SMR and XMLD measurements. We first show in Fig. 2a SMR signals of a control sample Fe2O3(12)/Pt(4) (units in nanometers), where the magnetic field (H) and current (I) is along x-axis and the spin polarization generated by the spin Hall effect of Pt is along y-axis. As expected, comparatively low resistance states at high magnetic fields reflect that the Néel vector (n) of Fe2O3 is perpendicular to H (I) due to the spin-flop at high fields and deviates towards H (I) at low fields, which is quite characteristic for negative SMR of AFM17–20. The resistance peaks at approximately µ0H = ±0.35 T owing to the deviation of n from the spin-flop state appears at a negative field as sweeping the field from positive to negative (black line), indicating that the Néel vector almost keeps the spin-flop state at zero-field16. Note that Fe2O3 with the thickness below tens of nanometers maintains easy-plane anisotropy without Morin transition16,18,20. Similar SMR signals are obtained in another control sample Cr2O3(4.4)/Fe2O3(4)/Pt(4) (Fig. 2b), where Fe2O3 was grown on a Cr2O3 buffer to ensure a closer scenario as the top Fe2O3 in the Fe2O3/Cr2O3/Fe2O3 junction. The SMR signals of the control samples are simulated and shown in Supplementary Note 2, where the hysteresis is due to the existence of Dzyaloshinskii-Moriya interaction (DMI) in Fe2O331. The antiferromagnetic Cr2O3 buffer possesses a high spin-flop field higher than 6 T9, which does not contribute to the observed SMR signals.
Figure 2c displays a representative HAADF-STEM (high-angle annular dark-field scanning transmission electron microscopy) image of the Fe2O3(12)/Cr2O3(4.4)/Fe2O3(4) cross-section, reflecting the epitaxial growth of the junction (Supplementary Note 3). Figure 2d presents the SMR curves of the Fe2O3/Cr2O3/Fe2O3 junction, which was covered by 4 nm-thick Pt. Four typical H [(i) → (iv)] are denoted in the inset. The most eminent feature is that two resistance peaks exist when sweeping H from positive to negative (black line) or reverse (red line), which is different from the SMR signals of a single Fe2O3 in Fig. 2a,b. The resistance peak appears before H = 0, which violates the principle of thermodynamics, indicating the existences of coupling effect. A low resistance is obtained for n ⊥ I (n is parallel to spin polarization) at the spin-flop state. As H is swept downward, the first resistance peak (high resistance state) at a positive H [µ0H = +0.3 T, (i)] reveals that n deviates from the spin-flop state and is unexpectedly aligned along n // H (I). This observation indicates that another effect suppresses the magnetic field effect. We attribute the overwhelming effect to the interlayer coupling between two Fe2O3 layers through the Cr2O3 spacer. The AFM coupling generates an orthogonal (90º) arrangement of n in two Fe2O3 layers. The coupling between net moment in Cr2O3 at high temperature and n in Fe2O3 is excluded (Supplementary Note 4). Based on both magnetic field (Fig. 2a,b) and angle dependent SMR measurements (Supplementary Note 5) in Fe2O3/Pt and Cr2O3/Fe2O3/Pt control samples, we find the top thinner Fe2O3 possesses a lower spin-flop field than its bottom thicker counterpart, in analogy to a soft ferromagnet with a small coercivity. Because of the relatively lower spin-flop field and smaller Zeeman energy of top thinner Fe2O3, the n in the top Fe2O3 has the priority to deviate from the spin-flop state as a result of the interlayer coupling, resulting in the resistance peak before zero-field. This is bolstered by the simulation based on calculating the energy profile of different magnetic configurations in Fig. 2e (Supplementary Note 2).
