$\widehat{P}\widehat{{T}}$ symmetry controlled magnetic order switching

Abstract

Precise electric control of magnetic order and anomalous Hall conductivity (AHC) is pivotal for spintronics. While electric-field control of magnetic order and AHC has been explored in magneto-electric materials, achieving precise and energy-efficient magnetic order switching between two $\hat{P}\hat{{T}}$ symmetry-connected magnetic states remains challenging. Here, we propose the utilization of the combined $\widehat{P}\widehat{{T}}$ symmetry that establishes a direct connection between electric polarization and magnetic orders, to electrically manipulate magnetic order and the AHC. Using 3MnB₂T₄·2B₂T₃ (B = Sb/Bi, T = Se/Te) as an example, we demonstrate that the $\widehat{P}\widehat{{T}}$ connected up-up-down (UUD) and up-down-down (UDD) states exhibit switchable magnetic configurations via electric polarization. The energy difference between the UUD and UDD states is linearly modulated by electric polarizations, enabling full control of the magnetic states via electric field, spontaneous polarization, or even weak sliding ferroelectricity. The findings demonstrate that $\widehat{P}\widehat{{T}}$ symmetry can be well utilized to design electric polarization-controlled magnetic orders and will find important applications in spintronics.

Introduction

Spontaneous magnetization is a key parameter in spintronic devices because the stability of the magnetization orientation ensures non-volatile data storage and access, while also affecting to some extent certain spin-related transport phenomena, such as the anomalous Hall effect (AHE). The primary methods for manipulating magnetization orientation are through external magnetic field or spin torques^1,2,3,4,5,6. However, generating an effective magnetic field or spin currents typically requires relatively large electric currents, which results in significant energy loss and poses limitations on the practical applications of spintronic devices.

The effective control of magnetic states and spin-dependent transport properties by electric field provides benefits of reducing energy consumption and ease of manipulation, making it pivotal in advancing future electronic devices^7,8,9. For example, the magnetic anisotropy of intermetallic compounds and localized flipping of ferromagnetic (FM) order in BiFeO₃ and Co_0.9Fe_0.1 can all be modified by an external electric field¹⁰. In addition, a reversible 180° magnetization direction change has been achieved in Co/PMN-PT heterostructures and bilayer CrI₃¹¹.

In addition, large magneto-electric (ME) couplings in magnetic, multiferroic materials and multiferroic heterojunctions can effectively couple electric polarization with magnetic orders, thereby enabling voltage modulations of the materials’ magnetic properties¹². However, in practice, there are limited types of multiferroic materials. For the realization of electronically controlled magnetism in multiferroic heterojunctions, it often necessitates precise control over stoichiometry, thickness, and interface structure, which can complicate realistic device design. As for traditional ME materials without spontaneous electric polarization like Cr₂O₃, the effect of electric polarization on magnetic order modulation is typically weak^13,14,15. Thus, despite these important advances in current research on electrically controlled magnetism, difficulties and challenges remain in realizing the complete regulation of magnetic state flipping by electric fields^16,17,18.

Fortunately, the recent emergence of two-dimensional (2D) magnetic materials^{19,20,21,22,23,24,25}, including magnetic topological materials^{26,27,28,29,30,31,32}, has provided new approaches to achieve an efficient magnetic order switching due to the existence of weak and modifiable interlayer magnetic couplings. 2D antiferromagnetic materials could exhibit an increased responsiveness to electric polarizations, attributed to their intrinsic combined symmetries. For instance, the joint $\widehat{O}\widehat{T}$ symmetry that arises from the combination of crystal symmetry ($\widehat{O}$) and the $\widehat{T}$ symmetry in antiferromagnets introduces additional degrees of freedom for electric-field-controlled interlayer magnetic state coupling. This is because an electric field can effectively break both $\widehat{O}$ symmetry and $\widehat{O}\widehat{T}$ symmetry. Among all types of joint symmetries, the $\widehat{P}\widehat{T}$ symmetry that links both the inversion and time-reversal symmetry is directly connected to electric polarizations, because internal or external electric polarization can efficiently disrupt $\widehat{P}$ symmetry, thus breaking the $\hat{P}\hat{T}$ symmetry as well.

