Manipulation of valley splitting for the WSe2/NiCl2 heterostructure by adjusting the interlayer spacing and constructing a NiCl2/WSe2/NiCl2 heterojunction

The electronic band structure and valley splitting of the WSe2/NiCl2 heterostructure have been investigated by density functional theory and Berry curvature calculations. We demonstrate that the valley polarization of monolayer WSe2 is induced due to the magnetic proximity effect caused by the single layer of ferromagnetic NiCl2. The magnitude of valley splitting depends on the stacking configurations of WSe2/NiCl2, and the maximum value of valley splitting reaches −11.87 meV. Large valley splitting can be achieved by adjusting the layer spacing and constructing a NiCl2/WSe2/NiCl2 heterojunction with Ni spins arranged in parallel between two NiCl2 sheets. The valley-contrasting Berry curvature between the K and K′ valleys suggests that the WSe2/NiCl2-based heterostructure could potentially be used as a valleytronic device to realize the valley-polarized anomalous Hall effect as both spin and valley filter.


Introduction
The success of research in graphene [1] has developed interest in the study of other single-layer 2D materials, such as silicene [2], black phosphorus [3], and transition metal dichalcogenides (TMDCs) [4][5][6]. TMDCs, such as MX 2 (M = Mo, W and X = S, Se), are semiconducting graphite analogues composed of a layer of atoms covalently bonded; the stacks of these layers are held together by van der Waals interactions [7,8]. Recently, the atomically thin 2D layered TMDCs have been extensively investigated due to their unique electronic properties and coupled spin-valley degrees of freedom [8,9]. The monolayer of TMDCs possesses a pair of inequivalent valleys in the vicinities of the vertices of hexagonal Brillouin zone (BZ) [10]. TMDCs are considered good candidates for the valley and the Berry phase-related physics, also called valleytronics, due to the strong spin-orbit coupling, valley-contrasting Berry curvature, and broken intrinsic inversion symmetry [11][12][13].
In valleytronic applications of monolayer TMDCs, the valley polarization between K and K should be introduced to exploit the valley degrees of freedom. The broken inversion symmetry of monolayer TMDCs could separate the paired valleys K and K in the momentum space; however, the energy degeneracy still remained because it is protected by the time-reversal symmetry [13]. A key factor to achieve valley polarization is to lift the valley degeneracy by breaking the time-reversal symmetry. Previous studies have shown many ways of lifting the K -K valley degeneracy. The optical pumping with circularly polarized light is one way to achieve valley polarization; however, optical pumping is difficult to robustly manipulate and inapplicable for practical valleytronic applications [14][15][16]. Other studies have shown that the magnetic field can be applied to lift the valley degeneracy. However, a large intensity of magnetic field is needed. For instance, the small valley splitting values in monolayer WSe 2 and MoSe 2 were approximately 0.2 meV T −1 [17,18] and 0.12 meV T −1 [19], respectively. Recent studies have shown that the proximity-induced Zeeman effect is an effective strategy to achieve considerable valley splitting with a large effective Zeeman field (EZF). A giant valley splitting has been theoretically predicted in MoTe 2 /EuO (440 T EZF) [20], MoS 2 /EuS (20 T EZF) [21], and MoS 2 /CoO (152 T EZF) [22]. The enhanced valley splitting in monolayer WSe 2 by exploiting the magnetic proximity effect from the EuS substrate was experimentally reported; the substrate provides the EZF of 12 T [23,24]. Xu et al also reported the valley splitting and polarization in the WSe 2 /CrI 3 heterostructure and detected a large EZF of approximately 13 T [25].
Monolayer NiCl 2 , which can be successfully fabricated from its layered bulk crystals, is a promising 2D intrinsic ferromagnetic (FM) semiconductor with an energy band gap of 2.4-2.8 eV [26,27], and the FM Curie temperature is 120 K [28]. A recent study found that the cleavage energy of NiCl 2 is 0.223 J m −2 , which is smaller than that of graphite, thereby implying that it can be easily exfoliated down to the monolayer [28]. In this work, we theoretically study the electronic and valleytronic properties of the WSe 2 /NiCl 2 heterostructure by first principle calculation. The results show that the K -K valley degeneracy is lifted with a large valley splitting (−11.87 meV). The K -K valley splitting depends on the stacking types that possess varying interlayer spacing. A large valley splitting can be achieved by building a NiCl 2 /WSe 2 /NiCl 2 heterojunction. The result of the calculated Berry curvature indicates that the valley Hall effects could occur in this WSe 2 /NiCl 2 heterostructure.

