Line vortices in a three-dimensional ordered Josephson medium

Two equilibrium configurations of a line vortex in a three-dimensional ordered Josephson medium are considered: (i) the vortex core is at the center of a cell and (ii) the vortex core is on a contact. Infinite systems of equations describing these configurations are derived. In going to a finite system, the currents far away from the center are neglected. A new technique for solving the finite system of equations is suggested. It does not require smallness of phase discontinuities at all vortex cells and, therefore, can be applied for any values of pinning parameter I down to zero. The structures and energies of both equilibrium states for isolated line vortices are calculated for any I from the range considered. For I >0.3, a vortex can be thought of as fitting a square of 5×5 cells. For lower I, the vortex energy can be expressed as a sum of the energies of the small discrete core and the quasi-continuous outside. The core energy is comparable to the energy of the outside and is a major contributor to the vortex energy when I is not too small. For any I, the energy of the vortex centered on the contact is higher than the energy of the configuration centered at the center of the cell.     

Pinning of linear vortices and possible distances between them in a 3D ordered Josephson medium with a nonzero structural factor

On the basis of the fluxoid quantization conditions, we derive a system of equations describing the current configuration of two interacting linear vortices in a 3D ordered Josephson medium in the entire range of possible values of structural factor b. The axes of these vortices are located in the middle row of an infinite strip with a width comprising 13 meshes. We propose a method for solving this system, which makes it possible to calculate the current configurations exactly. The critical values of pinning parameter I _d are calculated, for which two linear vortices can still be kept at a distance of d meshes between their centers in the entire range of possible values of parameter b. The formula describing the I _d(b) dependences for various values of d is derived. The dependences of the maximal pinning force F on parameter I for various values of b are analyzed. It is shown that for the same value of I, larger values of b correspond to larger maximal pinning forces.     

Pinning of vortices and penetration of the magnetic field into a periodically modulated long Josephson contact

The upper field of the Meissner regime, H _up, and overheat field H′_c1, above which vortices start penetrating into a Josephson contact, are calculated throughout the range of pinning parameter I. The stability of likely configurations is investigated. It is shown that H _up = H′_c1 at any I. The existence of a single vortex centered at the extreme cell in the contact is demonstrated to be a possibility. At I > 3.69, such a vortex may exist even in a zero magnetic field. At 1.48 < I < 3.69, this vortex can exist in an external field in the range from some H _v to H _up. At I < 1.48, the vortex cannot exist under any conditions. From the equality of H _up and H′_c1 at any I, the conclusion is drawn that penetration of vortices into any Josephson medium is conditioned by the need to satisfy flux quantization conditions. Here, not the forces of vortex pinning at defects in the medium but quantization requirements are of major importance, which are satisfied in specific quantum ways rather than by meeting equilibrium conditions for vortices, forces, etc.     

Interaction and pinning of plane vortices in a three-dimensional Josephson medium and possible distances between two isolated vortices

Within a continuous vortex model, exact expressions are obtained for the Josephson and magnetic energies of plane (laminar) vortices, as well as for the energy and force of pinning by cells in a three-dimensional Josephson medium. If the porosity of the medium is taken into account, the Josephson and magnetic energies of the vortex differ from those for the continuum case. The contributions to the pinning energy from the Josephson and magnetic energies have opposite signs. An algorithm for numerically solving a system of difference equations is proposed in order to find the shape and the energy of the vortex in its stable and unstable states. The continuous vortex model is shown to fail in predicting correct values of the Josephson and magnetic energy of the vortex, as well as of the pinning energy components. Expressions for the least possible distances between two isolated vortices are obtained for a small pinning parameter. Analytical results are in close agreement with computer simulation. An algorithm for numerically solving a system of difference equations is proposed in order to find the least possible distances between two isolated vortices when the pinning parameter I is not small. The minimal value of I at which the center-to-center distance N of the vortices equals three cells is 1.428; for N=2, I _min=1.947. At I>2.907, the vortices can be centered in adjacent cells.     

