Vortex Lattice In type-II Superconductors Research Articles

Extended Ginzburg-Landau theory of superconductivity from generalized momentum operator and position-dependent mass

The concept of generalized momentum operator which is motivated from the generalized uncertainty principle has been extensively investigated in quantum mechanics and various aspects of theoretical and applied physics. In this study, we have applied this formalism to Ginzburg-Landau theory of superconductvity and Abrikosov vortex lattice in type II-superconductors. After deriving the extended Ginzburg-Landau equations, we have discussed several independent structures of the auxiliary function of position operator in the generalized momentum operator and we have analyzed their features in both Ginzburg-Landau and London theories. Comparable properties to the basic formalism have been obtained even without the presence of the cubic nonlinear term in Ginzburg-Landau equations. However, not all structures of the auxiliary function of position operator result on the exclusion of the magnetic field from a superconductor when it's below its critical temperature. But, only specific forms may succeed to eliminate the magnetic field. This approach has been generalized by considered a position-dependent mass of the electric charge. Amazingly, for a position-dependent mass of hyperbolic soliton-like structure, the extended Ginzburg-Landau wave function is identical to the result obtained in the conventional formalism and besides, for specific correlations between the auxiliary function of the position operator and position-dependent mass, the exclusion of the magnetic field is conceivable. We have also discussed the Abrikosov vortex lattice solution based on the extended Ginzburg-Landau formalism with position-dependent mass of the electric charge. It was observed that, for a specific structure of the position-dependent mass and for a quantum number n=0, a transition between a type-II and type-I superconductor takes place if the Ginzburg-Landau parameter is κ=121+257≈2.0634. For large n, the problem depends on the asymptotic form of the Hermite polynomial and periodicity occurs if the electric charge is quantized. Further details are obtained and analyzed.

Observation of a topological defect lattice in the charge density wave of 1T-TaS2

Charge density wave (CDW) phases native to the transition metal dichalcogenide 1T-TaS2 are probed through a graphene layer covering its surface by using local scanning tunneling microscopy/spectroscopy and transport measurements. We show that over a wide range of temperatures, the graphene cover, while protecting the air-sensitive 1T-TaS2 from oxidation, does not obstruct access to the electronic properties of the CDW. At room temperature, where the system is in the nearly commensurate CDW phase, we observe contiguous domains of the CDW phase separated by a network of topological defects. These topological defects, dubbed CDW vortices, form a lattice with a quasi-long range translational order, which resembles the Bragg glass phase in Abrikosov vortex lattices in type-II superconductors. While the quasi-long range order in the CDW itself is not uncommon, its existence in the CDW vortex lattice is observed here for the first time. Finally, when reducing the temperature to 78 K into the commensurate CDW phase, we find that the graphene cover layer acquires properties consistent with a proximity-induced CDW.

The effect of Joule heating on the terahertz radiation in the superconductor media

Moving vortex lattice in type-II superconductors can result in the radiation of electromagnetic waves in the range of terahertz frequency. The vortex flow dynamics in superconductors follow the London equation. In this paper, the effects of the superconductor coherence length and the London penetration depth on the radiated power is investigated by solving the London equation in the presence of vortices. The results show that by decreasing the coherence length, the radiation power will significantly increase. Also, it is possible to obtain more radiated power with a sharp peak by increasing the penetration depth. Further investigation of the effect of Joule heating reveals that radiation power is almost independent of temperature when we are far from critical temperature. But on the other hand, near the critical temperature, the radiation power is strongly dependent on the temperature variation. Also, by increasing the bias current, the radiation power will augment, and the frequency of radiated peaks will change accordingly.

Transport evidence for a sliding two-dimensional quantum electron solid

Ignited by the discovery of the metal-insulator transition, the behaviour of low-disorder two-dimensional (2D) electron systems is currently the focus of a great deal of attention. In the strongly interacting limit, electrons are expected to crystallize into a quantum Wigner crystal, but no definitive evidence for this effect has been obtained despite much experimental effort over the years. Here, studying the insulating state of a 2D electron system in silicon, we have found two-threshold voltage-current characteristics with a dramatic increase in noise between the two threshold voltages. This behaviour cannot be described within existing traditional models. On the other hand, it is strikingly similar to that observed for the collective depinning of the vortex lattice in type-II superconductors. Adapting the model used for vortexes to the case of an electron solid yields good agreement with our experimental results, favouring the quantum electron solid as the origin of the low-density state.

