Hexagonal Vortex Lattice Research Articles

Flux creep regimes and vortex phase diagram in β-FeSe single crystals

We analyze the relationship between critical current densities (JC) and flux creep rates (S) in β-FeSe single crystals. This analysis was based on magnetization measurements. Additionally, we establish correlations with the recently reported magnetic field-induced geometrical deformation of the vortex lattice, transitioning from hexagonal to square shape due to a rhombic distortion [A. V. Putilov et al. Phys. Rev 99 (2019) 144514]. The results show that the magnetic field dependence of Jc displays distinct regimes, which is reflected by changes in S. The vortex dynamics is analyzed within the framework of the collective creep theory. S is characterized by low pinning energies and glassy exponents according to the expectation for small-bundles at low temperatures and magnetic fields where a hexagonal vortex lattice was reported. Conversely, we observe a systematic increase in S, resembling a shift from small to large vortex bundles, at magnetic fields corresponding to the rhombic distortion. Last, the relaxation rates exhibit significant values for magnetic fields where a square vortex lattice is expected, suggesting a potential crossover from elastic to plastic creep. Our findings highlight a direct relationship between vortex lattice deformations and a decrease in vortex pinning related to vortex-defect interactions.

Surface structure and multigap superconductivity of V3Si (111) revealed by scanning tunneling microscopy

V3Si, a classical silicide superconductor with relatively high T_{mathrm{C}} (∼16 K), is promising for constructing silicon-based superconducting devices and hetero-structures. However, real space characterization on its surfaces and superconducting properties are still limited. Here we report the first low-temperature scanning tunnelling microscopy (STM) study on cleaned V3Si (111) single crystal surface. We observed a sqrt{3} times sqrt{3} superstructure which displays mirror symmetry between adjacent terraces, indicating the surface is V-terminated and reconstructed. The tunneling spectrum shows full superconducting gap with double pairs of coherence peaks, but has a relatively small gap size with comparing to bulk T_{mathrm{C}}. Impurity induced in-gap state is absent on surface defects but present on introduced magnetic adatoms. Upon applying magnetic field, a hexagonal vortex lattice is visualized. Interestingly, the vortex size is found to be field dependent, and the coherence length measured from single vortex at low field is significantly larger than estimated value from bulk H_{c2}. These results reflect V3Si is a multi-band, s-wave superconductor.

Absence of hexagonal-to-square lattice transition in LiFeAs vortex matter

We investigated magnetic vortices in two stoichiometric LiFeAs samples by means of scanning tunneling microscopy and spectroscopy. The vortices were revealed by measuring the local electronic density of states (LDOS) at zero bias conductance of samples in magnetic fields between 0.5 and 12 T. From single vortex spectroscopy we extract the Ginzburg-Landau coherence length of both samples as $4.4\pm0.5$ nm and $4.1\pm0.5$ nm, in accordance with previous findings. However, in contrast to previous reports, our study reveals that the reported hexagonal to square-like vortex lattice transition is absent up to 12 T both in field-cooling and zero-field-cooling processes. Remarkably, a highly ordered zero field cooled hexagonal vortex lattice is observed up to 8 T. We argue that several factors are likely to determine the structure of the vortex lattice in LiFeAs such as (i) details of the cooling procedure (ii) sample stoichiometry that alters the formation of nematic fluctuations, (iii) details of the order parameter and (iv) magnetoelastic coupling.

Focal beam structuring by triple mixing of optical vortex lattices

On-demand generation and reshaping of arrays of focused laser beams is highly desired in many areas of science and technology. In this work, we present a versatile approach for laser beam structuring in the focal plane of a lens by triple mixing of square and/or hexagonal optical vortex lattices (OVLs). In the artificial far field the input Gaussian beam is reshaped into ordered arrays of bright beams with flat phase profiles. This is remarkable, since the bright focal peaks are surrounded by hundreds of OVs with their dark cores and two-dimensional phase dislocations. Numerical simulations and experimental evidences for this are shown, including a broad discussion of some of the possible scenarios for such mixing: triple mixing of square-shaped OVLs, triple mixing of hexagonal OVLs, as well as the two combined cases of mixing square-hexagonal-hexagonal and square-square-hexagonal OVLs. The particular ordering of the input phase distributions of the OV lattices on the used spatial light modulators is found to affect the orientation of the structures ruled by the hexagonal OVL. Reliable control parameters for the creation of the desired focal beam structures are the respective lattice node spacings. The presented approach is flexible, easily realizable by using a single spatial light modulator, and thus accessible in many laboratories.

