Anisotropic London Research Articles

Investigating the superconducting state of 2H−NbS2 as seen by the vortex lattice

2H−NbS2 is a classic example of an anisotropic multiband superconductor, with significant recent work focusing on the interesting responses seen when high magnetic fields are applied precisely parallel to the hexagonal niobium planes. It is often contrasted with its sister compound 2H−NbSe2 because they have similar onset temperatures for superconductivity, but 2H−NbS2 has no charge density wave whereas in 2H−NbSe2 the charge density wave order couples strongly to the superconductivity. Using small-angle neutron scattering, a bulk-sensitive probe, we have studied the vortex lattice and how it responds to the underlying superconducting anisotropy. This is done by controlling the orientation of the field with respect to the Nb planes. The superconducting anisotropy, Γac=7.07±0.2, is found to be field independent over the range measured (0.15 to 1.25 T), and the magnetic field distribution as a function of the applied magnetic field is found to be in excellent quantitative agreement with anisotropic London theory modified with a core-size cutoff correction, providing the first complete validation of this model. We find values of λab=141.9±1.5 nm for the in-plane London penetration depth, and λc∼1µm for the out-of-plane response. The field-independence indicates that we are primarily sampling the larger of the two gaps generating the superconductivity in this material. Published by the American Physical Society 2024

Adherence of the rotating vortex lattice in the noncentrosymmetric superconductor Ru7B3 to the London model

The noncentrosymmetric superconductor Ru7B3 has in previous studies demonstrated remarkably unusual behaviour in its vortex lattice (VL), where the nearest neighbour directions of the vortices dissociate from the crystal lattice and instead show a complex field-history dependence, and the VL rotates as the field is changed. In this study, we look at the VL form factor of Ru7B3 during this field-history dependence, to check for deviations from established models, such as the London model. We find that the data is well described by the anisotropic London model, which is in accordance with theoretical predictions that the alterations to the structure of the vortices due to broken inversion symmetry should be small. From this, we also extract values for the penetration depth and coherence length.

Pearl vortices in anisotropic superconducting films

The magnetic field of vortices in anisotropic superconducting films is considered in the framework of anisotropic London approach. It is found that at distances large relative to the core size, the magnetic field normal to the film surface may change sign. We find that the magnetic field attenuates at large distances as $1/r^3$ as it does in isotropic films, but the anisotropy induces an angular dependence to the supercurrents which causes the sign of the field to change for anisotropy parameters $\gamma=\lambda_2/\lambda_1>\sqrt{2}$ in some parts of the $(x,y)$ plane.

Anisotropic time-dependent London approach: Application to the ac response in the Meissner state

The anisotropic London equations taking into account the normal currents are derived and applied to the problem of the surface impedance in the Meisner state of anisotropic materials. It is shown that the complex susceptibility of anisotropic slab depends on the orientation of the applied microwave field relative to the crystal axes. In particular, the anisotropic sample in the microwave field is subject to a torque, unless the field is directed along with one of the crystal principle axes.

Probing the anisotropic vortex lattice in the Fe-based superconductor KFe2As2using small-angle neutron scattering

Using small angle neutron scattering, the anisotropy of the magnetic vortex lattice (VL), in the heavily hole-doped pnictide superconductor ${\text{KFe}}_{2}{\text{As}}_{2}$, was studied. Well-ordered VL scattering patterns were measured with fields applied in directions between $\mathbf{B}\ensuremath{\parallel}\mathbf{c}$ and the basal plane, rotating either towards $[100]$ or $[110]$. Slightly distorted hexagonal patterns were observed when $\mathbf{B}\ensuremath{\parallel}\mathbf{c}$. However, the scattering pattern distorted more strongly as the field was rotated away from the c axis. At low field, the arrangement of vortices is affected by the anisotropy of penetration depth in the plane perpendicular to the field. By fitting the distortion with the anisotropic London model, we obtain an estimate of $\ensuremath{\sim}$$3.4$ for the anisotropy factor $\ensuremath{\gamma}$ between the in-plane and c-axis penetration depths at the lowest temperature studied. The results further reveal VL phase transitions as a function of field direction. We discuss these transitions using the ``hairy ball'' theorem.

