Generalized Photo-Thermo-Microstretch Elastic Solid Semiconductor Medium Due to Excitation Process

A novel model in the theory of photo-thermoelasticity with microstretch properties is studied. The plasma-elastic-thermal plane waves are propagated in a linear isotropic generalized photo-thermo-microstretch elastic semiconductor solid medium. The photothermal excitation occurs in the context of the microinertia of microelement process during two dimensions (2D) deformation. The harmonic wave techniques are used to get the solutions for the basic variables. The analytical solution of the main physical fields; carrier intensity, normal displacement components, temperature, stress load force, microstress and tangential coupled stress can be obtained. Some graphics illustrated when using the plasma, thermal and mechanical load boundary conditions, which they apply at the outer free surface of the elastic medium. Some semiconductor materials as silicon (Si) and Germanium (Ge) are used to make the numerical simulation and some comparisons in different thermal memories are made. The main physical variables with new parameters are discussed theoretically and shown graphically.


Nomenclature λ, μ
Lame's parameters δ n The deformation potential difference T Absolute thermodynamic temperature (thermodynamic heat) T 0 Temperature in its natural state and satisfy T−T 0 T 0 < 1 γ = (3λ + 2μ + k)α t 1 The volume thermal expansion α t 1 The linear thermal expansion coefficient σ ij The stress tensor components ρ Medium density α t 1 , α t 2 Coefficients of linear thermal expansion e Cubical dilatation C e Specific heat at constant strain k The thermal conductivity D E The carrier diffusion coefficient τ The carrier lifetime E g The energy gap e ij Components of strain tensor , Two scalar functions j 0 The microinertia of microelement m ij Couple stress tensor α 0 , λ 0 , λ 1 Microstretch elastic constants τ 0 , ν 0 Thermal relaxation times φ Rotation vector φ * The scalar microstretch

