Topological nanophononic interface states using high-order bandgaps in the one-dimensional Su-Schrieffer-Heeger model

Topological interface states in periodic lattices have emerged as valuable assets in the fields of electronics, photonics, and phononics, owing to their inherent robustness against disorder. Unlike electronics and photonics, the linear dispersion relation of hypersound offers an ideal framework for investigating higher-order bandgaps. In this work, we propose a design strategy for the generation and manipulation of topological nanophononic interface states within high-order bandgaps of GaAs/AlAs multilayered structures. These states arise from the band inversion of two concatenated superlattices that exhibit inverted spatial mode symmetries around the bandgap. By adjusting the thickness ratio of the unit cells in these superlattices, we are able to engineer interface states in different bandgaps, enabling the development of versatile topological devices spanning a wide frequency range. Moreover, we demonstrate that such interface states can also be generated in hybrid structures that combine two superlattices with bandgaps of different orders centered around the same frequency. These structures open up avenues for exploring topological confinement in high-order bandgaps, providing a platform for unveiling and better understanding complex topological systems.

Figure 1

Left: Frequency of the band edges bounding the bandgap as a function of the parameter $\delta$. (a)–(d) Band inversion of the acoustic bandgap around 9.3 GHz (a), 18.6 GHz (b), 28 GHz (c), and 37.3 GHz (d). The mode symmetries are indicated by orange (symmetric) and blue (antisymmetric) lines. The thickness ratio of GaAs/AlAs is indicated by the vertical dashed line in each case. Right: The unit cell is displayed with light- and dark-gray colors, representing AlAs and GaAs, respectively. The band-edge modes are shown in the unit cells for each bandgap.

Figure 2

Top panels: Band inversion of the (a)–(c) third acoustic bandgap around 28 GHz, and (d)–(i) fourth bandgap around 37.3 GHz. Bottom panels: calculated acoustic reflectivity spectra for the concatenated DBRs with the corresponding GaAs/AlAs thickness ratio $\delta$ marked as dashed lines on the top panels.

Figure 3

Integrand of the Brillouin cross section for interface states in the third bandgap (a),(b), associated to the structures shown in Figs. 2 and 2(c), respectively; and the fourth bandgap (c)–(f), associated to the structures of Figs. 2, 2, 2, and 2, respectively. The vertical red line indicates the interface between the two DBRs.

Figure 4

(a) Resonant frequency in the third bandgap under random perturbations, with a uniform distribution of width $\mathrm{\Delta }\delta /\delta$. The acoustic resonance frequency stays trapped at the bandgap center for the topological mode (blue) but undergoes variations in the Fabry-Perot (orange). (b) Acoustic quality factor under random perturbations. For both types of resonators, the acoustic quality factor drops by a factor of 10. Similarly, (c) and (d) display the same comparisons for the fourth bandgap.

Figure 5

Engineering of topological interface states. (a) Band inversion of the acoustic bandgaps. The dots show the bandgap openings and symmetries at the given $\delta$. Inset: Symmetry of the two DBRs concatenated. There is inversion only for the fourth bandgap. (b) Schematic of the unit-cell configurations at the interface between the two DBRs, with the dark-blue and green colors representing GaAs and the corresponding light colors representing AlAs. (c) Simulated acoustic reflectivity spectra for two topological resonators formed by two concatenated DBRs embedded in GaAs with ${\delta }_{\mathrm{top}}=-0.33$ and ${\delta }_{\mathrm{bottom}}=+0.33$. Likewise in (d)–(f) ${\delta }_{\mathrm{top}}=-0.15$ and ${\delta }_{\mathrm{bottom}}=-0.85$; (g)–(i) ${\delta }_{\mathrm{top}}=-0.8$ and ${\delta }_{\mathrm{bottom}}=+0.2$; and (j)–(l) ${\delta }_{\mathrm{top}}=+0.4$ and ${\delta }_{\mathrm{bottom}}=+0.6$.

Figure 6

Hybrid topological acoustic resonator. (a),(b) Band inversion of the acoustic bandgaps associated to the two concatenated superlattices, named (a) S1 (green frame), and (b) S2 (blue frame). The third bandgap of S1 and the second bandgap of S2 share the same central frequency, around 28 GHz. (c) Simulated acoustic reflectivity of the two DBRs. The green (blue) line corresponds to the band inversion diagram displayed in panel (a) [panel (b)]. (d), (e) Simulated acoustic reflectivity spectra for different concatenated DBRs embedded in GaAs. The relative thickness ratio $\delta$ of the corresponding DBRs is marked by dashed lines on panels (a) and (b). (f), (g) Acoustic displacement ${\left|u\left(z\right)\right|}^2$ of the modes at the edge of the bandgaps plotted on top of the unit cell for the two superlattices. The green (blue) unit cell corresponds to the band-inversion diagram displayed in panel (a) [panel (b)], and the dark (light) colors represent GaAs and AlAs, respectively.