2D Multiferroics in As‐Substituted Bilayer α‐In2Se3 with Enhanced Magnetic Moments for Next‐Generation Nonvolatile Memory Device

Searching for multiferroic materials with ferromagnetic (FM) and ferroelectric (FE) properties holds promise for ultra‐high‐density and low‐energy‐consumption memory device applications, but 2D materials with both properties are rare. Herein, a general strategy to achieve nonvolatile electric field control of magnetism in the bilayer (BL) α‐In2Se3 by hole doping is proposed. By first‐principles calculations, it is demonstrated that hole doping can induce robust ferromagnetism in the bilayer α‐In2Se3 due to its unique flat Mexican‐hat‐shape valence band structure. Such band edges cause van Hove singularities (VHS), and proper hole doping can lead to time‐reversal symmetry breaking. The bilayer α‐In2Se3 exhibits ferromagnetism and ferroelectricity within a wide range of doping concentrations, resulting in an unexpected multiferroic phase. Furthermore, when the electrical polarization of α‐In2Se3 flips from downward to upward, it becomes non‐magnetic (NM) from ferromagnetic states in the As‐substituted bilayer α‐In2Se3, which can work as a nonvolatile magnetic storage unit. Remarkably, the As‐substituted bilayer α‐In2Se3 exhibits an enhanced magnetic moment of 1.2 μB per AsSe due to substantial charge transfer across the interface. Notably, the mechanism of electrically controlled magnetism is elucidated as the coupling among the Mexican‐hat‐like dispersion, ferromagnetism, and ferroelectricity. The findings offer a promising strategy for electrical writing and the magnetic reading memory device.


Introduction
2D ferromagnetic [1,2] and ferroelectric, i.e., multiferroic [3] magnetoelectric, materials are promising in the trend of the miniaturization of electronic devices for modern information synthesized by mechanical exfoliation, [20] physical vapor deposition (PVD), [21] and chemical vapor deposition (CVD). [22]In 2017, Ding et al. predicted the out-of-plane FE phase transition in the ML--In 2 Se 3 using density functional theory (DFT).[31] We and other groups previously proposed that proper hole-doping on the flat Mexican-hat-like valence band (VB) in ML--In 2 Se 3 could evoke a sizable room-temperature spin polarization. [13,32,33]More interestingly, Gong et al. proposed a bilayer VSe 2 /-In 2 Se 3 FM-FE VHS in which the polarization of -In 2 Se 3 layer could lift the spin degeneracy of the antiferromagnetic (AFM) coupled VSe 2 bilayer to half-metallic states. [34]So, it is promising to connect the FE character to the hole-dopinginduced magnetism in the layered -In 2 Se 3 .
In this paper, based on first-principles calculations, we predict that the FM ground state of the BL--In 2 Se 3 by As-substituted in selected Se sublayer.Remarkably, due to the out-of-plane FE character, the polarization flipping conducts to effectively swap the substitution sublayer, and the FM vanishes simultaneously.So, the As-doped BL--In 2 Se 3 is multiferroic.Furthermore, the charge transfer between two -In 2 Se 3 layers caused by the FE polarization can enhance the magnetic moments by ≈20%, up to 1.2  B per hole.Therefore, it provides a promising platform to exploit electrically-controlled magnetic switches for nonvolatile magnetic memory devices.

