Hole-Doped Nonvolatile and Electrically Controllable Magnetism in van der Waals Ferroelectric Heterostructures

Since the experimental preparation of intrinsic two-dimensional (2D) magnetic materials, such as Cr₂Ge₂Te₆,^[1] CrI₃,^[2] Fe₃GeTe₂,^[3] and VSe₂,^[4] tuning ferromagnetism (FM) in 2D systems by an electric field has been an intriguing long-term goal in materials research^[5–8] that still faces enormous challenges. For example, electric modulation of magnetic properties usually requires an ultra-strong electric field which is hardly met by the usual gate electrodes. One promising method is dielectric gating, which could achieve such a field but may affect the structure.^[9,10] With this method, carrier densities of 10¹⁵cm⁻² can be achieved in 2D materials.^[11,12] Another issue is that electrically tuned magnetism is volatile (maintaining the magnetic state requires continuous power, resulting in high energy costs) and/or irreversible (non-tunable). Therefore, it would be desirable to obtain nonvolatile electric-field-dependent magnetism in nonmagnetic (NM) materials.

On the other hand, ferroelectric (FE) materials with tunable broken inversion symmetry always exhibit a strong size effect, and the depolarization field provided by the surface charges is opposite to the internal polarization and will try to compensate the internal polarization.^[13–15] In particular, the polarization may disappear when the film thickness is very small, such as in BaTiO₃ and PbTiO₃. Therefore, as the dimensionality decreases to the nanoscale, the depolarization field becomes extremely strong and may cancel the polarization.^[16] Surprisingly, a series of atomically thin 2D FE materials, such as SnTe,^[17] CuInP₂S₆,^[18] Sc₂CO₂,^[19] and In₂Se₃,^[20] have recently been discovered through calculations and experiments.^[21,22] Due to their nanoscale size, 2D FE materials with switchable electrical polarization are suitable for next-generation data storage. In particular, FE In₂Se₃ exhibits out-of-plane and in-plane polarizations, conducive to practical device applications in high-density integration.^[23] Compared with other storage devices (e.g., FM memories), the nonvolatility of FE memory is advantageous for realizing long-term data preservation. Furthermore, to simultaneously enhance the data reliability and reduce disturbance of the reading operation, we would expect to separate the operations of writing and reading using multiferroic materials, for example, writing with an electric field and reading with a magnetic probe in an FE–FM material.

Besides some chance discoveries,^[24–27] single-phase 2D multiferroic materials with both FE and FM orders promise the control of disparate properties. However, they remain rarely reported because multiferroic materials are challenged by the contradictory preferences of different kinds of ferroic materials for metal ion d-orbitals, i.e., FE requires empty d-orbitals but FM usually comes from partially filled d-orbitals.^[28] Fortunately, different 2D flakes can stack into van der Waals heterostructures (vdWHs) because 2D materials have no surface dangling bonds and can be precisely stacked without leading to lattice mismatch. Therefore, using 2D FE materials in a vdWH provides an additional degree of freedom to control magnetism by flipping the polarization direction,^[29] i.e., one can electrically control the magnetism in an FE–FM coupled heterostructure, which is promising for extending applications in data storage devices. Recently, several studies have revealed that hole doping can cause magnetism in some 2D systems due to their inherent Mexican-hat-like band dispersion.^[30–34] The Mexican-hat-like dispersion provides a sharp van Hove singularity (VHS) near the Fermi level (E_F) on the density of states (DOS), leading to significant exchange splitting of the electronic states of the two spin channels under proper hole doping and resulting in a net magnetic moment and spontaneous polarization.^[35] Therefore, modulating FE states to control FM states in 2D vdWHs is promising for designs of nonvolatile memory devices.

In this Letter, we propose a design for realizing 2D multiferroics (coexistence of FE and FM) in FE heterostructures comprising only d⁰ semiconductors. We choose 2D InSe and FE In₂Se₃ monolayers to describe the principles, forming an InSe/In₂Se₃ vdWH in which both monolayers can almost hold the same band structures as those of the isolated individuals, respectively, and the band edge alignment couples directly to the polarization state. As the polarization is towards the InSe layer, the Mexican-hat-like band dispersion of the In₂Se₃ aligns on the top of the valence band (VB). Hence, under proper hole doping it evokes considerable magnetic polarization energy. In contrast, the reversed polarization pushes the VB maximum (VBM) of the InSe over the In₂Se₃ VB as the global edge of VBs. Therefore, much heavier doping is needed to evoke spin polarization with a rather tiny magnetic polarization energy. Therefore, with the FE reversion, the magnetic state bears an ON and OFF switch, which is promising for data storage devices.

