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Two dimensional vortex structures in a superconductor slab at low temperatures
Physica C 470 (2010) 225–230
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Physica C journal homepage: www.elsevier.com/locate/physc
Two dimensional vortex structures in a superconductor slab at low temperatures J. Barba-Ortega a,*, Ariel Becerra b, J. Albino Aguiar a a b
Departamento de Física, Universidade Federal de Pernambuco, 50670-901, Recife, PE, Brazil Grupo de Investigación INTEGRAR, Universidad de Pamplona, Pamplona, Colombia
a r t i c l e
i n f o
Article history: Received 29 August 2009 Received in revised form 19 December 2009 Accepted 21 December 2009 Available online 4 January 2010 Keywords: Ginzburg–Landau de Gennes extrapolation length Mesoscopics Superconducting slab
a b s t r a c t The dynamics of a two dimensional chain like structure of vortices is studied in the model of nonlinear time dependent Ginzburg–Landau equations (TDGL). The transition between different linear chains of vortices in a superconducting homogeneous slab with both surfaces in contact with a thin layer of metallic material is analyzed. The magnetization curve, vortex number, vortex configurations and modulus of the order parameter are studied as a function of the external magnetic field. We show how these vortex configurations are affected by the extrapolation length b (de Gennes boundary conditions), W due to the proximity effects in a mesoscopic sample of area dx dy, where dy = 60n(0) and dx varies discretely from 30n(0) to 12n(0). Possible connection with recent theoretical results in a two dimensional system of charged particles is discussed. Crown Copyright Ó 2009 Published by Elsevier B.V. All rights reserved.
1. Introduction During the last years, mesoscopic systems consisting of interacting particles in confined geometries or low dimensions attracted a lot of attention. Depending on the interacting potential, these systems crystallize in different lattice structures and exhibit a phase diagram with continuous and discontinuous structural transitions. Some important results in this topic from other authors are the differences founded between single-chain and multi-chain regimes as temperature increases in the analysis of the dynamics of quasi-one-dimensional chain like systems of charged particles at low temperatures [1,2]. Also is founded in thin superconductors films [3] that the vortex lattice consists of parallel linear chains of vortices and goes through transitions from n to n + 1 chain. The purpose of our study is to achieve a deeper understanding of the influence of the de Gennes extrapolation length on the vortex transition in a bi-dimensional sample. In particular, we considered a mesoscopic superconducting film which is immersed in a metallic medium in presence of an applied perpendicular uniform magnetic field He and here a superconductor–metal interface is always used, which will cause the superconductivity to be weaker at the edge of the sample and therefore the superconducting currents can flow out from the superconductor. Also we take the order parameter invariant and the magnetic field constant along z-axis. For this system we use the TDGL equations with the general boundary conditions for superconductors and the technique of gauge invariant variables to obtain the superconducting order parameter W, the magnetization curves M and the free energy G * Corresponding author. Tel.: +55 81 2126 7625; fax: +55 81 3271 0359. E-mail address: [email protected] (J. Barba-Ortega).
as a function of the de Gennes parameter. In this work the vortex configurations for a mesoscopic film of rectangular cross-section dx dy is studied, where dy = 60n(0) and dx varies from 30n(0) to 12n(0). Here n(0) is the coherence length at zero temperature. 1.1. Time dependent Ginzburg–Landau equations and boundary conditions The time dependent Ginzburg–Landau equations (TDGL) [1–5] which govern the superconductivity order parameter W and the vector potential A in the zero electric potential gauge are given by:
@W 1 ¼ ððir AÞ2 W þ ð1 TÞðjWj2 1ÞWÞ @t g @A ¼ ð1 TÞReðW ðir AÞWÞ j2 r r A @t
ð1Þ ð2Þ
Eqs. (1) and (2) were rescaled as follows: the order parameter W in units of W1(0) = (a/b)1/2, where a and b are two phenomenological parameters. Temperatures in units of the critical temperature Tc, lengths in units of the coherence length n(0), time in units of t0 = p⁄/96KBTc, A in units of Hc2(0)Tc(0), where Hc2(0) is the second thermodynamics critical field. The Gibbs free energy in units of G0=(aTc)2/b. The dynamical equations are complemented with the appropriate boundary conditions for the order parameter. The general boundary condition for superconductors, found by de Gennes h ^ is the unity vec^ i W, where n [6], is given by n hr þ ec A W ¼ i b tor, perpendicular to the superconductor surface; b is the extrapolation de Gennes parameter. For all simulations we used b > 0, which describes the superconductor/metal interface. The full discretization of the TDGL equations can be found in more detail in Ref. [7]. We use the UW method to solve the TDGL equations in a discrete
0921-4534/$ - see front matter Crown Copyright Ó 2009 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.12.051
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J. Barba-Ortega et al. / Physica C 470 (2010) 225–230
grid. Complex link variables Ux and Uy are introduced to preserve the gauge-invariant properties of the discretized equations. This new variables are related to A by [8]:
Z ux ¼ exp i
x Ax ðn; y; tÞdn ;
y
u ¼ exp i
x0
Z
!
