1. Introduction

Vibration control of structures has been the focus of studies [1-4]. The concept of directional waveguides in metamaterials is a widely researched topic [5,6]. In recent decades, topological insulators (TIs) or high-order topological insulators (HOTIs) have attracted extensive attention in the field of condensed matter physics owing to their robustness against defects, such as cavities, imperfections, and sharp edges, and the immunity of interface guided waves to backscattering [7]. More recently, the concept of TIs has been developed rapidly for applications in mechanical and acoustic metamaterials. In addition, the quantum Hall effect (QHE) [8,9], quantum spin Hall effect (QSHE) [10], and quantum valley Hall effect (QVHE) [11-13] have been used in phononic crystals (PCs) to realize a series of remarkable properties, such as waveguides [14-18], topological transport [19], rainbow-waveguides [20], and topological acoustic delay lines [21]. In particular, QVHE has attracted significant attention because it relies on breaking the spatial inversion symmetry, which easily opens Dirac cones in practical applications [22].

PCs based on the valley Hall effect comprise Dirac cones, which are formed by the intersection of two orders of frequencies in their band structures. The Dirac cone can be opened by breaking the space inversion symmetry to obtain a complete bandgap in the band structure of the valley PC. Valley PCs have topological edge modes that enable the propagation of acoustic waves at their edges within the bandgap frequency range. The acoustic waves that can propagate through the edges are called topological protected edge waves (TPEWs) [23].

In comparison with those based on the spin Hall effect and Hall effect, PCs based on the valley Hall effect can open the Dirac cone relatively easily. The application of valley TIs in acoustic systems has been studied extensively. However, owing to the fixed structural parameters, realizing dispersion tuning of acoustic topological states and reconfiguration of acoustic propagation routes is challenging. In recent years, there have been a few attempts to realize topological PCs. For example, Zhou et al. [24] applied a voltage to a membrane-based acoustic metamaterial which was sprayed with tiny and heavy particles to modulate its topological properties. Furthermore, Xia et al. [25] programmed different coding sequences by designing digital elements (0 or 1) to achieve programmable acoustic TIs. In valley topology acoustic systems, switchable asymmetric acoustic transmission can also be achieved by adding acoustic metasurfaces [26].

Scholars have studied wave propagation in chiral structures for several years [27,28]. Chiral structures are often used for the modulation of wave propagation for applications such as energy harvesting [29], waveguiding [6,30], antivibration [31], and negative refraction [32]. However, the implementation of TPEWs in chiral structures has not been extensively studied because chiral structures do not exhibit inversion symmetry. Recently, Wen et al. [33] proposed chiral edge states with higher robustness in comparison with that of the topological design of C6v. Inspired by the chiral metamaterials in the abovementioned studies, we constructed unit cells with chiral PCs to demonstrate the existence of topological edge states.

In this study, we propose valley PCs with chiral symmetry, which consist of a hexagonal plate and six straight ligaments. Based on the dispersion relation and edge mode of the model, we performed dispersion tuning and route reconfiguration of acoustic waves for valley PCs based on the valley Hall effect. The PCs enable flexible adjustment of their material parameters or topology to achieve dispersion tuning. We aimed to obtain robust TPEWs with designed routes and dispersion by analyzing the bandgap frequencies and edge mode frequencies using numerical simulations. Numerical analysis of the simulation model revealed that the valley PCs proposed in this study support TPEWs, and we demonstrated the propagation characteristics of TPEWs on different routes through frequency and time domains. The propagation properties of TPEWs were simulated in the frequency domain and were compared with the calculated transmission rate curves. Furthermore, to verify the robustness of TPEWs, we simulated the propagation characteristics of TPEWs on the interface routes with defects. Finally, we extended the analysis of the properties of TPEWs to the time domain to verify propagation immunity to backscattering.

