Competing magnetic and nonmagnetic states in monolayer ${\mathrm{VSe}}_2$ with charge density wave

The field of two-dimensional ferromagnets has been reinvigorated by the discovery of ${\mathrm{VSe}}_2$ monolayer grown on van der Waals substrates, which is reported to be ferromagnetic with a Curie point higher than 330 K. However, the ferromagnetic and nonmagnetic states of pristine monolayer ${\mathrm{VSe}}_2$ are highly debated. Here, employing density functional theory, Wannier function calculations, and the band unfolding method, we explore the electronic structure of monolayer ${\mathrm{VSe}}_2$ with a $\sqrt{3}\times \sqrt{7}$ charge density wave (CDW). Certain qualitative aspects of the calculated unfolded band dispersion and unfolded Fermi surface of monolayer ${\mathrm{VSe}}_2$ with the $\sqrt{3}\times \sqrt{7}$ CDW in the nonmagnetic state agree well with previous angle-resolved photoemission spectroscopy results, albeit with uncertainty about whether these experiments probed single or multiple domains. Specifically, we find that an isolated CDW domain naturally induces a strong breaking of the threefold symmetry of the electronic structure. In addition we find that, relative to the undistorted structure, the CDW structure shows a strong competition between nonmagnetic and various magnetic states, with an energy difference less than 5 meV/f.u. For the CDW structure in the antiferromagnetic state, the band dispersions and Fermi surface are similar to those in the nonmagnetic state, while the unfolded bands of the ferromagnetic CDW state display a sizable exchange splitting. These results indicate the possibility of various antiferromagnetic fluctuations in ${\mathrm{VSe}}_2$ to coexist and compete with ferromagnetic order and the experimentally reported CDW order. Our calculations build insights for exploring the interplay between magnetism and CDW behaviors more generally in transition metal dichalcogenides.

Figure 1

(a) The geometry of the normal, i.e., undistorted, $\sqrt{3}\times \sqrt{7}$ monolayer ${\mathrm{VSe}}_2$ supercell spanned by $A1=2a1+a2$ and $A2=-a1+2a2$, built from the primitive cell spanned by $a1$ and $a2$. (b) The relaxed structure of ${\mathrm{VSe}}_2$ ($\sqrt{3}\times \sqrt{7}$) supercell with CDW. The red and green balls in (a) and (b) denote the V and Se atoms, respectively. The light red and cyan balls in (b) denotes the normal structure, i.e., the undistorted structure. The non-spin-polarized (c) unfolded band structures, (d) density of states, and (e) Fermi surface of the normal monolayer ${\mathrm{VSe}}_2 \sqrt{3}\times \sqrt{7}$ supercell. The red (blue) color represents the V $d$ (Se $p$) character. The Fermi level is at 0 eV.

Figure 2

The non-spin-polarized unfolded (a) band structure and (b) Fermi surface of the monolayer ${\mathrm{VSe}}_2 \sqrt{3}\times \sqrt{7}$ CDW structure in the nonmagnetic state. The red (blue) color represents the V $d$ (Se $p$) character. The double sided black arrow in (a) indicates the CDW gap. The Fermi level is at 0 eV. The high symmetry points in the unfolded band structure are given by $M=(0,0.5,0)$ and $K=(-0.333,0.667,0)$.

Figure 3

(a) The frustrated antiferromagnetic order in the ${\mathrm{VSe}}_2 \sqrt{3}\times \sqrt{7}$ supercell. The red and blue balls denote V atoms in the spin-up and spin-down states, respectively. (b) The energy of the ${\mathrm{VSe}}_2 \sqrt{3}\times \sqrt{7}$ normal supercell and CDW structure in different magnetic states are denoted in blue and red respectively. The relevant antiferromagnetic states in (b) are displayed in (a) and (c). The energies in (b) are normalized to the energy of non-spin-polarized state in normal structure.

Figure 4

The (a) spin up and (b) spin down unfolded bands in the monolayer ${\mathrm{VSe}}_2 \sqrt{3}\times \sqrt{7}$ CDW structure with ferromagnetic order. The red (blue) color in the band structure represents the V $d$ (Se $p$) character. The Fermi level is at 0 eV. The high symmetry points in the unfolded band structure are given by $M=\left(0,0.5,0\right)$ and $K=(-0.333,0.667,0)$.

Figure 5

The spin up/down unfolded (a) band structure and (b) Fermi surface in the monolayer ${\mathrm{VSe}}_2 \sqrt{3}\times \sqrt{7}$ CDW structure with the antiferromagnetic order of AFM-B2, defined in Fig. 3. The red (blue) color in the band structures represents the V $d$ (Se $p$) character. The Fermi level is at 0 eV. The high symmetry points in the unfolded band structure are given by $M=(0,0.5,0)$ and $K=(-0.333,0.667,0)$.