⇦Chinese Physics Letters, 2020, Vol. 37, No. 10, Article code 107101 Coupling Stacking Orders with Interlayer Magnetism in Bilayer H-VSe$_{2}$ Aolin Li (李奥林)1, Wenzhe Zhou (周文哲)1, Jiangling Pan (潘江陵)2, Qinglin Xia (夏庆林)2, Mengqiu Long (龙孟秋)2,3, and Fangping Ouyang (欧阳方平)1,2,3* Affiliations 1State Key Laboratory of Powder Metallurgy, and Powder Metallurgy Research Institute, Central South University, Changsha 410083, China 2School of Physics and Electronics, and Hunan Key Laboratory for Super-Microstructure and Ultrafast Process, Central South University, Changsha 410083, China 3School of Physics and Technology, Xinjiang University, Urumqi 830046, China Received 9 July 2020; accepted 8 August 2020; published online 29 September 2020 Supported by the National Natural Science Foundation of China (Grant No. 51272291), the Distinguished Young Scholar Foundation of Hunan Province (Grant No. 2015JJ1020), the Young Scholar Foundation of Hunan Province (Grant No. 2016JJ3142), the Central South University Research Fund for Sheng-Hua Scholars, Central South University State Key Laboratory of Powder Metallurgy, and the Fundamental Research Funds for the Central Universities of Central South University.
*Corresponding author. Email: oyfp@csu.edu.cn Citation Text: Li A L, Zhou W Z, Pan J L, Xia Q L and Long M Q et al. 2020 Chin. Phys. Lett. 37 107101    Abstract Stacking-dependent magnetism in van der Waals materials has caught intense interests. Based on the first principle calculations, we investigate the coupling between stacking orders and interlayer magnetic orders in bilayer H-VSe$_{2}$. It is found that there are two stable stacking orders in bilayer H-VSe$_{2}$, named AB-stacking and A$^{\prime}$B-stacking. Under standard DFT framework, the A$^{\prime}$B-stacking prefers the interlayer AFM order and is semiconductive, whereas the AB-stacking prefers the FM order and is metallic. However, under the DFT+$U$ framework both the stacking orders prefer the interlayer AFM order and are semiconductive. By detailedly analyzing this difference, we find that the interlayer magnetism originates from the competition between antiferromagnetic interlayer super-superexchange and ferromagnetic interlayer double exchange, in which both the interlayer Se-4$p_{z}$ orbitals play a crucial role. In the DFT+$U$ calculations, the double exchange is suppressed due to the opened bandgap, such that the interlayer magnetic orders are decoupled with the stacking orders. Based on this competition mechanism, we propose that a moderate hole doping can significantly enhance the interlayer double exchange, and can be used to switch the interlayer magnetic orders in bilayer VSe$_{2}$. This method is also applicable to a wide range of semiconductive van der Waals magnets. DOI:10.1088/0256-307X/37/10/107101 PACS:71.70.Gm, 75.30.Et, 71.15.Mb, 71.30.+h © 2020 Chinese Physics Society Article Text The stacking-dependent properties in van der Waals (vdW) materials have caught wide attention for nanoelectronic applications. Although the interlayer coupling is weak in energy, it can cause drastic influence on the electronic properties by interlayer charge transfer,[1] wavefunction hybridization,[2] magnetic exchange,[3,4] and spin-orbit proximity,[5,6] etc. Recently, the atomically thin CrI$_{3}$ has been experimentally exfoliated and displays thickness-dependent magnetism due to the interlayer antiferromagnetic (AFM) coupling.[7] First-principles calculations have suggested that the interlayer coupling in bilayer CrI$_{3}$ can either be ferromagnetic (FM) or antiferromagnetic by switching the stacking orders.[3,4] This has been confirmed by recent experiments in which hydrostatic pressures were used to control the stacking orders and interlayer magnetic orders.[8,9] Despite the fascinating interlayer magnetism in CrI$_{3}$, the low Curie temperature ($\sim $45 K for the monolayer[7]) greatly restrict its application in spintronic. Seeking 2D materials with both room-temperature and stacking-dependent magnetism is of important value, but remains great challenging. Besides CrI$_{3}$, there are some other two-dimensional (2D) magnets that have been fabricated in laboratories, including Cr$_{2}$Ge$_{2}$Te$_{6}$,[10] Fe$_{3}$GeTe$_{2}$,[11] VSe$_{2}$,[12] MnSe$_{2}$,[13] etc.,[14,15] while stacking-dependent magnetism has few been reported in these materials. Instead, the stacking-induced magnetism has been found in some nonmagnetic materials.