Introduction

Since the discovery of graphene1, some other atomically thin 2D materials, such as transition metal dichalcogenides (TMDs)2,3,4,5,6 and 2D magnetic chromium triiodide (CrI3)7,8, have gained attention because of their fascinating properties for applications. As a matter of fact, TMDs have a long historical standing as semiconductors with layered structures9. Early in 2010, Mak et al.5 and Splendiani et al.3 have revealed independently, that a strong photoluminescence emerges when MoS2 crystal is thinned down to a monolayer, indicating an indirect to direct bandgap transition. Henceforth, there is a strong resurgence of interest in 2D TMDs, owing to the challenges and opportunities for applications in electronics10, optoelectronics11, and spintronics2,12,13.

Beyond individual materials, stacking different 2D materials into vdW heterostructures can give us more room to achieve improved functions14,15. Except for AA or AB stacking, 2D vdW twisted heterostructures16, in which one layer is rotated with respect to the other by an angle θ, are also interesting owing to their unexpectedly physical properties. For instance, Cao et al.17 reported unconventional superconductivity emerging in 2D twisted bilayer graphene, and Wu et al.18 showed topological insulators in 2D twisted TMDs homostructures. Similarly, Liao et al.19 experimentally confirmed a ~5 times enhancement of vertical conductivity in twisted MoS2/graphene heterostructures. These emerging twistronics merging spintronics could pave the way for future 2D material design and applications20.

In monolayer TMDs, a pair of degenerate but inequivalent energy valleys (K, K′) in momentum space, protected by time-reversal symmetry, become the third degree of freedom for information storage, other than charge and electron spin21. To break the time-reversal symmetry, one of the effective approaches is applying an external perpendicular magnetic field (B) on TMDs, which can induce valley polarization through the Zeeman effect. Li et al.22 obtained the magnitude of valley splitting of 0.12 meV T−1 in monolayer MoSe2, with B ranging from −10 to 10 T. Similarly, in monolayer WSe2, the valley splitting up to 0.25 meV T−1 is obtained23,24. However, a large external magnetic field is impractical for devices. Therefore, incorporating intrinsic 2D magnetic materials into vdW heterostructures, utilizing the large magnetic proximity effect (MPE), is an attractive alternative. It has been shown that MPE originating from 2D substrate can enhance the valley splitting25,26.

The recently fabricated magnetic monolayer CrI37,27 with an out-of-plane easy axis is a suitable candidate to generate MPE when forming 2D vdW heterostructures with TMDs28. Zhong et al.29 experimentally demonstrated that up to 3.5 meV valley splitting (which is equivalent to ~10 T external B in single-layer WSe2) has been achieved. Furthermore, this MPE can be tuned optically, which in turn alters the valley splitting of WSe230. Theoretical investigations also demonstrated that the valley degeneracy of WSe2 can be lifted in a vdW WSe2/CrI3 heterostructure because the time-reversal symmetry is broken by magnetic Cr ions31,32,33. Zhang et al.31 showed a modulated energy splitting from 0.31 to 1.04 meV, which is determined by interlayer distance and/or atom arrangement. Hu et al.32 confirmed that the valley splitting is sensitive to interlayer distance, increasing from 2.0 to 4.5 meV when the distance was decreased by 0.3 Å from its equilibrium value. In other words, in TMD/CrI3 heterostructures, MPE is sensitive to interlayer vdW interaction. So far, however, the role θ played in 2D vdW WSe2/CrI3 heterostructures is still unclear. How does the valley splitting depends on θ needs to be clarified.

In this work, we investigate a set of 2D vdW WSe2/CrI3 heterostructures with different θ and stacking patterns with an emphasis on the magnetic properties, such as the magnetic moment and magnetic anisotropy energy (MAE). After that, we studied the electronic properties of 2D vdW WSe2/CrI3 heterostructures by using a band unfolding method. The enhanced KK′ valley splitting of WSe2, having strong dependence on θ, is obtained. Finally, we analyzed the MPE effects through the difference of partial charge density, and deduced the equivalent B by a k·p model34.

