Ferromagnetic-antiferromagnetic coexisting ground state and exchange bias effects in MnBi4Te7 and MnBi6Te10

Natural superlattice structures MnBi2Te4(Bi2Te3)n (n = 1, 2, ...), in which magnetic MnBi2Te4 layers are separated by nonmagnetic Bi2Te3 layers, hold band topology, magnetism and reduced interlayer coupling, providing a promising platform for the realization of exotic topological quantum states. However, their magnetism in the two-dimensional limit, which is crucial for further exploration of quantum phenomena, remains elusive. Here, complex ferromagnetic-antiferromagnetic coexisting ground states that persist down to the 2-septuple layers limit are observed and comprehensively investigated in MnBi4Te7 (n = 1) and MnBi6Te10 (n = 2). The ubiquitous Mn-Bi site mixing modifies or even changes the sign of the subtle interlayer magnetic interactions, yielding a spatially inhomogeneous interlayer coupling. Further, a tunable exchange bias effect, arising from the coupling between the ferromagnetic and antiferromagnetic components in the ground state, is observed in MnBi2Te4(Bi2Te3)n (n = 1, 2), which provides design principles and material platforms for future spintronic devices. Our work highlights a new approach toward the fine-tuning of magnetism and paves the way for further study of quantum phenomena in MnBi2Te4(Bi2Te3)n (n = 1, 2) as well as their magnetic applications.

and field-cooled (FC) magnetic susceptibilities of H ∥ c (χ c ) of MnBi4Te7 crystals measured at 1 kOe, which indicate an PM to AFM transition at a Néel temperature of 12.1 K. An obvious bifurcation between the FC and ZFC curves was observed at temperatures below TN, which also help confirm the FM component in the AFM states. (b) The ZFC and FC magnetic susceptibilities of H ∥ c (χ c ) of MnBi6Te10 crystals measured at 100 Oe, showing a Néel temperature of 10.9 K, and a more distinguished bifurcation between FC and ZFC curves was observed below a temperature of around 8 K (which were also obseved in [1,2]). The bifurcations of the ZFC and FC curves at temperatures slightly below the Néel temperature indicate the FM-AFM coexisting magnetic order. Scrutiny of the M − H curve at 2 K under H ∥ c (Fig. S3a), we find that MnBi 4 Te 7 (MnBi 6 Te 10 ) undergoes a spinflip transition at a very low magnetic field and quickly enters the forced ferromagnetic (FM) state at about 0.25 T (0.21 T). That is in sharp contrast with MnBi 2 Te 4 , where the spin-flop transition occurs at about 3.5 T and its magnetic moment finally saturates under an external magnetic field larger than 8 T. We estimate the interlayer antiferromagnetic coupling J c and single-ion anisotropy D based on the Stoner-Wohlfarth model [3]. Unlike MnBi 2 Te 4 , which has comparable anisotropy and interlayer exchange energy where g =2 is the Landé g factor, S = 5/2, and z = 2 (z = 6) is the Mn nearest neighbors in adjacent septuple layers of MnBi 4 Te 7 (MnBi 6 Te 10  [4][5][6], indicating a greatly reduced interlayer coupling from MnBi 2 Te 4 to MnBi 4 Te 7 and MnBi 6 Te 10 . Thus, the magnetic moment will not tend to be perpendicular to the direction of the applied field to cause the spin-flop transition, but will flip to be parallel to the direction of the field under a small critical field. It is worth noting that at low temperatures, the MnBi 4 Te 7 and MnBi 6 Te 10 crystals show non-zero magnetization under zero field, and the magnetic reversal is completed through three sluggish spin-flip transitions (marked by the arrows in insets of Fig.  S3a). The nonzero moments of the plateaus imply that there may be some residual FM states in the AFM state. These inconspicuous steps disappear above 6 K in MnBi 4 Te 7 (above 8 K in MnBi 6 Te 10 ) ( Fig. S3b and Fig. S3c).
