A model heterostructure with engineered Berry curvature

Molecular-beam epitaxy enables ultrathin functional materials to be combined in heterostructures to create emergent phenomena at the interface. Magnetic skyrmions are an example of an exciting phase found in such heterostructures. SrRuO 3 and SrRuO 3 -based heterostructures have been at the center of the debate on whether a hump-like feature appearing in Hall resistivities is sufficient evidence to prove the presence of skyrmions in a material. To address the ambiguity, we synthesize a model heterostructure with engineered Berry curvature that combines, in parallel, a positive anomalous Hall effect (AHE) channel (a Sr 0.6 Ca 0.4 RuO 3 layer) with a negative AHE channel (a SrRuO 3 layer). We demonstrate that the two opposite AHE channels can be combined to artificially reproduce a “hump-like” feature, which closely resembles the hump-like feature typically attributed to the topological Hall effect and the presence of chiral spin textures, such as skyrmions. We compare our heterostructure with a parallel resistor model, where the inputs are the AHE data from individual Sr 0.6 Ca 0.4 RuO 3 and SrRuO 3 films. To check for the presence of skyrmions, we measure the current dependence, angle dependence, and minor loop dependence of R hump in the heterostructure. Despite the clear hump, no evidence of skyrmions is found.


I. INTRODUCTION
The ordinary Hall effect (OHE) is generated when an external magnetic field is applied to a conductive material. 1 Other than the OHE, there are many other Hall effects, including the anomalous Hall effect (AHE) 2 and the topological Hall effect (THE). 3 While the OHE requires an external magnetic field, the AHE and THE arise spontaneously. The anomalous Hall effect has extensively been studied, and it is known to emerge spontaneously in materials with broken time-reversal symmetry, for example, in ferromagnets. 2,4 Typically, the AHE is observed as a hysteretic "box-like" signal in the Hall measurement. The topological Hall effect emerges when electrons are able to acquire a Berry phase while moving through a material with a chiral spin texture, for example, magnetic skyrmions. 3 It is possible to measure magnetic skyrmions using techniques such as Lorentz transmission electron microscopy, 5 magnetic force microscopy, 6 and small-angle neutron scattering. 7 Nonetheless, the THE is typically inferred by the presence of a spontaneous "hump-like" feature that appears in the anomalous Hall effect measurement.
Recently, SrRuO 3 and SrRuO 3 -based heterostructures have become the focus of a debate surrounding the hump-like feature, THE, and skyrmions. 8 Matsuno et al. 9 published the discovery of an interface-driven THE in a SrRuO 3 -SrIrO 3 bilayer structure and theorized that the hump-like feature in their magnetotransport measurement was due to the presence of Néel-type skyrmions. Since this publication, a further investigation using magnetic force microscopy has shown small bubble domains to be present in ARTICLE scitation.org/journal/apm SrIrO 3 /SrRuO 3 bilayers made up of a 2 pseudocubic unit cell (u.c.) thick SrIrO 3 layer grown on top of a 10 u.c. thick SrRuO 3 layer, 10 where the growth direction of the bilayer is the [001] pseudocubic direction of the perovskites. Unfortunately, the evidence remains inconclusive that these bubbles are skyrmions. 8 Many other studies have reported similar hump-like features in SrRuO 3 thin films and heterostructures. [11][12][13] Meanwhile, others have argued that the hump can also appear as an artifact in materials or material systems that are inhomogeneous, [14][15][16] are strain-relaxed, 17 contain ruthenium vacancies, 18 or generally have two (or more) mixed, anomalous Hall effect channels of different magnitudes and opposite sign. [19][20][21][22] For example, if there is a large positive AHE channel and a small negative AHE channel, one can measure a net positive AHE signal with a hump-like feature on top. 8 This hump-like feature is indistinguishable from the THE hump, which is currently used as a signature measurement for skyrmion detection. Therefore, while it seems to be true that materials with skyrmions typically have the hump-like feature, the converse is not necessarily true: materials that exhibit the hump-like feature do not necessarily host skyrmions. Thus, more direct measurements are necessary in order to confirm the presence of skyrmions in a material.
In this work, we artificially reproduce the hump-like feature in Hall resistivity measurements by synthesizing a model heterostructure with engineered Berry curvature. The heterostructure is a bilayer, consisting of one layer with positive AHE sign and one layer with negative AHE sign, as shown in Fig. 1, of distinct magnitudes. In other words, the heterostructure in total has two major AHE channels, one in each layer. We are able to tailor the sign of the AHE in each layer of the heterostructure by controlling the amount of calcium substituted for strontium in the solid solution Sr 1−x CaxRuO 3 , as shown in Miao et al. 23 Thus, we can create a Berry curvature heterostructure by interfacing SrRuO 3 and Sr 0.6 Ca 0.4 RuO 3 , two isostructural materials, but with anomalous Hall effects of opposite sign. We demonstrate that each layer individually does not show the hump-like feature. Individually, these layers only show the box-like hysteresis associated with the AHE. Nonetheless, when the layers are combined in the heterostructure, the hump-like feature emerges. Using current dependence, angle dependence, and minor loop dependence measurements, we confirm that the humplike feature is not due to the presence of skyrmions. In addition, we utilize a parallel resistor model to demonstrate that the effect can be attributed to the mixing of two AHE signals and is not an emergent property of the interface. This experiment on a model interface represents a clear demonstration of how the hump-like feature can be artificially generated by multi-channel anomalous Hall effects, further supporting the premise that a hump-like feature is insufficient evidence to demonstrate the presence of skyrmions.

