Enlargement of band gaps in holey phononic crystal strips

Abstract

This study proposes an effective strategy to enlarge the band gaps of holey phononic crystal strips. The approach involves identifying each band and its key influencing factors, then implementing targeted synthetic adjustments like geometry optimization and embedment to reposition the relevant bands, and finally achieving larger band gaps. The main band gap can be enlarged from 47% to 64% with slight modifications. Furthermore, this optimal result is only achieved through the synergistic action of different methods, each of which has small or even counteractive individual effects. The design strategy starts from large-scale to small-scale adjustments, culminating in specific case-by-case adjustments. It presents new possibilities for enhancing phononic crystals.

Keywords

Phononic crystals band gaps wave control convex holey strip synergistic strategy

Introduction

Phononic crystals (PnCs) can manipulate the propagation of acoustic or elastic waves by virtue of their phenomenal merit of band gaps (Laude, 2015; Vasileiadis et al., 2021; Wang et al., 2020). Generally, the wider the band gaps, the more promising their applications and superior their performance becomes. As a result, one of the primary objectives for researchers in this field is to widen the band gaps. To achieve this objective, there are three main strategies that researchers can employ based on the nucleation mechanisms of band gaps. The first strategy involves enhancing Bragg scattering to induce destructive interference of waves with different phases (Jiang et al., 2017a, 2018; Jin et al., 2022). The second strategy intensifies the mismatch of physical properties (such as material, mass, and stiffness) between the components in a unit cell to provide strong local resonances (Coffy et al., 2015; Jiang et al., 2017b; Jin et al., 2021; Li et al., 2023a, 2023b). Finally, the third strategy combines the actions of the former two (Krushynska et al., 2017b; Lee and Iizuka, 2019). However, despite the substantial advancements made in the field of PnCs by exploring new configurations, only a few studies have examined strategies to broaden the band gaps within a particular configuration.

Compared to local resonant band gaps, widening Bragg band gaps is significantly more challenging due to the strict limitations imposed on geometry size. The lattice constant must be in the same order as the wavelength to achieve a wider Bragg band gap. Topology optimization is a popular method that has been used to discover novel topologies with wide band gaps (Dong et al., 2017; Li et al., 2019; Sharma et al., 2022; Zhang et al., 2021). However, most of the topologies obtained through this method are often complex and impractical, and their ramifications can be heavily influenced by the initially selected criteria. The optimization process typically involves minimizing or maximizing an objective function subject to various constraints. The interaction of these constraints can lead to intricate designs as the algorithm searches for optimal solutions within the specified criteria. Moreover, if the objective function or constraints are noisy or sensitive to small changes, the optimization algorithm may produce irregular shapes in an attempt to find the optimal solution. This sensitivity can lead to intricate designs that might be challenging to interpret intuitively. Additionally, topology optimization algorithms often do not take into account manufacturing constraints and considerations. Consequently, the resulting designs may not be directly manufacturable using traditional methods, making them seemingly impractical. Thus, there is a need to consider appropriate topological restraints to improve usability. On the other hand, when it comes to local band gaps, several popular compositions have been proposed, including pillar coatings (Coffy et al., 2015; Jin et al., 2016b, 2021) and the use of heavy and stiff inclusions in a light and soft matrix (Liu et al., 2000; Matlack et al., 2016). However, the strong heterogeneity of physical properties in a unit cell poses challenges in terms of fabrication and stability. Incidentally, some novel labyrinthine/space-coiling and bio-inspired fractal designs have also been proposed (Krushynska et al., 2017a; Liu et al., 2018; Man et al., 2019). However, these designs often lead to narrowed band gaps rather than widening them.

