^{1,a)}

### Abstract

The theory of the critical states of a vortex lattice in type-II superconductors is examined without any assumptions about the relative perpendicularity of the local magnetic fields and circulating currents in the sample. Such a theory has made it possible to solve a number of problems for thin films of superconductors in an external magnetic field oriented perpendicular to their surface: a theory of the shaking effect is constructed for rectangular superconducting plates and the critical states in samples with anisotropic pinning of the flux lines as well as in the presence of an order-disorder phase transition in a vortex lattice are studied. In addition, the critical states in a long superconducting strip in an inclined magnetic field are investigated.

I. INTRODUCTION

A. Types of critical states in type-II superconductors

B. Thin planar superconductors

II. CRITICAL STATE IN TYPE-II SUPERCONDUCTORS OF ARBITRARY SHAPE

A. Equations of the critical state

1. Taking account of the dependence of on

2. Taking account of flux-line pinning anisotropy

B. Critical state in thin planar superconductors

III. THEORY OF VORTEX-SHAKING IN THIN PLANAR TYPE-II SUPERCONDUCTORS

A. Shaking effect in a thin rectangular superconducting plate

1. Transverse shaking effect

2. Longitudinal shaking effect

3. General case

B. Generation of a constant electric field by an acmagnetic field in a superconducting strip

1. Plate in a parallel field

2. Strip in a perpendicular field

IV. CRITICAL STATES IN THIN PLANAR TYPE-II SUPERCONDUCTORS WITH ANISOTROPIC FLUX-LINE PINNING

A. Critical state in a superconducting strip with anisotropic pinning

B. Fishtail effect in thin planar superconductors

C. Critical state in a superconducting strip with an order-disorder transition in the vortex lattice

D. Determination of critical current density anisotropy from experimental data

V. CRITICAL STATES IN A THIN SUPERCONDUCTING STRIP IN AN INCLINED MAGNETIC FIELD

A. Bean critical states

B. Critical T states with constant

### Key Topics

- Magnetic fields
- 170.0
- Superconductors
- 97.0
- Critical currents
- 49.0
- Electric fields
- 44.0
- Alternating current power transmission
- 36.0

## Figures

The functions versus and versus (solid curves), shown schematically in the plane for fixed .^{45} The points of intersection of the lines correspond to double critical states where and . The directions of the electric field for the critical T and C states are also shown.

The functions versus and versus (solid curves), shown schematically in the plane for fixed .^{45} The points of intersection of the lines correspond to double critical states where and . The directions of the electric field for the critical T and C states are also shown.

Schematic figure explaining why in an anisotropic superconductor the direction in which a vortex starts to move can differ from the direction of the driving force.^{53} The ellipse shows the angular dependence of the maximum pinning force . The dashed lines are the projections of the forces acting along on some other directions. All these projections reach the ellipse at angles different from . The thick arrow shows the minimum force of this type, . For the vortex starts to move in the direction .

Schematic figure explaining why in an anisotropic superconductor the direction in which a vortex starts to move can differ from the direction of the driving force.^{53} The ellipse shows the angular dependence of the maximum pinning force . The dashed lines are the projections of the forces acting along on some other directions. All these projections reach the ellipse at angles different from . The thick arrow shows the minimum force of this type, . For the vortex starts to move in the direction .

Magnetic flux front penetrating into a thin rectangular superconducting plate with increasing magnetic field perpendicular to the plane of the plate.^{52} The top panel shows the two-dimensional curve which forms the equator of the three-dimensional front , shown in the bottom panel.

Magnetic flux front penetrating into a thin rectangular superconducting plate with increasing magnetic field perpendicular to the plane of the plate.^{52} The top panel shows the two-dimensional curve which forms the equator of the three-dimensional front , shown in the bottom panel.

Lines of the current in a thin rectangular superconducting plate with edge ratio 1:4 and isotropic pinning, , for two values of the applied magnetic field (solid lines) and (dashed lines).^{52} The arrows which are perpendicular to the current lines and whose length is proportional to indicate the direction of the flux lines on the surface of an infinitesimally thin superconductor as well as the flux lines lying at different depth in a plate of small but finite thickness.

