Odd-even layer-number effect and layer-dependent magnetic phase diagrams in MnBi2Te4

The intrinsic magnetic layered topological insulator MnBi2Te4 with nontrivial topological properties and magnetic order has become a promising system for exploring exotic quantum phenomena such as quantum anomalous Hall effect. However, the layer-dependent magnetism of MnBi2Te4, which is fundamental and crucial for further exploration of quantum phenomena in this system, remains elusive. Here, we use polar reflective magnetic circular dichroism spectroscopy, combined with theoretical calculations, to obtain an in-depth understanding of the layer-dependent magnetic properties in MnBi2Te4. The magnetic behavior of MnBi2Te4 exhibits evident odd-even layer-number effect, i.e. the oscillations of the coercivity of the hysteresis loop (at {\mu}0Hc) and the spin-flop transition (at {\mu}0H1), concerning the Zeeman energy and magnetic anisotropy energy. In the even-number septuple layers, an anomalous magnetic hysteresis loop is observed, which is attributed to the thickness-independent surface-related magnetization. Through the linear-chain model, we can clarify the odd-even effect of the spin-flop field and determine the evolution of magnetic states under the external magnetic field. The mean-field method also allows us to trace the experimentally observed magnetic phase diagrams to the magnetic fields, layer numbers and especially, temperature. Overall, by harnessing the unusual layer-dependent magnetic properties, our work paves the way for further study of quantum properties of MnBi2Te4.


Introduction
Recently, the research on the topological quantum materials has aroused tremendous interest and gained more and more attention in condensed matter physics [1][2][3][4][5] . Materials that combine magnetic and topological properties will reveal more exotic states, such as quantum anomalous Hall (QAH) insulators and axion insulators [6][7][8][9] . So far, such magnetic topological insulators (TIs) have been obtained by introducing magnetic atoms into TIs or using proximity effects in magnetic and topological materials heterostructures, however, the related exotic effects can only be observed at extremely low temperatures 6,10-12 . The recently discovered layered MnBi2Te4, showing an out-ofplane ferromagnetic coupling within the layer and antiferromagnetic coupling between the adjacent layers (A-type AFM), is found to be an intrinsic magnetic TI with antiferromagnetism [13][14][15][16][17][18] . The effective combination of antiferromagnetic order and nontrivial topological energy band makes MnBi2Te4 a promising material to discover novel topological phases and magnetic phase transitions by either controlling its crystal structures or applying magnetic fields [19][20][21][22] . Through complicated sample preparation processes, QAH and topological axion states were probed by lowtemperature electrical transport measurements in atomically thin flakes of MnBi2Te4 23,24 . However, comprehensively revealing the magnetic phase transitions of MnBi2Te4 under varying external magnetic field, temperature, and the number of layers has not been studied yet, which is of great significance for further exploration of the rich topological phenomena under different magnetic phases.
Polar reflective magnetic circular dichroism (RMCD) spectroscopy, which measures the differential absorption of left and right circularly polarized light induced by the out-of-plane magnetization of the sample (parallel to the light propagation), is a nondestructive optical method for measuring and imaging the magnetism of micro-sized flakes [25][26][27] . Owing to the small size of the laser spot (~2 μm in diameter), the RMCD spectroscopy measurements are less influenced by inhomogeneity of structural (the domain size of the MnBi2Te4 bulk was measured to be tens of μm 2 ) 28 , enabling subtle magnetic phases originating from finite-size effects in few-number (few-N) SLs MnBi2Te4 flakes to be detected. In addition, RMCD measurement does not require a complicated sample preparation process, which reduces fabrication-induced surface damages, and is very suitable for layer-dependent magnetic studies.
In this work, we utilize RMCD measurement (see the setup in Supplementary Fig. 1) to systematically study the magnetic properties of thin flakes, from single septuple layer (SL) to 9 SLs, and 25 SLs MnBi2Te4 under different applied magnetic fields and temperatures and drew their magnetic phase diagrams. The results show that for a single SL sample, the ferromagnetism is retained, and as the number of layers increases, the Néel temperature TN of the antiferromagnetic arrangement in adjacent layers increases simultaneously (from 15.2 K of 1 SL to 24.5 K of 25 SLs samples).
The magnetic behavior of MnBi2Te4 exhibits an evident odd-even layer-number oscillation. In the even-number (even-N) SLs MnBi2Te4, an anomalous magnetic hysteresis loop is observed, which is attributed to the thickness-independent surface-

