Ferromagnetism and Conductivity in Atomically Thin SrRuO 3

Atomically thin ferromagnetic and conducting electron systems are highly desired for spintronics, because they can be controlled with both magnetic and electric fields. We present ð SrRuO 3 Þ 1 − ð SrTiO 3 Þ 5 superlattices and single-unit-cell-thick SrRuO 3 samples that are capped with SrTiO 3 . We achieve samples of exceptional quality. In these samples, the electron systems comprise only a single RuO 2 plane. We observe conductivity down to 50 mK, a ferromagnetic state with a Curie temperature of 25 K, and signals of magnetism persisting up to approximately 100 K.


I. INTRODUCTION
The creation of an atomically thin ferromagnetic and conducting electron system has been a long-standing goal in science. If realized, it will combine the advantages of two-dimensional electron systems with those of magnetic materials, i.e., state control by electric and magnetic fields. Atomically thin transition-metal films can remain ferromagnetic [1][2][3][4], but these electron systems are stable only in a vacuum, limiting their impact. Moreover, these electron systems are not electrically isolated, because the substrates are metallic. Similarly, ferromagnetism has also been observed in hydrogen-doped graphene grown on graphite [5] or at single hydrogen adatoms on graphene [6]. Recently, ferromagnetism has been observed in isolated atomically thin van der Waals crystals of CrI 3 [7], bilayer van der Waals crystals of Cr 2 Ge 2 Te 6 [8], and epitaxially grown van der Waals crystals of MnSe 2 [9]. The small size of most of the van der Waals crystals, however, hampers the investigation of these materials, and, e.g., measurements of the electrical conductivity are challenging. Transitionmetal-oxide heterostructures circumvent these issues of stability [10] and crystal size and can be grown on insulating substrates, thus realizing isolated electron systems that can be electrically contacted and controlled. Most magnetic and conducting transition-metal-oxide materials, however, lose their functional properties well before the single-unit-cell layer thickness is reached; typically, a nonconducting and nonmagnetic dead layer is present [11][12][13][14][15][16].
SrRuO 3 is one of the oxide materials with the highest conductivity, and it is chemically inert. Therefore, it is widely used in applications such as electrodes of capacitors [17,18]. In addition, it is an itinerant ferromagnet with a saturation moment of 1.6 μ B =Ru and a Curie temperature T C of 160 K [19]. Band-structure calculations reveal a 1-eV Stoner splitting of the majority and minority spin bands, resulting in a 60% majority spin polarization [20]. As SrRuO 3 has low intrinsic disorder and its epitaxial growth is well understood [21][22][23], it is a good candidate for realizing a two-dimensional spin-polarized electron system.
Several studies have investigated the behavior of ultrathin SrRuO 3 films and SrRuO 3 superlattices [24][25][26][27][28][29][30][31][32]. In most studies, however, an insulating state is observed when the SrRuO 3 thickness is less than three unit cells. Additionally, this insulating state has been proposed to be antiferromagnetic [28]. Several theoretical studies agree with the antiferromagnetic and insulating ground state in ultrathin SrRuO 3 [33,34]. Nonetheless, one theoretical study concludes that ferromagnetism remains down to two-unit-cell-thick layers. In that work, films with a thickness of only one unit cell are predicted to be nonferromagnetic owing to surface-driven effects [35]. Based on this reasoning, these surface effects ought to be nonexistent in SrTiO 3 − SrRuO 3 superlattices. Indeed, it has been proposed that a one-unit-cell-thick SrRuO 3 layer, i.e., a single RuO 2 plane, remains metallic and is fully minority spin polarized if embedded in a SrTiO 3 lattice (Fig. 1) [36,37]. According to that proposal, the octahedral structure of the atomically thin SrRuO 3 is stabilized by structural coupling to the SrTiO 3 [36]. Recent experiments indeed show an enhancement of the T C by capping the SrRuO 3 with SrTiO 3 [38]. To test whether one-unit-cellthick SrRuO 3 is indeed magnetic and conducting if embedded with SrTiO 3 in a heterostructure, we fabricate high-quality ðSrRuO 3 Þ 1 − ðSrTiO 3 Þ 5 superlattices and single-unit-cell-thick SrRuO 3 samples that are capped with SrTiO 3 . These layers exhibit conductivity and ferromagnetism, in support of the proposal that for this atomically thin electron system a ferromagnetic ground state can be stabilized.