y
Ay ðx; g; tÞdg
yo
ð3Þ The link variable method is used since a better numerical convergence is obtained for high magnetic fields [9]. The TDGL Eqs. (1) and (2) can be written in the following form:
@W @ 2 ðux WÞ @ 2 ðuy WÞ y x þu þ ð1 TÞWð1 jW2 jÞ ¼u 2 @x @y2 @t @ðut WÞ t W J s ¼ ð1 TÞIm u @t
ð4Þ
x
x
@ Wi;j U i;j Wi;j 2Wi;j þ U i1;j Wi1;j þ ¼ @t ga2x U yi;j Wi;jþ1 2Wi;j þ U yi;j1 Wi;j1
ga2y
1T
g
ðWi;j Wi;j 1ÞWi;j
@U xi;j j2 ¼ ið1 TÞU xi;j ImðWi;j U xi;j Wiþ1;j Þ 2 U yi;j ðLi;j Li;j1 1Þ @t ay @U yi;j @t
¼ ið1 TÞU yi;j IððWi;j U xi;j Wi;jþ1 Þ
j2 a2x
ð6Þ
U yi;j ðLi;j Li;j1 1Þ 1
The open boundary conditions are: Wi;j ¼ ð1 ay b ÞW2;j U x1;j , and WNxþ1;j ¼ ð1 ay b1 ÞWNx;j U x1;j . In our simulation, the simple Euler method is used with a mesh spacing ax = ay = 0.3, for a rectangular samples of size dy = 60n(0) and dx takes the values dx = 30n(0), 25n(0), 18n(0), 12n(0) with fixed temperatures T = 0, j = 20. 2. Results
where t = (x, y) and Im indicates the imaginary part. The outline of this simulation procedure is as follows: the sample is divided into a rectangular mesh consisting of Nx Ny cells, with mesh spacing ax ay. To obtain the discrete equations let us define an arbitrary vertex point in the mesh by xi = (i 1)ax, yi = (i 1)ay as follows:
xi;j uxiþ1;j ¼ exp i U xi;j ¼ u ¼ exp i
Z
yiþ1
Z
!
xiþ1 xi
Ax ðn; yi Þdn ; !
Ay ðxi ; gÞdg ;
yi
¼ expðiax ay He Þ
yi;j uyi;jþ1 U yi;j ¼ u
Li;j ¼ U xi;j U yiþ1;j U xi;jþ1 U yi;j ð5Þ
Then the discretized version of the TDGL equations maintaining second order accuracy in space is given by:
The Fig. 1 shows the magnetization curve as a function of the applied magnetic field (A), the number of fluxoids quanta in the sample for b = 20n(0) for a sample dy = 60n(0) and dx = 12n(0) (B), and in (c) the sample geometry is depicted. The magnetization loop for a sample with dimensions dy = 60n(0), dx = 12n(0) and interface b = 20n(0) is characterized by a hysteresis behavior due to different circumstances in which vortices enter or are expelled from the sample through the surface barrier [3]. The surface barrier delays the incursion of vortices toward the sample interior. The peaks in the magnetization curve correspond to points of maximal penetration to the magnetic field. It is interesting to note that this behavior of the magnetic response, diamagnetic in part of the upward branch, is quite frequent in superconducting mesoscopic samples of different geometries
Fig. 1. Magnetization curve as a function of the applied magnetic field He, (B) number of fluxoids quanta in the sample for b = 20n(0) and dy = 60n(0), dx = 12n(0). (c) The sample geometry is depicted.
