Direct observation of vortex shells and magic numbers in mesoscopic superconducting disks

We have studied vortex configurations in mesoscopic superconducting disks using the Bitter decoration technique. For a broad range of vorticities L the circular geometry is found to lead to the formation of concentric shells of vortices. From images obtained on disks of different sizes in a range of magnetic fields we traced the evolution of vortex states and identified stable and metastable configurations of interacting vortices. Furthermore, the analysis of shell filling with increasing L allowed us to identify magic numbers corresponding to the appearance of consecutive new shells.     

Unconventional dynamics of vortex shells in mesoscopic superconducting corbino disks

The dynamics of vortex matter in mesoscopic superconducting Corbino disk is strongly influenced by the discrete vortex structure arranged in shells. While in previous works the vortex dynamics has been studied in large (macroscopic) and in very small mesoscopic disks (containing only few shells), in the intermediate-size regime it is much more complex and unusual, due to: (i) the competition between the vortex–vortex interaction and confinement and (ii) (in)commensurability among the vortex shells. We found that the interplay between these effects can result in a very unusual vortex dynamical behavior: (i) unconventional angular melting (i.e., propagating from the boundary, where the shear stress is minimum, towards the center) and (ii) unconventional dynamics of shells (i.e., the inversion of shell velocities with respect to the gradient driving force). This unusual behavior is found for different number of shells.     

Characteristic features of the melting of two-dimensional mesoscopic Wigner clusters

Two-dimensional Wigner microclusters in a semiconductor dot are studied. Their melting is investigated in detail and it is shown that, for typical mesoscopic clusters possessing a shell structure, melting occurs in two stages: orientational melting (rotation of the shells relative to one another) and total melting, where the shells start to overlap with one another and exchange particles. An example of a “magic” microstructure which has a triangular structure and melts in a single stage is presented. For this, the temperature dependences of various quantities characterizing cluster structure are investigated. The change in the distribution of cluster configurations over local minima of the potential energy with increasing temperature is investigated. At temperatures below the temperature of total melting, a cluster is always located near the configuration of a global minimum and, at temperatures above the temperature of complete melting, a cluster can be located with finite probability near configurations corresponding to various local minima of the potential energy. Fiz. Tverd. Tela (St. Petersburg) 41, 1499–1504 (August 1999)     

Structure, melting, and potential barriers in mesoscopic clusters of repulsive particles

This paper discusses two-dimensional mesoscopic clusters of particles that repel according to dipole, Coulomb, and logarithmic laws and are confined by an external parabolic potential. These models describe a number of physical systems, in particular, electrons in semiconductor structures or on a liquid-helium surface allowing for image forces, indirect excitons in coupled semi-conductor dots, and a small number of vortices in an island of a second-order superconductor or in superfluid helium. Two competing forms of ordering are detected in the particles in the mesoscopic clusters-the formation of a triangular lattice or of a shell structure. The temperature dependences of the potential energy, the mean-square radial and angular deviations, the radial and angular distributions of the particles, and the distribution of the particles over the local minima are studied. Melting in mesoscopic clusters occurs in two stages: at lower temperatures, there is orientation melting, from the frozen phase into a phase with rotational reorientation of “crystalline” shells with respect to each other; subsequently, a transition occurs in which the radial order disappears. Melting in dipole macroclusters occurs in a single stage. However, in Coulomb and logarithmic macroclusters, orientation melting occurs only for the outer pairs of shells. Orientation melting is also detected in three-dimensional Coulomb clusters. A connection is established between the character of the melting and the ratio of the energy barriers that describe the breakdown of the orientational and radial structure of a cluster. Zh. éksp. Teor. Fiz. 116, 2012–2037 (December 1999)     

Two-dimensional microclusters of vortices: Shell structure and melting

The melting of two-dimensional microclusters of “particles” which repel one another according to a logarithmic law and are confined by an external quadratic potential is investigated. The model describes Abrikosov vortices in a superconducting island of vortices in a rotating superfluid liquid and electrons in a semiconductor nanostructure surrounded by a low-permittivity medium. The structure of clusters and its dependence on temperature and melting are investigated. The melting of microclusters of vortices proceeds in two stages: 1. A transition from a frozen phase into a state corresponding to rotational reorientation of crystal shells relative to one another. 2. At a higher temperature, the radial order vanishes. This is connected with the fact that the barrier for rotation of the shells is much lower than the barrier for radial breakup of the shells. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 3, 268–273 (10 February 1997)     

Eigenstates of the linearlized Ginzburg–Landau equation for mesoscopic multiply-connected superconductor

Multiply-connected mesoscopic superconductors have rich structures of vortex systems that result from interference of order parameter. We studied magnetic field dependence of transition temperatures and vortex arrangements of finite sized honeycomb superconducting networks with 6-fold rotational symmetries. Near and above the lower critical field, vortices locate at center of the network. As increasing the field, vortices form a hexagon or hexagonal multi-shell structure. In higher field, order parameter damps exponentially from the central point of the network to the edge of the network.     

