Solving numerically the 3D non linear Ginzburg-Landau (GL) equations, we study equilibrium and nonequilibrium phase transitions between different superconducting states of mesoscopic disks which are thinner than the coherence length and the penetration depth. We have found a smooth transition from a multi-vortex superconducting state to a giant vortex state with increasing both the disk thickness and the magnetic field. A vortex phase diagram is obtained which shows, as function of the magnetic field, a re-entrant behavior between the multi-vortex and the giant vortex state. PACS number(s): 74.24. Ha, 74.60.Ec, 73.20.Dx Recently, mesoscopic superconductivity has attracted much attention in view of phase transitions in confined systems with sizes comparable to the coherence (ξ) and penetration (λ) lengths. While the type of bulk superconductors is only determined by the value of the GinzburgLandau parameter κ = λ/ξ, the experimental observations [1] and the numerical simulations [2,3] of magnetization of mesoscopic thin disks have shown that the type and the order of those transitions between different superconducting states and between the superconducting and the normal state depends crucially on the disk radius R and the thickness d. With increasing the disk radius the second-order reversible phase transition observed for small disk radii are replaced by first-order transitions with a jump in the magnetization. In previous theoretical investigations of superconductivity in such mesoscopic disks [2,3] only the giant vortex states with fixed total angular momentum L were considered with an axially symmetric order parameter. It is well known [4] that for type-II superconductors (κ > 1/ √ 2), the triangular Abrikosov vortex lattice is energetically favorable in the range H c1 < H < H c2 . Since the effective London penetration depth Λ = λ 2 /d increases considerably in thin disks with d ≪ λ one would expect the appearance of the Abrikosov multi-vortex state even in disks made from a material with κ < 1/ √ 2, like e.g. the Al disks studied in Refs. [2,3]. By analogy with classical particles confined by an external potential [5], the structure of a finite number of vortices should differ from a simple triangular arrangement and allow for different metastable states. Using the London approximation Fetter [6] calculated the critical field H c1 for flux penetration into a disk. For a superconducting cylinder the multi-vortex clusters, containing up to four vortices, were simulated by Bobel [7]. Using the method of images and the London approximation, Buzdin and Brison [8] have considered vortex structures in small R ≪ Λ disks and found a classical particle ringlike arrangement [5] of vortices. In the present Letter we study the transition from the giant vortex state to this multi-vortex configuration for thin superconducting disks within the nonlinear Ginzburg-Landau (GL) theory.We consider a superconducting disk immersed in an insulator media with a perpendicular uniform magnetic field H 0 . For thin disks (d ≪ ξ...

We present a study of the spectral properties like the energy spectrum, the eigenmodes and density of states of a classical finite system of twodimensional (2D) charged particles which are confined by a quadratic potential. Using the method of Newton optimization we obtain the ground state and the metastable states. For a given configuration the eigenvectors and eigenfrequencies for the normal modes are obtained using the Householder diagonalization technique for the dynamical matrix whose elements are the second derivative of the potential energy. For small clusters the lowest excitation corresponds to an intershell rotation. The energy barrier for such rotations is calculated. For large clusters the lowest excitation consists of a vortex/anti-vortex pair. The Lindeman melting criterion is used to calculate the order-disorder transition temperature for intershell rotation and intershell diffusion. The value of the transition temperature at which intershell rotation becomes possible depends very much on the configuration of the cluster, i.e. the distribution of the particles between the different shells. Magic numbers are associated to clusters which are most stable against intershell rotation.The specific heat of the cluster is also calculated using the Monte-Carlo technique which we compare with an analytical calculation where effects due to 1 anharmonicity are incorporated.