Introduction
The thermoelasticity theory, highly used in engineering structural material, has a great role in steel stress analysis and applied mechanic science. It can describe the mechanical solid behaviour of some common elastic materials such as coal, concrete and wood. But, it can't describe the mechanical behaviour of many synthetic materials of polymer and clastomer-type like polyethylene. It studies the thermal effect and its relation with stresses and strains that occur in elastic bodies. So, the linear micropolar elasticity theory can be obtained to achieve that. In this theory, body microstructure influence is significant and this influence shows waves can't exist in the elasticity classical theory. Body temperature change is caused not only due to the outer and internal heat sources, but also due to the deformations of process itself during the microinertia of microelement. In most of the previous studies, semiconductor materials could be thought of as an elastic body and, therefore, the thermoelasticity theory could be applied on it. But semiconductor materials have optical properties especially when they are exposed to the sun light or a laser beam. In this case, as a result of the high surface temperature of the semiconductor materials, its resistance decreases, allowing it to be a conductive material. But, the electrons become excited, which leads to the emergence of what is known as the carrier density or plasma density. Therefore, it must be taken into account the microinertia of microelement (due to the carrier-free charge) and the microstretch of the semiconductor material points can contract and stretch independently of their rotations and translations. The microstretch elastic semiconductor theory is different from the micropolar elasticity theory; there is an additional degree of freedom of the medium that is called the stretch and an additional coupled stress that is called microstretch vector. Microstretch elastic semiconductors are polymer composite materials. In this case, the overlap between the thermo-microstretch theory and photothermal excitation process can be studied. Misra et al. [1] studied the body microstructure when the thermoelastic interaction in isotropic homogeneous elastic half-space occurs using a generalized linear thermoelasticity theory. The microstructure effects are significant and this influence shows results of wave propagation can't exist in the classical theory of elasticity developed by Eringen and Şuhubi [2]. Eringen [3] introduced a new generalized theory of micropolar thermoelasticity theory, called the thermomicrostretch theory, for elastic solid bodies. This theory can be chosen as a special case of theory of micromorphic. Singh [4] studied the plane waves during the reflection and refraction processes through a liquid adjacent to elastic thermo-microstretch solid. Influence of the theory of two temperatures is used to study the reflection coefficient in a micropolar elastic medium when the energy dissipation is introduced [5]. The linear thermo-microstretch elastic solid theory is produced [6]. In this theory, the heat can transfer as thermal waves with a finite thermal speed. The plane wave is obtained during generalized thermo-microstretch theories for an elastic medium [7]. Othman et al. [8] illustrated the gravitational influence and hydrostatic initial stress when the generalized magneto-thermo-microstretch theory is used in different thermal memories. Othman and Abbas [9] studied the plane waves with some thermal memories in generalized thermo-microstretch when they used the numerical finite element method. Many applications of thermo-microstretch for porous media are done when the thermal radiation effect is studied during a casson fluid flow over a stretching sheet [10]. But, Rashidi et al. [11,12] investigated the unsteady convective heat and mass transfer in pseudoplastic nano-fluid over a stretching wall and used the analysis of heat transfer due to a stretching cylinder with a partial slip in the context of prescribed heat flux.
The photothermal method is introduced when a sample of intracavity spherical semiconductor material is exposed to a light laser beam [13]. A sensitive analysis of semiconductor material used the spectroscopy of photoacoustic when laser beams fall on it [14]. Tam et al. [15][16][17] studied many problems with applications in modern physics using the ultrasensitive laser spectroscopy. The 2D deformation, during photothermal transport interactions in an elastic semiconductor medium, is studied [18]. The electronic deformation mechanism and optical excitations are used to discuss the photoacoustic frequency transmission technique on the generalized thermoelastic vibrations [19,20]. Lotfy et al. investigated various problems in the photo-thermoelasticity theory with many applications in mechanical engineering [21][22][23][24][25][26][27]. Abbas et al. [28,29] studied the dual phase-lags theory of photothermal excitation processes with the interaction of a semiconductor medium. Many researchers studied when the physical properties of the semiconductor elastic medium depend on the temperature in different external fields [30][31][32][33]. Mondal and Sur [34] introduced wave propagation in an orthotropic elastic semiconductor during memory responses. The hyperbolic twotemperature theory is used to modify some models in photo-thermal-elastic interaction [35,36].
In all the above investigations, the influence of the thermo-microstretch theory was not taken into consideration when microinertia of microelement were neglected while studying the photo-thermoelasticity theory But in this work, the thermo-microstretch theory is applied during microinertia of microelement 2D deformation. In this paper, a linear theory for photothermal-elastic solids with an inner structure with its particles, and microstretch and plasma-thermal fields, is considered. The coupling between the thermomicrostretch theory and photo-thermoelasticity is investigated. In this case, the governing equations are taken in 2D (in the space (x, z)) deformation for the semiconductor medium. The microinertia of microelement are taken into consideration. The main physical variables (carrier intensity, displacement components, temperature field distribution, load force stress, microstress and tangential couple stress distribution) are studied in a generalized photo-thermo-microstretch elastic medium in different thermal relaxation times. The harmonic wave method, with some mechanicalthermal and plasma boundary conditions, is used with some algebraic techniques to get the complete solutions of the basic quantities. The obtained results are discussed compared graphically.

