Synthetic Weyl Points of the Shear Horizontal Guided Waves in One-Dimensional Phononic Crystal Plates

Abstract

Weyl physics in acoustic and elastic systems has drawn extensive attention. In this paper, Weyl points of shear horizontal guided waves are realized by one-dimensional phononic crystal plates, in which one physical dimension plus two geometrical parameters constitute a synthetic three-dimensional space. Based on the finite element method, we have not only observed the synthetic Weyl points but also explored the Weyl interface states and the reflection phase vortices, which have further proved the topological phase interface states. As the first realization of three-dimensional topological phases through one-dimensional phononic crystal plates in the synthetic dimension, this research demonstrates the great potential of applicable one-dimensional plate structural systems in detecting higher-dimensional topological phenomena.

1. Introduction

Weyl semimetals, which have linear intersections at the isolated momentum in three-dimensional (3D) materials, have attracted great research interests [1,2,3]. The linear intersection points named as Weyl points (WPs) are stable in the face of small disturbances [4,5,6]. Weyl semimetals exhibit various intriguing features, such as chiral anomalies and topological Femi interface states [5,6,7,8,9].

Researcher has introduced Weyl physics into sonic crystals [10,11,12,13,14], and the related topological surface states have also been extensively explored. Topological interface states usually occur at two insulator interfaces with different topological phases. Besides acoustic topological states, the fields of electromagnetic [15,16], plasmas [17], and elastic waves [18,19,20,21] have been investigated the topological states. However, the vast majority of Weyl crystals have a real 3D structure, which are complicated in manufacturing samples and detecting signals. The development of synthetic dimensions has been used to alleviate the problems encountered in real 3D space [22,23]. There are many ways to create synthetic dimensions, and the development of synthetic dimensions even provides a corresponding way to explore systems beyond 3D systems [24].

Different from previous studies [13,25,26,27], synthetic dimensions are used to replace purely physical dimension. By constructing two independent structural parameters in Weyl Hamiltonian with wave vector, one-dimensional (1D) phononic crystal (PC) plates are used to explore interesting Weyl physics. Different from the Dirac points, the WPs interacting with other bands, are immune to interference since the wave vector maintains a good quantum number [28,29]. The certain topological charge of each WP can be calculated by the Chern number [30]. Although the WPs are usually investigated in real 3D structures, which can also be studied theoretically in the synthetic dimension [31].

In this paper, synthetic WPs are observed, the transmission spectra (TS) and the reflection phases (RPs) with changing geometrical parameters are studied. The intrinsic vortex structure of the RP is confirmed, which provides a new representation for the topologically robust WPs. It is proved that the RP of the truncated one-dimensional (1D) phononic crystal (PC) plates shows the vortex structure around the synthetic WPs, which carries the same topological charge as the corresponding WPs. This study may facilitate several operational applications. For example, the interface states under strong constraints can enhance the elastic energy and thus can be combined into directional elastic antenna structures, providing a technical means for communication and detection.

2. Materials and Methods

As shown in Figure 1, the 1D PC plate is made of silicon, whose material parameters are as follows: mass density ρ = 2332 kg/m³, shear wave velocity v_t = 5840.9 m/s, and elastic constant C₄₄ = 79.56 GPa. Each unit cell contains four layers, whose characteristic length is t₁~t₄, where t₁ + t₃ = 2a/3 and t₂ + t₄ = a/3. The lattice constant is a = 9 mm, the height of pile h₁ = 3 mm and h₂ = 4.2 mm. The above geometric parameters remain unchanged in the discussion of this paper. Two additional geometrical parameters are defined as p = (t₁ − t₃)/(t₁ + t₃) and q = (t₂ − t₄)/(t₂ + t₄). Since the values of t₁~t₄ are positive, both parameters range from [−1,1], which forms a closed parameter space. A virtual 3D parameter space can be formed by these two parameters plus the wave vector, where WPs can be investigated.

When the Shear horizontal (SH) guided waves propagate in the 1D PC plates, the governing equation is expressed as:

$$-\rho {\omega }^2{U}_2=T_{12,1}+{({\mathrm{C}}_{44}{U}_{2,3})}_{,3},$$

(1)

where the subscripts 1, 2, and 3 represent the coordinates x, y, and z, U₂ is the displacement vector along the y-direction, C₄₄ is the elastic constant, and T₁₂ is the stress tensor component.

Based on the Bloch–Floquet theorem, the displacement in PC plates is given as:

$$u(r,t)=u_k(r)e^{i(k\cdot r+\omega t)},$$

(2)

where r = (x, y, z) is the position vector and k = (k_x, k_y, k_z) is the Bloch wave vector. Since the 1D PC plates are periodic in the x direction, one unit cell is considered. We substitute Equation (2) into (1) and discretize the model with sufficient elements. The eigenvalue equation in the finite element method (FEM) framework is:

$$(K(k)-\omega {(k)}^{2}M)u=0,$$

(3)

in which K and M stand for stiffness matrix and mass matrix. The band structures for the 1D PC plates can be obtained by scanning the points of the Brillouin zone.

Using the frequency domain study of COMSOL software 5.6, the TS and the RP can be calculated. For TS, the simulation model is shown in Figure 2. A uniform harmonic shear displacement is applied on the left side of the PC plate and the perfectly matched layers (PMLs) are used to avoid reflection. The energy density spectrum normalized by the flat panel is defined as:

$$I=\frac{\sqrt{{v_{2x}}^2+{v_{2z}}^2}}{\sqrt{{v_{1x}}^2+{v_{1z}}^2}}$$

(4)

in which v_1x and v_1z stand for the displacement of the receive point with PC plate, while v_2x and v_2z stand for the displacement of the receive point with flat panel.

