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

Junyi He,1 Q. Xie,1 and Gang Xu1, ∗

1Wuhan National High Magnetic Field Center and School of Physics,Huazhong University of Science and Technology, Wuhan 430074, China

(Dated: August 24, 2020)

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

I. INTRODUCTION

Ferromagnetic order in two-dimensional (2D) materi-als is a highly desirable property that provides a newphysical degree of freedom to manipulate spin behav-iors in spintronic devices1,2. Previously, magnetism in2D was mainly realized through depositing films ontomagnetic substrates, magnetic atoms adsorption, or dop-ing3–5. The shortcomings of these methods are obvious:(i) one does not have an ideal 2D system from depositingand it is impractical to integrate with spintronic devices,and (ii) disorder effects make the electronic propertieshard to design. Due to these drawbacks, 2D materialswith intrinsic magnetic order have been actively pursued.

The CrI36 and Cr2Ge2Te6

7 are the first two experimen-tally reported 2D materials exhibiting long-range ferro-magnetic (FM) order with the Curie temperatures Tc ∼45 K and 30 K, respectively6,7. These discoveries havestimulated numerous research interests on 2D magneticmaterials. Very recently, several materials with higher Tchave been experimentally and theoretically explored, in-cluding MnSex

8, Fe3GeTe29, and 1T-VSe2

10–14. Amongthem, the 1T-VSe2 is of particular interest since the bulk1T-VSe2 has a van der Waals (vdW) nature, which canbe easily exfoliated to few-layers thickness. This gives1T-VSe2 the advantage to be tailored and manipulatedfor nano spintronic devices at low cost.

However, the nature of the ground phase of 1T-VSe2 isstill under hot debate. Two groups have observed charge-density-wave (CDW) ground states and concluded thatmagnetic order is absent in the monolayer limit due tothe CDW suppression15,16. Wong et al. claimed that aspin frustrated phase was observed and the FM phasemust be attributed to extrinsic factors17. Chua et al.18

and Yu et al.12 suggested that the observed FM is notintrinsic, but caused by defects. Nevertheless, Bonilla etal.10 and many others11–14 have presented strong exper-imental evidences for intrinsic 2D magnetism in mono-layer 1T-VSe2, which also reported a NM to FM phasetransition from bulk to the monolayer limit19. This isin contrast with other 2D magnetic materials, where theFM phase is more stable in the bulk system.

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FIG. 1. (a) The crystal structure of bulk 1T-VSe2. d denotesthe interlayer distance. (b) and (c) are the top view and sideview of the VSe2 monolayer, respectively. (d) The total andprojected DOS of bulk 1T-VSe2 in the NM phase. (e) NMband structures of bulk VSe2 with V-3d orbitals projections.

In this paper, we study the ground-state propertiesof 1T-VSe2 by first-principles calculations. Through acomprehensive study of the density of states (DOS) andband structures of the bulk and few-layers 1T-VSe2, wereveal that the monolayer system has the strongest FMinstability 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 ofdz2 orbitals between interlayer V atoms split the saddlepoints away from the Fermi level and weaken the FM in-stability. The strongest FM instability in the monolayeris confirmed by the largest energy difference between theNM and FM phases and also verified by using the phe-nomenological Stoner theory20–23. We thus conclude thatthe room-temperature FM order in the monolayer 1T-VSe2 is intrinsic due to its unique electronic structures.Finally, we study the FM instability with respect to theinterlayer distance d [see Fig. 1(a)] and predict that it

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is possible to tune the NM to FM phase transition infew-layers 1T-VSe2 by enlarging the interlayer distanced. Our study provides an explanation to the origin of FMorder in monolayer 1T-VSe2 and also proposes a mecha-nism to tune the NM to FM phase transition in few-layers1T-VSe2.

II. COMPUTATIONAL DETAILS

VSe2 usually adopts the 2H and 1T structures. Whilethe 2H-VSe2 shows semiconducting behavior, the 1T-VSe2 is a metal24–26 and shows strong experimental ev-idence for FM ordering in the few-layers limit10. Differ-ent from the triangular prismatic crystal field in the 2Hstructure, 1T-VSe2 has an octahedral crystal structureand belongs to P 3m1 space group, where V atoms forma triangular lattice and each V atom occupies the centerof the octahedron surrounded by six Se atoms, as shownin Figs. 1(a), 1(b) and 1(c). As a result, each layer ofVSe2 is stoichiometric27. The bulk crystal is composedof an AA stacking of VSe2 sandwiches.

First-principles calculations based on density func-tional theory are carried out by using the Vienna abinitio simulation package (VASP)28. The Perdew-Burke-Ernzerhof functional29 is employed to treat the exchange-correlation interactions. The cutoff energy for wave func-tion expansion is set to 500 eV. We use 19×19×9 and21×21×1 Γ-centered k meshes to sample the Brillouinzone (BZ) in the bulk and slab calculations, respectively.Structures are optimized until the force on each atomis less than 0.001 eV/A. A vacuum layer of 15 A is setto minimize artificial interactions between layers in the

slab calculations. For the bulk calculations, the latticeconstants a = b = 3.356 A, and c = 6.105 A are used30.