Penetration of the magnetic field into a 3D ordered Josephson medium at very small values of the pinning parameter

The Meissner state of a 3D Josephson medium is analyzed for stability against small fluctuations of phase discontinuities at contacts. For any form of fluctuations, there exists value I ₀ of pinning parameter I such that the Meissner configuration remains stable if I < I ₀. Reasons why the configuration remains stable at small I are considered. Instability arises when the quadratic form of the second variation of Gibbs potential G is not a positively definite quantity. At small I, the contribution of the Josephson energy to G is small. The second variation of the magnetic energy, the other component of G, is always a positively definite quadratic form. Therefore, instability may arise only if I has a finite value. This statement holds true not only for the Meissner but also for any equilibrium configuration. At I < I ₀, stability persists up to the boundary of the Meissner state. Then, a sequence of plane vortices parallel to the boundary appears throughout the sample. Thus, vortices appearing at I < I ₀ are plane vortices rather than linear. The configurations of currents and the magnetic field profile inside the sample are calculated for I < I ₀. Calculation is based on analyzing the continuous variation of the current configuration toward a decrease in the Gibbs potential.     

Finite vortex numbers and symmetric vortex structures in a rotating trapped Fermi gas in the BCS-BEC crossover

The ground state of a three-dimensional (3D) rotating trapped superfluid Fermi gas in the BCS-BEC crossover is mapped to finite N _v -body vortex states by a simple ansatz. The total vortex energy is measured from the ground-state energy of the system in the absence of the vortices. The vortex state is stable since the vortex potential and rotation energies are attractive while the vortex kinetic energy and interaction between vortices are repulsive. By combining the analytical and numerical works for the minimal vortex energy, the 2D configurations of N _v vortices are studied by taking into account of the finite size effects both on xy-plane and on z-direction. The calculated vortex numbers as a function of the interaction strength are appropriate to the renew experimental results by Zwierlein in [High-temperature superfluidity in a ultracold Fermi gas, Ph.D. thesis, Massachusetts Institute of Technology, 2006]. The numerical results show that there exist two types of vortex structures: the trap center is occupied and unoccupied by a vortex, even in the case of N _v < 10 with regular polygon and in the case of N _v ≥ 10 with finite triangle lattice. The rotation frequency dependent vortex numbers with different interaction strengths are also discussed.     

Line vortex pinning and spacing in a three-dimensional ordered Josephson medium

A method of calculating the configuration of two line vortices interacting in a three-dimensional ordered Josephson medium and a minimal distance between them at a given pinning parameter is proposed. The axes of the vortices lie in the middle row of an infinite slab 9 or 13 cells thick with different conditions at the boundaries of the slab. Away from the centers of the vortices, the system of finite-difference equations becomes linear. Fluxoid quantization conditions in cells near the centers of the vortices serve as boundary conditions. An exact solution is approached by iterations in those phase discontinuities which cannot be considered small. This technique provides a much higher calculation accuracy and offers a wider domain of applicability than the earlier methods. Critical values I _d of the pinning parameter at which two initial vortices keep given spacing d between them are calculated. For various vortex configurations, maximal pinning forces are calculated as functions of the pinning parameter and the distance to the nearest vortices. It is shown that the pinning force decreases near parallel vortices and increases near antiparallel ones.     

