Modified London Equation Research Articles

Far-infrared electrodynamics of thin superconducting NbN film in magnetic fields

We studied a thin superconducting NbN film in magnetic fields up to 8 T above the zero-temperature limit by means of time-domain terahertz and scanning tunneling spectroscopies in order to understand the vortex response. Scanning tunneling spectroscopy was used to determine the optical gap and the upper critical field of the sample. The values obtained were subsequently used to fit the terahertz complex conductivity spectra in the magnetic field in the Faraday geometry above the zero-temperature limit. These spectra are best described in terms of the Coffey–Clem self-consistent solution of a modified London equation in the flux creep regime.

Magnetic structure of individual flux vortices in superconducting MgB2derived using transmission electron microscopy

Images of flux vortices in superconductors acquired by transmission electron microscopy should allow a quantitative determination of their magnetic structure but so far, only visual comparisons have been made between experimental images and simulations. Here, we make a quantitative comparison between Fresnel images and simulations based on the modified London equation to investigate the magnetic structure of flux vortices in MgB2. This technique gives an absolute, low-field (~30 Oe) measurement of the penetration depth from images of single vortices. We found that these simulations gave a good fit to the experimental images and that if all the other parameters in the fit were known, the penetration depth for individual vortices could be measured with an accuracy of +/- 5 nm. Averaging over 17 vortices gave a penetration depth in the ab plane of 113 +/- 2 at 10.8 K assuming that the entire thickness of the sample was superconducting. The main uncertainty in this measurement was the proportion of the specimen which was superconducting. Allowing for a non-superconducting layer of up to 50 nm thickness on the specimen surfaces gave a penetration depth in the range 100-115 nm, close to values of 90 +/- 2 nm obtained by small-angle neutron scattering and 118-138 nm obtained by radio-frequency measurements. We also discuss the use of the transport of intensity equation which should, in principle, give a model-independent measure of the magnetic structure of flux vortices.

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.

Topological Vortices in a Spin-Triplet Superconductor

Based on the complex three-component order parameter model of a spin-triplet superconductor, by using the ϕ-mapping theory, we derive a new equation describing the distribution of the magnetic field for vortices, which can be reduced to the modified London equation in the case of | ψ2|2 = |ψ3|2 = 0 and Wl1 = 1. A magnetic flux quantization condition for vortices in a spin-triplet superconductor is also derived, which is topological-invariant. Furthermore, the branch processes during the evolution of the vortices in a spin-triplet superconductor are discussed. We also point out that the sum of the magnetic flux quantization that those vortices carried is 2nΦ0 (Φ0 is the unit magnetic flux), that is to say, the sum of winding number is even, which needs to be proved by experiment.

The modified London equation, Abrikosov-like vortices and knot solitons in two-gap superconductors

We derive the exact modified London equation for the two-gap superconductor, and compare it with its single-gap counterpart. We show that the vortices in the two-gap superconductor are soft (or continuous) core vortices. In particular, we discuss the topological structure of the finite-energy vortices (Abrikosov-like vortices), and find that they can be viewed as the incarnation of the baby skyrmion stretched in the third direction. Besides, we point out that the knot soliton in the two-gap superconductor is the twisted Abrikosov-like vortex with its two periodic ends connected smoothly. The relation between the magnetic monopoles and the Abrikosov-like vortices is also discussed briefly.

Topological vortices in a two-gap superconductor

Based on the ϕ-mapping theory, we derive a new rigorous equation describing the distribution of the magnetic field for vortices in a two-gap superconductor, of which the so-called modified London equation is just a special case in a one-flavor limit. We explicitly investigate the London penetration depth, the Meissner and mixed states and Josephson effect. A magnetic flux quantization condition for vortices in a two-gap superconductor is also derived, from which it follows that in a two-gap superconductor there exist vortices which carry an arbitrary fraction of magnetic flux quantum. The branch processes during the evolution of the vortices in a two-gap superconductor are discussed.

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.