Results

Structural models and stability

Pristine WSe2 has a (hexagonal) sandwiched structure, in which one W layer is sandwiched between two S layers. Pristine CrI3 also has a sandwiched structure, with Cr layer being sandwiched between two I layers. We obtain the vdW WSe2/CrI3 heterostructure by placing a WSe2 layer on a CrI3 sheet, with θ ranging between 0 and 30˚. To find stable configurations, we slide the WSe2 layer relative to the CrI3 layer. Figure 1a, b shows an example of the atomic structures with a 0˚ twist angle. Here, we use ten points within a unit cell along with the zigzag (ZZ) and armchair (AC) directions, respectively. Figure 1c shows the energies along with the ZZ (upper) and AC (lower) directions, both suggesting that the starting structure is most stable. Here, the Cr atom is at the hexagonal center (HC) of WSe2, named Cr-HC hereinafter. In the following, we consider mainly the Cr-HC configurations for twisted WSe2/CrI3. However, we also consider other configurations, in particular, the less-stable Se-HC configurations where the Se atom is at the HC of Cr layer. The WSe2 (CrI3) monolayer belongs to the space group of P-6m2 (P-31m), which is reduced to P3 after stacking into the twisted heterostructures for either Cr-HC or Se-HC. The detailed atomic structures for the twisted Cr-HC structures can be found in the Supplementary Information.

Fig. 1: Structural model and stability of non-twisted WSe2/CrI3 heterostructure.
figure 1

a Top and b side views of 2D WSe2/CrI3 with θ = 0˚. Dashed lines indicate the unit cell. Blue arrows in panel a indicate sliding directions. c Energy as a function of sliding the WSe2 with respect to CrI3, along the ZZ and AC directions, respectively.

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For twisting, there are, in principle, many possibilities. Owing to the lattice constant difference (3.321 Å of WSe2 vs 7.002 Å of CrI3), most of them will result in huge supercells, making DFT calculations impossible. The lattice constant mismatch (δ) is defined as

$$\delta = \left( {a^{{{\mathrm{H}}}} - a^{{{\mathrm{M}}}}} \right)/a^{{{\mathrm{M}}}} \times 100\%$$
(1)

in which aH and aM are the lattice constants of the heterostructure and WSe2 (or CrI3) monolayer, respectively. Here, we select heterostructures with a mismatch threshold of 5.5% and have considered supercells up to 20 Å in lateral size. This leads to four heterostructures with θ = 0, 16.1, 23.4, and 28.1˚ (see Figs. 1 and 2). Detailed supercell parameters for the WSe2/CrI3 heterostructures are 2 × 2/1 × 1 (θ = 0˚), √13×√13/√3×√3 (θ = 16.1˚), √19×√19/2×2 (θ = 23.4˚), and √31×√31/√7×√7 (θ = 28.1˚). Details for the heterostructures are given in Table 1. It should be noted that both electronic structure of WSe235 and magnetic properties of CrI336 are sensitive to the lattice constant. For this reason, in this work we mainly have performed two limited sets of calculations: first using the lattice parameter of WSe2 and second using that of CrI3 for the heterostructure. Furthermore, we investigated the strain effects on the thermodynamic stability and electronic properties for the heterostructures with the relaxed lattice parameter.

Fig. 2: Structure models of twisted WSe2/CrI3 heterostructures.
figure 2

a θ = 16.1, b θ = 23.4, and c θ = 28.1˚.

