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Magnetic flux quantum in a Superconducting concave/convex disk

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2014 J. Phys.: Conf. Ser. 490 012218

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Magnetic flux quantum in a Superconducting

concave/convex disk

J Barba-Ortega1, Miryam R Joya1, Edson Sardella2

1Departamento de F́ısica, Universidad Nacional de Colombia, Bogotá, Colombia.
2Departamento de F́ısica, UNESP-Universidade Estadual Paulista, Bauru-SP, Brazil.

E-mail: jjbarbao@unal.edu.co

Abstract. The vortex state in a thin mesoscopic superconducting disk with a concave/convex
surface is found theoretically. It is assumed that the outer edge of the sample is in contact with
a metallic material. This configuration decreases the Bean-Livingston surface barrier energy,
that is, it allows the vortex entry into the sample at lower magnetic field. In this work, we
solve numerically the Ginzburg Landau equations with the metal/superconducting boundary
condition using the link variable method in polar coordinates. It is shown that for a determined
value for the deGennes parameter the superconductor becomes a type I superconductor and
the irregular surface allows the vortex giant formation. The value of the thermodynamical
properties decreases with this boundary condition and kind of surface.

1. Introduction
In the last years, mesoscopic superconducting heterostructures have been studied from both
experimental and theoretical point of view. The critical superconducting parameters can be
controlled for mesoscopic samples and the vortex matter can be influenced by their geometry,
boundary conditions and structural defects. In addition, a giant vortex state can occur in very
confined geometries [1, 2]. In superconducting disks the irregularity of the surface can be
present and they can act as pinning or antipinning centers. In previous works, we studied the
effect of weak defects on the vortex configurations in a circular geometry. We found that the
vortex configurations are strongly influenced by the geometry of the defects on the sample [3, 4].
In this paper we will study the effects of a concave/convex surface with a metallic interface on
the formation of vortices and how it influences the induced superconducting current for a thin
mesoscopic disk in the presence of a perpendicular external magnetic field. To this end, we use
the time dependent Ginzburg-Landau (TDGL) equations.

2. Theoretical Formalism
Many systems in superconductivity are described by the TDGL equations. These are a set of
two equations which couples the order parameter ψ and the vector potential A, which are the
two fundamental quantities describing the superconducting state. The non-dimensional version

2nd International Conference on Mathematical Modeling in Physical Sciences 2013 IOP Publishing
Journal of Physics: Conference Series 490 (2014) 012218 doi:10.1088/1742-6596/490/1/012218

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution
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Figure 1. Layout of the studied samples. Superconducting disk with (a) concave and (b)
convex surface.

of these equations is given by [5, 6, 7]:

∂ψ

∂t
= −(i∇+ A)2ψ + (1− T )ψ(1− |ψ|2) (1)

∂A

∂t
= Js − κ2∇× h (2)

(3)

Js = (1−T )Re
[
ψ̄(−i∇−A)ψ

]
is the supercurrent density measured in units of J0 = ~c2/8πeξ,

T is temperature in units of the critical temperature TC , order parameter is in units of ψ∞ of the
Meissner state, length is in units of coherence length at zero temperature ξ(0) and fields in units
the bulk upper critical field Hc2(0); κ = λ(0)/ξ(0) is the material dependent Ginzburg-Landau
parameter. In the limit of very thin disk, the first equation 1 for the variable surface of the
sample can be rewritten as:

∂ψ

∂t
= −1

τ
(i∇ + Ae) · τ (i∇ + Ae)ψ + (1− T )ψ(1− |ψ|2).

τ is just a function which describes the variability of the surface of the sample. The magnetic
field is considered nearly uniform inside the superconductor He = ∇ × Ae, where He is the
external applied field; in polar coordinates τ(r, θ) = 1 − 0.2(r/R)2 for the concave surface and
τ(r, θ) = 0.8+0.2(r/R)2 for the convex surface, where R is the radius of the disk. We have taken
a disk of radius R = 5ξ(0) (see Fig. 1). We will assume a metallic/superconducting interface,
that is, (−i∇−A)ψ · n = (i/b)ψ|n [7].

