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

Authors: Areej A. Almoneef; Shreen El-Sapa; Kh. Lotfy; A. El-Bary; Abdulkafi M. Saeed
Journal: Advances in Condensed Matter Physics
 (2022)

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## 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.


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## Body

## 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 ofx-axis) are introduced according to the heat-induced laser pulses energy in the following form: the carrier intensity Nx,t with plasma effect, the thermal effect with temperature distribution T, x,t, the elastic impact according to displacement distribution ux,t and the concentration of hole charges Hx,t where t 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]:(1)K1+τθ∂∂t∂2T∂x2+mnq∂2N∂x2+mhq∂2H∂x2−ρa1n∂N∂t+a1h∂H∂t−∂∂t+τqαΓα+1∂1+α∂t1+αρCeT+ρT0αnN+ρT0αhH+T0γ∂u∂x=ρa1ntnN+ρa1hthH+1+τq∂∂tpδe−Ωt+δx,(2)mqn∂2T∂x2+Dnρ∂2N∂x2−ρ1−a2nT0αn+ℑn∂∂t∂N∂t−a2nρCe∂T∂t+ρT0αh∂H∂t+T0γ∂∂x∂u∂t=−ρt1n1+ℑn∂∂tN,(3)mqh∂2T∂x2+Dhρ∂2H∂x2−ρ1−a2hT0αh+th∂∂t∂H∂t−a2hρCe∂T∂t+ρT0αn∂N∂t+T0γ∂∂t∂u∂x=−ρt1h1+th∂∂tH.The nonlocal equations of motion is formulated as follows [32]:(4)ρ1−ξ12∂2∂x2∂2u∂t2=2μ+λ∂2u∂x2−γ1+τθ∂∂t∂T∂x−δn∂N∂x−δh∂H∂x.where mnq,mqn,mhq, and mqh are the Peltier and Dufour-Seebeck-Soret-like polymer parameters. The diffusion for the electrons and holes represent by Dn and Dh, respectively, 0<α≤1 is the order of fractional parameter. The parameter τθ is first phase-lag of the temperature gradient and τq is the second time of the heat flux phase-lag. The life times of electrons and holes in equilibrium case are ℑn and th. The thermodiffusive holes and electrons parameters are αh and αn. The flux-like constants are aQn,aQh,aQ,an, and ah, where a1n=aQn/aQ, a1h=aQh/aQ, a2n=aQn/an, a2h=aQh/ah.The constitutive equation for nonlocal semiconductor according to the 1D deformation and electrons and holes coupling can be expressed as follows [36]:(5)1−ξ12∂2∂x2σij=σij′,σxx′=2μ+λe−γ1+τθ∂∂tT+δhH+δnN=σ.The fractional integral operator according to the Riemann–Liouville method can be presented as follows [38]:(6)IαΜt=1Γα∫0tt−℘α−1Μ℘d℘.For any continuous functionΜ℘, the fractional integral is Iα. The derivative ∂α/∂tα according to Caputo fractional for Μ℘ can be written in the following form [20–22]:(7)∂α∂tαΜx,t=Μx,t−Μ0,t,whenα⟶0,Iα−1∂Μx,t∂t,when0<α<1,∂Μx,t∂t,whenα=1.The main quantities can be transformed in the dimensionless case for more simplifications when the following dimensionless variables are introduced as follows [32]:(8)x′,u′,ξ1′=ω∗x,u,ξ1CT,t′,τq′,τθ′,ℑn′,th′,t1n′,t1h′=ω∗t,τq,τθ,ℑn,th,t1n,t1h,β2=CT2CL2,σxx′=σxx2μ+λ,N′=δnN2μ+λ,CT2=2μ+λρ,CL2=μρ,Ω′=ΩKρCeCT2,ω∗=Ceλ+2μK0,δn′,δh′=δnn0,δhh0γT0,T′=γT2μ+λ,H′=δhH2μ+λ,δ′=δKρCeCT.Ignoring the primes, according to equation (8) the governing equations (1)–(5) can be rewritten as follows:(9)1+τθ∂∂t∂2∂x2−∂∂t+τqαΓα+1∂1+α∂t1+αT+α1∂2∂x2−α2∂∂t+τqαΓα+1∂1+α∂t1+α−α3∂∂t−α4N+α5∂2∂x2−∂∂t+τqαΓα+1∂1+α∂t1+αα6−α7H−ε1∂∂t+τqαΓα+1∂1+α∂t1+α∂u∂x=1+τq∂∂tΓ1e−Ωt+δx,(10)∂2∂x2−α8∂∂tT+α9∂2∂x2−α10+ℑn∂∂tα11+1+tn∂∂tα11ℑnN−α12∂H∂t−α13∂∂x∂u∂t=0,(11)∂2∂x2−α18∂∂tT+α14∂2∂x2−α15+th∂∂tα16∂∂t+1+th∂∂tα17H−α19∂N∂t−α20∂∂t∂u∂x=0,(12)∂2∂x2−1−ξ12∂2∂x2∂2∂t2u−1+τθ∂∂t∂T^∂x−∂N∂x−α21∂H∂x=0,(13)σ=∂u∂x−1+τθ∂∂tT^+N−H,where(14)α1=mnqαtdnK,α2=T0αnCe,α3=a1nCe,α4=a1nγCeτn2μ+λ,α5=γmhqh02μ+λK,α6=T0αhKh0Ce,α7=a1hγω∗thK,α8=a2nKmqn,α9=Dnραtmqndn,α10=1−a2nT0αn,α11=αtKmqndnCe,α12=a2nγh0αhω∗mqn,(15)α19=a2hγT0αn2μ+λω∗mqhδn,α20=a2hγ2T0ω∗mqhρ,α21=δh2μ+λ,ε1=T0γ2ω∗ρK,Γ1=pδ1−τqΩρCeCT,(16)α13=a2nγ2T0ω∗ρmqn,α14=Dnh0γCT2mqh,α15=1−a2hT0αn,α16=γh0ω∗mqh,α17=γh0ω∗mqhτ1h,α18=a2hKmqh.At rest, the following initial conditions are considered to can be solved the problem mathematically, which can be represented as follows:(17)ux,t|t=0=∂ux,t∂t|t=0=0,Tx,t|t=0=∂Tx,t∂t|t=0=0,Hx,t|t=0=∂Hx,t∂t|t=0=0,Nx,t|t=0=∂Nx,t∂t|t=0=0.