Mathematical model and main equations
Eringen et al. [2,[37][38][39] constructed many models in generalized thermoelasticity. In this problem, linear isotropic properties of the generalized photo-thermomicrostretch semiconductor medium during the photothermal process were studied. The medium is taken in Cartesian coordinates (x, y, z) which originates at the external elastic surface y = 0 when the direction of the z-axis points vertically in the elastic half-space medium. When thermal waves occur on the external surface of the solid medium, the photothermal mechanism is obtained and the free carrier charges are generated (plasma wave propagation) [15]. In this case, the overlapping processes occur between the waves (plasma-thermal-elastic) during the elasticphoto-thermal-microstretch excitation [16]. The transmission of load across a differential element of the surface of a microstretch semiconductor elastic medium is described by a force vector, a couple stress vector and a microstress vector. The constitutive equations and field equations, when the body forces are absent in the context of 2D photo-thermo-microstretch theory (x, z), are given as (see geometry of the problem): Geometry of the problem The coupling between plasma wave distribution and thermal wave distribution during the photo-excited process of a microstraetch semiconductor is given by The equations of motion for a semiconductor material in photo-thermo-microstretch theory can be written as follows [16]: Heat conduction equation for a semiconductor medium in the photo-thermo-microstretch theory can be given as [15] follows: The constitutive relations in tensor form with two relaxation times for generalized photo-thermomicrostretch theory can be written as [17] follows: Constitutive relations for the generalized thermomicrostretch semiconductor elastic medium are given as follows: The relation between the strain and the displacement components relation can be written as follows: In the above equations, κ = ∂n 0 ∂T represents the thermal activation coupling parameter in a general case and γ 1 = (3λ + 2μ + k)α t 1 andγ 2 = (3λ + 2μ + k)α t 2 are the parameters depending on the mechanical source and thermo-microstretch properties, α t 2 is a coefficient of the linear thermal expansions. The displacement vector u can be analysed, rotation vector and the scalar microstretch function in 2D (x, z) can be expressed, respectively, as follows: The governing field Equations (2)(3)(4)(5) can be rewritten in 2D as follows: K[ where the thermal memories and n o , n 1 (constants) can be chosen according to the photo-thermo-microstretch theories (classical coupled theory (C-D), Lord and Şhulman (L-S) model and model of Green and Lindsay (G-L)).
To get the main fields in a dimensionless form, the following non-dimensional variables can be used: Using Equation (16), for the main governing equations (dropping prim), yields: The potential space-time functions as (x, z, t) and (x, z, t), can be used to simplify the main equations. These functions can take the non-dimensional form as follows: where (x, z, t) is the scalar function and = (0, ψ, 0) is the vector function (Helmholtz's theory in twodimensional (x, z)). Using Equation (35), the main field Equations (17)(18)(19)(20)(21)(22) in terms of the two potential space-time functions can be written as follows: where ,

Harmonic wave analysis
The wave propagation in semiconductor and thermoelastic materials has various applications in many fields of modern science and mechanical technology, namely, atomic physics, industrial engineering, thermal power plants, submarine structures, pressure vessel, aerospace, chemical pipes, and metallurgy. Many scientists have attempted to study the propagation of harmonic plane waves in semiconductor and elastic media.
The propagation of plane waves in classical thermoelasticity was discussed by Deresiewicz [40], Chadwick and Sneddon [41] and Chadwick [42]. The solution of the considered basic physical variables in 2D deformation can be decomposed in terms of harmonic waves (normal mode technique) as follows: where the componentsφ,ψ,φ * ,φ 2 ,σ iI ,m iI ,T,N are the amplitude of the main field functions (function of the distance x), ω represents a complex frequency and b is the wave number that is taken in the z-direction.
Using the normal mode method (Equation (30)) for Equations (17) and (24)- (28), yields: where Eliminatingφ 2 andψ between Equations (33) and (34), the fourth-order ordinary differential equation can be obtained is satisfied byφ 2 andψ as follows: The other quantitiesT,φ * ,N and¯ can be eliminated between equations (31), (32), (35) and (36), the following eighth-order ordinary differential equation can be obtained as in the following (they are satisfied byT,φ * ,N and¯ ) form: The solutions of the ordinary differential Equations (38) and (39) according to the linearity properties take the following form: and M n (b, ω) are some parameters (unknown) depending on the constants b (wave number) and ω = ω 0 + i ζ they can be determined when they apply the boundary conditions taken at the free surface of the medium. k 2 j , ( j = 1, 2 ) and l 2 n , ( n = 1, 2, 3, 4 ) represent the basic roots of the characteristic equation of Equations (38) and (39), respectively. Using the main Equations (31)- (36) to obtain the relations between j (b, ω) and j (b, ω), also the relations between the parameters, M n (b, ω), , M n (b, ω) and M n (b, ω) , the main fields can be represented in terms of j and M n as follows: where But, the expressions of displacement components (using Equation (23) into Equations (40) and (42)), force stress-strain, coupled stress (Equation (7)) and other quantities for the microstretch generalized photothermoelastic semiconductor medium take the following form:

Boundary conditions
To determine the six unknown parameters j and M n , some boundary conditions must be applied at the free surface (at the vertical plan) of the elastic semiconductor material. Boundary conditions vary between instantaneous mechanical (mechanical load) source when the medium is isolated thermally during recombination plasma processes at x = 0. In this case, the conditions at the boundary can be written as follows: Using boundary conditions Equation (57) the harmonic wave method (51-56) and (46)(47)(48)(49) can be applied; the following system of equations can be obtained: The above system of Equations (61-63) can be solved by using the matrix inverse or Cramer technique of the unknown parameters can be obtained.

