- Email: [email protected]
Linear arrangement of metallic and superconducting defects in a thin superconducting sample
Physica C 492 (2013) 1–5
Contents lists available at SciVerse ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Linear arrangement of metallic and superconducting defects in a thin superconducting sample J. Barba-Ortega a,⇑, Edson Sardella b,c, J. Albino Aguiar d Departamento de Fı´sica, Universidad Nacional de Colombia, Bogotá, Colombia UNESP – Universidade Estadual Paulista, Departamento de Fı´sica, Caixa Postal 473, Bauru, SP, Brazil c UNESP – Universidade Estadual Paulista, IPMet – Instituto de Pesquisas Metereológicas, CEP 17048-699 Bauru, SP, Brazil d Departamento de Fı´sica, Universidade Federal de Pernambuco, 50670-901 Recife, PE, Brazil a
b
a r t i c l e
i n f o
Article history: Received 8 February 2013 Received in revised form 20 April 2013 Accepted 22 April 2013 Available online 9 May 2013 Keywords: Vortex matter Ginzburg–Landau theory Metallic defects Superconducting defects
a b s t r a c t The vortex matter in a superconducting disk with a linear configuration of metallic and superconducting defects is studied. Effects associated to the pinning (anti-pinning) force of the metallic (superconducting) defect on the vortex configuration and on the thermodynamic critical fields are analyzed in the framework of the Ginzburg Landau theory. We calculate the loop of the magnetization, vorticity and free energy curves as a function of the magnetic field for a thin disk. Due to vortex–defect attraction for a metallic defect (repulsion for a superconducting defect), the vortices always (never) are found to be sitting on the defect position. Crown Copyright Ó 2013 Published by Elsevier B.V. All rights reserved.
1. Introduction Impurities, defects and size of the sample plays an important role in superconducting systems, beginning with the remarkable feature that defects strongly impact thermodynamics properties in conventional mesoscopics superconductors [1–4]. Many superconducting structures of different topologies have attracted attention of experimental and theoretical researchers as potential electronics components [5–12]. In the present paper, we calculate using the Ginzburg–Landau theory, the magnetization, free energy and vorticity for a thin disk with one, two and four linear metallic and superconducting defects, also we investigate the effect of the defects on the vortex configuration and critical fields. 2. Theoretical formalism We assume a thin mesoscopic superconductor disk immersed in an insulating medium in the presence of a perpendicular uniform magnetic field H0. We write the system of the time dependent Ginzburg–Landau equations in the following form [13–15]:
@w ¼ ði$ þ AÞ2 w þ wðjwj2 1Þ @t @A ¼ Re½wði r AÞw j2 $ r A @t ⇑ Corresponding author. Tel.: +573173141682. E-mail address: [email protected] (J. Barba-Ortega).
ð1Þ ð2Þ
T is the temperature in units of the critical temperature; lengths in units of n(T) the coherence length, and fields in units of Hc2(T), the second critical thermodynamic field; j is the Ginzburg–Landau parameter. In order to solve Eqs. (1) and (2) we used the link variables approach as it was adapted for circular geometries [6]. Let a thin superconducting disk domain be given by {(r, h, z) 2 R3: (r, h) 2 R2, jzj < ds(r, h)}, for all (r, h). rs(r, h)/s(r, h) is some function which describes the topology of the top surface of the disk and d is the thickness of the disk, with d n, k. According to Refs. [16– 18] the TDGL equations can be reduced to:
@w 1 ¼ ði$ þ A0 Þ sði$ þ A0 Þw þ wð1 jwj2 Þ @t s
ð3Þ
In Fig. 1 we appreciate an enhancement (suppression) of the superconductivity in the points where s > 1 (s < 1) is used, we can consider this region made of a superconducting material at higher critical temperature (metal or superconducting material at lower critical temperature). We solve the time dependent Ginzburg–Landau equations in order to obtain the vortex configurations in a mesoscopic superconducting disk of radius R = 6.5n(T) with a little central hole of radius ri = 0.05n(T). The magnetic field is nearly uniform inside the superconductor, that is, H0 ¼ $ A0 . We take the function s = 1 everywhere, except in points inside the disk where we use s = 0.8 (s = 1.3), simulating the presence of a metallic defect (superconducting at higher critical temperature defect). Due to the geometry of the computational mesh, the order parameter at H0 = 0.0 obey the equation: jw(r)j2 = 1.0 + r(jw(R)j2 1.0)/R, with
0921-4534/$ - see front matter Crown Copyright Ó 2013 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physc.2013.04.070
2
J. Barba-Ortega et al. / Physica C 492 (2013) 1–5
Fig. 1. Square modulus order parameter jwj2 at H0 = 0. Disk with a linear arrangement of metallic defects (left) and superconducting at higher critical temperature defects (right).
