# Plane and Surface Acoustic Waves Manipulation by Three-Dimensional Composite Phononic Pillars with 3D Bandgap and Defect Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Phononic Crystal and Theoretical Background

#### 2.1. Unit Cell Structure

^{®}. The PnC structure was designed in such a fashion that made it periodic in all three directions.

#### 2.2. Theory and Mathematical Framework

#### 2.3. Model Validation

## 3. Results and Discussion

#### 3.1. Plane Wave Propagation and Defect Analysis

#### 3.2. SAW Propagation and Defect Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The phononic crystal (PnC) unit cell structure and finite-length model: (

**a**) 3D, top and plan view; (

**b**) unit cell structure with perfect boundary condition (PBC) applied along all vertical edges and top-bottom; (

**c**) supercell model used for plane wave transmission curve study; (

**d**) unit cell structure utilized for surface acoustic wave (SAW) band structure study; (

**e**) finite-length model with SAW propagation at the surface of semi-infinite half-space can be observed; and (

**f**) supercell model adopted for SAW transmission, attenuation and defect analysis.

**Figure 3.**(

**a**) The 3D band structure of PnC unit cell structure, with irreducible Brillouin zone (IBZ) shown at the inset. (

**b**,

**c**) transmission curve for harmonic excitation applied in the x-y and the x-y-z directions. The BGs are highlighted. (

**d**) The vibration modes involved in the opening and closing of BGs.

**Figure 4.**The defect analysis (

**a**) supercell structure with defect height ${h}_{d}$; (

**b**) frequency response spectrum with defect frequencies for various ${h}_{d}$. At the defect frequency, wave energy is concentrated inside the defect, and wave propagation is isolated for other unit cell structures.

**Figure 6.**SAW band structures for (

**a**) ${h}_{2}=0.2a$; (

**b**) ${h}_{2}=0.4a$; (

**c**) ${h}_{2}=0.5a$ (parent structure); (

**d**) ${h}_{2}=0.6a$; and (

**e**) ${h}_{2}=0.8a$. One can deduce identical vibration modes as reported for the parent structure in Figure 5c.

**Figure 7.**SAW defect analysis for ${h}_{2d}=0.8a$; (

**a**) the schematic diagram in which defect location is highlighted, and the transmission curve is plotted. At defect position (

**b**,

**h**) SAW trapping (

**c**–

**e**,

**f**,

**g**) vibration isolation where partial SAW propagation in neighboring parent cells occur can be seen.

**Figure 8.**SAW defect analysis for ${h}_{2d}=0.2a$: (

**a**) the schematic diagram in which the defect location is highlighted, and the transmission curve is plotted. At defect position (

**b**,

**h**), SAW trapping (

**c**–

**e**,

**f**,

**g**) and vibration isolation with partial SAW propagation in neighboring parent cells can be seen.

Young Modulus E (GPa) | Density ρ (kg/m^{3}) | $\mathbf{Poison}\text{}\mathbf{Ratio}\text{}\mathit{\nu}$ | |
---|---|---|---|

Silicon (Si) | 166 | 2330 | 0.28 |

Tungsten (W) | 400 | 19,270 | 0.3 |

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**MDPI and ACS Style**

Muhammad; Lim, C.W.; Leung, A.Y.T.
Plane and Surface Acoustic Waves Manipulation by Three-Dimensional Composite Phononic Pillars with 3D Bandgap and Defect Analysis. *Acoustics* **2021**, *3*, 25-41.
https://doi.org/10.3390/acoustics3010004

**AMA Style**

Muhammad, Lim CW, Leung AYT.
Plane and Surface Acoustic Waves Manipulation by Three-Dimensional Composite Phononic Pillars with 3D Bandgap and Defect Analysis. *Acoustics*. 2021; 3(1):25-41.
https://doi.org/10.3390/acoustics3010004

**Chicago/Turabian Style**

Muhammad, C.W. Lim, and Andrew Y. T. Leung.
2021. "Plane and Surface Acoustic Waves Manipulation by Three-Dimensional Composite Phononic Pillars with 3D Bandgap and Defect Analysis" *Acoustics* 3, no. 1: 25-41.
https://doi.org/10.3390/acoustics3010004