Anisotropic London penetration depth and superfluid density in single crystals of iron-based pnictide superconductors

In- and out-of-plane magnetic penetration depths were measured in three iron-based pnictide superconducting systems. The “122” system was represented by electron-doped Ba(Fe 1− x Co x ) 2As 2 with the doping through the whole phase diagram with x ≈ 0.038, 0.047, 0.058, 0.074 and 0.10 ( T c ranged from 13 to 24 K) and by hole-doped (Ba 1− x K x )Fe 2As 2 with doping close to optimal, with measured x ≈ 0.45 ( T c ≈ 28 K) and an underdoped sample with x ≈ 0.15 ( T c ≈ 19 K). The “1111” system was represented by single crystals of NdFeAs(O 1− x F x ) with nominal x = 0.1 ( T c ≈ 43 K). All studied samples of both 122 systems show a robust power-law behavior, λ ( T ) ∝ T n , with the sample-dependent exponent n = 2–2.5, which is indicative of unconventional pairing. This scenario could be possible either through scattering in a S ± state or due to nodes in the superconducting gap. In the Nd-1111 system, the interpretation of the results is complicated by magnetism of the rare-earth ions. For all three systems, the anisotropy ratio, γ λ ≡ λ c / λ ab , was found to decrease with increasing temperature, whereas the anisotropy of the coherence lengths, γ ξ ≡ ξ ab / ξ c = H c 2 ⊥ c / H c 2 ‖ c , has been found to increase (both opposite to the trend in two-band MgB 2). The overall anisotropy of the pnictide superconductors is small, in fact much smaller than that of the cuprates (except YBa 2Cu 3O 7− x (YBCO)). The 1111 system is about two times more anisotropic than the 122 system. Our data and analysis suggest that the iron-based pnictides are complex superconductors in which a multiband three-dimensional electronic structure and strong magnetic fluctuations play important roles.

ANISOTROPY OF MULTILAYERED (CU,C)BA2CA3CU4OY SUPERCONDUCTORS STUDIED BY TORQUE MAGNETOMETRY

The magnetic torque of ( Cu , C ) Ba 2 Ca 3 Cu 4 O y ( ( Cu , C )-1234) aligned samples with various carrier concentrations has been measured under a magnetic field of 9 T at 80 K and 90 K. The carrier concentration was determined by Hall effect measurements. From angular dependent torque measurements, the anisotropy ratio γ was estimated using the 3D anisotropic London model. The γvalues decreased from 23 to 10 with an increase in the average Hall number per CuO 2 plane ( n H); however, these samples showed a nearly constant Tc of about 117 K. These results indicate that the anisotropy of ( Cu , C )-1234 strongly reflects the doping levels of the outer planes.

Small-angle neutron scattering measurements of the vortex lattice inCaC6

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.

Electron-optical phase shift of a Josephson vortex

The possibility of directly observing Josephson vortices in a superconducting material by transmission electron microscopy is here investigated. First, the anisotropic London equation for the magnetic field of a Josephsonvortex is solved in Fourier space. Then, from the knowledge of the magnetic field, the vector potential and the Aharonov-Bohm phase shift are derived. Finally, phase contrast image simulations are presented. It will be shown that, with current technology, the direct observation of a Josephson vortex is possible.

Current percolation and anisotropy in polycrystalline MgB(2).

The influence of anisotropy on the transport current in MgB(2) polycrystalline bulk samples and wires is discussed. A model for the critical current density is proposed, which is based on anisotropic London theory, grain boundary pinning, and percolation theory. The calculated currents agree convincingly with experimental data, and the fit parameters, especially the anisotropy, obtained from percolation theory agree with experiment or theoretical predictions.

Transverse-field components of the flux-line lattice in the anisotropic superconductorYBa2Cu3O7−δ

We report on detailed small-angle neutron scattering measurements with polarization analysis from the flux-line lattice in the anisotropic superconductor ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}.$ When the field was applied at an angle to the principal axes we have observed spin-flip neutron diffraction consistent with local transverse field components. From a detailed study of the angle dependence of the magnitude of the spin-flip neutron-scattered intensity we have found that the transverse-field components are larger than predicted by an anisotropic London model for reasonable mass anisotropy values. The transverse-field components are consistent with a reduced c-axis coherence length at low temperature.