Penetration of the magnetic field into a 3D ordered Josephson medium at small values of the pinning parameter

The current configurations and the profile of the magnetic field penetrating into a 3D ordered Josephson medium are calculated for I < I _C . The calculation algorithm (modified for finite-length samples) is based on analyzing the continuous variation of the configuration toward a decrease in the Gibbs potential. This algorithm makes it possible to find a configuration into which the Meissner state passes when I < I _C and an external field slightly exceeds H _max and trace the evolution of this configuration with a further rise in the field. At H > H _max, the magnetic field penetrates into the sample as a quasi-uniform sequence of plane vortices. When H is roughly equal to H ₀/2, where H ₀ is the outer field at which one fluxoid Φ₀ passes through each cell, the plane vortices disintegrate into linear ones centered in cells neighboring along the diagonal. As the field grows, the vortex pattern condenses: zero-fluxoid cells are gradually “filled” starting from the boundary. When the field approaches H ₀, a sequence of plane vortices centered in adjacent rows arises near the boundary. With a further increase in the field, sequences of linear vortices with a double fluxoid form at the boundary. Then, such a scenario is periodically repeated with a period H ₀ in the external field.     

Two dimensional vortex structures in a superconductor slab at low temperatures

The dynamics of a two dimensional chain like structure of vortices is studied in the model of nonlinear time dependent Ginzburg–Landau equations (TDGL). The transition between different linear chains of vortices in a superconducting homogeneous slab with both surfaces in contact with a thin layer of metallic material is analyzed. The magnetization curve, vortex number, vortex configurations and modulus of the order parameter are studied as a function of the external magnetic field. We show how these vortex configurations are affected by the extrapolation length b (de Gennes boundary conditions), Ψ due to the proximity effects in a mesoscopic sample of area d_x × d_y, where d_y = 60ξ(0) and d_x varies discretely from 30ξ(0) to 12ξ(0). Possible connection with recent theoretical results in a two dimensional system of charged particles is discussed.     

Pinning of planar vortices and magnetic field penetration in a three-dimensional Josephson medium

The behavior of planar (laminar) vortices in a three-dimensional, ordered Josephson medium as a function of the parameter I, which is proportional to the critical junction current and the cell size, is investigated with allowance for pinning due to the cellular structure of the medium. The minimum possible distances between two isolated vortices are calculated. A system of vortices formed in a sample in a monotonically increasing external magnetic field is analyzed. The minimum distance from the outermost vortex to the nearest neighbor is proportional to I ^−1.1. For I⩽1.3 each vortex contains a single flux quantum Φ₀, and the distance between them does not decrease in closer proximity to the boundary but remains approximately constant, implying that the magnetic field does not depend on the coordinate in the region penetrated by vortices. These facts contradict the generally accepted Bean model. The sample magnetization curve has a form typical of type II superconductors. Allowance for pinning raises the critical field H _c and induces a sudden jump in the curve at H=H _c. Zh. Tekh. Fiz. 67, 38–46 (September 1997)     

Possible spacing between linear vortices in a 3D ordered Josephson medium

A method is proposed for solving the nonlinear system of equations of fluxoid quantization for two interacting linear vortices. It is shown that the centers of the vortices may lie in adjacent cells only if the pinning parameter I > 0.91, in alternate cells if I > 0.44, and in each third cell if I > 0.25. These critical values are substantially lower than analogous values for planar vortices. It is shown that, as the value of I tends to zero, the minimal spacing between linear vortices does not increase indefinitely, but attains a certain finite value and then remains unchanged. This means that pinning of linear vortices cannot be ignored even for values of I quite close to zero. It is shown that two linear vortices with centers in the neighboring cells along a diagonal may coexist for indefinitely small values of I.     

Hysteresis in the behavior of a long periodically modulated Josephson junction in a magnetic field for not small values of the pinning parameter

The magnetization curve for a long periodically modulated Josephson junction is calculated using the approach based on analysis of the continuous change in the configuration in the direction of the decrease in the Gibbs potential upon cyclic variation of the external magnetic field for not small values of pinning parameter I. It is shown that unlike in the case of small I, when the hysteresis loop is a part of a certain universal curve, the segments of the loops corresponding to a decrease in h in the first and second quadrants (and symmetric to them) pass below the universal loop, the degree of deviation increasing with pinning parameter I. The properties of the hysteresis loops are considered for various amplitudes of the magnetic field variation on the basis of analysis of vortex configurations.     

