Confinement effect enhanced Stoner ferromagnetic instability in monolayer 1T-VSe2

Monolayer 1T-VSe2 has been reported as a room-temperature ferromagnet. In this work, by using first-principles calculations, we unveil that the ferromagnetism in monolayer 1T-VSe2 is originated from its intrinsic huge Stoner instability enhanced by the confinement effect, which can eliminate the interlayer coupling, and lead to a drastic increase of the density of states at the Fermi level due to the presence of Van Hove singularity. Our calculations also demonstrate that the Stoner instability is very sensitive to the interlayer distance. These results provide a useful route to modulate the nonmagnetic to ferromagnetic transition in few-layers or bulk 1T-VSe2, which also shed light on the enhancement of its Curie temperature by enlarging the interlayer distance.


I. INTRODUCTION
Ferromagnetic order in two-dimensional (2D) materials is a highly desirable property that provides a new physical degree of freedom to manipulate spin behaviors in spintronic devices 1,2 . Previously, magnetism in 2D was mainly realized through depositing films onto magnetic substrates, magnetic atoms adsorption, or doping [3][4][5] . The shortcomings of these methods are obvious: (i) one does not have an ideal 2D system from depositing and it is impractical to integrate with spintronic devices, and (ii) disorder effects make the electronic properties hard to design. Due to these drawbacks, 2D materials with intrinsic magnetic order have been actively pursued.
The CrI 3 6 and Cr 2 Ge 2 Te 6 7 are the first two experimentally reported 2D materials exhibiting long-range ferromagnetic (FM) order with the Curie temperatures T c ∼ 45 K and 30 K, respectively 6,7 . These discoveries have stimulated numerous research interests on 2D magnetic materials. Very recently, several materials with higher T c have been experimentally and theoretically explored, including MnSe x 8 , Fe 3 GeTe 2 9 , and 1T-VSe 2 10-14 . Among them, the 1T-VSe 2 is of particular interest since the bulk 1T-VSe 2 has a van der Waals (vdW) nature, which can be easily exfoliated to few-layers thickness. This gives 1T-VSe 2 the advantage to be tailored and manipulated for nano spintronic devices at low cost.
However, the nature of the ground phase of 1T-VSe 2 is still under hot debate. Two groups have observed chargedensity-wave (CDW) ground states and concluded that magnetic order is absent in the monolayer limit due to the CDW suppression 15,16 . Wong et al. claimed that a spin frustrated phase was observed and the FM phase must be attributed to extrinsic factors 17 . Chua et al. 18 and Yu et al. 12 suggested that the observed FM is not intrinsic, but caused by defects. Nevertheless, Bonilla et al. 10 and many others [11][12][13][14] have presented strong experimental evidences for intrinsic 2D magnetism in monolayer 1T-VSe 2 , which also reported a NM to FM phase transition from bulk to the monolayer limit 19 . This is in contrast with other 2D magnetic materials, where the FM phase is more stable in the bulk system. In this paper, we study the ground-state properties of 1T-VSe 2 by first-principles calculations. Through a comprehensive study of the density of states (DOS) and band structures of the bulk and few-layers 1T-VSe 2 , we reveal that the monolayer system has the strongest FM instability due to the presence of Van Hove singularity (VHS) originated from saddle points at the Fermi level. We also find that in the few-layers case, the couplings of d z 2 orbitals between interlayer V atoms split the saddle points away from the Fermi level and weaken the FM instability. The strongest FM instability in the monolayer is confirmed by the largest energy difference between the NM and FM phases and also verified by using the phenomenological Stoner theory 20-23 . We thus conclude that the room-temperature FM order in the monolayer 1T-VSe 2 is intrinsic due to its unique electronic structures. Finally, we study the FM instability with respect to the interlayer distance d [see Fig. 1 is possible to tune the NM to FM phase transition in few-layers 1T-VSe 2 by enlarging the interlayer distance d. Our study provides an explanation to the origin of FM order in monolayer 1T-VSe 2 and also proposes a mechanism to tune the NM to FM phase transition in few-layers 1T-VSe 2 .

II. COMPUTATIONAL DETAILS
VSe 2 usually adopts the 2H and 1T structures. While the 2H-VSe 2 shows semiconducting behavior, the 1T-VSe 2 is a metal [24][25][26] and shows strong experimental evidence for FM ordering in the few-layers limit 10 . Different from the triangular prismatic crystal field in the 2H structure, 1T-VSe 2 has an octahedral crystal structure and belongs to P3m1 space group, where V atoms form a triangular lattice and each V atom occupies the center of the octahedron surrounded by six Se atoms, as shown in Figs. 1(a), 1(b) and 1(c). As a result, each layer of VSe 2 is stoichiometric 27 . The bulk crystal is composed of an AA stacking of VSe 2 sandwiches.
First-principles calculations based on density functional theory are carried out by using the Vienna ab initio simulation package (VASP) 28 . The Perdew-Burke-Ernzerhof functional 29 is employed to treat the exchangecorrelation interactions. The cutoff energy for wave function expansion is set to 500 eV. We use 19×19×9 and 21×21×1 Γ-centered k meshes to sample the Brillouin zone (BZ) in the bulk and slab calculations, respectively. Structures are optimized until the force on each atom is less than 0.001 eV/Å. A vacuum layer of 15Å is set to minimize artificial interactions between layers in the slab calculations. For the bulk calculations, the lattice constants a = b = 3.356Å, and c = 6.105Å are used 30 .

