Unveiling the Anomalous Hall Response of the Magnetic Structure Changes in the Epitaxial MnBi2Te4 Films

Recently discovered as an intrinsic antiferromagnetic topological insulator, MnBi2Te4 has attracted tremendous research interest, as it provides an ideal platform to explore the interplay between topological and magnetic orders. MnBi2Te4 displays distinct exotic topological phases that are inextricably linked to the different magnetic structures of the material. In this study, we conducted electrical transport measurements and systematically investigated the anomalous Hall response of epitaxial MnBi2Te4 films when subjected to an external magnetic field sweep, revealing the different magnetic structures stemming from the interplay of applied fields and the material’s intrinsic antiferromagnetic (AFM) ordering. Our results demonstrate that the nonsquare anomalous Hall loop is a consequence of the distinct reversal processes within individual septuple layers. These findings shed light on the intricate magnetic structures in MnBi2Te4 and related materials, offering insights into understanding their transport properties and facilitating the implementation of AFM topological electronics.

−6 Owning to its intriguing topological and magnetic properties, MnBi 2 Te 4 exhibits various exotic states, such as quantum anomalous Hall insulators, 7 axion insulators, 8 Weyl semimetals, 2,3,9 etc., which are not only appealing in fundamental research but also have tremendous potential in realistic applications.
The tetradymite MnBi 2 Te 4 is a van der Waals material with septuple layers (SLs) stacked along the c-axis. 10The magnetic moment contributed by Mn atoms is A-type AFM ordered in the ground state, i.e., ferromagnetically ordered within each SL, while the neighboring SLs are antiferromagnetically coupled. 5,11The magnetic structure varies when subjected to an external magnetic field and influences the properties of AFM topological insulators in a profound manner.The arrangement of the magnetic order determines the symmetry of the system, thereby dictating the topology of the electronic structure.This, in turn, shapes the transport properties of the AFM topological insulators.As such, various exotic phases are realized in MnBi 2 Te 4 under different magnetic structures.The quantum anomalous Hall effect is realized in MnBi 2 Te 4 with an odd number of layers that hosts uncompensated surface magnetization. 7In contrast, the axion insulator phase is observed in even-layer MnBi 2 Te 4 , in which the magnetization of top and bottom surfaces lies in opposite directions. 8−15 In its bulk form, ferromagnetic MnBi 2 Te 4 with an out-of-plane magnetic order is predicted to be a type-II Weyl semimetal, 2 while it transitions to a type-I Weyl semimetal when magnetization direction is pointing inplane. 9Additionally, MnBi 2 Te 4 with a canted AFM order is proposed to host Mobius insulator and higher-order topological insulator phases. 16n transport measurements, the anomalous Hall effect is an effective probe for uncovering the magnetic structure of the material. 17The epitaxially grown MnBi 2 Te 4 films in different reports 4,18−21 exhibit certain common features in transport: the anomalous Hall resistance saturates at high magnetic fields as the system enters a ferromagnetic state, and at small fields, a nonsquare hysteresis loop is often observed.The nonsquare loop has been attributed to the coexistence of two different anomalous Hall components, which are contributed from the MnBi 2 Te 4 phase and a secondary phase, respectively, although the type of the secondary phase may differ in different reports. 18,20Notably, the magnitude of the anomalous Hall resistance contributed from the secondary phase is comparable to that from MnBi 2 Te 4 .On the other hand, it has been demonstrated via multiple characterization methods that the epitaxially grown films show a predominance of the MnBi 2 Te 4 phase, 22−24 the anomalous Hall contributed from the secondary phase should be small, if not absent at all.
In this work, we investigate the anomalous Hall response of the MnBi 2 Te 4 films upon subjecting them to an external field sweep.The anomalous Hall curves obtained from the 1 SL MnBi 2 Te 4 displayed a square hysteresis loop, indicative of a single flip process in the magnetization reversal.In the 3 SL MnBi 2 Te 4 , however, a nonsquare hysteresis loop in observed.The nonsquare behavior arises as a consequence of AFM coupling between the SLs.By examining the anomalous Hall resistance, we uncover the magnetic structures of MnBi 2 Te 4 during the magnetization reversal process, which was further corroborated by our micromagnetic simulations.
