Equilibrium topology of the intermediate state in type-I superconductors of different shapes

Band-resolved Caroli-de Gennes-Matricon states of multiple-flux-quanta vortices in a multiband superconductor

Superconductors are of type I or II depending on whether they form an Abrikosov vortex lattice. Although bulk lead (Pb) is classified as a prototypical type-I superconductor, we show that its two-band superconductivity allows for single-flux-quantum and multiple-flux-quanta vortices in the intermediate state at millikelvin temperature. Using scanning tunneling microscopy, the winding number of individual vortices is determined from the real space wave function of its Caroli-de Gennes-Matricon bound states. This generalizes the topological index theorem put forward by Volovik for isotropic electronic states to realistic electronic structures. In addition, the bound states due to the two superconducting bands of Pb can be separately detected and the two gaps close independently inside vortices. This yields strong evidence for a low interband coupling.

Magnetic molecules as local sensors of topological hysteresis of superconductors

Superconductors and magnetic materials, including molecules, are key ingredients for quantum computing and spintronics. However, only a little is known about how these materials interact in multilayer nanostructures like the hybrid architectures nowadays under development for such advanced applications. Here, we show that a single layer of magnetic molecules, Terbium(III) bis-phthalocyaninato (TbPc₂) complexes, deposited under controlled UHV conditions on a superconducting Pb(111) surface is sensitive to the topology of the intermediate state of the superconductor, namely to the presence and evolution of superconducting and normal domains due to screening and penetration of an external magnetic field. The topological hysteresis of the superconducting substrate imprints a local evolution of the magnetisation of the TbPc₂ molecules in the monolayer. Element and surface selective detection is achieved by recording the X-ray magnetic circular dichroism of the Tb atoms. This study reveals the impressive potential of magnetic molecules for sensing local magnetic field variations in molecular/superconductor hybrid devices, including spin resonators or spin injecting and spin filtering components for spintronics applications.

Topological magnetic hysteresis in single crystals of CeAgSb2ferromagnet

Closed-topology magnetic domains are usually observed in thin films and in an applied magnetic field. Here we report the observation of rectangular cross-section tubular ferromagnetic domains in thick single crystals of CeAgSb₂in zero applied field. Relatively low exchange energy, small net magnetic moment, and anisotropic in-plane crystal electric fields lower the domain wall energy and allow for the formation of the closed-topology patterns. The tubular domain structure irreversibly transforms into a dendritic pattern upon cycling the magnetic field. This transition between closed and open topologies results in a 'topological magnetic hysteresis'- the actual hysteresis in magnetization, not due to the imperfections and pinning, but due to the difference in the pattern morphology. Similar physics was suggested before in pure type-I superconductors and is believed to be a generic feature of other nonlinear single (present case), or two-phase (type-I superconductor) systems where the effects similar to demagnetization (shape-dependent macroscopic variation of properties) lead to pattern formation.

Analogy between periodic patterns in thin smectic liquid crystal films and the intermediate state of superconductors

Spontaneous breaking of symmetry in liquid crystal (LC) films often reveals itself as a microscopic pattern of molecular alignment. In a smectic-A LC, the emergence of positional order at the transition from the nematic phase leads to periodic textures that can be used as optical microarrays, templates for soft lithography, and ordering matrices for the organization and manipulation of functional nanoparticles. While both 1d and 2d patterns have been obtained as a function of the LC film thickness and applied fields, the connection has not been made between pattern formation and the peculiar critical behavior of LCs at the nematic-smectic transition, still eluding a comprehensive theoretical explanation. In this article, we demonstrate that an intense bend distortion applied to the LC molecular director while cooling from the nematic phase produces a frustrated smectic phase with depressed transition temperature, and the characteristic 1d periodic texture previously observed in thin films and under applied electric fields. In light of De Gennes' analogy with the normal-superconductor transition of a metal, we identify the 1d texture as the equivalent of the intermediate state in type I superconductors. The bend distortion is analog to the magnetic field in metals and penetrates in the frustrated phase as an array of undercooled nematic domains, periodically intermixed with bend-free smectic-A domains. Our findings provide fundamental evidence for theories of the nematic-smectic transition, highlighting the deep connection between phase frustration and pattern formation, and perspectives on the design of functional smectic microarrays.

Quantum dynamics of a single molecule magnet on superconducting Pb(111)

Magnetic materials interfaced with superconductors may reveal new physical phenomena with potential for quantum technologies. The use of molecules as magnetic components has already shown great promise, but the diversity of properties offered by the molecular realm remains largely unexplored. Here we investigate a submonolayer of tetrairon(III) propeller-shaped single molecule magnets deposited on a superconducting lead surface. This material combination reveals a strong influence of the superconductor on the spin dynamics of the single molecule magnet. It is shown that the superconducting transition to the condensate state switches the single molecule magnet from a blocked magnetization state to a resonant quantum tunnelling regime. Our results open perspectives to control single molecule magnetism via superconductors and to use single molecule magnets as local probes of the superconducting state.

