Manipulating individual vortices in a deterministic way is challenging, ideally, manipulation should be effective, local, and tunable in strength and location. Here, we show that vortices interact with strain fields generated by mechanical stress. We utilized this interaction to move individual vortices in thin superconducting films via local mechanical contact, without magnetic field or current. We used a scanning superconducting quantum interference device (SQUID) to image vortices and to apply local vertical stress with the tip of our sensor. Vortices were attracted to the contact point, relocated, and were stable at their new location. We show that vortices move only after contact and that more effective manipulation is achieved with stronger force and longer contact time. Mechanical manipulation of vortices provides a local view of the interaction between strain and nanomagnetic objects as well as controllable, effective, and reproducible manipulation technique.
We show in this paper that it is possible to induce superconducting vortices in a gold nanowire connected to superconducting electrodes. The gold nanowire acquires superconductivity by the proximity effect. The differential magnetoresistance of the nanowire beyond a critical magnetic field shows uniform oscillations with increasing field with a period of \phi0/(2\pir^2) (\phi0 = h/2e is the superconducting flux quantum, r = 35 nm is the radius of the nanowire). We demonstrate that these periodic oscillations are the signatures of the sequential generation and moving of vortices across the gold…
In this article an analytical and numerical study of superconducting thin film with ferromagnetic textures of elliptical geometries in close proximity is presented. The screening currents induced in the superconductor due to the magnetic texture are calculated. Close to the superconducting transition temperature $T_c$ the spontaneous creation of superconducting vortices becomes energy favorable depending on the value of the magnetization and the geometrical quantities of the magnetic texture. The creation of vortices by elliptic dots is more energy favorable than those created by circular ones…
We revisit the critical penetration of Pearl vortices in narrow superconducting flat rings cooled in magnetic fields. Scanning superconducting quantum interference device microscopy measurements showed how magnetic field penetrates and vortices are trapped in flat rings made of amorphous MoGe thin films. Counting the number of trapped vortices for each image, we found that the vortices are completely excluded from the ring annulus when the applied field $H$ is below a threshold field $H_{\rm{p}}$: Above this field, the vortices increase linearly with field. The obtained values of $H_{\rm{p}}$ …
Quantized magnetic vortices driven by electric current determine key electromagnetic properties of superconductors. While the dynamic behavior of slow vortices has been thoroughly investigated, the physics of ultrafast vortices under strong currents remains largely unexplored. Here we use a nanoscale scanning superconducting quantum interference device to image vortices penetrating into a superconducting Pb film at rates of tens of GHz and moving with velocities up to tens of km/s, which are not only much larger than the speed of sound but also exceed the pair-breaking speed limit of supercond…
We consider a two-dimensional array of ultra-small superconducting grains, weakly coupled by Josephson junctions with large charging energy. We start from an effective action based on a microscopic tunneling Hamiltonian, which includes quasiparticle degrees of freedom, and study the resulting dissipative dynamics of quantum vortices. The equation of motion for a single vortex is deduced, and compared with a commonly adopted phenomenological model.
Multi-quanta, or giant, vortices (GVs) are known to appear in very small superconductors near the superconducting transition due to strong confinement of magnetic flux. Here we present evidence for a new, pinning-related, mechanism for the formation of GVs. Using Bitter decoration to visualise vortices in small Nb disks, we show that confinement in combination with strong disorder causes individual vortices to merge into clusters or even GVs well below Tc and Hc2, in contrast to well-defined shells of individual vortices found in the absence of pinning.