Computational details
Our calculations, including geometric relaxation and electronic structure calculation, were performed by using density functional theory (DFT) on the basis of projector augmented wave implemented in the VASP package [29,30]. The exchange correlation potential was described with Perdew-Burke-Ernzerhof of the generalized gradient approximation; the van der Waals interaction was considered using the DFT-D2 method [31]. In structural optimization, consideration of spin-orbit coupling (SOC) has nearly no effect on the structural properties such as the bond length and bond angle by our test calculations. Therefore, in order to save computing resources and speed up the calculation, we did not consider SOC in structural optimization, but in the calculation of electronic structure. An energy cutoff of 450 eV was employed for the plane-wave basis set. A 15 × 15 × 1 k-sampling generated by the Monkhorst-Pack scheme for the BZ was adopted. During the structural relaxation, the energy convergent criterion was 10 −5 eV per unit cell. The force convergent criterion on all relaxed atoms was less than 0.02 eV Å −1 . We conducted test calculations for U values ranging from 3 eV to 6.5 eV with J = 0 eV to include the strong on-site Coulomb interaction in NiCl 2 [32]. The result indicated that the magnetic moment and lattice constant showed a weak dependence on the U value, while the band gap was sensitive to it. The values of 4.0 and 0 eV for U and J, respectively, were suitable for the further calculations of the electronic structure because the calculated band gap of 2.59 eV is in agreement with the experimental value of 2.4-2.8 eV of the parameters. The in-plane lattice constants of the optimized free-standing monolayer WSe 2 and NiCl 2 were 3.40 and 3.49 Å, respectively. These values are in good agreement with the previous theoretical study, thereby resulting in a 2% lattice mismatch of the WSe 2 /NiCl 2 heterostructure. In the Berry curvature calculation of the WSe 2 /NiCl 2 heterostructure, we used the maximally localized Wannier function method, as implemented in the WANNIER90 package, to construct real-space maximally localized Wannier functions (MLWFs) after obtaining the self-consistent ground state of the system under study [33].

Geometric structure of the WSe 2 /NiCl 2 heterostructure
We considered six possible stacking configurations (T 1 -T 6 ) in the WSe 2 /NiCl 2 heterostructure (figures 1(a)-(f)): (a) Se right above the midpoint of the Ni-Cl up bond; (b) Se over Ni and W over Cl dn ; (c) Se over Cl up and W over Ni; (d) Se over Cl up and W over Cl dn ; (e) Se over Cl dn and W over Ni; (f) Se over Ni and W over Cl up . Meanwhile, we defined d as the distance between the lower layer Se atoms and the Cl up atoms ( figure 1(a)). The top and side views of NiCl 2 and WSe 2 are shown in figures 1(h) and (g), respectively. The BZ with the high symmetry points and the k-path used for presenting the band structures are shown in figure 1(i). After fully relaxing the structures, we calculated the binding energies of the six configurations by using the following formula [34]: where E h and E w and E n are the total energies of the WSe 2 /NiCl 2 heterostructure and isolated monolayers WSe 2 and NiCl 2 , respectively; and N(N = 6) is the total number of atoms in the heterostructure. The calculated interlayer spacing, binding energies, and valley splitting are shown in table 1. The results of the  binding energies show that all the six stacking configurations can stably exist due to their negative values with only a little energy difference [34]. The binding energies and interlayer spacing of T 1 , T 2 , T 5 , and T 6 are significantly smaller than those of T 3 and T 4 , thereby indicating that the T 1 , T 2 , T 5 , and T 6 configurations are more stable and have a stronger interfacial interaction than the T 3 and T 4 configurations. Further, the phonon spectra were calculated for six possible stacking configurations (T 1 -T 6 ) as shown in figure S1 (https://stacks.iop.org/NJP/22/103061/mmedia). It is clear that the phonon dispersion of T 1 , T 2 , T 5 and T 6 has no imaginary frequencies, which indicates that T 1 , T 2 , T 5 and T 6 are dynamically stable. Therefore, in the following discussion we only consider the four configurations, T 1 , T 2 , T 5 and T 6 .

