiffiffi pffiffiffi pffiffiffi Here h ¼ 3a=2, x ¼ xi þ 3y i and y ¼ 3xi þ y i . The streamline function [17], defined as J ¼ r ð^zgÞ, for a vortex core at ðxi ; y i Þ is given by, gðr; ri Þ ¼
U0 4p
1 1 X X m¼1 n¼1
X I
" pffiffiffi 2 # 2 ðx xI 3maÞ þ y y I 3na ln pffiffiffi 2 ; 2 ðx xI 3maÞ þ y þ y I 3na ð2Þ
where the index I refers to the positions of the vortices and antivortices shown in Eq. (1). A schematic representation of the contour lines of the streamline function is presented in Fig. 1. It can be noticed that the current streamlines do not cross the equilateral triangle boundaries (in fact we are interested only in the vortex configurations inside one of such triangles). The rectangle also shown contains the vortices and antivortices with positions previously described [cf. Eq. (1)]. In order to model the shielding current density, we used the vector potential, Ash , calculated in Ref. [18] for an equilateral triangle, and that KJsh ¼ Ash . The equation of motion for the ith vortex is the Bardeen–Stephen equation [19] gvi ¼ g
dri ¼ Fi þ Ci ; dt
ð3Þ
where g U0 H c2 =qn is the viscous drag coefficient, PLvi is the vortex velocity, Fi ¼ U0 Ji ^z, Ji ¼ Jsh ðri Þ þ k¼1 Jk ðri Þ and Jk ðri Þ ¼ r ½^zgðr; rk Þjr¼ri obtained from Eq. (2) for the kth vortex. The vorticity L gives the number of fluxoids (or vortex cores) inside the superconducting triangle. The additional white noise term, with zero mean and obeying the fluctuation dissipation theorem hCa;i ðtÞCb;j ðt0 Þi ¼ 2gdab dij dðt t0 Þk B T accounts for fluctuations. Here, <>
Fig. 1. Contour lines of the streamline function defined in Eq. (2). A rectangle is drawn to help visualize the periodic structure and the vortex positions given in Eq. (1). Notice that the current obeys the equilateral triangular symmetry.
means average value, k B is the Boltzman’s constant, T is the ‘temperature’, and Greek and Italic indexes refer to vector components and vortex labels. The vortex equation of motion [Eq. (3)] was integrated in time by using the Euler method. For computational purposes, we used the indexes m and n appearing in Eq. (2) running from 15 to 15. The numerical procedure is described as follows: For a given value of the external magnetic field, H, L vortices are initially distributed at random inside the triangular domain (alternatively, when H started from zero, we placed the vortices along each of the triangle edges in order to investigate the entry of vortices); The ‘temperature’ was decreased slowly to let the system find a stable configuration; After finding the configuration, H was changed by an amount DH ¼ 0:1, and the process was repeated. This procedure allowed us to obtain vortex configurations in increasing and decreasing magnetic field. 3. Results We studied vortex configurations in equilateral triangles up to L ¼ 12. Examples of stable vortex states are shown in Figs. 2 and 3. The obtained configurations present usually twofold or threefold symmetry in response to the sample triangular geometry. The majority of the configurations with threefold symmetry have one or more vortex shells. The vortices within one of these shells are at a distance r ¼ const: pffiffiffi from the center of the triangle (which is located at 2a= 3 from the corners). In Fig. 2 vector plots of the current density for configurations from L ¼ 0 till L ¼ 5 are shown. Only the configurations with three vortices have threefold symmetry. The configurations with two and four vortices are symmetric
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Fig. 2. Vector plot of the current density for the configurations with L ¼ 0, L ¼ 1, L ¼ 2, L ¼ 3, and L ¼ 4 (clockwise direction), for H ¼ 0:1 (a), 1.2 (b), 7.0 (c), 8.0 (d), 9.0 (e), and 12.0 (f). Two different configurations with L ¼ 2, (c) and (d), are shown. The former presents a range of stability much larger than the latter.