Computational Methods. All calculations were performed by the spin-polarized density functional theory (DFT) method as implemented in the Vienna ab initio simulation package (VASP),^[36] and the Perdew–Burke–Ernzerhof (PBE) functional was used in the generalized gradient approximation (GGA)^[37] scheme. The k-point samples and energy cutoff for structural optimization were set to be 15 × 15 × 1 and 500 eV, respectively. The vdW interaction between layers was described by the DFT-D₃ method.^[38] Dipole correction was employed due to the asymmetric layer arrangement.^[39] Phonon dispersions were calculated with density functional perturbation theory using the PHONOPY code.^[40] Hole doping was simulated by removing electrons in a system with a compensated uniform negative background.

Results and Discussion. To realize memory devices with high speed and low energy consumption we propose a design of nonvolatile and switchable magnetic states based on 2D multiferroic vdWHs. These vdWHs should meet the following stringent conditions: (i) one of the materials is an FE material; (ii) the band alignment is tuned during polarization flipping; and (iii) one of the materials has a rather sharp DOS, for example, a VHS that can be easily realized by a Mexican-hat-like band dispersion, near the E_F. Guided by these strategies, we propose an InSe/In₂Se₃ FE vdWH as an example to demonstrate the feasibility of this design. Notably, d or f electrons are not necessary for this proposal.

We chose the experimentally accessible In₂Se₃ and InSe monolayers,^[41,42] as shown in Figs. 1(a) and 1(b), as the FE and auxiliary layers, respectively, forming a vdWH as shown in Fig. S1 and Table S1 in the Supplementary Material.^[43] The 2D FE In₂Se₃ contains five atomic sublayers in the Se–In–Se–In–Se order. The central Se atomic sublayer deviates from the middle position between the top and bottom Se atomic layers, resulting in out-of-plane polarization. Besides the FE property of In₂Se₃, it possesses a narrow VB with Mexican-hat-like band dispersion. InSe is chosen because of its semiconductor property and proper hexagonal lattice (i.e., its lattice parameters are similar to those of In₂Se₃). Although the lattice mismatch plays a trivial role in the vdWH, matched lattices can save on calculation expenses. InSe and In₂Se₃ are indirect bandgap semiconductors with bandgaps of 1.53 and 0.81 eV, respectively, as shown in Figs. 2(a) and 2(b), consistent with the previous results.^[21,44] The polarization state is marked as P_↑ and P_↓ when the central Se atom shifts close to the bottom and top layers, respectively. The polarization plays the role of a ‘handle’ that can modulate the electronic characteristics of the vdWH. Compared with the symmetrical InSe monolayer, the electrostatic potentials on both sides of FE In₂Se₃ are noticeably different, as shown in Fig. 2(c), leading to an electrostatic potential difference of 1.179 eV. In the vdWH, the InSe layer interacts with the P_↑ or P_↓ states of the In₂Se₃ layer, as depicted in Figs. 1(c) and 1(d), resulting in different band alignments. Given their relative positions, the two different polarization states (P_↑ or P_↓) are equal to the two cases in which the InSe layer contacts the In₂Se₃ layer from different sides. In addition, no imaginary mode occurs on the phonon spectra of the vdWH, illustrating good structural stability as shown in Fig. S2.^[43] The band structures are shown in Figs. 3(a) and 3(b). The VBM and conduction band minimum (CBM) vary slightly compared with the isolated InSe and In₂Se₃ monolayers, as shown in Fig. 2(d), due to the weak interlayer interaction.

Fig. 1.  Top (upper) and side (lower) views of (a) InSe and (b) In₂Se₃ monolayers. The magenta and green beads represent In and Se atoms, respectively. The orange dashed rhombi mark the unit cells. The two polarization states of InSe/In₂Se₃ vdWHs are shown in (c) and (d), respectively. The red double-headed arrows mark the interlayer distances, d.

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Fig. 2.  Band structures of (a) InSe and (b) In₂Se₃ monolayers (PDOS, partial density of states). (c) Average electrostatic potential of an In₂Se₃ monolayer. (d) Band alignment of InSe/In₂Se₃ (P_↑) and InSe/In₂Se₃ (P_↓) with respect to the vacuum level.