When compared to their two-dimensional (2D) and three-dimensional (3D) counterparts, one-dimensional (1D) PnCs that occupy less space have proven to be more suitable for devices with strict spatial requirements, such as high-integration MEMS (Bao et al., 2019; Feng et al., 2017; Hsu et al., 2011; Jiang et al., 2018; Workie et al., 2021). Figure 1 shows a schematic of 1D PnC strips serving as anchor support to control the external waves in MEMS. To improve the quality factor Q, reduce anchor loss, and shield extrinsic interference, both 1D and 2D PnCs have been widely introduced into MEMS. This has significantly improved the performance of MEMS devices. Additionally, due to their simple yet effective structures and dispersive curves, 1D PnC strips are extensively used to pioneer new physical phenomena such as optomechanics (Eichenfield et al., 2009; Gomis-Bresco et al., 2014), acousto-optic coupling (Pennec et al., 2014; Psarobas et al., 2010), and topological interface states (Fan et al., 2019; Li et al., 2020; Zhang et al., 2019), as well as new materials (Aly et al., 2018; Lou et al., 2018; Xue et al., 2023), methods (Cheng et al., 2021; Qian and Shi, 2017a, 2017b), and applications (Bergamini et al., 2014; Cai et al., 2020; Cao et al., 2019). After that, the verified results in 1D PnCs are then extended to 2D and 3D PnCs. However, it is worth noting that most 1D PnCs still use the simplest and most conventional constructions, and only a few can create wide band gaps.

Researchers have been working on ways to widen band gaps in 1D PnCs. Feng et al. (Feng et al., 2013) achieved a dramatical increase in band gap in an I-shaped holey PnC strip by changing the generating way from square lattice to hexagonal lattice, and the resultant BG% (ratio of the width to center frequency) increases from 11% to 48%. However, this approach is highly dependent on structural patterns and may only work in specific cases. Coffy et al. (Coffy et al., 2015) proposed a direct and credible method to enlarge the band gaps in pillared PnC strips. The core idea is to cut away the region with high-energy distribution and suppress the corresponding interfering eigenmodes, achieving the union of neighboring band gaps. Subsequently, based on this design, several interesting works were proposed, including the hollowed pillars and tunability endowed by fluid-filled pillars (Jin et al., 2016a, 2016b; Wang et al., 2017). However, this method seems to be applicable only for the pillared configuration, where the eigenmodes are initially well decoupled. For other structures like holey PnCs, where each geometric parameter is related to many eigenmodes, targeted cutting aimed at removing specific eigenmodes could cause a domino effect on the entire band structure. Therefore, there is a need to develop more effective and practical methods for widening band gaps in various 1D PnC structures.

In this study, we propose a systematic approach for widening band gaps in convex-like holey PnC strips. This type of PnC strip is selected as an example because its band gaps are split by various eigenmodes, making it a paradigm to embody the effectiveness of the proposed strategy. Our strategy involves identifying and categorizing each band, along with their primary influencing factors using the energy method, and then implementing targeted adjustments to the band structures. Specifically, we first carry out a large-scale approach by restructuring the overall band structure to optimize the band gaps. Next, we apply a small-scale approach to fine-tune the band gaps by adjusting congener bands. Finally, case-by-case modifications are imposed on specific single bands. Overall, by combining these different methods, we achieve a synergistic effect that results in the enlargement of the band gaps. Our proposed approach is not only effective for convex-like holey PnC strips but can also be applied to other types of PnC structures. We believe that this study will provide a valuable reference for the design and optimization of PnC-based devices with wide band gaps.

Band structures of PnC strips

Figure 2 provides an overview of the essential details of the chosen PnC strip, which comprises (a) a unit cell and its band structures and (b) selective eigenmodes. The strip’s geometry is determined by five independent parameters, namely, a₁, b, c, h, and d (or alternative parameters), and their combinations. To simplify the analysis and discussions, all the geometrical parameters are normalized by dividing them by the lattice constant a₂, which is set as 1 unit as the benchmark. The band structure shown is the outcome of the parameters (a₁/a₂, b/a₂, c/a₂, d/a₂, h/a₂) = (1, 0.82, 0.22, 0.65, 0.15), and single-crystal silicon is selected as the material for this computational example. The material’s properties considered in this analysis are density (ρ_silicon = 2330 kg/m³) and elasticity constants (D₁₁ = 165.7 GPa, D₁₂ = 64.1 GPa, and D₄₄ = 79.6 GPa). Since different materials with varying wave speeds will be introduced in the subsequent analyses, a single transverse speed c_t associated with one material is inadequate to achieve material agnostic. Therefore, the reduced frequency fa₂ is introduced, which is lattice-agnostic but cannot be further made material agnostic by merely dividing it by the wave speed. The selective eigenmodes shown in Figure 2(b) reveal the different modes that influence the band structure, and their categorization is crucial for widening the band gaps.

