The structure of the vortex states in a square mesoscopic superconductor is analyzed in detail using the numerical simulation within the time-dependent Ginzburg-Landau ͑TDGL͒ theory. Various vortex states ͑vortices, vortex molecules, multiquanta vortices͒ are observed and the magnetization curve is obtained. Different changes in vortex structures are identified with the peculiarities on the magnetization curve. Stability of a state consisting of vortices and antivortices is discussed. There has

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... been an exciting development 1 in the study of magnetic properties of mesoscopic superconductors initiated in pioneering works. 2-5 Small few fluxoid superconductors ͑FFS͒ reveal the exotic vortex formations-multiquanta vortices and vortex molecules-that can transform one into another via first or second order phase transitions. These exotic states appearing in small coherence length-size samples 2-5 are due to screening currents pushing vortices to the center of the sample. Thus one should expect the resulting vortex configurations to be very sensitive to the geometry of the sample. Indeed, Chibotaru et al. 1 showed that the conflict between, for example, the fourfold symmetry of the square sample and the three-quanta vortex configuration may result in the appearance of complex vortex-antivortex configurations ͑four vortices ϩ one centered antivortex͒ in the vicinity of upper critical magnetic field. While numerical studies of the vortex states in mesoscopic superconductors explained fairly well many of the observed features of FFS of different geometries, several fundamental questions, in particular, the mechanism of multiquanta vortex dissociation and the role of symmetry effects in formation of particular vortex configurations remain open. In our work we develop a regular numerical description of the vortex state of FFS, based on the timedependent Ginzburg-Landau ͑TDGL͒ theory and outline the symmetry approach to analytical studies of vortex molecules and complexes. Our simulations provide a possibility for detailed investigation of the nonlinear regime at low fields ͑far below the upper critical field͒. A large number of possible metastable states in the system is known to result in a strong dependence of vortex configurations on the initial conditions. In this case the evolution of the system with a change in magnetic field can be extremely sensitive to the details of the dynamic model. The use of the gauge-invariant timedependent approach allows us to control the dynamics of phase transitions between the vortex states, which is important for the analysis of the magnetization curve in realistic experimental conditions. In particular, our TDGL simulations allowed us to visualize the changes in the vortex arrangement and obtain different stationary vortex states in a square superconductor, and enables us to identify modifications in vortex arrangement with the peculiarities on the magnetization curve and its derivatives. Finally, we discuss a possibility of coexistence of vortices and antivortices in mesoscopic superconductors. The model. We investigate the structure of the vortex states ͑such as separated vortices, vortex molecules and multiquanta vortices͒ in a mesoscopic square superconductor, using the free energy functional for the order parameter ͑OP͒: where a(T)ϭ␣(TϪT c ), T c is the critical temperature, m* is the effective mass of electron, ⌽ 0 ϭបc/e is the flux quantum. Hereafter we use the following dimensionless units: ͉⌿ 0 ͉ϭͱ͉a͉/b for the order parameter ͑i.e., ͉⌿͉ϭ1 in a bulk superconductor with Hϭ0), ϭͱប 2 /(4m*͉a͉) for the length, 4ea 2 /(បb) for the current density ͓i.e., the dimensionless depairing current density j c ϭ2/(3ͱ3)͔, ⌽ 0 /(2) for the A field ͓i.e., the unit of magnetic field and local magnetization is H c2 ϭ⌽ 0 /(2 2 )͔. The dimensionless form for the superconducting current density is jϭIm͓⌿*ٌ͑ϪiA͒⌿͔. The magnetic field is perpendicular to the sample. We consider the sample of the size LӶ e f f ( e f f is the effective penetration depth of magnetic field͒. This allows to neglect the contributions to the magnetic field from supercurrents and thus to exclude Maxwell equations. Calculations. We calculate the OP distribution from the TDGL equations sequentially for different values of the magnetic field starting with Hϭ0 with a step ⌬Hϭ0.01. The dimensionless TDGL equations read u ͩ ‫ץ‬ ‫ץ‬t ϩi⌽ ͪ ⌿ϭ⌿Ϫ͉⌿͉ 2 ⌿ϩٌ͑ϪiA͒ 2 ⌿, Hereafter we use the units ប 2 n b/(8e 2 a 2 2 ) for time and 4ea 2 2 /(ប n b) for electric potential. Here n is the normal conductivity, u is the dimensionless characteristic time scale of the TDGL theory. The Laplace equation for electric po-