jw(R)j2 = 1.020 for s = 1.3, jw(R)j2 = 0.985 for s = 0.8 see Fig. 1. We consider the three following scenarios for ri < r 6 R: Case 1: half linear defect in h = 0, Fig. 1 up; Case 2: one linear defect in h = 0, p, Fig. 1 middle; Case 3: two linear defect in h = 0, p/2, p, 3p/2, Fig. 1 down. 3. Results and discussion We solve the time dependent Ginzburg–Landau equations in order to obtain the vortex configuration in a mesoscopic supercon-
ducting disk with defects distributed forming linear arrays. Figs. 2–4 shown the free energy, magnetization and vorticity loops, respectively as a function of the applied magnetic field for a superconducting disk with linear defects made of metallic (superconducting) materials for the three cases explained in Fig. 1. From these figures, we can see that the both the surface barrier field and the upper field are independent of the nature of defects [19]. The quantities N, M and G display a sequence of discontinuities and indicating that one or more vortices have entered the sample. Furthermore, the jumps in the curves are not regular for both types of defects. We can observe in these figures, that there are more vor-
J. Barba-Ortega et al. / Physica C 492 (2013) 1–5
3
Fig. 2. Energy as a function of the applied field for a disk with radius R = 6.5n(T): (up) case 1, half linear defect; (middle) case 2, one linear defect; (down) case 3, two linear defect (windows, layout of the defects position).
Fig. 3. Vorticity curves as a function of external magnetic field for the same cases of Fig. 2.
tex stationary states when the magnetic field is decreasing. Let us discuss the vortex configurations. Fig. 5 shows the case 1 for a superconducting linear defect simulated with s = 1.3. We can observe the entrance of one first vortex at H0 = 1.020Hc2(T) with entries of vortices from N = 1 to N + 2 all in non-stationary states except the last configuration (the last panel of the figure). However, for the same case 1 but now with a linear metallic defect (s = 0.8, Fig. 6), always the first vortex penetration occurs through the region next to the metallic defect due to the pinning force of the defects. the first vortex enter at H0 = 1.026Hc2(T) and the next two vortices enter to the sample exactly at the opposite region at H0 = 1.09Hc2(T) forming an equilateral triangle (this configuration repeat for the case 3 for a superconducting defect at H0 = 0.180Hc2(T) when the magnetic field is decreasing). Increasing the magnetic field we have N = 6, 7 vortices at H0 = 1.113Hc2(T), 1.120Hc2(T) in stable states, and finally we have N = 9 vortices forming an asymmetrical circular configuration in a non-stationary state at H0 = 1.14Hc2. The position of the defects is imposing the vortex configuration at low magnetic fields and for higher fields the geometry of the sample will prevail. The pinning (repulsive)
force of the metallic (superconducting) defects deform the circular vortex geometry so, the competition between the geometry of the sample, the number of defects and its nature led to novel vortex configurations. In Fig. 7 we show the contour plot of the order parameter for several vorticities for case 1 decreasing the magnetic field. We can see in (a) four vortices forming a square vortex configuration at H0 = 0.208Hc2(T) for a superconducting defect, (b) and (c) three vortices (two in the superconducting region and one in the center of the disk) at H0 = 0.208Hc2(T) for a metallic defect, and (d) one vortex in the center of the disk at H0 = 0.008Hc2(T). In Fig. 8 we show the contour plot of the order parameter for case 2 and linear metallic defect, decreasing the magnetic field for N = 9 at H0 = 0.373Hc2(T) (left) and N = 7 at H0 = 0.282Hc2(T) (right). In both cases one vortex remains in the center of the disk. For these cases we plot only stable vortex configurations. In Fig. 9 we show the contour plot of the order parameter for several vorticities for case 2 increasing the magnetic field for a superconducting defect (Fig. 9a and b) and for a metallic defect (Fig. 9c and d). In Fig. 9b seven vortices remain in the sample at H0 = 0.282Hc2(T) (in the downward branch of the magnetic field), six in the superconduc-
4
J. Barba-Ortega et al. / Physica C 492 (2013) 1–5
Fig. 5. Contour plot of the order parameter for case 1 and s = 1.3. Increasing the magnetic field. The symbols S and NS stand for stationary and non-stationary configurations, respectively.