When the impurity mean free path is short, only spin-polarized Cooper pairs which are non-locally and antisymmetrically correlated in time may exist in a half-metallic ferromagnet. As a consequence, the half-metal acts as an odd-frequency superconducting condensate. We demonstrate both analytically and numerically that quantum vortices can emerge in half-metals despite the complete absence of conventional superconducting correlations. Because these metals are conducting in only one spin band, we show that a circulating spin supercurrent accompanies these vortices. Moreover, we demonstrate that…
In superconducting thin films, engineered lattice of antidots (holes) act as an array of columnar pinning sites for the vortices and thus lead to vortex matching phenomena at commensurate fields guided by the lattice spacing. The strength and nature of vortex pinning is determined by the geometrical characteristics of the antidot lattice (such as the lattice spacing $a_0$, antidot diameter $d$, lattice symmetry, orientation, etc) along with the characteristic length scales of the superconducting thin films, viz., the coherence length ($ξ$) and the penetration depth ($λ$). There are at least tw…
We use a scanning nanometer-scale superconducting quantum interference device (SQUID) to image individual vortices in amorphous superconducting MoSi thin films. Spatially resolved measurements of the magnetic field generated by both vortices and Meissner screening satisfy the Pearl model for vortices in thin films and yield values for the Pearl length and bulk penetration depth at 4.2 K. Flux pinning is observed and quantified through measurements of vortex motion driven by both applied currents and thermal activation. The effects of pinning are also observed in metastable vortex configuration…
Using Ginzburg-Landau theory, we have performed detailed studies of vortices in the presence of artificial defect arrays, for a thin film geometry. We show that when a vortex approaches the vicinity of a defect, an abrupt transition occurs in which the vortex core develops a ``string'' extending to the defect boundary, while simultaneously the supercurrents and associated magnetic flux spread out and engulf the defect. Current induced depinning of vortices is shown to be dominated by the core string distortion in typical experimental situations. Experimental consequences of this unusual depinn…
We report a new nonlocal effect in vortex matter, where an electric current confined to a small region of a long and sufficiently narrow superconducting wire causes vortex flow at distances hundreds of inter-vortex separations away. The observed remote traffic of vortices is attributed to a very efficient transfer of a local strain through the one-dimensional vortex lattice, even in the presence of disorder. We also observe mesoscopic fluctuations in the nonlocal vortex flow, which arise due to "traffic jams" when vortex arrangements do not match a local geometry of a superconducting channel.
We study the Ginzburg-Landau equations in order to describe a two-dimensional superconductor in a bounded domain. Using the properties of a particular integrability point ($κ= 1/ \sqrt2$) of these nonlinear equations which allows vortex solutions, we obtain a closed expression for the energy of the superconductor. The presence of the boundary provides a selection mechanism for the number of vortices. A perturbation analysis around $κ= 1/ \sqrt2$ enables us to include the effects of the vortex interactions and to describe quantitatively the magnetization curves recently measured on small superc…
The magneto-transport of a superconducting/ferromagnetic hybrid structure consisting of a superconducting thin film in contact with an array of magnetic nanodots in the so-called "magnetic vortex-state" exhibits interesting properties. For certain magnetic states, the stray magnetic field from the vortex array is intense enough to drive the superconducting film into the normal state. In this fashion, the normal-to-superconducting phase transition can be controlled by the magnetic history. The strong coupling between superconducting and magnetic subsystems allows characteristically ferromagneti…
We comment on the work by Moler, et al., [Science 279, 1193 (1998)] concerning determination of the c-axis penetration depth in the single-layer compound Tl-2201 through measuring the magnetic flux due to isolated interlayer Josephson vortices.
We have generalized a quasiclassical model for Weyl semimetals with a tilted band in the presence of an externally applied magnetic field. This model is applicable to ballistic, moderately disordered, and samples containing a high density of nonmagnetic impurities. We employ this formalism and show that a self-biased supercurrent, creating a ϕ0-junction, can flow through a triplet channel in Weyl semimetals. Furthermore, our results demonstrate that multiple supercurrent reversals are accessible through varying junction thickness and parameters characterizing Weyl semimetals. We discuss the in…
Nucleation of vortices in a superconductor below the first critical field can be assisted by transverse sound in the GHz frequency range. Vortices will enter and exist the superconductor at the frequency of the sound. We compute the threshold parameters of the sound and show that this effect is within experimental reach.