J. Barba-Ortega et al. / Physica C 470 (2010) 225–230
and in a broad range of j values [10,11]. The first three maxima occur at He/Hc2(0) = 0.202, 0.258, 0.330. After each maximum, the curve turns downwards and remains smooth. Following by a sharp decrease in |Mz| from the maximal value, a sudden entry of magnetic flux into the sample takes place, at He/Hc2(0) = 0.230, 0.285, 0.405. In Fig. 1B the number N of fluxoids in the computational domain is plotted. The points cloud for He > 0.9Hc2(0) indicates that the sample has become normal. For He < 0.230 the sample remains in the Meissner state. Ten vortices enter at He = 0.235Hc2(0) and arrange themselves in an uniformly linear chain (Fig. 2a). In Fig. 1a–d, the values of the phase close to zero are given by red regions and those close to 2p by blue regions. The phase allows determine the number of vortices in a given region, by counting the phase variation along a closed path around this region. If the vorticity in this region is N, then the phase changes as Dh = 2pN. The number of vortices remains constant until a new vortex penetration event happens at He = 0.28Hc2(0). The system reaches another steady configuration at He = 0.315Hc2(0) with twenty vortices inside the sample (Fig. 2b). At He = 0.455Hc2(0) with 40 vortices, two chains appear forming a zigzag pattern Fig. 2c. At He = 0.52Hc2(0) a giant rectangle of depreciate superconductivity is observed, the vortices are overlapping, although they are not visible in the contour plot of the magnitude of order parameter, there is a change in the phase around the sample equal to Dh = 110, there are 55 vortices into the sample (Fig. 2d). At He = 0.25Hc2(0) the spatial distributions of the order parameter and its phase are shown in
227
Fig. 3a–d for four different values of b. We observe that the arrangement of the vortices is strongly affected by the value of b, clearly illustrating that it affects the superconductivity on the edge of the sample. The decreasing of |b| depreciates the superconductivity order parameter close to the sample edge. The number of vortices decreases into the sample for a same applied magnetic field, the region of the sample that remains superconductor decreases as decreases |b|. The magnetization curves for four homogeneous films of thickness 30n(0), 25n(0), 18n(0) and 12n(0) are shown in Fig. 4. We have Hfirst = 0.074Hc2(0) for dx = 30n(0), Hfirst = 0.083Hc2(0) for dx = 25n(0), Hfirst = 0.115Hc2(0) for dx = 18n(0) and Hfirst = 0.225Hc2(0) for dx = 12n(0), where Hfirst is the field at the entrance of the first vortices line. The results of N fluxoids and the energy curves G as functions of the applied external field are depicted in Fig. 3. The energy and the magnetization curves exhibit a series of discontinuities, each of them signals a rearrangement of vortex chains. As it is well know, we can appreciate that if we reduce the size of the superconductor, both, Hc2 and nucleation fields increase. Hc2 = 2.2; 1.7; 1.3; 1.0 for the dx/n(0) = 30, 25, 18, 12 films correspondingly (see Fig. 5). The vortex configurations show how the number of chains changes from n to n + 1 for all the samples. In Figs. 6 and 7 we can appreciate the vortex configuration for the dx = 25 and dx = 12 films. We can note the suppressed superconductivity at the border of the samples due to the presence of the metallic material. This thin layer of suppressed superconductivity is of the order
Fig. 2. Distribution of the order parameter modulus (above in each graph) and its phase (below in each graph) for b = 2.0n(0) in the intervals: (a) He = 0.23–0.275, (b) He = 0.315–0.350, (c) He = 0.455–0.475, (d) He = 0.520–0.550. Values of the phase close to zero are given by red regions and close to 2p by blue regions, (as well as Dh/2p, from 0 to 1). (For interpretation of colour references in this figure legend, the reader is referred to the web version of this article.)