Vortex-core properties and vortex-lattice transformation in FeSe

Low-temperature scanning tunneling microscopy and spectroscopy has been used to image the vortex core and the vortex lattice in FeSe single crystals. The local tunneling spectra acquired at the center of elliptical vortex cores display a strong particle-hole asymmetry with spatial oscillation, characteristic of the quantum-limit vortex core. Furthermore, a quasihexagonal vortex lattice at low magnetic field undergoes noticeable rhombic distortions above a certain field $\ensuremath{\sim}1.5$ T. This field ${H}^{*}$ also reveals itself as a kink in the magnetic field dependence of the specific heat. The observation of a nearly hexagonal vortex lattice at low field is very surprising for materials with an orthorhombic crystal structure and it is in apparent contradiction with the elliptical shape of the vortex cores. These observations can be directly connected to the multiband nature of superconductivity in this material, provided we attribute them to the suppression of superconducting order parameter in one of the energy bands. Above the field ${H}^{*}$ the superconducting coherence length for this band can well exceed the intervortex distance which strengthens the nonlocal effects. Therefore, in addition to multiple-band effects, other possible sources that can contribute to the observed evolution of the vortex-lattice structure include nonlocal effects which cause the field-dependent interplay between the symmetry of the crystal and vortex lattice or the magnetoelastic interactions due to the strain field generated by vortices.

Controllable beam reshaping by mixing square-shaped and hexagonal optical vortex lattices

In the present work we show experimentally and by numerical calculations a substantial far-field beam reshaping by mixing square-shaped and hexagonal optical vortex (OV) lattices composed of vortices with alternatively changing topological charges. We show that the small-scale structure of the observed pattern results from the OV lattice with the larger array node spacing, whereas the large-scale structure stems from the OV lattice with the smaller array node spacing. In addition, we demonstrate that it is possible to host an OV, a one-dimensional, or a quasi-two-dimensional singular beam in each of the bright beams of the generated focal patterns. The detailed experimental data at different square-to-hexagonal vortex array node spacings shows that this quantity could be used as a control parameter for generating the desired focused structure. The experimental data are in excellent agreement with the numerical simulations.

Far-field beam reshaping by manipulating the topological charges of hexagonal optical vortex lattices

In this work, we demonstrate experimentally the formation of eight different basic structures consisting of bright beams with flat phase fronts in the focus of a lens (i.e. in the artificial far-field). In all cases considered, the used input structure is a large, stable, hexagon-shaped optical vortex (OV) array composed of vortices with alternating topological charges (TCs). The TCs of one individual OV or of the complete OV lattice were erased/doubled in the cases of perfect superposition (on-site alignment) or are manipulated in phase in the cases of different offsets between the lattice elementary cells (off-site alignments). A dramatic reshaping of the input beam is observed in the far-field and it is shown to be in excellent agreement with the numerical simulations.

Vortex lattice and vortex bound states in CsFe2As2 investigated by scanning tunneling microscopy/spectroscopy

We investigate the vortex lattice and vortex bound states in CsFe$_2$As$_2$ single crystals by scanning tunneling microscopy/spectroscopy (STM/STS) under various magnetic fields. A possible structural transition or crossover of vortex lattice is observed with the increase of magnetic field, i.e., the vortex lattice changes from a distorted hexagonal lattice to a distorted tetragonal one at the magnetic field near 0.5 T. It is found that a mixture of stripelike hexagonal and square vortex lattices emerges in the crossover region. The vortex bound state is also observed in the vortex center. The tunneling spectra crossing a vortex show that the bound-state peak position holds near zero bias with STM tip moving away from the vortex core center. The Fermi energy estimated from the vortex bound state energy is very small. Our investigations provide experimental information to both the vortex lattice and the vortex bound states in this iron-based superconductor.

Magnetic field dependence of the density of states in the multiband superconductorβ−Bi2Pd

We present very low temperature scanning tunneling microscopy (STM) experiments on single crystalline samples of the superconductor $\beta$-Bi$_2$Pd. We find a single fully isotropic superconducting gap. However, the magnetic field dependence of the intervortex density of states is higher than the one expected in a single gap superconductor, and the hexagonal vortex lattice is locked to the square atomic lattice. Such increase in the intervortex density of states and vortex lattice locking have been found in superconductors with multiple superconducting gaps and anisotropic Fermi surfaces. We compare the upper critical field $H_{c2}(T)$ obtained in our sample with previous measurements and explain available data within multiband supercondutivity. We propose that $\beta$-Bi$_2$Pd is a single gap multiband superconductor. We anticipate that single gap multiband superconductivity can occur in other compounds with complex Fermi surfaces.