A. Sample growth
Thin films of SrRuO 3 and Sr 0.6 Ca 0.4 RuO 3 were grown on (100) (LaAlO 3 ) 0.29 -(SrAl 1/2 Ta 1/2 O 3 ) 0.71 (LSAT) substrates by oxide molecular-beam epitaxy (MBE) in a Veeco Gen10 MBE system. The two individual films were grown to 14 u.c. thick. The absorptioncontrolled growth was optimized utilizing the previously published thermodynamics of MBE (TOMBE) diagram. 24 Films were grown at a substrate temperature of 650 ○ C and at a chamber pressure of 10 −6 Torr in a distilled ozone environment (∼80% O 3 + 20% O 2 ). The substrate temperature was measured using an optical pyrometer operating at a wavelength of 1550 nm. The strontium (99.99%) and calcium (99.99%) source materials were evaporated from lowtemperature effusion cells, and an electron beam evaporator was used to deposit the ruthenium source material (99.99%). A bilayer heterostructure was also grown using the same conditions as were used for the two individual thin films on (100) LSAT. It is known that as the calcium concentration, or "x," is increased in the solid solution Sr 1−x CaxRuO 3 , the magnetic coercivity of the materials decreases. 23 Therefore, the coercivity of SrRuO 3 (x = 0) is larger than that of Sr 0.6 Ca 0.4 RuO 3 (x = 0.4). Nonetheless, to ensure that the as-grown heterostructure had sufficiently different coercivities in each of the layers, the x = 0.4 layer was grown thinner and the x = 0 layer was grown thicker than 14 u.c. to further enhance the disparity in coercivity. The nominal structure of the bilayer from the surface to the substrate is 10 u.c. of x = 0.4 on top of 18 u.c. of x = 0 on top of the (100) LSAT substrate, or when written in shorthand: Sr 0.6 Ca 0.4 RuO 3 [10 u.c.]/SrRuO 3 [18 u.c.]/LSAT(100). For the remainder of the paper, this bilayer sample will be referred to as the "heterostructure" for simplicity.

