Laser Short-Pulse Effect on Thermodiffusion Waves of Fractional Heat Order for Excited Nonlocal Semiconductor

Almoneef, Areej A.; El-Sapa, Shreen; Lotfy, Kh.; El-Bary, A.; Saeed, Abdulkafi M.

doi:https://doi.org/10.1155/2022/1523059

Advances in Condensed Matter Physics

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Abstract Introduction Discussion Conclusion Data Availability Ethical Approval Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 1523059 | https://doi.org/10.1155/2022/1523059

Laser Short-Pulse Effect on Thermodiffusion Waves of Fractional Heat Order for Excited Nonlocal Semiconductor

Areej A. Almoneef,¹Shreen El-Sapa,¹Kh. Lotfy,^2,3A. El-Bary,^4,5and Abdulkafi M. Saeed⁶

Academic Editor: Sergio Ulloa

Received04 Apr 2022

Revised04 Jun 2022

Accepted11 Jul 2022

Published12 Aug 2022

Abstract

In this work, the thermal effect of a laser pulse is taken into account when mechanical-thermodiffusion (METD) waves are studied. The nonlocal semiconductor material is used when interference between holes and electrons occurs. The fractional technique is applied on the heat equation according to the photo-thermoelasticity theory. The governing equations describe the photo-excitation processes according to the overlapping between the thermoelasticity and photothermal theories. The thermoelastic deformation (TD) and the electronic deformation (ED) for the dimensionless fields are taken in one dimension (1D). The Laplace transforms are applied to obtain the analytical solutions when some initial and boundary conditions are applied at the nonlocal surface. The complete nondimensional solutions of the main quantities are obtained according to some numerical simulation approximate during the inversion processes of Laplace transforms and Fourier expansion. The time-fractional order, nonlocal, and thermal memories are used to compare the wave propagations of the main fields and are discussed graphically for nonlocal silicon material.

1. Introduction

Semiconductors are materials whose properties were extensively studied at the end of the nineteenth century. With the development of technology and industrial development in the twentieth century, semiconductors have been widely used in most industries such as medical devices, electrical circuits, and solar cells to produce renewable energy. As it became clear to researchers interested in semiconductor implants that these materials are affected by their internal structure by increasing temperatures, especially when exposed to light or a beam of laser beams. The change in the internal composition of these materials depends on thermoelastic deformation (TED) and electronic deformation (ED). It is worth noting that during the study of semiconductors whose surfaces are exposed to light, it was found that the excited electrons are released towards the surface and move, causing the presence of an electron cloud called the carrier density (plasma), which is responsible for conducting the electric current during diffusion processes. Where the moving electrons during a recombination process leave behind holes, also in the case of movement in the form of a cloud. The photothermal (PT) theory can be applied to describe the governing equations in this case during the processes of diffusion of electrons and holes. Also, the theory of thermoelasticity can be used to describe the thermoelastic deformation processes.

The theory of thermal-elasticity appeared at the beginning of the twentieth century with some unacceptable assumptions in physics when it was assumed that waves propagating through an elastic medium propagate at infinite speeds (the uncoupled thermoelasticity theory). Biot [1] resolved this contradiction when he modified Fourier's law of heat conductivity by adding a relaxation time to the heat conduction equation. Biot [1] introduced the coupled-dynamic (CD) thermoelasticity theory with a finite speed of wave propagation. Lord and Shulman (LS) [2] and Green and Lindsay (GL) [3] modified the system of equations which describe the elastic bodies when they inserted some relaxation times into the motion equation and the heat equation. After that, Chandrasekharaiah [4,5] introduced the hyperbolic thermoelastic body to investigate the second sound effect. The reflection of waves is studied according to the thermoelasticity theory by Sharma [6]. On the other hand, Lotfy and Abo-Dahab [7] applied the two-temperature theory and normal mode analysis to study the elastic bodies in two-dimensional (2D) according to the thermal shock problem. The effect of gravity during the two-temperature theory is applied to investigate generalized thermoelastic materials [8].

In recent studies concerned with the study of semiconductors, the interference between holes and electrons must be taken into account. During the thermodiffusive processes of semiconductors, Maruszewski et al. [9–12] studied the interference between mechanical and thermal waves resulting from the effect of carrier’s charge fields and the short laser pulses. The PT theory is used when the mass concentration is diffusive from one place to another according to the heat flow of semiconductors [13]. The electron (carrier) density and hole carrier fields can be obtained using the PT technique during photo-generated transport processes [14]. The effect of electromagnetic radiation with noting long-transient effects and laser pulses impact is studied of semiconductor media [15]. Lotfy et al. [16–18] investigated many problems when using the photo-thermoelasticity theory with multiple applications for silicon semiconductor material. The Gaussian pulsed laser source is utilized to produce heat impact on semiconductor material [19]. The interaction between the magnetic field and holes\electrons fields of excited semiconductor media is considered when the photo-thermodiffusion waves is studied.

