Magnetic domain engineering in SrRuO 3 thin ﬁ lms

Magnetic domain engineering in ferromagnetic thin ﬁ lms is a very important route toward the rational design of spintronics and memory devices. Although the magnetic domain formation has been extensively studied, arti ﬁ cial control of magnetic domain remains challenging. Here, we present the control of magnetic domain formation in paradigmatic SrRuO 3 /SrTiO 3 heterostructures via structural domain engineering. The formation of structural twin domains in SrRuO 3 ﬁ lms can be well controlled by breaking the SrTiO 3 substrate symmetry through engineering miscut direction. The combination of x-ray diffraction analysis of structural twin domains and magnetic imaging of reversal process demonstrates a one-to-one correspondence between structural domains and magnetic domains, which results in multi-step magnetization switching and anomalous Hall effect in ﬁ lms with twin domains. Our work sheds light on the control of the magnetic domain formation via structural domain engineering, which will pave a path toward desired properties and devices applications.


INTRODUCTION
A wide range of perovskite oxide heterostructures provide a fertile playground for discovering emergent properties owing to their diverse capabilities to tune delicate coupling between spin, charge, and orbital degrees of freedom [1][2][3] . The two-dimensional electron/hole gas at LaAlO 3 /SrTiO 3 interface 4,5 , polar skyrmions in PbTiO 3 /SrTiO 3 superlattice 6 , chiral spin fluctuation in 2D SrRuO 3 ferromagnet, 7 and the newly discovered infinite layer nickelate superconductor 8 are just several remarkable examples out of many. Engineering the lattice structure through interfacial mismatch of lattice constant [9][10][11] and symmetry [12][13][14] is one key route toward desired electronic phases and functionalities. The key feature of the lattice of ABO 3 perovskite is the corners shared three-dimensional network of BO 6 octahedra where the lattice symmetry of a perovskite resides in. The rotation mode (in-phase, out-of-phase) and rotation amplitude of octahedra about the principle axes determine the lattice distortion and symmetry 15 . No matter what kinds of rotation patterns, they all share quite similar pseudocubic lattice with B site sitting in the center of oxygen octahedra and A site at the hole surrounded by eight octahedra to form ABO 3 stoichiometry. This structure character leads to easy and common formation of twin domains [16][17][18] . Furthermore, mismatch of rotation mode and rotation amplitude, which is widely observed in many heterointerfaces, can also result in the formation of twin domains 19,20 . Therefore, understanding the effect of twin domains is very crucial for perovskite heterostructures.
A lot of previous works have demonstrated the significant impact of domain walls on physical properties, such as local enhanced mobility of electron gas at LaAlO 3 /SrTiO 3 interface by SrTiO 3 (STO) tetragonal domain wall 16,21 and conductive ferroelectric domain walls in BiFeO 3 films 22,23 . Given that many perovskites exhibit crystalline anisotropy, e.g., orthorhombic perovskite Pbnm 11,12,24 , different domains in a film can exhibit quite different behaviors. As a result, a combined behavior from different individual domains could lead to complex properties. However, it is challenging to identify the behavior of individual domains with microscopic size and to clarify their impacts on the global properties. How the individual domains affect the film properties remains an open question and requires further investigation.
In this letter, we explore the interplay between structural twin domains and transport/magnetic properties in epitaxial SrRuO 3 films grown on (001) SrTiO 3 . The SrRuO 3 (SRO) is an important ferromagnetic metal widely used as oxide electrode materials in functional oxide heterostructures 24 . The recent discoveries of topological Hall effect [25][26][27] , chiral spin fluctuation 7 and skyrmion 27,28 bring the SRO back to the spotlight of condensed matter research. The SRO possesses orthorhombic structure (Pbnm) with a = 5.56 Å, b = 5.53 Å, and c = 7.84 Å 29 . When SRO is epitaxially grown on (001) STO, the c-axis can lie either in-plane (along one of the two principal directions of STO) or out-of-plane, leading to six possible orthorhombic domains as shown in Fig. 1a. By controlling the formation of twin domains through breaking the STO fourfold symmetry via exposing either (010) or (100) facets at substrate step edge (see Fig. 1b), we found that the transport and magnetic properties are strongly influenced by domain structures. Different structural domains within the SRO films have different magnetization reversal behaviors, resulting in multiple-step anomalous Hall effect (AHE). By geometric control of the number of different types of magnetic domains in SRO films through tuning substrate terrace directions, the number of the plateaus in magnetization and AHE can be manipulated. These intriguing phenomena were directly visualized using cryogenic magnetic force microscopy (MFM). Moreover, domain formation affects the sign of magnetoresistance. Previous study shows that stripe-like domains induced by controlled thickness inhomogeneity along terrace edge can cause two-channel AHE 30 . Our observation of terrace direction controlled structural inhomogeneity and thus multi-domain switching then further sheds the light on the crucial role of domain inhomogeneity in SRO transport and magnetic properties. What is more, the rational design of domain structures in perovskite heterostructures by growth symmetry breaking provides a smart route for properties and functionalities engineering.