Results and Discussion
The optimized geometric structure of BL--In 2 Se 3 is shown in Figure 1a in which the upper (L1) and lower (L2) layers are stacked together by vdW interaction, and the polarization is upwards.The presence of the out-of-plane (z-direction) electric polarization drives the charge to migrate across the interface between the two identical -In 2 Se 3 MLs, according to the planeaveraged electrostatic potential distribution along the z-direction, as shown in Figure 1b.To better understand the charge transfer, we also calculate the averaged differential charge density distribution along the z-direction: where  BL−−In 2 Se 3 (x, y, z),  L1 (x,y, z), and  L2 (x,y, z) are the charge densities at the position (x, y, z) of the BL--In 2 Se 3 , and the isolated L1 and L2 layers, respectively, that are superimposed in the BL unit cell.The positive and negative values of Δ(z) represent the accumulation (red) and depletion (blue) regions of electrons, respectively, as illustrated in Figure 1c.The integral of the differential charge density between the two -In 2 Se 3 layers reveals that 0.11 electrons transfer from the L2 to the L1 layers across the interface.Due to this small electron migration, the total electrostatic potential difference between the two sides of the BL--In 2 Se 3 is ≈2.123 eV which is smaller than twice that of the ML--In 2 Se 3 (1.179eV), as shown in Figure 1d.
Based on the HSE06 method, we carried out the energy band structure of BL--In 2 Se 3 , as shown in Figure 2b.Since the conduction band minimum (CBM) is located at the Γ Point, whereas the valence band maximum (VBM) is on the Γ-M path, it is a semiconductor with an indirect bandgap of 0.44 eV.By projecting to the six Se atomic sublayers, we find that the overall bands are formed by two sets of energy bands of the individual ML--In 2 Se 3 with an offset of ≈1 eV that is much bigger than the non-polarized bilayers.And there is only little mixing between the states from L1 to L2 layers.Furthermore, the lowest CB originates from the top Se sublayer in L1 marked in violet color, while the topmost two VBs, including the VBM, are mostly contributed by the middle (red) and bottom (blue) Se sublayers in L2.This is because the charge screening cannot cancel the vertical polarized electric field.Due to the existence of this out-of-plane polarization electric field in the -In 2 Se 3 layer, the work functions on the two sides are different, which will differently (even oppositively) affect the chemical potentials in the adjacent layers.Therefore, the band offset is much bigger than in the non-polarized bilayers.In other words, the electric field of one layer will elevate (or depress) the bands of the other layer in energy.For instance, as the electric field in the L2-In 2 Se 3 is away from the L1-In 2 Se 3 , the energy bands (e.g., CB and VB) of L1-In 2 Se 3 shift downwards.Meanwhile, the electric field in the L1-In 2 Se 3 is toward the L2-In 2 Se 3 , and so the bands in the L2-In 2 Se 3 shift upwards.Finally, the CB and VB of the -In 2 Se 3 of bilayers come from L1 and L2 layers respectively.This means that the polarization field depresses the band edges of L1 and elevates those of L2, resulting in a bandgap change of 1 eV and the rise of the VBM of L2 to become the global VBM, near the Γ Point.Meanwhile, saddle points continuously appear around the Γ Point, and the band dispersion keeps the flat Mexican-hat shape.Therefore, a sharp VHS appears near the VBM on the DOS, as illustrated in Figure 2b.When the hole doping shifts E F down to intersect the sharp peak around the VHS, the ground state is always favorable to the FM due to the strong electronic instability of the non-magnetic state. [12]ince proper hole doping can depress E F to the VHS, making the DOS(E F ) big enough to meet Stoner's criteria of D(E F )•U > 1 for FM transition, we may obtain spin polarization by doping holes in the BL--In 2 Se 3 .As shown in Figure 2a, the calculated magnetic moment (black line with solid circles) in BL--In 2 Se 3 , as a function of hole-doping concentration, remains zero as the hole concentration is below 0.5 holes per unit cell (hole per uc) due to the small dispersion of the valence band, which barely pulls down E F a little and DOS(E F ) does not meet Stoner's criteria.As hole concentrations up to 0.8 hole per uc, the magnetization energy (E M = E FM −E NM , red lines with open circles) is still trivial, although the magnetic moment of the FM state is non-zero up to 0.1  B per hole.Moreover, the magnetic moment sharply increases to ≈0.8  B per hole when the hole concentration is over 1.0 hole per uc.As the doping concentration increases up to 1.0 hole per uc, the magnetic moment becomes quasisaturated and only fluctuates little ≈0.8  B per hole.On the other hand, E M is negligible before 0.9 hole per uc and then linearly enhances as the doping concentration increases.When the hole concentration exceeds 1.5 hole per uc, E M is beyond −26 meV per hole, indicating that the FM state may be stable at ambient temperature.Thus, BL--In 2 Se 3 has promising potential in magnetic applications.Additionally, we also calculated the magnetic anisotropy energy (MAE) defined by MAE = E out-of-plane −E in-plane is ≈−1.37 meV, where E out-of-plane and E in-plane denote the total en-ergies of the states with the out-of-plane and in-plane spin alignments, respectively.Hence, the magnetic easy axis is perpendicular to the BL--In 2 Se 3 plane.
To gain deeper insight into the spin polarization by hole doping, we project energy band and DOS on the six Se atomic sublayers in the BL--In 2 Se 3 , as depicted in Figure 2b,c.It is revealed that the bottom (red line) and middle (blue line) Se sublayers of the L2 layer have the first and second most contributions to the VHS near below E F , respectively, and the L1 layer contributes another set of sharp peaks at −1.0 eV below.In contrast, the top Se sublayers of L1 and L2 are trivial in the VB peaks but dominant in the first and second CB peaks, respectively.Therefore, the injected holes by dielectric gating tend to enter the bottom two Se sublayers and then lift the spin degeneracy, without dopant compensation.And, if a Se atom is substituted by an As atom, forming an As Se defect, besides one hole is expected to be injected in the system, the trapping effect may not be negligible.According to the charge density analysis, as shown in Figure 3a-f, the hole tends to reside around the defect.This is the reason that spin polarization occurs only in the cases of As substitutions in the middle (As Se-middle ) and bottom (As Se-bottom ) Se sublayers in ML--In 2 Se 3 , while the substitution in the top Se-sublayer (As Se-top ) is NM 12 .In the scenarios of L2-As Se-bottom and L2-As Se-middle , The introduced holes are mainly localized near the valence band, with a high sharp peak on DOS.Therefore, according to the Stoner criterion, the holes introduced at this time will produce spontaneous magnetism.As listed in Table 1, similar to the ML--In 2 Se 3 case, the L1-As Se-top and L2-As Se-top cases of BL--In 2 Se 3 are also NM with trivial E M (of −0.05 or 0 meV), subject to the strong trapping effect of the hole around the As ions.Notably, L1-As Se-top and L2-As Se-top cases are the energetically most favorable.If one fabricates an As substituted BL--In 2 Se 3 sample, e.g., by the molecular beam epitaxial these two appear the biggest two yields, which will hinder the occurrence of the spin polarization.Then, if we apply a vertical electric field, and as the polarization flips, promisingly, the top and bottom surfaces effectively exchange because they are defined according to the polarization direction.So, the original L1-As Se-top and L2-As Se-top effectively become L1-As Se-bottom and L2-As Se-bottom which are magnetic according to Table 1.
Interestingly, the As Se-bottom and As Se-middle cases in both L1 and L2 have remarkable magnetic moments and E M .Interestingly, the L2-As Se-bottom and L2-As Se-middle cases have the first and second strongest magnetizations of ≈1.20 and 1.17  B per As Se with negative E M values of −109.39 and −29.02 meV, respectively.These magnetizations are intriguing, since the usual doping-induced magnetism always has a magnetic moment saturation ratio of no more than 1  B per dopant that contributes only one electron or hole.Likewise, the L1-As Se-bottom and L1-As Se-middle cases have fractional magnetic moments of 0.70 and 0.48  B per As Se , respectively.
