Ferroelectricity controlled chiral spin textures and anomalous valley Hall effect in the Janus magnet-based multiferroic heterostructure

Realizing effective manipulation and explicit identification of topological spin textures are two crucial ingredients to make them as information carrier in spintronic devices with high storage density, high data handling speed and low energy consumption. Electric-field manipulation of magnetism has been achieved as a dissipationless method compared with traditional regulations. However, the magnetization is normally insensitive to the electric field since it does not break time-reversal symmetry directly, and distribution of topological magnetic quasiparticles is difficult to maintain due to the drift arising from external fluctuation, which could result in ambiguous recognition between quasiparticles and uniform magnetic background. Here, we demonstrate that electric polarization-driven skyrmionic and uniform ferromagnetic states can be easily and explicitly distinguished by transverse voltage arising from anomalous valley Hall effect in the Janus magnet-based multiferroic heterostructure LaClBr/In2Se3. Our work provides an alternative approach for data encoding, in which data are encoded by combing topological spin textures with detectable electronic transport.


Introduction
Controlling magnetism by electric field is the fundamental avenue for realizing highefficiency and extremely low-power consumption memories and logic devices [1][2][3][4][5], such as electric field-tunable magnetoelectric random access memories (MERAMs) with low writing energy [6]. Multiferroic materials, with both ferromagnetism and ferroelectricity, are prospective candidates that can achieve electric-field modulation of ferromagnetism, magnetic direction, and magnetic anisotropy via the magnetoelectric coupling effect [7][8][9][10]. The topological magnetic quasiparticles, consisting of skyrmion and bimeron, have attracted intensive attention. Due to possess many advantages such as nonvolatility, small-size, and high mobility, these quasiparticles hold huge potential in spintronic devices for information technologies [11][12][13]. In 2016, it was reported that the skyrmion can be transformed to uniform ferromagnetism by electric field Fe/Ir(111) film with a scanning tunneling microscope (STM) [14]. Then, ferroelectrically tunable magnetic skyrmions have been demonstrated in ultrathin oxide BaTiO3/SrRuO3 heterostructure (HS) [15]. Compared with bulk or multilayered multiferroic materials, the two-dimensional (2D) van der Waals (vdW) magnetic/ferroelectric HSs promote much miniaturization, simpler structure, better tunability, and highly interface quality for spintronic devices [16]. Recently, writing and erasing of skyrmion and bimeron also has achieved in 2D multiferroic HSs MnBi2Se2Te2/In2Se3, LaCl/In2Se3, WTe2/CrCl3/CuInP2S6, and Fe3GeTe2/In2Se3, where MnBi2Se2Te2, LaCl, WTe2/CrCl3 and Fe3GeTe2 are vdW magnets with Dzyaloshinskii-Moriya interaction (DMI), while In2Se3 is an atomic FE material easily altered by out-of-plane (OOP) electric field [17][18][19][20]. The topological magnetic quasiparticles and ferromagnetic (FM) states are potentially encoded and stored as '1' and '0' bit carriers, respectively. However, the location of these quasiparticles is susceptible to external fluctuations, making it difficult to distinguish from FM state. This problem has been mentioned in the skyrmion-based racetrack-type memories, which need to be maintained skyrmion/bimeron distribution in the operational time scales [21,22]. Notably, the reversal of FE polarization induces different electronic states of 2D FE-based vdW HSs in reciprocal space.
For example, the switchable topological edge states, valley polarization, and metal-tosemiconductor phase transition are found in bilayer-Bi(111)/In2Se3 and β-Sb/In2Se3, HfN2/CrI3/In2Se3, and CrI3/Sc2CO2, respectively [23][24][25][26]. The band structures of a large variety of FE-aided vdW HS can be obviously regulated because the two surface of FE layer is nonequivalent induced by built-in electric field [27]. The above results imply that the variation of band structures controlled by FE polarization is hopefully used in unambiguously distinguishing spin configurations.
Here, employing the first-principles calculations and atomistic spin model simulations, we propose a type of magnetoelectric effect which can be realized in 2D Janus-based multiferroic vdW HS, LaXY/In2Se3 (X/Y=Cl, Br, I, X≠Y), where LaXY can be constructed by replacing one of halogen atom in 2H-LaX2 and considered as an ideal candidate to realize FEcontrolled electronic states and topological magnetic quasiparticles since it combines longrange ferromagnetism, robust valley polarization and inversion symmetry breaking-allowed DMI [28]