In Figure 2(a), the distribution of kinetic energy ratios across the three spatial axes is visualized using a color scale, which is defined as

e_{i} = \int_{V} {u_{i}}^{2} d V / \int_{V} (u_{x}^{2} + u_{y}^{2} + u_{z}^{2}) d V, i = x, y, z

(1)

where V represents the volume and u_i (where i = x, y, z) represents the displacement content. It provides essential information for understanding the polarization of elastic waves and the emergence of band gaps. Based on the symmetry distribution of energy (Hsu et al., 2010, 2011), we can identify and classify all bands into four distinct groups, including x-bending eigenmodes (B_x), y-bending eigenmodes (B_y), z-bending eigenmodes (B_z), and torsional eigenmodes (T). Moreover, B_x-eigenmodes and B_y-eigenmodes belong to in-plane modes, while B_z-eigenmodes and T-eigenmodes are out-of-plane modes. We observe three distinct band gaps with reduced frequency fa₂ spans of 1001–1275, 1797–1974, and 1986–3188, corresponding to BG% of 24%, 9.2%, and 47%, respectively. In Figure 2(b), we present the modal shapes from T₂ to B_z4 since these modes are worth more attention as they split the neighboring band gaps, especially the quasi-flat B_z4 with an almost zero group velocity. We define such modes as “interfering” throughout this paper. Naturally, we may question whether we can directly remove these interfering modes through optimization, similar to the approach adopted in reference (Coffy et al., 2015). However, as we discussed earlier, the cutting strategy that works in the pillared configuration is not suitable in such cases with a simple configuration for the lack of explicit structural decoupling. Therefore, we need to explore alternative strategies to eliminate or control the impact of these interfering modes effectively. Further investigations into these modes are necessary to develop robust and effective approaches.

Although interfering modes from T₂ to B_z4 cannot be avoided, it is advisable to shift their position to potentially enlarge the band gap. However, it sounds feasible that the enlargement of the band gap can also be achieved if the bands are repositioned well, but it is not a pushover to implement. Up to now, such works about the enlargement of band gaps through elaborate and targeted restructuring of band structures have not been reported. The key to properly shifting these interfering bands is to identify the main influencing factors of each band. In this case, the unit cell comprises only two structural components, namely, bulky lumps and deformable L-shaped connectors. The L-shaped connectors can be further divided into two types due to the periodic conditions applied in the x-direction.

Based on the modal shapes displayed in Figure 2(b), the L-shaped connectors undergo most of the deformation, while the lumps oscillate as rigid bodies. Therefore, it is possible to treat the deformable L-shaped connectors and rigid lumps separately. In addition, a more detailed analysis reveals that the mechanical behavior of L-shaped connectors can be decoupled into four distinct modes: axial extension, in-plane bending, out-of-plane bending, and torsion. This is consistent with the classification of eigenmodes, where the motion of L-shaped connectors in the in-plane (B_x- and B_y-) and out-of-plane (B_z- and T-) eigenmodes are, respectively, couplings of axial extension and in-plane bending, and couplings of out-of-plane bending and torsion. As a result, both the structural components and eigenmode bands can be implicitly distinguished. In other words, alternative implicit decoupling can be obtained through insights into mechanisms and rational classifications, even though there is no explicit decoupling.

Strategy for enlarging band gaps

In order to adjust the interfering bands, we identify four influential factors that could be modified: the length of the free side a₁, thickness h, the width of free lumps w₁ (with no periodic conditions), and embedment depth h_in at the elbows. These factors are chosen because they can all influence the mechanical behavior of the L-shaped connectors, which are found to be the most deformable components in the unit cell. Noteworthily, unlike the former three factors that only change the geometrical size, the embedment at the elbows h_in is specially introduced for eigenmode B_z4. It is observed that torsional eigenmodes T₃ and T₄ also show high-energy distribution at the elbows, but they are a byproduct of torsional lumps and not intrinsic eigenstates. To shift the B_z4 band, we implement the substitution of another material with a higher density at the crucial elbows to increase the mass and lower the eigenfrequency. Here, the material chosen for this purpose is lead, which has a density of ρ_lead = 11,340 kg/m³, Young’s modulus of E_lead = 16 GPa, and Poisson’s ratio of μ_lead = 0.44. The substitution of lead at the elbows increases the mass of the L-shaped connectors, which in turn lowers the eigenfrequency of the B_z4 band.

