Interfacial coupling induced critical thickness for the ferroelectric bistability of two-dimensional ferromagnet/ferroelectric van der Waals heterostructures
Xiaokun Huang ,1,*Guannan Li,2Chao Chen,1Xin Nie,1Xiangping Jiang,1,†and Jun-Ming Liu3
1School of Materials Science and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, 333001, China
2Department of Physics, Huaiyin Institute of Technology, Huaian 223003, China
3National Laboratory of Solid State Microstructures and Department of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
(Received 11 August 2019; revised manuscript received 5 December 2019; published 23 December 2019) The discovery of two-dimensional (2D) ferroic materials has stimulated substantial efforts in developing emergent functionalities by synthesizing van der Waals (vdW) heterostructures, and one promising effect is the nonvolatile electrical control of magnetism in magnetoelectric (ME) heterostructures consisting of coupled 2D vdW ferromagnetic and ferroelectric layers. In this paper, it is proposed that the asymmetric interfacial coupling in such heterostructures may seriously distort the ferroelectric double-well potential, thus destabilizing the ferroelectricity. We investigate this consequence in Fe3GeTe2/α-In2Se3 vdW heterostructure using the first-principles calculations. It is revealed that one of the two potential wells for ferroelectric monolayerα-In2Se3
is suppressed by the asymmetric interfacial coupling between electric polarization and the built-in electric field induced by intrinsic charge transfer, while the ferroelectric bistability can be recovered when theα-In2Se3layer is thicker than three unit cells. Therefore, this work presents an alternative mechanism of critical thickness for the 2D ferroelectricity in ME vdW heterostructures.
DOI:10.1103/PhysRevB.100.235445
I. INTRODUCTION
Since the discovery of graphene [1,2], large classes of two-dimensional (2D) van der Waals (vdW) materials with fascinating properties have generated a great deal of interest and opened a new frontier for materials research. Almost all the types of increasingly important physical properties in bulk materials, such as semiconductor, semimetal, and super- conductor, have been experimentally observed in numerous atomically thick 2D vdW materials [3–9], offering extensive opportunities for miniaturization of emerging electronic de- vices with novel functionalities.
As an important branch of the collective properties of condensed matters, ferroicity in 2D systems down to atomi- cally thick layers is of great significance, given the increasing concern with the limit of Moore’s law, which is being ap- proached. However, both ferromagnetism and ferroelectricity suffer from restrictions imposed by reduced dimensionality in the conventional framework of understanding, and remain elusive in the 2D materials library. It is believed that thermal fluctuations may easily destroy 2D magnetism in the isotropic Heisenberg model at finite temperatures, according to the Mermin-Wagner theorem [10], and the size effect originating from the depolarization field will disable the ferroelectricity if the system’s characteristic scale is below tens of nanometers [11–13]. These ideas have been challenged recently owing to the exciting breakthroughs and increasing evidence favoring
the existence of 2D ferroic orders [14–23]. Investigations on various 2D ferromagnets such as CrI3, Cr2Ge2Te6, and Fe3GeTe2have revealed that magnetocrystalline anisotropy is the key to countering the thermal fluctuations and stabilizing long-range ferromagnetic (FM) orders even in a monolayer [14–16]. Discoveries of 2D ferroelectric (FE) systems with out-of-plane polarizations such as In2Se3 [19] and MoTe2 [23], in which space-inversion symmetry is broken sponta- neously due to relative atomic displacements, show brand new intrinsic mechanisms for countering the depolarization field. In this sense, these 2D ferroic materials refresh our common perceptions and allow a platform on which the issue of the size limit can be rechecked from different points of view.
Nevertheless, these inspiring works do not undercut our concern with the stability of ferroicity in 2D systems. Instead, promising applications of 2D ferroic materials approaching their spatial lower limit make this concern even more critical.
In fact, not only the size effect but also some intrinsic/extrinsic perturbations that otherwise weakly influence the order pa- rameters in three-dimensional (3D) systems may now become comparable to the energy scale for the ground state, posing potential threats to the stability of 2D ferroicity. For example, in FE tunnel junctions based on perovskite-type oxide ferro- electrics, properties of the FE tunneling layer may be essen- tially related to the electronic environment at the interfaces via electrode-oxide bonding [24,25]. When the thickness of the FE barrier is reduced below a certain limit, the energy associated with the deleterious interface effects may overcome the energy gained due to FE ordering, and consequently the polarization may become either unstable or unswitchable.
