# Microelongated Thermo-Elastodiffusive Waves of Excited Semiconductor Material under Laser Pulses Impact

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## Abstract

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## 1. Introduction

## 2. Governing Equation

## 3. Analytical Solution Procedure

## 4. Applications

- (I)
- When $x=0$, the pulsing heat flow boundary condition may be represented by the thermally gradient temperature in the following ways:$${\frac{\partial T\left(x,t\right)}{\partial x}|}_{x=0}=-{q}_{0}\frac{{{\displaystyle {t}^{2}{e}^{-\frac{t}{{{\displaystyle t}}_{p}}}}}^{}}{16{t}_{p}^{2}}.$$

- (II)
- The load force $\Im $ may be expressed as follows when the Laplace transform is used, and it is regarded as a mechanical condition at the boundary $x=0$:$$\overline{\sigma}(0,s)=\overline{\Im}\Rightarrow {\displaystyle \sum _{i=1}^{5}{H}_{4i}{\mathsf{\Lambda}}_{i}}(s)=\Im .$$
- (III)
- The carriers are capable of obtaining the following value on the sample’s surface, $x=0$, as they disperse, where they have a chance of undergoing recombination. Consequently, the following equation might be used to express the carrier density boundary condition. In this, the plasma condition under the Laplace transform is derived using the carrier density diffusive and is shown as follows:$$\overline{N}(0,s)=\frac{\u019b{n}_{0}}{{D}_{e}}\overline{R}(s)\Rightarrow {\displaystyle \sum _{i=1}^{5}{H}_{1i}{\mathsf{\Lambda}}_{i}(s)}=\frac{\u019b{n}_{0}}{\tilde{s}{D}_{e}}.$$
- (IV)
- One may choose the microelongation condition as an elongation free on the surface $x=0$:$$\overline{\phi}=0\Rightarrow {\displaystyle \sum _{i=1}^{5}{H}_{3i}{\mathsf{\Lambda}}_{i}}(s)=0.$$
- (V)
- On the other hand, the equilibrium concentration situation for the hole charge carrier field at the boundary condition may be obtained in this instance by using the Laplace transform:

## 5. Inversion of the Laplace Transforms

## 6. Validation

#### 6.1. The Microelongated Thermoelasticity Models

#### 6.2. The Influence of Microelongation Parameters

#### 6.3. The Laser Pulses Impact

## 7. Numerical Results and Discussion

#### 7.1. The Photo-Thermoelasticity Models

#### 7.2. The Laser Pulses Effect

#### 7.3. The Effect of Elongation Parameters

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature (The Physical Quantities with Units)

$\lambda ,\mu $ | Elastic Lame’s parameters. |

${\delta}_{n}=(3\lambda +2\mu ){d}_{n}$ | Deformation potential difference between conduction and valence band. |

${d}_{n}$ | The electronic deformation coefficient ED |

${T}_{0}$ | Reference temperature in its natural state |

$\gamma =(3\lambda +2\mu ){\alpha}_{t}$ | Volume thermal expansion |

${\sigma}_{ij}$ | Stress tensor |

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

${\alpha}_{t}$ | Linear thermal expansion |

$p$ | The power intensity of the laser |

${C}_{e}$ | Specific heat at constant strain |

$K$ | Thermal conductivity of the semiconductor medium |

${t}^{n}$ | The electrons relaxation time |

${t}_{1}^{n}$ | The lifetime of photogenerated carriers |

${E}_{g}$ | Energy gap |

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

${m}_{nq},{m}_{qn},{m}_{hq},{m}_{qh}$ | Peltier–Seebeck and Dufour–Soret-like constants |

${a}_{Qn},{a}_{Qh},{a}_{Q},{a}_{n},{a}_{h}$ | The flux-like constants |

${\tau}_{\theta},{\tau}_{q}$ | Thermal and elastic relaxation times |

${\alpha}_{h},{\alpha}_{n}$ | Holes and electrons thermo-diffusive parameters |

${n}_{0}$,${h}_{0}$ | The equilibrium value of electrons and holes concentration |

${\delta}_{h}=(2\mu +3\lambda ){d}_{h}$ | The holes elastodiffusive parameter |

${d}_{h}$ | The coefficients of hole deformation |

$\tilde{s}$ | Recombination velocities |

${D}_{n},{D}_{h}$ | The diffusion coefficients of the electrons and holes |

${a}_{0},{\alpha}_{0},{\lambda}_{0},{\lambda}_{1}$ | Microelongational material parameters |

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

$\phi $ | The scalar microelongational function |

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

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

$\mathsf{\Omega}$ | The pulse parameter |

$\delta $ | The optical absorption coefficient |

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**Figure 2.**According to the two-temperature theory, the primary field distributions change with horizontal distance at various thermal memory (thermal relaxation durations) under the impact of laser pulses.