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Fig. 3. Distribution of the order parameter modulus (above in each graph) and its phase (below in each graph) for an applied magnetic field He = 0.25 and: (a) b = 2.0n(0), (b) b = 1.0n(0), (c) b = 0.5n(0), (d) b = 0.1n(0).
Fig. 4. Magnetization curves as functions of the applied external field for dx = 30n(0); 12n(0) (left) and dx = 25n(0); 18n(0) (right) for b = 1.0n(0),j = 20, T = 0.
n(0). For the dx = 12 film, the first vortex entrance occurs in a double chain, then, with increased field, a zigzag transition occurs and the double chain splits into single-chain. Further the 1 ? 2 ? 3 transition occurs until reach the normal state in He = Hc2. In this figure (Fig. 6) we illustrate how the vortices evolve in time. First, two vortex chains nucleate on the surfaces, then, increasing the magnetic field they align in a zigzag structure, since this configuration is degenerated, they stay in a significant range of the applied magnetic field before they form a single vortex chain. Further, the vortices organize in a well know numbers of chains. For film thickness dx = 25, six chain vortex are possible in the sample, with 1 ? 2 ? 3 ? 4 ? 5 ? 6 chain transitions until reach the normal state in He = 0.8Hc2, differently in comparison with results of Piacente et al. [1] for a systems of charged particles interacting through a screened Coulomb. There is an intermediate region where a four chain configuration has a lower energy. This difference can be due to substantial changes of the potential energy in
the system of charged particles within a very narrow range of temperatures [9]. 3. Conclusions The main contribution of this work is the consideration of the de Gennes boundary condition via the extrapolation length (b > 0) in the analysis of the dynamics of a two dimensional chain like vortex structure. We numerically solve the TDGL equations for a mesoscopic superconducting film immersed in a metallic material and in presence of a magnetic field. We observe a sequence of transitions as the magnetic external field increases, from single-chain structure to double chain structure, triple and so on. For all transitions in all studied samples the number of chains increases only by one unit from n to n + 1. Our results show similarities with theoretical [3] and experimental [10] works, where the peaks in the magnetization curve are associated with a stable
J. Barba-Ortega et al. / Physica C 470 (2010) 225–230
229
Fig. 5. Fluxoids number N (left) and free energy G (right) as functions of the applied external field for: (a) dx = 30n(0), (b) dx = 25n(0), (c) dx = 18n(0) and (d) dx = 12n(0) for b = 1.0n(0),j = 20, T = 0.
Fig. 6. Vortex configuration for dx = 12n(0) for: (a) He = 0.225Hc2(0), (b) He = 0.264Hc2(0), (c) He = 0.405Hc2(0), all with N = 16, (d) He = 0.629Hc2(0), N = 32, (e) He = 0.800Hc2(0), N = 48, (f) He = 0.864Hc2(0), N = 48 and (g) He = 1.0Hc2(0) in the normal state, T = 0 and j = 20. The spectrum from blue to red represents the values of the modulus of the order parameter (as well as Dh/2p, from 0 to 1). (For interpretation of colour references in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. Vortex configuration for dx = 25n(0) for: (a) He = 0.085Hc2(0), N = 14, (b) He = 0.146Hc2(0), N = 27 (c) He = 0.121Hc2(0), N = 54, (d) He = 0.220Hc2(0), N = 75 and (e) He = 0.252Hc2(0), N = 91, T = 0 and j = 20. The spectrum from blue to red represents the values of the modulus of the order parameter (as well as Dh/2p, from 0 to 1). (For interpretation of colour references in this figure legend, the reader is referred to the web version of this article.)
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configuration of the vortex lattice as parallel linear chains. Also, our results point out that, as we reduce the size of the superconductor, both, the nucleation and the first nucleation fields increase. References [1] G. Piacente, I.V. Schweigert, J.J. Betouras, F.M. Peeters, Phys. Rev. B 69 (2004) 045324. [2] G. Piacente, F.M. Peeters, Phys. Rev. B 72 (2005) 205208. [3] C. C de Souza Silva, J. Albino Aguiar, Physica C 354 (2001) 232; Physica C 404 (2004) 11.
