Giant vortex structures in mesoscopic spherical type-II superconducting samples (Q446046)

The paper is devoted to the study of magnetic properties of the mesoscopic spherical type-II superconducting samples embedded in different materials. With this aim the problem is considered in the framework of the non-linear Ginzburg-Landau approach with constant magnetic field throughout the sphere. By representing the free energy in spherical coordinates and considering the axially symmetric vortex state, in which the superconducting order parameter describing a giant vortex structure is an axially symmetric function, the order parameter is defined in the form of an orthogonal spherical function expansion. The solution obeys the general de Gennes boundary condition at the inclusion surface and the unknown coefficients are defined on the base of the boundary condition at the sample surface, where the ``extrapolation length'' depends on the temperature and the type of material contacting with a superconducting inclusion. Then, the minimization of the Ginzburg-Landau energy leads to a system of nonlinear algebraic equations for the expansion coefficients. Based on these expansion coefficients, the free energy is calculated. For the giant vortex state, the states with the lowest negative energy are considered analyzing the free energy as a function of the external magnetic field for spherical inclusions of different radii. This allows to present the dynamics of appearance and evolution of a giant vortex and the determination of the free energy features. Two cases of boundary conditions are considered numerically for the superconductor-vacuum interface and for a superconducting inclusion embedded in a metal matrix. It is shown that the surrounding material can determine possible types of the vortex states which can appear in the considered problem. In order to determine the upper critical field in the spherical mesoscopic superconductor, the linearized Ginzburg-Landau equation is solved in spherical coordinates with constant magnetic field throughout the sphere by repeating the above solution procedure. The equality to zero of the determinant of this system makes it possible to calculate the upper critical field depending on the inclusion radius and the type of boundary conditions. It is shown that the ``extrapolation length'' determines the number of possible vortex states which can be realized for the same inclusion radius.NEWLINENEWLINEReviewer's remark: Regrettably, it is necessary to note some careless formulations leading to difficulties in understanding. Because the parameter \(\rho\) is dimensionless, from expansion (2) and condition \(\rho=R\) on p.1300, it follows that the parameter \(R\) is also dimensionless. However, on pp. 1301--1306, it reads \(R=2\) (which is right) and \(R=2\xi\), where \(\xi\) is the coherence length (which is not right). Finally, in Fig.10, R/\(\xi\) can simply not be a dimensionless parameter.