As H sweeps to the negative side, the SMR signal decreases and a resistance valley appears at negative H (ii), which is almost the same as the location of resistance peak in Fig. 2a. The direction of n in the bottom Fe2O3 is n // H, and the interlayer coupling drives the Néel vector in the top Fe2O3 to n ⊥ H (I), again giving rise to the orthogonal configuration [(ii)’ in Fig. 2e]. In this case, the spin current is reflected at the interface between Pt/top Fe2O3, leading to the relatively low resistance. The SMR valley in Fe2O3/Cr2O3/Fe2O3 occurs at the magnetic field that is much larger than that of net moment reversal (Supplementary Note 6), excluding possible coupling between net moment in Cr2O3 at high temperature and n in Fe2O3 as well as artifacts due to positive SMR from weak ferromagnetism (caused by defects or uncompensated interface). Then n in the bottom Fe2O3 rotates towards the spin-flop state (n ⊥ H) due to the increasing negative H, and n in the top Fe2O3 deviates towards n // H (I) [(iii)’ in Fig. 2e], resulting in the absorption of spin current and the second resistance peak (iii). It should be clarified that the second peak can exist when the coupling energy is large enough to overcome the Zeeman energy of the top Fe2O3 at the valley (ii), otherwise the n (top Fe2O3) will maintain spin-flop state rather than deviating towards n // H. The magnitude of the second peak is smaller than the first one can be ascribed to less component of n along x-axis. In contrast, the SMR in inverted sandwich, Fe2O3(4)/Cr2O3(4.4)/Fe2O3(12) (Supplementary Note 7), does not present resistance peak signal before H = 0, demonstrating that the n in the 12 nm-thick Fe2O3 maintains spin-flop state rather than deviating towards H at low magnetic field because of the large Zeeman energy, indicating the existence of the orthogonal coupling.
Besides magneto-transport measurements, we further confirm the interlayer coupling by direct Néel vector characterizations. Fe L-edge XMLD spectra were used to detect the n of the top Fe2O3 (several nanometers-thick sensitivity) in the Fe2O3/Cr2O3/Fe2O3 junction, where 2 nm-thick Pt was deposited on top. The XMLD spectra were recorded at zero-field after applying a high magnetic field along the x-axis for the sake of the non-volatile feature of n in easy-plane Fe2O316. X-ray was vertically incident to the film and the polarized direction was parallel to the film plane. XMLD signals are obtained as XMLD = XAS⊥− XAS//, where XAS// and XAS⊥ denote the x-ray absorption spectroscopy (XAS) recorded with polarization parallel with x-axis (//) and y-axis (⊥), respectively. Corresponding data are presented in Fig. 2f, where L2-edge is highlighted because it is generally used for the Fe-based XMLD spectra analysis32,33. Remarkably, Fe L2-edge XMLD spectrum exhibits a zero–positive–negative–zero feature, which is quite a characteristic for the n along the parallel direction (n // x-axis)32,33, rather than the spin-flop direction (y-axis). The n (top Fe2O3) aligned along H confirms the interlayer coupling, which is also corroborated by a series of XMLD measurements with sample rotation (Supplementary Note 8). The scenarios differ dramatically for the control samples Fe2O3 and Cr2O3/Fe2O3, in which identical experiments were carried out, but an opposite polarity at L2-edge (Fig. 2g,h, respectively), namely zero–negative–positive–zero, was observed, suggesting that the n in Fe2O3 is mainly aligned along the spin-flop direction (n // y-axis) without interlayer coupling.
Temperature dependence of interlayer coupling. We now turn towards the temperature dependence of SMR measurements in Fe2O3/Cr2O3/Fe2O3/Pt samples. Figure 3a shows the SMR results at various temperatures (Supplementary Note 9). At a relatively high temperature (T = 270 K), there exists two resistance peaks as we discussed above for T = 300 K, but the intensity of the second peak is lower than that at T = 300 K. Such a tendency continues with further decreasing temperature to 250 K, producing a tiny peak (or just a protruding), accompanied by the absence the second peak at 200 K. Also visible is that the location of the first resistance peak shifts towards zero-field with decreasing T but maintains at positive H, reflecting n // H in the top Fe2O3 before zero-field. This behavior discloses that although the interlayer coupling persists at low temperatures, the coupling energy decreases, resulting in the dominant spin-flop state and the disappearance of the second resistance peak. This phenomenon is similar to the temperature dependence of the spin fluctuation around equilibrium position in Cr2O3 spacer13. And we also demonstrate that the interlayer coupling does not depend on the magnetic field direction and the magnitude of applied reading current (Supplementary Note 10 and 11).
