Ordered Vortex Lattice Research Articles

Vortex lattice state transitions in water surface waves

Surface waves can generate an ordered vortex lattice of fluid particles on the air-water interface. Here, the transitions of the lattice structure are observed through experiments and modelled analytically. A fully-immersed square waveguide system is positioned at the center of a larger bath which directs particles into counter-rotating vortices. Flow interactions with the bath free-surface alter the ordered structure, and a state transition is initiated by the increase of the waveguide immersion depth. It manifests as a change in the orbital momentum and spin properties of the vortices, and develops through the combination of reflected-wave attenuation and mean flow effects from the classical Stokes drift and Eulerian currents. Transitions from a perfect antiferromagnetic state to weakly-ordered ones are achieved, including a monopole vortex reminiscent of spectral condensates in thin fluid layers. Order in the disordered flow is uncovered through velocity field corrections after invoking time-scale separation. The lattice structures are modelled analytically through a superposition of mean flow contributions from the waveguide and bath modes. The results open a new platform for investigating the dynamics of wave and vortex lattice interactions, and devising new techniques to characterize complex flow phenomena on the air-water interface.

Circuit-QED-based measurement of vortex lattice order in a Josephson junction array

Superconductivity provides a canonical example of a quantum phase of matter. When superconducting islands are connected by Josephson junctions in a lattice, the low temperature state of the system can map to the celebrated XY model and its associated universality classes. This has been used to experimentally implement realizations of Mott insulator and Berezinskii--Kosterlitz--Thouless (BKT) transitions to vortex dynamics analogous to those in type-II superconductors. When an external magnetic field is added, the effective spins of the XY model become frustrated, leading to the formation of topological defects (vortices). Here we observe the many-body dynamics of such an array, including frustration, via its coupling to a superconducting microwave cavity. We take the design of the transmon qubit, but replace the single junction between two antenna pads with the complete array. This allows us to probe the system at 10 mK with minimal self-heating by using weak coherent states at the single (microwave) photon level to probe the resonance frequency of the cavity. We observe signatures of ordered vortex lattice at rational flux fillings of the array.

Signatures of two-step impurity-mediated vortex lattice melting in Bose-Einstein condensates

We simulate a rotating 2D BEC to study the melting of a vortex lattice in the presence of random impurities. Impurities are introduced either through a protocol in which vortex lattice is produced in an impurity potential or first creating the vortex lattice in the absence of random pinning and then cranking up the (co-rotating) impurity potential. We find that for a fixed strength, pinning of vortices at randomly distributed impurities leads to the new states of the vortex lattice. It is unearthed that the vortex lattice follows a two-step melting via loss of positional and orientational order. Also, the comparisons between the states obtained in two protocols show that the vortex lattice states are metastable states when impurities are introduced after the formation of an ordered vortex lattice. We also show that the existence of metastable states depends on the history of how the vortex lattice is created.

Onsager rule, quantum oscillation frequencies, and the density of states in the mixed-vortex state of cuprates

The Onsager rule determines the frequencies of quantum oscillations in magnetic fields. We show that this rule remains intact to an excellent approximation in the mixed-vortex state of the underdoped cuprates even though the Landau level index $n$ may be fairly low, $n\sim 10$. The models we consider are fairly general, consisting of a variety of density wave states combined with $d$-wave superconductivity within a mean field theory. Vortices are introduced as quenched disorder and averaged over many realizations, which can be considered as snapshots of a vortex liquid state. We also show that the oscillations ride on top of a field independent density of states, $\rho(B)$, for higher fields. This feature appears to be consistent with recent specific heat measurements [C. Marcenat, et al. Nature Comm. {\bf 6}, 7927 (2015)]. At lower fields we model the system as an ordered vortex lattice, and show that its density of states follows a dependence $\rho(B)\propto \sqrt{B}$ in agreement with the semiclassical results [G. E. Volovik, JETP Lett. {\bf 58}, 469 (1993)].