The theory of generalized microstretch-thermoelasticity
When the effect of carrier density N( r ,t) is vanished (i.e. N = 0), the problem can be discussed only in the generalized microstretch thermoelasticity theory [18].

The generalized photo-thermoelasticity theory
When the parameters of microstretch are neglected (i.e. α o = λ o = λ 1 = φ * = 0), the governing equations discuss the case of generalized photo-thermo-micro-polar elastic medium without a stretch.

Different theories of the microstretch photo-thermoelasticity
The problem is investigated during microstretch photothermoelasticity processes which depend on the thermal memories (thermal relaxation times) as follows [35]: [37]. (III) The G-L model is obtained when [38].

Discussion and numerical results
The silicone (Si) and germanium (Ge) materials are chosen as a semiconductor example to make the numerical simulation (using Maple program). To obtain the main results of the basic quantity fields numerically, the physical constants of semiconducting Si medium and Ge are chosen and given as follows [43][44][45][46]:  But, the physical constants of Ge material are given    α 0 = 0.779 × 10 −9 N, τ 0 = 0.00005s, ν 0 = 0.0005s.
The numerical computational are carried out when using real root parts of field distributions of the basic quantities as (thermal wave (thermo-dynamical temperature distribution), displacement distribution (strain wave), stress which describe the mechanical wave distribution, carrier density distribution (plasma wave), microstress and tangential couple stress, respectively). The investigation is done against the distance x at the plane z = −2 when the wave number b = 1, and P * = 1 in the context of the generalized photo-thermomicrostretch elastic GPTMSE medium. The complex constant ω can be ω = ω 0 + i ζ , where ω 0 , ζ are constants that can be taken as ω 0 = −2.5 and ζ = 0.05. The  real part of physical quantities fields is presented graphically in the numerical computation. Figures 1-7 show the main physical fields against the distance x under the effect of three different thermal relaxation times according to the C-D, L-S and G-N models at the same time when using the microstretch photo-thermoelasticity material constants. Figure 1 displays the displacement u = U, the behaviour of wave propagation for two theories L-S and G-N is the same when they increase in the interval 0 ≤ x ≤ 2.5. In the second range, the behaviours remain constant when the distance tends to infinity. Therefore, the wave propagation behaviour for the C-D theory increases smoothly in the same interval until reaching an equilibrium state at infinity.   shows the displacement w relative to the horizontal distance, the wave propagation behaviour for two theories (L-S and G-N) takes the same shape when they decrease in the interval 0 ≤ x ≤ 1.5. In the second range, they remain a constant when the distance tends to infinity. But, the wave propagation behaviour of the third curve (C-D theory) decreases smoothly in the interval 0 ≤ x ≤ 1.5 until reaching to an equilibrium state when the distance tends to infinity. Figure 3 exhibits the wave propagation behaviour of temperature T, all curves for three theories take the same behaviour when they increase in the interval 0 ≤ x ≤ 0.6 then decrease in the second range interval 0.6 ≤ x ≤ 1.8 and they increase and decrease again in the third interval. From this figure,  a very small difference appears between three theories. Figure 4 describes carrier density distribution N with respect to the distance x, the wave propagation behaviour for three theories in microstrach-photothermoelasticity (C-D, L-S and G-N) is the same when they satisfy the main conditions at the free boundary of the surface. All wave propagations of physical fields decrease in the interval 0 ≤ x ≤ 0.5 then increase in the second interval 0.5 ≤ x ≤ 3 and they decrease again in the third range. Figure 5 shows the normal stress σ zz distribution with horizontal distance x, the wave propagation behaviour for three models (C-D, L-S and G-N) has the same shape but, there is very small difference between two theories (L-S and G-N) and the third theory (CD) in the interval 0.3 ≤ x ≤ 0.6. That the wave      propagations decrease in the first interval 0 ≤ x ≤ 0.4, and then they increase in the second range 0.4 ≤ x ≤ 1 and they decrease and increase again in the reminder   interval until the behaviours remain constant in the last range when the distance tends to infinity. Figure 6 displays the rate change in stress component σ xz relative to the horizontal distance x. Figure 6 shows that σ xz always satisfy the surface boundary conditions for all   three models (C-D, L-S and G-N) [46]. In this figure, the behaviour of wave propagation for two theories (L-S and G-N) has the same shape when they decrease at the beginning as a straight line in the first interval 0 ≤ x ≤ 0.1 then they increase in the second interval until the behaviours remain constant when the distance tends to infinity. But, the wave propagation behaviour for the C-D theory decreases and increases in the same intervals until reaching an equilibrium state when the distance tends to infinity. There is very small difference between two theories (L-S and G-N) and the third theory C-D in the interval 0 ≤ x ≤ 1. Figure 7 displays the variation of microstress λ z distribution against distance x, the behaviour of wave propagation for three models (C-D, L-S and G-N) is the same when they increase in the first interval 0 ≤ x ≤ 0.2 then decrease in the second interval 0.2 ≤ x ≤ 0.9, then increase in the interval 0.9 ≤ x ≤ 2.5 and decrease in the last interval. Figures 8-14 study the comparisons made between the two elastic semiconductor media, Si and germanium (Ge) for the main physical quantities at the same time and conditions when using the microstretch photothermoelasticity material constants. From these figures, it is clear that the difference in physical constants of semiconductor Ge and Si materials have a significant impact on all the wave distributions when they are dimensionless (u, w, T, N, σ zz , σ xz and λ z ). Figures 15-22 show the effect of different values of parameters (ε 1 , ε 3 , ε 4 and ε 5 ) on some main physical quantities as displacement components u and w with the distance x for Silicon (Si) material under the L-S theory only [45]. The problem is studied in the presence of microstretch photo-thermoelasticity material constants. The wave propagation of displacement component u increases with increasing different values of parameters (ε 1 , ε 3 , ε 4 and ε 5 ). The wave propagation of displacement component w decreases with decreasing different values of parameters (ε 1 , ε 3 , ε 4 and ε 5 ). The volume thermal expansionγ has a great effect on parameter ε 1 when the silicone (Si) material is studied. Carrier diffusion parameter D E and carrier lifetime τ have an important role on parameter ε 3 . Carrier diffusion coefficient D E and thermal activation coupling parameter κ have a great influence on parameter ε 4 . The gab energy E g and photogenerated carrier lifetime τ have a great influence on parameter ε 5 .