[4] A.C. Bolech, G.C. Buscaglia, A. Lopez, Connectivity and Superconductivity, in: J. Berger, J. Rubinstein (Eds.), Springer, 2000. [5] J.J. Barba, C.C. Souza, L.R.E. Cabral, J. Albino Aguiar, Physica C 468 (2008) 718; J.J. Barba, J. Albino Aguiar, Physica C 469 (2009) 852; Physica C 469 (2009) 754. [6] D. Gropp, H.G. Kaper, G.K. Leaf, D.M. Levine, M. Palumbo, V.M. Vinokur, J. Comput. Phys. 123 (1996) 54. [7] E. Sardella, A.L. Malvezzi, Wilson Ortiz, Phys. Rev. B 74 (2006) 014512. [8] S.V. Yampolskii, F.M. Peeters, Phys. Rev. B 62 (2000) 9663. [9] I.V. Schweigert, V.A. Schweigert, F.M. Peeters, Phys. Rev. B 60 (1999) 14665. [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] V.V. Moshchalkov, X.G. Qiu, V. Bruyndoncx, Phys. Rev. B 55 (1997) 11793.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Two dimensional vortex structures in a superconductor slab at low temperatures J. Barba-Ortega a,*, Ariel Becerra b, J. Albino Aguiar a a b
Departamento de Física, Universidade Federal de Pernambuco, 50670-901, Recife, PE, Brazil Grupo de Investigación INTEGRAR, Universidad de Pamplona, Pamplona, Colombia
a r t i c l e
i n f o
Article history: Received 29 August 2009 Received in revised form 19 December 2009 Accepted 21 December 2009 Available online 4 January 2010 Keywords: Ginzburg–Landau de Gennes extrapolation length Mesoscopics Superconducting slab
a b s t r a c t The dynamics of a two dimensional chain like structure of vortices is studied in the model of nonlinear time dependent Ginzburg–Landau equations (TDGL). The transition between different linear chains of vortices in a superconducting homogeneous slab with both surfaces in contact with a thin layer of metallic material is analyzed. The magnetization curve, vortex number, vortex configurations and modulus of the order parameter are studied as a function of the external magnetic field. We show how these vortex configurations are affected by the extrapolation length b (de Gennes boundary conditions), W due to the proximity effects in a mesoscopic sample of area dx dy, where dy = 60n(0) and dx varies discretely from 30n(0) to 12n(0). Possible connection with recent theoretical results in a two dimensional system of charged particles is discussed. Crown Copyright Ó 2009 Published by Elsevier B.V. All rights reserved.
1. Introduction During the last years, mesoscopic systems consisting of interacting particles in confined geometries or low dimensions attracted a lot of attention. Depending on the interacting potential, these systems crystallize in different lattice structures and exhibit a phase diagram with continuous and discontinuous structural transitions. Some important results in this topic from other authors are the differences founded between single-chain and multi-chain regimes as temperature increases in the analysis of the dynamics of quasi-one-dimensional chain like systems of charged particles at low temperatures [1,2]. Also is founded in thin superconductors films [3] that the vortex lattice consists of parallel linear chains of vortices and goes through transitions from n to n + 1 chain. The purpose of our study is to achieve a deeper understanding of the influence of the de Gennes extrapolation length on the vortex transition in a bi-dimensional sample. In particular, we considered a mesoscopic superconducting film which is immersed in a metallic medium in presence of an applied perpendicular uniform magnetic field He and here a superconductor–metal interface is always used, which will cause the superconductivity to be weaker at the edge of the sample and therefore the superconducting currents can flow out from the superconductor. Also we take the order parameter invariant and the magnetic field constant along z-axis. For this system we use the TDGL equations with the general boundary conditions for superconductors and the technique of gauge invariant variables to obtain the superconducting order parameter W, the magnetization curves M and the free energy G * Corresponding author. Tel.: +55 81 2126 7625; fax: +55 81 3271 0359. E-mail address: [email protected] (J. Barba-Ortega).