II. SAMPLE GROWTH AND STRUCTURAL CHARACTERIZATION
The ðSrRuO 3 Þ 1 − ðSrTiO 3 Þ 5 superlattices are grown by reactive molecular-beam epitaxy (MBE) on (001) SrTiO 3 substrates, using the growth parameters listed in the Appendix. MBE enables excellent ruthenium stoichiometry control, including the minimization of ruthenium vacancies, the crucial ingredient for high-quality SrRuO 3 layers [39]. Superconducting films of the closely related compound Sr 2 RuO 4 with the highest transition temperature are also achieved by MBE [40]. For our samples, the growth parameters are optimized such that the correct ruthenium stoichiometry is obtained in thick films, as evidenced by the high residual resistivity ratio of 40 [41]. In independent deposition runs, we fabricate two ðSrRuO 3 Þ 1 − ðSrTiO 3 Þ 5 superlattice samples A and B, both of which have 20 repetitions of the building blocks. Furthermore, we grow a sample C that consists of a single unit-cell-thick layer of SrRuO 3 capped with 20 unit cells of SrTiO 3 .
We analyze the sample structure of the superlattices with scanning transmission electron microscopy (STEM) using both high-angle annular dark-field (HAADF) and electronenergy loss spectroscopy (EELS) imaging. Details of the EELS analysis are provided in Supplemental Material [42]. A representative image of sample B is shown in Fig. 2. Figure 2(a) shows the simultaneous annular dark-field image, Fig. 2(b) shows the ruthenium (red) and titanium (blue) color overlay, and Figs. 2(c) and 2(d) show the ruthenium and titanium signals, respectively. The ruthenium map and titanium map show that ruthenium is confined to single, two-dimensional layers. In this large field of view, step edges are apparent in the film [blue arrows in Fig. 2(c)].
Step edges are present in even in the best substrates, and, when the epitaxial film is deposited, the step edges form atomic-scale terraces and the step edges propagate up in the thin film. Often, these step edges run at angles through the film that are not parallel to the direction of the electron beam, so in the projection image of the thin sample we may see two partial layers of lowerintensity ruthenium. In this way, one single layer of ruthenium deposited over a step edge may appear in a projection image as an overlapping area with two SrRuO 3 layers with lower intensity. Because there is only one ruthenium layer shared between two rows, the HAADF and EELS ruthenium intensity near the step edges will be reduced compared to the single ruthenium layer. We observe this difference in the HAADF and EELS maps [Figs. 2(a) and 2(c), blue arrows]. Additionally, the titanium signal is also present but less intense in these partial ruthenium layers [ Fig. 2(d)]. If two layers of ruthenium were present, the ruthenium intensity in both layers would be the same as that of a single layer, and we would expect no titanium to be present in those layers. We do not see an accumulation of ruthenium forming two layers in this manner, even near the step edges in the FIG. 3. Lower-magnification HAADF-STEM images of the superlattices. Bright layers of SrRuO 3 are seen between five layers of SrTiO 3 . At the bottom of the images, the SrTiO 3 substrates are seen, and the dark layers at the top of the images are epoxy from the TEM sample preparation.
Step edges from the substrate are seen propagating through the film (blue arrows), showing 10-60 nm continuous single-layer sheets of ruthenium before encountering a step edge. There are a few regions (< 0.5%) that may contain ruthenium double layers (red arrows) and some regions with discontinuous ruthenium (orange arrow). film. These factors indicate that the ruthenium layers are largely continuous two-dimensional sheets and do not form double ruthenium layers. We also see some regions where the ruthenium layer does not look continuous [orange arrow, Fig. 2 Larger field-of-view images are shown in Fig. 3 of both samples A and B. Now we can interpret the HAADF STEM images with our understanding of the structure gained from the EELS spectroscopic images. We see in both samples that, where there are two overlapping layers of ruthenium, the intensity of the ruthenium is typically less and moving over what appears to be a step edge (e.g., see the blue arrows). There are some small regions (red arrows) that do appear to have two ruthenium layers, and, by analyzing several large field-of-view images, in sample A there is an upper bound of 0.4% two-layer SrRuO 3 , and in sample B there is an upper bound of 0.3% two-layer SrRuO 3 . We also find some regions of discontinuous SrRuO 3 where there appears to be no ruthenium layer (orange arrow). Because the overwhelming majority of the ruthenium layers are single ruthenium layers, these are remarkably well-ordered superlattices despite the occasional (and inevitable) presence of step edges. For both samples, the SrRuO 3 layers are continuous for approximately 10-60 nm before encountering a step edge.
