The self-energy of a moving vortex is shown do decrease with increasing velocity. The interaction energy of two parallel slowly moving vortices differs from the static case by a small term $\propto v^2$; the "slow" motion is defined as having the velocity $v<v_c =c^2/4πσλ$, where $σ(T)$ is conductivity of the normal excitations and $λ(T)$ is London penetration depth. For higher velocities, $v>v_c(T)$, the interaction energy of two vortices situated along the velocity direction is enhanced and along the perpendicular direction is suppressed compared to the static case.
We derive the exact modified London equation for the two-gap superconductor, 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 relat…
We consider non-Abrikosov vortex solutions in liquid metallic hydrogen (LMH) in the framework of two-component Ginzburg-Landau model. We have shown that there are three types of non-Abrikosov vortices depending on chosen boundary conditions at the core of vortices, namely, Neumann (N)-type, Dirichlet (D)-type and Gross-Pitaevskii (GP)-type vortices. The Neumann-type vortex has a non-vanishing condensation at the core, that is different from the ordinary vortex, and the magnetic flux could be reversed as well in LMH. Furthermore, we have obtained a new type of a neutral vortex which has no magn…
We calculate the local density of states of a vortex in a two-dimensional s-wave superconductor, in the presence of a uniform applied supercurrent. The supercurrent induces changes in the electronic structure for the isolated vortex as well as the vortex lattice, which agree with the recent measurements in 2H-NbSe2 [Maldonado et al., Phys. Rev. B 88, 064518 (2013)]. We find that the supercurrent polarizes the core states when the vortices are pinned. This shows that the transfer of momentum from the supercurrent to the bound states and the rigidity of the wave functions must be considered for …
The voltage shift RfIc with respect to the flux flow ohmic behavior of the current-voltage characteristic in type II superconductors is ascribed to the irreversible processes occurring when a vortex crosses defects. We include irreversible effects of the vortices-defects interaction into an effective law of motion. The obtained current-voltage characteristic at finite temperature is in agreement with experimental data.
Within the framework of the time-dependent Ginzburg-Landau theory we show that circularly polarized THz or far-infrared radiation induces a dc supercurrent that influences the dynamics of vortex-antivortex pair formation in a mesoscopic superconductor undergoing rapid thermal quenching. The Lorentz force arising from the supercurrents promotes vortex-antivortex separation and allows survival of the vortex polarity defined by the helicity of light. Based on this idea, we propose a two-stage irradiation scheme that provides a powerful method for controlled all-optical generation of Abrikosov vor…
We show that the strain-induced attraction between Abrikosov vortices has a non-core contribution overlooked up until now. This contribution is an example of the universal mechanism of soliton attraction in solids revealed in [Phys. Rev. B {\bf 66}, 14111 (2002)]. The resulting interaction energy is larger than that due to the vortex cores at least by a factor $\ln ^2 κ$. Consequently, the non-core contribution must be included, for instance, in discussions about the orientation of vortex lattices with respect to the crystal axes. It is shown to be also important when interpreting the thermal …
In systems combining type-II superconductivity and magnetism the non-stationary magnetic field of moving Abrikosov vortices may excite spin waves, or magnons. This effect leads to the appearance of an additional damping force acting on the vortices. By solving the London and Landau-Lifshitz-Gilbert equations we calculate the magnetic moment induced force acting on vortices in ferromagnetic superconductors and superconductor/ferromagnet superlattices. If the vortices are driven by a dc force, magnon generation due to the Cherenkov resonance starts as the vortex velocity exceeds some threshold v…
The electronic structure of a vortex line trapped by a planar defect in a type-II superconductor is analyzed within the Bogoliubov-de Gennes theory. The normal reflection of electrons and holes at the defect plane results in the topological transition in the spectrum and formation of a new type of quasiparticle states skipping or gliding along the defect. This topological transition appears to be a hallmark of the initial stage of the crossover from the Abrikosov to the Josephson vortex type revealing in the specific behavior of the quantized quasiparticle levels and density of states. The inc…
