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

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 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

## References

- Lord, H.; Shulman, Y. A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids
**1967**, 15, 299–309. [Google Scholar] [CrossRef] - Gordon, J.P.; Leite, R.C.C.; Moore, R.S.; Porto, S.P.S.; Whinnery, J.R. Long-transient effects in lasers with inserted liquid samples. Bull. Am. Phys. Soc.
**1964**, 119, 501–510. [Google Scholar] [CrossRef] - Kreuzer, L.B. Ultralow gas concentration infrared absorption spectroscopy. J. Appl. Phys.
**1971**, 42, 2934–2943. [Google Scholar] [CrossRef] - Mahdy, A.M.S.; Gepreel, K.A.; Lotfy Kh El-Bary, A.A. A numerical method for solving the Rubella ailment disease model. Int. J. Mod. Phys. C
**2021**, 32, 2150097. [Google Scholar] [CrossRef] - Todorović, D.M.; Nikolić, P.M.; Bojičić, A.I. Photoacoustic frequency transmission technique: Electronic deformation mechanism in semiconductors. J. Appl. Phys.
**1999**, 85, 7716–7726. [Google Scholar] [CrossRef] - Song, Y.; Todorovic, D.M.; Cretin, B.; Vairac, P. Study on the generalized thermoelastic vibration of the optically excited semiconducting microcantilevers. Int. J. Solids Struct.
**2010**, 47, 1871–1875. [Google Scholar] [CrossRef][Green Version] - Abbas, I.A.; Alzahrani, F.S.; Elaiw, A. A DPL model of photothermal interaction in a semiconductor material. Waves Random Complex Media
**2019**, 29, 328–343. [Google Scholar] [CrossRef] - Khamis, A.K.; El-Bary, A.A.; Lotfy Kh Bakali, A. Photothermal excitation processes with refined multi dual phase-lags theory for semiconductor elastic medium. Alex. Eng. J.
**2020**, 59, 1–9. [Google Scholar] [CrossRef] - Hosseini, S.M.; Zhang, C. Plasma-affected photo-thermoelastic wave propagation in a semiconductor Love–Bishop nanorod using strain-gradient Moore–Gibson–Thompson theories. Thin-Walled Struct.
**2022**, 179, 109480. [Google Scholar] [CrossRef] - Hosseini, S.M.; Sladek, J.; Sladek, V. Nonlocal coupled photo-thermoelasticity analysis in a semiconducting micro/nano beam resonator subjected to plasma shock loading: A Green-Naghdi-based analytical solution. Appl. Math. Model.
**2020**, 88, 631–651. [Google Scholar] [CrossRef] - Nunziato, J.W.; Cowin, S.C. A non-linear theory of elastic materials with voids. Arch. Ration. Mech. Anal.
**1979**, 72, 175–201. [Google Scholar] [CrossRef] - Cowin, S.C.; Nunziato, J.W. Linear theory of elastic materials with voids. J. Elast.
**1983**, 13, 125–147. [Google Scholar] [CrossRef] - Dhaliwal, R.S.; Wang, J. Domain of influence theorem in the theory of elastic materials with voids. Int. J. Eng. Sci.
**1994**, 32, 1823–1828. [Google Scholar] [CrossRef] - Jhorar, R.; Tripathi, D.; Bhatti, M.M.; Ellahi, R. Electroosmosis modulated biomechanical transport through asymmetric microfluidics channel. Indian J. Phys.
**2018**, 92, 1229–1238. [Google Scholar] [CrossRef] - Bhatti, M.M.; Zeeshan, A.; Tripathi, D.; Ellahi, R. Thermally developed peristaltic propulsion of magnetic solid particles in biorheological fluids. Indian J. Phys.
**2018**, 92, 423–430. [Google Scholar] [CrossRef] - Ainouz, A. Homogenized double porosity models for poro-elastic media with interfacial flow barrier. Math. Bohem.
**2011**, 136, 357–365. [Google Scholar] [CrossRef] - Svanadze, M. External boundary value problems of steady vibrations in the theory of rigid bodies with a double porosity structure. Appl. Math. Mech.
**2015**, 15, 365–366. [Google Scholar] [CrossRef] - Straughan, B. Stability and uniqueness in double porosity elasticity. Int. J. Eng. Sci.
**2013**, 65, 1–8. [Google Scholar] [CrossRef] - Eringen, A.C. Microcontinuum Field Theories: I. Foundations and Solids; Springer: New York, NY, USA, 1999. [Google Scholar]
- Ailawalia, P.; Singh, N. Effect of rotation in a generalized thermoelastic medium with hydrostatic initial stress subjected to ramp type heating and loading. Int. J. Thermophys.
