Nonlinear dynamics of domain walls with a vortex internal structure in magnetouniaxial films with planar anisotropy

The nonlinear and generally unsteady dynamics of domain walls with a vortex internal structure in a constant magnetic field H is investigated on the basis of the numerical solution of the Landau-Lifshitz equation for a 2D distribution of magnetization M in magnetic films with planar anisotropy taking into account exactly the main interaction, including the dipole-dipole interaction. It is shown that in addition to field H _c (bifurcation field) above which the motion of a wall becomes unsteady and its internal structure experiences global dynamic changes, there exists a field H₀ separating two steady motions of the wall with different structures. The data clarifying the physical origin of the nonlinear dynamic rearrangement of the wall structure are presented. New rearrangement mechanisms associated with the generation and attenuation of vortices as well as their tunneling through the central surface of the wall are established. The existence of subperiod oscillations of the wall velocity in a static field in addition to the oscillations associated with the precession of M around the easy magnetization axis is predicted. The period T of dynamic variations of the wall structure is studied, and an empirical formula is proposed for describing the singular behavior of the T(H) dependence near H=H _Hc with the critical index depending on the film parameters. The bifurcation process is studied, and a nonlinear dependence of the critical field H _c on the film thickness and the saturation magnetization is established. The possibility of direct experimental investigation of the dynamic rearrangement of the internal structure of the wall is indicated.     

Dynamics of the magnetic flux component normal to the layers of Bi<Subscript>2</Subscript>Sr<Subscript>2</Subscript>CaCu<Subscript>2</Subscript>O<Subscript>8</Subscript> in a tilted field

High-frequency losses in the strongly anisotropic layered superconductor Bi₂Sr₂CaCu₂O₈ are measured at 600 MHz under a magnetic field rocking about the ab plane. Anomalies in losses and hysteretic phenomena are found while performing periodic rocking, i.e., cycling the magnetic field component normal to the sample surface. Based on these observations, conclusions are drawn about the nature of magnetic-flux penetration into the superconductor. It is found that, in the range between 60 K and T _c , the dynamics of magnetic-flux vortex lines normal to the ab plane in the presence of a constant magnetic field applied parallel to this plane is governed by the critical penetration field H _c ^⊥* and the surface barrier in the presence of thermally activated vortex motion (giant flux creep). The dependences of H _c1 ^⊥* and the characteristic field of the surface barrier on the magnitude of the parallel magnetic field are measured.     

A swirling jet with vortex breakdown: three-dimensional coherent structures

The paper reports on shape of a three-dimensional coherent structure in a velocity field of a high-swirl turbulent jet with the bubble-type vortex breakdown. A set of the 3D instantenous velocity fields was measured by using the tomographic particle image velocimetry (tomographic PIV) technique and processed by the proper orthogonal decomposition (POD) method. The detected intensive coherent velocity component corresponded to a helical vortex core of the swirling jet and two secondary spiral vortices. The entire coherent structure was rotating around the jet axis in compliance with the direction of the flow swirl. From the 3D data it is concluded that the dynamics of the strsucture can be described by a traveling wave equation: Re[A(y, r)·e ^{i(mθ + ky - ωt)}] with the number of the spiral mode m = +1 for positively defined k and ω.     

Phase transitions and vortex structure evolution in two-level high-temperature granular superconductor YBa<Subscript>2</Subscript>Cu<Subscript>3</Subscript>O<Subscript>7 – δ</Subscript> under temperature and magnetic field

Temperature dependences of the resistivity ρ(T) of samples of granular high-temperature superconductor YBa₂Cu₃O_{7 – δ} are measured at various transverse external magnetic fields at 0 < H _ext < 1900 Оe in the temperature range from the upper Josephson critical temperature of “weak bonds” T _c2J to temperatures slightly exceeding the superconducting transition temperature T _c . Based on the data obtained, the behavior of the field dependences of the critical temperatures of superconducting grains and “weak bonds,” and temperature and field dependences of the magnetic contribution to the resistivity \(\left[ {\Delta \rho \left( {T,H} \right) = \rho {{\left( T \right)}_{{H_{ext}} = const}} - \rho {{\left( T \right)}_{{H_{ext}} = 0}}} \right]\). It is shown that the behavior of the magnetic contribution to the resistivity Δρ along the line of the phase transition related to the onset of the magnetic field penetration in the form of Abrikosov vortices into the subsystem of superconducting grains T _c1g (H _ext) is anomalous. The concepts on the magnetic flux redistribution between both subsystems of two-level HTSC near in the vicinity of T _c1g : the Josephson vortex decreases, and the Abrikosov vortex density increases.     

