**Effect of a superconducting defect on the Cooper pairs of a mesoscopic sample**

**Efecto de un defecto superconductor sobre los pares de Cooper de una muestra mesoscópica**

**Sindy Jessenia Higuera-Agudelo ^{a,b}, Heli Barba-Molina ^{c} & José José Barba-Ortega ^{d}**

^{a }*Departamento de Física, Universidad Nacional de Colombia, Bogotá, Colombia. sjhigueraa@unal.edu.co ^{b }Departamento de Física, Universidade Federal de Pernambuco, Recife, PE, Brasil ^{c }Universidad Cooperativa de Colombia, Bucaramanga, Colombia. heli.barba@campusucc.edu.co ^{d }Departamento de Física, Universidad Nacional de Colombia, Bogotá, Colombia. jjbarbao@unal.edu.co*

**Received: August 6 ^{th}, de 2013. Received in revised form: March 27^{th}, 2014. Accepted: April 4^{th}, 2014**

**Abstract **We investigate the vortex state in a very long prism of square cross section with a central square defect in the presence of an external perpendicular magnetic field. We considered that the inner defect edge is in contact with a thin superconducting layer at higher critical temperature and/or with a dielectric material, while the outer edge of the sample is in contact with the vacuum. We have evaluated the superconducting order parameter, magnetization and vorticity as a function of the size of the defect at the first vortex penetration field. Therefore we conclude and we are able to show that circular geometry of the vortices near to the defect is mildly modified by the enhanced superconductivity at the edge of the hole.

*Keywords*: Ginzburg-Landau; Superconducting; Mesoscopics; Square hole.

**Resumen **Se investiga el estado de vórtices en un cilindro largo de sección transversal cuadrada con un defecto cuadrado central, en presencia de un campo magnético externo aplicado perpendicular a su superficie. Consideramos que el borde del defecto está en contacto con una pequeña capa de material superconductor, a mayor temperatura crítica y/o con un material dieléctrico, mientras que el borde externo de la muestra está en contacto con el vacio. Evaluamos el parámetro de orden superconductor, magnetización y vorticidad como función del tamaño del defecto en el campo de penetración del primer vórtice. Mostramos que la geometría circular de los vórtices cerca al defecto es levemente modificada por el aumento de la superconductividad en los bordes del defecto.

*Palabras clave*: Ginzburg-Landau, Superconductor, Mesoscópicos, hueco cuadrado.

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**1. Introduction**

It is known that the properties of a mesoscopic superconductor are largely influenced by the boundary conditions, the geometry of the sample and by structural defects, therefore simple and complicated loop structures and networks have been theoretically [1-4] and experimentally studied [5] in some of these works, the authors found that in addition to the conventional vortex structures at the matching fields, a variety of vortex states can be stabilized by decreasing the pinning strength of the antidots, also when an antidot array is present the critical temperature is enhanced compared to a non patterned sample and distinct cusps in the phase boundary are found for different matching fields. Also, several authors report experimental results on the synthesis, the structural characterization, the ferroelectric behavior and the electronic properties of complex high temperature superconductors, the results reveal that the perovskite used, crystallizes in a rhomboidal structure, and has a ferroelectric hysteretic behavior at room temperature [6]. Two band or multi-band mesoscopics superconductors [7-10] and fractional vortices [11], present new and very interesting topics for theoretical and experimental study. In previous works, using the Ginzburg Landau formalism, we studied the effect of trench, holes, barrier and boundary conditions on the vortex configurations in circular and square geometries, we found that the lower and upper critical fields are independent of the geometry of the defect, and depend strongly on the boundary conditions [12-14]. In this paper we analyze the superconducting state in a long mesoscopic square cylinder with a central square defect in presence of an external magnetic field applied perpendicularly to its surface at the first vortex penetration field. We calculate magnetization, supercurrent, order parameter and vorticity for two different internal boundary conditions (superconducting/superconducting at higher critical temperature interface) and (superconducting/dielectric interface). We found that the first vortex penetration field does not depends on the size of the defects and that circular geometry of the vortices near to the defect is mildly modified by the enhanced superconductivity at the edge of the hole.

**2. Theoretical Formalism**

We take the order parameter and the local magnetic field invariant along the -direction. The time evolution was incorporated into the Ginzburg Landau equations in such a manner that their gauge invariance is preserved. Superconducting state is described in the time dependent Ginzburg-Landau theory (TDGL) by the order parameter that describe the superconducting electron density and the potential vector **A **related to the magnetic induction by ** **Also we take the case for electrical potential zero, the TDGL takes the form [13-15]:

Here, , is the normal current, and is the supercurrent. Eqs. (1) and (2) were rescaled as follows: in units of lengths in units of , the external applied magnetic field in units of in units of , temperatures in units of The dynamical equations are complemented with the appropriate boundary conditions for the order parameter:

is the de Gennes parameter and is the unity vector perpendicular to the surface of the superconductor. In this paper we analyze the superconducting state of a long mesoscopic cylinder of square transverse section of area with a central square defect of area in presence of an external magnetic field applied perpendicular to its surface. We considered and for two different internal boundary conditions and. The parameters used in our numerical solution were: grid spacing, for the computational mesh, constant temperature and Ginzburg-Landau parameter We ramp up the applied magnetic field adiabatically, typically in steps of . Also, we use the following criterion to obtain the stationary state: if the highest difference , for any vertex point in the mesh, is smaller than a certain precision , then we go over the next field; usually, this test is made over some thousands of times steps, i.e., . We have worked with a precision . Although the time dependent Ginzburg-landau equations can provide all the metastable states of a fixed field, in the present work we studied only the stationary state at the first vortex penetration field .

