Vortex imaging and vortex lattice transitions in superconducting Sr${}_2$RuO${}_4$ single crystals

Scanning Hall probe microscopy has been used to study vortex structures in very-high-quality single crystals of the unconventional superconductor Sr${}_2$RuO${}_4$ $(T_c\cong 1.5$ K). In none of our samples do we find credible evidence for the existence of the spontaneous fields or chiral domains predicted for the expected time-reversal symmetry-breaking order parameter. Even in our highest-quality samples we observe very strong vortex pinning and anomalous broadening of vortex profiles. The best samples also exhibit a clear field-driven triangular to square vortex lattice transition at low fields, as predicted by extended London theory calculations. In stark contrast, slightly less well-ordered samples exhibit pronounced vortex chaining/banding that we tentatively attribute to an extrinsic source of disorder.

Figure 1

Real part of ac susceptibility data measured through the critical temperature of the two crystals studied here. The inset shows the imaginary part of the ac susceptibility near $T_c$.

Figure 2

(a) SHPM images captured parallel to the $a$-$b$ face of sample 1 after field cooling from above $T_c$ in the indicated applied fields ($H\parallel c$ axis). Scan size $\approx$$14\times 14\mu$m; $T=330$ mK. (b) Statistical distribution of nearest-neighbor vortex distances for the image at $H=3.5$ Oe. The solid line is a Gaussian fit to the data ($\mathrm{mean}=2.95$ $\mu$m, $\sigma =0.8$ $\mu$m). (c) Line profile across a well-isolated vortex in an image captured at $H=-0.2$ Oe (see inset). The solid line is a fit to the Clem variational model with ${\xi }_v=66$ nm, $\lambda =165$ nm, $w=600$ nm, and $h=1.26$ $\mu$m.

Figure 3

SHPM images captured parallel to the $a$-$b$ face of sample 1 after field cooling from above $T_c$ in the indicated applied fields ($H\parallel c$ axis). Scan size $\approx$$14\times 14\mu$m; $T=330$ mK. The top row shows self-correlation plots of the raw images in the bottom row (vortices are white).

Figure 4

(a) SHPM images captured parallel to the $a$-$b$ face of sample 1 after field cooling from above $T_c$ in the indicated applied fields ($H\parallel c$ axis). Scan size $\approx$$14\times 14\mu$m; $T=330$ mK. (b) The square lattice spacing measured in the $x$- and $y$-scan directions plotted versus $({\Phi }_0/{\left|B\right|)}^{0.5}$. The solid line is a linear regression fit to these data. Also shown (open square) is the lattice spacing estimated from neutron diffraction experiments at $H=50$ Oe (Ref.18).

Figure 5

Composite SHPM images captured parallel to the $a$-$b$ face of sample 2 after field cooling from above $T_c$ in applied fields ($H\parallel c$ axis) (a) $H=2.4$ Oe ($\approx$$20\times 70\mu$m) and (b) $H=15.7$ Oe ($\approx$$45\times 70\mu$m). $T=330$ mK.

Figure 6

(a) SHPM image captured parallel to the $a$-$b$ face of sample 2 after field cooling from above $T_c$ in $H=16$ Oe ($H\parallel c$ axis). Scan size $\approx$$14\times 14\mu$m; $T=330$ mK. (b) Average of the $x$-scan lines in the region indicated in (a) showing the presence of vortex chains within the bands.

Figure 7

SHPM images captured parallel to the $a$-$b$ face of the same region of sample 2 after field cooling from above $T_c$ in the indicated applied fields ($H\parallel c$ axis). Vortices are white; scan size $\approx$$14\times 14\mu$m; $T=330$ mK. Grayscale ($\Delta B$) spans 1.98, 2.13, 1.89, 2.7, 2.55, and 0.89 G, respectively.

Figure 8

SHPM images captured parallel to the $a$-$b$ face of the same region of sample 2 after field cooling from above $T_c$ in the indicated applied fields ($H\parallel c$ axis). Scan size $\approx$$14\times 14\mu$m; $T=330$ mK.

Figure 9

Temperature dependence of SHPM images captured parallel to the $a$-$b$ face of the same region of sample 2 after field cooling from above $T_c$ in $H=2.4$ Oe ($H\parallel c$ axis). Scan size $\approx$$14\times 14\mu$m.

Figure 10

Simulations of the stray field along line scans perpendicular to an infinite chiral domain boundary ($x$-axis origin is at the center of the wall) based on the fitting protocol of Bluhm (Ref.12), who took $B_0$ to be 87 G. The fields have been calculated for various choices of $B_0$ and the noise floor of our measurements ($\Delta B\approx 0.04$ G) is indicated by horizontal dashed lines.