Anisotropy and multiband superconductivity in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ determined by small-angle neutron scattering studies of the vortex lattice

Despite numerous studies the exact nature of the order parameter in superconducting ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ remains unresolved. We have extended previous small-angle neutron scattering studies of the vortex lattice in this material to a wider field range, higher temperatures, and with the field applied close to both the $\langle 100\rangle$ and $\langle 110\rangle$ basal plane directions. Measurements at high field were made possible by the use of both spin polarization and analysis to improve the signal-to-noise ratio. Rotating the field towards the basal plane causes a distortion of the square vortex lattice observed for $\mathit{H}\parallel \langle 001\rangle$ and also a symmetry change to a distorted triangular symmetry for fields close to $\langle 100\rangle$.The vortex lattice distortion allows us to determine the intrinsic superconducting anisotropy between the $c$ axis and the Ru-O basal plane, yielding a value of $\sim 60$ at low temperature and low to intermediate fields. This greatly exceeds the upper critical field anisotropy of $\sim 20$ at low temperature, reminiscent of Pauli limiting. Indirect evidence for Pauli paramagnetic effects on the unpaired quasiparticles in the vortex cores are observed, but a direct detection lies below the measurement sensitivity. The superconducting anisotropy is found to be independent of temperature but increases for fields $\gtrsim 1$ T, indicating multiband superconductvity in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. Finally, the temperature dependence of the scattered intensity provides further support for gap nodes or deep minima in the superconducting gap.

Figure 1

Experimental geometry and data overview. (a) The coordinate system is defined with $z$ along $\mathit{H}$ and $y$ vertical in the Ru-O basal plane. The crystalline axes show the sample orientation used for the ILL experiments. The applied magnetic field $\mathit{H}$ is rotated away from the basal plane by an angle $\mathrm{\Omega }$. 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 ${\mathbf{Q}}_{\mathbf{\text{VL}}}$, and $h_z$ and $h_x$ are, respectively, the longitudinal and transverse Fourier component of the field modulation. (b) $H\text{−}\mathrm{\Omega }$ phase diagram showing data obtained using the ILL D33, ILL D22, and PSI SANS-I instruments. All measurements were performed with $\mathit{H}\perp \langle 100\rangle$, except for the SANS-I data where $\mathit{H}\perp \langle 110\rangle$. Solid symbols indicate where measurements were taken at several temperatures; all others were performed at base temperature ($\sim 50$ mK). The circles (red) indicate measurements using a polarized/analyzed SANS configuration. The upper critical field ($H_{c2}$) is the parametrization from Ref. [14].

Figure 2

Possible VL configurations. (a) SANS diffraction pattern for a field of 25 mT applied along the [001] axis. A square VL is observed, with Bragg peaks oriented along the $\langle 110\rangle$ axes. The position of the first order (larger circles) and second order (smaller circles) Bragg reflections are shown schematically for the two different experimental configurations: $\langle 100\rangle$ vertical (b) and $\langle 110\rangle$ vertical (c). Rotating the field towards the basal plane may simply cause the VL to distort due to the uniaxial anisotropy (d),(f) or also undergo a transition from a distorted square to a distorted triangular symmetry (e). Here a relatively small anisotropy ${\mathrm{\Gamma }}_{\text{VL}}=6$ was used for illustrative purposes. Only VL Bragg peaks on the vertical axis are observed in the SANS experiments (solid symbols), leading to an ambiguity in determining the anisotropy as indicated by the ellipses. As discussed in the text, the most likely VL configurations for the two different field directions are shown in (e) and (f).

Figure 3

Vortex lattice diffraction patterns as a function of magnetic field amplitude, rotation angle ($\mathrm{\Omega }$), and temperature. Positions in reciprocal space (horizontal and vertical axes) are normalized by the scattering vector for an isotropic triangular VL ($Q_0$), and data around the direct beam ($Q=0$) is masked off. The VL scattering vector is indicated in (f). In all cases, the open circles indicate the Bragg peak fitted centers on the detector, while the white lines indicate $Q/Q_0$ values determined from the rocking curves. Measurements as a function of rotation angle [(a)–(e), ILL D33] were performed at 1 T ($\mathit{H}\perp \langle 100\rangle$) and 50 mK, for positive $Q_y$ Bragg reflections only. The $\mathit{H}\perp \langle 100\rangle$ field sweep [(f)–(j), ILL D33] was performed at $\mathrm{\Omega }=0.9^{\circ }$ and 50 mK. Finally, the temperature scan [(k)–(o), PSI SANS-I] was performed at 0.4 T ($\mathit{H}\perp \langle 110\rangle$) and $\mathrm{\Omega }=2^{\circ }$. Temperature- and $\mathrm{\Omega }$-dependent measurements are shown using fixed intensity scales. For the field scan each panel was adjusted separately as the intensity decreases exponentially with increasing $H$. Panels (c) and (g) show the same data for $Q_y>0$.

