Tunable Magnetic Anisotropy and Dzyaloshinskii-Moriya Interaction in an Ultrathin van der Waals Fe3GeTe2/In2Se3 Heterostructure

The promise of future spintronic devices with nanoscale dimension, high-density, and low-energy consumption motivates the search for van der Waals heterostructure that stabilize topologically protected whirling spin textures such as magnetic skyrmions and domain walls. To translate these compelling features into practical devices, a key challenge lies in achieving effective manipulation of the magnetic anisotropy energy and the Dzyaloshinskii-Moriya (DM) interaction, the two key parameters that determine skyrmions. Through the first-principles calculation, we demonstrate that the polarization-induced broken inversion symmetry in the two-dimensional Fe3GeTe2/In2Se3 multiferroic heterostructure does cause an interfacial DM interaction. The strong spin-orbit coupling triggers the magnetic anisotropy of the Fe3GeTe2/In2Se3 heterostructure. The magnetic anisotropy and the DM interaction in Fe3GeTe2 can be well-controlled by the ferroelectric polarization of In2Se3. This work paves the way toward the spintronic devices based on van der Waals heterostructures. PACS: 63.20.dk, 74.78.Fk, 85.75.-d, 75.30.Gw


INTRODUCTION
Two-dimensional (2D) van der Waals (vdW) materials have many novel properties compared to their three-dimensional (3D) counterparts, such as topology, spin frustration, and magnetic skyrmion [1]. The vdW bulks can be cleaved/exfoliated down to the monolayer limit with their structural integrities and chemical stabilities retained [2], which brings great convenience to the building of heterostructures (HSs). When cleaved, the specific surface area of the vdW material greatly increases, which is suitable to be tuned by many kinds of experimental stimuli. These advantages would provide completely new platforms with all magnetic atoms participating in the magnetoelectric coupling, and largely reshape the landscape of 2D vdW multiferroics [3,4]. Inspired by the new physical picture of 2D materials, exploring the atom-thick vdW HS is the frontier of current spintronics research. The Dzyaloshinskii-Moriya (DM) interaction has recently received a lot of attention due to new findings on the magnetic skyrmions, spin waves, and chiral domain walls [5][6][7][8]. Magnetic skyrmion and domain wall are the smallest components that can be used in the next-generation magnetic race-track memories [9]. A critical constant of DM interaction D C = JK π is defined to evaluate the stability of the skyrmions [10], where J and K are the exchange coupling and magnetic anisotropy, respectively. The DM interaction and the magnetic anisotropy manipulated in 2D HS will greatly improve the practical applications of spintronic devices.
The coupling between ferroelectricity and ferromagnetism allows the control of spin via electric field or the control of charge via magnetic field, which leads to multifunctional performance of spintronic devices [11]. Thesore, the co-existence of these two ferroic orders in 2D materials is very attractive. The singlephase multiferroics are extremely rare because ferroelectricity arising from the off-center cations requires empty d-orbitals, while ferromagnetism usually results from partially occupied dorbitals [12]. Scientists have built various heterostructures from the ferroelectric (FE) and ferromagnetic (FM) materials. Not only the coexistence of ferroelectricity and ferromagnetism, but also the magnetoelectric couplings were achieved in these HSs [1,4,12]. Since the first prediction of 2D FE hydroxyl-decorated graphene in 2013, more and more 2D FE materials have been discovered, e.g., SnSe, SnS, GeS, Bi 2 O 2 Se, and Bi 2 O 2 Te (in-plane FE) [13,14], as well as MoS 2 , CuInP 2 S 6 , In 2 Se 3 , and MXenes (perpendicular FE) [13,15]. The mechanisms involved in these in-plane and perpendicular polarizations are the polar functional groups, the hopping of halogen adatoms, or polar phonon modes [11]. In recent years, dozens of 2D ferromagnets have been found experimentally or predicted theoretically [14,16]. The emerging of more novel 2D vdW ferroelectric materials and 2D vdW magnetic compounds indicates an enhanced probability to form multiferroic HSs.
