Small-angle neutron scattering study of the vortex lattice in superconducting ${\text{LuNi}}_2{\text{B}}_2\text{C}$

We present studies of the magnetic field distribution around the vortices in ${\text{LuNi}}_2{\text{B}}_2\text{C}$. Small-angle neutron scattering measurements of the vortex lattice (VL) in this material were extended to unprecedentedly large values of the scattering vector $q$, obtained both by using high magnetic fields to decrease the VL spacing and by using higher order reflections. A square VL, oriented with the nearest-neighbor direction along the crystalline [110] direction, was observed up to the highest measured field. The first-order VL form factor, $\left|F\left(q_{10}\right)\right|$, was found to decrease exponentially with increasing magnetic field. Measurements of the higher-order form factors, $\left|F\left(q_{hk}\right)\right|$, reveal a significant in-plane anisotropy and also allow for a real-space reconstruction of the VL field distribution.

Figure 1

(Color online) SANS diffraction pattern of the VL in ${\text{LuNi}}_2{\text{B}}_2\text{C}$ at 0.5 T and 2 K following a field cooling procedure. The image (a) is a sum of measurements as the sample is rotated around the vertical axis in order to satisfy the Bragg condition for reflections in the center-right part of the detector. The data are smoothed and shown on a logarithmic scale. Measurements at 6 T, where no scattering from the VL could be observed, were used for background subtraction. The axes show the orientation of the crystalline axes. An indexing of the peaks is shown in (b), with the cross indicating the origin of reciprocal space. The apparent difference in intensity of e.g., the $\left(\overline{1}2\right)$ and $\left(2\overline{1}\right)$ reflections compared to (12) and (21) is due to different Lorentz factors, for which the detailed reflectivity analysis have been corrected.

Figure 2

(Color online) Rocking curves at 0.5 T and 2 K for the ${\text{LuNi}}_2{\text{B}}_2\text{C}$ (10) and (32) VL reflections from Fig. 1. Note the different intensity scales for the two reflections. Error bars for the (10) reflection are not shown since they are smaller than the size of the data points. The intensity at each angular setting is obtained by summing the detector counts at the position of the specific Bragg reflection. The curves are Voigt fits to the data. The shoulder seen for the (32) reflection is unrelated to the VL, as discussed in the text.

Figure 3

(Color online) Field dependence of the VL (10) form factor for both the field cooled (FC) and zero-field cooled (ZFC) case. The fitted values of the penetration depth and coherence lengths are $\lambda =90.7\text{nm}$ and $=8.22\text{nm}$ for the London model $\left({\chi }^2=0.14\right)$ (Refs. 5, 36), $\lambda =61.9\text{nm}$ and $\xi =12.7\text{nm}$ for the Clem model $\left({\chi }^2=0.62\right)$ (Ref. 37), and $\lambda =104.1\text{nm}$ and $\xi =7.19\text{nm}$ for the Hao model $\left({\chi }^2=1.20\right)$ (Refs. 38, 39).

Figure 4

(Color online) VL form factor divided by the applied field versus scattering vector $q$ for all measured reflections. Curves through $\left|F\left(q_{10}\right)\right|$ and $\left|F\left(q_{11}\right)\right|$ are fits to the London model [Eq. (2) and Fig. 3].

Figure 5

(Color online) Field dependence of VL form-factor ratios $F\left(q_{11}\right)/F\left(q_{10}\right)$ (top) and $F\left(q_{20}\right)/F\left(q_{10}\right)$ (bottom). Note different scales. The upper line corresponds to the average value of $\left|F\left(q_{11}\right)/F\left(q_{10}\right)\right|$. The lower line is a fit to $\left|F\left(q_{20}\right)/F\left(q_{10}\right)\right|$.

Figure 6

(Color online) Real space magnetic field reconstruction from the measured VL form factors at 0.5 T and 2 K, using the London (a) and Ginzburg-Landau (b) sign schemes. In both cases an equidistant contour spacing of 5 mT was used, with the lowest contour at, respectively, 493 mT (a) and 486 mT (b). In both cases the amplitude of the field modulation is 48.9 mT or roughly 10% of the average (applied) magnetic field. Four VL unit cells are shown, with a vortex spacing of 64.3 nm corresponding to an applied field of ${\mu }_{0}H=0.5\text{T}$. As discussed in the text the true field reconstruction is with all form factors positive (a). Note that the field reconstruction images are rotated $45°$ with respect to Fig. 1, so that the {110}-directions are now horizontal/vertical. Panel (c) shows the magnetic field distribution function for the two different field reconstructions.

Figure 7

(Color online) Supercurrent density as a function of distance from the vortex center, along the VL nearest-neighbor vortex direction (crystalline [110] axis) and the VL diagonal (crystalline [100] axis). The inset shows the value of ${\xi }_J$ (distance of maximum current) in the basal plane. To visually emphasize the fourfold anisotropy, a circle with radius ${\xi }_J^{\left[100\right]}$ is also shown (dashed line).