Influence of an applied current on the vortex matter in a superconducting sample with structural defects

We show how the inclusion of a structural defect of determined geometry controls the vortex state in a square superconducting sample in the presence of an external magnetic field and a dc current. We simulated the defects by using the deformation parameter τ(x,y), solving the non-lineal time-dependent Ginzburg-Landau equations and using the link variable method, for four different geometries as possible options for the storage vortex, simulating the behavior of a capacitor. We found an exponential dependence of the current in which the first vortex penetrates the sample J→c as a function of the area of a square central defect in the sample. We also show the effect of the defects and the transport current on the magnetization, magnetic susceptibility, vorticity, and magnetic field at the first vortex entry into the sample H1 and the density of the superconducting electrons.

Phase Diagram and Superconductivity: New Insight into the Fundamentals of InSn Bimetallic Alloys

We report the crystal structure and superconducting phase diagram for In _pSn_{1- p} (0.01 ≤ p ≤ 0.99) bimetallic alloys. A weak electron-phonon coupling was observed in intergranular linked InSn superconductors over an infinite range mediated by high-energy phonons. An enhanced T_C(0) ∼ 6.2 K and critical field H_C(0) ∼ 2.7 kOe were determined from intermediate (γ-Sn + β-InSn) composite alloys attributed to internal strain possibly originating from thermal expansion effect of constituent phases.

Superconducting Properties in Arrays of Nanostructured β-Gallium

Samples of nanostructured β-Ga wires were synthesized by a novel method of metallic-flux nanonucleation. Several superconducting properties were observed, revealing the stabilization of a weak-coupling type-II-like superconductor (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}_{c}$$\end{document}Tc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx $$\end{document}≈ 6.2 K) with a Ginzburg-Landau parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa }_{GL}$$\end{document}κGL = 1.18. This contrasts the type-I superconductivity observed for the majority of Ga phases, including small spheres of β-Ga with diameters near 15 μm. Remarkably, our magnetization curves reveal a crossover field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{D}$$\end{document}HD, where we propose that the Abrikosov vortices are exactly touching their neighbors inside the Ga nanowires. A phenomenological model is proposed to explain this result by assuming that only a single row of vortices is allowed inside a nanowire under perpendicular applied field, with an appreciable depletion of Cooper pair density at the nanowire edges. These results are expected to shed light on the growing area of superconductivity in nanostructured materials.

Microfluidic manipulation of magnetic flux domains in type-I superconductors: droplet formation, fusion and fission

The magnetic flux domains in the intermediate state of type-I superconductors are known to resemble fluid droplets, and their dynamics in applied electric current is often cartooned as a "dripping faucet". Here we show, using the time-depended Ginzburg-Landau simulations, that microfluidic principles hold also for the determination of the size of the magnetic flux-droplet as a function of the applied current, as well as for the merger or splitting of those droplets in the presence of the nanoengineered obstacles for droplet motion. Differently from fluids, the flux-droplets in superconductors are quantized and dissipative objects, and their pinning/depinning, nucleation, and splitting occur in a discretized form, all traceable in the voltage measured across the sample. At larger applied currents, we demonstrate how obstacles can cause branching of laminar flux streams or their transformation into mobile droplets, as readily observed in experiments.

Routes to probe Bismuth induced strong-coupling superconductivity in bimetallic BiIn alloys

We report the observation of strong electron-phonon coupling in intergranular linked BiIn superconductors over an infinite range mediated by low-lying phonons. An enhanced superconducting transition temperature was observed from the magnetization, revealing a main diamagnetic Meissner state below T_C(0) = 5.86(1) K and a critical field H_C(0) = 1355(15) Oe with an In₂Bi phase of the composite sample. The electron-phonon coupling to low lying phonons is found to be the leading mechanism for observed strong-coupling superconductivity in the BiIn system. Our findings suggest that In₂Bi is in the strong-coupling region with T_C(0) = 5.62(1) K, λ_ep = 1.45, ω_ln = 45.92 K and α = 2.23. The estimated upper critical field can be well-described by a power law with α value higher than 2, consistent with the strong electron-phonon coupling.

Visualizing the morphology of vortex lattice domains in a bulk type-II superconductor

Alike materials in the solid state, the phase diagram of type-II superconductors exhibit crystalline, amorphous, liquid and spatially inhomogeneous phases. The multitude of different phases of vortex matter has thence proven to act as almost ideal model system for the study of both the underlying properties of superconductivity but also of general phenomena such as domain nucleation and morphology. Here we show how neutron grating interferometry yields detailed information on the vortex lattice and its domain structure in the intermediate mixed state of a type-II niobium superconductor. In particular, we identify the nucleation regions, how the intermediate mixed state expands, and where it finally evolves into the Shubnikov phase. Moreover, we complement the results obtained from neutron grating interferometry by small-angle neutron scattering that confirm the spatially resolved morphology found in the intermediate mixed state, and very small-angle neutron scattering that confirm the domain structure of the vortex lattice.