Figure 4

(a) Vortex lattice rocking curves at 0.75 T and 1.2 T for the Bragg peaks at positive $Q_y$ (upper half of detector, see Fig. 3) using a nonpolarized neutron beam. This shows the scattering intensity as a function of sample tilt angle $\phi$ relative to the rocking curve center ${\phi }_0$. Zeeman split peaks due to spin flip scattering are clearly seen, while non-spin-flip scattering at $\phi ={\phi }_0$ is not observed. For 1.2 T only the intensity around $(\phi -{\phi }_0)=0^{\circ }$ and $2^{\circ }$ was measured. Each peak is fitted by a Lorentzian. (b) Scattering geometry for the two different SF processes: Spin-up to spin-down (full lines) and spin-down to spin-up (dotted lines). The scattering angle ($2\theta =2{\phi }_0$) is the same in both cases, but different tilt angles (${\phi }_{1/2}$) are required to satisfy the Bragg condition.

Figure 5

Superconducting anisotropy. The vortex lattice anisotropy as a function of field orientation is shown separately for high (a) and low (b) fields. Lines are calculated using Eq. (3) and labeled with the corresponding value of ${\mathrm{\Gamma }}_{ac}$. The dashed lines represent a diverging ${\mathrm{\Gamma }}_{ac}$. In (a), the lines labeled ${\mathrm{\Gamma }}_{ac}=72.4$ and 92.8 are fits to Eq. (3) for the combined 1.0–1.1 T data ($\circ$ and $\square$) and 1.2–1.3 T data ($\diamond$, $▹$, and $◃$), respectively. In (b) the data points in the shaded region were averaged to obtain a measure of ${\mathrm{\Gamma }}_{ac}$. The solid lines in (a) and (b) show the previously obtained fit to Eq. (3) at intermediate fields of 0.5–0.7 T. [36] Panel (c) shows the superconducting anisotropy as a function of field from (a) and (b). The horizontal axis error bars indicate the range of fields included in the anisotropy fit. The solid line is a guide to the eye. Dashed lines indicate the upper critical field and $H_{\text{c2}}$ anisotropy in the zero temperature limit [49]. Panel (d) shows the temperature dependence of the VL anisotropy. The larger values for ${\mathrm{\Gamma }}_{\text{VL}}$ ($\diamond$) represent an average of measurements with $\mathrm{\Omega }=0.5^{\circ },0.7^{\circ }$, and $0.9^{\circ }$. Anisotropies for data with field orientation $H\perp \langle 110\rangle$ are calculated assuming a distorted square lattice, as shown in Fig. 2. The two horizontal dashed lines are the average of each data set, while the vertical dashed line is $T_{C2}(\mathrm{\Omega }=0,0.4\text{T})=1.35$ K [39]. The VL anisotropy expected for ${\mathrm{\Gamma }}_{ac}={\mathrm{\Gamma }}_{Hc2}\left(T\right)$ [49] and calculated by Eq. (3) is shown by the line ($\mathrm{\Omega }=2^{\circ }$) and shaded region ($\mathrm{\Omega }=0.5^{\circ }-0.9^{\circ }$).

Figure 6

Transverse form factor. The lines are guides to the eye for each field. Panel (a) shows high field data ($\ge 1$ T.) Panel (b) shows data for the low fields most strongly affected by the BLE. Both (a) and (b) were measured at base temperature (50 mK). Panel (c) compares data taken at 0.25 T for temperatures of 50 and 750 mK. The 50 mK data is the same as in (b). The dashed line is obtained by dividing the base temperature curve (full line) by 2.5. Panel (d) shows $h_x$ as a function of temperature. Here the $\mathit{H}\perp \langle 100\rangle$ values have been multiplied by 1.1 The vertical dashed line indicates $T_{C2}(\mathrm{\Omega }=0,0.4\text{T})=1.35$ K [39].

Figure 7

Fits to form factor temperature dependence. Data in all panels is the same as shown in Fig. 6 for $\mathit{H}\perp \langle 110\rangle$. Fits to an isotropic ($s$-wave) gap is shown in (a), both with the critical temperature as a free parameter (dashed line) or fixed to $T_{\text{c}}=1.21$ K (solid). Fits to a nodal gap is shown in (b), without (solid line) and with (dashed) nonlocal corrections. The best fit (c) is obtained for a gap with deep but nonzero minima (schematic). All fitting parameters are indicated on the plots.

Figure 8

Vortex lattice “misalignment” as defined in the text. The curves correspond to those in Fig. 6, divided by their corresponding applied fields and multiplied by a common scaling factor.