Sample preparation
The SRO films were epitaxially grown on TiO 2 -terminated (001) STO substrates using pulsed laser deposition (PLD). The substrates were atomic flat with visible terraces and the terrace edges were unidirectional (see Supplementary Fig. 1). The miscut angles of STO substrates varied from 0.06 to 0.16°. The STO substrates had different miscut direction (φ), which is defined as the angle between terrace edges normal and [100]. A high oxygen partial pressure of 0.25 mBar was used for growth in order to minimize the oxygen off-stoichiometry and to favor a bulk like structure 31 . The growth of SRO films was in situ monitored by reflection highenergy electron diffraction (RHEED) and exhibited layer by layer growth mode at first few layers followed by step flow growth.
Due to the terrace structure where either (100) or (010) facets are exposed, different SRO orthorhombic domains have different characteristic vertical interfaces with STO at step edge. As a result, the growth symmetry is broken and a specific domain is more favored. The structures of vertical interface between SRO domains and STO are summarized in Fig. 1b . Similar situation occurs for both Z x and Z y -domains, which have misaligned principal axes with that of STO at (001) interface. For Z x -domain, it has one more mismatch of inclined crystal plane with STO (100) plane. The different types of mismatch between the lattice of SRO domains and STO (inclined crystal planes angle vs. inclined crystal axes angle) break the growth symmetry, which shows that X-domain is more favored as illustrated below.
Miscut direction dependent domain structure Impacts of terrace edge on domain formation are first examined by growing an SRO film on unidirectional terrace nearly along [010] directions (φ ≈ 15°) as shown in Fig. 1c. For simplicity, SRO film with terrace direction φ is denoted as SRO-φ and the 15°u nidirectional terrace SRO sample is then denoted as SRO-15. The thickness of this film is 75 unit cell (u.c.). The orthorhombic domains were characterized by x-ray reciprocal space maps (RSMs) around (204) reflection of SRO film by azimuthally rotating the sample by 90°with respect to the surface's normal as shown in Fig. 1d. φ = 0°corresponds to [100]. The crystalline axes are intentionally defined in a way that the terrace ramping down direction is located in the first quadrant of coordination plane, which then allows us to identify the domain types and their relationship to terrace structures (see Supplementary Fig. 3). Due to the orthorhombic structure, six domains exhibit different diffraction patterns (see Supplementary Fig. 4). The downshift of (204) and upshift of (−204) with respect to (0 ± 24) peaks as shown in Fig. 1d suggest that the domain is X type. There are no visible impurity peaks, indicating single X-domain in SRO-15, consistent with the previous reports [32][33][34][35] . In sharp contrast, evident two groups of reflection peaks from two domains (X and Y) are observed for φ ≈ 35°SRO film with the same thickness of 75 u.c., where both (100) and (010) facets are exposed and comparable due to serrated step edges (see Fig. 1e, f). Using their peak intensities, the volume ratio of X to Y-domain is estimated to be 2:1. For Y-domain, it is observed that (024) peak shifts rightward and (0-24) peak shifts leftward, suggesting that the unit cell of Y-domain tilts away from STO (001) plane, forming an inclined angle between Y-domain and STO (001) plane. This inclined angle is determined to be 0.02°(more details, see supplementary It is worth to note that the intensity of (44 ± 4) o is only half of (620) o or (260) o for X-domain, hence the Z-domain has a very strong contribution to total peak intensity and its volume fraction is significant. The existence of Z-domain in this film is also indicated by transport and MFM measurements as we demonstrated in the following.
Impact of domain structure on anomalous Hall effect Impacts of the domain formation on properties are demonstrated in Fig. 2. For SRO-15 with a single X-domain, the AHE shows a typical square-like hysteresis loop and a sharp sign change at coercive field (see Fig. 2a). In sharp contrast, the SRO-35 film with twin domains shows multiple switching, leading to three Hall plateaus as indicated by green arrows in Fig. 2b. The anomalous sign change of the Hall resistance with reducing temperature (e.g., 50 K vs. 15 K in Fig. 2b), which is commonly seen in SrRuO 3 , is thought to arise from temperature-dependent Berry curvature in k-space driven by the change of band structure and magnetization 24,36 . The single switching vs. multi-switching behaviors are better illustrated by the first derivative of Hall resistance as shown in Fig. 2c, d. For SRO-15, only a single peak at each side of the magnetic field (positive field and negative field) is observed, whereas there are three peaks exhibited by SRO-35 at each side.
Such miscut direction controlled switching in AHE is also observed in much thicker films, e.g., 250 u.c. (see Supplementary Fig. 7), indicating the long-range impact of step edge on domain formation.

