Robust quantum anomalous Hall effect in a pentagonal ${\mathrm{MoS}}_2$ monolayer grown on CuI(001) substrates

Recently, the square planar ${\mathrm{MoS}}_2$ monolayer, which exhibits the Cairo pentagonal tiling (termed as $1\mathrm{P}\text{−}{\mathrm{MoS}}_2$), was identified as an intrinsic quantum anomalous Hall (QAH) insulator. However, there is a paucity of theoretical work concerning a suitable substrate to support its nontrivial electron transport properties, which is the prerequisite for practical applications. Here. we demonstrate that CuI(001) serves an excellent substrate candidate for epitaxial growth of the $1\mathrm{P}\text{−}{\mathrm{MoS}}_2$ sheet by showing that the intrinsic ferromagnetism and the QAH state of $1\mathrm{P}\text{−}{\mathrm{MoS}}_2$ remain unaltered in the $1\mathrm{P}\text{−}{\mathrm{MoS}}_2$/CuI(001) system. Further analyses of the strain effect on $1\mathrm{P}\text{−}{\mathrm{MoS}}_2$ reveal that the QAH is robust within a strain range from $-$ 2% to 2%. Our findings will inspire the experimental realization of QAH effects in two-dimensional (2D) pentagon-based materials.

Figure 1

Top and side views of (a) $\alpha$-type and (b) $\beta$-type penta-${\mathrm{MoS}}_2$, where the top and the bottom layers of S atoms are indicated by yellow and green balls, respectively. (c) Optimized planar $1\mathrm{P}\text{−}{\mathrm{MoS}}_2$ monolayer, where the magenta and yellow balls represent Mo and S atoms, respectively. (d) Total energy of penta-${\mathrm{MoS}}_2$ monolayers with different types of buckling as a function of lattice constant. The insets show the optimized structures of the penta-${\mathrm{MoS}}_2$ monolayers with $\alpha$- and $\beta$-type buckling.

Figure 2

(a) Optimized crystal structure of $1\mathrm{P}\text{−}{\mathrm{MoS}}_2/\mathrm{CuI}\left(001\right)$ surface from top view. (b) Schematic of the quantum transport device designed from a $1\mathrm{P}\text{−}{\mathrm{MoS}}_2/\mathrm{CuI}\left(001\right)$ heterostructure, where the purple arrows denote spin directions. (c) Fluctuation of total energy of the $1\mathrm{P}\text{−}{\mathrm{MoS}}_2/\mathrm{CuI}\left(001\right)$ system during the AIMD simulation. (d) Structure snapshots of the $1\mathrm{P}\text{−}{\mathrm{MoS}}_2/\mathrm{CuI}\left(001\right)$ system at the end of the simulation.

Figure 3

Temperature-dependent normalized magnetization for the $1\mathrm{P}\text{−}{\mathrm{MoS}}_2$ monolayer under 1.5% strain obtained from the Monte Carlo simulation. Blue dots indicate the simulated mean magnetization, and the blue solid line represents the corresponding fit based on Eq. (2).

Figure 4

Spin-polarized band structures without SOC for (a) spin-up and (b) spin-down channels for the 1P-MoS2 monolayer under 1.5% tensile strain. The symmetries of the band states along the $\mathrm{\Gamma }$-M path are labeled according to the irreducible representation of the ${\mathrm{C}}_{2v}$ point group with the aid of the Quantum Espresso package [50]. (c) Spin- and site-projected DOS. (d) 3D band structure for spin-up channel enlarged at the band crossing point (0.29, 0.29)$2\pi$/$a$.

Figure 5

(a) Band structure of the $1\mathrm{P}\text{−}{\mathrm{MoS}}_2/\mathrm{CuI}\left(001\right)$ surface with SOC considered, where magenta and cyan balls represent the contributions of $1\mathrm{P}\text{−}{\mathrm{MoS}}_2$ and CuI(001), respectively, while the size of balls is in proportion to the contribution. (b) Anomalous Hall conductivity as a function of the Fermi level for the $1\mathrm{P}\text{−}{\mathrm{MoS}}_2$ monolayer. (c–d) Edge densities of states of semi-infinite nanoribbon constructed based on isolated $1\mathrm{P}\text{−}{\mathrm{MoS}}_2$ for the left and right edges, respectively.

Figure 6

(a–b) Structural motifs under strains. (c)The variations of exchange energy and band gap as a function of strains. (d–e) The edge densities of states for the stretched $1\mathrm{P}\text{−}{\mathrm{MoS}}_2$ at $\varepsilon =-2\%$ and $\varepsilon =2\%$, respectively. Only the right-hand edges are shown here.