Dynamic instabilities in strongly correlated ${\text{VSe}}_2$ monolayers and bilayers

With the emergence of graphene and other two-dimensional (2D) materials, transition-metal dichalcogenides have been investigated intensely as potential 2D materials using experimental and theoretical methods. ${\mathrm{VSe}}_2$ is an especially interesting material since its bulk modification exhibits a charge-density wave (CDW), the CDW is retained even for few-layer nanosheets, and monolayers of ${\mathrm{VSe}}_2$ are predicted to be ferromagnetic. In this work, we show that electron correlation has a profound effect on the magnetic properties and dynamic stability of ${\mathrm{VSe}}_2$ monolayers and bilayers. Including a Hubbard-$U$ term in the density-functional-theory calculations strongly affects the magnetocrystalline anisotropy in the $1T-{\mathrm{VSe}}_2$ structure while leaving the $2H$-polytype virtually unchanged. This demonstrates the importance of electronic correlations for the electrical and magnetic properties of $1T-{\mathrm{VSe}}_2$. The Hubbard-$U$ term changes the dynamic stability and the presence of imaginary modes of ferromagnetic $1T-{\mathrm{VSe}}_2$ while affecting only the amplitudes in the nonmagnetic phase. The Fermi surface of nonmagnetic $1T-{\mathrm{VSe}}_2$ allows for nesting along the CDW vector, but it plays no role in ferromagnetic $1T-{\mathrm{VSe}}_2$. Following the eigenvectors of the soft modes in nonmagnetic $1T-{\mathrm{VSe}}_2$ monolayers yields a CDW structure with a $4\times 4$ supercell and Peierls-type distortion in the atomic positions and electronic structure. The magnetic order indicates the potential for spin-density-wave structures.

Figure 1

Structures of monolayer ${\mathrm{VSe}}_2$ with (a) octahedrally coordinated V as in $1T-{\mathrm{VSe}}_2$ and (b) trigonal-prismatically coordinated V as in the $2H-{\mathrm{VSe}}_2$ polymorph. Vanadium atoms are shown in light yellow and selenium atoms in dark green.

Figure 2

(a) Energy difference $\mathrm{\Delta }E$ between $1T$ and $2H-{\mathrm{VSe}}_2$ monolayers as a function of $U_{\text{eff}}$, exchange correlation, and van der Waals functional. Positive $\mathrm{\Delta }E$ indicates that $2H$ is more stable. (b) Magnetization $m$ of monolayer $1T-{\mathrm{VSe}}_2$ as a function of $U_{\text{eff}}$, exchange correlation, and van der Waals functional. (c) In-plane lattice parameters $a$ of monolayer $1T-{\mathrm{VSe}}_2$. The gray shades represent the range of experimental values found for ferecrystals.

Figure 3

Spin densities for ${\mathrm{VSe}}_2$ layers. (a) $1T-{\mathrm{VSe}}_2$ monolayer with ferromagnetic (FM) spin structure. (b) $2H-{\mathrm{VSe}}_2$ monolayer with FM spin structure. (c) $1T-{\mathrm{VSe}}_2$ monolayer with antiferromagnetic (AFM) spin orientation. (d)–(g) $1T-{\mathrm{VSe}}_2$ bilayer with AFM ordering (AFM 1–AFM 4). For $2H-{\mathrm{VSe}}_2$, AFM 3 and AFM 4 are identical. For AFM structures, light red and dark blue spin densities denote opposite spin orientations. The isosurface values are set to $0.01e/a_0^3$, where $a_0$ is the Bohr radius.

Figure 4

Spin-polarized band structures for $1T-{\mathrm{VSe}}_2$ (left) and $2H-{\mathrm{VSe}}_2$ (right) layers with $U_{\text{eff}}=\text{1.0}\text{eV}$. (a), (b) Ferromagnetic monolayer; (c), (d) ferromagnetic bilayer; (e), (f) bilayer with AFM 1 structure. Solid blue lines correspond to majority- and red dashed lines to minority-spin bands.

Figure 5

Orbital resolved majority-spin (top) and minority-spin (bottom) band structures of monolayer ${\mathrm{VSe}}_2$. (a) $2H-{\mathrm{VSe}}_2$ with $U_{\text{eff}}=\text{1.0}\text{eV}$; (b) $1T-{\mathrm{VSe}}_2$ with $U_{\text{eff}}=\text{0}\text{eV}$; and (c) $1T-{\mathrm{VSe}}_2$ with $U_{\text{eff}}=\text{1.0}\text{eV}$.

Figure 6

Fermi surfaces of ferromagnetic $1T-{\mathrm{VSe}}_2$ monolayers for $U_{\text{eff}}=\text{0}$ and $\text{1.0}\text{eV}$. Solid blue surfaces are from the majority-spin bands and red dashed surfaces are from the minority-spin bands. The edges of the Brillouin zone are shown by black solid lines, and the edges of the irreducible Brillouin zone are shown by black dashed lines.

Figure 7

Angular dependence of the magnetocrystalline anisotropy energy (MAE) with polar angle for monolayer ${\mathrm{VSe}}_2$. $0^{\circ }$ points along the positive $z$ axis.

Figure 8

Phonon-dispersion curves for spin-polarized $1T-{\mathrm{VSe}}_2$ (left) and $2H-{\mathrm{VSe}}_2$ (right) layers. (a), (b) Monolayer with $U_{\text{eff}}=\text{0}\text{eV}$; (c), (d) monolayer with $U_{\text{eff}}=\text{1.0}\text{eV}$; and (e), (f) bilayer with $U_{\text{eff}}=\text{1.0}\text{eV}$.

Figure 9

Phonon-dispersion curves for nonmagnetic $1T-{\mathrm{VSe}}_2$ (left) and $2H-{\mathrm{VSe}}_2$ (right) layers. (a), (b) Monolayer with $U_{\text{eff}}=\text{0}\text{eV}$; (c), (d) monolayer with $U_{\text{eff}}=\text{1.0}\text{eV}$; and (e), (f) bilayer with $U_{\text{eff}}=\text{1.0}\text{eV}$.

Figure 10

Fermi surface of a non-spin-polarized $1T-{\mathrm{VSe}}_2$ monolayer. The edges of the Brillouin zone are shown by black solid lines, and the edges of the irreducible Brillouin zone are shown by black dashed lines. Nesting vectors are shown in gray.

Figure 11

(a) Unit cell of the most stable distorted structure of monolayer $1T-{\mathrm{VSe}}_2$. The spin densities around the V atoms are shown in light blue and dark purple for the majority and minority spins, respectively. V atoms are dark green and Se atoms are light yellow. The isosurface values are set to $0.01e/a_0^3$, where $a_0$ is the Bohr radius. (b) Detailed view of the chains of hexagons and triangles with V-V distances given in angstroms. Distances of less than 3.30, 3.35, and 3.40 Å are shown in blue, orange, and brown, respectively. Larger distances are shown by black dashed lines. Se atoms are omitted for clarity. (c) Top view onto a $3\times 3$ supercell of the distorted $1T-{\mathrm{VSe}}_2$ monolayer. Connected V atoms are less than 3.4 Å apart. Small area triangular V clusters are highlighted blue (see the text). (d) Band structures of undistorted nonmagnetic and distorted $1T-{\mathrm{VSe}}_2$ monolayers. For the distorted structure, solid light blue lines represent majority-spin bands and dashed purple lines represent minority-spin bands.