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.     

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.     

Eigenschaften der Lösungen der nichtlinearisierten Ginsburg-Landau-Gleichungen in Zylindersymmetrie

Solutions of the nonlinear Ginzburg-Landau equations in cylindrical symmetry have been computed for a type I superconductor. From these solutions the behaviour of a circular cylinder of infinite length in a magnetic field parallel to its axis has been deduced. For a series of values of the magnetic field solutions are given in two cases. The first case was calculated with the assumption of no fluxoid frozen in (fluxoid quantum number n=0), whereas in the second case a vortex with fluxoid quantum numbern=1 was assumed on the axis of the cylinder. For both series of solutions investigation of the thermodynamic stability was carried out. This and further thermodynamic considerations led to the result that in a gedankenexperiment the transition from the normal to the superconducting state and vice versa can be performed in a reversible manner. The expulsion of the magnetic field from the sample during the reversible transition to the superconducting state (Meissner-Effect) is also described by the solutions. Further results are the existence of a supercooled state down to a magnetic fieldH _c2=κ√2H_cb and of a superheated state up to a fieldH _c1>H _cb. The value ofH _c1 depends on the radius of the cylinder. If a condensation to the superconducting state takes place at a fieldH ₀ whereH _c2<H ₀<H _cb, condensation withn=0 seems to be preferred in comparison to that withn=1.     

Interaction of vortices in type II superconductors and the behavior nearH c1 at arbitrary temperature

The interaction between well-separated Abrikosov vortices is calculated using Eilenberger's version of the Gorkov theory. Let λ and ξ be the decay lengths determining the asymptotic behavior of the magnetic field and the order parameter for an isolated vortex. Then the onset of attraction in the limit of large vortex separation is either given by λ=ξ or by λ becoming complex, whichever occurs first. It is shown that in general a first order transition atH _c1 takes place already in the region of asymptotic repulsion because of the appearance of a minimum in the interaction energy at finite vortex separation.     

Vortex states in soft magnets in two and three dimensions

The magnetization curves of arrays of near-spherical soft ferromagnetic particles are compared with those of quasi-two-dimensional dots with similar radius prepared by a rapid e-beam lithographic technique. Curves for the three-dimensional particles are anhysteretic and fit a M(H)/M_s=tanh(cμ₀H) law, whereas the two-dimensional arrays show irreversible segments in the first and third quadrants where the planar vortex state transforms to a collinear state by discontinuous rotation of magnetization about an axis perpendicular to the vortex axis. The additional symmetry of the spherical particle allows this rotation to occur continuously, without energy barriers due to the demagnetizing field.     

Distribution of the current density in the cylindrical sample of the type II superconductor

Deviation from a homogeneous distribution of the vortex line lattice and magnetic field induced by transport current near H_c2 in the cylindrical sample of type II superconductor in the mixed state without pinning is determined. The dependence of the critical current density on the position in the sample is calculated.     

Effects of damping constant on gyrotropic motion and switching of vortex core in permalloy disk

We investigate the influence of damping constant on the dynamics process of the magnetic vortex in submicron-size permalloy disks by micromagnetic simulations and analytical calculations. Both of them reveal that damping constant influences the trajectory of vortex core gyrotropic motion strongly. Comparing with the case of no damping constant, the steady-state trajectory of vortex core motion becomes ellipse as the amplitude of the oscillating magnetic filed is small. The ellipse becomes more slab-sided and tilting with increasing of damping constant, and the tilting direction is also dependent on the vortex core polarization. As the amplitude of the magnetic field increases to a value, the polarization of the vortex core will reverse and a new vortex with opposite polarization will be produced. With increasing of damping constant, the minimum oscillating magnetic field amplitude H_S0 that can reverse the polarization of the vortex core increases proportionally.     

Superconducting plate in a transverse magnetic field: New state

A model is proposed for describing Cooper pairs near the transition (in temperature and magnetic field) point when their spacing is larger than their size. The essence of the model is as follows: the Ginzburg-Landau functional is written in operator form in terms of field operators of the Bose type so that the average value of the density operator gives the concentration of Cooper pairs, and the same Ginzburg-Landau expression is obtained for the Bose condensate. The model is applied to a superconducting plate with a thickness smaller than the size of a pair in a transverse magnetic field near its upper critical value H _c2. A new state is discovered that is energetically more advantageous in a certain interval in the vicinity of the transition point as compared to the Abrikosov vortex state. The wavefunction of the system in this state is of the type of the Laughlin function used in the fractional quantum Hall effect (naturally, as applied to Cooper pairs as Bose particles in our case) and corresponds to a homogeneous incompressible fluid. The energy of this state is proportional to the first power of quantity (1 ? H/H _c2) in contrast to the energy of the vortex state containing the square of this quantity. The interval of the existence of the new state is the larger, the dirtier the sample.     

