Superconductors in nano-sized shape are investigated. Superconductor and also electrons which form superconductivity have quantum nature. This means when size of superconductor is small, there appear quantum effects. The critical temperature T_c becomes higher than that of a bulk superconductor. In the extremely dirty nano-sized superconductor, the critical temperature becomes much higher than that of a clean nano-sized superconductor. This is because a part of superconducting electrons concentrates in a localized state and has a high density of states. However, if the superconducting state remains localized below T_c , resistivity may not be zero just below T_c . Therefore we investigate how this localized state of superconductivity develops with decreasing temperature below T_c . We show this localized state extends to whole superconductor just below T_c .

Extended three-dimensional molecular method is introduced. In this method, a pinning potential of a columnar defect is extended to include the interlayer pinning as well as intra layer pinning. Using this method, we can obtain the critical current or critical driving force as a function of vortex number or external field.

Using the Bogoliubov-de Gennes equations with an impurity potential and the finite element method, we obtain vortex structures in a square dirty superconducting plate under an external magnetic field. Size of vortex core structure depends on the size of the impurity potential. In order to clarify origins of the relations of vortex core and impurity sizes, we examine quasi-particle structures, especially bound states around the vortex core. Then we show how microscopic structure affects the vortex structure that is pinned at the impurity.

We have developed three-dimensional molecular dynamics method, in which anisotropy of a superconductor and vortex core energy are included. We have applied it to a superconductor with splayed columnar defects. We have found that under a driving force due to an external current, vortices show wandering structures, which comes from the pinning force from splayed columnar defects. Especially, for weak core energy, vortices become soft or flexible. In an isotropic superconductor, these wandering structures disappear.

We have investigated superconducting state of a type I superconducting wire solving the Ginzburg Landau equation with finite element method. When a critical current is applied to a type I superconducting wire, superconductivity becomes inhomogeneous. We find superconductivity shows oscillatory structures which are similar to London proposed intermediate state. In the simulations, we impose two types of boundary conditions, phase-fixed and vector potential-fixed. Under both boundary conditions, we have obtained similar periodic structures.

In this paper, we investigate a local density of state (LDOS) in a two-dimensional nano-structured superconductor. We solve the Bogoliubov-de Gennes equations self-consistently with the two-dimensional finite element method. In the nano-structured superconductor, the LDOS as a function of energy has many discrete peaks. Discretization of the LDOS comes from discretization of energy levels due to the quantum confinement effect in the nano-structured system. When temperature increases, a width of a peak in the LDOS is spread to a large energy range and neighbor peaks are overlapped. On the other hand, for the fixed temperature, the behavior of the LDOS is different between nano-scaled rectangular and square systems. In the nano-scaled rectangular system, when only a length of a long side increases, a contribution of the quantum confinement effect from the long side is suppressed, while the contribution from a length of a short side remains large. Then, some peaks are left in the LDOS even when the length of the long side is very large. These peaks form a periodic structure and can be regarded as gaps in a multi gap structure due to the quantum confinement effect. On the other hand, energy levels in the square system tend to arrange equally. Then, in the square system, peaks in the LDOS which exist in the rectangular system are small. Also, the period between peaks in the LDOS in the square system is smaller than that in the rectangular system.

We study three-dimensional structures of vortices in a type II superconductor under a magnetic field by solving the Ginzburg-Landau equations. First, we show under a perpendicular magnetic field, two vortices are almost parallel but bend toward the center of superconductor due to the Meissner current. Second, we show under a tilted magnetic field, two vortices enter the superconductor from edge to edge and they are parallel to the magnetic field.

We have studied vortex states in a chiral helimagnet / superconductor bilayer system numerically. We consider a two-dimensional superconductor subsystem. An effect of a chiral helimagnet on a superconductor is taken as a magnetic field, which oscillates spatially. Solving the Ginzburg-Landau equations, we obtain various vortex states. Comparing free energies, we find the most stable vortex states under only the oscillating magnetic field H-CHM and under the H-CHM and the homogeneous applied magnetic field H-appl.

In nano-structured superconductors, it is important to consider a discreteness of energy levels instead of continuous energy levels due to the quantum confinement effect. This discreteness causes an appearance of many peaks in a density of state (DOS). In this paper, the effect of the discreetness of energy level or the effect of the quantum confinement on the DOS in superconducting systems with superconductor / normal metal (S/N) interfaces are focused. In particular, in nano-sized finite systems, a local density of state (LDOS) becomes strongly spatial dependent because of non-uniform spatial distributions of a gap energy and a pair amplitude. We investigate the pair amplitude and the LDOS by solving the Bogoliubov-de Gennes equations self-consistently. The proximity effect leads to a penetration of the pair amplitude into the normal metal region. Then, the pair amplitude in the normal metal decays non-monotonously because of the effect of the nano-sized finite system. On the other hand, the spatial-averaged LDOS plots as a function of the energy have many peaks in both superconductor and normal metal regions. Also, a contribution of the normal metal to the superconductor causes an appearance of peaks of the LDOS at the energy below the gap energy. In the SNS junction, when width of the normal metal increases, these peaks at the energy below the gap energy appear clearly.