We study the spatial distribution of supercurrent circulated around an Abrikosov vortex in an SF bilayer in perpendicular magnetic field. Within the dirty limit regime and circular cell approximation for the vortex lattice, we derive the conditions when the Usadel equations the F layer can be solved analytically. Using the obtained solutions, we demonstrate the possibility of reversal of direction of proximity induced supercurrents around the vortex in the F layer compared to that in the S-layer. The direction of currents can be controlled either by varying transparency of the SF interface or …
Abrikosov vortices in altermagnetic superconductors
Centro de Física de Materiales (CFM-MPC) Centro Mixto CSIC-UPV/EHU（材料物理中心（CFM-MPC）CSIC-巴斯克国立大学混合中心）；Donostia International Physics Center (DIPC)（圣塞巴斯蒂安国际物理中心）
We show that low-angle grain boundaries (GB) in high-temperature superconductors exhibit intermediate Abrikosov vortices with Josephson cores, whose length $l$ along GB is smaller that the London penetration depth, but larger than the coherence length. We found an exact solution for a periodic vortex structure moving along GB in a magnetic field $H$ and calculated the flux flow resistivity $R_F(H)$, and the nonlinear voltage-current characteristics. The predicted $R_F(H)$ dependence describes well our experimental data on $7^{\circ}$ unirradiated and irradiated $YBa_2Cu_3O_7$ bicrystals, from …
We present a theoretical and experimental study of the Fiske resonances in the current-voltage characteristics of "small" Josephson junctions with randomly distributed misaligned Abrikosov vortices. We obtained that in the presence of Abrikosov vortices the resonant interaction of electromagnetic waves, excited inside a junction, with the ac Josephson current manifests itself by Fiske steps in a current-voltage characteristics even in the absence of external magnetic field. We found that the voltage positions of the Fiske steps are determined by a junction size, but the Fiske step magnitudes d…
We show that Abrikosov vortices moving towards the surface of a superconductor emit electromagnetic radiation into free space. The frequency distribution of the radiated intensity displays a pronounced maximum at microwave frequencies around v_x/lambda, where lambda is the magnetic penetration length. Coherent motion of a lattice of flux lines leads to constructive interference and increases the strength of the radiated power by a large factor.
Many practical applications of high T$_c$ superconductors involve layered materials and magnetic fields applied on an arbitrary direction with respect to the layers. When the anisotropy is very large, Cooper pair currents can circulate either within or perpendicular to the layers. Thus, tilted magnetic fields lead to intertwined lattices of Josephson and Abrikosov vortices, with quantized circulation across and within layers, respectively. Transport in such intertwined lattices has been studied in detail, but direct observation and manipulation of vortices remains challenging. Here we present …
Equidistant resonance jumps in superconducting coplanar resonators driven by Abrikosov vortices
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Institute for Physics of Microstructures, Russian Academy of Sciences（俄罗斯科学院微结构物理研究所）；Dukhov Research Institute of Automatics (VNIIA)（杜霍夫自动仪器科学研究院）
Electrodynamic phenomena related to vortices in superconductors have been studied since their prediction by Abrikosov, and seem to hold no fundamental mysteries. However, most of the effects are treated separately, with no guiding principle. We demonstrate that the relativistic vortex worldsheet in spacetime is the object that naturally conveys all electric and magnetic information, for which we obtain simple and concise equations. Breaking Lorentz invariance leads to down-to-earth Abrikosov vortices, and special limits of these equations include for instance dynamic Meissner screening and the…
Abrikosov fluxonics, a domain of science and engineering at the interface of superconductivity research and nanotechnology, is concerned with the study of properties and dynamics of Abrikosov vortices in nanopatterned superconductors, with particular focus on their confinement, manipulation, and exploitation for emerging functionalities. Vortex pinning, guided vortex motion, and the ratchet effect are three main fluxonic ``tools'' which allow for the dynamical (pinned or moving), the directional (angle-dependent), and the orientational (current polarity-sensitive) control over the fluxons, res…
We study the impact of the non-analytic reconstruction of vortex cores on static vortex structures in weakly coupled superfluids. We show that in rotating two-dimensional systems, the Abrikosov vortex lattice is unstable to vortex core deformation: Each zero of the wave function becomes a cut of finite length. The directors characterizing the orientations of the cuts are themselves ordered in superstructures due either to surface effects or to interaction with shear deformations of the lattice (spiral structure). Similar instability may be also observable in clean superconducting films.