The MnBi 2 Te 4 films are grown using molecular beam epitaxy (MBE), 4,22 and the details are described in Materials and Methods.The crystal structure of MnBi 2 Te 4 is schematically shown in Figure 1a, where each SL layer consists of atomic layers stacked in the order Te−Bi−Te−Mn−Te−Bi− Te along the c-axis.The epitaxial growth process was monitored using a reflection high electron energy diffraction (RHEED).The RHEED image of 5 SL MnBi 2 Te 4 is presented in Figure 1b, revealing sharp and streaky patterns indicative of high-quality films.The high quality of the film was further confirmed by X-ray diffraction (XRD) measurement, as shown in Figure 1c.The XRD results exhibit clear peaks corresponding to MnBi 2 Te 4 , while the peaks associated with other secondary phases, such as the most commonly observed Bi 2 Te 3 phase, are absent, demonstrating the high purity of the sample.
The MBE grown films are patterned into Hall bars for transport measurement, and the schematic drawing of the measurement setup is shown in the inset of Figure 2a. Figure 2 presents the anomalous Hall resistance (ρ AH ) data measured under a perpendicular magnetic field for MnBi 2 Te 4 samples with thicknesses ranging from 1 SL to 4 SL (the corresponding field dependences of the longitudinal resistance (ρ xx ) and Hall resistance (ρ xy ) for the same samples can be found in Figures   Figure 2e summarizes the thickness dependence of the temperature evolution of the anomalous Hall value at zero magnetic field (ρ AH 0 ).Upon increasing temperature, ρ AH 0 decreases across all thicknesses.We define T c as the temperature at which ρ AH 0 exceeds 5% of its base temperature value.The values of T c obtained in this manner are consistent with those acquired from the Arrott plot, as presented in Figure S3.The thickness dependence of T c is shown in Figure 2f, where the monotonic rise in T c with increasing thickness demonstrates a stronger magnetic coupling in thicker samples.Interestingly, the value of ρ AH 0 at 1.6 K displays oscillatory behavior (Figure 2f).This even−odd oscillation is a direct consequence of the antiferromagnetic ordering inherent to MnBi 2 Te 4 .In an ideal even layer sample, ρ AH 0 would be zero due to the absence of net magnetization.However, achieving a perfect sample with an exact number of 2n layers of MnBi 2 Te 4  is challenging in the MBE growth.The growth, conducted at a relatively high substrate temperature to avoid the formation of Mn-doped Bi 2 Te 3 , 22 leads to the formation of the 2n + 1 layer before the complete coverage of the 2n layer.Consequently, odd-layer components unavoidably exist in macroscopic evenlayer films, contributing to anomalous Hall signals.Therefore, the value of ρ AH 0 is comparable in the 2 SL and 4 SL samples and is significantly smaller than that in the 3 SL sample, even though the 4 SL sample possesses the highest T c .For the same reason, the anomalous Hall curves of the 2 and 4 SL MnBi 2 Te 4 (Figure 2b,d) are qualitatively similar to those in the 3 SL sample (Figure 2c).
Considering that the anomalous Hall resistance in even-layer samples stems from the residual magnetization of the odd-layer component, our following discussion focuses on the anomalous Hall results of the 1 and 3 SL samples and the insights they provide in revealing magnetic structures of MnBi 2 Te 4 .Figure 3a  , where ρ 0 and H c0 are the amplitude and coercivity of the anomalous Hall loop, respectively, and H 0 is a constant (see fitted results in Figure S6).As the direction of the applied tilts, a hump structure emerges in the hysteresis loop.This behavior can be attributed to the combined effects of the Zeeman energy and perpendicular magnetic anisotropy, as illustrated in Figure 3b.Under the high fields where Zeeman energy dominates, the magnetization lies along the field direction.At zero field, the perpendicular magnetic anisotropy drives the magnetization vertically aligned.Under modest fields, the magnetization tilts in a manner that minimizes the total energy.The observed anomalous Hall response is further corroborated by the model calculation.The total energy for 1 SL MnBi 2 Te 4 is E total = E Zeeman + E ani = −MH cos(α − θ) + K sin 2 θ, wherein M is magnetization, H is the applied field, K is the uniaxial anisotropy energy constant, and θ and α are the tilting angles of M and H, respectively.The calculation results are presented in Figure 3c, displaying good agreement with the experiment results.Note that in the discussion of anomalous Hall resistance of the 1 SL MnBi 2 Te 4 , no contribution from any secondary phase is involved.
