Pseudospin-dependent phononic topological waveguide
3.3. Phononic dispersion analysis
Our underlying hexagonal lattice based phononic structures start from a constant lattice parameter, a
= 5.0µm and symmetric radius, R0 = 3.5µm that maintain translation symmetry in the primitive cell and
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in the enlarged supercell as shown in Figure 3.2b. In order to break translation symmetry in the primitive cell level whether maintain its symmetry in the enlarged supercell level, so-called manipulation of duality symmetry, we focused on the changing interaction between nearest cell. In our veinless structure, by changing circle to ellipse under same center position, we can enforce the inside interaction or outside interaction between atoms in the enlarged supercell. If we increase the radius in vertical direction and decrease it in horizontal direction as a1=R0+d and b1=R0-d (Figure 3.2a), the stiffness of structure between outside of the enlarged supercell increase then corresponding deformation in the supercell is relatively de-forced. On the contrary, if we decrease the radius in vertical direction and increase it in horizontal direction as a1=R0-d and b1=R0+d (Figure 3.2c), the stiffness of structure between inside radius is increase and the resistivity between nearest supercell become relatively free. Therefore, we define a breaking symmetry parameter as gd that describe the order of symmetry breaking by changing radii of diatomic positions simultaneously as gd = (b1-a1)/2 = 2d/R0 where R0 is the radius that makes the structure maintain its overall symmetry and d is radius changing value which makes the circle to ellipse in complementary fashion in order to sustain overall frequency range by preventing excessive frequency range variation.
Figure 3.2d-f demonstrate the frequency dispersion results which induced by symmetry breaking where d = ±0.2. For symmetry, gd = 0, double Dirac cones by zone folding calculation occurred at 86.117MHz at the Γ point of the Brillouin zone. These four-fold Dirac bands are degenerate with changing the value of gd. In case of gd = -0.057, lower bands and upper bands have distinct Dirac frequency at 85.61MHz and 87.586MHz that makes 1.976MHz-wide-bandgap while gd = 0.057, lower and upper bands have their extreme value at 85.772 MHz and 87.413MHz that makes 1.641MHz- wide-bandgap (blue box). These asymmetry bandgap emergences indicate not only Dirac cone degeneracy but also the band inversion between two symmetry breaking states. In Figure 3.3, we describe the continuous bandgap emergence with respect to changing value of gd. They tend to enlarge bandgap initially then increase the overall frequency region increase rapidly with local minimum frequencies at gd = ±0.057. Moreover, the visualization of vibration eigenmode reveals the band inversion between p states and d states. In Figure 3.3a, two-fold lower bands show dipole modes such as 𝑝y and 𝑝z modes and the other two-fold upper bands show quadratic modes such as 𝑑yz and 𝑑y|}z| that refers to trivial topological states. On the other hand, in Figure 4.3c, lower bands show quadratic modes and upper bands show dipole modes that reveals band inversion induced by duality symmetry breaking gd.
In veined structure, we implement breaking duality symmetry in a similar way of veinless structure by changing the thickness of inside vein and outside vein as shown in Figure 3.4b. If we decrease the
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thickness of outside vein as tb = t0-t and ta = t0+t (Figure 3.4a), corresponding stiffness between enlarged cell increase while inside veins have relatively lower stiffness and such stiffness change induced by thickness change has opposite tendency where tb = t0+t and ta = t0-t (Figure 3.4c). In the same way, we define this symmetry breaking as gt = (tb-aa)/2 = 2t/t0 where t0 is the thickness of vein which makes the structure intact translation symmetry. Figure 3.4d-f demonstrate the frequency dispersion band structure induced by changing gt where t = ± 0.2 in complementary fashion. In symmetry case, four-folding Dirac cone dispersion at Γ point in the Brillouin zone supports gapless frequency dispersion with Dirac frequency as 151.42MHz where t0 = 2.5µm and R0 = 3.0µm. The degeneracy of corresponding folding bands are initiated with changing gt. In case of gt = - 0.08, 3.48MHz-wide-bandgap emerges from 147.71MHz to 151.19MHz while 4.1MHz-wide-bandgap emerges from 150.43MHz to 154.53MHz in opposite gt value. The emergency of bandgap initiates from thickness change of vein and enlarged with respect to increasing the breaking symmetry ratio (Figure 3.5b). In positive gt which means the enforced stiffness inside cell, band inversion occurs as lower bands to d-states and upper bands to p-states that represent non-trivial topological states.
Although we observe the band inversion as well as bandgap emergence in both veinless and veined structure, the frequency variation is too rapid in both symmetry breaking direction and get out the Dirac frequency of symmetry structure easily. In order to retain not only bandgap region but also sufficient bandgap width from four-fold Dirac frequency, 86.117MHz (veinless) 151.42MHz (veined), we implement uncomplimentary change of gt and gd . For veinless structure, dispersion parametric studies are conducted by decreasing the lattice parameter, a = 4.5 m (Figure 3.6a), decreasing the radius of symmetry circle R0 = 3.25 m (Figure 3.6b), increasing the horizontal axis ratio of ellipse b/a = 5/3 (Figure 3.6c) and increasing the vertical axis ration of ellipse b/a = 3/5 (Figure 3.6d). As a result, we derive proper bandgap emerged frequency region, 5.872-wide-bandgap from 84.949MHz to 90.821MHz for b/a = 5/3 in trivial state and 5.128-wide-bandgap from 84.708MHz to 89.836MHz for b/a = 3/5 in non-trivial state. For veined structure, we remove the circle in lattice site in order to focusing the effect of vein structure and similar frequency dispersion parametric studies are described by removing circle in lattice position (Figure 3.7a), increasing the thickness ratio of outer vein, tb/ta = 3/2 (Figure 3.7b), increasing the thickness ratio of inner vein, tb/ta = 1/3 (Figure 3.7c) and tb/ta = 1/4 (Figure 3.7d). As a result, we manipulate relatively stable bandgap region without radius, 18.46-wide-bandgap from 165.63MHz to 185.409MHz for tb/ta = 1/1 in non-trivial state and 8.35-wide-bandgap from 167.81MHz to 176.16MHz for tb/ta = 1/4 in trivial state.
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