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Fig. 3.  Projected band structures for neutral (a) InSe/In₂Se₃ (P_↓) and (b) InSe/In₂Se₃ (P_↑). Doping concentration dependence of the magnetic moment for (c) InSe/In₂Se₃ (P_↓) and (d) InSe/In₂Se₃ (P_↑). The light blue region indicates large spin polarization energies. The inset of (c) is the spin density distribution in the FM state of InSe/In₂Se₃ (P_↓) at a hole density of 9.75 × 10¹⁴ cm⁻². (e) Spin-polarized band structure and (f) spin-polarized partial density of states (PDOS) of InSe/In₂Se₃ (P_↓) vdWH for a hole density of 9.75 × 10¹⁴ cm⁻². Blue (red) and cyan (orange) lines represent the contributions of the InSe (In₂Se₃) layer with the spin-up and spin-down bands, respectively. The inset of (e) is the differential charge density before and after the hole doping of InSe/In₂Se₃ (P_↓). The E_F is set to be zero and marked with a black dashed line.

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To further analyze the change in electronic properties, we plot the projected band structures of the InSe/In₂Se₃ vdWHs, as shown in Figs. 3(a) and 3(b). Although the bandgaps of InSe/In₂Se₃ vdWHs are slightly different in the two polarization directions, i.e., the bandgaps for InSe/In₂Se₃ (P_↓) and InSe/In₂Se₃ (P_↑) are 0.48 and 0.45 eV, respectively, the band alignments are distinctly different. For the InSe/In₂Se₃ (P_↓) vdWH, the CBM and VBM are contributed by the InSe and In₂Se₃ layers, respectively. Conversely, by reversing the polarization of In₂Se₃ (to P_↑), the band edge of the In₂Se₃ layer shifts downward, resulting in the VBM of the InSe layer being closer to E_F. Therefore, according to the design principles, flipping the polarization of the FE layer can switch band alignment.

Because the VBMs of both monolayers possess Mexican-hat-like dispersion, there is a one-dimensional-like VHS at the VB edge on the DOS spectrum. Therefore, as proper hole doping shifts E_F to the VHS, spin degeneracy will be lifted due to electronic interaction, similar to the previous report on GaSe.^[35] Therefore, we calculated the two FE polarization cases for the vdWH. The relevant results for InSe and In₂Se₃ monolayers are shown in Fig. S3.^[43] This displays the dependence of magnetic moment and magnetization energy on the number of doped holes, where one hole per unit cell corresponds to a doping concentration of approximately 7 × 10¹⁴ cm⁻². For the InSe/In₂Se₃ (P_↓) case, the In₂Se₃ layer dominates the VBM. Quite light doping could drive the system to spin polarization with tremendous magnetic polarization energy (E_p), defined as the energy difference between the FM and NM states, as depicted in Fig. 3(c), as in our previous report on a In₂Se₃ monolayer.^[45] Figure 3(c) shows that the magnetic moment suddenly emerges at a concentration of about 4.53 × 10¹⁴ cm⁻² and increases continuously as the hole doping level enhances gradually then rapidly saturates at about 0.5 μ_B/hole, indicating magnetization of the vdWH. As shown in the inset of Fig. 3(c), the spin density is rather delocalized and is mainly distributed around all top layer Se atoms of In₂Se₃, so the magnetic moment comes from itinerant holes due to the homogeneous distribution in the top Se sublayer of In₂Se₃. The large DOS at Fermi energy level indicates FM instability according to the Stoner criterion, and this itinerant magnetism can persist over a sizable range of doping concentrations. E_p becomes more negative as doping increases, indicating more robust spin polarization and magnetic stability. Noticeably, E_p remains more negative than −26 meV from the doping concentration of 9.75 × 10¹⁴ to 1.39 × 10¹⁵ cm⁻², implying big spin polarization of the InSe/In₂Se₃ vdWH. However, the InSe layer makes a major contribution to the VBM for the InSe/In₂Se₃ (P_↑) case. Under hole doping, the system can be spin-polarized but with trivial E_p, indicating an NM nature, as shown in Fig. 3(d), similarly with doping in GaSe. Consequently, as the FE polarization determines the manner of band alignment dominating the electronic structure and then determines the spin polarization, we can tune the FE polarization with an external electric field to modulate the electricity and magnetism in the vdWH.

