The magnetic vortices in superconductors usually repel each other. Several cases are discussed when the vortex interaction has an attractive tail and thus a minimum, leading to vortex clusters and chains. Decoration pictures then typically look like in the intermediate state of type-I superconductors, showing
lamellae or islands of Meissner state or surrounded by Meissner state, but with the normal regions filled with Abrikosov vortices that are typical for type-II superconductors in the mixed state. Such intermediate-mixed state was observed and investigated in detail in pure Nb, TaN and other materials 40 years ago; last year it was possibly also observed in MgB$_2$, where it was called "a totally new state" and ascribed to the existence of two superconducting electron bands, one of type-I and one of type-II. The complicated electronic structure of MgB$_2$ and its consequences for superconductivity and vortices are discussed. It is shown that for the real superconductor MgB$_2$ which possesses a single transition temperature, the assumption of two independent order parameters with separate penetration depths and separate coherence lengths is unphysical.
The problem of the vortex structure in the lattice modele of superconductors has been reduced to the nonlinear map problem characteristic for the fractal theory.
Magnetic response of a superconductor depends on the thermodynamic stability of vortex in the material. Here we show that the vortex stability has a close relation with the ratio of the magnetic field at the vortex core center to the thermodynamic critical field. This finding provides a new criterion for the classification of the superconductors according to their magnetic responses.
The vortex-vortex interaction potential in bulk superconductors is calculated within the Ginzburg-Landau (GL) theory and is obtained from a numerical solution of a set of two coupled non-linear GL differential equations for the vector potential and the superconducting order parameter, where the merger of vortices into a giant vortex is allowed. Further, the interaction potentials between a vortex and a giant vortex and between a vortex and an antivortex are obtained for both type-I and type-II superconductors. Our numerical results agree asymptotically with the analytical expressions for large…
The remarkable phenomenon of weak magnetization hysteresis loops, observed recently deep in the vortex-liquid state of a nearly two-dimensional (2D) superconductor at low temperatures, is shown to reflect the existence of an unusual vortex-liquid state, consisting of collectively pinned crystallites of easily sliding vortex chains.
It is shown that there are three kinds of vortices in a superconductor; a vortex with the fluxoid quantum 1 (VF), a new vortex with 0 (VZF), and a vortex with -1 (VAF), respectively. The Hall field via three vortices is studied and found to change its sign, depending of the relative strength of contributions via VF and VAF, similar to that of a system with two kinds of carriers. VZF is suggested to act as a domain wall between VF and VAF regions, accounting for the flux and antiflux regions observed in Nb films.
Abrikosov vortex melting in high-temperature superconductors of ErBa2Cu3O7-δ system is investigated using mechanical contactless method of energy dissipation.
We studied strong-coupling effect upon an isolated vortex in a two-dimensional chiral p-wave superconductor. We solved the Eilenberger equation for the quasiclassical Green's functions and the Eliashberg equation with single mode Einstein boson self-consistently. We calculated the free-energy of each obtained vortex, and found that a non-axisymmetric vortex metastably exists in some situation.
In a type II superconductor in a moderate magnetic field, the superconductor to normal state transition may be described as a phase transition in which the vortex lattice melts into a liquid. In a biaxial superconductor, or even a uniaxial superconductor with magnetic field oriented perpendicular to the symmetry axis, the vortices acquire elongated cross sections and interactions. Systems of anisotropic, interacting constituents generally exhibit liquid crystalline phases. We examine the possibility of a two step melting in homogeneous type II superconductors with anisotropic superfluid stiffn…
We establish the existence of topologically stable knot in two-gap superconductor whose topology $π_3(S^2)$ is fixed by the Chern-Simon index of the electromagnetic potential. We present a helical magnetic vortex solution in Ginzburg-Landau theory of two-gap superconductor which has a non-vanishing condensate at the core, and identify the knot as a twisted magnetic vortex ring made of the helical vortex. We discuss how the knot can be constructed in the recent two-gap $\rm MgB_2$ superconductor.
