DE GENNES PARAMETER DEPENDENCE ON SUPERCONDUCTING PROPERTIES OF MESOSCOPIC CIRCULAR SECTOR
DEPENDENCIA DEL PARÁMETRO DE DE GENNES SOBRE LAS PROPIEDADES SUPERCONDUCTORAS DE UN SECTOR CIRCULAR MESOSCÓPICO
JOSÉ JOSÉ BARBA ORTEGA
PhD,
Universidad Nacional de Colombia, Bogotá, jjbarbao@unal.edu.co
MIRYAM RINCÓN JOYA
PhD, Universidad
Nacional de Colombia, Bogotá, mrinconj@unal.edu.co
Received for review November 11th, 2011, accepted March 9th, 2012, final version March, 16th, 2012
ABSTRACT: We implemented an algorithm using the link variable
method to solve the time dependent Ginzburg-Landau
equation in a superconductor prism with circular geometry. The sample is
surrounded by a thin layer of another superconductor at higher critical
temperature and submitted to an external magnetic field applied perpendicular
to its plane. The boundary condition is taken into account with the de Gennes extrapolation length . We evaluate the
magnetization, vorticity, the first, and the third
critical thermodynamical fields as functions of the
external magnetic field and
parameter. We found that for these interfaces,
the third critical field
and magnetization are largely increased while
the first critical field
remains practically
constant.
KEYWORDS: de Gennes parameter, vortex configuration, magnetization
RESUMEN: Implementamos un algoritmo usando el método de variables de enlace para
resolver las ecuaciones Ginzburg Landau dependientes del tiempo en un prisma superconductor con geometría circular. La
muestra está rodeada por una pequeña capa de otro superconductor a mayor
temperatura crítica y sometida a un campo magnético externo aplicado
perpendicular a su plano. Las condiciones de contorno son tomadas en cuenta con
la longitud de extrapolación de de Gennes Evaluamos la magnetización, la vorticidad, el primero y tercer campos críticos
termodinámicos como función del campo
magnético externo y el parámetro
. Encontramos que para estas interfaces, el
tercer campo crítico termodinámico
y la magnetización aumentan grandemente
mientras el primer campo crítico
permanece prácticamente constante.
PALABRAS CLAVE: parámetro de de Gennes, configuración de vortices, magnetización
1. INTRODUCTION
The
properties and applications of superconductors are determined by their critical
parameters. By nanostructuring a superconductor, one
can modify the properties of an existing superconducting structure [1-6].
A way to modify, enhance, or suppress the properties of superconducting samples
can be realized by controlling the sample boundary conditions. Theoretically,
one can simulate different types of material by varying the de Gennes extrapolation length in the boundary conditions for the order
parameter. It is well known that the phenomenology of superconductivity can be
described by the time-dependent Ginzburg Landau (TDGL) equations [7-9]. In the present paper we use
the TDGL theory to study the magnetization, vortex
configuration, and the transition fields in a circular sector with arbitrary
shape (see Fig. 1 of [10]). We use an algorithm considering the boundary
conditions for the order parameter. Our procedure makes it possible to
generalize the algorithm to any circular geometry and
value. According to the choice of boundary
conditions, we will show that the superconductivity can be considerably
enhanced, and a new classification of type-I and type-II superconductor may
occur.
2. THEORETICAL FORMALISM
The
properties of the superconducting state are usually described by the complex
order parameter for which the absolute square value
represents the superfluid density, and the
vector potential A, which is related to the local magnetic field, as
. In
dimensionless units, the TDGL equations are given by
[11-14]:
Equations
(1) and (2) were rescaled as follows: in units
of
lengths in units of
,
in units of
in units of
,
temperatures in units of
we use
The
dynamical equations are complemented with the appropriate boundary conditions
for the order parameter:
where is the unity vector perpendicular to the
surface and directed outward the domain of the superconductor, and
is the mesh size. This domain is defined by
the internal and external radii, r and R,
respectively, and spans an angular width, which can vary from
(slit) to
(disc) (see Fig. 1 of [10]). (We will assume
that the current density normal to surface does not vanish at the interfaces.
We can show that the discrete implementation of this condition is as follows:
We unify
the boundary conditions upon introducing . For
convenience, this notation allows us to obtain a better analysis of the
results. For more details see [8].
3. RESULTS
The
parameters used in our numerical simulations were .
The area of the circular sector is for angular width
. The internal radius is
. Such that the external radius is given by
. We have
taken the length of the largest unit cell to be no larger than $
. Since
the order parameter varied most significantly over a distance
this choice
for the grid space is sufficient to pick the variations of
. We started from the Meissner state,
where
and
everywhere are taken as the initial
conditions. Then we let the time evolve until the system achieves a stationary
state. This is done by keeping the external applied magnetic field
which is
taken constant until the system achieves a stationary state. Next, we ramp up
the applied field by an amount
. The
stationary solution for
is then
used as the initial state to determine the solution for
, and so
on. Usually we started from zero field and increased
until the
superconductivity is entirely destroyed. We ramp up the applied magnetic field
adiabatically, typically in steps of
.
In Fig., 1 we determined the values of the magnetization for an external magnetic field as a function of
. At this
magnetic field, the superconducting sample is still in the Meissner state and will be more pronounced for bigger values of
. As one
can easily notice, the negative magnetization grows with
. We found
a linear behavior of
as a function of
, with
as the slope.
In Fig. 2, We present the magnetization as a function of the
external applied magnetic field for a circular sector for several values of
the
parameter.
These curves present a typical profile of a magnetization curve of a mesoscopic
superconductor and exhibit a series of discontinuities, in which each jump
signals the entrance of one or more vortices into the sample. Notice that the
number of jumps and the transition fields vary significantly with
.
Another
interesting feature is shown for the third critical field , the
critical field which marks the transition from surface superconductivity to the
normal state. In Fig. 3, we present the
phase diagram for 5 samples with
; we found
for these cases, respectively. The
superconductor/normal transition field
grows quickly with
The Cooper
pair density as a function of is shown
in Fig. 4. For these
values, we observed a power-law behavior, and
obtained slope
.
In Figs. 5 and 6, we depict for the stationary states with a vorticity equal to 3. We choose different values of the
parameter to illustrate the role played by the
boundary conditions on the properties of the interfaces. We use (a)
at
, and (b)
at
. In both cases, we have 4 vortices in the sample.
In the
superconductor/superconductor at higher critical temperature interface we see
that is strongly enhanced in a region near the
surface due to the shielding currents and the Cooper pairs originating from the
superconductor of a higher
.
4. CONCLUSIONS
In
summary, an algorithm has been devised for solving the time-dependent Ginzburg Landau equations for circular geometry and a
superconductor/superconductor at a higher critical temperature boundary
condition. We have presented some evidence that, if we choose the parameter accordingly, we can strongly enhance
superconductivity. Also, we found an analytical behavior for the magnetization
curve as a function of the
parameters; we obtained
,
and
for this
sample.
ACKNOWLEDGEMENTS
The authors would like to thank Edson Sardella from UNESP-Brasil and LSMA from UFPE-Brasil for their very useful discussions.
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