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Table 1 Structural information and properties of WSe2/CrI3 heterostructures for Cr-HC and Se-HC (bracket) configurations.
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The stability of WSe2/CrI3 is determined by their formation energy defined as

$$E_f({{{\mathrm{meV}}}}{\AA}^{ - 2}) = (E_{{{{\mathrm{CrI}}}}3} + E_{{{{\mathrm{WSe}}}}2} - E_{{{{\mathrm{CrI}}}}3/{{{\mathrm{WSe}}}}2})/S$$
(2)

where S is the cross-section of the supercell, ECrI3, EWSe2, and ECrI3/WSe2 are total energies of respective systems. Table 1 below and Supplementary Fig. 1 show that twisted WSe2/CrI3 have larger Ef than that of non-twisted ones, indicating that the twisted heterostructures are more stable. Noteworthy, by using the same lattice parameters for individual and heterostructure systems, the strain energy owing to lattice mismatch have been excluded here. The energy differences between the Cr-HC and Se-HC configurations with the same twist angle θ range between 0.01 and 4.68 meV Å−2, showing a high lubricity of 2D WSe2/CrI3 similar to that of layered graphene37. Also noted is that the spin-orbit coupling (SOC) part of the formation energy also depends on θ. Taking the Cr-HC with WSe2 lattice constant as an example, we obtain Ef_SOC = 0.37 (0˚), 0.67 (16.1˚), 0.61 (23.4˚), and 0.75 (28.1˚) meV Å−2, which is in line with the trend observed for Ef.

Magnetic properties and valley splitting

Table 1 also lists the magnetic anisotropy energies (MAE), defined as the energy difference between out-of-plane (E⊥) and in-plane (E//) easy-axis per Cr atom, i.e.,

$${{{\mathrm{MAE}}}} = (E_ \bot - E_{//})/N_{Cr}$$
(3)

For WSe2/CrI3 with WSe2 lattice constant, MAE decreases with increasing δ (from −5.14% of compressive strain to 3.36% of tensile strain), which is similar to monolayer CrI338, but it is independent of sliding between WSe2 and CrI3. For WSe2/CrI3 with a CrI3 lattice constant, on the other hand, only a slight fluctuation of MAE is observed.

Next, we consider the valley splitting, schematically shown in Fig. 3. With SOC, both the (high-lying) valence band (VB) and (low-lying) conduction band (CB) of WSe2 split into two subbands: one spin up and one spin down. In particular, the VB (CB) in the K valley will split into VB1 and VB2 (CB1 and CB2) and the VB′ (CB′) in the K′ valley will split into VB1′ and VB2′ (CB1′ and CB2′), denoted by dashed lines in Fig. 3b. When WSe2 is put on a magnetic CrI3 substrate, the spin-up bands (red) are shifted downward and the spin-down bands (blue) are shifted upward. The VB (CB) at K and K′ with different spins will split by ΔVB (ΔCB), shown in Fig. 3b. The effective KK′ valley splitting (∆KK′) is defined as

$${\Delta}_{{{{\mathrm{KK}}}}^{\prime} } = {\Delta}_{{{{\mathrm{CB}}}}} - {\Delta}_{{{{\mathrm{VB}}}}}$$
(4)
Fig. 3: Schematic diagrams of irreducible Brillouin zone and split of degeneracy at the KK′ valleys of WSe2.
figure 3

a Irreducible Brillouin zone of WSe2. b Split of degeneracy at the KK′ valleys. In (b), σ+ and σ−denote the right-hand and left-hand circularly polarized light under optical selection rules at the KK′ valleys. The dashed lines denote the B = 0 bands. Red and blue arrows denote spin up and spin down states, respectively. CB1 (CB2) and VB1 (VB2) denote the first (second) conduction and valence bands at the K valley, while CB1′ (CB2′) and VB1′ (VB2′) denote the corresponding bands at the K′ valley. ∆CB and ∆VB are the energy shifts of the conduction and valence band, respectively.