## 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 functionℝt is as follows:(18)Lℝt=∫0∞ℝtexp−stdt=ℝ¯s.The fractional derivative for the differentiable functionℝt in Laplace domain can be constructed as follows [39]:(19)L∂α∂tαℝt=sαLℝt.In this case, the main equations (10)–(13) according to Laplace transform of fractional derivative can be reformulated as follows:(20)q1D2−q2T¯+α5D2−q4H¯+α1D2−q3N¯−q5Du¯=Γ2e−δx,(21)D2−q7T¯+α9D2−q6N¯−q9Du¯−q8H¯=0,(22)D2−q10T¯+α14D2−q11H¯−q13Du¯−q12N¯=0,(23)D2−ℚu¯−q14DT^¯−ℤDN¯−ℂDH¯=0,(24)σ¯=α23Du¯−1+sτθT¯+N¯−H¯,where D=d/dxq1=1+τθs, q2=s+τqα/Γα+1s1+α, ℚ=s2/1+ξ12s2,(25)ℂ=α211+ξ12s2,(26)q3=α2s+τqαΓα+1s1+α+α3s+α4,(27)ℤ=11+ξ12s2,(28)Γ2=Γ11+τqss+Ω.q
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. The nonhomogenous ODEs (20)–(23) can be solved using the eliminating technique, as follows:(29)D8−∏1D6+∏2D4−∏3D2+∏4H¯,N¯,T¯,u¯x,s=Ce−δx,where C=Γ1δ2q2−s2/s+Ω.In this case, the coefficients of equation (29) take the following form:(30)∏1=−1α9α14q1−α1α14−α5α9−s2α9α14q1+s2α1α14+α5α9+α1α14q9q14+α5α9q13q14−α9α12q1q13−ℂα9α14q5q14−α1α12q9+α1α12q13+α1α14q7+α5α9q11+ℤα9α12q5−α9α14q2−α9q1q10−α14q1q6−α14q1q9−α1q8+α1q10+α5q6+α5q9−ℚα5q12−α5q13+α9q4+α14q3,(31)∏2=1α9α14q1−α1α14−α5α9−s2α1α14q7−ℂα5α9q11+s2α14α9q2+s2α9q1q6+s2α1q8−s2α1q10−s2α5q6+s2α5q12−s2α9q4−s2α14q3−α1α12q7q13+α1α12q9q11+α1q8q13q14−α1q9q10q14−ℤα5q6q13q14+α5q9q12q14+α9α12q2q13−α9α12q5q11−α9α4q13q14+α9q5q10q14+α12q1q6q13−α12q1q9q12−ℤα14q3q9q14+α14q5q6q14−α1q7q10+α1q8q11−α5q6q11+α5q7q12−α5q9q11+α9q2q10−α9q4q11+ℚα12q3q9−α12q3q13−α12q5q6+α12q5q12+α14q2q9−α14q3q7ℂ−α14q5q7+q1q6q10−q1q8q12−q1q8q13+q1q9q10+q3q8−q3q10−q4q6−q4q9+q4q12+q5q8+q4q13−q5q10,(32)∏4=s2q2q6q10−q2q3q12ℚ−s2q3q7q10+ℂq3q3q11−s2q4q6q11+s2ℤq4q7q12α9α14q1−α1α14−α5α9,(33)∏1=−1α9α14q1−α1α14−α5α9s2α1q7q11−ℂα1q8q11+s2α5q6q11−s2α5q7q12−s2α9q8−s2α2q10−s2q1q6q10+s2q1q8q12−s2q3q8+s2q4q6−s2q4q12−α12q2q6q13+ℤα12q2q9q11+α12q3q7−q3q8q13q14+q3q9q10q14+q4q6q13q14−ℚq4q9q12q14−q5q6q10q14+q5q8q12q14−q2q6q10+q2q8q12+q2q8q13−q2q9q10+q3q7q10−q3q8q11+q4q6q11−q4q7q12−q4q7q13+q4q9q11+q5q7q10−ℚq5q8q11.The