A Study of Photo-Thermoelastic Wave in Semiconductor Materials with Spherical Holes Using Analytical-Numerical Methods

Analytical and numerical solutions are two basic tools in the study of photothermal interaction problems in semiconductor medium. In this paper, we compare the analytical solutions with the numerical solutions for thermal interaction in semiconductor mediums containing spherical cavities. The governing equations are given in the domain of Laplace transforms and the eigenvalues approaches are used to obtained the analytical solution. The numerical solutions are obtained by applying the implicit finite difference method (IFDM). A comparison between the numerical solutions and analytical solution are presented. It is found that the implicit finite difference method (IFDM) is applicable, simple and efficient for such problems.


Introduction
In the surrounding nature, many materials are very important in industry especially, these materials have many applications in renewable energy. The semiconductor materials exist in abundance in the surrounding nature which have great economic importance in solar cells industry. When the semiconductor media are exposed to a focus of laser beams or a beam of sunlight, the surface electrons at free surface are thermally exited and they will be vibrated due to the thermal effect of laser beams. In this case, the electrons and holes will transport from one position to another and the free carriers photoexcited appearing with a weak electrical current.
On the other hand, the recombination processes during the photo-excited processes will be taken into account between the electrons (carrier density (plasma)) and holes. Much effort has been made for generalized theories of thermoelasticity in solving thermoelastic models instead of the classical decoupled and coupled theory of thermoelasticity. For decoupled thermoelasticity, the absence of any term reflecting elasticity in the thermal conduction equation does not appear to be real where due to the mechanical loading of an elastic body, the strain causes a change in the temperature field.
Moreover, the thermal conduction equation is of the parabolic type of the results of the propagation of thermal waves at an infinite speed, which also contradicts the real physical phenomena. Introduction of the strain rate term in the decoupled thermal conduction equation. The models of bodies explained the properties of the internal structure of medium when used the secondly law of thermodynamic with the development of semi-conductor integrated circuit technology and solid-state sensors technology have been widely used in several fields.
Previously, micro-mechanical structures of the thermoelasticity and plasma field are analyzed experimental and theoretical as in Todorovic et al. [1][2][3]. Abbas et al. [4] presnted th solutions of photo-thermal interaction in a semiconducting materials with cylindrical hole and variable thermal conduct i v i t y . H o b i n y a n d A b b a s [ 5 ] d i s c u s s e d t h e photothermoelasticity interaction in a two-dimension semiconducting plane under Green-Naghdi theory. Lotfy et al. [6] studied the electro-magnetic and Thomson effects through the photo-thermal transport process of semiconductor material. Lotfy et al. [7] duscussed the responses of Thomson and electro-magnetic influnces of semi-conductor material casued by laser pulses under photo-thermoelastic excitation. Abbas et al. [8] studied the analytical solution of plasma and thermoelastic wave photogenerated by a focused laser beam in a semiconducting medium. Lotfy et al. [9] investigated the influences of variable thermal conductivity in semiconductors mediums with cavity under fractional-order magnetophotothermal models. Alzahrani [10] investigated the effects of variable thermal conductivity in semi-conductor materiale. Alzahrani and Abbas [11] discussed the photothermoelasticity interactions in a two-dimension semiconductors mediums without energy dissipation. Hobiny and Abbas [12] presented a study on photo-thermal wave in an unbounded semiconductor material with cylindrical cavities. Das et al. [13] studied the electro-magneto-thermo-elastic analysis for a thin circular semiconductor material. Mondal et al. [14] studied the photo-thermoelastic waves propagation under the influence of magnetic field. Mondal and Sur [15] investigated the photo-thermo-elastic waves propagation in an orthotropic semiconductor with spherical cavities. Sarkar et al. [16] presented L-S theory for the propagation of the photo-thermal wave in a semiconductor nonlocal elastic material.
The FDM is a numerical technique that finds an approximate solution of a given problem. The concept of this method is replacing the derivatives that appear in the differential equation by an algebraic approximation. The unknowns of the approximated algebraic equations are the dependent variables at the grid points. Mukhopadhyay and Kumar [17] applied the finite difference technique to study the generalized thermoelasticity problem of annular cylinders with variable material properties. Abd-Alla et al. [18] studied the effects in a thermoelastic annular cylinder using the finite difference method. Patra et al. [19] used the finite difference technique to study the computational model on thermoelastic analysis with the magnetic field in a rotating cylinder. Abd-Alla et al. [20] studied the effects of nonhomogeneous in an isotropic cylinder under magnetic field. Abd-Alla et al. [21] studied the solutions of the transient coupled thermoelastic of an annular fins by the implicit finite-difference technique. Many researchers [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] used the many thermoelasticity theories to get the solutions of many problems.
This investigation is an attempt to get new numerical solutions of photo-thermoelastic interactions in semiconductor medium by applying the implicit finite difference method (IFDM). Numerical outcomes for the carrier density, the displacement, the temperature and the redial and hoop stresses distributions are presented graphically. Finally, the accuracy of the finite difference method was validated by the comparing between the numerical and the analytical solutions for all physical fields.

