## Introduction

The dream of a ferroelectric ferromagnet that is fully coupled at room temperature is the grand challenge of multiferroics and magnetoelectrics. Heteroepitaxy enlarges the design space to achieve this difficult but important goal, and examples abound of superlattices and interfaces at which exotic properties emerge1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19. Superlattices of the form (LuFeO3)m/(LuFe2O4)n that sport ferroelectric ferrimagnetism are prominent examples20. The layer indices run from 0 to 9 and for simplicity are denoted (m, n). One end member h-LuFeO3 is a polar, improper ferroelectric below 1020 K, and it orders antiferromagnetically at 147 K in a pattern in which symmetry allows a slight canting of the spins—giving rise to weak ferromagnetism21,22,23,24. The other end member LuFe2O4 is an antiferroelectric with a complex phase diagram, exemplified by a series of charge-ordering transitions above room temperature, a 240 K ferrimagnetic ordering temperature, and a structural transition near 170 K25,26,27,28,29,30. The crystal structures of the end members are shown in Supplementary Fig. 1. In certain members of the (LuFeO3)m/(LuFe2O4)n series, ferroelectric ferrimagnetism emerges with ordering temperatures up to 281 K20. Such a superlattice has a higher magnetic ordering temperature than either of its end members due to interface effects20. The microscopic nature of these interface effects and their connection to the more robust magnetism is highly under explored. At the same time, the charge-ordering pattern in the LuFe2O4 layer is complex, with both polar and nonpolar Fe double layers predicted to be essentially isoenergetic20,25,26. The symmetric vs. asymmetric displacement of the upper and lower Lu layers adjacent to LuFe2O4 is one characteristic that differentiates the polar vs. nonpolar Fe double layer charge-ordering candidates. Resolving these issues is crucial for determining the ground state. Here, we use a spectroscopic approach that builds upon magnetic property, polarization, and neutron scattering measurements20,21,23 to reveal the site-specific electronic structure of these engineered materials, unraveling both the microscopic origin of high-temperature magnetism in the (LuFeO3)m/(LuFe2O4)n family of materials and the charge-ordering pattern in the Fe bilayer of the (3, 1) superlattice. In addition to introducing a remarkably powerful and versatile technique for extracting spin and charge character at the interface in a homologoes series of multiferroic heterostructures, our work opens the door to similar approaches in other engineered materials as well as opportunities for the development of structure–property relationships and interface descriptors, potentially advancing a number of allied fields including spintronics and photonics.

## Results and discussion

### Uncovering the electronic excitations of different Fe centers in the (3, 1) superlattice

We begin with the (3, 1) superlattice because it is the most theoretically tractable. Figure 1a, b displays the crystal structure highlighting the (LuFeO3)3/(LuFe2O4)1 layer pattern along with a scanning transmission electron microscope (STEM) image of the film. Inversion symmetry in the LuFe2O4 layer is broken due to the rumpling imposed by the adjacent LuFeO3 layers20. This is because the pattern of Lu-layer distortions around the Fe double layer is asymmetric with both down/up/up and down/up/down displacements along c. Here, d represents the size of the Lu-layer displacement. One way to separate the role of the different metal sites—at least in principle—is by projecting out the contribution of various layers and their Fe centers. Figure 1c displays the spin-projected density of states for the Fe double layer in LuFe2O4, the adjacent monolayer and the central monolayer in LuFeO3. The six types of excitations (summarized in Table 1) provide for a site-specific analysis of magnetism in (LuFeO3)3/(LuFe2O4)1. This is an unusual amount of complexity for dichroic analysis of an iron-containing material. Fortunately, of these six excitations, only three are important due to the relative size of the matrix elements. For instance, the two charge-transfer excitations are quite strong in the linear absorption and magnetic circular dichroism because they involve Fe sites with different charges. The on-site Fe2+d-to-d excitation is important in the dichroic response due to the large Fe2+ density of states in the spin-down channel (Fig. 1c). This feature is also evident in the optical absorption of the LuFe2O4 end member27. That these structures occur in different energy regions allows us to separate closely related Fe-containing excitations in both the optical absorption and magnetic circular dichroism.

