Probing chiral superconductivity in Sr$_{2}$RuO$_{4}$ underneath the surface by point contact measurements

Sr$_{2}$RuO$_{4}$ (SRO) is the prime candidate for chiral $p$-wave superconductor with critical temperature $T_{c}(SRO)\sim$1.5 K. Chiral domains with opposite chiralities $p_{x}\pm ip_{y}$ were proposed, but yet to be confirmed. We measure the field dependence of the point contact (PC) resistance between a tungsten tip and the SRO-Ru eutectic crystal, where micrometer-sized Ru inclusions are embedded in SRO with atomic sharp interface. Ruthenium is an $s$-wave superconductor with $T_{c}(Ru)\sim$0.5 K, flux pinned near the Ru inclusions can suppress its superconductivity as reflected from the PC resistance and spectra. This flux pinning effect is originated from SRO \textit{underneath} the surface and is very strong. To fully remove it, one has to thermal cycle the sample above $T_{c}(SRO)$. This resembles the thermal demagnetization for a ferromagnet, where ferromagnetic domains are randomized above its Curie temperature. Another way is by applying alternating fields with decreasing amplitude, resembling field demagnetization for the ferromagnet. The observed hysteresis in magnetoresistance can be explained by domain dynamics, providing support for the existence of chiral domains. The origin of strong pinning \textit{underneath} the surface is also discussed.

Results from experiments on muon spin relaxation [6] and the polar Kerr effect [7] suggest that the superconducting order parameter (OP) breaks the time-reversal symmetry and forms chiral domains with two different chiralities (  p p i x y ), similar to the domains in a ferromagnet. The existence of such chiral domains has not been conclusively confirmed, although there is indirect evidence: domain wall pinning was assumed in order to interpret the strong flux pinning (zero flux creep at lower temperatures) in measurements of bulk magnetization relaxation [8][9][10]; and domain dynamics was also assumed to explain the field modulation of critical currents for corner junctions [11]. However, there is no direct evidence of the chiral domains. For example, an edge current around domain walls and sample edges, which should lead to measurable magnetic fields, has not been observed by local field imaging methods on etched mesoscopic disks [12][13][14][15], nor by early micro-Hall probe studies near the edge of an SRO crystal [16].

Experimental methods
Rod-like Ru inclusions on the surface of the cleaved SRO crystal can be seen clearly in the optical microscope image in figure 1(a), and more details are revealed in the scanning electron microscope image in figure 1(b), where surface degradation can be observed for the crystal that was exposed in ambient conditions for an extended period of time (water and carbon dioxide may react with SRO). Before the PC experiment, the surface is scratched with a ceramic knife to expose a relatively fresh surface.
The SRO single crystals are grown by floating-zone methods with Ru as self-flux [29] and with an excess amount of Ru, so that a eutectic phase is formed with embedded lamellar inclusions of Ru [24]. The resulting SRO-Ru crystal has an extended critical temperature for the superconducting transition (T c ) from the intrinsic 1.5 K to about 3 K, and this enhancement in T c is believed to originate from the interface between Ru inclusions and SRO, possibly due to lattice distortions and strain although the exact mechanism is not confirmed [24,[26][27][28]. In fact, it has been found that uniaxial pressure on pure SRO can enhance SC as well [22,[30][31][32]. Regarding the symmetry of the OP for this enhanced SC, a non-monotonic temperature dependence of the critical current (kinks near 1.5 K) and critical current switchings in so-called topological junctions suggested that the OP symmetry is different from that of pure SRO [33,34].
In figure 1(c) a schematic of the tungsten tip and the eutectic SRO-Ru crystal is shown. The point contact is made between a tungsten tip and the SRO crystal, which is fixed on a silicon chip and mounted on the attoCube nanopositioner stack. The tip and the nanopositioner stack are both secured on a metal housing that is suspended by springs from the end of a cold-insertable probe for a Leiden cryogen-free dilution fridge. The base temperature of the probe at the mixing chamber stage is about 0.1 K (12 mK) with (without) the point contact setup attached, while the base temperature for the sample stage is higher (∼0.3 K) due to thermal loads from the wirings for the sample and nanopositioners, as well as Joule heating during the measurements. Differential resistance (dV/dI) is measured with a standard lock-in technique with home-made battery-powered electronics to reduce external noise. For more details please refer to our previous study on pure SRO [22].