London theory of the crossing vortex lattice in highly anisotropic layered superconductors

A novel description of Josephson vortices (JVs) crossed by the pancake vortices (PVs) is proposed on the basis of the anisotropic London theory. The field distribution of a JV and its energy have been calculated for both dense ($a<\lambda_J$) and dilute ($a>\lambda_J$) PV lattices with distance $a$ between PVs, and the nonlinear JV core size $\lambda_J$. It is shown that the ``shifted'' PV lattice (PVs displaced mainly along JVs in the crossing vortex lattice structure), formed in high out-of-plane magnetic fields transforms into the PV lattice ``trapped'' by the JV sublattice at a certain field, lower than $\Phi_0/\gamma^2s^2$, where $\Phi_0$ is the flux quantum, $\gamma$ is the anisotropy parameter and $s$ is the distance between CuO$_2$ planes. With further decreasing $B_z$, the free energy of the crossing vortex lattice structure (PV and JV sublattices coexist separately) can exceed the free energy of the tilted lattice (common PV-JV vortex structure) in the case of $\gamma s<\lambda_{ab}$ with the in-plane penetration depth $\lambda_{ab}$ if the low ($B_x<\gamma\Phi_0/\lambda_{ab}^2$) or high ($B_x\gtrsim \Phi_0/\gamma s^2$) in-plane magnetic field is applied. It means that the crossing vortex structure is realized in the intermediate field orientations, while the tilted vortex lattice can exist if the magnetic field is aligned near the $c$-axis and the $ab$-plane as well. In the intermediate in-plane fields $\gamma\Phi_0/\lambda_{ab}^2\lesssim B_x \lesssim \Phi_0/\gamma s^2$, the crossing vortex structure with the ``trapped'' PV sublattice seems to settle in until the lock-in transition occurs since this structure has the lower energy with respect to the tilted vortex structure in the magnetic field ${\vec H}$ oriented near the $ab$-plane.

Superconducting anisotropy in Nd 1.85Ce 0.15CuO 4 single crystals

The magnetic torque of the Nd 1.85Ce 0.15CuO 4 crystal consists of three contributions in the superconducting state, i.e., (1) a torque of the anisotropic magnetic moment of the Nd 3+ ions, (2) a torque coming from the mass difference between m ab and m c and (3) a torque of the vortex lock-in states appeared when the field is applied parallel to the CuO 2 planes. We analyze a torque component arising from the mass anisotropy by using the anisotropic London model and obtain an anisotropy parameter γ c =28±3. The irreversibility line of Nd 1.85Ce 0.15CuO 4 is in excellent accordance with that of the La-based cuprate.

Vortex lines in films: Fields and interactions

General expressions are given for the magnetic field and energy of arbitrary arrangements of straight and curved vortices in an anisotropic superconductor film of finite thickness within anisotropic London theory. As examples we consider the magnetic field and interaction of straight perpendicular vortex lines in films of finite thickness.

Effect of oxygen stoichiometry on the out-of-plane anisotropy of YBa2Cu3O7- single crystals near optimal doping

We have performed magnetic measurements on pure detwinned YBa2Cu3O7- single crystals in order to investigate the influence of variations in the oxygen stoichiometry on the out-of-plane anisotropy ck in the region close to optimal doping (6.90 < 7- < 6.98). The anisotropy is estimated using the anisotropic London model from the period of the commensurability oscillations of the hysteretic magnetization as observed for fields applied parallel to the a-b-planes. For a variation in the oxygen deficiency by as much as 300%, we find that ck decreases but only slightly (less than 12%) with decreasing oxygen deficiency. The possibility of a temperature-dependent ck is also discussed.

Pinning of Vortices by a Cylindrical Defect in High-Temperature Superconductors

The uniaxial anisotropic London equation is used to study the interaction between curved flux lines and a cylindrical defect inside a high-temperature superconductor. The distribution of the magnetic fields inside the high-temperature superconducting defect (a non-superconducting defect) and the superconducting medium (outside the defect) are exactly determined. The energy of the interaction between the curved flux lines and the defect, in terms of the flux line segment position, is obtained. The dependence of the free energy of the system on σ (the ratio between the penetration depths of the defect and the superconducting medium) and on the radius of the cylindrical defect are established. Moreover, the magnetic pinning force of the flux lines as well as the critical current density are determined. A good agreement with previous results is achieved.