Moreover, we project the spin-polarized energy bands to the two layers, in the P_↓ case at a hole density of 9.75 × 10¹⁴ cm⁻², as shown in Fig. 3(e). The spin degeneracy of the energy band from the In₂Se₃ layer is lifted with a large exchange splitting of about 0.28 eV. As a result, the spin-up band (red line) is fully occupied and lies about 0.2 eV below E_F, while the spin-down band (orange line) intersected by E_F lies much higher. In contrast, the bands originating from the InSe layer are still degenerate. Since the VBs (blue and cyan lines) of the InSe layer are very dispersive, there is no exchange splitting of the holes residing on the InSe layer. As shown in the inset of Fig. 3(e), the holes are mainly doped on Se₁ and Se₅ atoms. Therefore, half of the holes gather in the InSe layer and the other half reside in the In₂Se₃ layer. Thus, the total magnetic moment saturates at about 0.5 μ_B/hole. According to our test calculations for the noncollinear magnetic moments, the magnetically easy axis is in the out-of-plane direction with a magnetic anisotropic energy of 0.2 meV. By considering the unit cell magnetic moment as an integral (this is a coarse approximation), we could examine the parallel and antiparallel magnetic configurations in a 2 × 2 supercell and find that the parallel configuration is about 99.6 meV more favorable than the antiparallel one. Based on the Heisenberg model, we could make a rough estimation of the nearest-neighbor exchange integral to be J = 24.91 meV and the Curie temperature to be T_C ∼ 84 K, by means of classic Monte Carlo simulations, as stated in Note III and Fig. S4 in the Supplementary Material.^[43]

We have also plotted the band structure and DOS for InSe/In₂Se₃ (P_↑) at the same hole density for comparison between P_↑ and P_↓. As shown in Fig. S5,^[43] there is no spin splitting in InSe/In₂Se₃ (P_↑). The semiconductor InSe auxiliary layer in this tunable multiferroic vdWH contributes to the global VBM on the P_↑ state of the In₂Se₃ layer, making the system NM (OFF) under light hole doping. Although it becomes spin polarized under heavier doping, E_p is too small to hold the magnetism. Therefore, the InSe/In₂Se₃ (P_↑) case is NM over the entire doping range. Hence, with the semiconductor InSe, nonvolatile and switchable magnetism under hole doping can be coded by setting the FE state of the In₂Se₃ layer. Emphatically, the magnetic state switches from ON to OFF when the polarization direction of In₂Se₃ alters from P_↓ to P_↑. Both states are robust even if the external electric field is removed, which benefits information nonvolatility and a reduction in energy consumption. Experimentally, the switchable behavior has been successfully realized in the 2D polar material WTe₂.^[46] Moreover, as shown in Fig. 4, based on the nudged elastic band calculation of migration paths, the energy barriers between P_↑ and P_↓ states could be decreased upon hole doping, which benefits FE phase transition. As the doping concentration reaches 1.4 holes/u.c., the magnetic moment of the system reaches saturation. At this point, the value of the energy barrier is 0.064 eV, which ensures that the higher-energy electrically polarized state is stable compared with the lower-energy electrically polarized state. Additionally, we have computed the polar atomic displacements in the InSe/In₂Se₃ vdWH (P_↓) as a function of charge-carrier concentration obtained via the background charge method. Figure S6 suggests that the polar phase can support a substantial number of charged defects while maintaining the polar structure. Therefore, we believe that the external electric field can accurately control the spin states by flipping the FE polarization of In₂Se₃ in the vdWH. The above analyses show that the magnetism-switching behavior of InSe/In₂Se₃ vdWH comes from two aspects: (i) the coupling at the InSe/In₂Se₃ vdWH interface renders different band alignments under different polarization states, and (ii) the Mexican-hat-like band edge dispersion can realize magnetism under hole doping. These two aspects are not exclusive to the InSe/In₂Se₃ vdWH. Therefore, modulating ON–OFF switchable and nonvolatile magnetism under hole doping by interface engineering may also be extended to other heterostructures. Additionally, implementing nonvolatile magnetic switching via an FE switch relies on two common features of FE materials, namely the FE bistable state and broken spatial-inversion symmetry, which shows that engineering semiconductor-aided multiferroic heterostructures is a general strategy for realizing nonvolatile magnetic switching.

Fig. 4.  (a) The minimum energy pathway of an InSe/In₂Se₃ vdWH from the P_↑ to the P_↓ phase. (b) The energy barrier as a function of hole doping levels.