The results of the impact of various factors on the band structures are presented in detail in Figures 3 and 4. The first factor, the length of the free side a₁, has a noticeable effect on all bands, causing them to shift downward as a₁ decreases, as shown in Figure 3(a). Specifically, the high-frequency bands show a larger downward shift with a decrease in a₁. The second factor, thickness h, only affects the out-of-plane modes, such as B_z-modes and T-modes, as shown in Figure 3(b). This can be attributed to that the thickness of the L-shaped connectors affects their z-bending and torsional stiffnesses, which monotonically increase with h, but show a weak correlation with x- and y-extension and bending. As a result, the out-of-plane modes are more sensitive to changes in thickness than the in-plane modes. The third factor, the width of the free lumps w₁, mainly impacts the in-plane modes and T₂ but has little significance in T₃ and B_z4, as shown in Figure 4(a). In particular, T₂ shows the largest downward shift with an increase in w₁. The fourth factor, embedding at the elbows, can effectively move B_z4 downward with little or no change to the surrounding bands. However, it may have side effects on other bands at the upper edge. Finally, h_in in Figure 4(b) can slightly widen the band gap, while the other factors are either unproductive or even counterproductive. In summary, the four influential factors can shift the related bands but are unable to effectively improve the band gaps. The study of the impact of these factors on the band structures provides insight into the mechanisms underlying the band gap formation.

Inspired by the ubiquitous synergistic phenomena that refer to the enhanced outcome resulting from the collaboration or interaction of different factors, exceeding the simple sum of their individual contributions, namely, 1+1>2, the synergistic effect of the above four influential factors is investigated. The approach involves dividing the design concept into three steps to gradually decrease the adjustable number of bands: (1) adjusting the overall layout using a₁, (2) locally adjusting with h and w₁, and (3) individually shifting with h_in. Figure 5 provides a panorama of the combined effect of a₁, w₁, and h_in, in which BG% is illustrated by color as a function of a₁/a₂, 2w₁/c, and h_in/h. It should be noted that the value of h/a₂ is fixed at 0.15 during the iterative calculation since its influence is relatively regular and can be predicted with ease.

The results presented in Figure 5 indicate that the synergistic effect of the three influential factors, a₁, w₁, and h_in, can easily lead to better band gap performance that surpasses both the initial maximum of 47% (the region between the green isosurfaces) and the sum of the second and third band gaps of 56% (region coated by red isosurface). The maximum band gap achieved is BG% = 64% at (a₁/a₂, 2w₁/c, h_in/h, h/a₂) = (1.02, 2, 0.8, 0.15). This corresponds to a 36% improvement in band gap performance compared to the initial maximum. More importantly, such improvement is achieved without imposing additional size consumption on the critical width of L-shaped connectors t = (a₂–c–d)/2, which is not only crucial for the mechanical behaviors of L-shaped connectors but also is the smallest size limiting the fabrication. Generally, achieving the theoretical maximum band gap is often associated with extreme geometrical sizes that can be challenging to realize in practice.

Moreover, the enlargement of band gaps is available over a wide tunable range of influential factors, providing great practicality and versatility for the design of metamaterials with tailored band gap properties. The underlying mechanism responsible for this band gap enlargement can be attributed to the nucleation mechanisms of the band gaps. The unique folding topology, consisting of deformable L-shaped connectors and rigid lumps with a significant mismatch in stiffness, induces both Bragg scattering and local resonance mechanisms. The implemented adjustments of a₁, w₁, and h_in, further enhance the mismatch within the unit cell, leading to the final enlargement of the band gaps.