We predict a resonant microwave absorption on collective vortex modes in a superclean d-wave superconductor. Energies of the collective modes are multiples of the distance between the exact quantum levels of bound states in the vortex core at lower temperatures and involve delocalized states for higher temperatures. We calculate the vortex mass in a d-wave superconductor as a response to a slow acceleration of the vortex. The universal flux-flow regime predicted by N. Kopnin and G. Volovik [Phys. Rev. Lett. 79, 1377 (1997)] is discussed in more detail.
We calculate the dynamic vortex mass for clean Fermi superfluids including both s- and d-wave superconductors as a response to a vortex acceleration. Assuming a finite quasiparticle mean free time, the vortex mass appears to be a tensor. The diagonal component dominates in the limit of long mean free time while the off-diagonal mass takes over in the moderately clean regime.
We study the effects of quantum vortex fluctuations in two-dimensional superconductors using a dual theory of vortices, and investigate the relevance to underdoped cuprates where the superconductor-insulator transition (SIT) is possibly driven by quantum vortex proliferation. We find that a broad enough phase fluctuation regime may exist for experimental observation of the quantum vortex fluctuations near SIT in underdoped cuprates. We propose that this scenario can be tested via pair-tunneling experiments which measure the characteristic resonances in the zero-temperature pair-field susceptib…
We propose a microscopic state for the vortex phase of BSCO superconductors. Around the vortex core or above H_{c2}, the d wave hole pairs form a checkerboard localized in the commensurate antiferromagnetic background. We discuss this theory in connection with recent STM experiments.
We show that under certain conditions multiband superconductors with broken time-reversal symmetry have a new vortex viscosity-generating mechanism which is different from that in conventional superconductors. It appears due to the existence of mixed superfluid phase-density mode inside vortex core. This new contribution is dominant near the time reversal symmetry breaking phase transition. The results could be relevant for three band superconductor $Ba_{1-x}K_{x}Fe_2As_2$.
The vortex transport entropy in the mixed state of the cuprate superconductors YBa$_2$Cu$_3$O$_{7-δ}$ and Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ at the maximum Nernst or Ettingshausen signal below $T_c$ is discussed. The vortex transport entropy per CuO$_2$ double layer in the cuprate epitaxial films is found to be close to the Boltzmann constant.
We describe a semi-analytic approach to the two-band Ginzburg-Landau theory, which predicts the behavior of vortices in two-band superconductors. We show that the character of the short-range vortex-vortex interaction is determined by the sign of the normal domain - superconductor interface energy, in analogy with the conventional differentiation between type-I and type-II superconductors. However, we also show that the long-range interaction is determined by a modified Ginzburg-Landau parameter $κ^*$, different from the standard $κ$ of a bulk superconductor. This opens the possibility for non…
Sharp increase of pinning force was observed at the 3D-2D phase transition in strongly anisotropic high temperature superconductors of the BiPbSrCaCuO system.
In superconductors with strong coupling between superconductivity and elasticity manifested in a strong dependence of transition temperature on pressure, there is an additional contribution to inter-vortex interactions due to the strain field generated by vortices. When vortex lines are along the $c$ axis of a tetragonal crystal, a square vortex lattice (VL) is favored at low vortex densities, because the vortex-induced strains contribution to the inter-vortex interactions is long range. At intermediate magnetic fields, the triangular lattice is stabilized. The triangular lattice evolves to th…
We present a geometry-based discussion of possible vortex configurations in the mixed state of anisotropic type-II superconductors. It is shown that, if energy considerations assign six nearest neighbors to each vortex, two distinct modifications of the vortex lattice are possible. It is expected that certain conditions lead to a first order phase transition from one modification of the vortex lattice to the other upon varying the external magnetic field.