B. Sample characterization
For the resistivity and Hall measurements, the SrRuO 3 [14 u.c.]/LSAT(100) film, the Sr 0.6 Ca 0.4 RuO 3 [14 u.c.]/LSAT(100) film, and the heterostructure were fabricated into Hall bars all with 20 × 50 μm 2 channels defined by photolithography. The channels were formed by argon ion milling, and platinum/titanium pads were subsequently sputtered onto the thin films to form ohmic contacts.
Transport measurements were performed in a quantum design physical property measurement system (PPMS). The AHE measurements of the 14 u.c. thick SrRuO 3 and 14 u.c. thick Sr 0.6 Ca 0.4 RuO 3 films were done in the aforementioned PPMS by measuring the transverse resistivity (ρxy) on the patterned Hall bars while applying a magnetic field perpendicular to the sample surface. For each AHE measurement at a given temperature, the applied magnetic field was ARTICLE scitation.org/journal/apm swept between −4 and 4 T. The measurement of the heterostructure was performed using these same parameters. Cross-sectional scanning transmission electron microscopy (STEM) specimens were prepared by standard focused ion beam (FIB) lift-out with a Thermo Fisher Scientific Helios G4 UX FIB. High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) images were acquired on an aberration-corrected FEI Titan Themis 300 operating at 300 kV with a 30 mrad probe convergence semi-angle and inner (outer) collection angle of 68 (340) mrad. Electron energy loss spectroscopy (EELS) mapping was performed under the same operating conditions on the same instrument equipped with a 965 GIF Quantum ER energy filter and a Gatan K2 Summit direct electron detector operated in the electron counting mode. The concentration maps for strontium, ruthenium, and calcium were obtained through a nonnegative least squares fit to the weighted sum of reference spectra for each element over a range of 270-361 eV. The reference components used and the resulting score maps are shown in Fig. S1 of the supplementary material. The reference component for strontium was obtained from the average signal summed over the substrate region of the original EELS map. References for ruthenium and calcium are from the Gatan EELS Atlas and acquired from measurements of RuO 2 and CaCoO 3 , respectively. The inelastic backgrounds of all spectra are modeled with exponential fits and subtracted prior to this analysis.

III. RESULTS AND DISCUSSION
A. Basic characterization of the thin films and heterostructure The 14 u.c. thick SrRuO 3 and Sr 0.6 Ca 0.4 RuO 3 thin films grown on LSAT(100) were confirmed to be phase pure by x-ray diffraction and were found to be metallic for the entire temperature range of 2.5-300 K, as shown in Figs. S2(a) and S2(c) of the supplementary material. 23 The two individual films were then measured in the Hall geometry described above. We find that the 14 u.c. thick SrRuO 3 film shows a negative AHE in both the low-temperature (T = 40 K) and the high-temperature (T = 80 K) measurements, as shown in Figs. 2(a) and 2(b), respectively. As for the 14 u.c. thick Sr 0.6 Ca 0.4 RuO 3 film, the Hall measurements at 40 and 80 K are shown in Figs. 2(c) and 2(d), respectively. This Sr 0.6 Ca 0.4 RuO 3 film exhibits a positive AHE signal at 40 K but no observable AHE signal was exhibited at 80 K (no observable hysteresis). Therefore, at 40 K, both films have an AHE signal, but of opposite sign and magnitude, and at 80 K, only the SrRuO 3 has an observable negative AHE signal. When comparing the AHE signals at 40 K, it is clear that the two films exhibit differences in the magnitude of the AHE signal and that the SrRuO 3 film has a much larger coercive field than the Sr 0.6 Ca 0.4 RuO 3 film. The combination of these two AHE signals provides an opportunity to create a Berry curvature heterostructure to investigate the hump-like feature in multi-channel AHE samples.
Specifically, we combined the SrRuO 3 film (negative AHE) with the Sr 0.6 Ca 0.4 RuO 3 film (positive AHE) in a bilayer heterostructure, which we grew on a (100) LSAT substrate. The concept of the heterostructure, which is to create an interface between two materials with Berry curvatures of opposite sign, is shown in Fig. 1. The heterostructure was characterized by STEM-EELS to confirm the quality of the interface, the chemical composition, and the microstructure of each of the layers. Figure 3(a) shows a HAADF-STEM image, demonstrating a sharp substrate-film interface and coherent crystallinity across both layers. The Sr 0.6 Ca 0.4 RuO 3 layer shows a slightly darker contrast than the SrRuO 3 layer due to the lower average Z of the A-sites in the doped structure (Ca = 20, Sr = 38). The layers are more clearly distinguished by EELS mapping, also shown in Fig. 3(a). The concentrations of spectral components assigned to calcium, strontium, and ruthenium are shown in an atomic-resolution two-dimensional score map and integrated parallel to the interface to produce vertical concentration profiles along the growth direction of the heterostructure. There appears to be a transition region of 1-2 u.c. at the interface between the x = 0 and x = 0.4 materials where the calcium is at a lower nominal concentration than the Sr 0.6 Ca 0.4 RuO 3 layer above it. Nevertheless, it is clear based on these measurements that there is a region of only strontium and ruthenium, which corresponds to the SrRuO 3 layer, and that in the Sr 0.6 Ca 0.4 RuO 3 layer, strontium, ruthenium, and calcium are all present.
We measured the resistance of the heterostructure as a function of temperature upon cooling, which is shown in Fig. 3(b). The heterostructure is metallic over the entirety of the temperature range shown. The first derivative with respect to temperature of the resistivity data is shown in the bottom half of Fig. 3(b), with the ferromagnetic Curie temperature (Tc) of each layer (x = 0 and x = 0.4) marked with an arrow, as determined in Miao et al. 23 and shown in Fig. S2 of the supplementary material. In SrRuO 3 , kinks in the resistivity vs temperature measurement are typically associated with Tc. In a bare SrRuO 3 film, there will be one kink in the data since there is only one ferromagnetic transition. Importantly, in Fig. 3(b), we find that there are two kinks, although the kink from the x = 0.4 layer is much less prominent due to the presence of two ferromagnetic layers with two different Tc values. SrRuO 3 has the higher Tc (T c0 ) of

