Vortex Lattice Anisotropy and Pauli Limiting in KFe$_2$As$_2$

In superconducting KFe$_2$As$_2$ (KFA), $H_{c2}$ (in Tesla) parallel to the basal plane is much larger than $T_c$ (in Kelvin) suggesting Pauli limiting. We have used small-angle neutron scattering (SANS) with $H$ applied close to the basal plane to study the vortex lattice (VL) anisotropy and scattered intensity in KFA. The VL anisotropy reflects the intrinsic anisotropy of the superconducting state ($\Gamma_{ac}$), and may differ significantly from the $H_{c2}$ anisotropy ($\Gamma_{H_{c2}}$) as recently seen in Sr$_2$RuO$_4$ [C. Rastovski $et$ $al.$, Phys. Rev. Lett. {\bf 111}, 087003 (2013)]. Our data indicates a field dependent VL anisotropy in KFA, increasing with $H$, possibly caused by multi-band effects. At high fields $\Gamma_{ac} > \Gamma_{H_{c2}}$, supporting Pauli limiting. Due to the moderately high anisotropy, both the longitudinal and transverse VL field modulation may contribute to the SANS signal, occurring as non-spin flip and spin flip scattering respectively. This allow an independent determination of possible Pauli paramagnetic effects (spin polarization of un-paired quasiparticles in the vortex cores) along the two different directions.