The bulk Fe 3 GeTe 2 (FGT) belongs to 2D vdW magnetic materials with a layered hexagonal lattice, where the weakly bonded Fe 3 Ge layers are sandwiched between two Te slabs. As a strongly correlated itinerant ferromagnet, FGT contains two inequivalent Fe coordinates: the Fe1 atoms form a hexagonal net, while the Fe2 atoms are bonded covalently with Ge. This compound has many excellent properties, for example: a mixed valence state (Fe 2+ )(Fe 3+ ) 2 (Ge 4− )(Te 2− ) 2 [17], a [001] oriented easy axis [18], and a FM configuration down to the monolayer regime [19,20]. In particular, FGT can be easily exfoliated along the [001] direction due to the weak vdW interaction [19]. Recently, Yi et al. [21] have demonstrated a new phase of FGT (namely a competing AFM configuration) and the FM slabs become AFM order below 152 K. Nevertheless, Tian et al. [22] reported that the AFM configuration originates from the movement of pinned magnetic domain walls. The abovementioned arguments indicate that the fundamental properties of FGT are still under development. The experimentally confirmed 2D vdW ferromagnets are mainly focused on three materials: CrI 3 , Cr 2 Ge 2 Te 6 , and Fe 3 GeTe 2 [23][24][25]. Even though the Curie temperature (T C ) of 230 K [17] is below (but not too far below) the room temperature, FGT show a much higher T C than the CrI 3 (T C = 61 K) and Cr 2 Ge 2 Te 6 (T C = 61 K) bulks [26,27]. The combination of high T C and the abovementioned properties makes FGT a promising candidate for exploring itinerant ferromagnetism in a truly 2D form.
Having the FM part at hand, we now investigate the FE part of our multiferroic HS. As an all-known fact, the perpendicular polarization is usually more important than the in-plane one because the perpendicular polarization can effectively penetrate the FM layers in a FE/FM HS, thus manipulating the magnetism of the FM slabs. The 2D vdW materials with perpendicular polarization are relatively rare, which is mainly due to the fact that incomplete screening of surface charges induced depolarization field opposes the spontaneous polarization of the vdW materials, thereby suppressing the ferroelectricity. Recently, In 2 Se 3 has been widely used in spintronic devices for memory and logic applications. Ding et al. [28] have reported that α-In 2 Se 3 is a room-temperature FE material (down to the stable single-layer regime) with intrinsic in-plane and controllable perpendicular polarizations, which may open avenues for developing new concepts of magnetoelectric devices.
Very recently, Huang et al. [29] have investigated the effect of asymmetric interfacial coupling on the FE stability of 2D Fe 3 GeTe 2 /In 2 Se 3 HS through the first-principles calculation, but some important physical properties such as the MAE and DM interaction are not included in their work. Besides, a large hole doping induced magnetic anisotropy reduction is demonstrated in Fe 3−x GeTe 2 layered structure [30]. They also found the sharp decrease in magnetic anisotropy energy (MAE) caused by the change of electronic structure. Nevertheless, the reason for the change of electronic structure, hence the change of MAE is still unclear. Motivated by recent developments, we now study the DM interaction and magnetic anisotropy manipulated by a ferroelectric polarization in a truly 2D form. Control the magnetism of 2D materials, as well as modulate the magnetic order and electronic spin are of great importance in novel spintronic devices since they have smaller size and lower energy consumption compared with traditional electronic devices. We have first demonstrated the reversible control the DM interaction and MAE by the FE polarization in the Fe 3 GeTe 2 /In 2 Se 3 HS, as well as its underlying mechanisms. A magnetoelectric coupling is expected in our FE/FM HSs, opening up the possibility to control the magnetic skyrmions by the FE-polarization.

MODELS AND METHODS
The freestanding FGT monolayer is composed of two Fe1 layers, two Te layers and one mixed layer of Ge and Fe2 atoms, as shown in Figure 1A. The Fe atoms in the FGT monolayer can be divided into two inequivalent Fe1 and Fe2 atoms with the valence states of (Te 2− )(Fe1 3+ )/(Ge 4− )(Fe2 2+ )/(Te 2− )(Fe1 3+ ) [31]. Excitingly, the exfoliation of mono-layer or few-layer Fe 3 GeTe 2 has been experimentally achieved by Fei et al. [32] and Deng et al. [2]. Due to the relatively strong FE polarization, we choose In 2 Se 3 as the substrate with its most stable structure [28] [see Figure 1B]. Figure 1C shows the coefficients of the DM interaction. Besides, the perpendicular FE polarization of the In 2 Se 3 quintuple layers (QLs) can be easily reversed through several kinetic pathways given in Ding et al. [28]. In this work, four In 2 Se 3 QLs are applied to form a substrate because the magnitude of the FE polarization reaches its maximum value at the thickness of 3 or 4 QLs [28]. Huang et al. [29] found that the energy difference between the up-polarization and down-polarization became a constant when the thickness of In 2 Se 3 is more than 4 QLs. This result proven that our In 2 Se 3 substrate is thick enough to get reliable results. The unit cell is chosen to be equal to that of ferromagnetic Fe 3 GeTe 2 with the experimental lattice parameters a, b that also commensurate to the In 2 Se 3 layers. In order to comprehensively describe the AFM configuration, we build a √ 2 × √ 2 supercell with the x-y planes of FGT and In 2 Se 3 changed from rhombus to rectangular, as shown in Figure 1D. Our HSs have idealized hard interfaces without any defects or adsorbates. More importantly, the calculation methods and detailed parameters are given in the Supplementary Information.

