Flux-line Liquid Research Articles

Flux Line Lattice Melting and the Formation of a Coherent Quasiparticle Bloch State in the UltracleanURu2Si2Superconductor

We find that in the ultraclean heavy-fermion superconductor URu(2)Si(2) (T_{c0}=1.45 K) a distinct flux line lattice melting transition with outstanding characters occurs well below the mean-field upper critical fields. We show that a very small number of carriers with heavy mass in this system results in exceptionally large thermal fluctuations even at sub-Kelvin temperatures, which are witnessed by a sizable region of the flux line liquid phase. The uniqueness is further highlighted by an enhancement of the quasiparticle mean free path below the melting transition, implying a possible formation of a quasiparticle Bloch state in the periodic flux line lattice.

Weak point disorder in strongly fluctuating flux-line liquids

We consider the effect of weak uncorrelated quenched disorder (point defects) on a strongly fluctuating flux-line liquid. We use a hydrodynamic model which is based on mapping the flux-line system onto a quantum liquid of relativistic charged bosons in 2+1 dimensions [P. Benetatos and M. C. Marchetti, Phys. Rev. B 64, 054518, (2001)]. In this model, flux lines are allowed to be arbitrarily curved and can even form closed loops. Point defects can be scalar or polar. In the latter case, the direction of their dipole moments can be random or correlated. Within the Gaussian approximation of our hydrodynamic model, we calculate disorder-induced corrections to the correlation functions of the flux-line fields and the elastic moduli of the flux-line liquid. We find that scalar disorder enhances loop nucleation, and polar (magnetic) defects decrease the tilt modulus.

Theory of disordered flux-line liquids

We study the equilibrium statics and nonequilibrium driven dynamics of flux-line liquids in the presence of a random pinning potential. Under the assumption of replica symmetry, we find in the static case using a replica Gaussian variational method that the only effect of disorder is to increase the tilt modulus and the confining ``mass'' of the internal modes of the flux lines, thus decreasing their thermal wandering. In the nonequilibrium driven case, we derive the long-scale, coarse-grained equation of motion of the vortices in the presence of disorder, which, apart from new Kardar-Parisi-Zhang nonlinearities, has the same form as the equation of motion for unpinned vortices, with renormalized coefficients. We also compute the structure factor of a disordered vortex liquid, and show that the disorder contributes an additive, Lorentzian squared term, to the structure factor of the center of mass mode, otherwise leaving the functional form of terms describing the internal modes unchanged. The expression of the static structure factor derived within our approach is consistent both with experimental data and with the standard theory of elasticity of vortices in high temperature superconductors.

Variational theory of flux line liquids

We formulate a variational (Hartree like) description of flux line liquids which improves on the theory we developed in an earlier paper [A.M. Ettouhami, Phys. Rev. B 65, 134504 (2002)]. We derive, in particular, how the massive term confining the fluctuations of flux lines varies with temperature and show that this term vanishes at high enough temperatures where the vortices behave as freely fluctuating elastic lines.

Entanglement in dilute flux-line liquids

We construct, within the framework of classical statistical mechanics, a mean-field theory of dilute flux-line liquids which goes beyond linear hydrodynamics. Within our approach, we find that interactions between vortices produce a massive term in the Hamiltonian of the internal modes of the flux lines which confines their transverse fluctuations. This suggests that the flux-line liquid, at least in the low density limit considered in this paper, might very well be in a weakly entangled state, where the average width $〈{u}^{2}{〉}^{1/2}$ of flux lines can be much larger than the average distance between the lines but does not diverge with the sample thickness L. Consequences on the physics of flux-line liquids are briefly discussed.

Collective modes influx line liquids

We study the collective modes of flux line liquids such asoccur in the type-II superconductors of recent interest composed oftwo-dimensional layered structures. Starting from the vortex-vortexinteraction and employing the dielectric formalism within the random-phase approximation, we find propagating sound modes in thelong-wavelength limit.

Nonlinear hydrodynamics and tilt modulus of flux-line liquids

In this paper we use non-Gaussian hydrodynamics to study the magnetic response of a flux-line liquid in the mixed state of a type-II superconductor. Both the derivation of our model, which goes beyond conventional Gaussian flux liquid hydrodynamics, and its relationship to other approaches used in the literature are discussed. We focus on the response to a transverse tilting field which is controlled by the tilt modulus, c44, of the flux array. We show that interaction effects can enhance c44 even in infinitely thick clean materials. This enhancement can be interpreted as the appearance of a disentangled flux-liquid fraction. In contrast to earlier work, our theory incorporates the nonlocality of the intervortex interaction in the field direction. This nonlocality is crucial for obtaining a nonvanishing renormalization of the tilt modulus in the thermodynamic limit of thick samples.