Fig. 4. Magnetization curve as a function of external magnetic field for the same cases of Fig. 2.
ting region and one in the center of the disk. In Fig. 9d we see four vortices in the superconducting region and one in the center of the disk at H0 = 1.114Hc2(T) (this last configuration repeat in the case 3 considering a metallic defect at H0 = 0.254Hc2(T) when the magnetic field is decreasing, but there is not any vortex in the center of the disk). In Fig. 9a and c we appreciate two vortices at H0 = 1.088Hc2(T) for each defect. In Fig. 10 we show the contour plot of the order parameter for N = 8 at H0 = 1.142Hc2(T) for case 3, (a) superconducting defect and (b) metallic defect, increasing the magnetic field.
4. Conclusions We have solved the Ginzburg–Landau equations assuming the magnetic field to be homogeneous and equal to external field, also inside the sample. We studied the effect of the nature of one internal linear defect on the vortex configuration of a mesoscopic superconducting disk. Our results have shown that the first and third thermodynamical fields are independent of the nature of the defect, but the vorticity at low fields changes strongly. Due to vortex–superconducting defect repulsion we have never observed a
Fig. 6. Contour plot of the order parameter for case 1 and s = 0.8. Increasing the magnetic field. The symbols S and NS stand for stationary and non-stationary configurations, respectively.
J. Barba-Ortega et al. / Physica C 492 (2013) 1–5
5
Fig. 10. Contour plot of the order parameter for case 3, increasing the magnetic field.
vortex sitting on defect position. However, the vortex–metallic defect interaction is attractive and the first vortex entry is always find siting on the defect position. Acknowledgements Fig. 7. Contour plot of the order parameter for case 1 decreasing the magnetic field.
This work was partially supported by the Brazilians agencies CNPq, CAPES, FAPESP, FACEPE (APQ 0589-105/08) and Colombian Agencies Colciencias and DIB. References
Fig. 8. Contour plot of the order parameter for case 2 and linear metallic defect, decreasing the magnetic field.
Fig. 9. Contour plot of the order parameter for case 2, increasing the magnetic field.
