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Using small-angle neutron scattering we have studied the superconducting vortex lattice (VL) phase diagram in MgB2 as the applied magnetic field is rotated away from the c axis and towards the basal plane. The field rotation gradually suppresses the intermediate VL phase which exists between end states aligned with two high symmetry directions in the hexagonal basal plane for H || c. Above a critical angle, the intermediate state disappears, and the previously continuous transition becomes discontinuous. The evolution towards the discontinuous transition can be parameterized by a vanishing twelvefold anisotropy term in the VL free energy., 6 pages, 4 figures; [v2]: minor revisions to text and figures; [v3] typos fixed, journal reference added

We introduce a phenomenological model for a pairwise repulsive interaction potential of vortices in a type-II superconductor, consisting of superimposed sixfold and twelvefold anisotropies. Using numerical simulations we study how the vortex lattice configuration varies as the magnitudes of the two anisotropic interaction terms change. A triangular lattice appears for all values, and rotates through ${30}^{\ensuremath{\circ}}$ as the ratio of the sixfold and twlevefold anisotropy amplitudes is varied, in agreement with experimental results. The transition causes the vortex lattice to split into domains that have rotated clockwise or counterclockwise, with grain boundaries that are ``decorated'' by dislocations consisting of fivefold and sevenfold coordinated vortices. We also find intradomain dislocations and defects, and characterize them in terms of their energy cost. We discuss how this model could be generalized to other particle-based systems with anisotropic interactions, such as colloids, and consider the limit of very large anisotropy where it is possible to create cluster crystal states.

For iron-sulfide (FeS), we investigate the correlation between the structural details, including its dimensionality and composition, with its magnetic and superconducting properties. We compare, theoretically and experimentally, the two-dimensional (2D) layered tetragonal (“t-FeS”) phase with the 3D hexagonal ("h-FeS") phase. X-ray diffraction reveals iron-deficient chemical compositions of t-Fe0.93(1)S and h-Fe0.84(1)S that show no low-temperature structural transitions. First-principles calculations reveal a high sensitivity of the 2D structure to the electronic and magnetic properties, predicting marginal antiferromagnetic instability for our compound (sulfur height of zS = 0.252) with an ordering energy of about 11 meV/Fe, while the 3D phase is magnetically stable. Experimentally, h-Fe0.84S orders magnetically well above room temperature, while t-Fe0.93S shows coexistence of antiferromagnetism at TN = 116 and filamentary superconductivity below Tc = 4 K. Low temperature neutron diffraction data reveals antiferromagnetic commensurate ordering with wave vector km = (0.25,0.25,0) and 0.46(2) µB/Fe. Additionally, neutron scattering measurements were used to find the particle size and iron vacancy arrangement of t-FeS and h-FeS. The structure of iron sulfide has a delicate relationship with the superconducting transition; while our sample with a = 3.6772(7) A is a filamentary superconductor coexisting with an antiferromagnetic phase, previously reported samples with a > 3.68 A are bulk superconductors with no magnetism, and those with a ≈ 3.674 A show magnetic properties.

Magnetic skyrmions are particle-like, topologically protected magnetization entities that are promising candidates for information carriers in racetrack-memory schemes. The transport of skyrmions in a shift-register-like fashion is crucial for their embodiment in practical devices. Recently, we demonstrated experimentally that chiral skyrmions in Cu2OSeO3 can be effectively manipulated by a magnetic field gradient, leading to a collective rotation of the skyrmion lattice with well-defined dynamics in a radial field gradient. Here, we employ a skyrmion particle model to numerically study the effects of resultant shear forces on the structure of the skyrmion lattice. We demonstrate that anisotropic peak broadening in experimentally observed diffraction patterns can be attributed to extended linear regions in the magnetic field profile. We show that topological (5-7) defects emerge to protect the six-fold symmetry of the lattice under the application of local shear forces, further enhancing the stability of proposed magnetic field driven devices.

