^{1,a)}

### Abstract

The solution of the linearized Ginzburg-Landau theory describing a periodic lattice of vortex lines in type-II superconductors with high inductions and first discovered by Abrikosov is generalized to nonperiodic vortex arrangements, e.g. lattices with a vacancy, surrounded by a relaxing vortex lattice, and periodically distorted lattices that are needed in the nonlocal theory of elasticity of a vortex lattice. Generalizations to lower magnetic inductions and three-dimensional arrangements of curved vortex lines are also given. It is shown how a periodic vortex lattice can be computed for bulk superconductors and for thick and thin films in a perpendicular field for all inductions and all Ginzburg-Landau parameters .

I. INTRODUCTION

II. ABRIKOSOV’S IDEAL VORTEX LATTICE NEAR

III. DISTORTED VORTEX LATTICE NEAR

IV. VORTEX LATTICEVACANCY NEAR

V. DISTORTED VORTEX LATTICE AWAY FROM

VI. CURVED VORTICES

VII. NONLOCAL ELASTICITY OF A VORTEX LATTICE

VIII. VORTEX ARRANGEMENTS AT LOW INDUCTIONS

IX. VORTEX LATTICE SOLUTION FOR ALL AND

X. VORTEX LATTICE FOR THIN AND THICK FILMS

### Key Topics

- Rotating flows
- 75.0
- Vortex lattices
- 51.0
- Elasticity
- 26.0
- Superconductors
- 16.0
- Lattice theory
- 13.0

## Figures

Order parameter , for the ideal vortex lattice (dashed line) and for a vortex lattice with a vacancy Eq. (15), with a simple relaxation field Eq. (19) (dotted line), and with a better relaxation field that minimizes the defect energy (solid line, see text).

Order parameter , for the ideal vortex lattice (dashed line) and for a vortex lattice with a vacancy Eq. (15), with a simple relaxation field Eq. (19) (dotted line), and with a better relaxation field that minimizes the defect energy (solid line, see text).

Contour lines of the order parameter Eq. (15) of a vortex lattice with one vacancy at and complete relaxation (see solid line in Fig. 1). The vortex displacements are indicated by short bold lines between dots.

Contour lines of the order parameter Eq. (15) of a vortex lattice with one vacancy at and complete relaxation (see solid line in Fig. 1). The vortex displacements are indicated by short bold lines between dots.

The magnetization curves for a triangular vortex lattice (solid lines, numerical result), which are identical to within the line thickness with those for a square lattice. Shown are versus (upper left triangle) and versus (lower right triangle). The dots show the fit, Eq. (59), good for .

The magnetization curves for a triangular vortex lattice (solid lines, numerical result), which are identical to within the line thickness with those for a square lattice. Shown are versus (upper left triangle) and versus (lower right triangle). The dots show the fit, Eq. (59), good for .

The magnetic field and order parameter of an isolated vortex line calculated from the Ginzburg–Landau theory for GL parameters , 5, and 20. For such large the field in the vortex center is twice the applied equilibrium field, .

The magnetic field and order parameter of an isolated vortex line calculated from the Ginzburg–Landau theory for GL parameters , 5, and 20. For such large the field in the vortex center is twice the applied equilibrium field, .

Two profiles of the magnetic field and order parameter along the axis (nearest neighbor direction) for triangular vortex lattices with lattice spacings (, bold lines) and (, thin lines). The dashed line shows the magnetic field of an isolated flux line from Fig. 4—the Ginzburg–Landau theory for .

Two profiles of the magnetic field and order parameter along the axis (nearest neighbor direction) for triangular vortex lattices with lattice spacings (, bold lines) and (, thin lines). The dashed line shows the magnetic field of an isolated flux line from Fig. 4—the Ginzburg–Landau theory for .

Contour lines of and for , .

Contour lines of and for , .

The magnetic field variance of a triangular FLL for plotted in units of as (solid lines) such that the curves for all merge near . The dashed lines show the same functions divided by such that they go to a finite constant 0.172 at . All curves are plotted versus to stretch them at small values and show that they go to zero linearly. The limit for very small is shown as two dot-dash straight lines for and . The top edge 0.383 shows the usual London approximation.

The magnetic field variance of a triangular FLL for plotted in units of as (solid lines) such that the curves for all merge near . The dashed lines show the same functions divided by such that they go to a finite constant 0.172 at . All curves are plotted versus to stretch them at small values and show that they go to zero linearly. The limit for very small is shown as two dot-dash straight lines for and . The top edge 0.383 shows the usual London approximation.

The shear modulus of a triangular lattice in bulk superconductors as a function of the reduced induction for GL parameters , 0.5, 0.6, 0.7, 0.707, 0.75, 1, 1.4, 2, 3, 5, 7, 10, 100 in units of . For the shear modulus is formally negative, , though vortices and a vortex lattice are energetically not favorable in bulk type-I superconductors.

The shear modulus of a triangular lattice in bulk superconductors as a function of the reduced induction for GL parameters , 0.5, 0.6, 0.7, 0.707, 0.75, 1, 1.4, 2, 3, 5, 7, 10, 100 in units of . For the shear modulus is formally negative, , though vortices and a vortex lattice are energetically not favorable in bulk type-I superconductors.

Magnetic field lines for a superconductor film calculated from the Ginzburg–Landau theory for a triangular vortex lattice. Shown is the example , , triangular lattice with vortex spacing (unit length) , film thickness . The left-hand side shows the field lines that would apply if the field inside the film would not change near the surfaces marked by the dashed lines. The right-hand side shows the correct solution. The density of the depicted field lines is proportional to .

Magnetic field lines for a superconductor film calculated from the Ginzburg–Landau theory for a triangular vortex lattice. Shown is the example , , triangular lattice with vortex spacing (unit length) , film thickness . The left-hand side shows the field lines that would apply if the field inside the film would not change near the surfaces marked by the dashed lines. The right-hand side shows the correct solution. The density of the depicted field lines is proportional to .

Profiles of the order parameter and magnetic field for the case of Fig. 9, film thickness . The solid lines show and at the center of the film and the dashed lines at the film surfaces. The dotted line indicates the average induction equal to the applied field .

Profiles of the order parameter and magnetic field for the case of Fig. 9, film thickness . The solid lines show and at the center of the film and the dashed lines at the film surfaces. The dotted line indicates the average induction equal to the applied field .

The shear modulus of a triangular vortex lattice in films with thickness , 0.32, 0.56, 1, 1.8, 3.2, 5.6, 10, and 32 plotted versus for . This is positive, i.e. the triangular vortex lattice is stable, for sufficiently thin films or for low inductions. For the bulk for the same is reached (dot-dash line), and for the bulk in the limit is reached (dashed line).

The shear modulus of a triangular vortex lattice in films with thickness , 0.32, 0.56, 1, 1.8, 3.2, 5.6, 10, and 32 plotted versus for . This is positive, i.e. the triangular vortex lattice is stable, for sufficiently thin films or for low inductions. For the bulk for the same is reached (dot-dash line), and for the bulk in the limit is reached (dashed line).

The magnetization of infinite films of thickness , 1, 3, 10, with a triangular vortex lattice generated by a perpendicular magnetic field . The plots show versus for , 0.707, 1, 1.5.

The magnetization of infinite films of thickness , 1, 3, 10, with a triangular vortex lattice generated by a perpendicular magnetic field . The plots show versus for , 0.707, 1, 1.5.

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