factorized method is used for nonhomogenous equation (29) as follows:(34)D2−m12D2−m22D2−m32D2−m42T^¯,u¯,N¯,H¯x,s=Ce−δx,where mi2i=1,2,3,4 represent the homogenous real roots when x⟶∞ in the following equation:(35)D2−m12D2−m22D2−m32D2−m42T^¯,u¯,N¯,H¯x,s=0.The general linear solutions of equation (34) can be expressed as follows:(36)T¯x,s=∑i=14Βise−mix+Ψe−δx,(37)N¯x,s=∑i=14H1iΒise−mix+f1se−δx,(38)u¯x,s=∑i=14H2iΒisexp−mix+f2se−δx,(39)H¯x,s=∑i=14H3iΒisexp−mix+f3se−δx,(40)σ¯x,s=∑i=14H4iΒisexp−mix+f4se−δx,where Ψ=Cδ8−∏1δ6+∏2δ4−∏3δ2+∏4.According to the undetermined coefficients method for ODE (34), the values of the coefficients fii=1,2,3,4 can be obtained in the following form:(41)f1=Ψc1δ4+c2δ2+c3c4δ4+c5δ+c6,(42)f2=−Ψ−δ2q1−q2q9δ−q5δδ−q7−δ2α5−q4q9δ+q5q8δ−−δα1−q3q9δ−q5δδ2α9−q6H3i−δ2α5−q4q9δ+q5q8δ,(43)f3=Ψδ2q1−q2q5δ+δ2α5−q4H2iq5δ+δ2α1−q3H1iq5δ,(44)f4=Ψα23δH2i−1+sτθH1i+1−α22H3i,(45)H1i=c1mi4+c2mi2+c3c4mi4+c5mi2+c6,H3i=mi2q1−q2q5mi+mi2α5−q4H2iq5mi+mi2α1−q3H1iq5mi,(46)H2i=−−mi2q1−q2q9mi−q5mimi2−q7−mi2α5−q4q9mi+q5q8mi−−mi2α1−q3q9mi−q5mimi2α9−q6H3i−mi2α5−q4q9mi+q5q8mi,H4i=α23miH2i−1+sτθH1i+1−α22H3i,(47)c1=α14q1q9−α5q9−α5q13+α14q5c2=α5q7q13+α5q9q11−α14q2q9+q5q7−q1q8q13−q1q9q10+q4q9+q13+q5q8−q5q10,c3=q2q8q13+q2q9q10−q4q7q13−q4q9q11+q5q7q10−q8q11,c4=α1α14q9−α5α9q13+α9α14q5,c5=−α1q8q13−α1q9q10+α5q6q13+α5q9q12+α9q4q13−α9q5q10−α14q3q9−α14q5q6,c6=q3q8q13+q3q9q10−q4q6q13−q4q9q12+q5q6q10−q8q12.The unspecific parameters in this problem areΒi,i=1,2,3,4 which depend on the parameter s and can be calculated from the following boundary conditions.