The STEM-EELS analysis of sample C is shown in Fig. 4. From the HAADF images, it appears that the SrRuO 3 is spread across two unit cells in the structure. This spreading is confirmed by EELS mapping, which shows that a significant fraction of the B sites in what is expected to be a ruthenium row are titanium. The ruthenium is spread across one, two, or three layers in projection. By integrating the B-site intensity column by column, we find an average ruthenium density of 1.19 AE 0.04 monolayers of ruthenium. This amount is slightly more than the nominal one monolayer that is found in the superlattice samples.  4.8 unit cells instead of five, and they are not expected to affect the properties of the SrRuO 3 layers (Supplemental Material [42]). Furthermore, we also measure reciprocalspace maps of sample A along the20L, the10L, and the 00L lines (Fig. 6). Relatively strong superlattice reflections are found for 3 < L < 4 along20L, for L < 3 and for L > 4 along10L, and for 3 < L < 4 along 00L. Here, L is normalized to the SrTiO 3 reciprocal lattice. The entire superlattice structure is coherently strained to the lattice parameter of the SrTiO 3 substrate. The mosaicity of the superlattice is smaller than 0.05°, as determined from the full width at half maximum of the rocking curves.

III. ELECTRICAL TRANSPORT
The temperature dependence of the resistivity ρ of samples A and B is shown in Fig. 7 together with literature data. In the temperature range between 2 and 300 K, the samples have a resistivity of approximately 1000 μΩ cm, corresponding to a sheet resistance of 25 kΩ for a single SrRuO 3 layer. This resistivity is higher than either the bulk or the thick-film resistivity [18] but significantly lower than the resistivities of the two-unit-cell-thick films and the ðSrRuO 3 Þ 1;2 − ðABO 3 Þ n superlattices of the previous studies [24][25][26][27][28][29][30]32]. The decreased resistivity is due partially to the superlattice structure and partially to the high structural quality of our samples. The samples show a minimum of the resistivity at 120 (sample A) and 80 K (sample B); see also Fig. 11(a) for a linear plot of the data. Below these temperatures, dρ=dT is negative, possibly owing to localization of the charge carriers. To shed light on the ground state of the system, we measure ρðTÞ down to 30 mK [ Fig. 3(a)]. The resistance of the superlattice samples increases continually with a decreasing temperature, in contrast to the theoretical predictions [36,37]. Nevertheless, a finite conductivity of the order of 10 μS remains at the lowest temperature. In this temperature range, a large difference between the samples is observed. Surprisingly, the sample found by our XRD measurements to have a higher structural quality also has the greater resistance.
To elucidate the presence of the magnetization in the samples, we first study the magnetoresistance (MR). The MR is generally parabolic in nonmagnetic conductors. In the ferromagnetic state of SrRuO 3 , in contrast, the domainwall resistance is known to be the dominant contribution to the MR [43]. With an increasing magnetic field, the density of the domain walls is reduced, and, therefore, a decrease of the resistance with an applied magnetic field is expected. Thus, the presence of a negative (nonparabolic) MR is considered to be a strong indication of ferromagnetism. Indeed, for nonmagnetic samples of two-and three-unitcell thickness, only a very small MR is found [28]. The MR of samples A and B is shown in Figs. 8(a) and 8(b) for perpendicular fields up to 5 T. Above 100 K, hardly any MR is observed, and below 100 K, the MR increases steadily with a decreasing temperature to about 12% at H ¼ 5 T. Furthermore, below 25 K, a hysteresis is observed in the curves, revealing the butterfly-loop characteristic of ferromagnetic ordering. The upper limit for the magnetic moment in SrRuO 3 is 4 μ B =Ru corresponding to  2/5 sl, Ref. [26] 2/2 sl, Ref. [30] 4 uc film, Ref. [27] 2 uc film, Ref. [29] 2 uc film, Ref. [28] 3/3 sl, Ref. [32] Temperature (K) 2 uc film, Ref. [32] 0.01 0.  [30], and fðSrRuO 3 Þ 3 − ðSrTiO 3 Þ 3 g 15 [32]. All samples are grown on SrTiO 3 substrates. a magnetization M of 0.8 T. This value is too small to explain the large hysteresis in the MR by the standard relation MR ¼ KðH þ MÞ 2 , where K is an appropriate constant.
