key: cord-0825912-c6la56hw authors: Huang, Kai; Shao, Ding-Fu; Tsymbal, Evgeny Y. title: Ferroelectric control of magnetic skyrmions in two-dimensional van der Waals heterostructures date: 2022-02-23 journal: Nano lett DOI: 10.1021/acs.nanolett.2c00564 sha: ceffee62750fb035e4009e1b6cbca74c0aa23de3 doc_id: 825912 cord_uid: c6la56hw Magnetic skyrmions are chiral nanoscale spin textures which are usually induced by Dzyaloshinskii-Moriya interaction (DMI). Recently, magnetic skyrmions have been observed in two-dimensional (2D) van der Waals (vdW) ferromagnetic materials, such as Fe$_{3}$GeTe$_{2}$. The electric control of skyrmions is important for their potential application in low-power memory technologies. Here, we predict that DMI and magnetic skyrmions in a Fe$_{3}$GeTe$_{2}$ monolayer can be controlled by ferroelectric polarization of an adjacent 2D vdW ferroelectric In$_{2}$Se$_{3}$. Based on density functional theory and atomistic spin-dynamics modeling, we find that the interfacial symmetry breaking produces a sizable DMI in a Fe$_{3}$GeTe$_{2}$/In$_{2}$Se$_{3}$ vdW heterostructure. We show that the magnitude of DMI can be controlled by ferroe-lectric polarization reversal, leading to creation and annihilation of skyrmions. Furthermore, we find that the sign of DMI in a In$_{2}$Se$_{3}$/Fe$_{3}$GeTe$_{2}$/In$_{2}$Se$_{3}$ heterostructure changes with ferroelectric switching reversing the skyrmion chirality. The predicted electrically controlled skyrmion formation may be interesting for spintronic applications. Magnetic skyrmions are topological magnetic quasiparticles exhibiting a whirling spin texture in real space [1] . These spin-chiral objects have recently attracted significant interest due to the rich physics and promising spintronic applications [2] [3] [4] [5] [6] . The formation of magnetic skyrmions usually requires a strong Dzyaloshinskii-Moriya interaction (DMI) -an exchange interaction between adjacent magnetic moments driven by structural asymmetry and spin-orbit coupling [7] [8] [9] . Bulk magnets hosting magnetic skyrmions are limited to those with chiral crystal structures that support a finite DMI [2, [10] [11] [12] [13] . The skyrmions in these chiral magnets are usually observed at low temperature, which limits potential applications. Alternatively, a strong DMI can be induced by symmetry breaking and strong spin-orbit coupling at interfaces in the multilayer films composed of magnetic and heavy-metal layers [2, [14] [15] [16] . This allows the formation of skyrmions at high temperature. Currently, the efforts are aimed at exploring the material systems producing stable magnetic skyrmions that can be conveniently manipulated by external stimulus. Typically, the generation of magnetic skyrmions requires an external magnetic field to adjust a delicate balance between the exchange, anisotropy, DMI, and Zeeman energy contributions controlling different types of spin textures. Using an electric field is however a more energy-efficient method to control skyrmions which can be used in low-power spintronics. It has been demonstrated that magnetic skyrmions can be created and annihilated by voltage-controlled exchange coupling or magnetic anisotropy [17, 18] . Another approach to electrically control skyrmions is to exploit switchable polarization of an adjacent ferroelectric material [19, 20] . This method has been shown to efficiently manipulate skyrmions in magnetic thin films interfaced with perovskite ferroelectrics [21] [22] [23] [24] . The recent discoveries of two-dimensional (2D) van der Waals (vdW) materials exhibiting spontaneous electric or magnetic polarizations [25-38] opened a new dimension in exploring and exploiting ferroic and topological properties of materials including magnetic skyrmions. In a 2D ferromagnet, the symmetry breaking required for DMI to be realized by the interfacial proximity effect in a vdW heterostructure [39] . This allows the stabilization of magnetic skyrmions in a device with minimum thickness and even in the absence of the external magnetic field, as has been demonstrated in the recent experiments [40] [41] [42] [43] . A nonvolatile control of magnetic skyrmions by electric field is desirable for device applications. The recently discovered 2D ferroelectric materials [30] [31] [44] [45] , such as In2Se3 [46] [47] [48] [49] , can be used to mediate this effect. Due to their switchable electric polarization [50] [51] [52] [53] [54] , the 2D ferroelectrics can provide a voltage tunable interface proximity effect to other 2D materials. In particular, the electric polarization of a 2D ferroelectric is expected to efficiently control DMI in an adjacent 2D ferromagnet. This is due to DMI being sensitive to the interface orbital hybridizations and charge transfer effects [55] . As a result, when heterostructured with 2D ferromagnets, 2D ferroelectrics can be used to mediate the electrical control DMI and thus skyrmion behaviors [56, 57] . Fe3GeTe2 is a representative 2D ferromagnetic metal [58] which can be used as a viable material to explore the control of skyrmions by ferroelectric polarization. Experimentally, the emergence of skyrmions has been demonstrated in Fe3GeTe2, under certain conditions [40] [41] [42] [43] . Theoretically, when interfaced with a 2D ferroelectric In2Se3 layer ( Fig. 1(a) ), different magnitudes of DMI are expected in Fe3GeTe2 depending ferroelectric polarization of In2Se3. This may lead to the reversable creation and annihilation of magnetic skyrmions. A Fe3GeTe2 monolayer can be further sandwiched between two identical ferroelectric In2Se3 layers ( Fig. 1(b) ). In this case, a simultaneous switching of polarization in both layers is equivalent to applying a symmetry operation to the whole system, which is expected to change the sign of DMI and thus reverse the chirality of magnetic skyrmions. In this work, using first-principles density functional theory (DFT) calculations, we demonstrate that DMI in a Fe3GeTe2 monolayer can be induced and controlled when it is interfaced with a 2D ferroelectric In2Se3 layer. In such a Fe3GeTe2/In2Se3 vdW heterostructure, we find that the reversal of ferroelectric polarization of In2Se3 switches the DMI from a large to small value. As a result, based on our atomistic spin-dynamics modeling, we predict the reversable creation and annihilation of skyrmions in the system. Further, we show that the DMI sign in a In2Se3/Fe3GeTe2/In2Se3 sandwich structure changes with ferroelectric polarization switching, resulting in the reversal of magnetic skyrmion chirality. Our results show a possibility of the nonvolatile control of magnetic skyrmions in monolayer Fe3GeTe2 by an electric field, which may be interesting for the potential application of skyrmion systems. Fe3GeTe2 is a hexagonal layered ferromagnetic metal with the easy axis along the [001] direction [59] [60] [61] . Each Fe3GeTe2 layer contains five atomic layers stacked with Te-FeⅠ-GeFeⅡ-FeⅠ-Te sequence, where FeⅠ denotes the upmost and downmost Fe atoms, and FeⅡ denotes the Fe atom in the central GeFe plane ( Fig. 2(a,b) ). In its bulk phase, adjacent Fe3GeTe2 monolayers are rotated by 180° with respect to each other, forming a centrosymmetric structure with a magnetic point group 6/ ′ ′ . The presence of inversion symmetry in bulk Fe3GeTe2 prohibits a finite DMI and hence magnetic skyrmions are not expected to emerge in bulk Fe3GeTe2. Recently, Fe3GeTe2 has been successfully exfoliated down to a monolayer [58, 59] . In the absence of the bulk interlayer stacking, monolayer Fe3GeTe2 is noncentrosymmetric and belongs to the magnetic point group 6 2 . Although the inversion symmetry is broken, the DMI in monolayer Fe3GeTe2 is still vanishing due to other symmetry operations [62] , such as (001) mirror plane reflection ( Fig. 2(a) ), and symmetry operation ∥ combining time reversal symmetry and two-fold rotation symmetry ∥ with respect to the three in-plane axes along the in-plane directions of the nearest Fe neighbors ( Fig. 2(b) ). The absence of DMI in monolayer Fe3GeTe2 can be understood by comparing the energies of two artificial magnetic configurations with opposite chirality. As schematically shown in Fig. 2 (c), the clockwise (CW) and counterclockwise (CCW) configurations of Fe3GeTe2 can be transformed to each other by a operation. This is due to reversing the in-plane components of the Fe moments but conserving the out-of-plane component, i.e. , , = − , − , . ( As a result, the DMI energy which is determined by the energy difference between the CW and CCW states [63] is zero due to the equal values of