Critical current oscillations in the intrinsic hybrid vortex state of SmFeAs(O,F).

In layered superconductors the order parameter may be modulated within the unit cell, leading to nontrivial modifications of the vortex core if the interlayer coherence length ξ(c)(T) is comparable to the interlayer spacing. In the iron pnictide SmFeAs(O,F) (T(c)≈50 K) this occurs below a crossover temperature T(⋆)≈41 K, which separates two regimes of vortices: anisotropic Abrikosov-like at high and Josephson-like at low temperatures. Yet in the transition region around T(⋆), hybrid vortices between these two characteristics appear. Only in this region around T(⋆) and for magnetic fields well aligned with the FeAs layers, we observe oscillations of the c-axis critical current j(c)(H) periodic in 1/sqrt[H] due to a delicate balance of intervortex forces and interaction with the layered potential. j(c)(H) shows pronounced maxima when a hexagonal vortex lattice is commensurate with the underlying crystal structure. The narrow temperature window in which oscillations are observed suggests a significant suppression of the order parameter between the superconducting layers in SmFeAs(O,F), despite its low coherence length anisotropy (γ(ξ)≈3-5).

Vortex lattice structure inBaFe2(As0.67P0.33)2via small-angle neutron scattering

We have observed a magnetic vortex lattice (VL) in ${\mathrm{BaFe}}_{2}({\mathrm{As}}_{0.67}{\mathrm{P}}_{0.33}){}_{2}$ (BFAP) single crystals by small-angle neutron scattering. With the field along the $c$ axis, a nearly isotropic hexagonal VL was formed in the field range from 1 to 16 T, and no symmetry changes in the VL were observed. The temperature dependence of the VL signal was measured and confirms the presence of (non-$d$-wave) nodes in the superconducting gap structure for measurements at 5 T and below. The nodal effects were suppressed at high fields. At low fields, a VL reorientation transition was observed between 1 and 3 T, with the VL orientation changing by ${45}^{\ensuremath{\circ}}$. Below 1 T, the VL structure was strongly affected by pinning and the diffraction pattern had a fourfold symmetry. We suggest that this (and possibly also the VL reorientation) is due to pinning to defects aligned with the crystal structure, rather than being intrinsic. The temperature dependence of the scaled intensity suggests that BFAP possesses at least one full gap and one nodal gap with circular symmetry. Judging from the symmetry, the node structure should take the form of an ``accidental'' circular line node, which is consistent with recent angle-resolved photoemission spectroscopy results [Y. Zhang, Z. R. Ye, Q. Q. Ge, F. Chen, J. Jiang, M. Xu, B. P. Xie, and D. L. Feng, Nature Physics 8, 371 (2012).].

Vortex structure and magnetic properties of BaFe2(As1−xPx)2 single crystals

Magnetic decoration of BaFe2(As1−xPx)2 revealed an hexagonal vortex lattice in overdoped single crystals (x>0.4, Tc=14-18 K). A disordered vortex lattice was observed in a crystals with optimal content of phosphorus (x ≈ 0.4, Tc = 25-30 K). Magneto-optical studies and global magnetization measurement of the optimally-doped crystal have led to an estimated critical current density of ~ 105A/cm2 at 5-12 K in fields up to 0.7 T. This high critical current density suggests strong pinning, probably due to defects, a result consistent with the observed disordered vortex lattice. The critical current density of the overdoped crystal is much smaller, i.e. ~ 104 A/cm2 at 5 K in fields up to 0.1 T, decreasing to virtually zero at 0.3 T.

Elastic contribution to interaction of vortices in uniaxial superconductors

The stress caused by vortices in tetragonal superconductors contributes to the intervortex interaction which depends on vortex orientation within the crystal, on elastic moduli, and is attractive within certain angular regions even in fields along the $c$ crystal axis. For sufficiently strong stress dependence of the critical temperature, this contribution may result in distortions of the hexagonal vortex lattice for $\bm H||\bm c$. In small fields it leads to formation of a square vortex lattice with a fixed $H$ independent spacing. This should be seen in the magnetization $M(H)$ as a discontinuous jump of magnetization at the transition from the Meissner to mixed states.

Vortex Lattice Studies inCeCoIn5withH⊥c

We present small angle neutron scattering studies of the vortex lattice (VL) in CeCoIn5 with magnetic fields applied parallel (H) to the antinodal [100] and nodal [110] directions. For H || [100], a single VL orientation is observed, while a 90 degree reorientation transition is found for H || [110]. For both field orientations and VL configurations we find a distorted hexagonal VL with an anisotropy, Gamma = 2.0 +/- 0.05. The VL form factor shows strong Pauli paramagnetic effects similar to what have previously been reported for H || [001]. At high fields, above which the upper critical field (Hc2) becomes a first-order transition, an increased disordering of the VL is observed.

