Odd-Even Layer-Number Effect and Layer-Dependent Magnetic Phase Diagrams in ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$

Recently reported with nontrivial topological properties and magnetic orders, ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ is an intrinsic, magnetic topological insulator which holds promise for exploring exotic quantum phenomena such as the quantum anomalous Hall effect. However, the layer-dependent magnetism of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$, which is fundamental and crucial for further exploration of related quantum phenomena in this system, remains elusive. Here, by using polar reflective magnetic circular dichroism spectroscopy, we show that few-layered ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ exhibits an evident odd-even layer-number effect, i.e., the oscillations of the coercivity of the hysteresis loop (at ${\mu }_0{H}_c$) and the spin-flop transition (at ${\mu }_0{H}_1$), concerning the Zeeman energy and magnetic anisotropy energy. Noticeably, an anomalous magnetic hysteresis loop is observed in the even-number septuple-layered ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$, which might be attributed to the thickness-independent surface-related magnetization. A linear-chain model is applied to elucidate this odd-even layer-number effect of the spin-flop field and to determine the evolution of the magnetic states when subjected to an external magnetic field. A mean-field method further allows us to fully map the ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ flake’s magnetic phase diagrams in the parameter space of the magnetic field, layer number, and, especially, temperature. By harnessing the unusual layer-dependent magnetic properties, our work paves the way for further study of quantum phenomena of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$.

Popular Summary

The recently discovered material ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ has the potential to be a powerful laboratory for exploring many exotic quantum phenomena that combine intrinsic magnetic structure with topological behavior, in which the electrons can move only along the surface of the material. To pave the way for further exploration, we study the magnetic properties of ultrathin ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ flakes and find a number of intriguing magnetic behaviors that depend on the number of atomic layers in the material. These findings pave the way for further study of the quantum properties of this material.

${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ is built up from layered sheets of atoms. The number of layers, along with the magnetic properties, can alter topological behavior. So, we set out to quantify how these three players interact. To do so, we systematically study the magnetic properties of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ flakes down to a single layer. These observations, combined with theoretical calculations, allow us to describe the essential magnetic parameters of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$, which, in turn, let us map out how the magnetic state evolves in response to an external field. In particular, we discover distinctive behavior that depends on whether the number of layers is odd or even, along with extraordinary magnetism in a sample with an even number of layers.

The complex magnetic states in samples of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ with different numbers of layers under varying external magnetic fields and temperatures should allow researchers to further manipulate the related quantum states of this material by harnessing the diverse magnetic phases and, plausibly, integrating ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ with other 2D materials.

Figure 1

Crystal structure and RMCD measurements of 1-SL to 9-SL ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ flakes. (a) Crystal structure of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. The septuple atomic layers are stacked through vdW force. The arrows on the Mn atoms denote the magnetic moment of each Mn ion. Without an external magnetic field, the neighboring ferromagnetic SLs couple antiferromagnetically, with an out-of-plane orientation. (b) Typical optical image of a 1-SL ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ on the gold substrate. (c) Height line profile of the single-SL ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$, along with the white dashed line in panel (b). The step height is 1.4 nm, consistent with the thickness of 1-SL ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. (d) Temperature-dependent RMCD measurements of the 1-SL ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. (e) RMCD measurements of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ flakes (from 1 SL to 9 SLs) at 1.6 K. The shaded areas highlight the thickness dependence of the low-field spin-flip and spin-flop phase transitions in odd-$N$ and even-$N$ SL samples. Note that the layer-dependent measurements are performed on the freshly exfoliated flake without PMMA coating.