Nevertheless, we obtain the magnetic moment is over 1  B per hole by ≈20% (17%) in the L2-As Se-bottom (L2-As Se-middle ) case, exhibiting a magnetic enhancement that is induced by the charge redistribution around the As center.In comparison, the similar substitution in the L1 layer only has an unsaturated magnetic moment, since the As Se-bottom and As Se-middle in L1 have to pull the deeper band up to E F against the intrinsic electronic field of the polarization and then lift the spin degeneracy.Intriguingly, the FE polarization flipping can effectively trigger the top-bottom exchange, i.e., the As Se-bottom of L2 becomes the As Se-top of L1, and then the magnetic state switches to the NM state.The details will be discussed later.In order to verify the stability of doped bilayer, such as L2-As Se-bottom , Ab initio molecular dynamics (AIMD) simulations is performed up to 10 ps, with a Nosé-Hoover thermostat.As shown in Figure 2d, there is only small fluctuations in total energy and no obvious structure distortion and bond broken occur till the last snapshot.Moreover, the geometric structures maintained their integrity, which demonstrates the thermal stability.
To unravel the details of magnetic enhancement, as an example, the energy band structure and DOS of the L2-As Se-bottom defect are shown in Figure 3g,h.Due to the As substitution, an impurity band forms in the bandgap with a sharp peak of DOS at the E F , which generates the magnetic moment based on the strong electronic instability.The Bader charge analysis was carried out to examine the effect of the interlayer charge migration on doping magnetism in L2-As Se-middle and L2-As Se-bottom defects.The results show that 0.113 and 0.153 electrons migrate from the L2 to L1 layers, respectively.Thus, the As substitution can introduce by an additional interlayer electron transfer besides the localized holes, i.e., the total unpaired charges are more than one hole per dopant, which is responsible for the magnetic moment exceeding 1  B per As Se .In a 3 × 3 supercell, we calculate the spin-polarized densities (SPDs) of a single As Se in the various sublayers, as illustrated in Figure 4.For the substitution defects without partially occupied d-or f-orbitals, the main contribution to SPD comes both from the defect center and the surrounding Se atoms region represented by the yellow bubbles in Figure 4.The As Se-bottom (As Se-middle ) can induce the sizable SPD on the Se atoms surrounding the As Se even in the neighbor middle (bottom) sublayer.Obviously, the SPD bubbles on Se around As are bigger in the case of substitution in L2 than those in L1.The extra SPD is caused by the additional electron migrations from L2 to L1 when substitution As in L2.So, the effective amount of doped charges in L2 is larger than one hole per As, which is the essential reason for the enhanced magnetic moment.Meanwhile, although the electrons enter L1 in tiny amounts, they can still depress the CB down to E F and appear slight spin polarization.Consequently, the effective total magnetic moment generated by As-doped holes and the additional holes (induced by the L2-As Se-bottom and L2-As Se-middle defects) is larger than those of the substitutions-in-L1 cases, as well as the full space doping or gate regulation.
As shown in Figure 5, we diploid the L2-As Se-bottom defected 3 × 3 supercell, including two As Se-bottom , to check the magnetic coupling between the defect-induced magnetic moment.After the structural relaxation, the total magnetic moment of parallelly-aligned (FM) spin configuration holds a magnetic moment of 2.28  B , exhibiting that the individual magnetic moment is still larger than 1  B per As Se , as shown in Table 2, while the antiparallelly-aligned one (AFM) is spin-degenerate with 0 magnetic moments.And the FM state is 31.68meV energetically more favorable than that of the AFM state, indicating that the ground state of the system is FM.
The magnetic ground state is in-plane magnetization under the L2-As Se-bottom substitution since the in-plane ordering is energetically more favorable than out-of-plane magnetism.The Curie temperature (Tc) is an important property of magnetic materials.Thus, the statistical Monte Carlo (MC) simulation based on the Heisenberg model is performed.This approach has been successfully used to predict the Tc in other 2D materials producing    values close to the experimental data.The spin Hamiltonian is defined as: where J is the nearest neighbor exchange parameter, evaluated by the energy difference between the FM and AFM states, i.e., the energy of the FM state is ≈31.7 meV per hole lower than that of the AFM state, as shown in Table 2. ‚ S are the spin operators, the summation ⟨ij⟩ runs over all nearest-neighboring As sites.D x and D y represent the magnetic anisotropy energy, which is calculated from the energy difference of magnetization along x/y and z-axis, with a value of 2.62 meV.Due to the big distances between the magnetic moments, only first neighbored magnetic exchanges are considered, so the energies of the FM and AFM states are (3) here, E 0 is the system energy without considering the spin degree of freedom.The value of J has been determined to be 12.63 meV.In the MC simulations, we used a (100 × 100) supercell to reduce the periodic constraints.5 × 10 6 loops are taken to achieve an average magnetic moment value.As shown in Figure 5, we obtain one L2-As substitution per 3 × 3 supercell, ≈10 13 hole cm −2 .The Tc of 131 K was read from the peak position of the specific heat defined as C V = (〈E 2 〉-〈E〉 2 )/(k B T 2 ).This value is higher than that of substitutions by transition metal (TM) atoms such as Fe and Mn. [35,36]This also means that compared to direct hole doping, [13] spontaneous magnetization can be achieved by replacing As atoms at a lower doping concentration.
To demonstrate the influence of hole doping on ferroelectric properties, two charge distributions  c (z) and   (z) are defined.They respectively describe the contribution of charge density to the electrical conductivity and polarization properties of the material.This approach has been employed to verify the existence of metallic ferroelectricity. [37]Here,  c (z) is considered as conduct-ing electron density, which is typically regarded as partial electron densities within energy range |E − E f | < 0.05 eV.Furthermore,   (z) is defined as: Here,  FE (r) and  PE (r) respectively denote the overall charge density of the material in the ferroelectric (FE) and paraelectric (PE) states.Therefore,   (z) is considered to describe the contribution of electrons to polarization.The Figure 6 illustrates the distribution of the two charge densities.we have discovered that these two types of charges are spatially separated.This implies that under low concentration hole doping, the ferroelectric properties can still be maintained Therefore, we could summarize a general principle to the realization of this kind of electric-field-controlled magnetic switching materials in fourfold, i.e., first, find a 2D FE material with a narrow (flat) VB or CB (always leading to a sharp peak, i.e., VHS, on DOS).Due to the FE polarization field, the atoms on the two surface always contributes to the different band edges, respectively, e.g., the top (bottom) anions contribute to CBM (VBM).Second, dope proper charges by substitutions of the atoms on the surface.Third, flipping the FE polarization can exchange the bandedge contributions of the sublayer that dopants reside on to realize the magnetism switch.Fourth, stack two or three monolayers together by vdW interactions to enhance the maximal magnetic moments.Employing this As-doped bilayer In 2 Se 3 material, we propose a prototype for a non-volatile magnetic switching device.First, according to our calculations, under a fixed FE polarization, the As-substitutions on the bottom Se sublayer of L2 (L2-As Se-bottom ) evoke the FM ordering, while, the As-substitutions on the top Se sublayer of L1 (L1-As Se-top ) are NM.Second, as an external vertical electric field drives the Se-middle atoms from the lower off-center position to the upper one, the FE states flips.If we vary the view along the direction of the polarization, the defects initially reside at L2-As Se-bottom equivalently becomes the defects at L1-As Se-top after a FE reversal.Consequently, the FM state of L2-As Se-bottom transits into the NM state of L1-As Se-top .This external electric field-induced FE reversal significantly modifies the magnetic configuration of the system, thereby achieving an electric field-controlled magnetic switch.