[1] I.V. Grigorieva, W. Escoffier, J. Richardson, L.Y. Vinnikov, S. Dubonos, V. Oboznov, Phys. Rev. Lett. 96 (2006) 077005. [2] M.L. Latimer, G.R. Berdiyorov, Z.L. Xiao, W.K. Kwok, F.M. Peeters, Phys. Rev. B (2012) 012505. [3] G.R. Berdiyorov, M.V. Milosevic, M.L. Latimer, Z.L. Xiao, W.K. Kwok, F.M. Peeters, Phys. Rev. Lett. 109 (2012) 057004. [4] P.W. Anderson, J. Phys. Chem. Solids 11 (1959) 26. [5] R. Geurts, M.V. Milosevic, F.M. Peeters, Phys. Rev. Lett. 97 (2006) 137002; R. Geurts, M.V. Milosevic, F.M. Peeters, Phys. Rev. B 75 (2007) 184511. [6] E. Sardella, P.N.L. Filho, A.L. Malvezzi, Phys. Rev. B 77 (2008) 104508. [7] J. Barba-Ortega, Edson Sardella, J. Albino Aguiar, Physica C 480 (2012) 118. [8] J. Barba-Ortega, Edson sardella, J. Albino Aguiar, Supercond. Sci. Technol. 24 (2011) 015001. [9] T. Puig, E. Rosseel, L. Van Look, M.J. Van Bael, V.V. Moshchalkov, Y. Bruynseraede, R. Jonckheere, Phys. Rev. B 58 (1998) 5744. [10] V. Bruyndoncx, J.G. Rodrigo, T. Puig, L. Van Look, V.V. Moshchalkov, R. Jonckheere, Phys. Rev. B 60 (1999) 4285. [11] J.R. Clem, in: K.D. Timmerhaus, W.J. O Sullivan, E.F. Hammel (Eds.), Low Temperature Physics, vol. 3, Plenum, New York, 1974, p. 102. [12] J. Barba-Ortega, Edson sardella, J. Albino Aguiar, E.H. Brandt, Physica C 479 (2012) 49. [13] W.D. Gropp, H.G. Kaper, G.K. Leaf, D.M. Levine, M. Palumbo, V.M. Vinokur, J. Comput. Phys. 123 (1996) 254. [14] G.C. Buscaglia, C. Bolech, A. Lòpez, in: J. Berger, J. Rubinstein (Eds.), Connectivity and Superconductivity, Springer, 2000. [15] A.K. Geim, S.V. Dubonos, I.V. Grigorieva, K.S. Novoselov, F.M. Peeters, V.A. Schweigert, Nature 407 (2000) 55. [16] Q. Du, M.D. Gunzburger, Physica D 69 (1993) 215. [17] Q. Du, M.D. Gunzburger, J.S. Peterson, Phys. Rev. B 51 (1995) 16194. [18] G.R. Berdiyorov, M.V. Milosevic, B.J. Baelus, F.M. Peeters, Phys. Rev. B 70 (2004) 024508. [19] J. Barba-Ortega, Edson Sardella, J. Albino Aguiar, Physica C 485 (2013) 107.
Contents lists available at SciVerse ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Linear arrangement of metallic and superconducting defects in a thin superconducting sample J. Barba-Ortega a,⇑, Edson Sardella b,c, J. Albino Aguiar d Departamento de Fı´sica, Universidad Nacional de Colombia, Bogotá, Colombia UNESP – Universidade Estadual Paulista, Departamento de Fı´sica, Caixa Postal 473, Bauru, SP, Brazil c UNESP – Universidade Estadual Paulista, IPMet – Instituto de Pesquisas Metereológicas, CEP 17048-699 Bauru, SP, Brazil d Departamento de Fı´sica, Universidade Federal de Pernambuco, 50670-901 Recife, PE, Brazil a
b
a r t i c l e
i n f o
Article history: Received 8 February 2013 Received in revised form 20 April 2013 Accepted 22 April 2013 Available online 9 May 2013 Keywords: Vortex matter Ginzburg–Landau theory Metallic defects Superconducting defects
a b s t r a c t The vortex matter in a superconducting disk with a linear configuration of metallic and superconducting defects is studied. Effects associated to the pinning (anti-pinning) force of the metallic (superconducting) defect on the vortex configuration and on the thermodynamic critical fields are analyzed in the framework of the Ginzburg Landau theory. We calculate the loop of the magnetization, vorticity and free energy curves as a function of the magnetic field for a thin disk. Due to vortex–defect attraction for a metallic defect (repulsion for a superconducting defect), the vortices always (never) are found to be sitting on the defect position. Crown Copyright Ó 2013 Published by Elsevier B.V. All rights reserved.
