Stability and magnetism of strongly correlated single-layer ${\mathrm{VS}}_2$

Single-layer transition metal dichalcogenides exhibit a variety of atomic structures and associated exotic electronic and magnetic properties. Density-functional calculations using the LDA+$U$ approximation show that single-layer ${\mathrm{VS}}_2$ is a strongly correlated material, where the stability, phonon spectra, and magnetic moments of the octahedral ($1T$) and the trigonal prismatic ($2H$) structures significantly depend on the effective Hubbard $U$ parameter, $U_{\mathrm{eff}}$. Comparison with the HSE06 hybrid density functional used as a benchmark indicates that $U_{\mathrm{eff}}=2.5$ eV, which consistently shows that the $2H$ structure is more stable than the $1T$ structure and a ferromagnetic semiconductor. The magnetic moments are localized on the V atoms and coupled ferromagnetically due to the superexchange interactions mediated by the S atoms. Calculations of the magnetic anisotropy show an easy plane for the magnetic moment. Assuming a classical $\mathit{XY}$ model with nearest neighbor coupling, we determine the critical temperature, $T_{\mathrm{c}}$, for the Berezinsky-Kosterlitz-Thouless transition of $2H$ single-layer ${\mathrm{VS}}_2$ to be about 90 K. Applying biaxial tensile strains can increase $T_{\mathrm{c}}$. Using Wannier interpolation, we evaluate the Berry curvature and anomalous Hall conductivity of $2H$ single-layer ${\mathrm{VS}}_2$. The coexistence of quasi-long-range ferromagnetic ordering and semiconducting behavior enables $2H$ single-layer ${\mathrm{VS}}_2$ to be a promising candidate for spintronics applications.

Figure 1

Energy difference $\mathrm{\Delta }E$ between $1T$ and $2H$ single-layer ${\mathrm{VS}}_2,\mathrm{\Delta }E=E_{1T}-E_{2H}$, as a function of $U_{\mathrm{eff}}$ calculated with the LDA and PBE functionals. The inset illustrates the atomic structures of $1T$ and $2H$ single-layer ${\mathrm{VS}}_2$. The red solid line labels the $\mathrm{\Delta }E$ calculated with the HSE06 functional.

Figure 2

Magnetic moment of $1T$ and $2H$ single-layer ${\mathrm{VS}}_2$ per formula unit calculated with the LDA+$U$ (solid lines) and PBE+$U$ (dotted lines) methods, where $U_{\mathrm{eff}}$ is varied from 0 to 5 eV.

Figure 3

Calculated phonon spectra of (a) $1T$ and (b) $2H$ single-layer ${\mathrm{VS}}_2$ with the LDA+$U$ ($U_{\mathrm{eff}}=2.5$ eV) method.

Figure 4

Orbital resolved (top) spin-up and (bottom) spin-down band structures of $2H$ single-layer ${\mathrm{VS}}_2$ calculated with the (a) LDA+$U$ ($U_{\mathrm{eff}}=2.5$ eV), (b) PBE+$U$ ($U_{\mathrm{eff}}=1.0$ eV), and (c) HSE06 methods.

Figure 5

Electron density corresponding to the valence band maximum states in the spin-up band structure illustrating the $d_{z^2}\left(a_1^{\prime }\right)$ character. The isosurface value is set to 0.01 $e/a_0^3$, where $a_0$ is the Bohr radius.

Figure 6

Angular dependence of MAE of $2H$ single-layer ${\mathrm{VS}}_2$ with the direction of magnetization lying (a) on the $xz$ or $yz$ plane and (b) on the $xy$ plane. The MAE is calculated with reference to the energy minimum when the magnetization locates on the $xy$ plane. We observe an easy plane for the spin oriented within the plane of the 2D material.

Figure 7

Comparison of the band structures obtained from the DFT+SOC calculation and the Wannier interpolation.

Figure 8

(a) Berry curvature and (b) anomalous Hall conductivity of $2H$ single-layer ${\mathrm{VS}}_2$.