Fractional Order Theory in a Semiconductor Medium Photogenerated by a Focused Laser Beam

In this paper, the fractional order theory has been applied for thermal, elastic and plasma waves to determine the carrier density, displacement, temperature and stress in a semiconductor medium. The thermal, elastic and plasma waves in a semi-infinite medium photogenerated by a focused laser beam were analyzed. The Laplace transformation is used to express the governing equation and solved analytically by applying eigenvalue approach methodology in that domain. A semiconducting material like as silicon was considered. According to the numerical results and graphics, the fractional order parameter and thermal relaxation time may play an important role in the behavior of all physical quantities.


INTRODUCTION
When a semiconductor with band gap energy g E is illuminated by a laser beam with an energy E higher than g , E an excitation process of electrons will take place. The excited electrons will transfer to an energy level from the valence band with energy of E g E above the conduction band edge. Then these photoexcited free carriers will relaxe to one of the unfilled levels nearby the conduction band bottom (transition of nonradiative). After relaxation there are electron and hole plasma which is followed by the formation of holeelectron pairs through the recombination process. In semiconductors there is a periodic elastic deformation produced by the photoexcited carriers known as electronic deformation. The electronic deformation may cause local tensions in the sample which can introduce plasma waves which are similar to the heat wave generated by local periodic elastic deformation. Considerable attention has been currently given to the surface waves of bounded plasma systems. The existence of plasma boundaries makes it possible for various surface wave modes to arise, which may in some cases have frequency spectra drastically different from those of volume-wave modes.
Understanding of transport phenomena in the solid through the development of spatially in situ resolved probes has great of attention. In the present work we try to measuring transport processes based on the principle of optical beam deflection using a photothermal approach that can be considered as an expansion of the photothermal deflection technique. Such a technique is contactless and directly yields the parameters of electronic and thermal transport at the semiconductor surface or at the interface and within the bulk of the semiconductor. Pure silicon is an intrinsic semiconductor and is used widely in semiconductor industry, for example, the monocrystalline Si is used to produce silicon wafers. In general, the conduction in the semiconductor (pure Si) is not the same as in metals. Both the electrons and holes are responsible about the conduction in semiconductors as well as the electrons that may be released from atoms by heat. Therefore electric resistance of semiconductors decreases with temperature increasing. Todorovic et al. [13] discussed theoretical and experimental results on microelectromechanical structures in plasma, thermal and elastic waves. These results give valuable information about carrier recombination and transport mechanisms in semiconductors. Also, the study includes the variations in propagating both plasma and thermal waves due to the linear coupling between heat and mass transport (i.e., thermos diffusion). The effects of thermoelastic and electronic deformations in semiconductors without considering the coupled system of the equations of thermal, elastic and plasma waves have been studied [46]. In addition, local thermoelastic deformations at the sample surface due to the excitation by a probe beam have been analyzed by Rosencwaig et al. [7], then Opsal and Rosencwaig [8] study the depth profiling of thermal and plasma waves in silicon. On the other hand, Song et al. [9,10] study in detail the generalized thermoelastic vibrations due to optically excited semiconducting microcantilevers. They concluded that the plane wave reflection in a semiconducting material is under the theory of generalized thermoelasticity and photothermal theory [11,12].
Many existing models of physical processes have been modified successfully by using the fractional calculus. .ractional order of weak, normal and strong heat conductivity under generalized thermoelastic theory were established by Youssef [13,14] who developed the corresponding variational theorem also. The theory was then used to solve the 2D problem of thermal shock using the Laplace and .ourier transforms [15]. Based on a Taylor expansion of fractional order of time, a new model of fractional heat equation was established by Ezzatt and Karamany [1618] and Sherief et al. [19]. Sherief and Abd El-Latief [20] studied the effect of the fractional order parameter and the variable thermal conductivity on a thermoelastic half-space. Due to thermal source, the effect of fractional order parameter on plane deformation in a thermoelastic medium was studied by Kumar et al. [21]. Recently, Abbas [22 25] studied the effect of fractional order on thermoelastic problems by using the eigenvalue approach.
The present work is an attempt to get a new picture of photothermoelastic theory with one relaxation time using the fractional calculus theories. Based on the eigenvalue approach and Laplace transformation, the governing nonhomogeneous equations are processed using an analytical-numerical technique. .rom the numerical results, the physical interpretations are given for the distribution of physical quantities observed in this study.