The $\hat{P}\hat{T}$ symmetry itself is quite prevalent in most antiferromagnetic materials, but it disables the observation of some effects, such as the efficient tuning of magnetic order and the related AHE. We have demonstrated that the $\hat{P}\hat{T}$ symmetry must be broken to induce a switchable AHE that varies with the direction of interlayer electric polarization in antiferromagnetic systems^33,34. In contrast, for FM or ferrimagnetic materials, the $\hat{P}\hat{T}$ symmetry has been overlooked in magnetic order modulations due to two main reasons. Firstly, it does not occur in FM or ferrimagnetic materials because of their net magnetic moments³⁵, and secondly, the studies in antiferromagnetic systems indicate that the $\hat{P}\hat{T}$ symmetry is useless in magnetic order tuning, which inhibits the attempt to the exploration of $\hat{P}\hat{T}$ symmetry in non-antiferromagnetic systems. However, we will demonstrate that $\hat{P}\hat{T}$ symmetry provides a fantastic scheme for tuning the magnetic order in certain magnetic systems in which two different magnetic states are connected with each other by the $\hat{P}\hat{T}$ symmetry. It is ascribed to the fact that the $\hat{P}\hat{T}$ symmetry could weaken the dominant role of magnetic exchange interactions, while simultaneously amplifying the dominance of demagnetization energy, which eventually allows for efficient switching of $\hat{P}\hat{T}$-connected magnetic states.

Thus, here we propose a universal scheme for electric polarization switching of magnetic order and anomalous Hall conductivity (AHC) in FM materials, stemming from the $\hat{P}\hat{T}$ connection between different magnetic states. By utilizing 3MnB₂T₄·2B₂T₃ family as an example, we design a trilayer structure and prove that their up-up-down (UUD) and up-down-down (UDD) magnetic states could be efficiently reverted because of $\hat{P}\hat{T}$ symmetry combination. The “up” and “down” here refer to magnetic orientations pointing vertically upward and downward, respectively, along the vertical axis of the MnB₂T₄ plane. More importantly, the energy difference between the UUD and UDD magnetic states is linearly modulated by vertical electric polarization, which greatly facilitates the realization of full control of magnetic states with either electric field, spontaneous polarization, or even weak sliding ferroelectricity.

Results

Free energy analysis of different magnetic states

To analyze the $\hat{P}\hat{T}$ symmetry connection effects on magnetic order modulations of 2D magnetic materials, we start firstly from simple bilayer MnB₂T₄ films. There are two main magnetic orders here, the A-type antiferromagnetic order (A-AFM) and the FM one. The $\hat{P}\hat{T}$ symmetry exists in the AFM state, but it is broken in the FM one. There is no symmetry connection between the AFM and FM state. Our simulation results in Fig. 1 demonstrate that, the energy difference between the AFM and FM state is almost independent of the external electric field.

Fig. 1: The energy difference between the FM and AFM under different electric fields.

a, c Side view of atomic structure of MnBi₂Te₄ and MnSb₂Se₄, and possible FM, and A-type AFM. b, d The energy difference between FM and AFM magnetic orders under varying external vertical electric fields. Positive (Negative) values of the electric field indicate that it is aligned with (opposite to) the direction of the FM order.

These results could be elucidated by assessing the free energy of magnetic systems, which is described by the following equation³⁶:

$$F={F}_{0}+A+{F}_{d}+{F}_{H}-{F}_{{ME}}$$

(1)

where F₀ is the magnetic-independent energy, A is the interlayer magnetic exchange energy, F_H is the external magnetic field energy, F_d is the demagnetization energy, and F_ME is the magnetoelectric coupling energy. Here, bilayer MnBi₂Te₄ (MBT-2L) and MnSb₂Se₄ (MSS-2L) films are utilized as examples to elucidate the energy disparity between the A-AFM and FM configurations. When neither an external magnetic field nor an electric field is included, both the F_H and F_ME can be neglected, and thus the energy difference between the A-AFM and FM state is mainly determined by A and F_d. The absolute values of A and F_d are calculated to be 8.38 × 10⁻⁴ J/m², 3.7 × 10⁻⁴ J/m², respectively (see SI for the calculation details).