Figure 2a, b summarizes the spectroscopic response of (LuFeO3)3/(LuFe2O4)1. The challenge that arises immediately —even upon cursory inspection of the dichroic spectra—is how to distinguish the different Fe contributions. One path forward is to employ the linear absorption spectrum, α(E), along with assignments from electronic structure calculations20,21,22,27,31,32 to determine characteristic excitation energies of each type of iron center. Figure 2a displays the optical absorption spectrum of the (3, 1) superlattice. Based upon our spin-projected density of states calculations (Fig. 1c and Table 1), the Fe-related excitations take place over a broad energy range. We can address the various site-specific Fe-related excitations by dividing the spectra into different energy regions and performing subtractions where necessary. As a reminder, the most important are (i) the Fe2+ → Fe3+ charge-transfer excitations in the spin-up and spin-down channels of the LuFe2O4 double layer and (ii) the Fe2+d → d on-site excitation in the spin-down channel of the LuFe2O4 double layer. These features are indicated in Fig. 2a.

Figure 2b displays the magnetic circular dichroism spectrum of the (3, 1) superlattice at full field. Access to the 25 T split helix magnet at the National High Magnetic Field Laboratory33 was crucial to this work, providing both direct optical access and a field high enough to saturate the magnetic state of interest. Two zero-field spectra are also included. They are not the same because the measurement pathway is hysteretic (25 T → 0 T  → −25 T → 0+ T → 25 T), and the ferrimagnetic film is not fully demagnetized when the field is removed. At full field, the dichroic spectra reveal a broad, asymmetric structure centered at 1.5 eV and a smaller lobe near 2.2 eV. Based upon Fig. 2a, we assign the 1.5 eV feature to the Fe2+ → Fe3+ charge-transfer excitation of the LuFe2O4 layer in the spin-down channel. The small 2.2 eV feature has a more complex origin because excitations are heavily mixed in this energy range. Our analysis shows that this structure emanates from a combination of charge-transfer excitations in both the spin-down and spin-up channels. The sign change at 2.1 eV is a reminder of how the spin-up channel density of states comes to dominate the response. There is another inflection point near 2.4 eV, above which Δα changes sign due to the way in which the spin-down channel Fe2+d → d excitation dominates the dichroic response. The on-site d-to-d excitations of Fe3+ have much lower intensity (Table 1). Importantly, features in the dichroic spectra Δα(E)MCD are directly proportional to net magnetization, and since we can analyze this effect at different energies, the response can be correlated with specific iron centers34,35,36.

### Revealing the role of each individual Fe center

In order to uncover the role of each type of Fe center, we take constant energy cuts of the dichroic spectra based upon the excitation of interest. For instance, constant energy cuts of the spectra at 1.33 eV reveal the behavior of the Fe2+ → Fe3+ charge-transfer excitation in the spin-down channel. Remarkably, a plot of ΔαMCD at 1.33 eV vs. magnetic field unveils an optical hysteresis loop (Fig. 2c). Because the charge-transfer excitation is located in the LuFe2O4 layer, we can explicitly connect the behavior of the Fe bilayer to the magnetic response. We also measure the dichroic spectra of the (3, 1) superlattice at different temperatures. Analysis again reveals optical hysteresis loops that close with increasing temperature (Fig. 2c).

We also consider how other Fe centers support high-temperature magnetism in the (3, 1) superlattice by taking cuts of ΔαMCD at several different energies (Fig. 2d, e). While the optical hysteresis loop at 1.33 eV has a traditional shape, the loop becomes irregular at higher energies due to mixing (Supplementary Fig. 4b, c). The challenge is to extract the response of each individual Fe center from the mixed state. We perform constant energy cuts at 1.8, 2.2, and 2.6 eV to address this issue. Details are discussed in the Supplementary Information (Supplementary Fig. 4). By subtracting the dichroic response at 1.8 eV from that at 2.2 eV, we can obtain the pure signal of the charge-transfer excitation in the spin-up channel (Fig. 2d). The direction of the hysteresis loop thus obtained is reversed because the spin-state changes from down to up. A similar analysis is applied to the 2.6 eV energy cut of the spectral data. Here, spin-up charge transfer is strongly mixed with spin-down Fe2+d-to-d on-site excitations. Subtraction yields the signature of the Fe2+ site (Fig. 2e). Note that the shape of the hysteresis loop returns to “normal” because the spin state flips again.