Disorder-induced two-step melting of vortex matter in Co-intercalatedNbSe2single crystals

Disorder induced melting, where the increase in positional entropy created by random pinning sites drives the order-disorder transition in a periodic solid, provides an alternate route to the more conventional thermal melting. Here, using real space imaging of the vortex lattice through scanning tunneling spectroscopy, we show that in the presence of weak pinning, the vortex lattice in a type II superconductor disorders through two distinct topological transitions. Across each transition, we separately identify metastable states formed through superheating of the low temperature state or supercooling of the high temperature state. Comparing crystals with different levels of pinning we conclude that the two-step melting is fundamentally associated with the presence of random pinning which generates topological defects in the ordered vortex lattice.

Partially ordered vortex lattices in the high-field low-temperature mixed state of quasi-two-dimensional organic superconductors

We report the results of high-field, low-temperature $\ensuremath{\mu}\mathrm{SR}$ measurements of the quasi-two-dimensional organic superconductors $\ensuremath{\kappa}\text{\ensuremath{-}}{(\mathrm{ET})}_{2}\mathrm{Cu}{(\mathrm{NCS})}_{2}$, and $\ensuremath{\kappa}\text{\ensuremath{-}}{(\mathrm{ET})}_{2}\mathrm{Cu}[\text{N}{(\mathrm{CN})}_{2}]\mathrm{Br}$. The $\ensuremath{\mu}\mathrm{SR}$ lineshapes for these compounds indicate the existence of partially ordered vortex lattice phases in the high-magnetic-field regime, up to 2.5 T for the former compound and 4 T for the latter compound. The observed sharp loss of order is found to be consistent with a vortex-lattice melting transition that is predicted by numerical simulations of weakly coupled layers of pancake vortices. It is argued that the robustness of the partially ordered vortex lattice phases could be due to strong flux-line pinning by a dilute ensemble of defects.

Vortex dynamics of rotating dipolar Bose–Einstein condensates

We study the influence of dipole–dipole interaction on the formation of vortices in a rotating dipolar Bose–Einstein condensate (BEC) of 52Cr and 164Dy atoms in quasi two-dimensional geometry. By numerically solving the corresponding time-dependent mean-field Gross–Pitaevskii equation, we show that the dipolar interaction enhances the number of vortices, while a repulsive contact interaction increases the stability of the vortices. Furthermore, an ordered vortex lattice of a relatively large number of vortices is found in a strongly dipolar BEC.

Exact analytic Gorkov-Ginzburg-Landau theory of type-II superconductivity in the magnetoquantum oscillations limit

A new Green's function representation is employed in a microscopic derivation of a Ginzburg-Landau theory of strongly type-II superconductivity at high magnetic fields. An exact analytical, physically transparent expression for the quartic term in the corresponding order parameter expansion is presented. The resulting expression reveals nonlocal contributions to the superconducting (SC) free energy, associated with highly coherent cyclotron motions of the paired electrons near the Fermi surface, which lead to singular couplings to the vortex lattice. A major part of these contributions arises from incoherent scattering by the spatially averaged pair potential, which is purely harmonic in the de Haas--van Alphen frequency. However, coherent scatterings by the ordered vortex lattice generate a large erratically oscillating (i.e., paramagnetic-diamagnetic) contribution to the SC free energy as a function of the magnetic field. Vortex lattice disorder, which tends to suppress this oscillatory component, is found to preserve the harmonic part of the SC free energy.

Absence of vortex lattice melting in a high-purity Nb superconductor

The state of the vortex lattice extremely close to the superconducting to normal transition in an applied magnetic field is investigated in high purity niobium. We observe that thermal fluctuations of the order parameter broaden the superconducting to normal transition into a crossover but no sign of a first order vortex lattice melting transition is detected in measurements of the heat capacity or the small angle neutron scattering (SANS) intensity. Direct observation of the vortices via SANS always finds a well ordered vortex lattice. The fluctuation broadening is considered in terms of the Lowest Landau Level theory of critical fluctuations and scaling is found to occur over a large H_{c2}(T) range.

Frustrated Classical Heisenberg andXYModels in Two Dimensions with Nearest-Neighbor Biquadratic Exchange: Exact Solution for the Ground-State Phase Diagram

The ground-state phase diagram is determined exactly for the frustrated classical Heisenberg model plus nearest-neighbor biquadratic exchange interactions on a two-dimensional lattice. A square- and a rhombic-symmetry version are considered. There appear ferromagnetic, incommensurate-spiral, "up-up-down-down" (UUDD), and canted ferromagnetic states, a nonspiral coplanar state that is an ordered vortex lattice, plus a noncoplanar ordered state (a "conical vortex lattice"). In the rhombic case, which adds biquadratic terms to the Heisenberg model used widely for insulating manganites, the UUDD state found is the E-type state observed; this, along with accounting essentially for the variety of ground states observed in these materials, shows that this model probably contains the long-sought mechanism behind the UUDD state.