In this connection, a major challenge has occurred to us.
Considering that 2D ferroic materials with diversified physical properties are usually integrated in vdW heterostructures for most device applications [26–30], the interfacial coupling in such heterostructures as an emerging ingredient detrimental to the stability of 2D ferroicity will become an urgent issue if the 2D ferroic layer is thin to the limit. This issue has not yet received much attention.
In this work, we consider magnetoelectric (ME) het- erostructures consisting of a 2D FM layer and a 2D FE layer.
In such heterostructures, it is believed that the ME coupling originating from carrier-mediated interfacial couplings be- tween two component layers makes it possible to modulate the magnetization near the interface if the FE polarization reversal is triggered by an electric field, offering a promising scheme for future storage devices. Recent first-principles studies pre- dicted polarization reversal induced magnetic transition [31], metal-insulator transition [32], and magnetization switching [33] in various 2D FM/FE bilayer heterostructures, suggesting that the interfacial ME coupling is quite effective. However, the effect of interfacial ME coupling on the ferroic stability of 2D component layers has not been demonstrated. Actually, it is believed that the heterointerface can further break the spatial rotation symmetry of the 2D FM layer, offering extrinsic mag- netocrystalline anisotropy which helps to stabilize magnetic orders. The heterointerface also breaks the mirror-inversion symmetry of the 2D FE layer, causing the interfacial coupling to be asymmetric given two different FE polarization states.
However, for 2D ferroelectricity, the asymmetric interfacial coupling as the origin of the ME coupling effect will be adverse for its stability to some extent.
In fact, from a phenomenological point of view, there is a conflict between the ME coupling effect and the FE bistability of the FM/FE vdW heterostructure. It is known that ME coupling relies on energy terms involving both polarization Pand magnetizationM. Since the ME coupling effect in the FM/FE vdW heterostructure originates from the asymmetric interfacial coupling, the ME coupling energy is variable under the inversion operation of polarizationP[34]. If one expects a remarkable ME coupling effect, then the energy difference between the two polarization states (P and –P) should be large. Consequently, the asymmetric energetics would seri- ously damage the FE bistability. As shown in Fig. 1, the asymmetry of the interfacial ME coupling will cause the FE double-well potential to distort. One of the two degenerate po- larization states would then lose its stability if the asymmetric interfacial coupling is strong enough to be comparable to the potential well depth. Such physics generally exists in FM/FE heterostructures. In 3D cases, the FE bistability is usually not concerned since the FE component is thick enough to protect it. As for 2D FM/FE vdW heterostructures, the FE component, down to the monolayer limit, is as thin as the characteristic length of the interfacial coupling. Consequences of the asym- metric interfacial coupling then become significant, raising concern about FE bistability.
In this work, we study the effect of asymmetric inter- facial coupling on the FE bistability of 2D FM/FE vdW heterostructures using first-principles calculations. We choose an Fe3GeTe2 layer plus an α-In2Se3 layer to constitute a representative FM/FE heterostructure. Fe3GeTe2 is a 2D FM
MEcoupling
Free 2D Ferroelectric 2D Ferromagnet
2D Ferroelectric
P P P
P
FIG. 1. The schematic diagram of the ferroelectric double-well potential energy surface. The red line corresponds to the free 2D ferroelectric. The red-blue hybrid line corresponds to the 2D FM/FE vdW heterostructure. The upward polarization state of the FE layer has lost the stability due to the strong asymmetric interfacial ME coupling in the heterostructure.
metal [16] and α-In2Se3 is a 2D FE semiconductor [35]
with interlocked in-plane and out-of-plane ferroelectricity [20,36]. They are both practically available and their Curie temperatures remain high even in monolayers [16,21]. We pay special attention to the thinnest heterostructure consisting of a Fe3GeTe2 monolayer and anα-In2Se3 monolayer. This choice is due to the fact that the FM monolayer is usually the best candidate to maximize the ME effect because most of the screening charges are located in the interface region [17].
