# Two-Temperature Semiconductor Model Photomechanical and Thermal Wave Responses with Moisture Diffusivity Process

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## Abstract

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## 1. Introduction

## 2. Basic Equations

## 3. Formulation of the Problem

## 4. Harmonic Wave Analysis

## 5. Applications

## 6. Numerical Results and Discussions

#### 6.1. The Effect of Two-Temperature Parameter

#### 6.2. The Impact of Thermoelastic Coupling Parameter

#### 6.3. The Effect of the Thermoelectric Coupling Parameter

#### 6.4. Influence of Reference Moisture

#### 6.5. The Comparison between Si and Ge Materials

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Lord, H.; Shulman, Y. A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids
**1967**, 15, 299–309. [Google Scholar] [CrossRef] - Green, A.E.; Lindsay, K. Thermoelasticity. J. Elast.
**1972**, 2, 1–7. [Google Scholar] [CrossRef] - Dhaliwal, R.S.; Sherief, H.H. Generalized thermoelasticity for anisotropic media. Q. Appl. Math.
**1980**, 38, 1–8. [Google Scholar] [CrossRef] [Green Version] - Green, A.E.; Naghdi, P. A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. London. Ser. A Math. Phys. Sci.
**1991**, 432, 171–194. [Google Scholar] - Green, A.E.; Naghdi, P. On undamped heat waves in an elastic solid. J. Therm. Stress.
**1992**, 15, 253–264. [Google Scholar] [CrossRef] - Green, A.E.; Naghdi, P. Thermoelasticity without energy dissipation. J. Elast.
**1993**, 31, 189–208. [Google Scholar] [CrossRef] - Chen, P.J.; Gurtin, M.E. On a theory of heat conduction involving two temperatures. Z. Angew. Math. Und Phys. ZAMP
**1968**, 19, 614–627. [Google Scholar] [CrossRef] - Chen, P.J.; Williams, W.O. A note on non-simple heat conduction. Z. Angew. Math. Und Phys. ZAMP
**1968**, 19, 969–970. [Google Scholar] [CrossRef] - Warren, W.E.; Chen, P.J. Wave propagation in the two temperature theory of thermoelasticity. Acta Mech.
**1973**, 16, 21–33. [Google Scholar] [CrossRef] - Singh, M.; Kumari, S. Influence of gravity and initial stress on Rayleigh wave propagation in magneto-thermoelastic medium. J. Math. Comput. Sci.
**2021**, 11, 2681–2698. [Google Scholar] - Youssef, H.M. Theory of two-temperature-generalized thermoelasticity. IMA J. Appl. Math.
**2006**, 71, 383–390. [Google Scholar] [CrossRef] - Ezzat, M.A.; El-Karamany, A.S.; Ezzat, S.M. Two-temperature theory in magneto-thermoelasticity with fractional order dual-phase-lag heat transfer. Nucl. Eng. Des.
**2012**, 252, 267–277. [Google Scholar] [CrossRef] - Shivay, O.N.; Mukhopadhyay, S. On the temperature-rate dependent two-temperature thermoelasticity theory. J. Heat Transf.
**2020**, 142, 022102. [Google Scholar] [CrossRef] - Bajpai, A.; Kumar, R.; Sharma, P.K. Analysis of wave motion and deformation in elastic plate based on two temperature theory of thermoelasticity. Waves Random Complex Media
**2021**, 1–22. [Google Scholar] [CrossRef] - Abouelregal, A.E.; Alanazi, R. Fractional Moore-Gibson-Thompson heat transfer model with two-temperature and non-singular kernels for 3D thermoelastic solid. J. Ocean. Eng. Sci. 2022, in press. [CrossRef]
- Sharma, J.N.; Sharma, Y.D.; Sharma, P.K. On the propagation of elasto-thermodiffusive surface waves in heat-conducting materials. J. Sound Vib.
**2008**, 315, 927–938. [Google Scholar] [CrossRef] - Aouadi, M.; Moulahi, T. Asymptotic analysis of a nonsimple thermoelastic rod. Discret. Contin. Dyn. Syst.-S
**2016**, 9, 1475. [Google Scholar] [CrossRef] [Green Version] - Lotfy, K.; Hassan, W. Normal mode method for two-temperature generalized thermoelasticity under thermal shock problem. J. Therm. Stress.
**2014**, 37, 545–560. [Google Scholar] [CrossRef] - Kumar, R.; Gupta, V. Wave propagation at the boundary surface of an elastic and thermoelastic diffusion media with fractional order derivative. Appl. Math. Model.
**2015**, 39, 1674–1688. [Google Scholar] [CrossRef] - Kumar, R.; Kansal, T. Propagation of Rayleigh waves on free surface of transversely isotropic generalized thermoelastic diffusion. Appl. Math. Mech.
**2008**, 29, 1451–1462. [Google Scholar] [CrossRef] - Kumar, R.; Kansal, T. Effect of rotation on Rayleigh waves in an isotropic generalized thermoelastic diffusive half-space. Arch. Mech.