In case of the A-AFM state, A is −8.38 × 10⁻⁴ J/m² and F_d is zero since there is no net magnetic moment in A-AFM states. The total free energy of A-AFM states is −8.38 × 10⁻⁴ J/m². For the FM state, A is 8.38 × 10⁻⁴ J/m², and the net magnetic moment of bilayer MnBi₂Te₄ is as high as ~10 μ_B. Correspondingly, the demagnetization energy F_d is 3.7 × 10⁻⁴ J/m². Hence, the free energy of the FM state is 12.08 × 10⁻⁴ J/m², and the FM is always a high-energy state with respect to A-AFM. When an external electric field is applied, the influence of the electric field on the strength and net magnetic moment of the interlayer magnetic coupling is rather small. The largest induced net magnetic moment is 0.2 μ_B and the estimated value of the introduced ME coupling energy, ${F}_{{ME}}$, falls within the range of $1\times {10}^{-9} \sim 1\times {10}^{-10}{\rm{J}}/{{\rm{m}}}^{2}$. Such value has only a negligible impact on the energy difference between the two magnetic configurations, making it unlikely for the electric field to significantly alter the magnetic order of bilayer MnBi₂Te₄. Therefore, the absence of direct symmetry connections between the AFM and FM states results in a significant interlayer exchange energy difference of $-8.38\times {10}^{-4}{\rm{J}}/{{\rm{m}}}^{2}$, which predominates the magnetic ground state irrespective of the external electric field.

Accordingly, to increase the electric field or electric polarization modulation effects on magnetic orders, it is crucial to reduce magnetic exchange interaction while simultaneously establishing symmetric connections between different magnetic states. Firstly, the impacts of reduced magnetic exchange energy on the modulation of electric polarization are analyzed. For the MnB_xT_y family, it is already known that the insertion of a nonmagnetic B₂T₃ layer (where B represents Bi or Sb, and T denotes Te or Se) into bilayer or multilayer MnB₂T₄ structures will significantly decreases the interlayer A-AFM magnetic coupling strength. For example, for the intercalated MnBi₂Te₄-Bi₂Te₃-MnBi₂Te₄ structure, the intercalation of Bi₂Te₃ layer can greatly reduce the magnetic exchange energy, A, from $8.38\times {10}^{-4}{\rm{J}}/{{\rm{m}}}^{2}$ in bilayer MnBi₂Te₄ to$\,2.40\times {10}^{-6}{\rm{J}}/{{\rm{m}}}^{2}$ in MnBi₂Te₄-Bi₂Te₃-MnBi₂Te₄. However, because there is still no symmetry relation between the AFM and FM states in MnBi₂Te₄-Bi₂Te₃-MnBi₂Te₄, the positive interlayer exchange energy ($2.40\times {10}^{-6}{\rm{J}}/{{\rm{m}}}^{2}$) as well as large demagnetization energy ($3.7\times {10}^{-4}{\rm{J}}/{{\rm{m}}}^{2}$) in FM state still dominate free energy of magnetic system, so that the FM state is always the high energy magnetic states, and the magnetic order still can hardly be modulated by external electric field (Fig. S1). Secondly, the connection between two types of magnetic orders through $\hat{P}\hat{T}$ symmetry can improve their response to electric polarizations.

Linear modulation effects of electric field on $\hat{{\boldsymbol{P}}}\hat{{\boldsymbol{T}}}$ symmetry connected magnetic states

We extend the aforementioned analysis to the intercalation structure of MnB₂T₄-B₂T₃-MnB₂T₄-B₂T₃-MnB₂T₄ (denoted as 3MnB₂T₄·2B₂T₃) in Fig. 2a to acquire the$\,\hat{P}\hat{T}$ symmetry connection between the UUD and UDD magnetic states (see Fig. S2 in SI). We will show that, because of the$\,\hat{P}\hat{T}$ symmetry connection here, the magnetic order, and the related energy bands as well as anomalous Hall conductivities, can be efficiently modulated and reversed by electric field or electric polarizations.

