3J2-2
Multichannel topological waveguide in
Kagome phononic crystal

multilayer

Yusuke Hata1‡ and Kenji Tsuruta1* (1Okayama Univ.)
1. Introduction
Conventional elastic devices, represented bysurface acoustic wave (SAW) filter, have 2dimensional structure. While it is quite stable tofabricate, it has several disadvantages. First, anadditional device cannot be mounted directly on topof the structure, which hinders further integration.
Second, it is difficult to hold multichannel andmultifunctionality in planar structure. On the otherhand, 3-dimensional elastic devices can achieve
multichannel and multifunctionality with higher
integration. In the present study we designed awaveguide perpendicular to substrate with layerstacked topological phononic crystal.1)
Topological phononic crystals (PnCs) areartificial acoustic/elastic structures to realize
efficientacoustic-/elastic-wave
propagations.
Topological insulators have robust boundary statesagainst bend, disorder, and defects.2) A lot of study
about robust propagation using topological PnCs
have been reported.3-7) Recently, multilayer
waveguides using topological PnCs have beenproposed and demonstrated in acoustic system.8,9)Multilayer system has not only valley (wave numberdependent) degree of freedom (DOF) but also layer
DOF, thus layer polarized states appear in thatsystem. Layer polarized states also lead tomultichannel features in the waveguides, e.g., layer
converter and wave splitter. Furthermore, multilayer
is compatible with the development of 3dimensional integrations. While the study on
multilayer topological PnCs in acoustic system israpidly increasing, the efforts to develop multilayer
topological PnCs in elastic system are limited in aspecific lattice.10)
In this study, we investigate topological boundarymodes with Kagome lattice, aiming to develop an
efficient multichannel waveguide through thetopological band design.
2. Bilayer Kagome phononic crystal
We propose a bilayer Kagome PnC to design
multichannel waveguides. Kagome lattice is second
order topological insulator;1,11,12) the topologicallyprotected states appear in the physical region of twoorder lower than the dimension of the system. In themultilayered systems, selective wave transmissions
between the layers can be achieved throughE-mail: ‡y_hata@s.okayama-u.ac.jp, *tsuruta@okayamau.ac.jp

Fig. 1 (a) Schematic figure of bilayer Kagome
lattice. Strengths of bonds (tLI, tLO, tUI, tUO)
determines topological states. (b) Dispersion
relation in N state, (c) L+ and L− states, and (d)V+ and V− states.
interconnections vertically implemented in themultichannel waveguides. The schematic drawing ofbilayer Kagome lattice is shown in Fig. 1(a). Sixsites are bridged by bonds with four different
strengths (red, blue, yellow, and green), and upperlayer and lower layer are connected with thin (graydots) bonds. By changing intracell- and intercellbonds in both upper and lower layer, one can control
topology of elastic modes in this structure. We nameeach state by those strengths. N (neutral) statesatisfies ȁ‫ݐ‬௅ூ ȁ ൌ ȁ‫ݐ‬௅ை ȁ ൌ ȁ‫ݐ‬௎ூ ȁ ൌ ȁ‫ݐ‬௎ை ȁ . L+ (L−)ȁ‫ݐ‬௅ூ ȁ ൐(layerpolarized)
statesatisfiesȁ‫ݐ‬௅ை ȁ ሺȁ‫ݐ‬௅ூ ȁ ൏ ȁ‫ݐ‬௅ை ȁሻ and ȁ‫ݐ‬௎ூ ȁ ൏ ȁ‫ݐ‬௎ை ȁ ሺȁ‫ݐ‬௎ூ ȁ ൐ȁ‫ݐ‬௎ை ȁሻ . V+ (V−) (valley polarized) state satisfiesȁ‫ݐ‬௎ூ ȁ ൏ȁ‫ݐ‬௅ூ ȁ ൏ ȁ‫ݐ‬௅ை ȁ ሺȁ‫ݐ‬௅ூ ȁ ൐ ȁ‫ݐ‬௅ை ȁሻandȁ‫ݐ‬௎ை ȁ ሺȁ‫ݐ‬௎ூ ȁ ൐ ȁ‫ݐ‬௎ை ȁሻ.
Dispersion relations of bilayer Kagome PnCs
with various states are shown in Fig. 1(b-d). Energy
is doubly degenerated at K point and around K point
in N state because of the symmetry protecting. If thesymmetry is broken, i.e., strengths of bonds arechanged, degeneracy is lifted up and bandgap opens.
By remaining some symmetry, Lʸ states still havedegenerate at K point.