The restructuring process of the widest band gap, depicted in Figure 6, involves a sequential adjustment of the influential factors. The process starts with the adjustment of a₁, which fine-tunes the overall band structure. Next, w₁ is adjusted, leading to a shift in B_z3, T₂, and the in-plane bands in blue. Finally, h_in is introduced to flatten T₃ and shift B_z4 and T₄. The synergistic effect of these three factors results in a significant enlargement of the band gap. However, once the band structures are reorganized through the synergistic effect of (a₁, w₁, h_in), the thickness h cannot further widen the band gap. This is because both the upper and lower edges are determined by out-of-plane eigenmodes, and the interfering bands B_z3, T₄, and B_z4 are highly overlapped. Therefore, only slight changes are sufficient to significantly broaden the band gaps while maintaining machinability and practicality. It is important to note that each band may be influenced by multiple factors, and the widest band gap is the result of a subtle balance between these factors. This study specifically focuses on the third/main band gap, and different strategies may be required to optimize other band gaps. Overall, the proposed approach provides a practical and versatile method for achieving significant improvements in the band gap performance.

Transmission spectra

Another important consideration for PnCs used in the suppression of vibration is the transmission spectrum, which can verify the band gaps in the band structure obtained from a unit cell and can evaluate the attenuating performance in a practicable structure with finite periods. A standard setup is used for this purpose, as shown in Figure 7. The signal excitation (loaded by polarized line/surface wave sources) and receiver (loaded by average displacement probe) are deployed at the two ends of the PnC strip, respectively. Moreover, to shield the influence of reflectible outgoing waves, perfectly matched layers have been added at the ends (Diatta et al., 2016). To characterize the response to different polarized waves, two kinds of excitations are loaded: one is pure x-, y-, and z-directional excitations and the other is a multidirectional mixed excitation that includes a mixture of x-, y-, and z-directions.

Figure 8 presents the band structure versus transmission spectra under different excitations. Figure 8(a) is the band structure with color representing z-polarization kinetic energy contents, and Figure 8(b) is the transmission spectra obtained from the simulation. First, all band gaps in Figure 8(a) have a one-to-one correspondence with the attenuating ranges in the mixed-source transmission spectrum in right of Figure 8(b), which mutually verifies their validity. The nuance is negligible. Second, in the pure x-, y-, and z-source transmission spectra, although the attenuating ranges show a good match with the band gaps, there are several points that need to be explained. In left of Figure 8(b), the peak “P1” corresponding to out-of-plane eigenmodes that should not have appeared appears in the pure x-source transmission spectrum. It is perhaps because these eigenmodes are supersensitive modes that are susceptible to indirect excitations. On the contrary, in middle of Figure 8(b), the peak “P2” corresponding to out-of-plane eigenmodes that should have appeared disappeared in the pure z-source transmission spectrum. It is perhaps because these eigenmodes are deaf modes that are unresponsive to specific excitations. Thus, the fairly well-matching results of both unidirectional and omnidirectional band gaps and transmission spectra theoretically prove the effectiveness of the proposed synergistic strategy. Ideally, the related experiments should be conducted to further verify the results. Considering the processing cost, photosensitive resin and copper can be used for the matrix and embedding materials. Generally, a scanning laser Doppler vibrometer is requisite, but we cannot afford it for the lack of funds in the short run. However, many previous works have finished similar experiments (Faiz et al., 2020; Miniaci et al., 2017, 2018; Yan and Gao, 2021). We think it is sufficient to prove the effectiveness of the proposed strategy. We also expect competent researchers to complete such an experiment.

Conclusion

We successfully widen the band gaps of a convex-like 1D PnC strip. With the identification and classification of every band, the interfering bands are repositioned by combined actions of geometry optimization and material substitution at critical locations. The importance of this work lies in the proposed concept more than in the presented results: the band gaps can be greatly expanded by the orderly superposition of different methods that do not work well alone. Furthermore, some slight modifications lead to a dramatic improvement in the band gaps, endowing the proposed concept with universal applicability. It is not confined to the specific model and methods shown in this work. Along the same lines, the performance of band gaps can be improved in much more cases with simple construction. We are convinced that this research will be beneficial for a more realistic and practical design.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant No. 52105575), the Fundamental Research Funds for the Central Universities (Grant No. QTZX23063), the Proof of Concept Foundation of Xidian University Hangzhou Institute of Technology (Grant No. GNYZ2023YL0302), the Aeronautical Science Foundation of China (Grant Nos. 2022Z073081001 & 20230018081023), and the Open Research Funds of National Key Laboratory of Strength and Structural Integrity (Grant No. ASSIKFJJ202301005).

ORCID iD

Shan Jiang

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