Emulating 2D Materials with Magnons

TL;DR

The study tackles how magnonic systems can emulate key physics of 2D materials, such as graphene-like Dirac points, kagome-derived flat bands, and topological states, within experimentally accessible microwave regimes. It combines micromagnetic simulations of a hexagonal anti-dot YIG thin film with a nine-band tight-binding–like model based on $s$- and $p$-like orbitals on the honeycomb lattice plus $s$ orbitals on the kagome lattice, revealing graphene-like bands and emergent topology. Key contributions include tunable band gaps via inversion symmetry breaking, valley-polarized edge modes along 1D phase boundaries, and spectrally isolated defect modes below the bulk continuum, all of which can be engineered and observed in magnonics. The results establish a versatile platform that portends magnon-based valleytronics, robust waveguiding, and defect-state functionalities at microwave frequencies, while offering generalizable design principles applicable to other wave systems through a Schrödinger-like description of spin waves.

Abstract

Spin waves (magnons) in 2D materials have received increasing interest due to their unique states and potential for tunability. However, many interesting features of these systems, including Dirac points and topological states, occur at high frequencies, where experimental probes are limited. Here, we study a crystal formed by patterning a hexagonal array of holes in a perpendicularly magnetized thin film. Through simulation, we find that the magnonic band structure imitates that of graphene, but additionally has some kagome-like character and includes a few flat bands. Surprisingly, its nature can be understood using a 9-band tight-binding Hamiltonian. This clear analogy to 2D materials enables band-gap engineering in 2D, topological magnons along 1D phase boundaries, and spectrally isolated modes at 0D point defects. Interestingly, the 1D phase boundaries allow access to the valley degree of freedom through a magnonic analog of the quantum valley-Hall insulator. These approaches can be extended to other magnonic systems, but are potentially more general due to the simplicity of the model, which resembles existing results from electron, phonon, photon, and cold atom systems. This finding brings the physics of spin waves in 2D materials to more experimentally accessible scales, augments it, and outlines a few principles for controlling magnonic states.

Figures (10)

• Figure 1: Spin waves in (a) 1D and (b) 2D. Each arrow indicates a macrospin and its color indicates the direction of the in-plane magnetization and hence the phase of the precession. System geometry: (c) The hexagonal anti-dot lattice with a perpendicular field, leaving a film similar to a honeycomb lattice. (d) (shaded) our rectangular supercell, with orthogonal lattice vectors $333 \text{ nm } \times 589 \text{ nm}$, and (dashed) the typical unit cell of the hexagonal lattice. (e) Corresponding Brillouin zone (BZ) for the rectangular unit cell, with a few special points $\mathbf{X}$ and $\mathbf{Y}$ marked. (f) relationship between special points in the typical BZ (dashed, black letters), and their placement in the rectangular BZ (shaded, colored letters).
• Figure 2: The effect of hole diameter $d$ on YIG films with lattice parameter $a=333\text{ nm}$, thickness $t=15\text{ nm}$. Band structures are plotted as projections in the $(k_x,f)$ plane for: (a) $d/a=0.0$ (no holes; equivalent to the free spin wave spectrum), (b) $d/a=0.4$, and (c) $d/a=0.8$. Rectangular unit cells appear as insets. $B_{\text{ext}} ||\hat{z}$ is varied to keep the lowest-frequency mode constant. (d) a 3D volumetric plot of the projection in (c), more clearly showing the isotropic character of flat bands. The $d/a=0.8$ geometry is the main subject of this study. To be more precise, the plots are the magnitude of the response $|\Psi(f,x,y,k_x,k_y)|$ to a $\delta$-like excitation, displayed as projections on to only a few axes: $(k_x,f)$ for 2D, or $(k_x,k_y,f)$ for 3D.
• Figure 3: Under periodic boundary conditions, a rectangular region (yellow) is continuously excited at the first flat band frequency of 2.08 GHz. Scaled profiles of $|\Psi|^2$ are plotted after 50 ns for the geometries corresponding to (a,b,c) in Fig. \ref{['Crystal:fig:Bands']}. The strong localization in (c) demonstrates the exceptional flatness of the band. Note the scale differences: the flat band mode is excited to $\approx1000 \times$ the density of its counterparts, making it potentially useful in easily reaching the nonlinear regime. (d) Zoomed-in plot of the mode in (c), with phase encoded in color. (left) Key for color plots and (right) the localized wavefunction of the kagome lattice, which this mode closely resembles. This is an example of how $\Psi$ can be useful to interpret as a wavefunction.
• Figure 4: (a) Magnonic band structure for $d/a=0.8$ and $B_{\text{ext}}=185\text{ mT}$ along a few paths in the rectangular BZ. (b) A few examples of Bloch function-like responses at the $\mathbf{\Gamma}$ point, and (c) their reconstructions using a small number of basis 'orbitals,' discussed in more detail in Sec. \ref{['Crystal:sec:TB_Analysis']}. (right) Key for color plots of $\Psi$. The main finding is that the Bloch functions are simple, suggesting a tight-binding model.
• Figure 5: (a) Simulated $k_x$-projected band structure as plotted in Fig. \ref{['Crystal:fig:Bands']}(c), and (b) its 9-band tight-binding model using fitting parameters listed in table \ref{['Crystal:tab:TBParams']}. (c) Basis orbitals for the TB model (left to right): $s,p_x,p_y$ orbitals on honeycomb sites, and $s$ orbitals on kagome sites. (d) Illustration of fitting parameters.