London Approximation Research Articles

Vibrational motion of a vortex lattice induced near the surface of a type-II superconductor by an external low-frequency ac magnetic field

The theoretical description and experimental study of a reversible vibrational vortex motion induced in a type-II superconductor by a weak low-frequency ac magnetic field are discussed. The case where the external magnetic field and vortex lines are parallel to the surface of the superconductor is considered. The equilibrium positions of vortices near a surface, the amplitude of their vibrations, and the contribution of this process to the dynamic permeability ${\ensuremath{\mu}}_{v}$ are calculated. In these calculations, the interactions of vortices with each other, with the surface, and with pinning centers are considered within the discrete vortex lattice model. The calculations in the London approximation are compared with the results obtained within a more accurate model, including the spatial variation of the order parameter in the vortex core by means of a variational function. It is shown that the structural parameters of the vortex lattice near the surface depend on the accuracy of the description of the intervortex interactions, but the contribution ${\ensuremath{\mu}}_{v}$ appears to be model independent. It is important that this conclusion is valid in the case of the exponential approximation for the interaction between vortex rows. In this approximation, the problem under consideration is analytically examined, and formulas describing the dependences of the parameters of the vortex lattice and the contribution ${\ensuremath{\mu}}_{v}$ on the external dc magnetic field and the characteristics of a superconductor are derived. The theoretical results are compared with experimental data for a $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{y}$ high-${T}_{c}$ superconducting single-crystal plate. The magnetic field is parallel to the plate, i.e., to the crystallographic $ab$ plane. A strong anisotropy of the properties and shape of the crystal is considered when analyzing the experimental data. A procedure for the determination of the experimental dependence ${\ensuremath{\mu}}_{v}(H)$ is described. It is shown that the reversible vibrations of vortices occur near the planes of the single-crystal plate in the direction of the crystallographic $c$ axis, whereas vortices enter its lateral ends and leave them (along the structural layers) even at a small magnetic field amplitude of about 1 Oe. It is found that the developed theoretical description reproduces the experimental data ${\ensuremath{\mu}}_{v}(H)$ in the entire studied range of temperatures and magnetic fields under the assumptions that pinning is isotropic and the length of interaction of vortices with pinning centers is close to the coherence length.

Low-energy excitations in helium-like dimer within an exact diagonalization approach

The non-relativistic quantum-mechanical problem on bound states of four interacting spinless electrons moving in the Coulomb field of two attractive point centers with Z0=2 spaced at fixed distance R0 (4He2 dimer within Born–Oppenheimer–Heitler–London approximation) has been solved rigorously through exact diagonalization (expansion on truncated orthonormalized basis) method. The four-spin-conditioned relativistic corrections (of order ∼1/c2) to the ground state level of the dimer have been calculated with exact diagonalization on spin cluster, the spectrum and eigenvectors of the spin problem have been obtained. It is shown that pair spin coupling is antiferromagnetic with exchange constant of 12 K (it provides antisymmetry of the spin-singlet ground-state wave function for isolated 4He atom with two fermions coupled within the unitary spatially symmetric shell), but within four-electron shell of 4He2 dimer this fact yields the quintet ground state of the four-spin cluster totally antisymmetric relative to pair permutations. The exchange within the interatomic bond depends on the interatomic distance, so that there is a coupling between spin and phonon degrees of freedom which leads to renormalization of phonon spectrum in condensed phase as compared to the corresponding spinless medium. This effect can be interpreted as a direct analog of translation-rotation interaction in molecular cryocrystals.