III. RESULTS AND DISCUSSION
We first calculate and plot the total and projected DOS of the NM bulk 1T-VSe 2 in Fig. 1(a), which are in good agreement with previous results [31][32][33][34] . The density at the Fermi level is about 2.9 states/eV, confirming its metallic nature. The projected DOS demonstrates that the states between −0.9 eV and 3.5 eV are mainly contributed by the V-3d orbitals. In an octahedron crystal field, the five 3d orbitals split into the lower t 2g and the upper e g manifolds, which mainly contribute to the DOS around −0.9 ∼ 1.4 eV and 1.9 ∼ 3.5 eV, respectively. Furthermore, due to the presence of a triangular field, the t 2g manifold splits into the lower a 1g (d z 2 ) and upper e g (d xy and d x 2 −y 2 ) orbitals. This picture is also verified from the projected band structures in Fig. 1(e), in which the d z 2 is the lowest 3d orbital that crosses the Fermi level and dominates the low-energy physics of the bulk 1T-VSe 2 .
We further plot the total and projected DOS of the NM monolayer 1T-VSe 2 in Fig. 2(a). Compared with the bulk DOS in Fig. 1(d), we notice that these two DOS plots share many similarities. For instance, they both have high densities around 1 eV and they both possess energy gaps at ∼1.4 eV. This is reasonable due to that 1T-VSe 2 is a layered vdW material. The interlayer coupling does not significantly alter the electronic structures. Nevertheless, a shape peak appears at the Fermi level E F in the monolayer case as shown in Fig. 2(a). The DOS at E F is about N (E F ) ≈ 6.4 state/eV, much higher than the bulk value. Such high DOS suggests the presence of VHS on the band structures. We thus plot the band structures along high symmetry k-path in Fig. 2(b) and observe that only the d z 2 band crosses the Fermi level. On the Γ − M path there is a maximum at Γ and on the Γ − K path a minimum at S appears. These characteristics indicate the presence of a saddle point. To show more details, we plot this band on the whole BZ in Fig. 2(c), where there are six saddle points on the Γ−K and Γ−K paths.  35 . Therefore, we have demonstrated that the high DOS and its divergent behavior are due to the presence of saddle-points VHS on the band structures.
We suggest that the VHS in monolayer 1T-VSe 2 may cause FM instability according to the phenomenological Stoner theory 20 , which states that the FM phase is favored when the Stoner criteria N ( f ) · I > 1 is satisfied.
Here N ( f ) is the DOS at the Fermi level in the NM state, and I is the Stoner parameter that measures the strength of the magnetic exchange interaction, which is related to the energy splitting between the spin-up and spin-down states in the FM phase via the following formulas 36 Here E 0 (k) is the energy of the NM phase, E σ (k) and n σ are the energy and number of electrons with spin σ (σ =↑ , ↓) in the FM phase, respectively. The total number of electrons is n = n ↑ + n ↓ . Since only the d z 2 band is responsible for the Stoner instability in monolayer 1T-VSe 2 , n ↑ and n ↓ can be estimated as 1. Therefore we have n = 2. Finally, the Stoner parameter I can be estimated as E ↓ (k) − E ↑ (k). Figure 3(a) presents the DOS of the spin-up and spindown states in the FM phase of the monolayer 1T-VSe 2 . By comparing with the NM results in Fig. 2(a), we observe that the sharp VHS peak splits into two peaks, which is a typical signature of the FM exchange interaction. We assume the exchange interaction is kindependent and use the energy difference of the VHS peaks to evaluate its average magnitude 36 , which gives I = 0.68 eV. Together with N (E F ) = 6.4 state/eV in the NM phase, we obtain a Stoner criterion N (E F ) · I = 4.3. This large value indicates a strong FM instability in the monolayer 1T-VSe 2 .
Recent experiments have shown that the monolayer 1T-VSe 2 exhibits FM order, while the bulk 1T-VSe 2 displays a NM property [10][11][12][13][14] . To study this transition, we show the evolution of the Stoner criterion N (E F ) · I with respect to the number of layers N in Fig. 3(b), from which a drastic decrease of N (E F ) · I from the mono- to the bilayer is observed, indicating the decrease of FM instability in the bilayer 1T-VSe 2 . We further plot the evolutions of N (E F ) and I with respect to N in Fig. 3(c), which clearly shows that the drastic decrease is due to the decrease of N (E F ) since I is insensitive to N [see the red curve in Fig. 3(c)]. We notice that when N ≥ 2, the Stoner criterion N (E F ) · I, density N (E F ), and Stoner parameter I