Lines of the current in a thin rectangular superconducting plate with edge ratio 1:4 and isotropic pinning, , for two values of the applied magnetic field (solid lines) and (dashed lines).^{52} The arrows which are perpendicular to the current lines and whose length is proportional to indicate the direction of the flux lines on the surface of an infinitesimally thin superconductor as well as the flux lines lying at different depth in a plate of small but finite thickness.

Geometry of a strip and applied magnetic fields (top inset). The flux lines, “stepping” from left to right through the section of a strip (which is regarded as an “infinite” plate) is shown at the times .^{74} The circles mark the points around which a flux line turns. Here , , which gives and ; is the shift of the flux line over a period of the ac field; is measured from an arbitrary point of the “plate.” The diagram on the right-hand panel shows the profile of the current across the thickness of the strip during the first and second half of the period.

Geometry of a strip and applied magnetic fields (top inset). The flux lines, “stepping” from left to right through the section of a strip (which is regarded as an “infinite” plate) is shown at the times .^{74} The circles mark the points around which a flux line turns. Here , , which gives and ; is the shift of the flux line over a period of the ac field; is measured from an arbitrary point of the “plate.” The diagram on the right-hand panel shows the profile of the current across the thickness of the strip during the first and second half of the period.

Shift of a flux line when the magnetic field increases from to .^{60} The thick solid lines are the projections of the flux line on the plane. As the slope of this line increases, it simultaneously shifts along from to . The arrows with the components and mark the projections of the shifts of the elements of a flux line. These arrows are perpendicular to the local currents (solid arrows) flowing in the plane at angle to the axis.

Shift of a flux line when the magnetic field increases from to .^{60} The thick solid lines are the projections of the flux line on the plane. As the slope of this line increases, it simultaneously shifts along from to . The arrows with the components and mark the projections of the shifts of the elements of a flux line. These arrows are perpendicular to the local currents (solid arrows) flowing in the plane at angle to the axis.

Direction of the currents flowing in the plane as a function of .^{60} Left-hand side: the angle between and the axis. Top: start of relaxation, . Bottom: .

Direction of the currents flowing in the plane as a function of .^{60} Left-hand side: the angle between and the axis. Top: start of relaxation, . Bottom: .

Orientation of the electric field , constructed on the basis of Eqs. (40)–(43).^{58} The deviation of the directions of from is greatest with weak currents , and the deviation vanishes when is directed along or . The length of the arrows is proportional to .

Orientation of the electric field , constructed on the basis of Eqs. (40)–(43).^{58} The deviation of the directions of from is greatest with weak currents , and the deviation vanishes when is directed along or . The length of the arrows is proportional to .

The relaxation of the magnetic moment of rectangular superconducting plates with different edge ratios: .^{58} Here and . The dashed lines correspond to the transverse and longitudinal shaking of the vortex medium.

The relaxation of the magnetic moment of rectangular superconducting plates with different edge ratios: .^{58} Here and . The dashed lines correspond to the transverse and longitudinal shaking of the vortex medium.

Profiles of the magnetic induction in a plate placed in a field and carrying a transport current for (a), (b), (c).^{61} Two limiting profiles are shown: (solid lines) and (dashed lines). In the cases (a) and (b) the central part of the profile is “frozen.” In the case (c) the magnetic flux enters the regions 1 and 2 from the left and the region 3 from the right during one-half cycle. In the other half-cycle the flux 1 exits to the left and the fluxes 2 and 3 exit to the right. Thus the flux 2 confined in the parallelogram intersects the plate from left to right.

Profiles of the magnetic induction in a plate placed in a field and carrying a transport current for (a), (b), (c).^{61} Two limiting profiles are shown: (solid lines) and (dashed lines). In the cases (a) and (b) the central part of the profile is “frozen.” In the case (c) the magnetic flux enters the regions 1 and 2 from the left and the region 3 from the right during one-half cycle. In the other half-cycle the flux 1 exits to the left and the fluxes 2 and 3 exit to the right. Thus the flux 2 confined in the parallelogram intersects the plate from left to right.