Also, the FE monolayer can maximize the adverse effect of the asymmetric interfacial coupling on the FE bistability. We show that one of the two polarization states of the monolayer α-In2Se3 loses its stability in such a bilayer heterostructure, while the FE bistability can be recovered when theα-In2Se3
layer is thicker than three unit cells. In addition, we discuss the effect of polarization reversal on the magnetism of the Fe3GeTe2layer.
II. COMPUTATIONAL DETAILS
Our first-principles calculations were performed us- ing density-functional theory (DFT) within the projector augmented wave method as implemented in the Vienna ab initio simulation package (VASP) [37–39]. The exchange and correlation functional was treated using the Perdew- Burke-Ernzerhof (PBE) parametrization of generalized gra- dient approximation (GGA) [40]. The projector augmented wave potentials explicitly include eight valence electrons for Fe(3d64s2), four for Ge(4s24p2), six for Te (5s25p4), 13 for In (4d105s25p1), and six for Se (4s24p4). To eliminate interactions between the heterostructure with its periodic im- ages, a vacuum spacer of around 1.5 nm was inserted in the superstructure. We used the DFT-D3 correction method to describe vdW interactions [41]. A dipole correction procedure perpendicular to the heterostructure was applied to eliminate the electric field in the vacuum region [42,43]. The-centered 24 × 24 × 1 Monkhorst-pack k-points mesh was used for Brillouin-zone integrations, and the linear tetrahedron method with Bloch corrections was used for total energy calcula- tions [44,45]. To have an accuracy sufficient to evaluate the total energy of the heterostructure, we have checked the
Fe
In Te Ge
Se (a) Fe3GeTe2
b In2Se3
P↓ P↑
c Fe3GeTe2/ In2Se3
FGT/ P↓ FGT/ P↑
d asymmetric interfacial coupling
P P
Fe3GeTe2In2Se3
FeII
FeI
Ebi Ebi
FIG. 2. (a,b) Top and side views of monolayer Fe3GeTe2 andα-In2Se3, respectively. (c) Side views of the Fe3GeTe2/α-In2Se3bilayer heterostructures. The black frame represents the unit cell. The shown structures exhibit the most energetically favorable interfacial stacking configuration for bothα-P↑andα-P↓states. (d) Schematic diagrams illustrate the mechanism of the asymmetric interfacial coupling. The red Prefers to the FE polarization andEbirefers to the built-in electric field induced by intrinsic charge transfer.
convergence with respect to cutoff energy. A large plane-wave cutoff energy of 600 eV was used to converge the total energy to within 1 meV per unit. The energy convergence criterion is 10−6eV. During structural relaxation, atomic positions were relaxed until the Hellman-Feynman forces were less than 0.005 eV/Å. Bader charge analysis was used to evaluate elec- tron distributions [46]. The climbing-image nudged elastic band method was used to calculate the kinetics pathways [47]. In addition, it should be noted that the GGA approach in dealing with band structures usually somewhat underesti- mates the band-gap size, while the hybrid exchange correla- tion functional such as B3LYP and B3PW helps to achieve better precision [48,49]. Nevertheless, the GGA approach is adequate to unveil the main physics addressed in this work;
i.e., the energetics and ferroelectricity, and thus we just focus on GGA calculations here.
III. RESULTS AND DISCUSSION A. Fe3GeTe2/α-In2Se3bilayer heterostructures Monolayer Fe3GeTe2is a quintuple layer (QL) consisting of five atomic sublayers [16]. As shown in Fig. 2(a), the stacking sequence can be written as Te-FeI-(FeIIGe)-FeI-Te per formula unit with two inequivalent Fe sites denoted as FeI and FeII. The Te and FeI sublayers are both triangular lattices, while the (FeIIGe) sublayer consists of two triangular sublattices, forming a graphenelike honeycomb lattice. The itinerant ferromagnetism originates from partially filled Fe-d orbitals, which dominate the band structures around the Fermi level.
