Valley-dependent phononic topological waveguide
4.4. Full-field simulations
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edge states are localized in interface consisting ARM edge (Figure 4.11c).
Moreover, the wave propagation through topologically protected edge states in this system is clearly established through visualization of edge states propagation modes. If we look into deformation change along the time period at a certain frequency 102MHz and 186MHz in corresponding bandgap region, forward propagating modes induced by ZIG-L (ZIG-S) emerged from near K (K’) valley whereas backward propagating modes induced by ZIG-L (ZIG-S) emerged from near K’ (K) valley are identified.
In more detail, for ZIG-L (ZIG-S), the A-state shows clockwise propagating mode while B-state shows counterclockwise propagating mode at first bandgap region, 102MHz. These counter propagating characteristics pushing the propagation of wave in same direction at interface edge with a large interval in momentum space from K to K’ ensures valley-dependent interfacial wave propagation that represents an analogue to the QVHE, implying backscattering immunity as well as unidirectional wave propagating system. This demonstrates a topologically protected waveguide where waves can travel unimpeded along all possible paths in a hexagonal lattice.
Additionally, the edge states of veined structure in same methods are discussed and analyze how the relatively large order of breaking SIS affect to topological phase transition as shown in Figure 4.12. For ZIG edge states structured by veined structure did not show edge state at the first bandgap region. In second bandgap region, they show edge state like dispersion curve, although, they cannot intersect bandgap region from lower bulk bands to upper bulk bands. In third bandgap region which has localized Berry curvature that guarantee topological phase transition, edge states occur as same way in veinless structure at last. Moreover, we can identify the effect of breaking SIS order in veinless structure. If we increase g as 0.46, R1=3.7µm and R2=2.3µm, relatively large bandgap width from 90.38MHz to 106.6MHz is obtained as shown in Figure 4.13a. However, corresponding ZIG edge states did not support topologically protected edge states as veined structure. They only can show vibration insulating characteristic such as normal insulator.
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ZIG-S interface edge are constructed with 10x15 unit cells as shown in Figure 4.14a and Figure 4.15a.
In case of ZIG-L (ZIG-S) edge, anti-symmetrical (symmetrical) force excitation is applied at the center of domain edge and only frequency in bandgap region, 101MHz and 186MHz from dispersion analysis, are propagate along the interface line as shown in Figure 4.14c and Figure 4.14f (Figure 4.15c and Figure 4.14f). Two opposite direction of propagation supported by K and K’ valley enables the equal frequency response in both upward and downward simultaneously. For the frequency in bulk bands in dispersion analysis, wave propagation occurs not only along line but also through entire geometry (Figure 4. 14b, d, e, g and Figure 4. 15b, d, e, g). ARM interface edge is constructed with 16x10 unit cells as shown in Figure 4.16a and show same propagation results (Figure 4.16 b-g). As we mentioned previous section, ZIG-L edge and ZIG-S edge are protected by their own propagation mode such as anti-symmetry and symmetry. If we excite ZIG-L (ZIG-S) with symmetrical (anti-symmetrical) force, wave did not propagate even in frequency in bandgap region as shown in Figure 4.17a and Figure 4.17b.
In ARM edge case, although both excitation type available (Figure 4. 16c,f and Figure 4.17c), the remaining extremely small bandgap region in the first bandgap region disturb wave propagation along ARM edge (Figure 4.17d) at 101.4MHz.
The wave scattering between two opposite propagations at K and K’ valley is protected by large separation in k-space. In order to verify this topological feature, we apply phased excitation near edge of domain as 𝐹I= 𝐹Iexp 𝑖𝜔𝑡 + Π , 𝐹/= 𝐹/exp 𝑖𝜔𝑡 + Π/3 , 𝐹J = 𝐹Jexp 𝑖𝜔𝑡 . This phased excitation emulates pseudospin valley transport what we observed in edge state analysis. For all edge case, valley dependent wave propagation corresponds to phased excitation shows one-way wave propagation refers to each valley state in both frequency bandgap region as shown in Figure 4.18 and Figure 4.19. Moreover, remarkable topological feature is unidirectional wave propagation by valley dependent counter-propagation. In order to clarify this, we composed four distinct domains consisting of two A states and two B states distinguished by ZIG-L, ZIG-S and two ARM edge states as shown in Figure 4.20a, which induces wave propagation in the directions marked in yellow along the interfaces.