and enforced by . Similarly, ∥ forbids the chiral magnetic configurations and results in zero DMI in monolayer Fe3GeTe2. These symmetry constraints can be broken if the top and bottom surfaces of the monolayer Fe3GeTe2 become asymmetric due to an interface proximity effect. Here, we consider the proximity of 2D ferroelectric In2Se3 interfaced with monolayer Fe3GeTe2. A monolayer In2Se3 contains five triangular lattices stacked with Se-In-Se-In-Se sequence. The central Se atom is located at one of the two asymmetric but topologically identical sites, associated with a finite out-of-plane polarization ⃗ pointing in opposite directions [34, 49] . Recent reports show that In2Se3 can be effectively used in vdW heterostructures to provide a nonvolatile control of 2D electronic structures despite of very weak interlayer coupling [50] [51] [52] [53] . Due to both having hexagonal atomic structures and similar in-plane lattice constants [60, 64 ] , Fe3GeTe2 and In2Se3 are well matched to construct commensurate vdW heterostructures. In such heterostructures, the and ∥ symmetries are broken, making DMI finite and thus the appearance of magnetic skyrmions possible. The DMI can be further controlled by ferroelectric polarization of In2Se3, resulting in variable behaviors of magnetic skyrmions. The band gap and ferroelectric polarization of In2Se3 are estimated to be 0.64 eV and 1.45 μC/cm , respectively, slightly different to these reported in Ref. [49] but do not influence the results in this work (see supporting information). We then construct a vdW heterostructure by attaching monolayer Fe3GeTe2 on top of monolayer In2Se3 (Fig. 3(a) ). To ensure that the Fe3GeTe2/In2Se3 stacking corresponds to the ground state, we calculate the total energy of the bilayer as a function of the lateral alignment of the two monolayers (see Supplemental Material for details). We find that the ground state corresponds to the stacking order where FeI atoms lie atop the nearest In atom (Fig. 3(a) ). Figures 3 (c,d) show the band structure of the freestanding monolayer Fe3GeTe2 in comparison to that of the bilayer Fe3GeTe2/In2Se3. We find that the Fe3GeTe2 electronic bands in the bilayer structure ( Fig. 3(d) ) do not change much compared to those in the freestanding Fe3GeTe2 (Fig. 3(c) ). This is due to negligible orbital hybridization across the interface reflecting weak vdW interactions. The bilayer bands exhibit a rigid shift toward higher energy resulting from a different chemical potential in the heterostructure. The electronic bands contributed by In2Se3 in the bilayer (denoted by colored dots) are also similar to those in its freestanding form (not shown). However, the conduction band minimum of In2Se3 is shifted below the Fermi level ( ) (Figs. 3(d) ), due to the electron charge transfer from Fe3GeTe2 to In2Se3. Polarization switching from up to down does not notably change the bands contributed by Fe3GeTe2. However, it shifts the In2Se3 bands to lower energy due to the field effect resulting from the polarization charge of In2Se3. Fig. 2(c) . (c) Exchange stiffness ( ), magnetic anisotropy ( ), and the DMI coefficient ( ). The single arrows (↑ or ↓) and double arrows (↑↑ or ↓↓) denote the polarization direction in Fe3GeTe2/In2Se3 and In2Se3/Fe3GeTe2/In2Se3, respectively. Figure 4 shows results of our calculations of the magnetic parameters for the Fe3GeTe2/In2Se3 bilayer. We find that the magnetic moments of the top FeI atom (denoted by FeI,t) and the central FeII atom do not change much compared to those in the freestanding Fe3GeTe2. On the contrary, the magnetic moment of the bottom FeI atom (denoted by FeI,b), which lies closer to the interface with In2Se3, is reduced by about ~0.1 compared to that in the freestanding Fe3GeTe2. This is due to the electron charge transfer from Fe3GeTe2 to In2Se3 not related to ferroelectric polarization. The polarization switching in In2Se3 does not produce notable changes in the magnetic moments due to the small polarization of In2Se3. This is consistent with the similar band structures contributed by Fe3GeTe2 in the Fe3GeTe2/In2Se3 bilayer for different polarizations. The difference in the magnetic moments of the FeI,t and FeI,b atoms reflects the broken mirror symmetry in the bilayer. As a result, the Heisenberg exchange coupling