Chern insulators without band inversion in $\mathrm{Mo}{\mathrm{S}}_2$ monolayers with $3d$ adatoms

Electronic and topological properties of $\mathrm{Mo}{\mathrm{S}}_2$ monolayers endowed with $3d$ transition metal (TM) adatoms (V-Fe) are explored by using ab initio methods and $k·p$ models. Without the consideration of the Hubbard $U$ interaction, the V, Cr, and Fe adatoms tend to locate on the top of the Mo atoms, while the most stable site for the Mn atom is at the hollow position of the Mo-S hexagon. After the Hubbard $U$ is applied, the most stable sites of all the systems become the top of the Mo atoms. Chern insulators without band inversion are achieved in these systems. The V and Fe adsorption systems are the best candidates to produce the topological states. The $k·p$ model calculations indicate that these topological states are determined by the TM magnetism, the $C_{3v}$ crystal field from the $\mathrm{Mo}{\mathrm{S}}_2$ substrate, and the TM atomic spin-orbit coupling (SOC). The special two-meron pseudospin texture is found to contribute to the topology. The apparent difference between the Berry curvatures for the V and Fe adsorption systems is also explored. Our results widen the understanding of the Chern insulators and are helpful for the applications of the $\mathrm{Mo}{\mathrm{S}}_2$ monolayers in the future electronics and spintronics.

Figure 1

Top (a) and side (b) views of the $\mathrm{Mo}{\mathrm{S}}_2$ ML. Three high-symmetry adsorption sites, labeled as $H$ (Hollow), $M$ (Mo-top), and $S$ (S-top), are shown in (a). The blue and yellow spheres represent the Mo and S atoms, respectively. The dashed lines illustrate the $3\times 3$ supercell.

Figure 2

The $3d$ and $4s$ partial DOSs of the TM atoms adsorbed on the $\mathrm{Mo}{\mathrm{S}}_2$ ML. For clarity, the $3d$ and $4s$ partial DOS are amplified by a factor of 10. The light gray curves give the total DOS of the supercells. The peaks which marked with $E_1,E_2$, and $A$ are the $3d$ orbital components for the TM adatoms under the $C_{3v}$ crystal field. The positive and negative DOS values correspond to the spin-up and spin-down states, respectively. The vertical dashed line in each panel indicates the Fermi level. The left and right columns correspond to the results without and with the Hubbard $U$ considered, respectively.

Figure 3

The band structures of the $\mathrm{Mo}{\mathrm{S}}_2$ ML with V, Cr, Mn, or Fe adsorbed. The Hubbard $U$ interaction is considered. The red and blue curves denote the spin-up and spin-down components, respectively. The upper and lower rows give the results without and with the consideration of the SOC. The $E_{\mathrm{F}}$ is set as zero.

Figure 4

(a) and (b) The enlarged $E_2$ and $E_1$ bands in Fig. 3, respectively. (c) The enlarged $E_2$ band in Fig. 3. The blue dashed curves denote the corresponding Berry curvatures (Ω) in atomic units (a.u.). The corresponding surface states of the nanoribbons with finite widths are given in (d), (e), and (f), respectively.

Figure 5

The 2D Berry curvature distributions (in a.u.) in the momentum space of the $E_2$ bands in the V (a) and Fe (b) adsorbed $\mathrm{Mo}{\mathrm{S}}_2$ MLs. The four high-symmetry k points are given.

Figure 6

The $k·p$ model band structures around the Γ point under the parameters (a) $\alpha =-0.1\mathrm{eV}·{\AA }^2,\beta =0.2\mathrm{eV}·\AA ,M=0.0\mathrm{eV},{\lambda }_{\mathrm{so}}=0.0\mathrm{eV}$; (b) $\alpha =-0.1\mathrm{eV}·{\AA }^2,\beta =0.2\mathrm{eV}·\AA ,M=-0.5\mathrm{eV},{\lambda }_{\mathrm{so}}=0.0\mathrm{eV}$; (c) and (d) $\alpha =-0.1\mathrm{eV}·{\AA }^2,\beta =0.2\mathrm{eV}·\AA ,M=-0.5\mathrm{eV},{\lambda }_{\mathrm{so}}=-0.02\mathrm{eV}$ for the spin-up and spin-down bands, respectively. The red solid curves (blue dashed curves) represent the spin-up (spin-down) bands. With the $E_{\mathrm{F}}$ located within the SOC induced gap, the obtained Berry curvatures (green solid curves) and the Chern numbers $C$ for the spin-up and spin-down bands are also given in (c) and (d). (e) and (f) Pseudospin textures around the Γ points for the spin-up and spin-down bands, respectively.

Figure 7

(a) The corresponding bands of Fig. 4 with the height $\left(h\right)$ of Fe adatom increasing from 0.93 Å to 1.5 Å. (b) Schematic diagram of the band evolution with the consideration of the atomic $\mathrm{SOC}\left({\lambda }_{\mathrm{so}}\right)$ and Rashba $\mathrm{SOC}\left({\lambda }_R\right)$.

Figure 8

The charge distributions of the $E_2$ bands in the Fe cases with the Fe atom at the normal height (a), (b) or enlarged height (c), (d). The upper and lower panels give the top and side views, respectively. The isosurface value is $0.002e/{a_0}^3$.