In order to link the microscopic response of the spin in the Fe double layer with the bulk magnetic properties20,21, we extract the coercive fields from the optical hysteresis loops (Fig. 2c–e) and plot these spectroscopically determined coercive fields (Hc) with those obtained from bulk magnetization, as a function of temperature (Fig. 2f). The trend in Hc is similar for all Fe centers, and the extracted coercive fields are in excellent agreement with bulk magnetization20,21. This demonstrates that a significant portion of the magnetism in the (3, 1) superlattice originates from the LuFe2O4 layer. In other words, the global coercive field is approximately equal to the local coercive field in the LuFe2O4 layer. We fit the temperature dependence of the coercive field in the (3, 1) superlattice with the Néel relaxation and Bean–Livingston models37,38, which relate Hc to the single-ion anisotropy (K), ferrimagnetic TC, and the power index n (Fig. 2f). Overall, this model is in reasonable agreement with our data, although n = 2.6 may indicate a slightly nonclassical response.

### Structure–property relations in the (LuFeO3)m/(LuFe2O4)n superlattices

In order to unravel the mechanism of high-temperature magnetism and the consequences of Lu-layer distortion on the electronic structure of the interface, we measured the dichroic response of the (7, 1) and (9, 1) superlattices and compared the results to those of the (3, 1) material. As a reminder, higher-order superlattices contain more LuFeO3 layers, which increases the amplitude of the asymmetric Lu-layer distortion and raises the Curie temperature TC (Fig. 1). In order to make the most effective comparison, we need to isolate the spectroscopic response of the interface. We begin by normalizing the magnetic circular dichroism spectra on a “per repeat unit” basis by taking the raw dichroic signal and dividing by the number of repeat layers (Eq. (3)). Next, we use the normalized spectra of two end members to construct a “composite spectrum” for each superlattice. The composite response is simply the dichroic signal generated from combining the per repeat unit end member spectra based upon the composition, as given by m and n (Eq. (4)). We extract the interface spectrum of each film by subtracting the composite response from the measured spectrum on a “per repeat unit” basis. A detailed discussion of this procedure is available in the “Methods” section and Supplementary Information. We immediately see that the interface spectra, ΔαMCD, Int., is significant—at least at certain energies. This indicates that additional magnetism arises from the LuFe2O4–LuFeO3 layer interaction.

Figure 3a summarizes the interface response of our set of superlattices. Remarkably, the interface spectra are nearly identical below 2 eV, demonstrating that magnetism emanating from the Fe2+ →  Fe3+ charge-transfer excitation in the spin-down channel is only minimally dependent upon the size of the Lu-layer distortion (or the number of LuFeO3 layers in the superlattice). The situation is different above 2 eV where, despite mixing with the spin-down channel transitions, the charge-transfer excitation in the spin-up channel dominates the dichroic response (Fig. 1c). This reveals that increased Lu-layer distortion selectively enhances the magnetic moment emanating from the spin-up channel Fe2+ →  Fe3+ charge-transfer excitation, which amplifies the LuFe2O4 layer magnetization, and therefore the dichroic signal. This analysis naturally raises the question of exactly how Lu-layer distortion impacts individual Fe sites in the LuFe2O4 bilayer. We unveil the interface behavior of each Fe center by taking constant energy cuts of ΔαMCD, Int. and plotting these values as a function of magnetic field. In addition to coercivity and related trends in the single-ion anisotropy (Fig. 3b), the optical hysteresis loops that we extract from the magnetic circular dichroism spectra of the interface yield a remnant value of the dichroism (ΔαMCD, rem.) that is proportional to remnant magnetization (Supplementary Fig. 4d–f). We can therefore reveal how superlattice periodicity affects local Fe site magnetization. Figure 3c displays ΔαMCD, rem. for the different Fe-related excitations as a function of the number of LuFeO3 layers. Because superlattice periodicity and the Lu-layer distortion are correlated, there is a relationship between ΔαMCD, rem. and the Lu-layer distortion as well. Above m = 3, the remnant signal from the charge-transfer excitation in the spin-up channel increases sharply—consistent with the theoretically predicted saturation moment in the LuFe2O4 layer20. By contrast, spin-down charge transfer and the Fe2+d → d excitation are relatively insensitive to the number of LuFeO3 layers (and the Lu-layer distortion). This behavior demonstrates that increasing magnetic moment in the LuFe2O4 layer emanates from rising Fe2+ and Fe3+ density of states in the spin-up channel of the higher-order superlattices. This conclusion arises from the corresponding changes in the dichroic spectra. At the same time, the trend provides a microscopic explanation for how high-temperature magnetism in these superlattices derives from Lu-layer distortion, as well as the more growth-oriented parameter of superlattice periodicity. We can understand in part why the enhanced magnetic moment emanates from the spin-up channel excitations by considering the charge-ordered state in greater detail.

