# Hall Current Effect of Magnetic-Optical-Elastic-Thermal-Diffusive Non-Local Semiconductor Model during Electrons-Holes Excitation Processes

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## Abstract

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## 1. Introduction

## 2. Basic Equations

## 3. The Mathematical Solutions

## 4. Boundary Conditions

## 5. Inversion Processes of the Laplace Transforms

## 6. Special Cases

#### 6.1. The Photo-Thermoelasticity Models

- When $0\le {\tau}_{\theta}<{\tau}_{q}$, in order to obtain the dual phase lag DPL model;
- When ${\tau}_{\theta}=0$, $0<{\tau}_{q}$, in order to obtain the Lord and Șhulman (LS) model;
- When ${\tau}_{\theta}={\tau}_{q}=0.0$, one obtains the coupled thermoelasticity (CT) model.

#### 6.2. Influence of Magnetic Field

#### 6.3. The Non-Local Thermoelasticity Theory without Electrons/Holes Interaction

#### 6.4. The Generalized Non-Local Magneto-Photo-Thermoelasticity Theory

#### 6.5. The Non-Local Semiconductor Medium

## 7. Numerical Results and Discussions

#### 7.1. The Photo-Thermoelasticity Models

#### 7.2. The Impact of Hall Current

#### 7.3. The Impact of Non-Local Parameter

#### 7.4. The 3D Graph

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$\lambda ,\mu $ | Counterparts of Lame’s parameters, |

${n}_{0}$ | Equilibrium carrier concentration (electrons concentration) |

${h}_{0}$ | Equilibrium holes concentration |

${T}_{0}$ | Absolute temperature |

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

${\sigma}_{\mathrm{ij}}$ | Components of the stress tensor |

$\mathsf{\rho}\hspace{1em}\hspace{1em}$ | Density of the medium |

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

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

${t}^{n},{t}^{h}$ | The electrons and holes relaxation times |

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

${\tau}_{q}$ | The elastic relaxation time |

${\tau}_{\theta}$ | Thermal relaxation time |

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

$K$ | The thermal conductivity of the medium |

${\tau}^{*}$ | The photogenerated carrier lifetime |

${E}_{g}$ | The energy gap of the medium of semiconductor |

${\delta}_{n}=(2\mu +3\lambda ){d}_{n}$ | The electrons elasto-diffusive parameter |

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

${d}_{n}$ | The coefficients of electronic deformation |

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

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

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

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

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**Figure 1.**The key physical distributions’ variation in relation to distance under the influence of Hall current and non-local parameters, as predicted by photo-thermoelasticity models.

**Figure 2.**In both the presence and absence of the Hall current effect, the main field distributions against distance vary, according to the DPL model in non-local case.

**Figure 3.**According to the DPL model with the influence of Hall current, the primary field distributions against distance change in both the presence and absence of the non-local parameter.

**Figure 4.**According to the DPL model and under the influence of Hall current, the principal fields in 3D plots for non-local Si material vary with variations in time and distance.

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 $ | $2330$ |

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

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

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

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

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

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

$\mathrm{H}/\mathrm{m}$ | ${\mu}_{0}$ | $4\pi \times {10}^{-7}$ |

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

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

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

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

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

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

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

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

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**MDPI and ACS Style**

Chteoui, R.; Lotfy, K.; El-Bary, A.A.; Allan, M.M.
Hall Current Effect of Magnetic-Optical-Elastic-Thermal-Diffusive Non-Local Semiconductor Model during Electrons-Holes Excitation Processes. *Crystals* **2022**, *12*, 1680.
https://doi.org/10.3390/cryst12111680

**AMA Style**

Chteoui R, Lotfy K, El-Bary AA, Allan MM.
Hall Current Effect of Magnetic-Optical-Elastic-Thermal-Diffusive Non-Local Semiconductor Model during Electrons-Holes Excitation Processes. *Crystals*. 2022; 12(11):1680.
https://doi.org/10.3390/cryst12111680

**Chicago/Turabian Style**

Chteoui, Riadh, Khaled Lotfy, Alaa A. El-Bary, and Mohamed M. Allan.
2022. "Hall Current Effect of Magnetic-Optical-Elastic-Thermal-Diffusive Non-Local Semiconductor Model during Electrons-Holes Excitation Processes" *Crystals* 12, no. 11: 1680.
https://doi.org/10.3390/cryst12111680