Supplementary Note II. Atomic force microscopy measurements of few-layer MnBi 4 Te 7 and MnBi 6 Te 10 flakes Before atomic force microscopy measurements, the PMMA covering on the MnBi 4 Te 7 and MnBi 6 Te 10 samples were removed using acetone, and then the samples were thoroughly rinsed with isopropanol (IPA). The residual particles on the surface were cleaned by the contact mode of the instrument to obtain accurate step thicknesses. The detailed results are shown in the following. The RMCD measurements of all the few-layer MnBi 4 Te 7 and MnBi 6 Te 10 samples at 2 K are presented in this section. There are two kinds of terminations in MnBi 4 Te 7 , namely MBT and MBT+BT at the outermost layer, which are drawn in green and orange colors, respectively (Fig. S6). For MnBi 6 Te 10 , there are three kinds of terminations, namely MBT, MBT+BT and MBT+BT+BT at the outermost layer. The BT+BT protected samples were presented in the main text, thus here we only show MBT and MBT+BT terminations drawn in green and orange colors (Fig.  S7).
The temperature-dependent RMCD measurements of MnBi 6 Te 10 and MnBi 4 Te 7 samples are also shown in this part. With the increasing temperature, the hysteresis loop of 1 SL MnBi 6 Te 10 shrinks and disappears at 10 K, indicating an FM to PM phase transition at 8-10 K (Fig. S8a). Compared with the intrinsic 1-SL MnBi 2 Te 4 (T C = 14.5 K), the decreased T C may be due to the increased Bi Mn antisite defects, as the the intralayer exchange coupling decreases with the increase of the average distance between the occupied intrinsic Mn atoms. The 6 SL-MnBi 6 Te 10 posseses similar behaviors as that of 3 SLs presented in the main text (Fig. S8b). For temperature-dependent measurements of MnBi 4 Te 7 samples (Fig. S9), we drew the descending and ascending curves in blue and orange, respectively. The temperature-dependent measurements all show increasingly pronounced multi-step spin-flip transitions, which are similar to MnBi 6 Te 10 . However, there are still differences between the MnBi 6 Te 10 and MnBi 4 Te 7 , which are discussed in the exchange bias sections in the main text.

0.5%
Temperature-dependent RMCD measurements of MnBi6Te10 flakes. Temperature-dependent RMCD measurements of (MBT + 2BT) (a) and (MBT + 2BT)6 (b) samples. With increasing temperature, the hysteresis loop of (MBT + 2BT) shrinks and disappears at 10 K, indicating an FM to PM phase transition. (b) RMCD sweeps for the (MBT + 2BT)6 flake at a temperature range that passes through its TN. The behaviors are very similar to those of (MBT + 2BT)3 flakes shown in Fig. 1d in the main text, with slightly different spin-flip fields. The mixture of the interlayer AFM and FM coupling endows rich hysteresis behaviors in van der Waals magnets. Different ratios between AFM and FM couplings would result in distinct hysteresis loops. Here, we build a fivelayer macrospin model to interpret the hysteresis behavior in two materials of MnBi 4 Te 7 and MnBi 6 Te 10 which have different AFM/FM ratios and exhibit distinct hysteresis loops.
MnBi 6 Te 10 MnBi 4 Te 7 For the case of MnBi 4 Te 7 that is supposed to have a high AFM/FM ratio and strong interlayer AFM coupling strength (Fig. S10a), the top two magnetic moments are set to be FM coupled while the rest are set to be AFM coupled. When the magnetic field decreases from the positive saturation magnetic field, some of the magnetic moments flips as the interlayer AFM coupling prefers magnetic moment in the neighboring layer to be antiparallel, forming a step in the hysteresis loop. Because the AFM coupling strength is relatively strong, the switching magnetic field is high and Zeeman energy would keep most of the magnetic moment parallel to the magnetic field. Hence, only the magnetic moment in the middle of AFM region would be flipped. When the magnetic field is reversed, the magnetic moment in the FM region is flipped. Due to the magnetic configuration on the FM/AFM boundary, the minor hysteresis loop of FM region is shifted to the right side, indicating a positive exchange bias (Fig. 4a in main text). Further increase of magnetic field would saturate all magnetic moment in the negative direction. The hysteresis behavior under the magnetic field sweeping in the positive direction is reciprocal to that under the magnetic field sweeping in the negative direction. The minor hysteresis loop of FM region is then shifted to the left side, indicating a negative exchange bias (Fig. 4a in main text).
