Magnetic-field-induced nonlocal effects on the vortex interactions in twin-free YBa${}_2$Cu${}_3$O${}_7$

The vortex lattice (VL) in the high-$\kappa$ superconductor YBa${}_2$Cu${}_3$O${}_7$, at 2 K and with the magnetic field parallel to the crystal c axis, undergoes a sequence of transitions between different structures as a function of applied magnetic field. However, from structural studies alone, it is not possible to determine precisely the system anisotropy that governs the transitions between different structures. To address this question, here we report new small-angle neutron scattering measurements of both the VL structure at higher temperatures and the field and temperature dependence of the VL form factor. Our measurements demonstrate how the influence of anisotropy on the VL, which in theory can be parameterized as nonlocal corrections, becomes progressively important with increasing magnetic field, and suppressed by increasing the temperature toward $T_c$. The data indicate that nonlocality due to different anisotropies plays an important role in determining the VL properties.

Figure 1

VL diffraction patterns obtained at 2 K, and in fields applied parallel to the crystal c axis, of (a) 0.5 T, (b) 5.0 T, and (c) 7.5 T. The axes indicated in (a) apply to all figures. In each image, dashed line patterns indicate the VL structures. Solid lines represent the basis vectors of the primitive cell, with the primitive cell opening angle of (a) $\phi$, (b) $\rho$, and (c) $\nu$. The images are constructed by summing together the diffraction patterns obtained at a series of angles about the horizontal and vertical axes. This allows the presentation of all the Bragg spots in a single picture. Statistical noise at the center of the patterns has been masked, and the data smoothed with a Gaussian envelope smaller than the instrument resolution. The VL in real space can be visualized by rotating the reciprocal VL by 90${}^{\circ }$ and adding an additional vortex at the origin.

Figure 2

The field dependence at 2 K of the primitive cell opening angle for the various VL structures. Inset schematics indicate the reciprocal VL primitive cell for each phase. The dashed line at 90${}^{\circ }$ indicates the opening angle of a perfectly square VL. The dashed-dotted line at 60${}^{\circ }$ indicates the opening angle of an isotropic hexagonal VL. The (dark) gray shaded portions indicate the field regions over which we observe first-order VL structure transitions. The error bars are sufficiently small to be masked by the data symbols. Some of these data were first presented in Ref. 40.

Figure 3

The field dependence at 2 K of the parameter $\eta$ determined for both the LFS and IFS. The (dark) gray shaded portion indicates the field region over which we observe a first-order transition between the LFS and IFS phases. The line corresponds to a linear fit of the data obtained in the IFS phase. Error bars not visible can be considered of order the size of the symbol. Some of these data were first presented in Ref. 40.

Figure 4

The predicted field dependencies of the VL opening angle $\beta$ for both the fourfold symmetric case (down-pointing triangles and diamonds) and the twofold symmetric case (up-pointing triangles). Figure taken from Ref. 15.

Figure 5

Numerical evaluations of the free energy carried out using the Kogan nonlocal theory in the mid-field region and for the twofold symmetric case (Ref. 15). Inset diagrams show the degenerate reciprocal VLs. Filled symbols denote the primitive cell, and empty symbols represent the additional spots required to complete the hexagonal structure. Dashed lines indicate horizontal and vertical symmetry planes. The parameters used for the calculations are the same as those in Ref. 15.

Figure 6

The field dependence of the primitive cell opening angle $\beta$ as calculated using the Kogan theory for the twofold symmetric case. In the mid-field region, we plot only the small-$\beta$ solution in order to compare with Fig. 4. The reciprocal VL structures are defined by the inset diagrams. The short- and long-dashed lines indicate the crystal symmetry planes for the fourfold case. For the twofold case, only the short-dashed line symmetry planes remain. The parameters used for the calculations are the same as those in Ref. 15.

Figure 7

The zero field in-plane anisotropies of superconducting YBa${}_2$Cu${}_3$O${}_7$ that are expected to influence the VL properties. (a) A schematic of the irreducible quadrant of the first Brillouin zone showing the main Fermi surfaces at $k_z=0$ (Refs. 70 and 71). (b) The predominantly $d_{x^2-y^2}$ order parameter combined with the $s$-wave admixture suggested by phase-sensitive measurements (Ref. 49).

Figure 8

The angular dependence of the diffracted intensity (rocking curve) for the top and bottom Bragg spots of the diffraction pattern shown in Fig. 1a. The dashed lines are fits of Lorentzian line shapes to the data, and the solid lines indicate the full width at half maximum (FWHM).

Figure 9

The field dependence at 2 K of the observed form factors $F\left(\mathbf{\text{q}}\right)$. Across the entire field range, we distinguish between the different types of Bragg spot according to the different magnitudes of the q vectors for each type. Each data point represents the average form factor for a certain spot type.

Figure 10

The field dependence at 2 K of the ratio between on-axis and off-axis form factors. For the LFS and HFS phases, the on-axis spots lie along ${\mathbf{a}}^{*}$, while for the IFS phase, the on-axis spots lie along ${\mathbf{b}}^{*}$. The black dashed line indicates a ratio of unity, which is expected in the anisotropic London limit.

Figure 11

A semilogarithmic graph showing the field dependence of the VL form factor for Bragg spots with a q vector lying off-axis. The fit of the data to the London model is also shown. The inset shows the model fit of the form factor data with $\mathbf{q}\parallel {\mathbf{a}}^{*}$.

Figure 12

The temperature dependence of the VL structure opening angles (a) $\phi$ and $\nu$ for the LFS and HFS phases respectively, and (b) $\rho$ for the IFS phase. The indicated angles correspond to those in Fig. 1. In both panels, the lines passing through for the data at each field are guides to the eye. Data are shown only for temperatures where the spot position on the detector could be determined accurately. Error bars not visible can be considered of order the size of the data point.

Figure 13

The temperature dependence of the form factor $\left|F\right(q\left)\right|$ for the two types of Bragg spot in (a) the LFS at 0.2 T, and (b) the IFS at 5.0 T. For each scan, we performed an OFC procedure down to base temperature, and then measured on heating the sample. In (a), the solid and dashed lines show fits to the data made using both a minimal and local $d$-wave model. In (b), the solid and dashed lines show fits to the data after extending the local $d$-wave model used in (a) with a nonlocal correction. Both models are described in the text. The inset of each figure shows the temperature dependence of the rocking curve FWHM as obtained from the rocking curve measurements. The dashed and dotted lines, respectively, indicate the mean value of the FWHM used to normalize the form factor values of the fixed angle data (see text for details).

Figure 14

The temperature dependence of the form factor ratio taken between the different types of Bragg spots at various fields. For the fields of 0.2 T and 8.0 T, taken in the LFS and HFS phases, respectively, the form factor ratio plotted at each temperature is $|F(\mathbf{q}\parallel {\mathbf{a}}^{*})|/|F(\mathbf{q}\neg \parallel {\mathbf{a}}^{*})|$. For the 5.0 T data taken in the IFS phase, the ratio is $|F(\mathbf{q}\parallel {\mathbf{b}}^{*})|/|F(\mathbf{q}\neg \parallel {\mathbf{b}}^{*})|$.

Figure 15

The (${\mu }_{0}H$,$T$) VL structure phase diagram for fields applied parallel to the c axis. Circle data points were obtained from measurements of the VL form factor, and correspond to the point where each relevant VL structure is measured to occupy 50$\%$ of the sample volume. Diamond data points correspond to estimates of the transition points as determined from structural measurements. The dotted and dashed lines are guides for the eye phase boundary lines determined from the form factor data only. The melting line is deduced from data presented in Ref. 94.