RESULTS AND DISCUSSIONS
As it can be seen in Table 1, the lattice constants of the FGT monolayer agree well with the experimental data (a = 0.4050 nm, b = 0.7014 nm) [33]. Our calculated magnetic moments of the freestanding FGT monolayer are also in agreement with the experimental data (m 1 = 2.18 µ B , m 2 = 1.54 µ B ) [34], which partially proved that our results are reliable. The lattice mismatch between FGT and In 2 Se 3 is only 1.2% for both a and b axes, which indicates that these two materials can form heterostructures perfectly. It is worth noting that 2D FGT retains its FM configuration not only in the freestanding state, but also in the heterostructures. The orientation of the FE polarization has little effect on the magnetic moments and magnetic order of the FGT/In 2 Se 3 HS. Our calculated exchange parameters J 1 and J 2 are shown in Figure 1C, which are in general agreement with Deng's results (J 1 =5.79 meV, J 2 = −6.48 meV) [2]. Although the in-plane exchange interaction J 1 is AFM, it is much weaker when compared to the FM interaction between the Fe1 and Fe2 atoms [2]. Hence the FM configuration can be retained in the freestanding FGT monolayer. When coupled with the In 2 Se 3 substrate, J 2 increases quickly. The sign of the exchange parameter for the FGT/In 2 Se 3 HS, found positive for J 1 and negative for J 2 , while J 2 is predominantly stronger than J 1 . The parameter J 2 is strong and negative so as to stabilize the FM order of the FGT/In 2 Se 3 HS.
The MAE is calculated as the total energy difference between the HS with in-plane and perpendicular magnetization axes, which can be written as [35] To obtain the MAE, we consider the SOC effect in our calculation. The calculated MAE is 0.39 meV per Fe atom, indicating strong easy axis anisotropy in the freestanding FGT monolayer. Our result is in good agreement with the data obtained by Zhuang et al. [19] (0.52 meV per Fe atom). The significant MAE exhibited by the freestanding FGT monolayer, which caused by strong SOC, suggests that the 2D FGT has potential for applications in magnetic data storage applications [19]. The FGT monolayer shows a perpendicular magnetic anisotropy in all cases involved, which agrees well with the experimental data of the bulk FGT [17]. The MAE of the HS is reduced by 28% and enhanced by 77% in the down-polarization and the up-polarization directions, respectively. Our calculated average orbital moment of Fe1 and Fe2 is ∼0.08 µ B , which is much smaller than that of an isolated Fe 2+ ion (2 µ B ) according to Hund's rule. The relatively sizable value of 0.08 µ B shows that orbital magnetic moment is not completely quenched and hence causing the strong SOC and a large MAE [19]. Therefore, we conclude that the strength of SOC is enhanced by the +P polarization, leading to the enhancement of MAE. Figure 2 shows the spin-resolved total and partial DOS of the fully relaxed FGT monolayer, in which the Fermi level (E F ) is set as zero. According to Figure 2A, we find that our system is metallic with several high peaks in the valence band (VB). The VB is mainly contributed by the non-localized Fe-3d, Ge-4p, and Te-5p states, while the Fe-3p, Ge-4s, and Te-5s states are small and negligible. The DOS just below the Fermi level is mainly contributed by the majority spin state of Te-4p, as well as the majority and minority states of Fe-3d. We observe the stronger Fe-Te hybridization and weaker Fe-Ge interaction mainly in the −4.5 -−2, eV and −5.5 -−4 eV region, respectively. According to the Stoner criterion [36], I·N(E F )>1 reflects the itinerant ferromagnetism of the material, where N(E F ) is the spin-averaged DOS at the E F and I is the Stoner parameter [19]. From Figure 2B, our calculated value of I·N(E F ) is 2.05, which reflects an itinerant ferromagnetic nature of the FGT monolayer. This is in agreement with the results given in Zhuang et al. [19]. Moreover, the ferromagnetic order of the freestanding FGT monolayer has also been confirmed in Table 1. Figure 3 shows the total DOS (TDOS) and partial DOS (PDOS) of the Fe 3 GeTe 2 /In 2 Se 3 HS for the -P case, which is useful in understanding the specific contributions of different atomic orbitals. We can see less TDOS near the E F in the freestanding FGT monolayer. When combined with the In 2 Se 3 substrate, the TDOS near the E F increases dramatically, inducing a sizable DOS at the Fermi level. Namely, the metallicity is much better in the Fe 3 GeTe 2 /In 2 Se 3 HSs than in the freestanding FGT monolayer. This novel feature is quite different from the case when the anisotropic SOC plays a decisive role in stabilizing the FM configuration in another famous 2D material CrI 3 [37]. In the CrI 3 monolayer, a spin gap at the center of the Brillouin zone can be found below the Curie temperature. The spin-up and spin-down DOSs are asymmetric, indicating significant spin polarization of our HS system. The spin polarizability can be defined as: where N(E F ) α and N(E F ) β are the DOS at Fermi level for the spin-up and spin-down channels, respectively [38]. The SP F of the freestanding FGT monolayer, the -P HS, and the +P HS are 34.01, 35.57, and 29.69%, respectively. This indicates a much weaken quantum confinement effect in our HS system than in the strained FGT nanoribbons (SP F = 45-85.6%) shown in Han et al. [38].