**Figure 3.**According to the DPL model, the main field distributions in two temperature fields under the impact of laser pulses and without them changing in terms of horizontal distance.

**Figure 4.**The main field distributions in distinct two temperature cases under the influence of laser pulses vary with horizontal distance in accordance with the DPL model.

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

${\mathrm{N}/\mathrm{m}}^{2}$ | $\lambda $ $\mu $ | $6.4\times {10}^{10}$ $6.5\times {10}^{10}$ |

${\mathrm{kg}/\mathrm{m}}^{3}$ | $\rho $ | $2330$ |

$\mathrm{K}$ | ${T}_{0}$ | $800$ |

$\mathrm{sec}\left(\mathrm{s}\right)$ | $\tau $ | $5\times {10}^{-5}$ |

${\mathrm{K}}^{-1}$ | ${\alpha}_{t}$ | $4.14\times {10}^{-6}$ |

${\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$ | $K$ | $150$ |

$\mathrm{J}/(\mathrm{kg}\mathrm{K})$ | ${C}_{e}$ | $695$ |

$\mathrm{m}/\mathrm{s}$ | $\tilde{s}$ | $2$ |

${\mathrm{vk}}^{-1}$ | ${m}_{qn}$ | $1.4\times {10}^{-5}$ |

${m}_{nq}$ | $1.4\times {10}^{-5}$ | |

${m}_{qh}$ | $-0.004\times {10}^{-6}$ | |

${m}_{hq}$ | $-0.004\times {10}^{-6}$ | |

$\mathrm{J}\cdot {\mathrm{m}}^{-2}$ | $p$ | ${10}^{11}$ |

${\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ | ${D}_{n}$ | $0.35\times {10}^{-2}$ |

${\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ | ${D}_{h}$ | $0.125\times {10}^{-2}$ |

${\mathrm{m}}^{2}/\mathrm{s}$ | ${\alpha}_{n}$ | $1\times {10}^{-2}$ |

${\mathrm{m}}^{2}/\mathrm{s}$ | ${\alpha}_{h}$ | $5\times {10}^{-3}$ |

$\mathrm{ps}$ | ${t}_{p}$ | $4$ |

Constants | ${q}_{0}$ $\Im $ | 3 2 |

$\mathrm{J}\cdot {\mathrm{m}}^{-2}$ | $p$ | ${10}^{11}$ |

${\mathrm{m}}^{2}$ | $j$ | $0.2\times {10}^{-19}$ |

${\mathrm{Nm}}^{-2}$ | ${\lambda}_{0}$ | $0.5\times {10}^{10}$ |

${\mathrm{Nm}}^{-2}$ | ${\lambda}_{1}$ | $0.5\times {10}^{10}$ |

${\mathrm{Nm}}^{-2}$ | $k$ | ${10}^{10}$ |

${\mathrm{K}}^{-1}$ | ${\alpha}_{\vartheta}$ | $0.017\times {10}^{-3}$ |

${\mathrm{Nm}}^{-2}$ | ${\alpha}_{0}$ | $0.779\times {10}^{-9}$ |

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## Share and Cite

**MDPI and ACS Style**

Tayel, I.M.; Lotfy, K.; El-Bary, A.A.; Alebraheem, J.; Mohammed, M.A.Y. Microelongated Thermo-Elastodiffusive Waves of Excited Semiconductor Material under Laser Pulses Impact. *Mathematics* **2023**, *11*, 1627.
https://doi.org/10.3390/math11071627

**AMA Style**

Tayel IM, Lotfy K, El-Bary AA, Alebraheem J, Mohammed MAY. Microelongated Thermo-Elastodiffusive Waves of Excited Semiconductor Material under Laser Pulses Impact. *Mathematics*. 2023; 11(7):1627.
https://doi.org/10.3390/math11071627

**Chicago/Turabian Style**

Tayel, Ismail M., Kh. Lotfy, Alaa A. El-Bary, Jawdat Alebraheem, and Mogtaba A. Y. Mohammed. 2023. "Microelongated Thermo-Elastodiffusive Waves of Excited Semiconductor Material under Laser Pulses Impact" *Mathematics* 11, no. 7: 1627.
https://doi.org/10.3390/math11071627