### Determining the charge-ordering pattern in (LuFeO3)3/(LuFe2O4)1

Because charge ordering is one of the highest energy scales in the system20,25,26,27, magnetism in the (LuFeO3)m/(LuFe2O4)1 superlattices depends intimately upon the charge-ordering pattern in the Fe double layer. In order to reveal the relative importance of these states and distinguish between them, we calculated magnetic circular dichroism of several different candidate charge-ordering patterns, using first-principles methods and compared the results to the experimental dichroic spectra. The (3, 1) material has a large supercell containing 132 atoms, so we began by testing our predictions against the end members. Importantly, we tested two different states for the LuFe2O4 parent compound: CO-I and CO-II. Here, CO-I is an antiferroelectric state in which Lu trimer distortion is forbidden by symmetry (Supplementary Fig. 6a)25,39. CO-II, on the other hand, allows Lu trimer distortion and has alternate Fe2+- and Fe3+-rich layers, the stacking of which breaks inversion and introduces ferroelectricity (Supplementary Fig. 6b)40. As discussed in the Supplementary Information, the computed spectra of both LuFe2O4 and LuFeO3 are in good agreement with our measurements (Supplementary Fig. 7a, b), and a CO-I pattern is identified in the LuFe2O4 end member25,39. We therefore extended this approach to the (3, 1) superlattice.

The charge-ordered state in (LuFeO3)3/(LuFe2O4)1 is more complicated than that in the LuFe2O4 end member due to Lu-layer distortion at the interface, which is induced by the LuFeO3 layer. As a result, the CO-II-type state is more stable than CO-I for m ≥ 3 superlattices. Based on a CO-II arrangement in the LuFe2O4 bilayer, theory predicts two possible charge-ordering patterns for the (3, 1) superlattice20. These candidates, termed CO-FE and CO-DOPED for reasons that will become clear below, are slight variations (subsets) of the aforementioned CO-II pattern. What differentiates these candidates is (i) polar vs. nonpolar character of the Fe double layer and (ii) symmetric vs. asymmetric Lu-layer displacement. The latter is closely associated with the phase shift across the ferroelectric domain wall in the superlattices41. We find that the single-domain type charge-ordering state (CO-FE) is ferroelectric (Fig. 4a). This is because the Lu-layer distorts in the same direction along c and with the same down/up/up pattern throughout the material. On the other hand, the Lu-layer displacement is asymmetric in the doped-type state (CO-DOPED). Here, our calculations predict a spontaneous electron transfer from Fe2+ sites in the bilayer to Fe3+ sites in the LuFeO3 layer (Fig. 4b). This leads to an Fe3+-rich bilayer, which increases magnetization in the LuFe2O4 slab—consistent with a larger coercive field and higher moment. It is the electron transfer that periodically reverses the Lu-layer distortion to create the asymmetric down/up/up and down/up/down pattern across the Fe bilayer. This changes the direction of electric polarization across each domain wall which acts to create a nonpolar Fe double layer and overall antiferroelectric state. As we shall see below, this is the state that corresponds most closely with experiment.