Waves excited from the red (blue) point propagate in a topologically-protected manner first along the ZIG-L (ZIG-S) edge, and subsequent along the ARM edge (Figure 4.20b). Although the opposite edge state ZIG-S (ZIG-L) has identical frequency dispersion, wave propagation through the opposite edge ZIG-S (ZIG-L) is locked since the ZIG-L edge state supports upward wave propagation while ZIG-S supports downward wave propagation. This propagating characteristic is analogous to pseudospin- dependent topological phase that ensures unidirectional wave propagation.
Another significant features of a topologically protected waveguide system is robustness against imperfections such as sharp edge corners and defects. We implement all possible edge path in structure hexagonal lattice array of our structure such as 60°, 90° and 120° angle transition. In order to confirm
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immunity to backscattering at sharp edge corners, four distinct types of sharp edge corners are employed to demonstrate the ability to support all possible wave propagation paths in a hexagonal lattice. A triangle shaped closed waveguide composed of two identical single ZIG (ZIG-L and ZIG-S) edges with 60° angles (Figure 4.21a), a parallelogram shaped closed waveguide composed of ZIG-L and ZIG-S edge with 120° angles (Figure 4.23a) and a square shaped closed waveguide composed of ZIG and ARM with 90° angles (Figure 4.22a) in 23x10 unit cells are adopted for verification of topologically protected wave propagation. The displacement field calculated from the frequency response for the excitation point (red or orange circle) with topologically-protected frequencies 101MHz and 186MHz shows stable and strongly localized wave propagation through the topologically-protected interface for all sharp angle corners (Figure 4. 21b-c, Figure 4.22 b-c and Figure 4.23 b-c). Interesting is that it looks wave propagation mode transition occurred between ZIG-L and ZIG-S. However, the wave propagation to ZIG-L to ZIG-S in same interface, A-B or B-A type, is wave transition occurred from K valley of ZIG-L to K’ valley of ZIG-S, so that mode transition is available in such case. As a result, the robustness to sharp bends enables arbitrarily complicated waveguides in practice. Moreover, in order to verify robustness to defects we deleted a single nanodrum near the path of ZIG-L in the triangular shaped closed waveguide (Figure 4.24a). The structure is subjected at both topologically-protected frequencies, 101MHz and 186MHz, and one way wave propagation to defect are applied by phased excitation. Under single atom defect and double atom defect in opposite section, the waveguide maintains their localized propagation (Figure 4.24a-b). In case of line defect, the localized propagation decreases slightly after passing line defect (Figure 4.24c) and extremely decrease under double atom defect extend in same section. This indiscrete double atom defect is very unusual defect condition, thus demonstrating robustness against imperfections by topologically protected states in general.
4.4.2. Real-time wave propagation
Finally, we verify the topologically-protected wave propagation using real-time finite element simulations via the commercial finite element code ABAQUS. In basic, we implement line path consisting of ZIG-L and ZIG-S interface with 7x14 unit cells as shown in Figure 4. 25a. Similar to previous sections, ZIG-L excited with anti-symmetric and ZIG-S excited at two distinct radii in ZIG edge in the center of domain with symmetric excitation. We detect the wave propagation in constant time step, period/20, 0.00049 s for 101MHz and 0.00024 s for 186MHz frequency excitation with a 10-count wave pulse in transverse direction. Figure 4. 25b-c and Figure 4.26b-c shows the real-time snapshot of wave propagation displacement field with same time period. In ZIG-L (ZIG-S), excitation from center propagate with anti-symmetry (symmetry) mode to ending edge of domain and stay in
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ending edge without reflection in both excitation frequency Recall the dispersion analysis in topological edge state section, propagation speed which corresponds to the gradient of edge states with respect to k is a little larger in case of ZIG-L (ZIG-S) edge than ZIG-S (ZIG-L) edge in lower (higher) frequency region. That is, the propagation in ZIG-L (ZIG-S) edge is faster than ZIG-S (ZIG-L) edge and reach to the ending edge quickly.
To demonstrate propagation over all available paths within the hexagonal lattice structure, we created an L-shaped path consisting of ZIG-L, ZIG-S, and ARM type edges as shown in Figure 4.27a. A 40- count wave pulse with frequency of 186MHz was used as the excitation on the ZIG-L edge. The wave pulse, which starts from the ZIG-S edge propagates through a 60° angle change to the next ZIG-S edge, after which a 120° angle change is made to propagate along the ZIG-L edge. Finally, a 90° angle change is made to propagate along the ARM edge. The snapshots of the wave propagation at different times in Figure 27b and Figure 28b clearly indicate the absence of backscattering at sharp corners, with the energy transported along the interface. This result is completely consistent with the previously discussed valley-dependent characteristics and frequency domain results, demonstrating robustness to sharp turns and corners and unidirectional wave propagation.