Formation of a quasi-uniform sequence of vortices in a periodically modulated finite-length Josephson contact

The configurations of currents and the profile of a magnetic field penetrating into a finite-length contact at I < I _C are calculated. The computational method is based on analyzing the continuous variation of the current structure leading to a decrease in the Gibbs potential. Such an approach makes it possible to find a configuration that sets in when an external field slightly exceeds H _max and trace the evolution of this configuration with increasing field. It is shown that at H > H _max boundary structures turn into quasi-uniform sequences of vortices the spacing between which oscillates about a mean value decreasing with increasing H. At some values of H, vortices with a number of fluxoids Φ₀ larger by unity start penetrating into the contact in the form of boundary sequences. As the field grows, they produce quasi-uniform sequences, etc. Vortices with the number of fluxoids Φ₀ differing by more than unity can fall into the contact at no field. The penetration of vortices with (k + 1)Φ₀ into a contact each cell of which contains kΦ₀ is fully identical to the penetration of vortices with one Φ₀ into the Meissner configuration. This statement is supported by the almost strict periodicity of mean induction b in the contact versus external field h dependence with a period of 1 along both axes and also by the form of the dependences of the magnetic field in the cells on the cell-boundary distance.     

Vortex configurations on a thin superconducting spherical shell in the presence of a magnetic dipole

In this work we study vortex configurations on a thin superconducting spherical shell of radius R and thickness d  (R?d?ξ)$(R?d?\xi )$ with a magnetic dipole inside it. The point magnetic dipole (with magnetic moment, m_z$m_z$) is oriented along one of the sphere main axis. It is placed a distance z₀$z_0$ from the center of the sphere. Due to the symmetry of the sample, there are vortices and antivortices pancakes on the shell resulting in zero total vorticity. The vortex interactions with the shielding currents produced by the external fields – as well as with other vortices – are calculated within the London limit. The vortex configurations are obtained by solving numerically the Bardeen–Stephen equation of motion for the vortices. For z₀≈0$z_0\approx 0$ the most frequent vortex configurations present equal arrangements of vortices and antivortices on the north and south hemispheres. For z₀≈0.5R$z_0\approx 0.5R$, the diversity of vortex configurations increases, with a higher number of configurations (in comparison to smaller z₀$z_0$) having different vortices and antivortices distributions on each shell hemisphere. We also study the equilibrium states dependence on the magnetic dipole strength and position.     

Localization of quasiparticles in a disordered vortex

We study the diffusive motion of low-energy normal quasiparticles along the core of a single vortex in a dirty, type-II, s-wave superconductor. The physics of this system is argued to be described by a one-dimensional supersymmetric non-linear σ model, which differs from the σ models known for disordered metallic wires. For an isolated vortex and quasiparticle energies less than the Thouless energy E_Th, we recover the spectral correlations that are predicted by random matrix theory for the universality class C. We then consider the transport problem of transmission of quasiparticles through a vortex connected to particle reservoirs at both ends. The transmittance at zero energy exhibits a weak localization correction reminiscent of quasi-one-dimensional metallic systems with symmetry index β = 1. Weak localization disappears with increasing energy over a scale set by E_Th This crossover should be observable in measurements of the longitudinal heat conductivity of an ensemble of vortices under mesoscopic conditions. In the regime of strong localization, the localization length is shown to decrease by a factor of 8 as the quasiparticle energy goes to zero.     

Finite number of vortices and bending of finite vortex lines in a confined rotating Bose-Einstein condensate

The minimal energy configurations of finite N_v-body vortices in a rotating trapped Bose-Einstein condensate is studied analytically by extending the previous work [Y. Castin, R. Dum, Eur. Phys. J. D 7, 399 (1999)], and taking into account the finite size effects on z-direction and the bending of finite vortex lines. The calculation of the energy of the vortices as a function of the rotation frequency of the trap gives number, curvature, configuration of vortices and width of vortex cores self-consistently. The numerical results show that (1) the simplest regular polynomial of the several vortex configurations is energetically favored; while the hexagonal vortex lattice is more stable than square lattice; (2) bending is more stable then straight vortex line along the z-axis for λ<1; (3) the boundary effect is obvious: compared with the estimation made under infinite boundary, the finite size effect leads to a lower vortex density, while the adding vortex bending results in a less higher density because of the expansion. The results are in well agreement with the other authors' ones.     

