Stability of the Meissner state in a 3D ordered Josephson medium

The stability of the Meissner state of a 3D Josephson medium against combinations of phase jump small fluctuations at contacts is considered. Expressions for the elements of the quadratic form matrix for the second variation of the Gibbs potential are derived. Overheat field values and forms of fluctuations causing instabilities are found. Ratio H _S1/H _S2, where H _S1 is the overheat field and H _S2 is the maximal field at which the Meissner state still exists, grows with increasing pinning parameter I, varying between 0.84 and 1. Almost at all pinning parameters, critical fluctuations represent rapidly decreasing (inward to the sample) periodic alternating-sign structures one cell wide. When the pinning parameter is very small (I < 0.1), such an instability is absent. In this range of I, ratio H _S1/H _S2 is close to unity.     

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.     

Development of the Meissner state instability in an ordered 3D Josephson medium

A new approach based on analysis of continuous configurational modification in the direction of a decrease in the Gibbs potential is proposed for computing the penetration of an external magnetic field in an ordered 3D Josephson medium. The configuration to which the Meissner state passes when the external field slightly exceeds the Meissner stability threshold is determined. This configuration contains a periodic sequence of linear vortices with centers lying in an alternating cell, parallel to the boundary, and located at a certain distance from it. A further increase in the field reveals that the 3D medium behaves like a long periodically modulated Josephson junction. However, the critical value I _C of the pinning parameter for a 3D medium, which lies in the interval 0.7–0.8, is lower than the analogous value I _C = 0.9716 for a long junction. The values of H _max for I < I _C , as well as the steepness of the decrease in the magnetic field at the boundary for I > I _C , are higher in the 3D medium than in a long junction. For very large values of I, the field penetrates the boundary region not as a 2D lattice of linear vortices, but as a 1D lattice of plane vortices, which are mathematically equivalent to the vortices in a long junction.     

Penetration of magnetic field into a long periodically modulated Josephson contact

A new approach to magnetic field profiling inside a Josephson contact is suggested. Its essence consists in analyzing continuous variation of a current configuration leading to a decrease in the Gibbs potential. With this approach, one can find a configuration into which the Meissner state turns when an external field slightly exceeds the upper boundary of the Meissner regime and trace the evolution of this configuration with increasing field. Calculations show that there exists critical value I _c of the pinning parameter in the range 0.95–1.00. This critical value separates two possible conditions of magnetic field penetration into the contact. At I > I _c, a near-boundary current configuration completely compensating for the external field inside the contact arises irrespective of the external field strength. At I < I _c, such a situation is observed only until the external field strength exceeds certain value H _max. Higher fields penetrate into the contact indefinitely deep. In nearboundary configurations, the magnetic field drops with increasing depth almost linearly. Its slope k has rational values, which remain constant within finite intervals of I. As I goes beyond a given interval, k rises stepwise and takes on another rational value. When an external magnetic field is switched on adiabatically, configurations with a maximal growth rate of the magnetic field are observed.     

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.     

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)     

Magnetic field penetration into a 3D ordered Josephson medium and applicability of the bean model

The results of calculation of penetration of an external magnetic field into a 3D ordered Josephson medium, based on analysis of modification of the configuration in the direction of the decrease in its Gibbs potential, are reported. When the external field slightly exceeds the stability threshold, the Meissner configuration is transformed into a periodic sequence of linear vortices, which are parallel to the boundary of the medium and are located at a certain distance from it. There exists a critical value I _C separating two possible regimes of penetration of the external magnetic field into the medium. For I > I _C , for any value of the external field, a finite-length boundary current configuration appears, which completely compensates the external field in the bulk of the sample. At the sample boundary, the field decreases with increasing depth almost linearly. The values of the slope of the magnetic field dependence are rational fractions, which remain constant in finite intervals of I. When the value of I exceeds the upper boundary of such an interval, the slope increases and assumes the value of another rational fraction. If, however, I < I _C , such a situation takes place only up to a certain value of external field H _max. For higher values, the field penetrates into the medium to an infinite depth. These results lead to the conclusion that the Bean assumptions are violated and that Bean’s model is inapplicable for analyzing the processes considered here.     

Equilibrium states of planar vortices in a three-dimensional josephson medium and meaning of the pinning-energy concept

Various ways of specifying the pinning-energy concept for planar vortices in a three-dimensional cellular Josephson medium are analyzed. It is shown that, for values of the pinning parameter I that are not small, a universal characteristic of vortex interaction with the lattice cannot be found, since the displacement of a vortex distorts its shape. At small values of I, the maximum pinning force can be chosen for such a characteristic. Two equilibrium states of a vortex are analyzed for stability. It is revealed that the state of higher energy is not inevitably unstable. A correct analysis of stability must be based on exploring a quadratic form that describes the energy of a current configuration. Such an investigation is performed for the equilibrium state of a vortex. At small values of the pinning parameter, the vortex state of higher energy is quasistable.     

