# A Novel Model of Semiconductor Porosity Medium According to Photo-Thermoelasticity Excitation with Initial Stress

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## Abstract

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## 1. Introduction

## 2. Mathematical Model and Basic Equations

- (i)
- According to [25,28], during the photo-excited process of semiconductor elastic medium, there is a link between thermal waves and plasma waves:$$\frac{\partial N}{\partial t}={D}_{E}{\nabla}^{2}N-\hspace{0.17em}\frac{N}{\tau}+\kappa \hspace{0.17em}T.$$The quantity $\hspace{0.17em}\kappa =\frac{\partial {n}_{0}}{\partial T}$ symbolizes the general case of the thermal activation coupling parameter.
- (ii)
- According to the photo-thermoelastic theory, the equations of motion for semiconductor materials under the influence of double porosity and initial stress can be expressed as follows [29]:$$\begin{array}{c}(\lambda +\mu +\frac{p}{2})\nabla (\nabla \cdot \overrightarrow{u})+(k+\mu -\frac{p}{2}){\nabla}^{2}\overrightarrow{u}+{\lambda}_{o}\nabla \varphi -\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\widehat{\gamma}(1+{v}_{o}\frac{\partial}{\partial t})\nabla T-{\delta}_{n}\nabla N=\rho \ddot{\overrightarrow{u}}\end{array}\}.$$
- (iii)
- The photo-thermoelastic theory’s heat conduction equation for semiconductor media under the influence of twofold porosity and initial stress can be written as [30]:$$K{\nabla}^{2}T-\rho {C}_{E}({n}_{1}+{\tau}_{o}\frac{\partial}{\partial t})\dot{T}-\widehat{\gamma}{T}_{o}({n}_{1}+{n}_{o}{\tau}_{o}\frac{\partial}{\partial t})\dot{e}+\frac{{E}_{g}}{\tau K}N=\widehat{\gamma}{T}_{o}\dot{\varphi}.$$
- (iv)
- The double porosity model provides a novel approach to the investigation of significant mechanical and civil design problems. When conducting a nondestructive evaluation (NDE) of composite materials and structures, the phenomenon of coexistence of porosity and thermoelasticity is crucial. These substances are frequently discovered in the earth’s reservoir and crustal rocks. According to the coupling nature of the thermal waves and the porous potentials, the porous (voids) equation can be given as [12]:$$\alpha {\nabla}^{2}\varphi -{\lambda}_{o}e-{\varsigma}_{1}\varphi -{\omega}_{o}\dot{\varphi}+\widehat{\gamma}T=\rho \psi \ddot{\varphi}.$$

## 3. A solution to the Problem

## 4. Boundary Conditions

## 5. Inversion of the Fourier-Laplace Transforms

## 6. Numerical Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$\lambda ,\hspace{0.17em}\hspace{0.17em}\mu \hspace{1em}\hspace{1em}\hspace{0.17em}$ | Lame’s parameters. |

${\delta}_{n}$ | The distinction between the valence band and the conduction band’s deformation potential. |

$T$ | Absolute temperature. |

${T}_{0}\hspace{0.17em}$ | Reference temperature when $\left|\frac{T-{T}_{0}}{{T}_{0}}\right|<1$. |

$\hat{\gamma}=(3\lambda +2\mu +k){\alpha}_{t}$ | The thermal expansion of volume. |

${\alpha}_{t}$ | The thermal expansion coefficient. |

${\sigma}_{\mathrm{ij}}$ | The stress tensor. |

$\mathsf{\rho}\hspace{1em}\hspace{1em}$ | The density. |

$\mathrm{e}$ | Cubical dilatation. |

${C}_{e}$ | Specific heat. |

$k$ | The thermal conductivity. |

${D}_{E}$ | The carrier diffusion coefficient. |

$\tau $ | Lifetime. |

$t$ | Time variable. |

${E}_{g}$ | The energy gap. |

${e}_{i}{}_{j}$ | The strain tensor. |

${u}_{i}$ | Displacement vector. |

$N$ | Carrier concentration. |

$p$ | The initial pressure. |

$\alpha ,{\lambda}_{o},{\varsigma}_{1},{\omega}_{o},{\widehat{\gamma}}_{1},\psi $ | The constants of voids. |

${\tau}_{0},{\nu}_{0}$ | Thermal memories. |

$\varphi $ | The change in volume fraction field. |

## Appendix A

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**Figure 1.**(

**a–f**) Variation of the main physical distributions w.r.t the distance for different relaxation times under the impact of porosity parameters.

