Limits on superconductivity-related magnetization in ${\text{Sr}}_2{\text{RuO}}_4$ and ${\text{PrOs}}_4{\text{Sb}}_{12}$ from scanning SQUID microscopy

We present scanning superconducting quantum interference device microscopy data on the superconductors ${\text{Sr}}_2{\text{RuO}}_4\left(T_c=1.5\text{K}\right)$ and ${\text{PrOs}}_4{\text{Sb}}_{12}\left(T_c=1.8\text{K}\right)$. In both of these materials, superconductivity-related time-reversal symmetry-breaking fields have been observed by muon spin rotation; our aim was to visualize the structure of these fields. However, in neither ${\text{Sr}}_2{\text{RuO}}_4$ nor ${\text{PrOs}}_4{\text{Sb}}_{12}$ do we observe spontaneous superconductivity-related magnetization. In ${\text{Sr}}_2{\text{RuO}}_4$, many experimental results have been interpreted on the basis of a $p_x\pm i{p}_y$ superconducting order parameter. This order parameter is expected to give spontaneous magnetic induction at sample edges and order parameter domain walls. Supposing large domains, our data restrict domain wall and edge fields to no more than $\sim 0.1\mathrm{\%}$ and $\sim 0.2\mathrm{\%}$ of the expected magnitude, respectively. Alternatively, if the magnetization is of the expected order, the typical domain size is limited to $\sim 30\text{nm}$ for random domains or $\sim 500\text{nm}$ for periodic domains.

Figure 1

Left: photograph of the ${\text{Sr}}_2{\text{RuO}}_4$ sample scanned in this work. The surface is a cleaved $ab$ surface. The left edges are polished and the right is a growth edge. Cleave terraces visible in the magnetic scans are circled. Middle: ${\text{PrOs}}_4{\text{Sb}}_{12}$. The right edge is a growth edge. Right: front end of the SQUID used in this work. The pick-up coil (of diameter $3.2\mu \text{m}$) and contact tip, the point where the SQUID contacts the sample, are indicated.

Figure 2

Three overlapping SQUID magnetometry scans of the ${\text{Sr}}_2{\text{RuO}}_4$ crystal pictured in Fig. 1, in earth’s $\sim 1/2\text{Oe}$ field. Single vortices are clearly visible. Across the entire scan range, vortices are lined up along upper-left-to-lower-right stripes. A closeup of a single vortex is shown at the upper right; the weak tail extending leftward from the vortex is an artifact of the SQUID imaging kernel.

Figure 3

(Color) Overlapping scans of ${\text{Sr}}_2{\text{RuO}}_4$ at 0.4 K; each is a separate thermal cycle. In none of these three scans does the spontaneous edge and domain-wall magnetization expected from chiral superconductivity appear. (a) A scan over the cleave terraces, of total height $\sim 700\text{nm}$, indicated in Fig. 1a, and (lower panel) a section along the dashed line. Note the broken color scale: the peak vortex signal far exceeds all other features. The terrace signal is an artifact of SQUID-sample interaction. The vortex tail, from a vortex $\sim 50\mu \text{m}$ beyond the right edge of the scan, is an imaging kernel artifact. (b) An area with fewer terraces. The “edge shadow” is an imaging kernel artifact. Other features include dipoles (magnetic inclusions) and some terraces. (c) A scan over the sample edge and (lower panel) a section along the dashed line. The rms signal in the solid and dotted boxes, after local plane subtraction, are $0.06{\mathrm{m\Phi }}_0$ and $0.05{\mathrm{m\Phi }}_0$, respectively.

Figure 4

(a)–(c) Expected signals compared with observed data on ${\text{Sr}}_2{\text{RuO}}_4$ for (a) a sample edge, (b) an isolated domain wall, and (c), domains periodic along one direction in the $ab$ plane. The expected signals were obtained by extension of the MS results to a scan height of $1.5\mu \text{m}$, using Eq. (1), followed by averaging over a $3.2\text{−}\mu \text{m}$-diameter pick-up coil. The data in (a) and (b) is along the dashed line in Fig. 3c, and the noise level indicated in (c) is the pixel-to-pixel noise in the boxes in Fig. 3c. (d) Hypothetical $c$-axis domain structures with the magnetization direction and equivalent edge currents indicated. Taking $\left|\mathbf{M}\right|=10\text{G}$, for the cases at left and right our scans indicate $L<20\text{nm}$ and 400 nm, respectively.

Figure 5

(Color online) (a) and (b) Simulated scans of a quasiperiodic domain structure in ${\text{Sr}}_2{\text{RuO}}_4$. The MS results are extended to $z=2.0\mu \text{m}$ using Eq. (1), then averaged over a $3.2\text{−}\mu \text{m}$-diameter pick-up coil. (c) Observed signal, from Fig. 3c. (d) and (e) Observed signal plus a scale factor times the simulation. In (e), domain-wall contributions are removed.

Figure 6

(Color) (a)–(c) Magnetic scans of the ${\text{PrOs}}_4{\text{Sb}}_{12}$ sample shown in Fig. 1. All scans are at 0.4 K, under different cooling fields, $H_a$. Vortices cluster at extended defects. The indicated vortices in (b) are used for scan height determination. In (c), $H_a=19\text{mOe}$ cancels the background $z$-axis field. Four vortices remain toward the left side of the scan and a fifth in a deep trench on the right side. Note the broken color scale: the vortex signal greatly exceeds other features. Away from the vortices, surface defects appear as an artifact of SQUID-sample interaction. A shadow $\sim 70\mu \text{m}$ left of the edge is an artifact of the imaging kernel and the edge. No magnetic features appear that could explain the $\sim 1.5\text{G}$ internal induction observed by $\mu \text{SR}$. After local plane subtraction, the rms signal in the solid box is $11{\mathrm{m\Phi }}_0$, and in the dotted box, $10{\mathrm{m\Phi }}_0$. (d) Sections along the solid and dashed lines in (c).

Figure 7

(Color) Scans of the same area of ${\text{PrOs}}_4{\text{Sb}}_{12}$, with the SQUID bias current reversed between the two scans. Features resulting from SQUID-sample interaction change sign while features resulting from static magnetic inductions do not.