Two scenarios can explain the observed MR. In the first scenario, the MR is attributed to the influence of the domain-wall resistance and, accordingly, the hysteresis to domain-wall pinning. Because the MR is observed reversibly in a large magnetic-field range, this model implies that the domain-wall creation energy is small and that pinning sites with varying trapping energies exist. In the second scenario, we have to assume that the samples phase separate into well-conducting ferromagnetic regions and poorly conducting nonmagnetic regions [44]. Then, the transport between the ferromagnetic regions depends on the relative orientation of the magnetic moments in the regions. A hysteresis in the MR results when the different regions reverse their magnetic moments at different values of the magnetic field. In this scenario, the negative dρ=dT at a low temperature can be explained if the transport between the ferromagnetic regions is dominated by thermally activated hopping. Both scenarios require the presence of ferromagnetism in the samples. The larger hysteresis is observed in sample A, the sample with the higher resistivity. This result is consistent with both scenarios: more domain-wall pinning due to an increased number of point defects and/or more electronic inhomogeneity resulting in a larger spread in switching fields. This understanding implies that the temperature at which the hysteresis disappears can be lower than T C , because that is merely the temperature at which thermal fluctuations exceed the domain-wall pinning or switching field distribution.
We next study the temperature dependence of the MR in greater detail. Figure 8(c) presents the temperature dependence of the resistivity at H ¼ 0 T, the resistivity at H ¼ 5 T, and the MR ½ρð0Þ − ρð5 TÞ=ρð0Þ. Even though the resistivities of our two samples are different in magnitude, their MRs show similar temperature dependence. The MR at H ¼ 5 T is 10%-12% at 4 K and disappears at approximately 100 K, close to the phase-transition temperature of SrTiO 3 . The MR due to domain-wall resistance and due to an inhomogeneous ferromagnetic network is expected to deviate from parabolic behavior [43,44]. We therefore plot the MR in Fig. 8(d) as a function of H 2 . At 120 K, the MR has a linear MRðH 2 Þ dependence; however, below 100 K, pronounced deviations from the linear behavior are observed, especially at small applied fields. We conclude that signatures of magnetism persist in the samples up to a temperature of approximately 100 K.
We also perform measurements on samples A and B in perpendicular magnetic fields up to 10 T at T ¼ 500 and 200 mK (Fig. 9). The samples show a large negative MR (about 30% at H ¼ 10 T) with a hysteresis loop that is very similar to the loops observed at higher temperatures.  The loops, however, close at higher fields, indicating an increase of the coercive field. We have not observed a saturation of the MR, even at fields of H ¼ 10 T. The observation of hysteretic MR at temperatures below 1 K, where the resistivity is orders of magnitude larger, suggests that the second scenario as discussed above (electronic inhomogeneity) is dominant at low temperatures, because it is quite unlikely that the domain-wall resistance has the same temperature dependence as the resistivity. In addition to the magnetoresistance measurements in a perpendicular field shown in Fig. 8, we measure the magnetoresistance of samples A and B in a parallel field. The data are shown in Fig. 10. Both samples A and B show magnetoresistance below 100 K, just as in the case of the perpendicular magnetic field. A difference in the shape of the MR curves is, however, observed [Figs. 8(a) and 8(b) and Figs. 10(a) and 10(b)]. The in-plane MR is larger, especially at a small field strength. This result is attributed to the magnetocrystalline anisotropy of SrRuO 3 thin films. The magnetic easy axis is estimated to be about 10°-15°a way from the out-of-plane direction [28]. Therefore, the switching of the in-plane component of the magnetization has a smaller energy barrier than the switching of the outof-plane component, resulting in stronger low-field MR. Furthermore, no hysteresis is observed in the parallel field. This result is another direct consequence of the smaller energies involved in the switching of the in-plane component of the magnetization. At a large magnetic-field strength, however, the out-of-plane MR and the in-plane MR are similar. The temperature dependence of the MR shows a similar trend for in-plane and out-of-plane magnetic fields [Figs. 8(c) and 10(c)], except for the two-step behavior seen in sample B for an in-plane magnetic field.  The latter is probably due to sample inhomogeneity. The MRðH 2 Þ plot for an in-plane magnetic field is shown in Fig. 10(d). A deviation from linear behavior is observed below 80 K, suggesting that ferromagnetic effects are governing the MR, consistent with the data for the outof-plane magnetic field shown in Fig. 8(d).