Astigmatic telescopic transformation of a high-order optical vortex

Properties of an optical vortex light beam formed after the astigmatic telescopic transformation of a circular Laguerre-Gaussian mode are considered both theoretically and experimentally. The beam evolution is found to be in conformity with the general notions on the high-order optical vortex symmetry breakdown. Upon propagation, the asymmetric beam shows a sort of rotation of its transverse profile in accord with the energy circulation in the original circular mode; this process is described on the base of the beam intensity moments and the vortex and asymmetry components of its orbital angular momentum. An l-charged optical vortex converts into |l| secondary first-order vortices positioned on a straight line crossing the beam axis. Orientation of this straight line in the beam cross section and spatial separation of the secondary vortex cores depend on the propagation distance. Morphology (orientation and anisotropy) of all the secondary vortices is the same and depends on the propagation distance; the anisotropy can be characterized by the vortex component of the beam angular momentum. At certain distance, relative separation of secondary vortices with respect to the beam transverse size reaches its maximum that corresponds to the minimum anisotropy of the vortices. The results can be useful in the context of current research of the optical vortex arrays.     

Velocity field vortices and Mermin-Ho vortices in two-component spinor BEC

In light of the φ-mapping topological current theory, two important vortex structures in two-component spinor BEC—the velocity field vortices and the Mermin-Ho vortices are discussed. It is revealed that these two different kinds of vortices are created respectively from the zero points of two different order parameter configurations in the condensates, and both their topological charges, locations and motions can be determined by the φ-mapping theory.     

Direct visualization of the vortex distributions in a superconducting film with a Penrose array of magnetic pinning centers: Symmetry induced giant vortex state

Using scanning Hall probe microscopy a direct visualization of the flux distribution in a Pb film covering a fivefold Penrose array of Co dots is obtained. We demonstrate that stable vortex configurations can be found for fields H ∼ 0.8H₁, H₁ and 1.6H₁, where H₁ corresponds to one flux quantum per pinning site. The vortex pattern at 0.8H₁ corresponds to one vacancy in one of the vertices of the thin tiles whereas at 1.6H₁ the vortex structure can be associated with one interstitial vortex inside each thick tile. Strikingly, for H = 1.6H₁ interstitial and pinned vortices arrange themselves in ring-like structures (“vortex corrals”) which favor the formation of a giant vortex state at their center.     

Contribution a la theorie atomique de la diffusion chimique

Here we present an analysis and a development of the atomic theory of chemical diffusion as proposed by Manning for a binary system a/b.The general expression for the flux of a tracer in a concentration gradient is first established. This expression of the flux is identified with that deduced in the phenomenological theory. Thus a relationship between the partial correlation factors of vacancies with each of the a and b species is obtained.The effect of “vacancy flow” can be described in terms of these correlation factors. Thus the vacancy flow on species A leads to a correlation of the vacancy jumps with species B and vice versa.We shall see that the Nernst-Einstein equation can be extended to the case of chemical diffusion and that the ratio of the intrinsic diffusion coefficients is equal to the ratio of the mean jump frequencies W_A and W_B.Also, the activation energies of intrinsic diffusion coefficients are related very simply to the activation enthalpies of atomic jumps.In conclusion, we shall see that chemical diffusion in a binary system a/b can be completely described if either the thermodynamic factor and the coefficients of self diffusion, or the thermodynamic factor and the coefficients of intrinsic diffusion are known as functions of the concentration.