Quantum anomalous Hall effect in ferromagnetic transition metal halides

The quantum anomalous Hall (QAH) effect is a novel topological spintronic phenomenon arising from inherent magnetization and spin-orbit coupling. Various theoretical and experimental efforts have been devoted in search of intrinsic QAH insulators. However, up to now, it has only been observed in Cr or V doped ${(\mathrm{Bi},\mathrm{Sb})}_2\mathrm{T}{\mathrm{e}}_3$ film in experiments with very low working temperature. Based on the successful synthesis of transition metal halides, we use first-principles calculations to predict that the $\mathrm{Ru}{\mathrm{I}}_3$ monolayer is an intrinsic ferromagnetic QAH insulator with a topologically nontrivial global band gap of 11 meV. This topologically nontrivial band gap at the Fermi level is due to its crystal symmetry, thus the QAH effect is robust. Its Curie temperature, estimated to be $\sim 360\mathrm{K}$ using Monte Carlo simulation, is above room temperature and higher than most two-dimensional ferromagnetic thin films. The inclusion of Hubbard $U$ in the Ru-$d$ electrons does not affect this result. We also discuss the manipulation of its exchange energy and nontrivial band gap by applying in-plane strain. Our work adds an experimentally feasible member to the QAH insulator family, which is expected to have broad applications in nanoelectronics and spintronics.

Figure 1

(a) Top and side views of the optimized 2D $\mathrm{Ru}{\mathrm{I}}_3$ monolayer. The dashed rhombus refers to the unit cell. (b) Different magnetic configurations. (c) Spin density (isovalue of $0.04\mathrm{e}/{\AA }^3$) and exchange path $J$. (d) Magnetic moment per formula unit as a function of temperature from Monte Carlo simulation.

Figure 2

(a) Band structure without SOC. Blue and red curves represent spin-up and spin-down bands, respectively. (b) Projected density of states. (c) Schematic diagram of the evolution from the atomic $d$ orbitals to the final states at the Г point. (d) Orbital-resolved spin-down bands around the Fermi level. Different colors represent proportional contributions of ${e_1}^{\downarrow }$ states and $a^{\downarrow }$ states. Thin black curves show the evolution of ${e_1}^{\downarrow }$ and $a^{\downarrow }$ states from Г to $K$.

Figure 3

(a) Band structure with SOC (left panel) and anomalous Hall conductance as a function of the relative chemical potential (right panel). Different colors in the band structure represent the $\langle s_z\rangle$. The Chern numbers of frontier bands are indicated. The quantized terrace of ${\sigma }_{xy}$ is highlighted by the red dashed oval. (b) $k$-resolved Berry curvature Ω($k$). The red dashed hexagon denotes the first Brillouin zone. (c) Tight-binding band structure of nanoribbon obtained by MLWFs show edge states (yellow) inside the gap of bulk bands (blue).

Figure 4

(a) Real and (b) imaginary parts of the optical conductivity ${\sigma }_{xy}$ with respect to photonic energy and chemical potential.

Figure 5

Nontrivial bulk energy gap and magnetic exchange parameter $J$ as functions of biaxial in-plane strain.

Figure 6

Band structures of ferromagnetic states for the $\mathrm{RuC}{\mathrm{l}}_3$ and $\mathrm{RuB}{\mathrm{r}}_3$ monolayers without and with SOC. The blue and red curves in the left panels denote spin-up and spin-down channels, respectively.