Advances in Mathematical Physics

Volume 2018, Article ID 8652151, 19 pages

https://doi.org/10.1155/2018/8652151

## Vortex Structures inside Spherical Mesoscopic Superconductor Plus Magnetic Dipole

Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, USA

Correspondence should be addressed to A. Ludu; ude.uare@audul

Received 30 July 2018; Accepted 11 October 2018; Published 1 November 2018

Academic Editor: Alexander Iomin

Copyright © 2018 A. Ludu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We investigate the existence of multivortex states in a superconducting mesoscopic sphere with a magnetic dipole placed at the center. We obtain analytic solutions for the order parameter inside the sphere through the linearized Ginzburg-Landau (GL) model, coupled with mixed boundary conditions, and under regularity conditions and decoupling coordinates approximation. The solutions of the linear GL equation are obtained in terms of Heun double confluent functions, in dipole coordinates symmetry. The analyticity of the solutions and the associated eigenproblem are discussed thoroughly. We minimize the free energy for the fully nonlinear GL system by using linear combinations of linear analytic solutions, and we provide the conditions of occurring multivortex states. The results are not restricted to the particular spherical geometry, since the present formalism can be extended for large samples, up to infinite superconducting space plus magnetic dipole.

#### 1. Introduction

The rapidly growing field of quantum computation requires nanoscale miniaturization of electronic circuits, way beyond the silicon era type of devices [1]. Therefore, the mesoscopic superconductors, having the size comparable to the coherence length or the magnetic penetration depth [2], are the prime candidates for construction of nanodevices among all other superconducting systems. Mesoscopic physics revealed a number of open fundamental problems like quantum confinement, quantum vortices and loops, spintronics, quantum dots, etc. [3] whose solutions can bring significant knowledge in fields like nanotechnology, synthesis of new materials, novel sensors, modern lithography, or molecular biology [4–6].

The most important feature of a mesoscopic superconductor is that its shape and size have an important effect on the interplay of the magnetic field and superconducting condensate. The properties of mesoscopic superconductors are very different compared to those of bulk superconductors. While in bulk superconductors penetrating vortices form a lattice due to the vortex-vortex repulsion, in mesoscopic superconductors there is a competition between the vortex lattice and the boundary which tries to impose its geometry on the vortex lattice. It is observed experimentally that flux quantum configurations have the same symmetry as the symmetry of the shape of the sample in a homogeneous magnetic field [7]. Such systems are studied via the Ginzburg-Landau (GL) model. The GL equations arise from the Euler-Lagrange equations for the free Gibbs energy for a mesoscopic superconductor sample in magnetic field. These equations must be solved under specific boundary conditions: the normal component of the superconducting current is equal to zero [8]. Near and below the transition temperature, theoretical calculations show that anti-vortex and giant-vortex can appear in order to maintain the symmetric vortex configuration.

The response of mesoscopic superconductor samples of different shapes (thin discs, spheres, cones, and rings) to an external magnetic field, as well as the effect of the geometry, has been theoretically [9–18] and experimentally [7] studied. In all these cases a constant external magnetic field is applied along the the revolution axis. The small volume to surface ratio of these mesoscopic structures brings new features not found in the bulk: there are two kinds of superconducting states, depending on the strength of the magnetic field, the sample geometry (external surface), and its size: giant and multivortex states. The giant vortex state has cylindrical symmetry and is the only kind stable in small size superconductors due to the confinement effect [13, 14]. If the size of the sample increases, such giant vortex states can break up into multivortices through saddle-point transitions [17, 18]. For three-dimensional objects (sphere or cone) the vortex lines need to intersect the surface perpendicularly in order to cancel the outward supercurrent component [9–12]. Consequently, the shape of the lines is strongly affected by the sample surface. For example, in the case of a mesoscopic sphere placed in uniform external magnetic field, the vortex lines are curved inside, packing denser in the equatorial plane, and spreading out towards the poles [10, 11].

In this paper we consider a different situation where the magnetic field is not anymore externally generated but is generated from inside the sample,* e.g.*, an infinitesimal magnetic dipole placed at the center of a mesoscopic sphere. Such a configuration can generate confined vortex loops. The topological transition between open and closed vortex loops is controlled by the geometry,* i.e.*, , and the central magnetic dipole strength.

The goal of this paper is to demonstrate the occurrence of multivortex structures in the superconducting sphere plus magnetic dipole configuration, especially below and around the transition temperature. Our approach is based on the GL model of free energy for a finite volume . Outside of this volume the Cooper pair density describing the superconducting phase, called the order parameter, is zero [5, 6, 9–18]. The free energy functional is given in the GL model by [10, 11]where is the Cooper pairs mass, is the quantum momentum operator in the presence of magnetic field, and is the intensity of the magnetic field. The temperature dependent coefficient function and the nonlinear term coupling constant are typical Landau second-order phase transition parameters [9–12]. For mesoscopic samples, one can neglect the term responsible of the expulsion of magnetic flux from the superconductor, that is last term in Eq. (1) [9–12].