Vortex molecules in thin films of layered superconductors

The seminal work of Abrikosov on the vortex lattices in type-II superconductors has triggered the creation and rapid development of the whole new field of condensed matter physics, namely, the physics of vortex matter in a large variety of superfluid and superconducting systems. Among these systems and compounds the layered superconductors are known to form a very important class which is associated with a number of fascinating new discoveries in vortex physics. The phenomenon of the vortex attraction in tilted magnetic fields provides one of the examples of such findings which affected theoretical and experimental research in the field for almost two decades. In our paper we review some recent advances in this direction focusing on the intervortex interaction and equilibrium vortex structures in thin films of layered superconductors in magnetic field tilted with respect to the layers. In such a case the magnetic field penetrates superconductor in the form of tilted vortices or a crossing array of Josephson vortices and pancake stacks. We study the interplay between two different long-range potentials: (i) attraction of tilted vortices or deformed stacks; (ii) the Pearl's repulsion. This interplay is responsible for the formation of the minimum in the total interaction energy and resulting decay of vortex chains in clusters. The number of Abrikosov vortices in these clusters (or vortex molecules) depends on field tilting angle and film thickness.

Electromagnetic Response of Superconductors with Pinned Vortices

Some features of the vortex lattice in type-II superconductors and of its pinning and thermally activated depinning are given. Manipulation of a pinned single vortex by the tip of a Magnetic Force Microscope is mentioned. Statics and dynamics of pinned vortices in thick and thin strips and disks, and in rectangular plates, can be computed from continuum theory.

Vortex lattices in type-II superconductors studied by small-angle neutron scattering

Here we review recent small-angle scattering studies of the vortex lattice in a range of type-II superconductors carried out by our group. Emphasis is placed on providing examples of the kind of information which can be obtained by such measurements, focusing in particular on studies of the vortex lattice structure and form factor in LuNi2B2C, TmNi2B2C, CeCoIn5 and Ba(Fe0.93Co0.07)2As2.

Critical states in thin planar type-II superconductors in a perpendicular or inclined magnetic field (Review)

The theory of the critical states of a vortex lattice in type-II superconductors is examined without any assumptions about the relative perpendicularity of the local magnetic fields and circulating currents in the sample. Such a theory has made it possible to solve a number of problems for thin films of superconductors in an external magnetic field oriented perpendicular to their surface: a theory of the shaking effect is constructed for rectangular superconducting plates and the critical states in samples with anisotropic pinning of the flux lines as well as in the presence of an order-disorder phase transition in a vortex lattice are studied. In addition, the critical states in a long superconducting strip in an inclined magnetic field are investigated.

Photonic band structure of isotropic and anisotropic Abrikosov lattices in superconductors

We have performed a numerical solution for band structure of an Abrikosov vortex lattice in type-II superconductors forming a periodic array in two dimensions for applications of incorporating the photonic crystals concept into superconducting materials with possibilities for optical electronics. The implemented numerical method is based on the extensive numerical solution of the Ginzburg–Landau equation for calculating the parameters of the two-fluid model and obtaining the band structure from the permittivity for both orthogonal polarizations, which depends on the above parameters and the frequency. This is while the characteristics of such crystals highly vary with an externally applied static normal magnetic field, leading to nonlinear behavior of the band structure, which also has nonlinear dependence on the temperature. The similar analysis for every arbitrary lattice structure is also possible to be developed by this approach as presented in this work. We also present some examples and discuss the results.

Precision photonic band structure calculation of Abrikosov periodic lattice in type-II superconductors

We have performed a numerical solution for band structure of an Abrikosov vortex lattice in type-II superconductors forming a periodic array in two dimensions for applications of incorporating the photonic crystals concept into superconducting materials with possibilities for optical electronics. The implemented numerical method is based on the extensive numerical solution of the Ginzburg–Landau equation for calculating the parameters of the two-fluid model and obtaining the band structure from the permittivity, which depends on the above parameters and the frequency. This is while the characteristics of such crystals highly vary with an externally applied static normal magnetic field, leading to nonlinear behavior of the band structure, which also has nonlinear dependence on the temperature. The similar analysis for every arbitrary lattice structure is also possible to be developed by this approach as presented in this work. We also present some examples and discuss the results.