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Note that to calculate the KK′ splitting, requires the calculation of energy levels at K (K′) in the unfolded Brillouin zone. This is done as follows: the wavefunction of the supercell (Ψ) can be mapped on to those of the primitive cell (Ψk)39,40,41 by

$${\Psi}_{{{\mathbf{k}}}}{{{\mathrm{ = }}}}\frac{1}{N}\mathop {\sum}\nolimits_R {\chi _{{{\mathbf{k}}}}^ \ast ({{{\mathbf{R}}}})\hat T_{{{\mathbf{R}}}}{\Psi}}$$
(5)

where N is the number of primitive cells contained in the supercell, k is the reduced wave vector inside the first Brillouin zone,

$${{{\mathbf{R}}}} = \mathop {\sum}\limits_j {n_ja_j}$$
(6)

is the integral multiple of lattice vectors \({{{\boldsymbol{a}}}}_j\), \(\hat T_{{{\mathbf{R}}}}\) is the translational operator for a translation R, and

$$\chi _{{{\mathbf{k}}}}\left( {{{\mathbf{R}}}} \right) = e^{i{{{\mathbf{k}}}} \cdot {{{\mathbf{R}}}}}$$
(7)

For a given region in WSe2/CrI3 such as WSe2 layer in the heterostructure, the relative weight (ρ) of layer projection in the range between Z1 and Z2 is given by

$$\rho = {\int}_{{{{\mathrm{Z}}}}_1}^{Z_2} {{\Psi}_{{{\mathbf{k}}}}^ \ast {\Psi}_{{{\mathbf{k}}}}dr}$$
(8)

as proposed by Chen and Weinert42. Figure 4 shows, for WSe2/CrI3 with WSe2 lattice parameter, the band structures for θ = 0° and 16.1°, (a, c) before and (b, d) after band unfolding. We see that although there is interaction between WSe2 and CrI3, the unfolded band structures of WSe2 look similar with a band gap of 1.24 ± 0.01 eV, close to that of monolayer WSe2.

Fig. 4: Band structures of WSe2/CrI3 heterostructures.
figure 4

a, c are the projected ones, in which green and blue denote CrI3 and WSe2 states, respectively. b, d are the unfolded ones for WSe2, in which red and blue arrows denote spin up and down states.

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Table 2 lists the components for valley splitting in WSe2. For monolayer WSe2, both ΔVB (ΔCB) and ΔKK′ are zero due to time reversal symmetry. When placed on CrI3, this symmetry is broken so valley splitting is generally observed. Consistent with previous calculations31,32, at θ = 0°, |ΔKK′ | = 0.43 and 0.27 meV for Cr-HC and Se-HC, respectively. We find the enhanced valley splitting of twisted heterostructures, for either Cr-HC or Se-HC. Compared with θ = 0°, the magnitude can also be magnitude larger (e.g., 0.43 meV vs 5.18 meV). With the parameter relaxation, as presented in Supplementary Table 1, the valley splitting of twisted WSe2/CrI3 heterostructure with Cr-HC is still larger than that of the non-twisted ones. We notice that the valley splitting without the twisting arises mainly from CB states, while for twisted ones, VB states also play an important role.

Table 2 Band splitting in WSe2 and WSe2/CrI3 heterostructures for Cr-HC and Se-HC (bracket) configurations with WSe2 lattice constant.
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The enhanced valley splitting in heterostructures may be understood in terms of an MPE. This can be seen in Table 2, which shows that for both K and K′, EVB decreases, but ECB is nearly a constant, with an increasing θ. One may note that both the VB and CB states of WSe2 are made of predominantly W d orbitals: \(d_{{{{\mathrm{x}}}}^2 - {{{\mathrm{y}}}}^2/{{{\mathrm{xy}}}}}\) for VB, and \(d_{{{{\mathrm{z}}}}^2}\) for CB. Figure 5a–d shows the differences in the partial charge density Δρ of the valence band maximum (VBM) states at K and K′. An obvious θ-dependence of Δρ is observed near Cr atoms. Figure 5e shows the planar averaged Δρ along z, \(\overline {{{{\mathrm{{\Delta}}}}}\rho } (z)\). It reveals how the states of WSe2 is affected by the proximity of Cr atoms inside CrI3. This analysis suggests that the VBM states are responsible for the enhanced valley splitting. Indeed, results in Table 2 support such a conclusion as it shows that valley splitting is mainly a result of a θ-dependent ΔVB.