Onsager loop transition and first-order flux-line lattice melting in high-Tcsuperconductors

Monte-Carlo simulations in conjunction with finite-size scaling analysis are used to investigate the $(H,T)$-phase diagram in uniaxial anisotropic high- $T_c$ superconductors, both in zero magnetic field and in intermediate magnetic fields for various mass-anisotropies. The model we consider is the uniformly frustrated anisotropic Villain Model. In zero magnetic field, and for all anisotropies considered, we find one single second order phase transition, mediated by an Onsager vortex-loop blowout. This is the superconductor-normal metal transition.A comparison with numerical simulations and a critical scaling analysis of the zero-field loop-transition yields the same exponent of the loop distribution function at the critical point. In the intermediate magnetic field regime, we find two anomalies in the specific heat. The first anomaly at a temperature $T_m$ is associated with the melting transition of the flux-line lattice. The second anomaly at a temperature $T_z$ is one where phase coherence along the field direction is destroyed. We argue that $T_m=T_z$ in the thermodynamic and continuum limit. Hence, there is no regime where the flux line lattice melts into a disentangled flux-line liquid. The loss of phase coherence parallel to the magnetic field in the sample is argued to be due to the proliferation of closed non-field induced vortex loops on the scale of the magnetic length in the problem, resulting in flux-line cutting and recombination. In the flux-line liquid phase, therefore, flux-lines appear no longer to be well defined entities. A finite-size scaling analysis of the delta function peak specific heat anomaly at the melting transition is used to extract the discontinuity of the entropy at the melting transition.This entropy discontinuity is found to increase rapidly with mass-anisotropy.

Superfluid bosons and flux liquids: disorder, thermal fluctuations, and finite-size effects

The influence of different types of disorder (both uncorrelated and correlated) on the superfluid properties of a weakly interacting or dilute Bose gas, as well as on the corresponding quantities for flux line liquids in high-temperature superconductors at low magnetic fields are reviewed, investigated and compared. We exploit the formal analogy between superfluid bosons and the statistical mechanics of directed lines, and explore the influence of the different “imaginary time” boundary conditions appropriate for a flux line liquid. For superfluids, we discuss the density and momentum correlations, the condensate fraction, and the normal-fluid density as function of temperature for two- and three-dimensional systems subject to a space- and time-dependent random potential as well as conventional point-, line-, and plane-like defects. In the case of vortex liquids subject to point disorder, twin boundaries, screw dislocations, and various configurations of columnar damage tracks, we calculate the corresponding quantities, namely, density and tilt correlations, the “boson” order parameter, and the tilt modulus. The finite-size corrections due to periodic vs. open “imaginary time” boundary conditions differ in interesting and important ways. Experimental implications for vortex lines are described briefly.

Properties of the vortex liquid in the classical and quantum regime

The properties of vortex matter in thin films of amorphous NbGe is investigated in regimes that have not been studied before. By designing a device of alternating strong and weak pinning stripes we were able to measure the shear viscosity of the 2D flux line liquid above the 2D melting line. Good agreement is observed with the predictions of the theory of Nelson and Halperin. By studying the properties of an 18 nm thick film down to very low temperatures we entered the regime where quantum fluctuations influence the spectrum of vortex vibrations. It is seen that melting at T = 0 takes place well below the transition to the normal state at B c2 (0). From resistivity measurements it follows that thermally activated creep in the liquid regime at relatively high temperatures crosses over to quantum creep at low temperatures. The crossover temperature as well as typical results for the energy barriers and Euclidean action are presented.

Tilt instabilities and multiple coexisting vortex orientations in flux-line liquids.

Phase diagrams for various tilt instabilities, as well as phase diagrams for possible coexistence of vortex lines with different orientations in a tilted magnetic field of anisotropic type-II superconductors, are mapped out and compared within anisotropic London theory. Double minima in the Gibbs energy as a function of vortex orientation in the case of noninteracting vortex liquids may be found in a regime of mass anisotropy where long-wavelength tilt instabilities do not occur, and are associated with coexisting species of flux lines of different orientations with respect to the crystal c^ axis. At larger mass-anisotropies, long-wavelength tilt instabilities do occur for tilt modes perpendicular to the ab planes. For tilt modes parallel to the ab planes, instabilities only occur for very large wave numbers, beyond the regime of applicability of London theory. The cutoff dependence on the phase diagrams for the isolated flux-line limit is investigated. It is essential that the elliptic shape of the vortex core is accounted for. When interactions are accounted for in a dense flux-line liquid, the tilt instabilities are removed. \textcopyright{} 1996 The American Physical Society.

AC permeability of the flux-line liquid in the anisotropic high-T/sub c/ superconducting crystals

AC properties of the flux-line liquid in the anisotropic high-temperature superconducting crystals in a parallel field are theoretically investigated. The ac responses in the simple flux-flow regime are analyzed from associated effective ac-magnetic permeability calculated on the hydrodynamic theory basis. The responses are studied as functions of anisotropic ratio and sample dimensions. The results illustrate the influence of the platelet crystal's size on permeability in the anisotropic superconductors while in the isotropic superconductors, the relationship of responses between a square rod and cylinder is found. It indicates that the permeability of a cylinder can be essentially replaced by that of a square rod and vice versa. The geometric effect on response is also elucidated in the isotropic superconductors.

Two different thermally activated flux-flow regimes in oxygen-deficient YBa2Cu3O7-x thin films.