It is shown that the order parameter $Δ$ induced in the normal part of superconductor-normal-superconductor proximity system is modulated in the magnetic field differently from vortices in bulk superconductors. Whereas $Δ$ turns zero at vortex centers, the magnetic structure of these vortices differs from that of Abrikosov's.
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 periodic lattice of Abrikosov vortices in chiral $p$ wave superconductor in general supports fermionic states with zero energy provided the intervortex distance is smaller than the critical one. The zero modes appear at the intersection with the Fermi level of electronic magnetic Bloch bands formed by the overlapping vortex core states. The Bloch bands are robust against lattice disorder induced by fluctuations of vortex positions and can transmit the energy flow across the lattice. The hallmark of zero modes on Bloch bands in electronic heat conductivity is discussed.
In superconductors penetrated by Abrikosov vortices the magnetic pressure and the inhomogeneous condensate density induce a deformation of the ionic lattice. We calculate how this deformation corrugates the surface of a semi-infinite sample. The effect of the surface dipole is included.
We take account of normal currents that emerge when vortices move. Moving Abrikosov vortices in the bulk and Pearl vortices in thin films are considered. Velocity dependent distributions of both normal and persistent currents are studied in the frame of time-dependent London equations. In thin films near the Pearl vortex core, these distributions are intriguing in particular.
Mass of Abrikosov vortices defied experimental observation for more than four decades. We demonstrate a method of its detection in high-temperature superconductors. Similarly to electrons, fluxons circulate in the direction given by the magnetic field, causing circular dichroism. We report the magneto-transmittance of a nearly optimally doped thin YBaCuO film, measured using circularly polarized submillimeter waves. The circular dichroism emerges in the superconducting state and increases with dropping temperature. Our results confirm the dominant role of quasiparticle states in the vortex cor…
Scanning $μ$SQUID force microscopy is used to study magnetic flux structures in single crystals of the layered spin triplet superconductor Sr$\_{2}$RuO$\_{4}$. Images of the magnetic flux configuration above the $\vec{a}\vec{b}$-face of the cleaved crystal are acquired, mostly after field-cooling the sample. For low applied magnetic fields, individual vortices are observed, each carrying a single quantum of flux. Above 1 gauss, coalescence of vortices is discovered. The coalescing vortices may indicate the presence of domains of a chiral order parameter. When the applied field is tilted from t…
By using the time-dependent Ginzburg-Landau theory, we show that extremely diverse experimental data on flux-flow resistivity in multiband superconductors can be qualitatively explained by a composite nature of Abrikosov vortices consisting of elementary fractional vortices in different bands. In composite vortices, the ratio of a core size to electric field relaxation length is found to vary in wide limits depending on system parameters. As a result, the flux-flow magneto-resistance can strongly exceed the single-component Bardeen-Stephen estimation provided that moving vortices generate elec…
Magnetic flux structures in single crystals of the layered spin triplet superconductor Sr$\_{2}$RuO$\_{4}$ are studied by scanning micro SQUID Force microscopy. Vortex chains appear as the applied field is tilted along the in-plane direction of the superconductor. The vortex chains align along the direction of the in-plane component of the applied magnetic field. The decoration of in-plane vortices by crossing Abrikosov vortices is observed: two vortex orientations are apparent simultaneously, one along the layers and the other perpendicular to the layers. The crossing vortices appear preferen…
The intervortex interaction is investigated in very anisotropic layered superconductors in tilted magnetic field. In such a case, the crossing lattice of Abrikosov vortices (AVs) and Josephson vortices (JVs) appears. The interaction between pancakes vortices (PVs), forming the AVs, and JVs produces the deformation of the AV line. It is demonstrated that in the result of this deformation a long range attraction between AVs is induced. This phenomenon is responsible for the dense vortex chains formation. The vortex structure in weak perpendicular magnetic field is the vortex chain phase, while i…
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…
Numerical calculations on a mesoscopic ring of a type II superconductor in the London limit suggest that an Abrikosov vortex can be trapped in such a structure above a critical magnetic field and generate a phase shift in the magnetoresistance oscillations. We prepared submicron-sized superconducting loops of single-crystal, type II superconductor NbSe$_2$ and measured magnetoresistance oscillations resulting from vortices crossing the loops. The free energy barrier for vortex crossing determines the crossing rate and is periodically modulated by the external magnetic flux threading the loop. …
Moving vortices in thin superconducting films are considered within the time-dependent London description. The dissipation due to out-of-core normal excitations for two vortices moving together turns out to have a minimum for the separation vector $\bm a$ parallel to the velocity and equal to $a_m \approx 2.2\, Λ$, where $Λ$ is the Pearl length. The minimum entropy production suggests that moving vortices should have a tendency to form chains along the velocity with a period of the order $a_m$.