Conclusion
In conclusion, based on the theoretical investigations of the electric-controllable magnetism caused by hole doping introduced by atomic substitution in BL--In 2 Se 3 .we proposed a general principle to achieve electromagnetic coupling in few-layer 2D multiferroic magnetoelectric materials with the enhancement of the magnetic moment.Our work shed new light on the design of novel multiferroic 2D materials.

Experimental Section
First-principles calculations based on density functional theory (DFT) [38] were implemented in Vienna Ab initio Simulation Package (VASP) [39] with the projector-augmented wave (PAW). [40]The Perdew-Burke-Ernzerhof (PBE) scheme within the generalized gradient approximation (GGA) [41] was used to deal with the exchange-correlation interaction of electrons.Because the PBE functional underestimated the semiconductor bandgap, the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional [42] was employed in the band structure calculations.After the convergence tests, the cutoff energy was set to 300 eV for the plane wave basis, and the 7×7×1 Monkhorst-Pack k mesh was chosen for the Brillouin-zone (BZ) sampling.A vacuum region of 15 Å was added to avoid the interactions between periodic layers.The convergence tolerances for energy and the residual force on each atom during structural relaxation were set to 10 −5 eV and 0.01 eV Å −1 , respectively.The lattice constant of the BL--In 2 Se 3 was converged to 4.106 Å, which is equal to the ML--In 2 Se 3 .The magnetic anisotropy energy (MAE) calculation was performed with a higher convergence level of 10 −6 eV.The empirical correction scheme of the DFT-D3 [43] approach within the Grimme scheme was adopted for the nontrivial vdW interactions.