The temperature dependence of the resistivity of sample C is shown in Fig. 11(a) together with the data from samples A and B. The resistivity of sample C has a similar temperature dependence as the other samples, and it is of similar magnitude. We investigate the possibilities of electrostatically modifying the sample using a back gate across the 1-mm-thick SrTiO 3 crystal. The lowtemperature resistivity is modified about 10% by applying positive and negative voltages of up to 200 V [ Fig. 11(b)]. The magnetotransport is shown in Figs. 11(c) and 11(d). The low-temperature magnetoresistance is 10% at H ¼ 5 T, a butterfly loop is observed for T ≤ 15 K when the field is applied perpendicular to the layer, and the magnetoresistance deviates from the standard parabolic dependence for T < ∼100 K. These results agree well with the magnetotransport of the superlattice samples. The application of the electrostatic potential does not change the shape and temperature dependence of the out-of-plane magnetotransport characteristics, indicating that the modification of the carrier density is too small to affect the ferromagnetic properties of the sample. We expect that larger changes in the sample properties can be achieved by using thinner gate dielectrics and electrolyte gating.

IV. MAGNETIZATION MEASUREMENTS
We now turn to direct magnetization measurements of the samples. The magnetocrystalline anisotropy of SrRuO 3 favors a predominantly out-of-plane magnetic moment for thin films [28]. Therefore, magnetic domain formation is expected to occur, resulting in a reduction of the global possible magnetic moment compared to that of a monodomain sample. The MR data also indicate that the samples are not monodomain. As the anisotropy field is very large, it is difficult to saturate the moment. This difficulty makes conventional magnetization measurements challenging.
We therefore use scanning superconducting-quantuminterference-device (SQUID) microscopy (SSM), which is a very sensitive local measurement technique that can directly image the magnetic domain structure [45,46]. Representative scans of samples A and B are shown in Figs. 12(a) and 12(b). A magnetic contrast is observed consisting of up and down domains in a bubblelike pattern, consistent with the expected out-of-plane anisotropy. The typical feature size of the magnetic flux pattern is 5-10 μm. The measured magnetic flux corresponds to a magnetic signal of 0.001-0.01 μ B =Ru in the superlattice, which is much smaller than the theoretical prediction [36] of a fully spin-polarized material. Because the area of the pickup loop is 3 × 5 μm 2 , any possible submicrometer domain structures are averaged out during the measurements. In addition, as the domain structure is not expected to be uniform across the stack of 20 magnetic layers, the measured magnetic signal is smaller than the magnetic moment inside the domains. Figure 12(c) shows the magnetization as imaged by scanning SQUID microscopy of sample C. A domain pattern is found that agrees well with the domain patterns observed in the superlattice samples. The magnitude of the magnetic signal corresponds to approximately 0.2 μ B =Ru, which is much larger than the moment observed in the superlattice samples. This result indicates that the magnetization in the different layers in the superlattice samples partially cancels out. For reference, a SrTiO 3 sample is imaged as well. A SrTiO 3 layer of 100 unit cells is grown using identical settings as the SrTiO 3 − SrRuO 3 superlattices. No magnetic signal is observed [ Fig. 12(d)], indicating that the magnetic signal in the SrRuO 3 samples is not due to impurities in the SrTiO 3 . Note that SSM detects only ferromagnetism from the surface of the sample, eliminating possible contributions from the back side of the sample. The magnetic domain patterns observed in samples A, B, and C clearly proves the samples to be ferromagnetic.
Additionally, we perform magnetic torque and magnetization measurements on the superlattice samples. Magnetic torque, τ ¼ μ 0 M × H, is sensitive only to anisotropic magnetic responses. Figure 13 shows torque curves obtained from the superlattice samples, together with the magnetic hysteresis loops obtained from the torque data. In these measurements, positive torque corresponds to a net magnetic moment in the in-plane direction and negative torque to a net magnetic moment in the out-of-plane direction when the magnetization is aligned with the projection of the field. Both samples show strong hysteretic behavior, characteristic of ferromagnetic ordering. As discussed in detail in the previous section, the magnetic anisotropy of SrRuO 3 favors a domain structure in which the magnetization vector is rotated away from the out-ofplane direction. This domain structure can generate a net magnetic moment in both the in-plane and the out-of-plane directions, depending on the volume fractions of the different domains. The magnetic hysteresis loop of sample A contains two contributions: an in-plane magnetic hysteresis with a switching field of H ¼ 4 T and an out-of-plane contribution that is linear in field and saturates above H ¼ 3 T. The saturation moments are 0.08 and 0.04 μ B =Ru for the in-plane and out-of-plane components, respectively. The magnetic hysteresis loop of sample B, in contrast, contains an out-of-plane hysteretic component and a linear, nonsaturating, in-plane component. The saturation magnetization of the out-of-plane component is 0.05 μ B =Ru. With an increasing temperature, the switching fields are reduced, and, for T > 25 K, the samples are no longer hysteretic. A smaller magnetic signal, however, persists up to higher temperatures (Supplemental Material [42]). As discussed in detail in Supplemental Material [42], the observed difference between the samples is attributed to the complicated magnetic domain structure and variations in domain pinning strength that affect the switching field distributions. We find that the saturation moment varies for different pieces of the samples between approximately 0.05 and 0.5 μ B =Ru (Supplemental Material [42]). The torque measurements clearly show the atomically thin SrRuO 3 layers to have a spontaneous magnetization and, therefore, to be ferromagnetic. The magnetic moment is too large to be explained by the small number of two-unit-cell-thick SrRuO 3 areas in the samples. Both the disappearance of the magnetic hysteresis at T ≈ 25 K and the observation of a magnetic signal for T > 25 K are in good agreement with the MR data.