Superconducting Properties of Indium Nanostructured in Pores of Thin Films of SiO<Subscript>2</Subscript> Microspheres

Samples of a superconducting indium nanocomposite based on a thin-film porous dielectric matrix prepared by the Langmuir–Blodgett method are obtained for the first time, and their low-temperature electrophysical and magnetic properties are studied. Films with thickness b ≤ 5 μm were made from silicon dioxide spheres with diameter D = 200 and 250 nm; indium was introduced into the pores of the films from the melt at a pressure of P ≤ 5 kbar. Thus, a three-dimensional weakly ordered structure of indium nanogranules was created in the pores, forming a continuous current-conducting grid. Measurements of the temperature and magnetic field dependences of the resistance and magnetic moment of the samples showed an increase in the critical parameters of the superconductivity state of nanostructured indium (critical temperature T_c ≤ 3.62 K and critical magnetic field H_c at T = 0 K H_c(0) ≤ 1700 Oe) with respect to the massive material (T_c = 3.41 K, H_c(0) = 280 Oe). In the dependence of the resistance on temperature and the magnetic field, a step transition to the superconductivity state associated with the nanocomposite structure was observed. A pronounced hysteresis M(H) is observed in the dependence of the magnetic moment M of the nanocomposite on the magnetic field at T < T_c, caused by the multiply connected structure of the current-conducting indium grid. The results obtained are interpreted taking into account the dimensional dependence of the superconducting characteristics of the nanocomposite.     

Visualization of the magnetic flux structure in phosphorus-doped EuFe<Subscript>2</Subscript>As<Subscript>2</Subscript> single crystals

Magnetic flux structure on the surface of EuFe₂(As_1-x P _x )₂ single crystals with nearly optimal phosphorus doping levels x = 0.20 and x = 0.21 is studied by low-temperature magnetic force microscopy and decoration with ferromagnetic nanoparticles. The studies are performed in a broad temperature range. It is shown that the single crystal with x = 0.21 in the temperature range between the critical temperatures T _SC= 22 K and T _C = (18 ± 0.3) K of the superconducting and ferromagnetic phase transitions, respectively, has the vortex structure of a frozen magnetic flux, typical for type-II superconductors. The magnetic domain structure is observed in the superconducting state below T _C. The nature of this structure is discussed.     

Enhancement of the intrinsic pinning by a magnetic field in a single crystal of high-temperature superconductor TmBa<Subscript>2</Subscript>Cu<Subscript>3</Subscript>O<Subscript><Emphasis Type="Italic">y</Emphasis></Subscript>

This paper presents a comparative study of the complex permeability μ* of single crystals of hightemperature superconductors RBa₂Cu₃Oy (R = Y, Tm) as a function of the magnetic field applied along the crystallographic ab-plane. Contributions to μ* from the oscillatory motion of vortices perpendicular to layers of the crystal lattice, μ_v, and realization of the critical state along the layers are obtained. It is found that, as the temperature approaches T _c, the behavior of the field dependences of μ_v and the critical current is substantially different for the given samples. This effect is related to the manifestation of an additional unusual mechanism of intrinsic pinning, which arises when Y₃₊ is replaced with the Tm³⁺ magnetic ion. The revealed specificities of the interaction of vortices with the magnetic ion layer suggest that, most probably, they have a magnetic nature and are not related to the variation in the condensation energy in the core of the vortex.     

Self-similar distribution in “giant” magnetic flux creep

The problem of magnetic field penetration into the half-space is considered in parallel geometry in an external magnetic field increasing with time in accordance with the law B(0, t, τ₀ = B _{c 1} (1 + t/τ₀) ^m , m ≥ 0, t ≥ 0 (τ ₀ is the time of magnetic flux redistribution and B _{c 1} is the lower critical field). It is assumed that the flow of vortices is thermally activated in the “giant” creep mode (i.e., for weak pinning creep and high temperatures). A model equation is derived for describing the magnetic flux evolution. Analytic formulas are obtained for the depth and velocity of magnetic field penetration. It is shown that the giant creep regime is stable for 0 ≤ m ≤ 1/2.     