Moving forward, we examine the anomalous Hall behavior in the 3 SL MnBi 2 Te 4 sample.The field dependence of ρ AH at the base temperature for the 3 SL MnBi 2 Te 4 sample is presented in Figure 4a.In contrast to the 1 SL case, the anomalous Hall curve of 3 SL MnBi 2 Te 4 exhibits a nonsquare hysteresis loop in which multiple steps are observed between two plateaus under saturated magnetization.That the nonsquare loop appears only in the 3 SL (as well as in 2 SL and 4 SL) sample but not the 1 SL one suggests it is not due to the existence of certain secondary phases in the material; rather, it indicates a more complicated magnetization reversal process in the multilayer MnBi 2 Te 4 .This complexity arises due to the AFM coupling between the SLs, which competes with ferromagnetic alignment forced by external magnetic field.Specifically, at high magnetic fields, the magnetization in all three SLs aligns along the applied field, while at zero magnetic field, the magnetization in the middle layer points in the opposite direction compared to the top and bottom layers, as schematically illustrated in Figure 4a.Therefore, during a full magnetization reversal process, the magnetization in the top and bottom layers flips once, while the magnetization in the middle layer undergoes three flips.
The anomalous Hall curve therefore can be decomposed into two components: ρ AH = ρ AH1 + ρ AH2 ; here ρ AH1 represents the contribution from the top and bottom layers and ρ AH2 is the contribution from the middle layer.The two components can be fitted with the expressions ρ AH1 = ( ) , respectively, to showcase the one-time and three-time flips of the reversal process in the corresponding layers.Here, ρ 0 is the amplitude of anomalous Hall contributed by a single SL, H c1 is the coercivity for the top/bottom layer, H c21 , H c22 , and H c23 are the field values when the magnetization of the middle layer undergoes a flip, and finally, H 1 , H 21 , H 22 , and H 23 are constants.The two components are displayed in Figure 4b,c, respectively.The fitted result of ρ AH is highlighted by dashed lines in Figure 4a.
To gain further insights into the intricate magnetic structure during the magnetization reversal process, we conducted micromagnetic simulations of 3 SL MnBi 2 Te 4 .The simulated result of the out-of-plane component of the magnetization is presented in Figure 4d, showing good agreement with the anomalous Hall resistance obtained from the transport measurements.Furthermore, we examine the out-of-plane component for each individual SL during the field sweep process.The contributions from the top/bottom and middle layers, denoted as M 1 and M 2 , respectively, are shown in Figure 4e,f.As expected, the magnetization in the top and bottom layers undergoes a single flip, while the magnetization in the middle layer experiences three successive flips throughout the entire reversal process.Notably, M 1 deviates from its saturation value in the field range of approximately 1 T < |B| < 3 T, indicating the canted magnetization in the top and bottom layers.This intriguing behavior is mirrored by the occurrence of the first flip in the middle layer within the same field range.The deviation can be attributed to the spin-flop transition as the magnetization transitions between the canted AFM state and AFM state. 13,18,25In Figure 4g, we present a schematic drawing of the magnetic structure for the entire reversal process.