Abrikosov's solution of the linearized Ginzburg-Landau theory describing a periodic lattice of vortex lines in type-II superconductors at large inductions, is generalized to non-periodic vortex arrangements, e.g., to lattices with a vacancy surrounded by relaxing vortices and to periodically distorted lattices that are needed in the nonlocal theory of elasticity of the vortex lattice. Generalizations to lower magnetic inductions and to three-dimensional arrangements of curved vortex lines are also given. Finally, it is shown how the periodic vortex lattice can be computed for bulk superconduct…
Study of the Abrikosov vortex motion in superconductors based on time-dependent Ginzburg-Landau equations reveals an opportunity to locally detect the values of the Aharonov-Bohm type curl-less vector potentials.
We show that the two-gap superconductor has two types of non-Abrikosov magnetic vortex, the D-type which has no concentration of the condensate at the core and the N-type which has a non-vanishing concentration of the condensate at the core, which may carry $4π$-flux, $2π$-flux, or a fractional flux, depending on the parameters of the potential. Furthermore, we show that the non-Abrikosov vortex is described by two types of topology. The integer flux vortex has the non-Abelian topology $π_2(S^2)$, but the fractional flux vortex has the Abelian topology $π_1(S^1)$. We show that the inclusion of…
The elementary vortex pinning potential is studied in a chiral p-wave superconductor with a pairing d=z(k_x + i k_y) on the basis of the quasiclassical theory of superconductivity. An analytical investigation and numerical results are presented to show that the vortex pinning potential is dependent on whether the vorticity and chirality are parallel or antiparallel. Mutual cancellation of the vorticity and chirality around a vortex is physically crucial to the effect of the pinning center inside the vortex core.
By measuring the dynamic and traditional magnetization relaxations we investigate the vortex dynamics of the newly discovered superconductor SmFeAsO_0.9F_0.1 with Tc = 55K. It is found that the relaxation rate is rather large reflecting a small characteristic pinning energy. Moreover it shows a weak temperature dependence in wide temperature region, which resembles the behavior of the cuprate superconductors. Combining with the resistive data under different magnetic fields, a vortex phase diagram is obtained. Our results strongly suggest that the model of collective vortex pinning applies to …
The quasiparticle density of states (DOS) in the vortex state has been probed by specific heat measurements under magnetic fields (H) for clean and dirty s-wave superconductors, Y(Ni1-xPtx)2B2C and Nb1-xTaxSe2. We find that the quasiparticle DOS per vortex is appreciably H-dependent in the clean-limit superconductors, while it is H-independent in the dirty superconductors as expected from a conventional rigid normal electron core picture. We discuss possible origins for our observations in terms of the shrinking of the vortex core radius with increasing H.
The transformation of the vortex lattice in a tetragonal superconductor which consists of its 45 degree reorientation relative to the crystal axes is studied using the nonlocal London model. It is shown that the reorientation occurs as two successive second order (continuous) phase transitions. The transition magnetic fields are calculated for a range of parameters relevant for borocarbide superconductors in which the reorientation has been observed.
Harnessing the properties of vortices in superconductors is crucial for fundamental science and technological applications; thus, it has been an ongoing goal to locally probe and control vortices. Here, we use a scanning probe technique that enables studies of vortex dynamics in superconducting systems by leveraging the resonant behavior of a raster-scanned, magnetic-tipped cantilever. This experimental setup allows us to image and control vortices, as well as extract key energy scales of the vortex interactions. Applying this technique to lattices of superconductor island arrays on a metal, w…
The quantization of magnetic flux in superconductors is usually seen as vortices penetrating the sample. While vortices are unstable in bulk type I superconductors, restricting the superconductor causes a variety of vortex structures to appear. We present a systematic study of giant vortex states in type I superconductors obtained by numerically solving the Ginzburg-Landau equations. The size of the vortices is seen to increase with decreasing film thickness. In type I superconductors, giant vortices appear at intermediate thicknesses but they do not form a well-defined vortex lattice. In the …
It is argued that the stress caused by vortex cores in the mixed state of superconductors may result in a field dependent contribution to the free energy and magnetization of measurable levels. For sufficiently strong stress dependence of the critical temperature, $\partial T_c/\partial p$, this contribution may result in the so-called "second peak" in the field dependence of the reversible magnetization, the effect often masked by vortex pinning and creep.
