Magnetization-Tunable Topological Phase Transitions in Ferromagnetic Kagome Monolayers of Co$_3$X$_3$Y$_2$ ($X=\mathrm{Sn},\mathrm{Pb}$; $Y=\mathrm{S},\mathrm{Se}$)

Abstract

The quantum anomalous Hall effect in magnetic kagome materials has emerged as a versatile platform for dissipationless electronic and spintronic devices. In this work, we demonstrate that the orientation of magnetic moments $\hat{m}(θ,φ)$ at lattice sites provides a practical tuning mechanism for engineering nontrivial topological phases in monolayer kagome ferromagnets. To elucidate the mechanism, we construct a symmetry-adapted minimal tight-binding model for kagome ferromagnets that includes intrinsic spin-orbit coupling (SOC) and the intrinsic Rashba SOC permitted by broken out-of-plane mirror symmetry between nearest-neighbor kagome sites and can capture the resulting topological phase diagram as a function of $\hat{m}(θ,φ)$. In particular, the restoration of in-plane mirror symmetry for specific values of $φ$ drives a topological phase transition upon varying the in-plane orientation of the moments $\hat{m}(θ= 90^{\circ}, φ)$. In contrast, for fixed $φ$, the transitions driven by varying $θ$ originate from the competition between Rashba SOC and intrinsic SOC. Density functional theory calculations for ferromagnetic kagome monolayers belonging to the Co$_3$X$_3$Y$_2$ family ($X=\mathrm{Sn},\mathrm{Pb}$; $Y=\mathrm{S},\mathrm{Se}$) support the predictions of the proposed minimal tight-binding model. These findings provide design guidelines for tunable topological phases in kagome materials.

Figures (6)

• Figure 1: Kagome lattice geometry. (a) The kagome lattice with in-plane mirrors (grey dotted lines). Red (blue) arrows represent anticlockwise (clockwise) hoppings within the associated triangles, indicating the favored paths for up (down)-spins due to intrinsic SOC. The surrounding atoms above (red) and below (blue) the kagome plane break the out-of-plane mirror symmetry. (b) Schematic of the origin of first-nearest-neighbor SOC effect. (c) Directional convention used for $\vec{\eta}_{\alpha\beta}$ in the intrinsic Rashba SOC term of the TB model.
• Figure 2: Topological properties of the TB model with parameters $t = 1.0$, $B = 1.5t$, $t_I = 0.2t$, and $t_R = 0.5t$. (a) First BZ of the kagome lattice with in-plane mirrors (grey dotted lines) and distinct high-symmetry points. (b) Topological phase diagram showing the Chern number $C$ for different bands as a function of $\hat{m}(\theta=90^{\circ}, \phi)$. (c) Band-crossing points due to ferromagnetic moment alignments $\hat{m}(\theta=90^\circ, \phi)$ for $\phi \in \{\phi_C\}$ indicating TPT. (d) Band structure and Chern number $C$ for each band at $\hat{m}(\theta=90^\circ, \phi=30^\circ)$. (e) Same as (d), but for $\hat{m}(\theta=90^\circ, \phi=90^\circ)$. (f) TPT of the fourth band due to the variation of $\hat{m}(\theta, \phi=\text{constant})$ for $\phi = 0^\circ, 30^\circ$ and $90^\circ$.
• Figure 3: Topological properties of the ferromagnetic Co$_3$Pb$_3$S$_2$ monolayer under in-plane moment variation. (a) Crystal structure showing in-plane mirrors (black dotted lines) and broken out-of-plane mirror symmetry (light-blue plane, coinciding with the kagome plane). (b) Topological phase diagram for in-plane moment variation ($\theta = 90^\circ$). (c) Electronic band structure for $\hat{m}(\theta = 90^{\circ}, \phi = 30^{\circ})$, with a Chern number $C = -1$ at $E_F$, as supported from the variation of the normalized Hall conductivity $\tilde{\sigma}_{xy}$. (d) Edge states for $\hat{m}(\theta = 90^{\circ}, \phi = 30^{\circ})$, with edge cuts perpendicular to the crystallographic axis $\vec{b}$. (e) Top-edge spectral function confirming $C = -1$ for the same edge cut. (f)–(h) Corresponding results for $\hat{m}(\theta = 90^{\circ}, \phi = 90^{\circ})$ with $C = +1$. (i) TPT for in-plane moment variation ($\theta = 90^\circ$), occurring cyclically at $\phi = 0^\circ$, $60^\circ$, and $120^\circ$, and their equivalents at $\phi = 180^\circ$, $240^\circ$, and $300^\circ$, tied to different $\Gamma$–$K$ paths parallel to the moment directions.
• Figure 4: TPT and Berry curvature of the top valence band within the first BZ for the ferromagnetic Co$_3$Pb$_3$S$_2$ monolayer under variation of moment $\hat{m}(\theta, \phi=$constant) for (a)$\phi = 0^\circ$, (b)$\phi = 30^\circ$, (c)$\phi = 60^\circ$, and (d)$\phi = 90^\circ$.
• Figure 5: Topological phase transitions of the TB model with parameters $t=1.0$ and $B=1.5t$ as a function of $t_R/t_I$ and the polar angle $\theta$ for fixed azimuthal angles $\phi=$(a)$0^{\circ}$, (b)$30^{\circ}$, (c)$60^{\circ}$, and (d)$90^{\circ}$.