Results
Layer-dependent ferromagnetism MnBi2Te4 is a layered ternary tetradymite compound with the space group of R3 m 29 , which consists of Te-Bi-Te-Mn-Te-Bi-Te SL stacking through van der Waals (vdWs) force. Below the TN, the spins of Mn 2+ ions couple ferromagnetically within the SL with an out-of-plane easy axis but have an antiferromagnetic exchange coupling between the adjacent SL (Fig. 1a), showing an A-type AFM order. The room-temperature Raman spectrum of the MnBi2Te4 crystal shows well-resolved Eg (47 cm −1 ), A1g (66 cm −1 ), E 2 g (104 cm −1 ), and A 2 1g (139 cm −1 ) Raman modes (see Supplementary Fig. 3a), consistent with previous reports 17,30 . The temperature-independence Raman spectra imply that there is no structure transition in the measured temperature range down to 2 K (see characterizations (see Supplementary Fig. 4 for details). The height line profiles of the 1 SL (Fig. 1c) and the stepped MnBi2Te4 flakes (Fig. 3b) indicate an SL thickness to be ~1.4±0.1 nm, consistent with previous reports 17,22 . Note all the optical and AFM images were obtained after removing PMMA unless otherwise specified.
The magnetic order of few-N SLs MnBi2Te4 was probed by RMCD microscopy as a function of the applied external magnetic field perpendicular to the sample plane. The RMCD signals were collected under a 0.25 µW 633 nm HeNe laser excitation with a spot size of ~ 2 μm (see results under a 532 nm CW laser excitation in Supplementary   Fig. 5). Fig. 1d shows the magnetic field dependence of the RMCD signals of 1 SL MnBi2Te4 at a temperature range from 1.6 K to 18 K. The nonzero RMCD signal at zero field and clear hysteresis loop confirm the ferromagnetism of 1 SL MnBi2Te4. As the temperature increases, the hysteresis loop shrinks and disappears at 18 K, indicating a ferromagnetic (FM) to a paramagnetic (PM) phase transition.
To study the layer-dependent magnetism, we investigated the behavior of thin flakes from 1 SL to 9 SLs under a magnetic field sweeping back and forth from +7 T to −7 T at 1.6 K. RMCD signals versus μ0H were shown in Fig. 1e. All measured odd-number (odd-N) SLs consistently show an FM behavior with a single hysteresis loop centered at μ0H = 0 T (highlighted by the grey shaded area in Fig. 1e), indicating its ferromagnetic feature due to an uncompensated layer. The coercive field μ0Hc odd increases monotonously with the thickness. In an odd-N SLs A-type AFM material, the Zeeman energy at a fixed magnetic field is proportional to the single uncompensated SL magnetization (invariant with the film thickness), while the anisotropy energy adds up with each SL (increases with the film thickness). Thus, a higher magnetic field is required for the Zeeman energy to overcome the anisotropy energy in the thicker odd-N SLs materials, resulting in a larger coercive field μ0Hc odd . Surprisingly, we also observed an anomalous magnetic hysteresis loop centered at μ0H = 0 T in even-number (even-N) MnBi2Te4 SLs, indicating a net magnetization, which is unexpected for an A-type AFM material. We note this anomalous FM response was also observed in Hall resistance measurements of 4 SLs MnBi2Te4 and was attributed to the possible substrate-induced top-bottom surface asymmetry or disorders in the sample 23 . The observed magnetic hysteresis loop is persistent in all the measured even-N SLs MnBi2Te4, and its coercive field, μ0Hc even , increases with the film thickness (highlighted by the pink shaded area in Fig. 1e). Coupled with the fact that the μ0Hc even is much larger than the μ0Hc odd regardless of the film thickness, we can conclude that the net magnetization in the even-N SLs samples is much smaller than those in the odd-N SLs samples and it is also not sensitive with the film thickness.