The traditional procedure for finding by minimizing the free energy functional Eq. (1) for constant volume consists in expanding the order parameter in a basis of eigenfunctions of the corresponding linearized GL problem [10, 11, 16–18] and numerically evaluating the expansion coefficients which minimize the full nonlinear functional Eq. (1).

In this paper we solve the GL linear problem analytically and investigate the properties of the eigenfunctions and spectrum. The contribution of the infinitesimal magnetic dipole will be approached in the dipole system of coordinates. The linearized GL equation factorizes in two ordinary differential equations, for the two orthogonal dipole coordinates, respectively. From the physical point of view such a factorization seems natural because far enough from the sphere surface, the abstract surfaces containing vortex lines follow the magnetic field stream lines, but these are exactly the dipole magnetic field lines. Along these surfaces the order parameter has slow variation or is practically constant. This gives sufficient physical reason for neglecting higher order terms in the dipole variable going along the magnetic field lines. This approximation allows integrating the two ordinary differential equations. The linear solution consists in a product of angular momentum eigenfunctions in the azymuthal coordinate, exponential function for one of the dipole coordinates, and a double confluent Heun function in the third coordinate. The final step is to come back to the fully nonlinear GL problem, and write as a linear combination of eigenfunctions of the linear GL problem, with arbitrary coefficients, and then minimize the free energy with respect to these coefficients.

We dedicate a large part of the present calculations to solve exactly the linearized GL equation and to ensure the completeness and orthogonality of the linear basis because near and below the transition temperature, even the linear GL equation is sufficient to describe multivortex states. The order parameter is very small in this range, and higher order terms of can be neglected. Nevertheless, at lower temperatures, the vortex configuration does not have to match the symmetry of the system, and higher order terms of GL equation cannot be negligible [19]. It is the contribution of these nonlinear terms which generates the multivortex states at lower temperatures.

The paper is organized as follows. In Section 2 we formulate the nonlinear and auxiliary linear GL problem and write the partial differential equation associated with the GL problem. In Section 3 we discuss the importance of the infinitesimal central magnetic dipole from a potential aspect, introduce the dipole coordinates, and obtain general form of the dipole equation in azimuthal symmetry plus dipole coordinates. In Section 4 we reduce the general dipole equation to a double confluent Heun equation (DCHE) by the help of a geometric approximation and a decoupling of the dipole orthogonal modes. We obtain analytic solutions for the DCHE as Heun series around the point at infinity, and we present some examples. In Section 5 we show how the dipole equation plus the physical boundary conditions can be brought to a Sturm-Liouville problem, and we solve the associated GL linear eigenproblem and find the eigenvalues spectrum. Examples of spectra for different configurations are presented. In Section 6 we describe how one can use the exact solutions and spectra of the linear GL problem to build approximate solutions for full nonlinear GL problem, by minimizing the free energy. We describe the procedure to identify multivortex states, give the sufficient criteria, and provide an example of equipotential surfaces with vortex structure inside the sphere.

#### 2. Ginzburg-Landau Equation for Dipole inside Sphere

In this section we obtain the governing ODEs for the GL problem given in Eq. (1) for a mesoscopic superconducting sphere in a local magnetic field produced by a infinitesimal dipole placed at its center . The Euler-Lagrange equations for the free energy functional Eq. (1), written in the absence of the last term in as explained above, are known as the GL equation [9–12]where the electromagnetic momentum is given bywhere is the vector potential, is the charge of the Cooper pair, and is the current density. The main approach for the nonlinear problem is to minimize the free energy, Eq. (1), with test functions chosen from a basis of eigenfunctions of the associated linear problem. The linearized GL equation (LGL) is obtained by neglecting the last term of Eq. (2)We rewrite Eq. (6) in the London electromagnetic gauge, , and we rescale the variables and . We follow the GL phenomenological model from [9–18] and use the temperature dependencewhere is the critical temperature in absence of magnetic field and is the coherence length. In the new variables the LGL Eq. (6) can be written in the form of a linear operator equationSince the dipole magnetic field has axial symmetry we can use any axially symmetric orthogonal coordinates in azimuth angle in the form . In this case the dependence in Eq. (8) can be separated, and has as eigenfunctions labeled by the angular momentum . This decomposes the space of solutions into subspaces parameterized by with square integrable function. The decomposition in Eq. (9) reduces Eq. (6) to a two-dimensional PDE. On each of these subspaces the operator has a particular form depending on denoted by , and for each such restricted operator we can attach an eigenvalue problemwhere is a provisional degeneracy label, which later on becomes the radial quantum number. Of course, and carry the dependence.

The vector potential associated with the infinitesimal (point-like) magnetic dipole can be expressed in a simple form in spherical coordinateswhere is the unit vector in the direction. The boundary conditions associated with this problem request the order parameter wave function to cancel at the origin and the flow at the mesoscopic volume surface to be zero (the Saint-James-de Gennes conditions) [9–12, 15–18]where represents the outward normal to the spherical surface defined by .