ARTICLE scitation.org/journal/apm
∼139 K, and as calcium is substituted, the Tc is lowered such that the Tc of Sr 0.6 Ca 0.4 RuO 3 (T c0.4 ) is ∼55 K. 23 The presence of the two Tc values is another confirmation that the two layers are chemically different, and it allows for the ferromagnetic ordering of the two layers to be tuned separately by varying the temperature. For instance, at 200 K, neither layer is ferromagnetic, while at 80 K, the SrRuO 3 layer is ferromagnetic and the Sr 0.6 Ca 0.4 RuO 3 layer is still in the paramagnetic state. At 40 K, both layers are ferromagnetically ordered. Another method for switching the layers on and off preferentially is by utilizing the difference in the two coercive fields (Hc). As mentioned before, SrRuO 3 (x = 0) has a higher coercive field than Sr 0.6 Ca 0.4 RuO 3 (x = 0.4); therefore, we can magnetize both layers and preferentially demagnetize and switch the x = 0.4 layer without influencing the magnetization of the x = 0 layer.

B. Hall measurements
We fabricated Hall bars on the heterostructure using the steps outlined in Sec. II and then measured the Hall effect of the heterostructure [see Fig. 4(a)]. Note that the so-called hump-like feature starts to appear as the heterostructure is cooled through T c0. 4 . The decrease in the hump signal as the sample is cooled below 40 K is difficult to explain. One observation is that the magnitude of the AHE signal, shown in Fig. 4(b), is decreasing with temperature starting at around 60 K when it is mostly negative. The hump is highest at 20-50 K and peaks at 40 K [see Fig. 4(c)], which is just under T c0.4 . Since the hump comes from the two AHE signals, most likely the decrease in the hump signal below 40 K can be derived from the decrease in the magnitude of the AHE signal.