Thus, we rule out the possibility of the net magnetization induced by impurities, defects, or disorders, whose magnitude will increase with the film thickness. We attributed the net magnetization in the even-N SLs MnBi2Te4 to the thicknessindependent surface-related magnetization. Recent first-principles calculations and STEM results indicated that the abundant intrinsic Mn-Bi and tellurium vacancy in the exfoliated surface would cause a spontaneous surface collapse and reconstruction in few-layer MnBi2Te4, which might be the origin of the surface magnetization 31 .
Under larger magnetic fields, both the odd-N (except for the 1 SL) and even-N SL s flakes undergo spin-flop transitions 14,17 and evolve into complete out-of-plane magnetization above the spin-flip transition fields (µ0H2) 32,33 . However, for the flakes with N ≥ 4, the spin-flip fields are too large that beyond the magnitude of the magnetic field we can apply 33 . The spin-flop transitions exhibit strong odd-even layernumber effects. The spin-flop fields (µ0H1) in the odd-N flakes are much larger than those in the even-N flakes, and it decreases (slightly increases) with the film thickness in the odd-N (even-N) samples. The magnetic phase transitions can be understood quantitatively using an antiferromagnetic linear-chain model, where the magnetization of each layer is represented by a "macro-spin" coupled to its nearest neighbor layers through the interlayer exchange energy . This simplification is effective when the intralayer ferromagnetic coupling is much stronger than the interlayer antiferromagnetic coupling 34 , and it is reasonable to assume uniform magnetization within the single layer at zero temperature. For different layers, the magnetization in the i-th layer can be fully described by the angle, , with the normal direction of the sample. When a perpendicular magnetic field is applied, the average energy per unit cell reads  Thickness -temperature magnetic phase diagram Then, we discuss the thicknessdependent temperature-driven phase diagram from the AFM phase to the PM phase.
The height line profiles ( Fig. 3b and Supplementary Fig. 4) helps to clarify the layer number from 1 SL to 9 SLs samples shown in Fig  Temperature -field phase diagrams In the above linear chain model, only the ground state is considered, which corresponds to zero temperature. However, "macrospin" approximation will no longer hold strictly at finite temperatures, so we propose a more precise energy expression (see Eq. (6) in methods), which includes the energy from each spin site and its interactions with every other site. By utilizing the meanfield (MF) method for intralayer interactions to simplify the model, the spin sites are "decoupled" and we can choose one representative spin in each layer to get the Nmoment energy, which can be written as, where represents effective interlayer interaction, ∥ represents effective intralayer interaction (see supplementary Eq. (14)), and represents the magnetic anisotropy. Using this method, we obtained the temperature -field phase diagrams of 2 SLs to 6 SLs samples (Fig. 4, and see more in Supplementary Fig. 11). From temperature -