Conclusion
The physical quantities (as temperature, displacements, stresses, carrier density, microstress and tangential couple stress) in generalized photo-thermo-microstretch elastic semiconductors for (Si and Ge) solid are studied at small time and in different thermal memories (C-D, L-S and G-N models). The thermal memories have a small effect of wave propagations of the main quantities field due to the effect of microstretch parameters. The comparisons between Si and Ge semiconductor materials are made when using the L-S model, the physical constants that depend on the type of material have a great significant on the wave distributions of the basic physical fields for compression between Si and Ge materials. The displacement component distribution through various values of ε 1 , ε 3 , ε 4 and ε 5 under the L-S theory is displayed graphically in the generalized photo-thermo-microstretch theory for Si material. The different values of the quantities ε 1 , ε 3 , ε 4 and ε 5 has a great impact on all the physical distributions of the basic fields.
The mechanical interaction between the thermal, plasma and mechanical fields in semiconductors has great practical applications in modern aeronautics, astronautics, modern chemical engineering (chemical mechanical planarization) and nuclear reactors. The classical photo-thermoelasticity theory is not adequate to model the behaviour of materials possessing an internal structure. Furthermore, the inner microstretch semiconductor elastic model is more realistic than the purely semiconductor elastic (photo-thermoelasticity) theory for studying the response of materials to external stimuli.

Disclosure statement
No potential conflict of interest was reported by the author(s).