The phase diagram of type-II superconductors exhibits a multitude of different phases, whose study can shed light on domain nucleation and morphology. Here the authors use neutron grating interferometry to investigate the nucleation and phase changes of an intermediate mixed state in a niobium superconductor.

The role of temperature in the magnetic irreversibility of type-I Pb superconductors

Evidence of how temperature plays a role in the magnetic irreversibility in the intermediate state of a cylinder and various disks of pure type-I superconducting lead is presented. Isothermal measurements of first magnetization curves and hysteresis cycles are analyzed in a reduced representation that defines an equilibrium state for flux penetration in all the samples and reveals that flux expulsion depends on temperature in the disks but not in the cylinder. The magnetic field at which irreversibility sets in along the descending branch of the cycle and the remnant magnetization at zero field are found to decrease with temperature. The contributions to irreversibility of the geometrical barrier and the energy minima associated with stress defects that act as pinning centers on normal-superconductor interfaces are discussed in an effort to account for the results obtained.

Magnetization measurements and Ginzburg-Landau simulations of micron-size β-tin samples: evidence for an unusual critical behavior of mesoscopic type-I superconductors

We describe investigations of the largely unexplored field of mesoscopic type-I superconductors. Micromagnetometry and 3D Ginzburg-Landau simulations of our single crystal β-tin samples in this regime reveal size- and temperature-dependent supercritical fields whose behavior is radically different from the bulk critical field H(c)(B). We find that complete suppression of the intermediate state in medium-size samples can result in a surprising reduction of the critical field significantly below H(c)(B). We also reveal an evolution of the superconducting-to-normal phase transition from the expected irreversible first order at low temperatures through the previously unobserved reversible first-order to a second-order transition close to T(c), where the critical field can be many times larger than H(c)(B). Finally, we have identified striking correlations between the mesoscopic H(c3) for nucleation of surface superconductivity and the thermodynamic H(c) near T(c). All these observations are entirely unexpected in the conventional type-I picture.

Systems biology beyond networks: generating order from disorder through self-organization

Erwin Schrödinger pointed out in his 1944 book "What is Life" that one defining attribute of biological systems seems to be their tendency to generate order from disorder defying the second law of thermodynamics. Almost parallel to his findings, the science of complex systems was founded based on observations on physical and chemical systems showing that inanimate matter can exhibit complex structures although their interacting parts follow simple rules. This is explained by a process known as self-organization and it is now widely accepted that multi-cellular biological organisms are themselves self-organizing complex systems in which the relations among their parts are dynamic, contextual and interdependent. In order to fully understand such systems, we are required to computationally and mathematically model their interactions as promulgated in systems biology. The preponderance of network models in the practice of systems biology inspired by a reductionist, bottom-up view, seems to neglect, however, the importance of bidirectional interactions across spatial scales and domains. This approach introduces a shortcoming that may hinder research on emergent phenomena such as those of tissue morphogenesis and related diseases, such as cancer. Another hindrance of current modeling attempts is that those systems operate in a parameter space that seems far removed from biological reality. This misperception calls for more tightly coupled mathematical and computational models to biological experiments by creating and designing biological model systems that are accessible to a wide range of experimental manipulations. In this way, a comprehensive understanding of fundamental processes in normal development or of aberrations, like cancer, will be generated.

Confinement effects on intermediate-state flux patterns in mesoscopic type-I superconductors

Intermediate-state flux structures in mesoscopic type-I superconductors are studied within the Ginzburg-Landau theory. In addition to well-established tubular and laminar structures, the strong confinement leads to the formation of (i) a phase of singly quantized vortices, which is typical for type-II superconductors and (ii) a ring of a normal domain at equilibrium. The stability region and the formation process of these intermediate-state structures are strongly influenced by the geometry of the sample.

Unconventional rotational responses of hadronic superfluids in a neutron star caused by strong entrainment and a Sigma- hyperon gap

I show that the usual model of the rotational response of a neutron star, which predicts rotation-induced neutronic vortices and no rotation-induced protonic vortices, does not hold (i) beyond a certain threshold of entrainment interaction strength nor (ii) in the case of nonzero Sigma(-) hyperon gap. I show that in both of these cases the rotational response involves the creation of phase windings in an electrically charged condensate. Lattices of bound states of vortices which result from these phase windings can (for a range of parameters) strongly reduce the interaction between rotation-induced vortices with magnetic-field carrying superconducting components.