The magnetization, where is the induction (the spatial average of the local magnetic field) is:

]]> (4)The number of vortices can be found integrating the supercurrent along a rectangle containing the superconductor. This leads us to:

(5)

Where N is the vorticity or number of vortices and is the total penetration flux.

**3. Results**

In Fig. 1 we plot (a) Magnetization curve - and (b) Vorticity as a function of the magnetic field, for a square sample with a central hole of area in contact with different materials characterized by (top) and (bottom). We can notice that the presence of the defect causes a noticeable drop of the first penetration field and leads to a qualitative change of the magnetization and vorticity curves. In the Meissner state the magnetization is a linear function of the applied field and in the Abrikosov state it has a series of jumps which indicate the nucleation of one or more vortices.

Figure 1. (Color online) (a) Magnetization curve and (b) Vorticity N as function of the magnetic field, for a square sample with a central hole of area and (top) and (bottom) . Dark and bright regions in the inset represent values of the modulus of the order parameter , superconducting (normal) state (as well as , from 0 to 1).

In Fig. 1(b) (top) we included (a) the superconducting electron density (b) the phase of order parameter and (c) the supercurrent density Js for a square sample at . We found a typical vortex configuration with vortices, they are arranged symmetrically, but it is not a stationary state, as it is well known, increasing the magnetic field the vortices goes to corners to the sample due to mutual repulsive force, forming configurations to minimize the internal energy of the system. We note also that the magnetic field for the first arrival vortices is for all the samples with the defect.

In Fig. 2 we plotted the superconducting electron density for a square sample with the inner defect edge in contact with (top) a superconducting material at higher critical temperature and (bottom) a dielectric material of area (left to right) at represent the vorticity in the hole. For we found that the first entry of vortices occurs for at for all cases. When we analyze for defects with , we have a square vortex configuration due to the geometry of the sample, with a small increase of the magnetic field, four first vortices are attracted quickly towards to the dielectric defect center forming a giant vortex with vorticity increasing the magnetic field, then four more vortices enter the sample, four sit in the hole and the other four sit in the superconductor region, although they are not visible in the contour plot of the magnitude of order parameter, also there is a change in the phase around the hole equal to (Fig. 3 (top)). The vortices inside the hole repel the vortices in the superconductor region, repulsion increases with the increase of for a constant magnetic field for and respectively. It is interesting to note that the presence of the superconducting layer in the defect acts with a repulsion force and repels the vortices deforming its own (top)). It is possible to include a new internal surface energy circular geometry for a distance of of the sample side for and for (green line in Fig. 2 barrier due to the presence of the defect with , this barrier will be greater for smaller values of This small vortex deformation is not present in a sample with a central hole in contact with a dielectric material, even when inside the defect there are vortices (Fig. 2 (bottom)).

]]> Figure 2. (Color online) Superconducting electron density for case (top) and (bottom for area (left to right) at represent the vorticity in the hole.In Fig. 3 we plot the phase of the order parameter and supercurrent density , for two cases: (up) and (down) with area of the defect (left to right) respectively. For we found , increases slowly with the defect size. Four vortices sit in the superconducting area, there is a change in the phase around the sample equal to and around the hole equal to .

Figure 3. (Color online) Superconducting order parameter phase and supercurrent density , (up) and (down) and area (left to right) at . Dark and bright regions represent values of the phase , from 0 to 1.

**4. Conclusions**

We studied the effect of a central square defect on the thermodynamical properties of a mesoscopic superconducting cylinder solving the time dependent Ginzburg-Landau equations. Our results have shown that the lower thermodynamic field varies slowly depending on the size of the defects, and is independent of the boundary condition. For these samples the presence of the superconducting material inside the defect acts like an antipinning center. The repulsive force of the antipinning center mildly changes the circular geometry of the vortices for a distance of of the sample side. If the inner defect edge is in contact with a thin superconducting layer at a higher critical temperature, the first critical field increases with the presence of the defect and the diamagnetism of the sample increases and will be more pronounced for smaller values of the deGennes length In our opinion these findings are important for the groups exploring the superconducting state in nano-engineered materials.

**Acknowledgement**

The authors would like to thank Edson Sardella UNESP - Bauru - Brazil and Cesar Barba for their very useful discussions.