2. Unit cells with valley Hall effects

The specific configuration of the model and selection of the Brillouin zone is illustrated in Fig. 1. Figure 1a depicts a chiral topological PC consisting of a hexagonal plate and six rectangular ligaments. The yellow part of the medium represents air. Each chiral PC consisting of a hexagonal plate and six rectangular columns was distributed in a graphene-like two-dimensional (2D) honeycomb lattice. The side length of the plate R was 5 mm, the length of the surrounding ligaments b was 20 mm, and the width t was 2 mm. The angle between the ligament and center plate φ was 35°. Each ligament extended from the center of the side length of the hexagonal plate. The lattice constant a = 52 mm, the material used was steel, Young’s modulus E = 200 GPa, Poisson’s ratio ν = 0.3, and density ρ = 7850 kg/m³.a Designed chiral unit cell composed of a hexagonal plate and six rectangular ligaments. b Brillouin zone. c-d Two types of disturbed unit cells with alternate distributions of ligaments of lengths 12 and 20 mm, respectively.

Floquet boundary conditions were added to simulate the periodicity of the unit cells through COMSOL Multiphysics, and the dispersion relation for valley PCs was obtained by sweeping the wave vector k along the boundaries of the first irreducible Brillouin zone, which is surrounded by blue lines in Fig. 1b. The vertex coordinates of the surrounded area were Γ (0, 0, 0), K (1/3, $-\sqrt{3}$/9, 0) and M (1/6, $-\sqrt{3}$/6, 0) in the reciprocal space.

Dirac cones are presented in the energy band structure of the chiral model. To realize the Dirac cones, we designed a unit cell consisting of two types of peripheral ligaments with different lengths and a central plate. We considered the difference between the two types of peripheral ligament lengths ∆b = b₁ − b₂ as the key geometrical parameter. Subsequently, we changed the length of specific ligaments and observed that the Dirac cone opened to form a complete bandgap. Using the abovementioned method, we can control the wave propagation route of the topological materials with different values of ∆b as a parameter to form a valley topological system.

Figure 1c-d depict a unit cell of Type I with ∆b = 8 mm(b₁ = 20 mm, b₂ = 12 mm) and a unit cell of Type II with∆b = −8 mm (b₁ = 12 mm, b₂ = 20 mm), respectively. We designed these two disturbed unit cells to lift the Dirac degeneracy and form topological interfaces. For the two perturbed unit cells, the theoretical valley Chern numbers at the corner points K and K′ were –1/2 (1/2) and 1/2 (–1/2), respectively, indicating opposite values. The unit cell of Type I could be obtained by rotating the unit cell of Type II by 60° without varying the structural parameters of the unit cell.

3. Analysis of band structure and mode shape

In this section, we adopted the pressure acoustics module in COMSOL Multiphysics to calculate the energy band structure of the unit cells. The wave vectors were extracted at the integrable Brillouin zone boundary to solve the eigenvalue problem. Figure 2a depicts the band structure of the chiral PCs that feature the Dirac degeneracies at the K points of the first and second frequencies (indicated by red circles). The propagation of acoustic waves in the chiral lattice was restricted to 2D propagation. Therefore, the thickness of the chiral unit cell did not affect the band structure. By breaking the inversion symmetry, the Dirac cone in the initial unit cell was opened and a complete bandgap was obtained (red dots in Fig. 2b), which emerged as two separated modes at the K and K′ points with opposite valley Chern numbers, as depicted in Fig. 2c. Therefore, the topologically protected edge state could theoretically be formed at the interface between the two types of unit cells, as illustrated in Fig. 1d.a Dispersion curves for chiral valley PCs where the length of the ligaments is b = 20 mm. b Dispersion curve for disturbed chiral valley PCs, with two types of ligaments of lengths b₁ = 20 mm, b₂ = 12 mm, respectively. c Two separated modes at the K and K′ points in the band structure shown in b, respectively. d Frequency variations in the bandgap with respect to the difference in length ∆b between the two types of ligaments. The blue area indicates the frequency range of the bandgap.

