Weyl Fermions in VI3 Monolayer

We report the presence of a Weyl fermion in VI3 monolayer. The material shows a sandwich-like hexagonal structure and stable phonon spectrum. It has a half-metal band structure, where only the bands in one spin channel cross the Fermi level. There are three pairs of Weyl points slightly below the Fermi level in spin-up channel. The Weyl points show a clean band structure and are characterized by clear Fermi arcs edge state. The effects of spin-orbit coupling, electron correlation, and lattice strain on the electronic band structure were investigated. We find that the half-metallicity and Weyl points are robust against these perturbations. Our work suggests VI3 monolayer is an excellent Weyl half-metal.

Up to now, a large number of WSMs have been reported, and some of which have been confirmed in experiments such as TaAs (Lv et al., 2015), NbP (Shekhar et al., 2015), and NbAs . These examples are all three-dimensional (3D) non-magnetic materials. Recently, WSMs in two-dimensional (2D) materials and magnetic materials have received increasing interests. For 2D WSMs, the interest arises from the promising applications in spintronic nanodevices (You et al., 2019). For magnetic WSMs, the interest comes from the novel interplay between the non-trivial band topology and the magnetic ordering (Xu et al., 2011;Kübler and Felser, 2016;Wang et al., 2016;Chen et al., 2019;He et al., 2019;Jin et al., 2020;Meng et al., 2020a,b,c). Recently, the VI 3 material (Kong et al., 2019;Tian et al., 2019;Huang et al., 2020;Long et al., 2020;Zhang et al., 2020), both in 3D and 2D, has attracted great attention. The 3D VI 3 material is a ferromagnetic insulator, which is a layered material (Kong et al., 2019). In addition, 3D VI 3 compound has a R3 phase at room temperature, and experimental evidence suggests that it may undergo a structure phase transition at 78 K (Huang et al., 2020). Importantly, 2D VI 3 had been proved by Long et al. (2020) to be a ferromagnetic half-metal and provides an excellent candidate material for electronic devices. In this work, we report that VI 3 monolayer is an excellent 2D WSM. We have systematically investigated the stability, magnetism, and band topology of VI 3 monolayer. We find VI 3 monolayer is dynamically stable and naturally has the ferromagnetic ordering. The band structure suggests VI 3 monolayer is a half-metal, which holds fully spin-polarized conducting electrons. Especially, there exists a band crossing near the Fermi level, which forms three pairs of Weyl fermions in the spin-up band structure. We have further investigated the effects of spin-orbit coupling (SOC), electron correlation, and lattice strain on the electronic band structure. In addition, the non-trivial edge states for the Weyl points are clearly identified. These results suggest VI 3 monolayer can serve as a good platform to investigate Weyl states in 2D.

COMPUTATIONAL METHODS AND DETAILS
The first-principles calculations in this work are performed by using the Vienna ab initio Simulation Package (Blochl, 1994;Kresse and Joubert, 1999). The exchange-correlation potential is adopted by the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof functional (Perdew et al., 1996). For the crystal structure of VI 3 monolayer, we built a vacuum with the thickness >18 Å to avoid potential interactions between layers. The cutoff energy is set as 500 eV. The Brillouin zone is sampled by a Monkhorst-Pack k-mesh with size of 15 × 15 × 1. To account for the Coulomb interaction, the GGA + U method is applied during our calculations (Anisimov et al., 1991). For the V-3d orbitals, the U value is chosen as 3eV. The phonon spectra are calculated by using the PHONOPY code (Togo et al., 2008).