Visualization of magnetic domains by MFM
In order to further reveal the relationship between the structural domains and magnetic domains, we performed the cryogenic MFM and in situ transport measurements in a Hall-bar geometry at various magnetic fields. Since the anisotropic field of SRO is extremely large, e.g., 12 T, the dipole-dipole interaction, which determines the magnetic domain structures in low-anisotropy materials, such as martensites 37 , can be negligible in SRO. Therefore, the structurally pinned magnetic domains would follow local anisotropy and the magnetic domains can be mapped to structural domains. The single switching of SRO-15 sample was confirmed by MFM. The switching process was so sharp that it was very challenging for MFM technique to capture the magnetic domain boundaries. Occasionally, the long domain boundary (>20 μm) sweeping across the film could be captured by MFM (see Supplementary Fig. 8). This suggests that the SRO-15 film has a relatively small amount of magnetic disorder and thus exhibits a weak pinning effect. In sharp contrast, the magnetization reversal process of SRO-35, which has triple switching steps in Hall measurements, shows a much stronger pinning effect. During the first switching process, as shown in Fig. 3a-f, over 50% of the area was reversed. The reversed region is likely related to X-domain, since the first switching magnetic field is very close to the SRO-15 coercive field and X-domain has the largest volume fraction. The second switching process was also visualized in MFM images, as indicated by black circles in Fig. 3g, h. The remaining pinned area, which survived at higher magnetic fields up to 0.8 T, consisted of bubble-like domains with an average size around 1 μm. Above 0.8 T, these magnetic bubbles were gradually reversed and the film was fully saturated at 1.6 T (see Fig. 3i-l). Via estimating the distribution of up and down domains, the M-H loop was estimated, as shown in Fig. 3m, which shows triple-step magnetic domain switching and thus is reasonably consistent with the Hall loop. The slight quantitative discrepancy is probably due to a spatial variation of structural twin domains over a length scale much larger than the scan size (~20 × 20 µm 2 ) of our MFM. The discrepancy could be reduced by sampling a larger area.
To further confirm the fact that structural domains pin the ferromagnetic domains, especially the minor Z or Y-domain in the film, which are hard to be investigated separately, we performed the MFM studies on a 250 u.c. SRO-30 sample, which was comprised of only X and Z-domains (see Supplementary Fig. 9). The Hall resistance of SRO-30 sample exhibited two-step switching, in good agreement with two types of structural domains. The volume percentage of Z and X-domain is quantified by the RSMs data. The ratio of Z to X-domain is around 0.12:1, which agrees with the ratio of two-step sizes in the anomalous Hall data (~0.1:1), suggesting again the correlation between magnetic and structural domains. Field-dependent MFM images taken on this 250 u.c. film also confirm this relationship. As shown in Fig. 4a-e, the majority of the film has been reversed during the first switching process, which is related to X-domain. Scattered magnetic bubble domains which remained at higher fields around 1.1 T and were entirely switched at 1.4 T then are assigned to the Z-domain (see Fig. 4f-i). This is further evidenced by the fact that the M-H loop estimated by MFM is quantitatively consistent with Hall loop as shown in Fig. 4j and also the fact that the isolated small bubble