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Usually, magnetic properties are strain sensitive. So, we examine the effect of biaxial strain on the doping-induced magnetization of the InSe/In₂Se₃ (P_↓) vdWH. We simulate the biaxial strain by changing the lattice constant, and the strain is defined as ε = (a – a₀)/a₀, where a (a₀) is the lattice constant of the strained (unstrained) InSe/In₂Se₃ (P↓) vdWH. We calculate the magnetic moment and E_p for various strains. The magnetic moment and E_p are strongly strain-dependent in magnetic transition, as shown in Figs. 5(a) and 5(b). The magnetism can be turned on at a lower doping concentration under tensile strain, i.e., the tensile strain can lose the constraint of the doping concentration to achieve magnetism. We calculate the biaxial-strain-dependent magnetic moment at three doping levels, 4.9 × 10¹⁴ cm⁻² (red), 7.0 × 10¹⁴ cm⁻² (blue) and 9.8 × 10¹⁴ cm⁻² (green), respectively, as presented in Fig. 5(c). The doping-induced magnetization is suppressed under compressive strain when the doping level is relatively low (4.9 × 10¹⁴ cm⁻²). Conversely, its magnetic moment rapidly increases to saturation under tensile strain. At a doping concentration of 9.8 × 10¹⁴ cm⁻², the magnetic moment retains saturation under stronger strains.

Fig. 5.  (a) Magnetic moment and (b) polarization energy under different strains. (c) Strain dependence of the hole doping magnetic moment at different concentrations. (d) Band structures of InSe/In₂Se₃ (P_↓) vdWH under 4% tensile strain, no strain and 2% compressive strain, respectively. The radius of the Mexican-hat band edge (k_r) is indicated by black double-headed arrows.

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To reveal the underlying mechanisms of the strain-induced variation in magnetism, we plot the band structures of InSe/In₂Se₃ (P_↓) vdWH under different strains, as shown in Fig. 5(d). Compared with the pristine one (cyan line) in Fig. 5(d), the radius of the Mexican-hat band edge (k_r) becomes larger under tensile strain (blue line). The band edge circle will extend in the Brillouin zone, enhancing and sharpening the DOS peak and increasing hole accommodation near E_F. As a result, a sharper DOS peak means that the Stoner criterion is more easily fulfilled under hole doping. In contrast, k_r becomes smaller when a compressive strain is applied (red line), thus weakening the magnetism. Because the orbital overlaps less (more) after the extension (compression), the VB becomes flatter (more dispersive) and easily (hardly) fulfills the Stoner criterion. In addition, the tensile strain also raises the energy band near the K point, which leads to a sharper and higher DOS peak close to the Fermi level. This explains why the onset of magnetism occurs at a lower hole doping concentration under tensile strain, as shown in Fig. 5(a). Conversely, under compressive strain, the appearance of magnetism requires a higher hole doping concentration. Therefore, mechanical modulation can strongly affect magnetism in the InSe/In₂Se₃ vdWH.

Furthermore, we propose a data storage device, as illustrated in Fig. 6, for nonvolatile magnetic switching applications based on this InSe/In₂Se₃ FE vdWH, with a 2D insulator such as hexagonal boron nitride (h-BN) as the substrate. Since the vdWH is made of atomically thin layered materials, the doping levels of our device could be electrostatically tuned by a gate. The properties of the InSe/In₂Se₃ vdWH, in either the FM or NM state under hole doping, are controlled by the polarization state (P_↓ or P_↑) of the In₂Se₃. As mentioned above, the data writing operation is performed by coding the FE polarization, similar to the case in an In₂Se₃-only device. In addition, to illustrate the influence of the substrate on the properties of the system, we also calculated the partial density of states of the InSe/In₂Se₃ (P_↓) vdWH on a single-layer h-BN substrate, as shown in the inset in Fig. 6. Remarkably, the effect of the h-BN substrate on the inherent characteristics of the InSe/In₂Se₃ (P_↓) vdWH is negligible. The In₂Se₃ layer accounts for the dominant proportion (red) in the VB peaks compared with those of the InSe (blue) and BN (magenta) layers, so the magnetic switching character can also be achieved with the BN substrate, and the magnetic moment does not substantially change. Therefore, an InSe/In₂Se₃-based data storage device can be prepared on an h-BN substrate with proper hole doping. The data reading process can be performed by checking the magnetoelectric signal differences from the damage-free current.^[47] Such a strategy is ideal for nano-spintronic devices with nonvolatility and low energy consumption.

Fig. 6.  Dependence of the magnetic moment of the h-BN/InSe/In₂Se₃ (P_↓) vdWH on doping concentration. Bottom inset: concept and sketch diagram of the data storage device based on the h-BN/InSe/In₂Se₃ vdWH. Upper right inset: partial density of states (PDOS) of the h-BN/InSe/In₂Se₃ (P_↓) vdWH. The light blue regions indicate large spin polarization energies.

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In summary, an intriguing strategy is proposed to introduce a semiconductor auxiliary to an FE layer to control the magnetic state. Interestingly, hole-doped InSe/In₂Se₃ transits from NM to FM when the polarization direction is flipped. Therefore, this approach is general for securing and engineering new material systems for nonvolatile data storage applications.