Unconventional magnetic phase diagram of cuprate superconductor La2−xSrxCuO4 at quantum critical point x=1/9

We propose a new magnetic phase diagram of La_2?xSr_xCuO₄ around a quantum critical point x=1/9 based on field-cooled magnetization measurements and critical fittings. A new phase boundary T_m2(H) is discovered which buries deeply below the first order vortex melting line in the vortex solid phase. The coupling between superconductivity and antiferromagnetism is found to be attractive below T_m2(H) while repulsive above. The attractive coupling between superconducting order and static antiferromagnetic order provides compelling experimental evidence that the antiferromagnetism microscopically coexists and collaborates with the high temperature superconductivity in cuprates.     

Two regimes of vortex penetration into platelet-shaped type-II superconductors

Vortex penetration into a thin superconducting strip of a rectangular cross section is considered at an increasing applied magnetic field H _a , taking an interplay between the Bean-Livingston and the geometric barriers in the sample into account. We calculate the magnetic field H _p at which the penetration begins and show that two regimes of vortex penetration are possible. In the first regime, vortices appearing at the corners of the strip at H _a = H _p immediately move to its center, where a vortex dome starts to develop. In the second regime, the penetration occurs in two stages. In the first stage, at H _a < H _p , tilted vortices penetrate into the edge regions of the strip, where novel domes are shown to be formed at the top, bottom, and lateral surfaces. In the second stage, at H _a = H _p , the vortex propagation to the center becomes possible. The difference between the regimes manifests itself in slightly different dependences of the magnetic moment of the strip on H _a .     

Structure of the Abrikosov vortex lattice in a thin superconducting film in a parallel magnetic field

The structure of a vortex lattice in thin (d<λ, where d is the film thickness and λ is the London penetration depth) superconducting films is investigated in a magnetic field parallel to the film surface. It is shown that the stable configuration has the form of discrete vortex rows whose number changes discretely with an increase in the applied magnetic field. The entry fields H _c1 ^(N) (d) for vortex rows are calculated for N=1, 2. It is shown that the structural transition in the vortex ensemble is a second-order phase transition. A simpler method (as compared to the Monte Carlo technique) is proposed for calculating the vortex lattice parameters.     

Vortex lattice in Bi2Sr2CaCu2O8+δ well above the first-order phase-transition boundary

Measurements of non-local in-plane resistance originating from transverse vortex-vortex correlations have been performed on a Bi₂Sr₂CaCu₂O_8+δ high-T_c superconductor in a magnetic field up to 9 T applied along the crystal c-axis. Our results demonstrate that a rigid vortex lattice does exist over a broad portion of the magnetic field-temperature (H-T) phase diagram, well above the first-order transition (FOT) boundary H_FOT(T). The results also provide evidence for the vortex lattice melting and vortex liquid decoupling phase transitions, occurring above the H_FOT(T).     

Tricritical point in inhomogeneous superconductors

We show that in large period modulated alloys the tricritical points curve begins at α_c = 1/(1 + δ, where δ is the modulation amplitude. In the domain of the 2nd order transition, which appears smaller than in the homogeneous case, H_c2¹(T) and H_c3¹(T) are calculated.     

On the Meissner effect in gauge theories

The phase structure of spontaneously broken scalar electrodynamics in an external electromagnetic field is analyzed. With no external field, the spectrum comprises a scalar boson of mass m_H and a vector boson of mass m_W. If m_H ≤ m_W, it is shown that in the tree approximation, as the external field is increased, a first order phase transition to a restored symmetry phase occurs, and the critical field strength is calculated. Below the critical point the external field is completely screened, this being the analogue of the Meissner effect in superconductivity. If m_H > m_W, a third phase, characterized by vortex solutions of the field equations, occurs. Quantum effects, such as pair production in an electric field, are considered at the one (and two) loop level in the massless theory (the Coleman-Weinberg model). The leading correction to the critical magnetic field strength is calculated, and it is shown that for an external electric field the phase transition does not exist.     