In addition to the H-dependent SMR we have explored the interlayer coupling between antiferromagnets by in-plane angle(α)-dependent SMR. Corresponding data of the Fe2O3/Cr2O3/Fe2O3/Pt sample at T = 300 K measured at two typical field of 0.5 T and 1 T are shown in Fig. 3b where α = 0º means H // I. For µ0H = 1 T, the SMR signals exhibit a negative polarity with the valley at α = 0º, which is a typical feature for the antiferromagnetic SMR at spin-flop state18,19. The situation differs dramatically for µ0H = 0.5 T. The SMR curve exhibits a positive polarity, indicating that the Néel vector of the top Fe2O3 maintains n // H due to the dominant interlayer coupling. This finding coincides with the results of field dependent SMR. In contrast, the polarity of SMR keeps negative in the control sample Cr2O3/Fe2O3/Pt (Supplementary Note 4), reflecting the antiferromagnetic feature of Fe2O3 and the absence of the interlayer coupling. Identical angle-dependent measurements were carried out in the Fe2O3/Cr2O3/Fe2O3/Pt sample with µ0H = 0.5 T at various temperatures. The polarity of SMR is positive at high temperatures (T = 350 and 300 K). With decreasing temperature to 250 K, the SMR signals become quite weak or even noisy, because of a competition between the interlayer coupling (n // H) and the H-induced spin-flop (n ⊥ H). This is accompanied by the typically negative SMR induced by the spin-flop with further decreasing temperature to 230 K and 200 K. The polarity of the control sample Cr2O3/Fe2O3/Pt is always negative at different temperatures (Supplementary Note 4), reflecting the absence of the interlayer coupling and further eliminating the existences of coupling between net moment in Cr2O3 and Néel vectors in top Fe2O3.
Interlayer coupling strength. It is significant to characterize the interlayer coupling strength. Considering that the existence of the first peak is the compromise between the interlayer coupling and spin-flop state, its location (µ0HCoupling) as a function of temperature for different Cr2O3 thicknesses (t) is summarized in Fig. 4a (Supplementary Note 12) to reflect the coupling strength. The first peak persists at a positive field for all of the measured SMR curves, suggesting the orthogonal antiferromagnetic interlayer coupling when the t ranges 3–4.4 nm. The maximum coupling strength increases with decreasing Cr2O3 thickness. For thin Cr2O3 (t = 3.0 and 3.5 nm), µ0HCoupling emerges from about 10 K, increases with increasing temperature and gets saturated at around 150 K. A plateau of the coupling strength exists from 150 K to 300 K. Then the coupling strength drops just above room temperature, which coincides with the spin fluctuation in Cr2O3 (bulk Néel temperature ~307 K)13. While for thick Cr2O3 (t = 4.1 and 4.4 nm), µ0HCoupling has an onset temperature of about 100 K, increases with increasing temperature and reaches the maximum just above room temperature, then drops, without showing a plateau. For thicker Cr2O3 (t = 6 and 12 nm), Fe2O3/Cr2O3/Fe2O3/Pt samples show almost the same SMR signals at 300 K as samples with only one Fe2O3 layer (Supplementary Note 13), indicating the absence of the interlayer coupling when the Cr2O3 layer is too thick. In addition, the interlayer coupling is observed in the Fe2O3/NiO/Fe2O3 junction with antiferromagnetic NiO spacer (Supplementary Note 14), indicating that the coupling effect is not restricted to a certain spaced material.
The distinct behavior for the samples with thin and thick Cr2O3 shows a significant role of spacer thickness. Since the out-of-plane anisotropy decrease rapidly with smaller sample size34, the most possible origin of the two distinct types of temperature dependence is that at low temperature, the moments in the thin Cr2O3 already have large in-plane component34–37, but the thick Cr2O3 have stable out-of-plane Néel order. The coupling arises from the fluctuating magnetic moments, hence thin Cr2O3 mediates coupling in low temperature, and thick Cr2O3 can only mediate coupling above 100 K.
