Semi-infinite Superconductor Research Articles

Superheating fields of semi-infinite superconductors and layered superconductors in the diffusive limit: structural optimization based on the microscopic theory

The superheating field of the Meissner state is thought to determine the theoretical field-limit of superconducting accelerator cavities. We investigate of semi-infinite superconductors and layered structures in the diffusive limit using the well-established quasiclassical Green’s function formalism of the BCS theory. The coupled Maxwell–Usadel equations are self-consistently solved to obtain the spatial distributions of the magnetic field, screening current density, penetration depth, pair potential, and . For a semi-infinite superconductor in the diffusive limit, we obtain at the temperature . Here, is the thermodynamic critical-field at the zero temperature. By laminating a superconducting film (S) with the thickness on a semi-infinite superconductor (Σ), we can engineer of the layered structure. When is the optimum thickness , can be larger than that of the simple semi-infinite superconductors made from the S and Σ materials: . The present study addresses the calculation of of the dirty heterostructure using the microscopic theory from beginning to end for the first time, which contributes to the microscopic understanding of the surface engineering for pushing up the accelerating gradient of superconducting cavities for particle accelerators.

Theory of Surface Andreev Bound States and Odd-Frequency Pairing in Superconductor Junctions

In this review, we mention about the theoretical achievement of physics of surface Andreev bound state (SABS) and odd-frequency pairing during this quarter century based on our research. Currently, topological superconductors with Majorana edge state have become a central issue in condensed matter physics. It is noted that Majorana edge state is nothing but a zero energy surface Andreev bound state (ZESABS) and the study of ZESABS has started in the context of the pairing symmetry of high Tc cuprate. We mention our obtained conductance formula of quasiparticle tunneling and Josephson current in unconventional superconductor junctions through ZESABS have already captured the essence of the charge transport profile via the Majorana fermion. We also mention that in the presence of ZESABS, odd-frequency pairing like spin-triplet s-wave or spin-singlet p-wave is hugely enhanced near the surface or interface. We further analyze the spectroscopy of the induced odd-frequency pairings by extending bulk-boundary correspondence in ZESABS. If we consider a semi-infinite superconductor with ZESABS, the induced odd-frequency pair amplitude can be expressed by a c-number defined in bulk, which we call spectral bulk-boundary correspondence. The odd-frequency pair amplitude is classified into singular part and regular one, the former part is proportional to $\sim 1/z$ (z is a complex frequency) reflecting the presence of ZESABS, and the latter one is proportional to $\sim z$ . The coefficient of the latter one diverges at the topological phase transition point. Analyzing the spectral behavior of the odd-frequency pairing is useful to understand the critical phenomena near topological transition points.

Superfluid flow in disordered superconductors with Dynes pair-breaking scattering: Depairing current, kinetic inductance, and superheating field

We investigate the effects of Dynes pair-breaking scattering rate $\Gamma$ on the superfluid flow in a narrow thin-film superconductor and a semi-infinite superconductor by self-consistently solving the coupled Maxwell and Usadel equations for the BCS theory in the diffusive limit for all temperature $T$, all $\Gamma$, and all superfluid momentum. We obtain the depairing current density $j_d(\Gamma, T)$ and the current-dependent nonlinear kinetic inductance $L_k(j_s, \Gamma, T)$ in a narrow thin-film and the superheating field $H_{sh}(\Gamma, T)$ and the current distribution in a semi-infinite superconductor, taking the nonlinear Meissner effect into account. The analytical expressions for $j_d(\Gamma,T)|_{T=0}$, $L_k(j_s, \Gamma, T)|_{T=0}$, and $H_{sh}(\Gamma, T)|_{T=0}$ are also derived. The theory suggests $j_d$ and $H_{sh}$ can be ameliorated by reducing $\Gamma$, and $L_k$ can be tuned by a combination of the bias current and $\Gamma$. Tunneling spectroscopy can test the theory and also give insight into how to engineer $\Gamma$ via materials processing. Implications of the theory would be useful to improve performances of various superconducting quantum devices.

Time-dependent Ginzburg-Landau treatment of rf magnetic vortices in superconductors: Vortex semiloops in a spatially nonuniform magnetic field.

