# Influence of Homo- and Hetero-Junctions on the Propagation Characteristics of Radially Propagated Cylindrical Surface Acoustic Waves in a Piezoelectric Semiconductor Semi-Infinite Medium

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## Abstract

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## 1. Introduction

## 2. Problem Description and Basic Equations

**,**and $\mathit{\epsilon}({\mathit{\epsilon}}^{L}\mathrm{o}\mathrm{r}{\mathit{\epsilon}}^{U})$ represent the elastic, piezoelectric and dielectric parameter tensors, respectively; ${\mathit{\mu}}^{c}$ (${\mathit{\mu}}^{pL}$ or ${\mathit{\mu}}^{nU}$) and ${\mathit{d}}^{c}$ (${\mathit{d}}^{pL}$

**or**${\mathit{d}}^{nU}$) are the (hole or electron) carrier migration and diffusion parameter tensors, respectively; $\overline{c}$ (${\overline{p}}^{L}$ or ${\overline{n}}^{U}$) is the steady (hole or electron) carrier concentration resulting from the doping, i.e., the doping concentration. Moreover, $c$ (${p}^{L}$ or ${n}^{U}$) is the (hole or electron) carrier concentration perturbation scalar; $q=1.602\times {10}^{-19}\mathrm{C}$ is the carrier charge; $\mathbf{\nabla}={\mathit{e}}_{\mathit{r}}\partial /\partial r+{\mathit{e}}_{\mathit{\theta}}\partial /r\partial \theta +{\mathit{e}}_{\mathit{z}}\partial /\partial z$ is the gradient operator in the cylindrical coordinate system; ${\mathit{e}}_{\mathit{r}}$, ${\mathit{e}}_{\mathit{\theta}}$ and ${\mathit{e}}_{\mathit{z}}$ are unit base vectors in the cylindrical coordinate system, of which the untrivial differentiations are $d{\mathit{e}}_{\mathit{r}}/d\theta ={\mathit{e}}_{\mathit{\theta}}$ and $d{\mathit{e}}_{\mathit{\theta}}/d\theta =-{\mathit{e}}_{\mathit{r}}$; $\mathit{S}({\mathit{S}}^{L}\mathrm{o}\mathrm{r}{\mathit{S}}^{U})$ is the strain tensor; $\mathit{E}({\mathit{E}}^{L}\mathrm{o}\mathrm{r}{\mathit{E}}^{U})$ is the electric field vector. Under the supposition of small deformations and the quasi-static electric field approximation, the strain tensor $\mathit{S}$ and electric field vector $\mathit{E}$ are

## 3. Two Equivalent Mathematical Models

#### 3.1. The First Mathematical Model

**,**and ${\mathit{J}}^{nJ}$ are the interface electric displacement, hole and electron carrier current density vectors, respectively [30,31]:

#### 3.2. The Second Mathematical Model

## 4. Numerical Results and Discussion

## 5. Concluding Remarks

- The physical properties of the homo- and hetero-junctions are closely related to the doping modes and concentrations in the PSC semi-infinite medium, of which the increase not only makes the homo- and hetero-junctions more like electrically imperfect interfaces but also enhance the influence of the homo- and hetero-junctions.
- The influence of the homo- and hetero-junctions on the dispersion and attenuation characteristics of radially propagated cylindrical SAW is closely related to the angular frequency. It is of great significance to consider the existence of homo- and hetero-junctions when studying the propagation of high-frequency SAW in PSC materials.
- The homo- and hetero-junctions can enhance the conversion efficiency from mechanical to electrical energy; adjusting doping modes and concentrations of PSC semi-infinite medium can regulate attenuation characteristics of cylindrical SAW and optimize the acoustic characteristics of SAW resonators, energy harvesters, and acoustic wave amplification formed by PSC semi-infinite medium.
- Since the dispersion and attenuation characteristics of cylindrical SAW are independent of the radial coordinate, the mathematical models in this paper provide theoretical guidance for studying the propagation characteristics of other types of SAW in PSC materials, such as the Rayleigh-type SAW. Since the influence of the homo- and hetero-junctions mainly concentrates on the attenuation characteristic of SAW, the discussion in this paper provides theoretical guidance for the design and manufacture of SAW resonators [3,4], energy harvesting [6,7], and acoustic wave amplification [8]. Due to the interface characteristic lengths, the first mathematical model is more complex than the second. The homo- and hetero-junctions are treated as functional gradient thin layers; the second mathematical model is more accurate than the first. Based on Figure 9 and Figure 11, the computational stability of the second mathematical model is weaker than that of the first. Considering the interface characteristic lengths are independent of the doping concentrations, the first mathematical model is more practical in studying wave motion problems. Based on Figure 6 and Figure 10, the relative changes of the radial wave speed and dimensionless attenuation coefficient are considerably lower than 1, which means that the homo- and hetero-junctions have less evident influence on the propagation characteristics of low-frequency radially propagated cylindrical SAW. Therefore, in subsequent work, we must establish a simpler mathematical model with better computational accuracy and stability to study the propagation characteristics of high-frequency SAW in the PSC materials, such as the PSC layered composite structures containing multi-layer homo- and hetero-junctions.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