Atomic resolution and vortex lattice studies of magnetic superconductors: A first approach in the nickel borocarbide TmNi 2B 2C

We present new scanning tunneling microscopy measurements in the superconductor TmNi 2B 2C. The topography shows in some areas flat surfaces, where atomic size modulations can be identified. We find a hexagonal vortex lattice between 0.15 T and 1.4 T, when the magnetic field is applied along the basal plane of the tetragonal crystal structure ( B ⊥ c ) , and a hexagonal to square transition around 0.15 T when the field is applied along the c-axis ( B ‖ c ) . Measured intervortex distance are smaller than expected at high field, due to the internal field being larger than the applied field.

Vortex properties of two-dimensional superconducting Pb films

Using low temperature scanning tunnelling microscopy/spectroscopy (STM/STS) we haveinvestigated the vortex behaviours of two-dimensional superconducting Pb films atdifferent thicknesses. STS at the vortex core shows an evolution of electronicstates with film thickness. Transition from the clean limit to the dirty limit ofsuperconductivity is identified, which can be ascribed to the decreased electronic meanfree path induced by stronger scattering from the disordered interface at smallerthicknesses. A magnetic field dependent vortex core size is observed even for such a low-κ superconductor. The weak pinning induced by surface defects leads to the formation of adistorted hexagonal vortex lattice.

Generic First-Order Orientation Transition of Vortex Lattices in Type II Superconductors

First order transition of vortex lattices (VL) observed in various superconductors with four-fold symmetry is explained microscopically by quasi-classical Eilenberger theory combined with nonlocal London theory. This transition is intrinsic in the generic successive VL phase transition due to either gap or Fermi velocity anisotropies. This is also suggested by the electronic states around vortices. Ultimate origin of this phenomenon is attributed to some what hidden frustrations of a spontaneous symmetry broken hexagonal VL on the underlying four-fold crystalline symmetry.

Fermi Surface and Order Parameter Driven Vortex Lattice Structure Transitions in Twin-FreeYBa2Cu3O7

We report on small-angle neutron scattering studies of the intrinsic vortex lattice (VL) structure in detwinned YBa2Cu3O7 at 2 K, and in fields up to 10.8 T. Because of the suppressed pinning to twin-domain boundaries, a new distorted hexagonal VL structure phase is stabilized at intermediate fields. It is separated from a low-field hexagonal phase of different orientation and distortion by a first-order transition at 2.0(2) T that is probably driven by Fermi surface effects. We argue that another first-order transition at 6.7(2) T, into a rhombic structure with a distortion of opposite sign, marks a crossover from a regime where Fermi surface anisotropy is dominant, to one where the VL structure and distortion is controlled by the order-parameter anisotropy.

Structure and degeneracy of vortex lattice domains in pure superconducting niobium: A small-angle neutron scattering study

High-purity niobium exhibits a surprisingly rich assortment of vortex lattice (VL) structures for fields applied parallel to a fourfold symmetry axis, with all observed VL phases made up of degenerate domains that spontaneously break some crystal symmetry. Yet a single regular hexagonal VL domain is observed at all temperatures and fields parallel to a threefold symmetry axis. We report a detailed investigation of the transition between these lush and barren VL landscapes, discovering new VL structures and phase transitions at high fields. We show that the number and relative population of VL domains is intrinsically tied to the underlying crystal symmetry. We discuss how subtle anisotropies of the crystal may generate the remarkable VLs observed.

Observations of the configuration of the high-field vortex lattice inYBa2Cu3O7: Dependence upon temperature and angle of applied field

We have mapped out the high-field phase diagram of the vortex lattice in lightly twinned ${\text{YBa}}_{2}{\text{Cu}}_{3}{\text{O}}_{7}$ by employing small-angle neutron scattering. This extends previous measurements [S. P. Brown et al., Phys. Rev. Lett. 92, 067004 (2004)] in which we reported a transition in local flux-line coordination from distorted triangular to square as a function of field applied parallel to the crystal $c$ axis. At high fields, we observe a transition back to triangular coordination with increasing angle between the applied field and $c$ axis or with increasing temperature. At lower fields, our results may be interpreted in terms of a field-induced reduction of the basal-plane $a\text{\ensuremath{-}}b$ anisotropy. Tilting the flux lines away from the $c$ axis reduces the pinning by twin planes, changes the orientation of the vortex lattice nearest-neighbor directions, and the transition between square and hexagonal vortex lattices becomes first order.