Calculation method
All first-principles calculations are performed using the density functional theory (DFT) as implemented in the Vienna ab initio Simulation Package (VASP) code [29][30][31]. The exchange and correlation functionals are treated by the generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof functional (PBE) [32,33]. The interaction between ions and electrons is described by projector-augmented wave (PAW) method [34]. After compared the magnetic parameters under the different cut-off energies (see Table S1 and Section S4 in Supplementary material), the plane-wave basis set in 420 eV. To avoid interaction between adjacent layers, the thickness of vacuum layer is set no less than 15 Å along the z direction. We employ the GGA+U method with U = 7 eV to treat the f electrons for La as reported in the previous studies [18]. Brillouin zone is sampled using Г-centered 24×24×1 Monkhorst-pack kpoint mesh. The electronic convergence is performed with a tolerance of 10 -7 eV. Optimized structures are fully relaxed until the force converged on each atom less than 10 -3 eV/Å. To check the stability of the Janus LaXY monolayer, Phonon dispersions are calculated with 4×4×1 and 5×5×1 supercell by using the PHONOPY code [35], and ab initio molecular dynamics (AIMD) are simulated with 4×4×1 supercell in the canonical NVT ensemble [36]. The vdW interaction between LaXY and In2Se3 monolayer are corrected by DFT-D3 [37]. To avoid the influence of external strain from sublattice [ Table S2 and Section S3 of supplementary material], by using 1×1 cell of In2Se3 to match 1×1 cell of LaXY, the lattice mismatch is 0.51%, 1.85%, and 3.09% for LaClBr/P, LaClI/P, and LaBrI/P, respectively. The maximally location Wannier function are calculated using WANNIER90 package code to obtain the Berry curvature and anomalous Hall conductivity [38]. More computational details are demonstrated in Supplementary material.  Figure 1(a) displays the top and side views of Janus LaXY (X/Y=Cl, Br, I, X≠Y), which contain two atomic planes with different halogen elements and form hexagonal network with point group C3v. Bulk LaBr2 is a hexagonal layered crystal with a space group of P63/mmc [39], and LaBr2 monolayer has been proposed as an ideal ferromagnetic semiconductor with Tc>200K [40,41], Which is higher than that of CrI3 monolayer (45K) [42] and Cr2Ge2Te6 bilayer (28K) [43]. The optimized lattice constant of LaClBr, LaClI, and LaBrI monolayer is listed in Table   1, and corresponding magnetic moment of La atom is 0.385, 0.374, and 0.371 μB, respectively.