Figure 5 presents the field dependence of ρ AH for 3 SL MnBi 2 Te 4 under different rotation angles.As the field direction changes from out-of-plane to in-plane, systematical changes are revealed in the anomalous Hall curves.Upon increasing θ, ρ AH s decreases monotonically, displaying a cosine dependence (Figure 5f).This indicates that in 3 SL MnBi 2 Te 4, the direction of magnetization is fully pinned by the applied field at high fields.As the field decreases to zero, the magnetization becomes AFM ordered along the vertical direction.As such, the anomalous Hall at zero field, ρ AH 0 , maintains almost the same value as θ varies between 0°and 80°.Interestingly, when θ = 90°, the value of ρ AH 0 is significantly reduced.This suggests that when pinned by an in-plane field, the magnetization does not fully restore the out-of-plane AFM alignment when the field is removed.As long as θ ≠ 90°, the out-of-plane component of the field breaks the symmetry and will force the magnetization to align along the vertical direction when approaching the zero field, resulting in a noticeable increase in ρ AH 0 .
As discussed above, the nonsquare hysteresis loop in the anomalous Hall curve reveals the complex evolution of the magnetic structure in the MnBi 2 Te 4 , which is subject to both the applied field and its intrinsic AFM ordering.Notably, the contribution from any secondary phase is not included in the discussion.−28 Given that MnBi 2 Te 4 naturally forms during the epitaxial growth of MnTe/Bi 2 Te 3 heterostructures or Mndoped Bi 2 Te 3 , it may be responsible for the nonsquare loops in the anomalous Hall effect of these materials as well.
In conclusion, we carried out transport measurement on epitaxial MnBi 2 Te 4 films with different thicknesses, and the anomalous Hall results reveal the intricate magnetization structure that arises due to the interplay of the applied field and its intrinsic AFM ordering.The observed nonsquare behavior in the anomalous Hall curves indicates the presence of multiple contributions from different SLs with distinct reversal processes.These findings are of significant importance as they provide insight into the intricate magnetic structures of MnBi 2 Te 4 , the exploration of which is crucial for understanding the exotic phases and unique properties exhibited by this material.Our work paves the way for the development of novel technologies based on AFM topological insulators.

■ MATERIALS AND METHODS
Sample Growth and Characterizations.The growth of the MnBi 2 Te 4 films was conducted in a home-built MBE chamber with a base vacuum rate of 1 × 10 −10 Torr.The semiinsulating Si (111) wafers were used as the substrate.Prior to the growth, Si(111) was flashed to 1200 °C to remove the oxides.During the growth, the substrate temperature was held at 240 °C, and high purity Mn (99.9999%),Bi (99.9999%), and Te (99.9999%) were evaporated using standard Knudsen cells.After the growth, a few layers of Bi 2 Te 3 were capped for protection.The growth was in situ monitored by an RHEED.The high-resolution XRD was performed using a PANalytical X'Pert Pro X-ray powder diffractometer with Cu Kα radiation (λ = 1.5406Å).
Transport Measurements.Transport measurements were conducted in an Oxford TeslatronPT system.A 1 μA AC current was sourced, and the voltage is picked up by standard lock-in amplifiers (SR830).The longitudinal and Hall resistances are symmetrized and antisymmetried, respectively.The rotation angle was calibrated by the ordinary Hall resistance.
Micromagnetic Simulations.Micromagnetic simulations were performed using the standard OOMMF 29 extensible solver (OXS).To achieve the construction of A-type AFM MnBi 2 Te 4 , we set the in-plane ferromagnetic intralayer coupling with positive exchange stiffness A in and the out-ofplane antiferromagnetic interlayer coupling with negative exchange stiffness A out .The system size in simulation is 100 × 100 × 3 nm 3 for a total of 3 layers with mesh size 5 × 5 × 1 nm 3 .The material parameters are set with intralayer exchange stiffness A in = 5 × 10 −14 J/m, interlayer exchange stiffness A out = −1 × 10 −13 , the saturation magnetization (M s = 1.4 × 10 5 A/ m) and the perpendicular anisotropy Ku = 2 × 10 4 J/m 3 .

Figure 1 .