Figure 4c compares the dichroic spectra for the (3, 1) superlattice with our calculations. As discussed above, two different CO-II-type charge-ordering patterns were imposed in the simulations, with the goal of distinguishing between them. These include CO-FE and CO-DOPED (Fig. 4a, b). Overall, the experimental spectrum is in agreement with the CO-DOPED model. This means that the Fe double layer is nonpolar, and the Lu-layer displacement is asymmetric. Comparison reveals very similar results below  ≈1.5 eV for both states. The model predictions separate above this energy—similar to what we find for the case of bulk LuFe2O4 (Supplementary Fig. 7b). Spectral signatures that distinguish the CO-DOPED model include the minimum near 1.5 eV and sign change near 2 eV. The overall agreement becomes less quantitative at higher energies—possibly due to additional complexity in the charge-ordering pattern due to charged ferroelectric domain walls or reduced measurement sensitivity as the absorption coefficient rises (Fig. 2a). In any case, all of our calculations in Figs. 1c and 4c implement this particular charge-ordering pattern and are internally consistent. The CO-DOPED model is likely to apply to the higher-order superlattices (m = 7 and 9) due to the stronger Lu-layer distortion, although calculations cannot be performed at this time due to the extraordinary size of the unit cells. Our finding for the nonpolar CO-DOPED model is consistent with real space HAADF-STEM images as well (Fig. 1b)20.

### Summary and outlook

By combining optical absorption spectroscopy, high-field magnetic circular dichroism, and first-principles calculations, we unravel the microscopic origin of high-temperature magnetism in the (LuFeO3)m/(LuFe2O4)1 superlattices (m = 3, 7, and 9) and, at the same time, reveal the charge-ordering pattern in the m = 3 member of this family of multiferroic materials. Analysis of the site-specific coercivity vs. temperature curves, obtained from constant energy cuts of the dichroic spectra, demonstrates that bulk magnetism derives principally from the LuFe2O4 layers. Magnetism emanating from the LuFe2O4 layer becomes more robust as the (3, 1) → (7, 1) → (9, 1) series progresses—a trend that correlates with increasing Lu-layer distortion. To understand this relationship more deeply, we extract the spectral signature of the interface for the (LuFeO3)m/(LuFe2O4)1 series (m = 3, 7, and 9). While the overall contribution of spin-down channel excitations is persistent over the sequence, enhanced Lu-layer distortion at the interface increases the contribution of the Fe2+ → Fe3+ charge-transfer excitation in the spin-up channel. This amplifies LuFe2O4 layer magnetization and pinpoints the role of Fe2+. Key to this discovery is the ability of magneto-optical spectroscopy to provide direct, microscopic, site-specific information about interface magnetism in a two-dimensional material with multiple magnetic centers. Comparison of the theoretically predicted magnetic circular dichroism with the experimental spectrum also establishes the nonpolar self-doped structure as the precise charge-ordering arrangement within the LuFe2O4 layer of the (3, 1) film, thus resolving controversy regarding the many different isoenergetic charge states. In addition to introducing a remarkably powerful and versatile spectroscopic decomposition technique for revealing microscopic spin and charge character at the interface of a multiferroic superlattice with many different iron centers in a site-selective manner, this work provides a pathway to link bulk and interface properties in other engineered materials.

In our view, superlattice multiferroics remain a huge untapped frontier that is very likely to yield high-performance room temperature fully coupled multiferroics. (LuFeO3)m/(LuFe2O4)n superlattices are just the tip of the iceberg. Synthesizing these interface multiferroics is challenging. But understanding the inner workings of these interface materials is in its infancy and the spectroscopic decomposition method that we report is a powerful means to learn about how they tick with site-specific understanding directly at the interface. Analogous opportunities exist to exploit interface materials to enhance spintronics and photonics. As a result, there is broad utility in revealing interface dynamics well beyond the multiferroics community.