[16,17] For example, in the twisted bilayer graphene,[16] bilayer silicone and germanene,[17] first-principles calculations have suggested that spontaneous magnetization can be induced by the stacking effects, but this magnetism seems not to be maintained at room temperature. Remarkably, Zhou et al. have reported a Bethe-Slater-Curve (BSC) like behavior in bilayer chromium dichalcogenides, in which the interlayer AFM order and FM order are preferred at shorter and longer distances, respectively.[18] Inspired by their work, we propose that the stacking-dependent magnetism may be realized in 2D magnets, as long as the interlayer distances fall into the FM and AFM regions for different stacking orders, respectively. In this work, we choose vanadium selenide (VSe$_{2}$) to explore the stacking-dependent magnetism, because of its strong magnetism and semiconducting properties for spintronic applications. There are two polymorphs for VSe$_{2}$, the semiconductive hexagonal (H) phase and the metallic octahedral (T) phase. The monolayer T-VSe$_{2}$ has been successfully synthesized in 2018, and its strong room-temperature ferromagnetism on van der Waals substrates is demonstrated.[12] Although atomically thin H-VSe$_{2}$ has not been synthesized in laboratory, its bulk material 3R-VSe$_{2}$ was reported very early.[19] The first principles calculation has suggested that the Curie temperature of monolayer H-VSe$_{2}$ can reach 541 K.[20] Recent studies have shown that the bilayer H-VSe$_{2}$ is of great potential in achieving all-electric valleytronics and spintronics.[21,22] All density functional theory (DFT) calculations in this work are performed with the Vienna ab initio Simulation Package (VASP).[23] Generalized gradient approximation (GGA) with the PBE formalism[24] and projected augmented wave (PAW) pseudopotentials[25] are used. The cutoff energy of the plane waves is set to 500 eV, and the $k$-points are sampled on a $21\times 21\times 1$ Monkhorst–Pack grid. For the structural relaxation, the tolerance of the Hellmann–Feynman forces is set as 1 meV/Å, and for self-consistent calculations, the energy convergence criterion is set as 10$^{-7}$ eV. A vacuum layer thicker than 10 Å along the $c$ axis is used to screen the interaction between neighbor repeated imagines. To describe the interlayer van der Waals interactions, the DFT-D2 method is used.[26] DFTs without and with Hubbard $U$ correction[27] for V-3$d$ orbitals are both performed, and the difference is carefully compared. Our results suggest that Hubbard $U$ correction only makes little effects on the structural stability and relative energies for different stacking orders, while can significantly affect the bandgap and interlayer magnetic coupling. The effect of spin-orbit coupling (SOC) is negligible for the results, thereby the DFT$+$SOC calculations are not considered in this work (see the Supplementary Material). Bilayer VSe$_{2}$ is usually considered to have an AB-stacking order as 2H-MoS$_{2}$, but other stacking order may also be stable. For example, Fig. 1(a) shows three possible stacking orders with high symmetry. In the AB-stacking, the V(Se) atoms of the upper layer cover the Se (V) atoms of the nether layer. In the A$^{\prime}$B-stacking, the V atoms of the upper layer cover the V atoms of the nether layer, while the Se atoms in different layers are staggered. In the AA$^{\prime}$-stacking, the Se atoms in different layers are directly head-to-head whereas the V atoms are staggered.
cpl-37-10-107101-fig1.png
Fig. 1. (a) Three high-symmetry stacking orders in bilayer H-VSe$_{2}$. (b) The sliding energy landscape, the blue dashed line represents the high-symmetry $[1\bar{1}0]$ direction. (c) The sliding energy along the high-symmetry $[1\bar{1}0]$ direction.