fluctuate slightly around their saturated values. These results prove that when the system transforms from the monolayer to bulk, only the monolayer exhibits a strong FM instability. Our result well explains the recent experiment by Bonilla et al., where a strong FM signal has been detected in the monolayer while the bilayer has a significantly weak FM signal comparable with the bulk 10 . This trend is also manifested by the energy difference ∆E = E F M − E N M between the FM and NM phases as depicted in Fig. 3(d), from which we observe that the maximal energy difference occurs in the monolayer case, and a drastic decrease of ∆E takes place from the monolayer to the bilayer.
To understand the drastic decrease of N (E F ) from the mono to the bilayer, we plot the NM total DOS of bilayer 1T-VSe 2 in Fig. 4(a). Due to the vdW nature, the bilayer DOS is very similar with that of the monolayer, except that two peaks emerge near the Fermi level at E 1 = −0.03 and E 2 = 0.08 eV [see the inset in Fig. 4(a)]. These two peaks also originate from the saddle points on the band structures as shown in Fig. 4(b). We notice that two bands are crossing the Fermi level. They are contributed by the d z 2 orbitals of the two V atoms in the bilayer unit cell. The upper and lower d z 2 bands are anti-bonding and bonding states, respectively. The energy difference of these two d z 2 bands at the Γ is about 0.44 eV, which gives an estimation of the interlayer coupling strength. The VHS splitting at the S point is de-termined by the two minima on the Γ − K path, which is about 0.11 eV, consistent with the two peaks on the DOS in Fig. 4(a). Such splitting is larger than that in typical vdW materials [37][38][39] . This is because that the VHS peak in 1T-VSe 2 is mainly contributed by the d z 2 orbitals, whose lobes from interlayer V atoms are head-to-head aligned along the z-direction and form relatively strong ddσ bonds. As a result, the VHSs no longer present at the Fermi level, and the N (E F ) is significantly reduced. Finally, the Stoner criterion N (E F ) · I decreases severely, and the FM instability is weakened.
To summarize, the transition from the bulk NM phase to the monolayer FM phase in 1T-VSe 2 can be understood as follows. In the bulk system, the coupling of d z 2 orbital between interlayer V atoms splits the VHSs away from the Fermi level. Especially from the bilayer to the monolayer, the enhanced confinement effect eliminates the interlayer d z 2 coupling. Thus the VHSs are pushed to the Fermi level, which leads to a drastic enhancement of the N (E F ) and the Stoner criterion N (E F )·I. Eventually, this enhanced N (E F ) causes a strong FM instability in the monolayer. In other words, the confinement effect in the monolayer 1T-VSe 2 prevents the interlayer coupling between the d z 2 orbitals of V atom and pushes the saddle-point VHS at the Fermi level, which results in a large Stoner criterion and leads to a stable FM ground state. Our numerical results can well explain recent experiments [10][11][12][13][14] .
Based on this understanding, we expect that the magnetic property of a few-layers 1T-VSe 2 can be tuned by the interlayer distance d 40 [see Fig. 1(a)]. The evo-lution of the VHS splitting ∆E d = E 2 − E 1 between the two peaks in the bilayer 1T-VSe 2 with the ratio ε = (d − d 0 )/d 0 is shown in Fig. 4(c). Here d 0 = 3.067Å is the interlayer distance of the bulk. The corresponding evolution of N (E F ) with ε is also shown in this figure. We observe that ∆E d monotonically decreases with the increase of d. This means the ddσ bond is weakened when the lobes of d z 2 orbitals in adjacent layers are moving apart. During this process, the N (E F ) monotonically goes up, reflecting an enhanced Stoner FM instability. The enhancement of the Stoner instability is also confirmed by the energy differences ∆E between the FM and NM phases in Fig. 4(d). We find that ∆E also monotonically decreases with ε, which indicates that the FM phase becomes more and more stable when d increases. This effect provides a useful route to control the NM to FM transition in few-layers 1T-VSe 2 through enlarging the interlayer distance d. It also sheds light on tuning the Curie temperature of 1T-VSe 2 through nanoengineering. Further experimental studies are highly desirable to verify these conjectures.

IV. ACKNOWLEGMENTS
This work is supported by the Ministry of Science and Technology of China (No. 2018YFA0307000) and the National Natural Science Foundation of China (No. 11874022).