III. RESULTS AND DISCUSSION

We first calculate and plot the total and projected DOSof the NM bulk 1T-VSe2 in Fig. 1(a), which are in goodagreement with previous results31–34. The density at theFermi level is about 2.9 states/eV, confirming its metallicnature. The projected DOS demonstrates that the statesbetween −0.9 eV and 3.5 eV are mainly contributed bythe V-3d orbitals. In an octahedron crystal field, thefive 3d orbitals split into the lower t2g and the upper egmanifolds, which mainly contribute to the DOS around−0.9 ∼ 1.4 eV and 1.9 ∼ 3.5 eV, respectively. Further-more, due to the presence of a triangular field, the t2gmanifold splits into the lower a1g (dz2) and upper e′g(dxy and dx2−y2) orbitals. This picture is also verifiedfrom the projected band structures in Fig. 1(e), in whichthe dz2 is the lowest 3d orbital that crosses the Fermilevel and dominates the low-energy physics of the bulk1T-VSe2.

We further plot the total and projected DOS of theNM monolayer 1T-VSe2 in Fig. 2(a). Compared withthe bulk DOS in Fig. 1(d), we notice that these two DOSplots share many similarities. For instance, they bothhave high densities around 1 eV and they both possessenergy gaps at ∼1.4 eV. This is reasonable due to that1T-VSe2 is a layered vdW material. The interlayer cou-pling does not significantly alter the electronic structures.Nevertheless, a shape peak appears at the Fermi level EFin the monolayer case as shown in Fig. 2(a). The DOS atEF is aboutN(EF ) ≈ 6.4 state/eV, much higher than thebulk value. Such high DOS suggests the presence of VHSon the band structures. We thus plot the band structuresalong high symmetry k-path in Fig. 2(b) and observe thatonly the dz2 band crosses the Fermi level. On the Γ−Mpath there is a maximum at Γ and on the Γ − K patha minimum at S appears. These characteristics indicatethe presence of a saddle point. To show more details, weplot this band on the whole BZ in Fig. 2(c), where thereare six saddle points on the Γ−K and Γ−K ′ paths. Fig-ure 2(d) zooms in the band structures around the saddlepoint S. The red and blue solid lines are the band dis-persions along Γ−K and its orthogonal directions, whichrepresent hole-like and electron-like dispersions, respec-tively. These dispersions evidently show the topology ofa saddle point35. Therefore, we have demonstrated thatthe high DOS and its divergent behavior are due to thepresence of saddle-points VHS on the band structures.

We suggest that the VHS in monolayer 1T-VSe2 maycause FM instability according to the phenomenologicalStoner theory20, which states that the FM phase is fa-vored when the Stoner criteria N(εf ) · I > 1 is satisfied.HereN(εf ) is the DOS at the Fermi level in the NM state,and I is the Stoner parameter that measures the strengthof the magnetic exchange interaction, which is related to

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FIG. 3. Results of the monolayer 1T-VSe2 in the FM phase.(a) The spin polarized DOS. E↑ and E↓ denote the energyshift of the VHS peaks, which are used to evaluate the energydifference between the spin-up and spin-down states. (b) Theevolution of the Stoner criterion with respect to the numberof layers N . (c) The evolutions of DOS at the Fermi levelN(EF ) and the Stoner parameter I with respect to N . (d)The energy difference ∆E = EFM −ENM as a function of N .

the energy splitting between the spin-up and spin-downstates in the FM phase via the following formulas36

E↑(k) = E0(k)− I n↑n,

E↓(k) = E0(k) + In↓n.

(1)

Here E0(k) is the energy of the NM phase, Eσ(k) and nσare the energy and number of electrons with spin σ (σ =↑, ↓) in the FM phase, respectively. The total number ofelectrons is n = n↑ + n↓. Since only the dz2 band isresponsible for the Stoner instability in monolayer 1T-VSe2, n↑ and n↓ can be estimated as 1. Therefore wehave n = 2. Finally, the Stoner parameter I can beestimated as E↓(k)− E↑(k).

Figure 3(a) presents the DOS of the spin-up and spin-down states in the FM phase of the monolayer 1T-VSe2.By comparing with the NM results in Fig. 2(a), we ob-serve that the sharp VHS peak splits into two peaks,which is a typical signature of the FM exchange in-teraction. We assume the exchange interaction is k-independent and use the energy difference of the VHSpeaks to evaluate its average magnitude36, which givesI = 0.68 eV. Together with N(EF ) = 6.4 state/eV in theNM phase, we obtain a Stoner criterion N(EF ) · I = 4.3.This large value indicates a strong FM instability in themonolayer 1T-VSe2.