Recently, many physical and engineering problems for polymer composites have been studied using fractional calculus applications. Many physics models have already been discussed and solved after presenting the theory of fractional calculus [20–22]. Different scientists [23–27] introduced many physical models according to the fractional calculus to modify the heat equation when the theory of photo-thermoelasticity is applied to semiconductors. Hobiny et al. [28] took into account the fractional time derivative of the heat equation to study skin tissue of the human body under the impact of induced laser irradiation. Abouelregal et al. [29] modified the heat conduction equation using the fractional derivative of the thermoelastic medium. On the other hand, Abouelregal et al. [30,31] used the time-fractional derivative technique to study the exciting viscoelastic two-temperature model for an infinite dipolar elastic body. El-Sapa et al. [32] studied the Moore-Gibson-Thompson model of an excited nonlocal semiconductor material when the stability investigations are taken into account. Abbas et al. [33–37] used the eigenvalue approach to study the effect of fractional order on the wave propagations of elastic bodies under the impact of magnetic fields with some numerical techniques.

In the previous studies, the fractional time derivative according to the thermal impact of short laser pulses was not taken into account for nonlocal semiconductor material during the electron and hole recombination thermodiffusive processes. In this work, the fractional time-order for the heat conduction equation under the influence of laser pulses is applied to discuss a novel photo-thermoelastic model of nonlocal material. During the photo-generated processes, the distributions of electron and hole charges subjected to the nonlocal semiconductor material are considered. The nonlocal governing equations are constructed in dimensionless 1D according to photo-excitation transport processes. Some surface conditions are utilized in the context of the Laplace transforms to get the main fields analytically. The complete solutions are gained numerically when some approximations are subjected to Fourier expansion in the inversion of the Laplace domain. The different values of fractional order and nonlocal parameters are used to make some numerical simulations to obtain the wave propagations of the physical fields graphically and discuss them.

2. Basic Equations

In equilibrium state, a homogenous nonlocal semiconductor medium is utilized which have a linear isotropic properties. The main physical fields in 1D (in the direction of -axis) are introduced according to the heat-induced laser pulses energy in the following form: the carrier intensity with plasma effect, the thermal effect with temperature distribution , , the elastic impact according to displacement distribution and the concentration of hole charges where is the time. The main equations according to the fractional time order for electrons-holes nonlocal semiconductor material with non-Gaussian laser pulse effect can be constructed as follows [9,11,12]:

The nonlocal equations of motion is formulated as follows [32]:where , and are the Peltier and Dufour-Seebeck-Soret-like polymer parameters. The diffusion for the electrons and holes represent by and , respectively, is the order of fractional parameter. The parameter is first phase-lag of the temperature gradient and is the second time of the heat flux phase-lag. The life times of electrons and holes in equilibrium case are and . The thermodiffusive holes and electrons parameters are and . The flux-like constants are , and , where , , , .

The constitutive equation for nonlocal semiconductor according to the 1D deformation and electrons and holes coupling can be expressed as follows [36]:

The fractional integral operator according to the Riemann–Liouville method can be presented as follows [38]:

For any continuous function , the fractional integral is . The derivative according to Caputo fractional for can be written in the following form [20–22]:

The main quantities can be transformed in the dimensionless case for more simplifications when the following dimensionless variables are introduced as follows [32]:

Ignoring the primes, according to equation (8) the governing equations (1)–(5) can be rewritten as follows:where

At rest, the following initial conditions are considered to can be solved the problem mathematically, which can be represented as follows:

3. The Mathematical Solutions

Laplace transforms have important uses in solving many engineering, physical and mathematical problems, especially those related to control system engineering. To convert the partial differential equations (PDEs) in physical problems to ordinary differential equations (ODEs) Laplace transforms are utilized. Laplace transform for the continuous function is as follows:

The fractional derivative for the differentiable function in Laplace domain can be constructed as follows [39]:

In this case, the main equations (10)–(13) according to Laplace transform of fractional derivative can be reformulated as follows:where , , ,

The nonhomogenous ODEs (20)–(23) can be solved using the eliminating technique, as follows:where .

In this case, the coefficients of equation (29) take the following form:

The factorized method is used for nonhomogenous equation (29) as follows:where represent the homogenous real roots when in the following equation:

The general linear solutions of equation (34) can be expressed as follows:where .