Structure and melting of dipole clusters

Two-dimensional clusters of particles, repelling due to dipole-dipole interactions and confined by an external parabolic potential, are considered. The model describes different physical systems, particularly electrons in semiconductor structures, or electrons above a drop of He near a metal electrode, a drop of colloid liquid etc. Two kinds of ordering are in competition in the clusters: a triangular lattice and a shell structure. The ground-state configurations corresponding to the local and global minima of the potential energy for clusters with N = 1 – 40 “particles” are calculated. The structure, the potential energy and the radial and angular r.m.s. displacements as functions of temperature are also calculated. Analysing these quantities the melting of clusters is studied. One- or two-stage melting occurs depending on the number of particles in the cluster. In the case of clusters consisting of two shells melting has two stages: at lower temperature reorientation of neighbouring shells (“orientational melting”) arises; at much higher temperatures the radial shell order disappears. In clusters consisting of more than two shells total melting occurs as a first-order one-stage transition (analogously to a dipole crystal). This is connected with the barrier of rotation being less than the barrier of interchange of particles between shells for small microclusters while the barriers are of equal order for clusters with a greater number of particles.     

Mesoscopic and macroscopic dipole clusters: Structure and phase transitions

A two dimensional (2D) classical system of dipole particles confined by a quadratic potential is studied. This system can be used as a model for rare electrons in semiconductor structures near a metal electrode, indirect excitons in coupled quantum dots etc. For clusters of N ≤ 80 particles ground state configurations and appropriate eigenfrequencies and eigenvectors for the normal modes are found. Monte-Carlo and molecular dynamic methods are used to study the order-disorder transition (the “melting” of clusters). In mesoscopic clusters (N < 37) there is a hierarchy of transitions: at lower temperatures an intershell orientational disordering of pairs of shells takes place; at higher temperatures the intershell diffusion sets in and the shell structure disappears. In “macroscopic” clusters (N > 37) an orientational “melting” of only the outer shell is possible. The most stable clusters (having both maximal lowest nonzero eigenfrequencies and maximal temperatures of total melting) are those of completed crystal shells which are concentric groups of nodes of 2D hexagonal lattice with a number of nodes placed in the center of them. The picture of disordering in clusters is compared with that in an infinite 2D dipole system. The study of the radial diffusion constant, the structure factor, the local minima distribution and other quantities shows that the melting temperature is a nonmonotonic function of the number of particles in the system. The dynamical equilibrium between “solid-like” and “orientationally disordered” forms of clusters is considered.     

Pinning-induced formation of vortex clusters and giant vortices in mesoscopic superconducting disks

Merged, or giant, multiquanta vortices (GVs) are known to appear in very small superconductors near the superconducting transition due to strong confinement of magnetic flux. Here we present evidence for a new, pinning-related, mechanism for vortex merger. Using Bitter decoration to visualize vortices in small Nb disks with varying degrees of disorder, we show that confinement in combination with strong disorder causes individual vortices to merge into clusters or even GVs well below Tc and Hc2, in contrast to well-defined shells of individual vortices found in the absence of pinning.     

A new vortex state with non-uniform vorticity in superconducting mesoscopic rings

A study is presented of the superconducting states in mesoscopic rings. On the basis of self-consistent solution of the Ginzburg-Landau equations, a new kind of vortex states with non-uniform vorticity is found for some cases to be thermodynamically more stable, than the solution with unique winding number for the whole ring. There are indications that the solution with non-uniform vorticity concerns a metastable state of a superconducting mesoscopic ring.     

Long-range nonlocal flow of vortices in narrow superconducting channels

We report a new nonlocal effect in vortex matter, where an electric current confined to a small region of a long and sufficiently narrow superconducting wire causes vortex flow at distances hundreds of intervortex separations away. The observed remote traffic of vortices is attributed to a very efficient transfer of a local strain through the one-dimensional vortex lattice (VL), even in the presence of disorder. We also observe mesoscopic fluctuations in the nonlocal vortex flow, which arise due to "traffic jams" when vortex arrangements do not match a local geometry of a superconducting channel.     

三维介观超导环的涡旋结构

在Ginzburg-Landan理论的框架下, 运用有限差分法研究了在圆环电流产生磁场下的介观超导圆环内的涡旋结构, 讨论了超导圆环尺寸和不同空间分布的磁场对涡旋形成的影响, 得到在一般超导圆环体内的基态多是巨涡旋态、而多涡旋态多以激发态形式存在的结论, 说明磁场一般从超导圆环的环孔穿过, 而很难穿过超导圆环体.     

Flow of a single magnetic vortex in a submicron-size superconducting Al disk controlled by radio-frequency currents

We report the first observation of a single-vortex flow in a mesoscopic superconductor. A flow of a single vortex is successfully controlled by an rf current superimposed on a dc current, evidence of which is provided by voltage steps in current-voltage (I-V) characteristics. Irrespective of the number of vortices confined to the disk, we unambiguously observe that when a single vortex inside the disk is driven out of the disk, another vortex enters the disk similarly to two balls colliding in billiards: only one vortex passes through the Al disk at the same time in mesoscopic systems.     