We have studied non-equilibrium phase transitions in the vortex lattice in superconducting MgB2, where metastable states are observed in connection with an intrinsically continuous rotation transition. Using small-angle neutron scattering and a stop-motion technique, we investigated the manner in which the metastable vortex lattice returns to the equilibrium state under the influence of an ac magnetic field. This shows a qualitative difference between the supercooled case which undergoes a discontinuous transition, and the superheated case where the transition to the equilibrium state is continuous. In both cases the transition may be described by an an activated process, with an activation barrier that increases as the metastable state is suppressed, as previously reported for the supercooled vortex lattice [E. R. Louden et al., Phys. Rev. B 99, 060502(R) (2019)]. Separate preparations of superheated metastable vortex lattices with different domain populations showed an identical transition towards the equilibrium state. This provides further evidence that the vortex lattice metastability, and the kinetics associated with the transition to the equilibrium state, is governed by nucleation and growth of domains and the associated domain boundaries., Comment: 27 pages, 10 figures. arXiv admin note: text overlap with arXiv:1812.05970

The vortex lattice (VL) in MgB2 is characterized by the presence of long-lived metastable states (MSs), which arise from cooling or heating across the equilibrium phase boundaries. A return to the equilibrium configuration can be achieved by inducing vortex motion. Here we report on small-angle neutron scattering studies of MgB2, focusing on the structural properties of the VL as it is gradually driven from metastable to equilibrium states (ESs) by an AC magnetic field. Measurements were performed using initial MSs obtained either by cooling or heating across the equilibrium phase transition. In all cases, the longitudinal correlation length remains constant and comparable to the sample thickness. Correspondingly, the VL may be considered as a system of straight rods, where the formation and growth of ES domains only occurs in the two-dimensional plane perpendicular to the applied field direction. Spatially resolved raster scans of the sample were performed with apertures as small as 80 μm, corresponding to only 1.2 × 106 vortices for an applied field of 0.5 T. These revealed spatial variations in the metastable and equilibrium VL populations, but individual domains were not directly resolved. A statistical analysis of the data indicates an upper limit on the average domain size of approximately 50 μm., New Journal of Physics, 21, ISSN:1367-2630

The temperature dependence of the London penetration depth $\mathrm{\ensuremath{\Delta}}\ensuremath{\lambda}(T)$ in the superconducting doped topological crystalline insulator ${\mathrm{Sn}}_{1\ensuremath{-}x}{\mathrm{In}}_{x}\mathrm{Te}$ was measured down to 450 mK for two different doping levels, $x\ensuremath{\approx}0.45$ (optimally doped) and $x\ensuremath{\approx}0.10$ (underdoped), bookending the range of cubic phase in the compound. The results indicate no deviation from fully gapped BCS-like behavior, eliminating several candidate unconventional gap structures. Critical field values below 1 K and other superconducting parameters are also presented. The introduction of disorder by repeated particle irradiation with 5 MeV protons does not enhance ${T}_{c}$, indicating that ferroelectric interactions do not compete with superconductivity.

Using small-angle neutron scattering, we investigated the behavior of a metastable vortex lattice state in MgB2 as it is driven towards equilibrium by an AC magnetic field. This shows an activated behavior, where the AC field amplitude and cycle count are equivalent to, respectively, an effective "temperature" and "time". The activation barrier increases as the metastable state is suppressed, corresponding to an aging of the vortex lattice. Furthermore, we find a cross-over from a partial to a complete suppression of metastable domains depending on the AC field amplitude, which may empirically be described by a single free parameter. This represents a novel kind of collective vortex behavior, most likely governed by the nucleation and growth of equilibrium vortex lattice domains., Comment: 5 pages plus 3 pages of supplemental material

We studied thermal and dynamic history effects in the vortex lattice (VL) near the order-disorder transition in clean ${\mathrm{NbSe}}_{2}$ single crystals. Comparing the evolution of the effective vortex pinning and the bulk VL structure, we observed metastable superheated and supercooled VL configurations that coexist with a hysteretic effective pinning response due to thermal cycling of the system. A novel scenario, governed by the interplay between (lower) elastic and (higher) plastic energy barriers, is proposed as an explanation for our observations: Plastic barriers, which prevent the annihilation or creation of topological defects, require dynamic assistance to be overcome and to achieve a stable VL at each temperature. Conversely, thermal hysteresis in the pining response is ascribed to low energy barriers, which inhibit rearrangement within a single VL correlation volume and are easily overcome as the relative strength of competing interactions changes with temperature.

We introduce a model of vortices in type-II superconductors with a four-fold anisotropy in the vortex-vortex interaction potential. Using numerical simulations we show that the vortex lattice undergoes structural transitions as the anisotropy is increased, with a triangular lattice at low anisotropy, a rhombic intermediate state, and a square lattice for high anisotropy. In some cases we observe a multi-$q$ state consisting of an Archimedean tiling that combines square and triangular local ordering. At very high anisotropy, domains of vortex chain states appear. We discuss how this model can be generalized to higher order anisotropy as well as its applicability to other particle-based systems with anisotropic particle-particle interactions., Comment: 7 pages, 7 figures