Whether the surface property correlates with that of the bulk underneath the surface is always a concern for surface probes. This is particularly critical for SRO since its surface undergoes reconstruction after cleaving and the SC may be destroyed; also, surface contamination may result in a dead layer. One way to circumvent this problem is to make PCs with hard tips, so the surface layer may be penetrated by the tip [35,36]. Using hard tungsten tips and our home-built point contact setup, we previously obtained a reproducible result for the SC gap (∼0.2 mV) for pure SRO [22], consistent with that estimated by weak coupling theory.
Since Ru is much softer than SRO and has no dead layer on it, it is not necessary to push the tip to penetrate the surface layer in order to make PCs on Ru inclusions, as plotted schematically in figure 1(d). When the tip is sharp and can penetrate the surface dead layer at the beginning of the experiment, the interface can be made directly between the tungsten tip and the SRO, with Ru inclusions nearby (this type is referred to as W/SRO-Ru). In figure 1(c) we also plot a cross-sectional view of the PC and the distance dependence of the superconducting OPs for SC in Ru and SRO, without considering intermixing of the two OPs. Results for a PC of this kind showing both OPs, labelled as PC-1, are presented later in figure 4 due to the complexity. Later in the experiment, the tip gets flattened and the interface is normally between the flattened tip and the bulging Ru inclusions (W/Ru-SRO), as shown in figures 1(d) and (e). Measurement results for two such PCs, labelled PC-2 (in the same run as PC-1) and PC-3 (in a later run), are presented in figures 2 and 3 respectively. For PC-1 we focus on the hysteresis of magnetoresistance (MR) and order parameters shown in point contact spectra; for PC-2 and PC-3 we focus on how to remove the pinned flux by processes similar to thermal demagnetization and field demagnetization for a ferromagnet (see figure 1(f)).
In previous PC studies, flux pinning in a conventional superconductor was only considered in a few cases [37,38]. MR hysteresis observed here was not reported in those previous PC measurements. It was usually assumed that making the point contact causes some damage and locally the SC is suppressed, thus vortices are trapped near the PC since the energy cost is lower. However, this is irrelevant here since for PCs on Ru inclusions we find that the origin of the flux pinning is not at the PC interface but within the SRO, i.e., flux pinning is not directly affected by any damage caused by making point contacts on Ru inclusions.

Results
For simplicity, we first describe the results for PC-2 and PC-3 where the tip gets flattened and the interface is normally between the flattened tip and the bulging Ru inclusions (W/Ru-SRO). The flux pinning effect and the difference between zero-field cooling (ZFC) and field cooling (FC) are presented, then methods to remove the flux pinning effect are introduced and compared with those in ferromagnetism.

PC-2 and PC-3, on Ru inclusions
During ZFC, as shown in figure 2(c) for PC-2 (figure 3(a) for PC-3), there is only one quick drop around 0.44 K (0.42 K), which is due to the superconductivity in Ru. The deviation from( ) T Ru 0.5 c K could be due to uncertainty of the PC temperature: the local temperature at the point contact may be different from that of the substrate, on which the thermometer is attached near the sample but not in direct contact. x y , is shown by the clockwise or counterclockwise arrows, and is not fully developed for zero-field cooling (ZFC). With randomized chirality, the effective M is close to zero. (e) For field cooling, as well as for ZFC but with field history, the polarization of the chiral domains is induced and remains when the field is reduced to zero. The magnetization states in (d) and (e) are indicated by black dots in the hysteresis loops in (f). Also in (f), we show (i) the initial magnetization curve (in green), which can be used for figure 2(e), (ii) the reversal curve (in brown) when we restart the field sweep routine at zero field in the reverse direction of the hysteresis loop, which can be used for figure 4(e), and (iii) the field demagnetization curve for figure 3(d). The two horizontal dashed lines in blue denote the critical fields (H c ) for Ru inclusions, within which the PC resistance shows a dip due to the recovered superconductivity.