the configurations with 5, 6, 7, 9, 10 and 12 vortices. The configurations with L ¼ 5 and L ¼ 6 have different symmetries, although neither of them has a vortex in the center of the triangle. For L ¼ 6, the vortices are at the same distance to the center of the triangle. In Fig. 3c–e, the configurations have one vortex in the center, with vortex shells around it. However, while for L ¼ 7 and L ¼ 10 the vortices are distributed in concentric circular shells (or a deformed shell) around the center of the triangle, for L ¼ 9 there is a mirror reflection symmetry with respect to the triangle middle axis. For the configuration with L ¼ 12, which shows a threefold symmetry, we observe three (nine) vortices in an inner (outer) shell. It should be pointed out that, although not shown here, configurations with different vorticities or with the same vorticity, but different distribution of vortices (for instance, a shell of five vortices with one in the center), have also been obtained. Fig. 4 depicts the distance of the vortices from the center of the triangle for the configurations L ¼ 1 ! 7 in decreasing magnetic field. As expected the distance increases with the decrement of H. Below certain values of H ¼ H ex (expulsion field), the curves present discontinuities. This is associated with the expulsion of one or more vortices from the sample (the vortex expulsion, as well as the vortex penetration, always occurs through the sides of the triangular domain), when the configuration ceases to be stable and the remaining vortices rearrange themselves. The values of the expulsion field H ex;L for the configuration with L vortices are: H ex;1 ¼ 1:0, H ex;2 ¼ 6:3, H ex;3 ¼ 7:6, H ex;4 ¼ 11:0, H ex;5 ¼ 15:4, H ex;6 ¼ 13:3, H ex;7 ¼ 14:1, H ex;8 ¼ 23:7, H ex;9 ¼ 25:5, H ex;10 ¼ 28:7, H ex;11 ¼ 31:0, and H ex;12 ¼ 33:1. Notice that H ex;2 refers to the configuration shown in Fig. 4b. The expulsion field for the one depicted in Fig. 4h (cf. Fig. 2d) is H ex;2 ¼ 6:9. Moreover, the configurations with L ¼ 6 and L ¼ 7 remain stable for H < H ex;5 . This is because the symmetry of the L ¼ 5 configuration is not suitable to fit inside a triangular geometry, which is not the case for the L ¼ 6 and L ¼ 7 configurations.
Fig. 3. Streamlines of the current density of the vortices for the configurations with L ¼ 5, L ¼ 6, L ¼ 7, L ¼ 9, L ¼ 10 and L ¼ 12 (clockwise direction), for H ¼ 16:0 (a), 15.0 (b), 17.0 (c), 26.0 (d), 32.0 (e), and 35.0 (f).
with respect to the triangle middle axis. Also, two different configurations with two vortices are shown in Fig. 2c and d. The former is stable for a wide range of magnetic field, while the latter appears after a transition in decreasing field from a higher vorticity configuration (e.g., when L ¼ 4 ! L ¼ 2, see Fig. 4). The current streamlines of the vortices (not including the magnetic field shielding currents) are depicted in Fig. 3, for
Fig. 4. Distance of vortices from the triangle center for L ¼ 1 ! 7. The respective configurations (with same colors) are depicted beside.
L.R.E. Cabral, J.A. Aguiar / Physica C 468 (2008) 722–725
For a ¼ 100n, we obtain that the magnetic field for vortex entry H en is much larger than H ex . For example, only at H en ¼ 18:3 are the first one or two vortices allowed to penetrate the sample. In increasing field, the configuration with L ¼ 3 is observed for H en 22:8, and a similar value is obtained for L ¼ 4. H en usually increases with L, but the exact value for H en seems to depend on the vortex arrangement of the previous configuration. 4. Conclusions We modeled a system of vortices inside a triangular mesoscopic superconductor within the London limit. The configurations obtained have twofold or threefold symmetry as expected from the sample geometry. We observe vortex configurations composed of shell of vortices. Regarding the barrier for vortex entry and exit, for the ratio a=n ¼ 100 there is a large difference between the entry and expulsion fields. This suggests a high surface barrier which confines a given vortex configuration. Nonetheless, further investigation is necessary. The results will be published elsewhere. Acknowledgements We would like to thank C.C. de Souza Silva for helpful discussions. This work was supported by the Brazilian Science Agencies CNPq and FACEPE. References [1] A.A. Abrikosov, Soviet Phys. JETP 5 (1957) 1174; A.A. Abrikosov, Phys. Chem. Solids 2 (1957) 199.
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