Monolayerα-In2Se3 also consists of five atomic sublay- ers [19]. Each atomic sublayer contains only one elemental
species, with atoms arranged in a triangular lattice. As shown in Fig.2(b), the stacking sequence of the five atomic sublayers in a QL is Se-In-Se-In-Se. Se atoms in both top and bottom sublayers reside on the hollow sites of the neighboring In atoms. Se atoms in the middle sublayer are vertically bonded to one side of the two neighboring In sublayers and have three Se-In bonds connecting with the other side. The spacing between the middle Se sublayer and the two In sublayers is different, breaking the centrosymmetry and leading to the spontaneous out-of-plane polarization. The polarization state is denoted as α-P↑ or α-P↓ when the middle Se atom is close to the lower or the upper In sublayer, respectively. The calculated magnitude of the out-of-plane electric dipole for monolayerα-In2Se3is 0.094eÅ per unit cell in this work.
The in-plane triangular lattice constants of 4.029 and 4.063 Å were obtained for free monolayer Fe3GeTe2and free monolayerα-In2Se3, respectively, which are consistent with their experimental bulk values [50,51]. The lattice mismatch is only 0.8%, allowing a perfect vertical stacking configuration.
Our designed bilayer heterostructures consist of a monolayer Fe3GeTe2unit cell on top of a monolayerα-In2Se3unit cell.
To find the energy minimum of the heterostructure, the interfa- cial stacking configuration should be taken into consideration.
Therefore, we performed a thorough search for the most en- ergetically favorable interfacial stacking configuration. Since both In2Se3 and Fe3GeTe2 monolayers have the threefold rotational symmetry, we only considered interfacial stacking configurations that retain this symmetry. Only six types of configurations satisfy the requirement. Crystal structures and calculated total energies of the 12 designed heterostructures (6 interfacial stacking configurations×2 polarization states) are shown in Fig. S1 and listed in Table S1 (see Supplemental
Material [52]). As shown in Fig. 2(c), the most energeti- cally favorable configuration is actually the same for both downward and upward polarization states, which are denoted as FGT/α-P↓ and FGT/α-P↑, respectively. The interfacial Te atoms reside on the hollow sites of the interfacial Se atoms, while vertical projections of the adjacent FeI atoms are located in other hollow sites of the interfacial Se atoms, facing the In atoms. Then we further introduced lateral relative displacements at the interface of this configuration to check whether or not the total energy could be reduced. It was found that all the displacements vanished and the most energetically favorable configuration was restored after fully structural relaxations. Thus, we believe that energy minima have been reached for both upward and downward polarization states.
In the following work, all the calculations were performed on this configuration.
We also performed calculations to check the stability of the FM order of monolayer Fe3GeTe2 in the bilayer het- erostructure from two aspects. On one hand, the calculated energy cost to flip the magnetic moment of one Fe ion is as high as hundreds of meV, suggesting that FM order is the most energetically favorable state. On the other hand, the calculated magnitude of magnetocrystalline anisotropy energy of monolayer Fe3GeTe2in the bilayer heterostructure, which is defined as the energy difference between the two states where the magnetization is perpendicular or parallel to the interface, is about 3.9 meV. This energy is even larger than that for a free Fe3GeTe2 monolayer which is 3.6 meV obtained in this work, suggesting that FM order in heterostructures can be even more stable than that in free monolayers. Due to these two reasons, the FM order of monolayer Fe3GeTe2 in the heterostructure is very robust, so we assume that the FM order remains as the most stable magnetic state of the heterostructures in this work, and we focus on the FM order in the following calculations.
B. Asymmetric interfacial coupling
Due to the asymmetry, the total energies of the bilayer heterostructures, given two different polarization states of the α-In2Se3 layer, are different. The calculated energy for the FGT/α-P↓ state is 62.7 meV lower than that for the FGT/α-P↑state, indicating that the interfacial coupling of the FGT/α-P↓state is stronger. The stronger interfacial coupling leads to a shorter interlayer distance between the two coupled monolayers. The calculated interlayer distances between the α-In2Se3 and Fe3GeTe2 monolayers are 2.915 and 3.042 Å for the FGT/α-P↓and FGT/α-P↑ states, respectively, also suggesting evident asymmetry of the coupling strength.