Figure 1.Growth and characterizations of MnBi 2 Te 4 .a, Schematics of the atomic structure of MnBi 2 Te 4 .b, RHEED images of a MnBi 2 Te 4 sample along high symmetry azimuths.c, XRD result of a 7 SL MnBi 2 Te 4 .The peaks corresponding MnBi 2 Te 4 to are highlighted by blue arrows.
Figure 1.Growth and characterizations of MnBi 2 Te 4 .a, Schematics of the atomic structure of MnBi 2 Te 4 .b, RHEED images of a MnBi 2 Te 4 sample along high symmetry azimuths.c, XRD result of a 7 SL MnBi 2 Te 4 .The peaks corresponding MnBi 2 Te 4 to are highlighted by blue arrows.

Figure 2 .
Figure 2. Thickness dependence of the anomalous Hall curves under perpendicular fields.a−d, Field dependence of ρ AH for MnBi 2 Te 4 films with 1 SL to 4 SL thickness.e, Temperature dependence of ρ AH 0 for MnBi 2 Te 4 films with different thicknesses.f, Thickness dependence of T c and ρ AH 0 at 1.6 K.

Figure 3 .
Figure 3. Anomalous Hall curves of 1 SL MnBi 2 Te 4 .a, Field dependence of ρ AH under different field angles for a 1 SL MnBi 2 Te 4 .All curves were taken at 1.6 K. Inset: angle dependence of ρ AH s , displaying a cosine relationship.b, Schematics of the magnetization direction under a tilted filed.c, Calculated results for the field dependence of ρ AH under different field angles.

Figure 4 .
Figure 4. Anomalous Hall curves of 3 SL MnBi 2 Te 4 under a perpendicular field.a, Field dependence of ρ AH for a 3 SL MnBi 2 Te 4 taken at 1.6 K.The field is applied perpendicular to the sample surface.The dashed line is the fitted results.ρ AH can be decomposed into two components contributed by b, the top and bottom layers, and c, the middle layers.These layers undergo different reversal process, as schematically shown by the arrows in a. d, Microsimulation results of field dependence of magnetization for a 3 SL MnBi 2 Te 4 .e, Simulated field dependence of magnetization for the top/bottom layer.f, Simulated field dependence of magnetization for the middle layer.g, Simulated magnetic structure during the reversal process.
presents the field dependence of ρ AH under different θ for the 1 SL MnBi 2 Te 4 , wherein θ is the angle between the directions magnetic field and the sample normal.(The corresponding field dependence of ρ xy for different θ is shown in Figure S4.)At high magnetic fields, ρ AH saturates for all θ.The saturation value of ρ AH , denoted as ρ AH s , decreases as θ increases, exhibiting a cosine relationship (Figure 3a inset).That ρ AH s is proportional to the out-of-plane component of the magnetic field suggests the direction of the magnetization is fully pinned by the applied field at high fields.The field dependent ρ AH curves in 1 SL of MnBi 2 Te 4 exhibit distinct hysteresis loops for different field directions.When the applied field is perpendicular to the sample (θ = 0), as discussed above, a square hysteresis loop is observed.The a n o m a l o u s H a l l r e s i s t a n c e c a n b e fi t t e d t o

Figure 5 .
Figure 5. Angle-dependent anomalous Hall curves in the 3 SL MnBi 2 Te 4 .a−e, Field dependence of ρ AH for a 3 SL MnBi 2 Te 4 under different field angles.All data were taken at 1.6 K. f, Field angle dependent anomalous Hall resistance at saturation field, ρ AH s , and zero field, ρ AH 0 , respectively.
Transport results for MnBi 2 Te 4 films of thickness from 1 SL to 4 SL at different temperatures; Arrott plots for MnBi 2 Te 4 films of thickness from 1 SL to 4 SL; angle dependent results for odd-layer MnBi 2 Te 4 films; fitted results for the anomalous Hall loop in the 1 SL MnBi 2 Te 4 ; angle dependent anomalous Hall results for 2 SL and 4 SL MnBi 2 Te 4 (PDF)