## 4. Boundary Conditions

In this part, we assume some boundary conditions taken on the surfacex=0 to determine the value of undefined parameters Βis. 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 surfacex=0 can be presented as follows [27,40]:(48)∂T¯0,s∂x=0⇒⇒∑n=14Βismi+δΨ=0.(II) The mechanical condition is selected when the traction is free on the nonlocal surfacex=0 as follows [41]:(49)σ¯0,s=0⇒⇒∑i=14H4iΒis+f4s=0.(III) During the electronic recombination and diffusion processes, the carriers’ fields density distribution can be selected on the nonlocal surface as follows [42]:(50)N¯0,s=s˜n0De⇒⇒∑i=14H1iΒis+f1s=s˜n0De,where De 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]:(51)H0,s=h0⇒⇒∑i=14H3iΒis+f3s=h0.When solving the system consisting of the previous four equations (48)–(51) in the inverse matrix method, the values of the variables Βis can be determined algebraically and thus we can know the linear solutions as a function of s [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℧¯x,s in the s domain is defined as follows:(53)℧x,t′=L−1℧¯x,s=12πi∫ξ−i∞ξ+i∞expst′℧¯x,sds.Equation (53) is rewritten in another form as follows:(54)℧x,t′=expξt′2π∫∞∞expiβt℧¯x,ξ+iβdβ.Fourier series expansion is utilized during the interval0,2t′ for ℧x,t′ in the closed form and the following relation can be obtained as follows:(55)℧x,t′=eξt′t′12℧¯x,ξ+Re∑k=1N℧¯x,ξ+ikπt′−1ξ.Here, i=−1ξ∈R, the real part is Re, the chosen large sufficient integer is N, and ξt′≈4.7 [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 (N=0) and holes (H=0) 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]:(56)K1+τθ∂∂t∂2T∂x2−∂∂t+τqαΓα+1∂1+α∂t1+αρCeT+T0γ∂u∂x=1+τq∂∂tpδe−Ωt+δx,(57)ρ1−ξ12∂2∂x2∂2u∂t2=2μ+λ∂2u∂x2−γ1+τθ∂∂t∂T∂x.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 at0τθ<τq, the Lord and Șhulman (LS) model is appeared when τθ=0, 0<τq, and the coupled thermoelasticity (CT) model is considered at τθ=τq=0.0 [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.H=0), 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]:(58)K1+τθ∂∂t∂2T∂x2+mnq∂2N∂x2−ρa1n∂N∂t−∂∂t+τqαΓα+1∂1+α∂t1+αρCeT+ρT0αnN+0γ∂u∂x=ρa1ntnN+1+τq∂∂tpδe−Ωt+δx,(59)mqn∂2T∂x2+Dnρ∂2N∂x2−ρ1−a2nT0αn+tn∂∂t∂N∂t−a2nρCe∂T∂t+T0γ∂∂x∂u∂t=−ρt1n1+tn∂∂tN,(60)ρ1−ξ12∂2∂x2∂2u∂t2=2μ+λ∂2u∂x2−δn∂N∂x−γ1+τθ∂∂t∂T∂x.


### 6.3. The Impact of Nonlocal Parameters

When the nonlocal parameter is neglected (ξ1=0.0), 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.


## 6.1. The Thermoelasticity Theories

To know the different models according to the theory of thermoelasticity only, the effect of each of electrons (N=0) and holes (H=0) 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]:(56)K1+τθ∂∂t∂2T∂x2−∂∂t+τqαΓα+1∂1+α∂t1+αρCeT+T0γ∂u∂x=1+τq∂∂tpδe−Ωt+δx,(57)ρ1−ξ12∂2∂x2∂2u∂t2=2μ+λ∂2u∂x2−γ1+τθ∂∂t∂T∂x.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 at0τθ<τq, the Lord and Șhulman (LS) model is appeared when τθ=0, 0<τq, and the coupled thermoelasticity (CT) model is considered at τθ=τq=0.0 [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.H=0), 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]:(58)K1+τθ∂∂t∂2T∂x2+mnq∂2N∂x2−ρa1n∂N∂t−∂∂t+τqαΓα+1∂1+α∂t1+αρCeT+ρT0αnN+0γ∂u∂x=ρa1ntnN+1+τq∂∂tpδe−Ωt+δx,(59)mqn∂2T∂x2+Dnρ∂2N∂x2−ρ1−a2nT0αn+tn∂∂t∂N∂t−a2nρCe∂T∂t+T0γ∂∂x∂u∂t=−ρt1n1+tn∂∂tN,(60)ρ1−ξ12∂2∂x2∂2u∂t2=2μ+λ∂2u∂x2−δn∂N∂x−γ1+τθ∂∂t∂T∂x.