We can make coordinate free general statements concerning the spectrum of the eigenfunction problem in Eq. (10). Following the Leinfelder-Simader theorem [20] and taking into account that the integral over of the forth power of the magnitude of the dipole vector potential is finite, we find out that the operator in Eq. (6) is essentially self-adjoint if applied on analytic functions. Moreover, from the Miller-Simon theorem [21], we know the eigenvalue problem of such type of magnetic multipole potential vector operators is given by a positive, unbounded from above, continuous spectrum. However, given the two-point boundary conditions, Eqs. (12), we expect the reduction of this spectrum to a discrete and bounded spectrum for energy.

#### 3. The Generalized Dipole Equation

In this section we present analytic solutions for the system Eqs. (6), (8), and (10) with boundary conditions given by Eq. (12). The presence of the magnetic dipole renders the spherical or the cylindrical coordinates unsuitable to factorize the LGL Eqs. (8) and (10) because of the terms mixing inside the square of the electromagnetic momentum. Attempts to solve similar equations in spherical coordinates in the presence of an electric dipole have been made. In [22], for example, the Schrödinger equation is separable, and the authors find exact wave functions in terms of Bessel and Mathieu functions for spherical modes in the presence of a dipolar external electric field. A similar generalized Coulomb problem for a class of general Natanzon confluent potentials is exactly solved in [23] by reducing the corresponding system to confluent hypergeometric differential equations. More recently, in [24], the authors succeeded to solve the eigenvalue wave equation for an electron in the field of a molecule with an electric dipole moment by expanding the solutions of a second-order Fuchsian differential equation with regular singularities in in a series of Jacobi polynomials with “dipole polynomial” coefficients.

In all these situations separation and integration of the Schrödinger equations are possible because in spherical (cylindrical) coordinates the resulting second-order ordinary linear differential equations are of Fuchsian type, with maximum three regular singularities. Such equations can be mapped into several types of hypergeometric differential equations.

However, the presence of the dipole magnetic field makes the situation more complicated because the order of singularity growths above the hypergeometric one, and consequently the dynamics is governed by differential equations of Heun, Hill, or Riemann type [25–27]. A similar situation occurs in the case of a charged particle moving on a sphere under a radial magnetic field and Coulomb force. The corresponding Schrödinger equation is transformed into a Heun equation in canonical form, and exact solutions are obtained in terms of series of hypergeometric functions. The separation of variables is still possible since the vector potential depends only on one spherical variable, in this case [28]. Another example of the same type of difficulty is the two-dimensional case of interaction of three particles with a perpendicular magnetic field [29]. Here the Schrödinger equations map into biconfluent Heun’s equations, too, through the higher order singularities induced by the Coulombian interaction between particles. Similar problems related to magnetic field (finite-gap potentials, Fokker-Planck, central two-point connection, generalized central potentials up to order , Hawking radiation, etc.) were approached in the literature and usually the resulting leading differential equation for the wave function reduces to one of Heun’s differential equations [30–34]. An interesting review and study on the use of the Heun’s type of differential equations as generalizations of the hypergeometric ones, in relation to supersymmetry, are given in [35], where a two-Coulomb-center problem is solved based on a self-adjoint separation of coordinates in prolate spheroidal coordinates.

Before moving to the dipole coordinates we perform a qualitative analysis of Eq. (6) in spherical coordinates. In spherical coordinates Eqs. (6) and (10) reduce to a linear Schrödinger equation in the formwhere we denoted , and the first two terms in the LHS are the radial and angular parts of the Laplace operator in spherical coordinates, respectively. We look for solutions of Eq. (13) in the subspaces described by Eq. (9), meaning that the wave functions have good quantum numbers for the angular momentum . We can write these wave functions in the formwith being an arbitrary positive integer. Eq. (13) reduces to a 2-dimensional Schrödinger equation with an effective potential in the formIn Figures 1 and 2 we present some equipotential contours of the effective potential at and . This potential has a repulsive tail that drops to zero towards infinity like . Depending on the potential generates a finite barrier in a neighborhood of the origin. The resulting effective potential valley produces a finite spectrum of energies for the linearized equation, hence a finite space of eigenfunctions available for the nonlinear GL equation. The parameter controls the barrier height. For the barrier reduces to zero, and actually the potential is almost everywhere attractive. This is normal, since without magnetic dipole the spectrum is continuous, and the only possible state is spherical isotropic without any vortex. For the barrier height increases and allows the accumulation of more eigenstates in the linear spectrum. This denser spectrum generates a higher probability of formation of open vortices that spring towards the sphere surface around the poles of the z-axis. Next to the origin the potential has an (attractive) infinite depth valley. For small dipole strength values the potential becomes pure repulsive, and for large values of the dipole strength the potential becomes almost attractive everywhere.