**2008**, 60, 421–443. [Google Scholar] - Kumar, R.; Kansal, T. Dynamic problem of generalized thermoelastic diffusive medium. J. Mech. Sci. Technol.
**2010**, 24, 337–342. [Google Scholar] [CrossRef] - Kumar, R.; Gupta, V. Problem of Rayleigh Wave Propagation in Thermoelastic Diffusion. J. Solid Mech.
**2016**, 8, 602–613. [Google Scholar] - Ezzat, M.A.; Abd Elall, M.Z. Generalized magneto-thermoelasticity with modified Ohm’s law. Mech. Adv. Mater. Struct.
**2010**, 17, 74–84. [Google Scholar] [CrossRef] - Othman, M.I.A.; Lotfy, K. On the Plane Waves in Generalized Thermomicrostretch Elastic Half-space. Int. Commun. Heat Mass Transf.
**2010**, 37, 192–200. [Google Scholar] [CrossRef] - Othman, M.I.; Lotfy, K.; Farouk, R. Generalized thermo-microstretch elastic medium with temperature dependent properties for different theories. Eng. Anal. Bound. Elem.
**2010**, 34, 229–237. [Google Scholar] [CrossRef] - Lotfy, K.; Othman, M. The effect of rotation on plane waves in generalized thermo-microstretch elastic solid with one relaxation time for a mode-I crack problem. Chin. Phys. B
**2011**, 20, 074601. [Google Scholar] [CrossRef] - Lotfy, K. Mode-I crack in a two-dimensional fibre-reinforced generalized thermoelastic problem. Chin. Phys. B
**2012**, 21, 014209. [Google Scholar] [CrossRef] - Othman, M.; Lotfy, K. The effect of magnetic field and rotation of the 2D problem of a fiber-reinforced thermoelastic under three theories with influence of gravity. Mech. Mater.
**2013**, 60, 120–143. [Google Scholar] [CrossRef] - Lotfy, K.; 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] - Youssef, H.; El-Bary, A. Theory of hyperbolic two-temperature generalized thermoelasticity. Mater. Phys. Mechs.
**2018**, 40, 158–171. [Google Scholar] - Fahmy, M.A. Boundary element modeling of 3T nonlinear transient magneto-thermoviscoelastic wave propagation problems in anisotropic circular cylindrical shells. Compos. Struct.
**2021**, 27, 114655. [Google Scholar] [CrossRef] - Fahmy, M.A.; Almehmadi, M.M.; Subhi, F.M.; Sohail, A. Fractional boundary element solution of three-temperature thermoelectric problems. Sci. Rep.
**2022**, 2, 6760. [Google Scholar] [CrossRef] - Fahmy, M.A. 3D Boundary Element Model for Ultrasonic Wave Propagation Fractional Order Boundary Value Problems of Functionally Graded Anisotropic Fiber-Reinforced Plates. Fractal Fract.
**2022**, 6, 247. [Google Scholar] [CrossRef] - Fahmy, M.A.; Alsulami, M.O. Boundary Element and Sensitivity Analysis of Anisotropic Thermoelastic Metal and Alloy Discs with Holes. Materials
**2022**, 15, 1828. [Google Scholar] [CrossRef] [PubMed] - Fahmy, M.A. Boundary element modeling of fractional nonlinear generalized photothermal stress wave propagation problems in FG anisotropic smart semiconductors. Eng. Anal. Bound. Elem.
**2022**, 134, 665–679. [Google Scholar] [CrossRef] - Hosseini, S.M.; Sladek, J.; Sladek, V. Application of meshless local integral equations to two dimensional analysis of coupled non-Fick diffusionelasticity. Eng. Anal. Bound. Elem.
**2013**, 37, 603–615. [Google Scholar] [CrossRef] - Abo-Dahab, S.; Lotfy, K. Two-temperature plane strain problem in a semiconducting medium under photothermal theory. Waves Random Complex Media
**2017**, 27, 67–91. [Google Scholar] [CrossRef] - Lotfy, K.; Sarkar, N. Memory-dependent derivatives for photothermal semiconducting medium in generalized thermoelasticity with two- Temperature. Mech. Time-Depend. Mater.
**2017**, 21, 519–534. [Google Scholar] [CrossRef] - Lotfy, K.; Elidy, E.S.; Tantawi, R.S. Photothermal Excitation Process during Hyperbolic Two-Temperature Theory for Magneto-Thermo-Elastic Semiconducting Medium. Silicon
**2021**, 13, 2275–2288. [Google Scholar] [CrossRef] - Lotfy, K.; Elidy, E.S.; Tantawi, R.S. Piezo-photo-thermoelasticity transport process for hyperbolic two-temperature theory of semiconductor material. Int. J. Mod. Phys. C
**2021**, 32, 2150088. [Google Scholar] [CrossRef] - Xiao, Y.; Shen, C.; Zhang, W.B. Screening and prediction of metal-doped α-borophene monolayer for nitric oxide elimination. Mater. Today Chem.
**2022**, 25, 100958. [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.**Represents the variations of main fields in this phenomenon according to the different values of two-temperature parameter under the effect of moisture field.