As the coupling field shifts towards H = 0 with decreasing temperature, the magnitude of second resistance peak, which is also related to the deviation of Néel vector in top Fe2O3, also changes (Fig. 3a). We then summarize in Fig. 4b the temperature dependent magnetoresistance (MR) for the Fe2O3/Cr2O3/Fe2O3/Pt samples with various t. MR is related to the magnitude of the second resistance peak and is defined as MR = [R(second peak)–R(lowest)]/R(lowest), where the lowest is the minimum of the SMR curves. It can be seen that the MR exhibits a similar temperature dependence as the coupling field, strongly suggesting that the MR is also relevant to the interlayer coupling. As we discussed above (Fig. 2d), the second resistance peak is the result of competition between interlayer coupling and spin-flop state in the top Fe2O3 at the valley. When the temperature is low, the coupling energy is relatively small as compared with the Zeeman energy. Therefore, the Néel vector in the top Fe2O3 maintains spin-flop state (n ⊥ H), causing the vanishment of the second peak. The vanishing temperature (MR = 0) is obviously higher than its counterpart for no coupling effect (µ0HCoupling = 0), such as T = 225 K and T = 125 K for t = 4.4 nm, respectively. With further increase of temperature, the coupling energy is enhanced and exceeds the Zeeman energy, resulting in the deviation of more n towards H and the resultant rapid rise of the second resistance peak (MR). As the spin correlation is partially destroyed at high temperatures (T > 340 K for t = 4.4 nm), the coupling is reduced, accompanied by the decreasing of MR. An analogical situation occurs for the other samples (t = 3.0, 3.5, and 4.1 nm), while the samples with thin (t = 3.0 and 3.5 nm) and thick (t = 4.1 and 4.4 nm) Cr2O3 are divided into two groups according to the temperature dependence of the coupling strength.
Analysis on the magnetic ordering. In the following we discuss the role of magnetic ordering in the Cr2O3 spacer in the interlayer coupling of Fe2O3. The energies related to the Cr2O3 magnetic ordering are the interfacial coupling F(Nt,Fe, Mt,Fe, Nt,Cr, Mt,Cr), F(Nb,Fe, Mb,Fe, Nb,Cr, Mb,Cr), and the magnetic energy U in the Cr2O3 spacer, where N is the Néel vector and M is the net magnetization, t, b label the top and bottom surfaces, respectively. For thinner sample, the exchange energy makes it harder to let (Nt,Cr, Mt,Cr) ≠ (Nb,Cr, Mb,Cr). Thus, the observed interlayer coupling which increases with decreasing Cr2O3 thickness is hard to be explained by the non-uniform distribution of the magnetic order in Cr2O3 in the thickness direction. Also, it is known that Cr2O3 is lack of inter-unit cell DMI31, which favor out-of-plane spiral spin structure and may cause (Nt,Cr, Mt,Cr) ≠ (Nb,Cr, Mb,Cr). If (Nt,Cr, Mt,Cr) = (Nb,Cr, Mb,Cr), assuming F(Nt,Fe, Mt,Fe, Nt,Cr, Mt,Cr) ≤ F(Nb,Fe, Mb,Fe, Nb,Cr, Mb,Cr), one can lower the total energy by rotating (Nb,Fe, Mb,Fe) towards (Nt,Fe, Mt,Fe). The process above is solid even when the interfacial coupling at different interfaces has a different magnitude, as long as the interfacial coupling has the same form. Thus, by considering the Cr2O3 magnetic ordering which is uniform in the film plane, the lowest energy state always has (Nb,Fe, Mb,Fe) = (Nt,Fe, Mt,Fe), i.e., no orthogonal interlayer coupling can be generated.