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**Figure 2.**Schematic of the functional gradient thin layer divided into a finite number of thin layers.

**Figure 3.**Distribution of steady charge carrier concentrations (

**a**,

**b**) and geometrical thicknesses (

**c**–

**e**) of the homo−junction with varying doping concentrations in the upper n−type ZnO/lower p−type ZnO semi−infinite medium.

**Figure 4.**The dispersion curves (

**a**) and attenuation curves (

**b**) of radially propagated cylindrical SAW in the upper n−type ZnO/lower p−type ZnO semi−infinite medium with variation of the $r$ coordinate.

**Figure 5.**Dispersion curves (

**a**,

**c**) and attenuation curves (

**b**,

**d**) of radially propagated cylindrical SAW in the upper n−type ZnO/lower p−type ZnO semi−infinite medium with varying doping concentrations.

**Figure 6.**Relative changes in the radial wave speed (

**a**,

**c**) and the dimensionless attenuation coefficient (

**b**,

**d**) of radially propagated cylindrical SAW in the upper n−type ZnO/lower p−type ZnO semi−infinite medium with varying doping concentrations.

**Figure 7.**Numerical comparations of the dispersion curves (

**a**,

**c**) and attenuation curves (

**b**,

**d**) of radially propagated cylindrical SAW in the upper n−type ZnO/lower p−type ZnO semi−infinite medium calculated by the two equivalent mathematical models.

**Figure 8.**Distribution of steady charge carrier concentrations (

**a**,

**b**) and geometrical thicknesses (

**c**–

**e**) of the hetero−junction with varying doping concentrations in the upper n−type GaN/lower p−type ZnO semi−infinite medium.

**Figure 9.**Dispersion curves (

**a**) and attenuation curves (

**b**) of radially propagated cylindrical SAW in the upper n−type GaN/lower p−type semi−infinite medium with varying doping concentrations.

**Figure 10.**Relative changes of the radial wave speed (

**a**) and the dimensionless attenuation coefficient (

**b**) of radially propagated cylindrical SAW in the upper n−type GaN/lower p−type ZnO semi−infinite medium with varying doping concentrations.

**Figure 11.**Numerical comparison of the dispersion curves (

**a**) and attenuation curves (

**b**) of radially propagated cylindrical SAW in the upper n−type GaN/lower p−type ZnO semi−infinite medium calculated by the two equivalent mathematical models.

${c}_{11}$ | ${c}_{13}$ | ${c}_{33}$ | ${c}_{44}$ | $\rho $ | ${e}_{15}$ | ${e}_{31}$ | ${e}_{33}$ | ${\epsilon}_{11}/{\epsilon}_{zz}^{0}$ | |||||||

ZnO | $209.7$ | $105.4$ | $211.2$ | $42.4$ | $5665$ | $-0.48$ | $-0.567$ | $1.32$ | $7.57$ | ||||||

GaN | $298.4$ | $142.5$ | $289.2$ | $23.1$ | $6095$ | $-0.31$ | $-0.52$ | $0.61$ | $9.5$ | ||||||