We investigate the mixed state properties in a type II multiband superconductor with uniaxial anisotropy under the Pauli paramagnetic effects. Eilenberger theory extended to a multiband superconductor is utilized to describe the detailed vortex lattice properties, such as the flux line form factors, the vortex lattice anisotropy, and magnetic torques. We apply this theory to ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ to analyze those physical quantities obtained experimentally, focusing on the interplay between the strong two-dimensional anisotropy and the Pauli paramagnetic effects. This study allows us to understand the origin of the disparity between the vortex lattice anisotropy ($\sim 60$) and the $H_{\mathrm{c}2}$ anisotropy ($\sim 20$). Among the three bands—$\gamma$ with the effective mass anisotropy $\sim 180, \alpha$ with $\sim 120$, and $\beta$ with $\sim 60$—the last one is found to be the major band, responsible for various magnetic responses while the minor $\gamma$ band plays an important role in the vortex formation. Namely, in a field orientation slightly tilted away from the two-dimensional basal plane those two bands cooperatively form the optimal vortex anisotropy which exceeds that given by the effective mass formula with infinite anisotropy. This is observed by small-angle neutron scattering experiments on ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. The pairing symmetry of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ realized is either spin singlet or spin triplet with the $d\text{-vector}$ strongly locked in the basal plane. The gap structure is that the major $\beta$ band has a full gap and the minor $\gamma$ band has a $d_{x^2-y^2}\text{-like}$ gap.

• Received 7 June 2015
• Revised 23 July 2015

DOI:https://doi.org/10.1103/PhysRevB.92.054505