Quasi-long range order mediated interlayer coupling. Having excluded the magnetic ordering which is uniform in the film plane as the origin of the interlayer coupling, we thus consider magnetic ordering which is non-uniform in the film plane, i.e., quasi-long range order (QLRO) enabled by fluctuations. It is known that in extremely thin magnets with in-plane moments, long-range magnetic order could be absent easily38, while spin correlation still exists39, which distinguishes from disorder paramagnetic phase [Fig. 4(e)]. Indeed, such QLRO emerges when the film gets thinner and the system drops from three to two dimension, consistent with the larger coupling strength in thinner films. Meanwhile, the temperature-dependent behavior of µ0HCoupling in Fig. 4a also strongly suggests that the interlayer coupling is correlated to the spin fluctuation in the Cr2O3 spacer. For QLRO state in the Cr2O3 sandwiched by two Fe2O3 layers, the parallel and perpendicular arrangements of the n in the two Fe2O3 layers means different boundary conditions. The perpendicular arrangement provides potential with \({D}_{2d}\) symmetry for QLRO rather than the \({D}_{2h}\) symmetry in parallel arrangement condition. Since the in-plane component of the moments point to all in-plane directions in QLRO state, the perpendicular arrangement provides more symmetric environment for such excitations than the parallel arrangement. Hence, we assume a smaller energy in the perpendicular arrangement, with the same entropy in the two arrangements since the entropy only depends on the QLRO itself40. This smaller energy makes the perpendicular arrangement favorable in the free energy, leading to the orthogonal interlayer coupling in the QLRO state.
Based on the above considerations, the schematic free energy diagrams at zero field for the thin and thick Cr2O3 cases are shown in Fig. 4(c) and (d), respectively. The ordered ground state has a free energy of F0, and all other states are represented by their free energy difference with the ground state F – F0. In thin Cr2O3 cases, no coupling exists in the ground state. At T1, QLRO emerges under the perpendicular condition, while the Cr2O3 under the parallel condition remains in the ground state. Therefore, the coupling emerges with a coupling strength represented by F0 – F(QLRO@90°) and increasing from T1 to T2. At T2, the Cr2O3 under the parallel condition switches from the ground state to the QLRO state, and the coupling strength saturates to F(QLRO@0°) – F(QLRO@90°). Then the Cr2O3 under the parallel condition switches from the QLRO state to paramagnetic state at T3, and the coupling strength F(paramagnet) – F(QLRO@90°) starts to decrease. The coupling strength finally vanishes at T4, where the Cr2O3 under the perpendicular condition switches to the disorder paramagnetic phase. Similar process happens to the thick Cr2O3 cases, with a larger T1 because the perpendicular anisotropy stabilizes the Néel order in the ground state. The coupling strength increases up to T3*, where the Cr2O3 under the parallel condition switches from the ground state directly to the disorder state, without entering the QLRO state. Then the coupling strength decreases and finally vanishes at T4.
The phase transition from the ordered ground state to the QLRO state can be phenomenally described by the following free energy
![](data:image/png;base64,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)
where T is the temperature, b is a parameter to stabilize the Néel Order, and λ is a parameter related to the entropy difference between the ground state and the QLRO. Δ reflects the energy difference between the ground state and the QLRO, with values Δ⊥ and Δ// for the perpendicular and the parallel conditions, respectively. Δ⊥ is smaller than Δ// and Δ// – Δ⊥ increases with decreasing Cr2O3 thickness. The Néel Order n and the volume of the QLRO state q satisfies
![](data:image/png;base64,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)
The first term in (1) describes the blue and red lines in Fig.4(c) and (d), and a combination of (1) and (2) gives the n-related part of the free energy ~ (Δ/λ–T) n2 + bn4, yielding a Néel temperature Δ/λ and n ~
near the Néel temperature40.The coupling energy Ec, the free energy difference under the two conditions (Supplementary Section 15),is displayed in Fig.4(f), obtaining a correspondence of 0.01 meV(per unit cell) to the coupling field 0.86 T, which semi-quantitatively agrees with the experimental curves (≤ 200 K).