Despite numerous studies the exact nature of the order parameter in superconducting ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ remains unresolved. We have extended previous small-angle neutron scattering studies of the vortex lattice in this material to a wider field range, higher temperatures, and with the field applied close to both the $\ensuremath{\langle}100\ensuremath{\rangle}$ and $\ensuremath{\langle}110\ensuremath{\rangle}$ basal plane directions. Measurements at high field were made possible by the use of both spin polarization and analysis to improve the signal-to-noise ratio. Rotating the field towards the basal plane causes a distortion of the square vortex lattice observed for $\mathbit{H}\ensuremath{\parallel}\ensuremath{\langle}001\ensuremath{\rangle}$ and also a symmetry change to a distorted triangular symmetry for fields close to $\ensuremath{\langle}100\ensuremath{\rangle}$.The vortex lattice distortion allows us to determine the intrinsic superconducting anisotropy between the $c$ axis and the Ru-O basal plane, yielding a value of $\ensuremath{\sim}60$ at low temperature and low to intermediate fields. This greatly exceeds the upper critical field anisotropy of $\ensuremath{\sim}20$ at low temperature, reminiscent of Pauli limiting. Indirect evidence for Pauli paramagnetic effects on the unpaired quasiparticles in the vortex cores are observed, but a direct detection lies below the measurement sensitivity. The superconducting anisotropy is found to be independent of temperature but increases for fields $\ensuremath{\gtrsim}1$ T, indicating multiband superconductvity in ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$. Finally, the temperature dependence of the scattered intensity provides further support for gap nodes or deep minima in the superconducting gap.

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.

We review superconducting vortex imaging in Co-doped BaFe 2 As 2 by Bitter decoration and small-angle neutron scattering (SANS). At all measured fields a highly disordered vortex configuration is observed, which is attributed to strong pinning. Further support of this conclusion comes from the absence of a Meissner rim in decoration images obtained close to the sample edge. The evolution of the SANS scattering vector with increasing applied field indicates vortex lattice domains of (distorted) hexagonal symmetry. This is consistent with the decoration images which show small, six fold coordinated ordered vortex domains. The SANS scattered intensity is found to decrease rapidly with increasing field, exceeding the rate expected from estimates of the upper critical field. This is consistent with the large degree of vortex “lattice” disorder.

We report structural evidence of dynamic reorganization in vortex matter in clean NbSe$_2$ by joint small angle neutron scattering and ac-susceptibility measurements. The application of oscillatory forces in a transitional region near the order-disorder transition results in robust bulk vortex lattice configurations with an intermediate degree of disorder. These dynamically-originated configurations correlate with intermediate pinning responses previously observed, resolving a long standing debate regarding the origin of such responses., Comment: 9 pages, 7 figures. To be published in Physical Review Letters

We present small-angle neutron scattering (SANS) measurements of the vortex lattice (VL) in single crystal MgB 2 ––a two-band superconductor. Bragg diffraction of neutrons visualizes the structure and orientation of the VL while the scattered intensity probes the super-carrier density. Superconductivity in the π-band is rapidly suppressed with increasing field with a corresponding decrease in scattered intensity and a re-orientation of the VL between 0.5 and 0.9 T. Both these observations are consistent with superconductivity in the π-band being weaker than in the σ-band ( Δ π Δ σ ), and a changing influence of different parts of the Fermi surface determining the VL orientation. SANS measurements of the VL for fields applied at an angle to the c -axis allow the penetration depth anisotropy, γ λ , to be estimated.

Single crystals of MgB2 with a size up to 1.5x0.9x0.2 mm3 have been grown with a high pressure cubic anvil technique. The crystal growth process is very peculiar and involves an intermediate nitride, namely MgNB9. Single crystals of BN and MgB2 grow simultaneously by a peritectic decomposition of MgNB9. Magnetic measurements in fields of 1-5 Oe show sharp transitions to the superconducting state at 37-38.6 K with width of ~0.5 K. The high quality of the crystals allowed the accurate determination of magnetic, transport and optical properties as well as scanning tunnelling spectroscopy (STS) and decoration studies. Investigations of crystals with torque magnetometry show that Hc2//c is very low (24 kOe at 15 K), while Hc2//ab increases up to 140 kOe at 15 K. The upper critical field anisotropy gamma = Hc2//ab/ Hc2//c was found to be temperature dependent (decreasing from 6 at 15 K to 2.8 at 35 K). The effective anisotropy gamma_eff, as calculated from reversible torque data near Tc, is field dependent (increasing roughly linearly from 2 in zero field to 3.7 in 10 kOe). The temperature and field dependence of the anisotropy can be related to the double gap structure of MgB2 with a large two-dimensional gap and small three-dimensional gap, the latter being rapidly suppressed in a magnetic field. Torque magnetometry investigations show a pronounced peak effect, indicating an order-disorder transition of vortex matter. Decoration experiments and STS visualise a hexagonal vortex lattice. STS spectra evidence two gaps (3 meV/6 meV) with direction dependent weight. Magneto-optic investigations with H//c show a clear signature of the smaller of the two gaps, disappearing in fields higher than Hc2//c.