as a function of the de Gennes parameter. In this work the vortex configurations for a mesoscopic film of rectangular cross-section dx dy is studied, where dy = 60n(0) and dx varies from 30n(0) to 12n(0). Here n(0) is the coherence length at zero temperature. 1.1. Time dependent Ginzburg–Landau equations and boundary conditions The time dependent Ginzburg–Landau equations (TDGL) [1–5] which govern the superconductivity order parameter W and the vector potential A in the zero electric potential gauge are given by:
@W 1 ¼ ððir AÞ2 W þ ð1 TÞðjWj2 1ÞWÞ @t g @A ¼ ð1 TÞReðW ðir AÞWÞ j2 r r A @t
ð1Þ ð2Þ
Eqs. (1) and (2) were rescaled as follows: the order parameter W in units of W1(0) = (a/b)1/2, where a and b are two phenomenological parameters. Temperatures in units of the critical temperature Tc, lengths in units of the coherence length n(0), time in units of t0 = p⁄/96KBTc, A in units of Hc2(0)Tc(0), where Hc2(0) is the second thermodynamics critical field. The Gibbs free energy in units of G0=(aTc)2/b. The dynamical equations are complemented with the appropriate boundary conditions for the order parameter. The general boundary condition for superconductors, found by de Gennes h ^ is the unity vec^ i W, where n [6], is given by n hr þ ec A W ¼ i b tor, perpendicular to the superconductor surface; b is the extrapolation de Gennes parameter. For all simulations we used b > 0, which describes the superconductor/metal interface. The full discretization of the TDGL equations can be found in more detail in Ref. [7]. We use the UW method to solve the TDGL equations in a discrete
0921-4534/$ - see front matter Crown Copyright Ó 2009 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.12.051
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J. Barba-Ortega et al. / Physica C 470 (2010) 225–230
grid. Complex link variables Ux and Uy are introduced to preserve the gauge-invariant properties of the discretized equations. This new variables are related to A by [8]:
Z ux ¼ exp i
x Ax ðn; y; tÞdn ;
y
u ¼ exp i
x0
Z
!
y
Ay ðx; g; tÞdg
yo
ð3Þ The link variable method is used since a better numerical convergence is obtained for high magnetic fields [9]. The TDGL Eqs. (1) and (2) can be written in the following form:
@W @ 2 ðux WÞ @ 2 ðuy WÞ y x þu þ ð1 TÞWð1 jW2 jÞ ¼u 2 @x @y2 @t @ðut WÞ t W J s ¼ ð1 TÞIm u @t
ð4Þ
x
x
@ Wi;j U i;j Wi;j 2Wi;j þ U i1;j Wi1;j þ ¼ @t ga2x U yi;j Wi;jþ1 2Wi;j þ U yi;j1 Wi;j1
ga2y
1T
g
ðWi;j Wi;j 1ÞWi;j
@U xi;j j2 ¼ ið1 TÞU xi;j ImðWi;j U xi;j Wiþ1;j Þ 2 U yi;j ðLi;j Li;j1 1Þ @t ay @U yi;j @t
¼ ið1 TÞU yi;j IððWi;j U xi;j Wi;jþ1 Þ
j2 a2x
ð6Þ
U yi;j ðLi;j Li;j1 1Þ 1
The open boundary conditions are: Wi;j ¼ ð1 ay b ÞW2;j U x1;j , and WNxþ1;j ¼ ð1 ay b1 ÞWNx;j U x1;j . In our simulation, the simple Euler method is used with a mesh spacing ax = ay = 0.3, for a rectangular samples of size dy = 60n(0) and dx takes the values dx = 30n(0), 25n(0), 18n(0), 12n(0) with fixed temperatures T = 0, j = 20. 2. Results
where t = (x, y) and Im indicates the imaginary part. The outline of this simulation procedure is as follows: the sample is divided into a rectangular mesh consisting of Nx Ny cells, with mesh spacing ax ay. To obtain the discrete equations let us define an arbitrary vertex point in the mesh by xi = (i 1)ax, yi = (i 1)ay as follows:
xi;j uxiþ1;j ¼ exp i U xi;j ¼ u ¼ exp i
Z
yiþ1
Z
!
xiþ1 xi
Ax ðn; yi Þdn ; !