To explore all possible stable stacking orders in bilayer H-VSe$_{2}$, we use the fully relaxed AB-stacking VSe$_{2}$ as the initial structure. When keeps the nether layer fixed, the upper layer is rigidly shifted along the (001) plane. The sliding energy $\Delta E_{\rm sliding}$ is defined as $$ \Delta E_{\rm sliding} =E_{\rm shift} -E_{\rm AB},~~ \tag {1} $$ where $E_{\rm AB}$ and $E_{\rm shift}$ represent the total energy of AB-stacking and the shifted configuration, respectively. The calculated sliding energy landscape is shown in Fig. 1(b), apart from the initial AB stacking, we find that the A$^{\prime}$B stacking is also energy-favored. Both AB and A$^{\prime}$B stackings belong to the $P\bar{3}M1$ space group and have the same crystallographic symmetries, their energy difference is within 3 meV per formula unit (f.u.). The AA$^{\prime}$-stacking is the most unstable in energy due to strong Pauli repulsion between the head-to-head interlayer Se atoms. Figure 1(c) shows the $\Delta E_{\rm sliding}$ along the high-symmetry $[1\bar{1}0]$ direction, the energy of AA$^{\prime}$-stacking is over 130 meV/f.u. higher than the two energy-favored stacking orders. The energy barrier between the AB- and A$^{\prime}$B-stackings is only $\sim $20 meV/f.u., similar to the values of bilayer CrI$_{3}$.[3,4] The Hubbard $U$ correction is considered in Fig. 1(c), which makes little impact on the results. The dynamical stability of AB- and A$^{\prime}$B-stacking has been checked by the phonon spectrum (see Fig. S1 in the Supplementary Material), which agrees well with the literature.[28] To understand the stacking effect of bilayer H-VSe$_{2}$, knowledge about the fundamental electronic nature of monolayer VSe$_{2}$ is necessary. Figure 2 shows the band structure of monolayer VSe$_{2}$ and its major orbital contributions. We are concerned with only two bands around the Fermi level ($E_{\rm F}$), the highest-occupied valence band (marked as $\alpha$-band) and the lowest-unoccupied conduction band (marked as $\beta$-band). Nominally, the V$^{4+}$ ion has an electronic configuration 3$d^{1}4s^{0}$. When spin polarization is not considered, a single $d$-band will cross the $E_{\rm F}$ under the hexahedral crystal field, which makes the monolayer VSe$_{2}$ metallic and a high density of states (DOS) at the $E_{\rm F}$ (see Fig. S2 in the Supplementary Material). After spin-polarization is considered as shown in Fig. 2, the single $d$-band spontaneously splits into the $\alpha$- and $\beta$-bands due to the Stoner instability. Under the DFT framework, the spin splitting energy $E_{\rm s}$ between the $\alpha$- and $\beta$-bands is about 1 eV at the $\varGamma$ point, which opens a small indirect bandgap $E_{\rm g}$ on the value of 0.25 eV. After considering the Hubbard $U$ correction, both the spin splitting energy and the bandgap increase with the increasing $U$ parameter (see Fig. S3 in the Supplementary Material). From Fig. 2 we can find that the $\alpha$- and $\beta$-bands are mainly contributed by V-$3d_{z^{2}}$ orbital around the $\varGamma$ point, while the Se-4$p_{z}$ orbitals have only a very small contribution ($\sim $3%). However, this small Se-4$p_{z}$ contribution plays a pivotal role in the band structure evolution when stacking into multilayers.
cpl-37-10-107101-fig2.png
Fig. 2. The band structure of monolayer H-VSe$_{2}$ (left) and its major orbital contribution. For clarity, the projection weight of Se-4$p$ orbitals is set as ten times the value of V-3$d$ orbitals.
cpl-37-10-107101-fig3.png
Fig. 3. (a)–(d) The band structures of bilayer H-VSe$_{2}$ with different stacking modes and magnetic orders. (e) The projected band structure for the interlayer and outerlayer Se-4$p_{z}$ orbitals with $U = 1$ eV. (f) The visualized wave functions for the eigenstates labeled in (e), where the contour surface is set as $5\times 10^{-4}\,e$/Å$^{3}$.
Figures 3(a)–3(d) show the band structures of bilayer VSe$_{2}$, the difference is fractional between AB- and A$^{\prime}$B-stacking orders, while it is obvious between the FM and interlayer AFM orders. In terms of the interlayer AFM order, both the $\alpha$- and $\beta$-bands are spin-degenerate, and the bilayers remain semiconductive. In terms of the FM order, a band splitting appears in the $\alpha$- and $\beta$-bands, which makes the FM bilayers change into metallic. The total energies of bilayer H-VSe$_{2}$ are listed in Table 1. We find that the AB-stacking prefers the FM order, while the A$^{\prime}$B-stacking prefers the interlayer AFM order. This suggests that a semiconductor-to-metal transition in bilayer H-VSe$_{2}$ can be achieved by switching either the stacking orders or the interlayer magnetic orders. However, the usage of Hubbard $U$ correction causes the bandgap not to be fully closed for the FM orders. For example, Fig. 3(e) shows that a bandgap on the value of $\sim $60 meV is opened for the FM bilayer with $U = 1$ eV. However, even without the gap closing, the interlayer magnetic orders can still cause a big difference in the conductivity or carrier concentration $\sigma$. The bandgap difference $\Delta E_{\rm g}$ between FM and interlayer AFM orders is about 350 meV. Using the relationship $\sigma \propto \exp(-E_{\rm g}/k_{_{\rm B}}T)$, we estimate that the carrier concentration of FM orders is about 10$^{5}$ times larger than the AFM orders at room temperature.