Shapiro voltage steps at voltages V_n=nV_0 (n integer) have been observed in the voltage-current characteristics of a superconducting film with a square lattice of perforating microholes (antidots)in the presence of radiofrequent radiation. These equidistant steps appear at the second matching field H_2 when the flow of the interstitial vortex lattice in the periodic potential created by the antidots and the vortices trapped by them, is in phase with the applied rf frequency. Therefore, the observation of Shapiro steps clearly reveals the presence of mobile intersitial vortices in superconduct…
We report an anomalous matching effect in MoGe thin films containing pairs of circular holes arranged in such a way that four of those pairs meet at each vertex point of a square lattice. A remarkably pronounced fractional matching was observed in the magnetic field dependences of both the resistance and the critical current. At the half matching field the critical current can be even higher than that at zero field. This has never been observed before for vortices in superconductors with pinning arrays. Numerical simulations within the nonlinear Ginzburg-Landau theory reveal a square vortex ic…
The screening currents induced in a superconducting film by a magnetic annulus whose magnetization is perpendicular to the superconductor are calculated. We show that close to the superconductor transition temperature $T_c$ particular values of the magnetization and radii of the annulus make the creation of superconducting vortices energy favorable. We also show that the magnetic annulus offers an alternative tool for vortex pinning in the superconductor. Statistical mechanical properties of this system are discussed.
Vortices in a narrow superconducting strip with a square array of pinning sites are studied. The interactions of vortices with other vortices and with external sources (applied magnetic field and transport current) are calculated via a screened Coulomb model. The edge barrier is taken into account and shown to have an important role on the system dynamics. Numerical simulations in this approach show that the field dependent magnetic moment presents peaks corresponding to history dependent configurations of the vortex lattice. Some effects of the edge barrier on the I-V characteristics are also…
We investigate a superconducting state irradiated by a laser beam with spin and orbital angular momentum. It is shown that superconducting vortices are created by the laser beam due to heating effect and transfer of angular momentum of light. Possible experiments to verify our prediction are also discussed.
The interaction between a superconducting vortex or antivortex in a superconducting film and a magnetic dipole with in- or out-of-plane magnetization is investigated within the London approximation. The dependence of the interaction energy on the dipole-vortex distance and the film thickness is studied and analytical results are obtained in limiting cases. We show how the short range interaction with the magnetic dipole makes the co-existence of vortices and antivortices possible. Different configurations with vortices and antivortices are investigated.
We demonstrate a unique prospect for inducing anisotropic vortex pinning and manipulating the directional motion of vortices using the stripe domain patterns of a uniaxial magnetic film in a the superconducting/ferromagnetic hybrid. Our observations can be described by a model, which considers interactions between magnetic charges of vortices and surface magnetic charges of domains resulting in the enhanced pinning of vortices on domain walls.
Pancake vortices in stacks of thin superconducting films or layers are considered. It is stressed that in the absence of Josephson coupling topological restrictions upon possible configurations of vortices are removed and various examples of structures forbidden in bulk superconductors are given. In particular, it is shown that vortices may skip surface layers in samples of less than a certain size R_c which might be macroscopic. The Josephson coupling suppresses R_c estimates.
In the present paper we develop an algorithm to solve the time dependent Ginzburg-Landau (TDGL) equations, by using the link variables technique, for circular geometries. In addition, we evaluate the Helmholtz and Gibbs free energy, the magnetization, and the number of vortices. This algorithm is applied to a circular sector. We evaluate the superconduting-normal magnetic field transition, the magnetization, and the superconducting density. Furthermore, we study the nucleation of giant and multi-vortex states for that geometry.
Here we describe a development of computer algorithm to simulate the Time Dependent Ginzburg-Landau equation (TDGL) and its application to understand superconducting vortex dynamics in confined geometries. Our initial motivation to get involved in this task was trying to understand better our experimental measurements on dynamics of superconductors with vortices at high frequencies leading to microwave stimulated superconductivity due to presence of vortex (Lara, et al., Scientific Reports, 59187 (2015)).