Limiting profiles of the magnetic induction in a strip placed in constant and ac fields perpendicular to the strip’s surface and carrying a transport current with ac field amplitudes (a) and (b).^{61} The solid and dashed lines correspond to and , respectively. In the case (a) the “frozen” flux region has just vanished and the flux lines once again do not intersect the strip, since the area is zero. In the case (b), during each field cycle the magnetic flux confined in the region intersects the strip from left to right. Inset: current profiles , which are identical in the cases (a) and (b).

Limiting profiles of the magnetic induction in a strip placed in constant and ac fields perpendicular to the strip’s surface and carrying a transport current with ac field amplitudes (a) and (b).^{61} The solid and dashed lines correspond to and , respectively. In the case (a) the “frozen” flux region has just vanished and the flux lines once again do not intersect the strip, since the area is zero. In the case (b), during each field cycle the magnetic flux confined in the region intersects the strip from left to right. Inset: current profiles , which are identical in the cases (a) and (b).

Magnetic field flux lines near a double ideally screened strip, shown at top-right (, , , the two thick lines in the main part of the figure), in a perpendicular magnetic field .^{61} The form of the film used in the experiments of Refs. 71 and 72 is shown schematically at the top-left ( and denote the voltage and current contacts).

Magnetic field flux lines near a double ideally screened strip, shown at top-right (, , , the two thick lines in the main part of the figure), in a perpendicular magnetic field .^{61} The form of the film used in the experiments of Refs. 71 and 72 is shown schematically at the top-left ( and denote the voltage and current contacts).

Bottom panel: model angular dependence , the relation (51), for the cases of internal pinning (solid line) and pinning by extended defects oriented parallel to the axis (dashed line).^{113} The top panel shows the corresponding dependences , the relation (52). The field and the current are measured in units of and the angle in degrees. The dependences are shown for the following cases as examples: , and , .

Bottom panel: model angular dependence , the relation (51), for the cases of internal pinning (solid line) and pinning by extended defects oriented parallel to the axis (dashed line).^{113} The top panel shows the corresponding dependences , the relation (52). The field and the current are measured in units of and the angle in degrees. The dependences are shown for the following cases as examples: , and , .

Some profiles and in a superconducting strip in the field for different dependences , Eq. (52), with the parameters and , and ∞.^{113} The quantities and are measured in units of . The dashed lines indicate the field and the point at which and . In the limit the field at increases sharply to the value and remains constant for .

Some profiles and in a superconducting strip in the field for different dependences , Eq. (52), with the parameters and , and ∞.^{113} The quantities and are measured in units of . The dashed lines indicate the field and the point at which and . In the limit the field at increases sharply to the value and remains constant for .

The curves for a superconducting disk, calculated using the relations (27), (64), and (65) for and several sets of the remaining parameters:^{117} , , (1); , , (2); , , (3); , , (4). The dashed line corresponds to the isotropic case with . Here , and is expressed in units of .

The curves for a superconducting disk, calculated using the relations (27), (64), and (65) for and several sets of the remaining parameters:^{117} , , (1); , , (2); , , (3); , , (4). The dashed line corresponds to the isotropic case with . Here , and is expressed in units of .

The distribution of the current in a thin strip with an order-disorder transition, which is described by the model (66).^{123} The lines and indicate the boundaries of the amorphous and ordered vortex phases, respectively. For there exists a mixture of two vortex phases with . The case shown corresponds to increasing external magnetic field , when .

The distribution of the current in a thin strip with an order-disorder transition, which is described by the model (66).^{123} The lines and indicate the boundaries of the amorphous and ordered vortex phases, respectively. For there exists a mixture of two vortex phases with . The case shown corresponds to increasing external magnetic field , when .

Profiles of the magnetic field and current in the strip for , and for the cases of increasing (a) and decreasing (b) .^{123}

Profiles of the magnetic field and current in the strip for , and for the cases of increasing (a) and decreasing (b) .^{123}

Magnetization loop (solid lines) for a strip in a perpendicular magnetic field (a) and for a plate in a parallel field (b) with .^{123} Note the different scales on the axis; this is because the total magnetic-flux penetration field for a strip differs strongly from that for a plate. The arrows indicate the direction of change of . The dashed lines show the first and second derivatives of the dimensionless magnetic moment with respect to . Inset: example of the magnetization half-loop for a plate.