K -K valley splitting in WSe 2 /NiCl 2
In this part, we study the valley splitting of the WSe 2 /NiCl 2 heterojunction. The K and K valleys for pristine single layer WSe 2 energetically degenerate (figure 2(a)). When WSe 2 is placed on a magnetic substrate, the K -K valley degeneracy is lifted due to the exerted exchange field ( figure 2(b)). The electrons in the two valleys can be selectively excited by the σ + and σ − photons because of the conservation of the angular momentum required by optical transition selection rules and the opposite valley angular  [18][19][20]. The valley splitting values in T 2 , T 5 , and T 6 configurations are larger than those in T 1 configurations. This finding suggests that the interlayer stacking pattern has a great influence on the valley splitting of WSe 2 /NiCl 2 . The monolayer WSe 2 maintains a semiconducting nature and valley characteristic in all the considered stacking patterns, and the band gap remains direct at the K and K points ( figure S2).
The valley splitting value of T 5 stack is the largest among the four stacking configurations. Hence, we calculated the charge density difference and density of states (DOS) for T 5 , and the results are shown in figures 3(a) and (c), respectively. The charge density difference Δρ is defined as follows: where ρ, ρ(WSe 2 ) and ρ(NiCl 2 ) represent the charge densities of the heterostructure, monolayer WSe 2 , and NiCl 2 , respectively. The side view of charge density difference of T 5 is shown in figure 3(c), where the yellow (blue) regions represent the net charge gain (loss). The charges at the interfaces are redistributed. The Se atoms close to the interface lose charges, while a significant charge accumulation occurred in the interfacial Cl atoms. Based on the Bader charge analysis, we find that the amount of 0.05e has been transferred from the WSe 2 to NiCl 2, showing slightly a covalent bonding character. The DOS show that the CB minimum is dominated by the states of Ni and Cl atoms, while the states of W and Se atoms mainly contribute to the VB maximum, exhibiting a quasi-II-type band alignment because the NiCl 2 DOS within 0.05-0.5 eV are involved in the band gap ( figure 3(a)). We further adjusted the interlayer spacing (d) and calculated the corresponding valley splitting for T 5 to examine the effect of interfacial distance on the valley splitting ( figure 3(d)). In the structural models, d takes nine values, namely, 2.9, 3.0, 3.1, 3.3, 3.5, 3.7, 3.9, 4.1 and 4.3 Å, around the most stable value

Valleytronic properties of the WSe 2 /NiCl 2 heterostructure
We address the character of the Berry phase of the WSe 2 /NiCl 2 heterostructure. This study takes T 5 as an example. Berry curvature Ω(k)is an odd function of k in the presence of time reversal symmetry and an even function in the presence of spatial inversion symmetry [36]. In WSe 2 /NiCl 2 , the time inversion symmetry is broken, thereby allowing nonzero values for Ω(k) and the valley contrasting properties. The nonzero Berry curvature can change the motion of carriers and make the system exhibit certain special transport properties, such as the valley Hall effect [37]. These effects enable valley polarization through electric or magnetic fields to be generated and detected, and the free storage and processing of information in the valley can be utilized [19,38]. According to the Kudo equation, the Berry curvature can be expressed as the summation of all occupied contributions [39,40]: where f n is the Fermi-Dirac distribution function, v x(y) is the velocity operator, and |ψ nk is the Bloch wave function with eigenvalue E n . To ensure the calculation accuracy of the Wannier base functions, we first plotted the tight binding band structure using MLWFs shown in figure 4(a). We found that the band dispersion coincided well with the DFT result ( figure 3(a)), indicating that the produced Wannier base functions were sufficiently localized and the accuracy of the calculation was ensured. Figures 4(b) and (c) demonstrate that the calculated Berry curvature at the K and K valleys of WSe 2 /NiCl 2 are nonzero and opposite, similar to monolayer WSe 2 . This finding suggests that the magnetic proximity effect induced by NiCl 2 maintains the valley-contrasting characteristic of monolayer WSe 2 . Under an in-plane longitudinal electric field, the Berry curvature will give rise to an anomalous transverse velocity v ⊥ for Bloch carriers (electrons or holes), v ⊥ ∼ E × Ω(k) [41], and the carriers at the K and K valleys will achieve opposite transverse velocity owing to the opposite signs of their Berry curvature. An illustration of the carrier movement is shown in the insets of figure 4(c), where the plane corresponds to the xy plane of the WSe 2 /NiCl 2 heterostructure. Compared with the band structure shown in figure 3(a), it is found that the spin-up holes at the K valley will move toward the upside in the presence of an in-plane external electric field due to their negative Berry curvatures. If the magnetic ordering of NiCl 2 is reversed from up to down ( figure 3(b)), the spin-down holes at the K valley will act as free carriers and move toward the downside since they have positive Berry curvatures. The hole carriers with K or K nature will accumulate at one transverse edge, and then a sizable voltage can be measured owing to the anomalous Hall effect. Thus, the 2D WSe 2 /NiCl 2 -based heterostructure could potentially be used as a valleytronic device to realize the valley-polarized anomalous Hall effect and filter carriers with certain spin and valley indexes.