Fig. 2: Electric field modulation effects on different magnetic order.

a Schematic of the electric field-controlled switchable anomalous Hall conductivity(middle) in 3MnB₂T₄·2B₂T₃ along with spin-resolved electronic band structures corresponding to the UUD (left) and UDD (right) magnetic states. The blue or brown rectangular shape in the diagram symbolizes the MnB₂T₄ layer, exhibiting an upward or downward magnetic moment. Conversely, the light blue shape denotes the incorporated B₂T₃ layer. The interlayer magnetic coupling could be modulated by varying electric polarization (−E and +E) between the top and bottom gate. The up (↑)-up (↑)-down (↓) (UUD) and up (↑) -down (↓)- down (↓) (UDD) magnetic order is connected by $\hat{P}\hat{T}$ symmetry, and the direction of the anomalous Hall current (the curved arrows) is flipped by the switching of electric polarization. b Side view of atomic structure of 3MnB₂Se_4·2B₂Se₃ (B = Bi, Sb). The yellow arrows indicate magnetic orientation in each magnetic layer, and the blue ones show the direction of electric field. c, d Energy difference between up (↑)-up (↑)-down (↓) and up (↑)-down (↓)-down (↓) magnetic states under different electric field in 3MnSb₂Se₄·2Sb₂Se₃ (MSS-5L), 3MnSb₂Se₄ (MSS-3L), 3MnBi₂Te₄·2Bi₂Te₃ (MBT-5L) and 3MnBi₂Te₄ (MBT-3L). Positive (negative) values indicate that the electric field is oriented in the upward (downward) direction.

It can be proved that the UUD or UDD magnetic order in 3MnB₂T₄·2B₂T₃ is the magnetic ground state instead of the traditional up-down-up (UDU) antiferromagnetic order in trilayer-MnB₂T₄ family. We utilize 3MnSb₂Se₄·2Sb₂Se₃ (hereafter referred to as MSS-5L) as an example to support above arguments. For simplicity, we only consider the nearest interlayer magnetic interaction in MSS-5L, and because of the intercalation of Sb₂Se₃, the absolute value of exchange energy ($A$) is calculated to be 1.6 × 10⁻⁶ J//m². For both the UUD and UDD magnetic orders, their demagnetization energy ${F}_{d}$ is calculated to be 6.36 × 10⁻⁴ J/m², and for the UDU magnetic order, the ${F}_{d}$ is calculated to be 6.40 × 10⁻⁴ J/m². The total free energy for UUD (UDD) is ${A-A+F}_{d}$, and it is 6.36 × 10⁻⁴ J/m², while for the UDU order, it is ${-A-A+F}_{d}$ and is equal to 6.37 × 10⁻⁴ J/m². Therefore, the UUD and UDD are degenerate magnetic ground states in free-standing 3MnSb₂Se₄·2Sb₂Se₃, and both have lower energy than the UDU state. Our DFT calculations on the magnetic ground states of MSS-5L in Table S1 also confirm that MSS-5L exhibits the UUD or UDD magnetic ground state, and it is consistent with above model analysis. Additionally, with the application of electric field, both the UUD and UDD magnetic order could become even more stable with respect to the UDU state (Fig. S3).

Symmetry analysis shows that for both the UUD and UDD magnetic states, the $\hat{P}\hat{T}$ symmetry is violated, implying a non-zero Berry curvature and non-zero intrinsic AHC that can be detected in experiments. Moreover, as it is schematically shown in Fig. 2a, because of the $\hat{P}\hat{T}$ symmetry connection between the UUD and UDD states, the sign of the Berry curvature, and the direction of the AHC are expected to be efficiently reversed by electric polarizations. All these results can be comprehended by examining the precise definitions of Berry curvature and AHC, which are presented in Eqs. (2), (3) and (4), respectively:

$$\varOmega \left(k\right)=\sum _{n}{f}_{{nk}}{\varOmega }_{{nk}}\,\left(k\right)$$

(2)

$${\varOmega }_{{nk}}(k)=-2{Im}{\sum }_{m\ne n}\frac{\left\langle n\left|\frac{\partial \widehat{H}}{\partial {k}_{x}}\right|m\right\rangle \left\langle m\left|\frac{\partial \widehat{H}}{\partial {k}_{y}}\right|n\right\rangle }{{\left({E}_{{nk}}-{E}_{{mk}}\right)}^{2}}$$

(3)

$${\sigma }_{{xy}}=-\frac{{{\rm{e}}}^{2}}{h}{\int }_{{BZ}}\frac{{{\rm{d}}}^{2}\kappa }{2\pi }\varOmega \left(k\right)$$