Magnetization loop (solid lines) for a strip in a perpendicular magnetic field (a) and for a plate in a parallel field (b) with .^{123} Note the different scales on the axis; this is because the total magnetic-flux penetration field for a strip differs strongly from that for a plate. The arrows indicate the direction of change of . The dashed lines show the first and second derivatives of the dimensionless magnetic moment with respect to . Inset: example of the magnetization half-loop for a plate.

Experimental dependences in a with (○) and without (△) extended defects along the axis.^{139} The solid line with shows the dependence (75) with , ; the dashed line shows the corresponding angular dependence of the critical current density , the relation (76).^{140} The dotted line shows in the case of isotropic pinning with . The current density is measured in units of and and in units of . The remaining solid lines give the computed dependences in an inclined magnetic field with and 1 (see Sec. V).

Experimental dependences in a with (○) and without (△) extended defects along the axis.^{139} The solid line with shows the dependence (75) with , ; the dashed line shows the corresponding angular dependence of the critical current density , the relation (76).^{140} The dotted line shows in the case of isotropic pinning with . The current density is measured in units of and and in units of . The remaining solid lines give the computed dependences in an inclined magnetic field with and 1 (see Sec. V).

The curves (dashed lines), relations (76), for , ((a) maximum of for ) and for , ((b) maximum of for ).^{147} The solid lines show the corresponding with , 0.3, 0.5, 1, 2, and 5. is measured in units of and and in units of .

The curves (dashed lines), relations (76), for , ((a) maximum of for ) and for , ((b) maximum of for ).^{147} The solid lines show the corresponding with , 0.3, 0.5, 1, 2, and 5. is measured in units of and and in units of .

The profiles of (a) and the current (b) in a strip, for which is described by the relation (76) with , .^{147} The fields and are switched on in the order written (third scenario). , 0.6, and 1.2. The magnetic fields and are given in units of . The dotted, dashed, dot-dash, and solid lines show , 0.5, 1, and 2, respectively. For comparison, the solid lines with dots indicate the profiles in the isotropic case . The dotted lines also correspond to the second field-switching scenario.

The profiles of (a) and the current (b) in a strip, for which is described by the relation (76) with , .^{147} The fields and are switched on in the order written (third scenario). , 0.6, and 1.2. The magnetic fields and are given in units of . The dotted, dashed, dot-dash, and solid lines show , 0.5, 1, and 2, respectively. For comparison, the solid lines with dots indicate the profiles in the isotropic case . The dotted lines also correspond to the second field-switching scenario.

The profiles of (a) and the current (b) in a strip with anisotropic pinning, which is described by the relation (76) with , (see Fig. 20).^{147} Here , 0.6; the dotted, dashed, dot-dash, and solid lines correspond to , 0.24, 0.35, and 0.6, respectively. The magnetic fields and are given in units of . For any the profiles are practically identical to those for , while for they are close to those for . The profiles for are also almost identical to the profiles for isotropic pinning with (solid lines with dots). The profiles and are identical for all three scenarios of switching on .

The profiles of (a) and the current (b) in a strip with anisotropic pinning, which is described by the relation (76) with , (see Fig. 20).^{147} Here , 0.6; the dotted, dashed, dot-dash, and solid lines correspond to , 0.24, 0.35, and 0.6, respectively. The magnetic fields and are given in units of . For any the profiles are practically identical to those for , while for they are close to those for . The profiles for are also almost identical to the profiles for isotropic pinning with (solid lines with dots). The profiles and are identical for all three scenarios of switching on .

The current in a strip with and for different increasing values of .^{45} The magnetic fields are given in units of .

The current in a strip with and for different increasing values of .^{45} The magnetic fields are given in units of .

The magnetic moment as a function of (top) and as a function of (bottom) for , 0.1, 0.2, 0.5, 1, and 5.^{45} The dots show the fit by the expression (81). Inset: versus (solid line with circles).

The magnetic moment as a function of (top) and as a function of (bottom) for , 0.1, 0.2, 0.5, 1, and 5.^{45} The dots show the fit by the expression (81). Inset: versus (solid line with circles).

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