3. Results and Discussion
The parameters used in our numerical simulations were: κ = 2.17, which is a value for to a thin
film of Nb with thickness d (assuming ξ(0) = 380 nm, Tc = 3.7K, d ≈ 60 nm), T = 0. The
largest unit cell size at the edge of disk was 0.25 × 0.25. We used the values b = 1.25ξ(0) and
b = 0.25ξ(0) for the external metallic interfaces. In Fig. 1 we plot the layout of the studied
samples, a superconducting disk with (panel (a)) concave and (panel (b)) convex surface. In
Fig. 2 we depict the magnetization −4πM (left), supercurrent density J (middle) and vorticity
N (right) and the contour plot of the order parameter |ψ| (insects) as a function of the magnetic
field for a disk with concave/convex surface and metallic interface simulated by two values of
b previously given, as a function of the magnetic field He. We can see from these figures that
the first critical field Hc1 for the nucleation of vortices dependents strongly on the boundary
conditions and slightly on the kind of surface. On the other hand, the second critical field Hc2

2nd International Conference on Mathematical Modeling in Physical Sciences 2013 IOP Publishing
Journal of Physics: Conference Series 490 (2014) 012218 doi:10.1088/1742-6596/490/1/012218

2



Figure 2. (Color online) Contour plot of magnetization, supercurrent and vorticity for a disk
with concave and convex surface and metallic interface b = 1.25ξ(0) and b = 0.25ξ(0), as a
function of the magnetic field He.

depends just of the boundary condition. We have Hc1 = 0.489, 0.513 and Hc2 = 1.086 for a
concave and convex surface, respectively, when we use b = 1.25ξ(0), and Hc1 = 0.715, 0.815
and Hc2 = 1.00 for a convex and concave surface respectively when we use b = 0.25ξ(0). In
the insects of Fig. 2 (left) we show the contour plot of the order parameter for (a) N = 0 at
He = 0.625 and (b) N = 2 at He = 0.98 for b = 0.25ξ(0), considering the stationary state
both surface geometry. In the insects of Fig. 2 (middle) we show the contour plot of the order
parameter for (a) N = 0 at He = 0.135 in a stationary state, (b) N = 3 at He = 0.488 for
b = 1.25ξ(0) in a non-stationary state and (c) N = 8 at He = 0.9 for both surfaces and b
values. Finally in the insects of Fig. 2 (right) we show the contour plot of the order parameter
for (a) N = 3 and (c,d) N = 6 for a convex surface and (b) N = 2 for a concave surface for
both values of b. Due to the boundary conditions the first three vortices enter in the sample at
lower magnetic field for b = 1.25ξ(0) case. By increasing the magnetic field three more vortices
enter into the sample one by one forming a giant vortex with vorticity N = 6. For the case of
b = 0.25ξ(0) the first two vortices enter into the sample and are attracted quickly towards to the
center of the disk forming a giant vortex with vorticity N = 2. Upon increasing the magnetic
field we have vortex transitions from N to N + 1. It is interesting to note that the presence of
the metallic interface changes the values of the first and second critical fields. Furthermore for
samples in contact with metallic material represented by b < 0.138ξ(0), no vortex can be formed
for any magnetic field and a continuous entrance of magnetic flux was observed.

4. Conclusions
We studied the effect of an irregular surface on the thermodynamical properties of a mesoscopic
superconducting disk in the presence of an external applied magnetic field by solving the time
dependent Ginzburg-Landau equations. We have taken two values for the deGennes parameter
b = 1.25 and b = 0.25 which simulate a metallic interface in a concave/convex surface. Our
results have shown that the first critical field dependent strongly on the values of b and slightly
on the kind of surface. In addition, the second critical field depends just on the values of b. We
found that for b = 0.138, the superconductor becomes a type I superconductor, since only allows
a continuous magnetic field penetration without any nucleation of vortices, even a single one.

Acknowledgements
The authors thanks Alejandro Barba for useful discussions. ES thanks FAPESP for financial
support.

2nd International Conference on Mathematical Modeling in Physical Sciences 2013 IOP Publishing
Journal of Physics: Conference Series 490 (2014) 012218 doi:10.1088/1742-6596/490/1/012218

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References
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[2] E. Sardella, and E. H. Brandt, Supercond. Sci. Technol. 23, 025015 (2010).
[3] J. Barba-Ortega, Edson Sardella, J. Albino Aguiar, Physica C, 485 107 (2013).
[4] J. Barba-Ortega, Edson Sardella, J. Albino Aguiar, Physica C, 492 1 (2013).
[5] P. G. de Gennes Superconductivity of Metals and Alloys, edit by Westview Press, (1999).
[6] M. Thinkham Introduction to Superconductivity, edit by McGraw Hill, New York, (1996).
[7] W. D. Gropp, H. G. Kaper, G. K. Leaf, D. M. Levine, M. Palumbo, and V. M. Vinokur, J. Comput. Phys.

123, 254 (1996).

2nd International Conference on Mathematical Modeling in Physical Sciences 2013 IOP Publishing
Journal of Physics: Conference Series 490 (2014) 012218 doi:10.1088/1742-6596/490/1/012218

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