Magneto-photo-thermoelastic interactions in a semiconductor solid cylinder due to a periodic decaying varying heat

In this article, a generalized photo-thermoelastic model with relaxation time (GPTE) was introduced and applied to an infinite semiconductor body in the form of a solid cylinder. The cylinder and its adjoining vacuum are constrained exposed to periodic decaying varying heat and subjected to a uniform axial magnetic field. Analytical formulas of the physical quantities of the problem were obtained using Laplace transformations. A numerical method is used to find the inverse Laplace transforms. The effect of phase-lags, the temperature frequency on the derived expressions have been illustrated graphically and discussed.


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Lamé's constants semiconductor gap energy thermal expansion coefficient thermal activation coupling parameter = (3 + 2 ) coupling parameter carrier density 0 environmental temperature Kronecker′s delta function = − 0 temperature increment displacement components absolute temperature body force components specific heat ⃗ ⃗ magnetic field 0 magnetic permeability current density ℎ ⃗ nduced magnetic field ⃗ induced electric field cubical dilatation 0 relaxation time stress tensor 1 lifetime of photogenerated electron strain tensor material density = (3 + 2 ) coefficient of electronic deformation Maxwell′s stress thermal conductivity

Introduction
Studying the interaction between stress and the magnetic field in a thermal body is very important because of its various applications in plasma physics, geophysics, and associated subjects. Particularly in nuclear fields, the gradients of very high temperature, as well as the magnetic fields created within nuclear reactors, influence the design and physical processes. Furthermore, the earth is due to the magnetic field in general, and its materials may be electrically conducting.
To remove the contradictions inherent in the coupled and uncoupled thermoelastic models that propagate the heat at infinite speed because of the parabolic heat equation, a generalized theory of thermoelasticity is introduced by Lord and Shulman [1] and Green and Lindsay [2]. Tzou [3] announced a new model of thermoelasticity where the Fourier law is exchanged by the approximation of a heat conduction equation that includes two different phase-lags (DPLs).
As technologies developed, semiconductor materials are widely used in advanced modern engineering. The study of wave propagation in bodies of semiconductors has academic significance and practical value. The wave propagation problem in a semiconductor solid is not recorded during the photothermal process yet. In addition, the progress of semiconductor processes and fabrication over the last decades have made great efforts. The photoacoustic effect theory was developed in semiconductors for the bass region, taking into account the acoustic and phonon drag by the electron-hole couple diffusion flux [4][5][6][7][8][9][10][11][12][13]. A classic theoretical model for photoacoustic influence (PA) has been studied in the semiconductor system, which includes space charge on the semiconductor surface, thermoelastic, thermo-diffusion, and electronic deformation effects [14,15]. Recently, Zenkour [16][17][18] presented a refined multi-phase-lags model for the thermomechanical response of microbeams or to treat photothermal waves of semiconductor half-space media.
The dynamic behavior of interactions between the magnetic field and temperature and pressure of the thermal solids is of great importance due to its wide uses in such diverse fields as geophysics, plasma physics, nuclear engineering, and related subjects. Using generalized thermal models, many researchers have made efforts to study thermomagnetic problems. Allam et al. [19] studied the magneto-thermoelasticity concerned with infinite solid with a spherical cavity when the properties of the material are variables. Abouelregal and Abo-Dahab [20] applied the DPL theory to study magneto-thermoelastic interaction in the inhomogeneous body with a spherical cavity. Zenkour et al. [21] discussed the Seebeck effect and magneto-thermoelastic interaction in the solid cylinder when the thermal conductivity is temperature-dependent. Deswal and Kalkal [22] used the statespace method to study a two-temperature model for magneto-thermoelasticity under initial stress. Bachher and Sarkar [23] displayed a two-dimensional problem of magneto-thermoelasticity due to a conduction equation with frictional heat. Singh and Kumar [24] applied the eigenvalue method to investigate the effect of rotating on the magneto-thermoelastic micropolar body. One team of researchers who studied generalized magneto-thermodynamic equations in the general case is Nayfeh and Nemat-Nasser [25], which they studied the wave propagation in bodies due to the effect of electromagnetic fields.
In the present work, the effect of photo-thermoelastic interactions for semiconductor solid cylinder is considered. The analytical solution is gained by using the techniques of the Laplace transform. To get the solutions in the physical domain, a numerical method is employed. Some outcomes are illustrated in graphs to discuss the coupling impact between the thermoelastic, magnetic, and plasma waves.
We assumed that the adjacent free space is saturated by an initial magnetic field ⃗ ⃗ . This produces an induced electric field ⃗ and brought the magnetic field ℎ ⃗ that satisfies electromagnetism Maxwell's equations and is suitable for slowly moving media:

Formulation of the problem
Let us consider due to symmetry, a one-dimensional problem of a thermally, homogeneous, isotropic, and electrically conducting cylinder of radius . So, all quantities are considered to be depending on the time and the distance . The outer surface of the cylinder is under a periodic and decaying time-dependent varying heat and traction free. The cylindrical coordinates system ( , , ) is used with -axis lying along the axis of the cylinder. The cylinder is initially at a constant uniform temperature 0 . In addition, for the regularity, all studied fields are considered to vanish as ⟶ 0.
The constitutive relations that given in Eq. (1) take the form where The electromagnetic dynamic equation of the solid cylinder is expressed as Since the cylinder is sited in a magnetic field with a constant intensity ⃗ ⃗ 0 = (0,0, 0 ), then from Eq. (7), we get For a perfect conductor, the radial component of Lorentz force brought by the magnetic field ⃗ ⃗ 0 is Thus, from Eqs. (8), (16) and (17), we get and Maxwell's stress in the forms From Eqs. (10), (11), (15), and (18), we obtain Applying the differential operator 1 ( ) to both sides of Eq. (19), we get In what tails we will utilize the accompanying dimensionless factors where 1 = √ +2 is the dilatational wave velocity in the half-space and 0 = √ 0 0 2 is the Alfven wave velocity of the medium. Utilizing non-dimensional factors (21), the fundamental equations take the structures

Initial and boundary conditions
Here we can express the initial conditions of the problem as We guess that the boundary of the cylinder = is compelled and is exposed to a heat fluctuating which periodically and decaying. So the accompanying boundary conditions are held:  The surface of the cylinder is exposed to a sinusoidal and decaying varying heat ( , ) = 0 e − /2 cos( ), > 0, where and 0 is constant and is a temperature frequency.
 The mechanical boundary condition as the displacement of the surface is constrained in the form ( , ) = 0.
(29) During the diffusion procedure, the carriers can arrive at the surface of the sample with a limited likelihood of recombination. Then the limit state of carrier density can be summarized below: where is the surface recombination velocity.
Presenting , ( = 1,2,3) into Eq. (35), one obtains where 1 2 , 2 2 and 3 2 are the roots of These roots are given by The solutions of Eq. (38) is expressed as where 0 ( . ) is the modified Bessel's function of the second kind of order zero and , ( = 1,2,3) are some parameters in terms of s. In the same manner, one obtains .
The non-dimensional Maxwell's stress is given by After applying the Laplace transform, the boundary conditions (28)-(30) take the forms  (48) Solving the system (42), one gets the estimations of constants, , ( = 1,2,3). Here, a precise and proficient numerical strategy dependent on a Fourier series expansion [25] is utilized to acquire the inversion of the Laplace transforms. In this strategy, any function in Laplace space can be altered to the time domain as where is a limited number of terms, Re is the real part, is an imaginary number unit and fulfills the connection ≅ 4.7 [26].

Numerical results
In this section, some numerical results have been computed numerically. The physical parameters used in the calculations are given at 0 = 298 K as follows [11]:

Case I: The effect of periodic frequency
In this case, all the studied physical field quantities with radial direction are displayed in Figs. (1)(2)(3)(4)(5)(6) for different values of the periodic frequency . The other physical parameters remain constant. For a thermal shock problem, = 0, and when the cylinder is due to a decaying varying heat we set = 5,10. Figs 1-6 illustrate that:    The periodic frequency has a clear effect on the studied fields.
 The temperature starts at the highest values at the surface of the cylinder due to the presence of decaying and periodic temperature and then its values gradually decrease in the opposite of increasing the radius .
 The distribution of heat when = 5,10 is less than in the case of = 0.    Displacement distribution is increasing in 0.9 ≤ ≤ 1 and rapidly decreases in the interval 0.9 ≤ ≤ 0.6 until it approaches zero.
 The values of displacement decrease with increasing the parameter .
 Stress Σ starts at the negative value and increases rapidly until it approaches zero.
 By increasing the angular frequency values, stress Σ values decrease.
 The second thermal stress Σ is the same behavior as the stress Σ .    Maxwell's stress M increasing in 0.85 ≤ ≤ 1 and rapidly decreases in the interval 0.85 ≤ ≤ 0.6 until it approaches zero  Like most previous distributions, carrier density starts with the greatest value at the surface and gradually decreases until it approaches zero, which is physically consistent.
 In generalized photo-thermoelasticity with relaxation time (GPTE) the thermoelastic and plasma waves propagate with infinite speeds.
 Finally, the field quantities depend not only on the state and space variables and , but also depend on the parameter of temperature frequency.

Case II: The effect of relaxation time
In the second case (Figs. (7-12)), two different values of relaxation time 0 are considered. two different models of thermoelasticity can be reached as special cases of the existing photothermoelasticity with phase lags model. The classical coupled model of photo-thermoelasticity (CPTE) introduced by Todorović [13,14] can be obtained can be attained as special cases of the current photo-thermoelasticity (GPTE) by taking 0 = 0. Figs. 7-12 represent variation in the values of temperature , the displacement , Maxwell's stress M , the thermal stresses stress Σ and Σ and the absolute carrier density against the radial distance , (0 ≤ ≤ 1) for = 5 and = 0.12. It can be observed that:

Case III: Influence of the photo-generated carrier lifetime parameter
The distributions of temperature , the displacement , the thermal stresses Σ and Σ , Maxwell's stress M and the absolute carrier density due to the influence of the photo-generated carrier lifetime parameter 1 with distance, are shown in Figs. 13-18. It is detected that the parameter of carrier lifetime 1 has significant impacts on all variables. We notice that:  The carrier lifetime parameter does not have a clear effect on the distribution of heat inside the solid cylinder.  The parameter 1 has a distinct effect on the distributions of displacement and carrier density , especially near the surface of the cylinder.  Carrier density distribution increases with increasing lifetime and this is also evident from the basic equations as the depends on 1 significantly (see Eq. 6).  There is a great similarity in the dependence of stresses on the parameter 1 near the surface of the cylinder.  Maxwell's stress M increases with increasing time in some intervals and decreases in other intervals. Figures 19-24 show the 3D curves to discuss the distributions of displacements, temperature change, stress forces, and carrier charge density with variations of distance and instant time . We can see that the instant time has significant impacts on all the studied fields. It has been observed that to play a vital role in the development of temperature, stresses, carrier charge density, and displacement fields.     In these figures, it is also observed that the values of Maxwell's stress M , displacement , radial and hoop stresses Σ and Σ , temperature distribution and carrier charge density in any fixed point ( , ) increase when increases. Figure 20 shows that displacement gratify the boundary condition at = and has a different response. The comparison of these figures shows the effect of instant time on the field variable. The field quantities depend not only on space , but also on the time .

Conclusions
In this article, we consider an infinitely long solid cylinder whose surface is due to a magnetic field and varying heat, and its surface traction free and constrained. We obtained the solutions using Laplace transformations and used an approximate numerical method to obtain the distributions in the physical field. From the numerical results, one can conclude that:  The presence of photo-thermal fields has affected all the considered physical variables.
 The thermoelastic stresses, displacement, carrier charge density, and temperature have a strong dependency on the periodic frequency parameter.
 In a generalized photo-thermoelasticity model with relaxation time, heat propagates as a wave with finite velocity instead of infinite velocity in the medium.
 The relaxation time parameter has a significant effect on all the field quantities.
 The models of coupled photo-thermoelasticity CPTE can extract as a special case.
 Also, the near behavior between GPTE and PTEPL models and differs from the CPTE model that indicates the origin of the photo-thermoelastic theory.
 The results we obtained are found that they were very close to the theoretical results.  The obtained results in this analysis should be useful to researchers who are working on a photothermal theory.  The results presented in this work are very important for researchers, scientists, and engineers, as well as those working in the field of solid body mechanics.  The method used in the present work applies to a wide range of problems in thermodynamics and photo-thermoelasticity.

Data Availability
Not Applicable.

Funding
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.