For the asymmetric interfacial coupling in the bilayer heterostructures, the underlying mechanism is believed to be twofold. On one hand, the out-of-plane electric polarization of theα-In2Se3layer induces charge redistributions across the interface to screen the polarization field. On the other hand, the intrinsic charge transfer that results from the chemical potential difference between the two monolayers leads to a net electric dipole across the interface and the built-in electric fieldEbi. These two factors contribute a phenomenological en- ergy term –EbiP,suggesting that the out-of-plane polarization of theα-In2Se3layer prefers to align alongEbi. If one assumes
that electrons transfer from Fe3GeTe2 to α-In2Se3, making the Fe3GeTe2 layer positively charged and establishing the downwardEbias illustrated in Fig.2(d), then the FGT/α-P↓
state will be the ground state.
To check this speculation, we plot the differential electron density distributions and quantitatively study the charge trans- fer. As shown in Figs.3(a)and3(b), much stronger charge re- distributions occur across the interface in the FGT/α-P↓state than in the FGT/α-P↑state. Using the Bader charge analysis, we observed electron transfers of 0.058e and 0.017e from Fe3GeTe2 to α-In2Se3 for the FGT/α-P↓ and FGT/α-P↑
states, respectively. The polarization reversal affects the amount of transferred electrons but does not change the di- rection of the built-in electric field, which suggests dominant intrinsic charge transfer effect due to the chemical potential drop across the heterointerface. Thus the speculation is con- firmed.
For further confirmation, the net dipole across the interface caused by the electron transfer should be checked. However, the interfacial electric dipole in this bilayer heterostructure could not be derived in a direct way. This is because the whole system is almost as thin as the characteristic length of the interfacial coupling, so we are unable to determine a precise definition of the interfacial electric dipole. Instead, we calculated the total electric dipoles of the bilayer het- erostructures. It should be mentioned that the in-plane electric polarization of the bilayer heterostructure cannot be defined since the Fe3GeTe2 layer is a ferromagnetic metal, while the out-of-plane electric polarization is well defined due to the presence of a vacuum region. We calculated the net dipole moment parallel to the z axis by directly integrating ρ times z over the whole superstructure, where ρ is the charge density andzis the coordinate. The calculated electric dipoles are –0.016eÅ and 0.109eÅ for the FGT/α-P↓and FGT/α-P↑states, respectively, while the calculated absolute value of the electric dipole for a free α-In2Se3 monolayer is 0.094eÅ. Owing to the interfacial coupling, the electric dipole has been changed by 0.078eÅ for the α-P↓state, which is much larger than 0.015eÅ for the α-P↑ state. The much larger change of the electric dipole also suggests stronger charge transfer effect in the FGT/α-P↓ state. Meanwhile, the changes of the total electric dipoles are both upward in the two polarization states, indicating that interfacial electric dipoles are upward regardless of polarization states. This result is more evidence to support our scenario as shown in Fig.2(d).
To study the underlying physics of the intrinsic charge transfer, we investigate the electronic structures of the FGT/α-P↓and FGT/α-P↑states. The calculated density of states (DOS) and band structures are shown in Figs.3(c)–3(h).