All the values in the hysteresis loop can be given by: where, m is magnetic moment of the macrospin, ∆E 1 , ∆E 2 , and ∆E 3 are the spin-flip barriers for each transition respectively. The strength of J AFM can be estimated by H E . For the case of MnBi 6 Te 10 that is supposed to have a low AFM/FM ratio and weak interlayer AFM coupling strength (Fig. S10b), one pair of magnetic moments is set to be AFM coupled while the rest are set to be FM coupled. Similar to the case of MnBi 4 Te 7 , when the magnetic field decreases from the positive saturation magnetic field, some of the magnetic moments flips as the interlayer AFM coupling prefers magnetic moment in the neighboring layer to be antiparallel, forming a step in the hysteresis loop. Because the AFM coupling strength is relatively weak, the switching magnetic field is low and the adjacent FM coupled magnetic moment is flipped as well. When the magnetic field is reversed, the magnetic moment in the FM region is flipped. As the third layer of magnetic moment is pointed to up and is FM coupled to the magnetic moments in the FM region, the minor hysteresis loop of FM region is shifted to the left side, indicating a negative exchange bias. Further increase of magnetic field would saturate all magnetic moment in the negative direction. The hysteresis behavior under the magnetic field sweeping in the positive direction is reciprocal to that under the magnetic field sweeping in the negative direction. The minor hysteresis loop of FM region is then shifted to the right side, indicating a positive exchange bias (Fig. 4b in the main text).
All the values in the hysteresis loop can be given by: where ∆E 1 , ∆E 2 , and ∆E 3 are the spin-flip barriers for each transition respectively. The strength of J FM can be estimated by H E . Therefore, the main features in the hysteresis loops of MnBi 4 Te 7 and MnBi 6 Te 10 can be interpreted by the five-layer macrospin model. In the real material, there is inhomogeneity in the distribution of AFM and FM coupling, which might cause the discrepancy of switching ratio between experimental hysteresis loop and macrospin model.  Fig. S11c) and Mn Te in QL (grey arrow). In addition, atomic electron energy loss spectroscopy (EELS) mapping shows clear Mn signals in the Bi layers in SL and also QL (Fig. S11d), demonstrating the existence of the Mn Bi antisite defects both in SL and QL. Supplementary Note VI. RMCD measurements of few-layer MnSb 2 Te 4 samples from three synthesized crystals To gain insight of the possible effects of the ubiquitous defects on magnetism, we turn to explore the magnetism in MnSb 2 Te 4 down to 1 SL. MnSb 2 Te 4 is isostructural to MnBi 2 Te 4 , and these two compounds have the same types of defects. MnSb 2 Te 4 crystals can be grown easily in a wide temperature range which makes it possible to tune the concentrations of lattice defects and hence the magnetism by varying the growth temperatures. As is expected, an evolution of A-type AFM to FM-AFM coexistence and finally to the FM ground state is observed with varying the Mn-Sb site-mixing concentration.

Supplementary Note VII. Discussions of the FM-AFM domain distribution
To evaluate the domain sizes of the FM and AFM components, we also characterize the magnetic spatial homogeneity by RMCD mapping in MnBi 4 Te 7 . In a typical RMCD-µ 0 H curve of a thick MnBi 4 Te 7 sample (Fig. S13a), we map the RMCD signals in a selected area (Fig. S13a, inset) under three selected magnetic fields (0.25 T, −0.1 T, and −0.25 T) corresponding to three different spin configurations (the plateaus around 0 T are hard to locate). The RMCD signals are uniform through the magnetic field sweep from state 1 to 3 (Figs. S13b-c) across the whole scanned area, indicating a homogeneous FM-AFM coexistence at a spatial resolution limited by the laser spot size of ∼ 2 µm. It should be noted that these spin-flip transitions, especially for the FM spin-flip transition at H c , are quite sharp despite the FM-AFM spatial inhomogeneity. In an inhomogeneous system, the sharp transition field suggests that its magnetic reversal is determined by the nucleation of reversed domain and the subsequent domain wall motion processes (i.e., the nucleation field is much higher than the propagation field) [7,8]. We take the most representative 3-SL MnBi 6 Te 10 as an example (Fig. S14a), we can see that the FM spin-flip transition (H c− ) is much sharper that the two AFM spin-flip transitions, and the second AFM spin-flip transition (H 2 f− ) is sharper than the first AFM spin-flip transition (H 1 f− ). To study these subtle differences, we need to consider the connectivity of the flipped components during these three spin-flip transitions. In the 3-SL sample, the two interlayer couplings would lead to four magnetic states (Fig. S14b), denoted by FF, FA, AF and AA (F for interlayer ferromagnetic coupling and A for interlayer antiferromagnetic coupling). To clarify the three spin-flip transitions discussed above and their connectivity, we performed a simplified simulation model that considered the spatial distributions of the four magnetic states in our sample.