Micromagnetic simulation of magnetic vortex cores in circular permalloy disks: Switching behavior in external magnetic field

Switching behaviors of magnetic vortex cores under external magnetic field in submicron circular permalloy disks have been systematically studied by using micromagnetic simulations. Simulation results show that the vortex core is stable in out-of-plane field even when it is located at the edge of the disk. The out-of-plane switching field H_sw is strongly dependent on the thickness of the disk. The core polarity and the vortex chirality can be modulated simultaneously on purpose by using a tilted field far smaller than the out-of-plane switching field H_sw. Moreover, it is found that the core polarities in asymmetric disks do not follow the direction of the z projection of the external saturation field.     

Topological objects in two-gap superconductor

We discuss the non-Abelian topological objects, in particular the non-Abrikosov vortex and the magnetic knot made of the twisted non-Abrikosov vortex, in two-gap superconductor. We show that there are two types of non-Abrikosov vortex in Ginzburg-Landau theory of two-gap superconductor, the D-type which has no concentration of the condensate at the core and the N-type which has a non-trivial profile of the condensate at the core, under a wide class of realistic interaction potential. We prove that these non-Abrikosov vortices can have either integral or fractional magnetic flux, depending on the interaction potential. We show that they are described by the non-Abelian topology π₂(S ²) and π₁(S ¹), in addition to the well-known Abelian topology π₁(S ¹). Furthermore, we discuss the possibility to construct a stable magnetic knot in two-gap superconductor by twisting the non-Abrikosov vortex and connecting two periodic ends together, whose knot topology π₃(S ²) is described by the Chern-Simon index of the electromagnetic potential. We argue that similar topological objects may exist in multi-gap or multi-layer superconductors and multi-component Bose-Einstein condensates and superfluids, and discuss how these topological objects can be constructed in MgB₂, Sr₂RuO₄, ³He, and liquid metallic hydrogen.     

Breakdown of the superheated Meissner state and spontaneous vortex nucleation in type II superconductors

The theory of local stability of the superheated Meissner state presented in an earlier paper is supplemented by an investigation of global stability in two dimensions. We conclude that flux penetration cannot be delayed beyond the fieldH _s1 where local instability sets in. Various new two-dimensional Ginzburg-Landau solutions, obtained numerically, are discussed. These include the lowest saddle point separating the Meissner from the normal and vortex state and a solution resembling the “nascent vortex state” whose existence was postulated recently by Walton and Rosenblum. Using these solutions the process of spontaneous vortex nucleation is discussed.     

Doping dependence of the vortex glass and sublimation transitions in the high-T<Subscript>c</Subscript> superconductor La<Subscript>2-x</Subscript>Sr<Subscript>x</Subscript>CuO<Subscript>4</Subscript> as determined from macroscopic measurements

Magnetization and ac-susceptibility measurements are used to characterize the mixed phase of the high-temperature cuprate superconductor La_2-xSr_xCuO₄ over a large range of doping (0.075 0.20). The first order vortex lattice phase transition line H_FOT(T), the upper critical field H_c2(T) and the second peak H_sp(T) have been investigated up to high magnetic fields (8 Tesla applied perpendicular to the CuO₂ planes). Our results reveal a strong doping dependence of the magnetic phase diagram, which can mainly be explained by the increasing anisotropy with underdoping. Within our interpretation, the first order vortex lattice phase transition is due to the sublimation (rather than melting) of the vortex lattice into a gas of pancake vortices, whereas the second peak is related to the transition to a more disordered vortex glass state.     

Abrikosov-to-Josephson vortex lattice crossover in heavy fermion CeCoIn5

We present torque magnetization measurements on the quasi-2D heavy fermion superconductor CeCoIn₅ at temperatures down to 20 mK and magnetic fields up to 18?T. At orientations with the magnetic field perpendicular to the conducting planes, a prominent vortex lattice peak effect is present at around 0.5H _c2. The peak effect gradually disappears upon rotating the field into the plane parallel orientation. We interpret the absence of the peak effect for the plane parallel case as a transformation of the Abrikosov lattice into a Josephson vortex state, favored by the Pauli paramagnetic limit in CeCoIn₅ together with the unusually large condensation energy. Additionally, we do not observe flux avalanches as found in organic superconductors and suggest that the complete absence of vortex activity in the plane parallel field orientation is crucial for the formation of Fulde–Ferrell–Larkin–Ovchinnikov superconductivity in CeCoIn₅.     

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.