V. CONCLUSIONS
In conclusion, we have shown atomically thin SrRuO 3 to be ferromagnetic and conducting if embedded in SrTiO 3 . The observation that conductivity and ferromagnetism is present in both the superlattices and in the single-layer sample indicates that the magnetization and conductivity originates from the properties of atomically thin layers of SrRuO 3 and do not arise because of coupling between the layers. In these samples, the electron systems comprise only a single RuO 2 plane. Magnetic hysteresis is observed for T < 25 K, and signals of magnetism persist up to approximately 100 K. Because the observed magnetic moment is only 3%-30% of the magnetic moment of bulk SrRuO 3 , we cannot exclude an inhomogeneous electron system with magnetic and nonmagnetic areas. These structures are a rare example of two-dimensional ferromagnetism and the first demonstration of two-dimensional ferromagnetism due to indirect exchange. They may therefore serve as a model system for further theoretical studies [4]. The conductance and T C of atomically thin SrRuO 3 is expected to increase with additional charge carrier doping [47], possibly resulting in a triplet superconducting ground state [48]. With recent advances in electricfield gating technology [49][50][51], we expect electric-field control of the conductivity and ferromagnetism to become possible.  The temperature is measured with a pyrometer at a measurement wavelength of 980 nm that detects the temperature of the platinum adhesion layer on the back side of the substrate. We obtain a singly terminated substrate surface by thermal annealing at 1300°C [52]. The one-monolayer-thick SrRuO 3 is grown in an adsorption-controlled regime, where the excess ruthenium evaporates off the film surface as RuO x (x ¼ 2 or 3) [23]. Strontium is evaporated from a lowtemperature effusion cell, titanium flux is provided by a Ti-Ball™ source [53], and ruthenium is evaporated from an electron beam evaporator. Before growth, the strontium and titanium fluxes are calibrated to an uncertainty of less than 1% using shuttered RHEED oscillations [54], while the ruthenium flux is calibrated using a quartz crystal microbalance (QCM). Postgrowth, the samples are cooled down below 100°C under the same ozone pressure in which they are grown.
Cross-sectional STEM specimens are prepared either by mechanical wedge polishing followed by Ar-ion milling [55] (sample A) or by focused ion beam (FIB) lift-out (samples B and C). FIB lift-out is performed using an FEI Strata 400 FIB with a final milling step of 2 keV to reduce surface damage. STEM and EELS data are recorded from cross-sectional specimens in the 100-keV NION UltraSTEM, a fifth-order aberration-corrected microscope optimized for EELS spectroscopic imaging with a probe size of approximately 1 Å, an EELS energy resolution of 0.4 eV, and a beam current of 100-200 pA. Large spectroscopic maps of the Ru-M 4;5 edge and the Ti-L 2;3 edge are acquired with an energy dispersion of 0.25 eV=channel with a Gatan Quefina dual-EELS spectrometer. For the large spectroscopic images, we integrate components of the spectra over energies corresponding to ruthenium and titanium after a linear combination of power-law background subtraction. Because of the close proximity of the Sr-M 2;3 and Ru-M 4;5 edges, and the Ru-M 2;3 and Ti-L 2;3 edges, we use small integration windows and principal component analysis (PCA) filtering to remove noise, keeping the six components of the spectra which capture all of the spatially varying components. The ruthenium is extracted from the tail of the M 4;5 edge, and the titanium is extracted from the first 1.5 eVof the Ti edge. To ensure PCA is returning no artifacts, we also use dual EELS to simultaneously map the Ru-L 2;3 edge and the Ti-L 2;3 edge, shown in Supplemental Fig. S1 [42]. The high energy of the Ru-L 2;3 edge makes it prohibitive to do large spectroscopic maps shown in the main text, although the mapping with the Ru-M 2;3 edge provides qualitatively similar results.