We have investigated the temperature dependence of the H parallel to c flux line lattice structural phase transition from square to hexagonal symmetry, in the tetragonal superconductor LuNi2B2C ( T(c) = 16.6 K). At temperatures below 10 K the transition onset field, H2(T), is only weakly temperature dependent. Above 10 K, H2(T) rises sharply, bending away from the upper critical field. This contradicts theoretical predictions of H2(T) merging with the upper critical field and suggests that just below the H(c2)(T) curve the flux line lattice might be hexagonal.

Small angle neutron scattering studies of the flux line lattice in LuNi{sub 2}B {sub 2}C and ErNi{sub 2}B {sub 2}C induced by a field parallel to the a axis reveal a first order flux line lattice reorientation transition. Below the transition the flux line lattice nearest neighbor direction is parallel to the b axis, and above the transition it is parallel to the c axis. This transition cannot be explained using nonlocal corrections to the London model. In addition, the anisotropy of the penetration depth {lambda} and the coherence length {xi} change at the transition.

We have discovered a new antiferromagnetic phase in TmNi2B2C by neutron diffraction. The ordering vector is Q(A) = (0.48,0,0) and the phase appears above a critical in-plane magnetic field of 0.9 T. The field was applied in order to test the assumption that the zero-field magnetic structure at Q(F) = (0.094,0.094,0) would change into a c-axis ferromagnet if superconductivity were destroyed. We present theoretical calculations which show that two effects are important: a suppression of the ferromagnetic component of the RKKY exchange interaction in the superconducting phase and a reduction of the superconducting condensation energy due to the periodic modulation of the moments at Q(A).

We review studies of the interactions between magnetic order and the flux line lattice (FLL) in the (RE)Ni2B2C intermetallic borocarbides for (RE)=Tm and Er using small angle neutron scattering (SANS) and magneto-transport. For (RE)=Tm the magnetic order and the FLL assume a common symmetry, sharing a phase transition at ∼2 kOe, despite an order of magnitude difference in periodicity. For (RE)=Er, the penetration depth λ and the coherence length ξ, both of which are derived from the FLL form factor, are modified near TN=6 K by a theoretically predicted weakly divergent pairbreaking. Finally, below 2.3 K, (RE)=Er shows a coexistence of weak ferromagnetism and superconductivity. This state reveals a highly disordered FLL and a striking increase in the critical current, both arising from the strong ferromagnetic pairbreaking.

Using small angle neutron scattering we have studied the square to hexagonal flux line lattice symmetry transition in different members of the borocarbide superconductors. The studies were performed using samples of ErNi 2 B 2 C, Lu(Ni 1− x Co x ) 2 B 2 C with cobalt doping levels x =1.5–9% and Y 0.64 Lu 0.36 Ni 2 B 2 C. We find that the onset field of the symmetry transition can be shifted more than an order of magnitude due to changes in the range of the non-local electrodynamics. Comparing the results to transport measurements of the electronic mean free path and the superconducting coherence length we find that the transition onset follows a model by V. Kogan et al., which includes non-local corrections to the London model due to the Fermi surface anisotropy of the borocarbides.

We have studied, using small angle neutron scattering, the flux line lattice square to hexagonal symmetry transition in single crystal $\mathrm{Lu}({\mathrm{Ni}}_{1\ensuremath{-}x}{\mathrm{Co}}_{x}{)}_{2}{\mathrm{B}}_{2}\mathrm{C}$. Low Co concentrations $(xl0.1)$, which reduce the mean free path and increase the coherence length, also move the structural transition to higher fields than in the undoped system. These data, quantitatively understood within the framework of a theory that includes nonlocal corrections to the London model due to the Fermi surface anisotropy, can be modeled using a simple ratio of the nonlocality range to the intervortex spacing.

