Field dependence of the superconducting basal plane anisotropy of TmNi${}_2$B${}_2$C

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 $H_{\mathrm{c}2}$.

Figure 1

SANS diffraction patterns showing the square, rhombic, and hexagonal VL phases in TmNi${}_2$B${}_2$C at $1.6$ K for applied fields of (a) $0.2$, (b) $0.35$, and (c) $0.5$ T, respectively. The images are obtained by summing measurements at multiple rotation and tilt angles to satisfy the Bragg condition for the different reflections. The scattered intensity is shown on a logarithmic false color scale to make strong and weak reflections simultaneously visible. The orientation of the crystalline axes and the magnetic field is shown in the inset to (a), where $\Omega$ is the angle between the field and the $c$ axis. The indexing of the VL Bragg peaks is shown for one quadrant in the schematics in (d)–(f) with the size of the circles indicating the intensity. For the rhombic and hexagonal VL phases, two domain orientations with an opening angle $\beta$ are observed, as shown by the black and red circles in (e) and (f). With increasing field, the VL Bragg reflections move to longer scattering vectors $q$ and decrease in intensity, making fewer of them visible.

Figure 2

Comparison of VLs at $0.5$ T ($\Omega ={10}^{\circ }$) and $1.6$ K. The diffraction patterns were obtained directly following a field-cooling procedure (FC) and after the application of a damped field oscillation with an initial amplitude of 25 mT (FCO). No higher-order VL reflections were observed due to shorter count times as compared to Fig. 1c.

Figure 3

Rocking curves for the square VL (10) and (03) Bragg reflections at $0.2$ T and $1.6$ K corresponding to Figs. 1a and 1d. Note the different axes for the two reflections. Each angle was counted for 9 min. For the (10) reflection, the error bars are smaller than the symbol size. The (10) reflection is fitted with a double Lorentzian function due to the irregular shape with a shoulder left of the main peak. The (03) reflections are fitted by a single Lorentzian.

Figure 4

VL form factor divided by the applied field vs scattering vector $q$ for all measured reflections at $0.2$, $0.35$, and $0.5$ T. For all fields and reflections, the error bars are smaller than the symbol size. The curves are fits to the London model, as described in the text. Full and dashed lines correspond to VL Bragg peaks along the crystalline [100] and [110] directions, respectively.

Figure 5

Real-space magnetic field reconstruction from the measured VL form factors at ${\mu }_{0}H=0.2$ T and $T=1.6$ K. (a) Contour plot of the magnetic field, showing four VL unit cells with a vortex spacing of 98 nm, corresponding to a magnetic induction $B=0.216$ T obtained from the magnitude of the scattering vector. Note that the image is rotated ${45}^{\circ }$ with respect to Fig. 1 such that $\left\{110\right\}$ directions are horizontal/vertical. The lowest contour corresponds to $B=198$ mT and the contour spacing is 15 mT. (b) Current density as a function of distance from the vortex center along the VL nearest-neighbor direction ([110]) and the unit cell diagonal ([100]). The inset shows the value of ${\xi }_J$ (distance of maximum current) in the basal plane. To emphasize the fourfold anisotropy, a circle with radius ${\xi }_J^{100}$ is shown by the dashed line.

Figure 6

(a) Angle dependence of the superconducting basal plane anisotropy ratio calculated from the VL form factors as described in the text. For each field, the curves show a fit to the anisotropy function given in Eq. (6). (b) The same results in polar coordinates.