K -K valley splitting in NiCl 2 /WSe 2 /NiCl 2
We have increased the heterostructure from two to three layers to achieve a huge valley splitting. A layer of WSe 2 exists between the two layers of NiCl 2 . According to our above calculated results, we found that T 1 is the most stable configuration and the valley splitting value of T 5 is the largest among the four stable configurations T 1 , T 2 , T 5 and T 6 . So, four structures (A 1 , A 2 , A 3 and A 4 ) are constructed on the basis of the above-mentioned T 1 and T 5 (A 1 and A 2 correspond to T 1 ; A 3 and A 4 correspond to T 5 ) (figure 5(a)). Two types of spin alignments of the NiCl 2 layers are considered to study the valley splitting properties. In the first case, the Ni spins are aligned in parallel between the two NiCl 2 sheets, denoted as ↑↑. The first and second arrows represent the identical Ni spins in the two NiCl 2 sheets. In the second case, Ni spins at the top and bottom NiCl 2 sheets are in an antiparallel alignment. We label this case as ↑↓. The calculated band structures of stacking configurations A 1 , A 2 , A 3 , and A 4 are shown in figure S3. The interlayer spacing and valley splitting of the four stacking configurations are illustrated in table 2. The valley splitting of ↑↓ is notably weaker than that of ↑↑. A 4 (↑↑) possesses the largest value (−19.94 meV) of ΔKK among the four configurations, and its band structure is shown in figure 5(b). In the parallel alignment ↑↑, the valley splitting for A 1 , A 2 , A 3 , and A 4 is distinctly larger than the corresponding double-layer heterostructure WSe 2 /NiCl 2 . This finding indicates that the magnetic effect of the two NiCl 2 sheet is more greatly enhanced than that of the single NiCl 2 sheet in WSe 2 /NiCl 2 . In the antiparallel alignment ↑↓, the valley splitting for A 2 , A 3 , and A 4 is notably smaller than corresponding double-layer heterostructure WSe 2 /NiCl 2 . This finding indicates that the magnetic effect of the two NiCl 2 sheets is greatly weakened. However, the magnetic field effect from the two NiCl 2 sheets does not completely disappear due to the unequal interlayer spacing between d 1 and d 2 (table 2); thus, the valley splitting still exist. In the A 1 stacking, the valley splitting of ↑↓ has increased compared with T 1 of WSe 2 /NiCl 2 . These results further verify the critical role of the stacking configurations in the valley splitting in WSe 2 /NiCl 2 . One may pretreat a trilayer NiCl 2 /WSe 2 /NiCl 2 sample in a magnetic field to ensure a parallel alignment of the Ni spins in the two NiCl 2 sheets to achieve an enhanced valley splitting. This prediction needs further experimental verification. In any case, the heterojunction of the sandwich structure can also be used to adjust the valley splitting.

Conclusions
In summary, we have calculated the structural properties, electronic properties, and Berry curvature of the WSe 2 /NiCl 2 and NiCl 2 /WSe 2 /NiCl 2 heterostructures by first principle calculations in combination with MLWFs. The results demonstrate that valley spitting exists in T 1 , T 2 , T 5 and T 6 stacking configurations for the heterostructure. This result shows that valley splitting is sensitive to the arrangement of atoms between the two layers of the heterostructure. The valley splitting of the T 5 configuration can be changed by adjusting the layer spacing. The valley splitting can also be adjusted by aligning the Ni spins of the two NiCl 2 sheets and constructing a heterostructure of a sandwich structure NiCl 2 /WSe 2 /NiCl 2 . The Berry curvature and spin splitting are opposite at the K and K valleys of the WSe 2 /NiCl 2 heterostructure, thereby enabling simultaneous polarization and locking of the valley and spin quantum degrees of freedom. The results demonstrate that the WSe 2 /NiCl 2 -based heterostructure can be widely used in next generation multifunctional valleytronic devices for the anomalous Hall effect as both spin and valley filter.