The flux line form factor in small angle neutron scattering and transport data determines the superconducting length scales and critical fields in single crystal ${\mathrm{ErNi}}_{2}{\mathrm{B}}_{2}\mathrm{C}$. For $H\ensuremath{\Vert}c$, the coherence length $\ensuremath{\xi}$ increases and the penetration depth $\ensuremath{\lambda}$ decreases when crossing ${T}_{N}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}6.0\mathrm{K}$, the N\'eel transition. The critical fields show corresponding anomalies near ${T}_{N}$. For $H\ensuremath{\perp}c$, the fourfold modulation of the upper critical field ${H}_{c2}$ is strongly temperature dependent, changing sign near ${T}_{N}$, and can be modeled using the anisotropy of the sublattice magnetization.

We have studied the flux line lattice in ${\mathrm{YNi}}_{2}{\mathrm{B}}_{2}\mathrm{C}$ and ${\mathrm{LuNi}}_{2}{\mathrm{B}}_{2}\mathrm{C}$, the nonmagnetic end members of the borocarbide superconductors using small angle neutron scattering and transport. For fields, $\mathbf{H}\ensuremath{\parallel}\mathbf{c}$, we find a square symmetric lattice which disorders rapidly above $H/{H}_{c2}\ensuremath{\sim}0.2$, well below the ``peak effect'' at $H/{H}_{c2}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.9$. The results for $H/{H}_{c2}l0.2$ can be understood within the collective pinning model, and are controlled by the tilt modulus ${c}_{44}$. For $H/{H}_{c2}g0.2$, the disordering appears to be associated with the field dependence of the shear modulus, ${c}_{66}$.

Small-angle neutron scattering and magnetic decoration both demonstrate a topological transition in the flux line lattice (FLL) in ErNi{sub 2}B{sub 2}C. The high-field square lattice slowly transforms into a hexagonal lattice via an area preserving [100] rhombohedral distortion below roughly 500Oe. The square FLL is aligned with the [110] direction of the tetragonal crystal, while the two domains of the hexagonal FLL are aligned with [100] and [010]. The differences in pinning for the two FLL topologies are reflected in the rf kinetic inductance. {copyright} {ital 1997} {ital The American Physical Society}

The dHvA oscillations in crystals of LuNi 2 B 2 C are known to persist deep inside the vortex state, across the region of the peak effect located at the edge of the irreversibility field normal phase boundary. This makes the identification of H c2 (T) ambiguous in magnetization hysteresis loops at temperatures at which the dHvA signal is prominent. One can overcome this difficulty if H c2 (T) value is ascertained from the measurement of isothermal quadrupole moment (Q) vs. field values using a vibrating sample magnetometer. In these crystals, one can witness dHvA oscillations up to about 8 K and we show that Q(T) data in LuNi 2 B 2 C can be used to construct the vortex phase diagram over the entire (H, T) range.

Recently, extensive vortex lattice metastability was reported in MgB2 in connection with a second-order rotational phase transition. However, the mechanism responsible for these well-ordered metastable vortex lattice phases is not well understood. Using small-angle neutron scattering, we studied the vortex lattice in MgB2 as it was driven from a metastable to the ground state through a series of small changes in the applied magnetic field. Our results show that metastable vortex lattice domains persist in the presence of substantial vortex motion and directly demonstrate that the metastability is not due to vortex pinning. Instead, we propose that it is due to the jamming of counterrotated vortex lattice domains which prevents a rotation to the ground state orientation., 5 pages, 4 fgures

Despite intense studies the exact nature of the order parameter in superconducting Sr2RuO4 remains unresolved. We have used small-angle neutron scattering to study the vortex lattice in Sr2RuO4 with the field applied close to the basal plane, taking advantage of the transverse magnetization. We measured the intrinsic superconducting anisotropy between the c axis and the Ru-O basal plane (~60), which greatly exceeds the upper critical field anisotropy (~20). Our result imposes significant constraints on possible models of triplet pairing in Sr2RuO4 and raises questions concerning the direction of the zero spin projection axis., Main paper: 5 pages, 4 figures. Supplementary material: 3 pages, 2 figures

The frequency- and wavevector-dependent conductivity tensor of a two-dimensional electron gas is determined within a semiclassical model containing two different characteristic times, a transport time and a lifetime. The velocity shift of a surface acoustic wave, caused by the piezoelectric coupling, is calculated as a function of magnetic field. The results are applicable under conditions where the frequency of the surface acoustic wave is comparable to the cyclotron frequency. The calculated velocity shift is compared to that observed for composite fermions in the quantum Hall regime near half-filling.