Banding of angular velocity in the current-driven vortex lattice of a type-II superconductor

We phenomenologically investigate a possible mechanism of the banding of angular velocity in a current-driven vortex lattice in type-II superconductors. It is shown that the plastic motion occurs above some critical value of the applied current, resulting in a splitting in the vortex lattice. The present model can qualitatively reproduce major features of the experimental results reported by Lopez et al. [Phys. Rev. Lett. 82, 1277 (1999)].

Generation of DC electric fields due to vortex rectification in superconducting films

The introduction of the manuscript reviews different mechanisms of generation of permanent electric fields by AC driven vortex lattices in type-II superconductors due to artificial symmetry breaking. The second part shows that superconducting Pb and Nb films (strips or crosses) with or without symmetric periodic pinning centers, subject to a magnetic field perpendicular to the film plane, also exhibit magnetically tunable AC current rectification. At low drive frequencies, not far below the critical temperature, the superconducting films work as one-dimensional rectifiers (i.e. generate an uniform DC electric field along the direction of the AC current) due to unavoidable small vortex pinning asymmetry. At higher frequencies, above 105Hz, rectification gradually becomes two-dimensional with a strongly non-uniform generated DC electric field. DC voltages, either longitudinal or transversal to the AC current are tunable with the applied magnetic field. The rectified voltage depends strongly on the temperature and the AC drive intensity. In superconductors with periodic pinning centers the rectified voltage varies periodically with the number of vortices per pinning center. Not far below Tc, the generated DC electric field is nearly opposite on the opposite film sides. The unusual two-dimensional character of rectification at high frequencies close to Tc could be qualitatively understood in terms of local rectification due the oppositely directed asymmetric edge barriers (Bean–Livingston type) or by slowly relaxing electric fields generated by local critical current excess. Anisotropic pinning effects represent a dominant contribution to the rectification further below Tc. Since in experiments on rectification in superconductors this electric field adds to the one due to, e.g., anisotropic vortex pinning, one has to take into account its presence when interpreting the rectification experiments.

Numerical Study of Dynamical Frictional Phenomena Based on the Two-Dimensional Frenkel-Kontorova Model with Impurities

We study numerically dynamical frictional phenomena using the Frenkel-Kontorova model with impurities. Even in the presence of randomness sliding states have a quasi long range order, which changes with the sliding velocity. Such an order affects the velocity dependence of the kinetic frictional force and produces peculiar transverse pinning effects. We discuss them in the relation with the moving Bragg glass theory developed for vortex lattices in type II superconductors. Frictional phenomena have common importance in various physical systems such as vortex lattices in type-II superconductors, charge and spin density waves, interfacial friction of solids and so on. 1),2) We investigate here dynamical frictional phenomena in two-dimensional elastic lattice systems based on the Frenkel-Kontorova (FK) model, 3) and focus especially on the kinetic friction and the transverse pinning effect. The FK model employed here consist of a discrete square lattice with harmonic interatomic force and a substrate lattice with impurities.The periodic potential due to the regular part of the substrate lattice has a two-dimensional sinusoidal form, of which principal axes coincide with those of the lattice on it.Impurities are distributed randomly in the substrate and the potential of each impurity has a Gaussian form.The amplitude of the periodic (impurity) potential is denoted by K(W ).We study overdamped dynamics of the FK model driven by an external force along one of the principal axes.Incommensurate systems are investigated here. The periodic potential pins the lattice even in an incommensurate case when the amplitude K is greater than a critical value Kc ≈ 0.23.The number of sites of a lattice is set to be N =8 9 2 (or 144 2 ) and the impurity density is fixed at about 0.21. A periodic boundary condition is imposed. The kinetic frictional force Fkin is plotted against the sliding velocity v in Fig.1. In the low velocity regime, v � 10, the kinetic frictional force is almost constant and not affected by the periodic potential.In the high velocity regime, however, it decreases with increasing velocity and depends on the periodic potential.The effectiveness of the periodic potential depends on the sliding velocity.This behavior will relate to the change of the order of the sliding lattice as mentioned later. We investigate next the response transverse to the sliding direction.According to the theory of the moving Bragg glass (MBG) developed by Giamarchi and Le Doussal, 4) a quasi long range order of the lattice exists in the transverse direction.