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.     

Josephson effect in superconducting constrictions with hybrid SF electrodes: Peculiar properties determined by the misorientation of magnetizations

Josephson current in SFcFS junctions with arbitrary transparency of the constriction (c) is investigated. The emphasis is on the analysis of the supercurrent dependencies on the misorientation angle θ between the in-plane magnetizations of diffusive ferromagnetic layers (F). It is found that the current-phase relation I(φ) may be radically modified with the θ variation: the harmonic I ₁ sin φ vanishes for a definite value of θ provided that, for an identical orientation of the magnetizations (θ = 0), the junction is in the “π” state. The Josephson current may exhibit a nonmonotonic dependence on the misorientation angle both for realization of the “0” and “π” state at θ = 0. We also analyze the effect of the exchange field induced enhancement of the critical current, which may occur in a definite range of θ. The text was submitted by the author in English.     

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.     

New model for systems of mesoscopic Josephson junctions

Quantum fluctuations of the phases of the order parameter in two-dimensional arrays of mesoscopic Josephson junctions and their effect on the destruction of superconductivity in the system are investigated by means of a quantum-cosine model that is free of the incorrect application of the phase operator. The proposed model employs trigonometric phase operators and makes it possible to study arrays of small superconducting granules, pores containing superfluid helium, or Josephson junctions in which the average number of particles n ₀ (effective bosons, He atoms, and so on) is small, and the standard approach employing the phase operator and the particle number operator as conjugate operators is inapplicable. There is a large difference in the phase diagrams between arrays of macroscopic and mesoscopic objects for n ₀<5 and U<J (U is the characteristic interaction energy of the particles per granule and J is the Josephson coupling constant). Re-entrant superconductivity phenomena are discussed. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 10, 649–654 (25 November 1997)     

Behavior of mixtures of iodine with anthracene,phenazine and thianthrene

The binary systems I₂-phenazine (I₂P), I₂-anthracene and I₂-thianthrene have been investigated over a wide range of concentrations (0.1 < x_I2 < 1) and over temperatures ranging from 0 to 60°C. EMF measurements were performed in the three systems employing galvanic cells with a silver-ion-conducting solid electrolyte in order to determine the possible compounds and their formation Gibbs energies. Only one compound of stoichiometry I₂P is present in the system I₂phenazine and its formation Gibbs energy can be expressed as follows: Δ_FG⁰(I₂P) = ?20830 + 19.7 T(J mol^?1), 273–333 K. Neither the anthracene nor the thianthrene form compounds or solid solutions in the studied ranges of concentrations and temperatures.     

Vortex rings moving into turbulence

We report on an experimental study of turbulent vortex rings injected with velocity U _v0 into a grid-generated turbulent flow (with RMS streamwise velocity u _*) and followed relative to the mean flow. The initial Reynolds number of the vortices varies from 4500 to 11,500. The turbulence was characterised by an intensity I_t =u _*/U _v0, which varied over the range 0<I_t <0.03. A mathematical model based on a stochastic model of the vortex core is developed to explain and interpret the results. The vortex radius grows diffusively in time with the rate of increase of the square of the vortex radius increasing linearly with I_t . As the vortices grow, they slow down sufficiently rapidly in a manner that they penetrate a finite distance into the turbulence. The vortex velocity, averaged over many experiments, showed an initial t ^?1 decay, consistent with Maxworthy’s experiments. The analysis and experiments show that such vortices ultimately only move a finite distance from their point of generation and this distance varies inversely with I_t .     

Order parameter quantum fluctuations in a two-dimensional system of mesoscopic Josephson junctions

The boson lattice Hubbard model is used to study the role of quantum fluctuations of the phase and local density of the superfluid component in establishing a global superconducting state for a system of mesoscopic Josephson junctions or grains. The quantum Monte Carlo method is used to calculate the density of the superfluid component and fluctuations in the number of particles at sites of the two-dimensional lattice for various average site occupation numbers n ₀ (i.e., number of Cooper pairs per grain). For a system of strongly interacting bosons, the phase boundary of the ordered superconducting state lies above the corresponding boundary for its quasiclassical limit—the quantum XY-model—and approaches the latter as n ₀ increases. When the boson interaction is weak in the boson Hubbard model (i.e., the quantum fluctuations of the phase are small), the relative fluctuations of the order parameter modulus are significant when n ₀<10, while quantum fluctuations in the phase are significant when n ₀<8; this determines the region of mesoscopic behavior of the system. Comparison of the results of numerical modeling with theoretical calculations show that mean-field theory yields a qualitatively correct estimate of the difference between the phase diagrams of the quantum XY-model and the Hubbard model. For a quantitative estimate of this difference the free energy and thermodynamic averages of the Hubbard model are expanded in powers of 1/n ₀ using the method of functional integration. Zh. éksp. Teor. Fiz. 113, 261–277 (January 1998)     