Ay ðxi ; gÞdg ;
yi
¼ expðiax ay He Þ
yi;j uyi;jþ1 U yi;j ¼ u
Li;j ¼ U xi;j U yiþ1;j U xi;jþ1 U yi;j ð5Þ
Then the discretized version of the TDGL equations maintaining second order accuracy in space is given by:
The Fig. 1 shows the magnetization curve as a function of the applied magnetic field (A), the number of fluxoids quanta in the sample for b = 20n(0) for a sample dy = 60n(0) and dx = 12n(0) (B), and in (c) the sample geometry is depicted. The magnetization loop for a sample with dimensions dy = 60n(0), dx = 12n(0) and interface b = 20n(0) is characterized by a hysteresis behavior due to different circumstances in which vortices enter or are expelled from the sample through the surface barrier [3]. The surface barrier delays the incursion of vortices toward the sample interior. The peaks in the magnetization curve correspond to points of maximal penetration to the magnetic field. It is interesting to note that this behavior of the magnetic response, diamagnetic in part of the upward branch, is quite frequent in superconducting mesoscopic samples of different geometries
Fig. 1. Magnetization curve as a function of the applied magnetic field He, (B) number of fluxoids quanta in the sample for b = 20n(0) and dy = 60n(0), dx = 12n(0). (c) The sample geometry is depicted.
J. Barba-Ortega et al. / Physica C 470 (2010) 225–230
and in a broad range of j values [10,11]. The first three maxima occur at He/Hc2(0) = 0.202, 0.258, 0.330. After each maximum, the curve turns downwards and remains smooth. Following by a sharp decrease in |Mz| from the maximal value, a sudden entry of magnetic flux into the sample takes place, at He/Hc2(0) = 0.230, 0.285, 0.405. In Fig. 1B the number N of fluxoids in the computational domain is plotted. The points cloud for He > 0.9Hc2(0) indicates that the sample has become normal. For He < 0.230 the sample remains in the Meissner state. Ten vortices enter at He = 0.235Hc2(0) and arrange themselves in an uniformly linear chain (Fig. 2a). In Fig. 1a–d, the values of the phase close to zero are given by red regions and those close to 2p by blue regions. The phase allows determine the number of vortices in a given region, by counting the phase variation along a closed path around this region. If the vorticity in this region is N, then the phase changes as Dh = 2pN. The number of vortices remains constant until a new vortex penetration event happens at He = 0.28Hc2(0). The system reaches another steady configuration at He = 0.315Hc2(0) with twenty vortices inside the sample (Fig. 2b). At He = 0.455Hc2(0) with 40 vortices, two chains appear forming a zigzag pattern Fig. 2c. At He = 0.52Hc2(0) a giant rectangle of depreciate superconductivity is observed, the vortices are overlapping, although they are not visible in the contour plot of the magnitude of order parameter, there is a change in the phase around the sample equal to Dh = 110, there are 55 vortices into the sample (Fig. 2d). At He = 0.25Hc2(0) the spatial distributions of the order parameter and its phase are shown in
227
Fig. 3a–d for four different values of b. We observe that the arrangement of the vortices is strongly affected by the value of b, clearly illustrating that it affects the superconductivity on the edge of the sample. The decreasing of |b| depreciates the superconductivity order parameter close to the sample edge. The number of vortices decreases into the sample for a same applied magnetic field, the region of the sample that remains superconductor decreases as decreases |b|. The magnetization curves for four homogeneous films of thickness 30n(0), 25n(0), 18n(0) and 12n(0) are shown in Fig. 4. We have Hfirst = 0.074Hc2(0) for dx = 30n(0), Hfirst = 0.083Hc2(0) for dx = 25n(0), Hfirst = 0.115Hc2(0) for dx = 18n(0) and Hfirst = 0.225Hc2(0) for dx = 12n(0), where Hfirst is the field at the entrance of the first vortices line. The results of N fluxoids and the energy curves G as functions of the applied external field are depicted in Fig. 3. The energy and the magnetization curves exhibit a series of discontinuities, each of them signals a rearrangement of vortex chains. As it is well know, we can appreciate that if we reduce the size of the superconductor, both, Hc2 and nucleation fields increase. Hc2 = 2.2; 1.7; 1.3; 1.0 for the dx/n(0) = 30, 25, 18, 12 films correspondingly (see Fig. 5). The vortex configurations show how the number of chains changes from n to n + 1 for all the samples. In Figs. 6 and 7 we can appreciate the vortex configuration for the dx = 25 and dx = 12 films. We can note the suppressed superconductivity at the border of the samples due to the presence of the metallic material. This thin layer of suppressed superconductivity is of the order
Fig. 2. Distribution of the order parameter modulus (above in each graph) and its phase (below in each graph) for b = 2.0n(0) in the intervals: (a) He = 0.23–0.275, (b) He = 0.315–0.350, (c) He = 0.455–0.475, (d) He = 0.520–0.550. Values of the phase close to zero are given by red regions and close to 2p by blue regions, (as well as Dh/2p, from 0 to 1). (For interpretation of colour references in this figure legend, the reader is referred to the web version of this article.)