(4)

where ${f}_{{nk}}$ is Fermi distribution function, and ${\Omega }_{{nk}}(k)$ is Berry curvature of the nth band. $\hat{H}$ is the Hamiltonian of the system, and ${E}_{{nk}}$ is the energy of the ${n}_{{th}}$ band at wave vector k, and |m〉, |n〉 are corresponding eigenstates of $\hat{H}$. The intrinsic AHC (${\sigma }_{{xy}}$) is calculated through the integration of Ω(k) in the whole first Brillouin zone (BZ). It is easy to obtain that $\hat{P}\hat{T}\varOmega \left(k\right)=-\varOmega \left(k\right)$ and $\hat{P}\hat{T}{\sigma }_{{xy}}=-{\sigma }_{{xy}}$, thus the magnetic order transition from the UUD to UDD will effectively reverse the direction of ${\sigma }_{{xy}}$. Moreover, this reversed ${\sigma }_{{xy}}$ could also be understood from the electronic structures of the UUD and UDD states. From $\hat{P}\hat{T}$ symmetry analysis, we can get that $\hat{P}\hat{T}{E}^{{up}}\left(k\right)={E}^{{down}}\left(k\right)$, where ‘up’ and ‘down’ indicate the spin orientations. Therefore, the spin direction-dominated energy bands reverse as the magnetic order changes from the UUD to UDD (Fig. 2a), and results in the opposite ${\sigma }_{{xy}}$ in 3MnB₂T₄·2B₂T₃.

More importantly, as demonstrated in Fig. 2c, because of the $\hat{P}\hat{T}$ symmetry connection and the reduced interlayer magnetic exchange energy in 3MnSb₂Se₄·2Sb₂Se₃, the energy difference between the UUD and UDD is linearly modulated by an external electric field. Specifically, under zero electric field, the UUD and UDD are degenerate magnetic states. However, when a non-zero electric field is applied in the upwards or downwards direction, the UUD or UDD state becomes the preferred magnetic state. The energy difference between the UUD and UDD magnetic states is linearly modulated by an external electric field.

Magnetoelectric coupling and demagnetization effects

To elucidate this linear electric field modulation effects on the energy difference between the $\hat{P}\hat{T}$ symmetry-connected UUD and UDD magnetic states, we delve deeper into the ME coupling effects here. This ME coupling effect alters the net magnetic moment, transitioning the net magnetic moment from M₀ at zero electric field to M_I under the external electric field, defining the induced magnetic moment $\Delta {M}_{E}$. This induced magnetic moment variation could impact the free energy of the systems in three aspects: (a) it generates the ME coupling effects, and contributes to the ${F}_{{ME}}$ term in Eq. (1) to reduce the free energy of the system; (b) it changes the interlayer exchange energy $A$, because of the induced magnetic moment; and (c) it also modifies the demagnetization energy, which could increase or reduce free energy of the system depending on the direction of $\Delta {M}_{E}$ with respect to initial net magnetic moment of ${M}_{0}$.

Firstly, for the ME coupling effects, it can be estimated that the ME coupling coefficient ${\alpha }_{{zz}}$ is approximately $-3.00\times {10}^{-13}{\rm{s}}/{\rm{m}}$ under electric fields of ±0.1 V/Å, with ME coupling energy ${F}_{{ME}}$ approximately $-2.20\times {10}^{-10}{\rm{J}}/{{\rm{m}}}^{2}$, see the section of the SI on estimating ME coupling energy for the exact calculation procedure. These values are significantly lower than the interlayer exchange energy ($\sim 1.0\times {10}^{-6}J/{m}^{2}$) and demagnetization energy ($\sim 1.0\times {10}^{-3}J/{m}^{2}$). Therefore, the ME coupling effect on magnetic order variations can be ignored here. Secondly, because of the $\hat{P}\hat{T}$ symmetry connection between the UDD and UUD states, the interlayer magnetic exchange energy is the same. Therefore, it also does not impact the relative stability between the UDD and UUD states.