## Methods

### Film growth and structural characterization

(LuFeO3)m/(LuFe2O4)1 (m = 3, 7, and 9) thin films were grown using reactive-oxide molecular-beam epitaxy on (111) (ZrO2)0.905(Y2O3)0.095 substrates. Lutetium and iron were evaporated from elemental sources and oxidized by a mixture of  ≈2% O3 and O2. The oxygen partial pressure was varied during the deposition to access the different iron valence states in LuFe2O4 (Fe2.5+) and LuFeO3 (Fe3+). The (LuFeO3)m/(LuFe2O4)1 (m = 3, 7, and 9) superlattices were grown as part of a full series of (LuFeO3)m/(LuFe2O4)n thin films to demonstrate consistent and reproducible trends in the ferroelectric and magnetic properties; characterization of the identical (LuFeO3)3/(LuFe2O4)1 film presented here by x-ray diffraction and bulk SQUID magnetometry is presented in ref. 20. The (3, 1), (7, 1), and (9, 1) superlattices, as well as the two end member films of LuFeO3 and LuFe2O4 were grown to a consistent number of iron layers to optimize the optical density and sensitivity for the transmission mode magnetic circular dichroism measurements. Cross-sectional TEM specimens were prepared using an FEI Strata 400 Focused Ion Beam with a final milling step of 2 keV to reduce surface damage. High-resolution HAADF-STEM images were acquired on an aberration-corrected 300 keV FEI Titan Themis with a probe convergence semi-angle of 30 mrad. Information on the Fe valence and oxygen stoichiometry of these films is available in Supplementary Information.

### Optical spectroscopy

We measured the ab-plane transmittance of the (3, 1), (7, 1), and (9, 1) superlattices, the LuFeO3 and LuFe2O4 end members, as well as the blank substrate using a λ-900 grating spectrometer covering the energy range from 1 to 6 eV. The linear absorption spectrum is calculated from measured transmittance as $$\alpha (E)=-\frac{1}{d}\mathrm{ln}\,(T(E))$$, where T(E) is the measured transmittance as a function of energy E and d is the sample thickness. The absolute absorption of (3, 1), (7, 1), and (9, 1) superlattices, as well as the end members were determined by subtracting the response of the substrate. Because the optical density of the films was optimized for magnetic circular dichroism spectroscopy rather than linear absorption, the excitations are not as pronounced as in prior work27,32. An open-flow cryostat provided temperature control (4.2–300 K).

### Magnetic circular dichroism spectroscopy

We measured the dichroic response of the superlattices (m = 3, 7, and 9), the LuFeO3 and LuFe2O4 end members and the (ZrO2)0.905(Y2O3)0.095 substrate between 0.8 and 2.8 eV. This is the energy window where our films transmit light. It is also the energy window where the most important excitations occur27,32. These experiments were performed at the National High Magnetic Field Laboratory using the 25 T split helix magnet33 in Faraday geometry along with a 240 W Xe lamp and a 0.25 m monochromator. We measured the difference in transmittance between left- and right-circularly polarized (LCP and RCP) light at various magnetic fields, and converted the result to absorbance difference, as discussed in detail below. Thus, the dichroic spectrum is the difference in absorption between LCP and RCP. A chopper was employed to increase the signal to noise ratio at a constant frequency, followed by a linear polarizer that was set to 45°. A photoelastic modulator was placed after the linear polarizer to convert the linearly polarized light into left or right circular polarized light periodically at a constant time interval $$\delta (t)=\lambda /4\sin (\omega t)$$. We did not need to keep the phase information, so an optical fiber was used to collect the light and route it to the detector. All signals were separated by lock-in amplifiers. The field sequence was chosen based upon the needed resolution, always within the +25 T → 0 T → −25 T → 0+ T → +25 T run pattern. The positive or negative sign of the magnetic field corresponds to the magnetic field direction and is parallel or antiparallel to the light propagation direction, respectively. The 0 and 0+ are both zero-field data; the sign denotes the sweep direction. Moreover, a training loop with this pattern was performed before each data collection run. The phase of the lock-in was set at full field. Magnetic circular dichroism spectra were taken at several different temperatures—from ~18 to 157 K for the (3, 1) and (7, 1) superlattices. For the (9, 1) superlattice, the temperature range was successfully increased to 218 K by adding an extra heater in the probe. Even so, we could not heat above this temperature.