Pinning of pancake vortices in a three-dimensional Josephson medium and structure of the vortex lattice in the “critical” state

A system of pancake vortices formed near the boundary of a sample in a monotonically increasing external magnetic field is calculated with allowance for pinning due to the cellular structure of the medium for various values of the pinning parameter I, which is proportional to the critical current of the junction and the cell diameter. The shortest distance from the outermost vortex to the nearest neighbor is proportional to I ⁻¹¹. It is shown that the pinning parameter has a critical value I _c separating two regimes with different types of critical states. For I<I _c the external magnetic field has a threshold value H _t(I), above which the field immediately penetrates the interior of the junction to an infinite distance. For I>I _c the magnetic field decays linearly from the boundary into the interior of the junction. The value obtained in the study, I _c=3.369, differs from the value of 0.9716 postulated by other authors. The dependence of the slope of the magnetic field profile near the boundary on I is determined. It is shown that the slope is independent of I in intervals 2πk<I<2πk+π. Fiz. Tverd. Tela (St. Petersburg) 39, 1958–1963 (November 1997)     

Josephson array of mesoscopic objects. Modulation of system properties through the chemical potential

The phase diagram of a two-dimensional Josephson array of mesoscopic objects (superconducting granules, superfluid helium in a porous medium, traps with Bose-condensed atoms, etc.) is examined. Quantum fluctuations in both the modulus and phase of the superconducting order parameter are taken into account within a lattice boson Hubbard model. Modulating the average occupation number n ₀ of the sites in the system (the “number of Cooper pairs” per granule, the number of atoms in a trap, etc.) leads to changes in the state of the array, and the character of these changes depends significantly on the region of the phase diagram being examined. In the region where there are large quantum fluctuations in the phase of the superconducting order parameter, variation of the chemical potential causes oscillations with alternating superconducting (superfluid) and normal states of the array. On the other hand, in the region where the bosons interact weakly, the properties of the system depend monotonically on n ₀. Lowering the temperature and increasing the particle interaction force lead to a reduction in the width of the region of variation in n ₀ within which the system properties depend weakly on the average occupation number. The phase diagram of the array is obtained by mapping this quantum system onto a classical two-dimensional XY model with a renormalized Josephson coupling constant and is consistent with our quantum path-integral Monte Carlo calculations. Zh. éksp. Teor. Fiz. 114, 591–604 (August 1998)     

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.     

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.     

Static characteristics of Josephson interferometers

This investigation deals with the range in operating currents for which a Josephson interferometer, sometimes also referred to as Superconducting QUantum Interference Device (SQUID), may remain in the zero-voltage Josephson condition. An interferometer consists of one or more inductive loops each of which contains two Josephson junctions or other weak links. Two types of current are considered. Gate currentI _gpasses the junctions in parallel. Control currentI _cgenerates magnetic flux via inductive coupling in the loops. Zero-voltage operation is possible within certain areas of theI _g,I _cplane. These areas are manifestations of flux-quantum states and their boundary lines are referred to as static characteristics. In view of the nonlinearity of the constituting equations, not all their formal solutions are physically realizable. A stability analysis yields criteria which permit the identification of realizable operating conditions. The static characteristics comprise operating conditions where the limit of stability is reached. To obtain the static characteristics, linearized equations may be utilized if theLI _o product, a measure for the size of an interferometer, is large compared to the flux quantumΦ ₀, whereL is the inductance per loop, andI _o the maximum Josephson current per junction. As a general method of solving system of transcendental equations, continuation is discussed. The utilization of continuation for obtaining interferometer characteristics is explained. It is shown that some changes in the gate-current feed arrangement are equivalent to shearing the characteristics in theI _g,I _cplane. Analytical results are given on extrema, inflexion points, and singularities in the shape of cusps which conceptually relate to the existence and connectivity of flux-quantum states. Experimental static characteristics are presented on two-and four-junction interferometers. They are in agreement with characteristics computed on the basis of simple lumped circuit models. Relevant circuit parameters are obtained from the experimental characteristics.