Finally, it can be proved that it is the demagnetization energy differences under electric fields that determines the energy difference between the UDD and UUD state. Specifically, the demagnetization energy (${F}_{d}$) can be expressed as ${F}_{d}=\frac{1}{2}{\mu }_{0}t{({M}_{0}+{\Delta M}_{E})}^{2}$ (Eq. S3 in SI), where ${\mu }_{0}$ and t are parameters, ${M}_{0}$ is saturation magnetization strength, and is identical for both the UUD and UDD states because of their $\hat{P}\hat{T}$ symmetry connection. ${\Delta M}_{E}$ is the magnetic moment induced by electric polarization, and it is rather small because of the above-mentioned weak ME coupling effects. Hence, the ${F}_{d}$ can be further expressed as ${F}_{d}=\frac{1}{2}{\mu }_{0}t{({M}_{0}+{\Delta M}_{E})}^{2}\cong \frac{1}{2}{\mu }_{0}t{M}_{0}^{2}+{\mu }_{0}t{M}_{0}{\Delta M}_{E}$. Therefore, the demagnetization energy variation under an electric field is $\Delta {F}_{d}={\mu }_{0}t{M}_{0}{\Delta M}_{E}$. Considering that$\,{\Delta M}_{E}=\frac{2{\alpha }_{{zz}}}{{\mu }_{0}}\Delta E$ due to the magnetoelectric coupling effects, $\Delta {F}_{d}=\,2{\alpha }_{{zz}}t{M}_{0}\Delta E$, and the demagnetization energy be linearly modulated by external electric field. It results in the linear modulation of energy difference between the UUD and UDD magnetic states under an external electric field (Fig. 2c). The corresponding ${\Delta M}_{E}$ induced by different electric field strengths of UUD and UDD system are shown in SI Fig. S2.

Energy barriers and electric field determined electronic structures and anomalous hall conductivity

Apart from the linear energy modulation between the $\hat{P}\hat{T}$ symmetry connected UUD and UDD magnetic states, we further calculate the switching energy barriers of magnetic moment in different layers of MSS-5L to understand the electric-field switchable magnetic orders between the UUD and UDD. Specifically, in the absence of an electric field, the $\hat{P}\hat{T}$-symmetry-connected UUD and UDD states are energetically degenerate. Only a 180° rotation of magnetic moments in the middle layer is required to switch the magnetic states between UUD and UDD. This avoids the large energy switching barrier for switching caused by the collective rotation of the magnetic order. As it is shown in Fig. 3a, the magnetic moment in the middle layer of MSS-5L can be easily reversed by crossing a rather low energy barrier of 0.05 meV, and it is consistent with its efficient orientation modulation under external electric field. The facile switching of the magnetic moment in the intermediate layer of 3MnB₂T₄·2B₂T₃ can be attributed to the following three factors: (i) the magnetic exchange interactions in 3MnB₂T₄·2B₂T₃ are relatively weak due to large interlayer separation of the magnetic layers; (ii) there exists a$\,\hat{P}\hat{T}$ symmetry connection between the UUD) and UDD magnetic states, resulting in identical anisotropy energy for both states; and (iii) magnetic frustration is present in both the UUD and UDD magnetic states, allowing the direction of the magnetic moment in the intermediate layer to align either upward (UUD) or downward (UDD).

Fig. 3: Energy barriers, electronic structures and anomalous Hall conductivities under different electric fields.

a Energy barriers for 180° magnetic moments reversing of different layers in MSS-5L(UUD). Top, Bottom and Middle layers demonstrate magnetic moment locations in MSS-5L. b Energy bands of MSS-5L under vertical electric field of 0.1 V/Å (left) and −0.1 V/Å (right), where gold solid lines represent spin-up dominated states and blue-purple lines represent spin-down dominated ones. c Berry curvatures in the 2D BZ, with the energy level 60 meV below the Fermi level, for electric fields pointing upwards (top) and downwards (bottom). d The AHC (${\sigma }_{{xy}}$) of MSS-5L as a function of electronic energy, depicted for electric fields pointing upwards (${E}_{u}$) and downwards (${E}_{d}$).