Table 1. The total energy of bilayer VSe$_{2}$ per unit cell with reference to the FM AB-stacking and the band gaps. For the DFT+$U$ calculations, an effective on-site $U$ on the value of 1.0 eV is used.
AB A$^{\prime}$B
FM AFM FM AFM
DFT Energy (meV) 0 0.3 $-1.1$ $-1.2$
Gap (eV) 0 0.15 0 0.14
DFT+$U$ Energy (meV) 0 $-3.3$ 2.1 $-2.0$
Gap (eV) 0.06 0.39 0.04 0.38
From Figs. 3(a) and 3(c), we can find that the band splittings are significant around the $G$ and $M$ points, where the Se-$p_{z}$ orbitals have a contribution to the bands (see Fig. 2), indicating that the interlayer Se-4$p_{z}$ orbitals play a crucial role in the band splitting. By analyzing the eigenstates of $\alpha$- and $\beta$-bands, we attribute this splitting to the interlayer covalent-like Se$\cdots$Se quasi-bonding states, which is first proposed by Zhao et al.[2] and has been recently confirmed by experiments.[29] The Bloch states for the split $\alpha$-band at the $G$ and $K$ points are labeled as $G_{1}$, $G_{2}$, and $K_{1}$ as shown in Fig. 3(e), and are visualized in Fig. 3(f). For the lower-energy $G_{1}$ point, the interlayer Se-4$p_{z}$ orbitals in adjacent layers are hybridized with each other, corresponding to the interlayer quasi-bonding state. For the higher-energy $G_{2}$ point, the interlayer Se-4$p_{z}$ orbitals are separated from each other, corresponding to the interlayer anti-bonding state. At the $K_{1}$ point, because the Se-4$p_{z}$ orbital contribution is absent, the wavefunction is separated at the vdW gap and the band splitting is suppressed. For the interlayer AFM orders, the spin degeneracy is preserved by its invariance under the combination of time-reversal operation and inversion operation, such that the band splitting is suppressed. Figure 3(e) also shows the contribution of interlayer/outerlayer Se-4$p_{z}$ orbitals to the $\alpha$- and $\beta$-bands. We can find that the inner Se-4$p_{z}$ orbitals mainly contribute to the upper bands, while the outer Se-4$p_{z}$ orbitals mainly contribute to the lower bands. In addition, our DFT calculations show that each Se ion has $\sim$$0.1\mu_{_{\rm B}}$ magnetic moment antiparallel to the one of V ions ($\sim $$1.1\mu_{_{\rm B}}$). The Pauli repulsion between interlayer Se ions will make the bilayers prefer to form an interlayer AFM order. To study the relationship between stacking orders and interlayer magnetic orders, we plot the landscape of interlayer exchange energy in Fig. 4(a). The interlayer exchange energy can be defined as the energy difference between the FM order and the interlayer AFM order per unit cell: $$ J_{\rm ex} = E ({\rm FM}) - E ({\rm AFM}),~~ \tag {2} $$ such that a negative value means that the bilayers prefers the FM orders, while a positive value means that the bilayers prefers the interlayer AFM order. In Fig. 4(a), the DFT results show obvious stacking-dependent magnetism in bilayer VSe$_{2}$. While the AB- and AA$^{\prime}$-stacking prefer the FM order, the A$^{\prime}$B-stacking prefers the interlayer AFM order. After considering the Hubbard $U$ correction, we find that all configurations favor the interlayer AFM state, such that the stacking orders and interlayer magnetic orders are decoupled. To understand this difference, a clear picture of the interlayer magnetic exchange coupling is necessary. Although direct exchange between V ions in different layers is impossible due to the big distance, the interlayer magnetic exchange can also be established through indirect ways, such as the super-superexchange (SSE)[4] and the double exchange (DE),[30,31] which are mediated by the intermediate Se ions and conduction electrons, respectively. We first consider the SSE mechanism that two spin-polarized electrons in different layers are coupled through a V–Se$\cdots$Se–V path, in which the Se$\cdots$Se represents the covalent-like quasi-bonding between interlayer Se atoms. This mechanism has successfully been used to explain the interlayer AFM order in the bilayer CrSe$_{2}$.[18] Considering the structural similarities of VSe$_{2}$ and CrSe$_{2}$, we believe that the SSE also plays an important role in bilayer VSe$_{2}$. The V $3d$ orbits of VSe$_{2}$ has the $t_{\rm 2g}^{1}e_{\rm g}^{0}$ occupied states under the hexahedral crystal field, containing both half-filled and empty V $3d$ orbitals. According to the Goodenough–Kanamori rule,[32,33] the SSE coupling can either be AFM or FM depending on the two sequential V–Se$\cdots$Se bond angles. Our results suggest that there are two typical bond angles, the nearly orthogonal one ($\theta_{1}\approx 93^{\circ}$) and the nearly collinear one ($\theta_{2}\approx 140^{\circ}$), as shown in Fig. 4(c). Their combination leads to three different SSE paths, i.e., ($\theta_{1}$, $\theta_{1}$), ($\theta_{1}$, $\theta_{2}$) and ($\theta_{2}$, $\theta_{2}$). Notice that the wavefunctions of interlayer Se ions are partly overlapped, the Pauli exclusion principle makes the two electrons in different Se-4$p_{z}$ orbitals tend to have an antiparallel spin arrangement. As a result, the symmetric paths, i.e., the ($\theta_{1}$, $\theta_{1}$) and ($\theta_{2}$, $\theta_{2}$) paths, prefer the interlayer AFM order, while the asymmetric path ($\theta_{1}$, $\theta_{2}$) prefers the FM order. Considering that the interlayer SSE is greatly weakened by the vdW gap, the intralayer coupling has higher priority in meeting the Goodenough–Kanamori rule. The spin arrangement in Se$\cdots$Se bond is forced to be parallel in the asymmetric path, which conflicts the Pauli exclusion principle. As a result, the overall effect of SSE coupling more likely to form the interlayer AFM order. Considering the arrangement of interlayer Se ions, they are staggered in the AB- and A$^{\prime}$B-stacking orders but are head-to-head in the AA$^{\prime}$-stacking order. According to the SSE mechanism, the interlayer coupling in the AA$^{\prime}$-stacking should be strongly antiferromagnetic. In Fig. 4(a), this is consistent with the DFT+$U$ results but is contrary to the DFT results. The interlayer exchange in the AA$^{\prime}$-stacking is ferromagnetic. Only considering the SSE mechanism can hardly explain this conflict. Notice that the major difference between DFT and DFT+$U$ calculations lies in the bandgap opening in the FM bilayers, the DE mechanism mediated by conduction electrons can give a reasonable explanation. As schematically plotted in Fig. 4(d), the interlayer electrons hopping makes the DE prefer the FM order due to lower kinetic energy rather than the interlayer AFM order. The strength of DE directly relies on the density of carriers in bilayer VSe$_{2}$. For the AA$^{\prime}$-stacking in DFT framework, the head-to-head Se-4$p_{z}$ orbitals cause an extraordinarily strong interlayer hybridization and a high DOS at Fermi level, such that the interlayer DE makes the AA$^{\prime}$-stacking are strong FM. On the whole, the interlayer magnetic coupling can be properly explained by the competition between the SSE and DE mechanisms.
cpl-37-10-107101-fig4.png
Fig. 4. (a) The landscape of the interlayer exchange energy without and with considering the Hubbard $U$ correction. (b) Schematic of the interlayer SSE interaction. (c) The calculated V–Se$\cdots$Se bond angles. (d) The interlayer DE interaction. (e) The change of interlayer exchange energy and band gaps with the increasing on-site Hubbard $U$ parameter.
To further validate this competitive mechanism, we plot the interlayer exchange energy $J_{\rm ex}$ as a function of $U$ in Fig. 4(e). When $U = 0$, the FM bilayer VSe$_{2}$ is metallic and the $J_{\rm ex}$ is close to zero, suggesting that the DE and the SSE are similar in strength, which makes the interlayer coupling be FM for the AB-stacking but be AFM for the A$^{\prime}$B-stacking. When $U$ increasing from 0 to 0.8 eV, the strength of DE decreases with the reduction of DOS at $E_{\rm F}$. As a result, the $J_{\rm ex}$ monotonically increases in this range. Notice that a bandgap is opened in bilayer VSe$_{2}$ when $U = 0.8$ eV, also in which the increase of $J_{\rm ex}$ stops. This suggests that the most important role of Hubbard $U$ correction is suppressing the DE coupling in bilayer VSe$_{2}$. Although the $U$ parameter can also strengthen the interlayer SSE in some degree by increasing electronic interactions, the effect should be limited when $U$ is not too large, because the interlayer SSEs are dominated by the low-localized interlayer Se-4$p_{z}$ orbitals. In the literature, an effective $U$ with a value of 1.16 eV is recommended,[22] which opens the bandgap and makes both AB- and A$^{\prime}$B-stackings have the interlayer AFM order. In the region of $U \in [0.8, 2.0]$ eV, the change of interlayer exchange energy is little, suggesting that the difference between the DFT+$U$ and DFT results is mainly caused by the suppressing of DE. We also calculated other possible magnetic orders, with their total energies much higher than the values of the FM order and interlayer AFM order. Furthermore, the Monte Carlo (MC) simulations were performed using the parameters obtained by our DFT+$U$ calculations to estimate the Curie temperature. We find that the interlayer coupling plays an important role in the magnetism of magnetic multilayers, even though it is very weak in energy. These results can be found in the Supplementary Material. To couple the stacking orders with different magnetic orders, one can use carrier doping to re-introduce the DE in bilayer VSe$_{2}$. Figure 5(a) shows the map of exchange energy $J_{\rm ex}$ as functions of the carrier density and the $U$ parameter. A moderate hole doping in the magnitude of 10$^{13}$ cm$^{-2}$ is enough to change the bilayers from the interlayer AFM order to FM order, whereas doping electrons can only slightly reduce the $J_{\rm ex}$. This difference can be understood by the distribution of additional charges after doping, as shown in Fig. 5(c). For the hole doping, a part of the carriers is distributed in the interlayer Se-4$p_{z}$ orbital, which can help to strengthen the interlayer exchange. For the electron doping, few additional carriers are distributed in the interlayer region, such that its effect is greatly limited.