between FeI,t and FeII ( ) is different from that between FeI,b and FeII ( ) (Fig. 4(b) ). We find that the changes of the Heisenberg exchange parameters and the derived exchange stiffness ( ) induced by ferroelectric switching are not significant ( Fig. 4(b,c) ) due to the small ferroelectric polarization in In2Se3. Similarly, change in the magnetic anisotropy parameter ( ) in response to ferroelectric switching is also not large (Fig. 4(c) ). The DMI coefficient can be calculated using [63] = − 4√3 ℎ , where , ℎ are the lattice constant and layer thickness, respectively. The finite is supported by the magnetic point group 3 ′1 of the Fe3GeTe2/In2Se3 heterostructure due to broken and ∥ symmetries. We find a large ↑ = 0.28 mJ/m (DMI for polarization of the In2Se3 is pointing upward) and a small ↓ = 0.06 mJ/m (DMI for polarization of the In2Se3 is pointing downward). Such a sizable change in the DMI induced by polarization switching is in contrast to the relatively small changes in other magnetic properties. This is due to DMI being very sensitive to the structural asymmetry, which can be effectively controlled by ferroelectric switching. Next, we construct a In2Se3/Fe3GeTe2/In2Se3 vdW heterostructure by adding an additional monolayer In2Se3 on the top of Fe3GeTe2/In2Se3 (Fig. 3(b) ). In order to ensure the low energy stacking at both interfaces, the top In2Se3 layer is obtained by applying a mirror reflection of the bottom In2Se3 monolayer. Figure 3 (e) shows the calculated band structure of the In2Se3/Fe3GeTe2/In2Se3 trilayer with polarizations pointing upward for both In2Se3 layers. We find that the electronic bands contributed by Fe3GeTe2 show negligible changes compared to those in the Fe3GeTe2/In2Se3 bilayer. At the same time, the bands contributed by In2Se3 can be considered as a superposition of these for Fe3GeTe2/In2Se3 with opposite polarizations (Figs. 3(c,d) ). Figure 4 shows the calculated magnetic parameters for the In2Se3/Fe3GeTe2/In2Se3 heterostructure. The magnitudes of the magnetic moments, Heisenberg exchange parameters, and magnetic anisotropy reveal slight changes compared to these of the Fe3GeTe2/In2Se3 structure. As expected, the magnetic moments (Fe , ) and (Fe , ) as well as the exchange constants and swap their values upon the simultaneous switching of polarizations of both In2Se3 layers. This is because such a switching is equivalent to applying ∥ operation to the system, which swap the moments and exchange parameters on top and bottom of the GeFeII plane. The DMI coefficient for polarization of both In2Se3 layers pointing upward (downward) is calculated to be ↑↑ ≈ 0.22 mJ/m ( ↓↓ ≈ −0.24 mJ/m ). We see that within the calculation accuracy, ↑↑ ≈ − ↓↓ , i.e. the sign of the DMI coefficient changes with polarization switching, which is due to ferroelectric switching being equivalent to the ∥ symmetry transformation. Comparing the DMI coefficients for the Fe3GeTe2/In2Se3 and In2Se3/Fe3GeTe2/In2Se3 structures (Fig. 4) , we find ↑↑ ≈ ↑ − ↓ . This result follows from the presence of two interfaces (top and bottom) in the In2Se3/Fe3GeTe2/In2Se3 trilayer, where at one interface the polarization is pointing toward the Fe3GeTe2 layer and at the other interface the polarization is pointing away from the Fe3GeTe2 layer. Therefore, the interfacial proximity effect on the DMI constant in the In2Se3/Fe3GeTe2/In2Se3 trilayer is a sum of the two contributions from two Fe3GeTe2/In2Se3 interfaces. The predicted ferroelectric switching of DMI in the Fe3GeTe2/In2Se3 and In2Se3/Fe3GeTe2/In2Se3 heterostructures indicates a possibility of the electric field control of magnetic skyrmions in these systems. To demonstrate this effect, we perform the atomistic spin-dynamics modeling using LLG equation Here is the Gilbert damping constant, is the gyromagnetic ratio, ⃗ = ⃗ ⃗ is the unit magnetization vector for each sublattice with the magnetization ⃗ . The magnetic field ⃗ = − ⃗ is determined by the spin Hamiltonian: where is the magnetic moment of a Fe atom, is the exchange coupling, ⃗ is the DMI vector, is the magnetic anisotropy energy per Fe atom, and is the direction of the easy axis. A supercell of Fe3GeTe2 monolayer with the size of 60 nm × 60 nm is used in our simulation. In this supercell, nonmagnetic Ge and Te atoms are ignored, only