According to the undetermined coefficients method for ODE (34), the values of the coefficients can be obtained in the following form:

The unspecific parameters in this problem are which depend on the parameter and can be calculated from the following boundary conditions.

4. Boundary Conditions

In this part, we assume some boundary conditions taken on the surface to determine the value of undefined parameters . Where Laplace transforms are applied to the boundary conditions in addition to taking nondimensional quantities. The boundary conditions are presented as follows:

(I) The isolated thermal boundary on the surface can be presented as follows [27,40]:

(II) The mechanical condition is selected when the traction is free on the nonlocal surface as follows [41]:

(III) During the electronic recombination and diffusion processes, the carriers’ fields density distribution can be selected on the nonlocal surface as follows [42]:where expresses the diffusion of electron charges coefficient.

(IV) On the other hand, during the diffusion of the holes, the following condition can be taken on the free nonlocal surface according to the holes fields recombination [12]:

When solving the system consisting of the previous four equations (48)–(51) in the inverse matrix method, the values of the variables can be determined algebraically and thus we can know the linear solutions as a function of [43].

5. Inversion Processes of the Laplace Transforms

To determine the complete solutions in the time domain of physical quantities, inverse Laplace transforms are used numerically, depending on the Fourier expansion and Riemann-sum approximation technique. The inverse Laplace transform of any function in the domain is defined as follows:

Equation (53) is rewritten in another form as follows:

Fourier series expansion is utilized during the interval for in the closed form and the following relation can be obtained as follows:Here, , the real part is , the chosen large sufficient integer is , and [44].

6. Validation

6.1. The Thermoelasticity Theories

To know the different models according to the theory of thermoelasticity only, the effect of each of electrons () and holes () is neglected. In this case, the governing equations (1)–(4) are reduced to the equations of heat and the equation of nonlocal motion in 1D as follows [7,8]:

The different models of the theory of thermoelasticity appear according to the difference in relaxation times as follows: the dual phase-lag (DPL) model is obtained at , the Lord and Șhulman (LS) model is appeared when , , and the coupled thermoelasticity (CT) model is considered at [1–3].

6.2. The Photo-Thermoelasticity Theory

The theory of photo-thermoelasticity shows when the effect of holes on the governing equations is neglected (i.e. ), and the main equations (1)–(4) are reduced to three equations that describe the nonlocal effect of plasma, heat, and the internal displacements of electrons as follows [38]:

6.3. The Impact of Nonlocal Parameters

When the nonlocal parameter is neglected (), the semiconductor medium is in local state. In this case, the governing equations describe the local photo-thermoelasticity models when the impact of holes is considered.

7. Numerical Results and Discussion

To illustrate the obtained results graphically, the physical constants of silicon (Si) material are used as an example of a polymeric semiconductor material. Through these physical inputs, it is possible to observe the wave propagation of the basic physical quantities inside the semiconductor material. To implement this numerical simulation with the help of MATLAB (2018), the physical constants of silicon in SI units shown in the following table (Table 1) can be used [45,46]:

7.1. The Influence of Fractional-Order Parameters

Figure 1(the first category) shows the wave propagations of the main physical distributions (thermal waves, mechanical waves, carrier density of holes and plasma waves) with the axial distance according to multiple values of fractional order parameters (, and (classical case)). The computational results are implemented according to the DPL model when the semiconductor medium is in the nonlocal case under the influence of laser pulses for Si material. The first subfigure displays the behavior of wave propagation of thermal (temperature) distribution against distance. From this subfigure, when the fractional order parameter , the magnitude of thermal distribution differs from the two unclassical cases when the fractional order parameters and . The thermal distribution satisfies the isothermal condition, which begins at the nonlocal surface at a positive real value with increases in the magnitude to reach the maximum value due to the thermal effect of the laser pulse. But in the second range, the thermal distribution decreases until it converges to the zero line to reach the steady-state. The behavior of mechanical wave propagation is shown in the second subfigure, which satisfies the mechanical condition at the nonlocal surface. The mechanical distribution of waves takes on a different behavior when (increases in the beginning to reach the maximum and the decreases to reach the minimum value in the second range) value than the distributions in the case of and . The holes carrier field distribution with the axial distance appears in the third subfigure. Where the wave propagations of holes field in the three cases of fractional order parameter take the same behavior but are different in magnitude and satisfy the holes recombination conditions. On the other hand, the fourth subfigure shows the wave propagation of carrier density (plasma) according to different values of the fractional order parameter. The plasma wave satisfies the plasma recombination conditions which starts at a real positive value at the nonlocal surface. Due to the photo-thermal-excitation at the surface, the distribution increases until it reaches the maximum value and then decreases and increases periodically in the second range with the increase in the distance until it converges to the zero line in the steady-state.