### MCD data treatment

In this work, we report the magnetic circular dichroism spectra in two different ways: as an absolute ΔαMCD for each superlattice or one that is normalized by the number of repeat units (which is just ΔαMCD/N). Here, N is the number of repeat units for the superlattices or end members. The latter rendering allows comparison of interface effects. Substrate correction to the magnetic circular dichroism spectrum is also important. As shown in Supplementary Fig. 2a, the MCD spectrum of the (ZrO2)0.905(Y2O3)0.095 substrate is not zero because of the weak ferromagnetism induced from the defects42. We therefore subtracted it from the dichroic response of the superlattices to obtain the true ΔαMCD (or ΔαMCD per repeat unit). Details are available in Supplementary Information.

The magnetic circular dichroism spectra are obtained using the signal from the lock-in referenced to the photoelastic modulator divided by the signal from the lock-in referenced to the chopper. The chopper frequency is set to 217 Hz to improve the signal to noise ratio. There is, however, still some magnetic field-dependent background signal in the raw data (including the natural circular dichroism and the signal due to the drift of the probe), which dramatically affects the data quality when the dichroic signal from the sample is low—for instance in a nonmagnetic or antiferromagnetic material. To reveal the pure magnetic circular dichroism spectrum (ΔαMCD), the field-induced background signal (Δαbackground) was subtracted from the total spectrum (Δαtotal) as: ΔαMCD = Δαtotal − Δαbackground. At a given field H, we isolate Δαbackground by averaging the positive and negative signals of the same field magnitude. This is because the Δαbackground for both the positive and negative fields is only dependent on the intensity (and not the sign). In contrast, ΔαMCD depends on both the sign and the intensity of the magnetic field—making it an odd function. As a result, at the field of interest, the following relations should apply:

$$\Delta {\alpha }_{MCD}=\frac{1}{2}\times (\Delta {\alpha }_{+H}-\Delta {\alpha }_{-H})$$
(1)
$$\Delta {\alpha }_{background}=\frac{1}{2}\times (\Delta {\alpha }_{+H}+\Delta {\alpha }_{-H}),$$
(2)

where Δα±H is the raw MCD signal from the measurement at a positive or negative magnetic field H, respectively. These equations indicate the pure MCD signal from the sample should be the average of the difference between the positive and negative fields. This method of analysis was applied to the (3, 1), (7, 1), and (9, 1) superlattices, as well as to the spectra of the two end members.

### Extracting the coercive field and interface response from the dichroic spectra

There are two aspects of the data treatment that deserve special mention. The first is the constant energy cuts of the MCD data. The second is the manner in which we extract the interface spectra. Constant energy cuts of ΔαMCD were used to reveal the behavior of specific Fe centers, and how the excitations of these centers contribute to the overall magnetic response. By taking fixed energy cuts of the dichroic spectra over the full +25 T → 0 T → −25 T → 0+ T → +25 T data set, we can generate optical hysteresis loops corresponding to the excitation of interest. For instance, cuts at 1.33 eV probe the Fe2+ → Fe3+ charge-transfer excitation in the spin-down channel, and the hysteresis loop generated by these iron centers. We can extract a site-specific value of the coercive field from this type of optical hysteresis loop.

We also sought to isolate the interface response for each of the (LuFeO3)m/(LuFe2O4)1 superlattices (m = 3, 7, and 9). This is important because high-temperature magnetism emanates from strain and rumpling at the interface. We could not, however, compare the measured MCD spectra directly because, even though each of the films were specifically designed to have a consistent number of Fe layers. This is because they have a different number of interfaces. To obviate this problem, we normalized the spectra by the number of repeat units. The MCD spectra per repeat unit is given by:

$$\Delta \alpha {(E)}_{{\mathrm{per-repeat}}}=\frac{\Delta \alpha {(E)}_{{\mathrm{MCD}}}}{N},$$
(3)

where N is the number of repeat units. This quantity contains the information about the interface that we seek, but the response of the LuFeO3 and LuFe2O4 layers has to be eliminated, in order to uncover it. To estimate the effect of the different LuFeO3 and LuFe2O4 layers in the absence of the interfaces, we created a composite spectrum and subtracted this quantity from the measured MCD spectrum per repeat unit. We construct the composite spectrum of a hypothetical superlattice as:

$$\Delta {\alpha}_{{\mathrm{composite}}}=m\times \frac{\Delta {\alpha}_{{\mathrm{LuFeO}}_{3}}}{N}+\frac{\Delta {\alpha}_{{\mathrm{LuFe}}_{2}{\mathrm{O}}_{4}}}{N},$$
(4)

where m = 3, 7, and 9. As mentioned above, ΔαInterface = ΔαMeasured − ΔαComposite on a “per repeat unit” basis. This process is discussed in detail in Supplementary Information and illustrated in Supplementary Fig. 3.

### First-principles electronic structure theory

Density functional theory + U (DFT + U) calculations were performed using the plane augmented wave method, as implemented in the Vienna Ab Initio Package (VASP), and selecting the Perdew–Burke–Ernzerhof form of exchange correlation functional. Lu 4f states were considered in the core and we set U = 4.5 eV and JH = 0.95 eV for the Fe 3d states. DOS calculations were performed on the relaxed (LuFeO3)3/(LuFe2O4)1 superlattice exhibiting a 2:1 ratio between Fe3+ and Fe2+ charges in each monolayer of the LuFe2O4 block. Specifically, in the Fe3+–Fe3+–Fe2+ layer of the LuFe2O4 block, the Fe3+ centers are antiparallel to each other; one of the Fe3+ spins is up, and the other is down. The Fe2+’s are in different layers and align ferromagnetically. The details of this structure which was found to be the lowest energy configuration among different charge orders were previously described in ref. 20. The DOS were calculated with a 4 × 4 × 2 k-point mesh and a kinetic energy cutoff of 500 eV. In order to probe the robustness of our results with respect to U, we also performed our calculations for a larger value, e.g., U = 5.5 eV. This introduces a global shift in the states above and below the Fermi level, but leaves the main features of the DOS unaltered. We also confirmed that this type of change in the value of U does not impact our transition assignments.

### Absorption and magnetic circular dichroism spectra calculation based on different charge-ordering patterns

The dichroic response can be modeled using the calculated matrix elements of the optical conductivity tensor as36,43:

$$\Delta {\alpha }_{{\mathrm{MCD}}}\approx \frac{d\omega }{2c}\Im ({\overleftrightarrow{n}}_{+}-{\overleftrightarrow{n}}_{-})\approx \frac{2\pi d}{c}\Im [\frac{{\overleftrightarrow{\sigma }}_{xy}}{{(1+\imath \frac{4\pi }{\omega }{\overleftrightarrow{\sigma }}_{xx})}^{1/2}}].$$
(5)

Here, $${\overleftrightarrow{n}}_{\pm }={({\overleftrightarrow{\epsilon }}_{xx}\pm {\overleftrightarrow{\epsilon }}_{xy})}^{1/2}$$ is the refractive index of RCP or LCP light arising from the dielectric function ϵ, d is the film thickness, and c is the speed of light. The theoretical predictions for both the parent compounds and the (3, 1) superlattice were made based on this equation. The dielectric functions were calculated using exact diagonalization as implemented in VASP. The theoretically predicted magnetic circular dichroism spectra were calculated based on the predicted DOS of different charge-ordering patterns in LuFe2O4 layers. The spin configuration considered in each case corresponds to the ferrimagnetic collinear arrangement of spins obtained from direct calculation of the magnetic ground state. They are mostly characterized by a ferromagnetic alignment of the Fe2+ spins and an antiferromagnetic alignment of the Fe3+ ones. The magnetic dichroism spectrum of the LuFeO3 was computed considering the noncollinear A2 magnetic phase, which has been determined to be the magnetic ground state for this system and corresponds to a 120° angle in-plane (with a small tilt in the z-direction) arrangement of the spins within the Fe-monolayers in LuFeO322. Because LuFeO3 (where the spins form a 120° noncollinear structure) is expected to provide a smaller contribution to Δα(E) than the LuFe2O4 layer (Supplementary Fig. 7a, b), we computed Δα(E)MCD of the two candidate charge-ordering models considering collinear spin structures in the LuFeO3 layer.