The resistivity measurements for T < 1 K are performed using a He 3 =He 4 dilution refrigerator and a low-frequency ac lock-in measurement technique with a 1-nA excitation. The high resistivity of sample A makes the measurements problematic. We do therefore not obtain reliable data of the magnetotransport of this sample at T ¼ 200 mK. It turns out that the temperature of the samples is not stable at low fields due to the relatively high sweep rates of the magnetic field used during the experiment. We attribute the thermal instability to magnetocaloric parts in the sample holder. The temperature varies up to 25 mK, and, since the samples have a strong temperature dependence of the resistivity in this temperature range, an error is made in the magnetoresistance measurements. We therefore exclude all data where the temperature is outside of a window of 2 (for the 200-mK sweep) and 10 mK (for the 500-mK sweep) from the analysis. We note that these temperature fluctuations do not affect the magnetic structure inside the samples. The resistivity and MR measurements for T > 2 K are performed with a Quantum Design physical properties measurement system using a 20 μA dc current excitation. The transport measurements determine the sheet resistance R sheet of the samples. The resistivity ρ of the samples is obtained from the sheet resistance by ρ ¼ R sheet ·d. Here, d is the total thickness of the conducting layer(s) that corresponds to either 20 times approximately 0.4 nm (samples A and B) or approximately 0.4 nm (sample C).
The SSM measurements are performed using a square pickup loop with an inner dimension of approximately 3 × 5 μm 2 . During the measurement, the pickup loop is scanned approximately 2 μm above the sample surface at a contact angle of approximately 10°. The SSM records the variation of magnetic flux threading the pickup loop, and the flux detected by the pickup loop is converted to a magnetic field by dividing by the effective pickup area of approximately 15 μm 2 . The typical flux sensitivity of the SSM is around 14 μΦ 0 Hz −1=2 , where Φ 0 ¼ 2 × 10 −15 T m 2 is the flux quantum and the bandwidth is 1000 Hz. As our SSM sensor has a 10-degree inclination, the measured magnetic stray-field component B z is almost perpendicular to the sample surface. The practical sensitivity during measurements is set by external noise sources and is estimated to be about 30 nT.
We perform torque magnetometry measurements with a homebuilt cantilever setup by attaching the samples to a thin beryllium copper cantilever. Under an external magnetic field H, the sample rotation is measured by tracking the capacitance between the metallic cantilever and a fixed gold film underneath using an AH2700A capacitance bridge with a 14-kHz driving frequency. To calibrate the spring constant of the cantilever, we track the angular dependence of the capacitance caused by the sample weight at a zero magnetic field.
We also explore the magnetization in the samples by SQUID measurements and by muon spin rotation. In these experiments, no (muon spin rotation) or only a weak (SQUID) magnetic signal is observed beyond the diamagnetic background of the SrTiO 3 substrate. It is not possible to attribute the magnetic signal observed in the SQUID measurements to a signal originating in the SrRuO 3 , because the signal could also arise from impurities in the SrTiO 3 . Therefore, we discuss only the magnetoresistance, SSM, and torque measurements here.  Fig. S1f, we see that there is a strong peak on the ruthenium row. There also seems to be some ruthenium present on the row below (row 2), although if we inspect the spatial distribution of the edge (e.g., Fig. S2e) it appears featureless, indicating it may be due to probe tails and beam spreading. The map of the Ru-L2,3 edge demonstrates that the ruthenium is confined to the single ruthenium layers, with minimal intensity and no structure outside these layers. This is also clear from inspecting a line profile of the ruthenium signal, Fig. S1(g), where the ruthenium is well localized to the atomic column of ruthenium. From these maps we find little intermixing between the two layers.

Ferromagnetism and conductivity in atomically thin
While unambiguous, doing mapping of the Ru-L2,3 edge on a large field of view is prohibited by the high energy and low scattering cross section of the edge. Mapping this edge with high signal-to-noise requires long integration times and high beam doses which may damage the specimen for larger field of view images. The lower energy ruthenium edge (Ru-M4,5) has a larger cross section and thus requires less dose to acquire high signal-to-noise data. However, the lower energy edge overlaps with Sr-M2,3 and C-K edges (see Fig. S2f), so care must be taken to separate the signals. Here we show our results from the low-energy edges qualitatively reproduce the results we have from the high-energy edge, as shown in Fig. S1. This implies there are no misleading artifacts in our large field of view spectroscopic maps, such as shown in the main text in Fig. 2 and in Fig. S2.