Nanoscale magnetization of a single vortex in d-wave superconductors

A finite-size scaling of the nanoscale magnetization m on size averaging R of a single vortex in d-wave bulk superconductor is developed using quasiclassical Eilenberger equations. Nanoscaling is anchoring around the linear London approximation for bulk superconductors. Comparing the results with those obtained in local nonlinear approach demonstrated the importance of the nonlocal contribution. Temperature dependences of two-point correlation function χ(T,R1,R2)=m(T,R2)/m(0,R2)−m(T,R1)/m(0,R1) with R2 > R1 and one-point function χ(T, R1 → ∞, R2) are calculated. It is found that χ(T, R1, R2), R2 > R1, is a nonmonotonous function of temperature and changes sign at high temperatures. This nonmonotonous temperature dependence can be understood as a result of competition between various effects i) Volovik effect and nonlocal corrections to superconducting electron density dominating in low temperature range, and ii) current-induced suppression of the order parameter dominating at high temperatures. The introduced nonmagnetic disorder greatly suppresses the low temperature nonlocal and nonlinear effects, leaving the order parameter effects to prevail in the whole temperature range. Nonlocal pairing and tunneling effects are investigated at the superconductor - normal metal border by considering a d-wave superconducting dot (d-dot) inside a normal diffusive metal. These effects result in a suppression of the supercurrent in the vortex core and are essential in nanodots with relatively small sizes. At sizes larger than a temperature dependent characteristic length the nanoscale physics transforms into bulk solution.

Vortex crossing, trapping, and pinning in superconducting nanowires of aNbSe2two-dimensional crystal

Nanowires of two-dimensional (2D) crystals of type-II superconductor ${\mathrm{NbSe}}_{2}$ prepared by electron-beam lithography were studied, focusing on the effect of the motion of Abrikosov vortices. We present magnetoresistance measurements on these nanowires and show features related to vortex crossing, trapping, and pinning. The vortex crossing rate was found to vary nonmonotonically with the applied field, which results in nonmonotonic magnetoresistance variations in agreement with theoretical calculations in the London approximation. Above the lower critical field ${H}_{c1}$ the crossing rate is also influenced by vortices trapped by sample boundaries or pinning centers, leading to sample-specific magnetoresistance patterns. We show that the local pinning potential can be modified by intentionally introducing surface adsorbates, making the magnetoresistance pattern a ``magnetofingerprint'' of the sample-specific configuration of vortex pinning centers in a 2D crystal superconducting nanowire.

Magnetostatic mechanism for control of chirality of magnetization distributions

It has been shown that the magnetostatic interaction in an inhomogeneous medium leads to the removal of the chiral degeneracy of magnetic distributions. Noncollinear states of two magnetic dipoles and a helical cycloid placed over a superconducting half-space have been considered as examples. The influence of a finite penetration depth of the magnetic field on the efficiency of removal of the chiral degeneracy has been studied in the framework of the London approximation.

Magnetization of layered superconductors with ferromagnetic nanorods

The magnetization of layered high-temperature superconductors (HTSC) with ferromagnetic nanorods as bulk pinning centers is studied in the 2D model of layered HTSC by using Monte Carlo method. Magnetic part of the interaction energy between a ferromagnetic cylinder of arbitrary radius and constant magnetization and an Abrikosov vortex was calculated in London approximation. The magnetization at different radius of magnetic defects was investigated. The results of calculations were compared with the results for extended nonmagnetic defects.

Mixed state and magnetization of a thin type II superconducting film in the applied parallel magnetic field: Variational technique of the inclusion of the contribution from the core of a vortex

The equilibrium vortex structure and magnetization for a thin superconducting film (with the Ginzburg-Landau parameter κ ≫ 1) in the magnetic field applied parallel to the film surface have been calculated with the inclusion of the spatial inhomogeneity of the order parameter within the core of a vortex by using a variational function. The variational approach allows obtaining more accurate values of the core self-energy in comparison to those found in the London approximation (including its changes with the growth of the vortex lattice density) and gives a more adequate picture of the interactions between vortices and of their interaction with the surface. The formula determining a correction to the London expression for the magnetic flux related to vortices in the film is derived. The comparison of the calculations performed using these two models demonstrates that the vortex core structure provides an important contribution if the film thickness is d < 50ξ. For such thin films, the novel size effect manifesting itself in a significant difference between the values of magnetization for the film and a bulk sample within the magnetic field range 0.08Hc2 < H < 0.5Hc2 has been found. It has also been demonstrated that if d decreases to d = 10ξ, the splitting of the central vortex row occurs as a second order phase transition.