Fig. 5: Partial charge density differences of the VBM states between K and K′ for different θ.
figure 5

a θ = 0, b 16.1, c 23.4, and d 28.1˚. Isosurface is set at 5 × 10-5 e Å-3. e Planar averaged partial charge density differences along z for states in (a–d). Inset is a zoom-in of the framed region at around z = 8 Å. f ΣΔρ and average magnetic moment of Cr atom M as functions of valley splitting ΔKK′.

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Figure 5f plots the sum of Δρ of Cr layer (ΣΔρ) as a function of ΔKK′. A nearly linear dependence is observed. Since Δρ is a measure of proximity effect, Fig. 5f thus suggests that MPE is the reason for enhanced ΔKK′. Figure 5f also shows that with an increase in valley splitting, the total magnetic moment of Cr also increases from 3.011 to 3.204 μB. Hence, the enhanced magnetic field is also a direct consequence of MPE via an enhanced valley splitting.

In the discussion above, we have chosen a lattice parameter, e.g., that of WSe2, to construct the supercells. It is desirable to consider the effect of strain. For this reason, Table 1 also shows the results when the lattice parameter is that of CrI3. In line with above results, ΔKK′ for the Cr-HC configuration shows an increase with θ, e.g., 1.97 meV (θ = 16.1°) < 3.69 meV (θ = 28.1°).

There are two important consequences due to strain: (1) a direct-to-indirect gap crossover between θ = 0 and 23.4°. To this end, recall that monolayer TMDs are direct gap semiconductors, in which both the VBM and conduction band minimum (CBM) are at the K (K′) valley of the Brillouin zone (see Fig. 3a)3,5. Previous experiments43,44 and theory35 also showed that the various physical properties of TMDs, including band gap and band edge positions in the Brillouin zone, can be tuned by applying an in-plane strain. In our calculations, the strain in WSe2 changes from 5.4% (tensile) at θ = 0° to −3.25% (compressive) at θ = 23.4°. In the former case, the CBM position is unchanged, but the VBM position moves from K (K′) to Γ. In the latter case, the VBM position is unchanged, but the CBM position moves to the Q (Q′) valley, which is about halfway between K (K′) and Γ, as shown in Fig. 3a. (2) The valley splitting for the Se-HC configuration vanishes at θ = 16.1°. This is shown in Table 1 where for θ = 16.1°, MAE = 0.27 meV Cr−1 is positive with an in-plane easy axis. The corresponding in-plane magnetic field can neither generate a Zeeman splitting nor lift the valley degeneracy. However, in this case, it is possible to obtain an easy axis component perpendicular to the 2D sheet by applying an external electric field (E-field). In the absence of the E-field, a small dipole of −0.016 eÅ is obtained, which is in line with a 0.0037 eCr−1 transfer from WSe2 to CrI3 by a Bader analysis45. Figure 6 shows the results for both dipole moment and KK′ valley splitting. A good linear relationship of the dipole moment with applied E-field is observed, and ΔKK′ can be tuned obviously by applying the external electric field. Taking E-field = −0.3 V Å−1 as an example, we find that MAE changes from 0.27 to −0.56 meV Cr−1, and ΔKK′ changed from 0 to −5.84 meV. This magnetoelectric effect here is reminiscent of what has been observed in bilayer CrI346, explaining the twist-induced enhancement of valley splitting in WSe2/CrI3 heterostructures.

Fig. 6: Dipole moment and KK′ valley splitting as functions of applied E-field.
figure 6

a Dipole moment of the CrI3/WSe2 heterostructure. b Valley splitting at KK′ valleys. Inset in b shows the direction of the field.