To further explore the difference between the two inequivalent Fe atoms, we show the PDOS of Fe1 and Fe2 in Figures 3B,C, respectively. Remarkably our HS system presents a metallic character, strictly confined in the FGT layer. The metallic FGT layer exhibits ferromagnetism, with a magnetic moment of 2.2 µ B /Fe atom. Both the majority and minority spin states of the five split 3d orbitals are non-zero states near the E F . They mainly assemble between −4 to 1 eV, while the smaller spindown states are around −3 to −0.5 eV, which produces large magnetic moments in Fe1 atoms. For the Fe2 atoms, there are more spin-down states between −3.5 to −0.5 eV than for Fe1 atoms, which compensate the difference between spin-up and spin-down states. This causes the Fe2 atoms to have lower magnetic moments than Fe1. In the VB region, the contribution of Fe atoms at the E F is mainly due to the Fe1 d xy , d yz , d xz , and d 2 x -2 y orbitals as well as the Fe2 d xz , d 2 x -2 y , and d 2 z orbitals (d xy and d yz orbitals are almost degenerated). The rest of the orbitals are either too small or too far from the Fermi level. In the CB region, the Fe-3d PDOS is mainly contributed by the spin-down channels. The TDOS shown in Figures 3A, 4A and the PDOS shown in Figures 3B,C, 4B,C are quite similar, especially in the valence band region.
It is also shown in Figures 3D,E, 4D,E that the Fe and Te atoms contributing to the DOS near E F , but Fe1 does more than others [39]. Contrastingly, the Ge atoms, which can hardly be seen near the E F , play a less significant role in the FGT/In 2 Se 3 HS. In Figures 3F-I, 4F-I, we can see that the upper VB is dominated by the states of selenium atoms and a slight contribution of indium atoms, while the lower VB and the CB are the results of strong hybridization of In and Se. The DOSs of the Layer 1 to Layer 4 are almost alike, but the -P DOS and +P DOS gradually move to the high energy and low energy regions, respectively. This is mainly due to the change of electric potential induced by polarization discontinuity. We found that all the insulating In 2 Se 3 layers have little effects on the magnetic properties of our HSs. For the Fe1 and Fe2 atoms, we find essentially the same features with minor changes in the hybridization peaks when the FE-P reverses. The very similar DOS for both the ±P cases indicating the almost unchanged magnetic properties of the FGT/In 2 Se 3 HSs. It is worthy of note that the DOS of FGT layer given by Huang et al. [29] is also metallic, and our DOS is in topological resemblance with the DOS shown in Huang et al. [29]. Figure 5 illustrates the charge density difference of the ferromagnetic Fe 3 GeTe 2 /In 2 Se 3 HS for both the ±P cases. We only show the charge of the FGT/In 2 Se 3 interface (not the whole HS) because the charge transfer mainly occurs near the interface. Here, the yellow (blue) region represents charge accumulation (depletion). For the -P case, there is obvious charge transfer between the Te-Se atoms than between the Te-Fe atoms. This reflects the importance of Te orbitals in bridging the exchangecoupling between the Fe atoms, which agrees well with the DOS result. For the +P case, the charge redistribution at the FGT/In 2 Se 3 interface are less obvious. This is mainly due to the fact that the positive FE polarization leads to an increment of the distance between the FGT layer and the In 2 Se 3 substrate, hence greatly reducing the interfacial charge transfer. On the one hand, 2D vdW ferromagnetic materials with the FM orders are of great scientific interesting and technological importance for the next-generation storage devices. On the other hand, the DM interaction plays a key role in stabilizing the chiral domain walls and magnetic skyrmions in magnetic thin films with broken inversion symmetry. Therefore, it is important  to control the DM interaction in 2D vdW materials with their FM orders unchanged. In order to obtain the coefficients of DM interaction, we have built a 2 × 2 × 1 supercell of our FGT/In 2 Se 3 HS containing 24 Fe atoms. As shown in Figure 1C, the coefficients D (D 1 , D 2 , D 3 , D 4 , D 5 , and D 6 ) can be obtained by mapping different spin