The electronic structures, Berry curvatures, as well as anomalous hall conductivities (${\sigma }_{{xy}}$) are calculated for the UUD and UDD magnetic states to analyze electric field modulation effects on MSS-5L. The energy bands in Fig. 3b show that, because of the $\hat{P}\hat{T}$ symmetry connection ($\hat{P}\hat{T}{E}^{{up}}\left(k\right)={E}^{{down}}\left(k\right)$), the magnetic state transition from the UUD to UDD configuration has effectively altered the relative positions of spin-up dominant states and spin-down dominant states within the energy bands. Moreover, due to$\,\hat{P}\hat{T}\varOmega \left(k\right)=-\varOmega \left(k\right)$ and $\hat{P}\hat{T}{\sigma }_{{xy}}=-{\sigma }_{{xy}}$ symmetry connections, electric field also efficiently modulates Berry curvature and reverses the direction of AHC (Fig. 3c, d). The 2D hexagonal distribution of $\varOmega \left(k\right)$ across the entire first BZ also prominently demonstrates the combined ${C}_{3z}\hat{T}$ symmetry (the ${C}_{3z}$ represents 120° rotation around the vertical axis) in MSS-5L. This results in red iso-surface (indicating positive values) or blue one (indicating negative values) recovered after 120° rotation in the BZ.

Additionally, asymmetric nonzero Berry curvature gives rise to intrinsic AHC (${\sigma }_{{xy}}$) and it is given in Fig. 3d. The switchable AHC is achieved through the magnetic order variation from the UUD to UDD, which could be easily achieved through electric field modulation (Fig. 2c). Considering the nearly linear tunability of UUD and UDD magnetic orders in MSS-5L under electric fields, we aim to utilize weak electric polarization to achieve a non-volatile control over spin-related transport properties. It is widely recognized that sliding ferroelectricity could exist in 2D vdW materials^37,38,39,40. However, it may be too weak to be applied in magnetic states modulation. Here, we prove that the introduction of interlayer sliding ferroelectricity in MSS-5L through vdW assembly could efficiently reverse the magnetic order, and so efficiently modules the topological transport properties (Fig. S4).

Apart from weak sliding ferroelectric polarization, we further show that interlayer magnetic coupling and the corresponding transport properties of MSS-5L could also be achieved by constructing In₂Se₃/MSS-5L/In₂Se₃ multiferroic heterostructures, where In₂Se₃ is a prototypical 2D ferroelectric material due to its electric field switchable spontaneous polarization. We firstly conduct systematic calculations on various stacking configurations of MnSb₂Se₄/In₂Se₃ heterostructures, as shown in Fig. S6, to identify the most stable interface between MnSb₂Se₄ and In₂Se₃. Next, as it is shown in Fig. 4a, symmetry analysis shows that$\,\hat{P}\hat{T}$ symmetry is conserved in this multiferroic heterostructure. Therefore, the magnetic ground state of MSS-5L in In₂Se₃/MSS-5L/In₂Se₃ hybrid structure is totally determined by the direction of ferroelectric polarization. When the ferroelectric polarization direction is upwards, the magnetic ground state of MSS-5L is in the UUD order; conversely, when the direction is downwards, the magnetic ground state is the UDD one. Therefore, the magnetic order, along with the corresponding AHC can be efficiently reversed by the electric polarization. All results are given in Fig. 4b-d, and the band structure of +PP in Fig. 4b shows that because of the generation of built-in electric field of In₂Se₃, the band gap of the MSS-5L is reduced to 0.1 eV. The Berry curvatures for both +PP and −PP at E − E_f = −0.05 eV energy level are given in Fig. 4c. At the same K point, the value of Berry curvature of +PP state is opposite to that of −PP ones, and because of this, the direction of AHC can be totally controlled (as shown in Fig. 4d) by the spontaneous electric polarization of In₂Se₃.

Fig. 4: Ferroelectric polarization effects on magnetic orders in the heterostructure.

a Schematic structure of In₂Se₃/MSS-5L/In₂Se₃ vdW heterostructure. The yellow arrows indicate the magnetic moment directions of the Mn atom, and blue ones represent the direction of electric polarization. b Band structure of In₂Se₃/MSS-5L/In₂Se₃ heterojunction with electric polarization of both top and bottom In₂Se₃ layer pointing upwards (+PP) or downwards (−PP). c Berry curvatures of the +PP and −PP states calculated at energy level of E − E_f = 0.05 eV (E_f is the Fermi level), respectively. d Anomalous Hall conductivities of the +PP and −PP states calculated at energy level of E − E_f = 0.05 eV (E_f is the Fermi level), respectively. e Energy barriers for the flipping of electric polarizations in In₂Se₃.

