Intrinsic Quantum Anomalous Hall Effect in the Kagome Lattice ${\mathrm{Cs}}_2{\mathrm{LiMn}}_3{\mathrm{F}}_{12}$

In a kagome lattice, the time reversal symmetry can be broken by a staggered magnetic flux emerging from ferromagnetic ordering and intrinsic spin-orbit coupling, leading to several well-separated nontrivial Chern bands and intrinsic quantum anomalous Hall effect. Based on this idea and ab initio calculations, we propose the realization of the intrinsic quantum anomalous Hall effect in the single layer ${\mathrm{Cs}}_2{\mathrm{Mn}}_3{\mathrm{F}}_{12}$ kagome lattice and on the (001) surface of a ${\mathrm{Cs}}_2{\text{LiMn}}_3{\mathrm{F}}_{12}$ single crystal by modifying the carrier coverage on it, where the band gap is around 20 meV. Moreover, a simplified tight binding model based on the in-plane $dd\sigma$ antibonding states is constructed to understand the topological band structures of the system.

Figure 1

(a) Schematic of the Mn kagome lattice and the antibonding $d$ orbitals near the Fermi level. An effective staggered magnetic flux $\varphi$ in the triangles and $-2\varphi$ in the hexagons of the kagome lattice emerge due to the FM ordering and the intrinsic SOC. (b) The unit cell of ${\mathrm{Cs}}_2{\text{LiMn}}_3{\mathrm{F}}_{12}$ with space group $R\overline{3}$, which contains three primitive cells. (c), The crystal structure of single layer ${\mathrm{Cs}}_2{\mathrm{Mn}}_3{\mathrm{F}}_{12}$.

Figure 2

(a) The first Brillouin zones of ${\mathrm{Cs}}_2{\text{LiMn}}_3{\mathrm{F}}_{12}$ with space group $R\overline{3}$. The four types of inversion-invariant points are $\mathrm{\Gamma }(0,0,0)$, $Z(\pi ,\pi ,\pi )$, $F(\pi ,\pi ,0)$, and $L(\pi ,0,0)$. The blue hexagon shows the 2D Brillouin zones for the single layer ${\mathrm{Cs}}_2{\mathrm{Mn}}_3{\mathrm{F}}_{12}$ and slab, in which the high-symmetry $k$ points $\overline{\mathrm{\Gamma }}$, $\overline{M}$, and $\overline{K}$ are labeled. (b) Spin resolved DOS and PDOS of the bulk ${\mathrm{Cs}}_2{\text{LiMn}}_3{\mathrm{F}}_{12}$. (c) FM band structures of ${\mathrm{Cs}}_2{\text{LiMn}}_3{\mathrm{F}}_{12}$ without SOC, in which the red circles, blue squares, and brown diamonds denote the projections on the $d_{z^2}$, $d_{x^2-y^2}\&d_{xy}$, and $d_{yz}\&d_{zx}$ orbitals of Mn, respectively. (d) FM band structures of ${\mathrm{Cs}}_2{\text{LiMn}}_3{\mathrm{F}}_{12}$ with SOC, in which the parities of the topmost three valence bands at eight inversion-symmetric points are marked. The inset is the enlargement around the $K$ point to show the fully opened gap. All Fermi levels are defined at 0 eV.

Figure 3

(a) TB band structures with $t_1=0.25\mathrm{eV}$, $t_2=0$ (blue), and $t_2=0.4t_1$ (red), respectively. (b) FM band structures of single layer ${\mathrm{Cs}}_2{\mathrm{Mn}}_3{\mathrm{F}}_{12}$ with (red) and without (blue) SOC calculations. (c) Enlargement of (b) around the $K$ point to show the Dirac point (without SOC) and topologically nontrivial gap (with SOC) clearly. (d) Calculated edge state for the semi-infinite boundary of the 2D kagome lattice ${\mathrm{Cs}}_2{\mathrm{Mn}}_3{\mathrm{F}}_{12}$. All Fermi levels are defined at 0 eV.

Figure 4

(a) FM band structures of the slab constructed by three layers of ${\mathrm{Cs}}_2{\text{LiMn}}_3{\mathrm{F}}_{12}$ with SOC. The inset is the enlargement around the $K$ point to show the topologically nontrivial gap (4 meV) clearly. (b) Calculated edge state for the slab constructed by three layers of ${\mathrm{Cs}}_2{\text{LiMn}}_3{\mathrm{F}}_{12}$. The inset is the enlargement around the Fermi level to show the insulating gap (4 meV) and chiral edge state clearly. All Fermi levels are defined at 0 eV.