C. Comparison of the model and the heterostructure
To further analyze the hump-like feature of the heterostructure sample, we can compare it to a parallel resistor model in which  Figs. 2(a)-2(d), and mentioned previously. We took the AHE data from the two individual films and included them in a parallel resistor model to mimic the bilayer heterostructure. This simulation is shown in Figs. 5(a) and 5(b). Here, we are simulating how the AHE data would look if we assume there are no significant emergent properties at the interface, or in other words, it is a clear method of combining two AHE signals. We compare this simulation to our heterostructure data to understand how the hump-like feature emerges. Is it an interface effect, or can it be explained by simply combining two AHE signals? By comparing the model, Figs. 5(a) and 5(b), with the actual heterostructure, Figs. 5(c) and 5(d), at the two temperatures, hump (T = 40 K) vs no hump (T = 80 K), it is clear that the model and the actual heterostructure are qualitatively very similar. First, both show the hump at 40 K and not at 80 K. Therefore, in both the model and the heterostructure, the hump-like feature can be turned on and off with temperature. Second, the net sign of the AHE is negative for both the heterostructure and the model. From this, one can conclude that the same hump that is shown in the heterostructure can emerge by simply mixing two AHE signals of opposite sign. We think that the slight differences in the absolute amplitude and the shape of the hump between the model and the heterostructure could come from defects or slight calcium intermixing at the interface between the x = 0 and x = 0.4 layers. In addition, as mentioned in the supplementary material, the model is a simple parallel resistor model and is meant only to validate the concept of two parallel resistors of opposite AHE sign, yielding a hump-like feature in the Hall measurement.

D. Ruling out skyrmions
In order to rule out the possibility that the heterostructure contains skyrmions, we followed the method described by Kan et al. 14 and measured the dependence of R hump on the current density, the canting angle of the magnetic field, and minor field loops. 17 The current-dependent Hall measurements are shown in Fig. 6(a) at three different currents. These measurements basically overlap. To show this more clearly, we plot R hump at various current densities in Fig. 6(b). There is no current dependence on R hump . If magnetic skyrmions were present in this heterostructure, one would expect R hump to be strongly dependent on current and to decrease at higher current densities. 25 Next, we measured the dependence of the Hall resistivity at various field canting angles (θ), as shown in Fig. 6(c). Similar to the current dependence, when R hump is plotted as a function of the canting angle [ Fig. 6(d)], there seems to be little to no dependence up to angles as high as 50 ○ . Ohuchi et al. 26 showed in EuO that the THE is strongly dependent on the tilting of the applied magnetic field, which we do not observe in our heterostructure measurements. Finally, the dependence of the Hall resistivity on the minor field loops is measured and shown in Figs. 6(e) and 6(f). Here, a closed-loop field sweep is applied, where each loop starts at the same maximum, Bmax, of 4 T to saturate the AHE and potential THE fully. The applied magnetic field is then swept to impart the desired B min . The B min is increased (from a negative magnetic field toward a 0 magnetic field) in each subsequent loop. Minor field loops have been used similarly  Fig. 2) are input into a parallel resistor model simulation, the result of which is shown in (a) and (b), and compared to the actual bilayer heterostructure at the same temperatures in (c) and (d). Indeed, as compared to the experimental bilayer data, the simulation shows a similar hump, at a similar applied field, and of similar magnitude. In addition, the net sign of the AHE is negative for both the model and the heterostructure.
in previous publications. 14,17 It is apparent in Fig. 6(e) that the hump-like feature depends on the path taken in the various loops, and R hump falls off dramatically when B min is not fully saturating the potential THE in the negative direction, i.e., when |B min | < |B hump, saturation |. Therefore, the hump-like feature in our heterostructure is hysteretic, and as Kan et al. 14 note, hysteresis is not congruent with the skyrmion picture.

IV. CONCLUSION
In summary, we have artificially created a hump-like feature in a model heterostructure with engineered Berry curvature and demonstrated that this hump-like feature is the direct result of mixing two AHE channels of opposite sign and not caused by the presence of skyrmions. Our bilayer model is a simplified case, in which we assume that the AHE properties change sharply across the interface. In reality, however, all samples are inhomogeneous to some degree, and therefore, the heterostructure should have a more continuous change in AHE properties near the interface than our simulation does. Furthermore, in the simulation of this heterostructure, the two layers are in parallel. In a real material, however, there are "puddles" of differing AHE signals that are nested together, most likely yielding a mixture of series and parallel resistors. Nevertheless, we expect that our simple model can still be applied to better understand real materials where the hump-like feature is observed. Berry curvature heterostructures, like the one at the heart of this study, could enable the exploration of more exotic states at interfaces in the future.

SUPPLEMENTARY MATERIAL
Additional data and characterization relevant to this article and referenced in the main text-including Figs. S1 and S2-are provided in the supplementary material.