For PC-2, conductance enhancement at small bias corresponds to SC in Ru, which can be suppressed by raising the temperature above 0.5 K or by ramping the field above 200 Oe (see figures 2(a) and (b), as well as the illustration of initial magnetization in figure 1(f)), suggesting that this conductance enhancement is indeed sustained by SC in Ru inclusions. Note that after the field is ramped from 200 Oe back to zero, the conductance enhancement is still suppressed, as shown in the inset of figure 2(b), suggesting remnant flux or flux pinning in SRO (see figures 1(e) and (f)).
Alternatively, flux pinning is demonstrated by the hysteresis behaviour in MR at zero bias, as shown in figure 2(e). This hysteresis reminds us of the magnetization curve of a ferromagnet (see, e.g., the textbook [39], especially chapter 16 where the same terminology can be used for ferromagnetism and superconductivity).  Oe. The inset shows the difference in the PC spectra without field history (blue) and those with field history (green), both at T=0.38 K and zero field. (c) Thermal demagnetization: after SC was suppressed by ramping H ⊥ at low temperature, the sample was warmed at zero field to different cycling temperatures T cycle , then zero-field cooled. T c is shifted with different T cycle . Inset: R(T) curves up to 1.5 K. Another feature of the MR curves is that after ZFC there is a small drop in resistance at around 100 Oe and then a quick increase to the normal state at a field close to 200 Oe, indicating first enhancement and then suppression of SC in Ru. The initial enhancement with increasing field could be due to some proximity effect from SRO, which may reduce the flux in the Ru ( figure 1(d)), or is possibly related to the spontaneous flux proposed at the interface between an s-wave superconductor and a p-wave superconductor [25]. By further increasing the field close to H coer , the chiral domains get polarized since each chirality prefers a certain polarity of the field, and the resulting large M leads to total suppression of SC in Ru inclusions, even after the field is reduced to zero (figure 1(e)).
With a small field sweeping amplitude the dips in resistance near H coer are observed, as shown in figure 2(e) by the green line from 0.2 to -0.2 kOe, and the red line from -0.2 to 0.5 kOe. But with increasing field sweeping amplitude, the resistance dips were not observed for PC-2. This is probably because the growth of chiral domains is affected by the rate and amplitude of the sweeping field, and the bigger domains (or stronger H with increasing field amplitude at 0.35 K, after ZFC. After the field is ramped to 3.7 kOe, the superconducting state at zero field is suppressed. For clarity a shift of +0.2 Ω has been made for consecutive curves in (c) and (f). pinning force) induced by higher fields cause a switching of the polarization that is too fast to resolve (the magnetization curve is still there but the probe is too slow to follow). Disappearance of the dip was also observed for another two PCs in the same run; see supplementary figure S2. Above H coer there is almost no change in resistance (in figure 2(e)), indicating no influence of SRO on PC resistance.
Resemblance to thermal demagnetization. We further describe how such flux pinning can be removed. It was proposed that chiral domains behave like ferromagnetic domains, thus it is natural to consider how the polarization (magnetization) of the chiral (ferromagnetic) domains may be randomized. A ferromagnet can be demagnetized by thermal or cyclic field methods [39]. For thermal demagnetization, the ferromagnetic domains are randomized after the sample is heated above its Curie point and cooled in the absence of a field.
Similarly, the suppressed SC at lower temperatures (due to a large M, see figures 1(e) and (f)) can indeed be recovered by thermal cycling to( ) T SRO 1.5 c K and then cooling in the absence of a field. As shown in  is still suppressed. This proves that the suppression of SC in Ru is due to remnant magnetization (chiral domain polarization) in SRO. T cycle may not reach the enhanced T c for the 3 K phase for the following possible reasons: firstly, the volume of the 3 K phase with enhanced T c is small and thus has a negligible influence on the PC; secondly the OP of the 3 K phase may not be chiral p-wave [24] so there is no intrinsic vortex pinning mechanism.