Strong shift of the irreversibility line in high-Tcsuperconductors upon vortex shaking with an oscillating magnetic field

Torque magnetometry is a powerful method for probing superconducting anisotropy in the mixed state. In order to use the three-dimensional anisotropic London model to analyze torque data, the vortex lattice must be in a reversible state, a state normally restricted to a narrow range close to the upper critical-field boundary ${H}_{c2}(T)$ because of large pinning effects that set in at lower temperatures $T.$ We show that the application of an additional oscillating magnetic field perpendicular to the main field $B$ leads to a fast depinning of the vortex lattice. This vortex-shaking process dramatically extends the reversible domain in the $(H,T)$ phase diagram, and thus the range in which torque investigations can be made.

A simple method to determine the mass parameter in uniaxial superconductors

Using the anisotropic London theory we investigate some properties of superconductors in the limit of large values of the magnetic induction. In this work we proposed a method of obtaining the average mass parameter using torque magnetometry. We consider the region where the pinning and the geometry of the sample are not important. Thus, the only relevant quantity is the angle associated to the biggest torque, θ max .

The magnetic field and energy of an Abrikosov vortex in an anisotropic London superconductor

The behavior of a straight Abrikosov vortex in an anisotropic uniaxial London superconductor is studied. Analytical expressions are derived that approximately describe the magnetic field in three regions: the asymptotic region, where the distance r from the vortex line is greater than λΓ (λ is the London length and Γ is the anisotropy constant), the intermediate region λ<r<λΓ, and the region r<λ. It is found that in the intermediate region with high anisotropy the component of the magnetic field along the vortex line changes sign for a certain interval of angles between the vortex line and the anisotropy axis. Because of this the interaction of parallel vortices whose plane is parallel to the anisotropy axis has a minimum and a maximum. This means that numerous metastable vortex lattices can exist. Additional terms in the vortex self-energy are obtained, and although they are smaller than the leading logarithmic term, they display a different dependence on the angle between the vortex line and the anisotropy axis.

Instabilities in the flux-line lattice of anisotropic superconductors

The stability of the flux-line lattice has been investigated within anisotropic London theory. This is a full-scale investigation of instabilities in the ``chain'' state, the equilibrium lattice that is similar to the Abrikosov lattice at large fields but crosses over smoothly to a pinstripe structure at low fields. By calculating the normal modes of the elasticity matrix, it has been found the lattice is stable at large fields, but that instabilities occur as the field is reduced. The field at which these instabilities first arise, ${\mathrm{b}}^{\mathrm{*}}$(\ensuremath{\epsilon},\ensuremath{\theta}), depends on the anisotropy \ensuremath{\epsilon} and the angle \ensuremath{\theta} at which the lattice is tilted away from the c axis. These instabilities initially occur at wave vector ${\mathbf{k}}^{\mathrm{*}}$(\ensuremath{\epsilon},\ensuremath{\theta}). The dependence of ${\mathbf{k}}^{\mathrm{*}}$ on \ensuremath{\epsilon} and \ensuremath{\theta} is complicated, but the component of ${\mathbf{k}}^{\mathrm{*}}$ along the average direction of the flux lines, ${\mathrm{k}}_{\mathrm{z}}$, is always finite. For rigid straight flux lines, the cutoff necessary for London theory has been ``derived'' from Landau-Ginzburg theory, where the shape of the vortex core is known. However, for investigating instability at finite ${\mathrm{k}}_{\mathrm{z}}$ it is necessary to know the dependence of the cutoff on ${\mathrm{k}}_{\mathrm{z}}$, and we have used a cutoff suggested by Sudb\o{} and Brandt. The instabilities only occur for values of the anisotropy \ensuremath{\epsilon} appropriate to a material like BSCCO, and not for anisotropies more appropriate to YBCO. The lower critical field ${\mathrm{H}}_{\mathrm{c}1}$(\ensuremath{\varphi}) is calculated as a function of the angle \ensuremath{\varphi} at which the applied field is tilted away from the crystal axis. The presence of kinks in ${\mathrm{H}}_{\mathrm{c}1}$(\ensuremath{\varphi}) is seen to be related to instabilities in the equilibrium flux-line structure.