We apply time-dependent Ginzburg-Landau (TDGL) numerical simulations to study the finite frequency electrodynamics of superconductors subjected to an intense rf magnetic field. Much recent TDGL work has focused on spatially uniform external magnetic fields and largely ignores the Meissner state screening response of the superconductor. In this paper, we solve the TGDL equations for a spatially nonuniform magnetic field created by a point magnetic dipole in the vicinity of a semi-infinite superconductor. A two-domain simulation is performed to accurately capture the effect of the inhomogeneous applied fields and the resulting screening currents. The creation and dynamics of vortex semiloops penetrating deep into the superconductor domain are observed and studied, and the resulting third-harmonic nonlinear response of the sample is calculated. The effect of pointlike defects on vortex semi-loop behavior is also studied. This simulation method will assist our understanding of the limits of superconducting response to intense rf magnetic fields.

Phase crystals

Superconductivity owes its properties to the phase of the electron pair condensate that breaks the $U(1)$ symmetry. In the most traditional ground state, the phase is uniform and rigid. The normal state can be unstable towards special inhomogeneous superconducting states: the Abrikosov vortex state, and the Fulde-Ferrell-Larkin-Ovchinnikov state. Here we show that the phase-uniform superconducting state can go into a fundamentally different and more ordered non-uniform ground state, that we denote as a phase crystal. The new state breaks translational invariance through formation of a spatially periodic modulation of the phase, manifested by unusual superflow patterns and circulating currents, that also break time-reversal symmetry. We list the general conditions needed for realization of phase crystals. Using microscopic theory we then derive an analytic expression for the superfluid density tensor for the case of a non-uniform environment in a semi-infinite superconductor. We demonstrate how the surface quasiparticle states enter the superfluid density and identify phase crystallization as the main player in several previous numerical observations in unconventional superconductors, and predict existence of a similar phenomenon in superconductor-ferromagnetic structures. This analytic approach provides a new unifying aspect for the exploration of boundary-induced quasiparticles and collective excitations in superconductors. More generally, we trace the origin of phase crystallization to non-local properties of the gradient energy, which implies existence of similar pattern-forming instabilities in many other contexts.

Transformation of the polarization of the electromagnetic waves reflected from the layered superconductors in an external dc magnetic field

The reflection of transverse-electric and transverse-magnetic polarized waves from the surface of a semi-infinite layered superconductor in the presence of external dc magnetic field is studied theoretically. The superconducting layers are assumed to be perpendicular to the boundary of the sample. Due to a strong anisotropy of the irradiated superconductor surface, a transformation of the wave polarization occurs upon the reflection. It is shown that, despite a relatively small penetration depth of the dc magnetic field, it qualitatively affects the field distribution of the electromagnetic wave and, therefore, the reflection and transformation coefficients. Thus, the external dc magnetic field can serve as an efficient tool to control the transformation of the wave polarization. The analytical expressions for the reflection and transformation coefficients are obtained and the parameters at which the most effective transformation of the transverse-electric to transverse-magnetic waves and vice versa occurs are found.

THE FORCE BETWEEN A MONOLAYER OF MAGNETIC PARTICLES AND A SUPERCONDUCTOR IN THE MIXED STATE

The interaction between a monolayer of fine ferromagnetic particles and a semi-infinite superconductor has been investigated in the mixed state. The frozen and diamagnetic images model was employed to calculate the levitation force as a function of the levitation height as well as the temperature of the monolayer under the zero-field-cooled (ZFC) and the field-cooled (FC) conditions. Results showed the well-known monotonic decrease of the levitation force as a function of the levitation height while it increases rapidly as a function of temperature up to saturation. As a result of the first order approximation used in our calculations in which the interaction was represented by the forces between the magnetic dipoles and their images, the levitation force was dominated by the diamagnetic properties rather than the flux pinning effects of the superconductor for small values of levitation heights compared to the initial field cooling height.

Force between a magnetic tip and an inhomogeneous superconductor at zero field

The force exerted by a semi-infinite inhomogeneous superconductor with a planar interface to vacuum on a magnetic tip is studied theoretically in the absence of external magnetic fields. It is shown that the force has a contribution from inhomogeneities due to material defects with unique characteristics. Defects are taken into account in the London limit by allowing the mass parameter to vary spatially. The contribution from defects to the force is calculated analytically to first order in the deviation of the mass parameter from its constant value for the homogeneous superconductor, assuming that the tip is a point dipole perpendicular to the interface, and that it does not spontaneously create vortex matter. Random point defects and linear localized defects are considered phenomenologically. For each defect type the force dependence on the dipole position coordinates is obtained, and the force magnitudes are estimated numerically. The predictions for the dependence of the linear defect force on the dipole lateral position are found to agree qualitatively with experiment.

