Topological analysis

Valley-dependent phononic topological waveguide

4.3. Topological analysis

4.3.1. Topological invariants

In order to have deeper understanding of the topological transition, we introduce the viewpoint of a nonzero valley Chern number. Through the 𝑘 ∙ 𝑝 perturbation method, we can obtain the effective Hamiltonian⁹⁸

H_¿ 𝛿𝑘 = 𝑣_Á𝛿𝑘_y𝜎_y+ 𝑣_Á𝛿𝑘_z𝜎_z+ 𝑚𝑣_Á^/𝜎_t (4.1) where 𝑣_Á is the Dirac velocity of the conic dispersion. 𝛿𝑘 is the momentums deviation from the valley center K. 𝜎 are the Pauli matrices of the vortex pseudospins. More importantly, the mass term expressed as

𝑚 = ∆𝜔

2𝑣_Á^/ (4.2)

characterizes the different valley Hall insulators separated by the Dirac semimetal phase with m= 0 in the phase diagram. And ∆𝜔 is the bandwidth for the opening of the full band gap, which is evolved with g as shown in Fig. 2(a). It is indicated that the effective Hamiltonian is strongly dependent on the parameter g. Utilizing the eigenvector from the effective Hamiltonian in Eq. (4.1), the local Berry

34 curvature centered at K valley can be calculated as

¿ 𝛿𝒌 = 𝑚𝑣_Á

2 𝛿𝑘^/+ 𝑚^/𝑣_…^{/ J//} (4.3)

Therefore, the topological charges of the first band can be calculated by integrating the local Berry curvature in half of the Brillouin zone

C_𝑲=1

2𝑠𝑔𝑛 𝑚 (4.4)

The Chern number of K′ valley can be similarly derived from time-reversal symmetry. Thus, the K′

valley also carries the equal topological charges with opposite signs. Obviously, when the parameter g satisfies the conditions of g< 0 or g> 0 , each valley carries a nonzero topological charge with the opposite sign, which leads to the nonzero topological invariant: valley Chern number, Cv= CK− CK′, has non-zero value.

For g< 0 org> 0, the topological charge distributions at the K and K′ valleys are totally different.

Ideally, for g< 0, the topological charge at the K valley is +1/2 and the topological charge at the K′

valley is −0.5, which results in a positive valley Chern number, Cv= (CK− CK′) = 1. On the contrary, for g> 0, the topological charge at the K valley is −0.5 and the topological charge at the K′ valley is +0.5, which makes a negative valley Chern number Cv= (CK− CK′) = -1. Therefore, the topological phase transition accompanied with the change of topological charge and valley Chern number occurs when the g is equal to zero, which gives potential to valley dependent edge states by interfacing two hexagonal lattices with opposite g parameters. This topological invariant known as the Chern number, which is calculated by numerically integrating the Berry curvature over a small region near the K and K’ points (see Simulation Methods in Chapter 2). The Berry curvatures calculated in the 1^st Brillouin zone at k_x = -2p/3 are highly localized at the K and K’ points with opposite sign in case of veinless structure as shown in Figure 4.5. These valley-dependent topological invariants are also reversed for the lower and upper bands in both bandgap regions (Figure 4.5a and 4.5b). Since the valley Chern number, defined as C_v = C_k - C_k’, is non-zero and has opposite value, this valley-dependent Berry curvature shows that breaking SIS ensures a QVHE-like topological phase transition.

However, in case of veined structure shown in Figure 4.6, calculated Berry curvatures have opposite value in distinct valley K and K’ but the intensity of localization is not strong compared to veinless structure at the first and second bandgap region (Figure 4.6b,c). As the frequency region increase, the Berry curvature start to slightly deflect near K and K’ and show highly localization at valley point in

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third high frequency region. Again, relatively large degree of breaking SIS attributes to this result.

Because of large bandgap width in the first and second bandgap region of veined structure which initiated from relatively large degree of breaking SIS, corresponding frequency dispersion bands loss its strong Dirac characteristic at K point (see band 1, band 2, band 4 in Figure 4.2). That is, large order of breaking SIS did not guarantee topological phase transition unconditionally even though they have two opposite pseudospins at valley point.