We study the Abrikosov-Nielsen-Olesen multi-vortices. Using a numerical code we are able to solve the vortex equations with winding number up to n=25,000. We can thus check the wall vortex conjecture previously made in hep-th/0507273 and hep-th/0507286. The numerical results show a remarkable agreement with the theoretical predictions.
We study force-free configurations of Abrikosov flux lines in the line-liquid and line-crystal limit, near the melting transition at H_m. We show that the condition for zero force configurations can be solved by appealing to the structure of chiral liquid crystalline phases.
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 …
We demonstrate that, in contrast to the single-component Abrikosov vortex, in two-component superconductors vortex solutions with exponentially screened magnetic field exist only in exceptional cases: in the case of vortices carrying an integer number of flux quanta, and in a special parameter limit for half-quantum vortices. For all other parameters the vortex solutions have delocalized magnetic field with a slowly decaying tail. Furthermore, we demonstrate a new effect which is generic in two-component systems but has no counterpart in single-component systems: on exactly half of the paramet…
Unconventional mixed state in the nematic superconductor LiFeAs
CNR-SPIN（意大利国家研究委员会自旋物理研究所）；Division of Mathematics, University of Dundee（邓迪大学数学系）；Maxwell Institute, University of Edinburgh（爱丁堡大学麦克斯韦研究所）；School of Mathematics, University of Leeds（利兹大学数学学院）；Leibniz-Institute for Solid State and Materials Research, IFW-Dresden（德累斯顿IFW莱布尼茨固体与材料研究所）；PSI Center for Neutron and Muon Sciences CNM（瑞士保罗谢尔研究所中子与缪子科学中心）；Laboratorium für Festkörperphysik, ETH Zürich（苏黎世联邦理工学院凝聚态物理实验室）
Strains in superconductors due to moving vortices and vortex lattices are discussed. It is shown that the energy stored in elastic strains increases with vortex velocity. For moving vortex lattices, the elastic energy depends on velocity orientation relative to the lattice having minimum for the velocity directed along one of the unit cell vectors of the moving lattice. It is shown that for supersonic motion, the vortex induced stress field has a shape similar to supersonic shock waves.
The magnetic field $h_z$ of a moving Pearl vortex in a superconducting thin-film in $(x,y)$ plane is studied with the help of time-dependent London equation. It is found that for a vortex at the origin moving in $+x$ direction, $h_z(x,y)$ is suppressed in front of the vortex, $x>0$, and enhanced behind ($x<0$). The distribution asymmetry is proportional to the velocity and to the conductivity of normal quasiparticles. The vortex self-energy and the interaction of two moving vortices are evaluated.
The dynamical phase diagrams for vortices in clean films, driven by an uniform force, and interacting with periodic pinning resulting from a columnar-defect lattice are investigated by numerical simulations of a London model, and other considerations. Dynamical phases are identified according to the spatial symmetry, and dynamical phase diagrams are studied as a function of the driving force magnitude and direction, and on the temperature, for typical vortex densities.
Moving vortex matter, driven by transport currents independent of time, in which vortices and anti-vortices coexist is investigated theoretically in thin superconducting films with nanostructured defects. A simple London model is proposed for the vortex dynamics in films with periodic arrays of nanomagnets or cylindrical holes (antidots). Common to these films is that vortex anti-vortex pairs may be created in the vicinity of the defects by relatively small transport currents, because it adds to the current generated by the defects - the nanomagnets screening current, or the antidots backflow …
In the frame of an exactly solvable model we calculate electric and magnetic fields created by uniformly moving lattice of Josephson vortices driven by the transport current in the magnetic field parallel to conducting layers. The distribution of fields and currents is strongly modified at large velocities of the vortex lattice, plasma oscillations of superconducting electrons being excited by flux flow at low temperatures. This modifies the flux flow resistivity and results in interference features in I-V curves, thus regions of negative differential conductivity appear in which the uniform m…
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…
The vortex system in a high-T_c superconductor has been studied numerically using the mapping to 2D bosons and the path-integral Monte Carlo method. We find a single first-order transition from an Abrikosov lattice to an entangled vortex liquid. The transition is characterized by an entropy jump dS = 0.4 k_B per vortex and layer (parameters for YBCO) and a Lindemann number c_L = 0.25. The increase in density at melting is given by dρ= 6.0*10^{-4} / λ(T)^2. The vortex liquid corresponds to a bosonic superfluid, with ρ_s = ρeven in the limit λ-> \infty.
