Electronic structure of $A$V$_3$Sb$_5$ kagome metals

TL;DR

This work develops an extended Slater-Koster multiorbital tight-binding framework to faithfully describe the low-energy electronic structure of A V$_3$Sb$_5$ kagome metals, addressing the limitations of one-orbital and naive multiorbital models. By constructing a symmetry-respecting basis, rotating to a $C_3$-symmetric form, and introducing hybrid orbitals with a tunable mixing angle, the authors reproduce the DFT band structure, including the sublattice-polarized double p-type van Hove singularities and the rotated Fermi surface. The extended model relies on six independent interorbital hoppings $t^{}$ and orbital-dependent crystal fields $_$, pinned by p-vHS energies, and reveals that interorbital hopping between mirror-even/odd d orbitals is essential, giving rise to mirror interorbital flat bands and a hybridization mechanism that yields a realistic, one-orbital-like dispersion through the double vHS. This framework provides a robust starting point for incorporating electron-electron interactions in kagome metals and can be readily applied to related 135 compounds, thereby advancing understanding of correlated and topological phenomena in these systems.

Abstract

The kagome metals $A$V$_3$Sb$_5$ ($A=$ K, Cs, Rb) have become a fascinating materials platform following the discovery of many novel quantum states due to the interplay between electronic correlation, topology, and geometry. Understanding their physical origin requires constructing effective theories that capture the low-energy electronic structure and electronic interactions. While the band structure calculated by density functional theory (DFT) broadly agrees with experiments in the unbroken symmetry phase, the multiorbital nature challenges a proper understanding of the band structure and its description by tight-binding models. Here, we point out the unusual and puzzling properties of the DFT electronic structure, including the sublattice type of the van Hove singularities, the geometric shape of the Fermi surface, and the orbital content of the low-energy band dispersion, which cannot be described by the commonly used one-orbital or multiorbital kagome tight-binding models. We address these fundamental puzzles and develop an extended Slater-Koster formalism that can successfully resolve these issues. We discover the important role of site-symmetry and interorbital hopping structure and provide a concrete multiorbital tight-binding model description of the electronic structure for $A$V$_3$Sb$_5$ and the family of ``135'' compounds with other transition metals. This is a crucial step toward studying the effects of electron-electron interactions for the correlated and topological states in kagome metals and superconductors.

Figures (9)

• Figure 1: Crystal structure and DFT band structure of $A$V$_3$Sb$_5$. (a) Crystal Structure of $A$V$_3$Sb$_5$ with mirror plane $\sigma_h$ along the kagome plane. (b) The Fermi surfaces and (e) 3D DFT band structure for $\sigma_h$ mirror odd $d_{xz}, d_{yz}$ orbitals (green line) and $\sigma_h$ mirror even $d_{xy}, d_{x^2-y^2}$ and $d_{z^2}$ orbitals (blue line) on the V atoms of CsV$_3$Sb$_5$. The 3D Brillouin Zone and high-symmetry path is shown in Appendix. (c) The single-layer 2D kagome lattice made of V atoms with important symmetry elements indicated. (d) The Fermi surface and (f) band structure of the one-orbital model with $t=-1.0$ eV, $\mu=0.0$ eV, where the contents of different sublattices are marked in cyan, magenta and yellow.
• Figure 2: Comparison of DFT calculation and extended SK model results. (a) DFT band structure at $k_z=0$ for $d_{x^2-y^2}, d_{z^2}$ and $d_{xy}$ orbitals on kagome lattice formed by V atoms. Extended SK model results with parameters: $\mu_1 = 0.85, \mu_2 = -0.084, \mu_3 = -0.75$ pinning all three p-vHS. (b) Only 1 nearest neighbor interorbital hopping $t^{23}_\text{\nth{1}} = -0.38, t^{31}_\text{\nth{1}} = -0.35, t^{12}_\text{\nth{1}} = -0.48$ and no intraorbital hopping $t^{\alpha\alpha}_{\nth{1}/\nth{2}}$. (c) Intraorbital hopping for $d_{h_{1/2}}$: $t^{11}_\text{\nth{1}} = -0.05, t^{22}_\text{\nth{1}} = -0.3, t^{11}_\text{\nth{2}} = 0.1, t^{22}_\text{\nth{2}} = 0.2$ and 2 nearest neighbor interorbital hopping $t^{23}_\text{\nth{2}} = 0.12, t^{12}_\text{\nth{2}} = 0.0, t^{31}_\text{\nth{2}} = 0.05$ are added to TB in (b). (d) Intraorbital hopping for $d_{h_{3}}$$t^{33}_\text{\nth{1}} = -0.05, t^{33}_\text{\nth{2}} = -0.25$ are added to previous parameters. (e) The electron density distribution of the p-vHS2 and the p-vHS3 calculated by DFT. p-vHS1 is not presented due to its relatively high energy. (g) The wavefunction distribution of the p-vHS2 and the p-vHS3 calculated by extended SK model (positive sign in light red and negative sign in light blue). The Fermi surfaces calculated by (f) DFT and (h) the extended SK model with orbital contents marked in red, green and blue for $d^r_{x^2-y^2}$, $d^r_{z^2}$ and $d^r_{xy}$, respectively. Units of parameters are in eV.
• Figure 3: Change of orbital basis by linear combination. Wavefunction distribution with positive sign marked in light red and negative sign in light blue.
• Figure 4: Band structure formed by different Hamiltonian blocks. (a) First nearest neighbor (1 nn) non-zero interorbital $H^K$ hopping with $t^{12}=1.0$ eV with other zero hopping. Three FB due to isolated $d_{h_3}$ orbitals with no hopping exist at $\mu_{3}$. (b) 1 nn non-zero interorbital $H^{AK}$ hopping with $t^{31}=1.0$ eV with other zero hopping. Two mirror interorbital FB consist of $d_{h_1}$ and $d_{h_3}$ with p-vHS pinned at $E=0.0$ eV. Three FB due to isolated $d_{h_2}$ orbitals with no hopping exist at $\mu_{2}$. (c) 1 nn intraorbital hopping $t^{22}_\text{\nth{1}}$ and (d)2 nn intraorbital hopping $t^{33}_\text{\nth{2}}$ are added to Hamiltonian in (b) with $t^{31}=1.0$ eV. For (a-d), all crystal fields are set to zero: $\mu_{1}=\mu_{2}=\mu_{3}=0.0$ eV. (e) Different crystal fields are added: $\mu_{1}=0.85$ eV, $\mu_{2}=-0.084 eV, \mu_{3}=-0.75$ eV, with interorbital 1 nn $t^{23}=t^{31}=-0.5$ eV with other zero hopping. The dispersive bands are doubly degenerate.
• Figure 5: Comparison of orbital contents for DFT calculation and extended SK model. The orbital contents of sublattice-2 (red: $d_{xy}$, green: $d_{x^2-y^2}$ and blue: $d_{z^2}$) of band structure calculated by (a) Extended SK model and (b) DFT. Sublattice-2 is chosen because the p-vHS wavefunctions along the chosen high-symmetry path is localized on sublattice-2.