Figure 2

Odd-even layer-number effect and magnetic-state evolution of thin ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ flakes. (a,d) Zoomed-in RMCD measurements of all investigated odd-$N$ samples (a) and even-$N$ samples (d), showing a prominent odd-even layer-number effect in ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. In odd-$N$ SLs, the spin-flop transition occurs at about 4.5 T. The blue dashed arrow represents the spin-flop field in each thickness. In even-$N$ SLs, the spin-flop transition occurs at about 2.5 T, which is marked by the red dashed arrow. For 6 SLs and 8 SLs, the RMCD signals exhibit a second transition marked by the grey dashed arrow. (b,e) Spin-flop field versus $N$ for odd-$N$ SL (b) and even-$N$ SL (e) samples. The green circles with error bars denote the experimental results extracted from the RMCD measurements, while orange pentagons denote the calculated theoretical values of spin-flop fields using the parameters of ${\mu }_0{H}_J=5.10\mathrm{T}$ and ${\mu }_0{H}_K=1.58\mathrm{T}$. The dashed lines indicate the evolution of the spin-flop field (green for experiment and orange for theory). For the even-$N$ samples, there is a small deviation of ${\mu }_0{H}_1^{\mathrm{even}}$ between the theory and experiment, which may result from the nonzero magnetization in their ground AFM states at zero field. (c,f) Magnetic-state evolution under the applied magnetic field in 3-SL (c) and 2-SL (f) ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ obtained from the antiferromagnetic linear-chain model. Different colors denote the evolution of each layer in the 3-SL (c) and 2-SL (f) samples, and the arrow orientations indicate the detailed spin alignments. Dashed lines reveal the coincident evolution of different layers. (g) Magnetic-state evolution of the 6-SL sample under external magnetic field. Different colors represent the spin arrangement of different layers. (h) Detailed spin alignments of layers 1 to 6, with the corresponding colors of (g) at several representative external magnetic fields. (i) Calculated total magnetization along the $z$ axis (defined as $M=M_s{\sum }_{i=1}^N\cos {\varphi }_i$) as a function of the external field for 6-SL ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. The magnetization change at ${\mu }_0{H}_1^{\prime }$ is almost invisible compared to that at ${\mu }_0{H}_1$ and the coherent increase at ${\mu }_0{H}_s$.

Figure 3

Temperature-dependent RMCD measurements with different thicknesses and the thickness-temperature phase diagram of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. (a) Optical image of the stepped ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ flakes with 1 SL to 9 SLs. (b) Line height profile of the ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ flakes, along with the white dashed line in panel (a). (c) RMCD sweeps for the 2-SL sample at a temperature range that passes through its $T_N$. The spin-flop field (${\mu }_0{H}_1$) decreases as the temperature increases and eventually vanishes at about 20 K. (d) RMCD sweeps for the 3-SL sample at a temperature range that passes through its $T_N$. The coercive field and remnant RMCD signal of the magnetic loop decrease as the temperature increases, and both eventually vanish at about 22 K. (e) Remnant RMCD signal as a function of temperature for the selected few-$N$ SL flakes (1 SL, 3 SLs, 5 SLs, and 25 SLs). The solid lines are least-squares criticality fits with the form $(1-T/T_N{)}^{\beta }$, and the black dashed line represents zero RMCD signal. (f) Layer number-temperature phase diagram of the ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ flakes. PM denotes the region where the flake is paramagnetic; A-type AFM denotes the region where adjacent ferromagnetic SLs couple antiferromagnetically with each other.

Figure 4

Temperature-field phase diagrams of 2-SL and 3-SL ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. (a) Temperature-field phase diagram of 2-SL ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ determined by MF methods, which is in coincidence with the RMCD measurements. The white circles and triangles represent the calculated spin-flop field ${\mu }_0{H}_1$ and spin-flip field ${\mu }_0{H}_2$, respectively, at various temperatures, showing the boundaries of the A-type AFM/CAFM phase and CAFM/FM phase. The experimental data points are represented using grey spheres and triangles with corresponding error bars. (b) Temperature-field phase diagram of 3-SL ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$, showing the phase boundaries between the uncompensated A-type AFM/CAFM phase and CAFM/FM phase. The scale bar from pink to blue in both graphs shows the magnetization values of $M/M_s$ from zero to 3.0. The inset orange ($\uparrow$) and green ($\downarrow$) arrows denote the spin orientation of each SL.