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Fig. 3. Distribution of the order parameter modulus (above in each graph) and its phase (below in each graph) for an applied magnetic field He = 0.25 and: (a) b = 2.0n(0), (b) b = 1.0n(0), (c) b = 0.5n(0), (d) b = 0.1n(0).
Fig. 4. Magnetization curves as functions of the applied external field for dx = 30n(0); 12n(0) (left) and dx = 25n(0); 18n(0) (right) for b = 1.0n(0),j = 20, T = 0.
n(0). For the dx = 12 film, the first vortex entrance occurs in a double chain, then, with increased field, a zigzag transition occurs and the double chain splits into single-chain. Further the 1 ? 2 ? 3 transition occurs until reach the normal state in He = Hc2. In this figure (Fig. 6) we illustrate how the vortices evolve in time. First, two vortex chains nucleate on the surfaces, then, increasing the magnetic field they align in a zigzag structure, since this configuration is degenerated, they stay in a significant range of the applied magnetic field before they form a single vortex chain. Further, the vortices organize in a well know numbers of chains. For film thickness dx = 25, six chain vortex are possible in the sample, with 1 ? 2 ? 3 ? 4 ? 5 ? 6 chain transitions until reach the normal state in He = 0.8Hc2, differently in comparison with results of Piacente et al. [1] for a systems of charged particles interacting through a screened Coulomb. There is an intermediate region where a four chain configuration has a lower energy. This difference can be due to substantial changes of the potential energy in
the system of charged particles within a very narrow range of temperatures [9]. 3. Conclusions The main contribution of this work is the consideration of the de Gennes boundary condition via the extrapolation length (b > 0) in the analysis of the dynamics of a two dimensional chain like vortex structure. We numerically solve the TDGL equations for a mesoscopic superconducting film immersed in a metallic material and in presence of a magnetic field. We observe a sequence of transitions as the magnetic external field increases, from single-chain structure to double chain structure, triple and so on. For all transitions in all studied samples the number of chains increases only by one unit from n to n + 1. Our results show similarities with theoretical [3] and experimental [10] works, where the peaks in the magnetization curve are associated with a stable
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Fig. 5. Fluxoids number N (left) and free energy G (right) as functions of the applied external field for: (a) dx = 30n(0), (b) dx = 25n(0), (c) dx = 18n(0) and (d) dx = 12n(0) for b = 1.0n(0),j = 20, T = 0.
Fig. 6. Vortex configuration for dx = 12n(0) for: (a) He = 0.225Hc2(0), (b) He = 0.264Hc2(0), (c) He = 0.405Hc2(0), all with N = 16, (d) He = 0.629Hc2(0), N = 32, (e) He = 0.800Hc2(0), N = 48, (f) He = 0.864Hc2(0), N = 48 and (g) He = 1.0Hc2(0) in the normal state, T = 0 and j = 20. The spectrum from blue to red represents the values of the modulus of the order parameter (as well as Dh/2p, from 0 to 1). (For interpretation of colour references in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. Vortex configuration for dx = 25n(0) for: (a) He = 0.085Hc2(0), N = 14, (b) He = 0.146Hc2(0), N = 27 (c) He = 0.121Hc2(0), N = 54, (d) He = 0.220Hc2(0), N = 75 and (e) He = 0.252Hc2(0), N = 91, T = 0 and j = 20. The spectrum from blue to red represents the values of the modulus of the order parameter (as well as Dh/2p, from 0 to 1). (For interpretation of colour references in this figure legend, the reader is referred to the web version of this article.)
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