On the basis of the time-dependent Ginzburg-Landau equation we studied the dynamics of the superconducting condensate in a wide two-dimensional sample in the presence of a perpendicular magnetic field and applied current. We could identify two critical currents: the current at which the pure superconducting state becomes unstable ($J_{c2}$ \cite{self1}) and the current at which the system transits from the resistive state to the superconducting state ($J_{c1}<J_{c2}$). The current $J_{c2}$ decreases monotonically with external magnetic field, while $J_{c1}$ exhibits a maximum at $H^*$. For suf…
A monopole harmonic superconductor is a novel topological phase of matter with topologically protected gap nodes that result from the non-trivial Berry phase structure of Cooper pairs. In this work we propose to realize a monopole superconductor by the proximity effect between a time-reversal broken Weyl semi-metal and an $s$-wave superconductor. Furthermore, we study the zero-energy vortex bound states in this system by projection methods and by exact solutions. The zero modes exhibit a non-trivial phase winding in real space as a result of the non-trivial winding of the order parameter in mo…
We address a recent experiment in which a strong decrease of the resistance of a superconducting film has been observed when a remote unbiased gate was placed above the film. Here we explain the experimental finding as a suppression of the vortex tunneling due to the Orthogonality Catastrophe of the electrons inside the gate. We interpret the change in the resistance of the film as a "metal-insulator" transition in the system of vortices induced by the gate.
We numerically solve the coupled Landau-Ginzburg-Maxwell equations for a model of a spin triplet px + ipy superconductor in which whole or half-quanta of flux thread through a hole. We recover the pattern of stable and unstable regions for the half-flux quanta observed in a recent experiment.
Microwave resonators with high quality factors have enabled many recent breakthroughs with superconducting qubits and photon detectors, typically operated in shielded environments to reduce the ambient magnetic field. Insufficient shielding or pulsed control fields can introduce vortices, leading to reduced quality factors, although increased pinning can mitigate this effect. A narrow slot etched into the resonator surface provides a straightforward method for pinning enhancement without otherwise affecting the resonator. Resonators patterned with such a slot exhibited over an order of magnitu…
We demonstrate a method for manipulating small ensembles of vortices in multiply-connected superconducting structures. A micron-size magnetic particle attached to the tip of a silicon cantilever is used to locally apply magnetic flux through the superconducting structure. By scanning the tip over the surface of the device, and by utilizing the dynamical coupling between the vortices and the cantilever, a high-resolution spatial map of the different vortex configurations is obtained. Moving the tip to a particular location in the map stabilizes a distinct multi-vortex configuration. Thus, the s…
Giant vortices with higher phase-winding than $2π$ are usually energetically unfavorable, but geometric symmetry constraints on a superconductor in a magnetic field are known to stabilize such objects. Here, we show via microscopic calculations that giant vortices can appear in intrinsically non-superconducting materials, even without any applied magnetic field. The enabling mechanism is the proximity effect to a host superconductor where a current flows, and we also demonstrate that antivortices can appear in this setup. Our results open the possibility to study electrically controllable topo…
Spontaneous nucleation and the consequent penetration of vortices into thin superconducting films and wires, subjected to a magnetic field, can be considered as a nonlinear stage of primary instability of the current-carrying superconducting state. The development of the instability leads to the formation of a chain of vortices in strips and helicoidal vortex lines in wires. The boundary of instability was obtained analytically. The nonlinear stage was investigated by simulations of the time-dependent generalized Ginzburg-Landau equation.
We propose a simple experiment to determine whether vortices persist above the superconducting transition temperature Tc in the pseudogap phase of high temperature cuprate superconductors. This involves using a magnetic dot to stabilize a vortex in a thin cuprate film beneath the dot. We calculate the magnetic field profile as a function of distance from the dot if a vortex is present, and discuss possible measurements that could be done to detect this. Finally, we comment on the temperature range where a stable vortex should be observable.