Phase transition to a square vortex lattice in type-II superconductors with fourfold anisotropy

We investigate the stability of the square vortex lattice which has been recently observed in experiments on the borocarbide family of superconductors. Taking into account the tetragonal symmetry of these systems, we add fourfold symmetric fourth-derivative terms to the Ginzburg-Landau(GL) free energy. At $H_{c2}$ these terms may be treated perturbatively to lowest order to locate the transition from a distorted hexagonal to a square vortex lattice. We also solve for this phase boundary numerically in the strongly type-II limit, finding large corrections to the lowest-order perturbative results. We calculate the relative fourfold $H_{c2}$ anisotropy for field in the $xy$ plane to be 4.5% at the temperature, $T_c^{\Box}$, where the transition occurs at $H_{c2}$ for field along the $z$ axis. This is to be compared to the 3.6% obtained in the perturbative calculation. Furthermore, we find that the phase boundary in the $H-T$ phase diagram has positive slope near $H_{c2}$.

Precision Ginzburg-Landau Solution of Ideal Vortex Lattices for Any Induction and Symmetry

A method is presented that solves the Ginzburg-Landau equations for the ideal Abrikosov vortex lattice in type-II superconductors with high precision for arbitrary induction, Ginzburg-Landau parameter, and vortex lattice symmetry. This iteration procedure excels previous one-dimensional circular cell methods and approximate variational methods, and is easily adapted to anisotropic and unconventional superconductors. Selected results are given for the order parameter, the form factors of the periodic magnetic field measurable by neutron scattering, reversible magnetization curves, and the shear modulus of the vortex lattice, which could not be obtained by previous methods.

Strong-Pinning Effects in Low-Temperature Creep: Charge-Density Waves inTaS3

Nonlinear conduction in the quasi-one dimensional conductor o-TaS_3 has been studied in the low-temperature region down to 30 mK. It was found that at temperatures below a few Kelvins the current-voltage (I-V) characteristics consist of several branches. The temperature evolution of the I-V curve proceeds through sequential freezing-out of the branches. The origin of each branch is attributed to a particular strong pinning impurity type. Similar behavior is expected for other physical systems with collective transport (spin-density waves, Wigner crystals, vortex lattices in type-II superconductors etc.) in the presence of strong pinning centers.

Renormalization group analysis of weak collective pinning

The effect of impurity disorder on vortex lattices in type-II superconductors is studied within the framework of the weak collective pinning theory. Using a renormalization group approach, we calculate the sizeR c of collectively pinned vortex bundles in the regimea o <R c <λ (small bundles) for isotropic superconductors. We assume weak disorder and neglect thermal effects. Our findings quantitatively refine the lowest-order perturbation result,R c ∞\(\sqrt B \), obtained by Larkin and Ovchinnikov in 1979 (B is the magnetic induction). In particular, we find a different numerical constant in the exponential as well as a new algebraic prefactor. These improvements provide us with a more accurate functional dependence of the critical current on the magnetic field.

Linear response and instability of vortex lattice in type-II superconductors

The wave vector dependent linear response of the vortex lattice in type-II superconductors is studied by calculating the superfluid density and is shown to have a singular behavior with respect to the long wave length perturbation of vector potential. The calculations are carried out in terms of the nonlinear elastic theory of the vortex lattice, which is, in principle, a low temperature expansion, and the terms up to the first order in temperature are taken into account, which is beyond the usual Gaussian elastic theory. The superfluid density is found to behave as ~ kBT ¦k⊥¦2 log ¦k⊥¦,for the small wave vector,k⊥,perpendicular to the external magnetic field, where Tand kBare the temperature and the Boltzmann constant, respectively. This behavior causes the divergence of the magnetic susceptibility at a critical wave vector, kc.We show that this behavior is associated with an instability of vortex lattice to the externally applied perturbation.

