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Spin-Orbit Driven Topological Phases in Kagome Materials

Chi Wu, Tiantian Zhang

Abstract

Kagome materials have garnered substantial attention owing to their diverse physical phenomena, yet canonical systems such as the AV$_3$Sb$_5$ family exhibit poor $Z_{2}$-type topological properties, spurring an urgent quest for kagome platforms hosting ideal topological states. Recently, Zhou et al. proposed the kagome-type IAMX family, which exhibits distinctive ideal topological states; however, their analysis is primarily restricted to the spinless approximation. In this work, we model relativistic effects in the IAMX family, demonstrating that tuning the spin-orbit coupling (SOC) strength drives topological phase transitions and induces novel topological states, resulting in a rich phase diagram. The configuration of topological surface states evolves continuously as the SOC strength is modulated, consistent with the evolution of the topological phase transition. This suggests a viable route toward designing multi-functional topological devices. First-principles calculations performed on three specific IAMX compounds confirm that SOC governs their topological phases, in complete accord with our model analysis.

Spin-Orbit Driven Topological Phases in Kagome Materials

Abstract

Kagome materials have garnered substantial attention owing to their diverse physical phenomena, yet canonical systems such as the AVSb family exhibit poor -type topological properties, spurring an urgent quest for kagome platforms hosting ideal topological states. Recently, Zhou et al. proposed the kagome-type IAMX family, which exhibits distinctive ideal topological states; however, their analysis is primarily restricted to the spinless approximation. In this work, we model relativistic effects in the IAMX family, demonstrating that tuning the spin-orbit coupling (SOC) strength drives topological phase transitions and induces novel topological states, resulting in a rich phase diagram. The configuration of topological surface states evolves continuously as the SOC strength is modulated, consistent with the evolution of the topological phase transition. This suggests a viable route toward designing multi-functional topological devices. First-principles calculations performed on three specific IAMX compounds confirm that SOC governs their topological phases, in complete accord with our model analysis.

Paper Structure

This paper contains 7 sections, 7 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. IAMX family: Crystal structure and electronic band structure
  3. Results and discussion
  4. The $k\cdot p$ effective Hamiltonian for IAMX family
  5. Topological phase diagrams based on the $k\cdot p$ model
  6. SOC-enriched topological phases in the IAMX family
  7. Conclusion

Figures (5)

  • Figure 1: (a) The crystal structure of the IAMX family, where IA, M, and X atoms are denoted by green, yellow, and purple spheres, respectively. The X atoms arrange in a honeycomb lattice configuration within layer $\beta$, whereas the M atoms organize into a distorted kagome lattice structure in layer $\alpha$. (b) The bulk Brillouin zone (BZ) and its projection onto the (001) surface are depicted. The three mirror planes orthogonal to the $\mathrm{M}_{001}$ plane are illustrated on the right. (c) Schematic energy bands for the system with stacked kagome and honeycomb lattice. (d) The projected band structures (PBSs) in the mirror $k_z=0$ for LiYC, LiNdGe and KLaPb from DFT, wherein the yellow segments represent contributions from the $d_{z^{2}}$ orbitals (kagome layer), and the purple segments indicate the origin from the $p_z$ orbitals (honeycomb layer).
  • Figure 2: The energy bands and topological invariants derived from the minimal four-band $\boldsymbol{k} \cdot \boldsymbol{p}$ model are presented for varying strength of SOC ($\lambda$). From left to right, $\lambda$ increases sequentially from zero. Specific values for $\lambda$ are shown in (a)-(d), leading to the emergence of distinct topological phases (here $\eta$ is assigned a value of 3). (a) Nodal ring semimetal (NRSM), featuring a single nodal ring (highlighted in green) residing within the $k_z$=0 plane, (b) Topological insulator (TI), (c) The critical state, with the degeneracy points denoted by black dots, and (d) Weyl semimetal (WSM), characterized by six pairs of Weyl points symmetrically positioned on either side of the $k_z$=0 plane.
  • Figure 3: Phase diagrams for the minimal $\boldsymbol{k} \cdot \boldsymbol{p}$ model. (a) Topological phase diagram as a function of the energy gap on the $k_z=0$ plane and the strength of SOC ($\lambda$). The gap closure serves as the signature of the topological phase transition. (b) The phase diagram as a function of $S_1$ and $S_2$ parameters, which corresponds to the first- and second-order SOC term, respectively. The curve represented by $\eta \lambda^{2}$ initiates at zero and traverses both the Weyl semimetal (WSM) and topological insulator (TI) phases (here $\eta$=3). (c) Topological phase as a function of SOC ($\lambda$) and $M_1$. Following the relationship in Eq. (\ref{['eq:seven']}), the approximate parameter and phase location for LiYC, LiNdGe, and KLaPb are denoted with ①②③.
  • Figure 4: The projected surface states on the (001) surface BZ along $\overline{M}-\overline{\Gamma}-\overline{M}$ for the minimal $\boldsymbol{k} \cdot \boldsymbol{p}$ model, with the top row showing 3D schematics of the surface states for each topological phase (the vertical dimension representing energy). These three states correspond to distinct combinations of $S_{1}$ and $S_{2}$, demonstrating the evolution of surface electronic structure under varying SOC strength. (a) Surface states in the absence of SOC, corresponding to the NRSM phase. (b) Surface states in the WSM phase. The upper portion shows the helicoid structure associated with a pair of Weyl points of opposite chirality, representing the fundamental building block of surface states in a WSM. (c) Surface states in the sTI phase, realized with larger $S_{1}$.
  • Figure 5: The projected surface states along [001] direction and their corresponding Fermi surfaces for three of the IAMX family materials from DFT+TB. In these figures, the bulk states are depicted in black and dark red, whereas the surface states are highlighted in gold. The surface spin texture for LiNdGe and KLaPb are depicted with orange arrows. (a) The Fermi surface and drumhead surface state for LiYC. (b) The Fermi arcs and helicoid surface states for LiNdGe, wherein each white dot represent a pair of Weyl fermions with opposite chirality. (c) The Fermi surface and Dirac surface states for KLaPb.