By analytically solving the Bogoliubov-de Gennes equations we study the fermion bound states at the center of the core of a vortex in a two-dimensional superconductor. We consider three kinds of 2D superconducting models: (a) a standard type II superconductor in the mixed state with low density of vortex lines, (b) a superconductor with strong spin-orbit coupling locking the spin parallel to the momentum and (c) a superconductor with strong spin-orbit coupling locking the spin perpendicular to the momentum. The 2D superconducting states are induced via proximity effect between an $s$-wave supe…
A scanning micro superconducting quantum interference device (microSQUID) microscope is used to directly image vortices in a superconducting Al thin film. We observe the temperature dependence of the vortex distribution in a regular defect (hole) array patterned into the Al film. The first direct observation of the localized superconducting state around the holes is shown as well as the effect of the hole size on nucleation of the superconducting state.
We have studied two nanomagnet systems with strong (Co/Pd multilayers) and weak (NdCo alloy films) stray magnetic fields by probing the out-of-plane magnetic states with superconducting vortices. The hybrid samples are made of array of nanomagnets embedded in superconducting Nb thin films. The vortex motion detects relevant magnetic state features, since superconducting vortices are able to discriminate between different magnetic stray field strengths and directions. The usual matching effect between the superconducting vortex lattice and the periodic pinning array can be quenched by means of …
The paper by G. R. Berdiyorov, M. V. Milosevic, and F. M. Peeters [Phys. Rev. B 81, 144511 (2010)] studies theoretically the dynamic properties of a superconducting loop. The authors claim that their consideration of asymmetric loop relates to our experimental results in this field. We point out that this claim is incorrect and explain shortly the true paradoxical essence of the results of our measurements of asymmetric superconducting loop.
The magnetic field of vortices in anisotropic superconducting films is considered in the framework of anisotropic London approach. It is found that at distances large relative to the core size, the magnetic field normal to the film surface may change sign. We find that the magnetic field attenuates at large distances as $1/r^3$ as it does in isotropic films, but the anisotropy induces an angular dependence to the supercurrents which causes the sign of the field to change for anisotropy parameters $γ=λ_2/λ_1>\sqrt{2}$ in some parts of the $(x,y)$ plane.
A plasma wave propagates inside an anisotropic superconducting film sandwiched between two semi-infinite non-conducting bounding dieletric media. Along the c-axis, perpendicular to the film surfaces, an external magnetic field is applied. We show how vortices, known to cause dissipation and change the penetration depth, affect the propagative mode. We obtain the complex wave number of this mode and, using YBCO at 4 K as an example, determine a region where vortex contribution is dominant and dissipation is small.
We have investigated the confinement of 3-D vortices in specific cases of Type-II ($κ= 2$) nano-superconducting devices. The emergent pattern of vortices greatly depends on the orientation of an applied magnetic field (transverse or longitudinal), and the size of the devices (a few coherence lengths $ξ$). Herein, cylindrical geometries are examined. The surface barriers become very significant in these nano-systems, and hence the characteristics of the vortices become highly sensitive to the shape of the system and direction of an applied field. It is observed that nano-cylindrical superconduc…
The superconducting state is formed by the condensation of a large number of Cooper pairs. The normal state electronic properties can give significant influence on the superconducting state. For usual type-II superconductors, the vortices are cylinder like with a round cross-section. For many two dimensional superconductors, such as Cuprates, 2H-NbSe$_2$ etc., albeit the in-plane anisotropy, the vortices generally have a round shape. In this paper we report results based on the scanning tunneling microscopy/spectroscopy measurements on a newly discovered superconductor Ta$_4$Pd$_3$Te$_{16}$. T…
Vortices in superconductors driven at microwave frequencies exhibit a response related to the interplay between the vortex viscosity, pinning strength, and flux creep effects. At the same time, the trapping of vortices in superconducting microwave resonant circuits contributes excess loss and can result in substantial reductions in the quality factor. Thus, understanding the microwave vortex response in superconducting thin films is important for the design of such circuits, including superconducting qubits and photon detectors, which are typically operated in small, but non-zero, magnetic fie…