.ORMULATION O. THE PROBLEM
A homogeneous semiconducting material is considered. The theoretical analysis of the transport processes in a semiconductor material involved in the study of coupled thermal, plasma and elastic waves simultaneously. The main variables are the distribution of temperature T(r, t), the density of carriers n(r, t) and the elastic displacement components ( , ). i u t r When an ultrafast laser Q(r, t) falls on an isotropic, elastic and homogeneous semiconductor, the governing equations of motion, plasma and heat conduction under fractional order theory can be described as [2, 26 29 the stressstrain relations are expressed as where ρ is the medium density, 0 τ is the thermal relaxation time (for semiconductor σ are the stress components, K is the thermal conductivity, e D is the carrier diffusion coefficient, g , E is the excitation energy, τ is the photogenerated carrier lifetime, and 0 n δ = ∂ ∂Θ is the coupling parameter of thermal activation [30], e c is the specific heat at constant strain, g E is the semiconductor energy gap, is the function of temporal modulation of the intensity of laser beam, t is the time, and r is the position vector. The different parameter values with a wide range 0 < ν ≤ 1 cover both conductivity, ν = 1 for normal conductivity and 0 < ν < 1 for low conductivity. Lets consider an isotropic, homogeneous, semiconductor medium, occupying the region x ≥ 0 where all the state functions depend only on the time t and the variable x. The x axis is taken perpendicular to the bounding plane of the halfspace pointing inwards. Therefore, the above equations may be take the form:

INITIAL AND BOUNDARY CONDITIONS
During recombination and transport processes (bulk and surface) of the photogenerated carriers at the surface where x = 0 which is constrained so, the boundary conditions for the carrier density, heat flux and displacement can be written as where 0 s is the speed of surface recombination and the state of the medium is initially quiescent, i.e.
.or the laser pulse, the temporal profile is non-Gaussian, which can be expressed as: According to [9], the laser source Q(x, t) can be described as , 2 where 0 I is the energy absorbed, R is the reflectivity of the sample surface, p t is the pulse rise time. Let us define the Laplace transformation for a function G(x, t) as Then for both sides Eqs. (14)(18) using the initial conditions (19), we obtain  , , ,  .
The solutions of Eq. (29) are the eigenvalues of matrix B that take the form The general solution of the nonhomogeneous equations (28) consists of the sum of the complementary solution c ( , ) V x s and the particular solution p ( , ). V x s The complementary solution has the following form: , , , Due to the conditions of regularity of the solution, the nature exponential growth of the space variable x has been discarded at infinity and the constants 1 2 , B B and 3 B can be calculated using the boundary condi-tions (27). Hence, the field variables have the general solutions with respect to x and s in the form: .or the general solutions of the temperature, carrier density, displacement and stress distributions, numerical inversion method was adopted. Based on Stehfest [34], the numerical inversion method was used. In this method, the inverse G(x, t) of the Laplace transformation ( , ) G x s is approximated by the relation 1 ln 2 ln 2 ( , ) , , where j V is given by the following equation: , , , .
The described numerical techniques were used in the context of the generalized photothermal theory under the fractional order derivative. By using the relation between the variable and its nondimensional form, the variables T, u, n and xx σ are taken in the dimensional forms and displayed graphically as in .igs. 1, 2. The calculations were performed for t = 4.447 ps.
.igures 1a1d show the variation of the temperature T, the displacement u, the carrier density n and the stress xx σ with respect to the distance x for different values of the fractional order parameter ν when 0 τ = 0.1. It observed that the dotted line refer to the normal conductivity while the solid and dashed lines refer to the low conductivity. .rom this results, the fractional parameter ν has a significant effect on all the physical quantities. The effect of thermal relaxation time on the variation of the carrier density n, the temperature T, the displacement u and the stress xx σ was depicted in .ig. 2. The results demonstrate the difference between the coupled photothermoelastic theory and the generalized photothermoelastic theory with one relaxation time. In this problem, the important phenomenon noticed where the medium is unboundedthe solution of any of the considered function for the generalized theory vanishes identically outside a bounded region of space. .igures 1a and 2a display the variation of carrier density along the distance x. It is noticed that n has a highest value for x = 0 and decreases with the increasing in distance x until attaining the equilibrium carrier concentration 0 (n = 10 20 m 3 ) for x = 1000 nm. .igures 1b and 2b show the variation of temperature with respect to the distance x. It is observed that the temperature starts with its maximum value at x = 0 and decreases gradually with increases the distance x to close to the reference temperature 0 (T = 300 K) beyond a wave front for the generalized photothermal theory, which satisfies our theoretical boundary conditions. .igures 1c and 2c represent the variation of displacement versus x. It was observed that the displacement starts from zero which satisfies the boundary condition then it reaches a peak value at a particular location proximately close to the surface and then continuously decreases to zero. .igures 1d and 2d show the variation of stress along the distance x. The magnitude of stress always starts from maximum values and then decreases with distance x and terminates at the zero value.

CONCLUSION
This work investigates the effects of fractional order and thermal relaxation time on the plasma, thermal, and elastic waves in a semiconductor medium. Analytical expressions for temperature, carrier density, displacement and stress in the medium have been obtained. Results carried out in this paper can be used to design various semiconductor elements for the coupled of plasma, thermal, and elastic waves and other fields in the materials science, physical engineering, and design of new materials to meet special engineering requirements