Variational calculations of vortex structures in bulk and mesoscopic superconductors Walter Pogosov, Laboratorium voor Vaste-Stoffysica en Magnetisme, K. U. Leuven In collaboration with: A. L. Rakhmanov and K. I. Kugel, Institute for Theoretical and Applied Electrodynamics, Moscow (Russia); E. H. Brandt, Max Planck Institute for Metals Research, Stuttgart (Germany); D. S. Goluboviĉ, M. Morelle, and V. V. Moshchalkov, Laboratorium voor Vaste-Stoffysica en Magnetisme, K. U. Leuven (Belgium).

1. Introduction 2. Vortex lattice in the bulk superconductors 3. Phase diagrams of mesoscopic samples 4. Conclusions plan

1. Introduction Ginzburg-Landau equations: Energy: Ginzburg-Landau theory

Lenghscales, ksi, lamdba, type I, typeII, GL parameter coherence length – the length scale of the order parameter variation London penetration depth – the length scale of the magnetic field variation Type I superconductors: first order phase transition to the normal state at increasing field Type II superconductors: second order phase transition to the normal state via the creation of Abrikosov mixed state Ginzburg-Landau parameter

The modulus of the order parameterThe magnetic field magnetic flux quantum The structure of the Abrikosov vortex

2. Vortex lattice in bulk superonductors H c1 and H c2 – the low and the upper critical fields

, = 0.50 = Abrikosov results for low and high fields: Abrikosov results

Variational Wigner-Seitz approximation Energy: f, a, and h are the dimessionless modulus of the order parameter, vector potential, and the magnetic field.

Ginzburg-Landau equations: Boundary conditions:

Local magnetic field within the Wigner-Seitz cell: Trial function for the modulus of the order parameter:

coefficientscoefficients

Energy:

Variational parameters:

Order parameter

Cooper pairs density

Virial theorem: Magnetization:

Field dependence:

Magnetization at high values

Magnetization at low values

The solid and dotted lines show the theoretical results at = 70 and = 80. The circles and triangles give the experimental data for YBa 2 Cu 4 O 8 [J. Sok, et al., Phys. Rev. B 51, 6035 (1995)] and Nd 1.85 Ce 0.15 CuO 4– [A. Nugroho et al., Phys. Rev. B 60, (1999)], respectively. The inset shows the magnetization plotted versus the logarithm H at = 100. Experiment

3. Phase diagrams of mesoscopic samples Vortex phases in the cylindrical geometry L - vorticity Cylindrically symmetric: -Meissner state (L = 0), -single vortex state (L = 1), -giant vortex states (L > 1), Without the cylindrical symmetry: -multivortex states (L > 1).

Ginzburg-Landau equations: The de Gennes boundary condition

Model G = G b + G s

Perturbation theory:

Furrier expansion The vortex cluster with L vortices, situated in a ring: mixture of two components with k 1 = 0 and k 2 = L. The vortex cluster with L - 1 vortices in a ring and one vortex on the center: mixture of two components with k 1 = 1 and k 2 = L - 1.

Trial function:

Energy: Magnetization:

b = 1b = 2.5 b = 5 Phase diagrams

The dimensionless modulus of the order parameter inside the mesoscopic cylinder in giant vortex state with L = 3 at b = 1, R = 4.625, h e = 0.75 (a) and, R = 3.6, h e = 1 (b). Order parameter

The equilibrium magnetization of the cylinder with radius versus applied field at b = 1, = 5. Jumps in the magnetization in Fig. (a) correspond to the transitions between the states with different vorticity. Fig. (b) shows the behavior of the magnetization in the vicinity of the second order phase transition at from the multivortex state with 2 vortices to the giant vortex phase with vorticity L = 2. Solid lines correspond to the equilibrium magnetization, dashed line shows the magnetization of the metastable giant vortex phase. Magnetization:graphs1

The equilibrium field dependence of the magnetization of the cylinder with radius 4.05 at, = 5. In Fig. (b) the magnetization is plotted versus applied field near the transitions from the multivortex states to the giant vortex phases. Solid lines correspond to the equilibrium magnetization, dashed line shows the magnetization of the metastable giant vortex phase. Magnetization:graphs2

Phase boundary of mesoscopic disc with the dot on the top

4. Conclusions 1.The variational method was applied to solve the Ginzburg-Landau equations for the regular flux-line lattice. The Wigner-Seitz approximation was used. The comparison with the results of exact numerical solution of the problem revealed a good accuracy of our method. An analytical expression for the field dependence of the magnetization was proposed. W.V. Pogosov, A.L. Rakhmanov, and K.I. Kugel "Magnetization of Type-II Superconductors in the Range of Fields : Variational Method", Zh. Eksp. Teor. Fiz. 118, 676 (2000) [JETP 91, 588 (2000)]. W.V. Pogosov, K.I. Kugel, A.L. Rakhmanov, and E.H. Brandt "Approximate Ginzburg-Landau solution for the regular flux-line lattice. Circular cell method", Phys. Rev. B 64, (2001).

2.The Ginzburg-Landau equations were solved for different vortex phases in mesoscopic cylinder. A perturbation theory was used with respect to parameter 1/. The equilibrium vortex phase diagrams of the cylinder were obtained in the plane of the cylinder radius and the applied field at different values of ''extrapolation length'' b. We analyzed the evolution of the phase diagram and magnetization with change of the ''extrapolation length''. Also studied is the phase boundary of the mesoscopic disc with the magnetic dot on the top. W. V. Pogosov "Vortex Phases in Mesoscopic Cylinders With Suppressed Surface Superconductivity", Phys. Rev. B 65, (2002). D. S. Goluboviĉ, W. V. Pogosov, M. Morelle, and V. V. Moshchalkov Nucleation of superconductivity in Al mesoscopic disc with magnetic dot, submitted to Appl. Phys. Lett. (2003). 3. Variational method based on trial functions for the order parameter is a powerfool and efficient tool allowing to avoid difficulties of the srtaightforward integration of the Ginzburg-Landau equations.