**2009**, 30, 2078–2097. [Google Scholar] [CrossRef] - Abbas, I.A.; Kumar, R. Response of thermal source in initially stressed generalized thermoelastic half-space with voids, J. Comput. Theoret. Nanosci.
**2014**, 11, 1472–1479. [Google Scholar] [CrossRef] - Bachher, M.; Sarkr, N.; Lahiri, A. Generalized thermoelastic infinite medium with voids subjected to an instantaneous heat sources with fractional derivative heat transfer. Int. J. Mech. Sci.
**2014**, 89, 84–91. [Google Scholar] [CrossRef] - Bachher, M.; Sarkr, N.; Lahiri, A. Fractional order thermoelastic interactions in an infinite voids material due to distributed time-dependent heat sources. Meccanica
**2015**, 50, 2167–2178. [Google Scholar] [CrossRef] - Eringen, A.C. Linear theory of micropolar elasticity. J. Math. Mech.
**1966**, 15, 909–923. [Google Scholar] - Hobiny, A.; Abbas, I. A GN model on photothermal interactions in a two-dimensions semiconductor half space. Results Phys.
**2019**, 15, 102588. [Google Scholar] [CrossRef] - Marin, M.; Lupu, M. On harmonic vibrations in thermoelasticity of micropolar bodies. J. Vibrat. Control
**1998**, 4, 507–518. [Google Scholar] [CrossRef] - Green, A.E.; Lindsay, K.A. Thermoelasticity. J. Elast.
**1972**, 2, 507–518. [Google Scholar] [CrossRef] - Todorovic, D.M. Plasma, thermal, and elastic waves in semiconductors. Rev. Sci. Instrum.
**2003**, 74, 582–588. [Google Scholar] [CrossRef] - Tam, A.C. Applications of photoacoustic sensing techniques. Rev. Mod. Phys.
**1986**, 58, 381–389. [Google Scholar] [CrossRef] - Tam, A.C. Ultrasensitive Laser Spectroscopy; Academic Press: New York, NY, USA, 1983; pp. 1–108. [Google Scholar]
- Tam, A.C. Photothermal Investigations in Solids and Fluids; Academic Press: Boston, MA, USA, 1989; pp. 1–33. [Google Scholar]
- Honig, G.; Hirdes, U. A method for the numerical inversion of Laplace transforms. J. Comput. Appl. Math.
**1984**, 10, 113–132. [Google Scholar] [CrossRef][Green Version] - Marin, M.; Stan, G. Weak solutions in Elasticity of dipolar bodies with stretch. Carpath. J. Math.
**2013**, 29, 33–40. [Google Scholar] [CrossRef] - Mandelis, A.; Nestoros, M.; Christofides, C. Thermoelectronic-wave coupling in laser photothermal theory of semiconductors at elevated temperatures. Opt. Eng.
**1997**, 36, 459–468. [Google Scholar] [CrossRef] - Hobiny, A.; Abbas, I. A study on photothermal waves in an unbounded semiconductor medium with cylindrical cavity. Mech. Time-Depend Mater.
**2016**, 6, 1–12. [Google Scholar] [CrossRef] - Lotfy Kh Hassan, W.; El-Bary, A.; Kadry, M. Response of electromagnetic and Thomson effect of semiconductor mediu due to laser pulses and thermal memories during photothermal excitation. Results Phys.
**2020**, 16, 102877. [Google Scholar] [CrossRef] - Liu, J.; Han, M.; Wang, R.; Xu, S.; Wang, X. Photothermal phenomenon: Extended ideas for thermophysical properties characterization. J. Appl. Phys.
**2022**, 131, 065107. [Google Scholar] [CrossRef]

**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 $ | $3.688\hspace{0.17em}\times \hspace{0.17em}{10}^{-5}$ |

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

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

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

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

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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