For the RP calculation, the left PML layer is replaced with a hard boundary condition. With the same excitation, the RP can also be obtained from the rigid boundary.

3. Results

Starting from the special case, the synthesis parameters are selected as p = q = 0, in other words, t₁ = t₃, and t₂ = t₄. As shown in Figure 1b, this case simply doubles the size of the unit cell and the Brillouin zone is folded. Such band folding results in a linear crossover of the first (third) and second (fourth) along the wave vector direction at k = ±π/a. Away from the point (p = 0 and q = 0), band gaps are formed since the degeneration resulting from the band folding is removed, as exemplified by the SC with (p = 0.76 and q = 0) (blue dotted lines). It can be inferred from Figure 1a that the increase and decrease of p (or q) from p = 0 (or q = 0) is not symmetric in the one-dimensional structure. Therefore, the dispersion relation ω (p, q, k) is expanded at p = q = 0, the first-order terms Δp and Δq will play the dominant role and give linear dispersion in the p and q directions. Based on the above analysis, the intersections at (0, 0, ±π/a) are WPs in this synthetic dimension.

The intersection between two adjacent bands can also form at the wave number k = 0 or k = ±π/a when p = 0 or q = 0, as guarded by space–reflection symmetry of the 1D PC plate. Let me give an example, an accidental linear degeneracy can also form at k = 0 with p = 0.76 and q = 0 as shown in Figure 1b. These degenerate points are also WPs in this synthetic dimension, which are further verified by the conic intersections in the bulk dispersions depicted in p-q space, as shown in Figure 1c.

It is noted that the mirror symmetry of our framework ensures that when a WP exists at (p, q, k), there must be corresponding WPs with the same frequency and the same topological charge at (±p, ±q, ±k). The main difference between our system and the real Weyl crystals is that the total charge does not vanish because the synthesis parameters are not periodic.

To further confirm the synthesized WPs, the frequency-dependent TS through 1D PC plates with different geometrical parameters are investigated. Here we pay attention to the frequency range around the WP formed by the first and second bands. Figure 3a–c shows 95 independent phononic crystal models (including 19 different p and five different q) and each model consists of 10 periods. The simulation results shows that when q gradually approaches 0, the bandwidth of the transmission gap decreases and disappears. It shows obvious linear crossover near the expected frequency and structural parameters. It is also worth noting that the band structures with negative q are the same as the band structures with the positive “q”, which leads to the same TS.

The RP of a PC plate, which is surrounded by a perfectly matched layer and a rigid boundary, is also studied. Regarding the effective Hamiltonian, the non-trivial topology of the synthetic WP results in the formation of an eddy current-like RP structure in the parameter space, where the number of quantized phase windings is identical with the corresponding Weyl charge [2]. The distributions of RP are shown Figure 3a, where the frequency 0.306 MHz of the Weyl point between band 1 and band 2 is used. The phase distribution in the p-q space clearly shows a vortex. The reflection simulation proved the non-trivial phase distribution.

Robust RP singularity is a new characteristic of topologically stable PC plates. At the same time, the vortex phase structure ensures the presence of interface states, and each WP is associated with an interface state. In Figure 4b, the energy density spectrum of the phononic crystal plate with seventeen periods at (p, q) = (0.35, 0.45), normalized by the flat panel is calculated. There is a significant peak value appearing at 0.305 MHz, which is in line with the interface transition frequency in Figure 4a, which verifies the topologically non-trivial interface state.

To visualize the presence of topological interface states, the displacement distribution of the PC plate at a Weyl frequency of 0.305 MHz with (p, q) = (0.35, 0.45) is plotted in Figure 5. We can visually and clearly see that the wave motion is concentrated between the rigid boundary and the PC plate, which is a typical property of the topological interface state. This strongly confined property has potential applications in designing elastic sensors and harvesting vibration energy. In order to quantitatively show the large differences in displacement amplitude, the distribution of displacement amplitude along the x-direction is also given in Figure 5.

Finally, the TR for detecting the WP are conceived as follows. The sample can be made by laser cutting technique. The ultrasonic experiments can be used to obtain the TR. The detailed test procedure is as follows. First, a PZT transducer bonded on the upper surface is driven by a short duration pulse generated by the generator at the left side. Second, a fiber Bragg grating sensor bonded on the upper surface is used to receive wave signals at the right side. Third, the collected signals are processed in the oscilloscope and then the TR can be obtained.

4. Discussion

The synthetic WPs of the SH guided waves in 1D PC plates are investigated. Here a virtual 3D space is formed by the Bloch wave vector plus the two additional parameters. Base on the zone folding mechanism, the WPs can be obtained. Furthermore, we also obtain the WP by changing one structural parameter. In order to confirm the WP, the conic intersection in the bulk dispersions is also presented in p-q space. To further confirm the synthesized WPs, the frequency-dependent TS through 1D PC plates with different structural parameters are investigated. Besides, the RP of a semi-infinite SC with a rigid boundary is also studied, and a vortex as the signature of Weyl physics is found in the phase distribution. The vortex phase structure ensures the presence of interface states, and the interface state is also shown in the TS. This study may facilitate several operational applications. For example, the interface states can enhance the elastic energy and thus can be combined into directional elastic antenna structures, providing a technical means for communication and detection.