We firstly generate a 200×200 matrix with randomly distributed FF, FA, AF, and AA magnetic states (each site represents a small uniform spin structure), as shown in Fig. S14c. Following the interlayer coupling and spin-flip rules under an external magnetic field, the spins in these four magnetic states will flip at H Using these connectivity laws of the spin flips at H 1 f− , H C− and H 2 f− (Fig. S14h), we can traverse the matrix in Fig. S14c and generate the connectivity matrices of the three spin-flip transitions, respectively. Specifically, at each spin-flip transition, for a magnetic state with coordinates (i, j) in the 200×200 matrix, if it has connectivity with the magnetic state at (i − 1, j) or (i, j − 1), it will be given the same value as the connected magnetic state, otherwise it will be given a new value (the specific value has no specific meaning). This assignment process will result in three new matrices with connected regions possessing the same values (color) and unconnected regions with new different values. Meanwhile, states that do not flip (such as FF state at the H 1 f− and H 2 f− transitions, and AA state at the H C− transition) are shown in white. In this way, we can obtain a connectivity map for each spin-flip transition (Fig.  S14i-k), where colored regions represent flipped states and white regions represent states that do not involve in that transition.
We can see that the connectivity map at H 1 f− shows a random discontinuous distribution (Fig. S14i), while the connectivity map at H C− shows roughly a single color of connectivity (Fig. S14j). Because of its excellent connectivity, for spin-flip transition of the FM component at H C− , even though the coupling strengths of each region are not exactly the same, once the reverse domain is nucleated, the DW will propagate across the sample to flip most of the FM component. This results in a sharp transition during the FM component reversal, which is in good agreement with what we observed experimentally. For spin-flip of the AFM component at H 1 f− (Fig. S14i), due to the disconnectivity and discrete distributions of the individual flipped regions, the process of nucleation and DW propagation will occur successively in each small area because of the inhomogeneous AFM interlayer coupling strength, resulting in a smeared transition (with a broad distribution of spin-flip field) compared with the H C− for FM component. We should also note that from the RMCD results (Fig. S14a), the transition at H 2 f− is sharper than that at H 1 f− , but still more sluggish than that at H C− , which can also be explained by the corresponding connectivity. The connectivity map at H 2 f− shows that its flipped regions are generally connected, but there are still some isolated island-liked connected regions scattered (Fig. S14k). Given the agreement between the experimental observations and the simulation results from our model, we believe that the magnetic reversal is determined by the nucleation of reversed domain and the subsequent DW motion processes, which indeed requires connectivity between flipped magnetic states for the continuous motion of the DW. Connectivity map at H f- The crystal structure of the as-grown MnBi 4 Te 7 (Table. I) and MnBi 6 Te 10 (Table. II) crystals were elucidated via single-crystal X-ray diffraction (SCXRD) measurement. Since the cation antisite defects are ubiquitous in MBT-type compounds [9][10][11], antisite mixing of Bi and Mn was taken into account in the structural refinement by allowing Bi to occupy Mn site and vice versa. The results indicate that with the intercalated Bi 2 Te 3 layers, the Mn Bi antisites increase from 0.5 % to 3 % and the Bi Mn antisites also increase from 26 % to 49 % from MnBi 4 Te 7 to MnBi 6 Te 10 . Combing with the STEM results, we can conclude the prevalent antisite defects distributed through the whole crystals.  To explore the stability of the exchange bias effect of the FM component in MBT systems, here, we take MnBi 6 Te 10 for an example. The large-field full hysteresis loop (grey data in Fig. S15) is plotted as reference for the minor hysteresis loops of the FM components. Historically polarized by a large positive saturation magnetic field, the minor hysteresis loop of the FM component shifts to the left side as we discussed in the main text. For magnetic field sweeping back and forth four times, all the minor loops overlap each other, showing no training effect and confirming the stability of the coupling between the AFM and FM components.