We investigate the vortex structure in chiral p-wave superconductors by the Bogoliubov-de Gennes theory on a tight-binding model. We calculate the spatial structure of the pair potential and electronic state around a vortex, including the anisotropy of the Fermi surface and superconducting gap structure. The differences of the vortex structure between $\sin p_x + {\rm i} \sin p_y$-wave and $ \sin p_x - {\rm i} \sin p_y$-wave superconductors are clarified in the vortex lattice state. We also discuss the winding $\mp 3$ case of the $\sin{(p_x+p_y)} \pm {\rm i} \sin{(-p_x+p_y)}$-wave superconduct…
We consider bound states at the vortex core of a non-centrosymmetric superconductor. We show that, despite the mixing of singlet and triplet order parameters, zero energy states survive within certain parameter space as in vortices of some chiral p-wave states.
The relaxation iterative method is used to minimize Ginzburg-Landau model for the two-band superconductor with Josephson-coupling. A stable spontaneous vortex-antivortex pair with long range order has been revealed. Our result appears due to the Josephson-coupling effect, which leads to a linearly-dependence of total free energy on the distance between vortex-antivortex pair, hence accounts for this phenomenon.
We present the solution to London's equations for the magnetic fields of a vortex oriented parallel to the plane, and normal to a crystal face, of a layered superconductor. These expressions account for flux spreading at the superconducting surface, which can change the apparent size of the vortex along the planes by as much as 30%. We compare these expressions with experimental results.
We explain 90 degree reorientation in the vortex lattice of borocarbide superconductors on the basis of a phenomenological extension of the nonlocal London model that takes full account of the symmetry of the system. We propose microscopic mechanisms that could generate the correction terms and point out the important role of the superconducting gap anisotropy.
We develop an alternative method to solve the Eilenberger equations numerically for the vortex-lattice states of type-II superconductors. Using it, we clarify the magnetic-field and impurity-concentration dependences of the magnetization, the entropy, the Pauli paramagnetism, and the mixing of higher Landau levels in the pair potential for two-dimensional $s$- and $d_{x^2-y^2}$-wave superconductors with the cylindrical Fermi surface.
We theoretically predict that the critical current $I_c$ and magnetization $M$ of hybrid superconductor/normal-metal (SN) strip may have nonmonotonous dependence on perpendicular magnetic field - so called peak effect. In contrast to familiar peak effect, which is connected either with vortex entry to the superconductor or with peculiarities of vortex pinning, the found phenomenon exists at low fields, in the vortex-free (Meissner) phase. We argue that the effect appears at specific parameters of studied hybrid structure when its in-plane current-supervelocity relation has two maxima. We expec…
The vortex dynamics of a d+is-wave superconductor is studied numerically by simulating the time-dependent Ginzburg-Landau equations. The critical fields, the free flux flow, and the flux flow in the presence of twin-boundaries are discussed. The relaxation rate of the order parameter turns out to play an important role in the flux flow. We also address briefly the intrinsic Hall effect in d- and d+is-wave superconductors.
The first integrals of the Ginzburg-Landau equations for a vortex-free state of superconductors with different mixed symmetries of the order parameter are found. The general boundary conditions for the order parameter at the ideal interface between the superconductor and vacuum are derived. Based on these integrals and boundary conditions, we analyze the stability criteria for vortex-free state in unconventional superconductors. The threshold field above which the Abrikosov vortices can enter the superconductor is found to be higher or equal to the thermodynamic critical field for all states u…
We present a microscopic approach to the quantum tunneling of vortices. The formalism characterizes the rate at which a many-body superconducting state with a vortex in one location makes a transition to a second many-body superconducting state with a vortex in a second location. The theory is utilized to study the effect of a d-wave order parameter upon the directionality of vortex tunneling.
The magnetic field probability P(B) is calculated from the Ginzburg-Landau theory for various lattices of vortex lines in type-II superconductors: Ideal triangular lattices, lattices with various shear strains and with a super lattice of vacancies, and lattices of short vortices in films whose magnetic field ''mushrooms'' near the surface.