Another dataset from the spectroscopic imaging of sample A is shown in Fig. S2. onset of the titanium edge should not be mixed with ruthenium signal, so we used a very small integration window at the onset of the edge to obtain titanium. Because of the small integration windows necessary, we used PCA filtering to remove noise, keeping six components of the spectra. Five of the six components had noticeable spatially varying features, and we kept the sixth component to ensure we did not throw away additional information. Figure S2b shows the ruthenium signal obtained by this method. Like in Fig. S1, which used the Ru-L2,3 edge instead of the Ru-M2,3 edge, the ruthenium signal is confined to single atomic layers of ruthenium. Figure S2c corresponds to the titanium signal, which is similarly well ordered. The color overlay shown in the main text is reproduced in Fig. S2d. The signal to noise ratio in the spectroscopic maps is 15 for titanium and 13 for ruthenium, and we clearly resolve both signals with atomic precision. Representative spectra that are averaged over a titanium column and a ruthenium column are shown in Fig. S2f, showing the overlapping edges. Reference spectra for the edges from the Gatan EELS atlas S1 are shown in Fig. S2g for reference. We could not determine the valence state of Ru from the fine structure of the spectra, because the Ru-M4,5 edge overlaps with Ti and C edges and the Ru-L2,3 edge is too high in energy. The latter does not enable us to obtain the high-signal-to-noise-ratio spectrum required for valence determination without damaging the sample. In contrast, we determined the valence state of Ti to be 4+ without any position dependence. Figure S3 shows a close-up of the 00(0+) and 00(0+2) superlattice reflections for both samples A and B. Next to the 00(0+) superlattice reflection, sample A shows some Kiessig fringes. The fringes correspond to the expected thickness of 20 (SrRuO3)1-(SrTiO3)5 repeat units. We did not observe the fringes in sample B. Also in sample A, the fringe visibility is much smaller than that expected for a perfect heterostructure. We have performed simulations with the Panalytical reflectivity software. The expected reflectivity curve for a perfect 20 times superlattice is shown as the top line in the set simulations 1. Adding interface roughness to the model does not affect the fringe visibility much. Even at a roughness of 0.4 nm per interface (much more than that deducible from the STEM data), the fringes should be clearly resolved, simulations 1 bottom line. Therefore, interface roughness cannot explain the lack of fringe visibility. The main effect of the roughness is to reduce the overall intensity at higher momentum transfer. We observe an intensity ratio of about 10 between the 00(0+) and 00(0+2) superlattice reflections. This is consistent with an interface roughness smaller than 0.2 nm.

X-ray diffraction
As discussed in the main text, the superlattice reflections deviate from the expected peak positions, indicating the average thickness of the repeat unit is somewhat less than 6 unit cells. We have therefore performed simulations of a heterostructure with an 80% sample volume of (SrRuO3)1-(SrTiO3)5 and a 20% sample volume of (SrRuO3)1-(SrTiO3)4. The precise stacking of the model is as follows: 4 times

Additional torque data
Additional torque magnetometry data is shown in Fig. S4. Figures S4a and S4b show the magnetic field dependence of the torque and derived magnetization of sample A for a larger temperature range than that depicted in Fig. 13. The magnetic hysteresis disappears at temperature above approximately 30 K, but magnetic signal remains present up to much higher temperatures. Figures S4c-h show torque data obtained from different pieces of sample B. Pieces B1 and B2 show no hysteresis but a magnetic signal that disappears at temperatures higher than 25 K. Piece B3 shows hysteresis and measurements with the field applied close to the in-plane direction also show magnetic signal persisting up to approximately 200 K. Taken together, all pieces are ferromagnetic but they have different switching behavior. As discussed in the manuscript, the magnetic anisotropy favours a domain structure with both in-plane and out-of-plane components. The energy barriers for the switching of these components clearly differs from piece to piece. The temperature dependence of the data that shows a hysteretic part at low temperatures and magnetic signal persisting up to high temperature is in good agreement with the MR measurements.       . (g,h) piece B3. Here H is applied 20 degrees away from the out-of-plane direction.