configurations on the Hamiltonian H given in equation (1) [40]. For our rectangular HS along the z-axis, the D x and D y will cancel each other out due to the C 3v symmetry of Fe atoms. Here only the D z is calculated for simplicity. We first set all the spins along the z-axis, and then change the spins of two neighbors Fe positions 1 and 2 (S 1 and S 2 ) into four configurations: (i) S 1 = (0, S, 0), S 2 = (0, 0, S); (ii) S 1 = (0, S, 0), S 2 = (0, 0, -S); (iii) S 1 = (0, -S, 0), S 2 = (0, 0, S); (iv) S 1 = (0, -S, 0), S 2 = (0, 0, -S); where S represents the magnetic moment of Fe [41]. These energies of the four spin states are described as E 1 , E 2 , E 3 , and E 4 . Then, the coefficient D can be determined by D = (E 1 +E 4 -E 2 -E 3 )/(4S 2 ) [41]. There are a large number of literatures calculate the coefficients of the DM interaction using VASP [42][43][44]. Although the coefficients are very small, these works confirm the accuracy of the VASP results. In this work, only the DM interactions between the nearest-neighboring Fe ions are considered.
To validate the plausible mechanism of ferroelectricallydriven DM interaction, we performed first-principles calculation on the FGT/In 2 Se 3 HS (see Supplementary Information for  details). The coefficients of the DM interaction D 1 -D 6 for firstnearest neighbors are listed in Table 2, resulting in a magnitude D of −0.0224 (−0.0185) meV for the HS at the -P (+P) case, that is about two orders of magnitude smaller than the exchange parameter J 1 . When the FE polarization changes from +P to -P, the D 2 and D 3 change from positive to negative, and other coefficients D 1 , D 4 , D 5 , D 6 , and D [D=(D 1 +D 4 +D 5 )/3] vary −24, 102, −92, −65, and −58%, respectively. This indicates that the FE polarization effects on the in-plane D 1 , D 6 and hence the average D is moderate. These DM interaction energies  may be result of the broken inversion symmetry and the strong SOC effect [45,46]. Our FGT/In 2 Se 3 HS has the polarizationdependent DM interaction, which can be used in skyrmionbased racetrack memory. Although our results are small, there are many ways to enhance the DM interaction in experiments. For example, one way is enhancing the spin-orbit coupling [47], and doping of heavy atoms is another way [48]. Using criticality analysis, Tan et al. [20] have demonstrated that the coupling length between vdW layers in Fe 3 GeTe 2 is estimated to be 5 layers. Wu et al. [45] have confirmed that the increasing of the thickness of FGT up to 4-60 layers will lead to greatly enhanced DM interaction. In short, we have demonstrated that the strong SOC in the In 2 Se 3 layers does induce an interfacial DM interaction at the interface with FGT, and have achieved the robust manipulation of DM interaction in the Fe 3 GeTe 2 /In 2 Se 3 heterostructure. Enhancing the DM interaction through the above-mentioned schemes can make our results more valuable in potential application of spintronics.

CONCLUSION
In this work, the vdW material In 2 Se 3 with both in-plane and perpendicular spontaneous polarization and the vdW ferromagnetic compound Fe 3 GeTe 2 are used to form a twodimensional artificial 2D heterostructure. Through the firstprinciples calculation, we have demonstrated the magnetoelectric coupling in the two-dimensional Fe 3 GeTe 2 /In 2 Se 3 system. This ferroelectric polarization of In 2 Se 3 can manipulate the magnetic anisotropy and the DM interaction in Fe 3 GeTe 2 . When the FE polarization is switched from -P to +P, the DM interaction of the heterostructure varies moderately, which is affected by the strong spin-orbit coupling. Through the reorientation of the polarization, we also realized the robust control of DM interaction which originates from the broken inversion symmetry. The spin polarizations shown in the density of states reflect spin-polarized states in the FGT/In 2 Se 3 system, which are important in spintronic devices. We hope that our results can promote the research on the spintronic devices with low power consumption, non-volatile, and high-speed.

DATA AVAILABILITY STATEMENT
All datasets generated for this study are included in the article/Supplementary Material.