Kagome Metamaterial Fundamentals

• Kagome metamaterial is a synthetic composite structure based on a two-dimensional kagome lattice, characterized by flat bands, Dirac cones, and van Hove singularities.
• The engineered structure exploits plasmonic localization and superdimensional resonance to enable robust slow-light and sensing applications in photonics and acoustics.
• Its design principles extend to quantum materials and mechanical systems, offering tunability for topological devices, high-performance sensors, and reconfigurable resonators.

A Kagome metamaterial is a synthetic composite structure engineered to exploit the unique geometric, electronic, photonic, or mechanical properties emergent from the kagome lattice—a two-dimensional network of corner-sharing triangles. This class of metamaterials spans electromagnetic, acoustic, and mechanical domains, exhibiting haLLMark phenomena such as flat bands, Dirac cones, van Hove singularities, higher-order topology, superdimensional resonance, and tunable electronic instabilities. Kagome metamaterials enable diverse applications ranging from terahertz photonics and topological devices to quantum materials and high-performance sensors.

1. Kagome Lattice Geometry and Fundamental Properties

The kagome lattice’s defining motif—a triangular network with three atoms per unit cell—generates a band structure characterized by:

• Dirac cones at the $K$-point of the Brillouin zone, yielding topologically protected crossing points.
• Saddle points at the $M$-point, responsible for van Hove singularities and enhanced density of states.
• Flat bands resulting from destructive interference and high degeneracy, typically localized in real space.

A minimal tight-binding Hamiltonian on the kagome lattice,

$$H = -t \sum_{\langle i,j \rangle} c_{i}^\dagger c_{j}$$

captures these essential features: linear dispersions, saddle point, and a non-dispersive flat band. The unique topology underpins exotic physical phenomena, including localization, quantum Hall effects, topological insulator phases, and enhanced correlations (Peng et al., 2021).

2. Flat Bands and Plasmonic Localization

A seminal implementation of kagome metamaterials is the metallic bar–disk resonator network (KBDR) in the terahertz regime (Nakata et al., 2012). Here:

• Metallic disks (radius $r = 145$ µm) are interconnected by bars ($l = 800$ µm, $d = 10$ µm; thickness $h = 30$ µm), forming an exact kagome pattern.
• The system exhibits a plasmonic flat band: the resonant frequency is angle-independent, corresponding to zero group velocity over the entire Brillouin zone. Localization is topologically protected by the kagome geometry.
• The adjacency matrix $A_{ij}$ of disk connectivity yields eigenmodes confined to hexagonal loops ($\lambda = -2$); coupling between loops vanishes.
• The excitation spectrum is governed by a set of coupled equations,

$$\sum_{j} A_{ij} \tilde{q}_{j} = \frac{4 - (\omega/\omega_{0})^{2}}{1 + \kappa (\omega/\omega_{0})^{2}} \tilde{q}_{i}$$

with $\omega_{0} = 1/\sqrt{LC}$ and coupling $\kappa = C_{M}/C$.

Experimental terahertz transmission studies demonstrate that in the "perpendicular" configuration (electric field orthogonal to wavevector), transmission minima linked to the flat band persist for all incident angles, verifying the nondispersive character.

Implications: Such flat bands enable slow-light devices, enhanced light–matter interaction, and potential routes to many-body effects, with robustness dictated by the topological design.

3. Kagome Metamaterials in Photonic, Mechanical, and Acoustic Systems

Photonic Crystal Fibers

Kagome hollow-core photonic crystal fibers (HC-PCFs) guide light via “inhibited coupling” rather than conventional photonic bandgap effects (Fsaifes et al., 2016). Notable characteristics include:

• Large mode-field diameter (up to 40 µm), promoting high free-space coupling efficiency (>90%).
• Negligible power fraction in silica ($\eta \sim 10^{-4}$ to $10^{-2}$), nearly eliminating Kerr nonlinearity ($$\Delta\dot{\theta} = \frac{c n_{2}}{D n_{0}} \Delta{P} \eta / \sigma$$).
• Broad transmission bandwidth from UV to IR.

Test ring resonators built from Kagome HC-PCFs achieve high finesse (~30), contrasts (up to 89%), and minimal Kerr-induced bias, validating their utility for high-precision rotation sensing.

Acoustic Metamaterials & Higher-Order Topology

Kagome lattices in acoustic metamaterials offer realizations of higher-order topological insulators (HOTIs) (Xue et al., 2018):

• Breathing Kagome lattice: two adjustable nearest-neighbor couplings ($t_1, t_2$) produce quantized Wannier centers that shift between triangle types upon tuning $t_1 / t_2$.
• Second-order TI: only zero-dimensional (0D) corner states appear, depending both on bulk topology (Wannier centers) and corner geometry. Acute-angle corners localize topologically protected states.
• Polarization defined as

$$P_{i} = -3S \int_{\text{BZ}} A_{i}(\mathbf{k}) d^{2}k$$

with $A_{i}$ the Berry connection.

This geometry-dependent topological protection enables reconfigurable local resonances, resilient to disorder.

Mechanical Metamaterials

The kagome lattice supports unique periodic mechanisms and Guest-Hutchinson (GH) modes (Li et al., 2022). Explicit formulas relate two-periodic mechanisms to infinitesimal GH modes, with necessary conditions established for their correspondence. This elucidates the mechanical flexibility and tunability central to lattice metamaterial design.

