Intrinsic Quantum Anomalous Hall Effect with In-Plane Magnetization: Searching Rule and Material Prediction

So far, most theoretically predicted and experimentally confirmed quantum anomalous Hall effects (QAHEs) are limited in two-dimensional (2D) materials with out-of-plane magnetization. In this Letter, starting from 2D nodal-line semimetal, a general rule for searching QAHE with in-plane magnetization is mapped out. Because of spin-orbital coupling, we found that the magnetization will prefer an in-plane orientation if the orbital of degenerate nodal-line states at the Fermi level have the same absolute value of magnetic quantum number. Moreover, depending on the broken or conserved mirror symmetry, either a QAHE or 2D semimetal can be realized. Based on first principles calculations, we further predict a real material of monolayer LaCl to be an intrinsic QAHE with in-plane magnetization. By tuning the directions of in-plane magnetization, the QAHE in LaCl demonstrates a threefold rotational symmetry with a Chern number of either $+1$ or $-1$, and the transition point is characterized by a 2D semimetal phase. All these features are quantitatively reproduced by tight-binding model calculations, revealing the underlying physics clearly. Our results greatly extend the scope for material classes of QAHE and hence stimulate immediate experimental interest.

Figure 1

Schematic rule for searching QAHE with in-plane magnetization. Different topological phases are determined by the direction of magnetization, mirror symmetry, and SOC.

Figure 2

(a) Top view of monolayer LaCl and angle of in-plane magnetization. (b) Exfoliation energy of monolayer LaCl. Inset is side view of bulk LaCl and interlayer distance. (c) Spin-polarized ferromagnetic band structure of monolayer LaCl without SOC. Red and blue colors denote spin-up and and -down bands, respectively. (d) 3D band around $\mathrm{\Gamma }$ point near the Fermi level in (c).

Figure 3

(a) Band structure of monolayer LaCl with SOC for in-plane magnetization along $\varphi =0°/180°$. (b) Schematic two degenerate points (red dot) on the mirror plane (dashed orange line) for in-plane magnetization (blue arrow) perpendicular to the mirror plane in (a). (c)–(f) are the same as (a),(b), but for in-plane magnetization along $\varphi =60°/240°$ and $120°/300°$, respectively. (g) Band structure of monolayer LaCl with SOC for in-plane magnetization along $\varphi =30°$, as denoted by the inset arrows. (h) 1D topological edge state for (g), showing QAHE with in-plane magnetization. The inset is schematic propagating direction for left and right edge states. (i),(j) are the same as (g),(h), but for in-plane magnetization along $\varphi =90°$.

Figure 4

(a)–(f) Berry curvature and Chern number of monolayer LaCl for in-plane magnetization along $\varphi =30°$, 90°, 150°, 210°, 270°, and 330°, respectively. The arrow denotes direction of magnetization, and the shadow region highlights unit cell chosen for different configurations. (g) Schematic QAHE measurement by varying the direction of in-plane magnetization. (h) Quantized Hall conductivity vs direction of in-plane magnetization.

Figure 5

(a) DFT and Wannier fitted band structure of monolayer LaCl without magnetization and SOC. (b) Top and side views of two fitted WFs. (c) TB band structure with in-plane magnetization along $\varphi =30°$. (d) 1D ribbon band structure for (c). Red and blue colors denote left and right edge states, respectively. (e) Berry curvature and Chern number for (c). (f)–(h) are the same as (c)–(e), but for in-plane magnetization along $\varphi =90°$. The TB parameters are $t=1.0\mathrm{eV}$, ${\lambda }_I=0.03\mathrm{eV}$, ${\lambda }_R=-0.03\mathrm{eV}$, and $t_M=-2.0\mathrm{eV}$.