1. Introduction Impurities, defects and size of the sample plays an important role in superconducting systems, beginning with the remarkable feature that defects strongly impact thermodynamics properties in conventional mesoscopics superconductors [1–4]. Many superconducting structures of different topologies have attracted attention of experimental and theoretical researchers as potential electronics components [5–12]. In the present paper, we calculate using the Ginzburg–Landau theory, the magnetization, free energy and vorticity for a thin disk with one, two and four linear metallic and superconducting defects, also we investigate the effect of the defects on the vortex configuration and critical fields. 2. Theoretical formalism We assume a thin mesoscopic superconductor disk immersed in an insulating medium in the presence of a perpendicular uniform magnetic field H0. We write the system of the time dependent Ginzburg–Landau equations in the following form [13–15]:
@w ¼ ði$ þ AÞ2 w þ wðjwj2 1Þ @t @A ¼ Re½wði r AÞw j2 $ r A @t ⇑ Corresponding author. Tel.: +573173141682. E-mail address: [email protected] (J. Barba-Ortega).
ð1Þ ð2Þ
T is the temperature in units of the critical temperature; lengths in units of n(T) the coherence length, and fields in units of Hc2(T), the second critical thermodynamic field; j is the Ginzburg–Landau parameter. In order to solve Eqs. (1) and (2) we used the link variables approach as it was adapted for circular geometries [6]. Let a thin superconducting disk domain be given by {(r, h, z) 2 R3: (r, h) 2 R2, jzj < ds(r, h)}, for all (r, h). rs(r, h)/s(r, h) is some function which describes the topology of the top surface of the disk and d is the thickness of the disk, with d n, k. According to Refs. [16– 18] the TDGL equations can be reduced to:
@w 1 ¼ ði$ þ A0 Þ sði$ þ A0 Þw þ wð1 jwj2 Þ @t s
ð3Þ
In Fig. 1 we appreciate an enhancement (suppression) of the superconductivity in the points where s > 1 (s < 1) is used, we can consider this region made of a superconducting material at higher critical temperature (metal or superconducting material at lower critical temperature). We solve the time dependent Ginzburg–Landau equations in order to obtain the vortex configurations in a mesoscopic superconducting disk of radius R = 6.5n(T) with a little central hole of radius ri = 0.05n(T). The magnetic field is nearly uniform inside the superconductor, that is, H0 ¼ $ A0 . We take the function s = 1 everywhere, except in points inside the disk where we use s = 0.8 (s = 1.3), simulating the presence of a metallic defect (superconducting at higher critical temperature defect). Due to the geometry of the computational mesh, the order parameter at H0 = 0.0 obey the equation: jw(r)j2 = 1.0 + r(jw(R)j2 1.0)/R, with
0921-4534/$ - see front matter Crown Copyright Ó 2013 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physc.2013.04.070
2
J. Barba-Ortega et al. / Physica C 492 (2013) 1–5
Fig. 1. Square modulus order parameter jwj2 at H0 = 0. Disk with a linear arrangement of metallic defects (left) and superconducting at higher critical temperature defects (right).
jw(R)j2 = 1.020 for s = 1.3, jw(R)j2 = 0.985 for s = 0.8 see Fig. 1. We consider the three following scenarios for ri < r 6 R: Case 1: half linear defect in h = 0, Fig. 1 up; Case 2: one linear defect in h = 0, p, Fig. 1 middle; Case 3: two linear defect in h = 0, p/2, p, 3p/2, Fig. 1 down. 3. Results and discussion We solve the time dependent Ginzburg–Landau equations in order to obtain the vortex configuration in a mesoscopic supercon-
ducting disk with defects distributed forming linear arrays. Figs. 2–4 shown the free energy, magnetization and vorticity loops, respectively as a function of the applied magnetic field for a superconducting disk with linear defects made of metallic (superconducting) materials for the three cases explained in Fig. 1. From these figures, we can see that the both the surface barrier field and the upper field are independent of the nature of defects [19]. The quantities N, M and G display a sequence of discontinuities and indicating that one or more vortices have entered the sample. Furthermore, the jumps in the curves are not regular for both types of defects. We can observe in these figures, that there are more vor-
J. Barba-Ortega et al. / Physica C 492 (2013) 1–5
3
Fig. 2. Energy as a function of the applied field for a disk with radius R = 6.5n(T): (up) case 1, half linear defect; (middle) case 2, one linear defect; (down) case 3, two linear defect (windows, layout of the defects position).
