Pseudospin-dependent phononic topological waveguide
3.4. Topological analysis
23
24 C±=1
2[𝑠𝑔𝑛 𝑀 + 𝑠𝑔𝑛(𝐵)] (3.3)
Since 𝐵 has typically negative value, the spin Chern number depends on the sign of 𝑀. Therefore, the bandgap emergency without band inversion has zero spin Chern number, so-called topological trivial state while the band inversion emergency between p states and d states in non-trivial state give negative sign of 𝑀 and following spin Chern number has value of ±1.
This topological invariant, the spin Chern number, which is calculated by numerically integrating the Berry curvature over a Brillouin zone (see Simulation Methods in Chapter 2). The Berry curvatures calculated in the Brillouin zone band 2, band 3 corresponds to inversed d states (Figure 3.8a) and band 4, 5 corresponds to inversed p states (Figure 3.8b) in non-trivial states. Two opposite berry curvature in each bands 2, 3 and band 4, 5 stand for the existence of pseudospin states and the inversion between band 2, 3 and band 4, 5 signify the topological phase transition due to band inversion at Γ point. The numerical integration of each inversed peak at Γ point gives π (band 3 and band 5) and -π (band 2 and band 4) that represent Chern number as +1/2 and -1/2 and thus the spin Chern number obtained as Cs =
±1. Therefore, a QSHE-like topological phase transition with two distinct pseudospin states is guaranteed.
3.4.2. Topological protected edge states
The abovementioned band inversion and topological phase transition induced by breaking translation symmetry of the enlarged supercell under zone folding technique enables the realization of topologically protected edge states based on QSHE analogy. In order to validate the presence of edge states which consist of topological trivial and non-trivial structure, we implement one dimensional frequency dispersion calculations. Figure 3.9a and 3.9b shows the two possible lattice structure that connects trivial and non-trivial phases: the zigzag types (Figure 3.9a) and the armchair type (Figure 3.9b). These interfaces marked as blue and red interaction simultaneously in both illustrations can be applied directly to veinless structure by simple superposition of the connecting components (Figure 3.9c), however, the connecting components show incompatible interface in veined structure (Figure 3.9d). If we consist the interface as it is, veined structure induces the structure inconsistency at interface. Then additional manipulation at the interface is inevitable and topological edge states have dependency of the interface construction. In this reason, we consider the edge states of veinless structure in following topological analysis.
25
For zigzag edge, we construct an 8x1 supercell consists of the four trivial type and four non-trivial type for frequency analysis with Bloch periodic boundary condition in one dimensional k-space, kx, as shown in Figure 3.10a. In order to reduce the effect of localization of fixed edge at end side, we modify the ends of trivial edge. The dispersion results described in Figure 3.10b demonstrate the existence of topological edge states in zigzag interface (bold line) and looks like intersecting the corresponding bandgap regions at Γ point (kx = 0) which is obtained by previous section, 84.949MHz to 88.566MHz.
Although the small bandgap emerges from 87.343MHz to 87.771MHz in edge states which is induced by local C6 symmetry, the relatively small bandgap width 0.428MHz-wide and strong localization at interface in Γ point described in Figure 3.10c ensure the topologically protected zigzag edge. More detail, the edge states seems to form a convex and concave shape of dispersion with symmetry in Γ point but the corresponding eigenvector, the vibration mode at a specified frequency in bandgap region reveals the topological phase transition occurred at Γ point. In order to verify the topological phase transition, we consider four points in edge state, red-blue marked and border respectively (Figure 3.10b).
At lower band of the edge state, highlighted with blue border, the topological edge state shows forward propagating modes in negative kx (red marked) while shows backward propagating modes in positive kx (blue marked). On the other hand, the edge state shows backward propagating modes in negative kx
(blue marked) while shows forward propagating modes in positive kx (blue marked) in the upper edge state band. That is, the topological phase transition emerges at Γ point and each phase show counter- propagation mode that is analogous to two pseudospins with opposite direction in QSHE. Moreover, the localization of propagating transverse vibration shows the two vibration mode, symmetry mode 𝑆 = (𝑝y+ 𝑑y|}z|)/ 2 and anti-symmetry mode 𝐴 = (𝑝z+ 𝑑yz)/ 2 , and corresponding pseudospins can be expressed as 𝑆 + 𝑖𝐴 and 𝑆 − 𝑖𝐴 respectively with respect to the evolution of wave propagation.
For armchair edge, similar topological phase transition can be obtained in same way in zigzag edge.
Again, 8x1 unit cell consisting of trivial and non-trivial topology is constructed and the length in kx
direction increase compare to zigzag edge because of the rotation of unit cell in armchair edge as shown in Figure 3.11a. It seems to occur strong localization and emerge perfectly extended topological phase transition at Γ point in corresponding bandgap region with 87.57MHz (Figure 3.11b). Furthermore, the visualization of eigenmode shows highly localization of deformation near interface and counter- propagating wave propagation characteristic as same in zigzag edge states. These counter propagating characteristics in both zigzag and armchair edge under overall k-space by pushing the propagation of wave in uni-direction at interface edge ensures the spin-dependent interfacial wave propagation that conclude an analogue to the QSHE, implying immunity to backscattering induced by defect and boundary as well as unidirectional one-way wave propagating system. This demonstrates a
26
topologically protected waveguide where waves can travel unimpeded along all possible paths in a hexagonal lattice.