Dipole magnetic flux in two-gap superconducting thin film in Ginzburg–Landau theory

In two-gap superconducting thin film, there are two kinds of pancake vortices: one is arisen from the vortex in the super-current of band I (it has greater gap) and the screening electrical current; another is arisen from the vortex in the super-current of band II and the screening electrical current. In London approximation, both carry fractional quantum magnetic flux ϕ01 and ϕ02 respectively. Except the change of the superconducting condensation energy in the region of pancake vortex, the energy of the pancake vortices include the magnetic energy and the Josephson type coupling energy between the Cooper pair of band I and II. Because the inter-bands Josephson coupling energy is infinite increase with the size of the film, the fractional magnetic flux cannot appear alone except in the boundary regions. The element for the investigation of the vortex in two-gap superconducting thin film is the pairs (I,s)+(II,s) and the pairs (II,+1)+(II,−1). We have discussed the magnetic interaction, the Josephson coupling interactions and the structure character of these pairs in this paper.

Magnetization of Layered High-temperature Superconductors with Extended Ferromagnetic Defects

The magnetization of layered high-temperature superconductors (HTSC) with ferromagnetic nanorods as bulk pinning centers is studied in the 2D model of layered HTSC by using a Monte Carlo method. Magnetic part of the interaction energy between a ferromagnetic cylinder of arbitrary radius and constant magnetization and an Abrikosov vortex was calculated in London approximation. The periodic and non-periodic lattices of magnetic defects were considered. The magnetization was investigated at different radii of magnetic defects. The results of calculations were compared with the results for extended nonmagnetic defects.

Phenomenological correlations in high-temperature superconductors

An interpretation of the quadratic parameter of the Ginzburg-Landau theory of superconductivity is presented in this paper. The negative term in the potential, which allows the spontaneous symmetry breaking, is interpreted as a direct contribution from the energy gap at the Fermi surface to the effective potential. As a result, in the London approximation of the Ginzburg-Landau theory for type-II superconductors, a strong correlation is predicted and observed between the upper critical field at zero kelvin and the critical temperature in high-temperature superconductors.

Effect of magnetic nanoparticles on the mechanical properties of type-ii superconductors

In this paper, the effect of magnetic nanoparticles on the mechanical properties of a type-II superconductor is investigated both theoretically and numerically. Magnetic part of the pinning force associated with the interaction between a finite-size spheroidal magnetic inclusion and an Abrikosov vortex is calculated in the London approximation. It is found that the size and shape of magnetic nanoparticles result in different enhancements of vortex pinning in large-κ type-II superconductors. Meanwhile, the screening current induced by a magnetic spheroid suffer the action of Lorentz force, which will lead to prestress in the superconductor, so further numerical calculations are needed to explore the interaction between the spheroidal magnetic particle and superconductor. The distribution of displacement, stress and strain in the superconductor are finally obtained. It is shown that different sizes and shapes of nanoparticles also can change the distributions of these quantities.

Magnetic and transport currents in thin film superconductors of arbitrary shape within the London approximation

We present a numerical method to calculate the current distribution and the magnetic field in a superconducting thin plate within the London approximation. The superconductor can have any two-dimensional shape, including multiply connected ones. The modeling can take into account transport currents fed on the superconductor and externally applied fields, and is valid for any value of the London penetration depth λ. From this modeling, we present current and field distributions for several geometries of the superconductor (including corners, turns, and holes), comparing them with these of a straight strip and with previous results in some limits. We show how the current density accumulates in the inner corners of a turn and how this accumulation depends on λ. We also study how far the presence of a turn or hole in a straight strip modifies the current (and field) distribution observing significant differences depending on λ, on the geometry of the turns and on the external conditions to which the superconductor is subjected. All these results may have implications in the design of single photon detectors, superconducting based mass spectrometers, as well as in the ability of tuning magnetic traps using thin superconducting films, among other applications.

Magnetic Measurements and Kinetic Energy of the Superconducting Condensate in SmBa2Cu3O7−δ

We report in-field kinetic energy results in the temperature region closely below the transition temperature of two differently prepared polycrystalline samples of the superconducting cuprate SmBa$_{\text{2}}$Cu$_{\text{3}}$O$_{7-\delta}$. The kinetic energy was determined from magnetization measurements performed above the irreversibility line defined by the splitting between the curves obtained according the ZFC and FC prescriptions. The results are analyzed in the intermediate field regime where the London approximation can be used for describing the magnetization. From the analysis, estimations were carried out for the penetration depth and the upper critical field of the studied samples.The difference between the kinectic energy magnitudes for the two studied samples is ascribed to effects from granularity.