7.2. The Photo-Thermoelasticity Models

Figure 2 (the second group) illustrates the variation of wave propagations of the dimensionless main fields versus the axial distance according to the different values of relaxation times. The difference in relaxation times leads to the emergence of different models of the theory of photo-thermoelasticity when the hole field effect is taken into account. All calculations are implemented for a small time when the fractional order parameter is with the nonlocal parameter. From this figure, it is clear that the wave propagation for the distribution of physical quantities depends largely on the difference in the values of relaxation times in the context of the electronic-hole excitation. On the other hand, the behavior of wave propagation takes almost the same behavior as in Figure 1 but is different in the magnitudes according to the different values of thermal memories.

7.3. The Influence of Nonlocal Parameters

Figure 3 (the third group) explains the comparison between the behavior of wave propagations of the main destitutions at various axial distances in three cases of nonlocal parameters. The first case describes the local semiconductor medium case when , but the other two cases describe the nonlocal semiconductor medium when and. The evaluations in this group are implemented according to the DPL model for fractional order parameter . The physical field distributions have the same behavior described in Figure 1. Indeed, the effect of nonlocal parameters on the distribution of wave propagation is clearly shown.

7.4. The Influence of Instant Time

Figure 4 illustrates the changes in the behavior of main physical fields as functions of distance and time in 3D plots. The numerical calculations were carried out in the context of the DPL model for the nonlocal semiconductor medium at when the fractional parameter is . From this category, when the values of time and the distance increase, the field distributions of main physical quantities decrease tile converge to zero line to reach the equilibrium state or limited propagation speeds.

7.5. The Validation of the Results

The present results, in the absence of both the nonlocal parameter and the fractional order differentiation, are in agreement with the results in [45]. On the other hand, the present work in the absence of the laser pulse parameters is in agreement with the results in [38]. In the absence of the effect of the nonlocal parameter and the hole effect, the results are consistent with those presented in [27]. In the absence of the effect of the nonlocal parameter, fractional order, and hole effect, the results are consistent with those presented in [14].

8. Conclusion

In this work, a new model was presented that describes the interaction between electrons and holes in a nonlocal semiconductor material in the context of the thermo-diffusion processes. In this model, the thermal effect of a beam of laser rays according to a non-Gaussian laser pulse falling on the outer nonlocal surface of the material was taken into account, in addition to the effect of the fractional differential on the heat conductivity equation. As a result of the difficulty of this model and the complex calculations, few researchers were interested in studying such models that took into account the interference between holes and electrons, so the governing equations were taken in one dimension. Through the obtained results and analysis, it became clear that the change in the fractional order parameters, relaxation times, and nonlocal parameters significantly affects the distribution of wave propagation when studying nonlocal semiconductors. There is great importance to the study of such models by scientists and engineers in the development of photo-thermoelastic science. There are many industrial applications that can benefit from this model, such as the industries of modern medical devices and sensors in electric cars.

Nomenclature

:	Lame’s elastic parameters
:	Equilibrium carrier concentration (electrons concentration)
:	Equilibrium holes concentration
:	Absolute temperature
:	The volume coefficient of thermal expansion
:	Components of the stress tensor
:	Density of the medium
:	The coefficient of linear thermal expansion
:	Cubical dilatation
and :	The thermal relaxation times (phase-lag)
:	Specific heat at constant strain of the medium
:	The thermal conductivity of the medium
:	The photo-generated carrier lifetime
:	The energy gap of the medium of semiconductor
:	The parameter of electrons elastodiffusion
:	The parameter of holes elastodiffusion
:	The coefficients of electronic deformation
:	The coefficients of hole deformation
:	The power intensity
:	The plasma recombination speed
:	The optical absorption coefficient
:	The nonlocal scale parameter
:	Pulse parameter.

Data Availability

Data are available upon request to the corresponding author.

Ethical Approval

This study and all procedures performed involving human participants were in accordance with the ethical standards.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

Kh. Lotfy contributed in conceptualization and methodology. El-Sapa contributed to software provision and data curation and wrote the original draft. A. El-Bary performed supervision, visualization, and investigation. Abdulkafi M. Saeed contributed to software provision and validation. Areej A. Almoneef contributed to writing, reviewing, and editing.

Acknowledgments

The authors extend their appreciation to Princess Nourah Bint Abdulrahman University for funding this research under researchers supporting project number PNURSP2022R154, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia.

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