## 6.3. The Impact of Nonlocal Parameters

When the nonlocal parameter is neglected (ξ1=0.0), 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 (Table1) can be used [45,46]:Table 1
The physical parameters of the laser and SI medium in SI unit.
Unit
Symbol
SI
N
/
m
2
λ
6.4
×
10
10
μ
6.5
×
10
10
k
g
/
m
3
ρ
2330
K
T
0
800
sec
s
τ
5
×
10
−
5
m
2
/
s
D
e
2.5
×
10
−
3
J
p
10
13
e
V
E
g
1.11
K
−
1
α
t
4.14
×
10
−
6
W
m
−
1
K
−
1
K
150
J
/
k
g
K
C
e
695
m
/
s
s
˜
2
Ω
6
v
k
−
1
m
q
n
1.4
×
10
−
5
m
n
q
1.4
×
10
−
5
m
q
h
−
0.004
×
10
−
6
m
h
q
−
0.004
×
10
−
6
m
2
s
−
1
D
n
0.35
×
10
−
2
m
2
s
−
1
D
h
0.125
×
10
−
2
m
2
/
s
α
n
1
×
10
−
2
m
2
/
s
α
h
5
×
10
−
3
m
−
1
δ
3


### 7.1. The Influence of Fractional-Order Parameters

Figure1(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 (α=0.3, α=0.7 andα=1 (classical case)). The computational results are implemented according to the DPL model when the semiconductor medium is in the nonlocal case ξ1=0.50 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 α=1, the magnitude of thermal distribution differs from the two unclassical cases when the fractional order parameters α=0.3, and α=0.7. 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 α=1 (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 α=0.3 and α=0.7. 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.Figure 1
The wave propagations of the main physical distributions against the distance according to different values of fractional order parameters subjected to the DPL model whenξ1=0.50.


### 7.2. The Photo-Thermoelasticity Models

Figure2 (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 t=0.0001 when the fractional order parameter is α=0.7 with the nonlocal parameterξ1=0.50. 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.Figure 2
The wave propagations of the main physical distributions against the distance according to different values of thermal memories (CT, LS, and DPL models) subjected to fractional order parameterα=0.7 when ξ1=0.50.


### 7.3. The Influence of Nonlocal Parameters

Figure3 (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 ξ1=0.0, but the other two cases describe the nonlocal semiconductor medium when ξ1=0.25 andξ1=0.50. The evaluations in this group are implemented according to the DPL model for fractional order parameter α=0.7. 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.Figure 3
The wave propagations of the main physical distributions against the distance according to different values of nonlocal parameter subjected to the DPL model when fractional order parameterα=0.7.


### 7.4. The Influence of Instant Time

Figure4 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 ξ1=0.50 when the fractional parameter is α=0.7. 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.Figure 4
The variation of 3D graph of field distributions verses the time and the distances according to the DPL model whenα=0.7 and ξ1=0.50.


### 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].


## 7.1. The Influence of Fractional-Order Parameters

Figure1(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 (α=0.3, α=0.7 andα=1 (classical case)). The computational results are implemented according to the DPL model when the semiconductor medium is in the nonlocal case ξ1=0.50 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 α=1, the magnitude of thermal distribution differs from the two unclassical cases when the fractional order parameters α=0.3, and α=0.7. 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 α=1 (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 α=0.3 and α=0.7. 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.Figure 1
The wave propagations of the main physical distributions against the distance according to different values of fractional order parameters subjected to the DPL model whenξ1=0.50.


## 7.2. The Photo-Thermoelasticity Models

Figure2 (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 t=0.0001 when the fractional order parameter is α=0.7 with the nonlocal parameterξ1=0.50. 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.Figure 2
The wave propagations of the main physical distributions against the distance according to different values of thermal memories (CT, LS, and DPL models) subjected to fractional order parameterα=0.7 when ξ1=0.50.


## 7.3. The Influence of Nonlocal Parameters

Figure3 (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 ξ1=0.0, but the other two cases describe the nonlocal semiconductor medium when ξ1=0.25 andξ1=0.50. The evaluations in this group are implemented according to the DPL model for fractional order parameter α=0.7. 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.Figure 3
The wave propagations of the main physical distributions against the distance according to different values of nonlocal parameter subjected to the DPL model when fractional order parameterα=0.7.


## 7.4. The Influence of Instant Time

Figure4 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 ξ1=0.50 when the fractional parameter is α=0.7. 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.Figure 4
The variation of 3D graph of field distributions verses the time and the distances according to the DPL model whenα=0.7 and ξ1=0.50.


## 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.



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*Source: 1523059-2022-08-12.xml*