**Figure 2.**(

**a–f**) Variation of the main physical distributions w.r.t the distance for different two cases of porosity under the impact of GL model.

**Figure 3.**(

**a–f**) Variation of the main physical distributions w.r.t the vertical and horizontal distance under the impact of porosity according to the GL model.

**Figure 4.**(

**a–d**) The main physical wave’s fronts distribution at different dimensionless time instants.

Unit | Symbol | Value | Unit | Symbol | Value |
---|---|---|---|---|---|

$N/{m}^{2}$ | $\lambda $ $\mu $ | $3.64\times {10}^{10}$ $5.46\times {10}^{10}$ | $N$ | $p$ | $100$ |

$kg/{m}^{3}$ | $\rho $ | $2330\hspace{0.17em}$ | $J/kg\hspace{0.17em}K$ | ${C}_{e}$ | $695\hspace{0.17em}$ |

$\mathrm{K}$ | ${T}_{0}\hspace{1em}$ | $800$ | $\hspace{0.17em}\frac{m}{s}$ | $\tilde{s}$ | $2$ |

$s$ | $\tau \hspace{0.17em}$ | $5\times \hspace{0.17em}{10}^{-5}$ | ${m}^{2}$ | $\psi $ | $1.753\hspace{0.17em}\times {10}^{-15}$ |

${m}^{3}$ | ${d}_{n}\hspace{0.17em}$ | $-9\hspace{0.17em}\times \hspace{0.17em}{10}^{-31}$ | $N$ | $\alpha $ | $3.688\hspace{0.17em}\times \hspace{0.17em}{10}^{-5}$ |

${m}^{2}/s$ | ${D}_{E}\hspace{0.17em}$ | $2.5\times {10}^{-3}$ | $N{m}^{-2}$ | ${\varsigma}_{1}$ | $1.475\hspace{0.17em}\times \hspace{0.17em}{10}^{10}$ |

$eV$ | ${E}_{g}\hspace{0.17em}$ | $1.11$ | $N{m}^{-2}$ | ${\lambda}_{o}$ | $1.13849\hspace{0.17em}\times \hspace{0.17em}{10}^{10}$ |

${K}^{-1}$ | ${\alpha}_{t}\hspace{0.17em}$ | $4.14\hspace{0.17em}\times {10}^{-6}$ | $N{m}^{-2}{\mathrm{deg}}^{-1}$ | $m$ | $2\times \hspace{0.17em}{10}^{6}$ |

$\hspace{0.17em}W{m}^{-1}{K}^{-1}$ | $k\hspace{0.17em}$ | $150\hspace{0.17em}$ | $N{m}^{-2}{s}^{-1}$ | ${\omega}_{o}$ | $0.0787\hspace{0.17em}\times \hspace{0.17em}{10}^{-3}$ |

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**MDPI and ACS Style**

Raddadi, M.H.; El-Bary, A.; Tantawi, R.S.; Anwer, N.; Lotfy, K.
A Novel Model of Semiconductor Porosity Medium According to Photo-Thermoelasticity Excitation with Initial Stress. *Crystals* **2022**, *12*, 1603.
https://doi.org/10.3390/cryst12111603

**AMA Style**

Raddadi MH, El-Bary A, Tantawi RS, Anwer N, Lotfy K.
A Novel Model of Semiconductor Porosity Medium According to Photo-Thermoelasticity Excitation with Initial Stress. *Crystals*. 2022; 12(11):1603.
https://doi.org/10.3390/cryst12111603

**Chicago/Turabian Style**

Raddadi, Merfat H., A. El-Bary, Ramdan. S. Tantawi, N. Anwer, and Kh. Lotfy.
2022. "A Novel Model of Semiconductor Porosity Medium According to Photo-Thermoelasticity Excitation with Initial Stress" *Crystals* 12, no. 11: 1603.
https://doi.org/10.3390/cryst12111603