We have solved the Bogoliubov-de Gennes equation with a random impurity potential V imp ( r ) under an external magnetic field with the finite element method and obtained the local density of states and the order parameter around a vortex. From these results, we find two properties of a vortex structure in a dirty superconductor. First, the order parameter structure around the vortex is deformed by the impurity potential. Second, distributions of bound states is also deformed, but these deformations of the order parameter and bound states do not correspond with each other.

It is known that the superconducting properties are improved by adding nanorods to a superconductor. As increasing nanorods into superconductor, the pinning force becomes stronger. Using the molecular dynamics method, we investigate vortex motions in a superconductor with a nanorod array. We obtain trajectories of vortices and standard deviation of vortex positions. We find peculiar temperature dependence of vortex motion.

When a voltage is applied to coupled Josephson junctions, an ac current flows because of the Josephson effect. Due to this ac current, an electromagnetic (EM) wave is emitted from junctions. For the cuprate High-Tc superconductor, frequency of the EM wave reaches to the THz regime since the energy gap is large. In order to simulate the EM wave in the superconductors, we solve the Josephson junction relations and the Maxwell equations simultaneously. Especially, we consider coupling of junctions including capacitive and inductive couplings, and spatial variations of phase differences of order parameters inside of junctions. We shows time developments of distributions of the phase differences, magnetic and electric fields in junctions in the transient state.

We study vortex penetration into two-layer structures of superconducting plates under a perpendicular magnetic field. We solve the heat transport equation and the Maxwell equations with the current-voltage relation for superconductor, simultaneously, and obtain magnetic flux and current densities. We show how magnetic flux structure depends on the structure, especially distance of two-layer of superconductors.

It is shown that in nano-structured s-wave superconductor, transition temperature T-c depends on random impurity potentials. This means violation of the Anderson's theorem, which states that nonmagnetic impurity does not affect the T-c of s-wave superconductor. We determine the impurity effects on T-c for nano-structured superconductor, using the finite element method to solve the Bogoliubov-de Gennes equations under spatially a random impurity potential. We show that some impurity potentials increase T-c but other impurity potentials decrease T-c. We find that the superconductor with localized order parameter shows increased T-c, which is contrary to expectation. Our results show that a dirty superconductor does not always mean weak superconductivity.

Enhancement of superconducting critical temperature T-c for nano-structured square and rectangular plates is investigated, using the Gor'kov equations and the finite element method. T-c increases with decreasing system size and shows sawtooth behavior as the function of system size. We find the enhancement of T(c )becomes larger for narrower rectangular plate. We discuss mechanisms of these behaviors of T-c, investigating quasi-particle eigen-energies and spatial order parameter distributions.

Vortex lattice melting in a dirty mesoscopic square superconducting plate is studied using the molecular dynamics (MD) method. We include an impurity potential in the MD method and investigate vortex number and impurity number dependences of the melting transition temperature. We find strong vortex number dependence and weak pinning potential dependence of the melting temperature.

Superconducting critical temperature Tc for a dirty nanosized square s-wave superconducting plate becomes five times higher than that of a pure bulk superconductor. Tc is obtained by solving the Bogoliubov-de Gennes equations with the finite element method. As the dirty superconductor, we consider a superconductor with a spatially random potential for superconducting electrons. We show that the enhancement of Tc comes from cooperation of size effects and impurity effects, both of which increase the local density of the Cooper pairs.

We study vortex penetration into two-layer structures of superconducting plates under a perpendicular magnetic field. We solve the heat transport equation and the Maxwell equations with the current-voltage relation for superconductor, simultaneously, and obtain magnetic flux and current densities. We show how magnetic flux structure depends on the structure, especially distance of two-layer of superconductors.

We have investigated vortex states in two-dimensional superconductors under a oscillating magnetic field from a chiral helimagnet. We have solved the two-dimensional Ginzburg-Landau equations with finite element method. We have found that when the magnetic field from the chiral helimagnet increases, vortices appear all at once in all periodic regions. This transition is different from that under the uniform magnetic field. Under the composite magnetic field with the oscillating and uniform fields (down-vortices), vortices antiparallel to the uniform magnetic field disappear. Then, the small uniform magnetic field easily remove down-vortices.