It is seen that the out-of-plane electric polarization of the α-In2Se3layer has a remarkable field effect on the band align- ment. The gaps between the conduction band minimum of the α-In2Se3layer and the Fermi level are about 0.22 and 0.72 eV for the FGT/α-P↓ and FGT/α-P↑ states, respectively. The band structures can explain why so few electrons transfer from Fe3GeTe2 to α-In2Se3. The Fermi level derived from the Fe3GeTe2 layer is located between the conduction band minimum and the valence band maximum of the α-In2Se3
layer. Electrons are supposed to transfer from the higher
-0.04 -0.02 0.00 0.02 0.04
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5 10 15 20 25
-0.04 -0.02 0.00 0.02 0.04
-3 -2 -1 0 1 2 3
-10 -5 0 5 10
-2 -1 0 1 2
•Se
•In
•Fe
•Ge
•Te
(h) (g)
(f)
(d) (c) (e)
P→ Fe3GeTe2
In2Se3
(e/Å )
(a)
DOS(States/eV)
Fe3GeTe2
In2Se3-P↑
E n erg y (eV)
(b)
In2Se3 Fe3GeTe2(e/Å )
z(Å)
←P
DOS( St at es/ eV)
Energy(eV)
Fe3GeTe2
In2Se3-P↓
Ene rgy(e V)
Spin↓ Spin↑
M K Γ
Γ
Γ M K Γ
FIG. 3. Electronic structures of the Fe3GeTe2/α-In2Se3bilayer heterostructure for the FGT/α-P↑and FGT/α-P↓states. (a,b) Integrals of differential electron density distributions in theabplane for the FGT/α-P↑and FGT/α-P↓states, respectively.ρ=ρFGT/α-In2Se3−ρFGT− ρα-In2Se3. The solid dots with different colors along the “ρ=0” line represent vertical positions (z) of different atoms. (c,d) Layer-resolved DOS for the FGT/α-P↑and FGT/α-P↓states respectively. (e,f) and (g,h) Layer-resolved band structures for the FGT/α-P↑and FGT/α-P↓ states, respectively. Left panels represent spin-up channels and right panels represent spin-down channels. Blue represents contributions from the Fe3GeTe2layer and red represents the In2Se3layer. The Fermi levels are set as 0 eV for (c–h).
energy level side (Fe3GeTe2) to the lower side (α-In2Se3).
However, transferred electrons cannot fill in the fully occupied valence bands of theα-In2Se3layer. Instead, they accumulate in the interface region and hybridize with empty orbitals of the interfacial Se atoms, leading to the crossing of conduction bands with the Fermi level. As the conduction bands of the α-In2Se3 layer are much closer to the Fermi level in the FGT/α-P↓state than in the FGT/α-P↑state, more electrons occupy the hybridized orbitals of the interfacial Se atoms in the FGT/α-P↓state, leading to larger intrinsic charge transfer effect. So far, the mechanism of the asymmetric interfacial coupling has been comprehensively revealed.
C. Kinetics of polarization reversal
Now we are in a good position to discuss whether the bistable polarization states for the heterostructure are avail- able or not. On one hand, we have checked the energy difference between the FGT/α-P↓ and FGT/α-P↑ states.
On the other hand, we further track the kinetics pathway of the polarization reversal process from the FGT/α-P↓ state to the FGT/α-P↑ state. To calculate the kinetics pathway of the Fe3GeTe2/In2Se3 bilayer heterostructure, we used the same transformation motions as the polarization rever- sal of the free α-In2Se3 monolayer. For the free α-In2Se3 monolayer, the most effective kinetics pathway connecting two degenerate polarization states (α-P↓andα-P↑) was re- vealed via a three-step motion, which involves two degenerate
β-In2Se3states as the intermediate states of the polarization reversal process [19]. Theβ-In2Se3is a metastable structure, which can be derived by shifting the middle Se sublayer of the highly symmetric nonpolar β structure slightly away (Fig. S2, Supplemental Material [52]); thus the centrosym- metry is broken [53]. Theβ-In2Se3 structures with the up- ward and downward polarizations are denoted as the β-P↑
and β-P↓ states, respectively. We obtained structures and total energies of the FGT/β-P↓and FGT/β-P↑states after fully structural relaxations. Then, these two metastable states were used as the intermediate states to calculate the kinetics pathway of the polarization reversal process for the bilayer heterostructure.
Figure 4(a) shows the kinetics pathway from the FGT/α-P↓state to the FGT/α-P↑ state and the pathway is seriously asymmetric. The energy of the FGT/α-P↑state is even higher than those of the two intermediate states; the energy difference between the FGT/α-P↑ and FGT/β-P↓
states is up to 54.6 meV. In this case, we deem that the FE bistability of theα-In2Se3layer in the bilayer heterostructure has been damaged. Meanwhile, the energy barrier between the FGT/α-P↑ state and the FGT/β-P↓ state at the local valley is 19.5 meV, much smaller than 28.2 meV for the case of the free α-In2Se3 monolayer as shown in Fig. 4(b), suggesting the seriously weakened dynamical stability of the FGT/α-P↑ state. Thus, one can argue that the monolayer α-In2Se3in the bilayer heterostructure is unlikely to preserve the room-temperature 2D ferroelectric bistability.
