Pauli Paramagnetic Effects on Vortices in Superconducting ${\mathrm{TmNi}}_2{\mathrm{B}}_2\mathrm{C}$

The magnetic field distribution around the vortices in ${\mathrm{TmNi}}_2{\mathrm{B}}_2\mathrm{C}$ 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 $0.6H_{c2}$ 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.

Figure 1

Measured magnetic induction versus applied field at 1.6 K for ${\mathrm{TmNi}}_2{\mathrm{B}}_2\mathrm{C}$ (squares) and nonmagnetic ${\mathrm{LuNi}}_2{\mathrm{B}}_2\mathrm{C}$ (circles). The top left inset show a VL diffraction pattern obtained at 0.2 T and 1.6 K. The bottom right inset show a schematic of the diffraction pattern, indicating the VL scattering vectors and opening angle. Open and solid circles represent peaks belonging to different domain orientations, while ×’s denote higher order reflections.

Figure 2

Field dependence of the measured VL form factor in ${\mathrm{TmNi}}_2{\mathrm{B}}_2\mathrm{C}$ at 1.6 K. The lines show the form factor calculated using Eq. (3) with $\xi =210\AA$ and $\lambda =780\AA$ (dotted line) or 600 Å (dashed line).

Figure 3

Comparison of measured and calculated VL form factors in ${\mathrm{TmNi}}_2{\mathrm{B}}_2\mathrm{C}$ at $T=1.6$, 3.5, and 5.5 K. The curves were calculated using the model described in the text, for $T=0.16T_c$ and $\mu =1.71$ (A), $T=0.35T_c$ and $\mu =1.28$ (B), and $T=0.50T_c$ and $\mu =0.86$ (C).

Figure 4

Spatial structure of internal field $B(\mathbf{r})$ within a unit cell of the square vortex lattice. Here $\overline{B}=0.1=0.36H_{c2}(0.5T_c)$, $\kappa =6.2$, $T=0.16T_c$, and $\mu =0.02$ (a) and 1.71 (b). A profile of the field distribution is shown in (c).