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Schweigert, Schweigert, and Peeters Reply: Rinn and Maass [1] claim that the Brownian dynamics (BD) simulation results of Schweigert et al. [2] are analyzed incorrectly. Furthermore, they claim that the definition for the intershell diffusion coefficient ͑D u ͒ used by Schweigert et al. [2] makes sense only when the particles remain in the same shell.The whole misunderstanding is based on the fact that Rinn et al. [1] believe that one needs to follow the trajectory of each individual particle in order to calculate D u . Within such an approach, one is in trouble when a particle jumps from one shell to another shell. In order to remove this switching of particles between shells they analyzed their data in two different ways: (1) ignoring shell jumps (open symbols in Fig. 2 of Ref. [1]); and (2) taking care of shell jumps (solid symbols in Fig. 2 of Ref. [1]).When Rinn et al. ignored shell jumps they found a very large "unrealistic" reduction of D u with decreasing G , 20. Notice that for G , 20 the diffusion coefficient D u attains values which are even smaller than in the G . 100 region, where the rigid crystal phase sets in. On physical grounds, this makes no sense.In their second approach, Rinn et al. calculated D u by "taking care of shell jumps." It is not clear what they mean with this and how they calculated D u . Did they remove the particles which performed a shell jump from their calculation of D u ? If so, it is not surprising that the results are different from those of Schweigert et al. [2]. As explained in Refs. [2,3], the radial fluctuations (and shell jumps) are essential for the stabilization of the intershell (or angular) diffusion in the reentrant region. The numerical results of Rinn et al. (solid symbols in Fig. 2 of Ref. [1]) saturate for G , 20 which is hard to understand physically.

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The plasma crystal formed by monodisperse particles trapped in the sheath of an rf discharge is known to show vertically aligned structures. Here, oscillations of the aligned particles are found below a threshold value of gas density as a precursor of the melting transition. Attractive forces due to the formation of a positive space-charge region below the upper particle are calculated from Monte Carlo simulations of ion trajectories in the sheath. The alignment as well as the oscillations of the plasma crystal are explained by a simple model based on the asymmetry of the forces. ͓S1063-651X͑96͒50307-7͔ PACS number͑s͒: 52.25. Vy, 52.35.Ϫg, 62.30.ϩd The formation of Wigner crystals in dusty plasmas has attracted much interest very recently. Dust particles immersed in a plasma interact by means of the Coulomb repulsion of the particle's charges acquired by electron and ion currents. Ikezi ͓1͔ theoretically predicted plasma conditions under which these particles should form regular lattices, the so-called plasma crystal. Experimentally these crystals were found by Chu and co-workers ͓2-4͔ in a magnetron rf discharge with trapped discharge-grown SiO 2 particles. Thomas et al. ͓5͔ and Melzer et al. ͓6,7͔ found plasma crystals in parallel plate rf discharges where dust particles intentionally added to the plasma are trapped in the sheath of the lower electrode, where mainly the upward-directed field force balances the gravitational force on the particles. The dust grains arrange in a flat crystal with a diameter of a few hundred interparticle distances and a thickness of up to 20 layers, with usual two-dimensional ͑2D͒ hexagonal order in the plane. In the vertical direction the particles are found to be aligned ͓4,7,8͔.In this paper we show that the alignment can be explained by nonreciprocal attractive forces on the particles due to ion streaming motion. These forces overcome the dust Coulomb repulsion. They are also responsible for the onset of particle oscillations about the aligned positions, which are compared with experimental findings.The measurements were performed in a parallel plate rf discharge at 13.56 MHz and a power input of 12 W with the lower electrode powered and the upper grounded. The discharge was operated in helium at pressures ranging from 30 to 150 Pa. Monodisperse spherical melamine/formaldehyde particles of 4.8 m and 9.4 m diameter were added to the plasma. The choice of different particle sizes provides a change of the gravitational force by a factor of about 8. The dust crystal is illuminated by a vertical or horizontal laser fan and is observed in scattered light with a video camera. The experimental setup has been decribed in detail in ͓7͔. The charge on the dust particles is determined from the resonance frequency in the potential well formed by the gravitational and electrical forces ͓6-8͔. The measured charge is Z Ϫ ϭ15 000 elementary charges corresponding to a surface potential of ϭϪ5 V for the 9.4-m particles (Z Ϫ ϭ3600,ϭϪ2.2 V for the 4.8-m particles͒. Figure 1 shows a side view of...

Solutions of Ginzburg-Landau equations coupled with three-dimensional Maxwell equations reveal an intriguing magnetic response of small superconducting particles, qualitatively different from the twodimensional approximation but in agreement with recent experiments. Depending on the radius and thickness, first or second order transitions are found for the normal to superconducting state. For a sufficiently large radius of the disk, several transitions in the superconducting phase are obtained which correspond to different angular momentum giant vortex states. The incorporation of the finite thickness in the calculation is crucial in order to obtain agreement with the position and the size of these jumps, and the line shape and magnitude of the magnetization curves. [S0031-9007(97)

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