This flux pinning effect is very symmetric to external fields. Starting from the same T cycle of 1.5 K, cooling with opposite fields (± 50 Oe, ± 75 Oe, ± 100 Oe etc, as shown in figure 2(d)) gives similar R(T) curves. These FC curves are also similar to those with different T cycle (compare figure 2(c)). Such a resemblance corroborates the existence of remnant magnetization when < T 1.5 cycle K. Resemblance to field demagnetization. The other option to revert to the magnetic 'virgin' state for the ferromagnet is to follow the field demagnetization procedure [39], by applying alternating fields of decreasing amplitude, which makes the domains smaller and/or randomly aligned. For PC-2, we only tried with increasing amplitude of sweeping fields. Later, in another run with the same crystal, we made PCs on Ru inclusions similar to PC-2, and we label one of them as PC-3. We start from large alternating fields and then reduce the amplitude; also we minimize the waiting time between the measurements of two consecutive data points, and reduce the step of field ramping, so the resistance drop is not too fast to register. Then the dips in resistance can be observed repeatedly, as shown in figure 3(c). The dip position now changed to around±270 Oe, higher than±200 Oe for PC-1 and PC-2, and the dip is much narrower, suggesting faster domain dynamics (narrow resistance dips were also reported for a magnetic insulator, reflecting magnetic domain dynamics [40]). Moreover, when the amplitude is further decreased from  -  - 0.4 0.3 0.2 0.1 0 kOe, as shown in figure 3(d), the resistance decreases and reaches a similar value to that at the bottom of the resistance dip near H coer , suggesting that the local remnant field M is minimized. As illustrated in figure 1(f), here there are no smaller loops (this is different from the analogy of a ferromagnet) and the only field to observe resistance dips is near H coer .
For comparison, thermal demagnetizaton and field cooling with different fields are also shown for PC-3 in figures 3(a) and (b) respectively. Similar to that for PC-2 (figures 2(c) and (d)), here the remnant magnetization can be completely removed after thermal cycling to 1.5 K, and for field cooling there is excellent symmetry for both field polarities.
Parallel field MR. The parallel field MR data are less well understood, but still are briefly presented here for completeness (additional results of tilted field and in-plane anisotropic field MR are not included here, for simplicity). SRO is a layered superconductor with very different in-plane( ) is not expected to affect SC in SRO. However, for PC-2 on Ru inclusions (W/Ru-SRO) we do observe resistance dips in the parallel field MR (see figure 2(f)), with even smaller H coer , and the dip is also broader, about 200 Oe. We note that this is larger than H c of the elemental superconductor Ru, which is about 25 Oe at 0.4 K (70 Oe at zero temperature) [41,42]. Another difference compared with perpendicular field MR is that with smaller amplitudes of the sweeping field the resistance dip appears on the opposite side after sweeping across zero field (to depolarize the remnant field at H coer ), whereas with larger sweeping fields the resistance dip appears on the same side towards zero field, e.g., the dip appears near −40 Oe for - 0.7 0.7 kOe, and at around 60 Oe for  -0. 7 1.4 kOe. Parallel field MR for PC-is also shown in figures 3(e) and (f) for decreasing and increasing amplitude respectively. The resistance minimum near zero field gets lower for consecutive sweeps with decreasing amplitude, which bears some similarity to the MR with H ⊥ (figure 3(d)). For field sweeping with increasing amplitude, similar to that for PC-2 (figure 2(f)), at smaller field amplitudes the resistance minimum appears on the opposite side after crossing zero field, whereas at larger amplitude it moves towards zero field. Additionally, for field cooling, || H has a similar but smaller suppression effect on SC of Ru inclusions (as shown in supplementary figure S3). One may think that there is finite H ⊥ component at the interface due to the inclined Ru/SRO interface, since the Ru inclusions are not aligned with any crystal orientation.

