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, email@example.com
MIRYAM RINCÓN JOYA
PhD, Universidad Nacional de Colombia, Bogotá, firstname.lastname@example.org
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
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 ). 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 ). (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 .
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 .
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.
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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