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**References**

**[1]** Moshchalkov, V. V., Gielen, L., Strunk, C., Jonckheere, R., Qiu, X., Haesendonck, C. and Bruynseraede, Y., Effect of sample topology on the critical fields of mesoscopic superconductors. Nature, 373 pp. 319-322, 1995. [ Links ]

**[2]** Berdiyorov, G. R., Yu, S. H., Xiao, X. L., Peeters, F. M., Hua, J., Imre, A., and Kwok, W. K., Effect of sample geometry on the phase boundary of a mesoscopic superconducting loop. Physical Review B, 80, 064511, 2009. [ Links ]

**[3]** Berdiyorov, G. R., Milosevic, B. J. and Peeters F. M., Stability and transition between vortex configurations in square mesoscopic samples with antidots. Physical Review B, 68, 174521, 2003. [ Links ]

**[4]** Zha, G. Q., Milosevic, M. V., Zhou, S. and Peeters F. M., Influence of impurities and surface defects on the flux-induced current in mesoscopic d-wave superconducting loops. Physical Review B, 84, 132501, 2011. [ Links ]

**[5]** Van de Vondel, J., Silhanek, A. V., Metlushko, V., Vavassori, P., Ilic, B. and Moshchalkov., V. V. Self-organized mode-locking effect in superconductor/ferromagnet hybrids. Physical Review B, 79, 054527, 2009. [ Links ]

**[6]** Landinez, D. A., Deluque, C. E. and Roa, J., Electronic, structural and ferroelectric properties of the double perovskite. DYNA, 81, (183), pp. 126-131, 2014. [ Links ]

**[7]** Geurts, R., Milosevic, M. V. and Peeters. F. M., Vortex matter in mesoscopic two-gap superconducting disks: Influence of Josephson and magnetic coupling. Physical Review B, 81, 214514, 2012. [ Links ]

**[8]** Vakaryuk, V., Stanev, V., Lee, W. C. and Levchenko, A., Fractional Quantization of the Magnetic Flux in Cylindrical Unconventional Superconductors. Physical Review Letters 111, 04700, 2013. [ Links ]

**[9]** Renosto, S. T., Consoline, H., Dos Santos, C. A. M., Aguiar, J. A., Jung, S., Vanacken, J., Moshchalkov, V .V., Fisk, Z. and Machado, A. J. S., Evidence of multiband behavior in the superconducting alloy . Physical Review B, 87, 174502, 2013. [ Links ]

**[10]** Geurts, R., Milosevic, M. V., Aguiar, J. A., and Peeters, F. M., Enhanced stability of vortex-antivortex states in two-component mesoscopic superconductors. Physical Review B, 87, 024501, 2013. [ Links ]

**[11]** Loder, F., Kampf, A. p. and Kopp, T., Topological defect-phase soliton and the pairing symmetry of a two-band superconductor: Role of the proximity effect Physical Review Letters, 109, 227003, 2012. [ Links ]

**[12]** Barba, J. J., Gonzalez J. D. and Sardella E., Superconducting State of a Disk with a Pentagonal/Hexagonal Trench/Barrier. Journal of Low Temperature Physics, 174, pp 96-113, 2014. [ Links ]

**[13]** Barba, J. J., Confinement effects on mixed state in superconducting prisms. DYNA, 78, (168), pp. 158-163, 2011. [ Links ]

**[14]** Barba, J. J., Sardella E. and Aguiar, J. A., Superconducting boundary conditions for mesoscopic circular samples Superconductor Science and Technology, 24, 015001, 2011. [ Links ]

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**S. J. Higuera-Agudelo.**Received a BSc. Physics in 2009, from the Universidad Pedagógica y Tecnológica de Tunja, Colombia, and an MSc degree in Physics in 2014, from the Universidad Nacional de Colombia, Bogotá, Colombia. Currently, she is a PhD candidate in Physics in the Universidade Federal de Pernambuco, Recife, Brasil. Her research interests include computational simulations in one and two band superconducting mesoscopic systems.

**H. Barba-Molina. **Received a BSc. Veterinary and Zootechnician surgeon in 1975, from the Universidad Nacional de Cordoba, Monteria, Colombia, a vocational guidance and occupational specialist degree in 2009 from the Universidad Francisco de Paula Santander, Cucuta, Colombia. Currently, he is a Full Professor in Veterinary and Zootechnic Faculty in the Universidad Cooperativa de Colombia, Bucaramanga. His research interests include nonconventional productions, production in minor species.

**J. J. Barba-Ortega. **Received a BSc. Physics in 2000 and an MSc degree in Physics in 2003, from the Universidad Industrial de Santander, Bucaramanga, Colombia. He reveived his PhD degree in Physics in 2007 and from 2007 to 2009, he obtained Post-doctoral experience from the Universidade Federal de Pernambuco, Recife, Brasil. Currently, he is a Full Professor in the Physics Department in the Universidad Nacional de Colombia, Bogotá. His research interests include computational simulations in superconducting mesoscopics and low dimension semiconducting systems including numerical methods.