4. Superdimensional Resonance and Density of States Engineering

Kagome metamaterials can be integrated with "superdimensional resonator" concepts to manipulate wave concentration and local density of states (Greenleaf et al., 2014):

• By tailoring the metamaterial’s governing equation to resemble degenerate Schrödinger operators,

$$\left( \frac{\partial^2}{\partial x^2} + x^{2r} \frac{\partial^2}{\partial y^2} + \omega^2 \right) u(x,y) = 0$$

the eigenfrequency count scales as $N(\omega) \sim \omega^{r+1}$ ($r \geq 2$), greatly exceeding the 2D Weyl law.

• Embedding such resonators in each kagome cell augments resonance density, wave concentration, and effective "dimensionality," facilitating applications in antennas and broadband devices.

Challenges involve realizing spatially graded parameters and mitigating cell-to-cell coupling effects.

5. Kagome Physics in Electronic and Quantum Materials

Recent quantum material research reveals that kagome-derived d-orbital bands are intricately intertwined with p-orbital bands from additional sublattices (Tsirlin et al., 1 May 2025):

• In AV$_3$Sb$_5$, FeGe, RV$_6$Sn$_6$, LaRu$_3$Si$_2$, p-bands from elements such as Sb, Ge, Sn, Si form distinct Fermi surfaces or hybridize with d-bands, critically influencing charge density wave (CDW) and superconducting instabilities.
• CDW order parameters ($\Delta_{\rm CDW}(\mathbf{q})$) and superconducting gaps ($\Delta_{\rm SC}$) involve carriers from both d- and p-bands, with pressure tuning and chemical substitution modulating the interplay.
• The enhanced density of states near van Hove singularities ($\rho(\varepsilon) \propto \ln|D/\varepsilon|$) is augmented or shifted by p-band effects.

A comprehensive microscopic understanding necessitates treating both d- and p-bands equivalently, with kagome p-sublattices acting as critical handles for tuning emergent states.

6. Tunability, Anisotropy, and Advanced Applications

Surface States and Interlayer Coupling

Angle-resolved photoemission spectroscopy (ARPES) studies on RV$_6$Sn$_6$ demonstrate (Peng et al., 2021, Ding et al., 2023):

• Nearly ideal 2D kagome surface band structures (Dirac cone, saddle point, flat band).
• "W"-like kagome surface states with intensity controlled by interlayer coupling and the choice of rare-earth R-element, modeled perturbatively as $H_1 \sim -J_H \sqrt{dG}$.
• Tunability by interlayer electron hopping opens platforms for quantum device development and topological electronics.

Optical Anisotropy

In FeSn, pronounced optical conductivity anisotropy is observed (Ebad-Allah et al., 18 Apr 2024), contradicting strict 2D conduction models:

• Out-of-plane conductivity ($\sigma_{zz}$) exceeds in-plane ($\sigma_{xx}$) below 1.1 eV.
• Kubo–Greenwood formula: $$\sigma_{k,\alpha\beta}(\hbar\omega) = \frac{i e^2}{\hbar V} \sum_{k,n,m} (f_{m,k} - f_{n,k}) \frac{(\epsilon_{m,k} - \epsilon_{n,k})}{(\epsilon_{m,k} - \epsilon_{n,k} - (\hbar\omega + i\eta))} A^{(H)}_{nm,\alpha}(k) A^{(H)}_{nm,\beta}(k)$$ demonstrates that interlayer Fe–Sn excitations dominate the low-energy optical response.
• This necessitates considering full 3D crystal anisotropy in kagome metamaterial charge dynamics.

Two-Dimensional Kagome Monolayers

In MoTe$_{2-x}$ kagome monolayers (Dai et al., 26 Aug 2024), mirror twin boundary (MTB) loops create two kagome band sets (KBS), with occupancy tunable by geometric parameters. Selected configurations exhibit topological (non-zero $Z_2$) phases or metallicity with Stoner ferromagnetism, highlighting the versatility in engineering 2D kagome-based metamaterials for quantum, spintronic, and correlated-electron applications.

Photonic Topological Metasurfaces and Leaky-Wave Antennas

Kagome metasurfaces in spin photonic topological designs (Abtahi et al., 5 Feb 2025):

• Block backscattering, support robust edge states, and allow propagation along sharp turns and fractal boundaries (e.g., Koch snowflake interfaces).
• Kagome unit cells furnish a 33% larger topological bandgap and continuous edge modes compared to hexagonal or rhombic alternatives.
• Application in X-band leaky-wave antennas: armchair interfaces enable two forward and two backward scanning beams (each with a ~50° scan) across 8.8–11.1 GHz. The electric field remains confined to the edge within the operational bandgap, providing directional and frequency agile performance.

7. Perspectives and Future Directions

Kagome metamaterials are a convergent domain where lattice topology meets engineered functionality:

• Wave localization, slow-light, and sensing: Flat-band physics delivers zero-group-velocity modes for delayed or localized electromagnetic/acoustic energy.
• Topological devices: HOTIs, reconfigurable resonators, and robust edge modes enable backscattering-immune and defect-tolerant platforms.
• Quantum matter and electronic instabilities: The interplay between d- and p-orbitals underlies tunable CDW, superconductivity, and magnetic order, broadening the scope of quantum materials.
• Superdimensional and anisotropic designs: Control over density of states, resonance spectra, and transport directionality expands device applications to antennas, sensors, and photonic circuits.
• Customizable 2D systems: MTB-engineered monolayers provide programmable topological and magnetic functionalities.

This suggests that future research will increasingly focus on multi-orbital engineering, interlayer coupling modulation, and topology-driven design strategies to realize kagome metamaterials with tailored properties for photonic, quantum, mechanical, and electronic technologies.