Recent experiments have shown that the monolayer1T-VSe2 exhibits FM order, while the bulk 1T-VSe2 dis-plays a NM property10–14. To study this transition, weshow the evolution of the Stoner criterion N(EF ) ·I withrespect to the number of layers N in Fig. 3(b), fromwhich a drastic decrease of N(EF ) · I from the mono-

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FIG. 4. Results of the bilayer 1T-VSe2. (a) and (b) arethe NM DOS and band structures, respectively. The insetof (a) shows the detail of the two peaks. (c) The blue curveshows the evolution of the two peaks’ energy splitting ∆Ed

with the ratio of interlayer distance ε. The red curve showsthe N(EF ) as a function of ε. (d) shows the energy difference∆E = EFM − ENM as a function of ε.

to the bilayer is observed, indicating the decrease of FMinstability in the bilayer 1T-VSe2. We further plot theevolutions of N(EF ) and I with respect to N in Fig. 3(c),which clearly shows that the drastic decrease is due to thedecrease of N(EF ) since I is insensitive to N [see the redcurve in Fig. 3(c)]. We notice that when N ≥ 2, theStoner criterion N(EF ) · I, density N(EF ), and Stonerparameter I fluctuate slightly around their saturated val-ues. These results prove that when the system transformsfrom the monolayer to bulk, only the monolayer exhibitsa strong FM instability. Our result well explains the re-cent experiment by Bonilla et al., where a strong FMsignal has been detected in the monolayer while the bi-layer has a significantly weak FM signal comparable withthe bulk10. This trend is also manifested by the energydifference ∆E = EFM −ENM between the FM and NMphases as depicted in Fig. 3(d), from which we observethat the maximal energy difference occurs in the mono-layer case, and a drastic decrease of ∆E takes place fromthe monolayer to the bilayer.

To understand the drastic decrease of N(EF ) from themono to the bilayer, we plot the NM total DOS of bi-layer 1T-VSe2 in Fig. 4(a). Due to the vdW nature, thebilayer DOS is very similar with that of the monolayer,except that two peaks emerge near the Fermi level atE1 = −0.03 and E2 = 0.08 eV [see the inset in Fig. 4(a)].These two peaks also originate from the saddle points onthe band structures as shown in Fig. 4(b). We noticethat two bands are crossing the Fermi level. They arecontributed by the dz2 orbitals of the two V atoms inthe bilayer unit cell. The upper and lower dz2 bands areanti-bonding and bonding states, respectively. The en-ergy difference of these two dz2 bands at the Γ is about0.44 eV, which gives an estimation of the interlayer cou-pling strength. The VHS splitting at the S point is de-

4

termined by the two minima on the Γ−K path, which isabout 0.11 eV, consistent with the two peaks on the DOSin Fig. 4(a). Such splitting is larger than that in typicalvdW materials37–39. This is because that the VHS peakin 1T-VSe2 is mainly contributed by the dz2 orbitals,whose lobes from interlayer V atoms are head-to-headaligned along the z-direction and form relatively strongddσ bonds. As a result, the VHSs no longer present atthe Fermi level, and the N(EF ) is significantly reduced.Finally, the Stoner criterion N(EF ) ·I decreases severely,and the FM instability is weakened.

To summarize, the transition from the bulk NM phaseto the monolayer FM phase in 1T-VSe2 can be under-stood as follows. In the bulk system, the coupling of dz2orbital between interlayer V atoms splits the VHSs awayfrom the Fermi level. Especially from the bilayer to themonolayer, the enhanced confinement effect eliminatesthe interlayer dz2 coupling. Thus the VHSs are pushedto the Fermi level, which leads to a drastic enhancementof the N(EF ) and the Stoner criterion N(EF )·I. Eventu-ally, this enhanced N(EF ) causes a strong FM instabilityin the monolayer. In other words, the confinement effectin the monolayer 1T-VSe2 prevents the interlayer cou-pling between the dz2 orbitals of V atom and pushes thesaddle-point VHS at the Fermi level, which results in alarge Stoner criterion and leads to a stable FM groundstate. Our numerical results can well explain recent ex-periments10–14.

Based on this understanding, we expect that the mag-netic property of a few-layers 1T-VSe2 can be tunedby the interlayer distance d40 [see Fig. 1(a)]. The evo-

lution of the VHS splitting ∆Ed = E2 − E1 betweenthe two peaks in the bilayer 1T-VSe2 with the ratioε = (d− d0)/d0 is shown in Fig. 4(c). Here d0 = 3.067 Ais the interlayer distance of the bulk. The correspond-ing evolution of N(EF ) with ε is also shown in this fig-ure. We observe that ∆Ed monotonically decreases withthe increase of d. This means the ddσ bond is weak-ened when the lobes of dz2 orbitals in adjacent layers aremoving apart. During this process, the N(EF ) monoton-ically goes up, reflecting an enhanced Stoner FM insta-bility. The enhancement of the Stoner instability is alsoconfirmed by the energy differences ∆E between the FMand NM phases in Fig. 4(d). We find that ∆E also mono-tonically decreases with ε, which indicates that the FMphase becomes more and more stable when d increases.This effect provides a useful route to control the NM toFM transition in few-layers 1T-VSe2 through enlargingthe interlayer distance d. It also sheds light on tuning theCurie temperature of 1T-VSe2 through nanoengineering.Further experimental studies are highly desirable to ver-ify these conjectures.

IV. ACKNOWLEGMENTS

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

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