Figure 2b depicts the dispersion relation of the chiral TI with ∆b = ±8 mm. The Dirac cone formed by the intersection of the first-order frequency band and second-order frequency band was opened to form a complete bandgap, with a frequency range of 2346-3316 Hz.

Figure 2d plots the variation curve of the bandgap width against the difference in length ∆b = b₁ − b₂. It can be concluded that the bandgap width increases as $\left|\text{Δ}b\right|$ varies from zero. It can be hypothesized that as the value of $\left|\Delta b\right|$ increases, the bandgap becomes wider. For example, when ∆b = 8 mm, the bandgap has a range of 2346-3316 Hz, while the band structure at ∆b = 16 mm indicates that the corresponding bandgap has a range of 2314-3767 Hz, as depicted in Fig. 2d.

The valley Hall interface state can be realized in these valley PCs. We selected a supercell with 8 Type I and Type II unit cells to form an interface between the two types, which is denoted by a red line in Fig. 3a. Unit cells of Type I and Type II were selected to form this supercell, which has a definite interface state that can avert strong intervalley mixing. Figure 3b depicts the dispersion relation of the supercell. The gray curves in the figure depict the bulk state, and the valley Hall interface state is denoted by the red curve.a Supercell with 8 unit cells of different types (Type I and Type II) along the e₁ direction. Periodic boundary conditions are applied in the e₁ direction. b Dispersion curves for the supercell composed of two types of unit cells, b₁ = 12 mm, b₂ = 20 mm and b₁ = 20 mm, b₂ = 12 mm. c Frequency range curves of interface mode under different ∆b values. d Acoustic pressure field on the interface of the supercell in a.

The interface mode can be tuned by changing the value of $\left|\text{Δ}b\right|$. Figure 3c depicts the effect of variation in the value of $\left|\text{Δ}b\right|$ on the frequency range of the interface mode. The frequency of the interface mode increases as $\left|\text{Δ}b\right|$ decreases.

4. Characteristics of robust TPEWs under different interface routes

We constructed a 16 × 14 system consisting of chiral unit cells using COMSOL Multiphysics and demonstrated the existence of acoustic interface modes in the system in the time and frequency domains and the robust routing of the TPEWs.

Since there exists an interface mode between the Type I and Type II chiral unit cells, and they can be converted into each other by rotation, the tunability of the chiral unit cell can be used to design designable topological interface routes. We combined the Type I and Type II chiral unit cells to form a plate structure with a definite interface. Since the total Chern number of the system varies, we can generate acoustic valley interface modes with desired propagation routes on the interfaces of the plate. The edges of the chiral plate system are covered with perfect match layers to prevent reflection and fully absorb the reflections from the surface boundaries. In the simulations, we considered three types of interface routes: a straight-line interface route, Z-shaped interface route, and U-shaped interface route. We first considered the propagation characteristics of TPEWs in the frequency domain. As Fig. 4a illustrates, a plane sound wave excitation at a frequency of 2.5 kHz was applied on the left side of the system in Fig. 4a and the receiving sources A and B were positioned at the right side to obtain the acoustic pressure fields of the waveguide state of the plate. We set Points A and B on the same side of the system. Point A denotes the end of the system interface route to detect the edge mode, and Point B was randomly set at a location away from Point A to detect the bulk mode existing in the structure. In addition, if the system has only edge modes in the frequency range of the input TPEWs, the TPEWs will only propagate along the system interface route, and then the sound pressure field intensity detected at Point B will approach zero. Figure 4b-d depict the acoustic pressure fields of the three types of routes at a frequency of 2.5 kHz. The pressure fields indicate the presence of TPEWs in the chiral topological systems with different interface routes, and the scattering is definite even at the sharp corner. To demonstrate the properties of TPEWs appropriately, we obtained the transmission spectra according to the expression T =20log(p/p₀) for Fig. 4b-d, respectively. Subsequently, we set up two different locations of receiving Points A and B in each system to obtain two types of transmission curves. The results are depicted in Fig. 4e-g. In the expression, T denotes the transmission for configurations, and p and p₀ denote the acoustic pressures of the receiving point and exciting point, respectively.a Interface configuration of valley topological PCs. b-d Simulation of a displacement field at 2.5 kHz along a straight interface route, Z-shaped route, and U-shaped route, respectively. e-g Simulation of the transmission spectra of the TPEWs in b-d. h Simulation of a displacement field at 2.5 kHz along cross routes. i Acoustic zigzag transmission of robustness against cavities in the valley topological PCs.