Fig. 3. Vorticity curves as a function of external magnetic field for the same cases of Fig. 2.
tex stationary states when the magnetic field is decreasing. Let us discuss the vortex configurations. Fig. 5 shows the case 1 for a superconducting linear defect simulated with s = 1.3. We can observe the entrance of one first vortex at H0 = 1.020Hc2(T) with entries of vortices from N = 1 to N + 2 all in non-stationary states except the last configuration (the last panel of the figure). However, for the same case 1 but now with a linear metallic defect (s = 0.8, Fig. 6), always the first vortex penetration occurs through the region next to the metallic defect due to the pinning force of the defects. the first vortex enter at H0 = 1.026Hc2(T) and the next two vortices enter to the sample exactly at the opposite region at H0 = 1.09Hc2(T) forming an equilateral triangle (this configuration repeat for the case 3 for a superconducting defect at H0 = 0.180Hc2(T) when the magnetic field is decreasing). Increasing the magnetic field we have N = 6, 7 vortices at H0 = 1.113Hc2(T), 1.120Hc2(T) in stable states, and finally we have N = 9 vortices forming an asymmetrical circular configuration in a non-stationary state at H0 = 1.14Hc2. The position of the defects is imposing the vortex configuration at low magnetic fields and for higher fields the geometry of the sample will prevail. The pinning (repulsive)
force of the metallic (superconducting) defects deform the circular vortex geometry so, the competition between the geometry of the sample, the number of defects and its nature led to novel vortex configurations. In Fig. 7 we show the contour plot of the order parameter for several vorticities for case 1 decreasing the magnetic field. We can see in (a) four vortices forming a square vortex configuration at H0 = 0.208Hc2(T) for a superconducting defect, (b) and (c) three vortices (two in the superconducting region and one in the center of the disk) at H0 = 0.208Hc2(T) for a metallic defect, and (d) one vortex in the center of the disk at H0 = 0.008Hc2(T). In Fig. 8 we show the contour plot of the order parameter for case 2 and linear metallic defect, decreasing the magnetic field for N = 9 at H0 = 0.373Hc2(T) (left) and N = 7 at H0 = 0.282Hc2(T) (right). In both cases one vortex remains in the center of the disk. For these cases we plot only stable vortex configurations. In Fig. 9 we show the contour plot of the order parameter for several vorticities for case 2 increasing the magnetic field for a superconducting defect (Fig. 9a and b) and for a metallic defect (Fig. 9c and d). In Fig. 9b seven vortices remain in the sample at H0 = 0.282Hc2(T) (in the downward branch of the magnetic field), six in the superconduc-
4
J. Barba-Ortega et al. / Physica C 492 (2013) 1–5
Fig. 5. Contour plot of the order parameter for case 1 and s = 1.3. Increasing the magnetic field. The symbols S and NS stand for stationary and non-stationary configurations, respectively.
Fig. 4. Magnetization curve as a function of external magnetic field for the same cases of Fig. 2.
ting region and one in the center of the disk. In Fig. 9d we see four vortices in the superconducting region and one in the center of the disk at H0 = 1.114Hc2(T) (this last configuration repeat in the case 3 considering a metallic defect at H0 = 0.254Hc2(T) when the magnetic field is decreasing, but there is not any vortex in the center of the disk). In Fig. 9a and c we appreciate two vortices at H0 = 1.088Hc2(T) for each defect. In Fig. 10 we show the contour plot of the order parameter for N = 8 at H0 = 1.142Hc2(T) for case 3, (a) superconducting defect and (b) metallic defect, increasing the magnetic field.
4. Conclusions We have solved the Ginzburg–Landau equations assuming the magnetic field to be homogeneous and equal to external field, also inside the sample. We studied the effect of the nature of one internal linear defect on the vortex configuration of a mesoscopic superconducting disk. Our results have shown that the first and third thermodynamical fields are independent of the nature of the defect, but the vorticity at low fields changes strongly. Due to vortex–superconducting defect repulsion we have never observed a
Fig. 6. Contour plot of the order parameter for case 1 and s = 0.8. Increasing the magnetic field. The symbols S and NS stand for stationary and non-stationary configurations, respectively.