Pattern formation at the traveling liquid-crystal twist grain boundary-smectic a interface

Pattern formation at phase boundaries moving in a temperature gradient is one of the major areas of nonequilibrium physics attracting considerable attention. While most of the early work concentrated on the moving solid-liquid interface, now the focus has changed to phase transitions characterized by broken continuous symmetry. Most recently we investigated consequences to interfacial patterns of a chirality-induced equilibrium length. Here we study patterns at another chiral interface where one of the phases has a chirality-induced defect lattice, the twist grain boundary (TGB) phase. The TGB state is analogous to the vortex lattice in Type-II superconductors predicted by the Gennes’ analogy between the nematic (N)-smectic A (A) transition and the normal-superconducting transition. In this analogy, a cholesteric A transition is analogous to the normal-superconducting transition in an external magnetic field and a theory has been developed for its analogous vortex lattice, the TGB phase, when this transition is Type II. We study patterns formed at the traveling TGB-A phase boundary. Different patterns are observed depending on whether TGB grows into A or A into TGB. Indeed, this maybe the first time a steady-state pattern is observed in directional melting (i.e. TGB growing into A). As these patterns have a broad band of wavelengths, they are difficult to characterize physically. Thus, we introduced a novel analysis (most simply but not rigorously described as) measuring the fractal dimension of the patterns at these traveling interfaces. Two lengths emerged from this analysis: a longer one set by sample thickness and a shorter one set by the smallest TGB unit that can grow into an oriented smectic A phase. We invoke our old dynamic arguments to account for why TGB cannot propagate at a second-order TGB-cholesteric phase transition so it is eventually squeezed out leaving behind a direct cholesteric-A transition.

Multicritical behavior of Abrikosov vortex lattices near the cholesteric–smectic-A–smectic-C*point

The mean-field phase diagram of the cholesteric--smectic-A--smectic-${\mathrm{C}}^{\mathrm{*}}$ (${\mathit{N}}^{\mathrm{*}--}$Sm-A--Sm-${\mathit{C}}^{\mathrm{*}}$) point is derived within the framework of the chiral Chen-Lubensky model. We show that on the Sm-${\mathit{C}}^{\mathrm{*}}$ side of the phase diagram, one or two additional twist-grain-boundary (TGB) (or chiral-smectic) phases will occur. These additional chiral-smectic phases, the ${\mathrm{TGB}}_{\mathit{C}}$ and ${\mathrm{TGB}}_{\mathit{C}}^{\mathrm{*}}$ phases, are highly dislocated versions of smectic-C and smectic-${\mathit{C}}^{\mathrm{*}}$ phases, respectively. According to the de Gennes superconductor analogy, the grain-boundary phase which occurred on the smectic-A side of the phase diagram (known as the ${\mathrm{TGB}}_{\mathit{A}}$ or chiral Sm-${\mathit{A}}^{\mathrm{*}}$ phase) is the analog of the Abrikosov vortex lattice in type-II superconductors. However, the ${\mathrm{TGB}}_{\mathit{C}}$ and ${\mathrm{TGB}}_{\mathit{C}}^{\mathrm{*}}$ phases would correspond to Abrikosov vortex lattices in a hypothetical superconductor where the Ginzburg parameter \ensuremath{\kappa} is negative and the photon has Bose condensed. These additional phases are predicted to occur near the recently observed ${\mathit{N}}^{\mathrm{*}--}$Sm-${\mathit{A}}^{\mathrm{*}--}$Sm-${\mathit{C}}^{\mathrm{*}}$ multicritical point. We discuss the ${\mathit{N}}^{\mathrm{*}\mathrm{\ensuremath{-}}}$${\mathrm{TGB}}_{\mathit{C}}$, ${\mathrm{TGB}}_{\mathit{A}\mathrm{\ensuremath{-}}}$${\mathrm{TGB}}_{\mathit{C}}$, and ${\mathrm{TGB}}_{\mathit{A}\mathrm{\ensuremath{-}}}$${\mathrm{TGB}}_{\mathit{C}}^{\mathrm{*}}$ phase transitions. Because of the twist, the XY-like Sm-A--Sm-C is (usually) replaced by the Ising-like ${\mathrm{TGB}}_{\mathit{A}\mathrm{\ensuremath{-}}}$${\mathrm{TGB}}_{\mathit{C}}$. However, if the Sm-${\mathit{C}}^{\mathrm{*}}$ helicoidal pitch length is somewhat smaller than the cholesteric pitch length, then the ${\mathrm{TGB}}_{\mathit{A}\mathrm{\ensuremath{-}}}$${\mathrm{TGB}}_{\mathit{C}}$ transition would be replaced by the ${\mathrm{TGB}}_{\mathit{A}\mathrm{\ensuremath{-}}}$${\mathrm{TGB}}_{\mathit{C}}^{\mathrm{*}}$ transition.