0 20 40 60 80
0 20 40 60 80
-0.12 -0.08 -0.04 0.00 0.04 0.08
0.12
(b)
Fe3GeTe2/In2Se3
In
2Se
3phase
−P↓
'−P↓ '−P↑
−P↑
-0.12 -0.08 -0.04 0.00 0.04 0.08 0.12
electric dipole
Electric Dipole(eÅ)
free monolayer In2Se3
reaction coordinates
Energy(meV)
kinetics pathway
reaction coordinates (a)
FIG. 4. Calculated kinetics of the polarization reversal for (a) Fe3GeTe2/In2Se3 and (b) free In2Se3 monolayer. In both (a,b), the left longitudinal coordinate axis is for the energy and the right one is for the electric dipole. It should be mentioned that the kinetics pathway of the free In2Se3monolayer was first given by Dinget al.[19], where the barrier height is different from here. This is because vdW corrections were taken into account for monolayer calculations in our work.
D. Critical thickness
As the thickness of the In2Se3layer in the heterostructure increases, the detrimental effect of the asymmetric interfacial coupling on the FE bistability diminishes, and the out-of- plane ferroelectricity of the heterostructure tends to approach the bistable nature of theα-In2Se3thick film. When the In2Se3
layer is above a certain critical thickness, the FGT/α-P↑
state will have lower energy than the FGT/β-P↓ state, in which case the FE bistability is supposed to be recovered.
To estimate the critical thickness, we calculated the energies of heterostructures with thickness of the In2Se3layer varying from 1 to 4 QLs. Bulkα-In2Se3 has two possible stacking modes, the 2H type and 3R type [51]. Structures and total energies of the 2H type and 3R type are both shown in Fig. S3 and listed in Table S2 (Supplemental Material [52]). We focus on the 3R type in the following work as it is energetically more favorable than the 2H type.
The calculated energies of the heterostructures with an N-QL (N =1–4) In2Se3layer are shown in Fig.5(a). To in- tuitively illustrate the critical thickness, the calculated energy data of the three states (two polarization states and one in- termediate state between them) are displayed by plotting four sets of polylines that correspond to four differentNvalues of the In2Se3layer. When the In2Se3layer in the heterostructures is thicker than 3 QLs, the energy of the FGT/α-P↑ state is lower than that of the FGT/β-P↓state and the FE bistability can be deemed to be recovered. As is shown in Fig. 5(b), the energy difference between the FGT/α-P↑and FGT/α-P↓
states increases as the In2Se3layer becomes thicker and does not reach the saturation value (∼160 meV) until the In2Se3 layer is thicker than about 6 QLs (∼6 nm). The saturation rate is slow because the interfacial coupling is a long-range Coulomb interaction. From the viewpoint of energetics, 6 QLs may represent a characteristic scale of the In2Se3 layer that can be used to distinguish between the 2D or 3D nature of such heterostructures.
E. Magnetoelectric coupling
Finally, we discuss the ME coupling effect. The calcu- lated magnetic moments (μB/unit) of the Fe3GeTe2/(1 QL) α-In2Se3 bilayer heterostructure are 6.188 and 6.24 for the FGT/α-P↓ and FGT/α-P↑ states, respectively. Change of the magnetic moment induced by the polarization reversal is 0.052μB, which is mainly attributed to the FE field effect on the distributions of the spin-polarized DOS derived from Fe3GeTe2. The magnetic response increases linearly with the thickness of the In2Se3layer varying from 1 to 4 QLs as shown in Fig.5(c). We can roughly estimate, according to the growth rate, that its saturation value is no more than 0.2μB(3%).
Although the ME effect seems modest, the associated energy difference (EFGT/α=EFGT/α-P↑−EFGT/α-P↓) as a result of the asymmetric interfacial coupling is sufficiently large to destroy the FE bistability of theα-In2Se3 layer thinner than 4 QLs (∼4 nm). Therefore, it may be a major challenge to reconcile the ME coupling strength and the thickness of the FE layer when designing 2D FM/FE vdW heterostructures.