Mathematical Model
The governing formulations under the coupled photo-thermal theory for an isotropic semiconductor medium in the absence of the body force and the thermal source are presented as in [38][39][40]: We consider an unbounded semiconducting material containing spherical cavities. Its state can be expressed in terms of the space variable r and the time t which occupying the region a ≤ r < ∞. Due to symmetry involved in the problem, only the radial displacement u r = u(r, t) un-vanishing, hence the formulations (1)-(4) are expressed as:

Initial and Boundary Conditions
In this problem, the initial conditions are given by While the boundary conditions can be given by [41].
Now, for appropriatenes, the dimensionaless physical fields can be given by where ω ¼ ρc e K and c 2 ¼ λþ2μ ρ . By using the variables of nondimensional forms (14), the basic relations with the neglecting of the dashes can be written by: where
Hence, the following system are obtained  (23) and (24) with respect to r and using in combination Eq. (22), which can be written as: Now, it is possible to solve the coupled differential Eqs. (28), (29) and (30) The matrix A has its characteristic formulation by The eigenvalues of matrix A are the three roots of Eq. (32) which are named here η 1 , η 2 , η 3 . Hence, the corresponding eigenvectors X = [X 1 , X 2 , X 3 ] can be calculated as: The solution of Eq. (31) which is bounded as r → ∞ are expressed by where m i ¼ ffiffiffiffi η i p , K 3/2 is the modified of Bessel's function of order 3 2 , A 1 , A 2 and A 3 are constants that can be calculated by using the problem boundary conditions. The numerical inversion method adopted the final solutions of the temperature, the displacement, the carrier density and the stress distributions. The Stehfest approach [47] can be given by with where G is the term numbers.

Numerical Method
The basic relations obtained are linear partial differential equations. For the solutions problem, the implicit finite difference method (IFDM) is used. The solutions domain 0 ≤ t ≤ t f , a ≤ r ≤ R f , are replaced by grids described by the set of nodes points (t s , r m ), in which t s = sk, s = 0, 1, 2, …. . , S and r m = mh, m = 0, 1, 2, …. . , M. Therefore, k ¼ M are taken as the time step and mess width respectively. For the time derivatives and the space derivatives, the derivatives are replaced the central differences. Thus, the approximations of finite difference method for the system of partial differential equations with respect to the independent variables: ∂f The Eqs. (15,19) are then replaced by the implicit finite difference equations by

Numerical Results and Discussions
To make the full discussion for this phenomenon, the numerical simulation by using an element which belong to the semiconductor family. Therefore, the silicon (Si) material can be used as an example of semiconductor with physical constants of it. Si material has many applications in plasma modern physics and industrials technology. The physical constants in SI unit of photo-thermoelasticity of Si are given by the following [48]: The influence of the exponent of the decayed heat flux Ω in isotropic, linear semiconductors in the context of the coupled photo-thermoelastic theory are very important for researchers. The physical quantities in this problem subjected to the exponent of the decayed heat flux Ω are obtained graphically and discussed in 2D plotted which illustrated in Figs. 1-5. The numerical techniques, outlined above, were used for the distributions of the temperature, the variation of redial displacement, the variation of carrier density, the variations of radial and hoop stresses with respect to the r-direction under coupled photo-thermal model. Figure 1 displays the temperature variation via the redial distance r. It is observed that the temperature equivalent to the constant temperature T 1 = 1 when Ω = 0. While the temperature equal to e −1.5 * 0.5 = 0.4724 for t = 0.5 and Ω = 1.5 which satisfy the problem boundary conditions on the internal surface of cavity r = 1 then the temperature decreases with the increasing of the redial distance r till it closes to zeros. Figure 2 show the carrier density variations via the radial distances r. It is noticed that the carrier density starts with its maximum value on the internal surface of cavity r = 1 then the carrier density gradually decreases with the increasing of the redial distance r till it come to zeros values. Figure 3 depicts the displacement variations via the radial distance r. It is observed that the displacement attains maximum negative Fig. 1 The temperature variation via the distance with and without the exponent of the decayed heat flux Ω Fig. 2 The carrier density variation via the distance with and without the exponent of the decayed heat flux Ω Fig. 3 The redial displacement variations via the distance with and without the exponent of the decayed heat flux Ω Fig. 4 The redial stress variations via the distance with and without the exponent of the decayed heat flux Ω values then it increases gradually up to it attains a peak value at a particular location proximately close to the surface and then continuously decreases to zero. Figure 4 depict the redial stress variation via the redial distance r. It is noticed that it start from zeros values which satisfied the problem boundary conditions. Figure 5 depict the hoop stress variations via the redial distance r. It is observed that the hoop stress attains some negative values then the magnitudes of stress decrease gradually to zeros values. The compressions between the solutions, one can conclude that considering the coupled photo-thermal model have major effects on the physical quantities distributions. The increasing of the exponent of the decayed heat flux Ω reduces to the physical quantities magnitudes. Otherwise, Figs. 1-5 illustrates the solutions obtained numerically by the implicit finite difference method (IFDM) overlaid onto the solutions obtained analytically. The accuracy of the implicit finite difference method (IFDM) formulation was validated by comparing the analytical and numerical solutions for the field quantities.