Thickness-dependent pinning in a superconductor thin film

The dependence of critical current density jc on film thickness (d) is calculated for the simple configuration of a semi-infinite finite-thickness superconductor film and a single vortex. This film-thickness dependence reflects the impact of vortex broadening in a thin film on its pinning by the film’s edge or an internal surface. The calculated film-thickness variation of jc in the range d∕λ<2, where λ denotes the penetration depth, is found to be proportional to d−1∕2. A parameter-free estimate of the ensuing critical current is within a factor of 3 of observed values in YBa2Cu3O7 coated conductors. It is argued that this type of film-thickness scaling applies beyond the simple configuration considered here.

Generalized painlevé equation and superconductivity. asymptotic behavior of unbounded solutions

Consider the nonlinear scalar differential equation 1 p(t) (p(t)y′(t))′=−(t)f(t,y(t),p(t)y′(t)) , where p and q are positive on (0, 1), “singular” at t = 0. 1 and/or y = 0 and f ϵ C( R + x R + x R +, R +), associated to the boundary conditions x(0)=a⩾0, lim t→∞ p(t)x′(t)=b⩾0 . We prove the existence of a global, positive and strictly increasing solution x = x( t) of this BVP, such that its “derivative” y = p( t) x( t) is also a positive and strictly decreasing map, under a natural growth in f of “superlinear” type. Our approach is based on the analysis of the corresponding vector field on the face-plane ( x, px′) and the well-known Knesser's type (continuum) technique. As an application, we study the generalized Painlevé equation in a semi-infinite interval (0, +∞) x″= x 2n+1 − (t−c) 2k+1x 2(n−k)−1 which, in turn, models a superheating field attached to a semi-infinite superconductor. Namely, we prove the existence of a (global) strictly positive solution satisfying x′(0)=0, lim t→∞ x(t) t β = M, and lim t→∞ t βx′ (t) = βM , where 0 < β < 1, M > 0, and n, k ϵ N with k < n are arbitrary constants.

Magnetic response of hard type-II superconductors with a semicircular indentation

The magnetic response of superconductors with a semicircular indentation in parallel field is studied. The indentation is placed in a long superconductor wedge. The Meissner and vortex fields are calculated for any indentation radius in the London background. This allowed us to obtain the energy and the forces acting on a vortex in a given position inside the superconductor. The entrance magnetic field dependence on the indentation radius is calculated and compared with the entrance field for an indentation free semi-infinite superconductor. The role of the indentation on the surface barrier and on the vortex dynamics is also discussed.

Spontaneous spin-polarized currents in superconductor-ferromagnetic metal heterostructures

We study a simple microscopic model for thin, ferromagnetic, metallic layers on semi-infinite bulk superconductor. We find that for certain values of the exchange spliting, on the ferromagnetic side, the ground states of such structures feature spontaneously induced spin polarized currents. Using a mean-field theory, which is selfconsistent with respect to the pairing amplitude $\chi$, spin polarization $\vec{m}$ and the spontaneous current $\vec{j}_s$, we show that not only there are Andreev bound states in the ferromagnet but when their energies $E_n$ are near zero they support spontaneous currents parallel to the ferromagnetic-superconducting interface. Moreover, we demonstrate that the spin-polarization of these currents depends sensitively on the band filling.

A Semi-Infinite Superconductor with Monotonous Tc Variation

The modified London equation for an inhomogeneous superconductor is used to study the magnetic field behavior as well as the surface barrier structure of semi-infinite type-II superconductors with monotonous T c variation. We developed a time dependent modified London equation for an inhomogeneous superconductor, and extented this model to study the effects of the surface barrier on the double critical state model of the system. Analytical expressions for the penetrating field, the flux line fields, the superheating field and the effective penetration depth are obtained.