occupies nearly 10% volume fraction. Comparing the three domains and their switching fields in 75 u.c. SRO-35 with those in 250 u.c. SRO-30, we can conclude that (1) Fig. 10), further indicating one-to-one corresponding between structural domains and magnetic domains. We also notice that the domains are irregular, which are similar with previously reported domain structures revealed microscopically by transmission electron microscopy 32,38 . This irregular shape of domains may arise from complex interplay between domain wall energy, miscut direction, and thermal dynamics during growth. The agreements between x-ray RSMs, transport and MFM data demonstrate that the multiple-step behavior in the magnetic reversal process is intimately related to the populations of structural domains.
Origin of multi-step switching The multiple-step switching shown in Hall resistance and magnetization data is a result of domain-dependent coercive fields. The orthorhombic SRO films, regardless of the type of domains, exhibit uniaxial magnetocrystalline anisotropy. Experiments on twin free (110) o orientated SRO films indicated that the  24,39,40 . Prior Lorentz TEM studies showed that the easy axis is close to [100] o , which is the long axis of ab plane, irrespective of the type of domains 41 . It has been further found that the easy axis orientation is sensitive to the variation of rotation and tilt of RuO 6 octahedra 13 . For Z-domain, the easy axis lies closer to in-plane direction, leading to an out-ofplane hard axis and a much bigger coercive field than X or Ydomain. The temperature-dependent coercive fields are shown in Fig. 5a. The coercive fields are obtained from the peaks in dR xy /dH vs. H curves as illustrated in Fig. 2. For SRO-15, there is only one coercive field from single X-domain, while for SRO-35, three coercive fields arise from X, Y, and Z-domain, respectively. With respect to SRO-30, two coercive fields from X and Z-domain are observed. Comparing these values, the coercive fields can be categorized into three groups (H C1 , H C2 , and H C3 ) as indicated by black, blue and red circles in Fig. 5a. The group-1 which has the lowest coercive field is attributed to X-domain. The group-3, which has the largest value, is attributed to Z-domain as illustrated from SRO-30 sample. In between is the group-2, which is then attributed to the remaining tilted Y-domain. The Y-domain though is also (110) orientated, it tilts away from STO (001), which as a result will lead to different structure distortion from nontilted X-domain due to different interface structure coupling effect. 13  Additionally, the size of Y-domain is much smaller than X-domain (see Fig. 3). The X-domain can create a vertical interface with Y-domain and thus can modify the structure of Y-domain as well. The different structure distortion and smaller domain size than X-domain could explain the slightly bigger coercive field of Y-domain than X-domain. Quantitative understanding of the unit tilting effect on magnetic anisotropy will require further investigation. The slightly different structure and coercive field between X and Y-domain could cause a little discrepancy of domain ratio estimated from RSMs than that estimated from AHE data and MFM image shown in Fig. 3m. Partial Y-domain could be dragged by already switched X-domain and switched earlier before H C2 , leading to an underestimated Y to X-domain ratio. Note that the coercive field of 75 u.c. SRO-35 is higher than those of 250 u.c. SRO-30 and 75 u.c. SRO-15 in each sub-group, likely due to the smaller domain size in SRO-35.