Results and discussions
The structural stability, including the phonon dispersions ( Figure S1 shows that the other two monolayers are dynamically stable except for LaClI with small imaginary frequency around Г point) and molecular dynamic simulations ( Figure S2 shows that LaXY monolayers are thermal stability), are demonstrated in Section S1 and S2 of Supplementary material. In2Se3 is a 2D vdW room-temperature FE material [44][45][46] and Figure 1  (MoS2/In2Se3) illustrate the experimental feasibility of construction of LaXY/In2Se3 multiferroic HSs [47][48][49][50]. The LaXY monolayers perform in-plane anisotropy, ferromagnetic state and clockwise chirality DMI [see Table S2]. As described in Section S3 of Supplementary material, we consider twelve stacking configurations of LaXY/P↓ [ Figure S3] and calculate the corresponding total energy [ Figure S4]. We find that the P↓ state is always energetically more stable than P↑ state, which is consistent with previously generic trend [27]. The magnetic properties can be regulated by changing the interfacial coupling, when revers the OOP polarization of In2Se3 in the LaXY/In2Se3 HS. In Table 1, the equilibrium interlayer distance of LaXY/P↓ is lower than P↑ due to the stronger interfacial coupling between LaXY and In2Se3 in P↑. Top and side views of the most stable structure of LaClBr/P↓ and LaClBr/P↑ are shown in Figure 1(c), where the electric polarization is -0.103 and 0.113 eÅ/u.c., respectively. The polarization of LaClBr/In2Se3 is very close to that of In2Se3 monolayer (0.1 eÅ/u.c.), indicating that the ferroelectricity of HS mainly comes from In2Se3 layer. In experiment, an OOP electrical field around 1 V/nm can flip the perpendicular polarization direction [51], which implies that the reversal of polar orientation is achieved using a large enough electric field in LaXY/In2Se3 HS. Table 1  In order to investigate the magnetic properties of LaXY/In2Se3, we adopt the following spin Hamiltonian: where is a unit vector representing the orientation of the spin of the ith La atom, and 〈 , !〉 represents the nearest-neighbor (NN) La atom pairs. The magnetic parameters K, J, and in the first three terms represent the magnetic anisotropy, exchange coupling, and interatomic DMI, respectively. And the and in the last term represent the magnetic moment of La and external magnetic field, respectively. Here we adopt the sign convention that K>0 and K<0  Table S3 and Section S3 i.e. LaClBr layer is switched along z direction, DMI chirality is consequently reversed according to Maryia rules [52]. That indicates the interface contact has a great influence on these systems, then we use the most stable structure in the following calculations [ Figure S4].
The calculated details about these parameters are explained in Section S4 of Supplementary material, and the results are shown in Table 1  ∆E∝4cosθ, where the θ is the angle between the direction of spin and +z axis [17]. That means the valley splitting will gradually disappear as spin rotates from OOP to IP, which has been confirmed by first-principles calculations in Figure 3 (2) where -. is Fermi-Dirac distribution function, 5 ( ) is velocity operator along x(y) direction, and 2 .3 is Bloch wave function with eigenvalue En. Figure 3 For LaClBr/In2Se3, the PMA of La is slightly larger than IMA of Br, while for LaClI(LaBrI)/In2Se3, both the La and X atoms contribute the IMA and the strength is enhanced significantly in P↑ compared with P↓. As shown in Table S4, the DMI is reduced when the polarization of In2Se3 is changed to zero by placing all atoms at the local centrosymmetric positions, which means the electric field produced by polarization rather than the interfacial hybridization has the great importance in determining DMI. We further elucidate the layerresolved SOC energy difference ∆ESOC between the CW and ACW configurations for  when FE polarization is changed from down to up [56]. We further calculate magnetic parameters of K, J, and ∥ for LaClBr monolayer under 0.035e hole doping [ Table S2]. These parameters are very close to that of LaClBr/P↓ HS, which indicate that the versatile properties in LaXY/In2Se3 is mainly derived from charge transfer between interface. Moreover, the planer-average charge-density difference shows that the charge is mainly transferred in In2Se3 layer of LaClBr/P↓ but restricted in the interface of LaClBr/P↑ [ Figure S12].

Conclusion
In summary, we systematically investigate the chiral spin configurations in real space and

Section S1: Phonon calculation
The dynamical stability of the two-dimensional Janus LaXY monolayer is checked using the finite displacement method. A 4×4×1 and 5×5×1 supercell and a 5×5×1 k-point grid is adopted to compute the force constants. The force constant matrices and phonon dispersion are calculated implementing by the PHONOPY code [1]. The calculated phonon band structures of LaXY are shown in Figure S1. For LaClBr and LaBrI, it is clearly seen that no imaginary frequency is present, indicating that the systems are dynamically stable. For LaClI, the acoustic phonon modes display negative frequencies around the Г-point, which is related to in-plane bending of the two different halogen atomic planes [2,3].