The magnetic phase diagram of the superconductor ErNi2B2 C( Tc ¼ 11 K and TN ¼ 6 K) has been studied by neutron diffraction as a function of temperature and magnetic field applied along the symmetry directions [0 1 0], [1 1 0] and [0 0 1] of the tetragonal crystal structure. A series of commensurate magnetic structures, consistent with a transversely polarized spin–density wave with modulation vectors Q ¼ n=ma � ð0:55 6 n=m < 0:60Þ and the spins along b � , have been observed. The experimental data are compared with the results of a mean-field model that has been established from an analysis of bulk magnetization and zero-field neutron diffraction data. The model accounts for

The superconductor TmNi${}_{2}$B${}_{2}$C possesses a significant fourfold basal plane anisotropy, leading to a square vortex lattice (VL) at intermediate fields. However, unlike other members of the borocarbide superconductors, the anisotropy in TmNi${}_{2}$B${}_{2}$C appears to decrease with increasing field, evident by a reentrance of the square VL phase. We have used small-angle neutron scattering measurements of the VL to study the field dependence of the anisotropy. Our results provide a direct, quantitative measurement of the decreasing anisotropy. We attribute this reduction of the basal plane anisotropy to the strong Pauli paramagnetic effects observed in TmNi${}_{2}$B${}_{2}$C and the resulting expansion of vortex cores near ${H}_{\mathrm{c}2}$.

The vortex lattice (VL) symmetry and orientation in clean type-II superconductors depends sensitively on the host material anisotropy, vortex density and temperature, frequently leading to rich phase diagrams. Typically, a well-ordered VL is taken to imply a ground-state configuration for the vortex-vortex interaction. Using neutron scattering we studied the VL in MgB(2) for a number of field-temperature histories, discovering an unprecedented degree of metastability in connection with a known, second-order rotation transition. This allows, for the first time, structural studies of a well-ordered, nonequilibrium VL. While the mechanism responsible for the longevity of the metastable states is not resolved, we speculate it is due to a jamming of VL domains, preventing a rotation to the ground-state orientation.

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 II 100], a single VL orientation is observed, while a 90° reorientation transition is found for H II [110]. For both field orientations and VL configurations we find a distorted hexagonal VL with an anisotropy, Γ=2.0±0.05. The VL form factor shows strong Pauli paramagnetic effects similar to what have previously been reported for H II [001]. At high fields, above which the upper critical field (Hc2) becomes a first-order transition, an increased disordering of the VL is observed.

The heavy fermion superconductor UPt$_3$ is thought to have odd-parity, a state for which the temperature dependence of the spin susceptibility is an important signature. In order to address conflicting reports from two different experiments, the NMR Knight shift and measurements of the anisotropy of the upper critical field, we have measured the bulk susceptibility in a high quality single crystal using polarized-neutron diffraction. A temperature independent susceptibility was observed for $H||a$ through the transitions between the normal state and the superconducting A-, B- and C-phases, consistent with odd-parity, spin-triplet superconductivity., Comment: 5 pages, 4 figures. Submitted to Physical Review Letters

We compare the results of small angle neutron scattering on the flux line lattice (FLL) obtained in the borocarbide superconductor LuNi2B2C with the applied field along the c- and a-axes. For H‖c the temperature dependence of the FLL structural phase transition from square to hexagonal symmetry was investigated. Above 10 K the transition onset field. H 2(T), rises sharply, bending away from H c2(T) in contradiction to theoretical predictions of the two merging. For H‖a a first order FLL reorientation transition is observed at H tr=3–3.5 kOe. Below H tr the FLL nearest neighbor direction is parallel to the b-axis, and above H tr to the c-axis. This transition cannot be explained using nonlocal corrections to the London model.

We present small-angle neutron scattering (SANS) and Bitter decoration studies of the superconducting vortices in $\text{Ba}{({\text{Fe}}_{0.93}{\text{Co}}_{0.07})}_{2}{\text{As}}_{2}$. A highly disordered vortex configuration is observed at all measured fields and is attributed to strong pinning. This conclusion is supported by the absence of a Meissner rim in decoration images obtained close to the sample edge. The field dependence of the magnitude of the SANS scattering vector indicates vortex lattice domains of (distorted) hexagonal symmetry, consistent with the decoration images which show primarily sixfold coordinated vortex domains. An analysis of the scattered intensity shows that this decreases much more rapidly than expected from estimates of the upper critical field, consistent with the large degree of disorder.

Many superconducting materials allow the penetration of magnetic fields in a mixed state in which the superfluid is threaded by a regular lattice of Abrikosov vortices, each carrying one quantum of magnetic flux. The phenomenological Ginzburg-Landau theory, based on the concept of characteristic length scales, has generally provided a good description of the Abrikosov vortex lattice state. We conducted neutron-scattering measurements of the vortex lattice form factor in the heavy-fermion superconductor cerium-cobalt-indium (CeCoIn 5 ) and found that this form factor increases with increasing field—opposite to the expectations within the Abrikosov-Ginzburg-Landau paradigm. We propose that the anomalous field dependence of the form factor arises from Pauli paramagnetic effects around the vortex cores and from the proximity of the superconducting state to a quantum critical point.