We have measured the temperature dependence of the longitudinal resistivity of an oxygen-deficient YBa{sub 2}Cu{sub 3}O{sub 6.45} thin film in perpendicular applied magnetic fields {ital B}{lt}12 T. A crossover from a three-dimensional (3D) flux-line liquid to a quasi-2D liquid of vortices is identified which is characterized by a crossover line {ital B}{sub cr}=136.13{sup *}(1{minus}{ital T}/{ital T}{sub {ital c}})/{ital T} (T{sub c}=30.35 K, {ital B}{sub cr} in T). In both 3D and 2D regimes, the dissipations are thermally activated. Above the crossover temperatures, the vortices are in a quasi-2D liquid state and the dissipation is dominated by the motion of edge dislocation pairs or vortex cutting and the activation energy is proportional to (1{minus}{ital T}/{ital T}{sub {ital c}})ln{ital B}. Below the crossover temperatures and above the irreversibility line, the vortices are in a 3D line liquid state, and the dissipation is governed by plastic deformation of the vortices through double kink nucleations in the vortices with activation energies {ital U}{sub 0}{proportional_to}(1{minus}{ital T}/{ital T}{sub {ital c}}){ital B}{sup {minus}0.46}.

Ac response of the flux-line liquid in the high-Tc superconductors.

We use a hydrodynamics theory to discuss the response of a viscous flux-line liquid to an ac perturbation applied at the surface of the sample. The theory incorporates viscoelastic effects and describes the crossover between liquidlike and solidlike response of the vortex array as the frequency of the perturbation increases. A large viscosity from flux-line interactions and entanglement leads to viscous screening of surface fields. As a result, two frequency-dependent length scales are needed to describe the penetration of an ac field. For large viscosities the imaginary part of the ac permeability can exhibit, in addition to the well-known peak associated with flux diffusion across the sample, a low-frequency peak corresponding to the transition from solidlike to liquidlike behavior.

Kinetic theory of flux-line hydrodynamics: Liquid phase with disorder.

We study the Langevin dynamics of flux lines of high--T$_c$ superconductors in the presence of random quenched pinning. The hydrodynamic theory for the densities is derived by starting with the microscopic model for the flux-line liquid. The dynamic functional is expressed as an expansion in the dynamic order parameter and the corresponding response field. We treat the model within the Gaussian approximation and calculate the dynamic structure function in the presence of pinning disorder. The disorder leads to an additive static peak proportional to the disorder strength. On length scales larger than the line static transverse wandering length and at long times, we recover the hydrodynamic results of simple frictional diffusion, with interactions additively renormalizing the relaxational rate. On shorter length and time scales line internal degrees of freedom significantly modify the dynamics by generating wavevector-dependent corrections to the density relaxation rate.

Translational correlations in the vortex array at the surface of a type-II superconductor.

We discuss the statistical mechanics of magnetic flux lines in a finite-thickness slab of type-II superconductor. The long wavelength properties of a flux-line liquid in a slab geometry are described by a hydrodynamic free energy that incorporates the boundary conditions on the flux lines at the sample's surface as a surface contribution to the free energy. Bulk and surface weak disorder are modeled via Gaussian impurity potentials. This free energy is used to evaluate the two-dimensional structure factor of the flux-line tips at the sample surface. We find that surface interaction always dominates in determining the decay of translational correlations in the asymptotic long-wavelength limit. On the other hand, such large length scales have not been probed by the decoration experiments. Our results indicate that the translational correlations extracted from the analysis of the Bitter patterns are indeed representative of behavior of flux lines in the bulk.

Dynamics of flux-line liquids in high-T c superconductors

A hydrodynamic theory of the flux line lattice and the isotropic and hexatic flux line liquids in high- T c superconductors is presented. Weak microscopic pinning centers are described within the flux flow model of Bardeen and Stephen, while strong macroscopic pinning centers set the boundary conditions for the flow. A large intervortex viscosity, which we attribute to entanglement, allows the effects of a few strong pins to propagate large distances in the fluid phases. In thin films, the diverging viscosity associated with a continuous freezing transition should produce similar effects. We propose a simple experiment to observe this pinning length scale and distinguish between various theories of the resistivity drop. The hydrodynamic modes of flux line solids, hexatics and liquids are discussed. Weak pinning shows up as an elastic peak in the dynamic correlation functions. The macroscopic flow of a flux liquid driven by a uniform external current in the presence of macroscopic obstacles of specified geometry is also described.

Correspondence between two-dimensional bosons and a bulk superconductor in a magnetic field.

We study the partition function of two-dimensional lattice bosons at $T=0$ and derive a dual representation which is isomorphic to a bulk superconductor with fluctuating-gauge field in an applied magnetic field. This allows us to relate boson ground states to thermodynamic phases of the superconductor. A density-wave Bose insulator corresponds to an Abrikosov flux-lattice phase, whereas boson superfluidity implies a nonsuperconducting flux-line liquid phase. By analogy with a boson supersolid, we suggest the possibility of an exotic Abrikosov flux-lattice phase with no superconducting long-ranged order.