Fe honeycomb lattice is kept. A round ferromagnetic domain is initially set in the center of the supercell, where the magnetic moments are pointing along +z/-z directions inside/outside the domain wall. The magnetic configuration of the system is then relaxed to the equilibrium state. The round domain can gradually relax to a stable skyrmion or shrink and eventually disappear in the background depending on the magnetic parameters used in the simulation. We find no magnetic skyrmion emerging in our atomistic modeling with the magnetic parameters given in Fig. 4 . This can be explained from the expected radius ( ) of a skyrmion [65] = 16 − . For the calculated magnetic parameters in Fig. 4 , we obtain the largest to be ~0.03 nm, which is too small to observe. This is due to a very large magnetic anisotropy predicted theoretically for Fe3GeTe2 [61] compared to the recent experimental measurements where magnetic skyrmions were observed in various Fe3GeTe2 based systems [40] [41] [42] [43] . It is known that the magnetic anisotropy of Fe3GeTe2 can be strongly suppressed by many factors such as doping and temperature [66] [67] [68] . Indeed, if we use the calculated and parameters and the radius of the magnetic skyrmions ~100 nm reported experimentally [40] [41] [42] [43] , a much weaker magnetic anisotropy ~0.008 MJ/m is estimated by Eq. (4). We thus assume a moderate anisotropy value of ~0.04 MJ/ m for our modelling to qualitatively demonstrate ferroelectric effect on the appearance of skyrmions in the considered systems. With this value of K, the results of our atomistic spin-dynamics modeling predict for the Fe3GeTe2/In2Se3 heterostructure, that a Néel-type skyrmion of about 12 nm in diameter emerges when the polarization is pointing upward (Fig. 5(a) ) and disappears when the polarization is pointing downward (Fig. 5(b) ). Such creation and annihilation of a magnetic skyrmion in this heterostructure is due to the change of DMI with reversal of ferroelectric polarization of In2Se3. For the In2Se3/Fe3GeTe2/In2Se3 heterostructure, we obtain a skyrmion of about 6 nm in diameter (Fig. 5(c) ). As expected, its chirality is reversed with reversal of polarization of In2Se3 (Fig. 5(d) ) due to the change of the DMI sign. Previous investigations showed that magnetic skyrmions can be moved at a high speed by a moderate electric current, indicating promising spintronic applications [1] . A well-controlled writing and erasing process is required prior to such applications. The creation and annihilation of skyrmions by voltage is desirable for this purpose, due to low energy consumption. Our work demonstrates that the required functional behavior can be achieved in a Fe3GeTe2/In2Se3 system where the interfacial proximity of a 2D magnet and a 2D ferroelectric induces DMI dependent on ferroelectric polarization. The controllable chirality of skyrmions in a In2Se3/Fe3GeTe2/In2Se3 heterostructure may be interesting for programmable skyrmion-based memories and logic [69] . The proposed approach is not limited to the Fe3GeTe2/In2Se3 system, but should be valid for vdW heterostructures constructed from other 2D ferroics. In conclusion, based on first-principles DFT calculations and atomistic spin-dynamics modelling, we have demonstrated that the Dzyaloshinskii-Moriya interaction in monolayer Fe3GeTe2 can be induced and controlled by ferroelectric polarization of an adjacent ferroelectric In2Se3. For a Fe3GeTe2/In2Se3 vdW heterostructure, we have predicted that the reversal of ferroelectric polarization of In2Se3 can switch the DMI of the system from a large to small value, resulting in the reversible skyrmion creation and annihilation. For a In2Se3/Fe3GeTe2/In2Se3 vdW heterostructure, we have shown that the DMI sign was changed with ferroelectric switching, resulting in the reversal of chirality of the magnetic skyrmions. Our work demonstrated a possibility of the nonvolatile control of magnetic skyrmions in monolayer Fe3GeTe2 by an applied electric field, which is promising for the potential device application of skyrmion systems. 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This work was supported by the by the EPSCoR RII Track-1 (NSF Award OIA-2044049) program. Computations were performed at the University of Nebraska Holland Computing Center.