Connectivity and superconductivity

Cover
......Page 1
Lecture Notes in Physics: Monographs 62......Page 2
Connectivity and Superconductivity......Page 4
Preface......Page 6
Table of Contents......Page 8
List of Contributors......Page 14
In the Memory of Shlomo Alexander......Page 16
1.1 Introduction......Page 18
1.2 Introduction to the Ginzburg –Landau Model......Page 19
1.3 Permanent Currents......Page 20
1.4 The Little–Parks Paradigm......Page 23
1.5 Ginzburg–Landau Model on Graphs......Page 24
1.6 The Zero Set......Page 32
1.7.1 Josephson Junctions......Page 35
References......Page 36
2.1 Introduction......Page 38
2.1.1 The Ginzburg–Landau Theory......Page 39
2.1.2 de Gennes–Alexander Linear Approach for Micronetworks......Page 41
2.1.3 Symmetry Analysis in the dGA Theory......Page 45
2.2.1 The Yin-Yang......Page 49
2.2.2 The Double Yin-Yang......Page 51
2.2.3 Wheatstone Bridge......Page 53
2.2.4 The Basic Triangular Gasket......Page 55
2.3.1 Geometrical Properties of Ladder Systems......Page 58
2.3.2 Nodal Equations and Phase Transition Boundaries......Page 61
2.3.3 Nodal Order Parameters......Page 65
2.3.4 Current Distribution near the Phase Transition......Page 67
2.3.5 Some Features of the Ladders......Page 68
2.3.6 Square and Triangular Networks......Page 70
2.3.7 Incommensurate,Fractal and Disordered Systems......Page 72
2.4 Beyond de Gennes–Alexander Theory: Inside the Phase Diagram......Page 74
References......Page 76
3.1 Zero Sets in Superconductivity......Page 78
3.2 Bifurcation from Normal Solutions.Preliminaries......Page 80
3.3 The Linear Schrödinger Operator with Magnetic Field......Page 81
3.4 The Operator K......Page 82
3.5 Bifurcation for Special Spaces......Page 84
3.6 The Toy Model......Page 85
3.7 Variation of the Ground State Energy......Page 87
3.8 Nodal Sets......Page 88
3.9 A Twofold Riemannian Covering Manifold......Page 92
3.10 About the Multiplicity......Page 94
3.11 About the Maximality at Circulation 1/2......Page 95
3.12 The Case of 3 Holes......Page 98
References......Page 99
4.1.1 Ginzburg –Landau theory......Page 102
4.1.2 Fluxoid Quantization......Page 103
4.1.3 Type-I versus Type-II Superconductivity......Page 104
4.1.4 Surface Nucleation of Superconductivity......Page 106
4.1.5 Solution of the Linear GL Equation in Cylindrical Coordinates......Page 109
4.2 Connectivity Effects in Bulk Superconductors......Page 111
4.3.1 Boundary Conditions......Page 116
4.3.2 ‘Mesoscopic’ Superconductors......Page 117
4.3.3 Nucleation of Superconductivity in a Loop......Page 118
4.3.4 Nucleation of Superconductivity in a Disk......Page 120
4.3.5 Nucleation of Superconductivity Around an Antidot......Page 122
4.3.6 Nucleation of Superconductivity in a Wedge......Page 123
4.4 Dimensional Crossover in a Loopof Finite Width......Page 124
4.5 Superconducting Micronetworks:Increasing Connectivity by Adding Loops in One Dimension......Page 129
4.6 Perforated Microsquares......Page 135
4.7 Superconducting Films with an Antidot Lattice......Page 139
4.7.1 Commensurability Effects at Low Magnetic Fields......Page 141
4.7.2 The η -enhancement in High Magnetic Fields......Page 146
Conclusions......Page 149
References......Page 150
5.1 Introduction:The Price of Connectivity......Page 153
5.2.1 One Loop......Page 155
5.3 Perturbational Approach......Page 157
5.3.2 Weakly Nonuniform Cross Section......Page 159
5.4 Rings with Finite Width......Page 162
5.4.1 Order of the Normal – Superconducting Transition......Page 164
5.4.2 Nonuniform Cross Section – Nondegenerate Perturbation......Page 168
5.4.3 Degenerate Perturbation......Page 170
5.4.4 Critical Points......Page 173
5.4.5 Vortices......Page 177
5.4.6 Inhomogeneous Fields......Page 179
5.4.7 Experimental and Numerical Evidence......Page 182
5.4.8 Josephson Junctions......Page 186
5.5 Main Conclusions......Page 187
References......Page 188
6.2 Onset of Currents -Trapped Fluxes......Page 189
6.2.1 The Transition Temperature......Page 191
6.2.2 The Case of an Annulus......Page 196
6.3 Temperature Below T_c......Page 197
6.4.3 The Case Where ε Is Small......Page 198
References......Page 201
7.1 Formulation of the Problem......Page 203
7.2 Analysis of Onset in a Disc......Page 205
7.3 Analysis of Onset in a General Bounded Domain......Page 209
7.5 Infinite Curvature – Domains with Corners......Page 212
References......Page 213
8.1 Introduction......Page 215
8.2.1 Discretization of the TDGL Equations......Page 217
8.2.2 External Boundary Conditions......Page 220
8.2.3 Boundary Conditions at Holes......Page 221
8.2.4 Discretization of the Free Energy......Page 222
8.3 Examples......Page 223
8.4 Concluding Remarks......Page 227
References......Page 228
9.1 Introduction......Page 230
9.2 Theoretical Framework for Discussing Defect Formation......Page 231
9.3 Conventional Models of Formation of Vortices and Antivortices......Page 232
9.4 A New Mechanism of Defect-Antidefect Production......Page 235
9.5 Resonant Production of Vortex-Antivortex Pairs......Page 237
9.6 Conclusion......Page 243
References......Page 244
10.1 The Free Bose Gas......Page 245
10.2 The Interacting Bose Gas......Page 246
10.3 Bose Gas with Internal Degrees of Freedom......Page 247
10.4 Cooper Pairing:s-Wave Case......Page 248
10.5 Cooper Pairs with Internal Degrees of Freedom......Page 251
References......Page 253
11.2 The Role of the Electromagnetic Potentials in Quantum Mechanics......Page 254
11.3 The Aharonov–Bohm Gedanken Experiments......Page 256
11.5 Roy ’s Theorems......Page 257
11.6 Crucial Experiments......Page 259
References......Page 261
12.1 Introduction......Page 263
12.2 The Case of a Normal-Metal/Superconductor Interface......Page 265
12.3 Superconducting Networks......Page 267
12.4 Granular Superconductors......Page 269
Further Reading......Page 270