Lawrence–Doniach model in London approximation as a system of 2D Coulomb particles of two kinds

Thermodynamical properties of layered superconductors with Josephson coupling in zero magnetic field are discussed. It is shown that the Lawrence–Doniach model of layered superconductors in the London approximation can be presented as a system of classical Coulomb particles of two kinds. This representation includes both Josephson and 2D vortex subsystems in absolutely symmetrical way. The model was analyzed by means of the real-space renormalization group approach and in the mean field approximation. It is found that there is no a second order phase transition in the system. Instead of this the model demonstrates the qualitative change of the system properties which can look as a phase transition for experimental purposes.

Structural phase transition in the mixed state of superconducting films in a parallel magnetic field

A thin film of a second-kind superconductor in a magnetic field parallel to the surface of the film is considered in the London approximation. It has been shown that if bulk pinning is absent and the suppression of super-conductivity by the magnetic field is negligible, the splitting of a vortex chain in the film occurs as a structural phase transition either of the first or second order, depending on the ratio of the thickness of the film d to the penetration depth of the magnetic field λ. The ratio d/λ, and thereby the character of the transition in the vortex lattice, can be changed by varying the temperature. The corresponding critical thicknesses of films and field ranges in which this effect can be observed experimentally have been calculated.

Orientational phase transitions of a lattice of magnetic dots embedded in a London-typesuperconductor

In recent experiments, structured arrays of ferromagnetic nanoparticles in the bulk ofa superconductor have been fabricated. We present the theory of orientationalphase transitions in a planar regular lattice of nanoscale ferromagnetic particlesembedded in a superconductor. In the London approximation, we show that theinteractions between ferromagnetic particles can lead to either a parallel or antiparallelspin alignment depending on the ratio of the interparticle distance, the Londonpenetration depth and the temperature. The extension of the results to systemsof ferromagnetic nanoparticles with more complicated geometries is discussed.

Modified London Equation in N-gap Superconductors

Vortices in the N-gap (N ≥ 1) superconductor is studied by Dirac δ function. The topological structure of vortices can be characterized by Brouwer degree and Hopf index, i.e. (β1η1, …, βNηN). The magnetic flux and circulation of vortices without London approximation in the N-gap superconductor are calculated. The universal modified London equation is given for the case that each individual condensate is conserved or not conserved.

MAGNETIC RELAXATION OF SUPERCONDUCTING QUANTUM DOT AND TUNNELING OF ELECTRON IN A MAGNETIC FIELD

Quantum tunneling of vortices had been found to be an important novel phenomena for description of low temperature creep in high temperature superconductors (HTSCs). We speculate that quantum tunneling may be also exhibited in mesoscopic superconductors due to vortices trapped by the Bean-Livingston barrier. The London approximation and method of images is used to estimate the shape of the potential well in superconducting HTSC quantum dot. To calculate the escape rate we use the instanton technique. We model the vortex by a quantum particle tunneling from a two-dimensional ground state under magnetic field applied in the transverse direction. The resulting decay rates obtained by the instanton approach and conventional WKB are compared revealing complete coincidence with each other.

QUANTUM DOTS BASED ON PARABOLIC QUANTUM WELLS: IMPORTANCE OF ELECTRONIC CORRELATIONS

We present measurements and theoretical interpretation of the magnetic field dependent excitation spectra of a two-electron quantum dot. The quantum dot is based on an Al x Ga 1-x As parabolic quantum well with effective g⋆-factor close to zero. Results of tunneling spectroscopy of the four lowest states are compared to exact diagonalization calculations and a generalized Heitler–London approximation and good agreement is found. Electronic correlations, associated with the formation of an H 2-type Wigner molecule, turn out to be very important in this system.

Interaction between Superconducting Vortices and a Bloch Wall in Ferrite Garnet Films

A theoretical model for how Bloch walls occurring in in-plane magnetized ferrite garnet films can serve as efficient magnetic micromanipulators is presented. As an example, the walls' interaction with Abrikosov vortices in a superconductor in close contact with a garnet film is analyzed within the London approximation. The model explains how vortices are attracted to such walls, and excellent quantitative agreement is obtained for the resulting peaked flux profile determined experimentally in NbSe(2) using high-resolution magneto-optical imaging of vortices. In particular, this model, when generalized to include charged magnetic walls, explains the counterintuitive attraction observed between vortices and a Bloch wall of opposite polarity.