A simulation study using the time-dependent Ginzburg-Landau theory is performed for the vortex state in two-component superconductors, such as PrOs_4_Sb_12_. We investigate the flux flow and the pinning of the vortex sheet structure. We find domain wall that traps half flux-quantum vortices and moves with the flux flow. In the pinning case, we observe an emitting process of a conventional vortex from the vortex sheet by combining a pair of half flux-quantum vortices.
We study effects of a chiral helimagnet (CHM) on vortex states in a superconductor, solving the Ginzburg-Landau equations in a chiral helimagnet/superconductor bilayer system. We found that vortices form a periodically modulated triangular lattice, because the magnetic field from the chiral helimagnet $H_{CHM}$ oscillates spatially. An increase of a critical current is expected, because vortices are pinned by the $H_{CHM}$.
The Josephson-plasma and vortex modes in layered superconductors have been studied theoretically for low magnetic fields parallel and perpendicular to layers. The two modes belong to the same lowest-frequency branch of collective-mode spectrum localized near vortices. This is the Josephson-plasma resonance if pancakes are strongly pinned and cannot move. Otherwise the lowest-frequency mode is a vortex mode governed by pancake pinning and the vortex mass. The latter is strongly enhanced by wandering of a vortex line around its average direction. The recently observed jump of the magnetoabsorpti…
We present a phenomenological model for the vortex dynamics in the peak effect region of weakly pinned superconductors. We explain the history dependent dynamic response of the metastable vortex states subjected to a transport current and the hysteretic voltage-current characteristics observed in the vicinity of peak effect in weakly pinned superconductors. A strong variation in voltage current characteristics with the current sweep rate and the anomalous dependence of critical current density $J_c$ on the magnetic field sweep rate have also been accounted for by this model.
We numerically show that the zero-energy Majorana surface states are suppressed around a vortex in the three-dimensional topological superconductors such as Cu$_{x}$Bi$_{2}$Se$_{3}$ and Sn$_{1-x}$In$_{x}$Te. On the other hand, the zero-energy Majorana bound states along the vortex line are robust against cut by the surface. The suppression of the surface bound states is similar to that with a magnetic impurity on the surface of the topological insulator. The suppression of the surface-bound states around a vortex can be observed as the unconventional energy dependence of imaging of the scannin…
We study the collective behavior of Majorana modes in the vortex state of chiral $p$-wave superconductors. Away from the isolated vortex limit, the zero-energy Majorana states communicate with each other on a vortex lattice, and form a coherent band structure with non-trivial topological character. We revealed that the topological nature of Majorana bands changes sensitively via quantum phase transitions in the two-dimensional (2D) systems, as sweeping magnetic field or Fermi energy. Through the dimensional reduction, we showed the existence of generic superconducting Weyl phase in a low magne…
In type-1.5 superconductors, vortices emerge in clusters, which grow in size with increasing magnetic field. These novel vortex clusters and their field dependence are directly visualized by scanning SQUID microscopy at very low vortex densities in MgB2 single crystals. Our observations are elucidated by simulations based on a two-gap Ginzburg-Landau theory in the type-1.5 regime.
Analogous to the various states of condensed matter exhibiting crystalline, amorphous, liquid and spatially inhomogeneous phases, such structures are likewise observed in superconducting vortex matter.
The vortex lattice with the superconducting and normal state charge carriers fractions may be regarded as three independent subsystems mutually connected by interactions. The equation of motion for these three subsystems must be solved simultaneously. In this way a new consistent theory of the vortex dynamics is obtained and the magnetoconductivity calculated.
We numerically study the vortex core structure in a noncentrosymmetric superconductor such as CePt3Si without mirror symmetry about the xy plane. A single vortex along the z axis and a mixed singlet-triplet Cooper pairing model are considered. The spatial profiles of the pair potential, local density of states, supercurrent density, and radially-textured magnetic moment density around the vortex are obtained in the clean limit on the basis of the quasiclassical theory of superconductivity.