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The k·p model

To estimate the magnitude of the equivalent B and understand the Zeeman effect induced by CrI3, we use a k·p model34, in which the interaction energy is divided into spin and orbital contributions:

$${{{\mathbf{H}}}}_B = g_0\mu _B{{{\mathbf{B}}}} \cdot {{{\mathbf{S}}}} + \mu _B{{{\mathbf{B}}}} \cdot {{{\mathbf{L}}}}$$
(9)

where \(g_0\) = 2 is the Landé factor, \(\mu _B = e\hbar /2m_0\) is the Bohr magneton, \(m_0\) is the electron mass, S = \(\sigma /2\) is the spin operator with \(\sigma\) being the Pauli matrices, and L is the orbital angular momentum operator. As mentioned earlier, in the K (K′) valley the CBM state mainly consists of W \(d_{{{{\mathrm{z}}}}^2}\) orbital with LZ = 0, whereas the VBM state mainly consists of W \(d_{{{{\mathrm{x}}}}^2 - {{{\mathrm{y}}}}^2/{{{\mathrm{xy}}}}}\) orbital with LZ = 2. From the DFT computation, the spin projection around CB and VB band edges remains mostly out of plane with little in-plane tilting as shown in Supplementary Table 2. Therefore, the Rashba interaction is neglected in the present k·p model. The value \(\mathop {\sum}\nolimits_{i,j} {VBM_i{{{\mathrm{|}}}}{{{\boldsymbol{H}}}}_B{{{\mathrm{|}}}}CBM_j}\), where i and j run over the four levels near the Fermi level as shown in Fig. 3b, can be directly calculated via DFT, from which we deduce the equivalent B. For WSe2/CrI3 heterostructures with the Cr-HC configuration and WSe2 lattice parameter, we obtain B = 2.1 T for θ = 0°, 7.4 T for θ = 16.1°, and 20.9 T for θ = 23.4°. It shows that not only the magnetic field strength is sufficiently strong but also twisting can be an effective way to amplify the MPE at interfaces.

Discussion

In summary, using first-principles calculation, we perform a systematic study on the structural stability, electronic and magnetic properties of 2D vdW WSe2/CrI3 heterostructures. We show that, compared to the non-twisted structure, the twisted ones are more stable, as evidenced by their higher interfacial binding energies. Our study also reveals an MAE of the heterostructures, which is controlled by strains of the CrI3 layer. The MPE can be understood in terms of the charge density difference of the VBM states at K and K′. More importantly, in twisted heterostructures, there is an order of magnitude enhancement of K-K′ valley splitting, which can also be tuned by applying an external electric field. With the help of a k·p model, we determine the equivalent B field due to twisting as the origin of an enhanced MPE.

Methods

First-principles calculations were performed using the Vienna ab initio simulation package (VASP) based on the density functional theory (DFT)47. We employed the Perdew–Burke–Ernzerhof (PBE)48 functional, within the projector augmented wave (PAW)49 approach, for exchange-correlation potential and energy. The plane-wave energy cutoff was set at 500 eV. vdW interactions between CrI3 and WSe2 were included by employing Grimme’s semiempirical DFT-D3 scheme50. To avoid interaction between periodic images in the supercell approach, we used a vacuum space larger than 30 Å. The atomic structures were fully relaxed until the Feynman–Hellman force on each atom is less than 0.02 eV Å−1. The electronic self-consistent convergence criterion was set at 10−7 eV. The K-point mesh51 in the Brillouin zone was sampled with the density of 0.03 Å. To determine the magnetic anisotropy, the spin-orbit coupling (SOC) was included in our DFT calculations. In addition, we performed electronic structure calculations with GGA + U methods52 described by Dudarev, in which the on-site Coulomb parameter U and the moderate exchange parameter J were set to 2.7 and 0.7 eV53, respectively. The k-projection method42 for interfaces modeled by supercells within the framework of the first-principles method was used to obtain the unfolded electronic band structures.