Anisotropy of the Superconducting State in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$

Despite intense studies the exact nature of the order parameter in superconducting ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ remains unresolved. We have used small-angle neutron scattering to study the vortex lattice in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ with the field applied close to the basal plane, taking advantage of the transverse magnetization. We measured the intrinsic superconducting anisotropy between the $c$ axis and the Ru-O basal plane ($\sim 60$), which greatly exceeds the upper critical field anisotropy ($\sim 20$). Our result imposes significant constraints on possible models of triplet pairing in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ and raises questions concerning the direction of the zero spin projection axis.

Figure 1

Experimental geometry. (a) The coordinate system is defined with $z$ along $\mathit{H}$ and $y$ in the Ru-O basal plane (along $\mathit{b}$). The applied magnetic field $\mathit{H}$ is rotated away from the Ru-O plane (spanned by $\mathit{a}$ and $\mathit{b}$) by an angle $\Omega$. Neutron spins ($\sigma$) are parallel or antiparallel to the magnetic field. The incident neutron beam is in the $yz$ plane, at an angle $\phi$ relative to the field direction. The observed VL scattering vector is denoted $\mathit{Q}$ and the longitudinal and transverse component of the field modulation by $h_z$ and $h_x$, respectively. (b) Diffraction pattern showing spin-flip scattering from the VL due to the transverse field modulation ($h_x$). The two Bragg peaks correspond to $\pm \mathit{Q}$ in (a). No background subtraction was performed, but a small remnant of the undiffracted beam close to $Q=0$ due to the finite flipping ratio ($\sim 8$) is masked off.

Figure 2

Vortex lattice rocking curves showing the scattered intensity as a function of the angle $\phi$, for an unpolarized neutron beam. Error bars are equal to or smaller than the symbols. Two maxima are observed for both the bottom (${\phi }_{1/3}$) and top (${\phi }_{2/4}$) VL Bragg reflections (rocking curves obtained with a polarized neutron beam can be found in the Supplemental Material [15]). The intensity was normalized to the incident neutron flux. The curves are fits to the data as described in the text.

Figure 3

Vortex lattice form factor at 40 mK as a function of applied field and angle $\Omega$ with the $a$ axis. The statistical error is roughly the size of the symbols. The solid lines are guides to the eye. The dashed line shows an extended London model fit to the 0.5 T data as discussed in the text, with ${\lambda }_{ab}=167\mathrm{nm}$, ${\xi }_{ab}=66\mathrm{nm}$, $c=1/4$, and ${\Gamma }_{\mathrm{ac}}=58.5$.

Figure 4

Vortex lattice anisotropy. (a) Schematic of VL Bragg reflections lying on an ellipse with major-to-minor axis ratio, ${\Gamma }_{\mathrm{VL}}=6$. Only the filled (red) peaks are observed. The reciprocal space area of the ellipse is $\pi Q_0^2=8{\pi }^3{\mu }_{0}H/\sqrt{3}{\Phi }_0$. (b) Measured VL anisotropy at 40 mK as a function of applied field and angle with the $a$ axis ($\Omega$). Except where shown explicitly, the statistical error is the size of the symbols. The lines show the VL anisotropy calculated using Eq. (2) and ${\Gamma }_{\mathrm{ac}}=20$ (dotted line), 58.5 (dashed line), and $\infty$ (full line).