Vortex State In The Three-dimensional Mesoscopic Superconducting Rings

Recent progress in microfabrication made it possible to manufacture three-dimensional mesoscopic superconductor. Development of computer promoted the progress of numerical simulation methods, and it led to the corresponding theoreti-cal research possible. Mesoscopic superconductor thus aroused people's great interest, and mesoscopic superconducting vortex dynamics gradually formed a frontier area of condensed matter physics.A mesoscopic sample was such that its size was comparable to the coherence lengthξand penetration depthλ. In this case the sample shape and sizes affected sig-nificantly the properties of superconductor. For example, the quantum size effect could be seen, and it was shown that many of the characteristics of mesoscopic superconduc-tors was different from the macrosopic.Main results were summarized as follows in the paper:1. The theoretical basis and the research methods In the general Landau theory of phase transitions of the second kind, the total free energy of a superconducting body included even powers of the superconducting electron wave functionψ. By applying a variational method, a pair of coupled differential equations forψ(r→) and the vector potential A(r→) and the boundary conditions were obtained. With the Ginzburg-Landau equations, we discussed the physical meaning of the superconducting characteristic parameters. Then we solved the first Ginzburg-Landau equation by numerical simulation method. Namely, Discrete linearized first Ginzburg-Landau equation were obtained by using a finite-difference technique. The eigenvalues and eigenfunctions were evaluated numerically by solving the linearized first Ginzburg-Landau equation. With the eigenvalues and eigenfunctions we further obtained the minimum free energy of system and the Corresponding superconducting wave function.2. The vortex state properties of three-dimensional mesoscopic superconductors We studied the vortex state properties from two cases. In the first case, the Cooper-pair density was axially symmetric inside the sample such as the Meiss-ner state and the giant vortex state (GVS). This case existed in the small ring. The characteristic of giant vortex was investigated with the Cooper-pair density and phase of the order parameter. Through eigenvalues and free energy, we stud-ied the stable and metastable state of giant vortex, and the relationship between the maximum vorticity and size of the ring. We got the phase diagram for gi-ant vortex states as the inner radius and ring radius. Our research showed that: When the ring radius r fixed, The Meissner state existed only in the ring of small inner ring radius. Despite the existence of the middle ring hole, the magnetic field could not penetrate the middle circle hole of ring. The superconducting proximity effect was revealed in three-dimensional mesoscopic ring. In addition, the well-known Little-Parks oscillations occurred for large inner ring radius R and small ring radius r. In the second case, the axially was broken and a vor-tex cluster was formed inside the sample. It was called as the multivortex state (MVS), and this state usually appeared mainly at larger sample size. The stable multivortex state might exist as the ground states. As the magnetic field changed, multi-vortex state was manifested mainly paramagnetic effect in low field region, meanwhile multi-vortex state could show diamagnetic effect in high-field region.3. The phase transition between vortex states and the intermittent superconductivity The phase transitions between giant vortex and multivortex state was studied through the saddle point state. Due to the presence of the surface barrier, the changes from L state to the L+1 states didn't occur in the magnetic field of the two intersection points of free energy curve, but in the larger magnetic field. Sim-ilarly, state L+1 to state L was also in the less magnetic field. When the inner ring radius R and ring r were relatively small (about 0.5 or so), with the magnetic field increasing, the superconducting state and normal state would cyclically ap-pear. Namely, It was called as intermittent superconductivity. The dimension phase diagram with the maximum vorticity and existing normal state between two superconducting states was given. There area of intermittent superconduc-tivity showed jagged. And it was important that the point of intersection of the curves of intermittent superconductivity was in the (0; 0.5ξ). This could be well understood by London Limit. Further studies had shown that those area that had not originally had intermittent superconductivity occurred intermittent supercon-ductivity in turn 0-1,1-2,…with the surface enhancement of superconductivity. Furthermore, the field range of the normal state became increasing.4. Time-dependent Ginzburg-Landau equation With variational free energy, a simplified Ginzburg-Landau equations with a single component or two-component and boundary conditions was given. By the time-dependent Ginzburg-Landau equation, we simulated how vortices en-tered and settled in stable arrangements. By solving a single component Time-dependent Ginzburg-Landau equations, we found that there existed the multi-vortex state in the wide ring, while the giant vortex state in the narrow ring. In addition, with two-component time-dependent Ginzburg-Landau equations, char-acteristics and spatial distribution of the s wave and d wave and the relationship between s wave and d wave in Dual-energy band structure of the samples were obtained in the external magnetic field. When the sample temperature was be-tween Critical temperature of s wave and d wave, one was in the superconducting state and the other in quasi-particle state.