Discussion
In summary, we examined layer-dependent magnetism in atomically thin intrinsic magnetic TI MnBi2Te4 flakes with varying temperature and applied magnetic field using RMCD spectroscopy. An evident odd-even layer-number effect was observed in reported in previous work in details 35 . Then, a layer of PMMA was spin-coated on the MnBi2Te4 flakes for protection.

RMCD measurements
The RMCD measurements were performed based on the Attocube closed-cycle cryostat (attoDRY2100) down to 1.6 K and up to 9 T in the out-of-plane direction. The sample was moved by an x-y-z piezo stage (Piezo Positioning Electronic ANC300). A 633 nm HeNe laser with linear polarization was coupled into the system using free-space optics (see Supplementary Fig. 1 for details). Raman spectroscopy Raman spectra of thick MnBi2Te4 flake were obtained using the WITec alpha300 confocal innovation system at room temperature. A 532 nm laser was focused by a 50× (0.55 NA, Zeiss) objective onto the sample and the resultant Raman signals were detected using a spectrometer with a 1800g/mm grating coupled with a charged coupled device (CCD). The temperature-dependent low-frequency Raman spectra were obtained using free-space optics base on the Attocube closed-cycle cryostat (attoDRY2100). The detailed optical setup is represented in Supplementary   Fig. 3.
Antiferromagnetic linear-chain model The core of the antiferromagnetic linearchain model is to represent the spin moment in a single layer by one equivalent spin, which is coupled antiferromagnetically to its neighboring equivalent spins. At zero temperature, this simplification is valid because the intralayer exchange interactions are much stronger than the interlayer ones, and the ferromagnetic intralayer interaction ensures a uniform magnetization within a single layer for the ground state.
Denote the magnetization per unit cell in the -th layer as ⃗ , the average energy per unit cell reads where is the interlayer antiferromagnetic coupling, is the saturation magnetization per unit cell of a single layer, 0 is the easy-axis anisotropy energy, and H denotes the applied magnetic field. Here, the anisotropy includes both magnetocrystalline anisotropy ( , from spin-orbit coupling in the material) and the shape anisotropy ( , associated with magnetostatic interactions). Actually However, when the integral dimension is larger than 5, Monte Carlo integration methods must be employed. One Monte Carlo step represents a random change of the spin direction, and the energy difference and temperature determine whether the change is accepted. Iterations are performed to meet the self-consistent condition Eq. There might be more than one locally stable solutions to Eq. (7), but only one of them will finally be obtained after the iteration. Therefore, to ensure the repetitiveness of this approach, we change the magnetic field from very high (for example, 12 T, when all the magnetic moments are aligned in the same direction) to zero and then back to

S2. Temperature-dependent Raman spectra of the thick MnBi 2 Te 4 flakes
Temperature-dependent Raman spectroscopy was utilized to explore the temperaturedependent lattice structure transition or spin-phonon coupling. Supplementary Fig. 2 shows the schematic of the optical setup used to measure the temperature-dependent Temperature-dependent Raman spectra at a temperature range that passes through its

TN. The four Raman signatures show indistinguishable differences in our experimental
configuration. The silicon's Raman signature at 520 cm −1 was used as the reference.

S3. Atomic force microscopy (AFM) measurements of the exfoliated MnBi 2 Te 4 flakes
Before the AFM measurements, the PMMA covered on the MnBi2Te4 was removed using acetone, and then the sample was thoroughly rinsed with isopropyl alcohol (IPA). The detailed AFM images and height profiles of the sample measured in the main text are shown in Supplementary Fig. 4. SLs (combing AFM data from the main text) and a ~25 SLs flake on the Au substrate.

S5. Antiferromagnetic linear-chain model Part Ⅰ. Description of the model and its basic properties
As discussed in the main text, at the zero-temperature limit, we can assume a uniform magnetization within each layer and write the magnetization in the i-th layer as ⃗ sin , 0, cos . For an N-layer system, the magnetic energy, corresponding to Eq. (1) in the main text, is are both finite, and a sudden change of magnetization happens at H1. Since the hysteresis loop is relatively small, and the barrier between two locally stable states (when < H < ) is relatively low, it is reasonable to assume that only the ground state is dominant, and no loop or only a small loop (also observed in the previous works 5, 6 for MnBi2Te4) can be observed near H1.
All results show that this simple model works surprisingly well for all three regions.
At the end of this part, we describe an alternative method for calculating the evolution of magnetic states. It is noticeable that the most time-consuming step in the above algorithm is to find all the solutions to a set of equations because most numerical methods for solving equations or finding the minimum value rely on the given initial state to find a solution. To take advantage of such methods, we change the magnetic field from a very high magnitude (when all the magnetic moments are aligned in the same direction) step by step to zero and then return to the high magnitude step by step. In each step, we search the local minimum value of the energy function (1)  Based on our experimental data, the summation over N in Supplementary Eq. (7) from N = 2 to N = 9, and the uncertainty for each N is also estimated from our experimental data.
Given the above fitting method, for MnBi2Te4, this model yields μ0 = 5.10 T and as FM , is tridiagonal and can be explicitly calculated as For a specific , the eigenvalues of this matrix read FM cos , 1,2, ⋯ , 1 9 Therefore, it is obvious that at a sufficiently high field, all eigenvalues FM are positive, and thus FM is positive definite. Since the minimum eigenvalue corresponds to 1, it is straightforward that the spin-flip field is given by: where FM vanishes.
For N = 2 and N = 3, this result is consistent with our experimental results, which is also verified in the main text. However, for N ≥ 4, the spin-flip field is so large that it exceeds the magnetic field range we can apply experimentally. Moreover, due to the smooth transition process at H2, the spin-flip field is hard to be accurately determined experimentally. Hence, H2 is not used in the fitting of and .