Magnetic field of an in-plane vortex inside and outside a layered superconducting film

In the present work we study an anisotropic layered superconducting film of finite thickness. The film surfaces are considered parallel to the $bc$ face of the crystal. The vortex lines are oriented perpendicular to the film surfaces and parallel to the superconducting planes. We calculate the local field and the London free energy for this geometry. Our calculation is a generalization of previous works where the sample is taken as a semi-infinite superconductor. As an application of this theory we investigate the flux spreading at the superconducting surface.

On a family of solutions of the second Painlevé equation related to superconductivity

In a recent paper devoted to the study of the superheating field attached to a semi-infinite superconductor, Chapman [1] constructs a family of approximate solutions of the Ginzburg–Landau system. This construction, based on a matching procedure, implicitly uses the existence of a family of solutions depending on a parameter c ∈ IR of the Painleve equation in a semi-infinite interval (0, +∞) formula here with a Neumann condition at 0 formula here and having a prescribed behaviour at +∞ formula here In this paper we prove the existence of such a family of solutions and investigate its properties. Moreover, we prove that the second coefficient in Chapman's expansion of the superheating field is finite.

An Inverse Problem for the ac Magnetic Force Microscopy of Superconductors

A basic inversion problem for the magnetic force microscopy (MFM) of a semiinfinite superconductor in the Meissner state is formulated, ac Detection is assumed, with the MFM tip oscillating at angular frequency Ω. The coupling of all electrodynamic fields is treated, including a normal-current density in the superconductor. Under certain assumptions on the tip and superconductor geometry, a unique penetration depth λ(z) can be recovered from onedimensional force gradient measurements. This development opens new possibilities for the nondestructive evaluation of superconducting crystals and films.

Levitation force on a permanent magnet over a superconducting plane: Modified critical-state model

We consider a model system of a permanent magnet above a semi-infinite superconductor, introduce a modified critical-state model, and carry out derivations of the levitation force acting on the magnet. A key feature of the modification allows the current density to be less than the critical value. Theoretical results show an exponential relationship between the force acting on the magnet and the distance between the magnet and the superconductor. Analytical expressions are developed for permanent magnets in the form of a point dipole, a tip of a magnetic force microscope, and a cylindrical magnet. In the latter case, the exponential relationship has been observed in numerous experiments but without previous interpretation.

Magnetic levitation force of semi-infinite type-II superconductors.

The levitation force acting on a point magnetic dipole above a semi-infinite type-II superconductor in both the Meissner and mixed states is studied. A formalism is developed for axisymmetric problems using London theory. The magnetostatic interaction energy and corresponding force can be put into closed form for such problems for arbitrary height of the magnetic point source. The results for stray fields have ready application to magnetic-force microscopy (MFM) with point probes. The results are useful for a range of experiments including the low-temperature MFM imaging of vortices and decoration measurements. Special cases include earlier results and previous approximations are obviated.

Some wave phenomena in elastic superconductors

This article formulates some wave phenomena in elastic superconductors, in which the effect of interaction of elastic waves with electromagnetic fields are taken into account. The effect of normal conduction current is also taken into consideration. It is shown that, due to the effect of dynamic motion, transverse magnetic waves may be induced by a transverse elastic wave in an elastic superconductor. Magnetic surface waves may also be induced by the Rayleigh elastic surface wave in the superconductor. The magnitude of the magnetic induction field associated with the induced magnetic wave can be significant at a certain excitation frequency. Furthermore, by considering the coupling effect between the elastic wave and the electromagnetic field, a study is given to investigate electromagnetoelastic surface waves in a semi-infinite elastic superconductor with the aid of the theoretical formulation given in the article. It is shown that the characteristics of the coupled wave in a superconductor can be quite different from that in a perfect conductor due to the presence of the Meissner effect in the superconductor.

Magnetic force microscopy signal of flux line above a semi-infinite type II-superconductor

We have examined a single flux line in the semi-infinite type-II superconductor. The stray magnetic field of the flux line has been calculated. We have found that the vertical force exerted on a magnetic force microscopy (MFM) tip from the flux line is measurable by currently existing MFM. Two types of magnetic tips were taken into consideration, solid and thin film tips. For example, with a Cobalt film of the thickness of 100 nm and 30 nm on a tip, we found a vertical force of 4*10−10 N and 1.5*10−10 N, respectively. The lateral force exerted on a tip by the flux line was also calculated. The lateral force must be small enough to prevent the flux line from becoming depinned.