# Excited Non-Local Microelongated Semiconductor Layer Thermal-Optical Mechanical Waves Affected by Rotational Field

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## Abstract

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## 1. Introduction

## 2. Mathematical Model and Main Equations

## 3. Solutions to the Problem

## 4. Boundary Conditions

## 5. Discussion and Numerical Results

#### 5.1. Impact of Thermal and Elastic Memories

#### 5.2. Impact of the Laser Pulse Rise-Time Parameter

#### 5.3. Impact of Rotation Parameter

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

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

${\delta}_{n}=(3\lambda +2\mu ){d}_{n}$ | The deformation potential difference. |

$\underset{\_}{n}$ | Unit vector in the direction of y-axis. |

${T}_{0}\hspace{0.17em}$ | Reference temperature in its natural state. |

$\hat{\gamma}=(3\lambda +2\mu ){\alpha}_{{t}_{1}}$ | The volume thermal expansion. |

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

$\rho \hspace{1em}\hspace{1em}$ | The density of the microelongated sample. |

${\alpha}_{{t}_{1}}$ | Coefficients of linear thermal expansion. |

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

${C}_{e}$ | Specific heat of the microelongated material. |

$K$ | The thermal conductivity. |

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

$\tau $ | The carrier lifetime. |

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

${e}_{i}{}_{j}$ | Components of strain tensor. |

$\Pi ,\Psi $ | Two scalar functions. |

${j}_{0}$ | The microinertia of microelement. |

${a}_{0},\hspace{0.17em}{\alpha}_{0},{\lambda}_{0},{\lambda}_{1}$ | Microelongational material parameters. |

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

$\phi $ | The scalar microelongational function. |

${m}_{k}$ | Components of the microstretch vector |

$s={s}_{kk}$ | Stress tensor component |

${\delta}_{i}{}_{k}$ | Kronecker delta |

$\underset{\_}{\Omega}=\Omega \underset{\_}{n}$ | Angular velocity |

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**Figure 2.**(

**a–f**). The variation of the main physical fields versus the horizontal distance according to the differences in thermal relaxation times under the effect of rotation parameter.

**Figure 3.**(

**a–f**). The variation of the main physical fields versus the horizontal distance according to the differences in the laser pulse rise-time parameters under the effect of rotation parameter.

**Figure 4.**(

**a–f**). The variations the main physical fields versus the horizontal distance in case of the GL model with rotation field influence and without rotation field influence.

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

${\mathrm{Nm}}^{-2}$ | $\lambda $ $\mu $ | $3.64\hspace{0.17em}\times \hspace{0.17em}{10}^{10}$ $5.46\times {10}^{10}$ | $\mathrm{sec}\hspace{0.17em}(\mathrm{s})$ | ${\tau}_{0}$, ${\nu}_{0}$ | $0.00005$, $0.0005$ |

${\mathrm{kg}/\mathrm{m}}^{3}$ | $\rho $ | $2330\hspace{0.17em}$ | $\hspace{0.17em}{\mathrm{m}}^{3}$ | ${d}_{n}$ | $-9\hspace{0.17em}\times \hspace{0.17em}{10}^{-31}$ |

$\mathrm{K}$ | ${T}_{0}$ | $800$ | ${\mathrm{m}}^{2}$ | $j$ | $0.2\times {10}^{-19}$ |

$\mathrm{sec}\hspace{0.17em}(\mathrm{s})$ | $\tau \hspace{0.17em}$ | $5\times \hspace{0.17em}{10}^{-5}$ | $\mathrm{N}$ | ${\alpha}_{0}$ | $0.779\times {10}^{-9}$ |

${\mathrm{K}}^{-1}$ | ${\alpha}_{{t}_{1}}$ | $0.04\times \hspace{0.17em}{10}^{-3}\hspace{0.17em}$ | ${\mathrm{Nm}}^{-2}$ | ${\lambda}_{0}$ | $0.5\times {10}^{10}$ |

$\hspace{0.17em}{\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$ | ${K}_{0}$ | $150\hspace{0.17em}$ | ${\mathrm{Nm}}^{-2}$ | $k$ | ${10}^{10}$ |

$\mathrm{J}/(\mathrm{kg}\hspace{0.17em}\mathrm{K})$ | ${C}_{e}$ | $695\hspace{0.17em}$ | ${\mathrm{Nm}}^{-2}$ | ${\lambda}_{1}$ | $0.5\times {10}^{10}$ |

${\mathrm{m}}^{2}/\mathrm{s}$ | ${D}_{E}$ | $2.5\times {10}^{-3}\hspace{0.17em}$ | ${\mathrm{K}}^{-1}$ | ${\alpha}_{{t}_{2}}$ | $0.017\times \hspace{0.17em}{10}^{-3}\hspace{0.17em}$ |

$\hspace{0.17em}\mathrm{m}/\mathrm{s}$ | $\tilde{s}$ | $2$ | ${\mathrm{m}}^{-3}$ | ${\tilde{n}}_{0}$ | $\hspace{0.17em}\hspace{0.17em}{10}^{20}\hspace{0.17em}$ |

$\mathrm{sec}\hspace{0.17em}(\mathrm{s})$ | $t$ | $0.001$ | $\mathrm{eV}$ | ${E}_{g}$ | $1.11\hspace{0.17em}$ |

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**MDPI and ACS Style**

El-Sapa, S.; Alhejaili, W.; Lotfy, K.; El-Bary, A.A.
Excited Non-Local Microelongated Semiconductor Layer Thermal-Optical Mechanical Waves Affected by Rotational Field. *Crystals* **2023**, *13*, 116.
https://doi.org/10.3390/cryst13010116

**AMA Style**

El-Sapa S, Alhejaili W, Lotfy K, El-Bary AA.
Excited Non-Local Microelongated Semiconductor Layer Thermal-Optical Mechanical Waves Affected by Rotational Field. *Crystals*. 2023; 13(1):116.
https://doi.org/10.3390/cryst13010116

**Chicago/Turabian Style**

El-Sapa, Shreen, Weaam Alhejaili, Khaled Lotfy, and Alaa A. El-Bary.
2023. "Excited Non-Local Microelongated Semiconductor Layer Thermal-Optical Mechanical Waves Affected by Rotational Field" *Crystals* 13, no. 1: 116.
https://doi.org/10.3390/cryst13010116