Order-disorder transition in the vortex lattice induced by anisotropy of the YBaCuO superconductor

Pinning force Fp variation with the magnetic field H and the angle α = ∠H, ab is investigated in crystals having different concentrations of point defects npd. Comparison between the functions Fp (H) and Fp (α) strongly suggests that the minimum position in the Fp (α) -curve corresponds to the angle αOD, which separates the ordered vortex lattice (VL) formed at α < αOD and the disordered VL formed at α > αOD. This conclusion is confirmed by variation in the E(J)-curve shape in the vicinity of the angle αOD, and by hysteresis behavior of the E(J)-curves measured for increased and decreased transport current at angles α > αOD. This order-disorder transition arises, because the elastic energy of the disordered VL decreases with an increased α, while the pinning energy is independent of α.

Dynamic ordering and instability of the vortex lattice in Nb films exhibiting moderately strong pinning

We study the nonlinear effects in the current-voltage characteristics of Nb superconducting thin films, induced both by the current dependence of the pinning force and by the electric-field dependence of the flux-flow resistance of Abrikosov vortices driven by high electric currents. Despite of the quite strong pinning in our samples, by increasing the bias current, in a temperature dependent magnetic-field range, the moving vortex system undergoes a dynamical transition from a disordered to an ordered vortex lattice. Such transition leads to a quite sharp reduction of the dynamic pinning force corresponding to a peak in the current dependence of the differential flux-flow resistance. On the same samples, in a higher voltage-velocity regime, for a different temperature dependent magnetic-field range, an instability of the moving vortex lattice also occurs, with a sudden jump in the $I\text{\ensuremath{-}}V$ curve from the non linear flux-flow branch to the linear normal resistive state. Within the Larkin-Ovchinnikov velocity dependence of the flux-flow resistance, this flux-flow instability has been studied as function of the magnetic field and temperature in order to get out its nonequilibrium electronic nature. Finally, we propose a dynamic phase diagram to describe the vortex lattice motion driven by high electric currents in the presence of such a disordered pinning landscape.

VORTEX PHASE DIAGRAM AND DYNAMICS NEAR T = 0 IN THE LIQUID PHASE OF A THICK a-MoxGe1-x FILM

We report on the equilibrium vortex phase diagram and vortex dynamics driven by dc current in the low-temperature liquid phase of thick amorphous (a-) Mo x Ge 1-x and a- Mo x Si 1-x films containing weak and moderately strong pinning, respectively. The solid state for the a- Mo x Ge 1-x film is an ordered vortex lattice or Bragg glass, which is different from the disordered vortex glass for the a- Mo x Si 1-x films. We find that the liquid state is similar to each other, involving the quantum-vortex-liquid (QVL) phase at low temperature. The width of the QVL phase is not enhanced by reduced pinning strength. In both systems we detect the unusual vortex flow in (near) the QVL phase.

Vortex phase diagram and dynamics at low temperature in a thick a-MoxGe1−x film

We study the vortex phase diagram and dynamics in the low-temperature liquid phase of a thick amorphous a-MoxGe1−x film with weak pinning. The vortex solid state is ordered vortex lattice or Bragg glass, which is different from the disordered vortex glass of a-MoxSi1−x films with stronger pinning. We find, however, that the vortex liquid state is similar to that for the a-MoxSi1−x films, involving the quantum-vortex-liquid phase at low temperature, where we again detect the unusual vortex flow.

Simulations of Dynamical Ordering in Pinned Vortex Systems

We model a vortex system in a sample with bulk pinning and superficial pinning generated by a magnetic decoration. We perform a sequence of finite temperature numerical experiments in which external forces are applied to obtain, a dynamically ordered vortex lattice. We analyze the final structures and the behavior of the total energy of the system.