Systematic Study of Vortex Pinning and a Liquid–Glass Phase Transition in BaFe<Subscript>2–<Emphasis Type="Italic">x</Emphasis></Subscript>Ni<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>As<Subscript>2</Subscript> Single Crystals

The vortex pinning and liquid-glass transition have been studied in BaFe_2–xNi _x As₂ single crystals with different doping levels (x = 0.065, 0.093, 0.1, 0.14, 0.18). We found that Ni-doped Ba-122 has rather narrow vortex-liquid state region. Our results show that the temperature dependence of the resistivity as well as I?V characteristics of Ni-doped Ba-122 is consistent with 3D vortex-glass model. It was found that -pinning gives the main contribution to overall pinning in 122 Ni-doped system. The vortex phase diagrams for different doping levels were built based on the obtained data of temperature of the vortex-glass transition T_g and the upper critical magnetic field H_c2.     

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.     

Geometrical surface vortex pinning in superconducting films

A model is proposed for vortex pinning in a superconducting film with a rough surface. The model relates the critical current to the steepness of the surface relief and, at a high vortex concentration, to the distance between neighboring steepness maxima on the paths of vortex motion. The dependence of the critical current density on the thickness of a high-T_c superconducting film is measured in a weak magnetic field. Its behavior can be explained by the pinning at the stepped surface relief.     

Self-similar magnetic structures and magnetic flux “Giant” creep

The penetration of a magnetic flux into a type-II high-T _c superconductor occupying the half-space x > 0 is considered. At the superconductor surface, the magnetic field amplitude increases in accordance with the law b(0, t) = b ₀(1 + t)^m (in dimensionless coordinates), where m > 0. The velocity of penetration of vortices is determined in the regime of thermally activated magnetic flux flow: v = v ₀exp?ub;?(U _0/T )(1-b?b/?x)?ub;, where U ₀ is the effective pinning energy and T is the thermal energy of excited vortex filaments (or their bundles). magnetic flux “Giant” creep (for which U ₀/T? 1) is considered. The model Navier-Stokes equation is derived with nonlinear “viscosity” v ∝ U ₀/T and convection velocity v _f ∝ (1 ? U ₀/T). It is shown that motion of vortices is of the diffusion type for j → 0 (j is the current density). For finite current densities 0 < j < j _c, magnetic flux convection takes place, leading to an increase in the amplitude and depth of penetration of the magnetic field into the superconductor. It is shown that the solution to the model equation is finite at each instant (i.e., the magnetic flux penetrates to a finite depth). The penetration depth x _eff ^A (t) ∝ (1 + t)^{(1 + m/2)/2} of the magnetic field in the superconductor and the velocity of the wavefront, which increases linearly in exponent m, exponentially in temperature T, and decreases upon an increase in the effective pinning barrier, are determined. A distinguishing feature of the solutions is their self-similarity; i.e., dissipative magnetic structures emerging in the case of giant creep are invariant to transformations b(x, t) = β^m b(t/β, x/β^{(1 + m/2)/2}), where β > 0.     

Resonance attenuation of ultrasonic waves in a superconductor with a moving vortex structure

Equations describing the interaction of ultrasonic waves with a moving vortex structure are derived. The addition to attenuation and the relative change in the velocity of longitudinal ultrasonic waves due to this interaction are calculated. It is found that when a longitudinal ultrasonic wave propagates along the direction of motion of the vortex structure and the velocity V of the structure is equal to half the velocity of the wave, then anomalous acoustic attenuation occurs and the contribution from the ultrasound-vortex interaction to the velocity of the ultrasonic wave vanishes. It is shown that if the vortex structure moves at a sufficiently high velocity, then (in contrast to the case of the structure at rest) a weakly damping collective mode propagating with velocity 2V arises in the structure. It is this mode that is responsible for anomalous attenuation of longitudinal ultrasonic waves.     

Magnetic field and doppler velocity correlation in quiescent solar prominences

Analyzing statistically the magnetic and Doppler velocity data for 85 quiescent prominences observed in 1983–1987 by Nikolsky's magnetograph, we came to the following preliminary conclusions:

The average longitudinal magnetic field of the prominence determines the dynamic velocity of the latter: the stronger magnetic fields correspond to the higher Doppler velocities.

A longitudinal magnetic field less than 25 G allows the material to move with arbitrary velocity within the limits of several kilometers per second. A magnetic field higher than 25 G suppresses such movements.

The horizontal length of the flux tube exceeds its vertical part by 1.5 orders of magnitude (the upper limit).