PC-1, near the SRO-Ru interface
For PC-1, the sharp tip penetrates through the dead layer (see figure 1(c) for illustration), so SC in both SRO and Ru may be probed. The SC in SRO is indicated by the gradual drop in resistance starting from about 2.3 K to 0.57 K ( figure 4(a)), and the quick drop below 0.57 K is due to SC of Ru, so the contact type is W/SRO-Ru. Note that the SRO/Ru interface does not contribute to the PC resistance, since the interface resistance is usually in the mΩ range (the interface area is usually of the order of 10 mm 2 [33,34]), much smaller than the PC resistance.
With two OPs involved, we expect to see something different in the point contact spectra at finite bias. As can be inferred from the temperature and field dependence of the point contact spectra (figures 4(b)-(d)), the dips in conductance at around±0.5 mV reveal information on the OP of SRO (see supplementary figures S4 and S5 for a similar feature for PC-2 and PC-3). Thedips in conductance at ±0.5 mV and the broad zero-bias conductance hump are suppressed when temperature is raised to 2.5 K or the out-of-plane field (H ⊥ ) is increased to 10 kOe. This is in striking contrast to the case for PC-2, where a similar feature is suppressed at 0.45 K and 200 Oe (the local field becomes larger than ( ) H Ru c ). This suggests that for W/Ru-SRO type contacts, (PC-2 and PC-3) while the PC spectra are influenced by SC in SRO (large gap feature), this feature is still sustained by the conventional SC in Ru. Note that the gap value for elemental Ru is about 0.07 mV using the mean-field estimation with T c = 0.5 K, which is much smaller than 0.5 mV observed here.
The most noticeable feature in addition to the spectra of PC-2 is a deflecting point at around±0.125 mV, as marked by the blue vertical dashed line, for the blue curve in figure 4(d). This feature is observed for the PC spectra of PC-1 at zero field with field history (finite M). When there is no field history (ZFC), the spectrum is different. As shown by the green curve in figure 4(d), an additional conductance dip evolves at±0.185 mV, as marked by the green vertical dashed line. The highest conductance without field history (6.47 Ω) is close to that shown in the ZFC R(T) curve (6.5 Ω, figure 4(a)), as this differential conductance was measured right after ZFC. After field ramping, the highest conductance is reduced (7.06 Ω), and the double dips at around±0.185 mV and±0.5 mV reduce to dips at±0.5 mV only, indicating that the OP in Ru is suppressed. So this is again consistent with Ru inclusions serving as a local probe for M.
Hysteresis in magnetoresistance. For PC-1 there is clearly a MR hysteresis in the perpendicular applied field ( figure 4(e)), similar to what we observed previously for PC on pure SRO [22]. The difference is that here there are additional dips in resistance at  H 200 coer Oe due to Ru inclusions, instead of a rounded valley for pure SRO (see supplementary figure S1 for comparison; for pure SRO, there are even Barkhausen-type jumps similar to the real magnetic domain dynamics). Such similar hysteresis confirms that the strong flux pinning is not due to the additional Ru inclusions but originates from SRO itself.
The presence of the resistance dip depends on the field history. Here is a good example: if the sweeping direction is reversed at zero field, as can be inferred from the hysteresis loops in figure 1(f), then there is no M = 0 point in that positive ramping direction until the field sweeps back again and crosses zero. As shown in figure 4(e), the field sweep follows the same routine,   -  0 1 1 1 0. So the last sweep is  1 0 kOe. Now we restart the field sweep at zero field,  0 1 kOe, opposite to the last sweeping direction; then there is no resistance dip at 200 Oe. When we sweep downward,  -1 1 kOe, the resistance dip appears at −200 Oe, both consistent with our model.
The lowest resistance at the position of the resistance dip is around 6.9 Ω, still larger than the resistance observed during ZFC (6.5 Ω, see figure 4(a)). This may be understood if the suppression of SC in Ru is only partially reduced. One possible scenario is that, although the total M for multiple domains is zero at H coer , there could locally be inhomogeneous flux where local M is nonzero. For ZFC, domains were not magnetized (or not trained, as indicated by the dashed arrows in figure 1(d)) so there is no net flux.