Topological structure of the London equation

A topological current is shown to exist in superconductivity. Using the $\ensuremath{\varphi}$-mapping topological current theory, we have obtained a rigorous London equation with topological current, which reveals the inner relation between the condensate wave function $\ensuremath{\psi}$ and the induction $\stackrel{\ensuremath{\rightarrow}}{B}$. It is also shown that the equation is characterized by the Hopf index and Brouwer degree of $\ensuremath{\varphi}$ mapping. As a result, the so-called modified London equation, which describes the distribution of the magnetic field for isolated vortices, is only a special case of the equation. Our theory, obtained without using any particular models or hypotheses, can also be applied to the illustration of a vortex-antivortex pair, vortex ring, and multicharged vortex.

Temperature dependence of the Meissner penetration length in submicron vortex-free and vortex-trapped YBaCuO particles: test for applicability of the modified London equation

To test the applicability of the modified London equation (MLE) the method based on comparison of the temperature dependencies of the penetration length λ ab in the Meissner and the vortex states is proposed. For experimental testing we investigate the magnetic properties of submicron superconducting YBaCuO particles prepared by a citrate gel modification of the sol-gel technique. It is found that the dependence λ ab 2(0) λ ab 2(T) obeys a power law with the exponent n = 2.0 ± 0.2 and n = 2.5 ± 0.05 in the Meissner and the vortex states, respectively. This difference indicates that MLE is not valid in the vortex state at least in the scale of the used particles i.e. of some hundred Å. Possible reasons for this are discussed.

Power Losses of a Type‐II Superconducting Slab of Arbitrary Thickness

AbstractThe distribution of the internal and external magnetic vector potentials of systems of a type‐II superconducting infinite slab of arbitrary thickness subjected to (i) a vibrating wire (between two vibrating wires) carrying dc currents, (ii) a normal conducting wire (between two wires) carrying ac currents, are obtained. A modified London equation is developed to study these systems in case the slab is moving. The power losses on the surface of the slab and its two constituent terms due to dissipated and reactive energies are calculated. The rcactive term is found to be either capacitivel or inductively reactive. It is found that the power losses are dependent on frequency, London penetration depth, permeability, conductivity, velocity, and the distance between the normal conductors and the surfaces of the superconducting slab. Moreover, the power losses decrease as the distance between the normal conductors and the surfaces of the superconducting slab decreases, and increase as frequency, London penetration depth, permeability, conductivity, velocity increase. These analytical solutions can be used for testing the superconducting slabs in motion and the related problem of the losses of moving superconductors. In the limiting case, a satisfactory agreement with previous results is achieved.

Critical temperature profile determination using a modified London equation for inhomogeneous superconductors

A modified form of London's equation describing the penetration of magnetic flux into an inhomogeneous superconductor is derived from the Ginzburg-Landau theory. This equation takes into account spatial variation of the magnetic penetration depth. The magnetic flux penetration into an inhomogeneous superconducting slab is calculated and a method is described whereby the spatial variation of the critical temperature can be obtained from an inductiveTctransition. The consequences of using other sample geometries and the relevance of this work to the characterization of practical superconducting wires is discussed.

Josephson tunneling studies of magnetic screening in proximity-superconducting silver

The diamagnetic properties of silver backed by a thick lead layer have been studied as a function of normal-metal thickness and of temperature. Data were obtained by measuring the period of the magnetic field dependence of the critical Josephson current in $S\ensuremath{-}I\ensuremath{-}N\ensuremath{-}{S}^{\ensuremath{'}}$ tunnel junctions. Strong screening has been seen at low temperatures and a thickness-independent penetration depth is indicated as the temperature goes to zero. This characteristic penetration depth is observed to be on the order of 1500 \AA{} for proximity-effect silver. We have used a modified London equation for the case of a spatially varying pair amplitude and have numerically solved for the magnetic field profile in this system. The spatial dependence of the pair amplitude obtained from Landau-Ginzburg theory produces good agreement with thickness- and temperature-dependence data.

Flux Quantization and the Current-Carrying State in a Superconducting Cylinder

London's phenomenological theory is modified to include quantization of angular momentum in the boson gas model of a superconductor. The modified London equation is solved in ring geometry to find the magnetic field in the penetration region. The unit of flux quantization is obtained for all thicknesses of the superconductor. The relation between angular momentum and kinetic energy in the current-carrying state is discussed briefly.