薄介观超导环中的涡旋电荷分布

本文用Ginzburg-Landau理论研究在薄介观超导环中的涡旋电荷分布．对于巨涡旋态，我们发现随着外场的增加，内半径附近的电荷会从负号变为正号．本文表明，是顺磁迈斯纳效应和抗磁迈斯纳效应的竞争决定了涡旋电荷的分布．     

磁场作用下超导圆环的涡旋演化

利用Ginzburg-Landau理论模拟在外磁场作用下超导圆环的涡旋演化,讨论了外磁场、材料参数以及圆环内外径对涡旋进入超导圆环体以及涡旋达到稳定分布的影响.研究结果表明:外磁场越大,材料参数κ越大,圆环环身越宽,则超导圆环体内容纳的涡旋就越多.当磁场较小时,涡旋只由内边界进入超导体,当磁场足够大时,涡旋则先由外边界,然后再从内边界进入超导体.     

Magnetic response of mesoscopic superconducting rings with two order parameters

The magnetic response and fluxoid transitions of superconducting aluminum rings of various sizes, deposited under conditions likely to generate a layered structure, show good agreement with a two-order-parameter Ginzburg-Landau model. For intermediate couplings, we find metastable states that have different phase winding numbers around the ring in each of the two order parameters. Those states, previously theoretically predicted, are analogous to fractional vortices in singly connected samples with two-order-parameter superconductivity. Larger coupling locks the relative phase so that the two order parameters are only manifest in the temperature dependence of the response. With increasing proximitization, this signature gradually disappears.     

Structure and melting of dipole clusters

Two-dimensional microclusters made up of particles repelled by the dipole law and confined by an external quadratic potential are considered. The model describes a number of physical systems, in particular, electrons in semiconductor structures near a metallic electrode, indirect excitons in coupled semiconductor dots etc. Two competing types of particle ordering in clusters have been revealed: formation of a triangular lattice and of a shell structure. Equilibrium configurations of clusters with N=1–40 particles are calculated. Temperature dependences of the structure, potential energy, and mean-square radial and angular displacements are studied. These characteristics are used to investigate cluster melting. Melting occurs in one or two stages, depending on N. Melting of a two-shell microcluster takes place in two stages: at low temperatures—from the frozen phase to a state with rotationally reoriented “crystalline” shells with respect to one another, followed by a transition involving breakdown of radial order. Melting in a cluster made up of a larger number of shells occurs in one stage. This is due to the fact that the potential barrier to intershell rotation is substantially lower than that to particle jumping from one shell to another for small N, and of the same order of magnitude for large N. A method is proposed for predicting the character of melting in shell clusters by comparing the potential barriers for shell rotation and intershell particle jumping. Fiz. Tverd. Tela (St. Petersburg) 40, 1379–1386 (July 1998)     

Two-dimensional Abrikosov-vortex microclusters: Shell structure and melting

Melting of two-dimensional Abrikosov-vortex microclusters in a type-II superconductor island with thickness less than the coherence length has been studied. Equilibrium configurations corresponding to local and global minima of potential energy for clusters with N=1–50 particles are calculated. The temperature dependences of the structure and of mean-square radial and angular vortex displacements are investigated. It is shown that vortex microclusters melt in two stages: first the frozen-out phase transfers to a state corresponding to rotational reorientation of crystalline shells with respect to one another, followed by a transition to a state with no radial order at a substantially higher temperature. The reason for this is that the barrier to shell rotation is significantly lower than that to radial breakdown of shells. Fiz. Tverd. Tela (St. Petersburg) 39, 1005–1010 (June 1997)     

Effects of geometrical symmetry on the vortex in mesoscopic superconductors

We first systematically study the multivortex states in mesoscopic superconductors via self-consistent Bogoliubov-de Gennes equations. Our work focuses on how the geometrical symmetry affects the penetration and arrangement of vortices in mesoscopic superconductors and find that the key parameter determining the entrance of the vortex is the current density at the hot spots on the edge of sample. Through determining the spatial distribution of hot spots, the geometrical symmetry of the superconducting sample influences the nucleation and entrance of vortices. Our results propose one possible experimental approach to control and manipulate the quantum states of mesoscopic superconductors with their topological geometries, and they can be easily generalized to the confined superfluids and Bose-Einstein condensates.     

Crystal-lattice coupling to the vortex-melting transition in YBa(2)Cu(3)O(7-delta)

Distinct discontinuities in the thermal expansion of the crystal lattice are observed at the melting transition of the vortex lattice in a naturally untwinned reversible YBa(2)Cu3O(7-delta) single crystal using high-resolution dilatometry. This coupling between the vortex transition and the crystal lattice demonstrates that the crystal lattice is more than a mere host for the vortices, and it is attributed to a strong pressure dependence of the superconducting transition temperature and thus to the condensation energy at the vortex-melting temperature.