cpl-37-10-107101-fig5.png
Fig. 5. (a) The exchange energy of AB-stacking as a function of doping density and Habburd $U$. (b) The difference of $J_{\rm ex}$ between AB- and A$^{\prime}$B-stacking, where the solid and dashed lines represent $J_{\rm ex}=0$ for AB- and A$^{\prime}$B-stackings, respectively. (c) The distribution of additional charges after electron and hole doping, where the contour surface is set as 10$^{-4}\, e$/Å$^{3}$.
Since the $J_{\rm ex}$ is similar for the AB- and A$^{\prime}$B-stackings, we only plotted the $J_{\rm ex}$ for AB-stacking in Fig. 5(a). Instead, we plot the difference of $J_{\rm ex}$ between AB- and A$^{\prime}$B-stackings in Fig. 5(b). The solid and dashed lines represent the critical condition of AFM-to-FM transition for the AB- and A$^{\prime}$B-stackings, respectively. A hole doping within $5\times 10^{13}$ cm$^{-2}$ can be used to couple the stacking orders with different interlayer magnetic orders. The required gate voltage can be estimated by a simple capacitor model: $$ V=\frac{Q}{C}=\frac{Q}{\varepsilon A/t}=\frac{e\sigma t}{\varepsilon_{0} \varepsilon_{\rm r} }, $$ where $e$ is the electron charge, $\sigma$ is the doping density, $t$ is the thickness of the dielectric layer, $\varepsilon_{0}$ and $\varepsilon_{\rm r}$ represent the vacuum and relative permittivity, respectively. Using the same parameters with a previous report (5 nm HfO$_{2}$, $\varepsilon_{\rm r} =25$),[34] the required voltage to control the interlayer magnetism is estimated to be $\sim $1.8 V.
cpl-37-10-107101-fig6.png
Fig. 6. (a) The total energy and pressure as a function of interlayer distance $d$. (b) The interlayer exchange energy $U$ is set to 1 eV.
As the pressure-controlled interlayer magnetism in bilayer CrI$_{3}$ has been achieved by recent experiments,[8,9] we have also explored this possibility in bilayer VSe$_{2}$. The vertical pressure in bilayer VSe$_{2}$ can be written as $P=S^{-1} \times dE_{\rm tot}/dl$, where $S$ is the surface area, $E_{\rm tot}$ is the total energy and $l$ is the interlayer distance. Figure 6(a) shows the total energy as a function of interlayer distance $l$. We find that there is a very tiny difference between the equilibrium distance of AB-stacking is imperceptibly larger than the one of A$^{\prime}$B-stacking, on the value of 0.03 Å. Figure 6(b) shows the exchange energy as a function of interlayer distance, the interlayer magnetism of A$^{\prime}$B-stacking is more sensitive to $l$ than the one of AB-stacking. Interestingly, different from the Bethe–Slater–Curve-like behavior reported in bilayer CrSe$_{2}$ that only a single inflection point is observed,[18] the $l$–$J_{\rm ex}$ curves of bilayer VSe$_{2}$ have double inflection points. This difference is closely relevant to the semiconductor-to-metal transition in bilayer VSe$_{2}$. According to the inflection points, we can divide the curves into three regions, as marked in Fig. 6(b). In the region I, the wavefunctions of interlayer Se-4$p_{z}$ orbitals are highly overlapped, the strong Pauli repulsion makes the bilayers prefer the interlayer AFM order. In the region II, the Pauli repulsion is weakened due to the increase of interlayer distance, whereas the bilayers are still metallic, such that the DE coupling can make the bilayers prefer the FM order. In region III, as the interlayer hopping being further weakened, a bandgap is opened and the DE is fully suppressed. The SSE mechanism makes the bilayers prefer the interlayer AFM order. In conclusion, based