Figure 1 .
Figure 1.a) Top and side views of the crystal structure of BL--In 2 Se 3 .The red dashed lines indicate the unit cell.The solid black lines represent 3 × 3 supercell.b) Plane-averaged electrostatic potential of SL--In 2 Se 3 , c) differential charge density Δ(z), and d) Plane-averaged electrostatic potential along the z direction of the BL--In 2 Se 3 .E eff represents the net effective electric field of BL--In 2 Se 3 .

Figure 2 .
Figure 2. a) Magnetic moment (left-handed axis) and spin polarization energy (right-handed axis) of BL--In 2 Se 3 as a function of doping level.b) Projected band structures c) PDOS concerning different Se atoms of BL--In 2 Se 3 based on HSE06 calculations and d) Molecular dynamics simulation of L2-As Se-bottom at 300 K up to 10 ps, in which the insect showing last snapshot.

Figure 3 .
Figure 3.The hole distribution of BL--In 2 Se 3 with a) L1-As Se-top defect, b) L1-As Se-middle defect, c) L1-As Se-bottom defect, d) L2-As Se-top defect, e) L2-As Se-middle defect, and f) L2-As Se-bottom defect.The blue and yellow bubbles represent the distribution of holes and electrons, respectively.Projected g) band structure and (f) DOS of L2-As Se-bottom defects based on HSE06 calculation, the sizes of the green circles represent the weights of the As Se states.

Figure 5 .
Figure 5.The spin polarization density (↑-↓) of a) AFM order and b) FM order of BL--In 2 Se 3 diploid supercell with L2-As Se-bottom defect.The isosurface level is 0.005 (Å −3 eV −1 ) and c) Variation of the calculated magnetization and specific heat versus temperature.

Figure 6 .
Figure 6.a) side views of partial electron density within energy range |E − E f | < 0.05 eV. and b) blue circles indicate the reduced conducting electron density  c (z), and red lines indicate the reduced polarization electron density  (z).

Table 1 .
The energies of FM and NM phases, the spin polarization energies, and the total magnetic moment of the FM phase of As Se defects in a 3×3 BL--In 2 Se 3 supercell.

Table 2 .
Total energies and magnetic moments of a diploid supercell of BL--In 2 Se 3 with L2-As Se-bottom defect under FM and AFM.