There is an angle of 10°between the horizontal component of the velocity vector in quiescent prominences and the long axis of the filament.

The maximum velocity in quiescent prominences is about 7 km/s.

     

Entanglement in Mixed-Spin (1/2, 3/2) Heisenberg XXZ Model with Dzyaloshinskii-Moriya Interaction

In this paper, the entanglement in a mixed-spin (1/2, 3/2) Heisenberg XXZ model with Dzyaloshinskii-Moriya (DM) interaction in an inhomogeneous external magnetic field is studied. We not only calculate the ground-state entanglement but also investigate the behaviors of quantum phase transition following the changes of DM interaction and nonuniform magnetic field. More importantly, we note that the DM interaction improves the critical magnetic field B _c , the critical temperature T _c and broadens the region of entanglement.     

New omega vortex identification method

A new vortex identification criterion called W-method is proposed based on the ideas that vorticity overtakes deformation in vortex.The comparison with other vortex identification methods like Q-criterion and λ_2-method is conducted and the advantages of the new method can be summarized as follows:(1) the method is able to capture vortex well and very easy to perform;(2) the physical meaning of W is clear while the interpretations of iso-surface values of Q and λ_2 chosen to visualize vortices are obscure;(3)being different from Q and λ_2 iso-surface visualization which requires wildly various thresholds to capture the vortex structure properly, W is pretty universal and does not need much adjustment in different cases and the iso-surfaces of W=0.52 can always capture the vortices properly in all the cases at different time steps, which we investigated;(4) both strong and weak vortices can be captured well simultaneously while improper Q and λ_2 threshold may lead to strong vortex capture while weak vortices are lost or weak vortices are captured but strong vortices are smeared;(5) W=0.52 is a quantity to approximately define the vortex boundary. Note that, to calculate W, the length and velocity must be used in the non-dimensional form. From our direct numerical simulation, it is found that the vorticity direction is very different from the vortex rotation direction in general 3-D vortical flow,the Helmholtz velocity decomposition is reviewed and vorticity is proposed to be further decomposed to vortical vorticity and non-vortical vorticity.     

Size Dependences of the Magnetic Properties of Superconducting Lead–Porous Glass Nanostructures

Superconducting structures Pb–PG formed by filling a porous glass matrix with the lead from melt under pressure have been investigated. Samples with characteristic pore structure diameters of d ≈ 7, 3, and 2 nm have been studied. It has been found that the critical temperature of the superconducting transition in the samples under study is similar to the corresponding value T_c ≈ 7.2 K for bulk lead. At the same time, it has been observed that the critical magnetic field of the nanocomposites, which attains H_c(T = 0 K) ≈ 165 kOe for Pb–PG (3 nm), exceeds several times the value H_c(0) = 803 Oe for bulk lead. The low-temperature magnetic- field dependences of magnetic moment M(H) contain quasi-periodic flux jumps, which vanish with a decrease in the lead nanostructure diameter. A qualitative model of the observed effects is considered.     

The effect of the spatial variation of the order parameter on the tunneling density of states of superconducting alloys

In the first part of the paper we derive expressions of the Ginzburg-Landau (GL) type for the local tunneling density of states of superconducting alloys. These expressions are quite generally applicable at high excitation energies. One can see immediately that the density of states,N(r, ω), at any positionr and high energiesω is always larger than the local BCS density of states if the space dependence of the order parameter is governed by the GL-equation. This effect is largest for long mean free pathsl. In the second part of the paper we calculate the spatial average of the density of states,¯N, at all energiesω for a lattice of vortex lines in a magnetic field slightly below the upper critical field. The resulting curve of [¯N? N(0)]/N(0) versus co shows no gap and has a zero at about the gap value in zero field. Its value at ω=0 depends onl like ln(ξ₀/l) for l?ξ₀ [N(0) denotes the normal density of states, and ξ₀ is the BCS coherence length].     

Domain structure in centroantisymmetric antiferromagnets

The domain structure of an antiferromagnet whose magnetic-symmetry group contains a center of antisymmetry is studied theoretically. The magnetoelectric effect and weak ferromagnetism are shown to coexist in a domain wall. It is established that when the inhomogeneous magnetic moment interacts with a sufficiently strong magnetic field H ‖ C ₃, a multidomain state with an odd number of 180° domain walls becomes energetically favorable. The critical field for the transition from a single-domain state to a multidomain state is found. It is shown that domain reversal occurs when the magnetic field H is reversed.