Domain regime in two-dimensional disordered vortex matter

A detailed numerical study of the real space configuration of vortices in disordered superconductors using 2D London-Langevin model is presented. The magnetic field $B$ is varied between 0 and $B_{c2}$ for various pinning strengths $\Delta$. For weak pinning, an inhomogeneous disordered vortex matter is observed, in which the topologically ordered vortex lattice survives in large domains. The majority of the dislocations in this state are confined to the grain boundaries/domain walls. Such quasi-ordered configurations are observed in the intermediate fields, and we refer it as the domain regime (DR). The DR is distinct from the low-field and the high-fields amorphous regimes which are characterized by a homogeneous distribution of defects over the entire system. Analysis of the real space configuration suggests domain wall roughening as a possible mechanism for the crossover from the DR to the high-field amorphous regime. The DR also shows a sharp crossover to the high temperature vortex liquid phase. The domain size distribution and the roughness exponent of the lattice in the DR are also calculated. The results are compared with some of the recent Bitter decoration experiments.

Rapidly rotating Bose-Einstein condensates in and near the lowest Landau level.

We create rapidly rotating Bose-Einstein condensates in the lowest Landau level by spinning up the condensates to rotation rates Omega > 99% of the centrifugal limit for a harmonically trapped gas, while reducing the number of atoms. As a consequence, the chemical potential drops below the cyclotron energy 2 variant Planck's over 2pi Omega. While in this mean-field quantum-Hall regime we still observe an ordered vortex lattice, its elastic shear strength is strongly reduced, as evidenced by the observed very low frequency of Tkachenko modes. Furthermore, the gas approaches the quasi-two-dimensional limit. The associated crossover from interacting- to ideal-gas behavior along the rotation axis results in a shift of the axial breathing mode frequency.

Vortex imaging in magnesium diboride withH⊥c

We have performed scanning tunneling spectroscopy imaging of vortices in ${\mathrm{MgB}}_{2}$ with both the tunneling current and the applied field perpendicular to the crystalline c axis. This revealed a well ordered vortex lattice with a relatively low anisotropy of $1.19\ifmmode\pm\else\textpm\fi{}0.04.$ This is significantly below the upper-critical-field anisotropy, but in good agreement with theoretical predictions for the penetration depth anisotropy, which at low fields is expected to be dominated by the $\ensuremath{\pi}$ band. The value of the anisotropy is smaller than estimated from small-angle neutron-scattering measurement.

Ordered Vortex Lattice and Intrinsic Vortex Core States in Bi2Sr2CaCu2OxStudied by Scanning Tunneling Microscopy and Spectroscopy

The ordered vortex lattice in Bi 2 Sr 2 CaCu 2 O x (overdoped, T c =83 K) has been observed for the first time at 4.2 K in 8 T by scanning tunneling spectroscopy (STS). The vortex lattice is short-range ordered in the length scale of 100 nm. The vortices form an almost square lattice with the sides parallel to the diagonal direction of the CuO 2 square lattice, that is, the nodal direction of the d x 2 - y 2 superconductor. In all of the vortex cores of the ordered lattice, the localized states are observed at ±9 meV symmetrically in the superconducting gap and are clearly determined to be intrinsic to the vortex in Bi 2 Sr 2 CaCu 2 O x . The intensity is found to be electron–hole asymmetric.

Flux-lattice melting in two-dimensional disordered superconductors

The flux line lattice melting transition in two-dimensional pure and disordered superconductors is studied by a Monte Carlo simulation using the lowest Landau level approximation and quasi-periodic boundary condition on a plane. The position of the melting line was determined from the diffraction pattern of the superconducting order parameter. In the clean case we confirmed the results from earlier studies which show the existence of a quasi-long range ordered vortex lattice at low temperatures. Adding frozen disorder to the system the melting transition line is shifted to slightly lower fields. The correlations of the order parameter for translational long range order of the vortex positions seem to decay slightly faster than a power law (in agreement with the theory of Carpentier and Le Doussal) although a simple power law decay cannot be excluded. The corresponding positional glass correlation function decays as a power law establishing the existence of a quasi-long range ordered positional glass formed by the vortices. The correlation function characterizing a phase coherent vortex glass decays however exponentially ruling out the possible existence of a phase coherent vortex glass phase.