Regarding this dilemma, we propose two perspectives for achieving appreciable ME effect in ultimately thin FM/FE vdW heterostructures. On one hand, one can modulate the magnetic exchange interactions in 2D ferromagnets through chemical engineering such as ion doping, to make the 2D magnetism reside near the ferromagnetic-antiferromagnetic boundary in the phase diagram where magnetic order depends on charge doping, so that interfacial charge variations due to the polarization reversal could trigger a magnetic recon- struction, achieving a substantial magnetic response. On the other hand, the energy difference between the two polarization states should be reduced to protect the FE bistability. For this purpose, a FM/FE/X-type heterostructure is suggested, in whichXcan be a monolayer vdW metal or semiconductor.
In this way, one can expect the two unbalanced energies aris- ing from asymmetric interfacial couplings derived from the FM/FE interface (EFM/FE) and the FE/Xinterface (EFE/X)
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1 2 3 4
0.06 0.08
Energy(meV)
0.10N=1 N=2 N=3 N=4
β′↓ α↑
α↓
DFT results fitting curve extrapolation curve
E(α↑)−E(α↓)(meV)
In2Se3thickness N (QLs)
•
(a) (b)
M(α↑)−M(α↓)(B)
In2Se3thickness N (QLs)
(c)
FIG. 5. (a) Calculated energies of the Fe3GeTe2/[(N-QL,N=1–4) In2Se3] heterostructures in the FGT/α-P↓, FGT/β-P↓, and FGT/α-P↑states, respectively. (b) Energy difference between the FGT/α-P↑and FGT/α-P↓states forN=1–8. The four black solid dots represent the DFT calculated results forN=1–4, while the red solid line represents the best fitting curve of the four points. The math formula used to fit the curve isa(e−N/b)+c;a, b, care fitting parameters. The dashed line is obtained by extrapolating from the fitting curve, representing the estimated energy differences forN=5–8. (c) Difference of calculated magnetic moments between the FGT/α-P↑and FGT/α-P↓states.
to cancel each other when the polarization is reversed. If an appropriateXlayer is chosen, asymmetry of energetics in the FM/FE/X heterostructure will be much smaller than that in the FM/FE heterostructure, so that the critical thickness can be reduced.
IV. CONCLUSIONS
In summary, we investigate the effect of the asym- metric interfacial coupling on the FE bistability of 2D Fe3GeTe2/α-In2Se3 vdW heterostructures by using first- principles calculations. It is revealed that the asymmetric interfacial coupling between electric polarization and built-in electric field suppresses one of the two potential wells for ferroelectric monolayerα-In2Se3 in the bilayer heterostruc- ture, thus destabilizing the ferroelectricity. When the In2Se3
layer is thicker than three unit cells, the FE bistability can be recovered. Thus, we propose that the asymmetric interfacial coupling leads to a critical thickness for the 2D ferroelec- tricity of ME heterostructures. In addition, we propose two perspectives for achieving appreciable ME effect in ultimately
thin FM/FE vdW heterostructures. One is to modulate the magnetic exchange interactions to achieve magnetic recon- structions triggered by polarization reversal, the other one is to choose an appropriateXlayer and build a FM/FE/X-type heterostructure. Our results help to achieve a fundamental understanding of FE bistability not only in 2D FM/FE vdW heterostructures but also in other asymmetric heterostructures involving 2D vdW ferroelectrics.
ACKNOWLEDGMENT
This work was financially supported by grants from the National Natural Science Foundation of China (Grants No.
51762024, No. 51562014, No. 51862016, and No.51602135), the Natural Science Foundation of Jiangxi Province (Grant No. 20192BAB212002), and the Foundation of Jiangxi Provincial Education Department (Grant No. GJJ 180739).
Part of the numerical calculations were carried out in the High Performance Computing Center (HPCC) of Nanjing University. We are grateful to the three referees for their constructive comments and suggestions. We also thank Prof.
Weiyi Zhang and Dr. Rui Lyu for helpful discussions.
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