Small-angle neutron scattering has been applied to study the vortex lattice in the intercalated graphite superconductor CaC6 (T-c=11.3 K). Scattering from the vortex lattice is in the form of a ring, most likely reflecting the absence of in-plane orientational order of the pyrolitic graphene planes. The temperature and field dependence of the scattered intensity allows the in-plane zero temperature value of the coherence length [xi=350(25) angstrom] and the London penetration depth [lambda=500(30) angstrom] to be estimated. Measurements with the applied field at 70 degrees to the c axis directly reveal the penetration depth anisotropy, gamma(lambda)=5.1(4), which, unlike MgB2, is equal to the anisotropy of the coherence length deduced from magnetization measurements. The orientation of the vortex lattice is fixed relative to the rotation axis of the crystal as predicted by anisotropic London theory.

The magnetic field distribution around the vortices in TmNi2B2C in the paramagnetic phase was studied experimentally as well as theoretically. The vortex form factor, measured by small-angle neutron scattering, is found to be field independent up to 0.6 Hc2 followed by a sharp decrease at higher fields. The data are fitted well by solutions to the Eilenberger equations when paramagnetic effects due to the exchange interaction with the localized 4f Tm moments are included. The induced paramagnetic moments around the vortex cores act to maintain the field contrast probed by the form factor., Comment: 4 pages, 4 figures

Using small-angle neutron scattering, we have studied the flux-line lattice (FLL) in the superclean, high-kappa superconductor CeCoIn5. The FLL undergoes a first-order symmetry and reorientation transition at approximately 0.55 T at 50 mK. In addition, the FLL form factor in this material is found to be independent of the applied magnetic field, in striking contrast to the exponential decrease usually observed in superconductors. This result is consistent with a strongly field-dependent coherence length, proportional to the vortex separation.

X-ray diffraction studies of fine polycrystalline samples of MgB$_2$ in the superconducting state reveal that crystals orient with their \emph{c}-axis in a plane normal to the direction of the applied magnetic field. The MgB$_2$ samples were thoroughly ground to obtain average grain size 5 - 10 $\mu$m in order to increase the population of free single crystal grains in the powder. By monitoring Bragg reflections in a plane normal to an applied magnetic field we find that the powder is textured with significantly stronger (\emph{0,0,l}) reflections in comparison to (\emph{h,k,0}), which remain essentially unchanged. The orientation of the crystals with the \emph{ab}-plane parallel to the magnetic field at all temperatures below $T_c$ demonstrates that the sign of the torque under magnetic field does not alter, in disagreement with current theoretical predictions.

Using small-angle neutron scattering, we have measured the misalignment between an applied field of $4\phantom{\rule{0.3em}{0ex}}\mathrm{kOe}$ and the flux-line lattice in $\mathrm{Mg}{\mathrm{B}}_{2}$, as the field is rotated away from the $c$ axis by an angle $\ensuremath{\theta}$. The measurements, performed at $4.9\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, showed the vortices canting towards the $c$ axis for all field orientations. Using a two-band/two-gap model to calculate the magnetization, we are able to fit our results yielding a penetration depth anisotropy, ${\ensuremath{\gamma}}_{\ensuremath{\lambda}}=1.1\ifmmode\pm\else\textpm\fi{}0.1$.

Using small-angle neutron scattering we have studied the magnetic flux line lattice in YNi2B2C with the field rotated 30° away from the crystalline c-axis. Previously we have reported on a square to hexagonal symmetry transition of the flux line lattice below 1 kOe for H|c . We find that the rotation of the field shifts the transition to higher fields in agreement with theoretical prediction. Two different directions in the ab-plane were chosen as axes of rotation. Rotating the field around [1,1,0] shifts the onset of the transition up to 3 kOe. Rotating the field around [1,0,0] stabilizes the hexagonal flux line lattice up to the highest measured field of 30 kOe. The difference between the two axes is also reflected in the flux line lattice distortion.