S7. Temperature-dependent RMCD measurements
In addition to the thicknesses mentioned in the main text (1 SL, 2 SLs, 3 SLs, and 25 SLs), we also measured the temperature-dependent RMCD signals of 4 SLs, 5 SLs, and 6 SLs MnBi2Te4. As the temperature increases, the spin-flop field becomes smaller and eventually disappears at a specific temperature (that is how we define TN in even-N samples). Here, instead of completely solving the problem of this thermodynamics system, we perform a mean-field (MF) approximation to the intralayer interactions, which is effective if the temperature is not too high because the interlayer interactions are far smaller than the intralayer interacrtions 2 . In the finite-temperature linear chain model 1  After using a similar formulism in Supplementary Information S5 Part I, this energy function will yield the same result as Supplementary Eq. (2), showing that these two models are in general the same at zero temperature.

Supplementary
Part Ⅲ . The temperature -field phase diagram For a fixed layer number N and temperature T, the magnetization-field curve (M -H curve) can be calculated by using the method described in Methods in the main text, and then the total magnetization (along the -direction) is expressed as: The magnetic field μ0H changes from 12 T to zero and then returns to 12 T in steps of 0.05 T. The M -H curve from zero temperature up to over 24 K is summarized to draw the phase diagram, in which temperature interval is as small as 0.5 K. Using such methods, the field-temperature phase diagrams for N = 2 and N = 3 are shown in With the complicated hysteresis loops removed, we can define the spin-flop and spinflip fields. At a certain temperature, when the magnetic field is small enough, the state was AFM. As the magnetic field increases, the magnetization gradually increases, corresponding to a non-zero susceptibility, until it jumps to another state (CAFM state) with far larger magnetization at the spin-flop field H1. Then, the magnetization evolves rapidly until reaching the FM state, leaving only the paramagnetic background (see M -H curves in Supplementary Fig. 12 at temperatures near TN). At  At very low temperatures, the slope change at the spin-flip field H2 (marked by cyan arrows in Supplementary Figure 12) is very sharp, and a clear discontinuity of the second derivative is seen in the curves. However, at higher temperatures, the slope change becomes remarkably smoother, and it turns out that it is difficult to determine the spin-flip field H2 at high temperatures. For higher temperatures, the paramagnetic signal in the FM region is also larger. These effects explain why H2 is shallowed by the paramagnetic background and cannot be accurately obtained in our measurements at temperatures above ~ 14 K. H1 is marked by a black arrow, and the spin-flip field H2 is marked by a cyan arrow.
In summary, our MF model agrees quantitively well with all of our experiment data, as long as the temperature is below the Néel temperature, and helps us understand many characteristics in the experimental curves qualitatively. However, due to the experimental limitations, the verification of the spin-flip field is hard to perform quantitively, where a larger magnetic field and more precise techniques are needed.
Additionally, to understand the behavior of this system around the Néel temperature, other theoretical models (such as Monte Carlo simulations 2 directly employed to the original energy Supplementary Eq. (11)), might be applied.