4.3.2. Topologically protected edge states

To verify the presence of topologically protected edge states and validate topological phase transition what we discussed previous sections, we conducted frequency dispersion simulations to one dimensional supercell consisting of topologically phase transition edge not the primitive cell. In graphene like diatomic structure, there exist three types of edges that are possible for the hexagonal lattice structure (Figure 4.7): the two zigzag types (ZIG-L and ZIG-S) and armchair type (ARM). For clarity, the structures corresponding to g = -0.13 and g = 0.13 are labeled as A type (Yellow region) and B type (Green region) such that ZIG-L and ZIG-S edge states are defined as interface of A-B type and B-A type respectively. Specifically, a ZIG-L edge contains two lattice structures with R1, the larger part, and a ZIG-S edge contains two lattice structure with R2 while a ARM edge contains both four lattice structure with R1 and R2.

For ZIG-L edge , we created an 8x1 supercell composed of an A-B type, four hexagonal A type primitive unit cells– four hexagonal B type primitive unit cells, for the frequency dispersion calculation with Bloch periodic condition in k_x direction as shown in Figure 4.8a. Edge states marked as magenta solid lines are states that cross the corresponding bandgap regions for the first bandgap, 97.8 to 103.2 MHz, and the second bandgap region, 181.3 to 189.5MHz. They intersect these bandgap regions from distinct bulk bands to another bulk bands near the valley point K (K’), kx = 2/3 (-2/3), with 100.77MHz at the first frequency region and 185.42MHz at the second frequency region respectively (Figure 4.8b).

In detail, at the valley point K (K’), the edge states have positive (negative) gradient in the first bandgap region and negative (positive) gradient in the second bandgap region with respect to kx. These opposite gradients demonstrate that two distinct valleys have incompatible wave propagation characteristic, i.e.

forward-propagating at K and backward-propagating at K’ with direction of kx in the first bandgap region. In order words, the edge state guarantees two counter propagating characteristics simultaneously with large separation in k-space, so-called valley-dependent wave propagation. The deformed mode shapes at valley frequencies show that these anti-symmetry mode ZIG-L edge states are strongly

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localized within a single hexagonal lattice constant away from the interface at both 100.77MHz and 185.42MHz respectively, which facilitates topologically protected wave propagation through the interface (Figure 4.8c and 4.8d).

Similar frequency dispersion results can be obtained for ZIG-S edge, B-A type, sequence of four hexagonal B type primitive unit cells– four hexagonal A type primitive unit cells, as shown in Figure 4.9a. The edge states of ZIG-S intersect the corresponding bandgap region with 102,.56MHz and 188.65MHz respectively at the valley point K and K’ and k-space gradient of edge state at each valley has opposite value (Figure 4.9b). The deformed vibration mode shaped at valley show symmetry mode ZIG-S edge states and highly localized near interface as before (Figure 4.9c and 4.9d). There exist two differences between ZIG-L and ZIG-S edge states. First, the edge states in same bandgap region have opposite convexity that indicates forward propagation domination at K valley of ZIG-L edge while backward propagation domination at K valley of ZIG-S edge. Second, the vibration modes localization at interface show contrasting mode, anti-symmetry mode for ZIG-L edge and symmetry mode for ZIG- S. This observation indicates that wave propagation between ZIG-L to ZIG-S at same valley is invalid.

Although each edge state intersects same frequency range and support the presence of a topologically non-trivial interface connecting the A-B or B-A states with corresponding frequency region, anti- symmetry vibration mode cannot propagate through pure ZIG-S edge and symmetry mode cannot propagate through pure ZIG-L edge, so that one-way wave propagation protected by valley dependency is available in our structure.

In the case of ARM edge states, a supercell composed by the A-B interface is adopted for the frequency dispersion calculation in the same way as the ZIG edges as shown in Figure 4.10a, the edge states intersect corresponding bandgap frequency region at G point not K point because of its symmetry structure. Since the rotation of Brillouin zone occurs in case of ARM edge lattice structure, the Dirac points and intersecting point of edge states in momentum space are shifted to the G point from the K point. Hence, the topological phase transition occurs at G point and wave propagation has opposite value within positive G and negative G as shown in Figure 4.10b. The blue (red) solid line represents forward (backward) propagation in both bandgap frequency regions. It has gapless phase transition in high frequency bandgap region and show localization as shown in Figure 4.10c, however, there exist a relatively large bandgap in low frequency region. Since the intrinsic ARM edge has two large radii and two small radii simultaneous, the order of breaking C6 symmetry is too large compared to ZIG-L and ZIG-S edge. Although intrinsic ARM edge states at low bandgap region still support topological phase transition, we add lattice components with radius R₀ = 3µm in order to enhance symmetry and enforce gapless phase transition as shown in Figure 4.11a. The relatively balanced order of breaking C6

symmetry enables decreasing bandgap at the first frequency region (Figure 4.11b) and corresponding

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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.