Section S2: Molecular dynamics simulation
To verify the thermodynamic stability of LaXY monolayer, we perform ab initio molecular dynamics (AIMD) simulations in the canonical NVT ensemble with Nosé thermostat [4]. A 4×4×1 supercell and a 5×5×1 k-point mesh is used. The simulations are conducted for 10 ps at 500K, and each time step is set to 1 fs. Figure S2 shows the evolution of the total energy and temperature during the AIMD simulations. And the structures ( Figure S2) after simulations for 10 ps are still maintained at initial phase.

Section S3: LaXY/In2Se3 heterostructure (HS) configurations
From the previous research, each atomic layer in In2Se3 monolayer only one element, which the atoms arranged in an equilateral triangular lattice and initially places on one of the three sublattice sites, A, B, or C, as shown in Figure S3. We mainly study the ferroelectric modulation of the topological magnetic phase and electronic states of LaXY layers, then fixing the lattice of LaXY can avoid the influence of external strain degree. As the lattice of In2Se3 layer is fixed, for example, LaClBr layer is stretched 0.51% to match In2Se3. The calculated magnetic parameters of LaClBr monolayer under 1% tensile strain are similar with the pristine LaClBr (Table S2). The same approach is also adopted in the multiferroic van der Waals heterostructures. For example, electric polarization manipulates a variety of properties of the magnetic anisotropy of the CrGeTe3 in CrGeTe3/In2Se3 HS, the conductivity of the CrI3 in the CrI3/Sc2CO2 HS, and the magnetic order of bilayer CrI3 in the bilayer CrI3/Sc2CO2 HS [5][6][7].
Considering structural symmetries of In2Se3 and LaXY, we propose 12 stacking configurations of LaXY/In2Se3 HSs by shifting or rotating the position of LaXY layer. The site of each atom in LaXY is provided at the top of each stacking sequence. The total energies of 12 optimized structures for LaXY/P↓, and the energy for LaXY/P↑ corresponding to the most stable stacking configuration is labeled in Figure S4. We find In2Se3 prefers to contact the heavier elements-  (Table S3). When the interface contact changes from Br-Se to Cl-Se, the DMI chirality is reversed. The calculated layerresolved SOC energy difference ∆G HIJ of Br-Se and Cl-Se contacts (see Figure S5) indicate that the DMI mainly origin from Br layer and the sign of ∆G HIJ is opposite at different cases, which responds for the inverse chirality of DMI. , where the ∥ >0 and ∥ <0 favor spin configuration with anticlockwise and clockwise, respectively [10]. The calculated DMI strength of LaClBr using four state method and CDED approach is -0.015 and -0.012 meV, respectively, and both favor spin configuration with clockwise chirality. We further calculate the basic properties of LaClBr/P↓ with different cutoff energies (see Table S1). One can see that the results of K, J, and ∥ is very similar when the different cutoff energy is adopted. Notably, all calculations are proceeded under the plane-wave basis set in 420 eV.

Section S5: Atomistic spin model simulation-VAMPIRE
All atomistic spin model simulations of LaXY/In2Se3 HSs are performed using the VAMPIRE package [11]. For topological spin textures of these heterostructures, a 300 nm×300 nm square system is created with periodic boundary. To describe the dynamic of atomic spins, the Landau-Lifshitz-Gilbert (LLG) equation is adopted. The LLG can be expressed as: where is a unit vector representing the direction of the magnetic spin moment of site i, j is the gyromagnetic ratio, and the damping constant λ is set to 0.2. f Agg is the net magnetic field on each spin and is obtained from where H is spin Hamiltonian of LaXY/In2Se3 HSs.  Figure S1. Calculated phonon band structure for the LaClBr, LaClI, and LaBrI monolayer.