on first-principles calculations, we have studied the coupling between stacking orders and magnetic orders in bilayer H-VSe$_{2}$. We find that there are two stable stacking orders, AB-stacking and A$^{\prime}$B-stacking. The interlayer Se-4$p_{z}$ orbitals play a crucial role in the magnetic-order-dependent band splitting and interlayer exchange coupling. We have demonstrated that the interlayer magnetism is determined by the competition between the interlayer super-superexchange and the double exchange. Hole doping can significantly modulate the interlayer double exchange, therefore it can be used to control the AFM-to-FM transition in the bilayers and to lock the stacking modes with the magnetic orders. Our work gives a further understanding of the interlayer magnetism in TMDCs bilayers, and it may provide certain guidelines for the study of magnetic multilayers and heterostructures. References Bonding in 2D Donor–Acceptor HeterostructuresExtraordinarily Strong Interlayer Interaction in 2D Layered PtS 2Stacking tunable interlayer magnetism in bilayer CrI 3 Stacking-Dependent Magnetism in Bilayer CrI 3Trivial and inverted Dirac bands and the emergence of quantum spin Hall states in graphene on transition-metal dichalcogenidesGeometry and electronic structure of monolayer, bilayer, and multilayer Janus WSSeLayer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limitBethe-Slater-curve-like behavior and interlayer spin-exchange coupling mechanisms in two-dimensional magnetic bilayersPressure-controlled interlayer magnetism in atomically thin CrI3Switching 2D magnetic states via pressure tuning of layer stackingDiscovery of intrinsic ferromagnetism in two-dimensional van der Waals crystalsGate-tunable room-temperature ferromagnetism in two-dimensional Fe3GeTe2Strong room-temperature ferromagnetism in VSe2 monolayers on van der Waals substratesRoom Temperature Intrinsic Ferromagnetism in Epitaxial Manganese Selenide Films in the Monolayer LimitLarge-Gap Magnetic Topological Heterostructure Formed by Subsurface Incorporation of a Ferromagnetic LayerSynthesis of sub-micro-flakes CrSe2 on glass and (110) Si substrates by solvothermal methodElectrically Controllable Magnetism in Twisted Bilayer GrapheneIntrinsic magnetism and spontaneous band gap opening in bilayer silicene and germaneneSelf-Assembled Nanofold Network Formation on Layered Crystal Surfaces during Metal IntercalationElectronic and Magnetic Properties of Vanadium Dichalcogenides Monolayers Tuned by HydrogenationElectrical control of the anomalous valley Hall effect in antiferrovalley bilayersElectrically induced 2D half-metallic antiferromagnets and spin field effect transistorsEfficient iterative schemes for ab initio total-energy calculations using a plane-wave basis setGeneralized Gradient Approximation Made SimpleProjector augmented-wave methodImproved Description of the Structure of Molecular and Layered Crystals: Ab Initio DFT Calculations with van der Waals CorrectionsElectron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U studyDynamic instabilities in strongly correlated VSe 2 monolayers and bilayersX-ray electron density investigation of chemical bonding in van der Waals materialsDouble exchange and the cause of ferromagnetism in doped manganitesDouble-exchange mechanism for CrO 2 Antiferromagnetism. Theory of Superexchange InteractionTheory of the Role of Covalence in the Perovskite-Type Manganites [ La ,   M ( II ) ] Mn O 3 Charge Mediated Reversible Metal–Insulator Transition in Monolayer MoTe 2 and W x Mo 1– x Te 2 Alloy