Finally, the switching pathway of electric polarization in Fig. 4e shows that the energy barrier between opposite polarization directions is approximately 81.4 meV/f.u.. This suggests that the manipulation of magnetic order and the orientation of AHC via an electric field is modest and feasible. In addition, we also calculated the heterojunction constructed only at the top or bottom of MSS-5L (Fig. S7). The same magnetic order as that under the directly applied electric field can be obtained only if the polarization direction of the top-layer or bottom-layer In₂Se₃ film is pointing to MSS-5L, i.e., intrinsic ferroelectric polarization of In₂Se₃ acts as an equivalent electric field, otherwise, the depolarization field will act as an equivalent electric field, which further confirm the phenomenon of the electric polarization direction dependent magnetic order UUD/UDD in MSS-5L.

Discussions

In this work, by combining symmetry analysis, model calculation and first-principles simulations, we demonstrate a universal mechanism linking $\hat{P}\hat{T}$ symmetry to fully electric-polarization-controlled and switched magnetic order, as well as AHE. By utilizing 3MnB₂T₄·2B₂T₃ family as an example, we prove that the $\hat{P}\hat{T}$ symmetry connected UUD and UDD magnetic states can be efficiently switched by electric polarization. Importantly, the energy difference between UUD and UDD could be linearly modulated by the co-directional electric field. It facilitates efficient modulation of magnetic orders under sliding ferroelectricity or spontaneous polarization in 3MnB₂T₄·2B₂T₃ related heterostructures. Furthermore, because of the $\hat{P}\hat{T}$ symmetry connection between the UUD and UDD magnetic states, the directions of the AHC in 3MnB₂T₄·2B₂T₃ could also be effectively reversed by the electric field. This investigation into the feasibility of utilizing the $\hat{P}\hat{T}$ symmetry to manipulate non-volatile magnetic order switching and spin-related properties offers new possibilities in the design of future spintronic devices.

Methods

Computational details

We perform all density functional theory (DFT) calculations by applying the Vienna Ab initio Simulation Package (VASP)⁴¹. The exchange and correlation effects are treated within the generalized gradient approximation (GGA)⁴². The GGA + U method with U_eff = 4 eV on Mn 3d orbitals is applied in all calculations^43,44. The value of U has been tested by Li et al, and for MnB_xT_y system and it was found that U_eff = 4 eV provides the band structures that match well with the experiments³¹. Moreover, the U_eff = 4 eV has been widely utilized in numerous theoretical studies to investigate the quantum transport properties of MnB_xT_y^33,45,46. In addition, we have systematically investigated the impact of Hubbard parameters (U_eff) on relative stabilities of different magnetic orders, as well as on the energy barriers required for 180° reversal of magnetic moments. The computational results demonstrate that, within the range of 3.5 eV–5.5 eV, the UUD/UDD magnetic orders are consistently magnetic ground states (Fig. S8). The energy barriers for the reversal of magnetic moment in the middle-layer maintain the lowest one with respect to the top and bottom layer (Fig. S9a-e) in MSS-5L, with values fluctuating between 0.04 and 0.07 meV (Fig. S9f). The plane-wave energy cutoff is set to 500 eV in all calculations. A 15 Å vacuum region was used between adjacent plates to avoid the interaction between neighboring periodic images. The vdW corrections, as parameterized within the semiempirical DFT-D3 method⁴⁷, are used in the calculations. The systems are fully relaxed until the residual force on each atom is less than 0.01 eV/Å. Electronic minimization is performed with a tolerance of 10⁻⁷ eV. The kinetic pathways of transitions between different polarization states were calculated using the climbing image nudge elastic band method^48,49. Unless mentioned in the text, spin-orbit coupling is always included in the calculations of the electronic and transport properties. The out-of-plane electric polarization of polar-stacked 3MnSb₂Se₄·2Sb₂Se₃ films is calculated using the dipole correction method⁵⁰. The maximally-localized Wannier functions are used to obtain the tight-binding Hamiltonians within the Wannier90 code. The Berry curvature and the AHC are calculated using the Wanniertools code⁵¹.