MAGNETISM and superconductivity are manifestations of two different ordered states into which metals can condense at low temperatures. In general these states are mutually exclusive1; they do not coexist at the same place in a sample. The study of the interplay between these properties has recently been revitalized by the discovery2,3 of a class of compounds with formula RNi2B2C (where R is a rare-earth element) which are both antiferromagnetic and superconducting at sufficiently low temperature4. It has been suggested5 that magnetic and superconducting order can coexist in these materials on an atomic scale. Here we use small-angle neutron scattering to study the structure of the superconducting vortex lattice in ErNi2B2C. Our results show that the development of magnetic order causes the vortex lines to disorder and rotate away from the direction of the applied magnetic field. This coupling of superconductivity and magnetism provides clear evidence for microscopic coexistence of magnetic and superconducting order, and indicates that magnetic superconductors may exhibit a range of unusual phenomena not observed in conventional superconductors.

We present a $(B,T)$-phase diagram of the magnetic superconductor ${\mathrm{TmNi}}_{2}{\mathrm{B}}_{2}\mathrm{C}$ obtained by neutron scattering. The measurements were performed in magnetic fields up to 6 T applied along the crystalline a axis. The observed phases are characterized by three ordering vectors, ${\mathbf{Q}}_{\mathrm{F}}=(0.094,0.094,0),{\mathbf{Q}}_{\mathrm{AI}}=(0.483,0,0),$ and ${\mathbf{Q}}_{\mathrm{AII}}=(0.496,0,0),$ all with the magnetic moment along the c axis. In zero and low fields the Tm $4f$-moments order in a long wavelength transverse spin density wave with $\mathbf{Q}={\mathbf{Q}}_{\mathrm{F}}.$ The magnetic ${\mathbf{Q}}_{\mathrm{AI}}$ structure is stabilized by an applied field of 1 T and a transition to ${\mathbf{Q}}_{\mathrm{AII}}$ is observed at 4 T. For both transitions there is a broad temperature and field range of overlap between the different states. Surprisingly, we observe that the ${\mathbf{Q}}_{\mathrm{A}}$ phases persist to increasingly higher temperatures when the field is increased. Doping with Yb has been introduced to partly suppress superconductivity. In $({\mathrm{Tm}}_{0.90}{\mathrm{Yb}}_{0.10}){\mathrm{Ni}}_{2}{\mathrm{B}}_{2}\mathrm{C}$ the ${\mathbf{Q}}_{\mathrm{F}}\ensuremath{\rightarrow}{\mathbf{Q}}_{\mathrm{AI}}$ phase transition is also observed but at a larger transition field compared to the undoped compound. In $({\mathrm{Tm}}_{0.85}{\mathrm{Yb}}_{0.15}){\mathrm{Ni}}_{2}{\mathrm{B}}_{2}\mathrm{C}$ the ${\mathbf{Q}}_{\mathrm{F}}$ phase persists up to at least 1.8 T. The magnetic correlation length of the ${\mathbf{Q}}_{\mathrm{AI}}$ phase in ${\mathrm{TmNi}}_{2}{\mathrm{B}}_{2}\mathrm{C}$ measured parallel and perpendicular to the magnetic field, is constant within 10% at all fields and temperatures.

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.

Using small-angle neutron scattering, we have imaged the magnetic flux line lattice (FLL) in the d-wave heavy-fermion superconductor CeCoIn5. At low fields we find a hexagonal FLL. Around 0.6 T this undergoes what is most likely a first-order transition to square symmetry, with the nearest neighbors oriented along the gap node directions. This orientation of the square FLL is consistent with theoretical predictions based on the d-wave order parameter symmetry.

We present neutron scattering from the flux line lattice (FLL) in MgB2. Between 0.5 and 0.9 T the FLL undergoes a 30 degrees reorientation, and simultaneously the scattered intensity falls sharply consistent with the weaker superconducting pi band being suppressed with increasing field. We speculate that the pi and sigma bands favor different FLL orientations, and that the reorientation is driven by the suppression of the pi band. When the c axis of the crystal is rotated 45 degrees to the applied field the penetration depth anisotropy could be measured, and rises both as a function of applied field and temperature.

Spanning a broad range of physical systems, complex symmetry breaking is widely recognized as a hallmark of competing interactions. This is exemplified in superfluid 3He which has multiple thermodynamic phases with spin and orbital quantum numbers S = 1 and L = 1, that emerge on cooling from a nearly ferromagnetic Fermi liquid. The heavy fermion compound UPt3 exhibits similar behavior clearly manifest in its multiple superconducting phases. However, consensus as to its order parameter symmetry has remained elusive. Our small angle neutron scattering measurements indicate a linear temperature dependence of the London penetration depth characteristic of nodal structure of the order parameter. Our theoretical analysis is consistent with assignment of its symmetry to an L = 3 odd parity state for which one of the three thermodynamic phases in non-zero magnetic field is chiral.