[1] Woomer A H, Druffel D L, Sundberg J D, Pawlik J T and Warren S C 2019 J. Am. Chem. Soc. 141 10300
[2] Zhao Y D, Qiao J S, Yu P, Hu Z X, Lin Z Y, Lau S P, Liu Z, Ji W and Chai Y 2016 Adv. Mater. 28 2399
[3] Jiang P H, Wang C, Chen D C, Zhong Z C, Yuan Z, Lu Z Y and Ji W 2019 Phys. Rev. B 99 144401
[4] Sivadas N, Okamoto S, Xu X, Fennie C J and Xiao D 2018 Nano Lett. 18 7658
[5] Gmitra M, Kochan D, Hogl P and Fabian J 2016 Phys. Rev. B 93 155104
[6] Zhou W Z, Chen J Y, Yang Z X, Liu J W and Ouyang F P 2019 Phys. Rev. B 99 075160
[7] Huang B, Clark G, Navarro-Moratalla E, Klein D R, Cheng R, Seyler K L, Zhong D, Schmidgall E, McGuire M A, Cobden D H, Yao W, Xiao D, Jarillo-Herrero P and Xu X D 2017 Nature 546 270
[8] Zhou X Y, Wang C, Zhou L W, Pan Y H, Lu Z Y, Wan X G, Wang X Q and Ji W 2020 Phys. Rev. B 102 020402
[9] Li T X, Jiang S W, Sivadas N, Wang Z F, Xu Y, Weber D, Goldberger J E, Watanabe K, Taniguchi T, Fennie C J, Mak K F and Shan J 2019 Nat. Mater. 18 1303
[10] Song T C, Fei Z Y, Yankowitz M, Lin Z, Jiang Q N, Hwangbo K, Zhang Q, Sun B S, Taniguchi T, Watanabe K, McGuire M A, Graf D, Cao T, Chu J H, Cobden D H, Dean C R, Xiao D and Xu X D 2019 Nat. Mater. 18 1298
[11] Gong C, Li L, Li Z L, Ji H W, Stern A, Xia Y, Cao T, Bao W, Wang C Z, Wang Y A, Qiu Z Q, Cava R J, Louie S G, Xia J and Zhang X 2017 Nature 546 265
[12] Deng Y J, Yu Y J, Song Y C, Zhang J Z, Wang N Z, Sun Z Y, Yi Y F, Wu Y Z, Wu S W, Zhu J Y, Wang J, Chen X H and Zhang Y B 2018 Nature 563 94
[13] Bonilla M, Kolekar S, Ma Y J, Diaz H C, Kalappattil V, Das R, Eggers T, Gutierrez H R, Phan M H and Batzill M 2018 Nat. Nanotechnol. 13 289
[14] O'Hara D J, Zhu T C, Trout A H, Ahmed A S, Luo Y K, Lee C H, Brenner M R, Rajan S, Gupta J A, McComb D W and Kawakami R K 2018 Nano Lett. 18 3125
[15] Hirahara T, Eremeev S V, Shirasawa T, Okuyama Y, Kubo T, Nakanishi R, Akiyama R, Takayama A, Hajiri T, Ideta S, Matsunami M, Sumida K, Miyamoto K, Takagi Y, Tanaka K, Okuda T, Yokoyama T, Kimura S, Hasegawa S and Chulkov E V 2017 Nano Lett. 17 3493
[16] Tang Q K, Liu C Y, Zhang B B and Jie W Q 2018 J. Solid State Chem. 262 53
[17] Gonzalez-Arraga L A, Lado J L, Guinea F and San-Jose P 2017 Phys. Rev. Lett. 119 107201
[18] Wang X Q and Wu Z G 2017 Phys. Chem. Chem. Phys. 19 2148
[19] Spiecker E, Schmid A K, Minor A M, Dahmen U, Hollensteiner S and Jager W 2006 Phys. Rev. Lett. 96 086401
[20] Pan H 2014 J. Phys. Chem. C 118 13248
[21] Tong W Y and Duan C G 2017 npj Quantum Mater. 2 47
[22] Gong S J, Gong C, Sun Y Y, Tong W Y, Duan C G, Chu J H and Zhang X 2018 Proc. Natl. Acad. Sci. USA 115 8511
[23] Kresse G and Furthmüller J 1996 Phys. Rev. B 54 11169
[24] Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
[25] Blöchl P E 1994 Phys. Rev. B 50 17953
[26] Bucko T, Hafner J, Lebegue S and Angyan J G 2010 J. Phys. Chem. A 114 11814
[27] Dudarev S L, Botton G A, Savrasov S Y, Humphreys C J and Sutton A P 1998 Phys. Rev. B 57 1505
[28] Esters M, Hennig R G and Johnson D C 2017 Phys. Rev. B 96 235147
[29] Kasai H, Tolborg K, Sist M, Zhang J W, Hathwar V R, Filso M O, Cenedese S, Sugimoto K, Overgaard J, Nishibori E and Iversen B B 2018 Nat. Mater. 17 249
[30] Zhao G M 2000 Phys. Rev. B 62 11639
[31] Schlottmann P 2003 Phys. Rev. B 67 174419
[32] Anderson P W 1950 Phys. Rev. 79 350
[33] Goodenough J B 1955 Phys. Rev. 100 564
[34] Zhang C X, Santosh K C, Nie Y F, Liang C P, Vandenberghe W G, Longo R C, Zheng Y P, Kong F T, Hong S, Wallace R M and Cho K 2016 ACS Nano 10 7370