Figure 4e-g clearly depict that the TPEWs are concentrated on the interface routes. The transmission of TPEWs is stable within the bandgap frequency range of –20-0 dB, indicating that the sound pressures are in the same order of magnitude at the exciting point and receiving Points A and B, respectively. This is strong evidence that the TPEWs propagate with minor loss on the interface routes.

The frequency of the acoustic excitation source is within the bandgap range and interface mode frequency of Type I or Type II unit cells. We regulated the interface route of the plate system by converting the Type I and Type II unit cells into each other to achieve the modulation of the propagation route of the TPEWs.

Figure 4h depicts the propagation behavior of an acoustic wave on cross routes. The interface between Type I and Type II unit cells can be divided into Type I-II interface and Type II-I interface, where interface modes have opposite chirality. Figure 4h indicates that when the acoustic wave passes through the intersection of the routes, it does not propagate rectilinearly. In contrast, it propagates in a zigzag manner. This selective propagation is determined by the interface mode of the structure at the time of incidence. Since the interface mode in the acoustic TI is locked in the K valley, the acoustic wave can propagate through the interface routes where the chirality of interface modes is the same.

An important characteristic of TPEWs is their robustness against defects of interface routes, such as sharp corners and cavities, during propagation. We measured the acoustic pressure field of the plate system when the interface route had a cavity, as depicted in Fig. 4i. TPEWs can be effectively propagated in a system with defective interfaces even with an increase in acoustic scattering.

To highlight the interface modes of TPEWs and their immunity to backscattering at sharp corners, we supplemented the simulation results with a comparison in the time domain. Figure 5 illustrates the propagation of TPEWs at each time point along the Z-shaped and U-shaped routes with corners depicted in Fig. 4.Time domain simulation shows the propagation of plane acoustic excitation along routes with corners. a-d Wave propagation along the sharp shape route at 2, 4, 6, and 8 ms, respectively. e-h Wave propagation along the U-shaped route at 4, 6, 8, and 10 ms, respectively. The frequency of the plane acoustic excitation field is 2.5 kHz..

Figure 5 illustrates in detail that the TPEWs are immune to backscattering at corners during the propagation of acoustic waves on the routes. In conclusion, we demonstrated that the chiral model comprises topological edge states and can be tuned by varying the turning angle and the difference in ligament lengths.

5. Conclusion

We constructed valley PCs with different values of ligament lengths and used the difference between the ligament lengths to open the Dirac cone. This chiral structure exhibits a dispersion relationship and allows designable waveguide routes. A Dirac cone is observed when the six ligaments around the hexagonal plate are of the same length. The difference in the lengths of the ligaments ∆b can be varied to open the Dirac cone to form a bandgap. We plotted the curves of the effect of ∆b on the width of the bandgap and found that the bandgap width increases with the increase of ∆b. We successfully adjusted the dispersion relationship through simulation, proving the robustness of the TPEWs in the system propagation routes and designed propagation routes. Furthermore, we demonstrated the propagation characteristics of the TPEWs through simulation in the frequency and time domains. The reconfigurability of the interface routes was achieved by adjusting the lengths of the ligaments and rotating each unit cell in the system. Our proposed chiral system makes the wave propagation routes designable, and the method of opening the Dirac cone without relying on breaking C6v symmetry provides a new idea for implementing topologically protected edge waves.