Monolayer ${\mathrm{V}}_2{MX}_4$: A New Family of Quantum Anomalous Hall Insulators

We theoretically propose that the van der Waals layered ternary transition metal chalcogenide ${\mathrm{V}}_{2}M{X}_4$ ($M=\mathrm{W}$, Mo; $X=\mathrm{S}$, Se) is a new family of quantum anomalous Hall insulators with sizable bulk gap and Chern number $\mathcal{C}=-1$. The large topological gap originates from the deep band inversion between spin-up bands contributed by $d_{xz}$, $d_{yz}$ orbitals of V and spin-down band from $d_{z^2}$ orbital of $M$ at the Fermi level. Remarkably, the Curie temperature of monolayer ${\mathrm{V}}_{2}M{X}_4$ is predicted to be much higher than that of monolayer ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. Furthermore, the thickness dependence of the Chern number for few multilayers shows interesting oscillating behavior. The general physics from the $d$ orbitals here applies to a large class of ternary transition metal chalcogenide such as ${\mathrm{Ti}}_2{WX}_4$ with the space group $P\text{−}42m$. These interesting predictions, if realized experimentally, could greatly promote the research and application of topological quantum physics.

Figure 1

(a),(b) Atomic structure of monolayer ${\mathrm{V}}_2{MX}_4$ from top and side views. The Wyckoff positions $1a$ and $2e$ are displayed (notation adopted from Bilbao crystallographic server [66, 67, 68, 69, 70, 71, 72]). The magnetic ground state of the 2D materials class is FM along the $z$ direction with spin magnetic moment $2.6{\mu }_B$ per V atom. The key symmetry operations of $P\text{−}42m$ include $S_{4z}$, $C_{2x}$, and $C_{2z}$ rotations, where $S_{4z}\equiv \mathcal{I}{C}_{4z}$ and $\mathcal{I}$ is inversion symmetry. (c) Brillouin zone. (d) Crystal field splitting and schematic diagram of the FM kinetic exchange coupling between the V atoms.

Figure 2

Electronic structures and topological properties of monolayer ${\mathrm{V}}_2{\mathrm{WS}}_4$, ${\mathrm{V}}_2{\mathrm{MoS}}_4$, and ${\mathrm{Ti}}_2{\mathrm{WS}}_4$. (a)–(c) ${\mathrm{V}}_2{\mathrm{WS}}_4$, (d)–(f) ${\mathrm{V}}_2{\mathrm{MoS}}_4$, (g)–(i) ${\mathrm{Ti}}_2{\mathrm{WS}}_4$, The band structure with SOC; topological edge states (ES) calculated along the $x$ axis; and anomalous Hall conductance ${\sigma }_{xy}$ as a function of Fermi energy, respectively. The shaded regions in (a),(d),(g) denote the topological gap.

Figure 3

(a) The $d$-orbital projection band structures without SOC of monolayer ${\mathrm{V}}_2{\mathrm{WS}}_4$, only those bands which are related to band inversion are shown. (b) The band structure for FM along the $z$ and $x$ axis of ${\mathrm{V}}_2{\mathrm{WS}}_4$ with SOC. (c) Energy and momentum dependence of the edge local density of state along the $y$ axis under the FM-$x$ state of ${\mathrm{V}}_2{\mathrm{WS}}_4$. (d) Dependence of the bulk gap along the high symmetry line and $\mathcal{C}$ on the spin orientation quantified by a polar angle $\theta$, where $\theta =0,\pi /2,\pi$ denote the $+z,+x,-z$ directions, respectively.

Figure 4

(a),(b) The $d$-orbital projection band structure of the tight-binding model with and without SOC. The irreducible representation at high-symmetry points of the Brillouin zone boundary and the orbital compositions (color of the bands) are consistent with that in Fig. 3. (c)–(e), The Wilson loop for the lowest one, two and three bands in (b), respectively. (f) Eigenvalues of $S_{4z}$ at the $\mathrm{\Gamma }$ and $M$ points and $C_{2z}$ at the $X$ point of bands $A$ and $B$. Chern number $\mathcal{C}$ is calculated by Eq. (1).