The magnetic field of moving vortices in anisotropic superconductors is considered in the framework of time-dependent London approach. It is found that at distances large relative to the core size, the field may change sign that alludes to a non-trivial intervortex interaction which depends on the crystal anisotropy and on the speed and direction of motion. These effects are caused to the electric fields and corresponding normal currents which appear due to the moving vortex magnetic structure. We find that the motion related part of the magnetic field attenuates at large distances as $1/r^3$ …
We use analytic and numerical methods to analyze the dynamics of vortices following the quench of a Type-II superconductor under the application of an external magnetic field. In three dimensions, in the absence of a field, the spacing between vortices scales with time t with an exponent phi=0.414 +/- 0.01, In a thin sheet of superconductor, the scaling exponent is phi=0.294 +/- 0.01. When an external magnetic field h is applied, the vortices are confined with respect to the length scale of the Abrikosov lattice, leading to a crossover between the power-law scaling length scale and the lattice…
The coherent nonlinear dynamics of Abrikosov vortices in asymmetric pinning nanolandscapes is studied by theoretical modeling and combined microwave and dc electrical resistance measurements. The problem is considered on the basis of a single-vortex Langevin equation within the framework of a stochastic model of anisotropic pinning. When the distance over which Abrikosov vortices are driven during one half ac cycle coincides with one or a multiple of the nanostructure period, Shapiro steps appear in the current-voltage curves (CVCs) as a general feature of systems whose evolution in time can b…
Current - voltage characteristics in magnetic field for layered superconductor with intrinsic pinning of Josephson vortices
M.N. Mikheev Institute of Metal Physics, Ural Branch, Russian Academy of Sciences（俄罗斯科学院乌拉尔分校 M.N. 米凯耶夫金属物理研究所）；National Research Nuclear University MEPhI（国立核研究大学莫斯科工程物理学院）
Abrikosov vortex lattice dynamics in a superconductor with weak defects is studied taking into account gyroscopic (Hall) properties. It is demonstrated that interaction of the moving lattice with weak defects results in appearance of the additional drag force, $F$, which is $F(V)\propto \sqrt V$ at small velocities, while at high velocities $F(V)\propto 1/\sqrt V$. Thus, the total drag force can be a nonmonotonic function of the lattice translational velocity. It leads to the velocity dependence of the Hall angle and a nonlinear current-voltage characteristic of the superconductor. Due to the …
We consider excited vortex states, which are vortex states left inside a superconductor once the external applied magnetic field is switched off and whose energy is lower than of the normal state. We show that this state is paramagnetic and develop here a general method to obtain its Gibbs free energy through conformal mapping. The solution for any number of vortices in any cross section geometry can be read off from the Schwarz - Christoffel mapping. The method is based on the first order equations used by A. Abrikosov to discover vortices.
In part (I) of this two paper series on stripe fractionalization, we argued that in principle the `domain wall-ness' of the stripe phase could persist in the spin and charge disordered superconductors, and we demonstrated how this physics is in one-to-one correspondence with Ising gauge theory. Here we focus on yet another type of order suggested by the gauge theory: the quantum spin nematic. Although it is not easy to measure this order directly, we argue that the superconducting vortices act as perturbations destroying the gauge symmetry locally. This turns out to give rise to a simple examp…
We consider the structure of Josephson vortices approaching the junction boundary with vacuum in large area Josephson junctions with the Josephson length $λ_J$ large relative to the London penetration depth $λ_L$. Using the stability argument for one-dimentional solitons with respect to 2D perturbations, it is shown that on the scale $λ_J$ the Josephson vortices do not spread near the boundary in the direction of the junction. %, which is in a striking difference with behavior of Abrikosov vortices exiting superconductors. The field distribution in vacuum due to the Josephson vortex is evaluat…