3. Boundary mode with various types of edge
shapesThere are three types of edge shapes in Kagome
lattice. In this study, we focus on a straight boundary(Fig. 2(a)). We investigate topological boundaryFig. 2 (a) Straight boundary. (b) Dispersion
relation of supercell with L+ and L− region. Redand blue lines denote polarized boundary modes
in upper and lower layer. (c) Dispersion relation ofsupercell with V+ and V− region. Green lines show
non-polarized boundary modes. In (b) and (c),gray dots show bulk and trivial boundary modes.

modes at straight edge shape with finite element
method (FEM) simulations. We name each region bywidths of bonds ሺ‫ݓ‬௅ூ ǡ ‫ݓ‬௅ை ǡ ‫ݓ‬௎ூ ǡ ‫ݓ‬௎ை ሻሾሿ ; L+:(1.5, 0.5, 0.5, 1.5), L−: (0.5, 1.5, 1.5, 0.5), V+: (0.5,1.5, 0.5, 1.5), and V−: (1.5, 0.5, 1.5, 0.5).
The dispersion relations and boundary modes areshown in Fib. 2(b, c). There are polarized modes inupper layer from the top of bulk bands and lower
layer from the bottom of bulk bands at a straightboundary between L+ and L− region (Fig. 2(b)).
Group velocity of each mode has opposite sign. On
the other hand, there are non-polarized modes fromthe top of bulk bands at a straight boundary between
V+ and V− region (Fig. 2(c)). Group velocity of eachmode has same sign, and these frequencies arealmost the same as polarized modes in upper layer
between L+ and L− region (Fig. 2(b)). Slightdifference of these frequencies can be attributed tothe symmetry breaking. In Fig. 2(b), polarizedmodes in upper and lower layer are inverted if thepositions (red and gray region in Fig. 2(a)) of L+ andL− region are inverted. This is a natural consequence
considering that the operation of inversion of region
is equivalent to rotate 180 along boundary axis.
4. Multichannel waveguides
In this section, we design a multichannel
waveguide using topological boundary modes
demonstrated in previous section. Here the FEM
simulations are conducted in frequency domain.
Displacement field of a layer converter is shownin Fig. 3. L+/L− regions are in left and right side andfront-back position of L+ and L− is opposite. V+/V−
region is in middle side. Pressure of 1Pa at 19.5 kHz
along x direction is applied to a straight boundary ofleft side of the upper layer. First, the wave
propagated in upper layer at left side, following withdispersion relation of Fig. 2(b). Second, both layers
vibrated at middle side, following with Fig. 2(c).
Finally, path of elastic wave changed to lower layer
at right side because front-back position of L+ and L−
region was opposite to left side.

5. Conclusion
We proposed bilayer Kagome PnCs which havelayer DOF and investigate topological boundarymodes with a straight edge shape. It is revealed that
there are polarized modes in upper and lower layer
at a boundary between L+ and L− region. In contrast,
there is no polarized modes at a boundary between
V+ and V− region. Finally, we demonstrated amultichannel waveguide which converts wave fromone layer to another layer by FEM simulations. The
present study paves the way to design
multifunctional 3-dimensional elastic devices.

Fig. 3 Displacement filed of an elastic layer
converter with a bilayer Kagome PnC. Inputand output area show with arrows.

Acknowledgment
This work was supported by Japan Society for thePromotion of Science grant number JP21H05020.
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