Spin-orbit coupling (SOC) is known to play an important role in superconductor/ferromagnet heterostructures. Here we demonstrate that SOC results in the spontaneous generation of vortices in an \textit{s}-wave superconductor placed below a ferromagnetic metal with intrinsic or interfacial SOC. By using the generalized London and Ginzburg-Landau theories, we calculate the supercurrent amplitude in various types of S/F heterostructures at and below the superconducting critical temperature. In particular, we show that even in the limit of weak SOC, vortex-antivortex pairs are stabilized and that …
Inspired by the seminal, ground-breaking work of Abrikosov in 1957, we developed a new approximation to the interaction between two widely separated superconducting vortices. In contrast with Abrikosov's, we take into account the finite size of the vortices and their internal magnetic profile. We consider the vortices to be embedded within a superconducting, infinitely long hollow cylinder, in order to simplify the symmetry and boundary conditions for the mathematical analysis. We study this system in the context of a magnetic Ginzburg-Landau functional theory, by solving for the magnetic fiel…
Vortices play a crucial role in determining the properties of superconductors as well as their applications. Therefore, characterization and manipulation of vortices, especially at the single vortex level, is of great importance. Among many techniques to study single vortices, scanning tunneling microscopy (STM) stands out as a powerful tool, due to its ability to detect the local electronic states and high spatial resolution. However, local control of superconductivity as well as the manipulation of individual vortices with the STM tip is still lacking. Here we report a new function of the ST…
Direct observation of vortex states confined in mesoscopic regular triangle dots of amorphous MoGe thin films was made with a scanning superconducting quantum interference device microscope. The observed magnetic images illustrate clearly how vortices are distributed over the triangle dots by forming not only commensurate triangular clusters, but also unique patterns imposed by incommensurability. We discuss the results in terms of vortex shells and construct the packing sequence of vortices in the multiple shell structure.
We consider a thin superconducting film with a magnetic dot with permanent magnetization (normal to the film) placed on it by a method based on London-Maxwell equations. For sufficiently high dot magnetization a single vortex appears in the ground state. Further increase of magnetization is accompanied with the appearance of antivortices and more vortices in the film. We study analytically conditions for the appearance of a vortex--antivortex pair for a range of parameters. The phase diagram with diversity of vortex--antivortex states is calculated numerically. When appear in the ground state,…
Interaction between a Bloch wall in a ferrite-garnet film and a vortex in a superconductor is analyzed in the London approximation. Equilibrium distribution of vortices formed around the Bloch wall is calculated. The results agree quantitatively with magneto-optical experiment where an in-plane magnetized ferrite-garnet film placed on top of NbSe2 superconductor allows observation of individual vortices. In particular, our model can reproduce a counter-intuitive attraction observed between vortices and a Bloch wall having the opposite polarity. It is explained by magnetic charges appearing due…
Stable vortex states are studied in large superconducting thin disks (for numerical purposes we considered with radius R = 50 ξ). Configurations containing more than 700 vortices were obtained using two different approaches: the nonlinear Ginzburg-Landau (GL) theory and the London approximation. To obtain better agreement with results from the GL theory we generalized the London theory by including the spatial variation of the order parameter following Clem's ansatz. We find that configurations calculated in the London limit are also stable within the Ginzburg-Landau theory for up to ~ 230 vor…
We show that the entropy per quantum vortex per layer in superconductors in external magnetic fields is bounded by the universal value k_B ln 2, which explains puzzling results of recent experiments on the Nernst effect.
In superconductors, the search for special vortex states such as giant vortices focuses on laterally confined or nanopatterned thin superconducting films, disks, rings, or polygons. We examine the possibility to realize giant vortex states and states with non-uniform vorticity on a superconducting spherical nanoshell, due to the interplay of the topology and the applied magnetic field. We derive the phase diagram and identify where, as a function of the applied magnetic field, the shell thickness and the shell radius, these different vortex phases occur. Moreover, the curved geometry allows th…
Ratchet motion of superconducting vortices, which is a directional flow of vortices in superconductors, is highly useful for exploring quantum phenomena and developing superconducting devices, such as superconducting diode and microwave antenna. However, because of the challenges in the quantitative characterization of the dynamic motion of vortices, a phase diagram of the vortex ratchet motion is still missing, especially in the superconductors with low dimensional structures. Here we establish a quantitative phase diagram of the vortex ratchet motion in a highly anisotropic superlattice supe…