J. Barba-Ortega et al. / Physica C 492 (2013) 1–5
5
Fig. 10. Contour plot of the order parameter for case 3, increasing the magnetic field.
vortex sitting on defect position. However, the vortex–metallic defect interaction is attractive and the first vortex entry is always find siting on the defect position. Acknowledgements Fig. 7. Contour plot of the order parameter for case 1 decreasing the magnetic field.
This work was partially supported by the Brazilians agencies CNPq, CAPES, FAPESP, FACEPE (APQ 0589-105/08) and Colombian Agencies Colciencias and DIB. References
Fig. 8. Contour plot of the order parameter for case 2 and linear metallic defect, decreasing the magnetic field.
Fig. 9. Contour plot of the order parameter for case 2, increasing the magnetic field.
[1] I.V. Grigorieva, W. Escoffier, J. Richardson, L.Y. Vinnikov, S. Dubonos, V. Oboznov, Phys. Rev. Lett. 96 (2006) 077005. [2] M.L. Latimer, G.R. Berdiyorov, Z.L. Xiao, W.K. Kwok, F.M. Peeters, Phys. Rev. B (2012) 012505. [3] G.R. Berdiyorov, M.V. Milosevic, M.L. Latimer, Z.L. Xiao, W.K. Kwok, F.M. Peeters, Phys. Rev. Lett. 109 (2012) 057004. [4] P.W. Anderson, J. Phys. Chem. Solids 11 (1959) 26. [5] R. Geurts, M.V. Milosevic, F.M. Peeters, Phys. Rev. Lett. 97 (2006) 137002; R. Geurts, M.V. Milosevic, F.M. Peeters, Phys. Rev. B 75 (2007) 184511. [6] E. Sardella, P.N.L. Filho, A.L. Malvezzi, Phys. Rev. B 77 (2008) 104508. [7] J. Barba-Ortega, Edson Sardella, J. Albino Aguiar, Physica C 480 (2012) 118. [8] J. Barba-Ortega, Edson sardella, J. Albino Aguiar, Supercond. Sci. Technol. 24 (2011) 015001. [9] T. Puig, E. Rosseel, L. Van Look, M.J. Van Bael, V.V. Moshchalkov, Y. Bruynseraede, R. Jonckheere, Phys. Rev. B 58 (1998) 5744. [10] V. Bruyndoncx, J.G. Rodrigo, T. Puig, L. Van Look, V.V. Moshchalkov, R. Jonckheere, Phys. Rev. B 60 (1999) 4285. [11] J.R. Clem, in: K.D. Timmerhaus, W.J. O Sullivan, E.F. Hammel (Eds.), Low Temperature Physics, vol. 3, Plenum, New York, 1974, p. 102. [12] J. Barba-Ortega, Edson sardella, J. Albino Aguiar, E.H. Brandt, Physica C 479 (2012) 49. [13] W.D. Gropp, H.G. Kaper, G.K. Leaf, D.M. Levine, M. Palumbo, V.M. Vinokur, J. Comput. Phys. 123 (1996) 254. [14] G.C. Buscaglia, C. Bolech, A. Lòpez, in: J. Berger, J. Rubinstein (Eds.), Connectivity and Superconductivity, Springer, 2000. [15] A.K. Geim, S.V. Dubonos, I.V. Grigorieva, K.S. Novoselov, F.M. Peeters, V.A. Schweigert, Nature 407 (2000) 55. [16] Q. Du, M.D. Gunzburger, Physica D 69 (1993) 215. [17] Q. Du, M.D. Gunzburger, J.S. Peterson, Phys. Rev. B 51 (1995) 16194. [18] G.R. Berdiyorov, M.V. Milosevic, B.J. Baelus, F.M. Peeters, Phys. Rev. B 70 (2004) 024508. [19] J. Barba-Ortega, Edson Sardella, J. Albino Aguiar, Physica C 485 (2013) 107.