For parallel field MR, as shown in figure 4(f), the data are less well understood than in the case of PC-2 and PC-3. There is no hysteresis, nor resistance dips at H coer . Instead, the resistance shows a broad minimum near zero field, and it is even possible to induce a sharp resistance drop near zero field. The broad minimum is consistent with the field dependence of a conventional superconductor. But if compared with the resistance in the perpendicular field MR (figure 4(f)), this change in resistance mostly corresponds to that for SRO, and only the sudden drop in resistance may be related to the Ru inclusion. In the scenario of domain dynamics, this may be interpreted as the chiral domains with out-of-plane polarization being randomized by || H .

Discussion
Clearly there is strong vortex pinning in SRO, but its origin is still not certain. There are a few theoretical proposals available to understand the vortex state in SRO. First, Sigrist and Agterberg studied the role of chiral domain walls in the vortex creep dynamics [43], which was used to explain the zero flux creep observed by Mota's group [8][9][10]. In this picture the domain walls are pinned at impurities and lattice defects so they do not move easily (this is how the domain picture in figures 1(d) and (e) is derived). Second, Garaud et al [44] consider SRO as a type-1.5 superconductor with long-range attractive and short-range repulsive intervortex interaction. This is used to explain the vortex coalescence observed by scanning Hall probe microscopy [14] and possible clusters of vortices nucleating within a Meissner-like state implied by muon spin rotation (μSR) measurements [45]. Third, Ichioka et al [46] used the time-dependent Ginzburg-Landau theory to study the magnetization process and found that with increasing magnetic fields, the domain walls move so that the unstable domains shrink to vanishing size, and the single-domain structure is realized at higher fields. Along these lines there are theories of doubly quantized vortices and other exotic behaviours that may lead to a broken field with nonzero chirality degeneracy [47,48]. Note that compared to the first proposal, chiral domain wall pinning is not emphasized, which to some extent suggests pinning by domains themselves and is probably more relevant to our observations here.
The proposal of chiral domain wall pinning was developed to understand the systematic experimental results on bulk magnetization relaxation obtained by Mota's group [8][9][10], where a novel strong flux pinning (even zero flux creep at the lowest temperatures) was found, and the higher the cycling magnetic field, the stronger the pinning effect. This is considered as indirect evidence of chiral domains. In our work the MR hysteresis also suggests strong pinning, but there are a few differences: (1) The previous scenario of chiral domain wall pinning seems inconsistent with the cycling field effect, since with higher cycling field, 'polarization' of the chiral domains in SRO is enhanced and domain walls are reduced, resulting in less pinning. Here we propose that the domains themselves can provide strong pinning (once they are formed with field history), and we compare the chiral domain dynamics to that of a ferromagnet; by assuming a local 'magnetization' (M) due to chirality polarization, the H coer is a natural explanation for the necessity of a high cycling field. (2) Previously the relaxation could not be measured at zero field, but here we can measure the point contact spectra in the ZFC situation, which probes the 'virgin' state without magnetization. (3) Previously the focus was the regime of zero flux creep at the lowest temperatures (50 mK and lower), while here the focus is the domain dynamics at higher temperatures (but still much lower than ( ) T SRO c ). (4) Previously, strong pinning for both || H ab andĤ ab was observed, but here only strong hysteresis forĤ ab is observed. The chiral domain walls should exist only in the ab plane if the 2d γ band is the active superconducting band, thus it is not clear whether the strong pinning for || H ab is due to the same mechanism. A further experiment, e.g., a study of ac susceptibility on the in-plane metastable vortex state [49], maybe helpful to investigate this issue.
