# Simulation and Optimization of Piezoelectric Micromachined Ultrasonic Transducer Unit Based on AlN

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design and Modeling

#### 2.1. Piezoelectric Effect

#### 2.2. Model Building

_{2}is the insulating layer and Si is the substrate (the materials involved in the structure can be obtained from the material library using COMSOL). A tiny part of the silicon at the lower end of the center of the transducer structure is etched away, effectively reducing the thickness of the active center region such that the device becomes a thin-film composite PMUT transducer. In this study, we used a 2D axisymmetric model and a 3D model in which a perfectly matched layer (PML) area added outside the water area effectively expanded the propagation range and simulated the propagation and absorption effects of sound waves at the boundary. The basic parameters of the AlN material required for the simulation are mentioned in the following. The density is 3250 kg/m

^{3}.

_{c}(η

_{c}= 0.001) and dielectric loss η

_{εs}(η

_{εs}= 0.001) are added. The device is subsequently meshed. The most prominent mesh element is specified as 1/5 of the corresponding minimum wavelength, and the sweep feature is used to partition the PML domain(as shown in Table 1 and Figure 3). The model’s dense solid area and relatively sparse mesh density distribution settings in the water area are completed. Subsequently, a “boundary layer mesh” is applied to set smooth transition boundaries to accurately simulate the performance of computing devices and reduce the computation time to a certain extent. Finally, the study type is added and the results are viewed. The study type is the frequency domain, with 1D/2D/3D plots available to view various performance attributes in the results.

## 3. Performance Simulation and Optimization Analysis

#### 3.1. Modal Analysis

_{v}is the density, σ is Poisson’s ratio, k is the wavenumber, and ω is the angular frequency. A and B A and B are undetermined constants that are determined using the boundary conditions of the disk. a is the radius of the upper electrode. The bending moment and transverse shear force at the boundary of the flexural vibration disk are zero [23]. Thus, the following three equations can be obtained:

_{1}= X

_{2},

_{3}= X

_{4}; through the material parameters of the given disc, the relationship between the geometric size of the element and the resonance frequency can be obtained,, and the resonance frequency of the model can be obtained from this relationship.

#### 3.2. Frequency-Domain Analysis and Optimization

_{front}is the on-axis intensity at a certain point in the axial direction, I

_{ave}is the average intensity of the monopolar source at a certain point in the axial direction, and DI represents the directivity index. Figure 9 shows the directivity index of the array element (red curve) with an optimized electrode distribution compared with the directivity of the clamped element (black curve). Notably, both directivity indices are similar when the transducer is operating in piston mode at lower frequencies. With an increase in the frequency, the former is evidently softer in mode switching and the trend of the curve in the second half is stable.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Mesh structure of (

**a**) transducer model (including water domain and PML); (

**b**) 1/4 block clamp structure array element; (

**c**) 1/4 block optimized structure array element.

**Figure 4.**The first three modes of bending vibration: (

**a**) the first-order bending vibration mode; (

**b**) the second-order mode; (

**c**) the third-order mode.

**Figure 5.**Two-dimensional axisymmetric diagram of the sound field: (

**a**) 2D sound pressure map (Pa); (

**b**) 2D map of the SPL (dB); (

**c**) 2D height map of sound pressure; (

**d**) sound wave radiation profile.

**Figure 6.**(

**a**) One-quarter of the modified unit model; (

**b**) radiation pattern of the clamped structure; (

**c**) radiation pattern of the double-slit structure.

**Figure 7.**(

**a**) Radiation pattern at an external field radius of 0.5 mm; (

**b**) radiation pattern at an external field radius of 50 mm.

**Figure 8.**(

**a**) SPL and stress distributions at the acoustic–structure interface; (

**b**) variation curve of the radiated power with radius at two different frequencies.

Parameter | Clamped Structure/Optimized Structure | The Transducer |
---|---|---|

Maximum element size | C_{ref}/f_{max}/6 | 11.3 µm |

Domain elements | 78,355/96,019 | - |

Boundary elements | 16,768/24,106 | - |

Edge elements | 1142/1606 | - |

Number of boundary layers | - | 1 |

Number of PML layers | - | 8 |

_{ref}: typical wave speed of PML.

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**MDPI and ACS Style**

Su, X.; Ren, X.; Wan, H.; Jiang, X.; Liu, X.
Simulation and Optimization of Piezoelectric Micromachined Ultrasonic Transducer Unit Based on AlN. *Electronics* **2022**, *11*, 2915.
https://doi.org/10.3390/electronics11182915

**AMA Style**

Su X, Ren X, Wan H, Jiang X, Liu X.
Simulation and Optimization of Piezoelectric Micromachined Ultrasonic Transducer Unit Based on AlN. *Electronics*. 2022; 11(18):2915.
https://doi.org/10.3390/electronics11182915

**Chicago/Turabian Style**

Su, Xin, Xincheng Ren, Haoji Wan, Xingfang Jiang, and Xianyun Liu.
2022. "Simulation and Optimization of Piezoelectric Micromachined Ultrasonic Transducer Unit Based on AlN" *Electronics* 11, no. 18: 2915.
https://doi.org/10.3390/electronics11182915