**Figure 2.**The variation of physical field distributions with distance at different values of thermoelectric coupling parameter ${\epsilon}_{3}$ under the effect of moisture and ${\epsilon}_{1}=0.001$.

**Figure 3.**The variation of physical field distributions with distance at different values of thermo- elastic coupling parameter ${\epsilon}_{1}$ under the effect of moisture field and ${\epsilon}_{3}=-7.8\times {10}^{-36}$.

**Figure 4.**The variation of physical field distributions with distance at different values of the reference moisture ${m}_{0}$ under the effect of magnetic field when ${\epsilon}_{1}=0.001$ and ${\epsilon}_{3}=-7.8\times {10}^{-36}$.

**Figure 5.**The comparison between Si and Ge materials of physical field distributions with distance under the effect of moisture field when ${\epsilon}_{1}=0.001$ and ${\epsilon}_{3}=-7.8\times {10}^{-36}$.

Units | Symbol | Si | Ge |
---|---|---|---|

$\lambda $ | $6.4\times {10}^{10}$ | $0.48\times {10}^{11}$ | |

$\hspace{0.17em}\mathrm{N}/{\mathrm{m}}^{2}$ | $\mu $ | $6.5\times {10}^{10}$ | $0.53\times {10}^{11}$ |

$\mathrm{kg}/{\mathrm{m}}^{3}$ | $\rho $ | $2330\hspace{0.17em}$ | $5300$ |

$\mathrm{K}$ | ${T}_{0}$ | $800$ | $723$ |

$\mathrm{s}$ | $\tau \hspace{0.17em}$ | $5\times {10}^{-5}$ | $1.4\times {10}^{-6}$ |

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

${\mathrm{m}}^{3}$ | ${d}_{n}\hspace{0.17em}$ | $-9\times {10}^{-31}$ | $-6\hspace{0.17em}\times {10}^{-31}$ |

$\mathrm{eV}$ | ${E}_{g}\hspace{0.17em}$ | $1.11$ | $0.72$ |

${\mathrm{K}}^{-1}$ | ${\alpha}_{t}\hspace{0.17em}$ | $4.14\times {10}^{-6}$ | $3.4\times {10}^{-3}$ |

$\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ | $k\hspace{0.17em}$ | $150\hspace{0.17em}$ | $60$ |

$\mathrm{J}/(\mathrm{kg}\hspace{0.17em}\mathrm{K})$ | ${C}_{e}$ | $695\hspace{0.17em}$ | $310$ |

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

${D}_{T}$ | $\frac{k}{\rho {C}_{e}}$ | $\frac{k}{\rho {C}_{e}}$ | |

$({\mathrm{m}}^{2}(\%{\mathrm{H}}_{2}\mathrm{O})/\mathrm{s}(\mathrm{K}))$ | ${D}_{T}^{m}$ | $2.1\times {10}^{-7}$ | $2.1\times {10}^{-7}$ |

$({\mathrm{m}}^{2}\mathrm{s}(\mathrm{K})/(\%{\mathrm{H}}_{2}\mathrm{O}))$ | ${D}_{m}^{T}$ | $0.648\times {10}^{-6}$ | $0.648\times {10}^{-6}$ |

Reference moisture | ${m}_{0}$ | $10\%$ | $10\%$ |

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

$\mathrm{cm}/\mathrm{cm}(\%{\mathrm{H}}_{2}\mathrm{O})$ | ${\alpha}_{m}$ | $2.68\times {10}^{-3}$ | $2.68\times {10}^{-3}$ |

$(\mathrm{kg}/\mathrm{msM})$ | ${k}_{m}$ | $2.2\times {10}^{-8}$ | $2.2\times {10}^{-8}$ |

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**MDPI and ACS Style**

Alhashash, A.; Elidy, E.S.; El-Bary, A.A.; Tantawi, R.S.; Lotfy, K.
Two-Temperature Semiconductor Model Photomechanical and Thermal Wave Responses with Moisture Diffusivity Process. *Crystals* **2022**, *12*, 1770.
https://doi.org/10.3390/cryst12121770

**AMA Style**

Alhashash A, Elidy ES, El-Bary AA, Tantawi RS, Lotfy K.
Two-Temperature Semiconductor Model Photomechanical and Thermal Wave Responses with Moisture Diffusivity Process. *Crystals*. 2022; 12(12):1770.
https://doi.org/10.3390/cryst12121770

**Chicago/Turabian Style**

Alhashash, Abeer, E. S. Elidy, A. A. El-Bary, Ramdan S. Tantawi, and Khaled Lotfy.
2022. "Two-Temperature Semiconductor Model Photomechanical and Thermal Wave Responses with Moisture Diffusivity Process" *Crystals* 12, no. 12: 1770.
https://doi.org/10.3390/cryst12121770