RESULTS AND DISCUSSION
Before studying the structure and electronic band of the monolayer VI 3 , we want to point out that the layered compound VI 3 has already been synthesized by chemical vapor transport method experimentally. The specific synthesis process can be found in Kong et al. (2019). In addition, monolayer VI 3 is very promising to be exfoliated from its bulk phase (Miro et al., 2014).
Next, we study the structure of single-layer VI 3 . Figures 1A,B show the top and side views of the geometric structure of VI 3 monolayer. Figure 1C shows the Brillouin zone (BZ) of monolayer VI3. In the structure, each V atom bonds with six I atoms, forming the hexagonal structure. From the side view, we can observe that the material has a triple-layered form, with one V layer sandwiched by two I layers. The circled regions in (A) and (B) show the unit cell of VI 3 monolayer, which contains two V atoms and six I atoms. The Cr 2 C compound shows a hexagonal structure with space group P3M1. The lattice constant of VI 3 monolayer is a = b = 7.13 Å. The bond length of V-I is 2.80 Å, and that of I-I is 4.13 Å. These values are very close to the reported results of He et al. (2016). We have calculated the phonon spectrum of VI 3 monolayer, as displayed in Figure 1D. We find no negative frequency phonons in all the highly symmetric k paths. This suggests that the VI 3 monolayer can be stable.
Before studying the band structure of VI 3 monolayer, we first verify its magnetic ground state. In VI 3 monolayer, the magnetic moments are mainly contributed by the 3d transition element V. Here, we consider three potential magnetization directions of V moment, including [001], [010], and [100]. Our calculation results show that the [001] magnetization direction has the lowest energy. Then, in the [001] direction, we considered four magnetic configurations including ferromagnet (FM), Néel antiferromagnet (AFM), stripe AFM, and zigzag AFM. Our calculation results show that the energy of FM is lower than that of other magnetic structures in VI 3 monolayer. The total magnetic moment is 4 µ B per unit cell, which is almost contributed by the V atoms. In addition, we have calculated the exchange energy E of VI 3 monolayer, which is approximately 28 meV. Then, we can estimate the Curie temperature (T c ) according to the following equation: In (1), the parameter "C" represents the number of magnetic atoms in unit cell, and "k B " represents the Boltzmann constant. We calculated that the Curie temperature T c value of VI 3 monolayer is 106 K, which is comparable with the Monte Carlo simulations (98 K) (He et al., 2016).
Here, we discuss the electronic band structure of VI 3 monolayer. At first, we did not consider SOC in the calculations. The band structures are shown in Figures 2A,B. In the spinup band structure, we find one band crosses the Fermi level, manifesting a metallic signature (Figure 2A). In the spin-down band structure, we can find a band gap around the Fermi level, manifesting the insulating signature ( Figure 2B). Therefore, VI 3 monolayer is in fact a half-metal. In particular, the half-metal gap is about 2.98 eV (Figure 2B), which is much layer than previously reported half-metals in 2D including YN 2 (1.35eV) (Liu et al., 2017) and Na 2 C (0.77eV) monolayer (Ji et al., 2018). In the spinup band structure, we notice there shows a band crossing point (P) in the M-K path, slightly below the Fermi level ( Figure 2B). By checking the density of states, we find the states near the Fermi level mostly come from the d orbitals of V atom. Figure 3A shows the orbital-projected band structure near the crossing point P. We can observe the band inversion of the two bands, suggesting the non-trivial band topology in VI 3 monolayer. Then we take into account the SOC effect in the calculation. The comparison of band structure between SOC and without SOC is shown in Figures 3B,C. We can find that the band crossing retains under SOC, even though its position has slightly changed. Without SOC, the crossing point locates at 0.04 eV below the Fermi level; under SOC, it locates at 0.08 eV below the Fermi level. Because the band crossing happens in two bands, the band crossing in fact forms Weyl points. To be noted, in VI 3 monolayer, the time reversal is broken, but the inversion symmetry is retained; hence, the Weyl points are time-reversal-breaking Weyl points.
In Figure 3D, we show the 3D plotting of band structure near the Weyl point. We can find that the Weyl cone is titled. Because of the preserved inversion symmetry, there are in total three pairs of such Weyl points in the system. From the symmetry analysis, the Weyl points in monolayer VI 3 are protected by the C 3v symmetry. It is worth noticing that Weyl points have many special physical phenomena, such as modified anomalous Hall conductivity, direction-dependent chiral anomaly, momentum space Klein tunneling (Koshino, 2016;O'Brien et al., 2016;Yu et al., 2016;Zyuzin and Tiwari, 2016). However, the Weyl fermions have been rarely found in 2D materials. Therefore,  the VI 3 monolayer reported here can be a good platform to study the Weyl fermions in 2D. In addition, in Figure 3E, the orbital projection is performed on (010) surface, we have also identified the Fermi arc edge states of the Weyl points, as shown in Figure 3F. This further verifies the non-trivial band topology in VI 3 monolayer.
Finally, we discuss the effects of electron correlation and lattice strain on the half-metal band structure and the Weyl points. In Figure 4A, we show the positions of the conduction band minimum and the valence band maximum with shifting the Uvalues of V atom from 0 to 6 eV. The results show that VI 3 monolayer is always a ferromagnetic half-metal, and the spindown band gap will increase with increasing the U-values. For the Weyl points, we find they can exist when U-values are at 2-4 eV. Figure 4B shows the positions of the Weyl points at different U-values. Similarly, in Figures 4C,D, we show the strain effects on the half-metal band structure and the Weyl points. We can find that the half-metal band structure can retain from 4% compressive stain to 4% tensile strain ( Figure 4C). Meanwhile, as shown in Figure 4D, we find the Weyl points can exist from 1% compressive stain to 2% tensile strain ( Figure 4D). These results show that the half-metal band structure and the Weyl points are in some degree robust against electron correlation effects and lattice strain.

SUMMARY
We have reported the Weyl fermion in 2D VI 3 monolayer. The phonon spectrum suggests VI 3 monolayer is dynamically stable. We have verified that VI 3 monolayer has the ferromagnetic ground state. In the ground state, we find VI 3 monolayer has a half-metal band structure with the half-metallic gap as large as 2.98 eV; thus, the conducting electrons can be fully spin-polarized. Very interestingly, VI 3 monolayer shows three pairs of Weyl points near the Fermi level, locating in the spin-up band structure. Importantly, the three pairs of Weyl points show clear Fermi arcs on the edge. Moreover, we verify that the half-metal band structure and the Weyl points in VI 3 monolayer are robust against proper electron correlation effects and lattice strain. These properties make VI 3 monolayer have promising applications in spintronic nanodevices.

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

AUTHOR CONTRIBUTIONS
This project was conceived by XZ and GL. TJ, HZ, and CL completed theoretical model and first principles calculations with the help from XZ and XD. TJ and WM performed the data analysis and designed the manuscript with the assistances from XZ and GL. This manuscript was written by TJ and XZ with important inputs from GL and XD. This project was supervised by GL. All authors discussed the results and commented on the manuscript.