The second model proposed to understand vortex clustering on the SRO surface is type-1.5 superconductivity, which was first named after the observation of vortex clustering on the surface of MgB 2 [50,51], a two-band superconductor that has two weakly coupled order parameters with k < 1 2 1 and k > 1 2 2 . In fact, for single-band superconductors with k~1 2, there was also such a long-range attractive force [52]. For SRO, the in-plane k ab = 2.3 and out-of-plane k c = 46 are both in the type II regime [2]. So to apply this theoretical model, Garaud et al [44] assume there are several coherence lengths in multicomponent superconductors [44], and find that type-1.5 behavior can occur in multiband chiral Ginzburg-Landau theories for SRO. This may explain the clustering of vortices imaged by a scanning field probe at low fields [13,14] and the bulk Meissner-like state implied by the μSR measurements [45]. However, type-1.5 superconductivity alone cannot explain the MR hysteresis observed here for several reasons: (1) Similar MR hysteresis has not been found in the point contact measurements for MgB 2 [53]. (2) The difference between types 1.5 and 2 is usually in the low-field region, but here the temperature and field ranges are outside the typical regions for Ginzburg-Landau theories. The dynamics here may involve only fully penetrated vortex domains instead of vortex domains mixed with Meissner-like domains in the low-field regime. (3) For a system close to type I, one does not expect to see hysteresis in M(H).
The last model is pinning by chiral domains themselves. This model can explain the striking similarity to the ferromagnetism here. After ZFC, the applied field leads to the formation of chiral domains similar to ferromagnetic domains, which themselves become high-energy barriers for flux, instead of resorting to domain wall pinning. This is also consistent with the μSR experiment, in which a large fraction of the volume is vortexfree until the field is ramped to above 100 Oe. By assigning the quite large observed H coer to be the flip field for chiral domains, we have to abandon the previous belief that chiral domains flip easily [16,54]. In fact, H coer is much larger than H c1 measured by local magnetization hysteresis loops with a Hall probe [16], and is close to the thermal dynamic critical field [2].
The proposed chiral domains should not be mixed with conventional vortex domains, since it is possible to push conventional vortices of different vorticity into the chiral domains with a preferred vorticity but with a different energy cost. And the strong vortex pinning by chiral domains is absent for conventional vortex domain pinning by domain walls. From the MR results here, there is another feature that probably points to unconventional vortex pinning, i.e., for regular vortex pinning the MR usually does not show exact symmetry with respect to zero field [11], but here it does. The scenario of chiral domain wall pinning was also suggested for UPt 3 [10], but in later experiments it seems that a single domain without domain walls was inferred [55,56]. This can be explained if pinning is by domains themselves, and not by domain walls. There was also a disparity regarding the size of chiral domains [57], which was estimated to be around 100 μm or larger in measurements of the polar Kerr effect [7], and about 1 μm in measurements of the critical current for corner junctions [11]. This can be reconciled if the domain size is determined by the internal defects, which are very sample-specific because of sample growth parameters, and then it may also be determined by the alignment in a multiple domain assembly, as drawn in the schematic illustration in figure 1.
As an additional note, it seems difficult to distinguish by the MR hysteresis alone the effect of strong flux pinning from the possible coexistence of ferromagnetism and superconductivity; the latter was proposed for the interface superconductivity between LaAlO 3 and SrTiO 3 [58], also an intriguing subject. From the aspect of the breaking of time-reversal symmetry, the difference between ferromagnetism and equal-spin triplet pairing is probably that the former has a static ferromagnetic order parameter while the latter does not.

Conclusion
By considering the lamellar inclusions of Ru embedded in single-crystal SRO as a local magnetization sensor, we found a new method to probe the local flux underneath the surface. The observed strong MR hysteresis with applied field perpendicular to the ab plane, and various field dependences (thermal demagnetization, field demagnetization, coercive field etc) indicate a striking similarity to ferromagnetic domains. Such a similarity provides indirect evidence of chiral domains and domain dynamics. We also discussed possible intrinsic pinning mechanisms, including chiral domain wall pinning [43] and type-1.5 superconductivity [50,51], though both seem to have some difficulties in explaining the hysteresis. One remaining proposal is pinning by chiral domains themselves, which is new and needs further investigation. Besides the zero-bias point contact resistance, the point contact spectra at finite bias manifest the order parameters of both Ru and SRO, and thus might be helpful for understanding the interaction between order parameters of different symmetries, although more investigation is needed. Additional experimental investigation in this direction includes possibly scanning point contact measurement to check the proximity effect near Ru inclusions, and PCs with ferromagnetic or s-wave superconducting tips.