Impact of domain formation on magnetoresistance
The domain formation is found to fundamentally change the magnetoresistance. Typical negative magnetoresistance arising from the suppression of spin fluctuation 42 is observed in SRO-15 and the hysteresis loop is caused by ferromagnetic domain reversal (see Fig. 5b and Supplementary Fig. 11). Differently, the SRO-35 shows positive MR at low-field (<2.6 T) and triple-step switching (see Fig. 5c). This positive MR feature is also seen in Hallbar geometry with I//[100] (see Supplementary Fig. 12). The lowfield positive MR could arise from Z-domain. As reported by Gunnarsson et al., positive MR arises when field is parallel to [001] o direction 34 , which has been suggested to originate from antilocalization effect or anisotropic magnetoresistance owing to relatively large spin-orbit coupling in SRO. Therefore, the Z-domain whose [001] o is along out-of-plane direction then will produce positive MR. For SRO-35, it contains~14% of Z-domain as estimated from AHE data (see Fig. 2b). The existence of Z-domain correlates with low-field positive MR. The triple-step switching in MR for SRO-35 should arise from magnetic domain wall resistance. Since different structurally pinned magnetic domains have different coercive fields, the magnetic domain reversal process will induce magnetic domain walls and thus domain wall resistance.

DISCUSSION
In summary, the geometric control of domain formation and physical properties are demonstrated in SRO thin films grown on (001) STO. Structurally twin free and twinned SRO films can be deliberately controlled by substrate miscut direction. Using cryogenic MFM to visualize the domain behavior in magnetization reversal process, we show that the local coercive fields correlate with the orthorhombic twin domains. This correlation leads to multiple magnetization switching steps in multi-twin-domain SRO films. More interestingly, introducing the [001] o domain (Z-domain) results in positive low-field magnetoresistance, which is otherwise absent. The demonstration of a close link between magnetic domains and structural twin domains provides deep insight into the domain formation and its importance in determining physics properties. The controlled domain formation by tuning substrate miscut geometry also implies an additional avenue toward rational materials design for desired functionalities in future spintronic applications.

Thin-film synthesis and characterization
The SrRuO 3 (SRO) thin films were epitaxially grown on TiO 2 terminated (001) STO substrates using PLD technique. The single TiO 2 terminated STO substrates were achieved by standard buffer HF etching for 30 s and subsequent annealing at 950°C for 90 min. The substrate temperature and oxygen partial pressure during growth were 650°C and 0.25 mBar, respectively. The laser fluence and repetition rate were 2 J/cm 2 and 1 Hz, respectively. In situ RHEED was used to monitor the growth. The substrates and films surface morphology was characterized by atomic force microscopy (AFM). Lattice structure of the films was characterized by PANalytical X'Pert Materials Research Diffractometer in high resolution mode.

Transport and magnetic properties measurement
The Hall resistance and magnetoresistance were measured by Quantum Design Physical Property Measurement System using van der Pauw method. All the Hall data were antisymmetrized.

MFM and in situ transport measurement
The MFM experiments were performed in an in-house-built cryogenic AFM. The tips for the MFM were coated by a nominally 100 nm Co film using ebeam evaporation. MFM images were taken in a constant-height mode with the scanning plane~80 nm above the sample surface. To avoid the relaxation effect (domain wall creeping) near H C and to minimize the stray field effect of the MFM tips, all MFM images were taken at a low magnetic field of~0.05 T after the magnetic field was ramped to the desired values. The MFM signal, the change of cantilever resonant frequency, is proportional to the out-of-plane stray field gradient. Electrostatic interaction was minimized by nulling the tip-surface contact potential difference. Dark (bright) regions in the MFM images represent up (down) ferromagnetic domains, where magnetizations are parallel (antiparallel) with respect to the positive external field. The in situ Hall resistance and longitudinal resistance were measured in a Hall-bar geometry by standard lock-in techniques with an alternating current of 40 μA modulated at 314 Hz. All Hall data were antisymmetrized.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.