We calculate various thermodynamic quantities of vortex liquids in a layered superconductor by using the nonperturbative parquet approximation method, which was previously used to study the effect of thermal fluctuations in two-dimensional vortex systems. We find there is a first-order transition between two vortex liquid phases which differ in the magnitude of their correlation lengths. As the coupling between the layers increases,the first-order transition line ends at a critical point. We discuss the possible relation between this critical end-point and the disappearance of the first-order …
We study the effect of the superconducting gap nodes on the vortex lattice properties of high temperature superconductors at very low temperatures. The nonlinear, nonlocal and nonanalytic nature of this effect is shown to have measurable consequences for the vortex lattice geometry and the effective penetration depth in the mixed state as measured by muon-spin-rotation experiments.
Vortices carrying fractions of a flux quantum are predicted to exist in multiband superconductors, where vortex core can split between multiple band-specific components of the superconducting condensate. Using the two-component Ginzburg-Landau model, we examine such vortex configurations in a two-band superconducting slab in parallel magnetic field. The fractional vortices appear due to the band-selective vortex penetration caused by different thresholds for vortex entry within each band-condensate, and stabilize near the edges of the sample. We show that the resulting fractional vortex config…
We have investigated vortex states in chiral helimagnet/superconductor bilayer systems under an applied external magnetic field Happl, using the Ginzburg-Landau equations. Effect of the chiral helimagnet on the superconductor is taken as a magnetic field HCHM, which is perpendicular to the superconductor and oscillates spatially. For Happl = 0 and weak HCHM, there appear pairs of up-and down-vortices. Increasing Happl, down-vortices gradually disappear, and number of up-vortices increases in the large magnetic field region. Then, up-vortices form parallel, triangular, or square structures.
As shown in recent experiments [Auslaender et al., Nature Physics 5, 35 (2009)] magnetic force microscopy permits one not only to image but also to manipulate an individual vortex in type-II superconductors, and this manipulation provides a new powerful tool to study vortex dynamics and pinning. We derive equations that describe the deformation of an individual vortex in an anisotropic biaxial type-II superconductor under the action of the microscope's magnetic tip. These equations take into account the driving force generated by the tip, the elastic force caused by the vortex deformation, and…
Contrary to recent theoretical prediction, we show that the magnetic flux of a vortex in SU(2) model of two-gap superconductor is quantized in units of 2π/g, not 4 π/g. For the U(1) version of this model, the flux is quantized in units of 2 πα/g where 0 < α< 1. The parameter αdepends on the masses and concentrations of the Cooper pairs in the two condensates.
The elementary vortex pinning potential is studied in unconventional superconductors within the framework of the quasiclassical theory of superconductivity. Numerical results are presented for d-, anisotropic s-, and isotropic s-wave superconductors to show explicitly that in unconventional superconductors the vortex pinning potential is determined mainly by the loss of the condensation energy in bulk due to the presence of the pinning center, i.e., by the breakdown of the Anderson's theorem. It is found that the vortex pinning energy in the d-wave pairing case is 4 -- 13 times larger than tho…
Thermodynamic stability of composite vortex in a two-component superconductor is investigated by the Ginzburg-Landau theory. The predicted nature of these vortices has recently attracted much attention. Here we consider axially symmetric quantized vortex and show that the stability of vortex depends on three independent dimensionless parameters: $κ_{1}$, $κ_{2}$, $κ_ξ$, where $κ_{i}(i=1,2)$ is the Ginzburg-Landau parameter of individual component, $κ_{ξ}=ξ_{1}/ξ_{2}$ is the ratio of two coherence lengths. We also show that there exists thermodynamic stable vortex in type-1+type-2 or type-2+typ…
A density-functional approach is used to calculate the inhomogeneous vortex density distribution in the flux liquid phase at the planar surface of a layered superconductor, where the external magnetic field is perpendicular to the superconducting layers and parallel to the surface. The interactions with image vortices are treated within a mean field approximation as a functional of the vortex density. Near the freezing transition strong vortex density fluctuations are found to persist far into the bulk liquid. We also calculate the height of the Bean-Livingston surface barrier.
