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Photo Thermal Diffusion of Excited Nonlocal Semiconductor Circular Plate Medium with Variable Thermal Conductivity

Photo Thermal Diffusion of Excited Nonlocal Semiconductor Circular Plate Medium with Variable... Hindawi Advances in Condensed Matter Physics Volume 2023, Article ID 1106568, 12 pages https://doi.org/10.1155/2023/1106568 Review Article Photo Thermal Diffusion of Excited Nonlocal Semiconductor Circular Plate Medium with Variable Thermal Conductivity 1 2,3 4,5 2,3 Shreen El-Sapa, Khaled Lotfy , Alaa A. El-Bary, and M. H. Ahmed Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt Department of Mathematics, Faculty of Science, Taibah University, Madinah, Saudi Arabia Arab Academy for Science, Technology and Maritime Transport, P.O. Box 1029, Alexandria, Egypt Council of Future Studies and Risk Management, Academy of Scientifc Research and Technology, Cairo, Egypt Correspondence should be addressed to Khaled Lotfy; khlotfy_1@yahoo.com Received 18 November 2022; Revised 12 January 2023; Accepted 17 April 2023; Published 26 April 2023 Academic Editor: Sergio Ulloa Copyright © 2023 Shreen El-Sapa et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. To examine the efects of the nonlocal thermoelastic parameters in a nanoscale semiconductor material, a novel nonlocal model with variable thermal conductivity is provided in this study. Te photothermal difusion (PTD) processes in a chemical action are utilized in the framework of the governing equations. When elastic, thermal, and plasma waves interact, the nonlocal continuum theory is used to create this model. For the main formulations to get the analytical solutions of the thermal stress, displacement, carrier density, and temperature during the nanoscale thermo-photo-electric medium, the Laplace transformation approach in one dimension (1D) of a thin circular plate is utilized. To create the physical felds, mechanical forces and thermal loads are applied to the semiconductor’s free surface. To acquire the full solutions of the research areas in the time-space domains, the inverse of the Laplace transform is applied with several numerical approximation techniques. Under the impact of nonlocal factors, the principal physical felds are visually depicted and theoretically explained. presence of mass difusion, particularly in the aerospace, 1. Introduction electronics, and integrated circuit industries. High- Nanotechnology is currently and in the future will be one of performance nanostructures, such as nanotubes, nano- the most crucial cornerstones of human existence. Tis flms, and nanowires, have been extensively used as reso- signifcant technology is expanding quickly, and several nators, probes, sensors, transistors, actuators, etc. with the scientists are engaged in this fascinating sector. Several of the fast development of nanomechanical electromechanical physical characteristics of elastic materials may vary systems (NEMS) technologies. It is crucial to comprehend the precise characterizations of these nanostructures’ ther- depending on the temperature. Many difculties arise in researching elastic materials without taking varying heat mal and mechanical characteristics. conductivity into account. When thermal conductivity Semiconductor materials (such as silicon) are an ex- varies, particularly in response to temperature, it becomes cellent research subjects for this phenomenon, particularly essential. Termo-difusion is the relationship between mass when subjected to laser or falling light beams. On the difusion and changing thermal conductivity. Termo- surface, the excited electrons will produce a charge known as difusion happens when particles move from an area of free carriers (plasma waves). According to the quantity of greater concentration to an area of lower concentration as light descending, the plasma density is employed to regulate a result of a temperature change. Modern engineering has the difusion [1–3]. Numerous publications [4–6] failed to several uses for the study of thermal conductivity in the take into account the coupling between thermal-elastic 2 Advances in Condensed Matter Physics waves and plasma waves during the deformation process in waves), carrier density (plasma waves), and difusive ma- semiconductor materials. Recently, several authors terial concentration (mass difusion). When the thermal employed photoacoustic spectroscopy to detect photo- activation coupling value κ for the nonlocal medium is thermal events when a laser beam struck a semiconductor nonzero, the photothermal difusion transport process takes [7, 8]. Semiconductors’ temperature, carrier intensity, and place. It makes use of cylindrical coordinates (r,ψ, z). When thermal difusion are measured using the photothermal a very thin circular plate is taken into account, all quantities phenomena [9–13]. When thermal waves propagate, gen- are independent of ψ and z because of the symmetry of the erating elastic oscillation, and plasma waves are formed by axis z. Elastic-plasma-thermal-difusion wave overlapping photo-excited free carriers, directly creating a periodic processes’ governing equations are presented as [34, 35], the elastic deformation as well [14–16], the interaction between photo-electronic equation is as follows: the elastic-thermal-plasma waves occurs. Without consid- zN(r, t) N(r, t) (1) ering the impact of changing thermal conductivity, several � D N (r, t) − + κ T(r, t). E ,ii zt τ issues in generalized thermoelasticity have been explored [17–25]. Later, a lot of writers studied generalized ther- Equations for thermal difusion in the photothermal moelasticity in many areas using variable thermal con- difusion process transport are as follows: ductivity. Te thermal-mechanical behavior of the medium z K may be afected by the deformation of elastic media 􏼐KT (r, t)􏼑 � 􏼒 T(r, t) + β T u (r, t) + cT C􏼓 ,i 1 0 ,i 0 ,i zt k depending on temperature [26–28]. Abbas [29–33] studied (2) many problems of the fber-reinforced anisotropic ther- moelastic medium in two dimensions with fractional − N(r, t). transient heating according to many mathematical methods. Te nonlocal thermoelastic model with variable thermal If there is no body force, the equations of motion for conductivity (which may be considered as a linear function nonlocal medium may be expressed as follows[34]: of temperature) is utilized in the current study using z u 2 2 a theoretical method. Te process of photo-thermal- ρ 􏼐1 − ξ ∇ 􏼑 � μu (r, t) +(μ + λ) u (r, t) i,jj i,jj zt difusion interactions in semiconductor nanoscale media is investigated. Te variation in temperature caused by the − β T (r, t) − β C (r, t) − δ N (r, t). 1 ,i 2 ,i n ,i light beam impacting the nonlocal semiconductor medium is the basis for the variable thermal conductivity. Te (3) chemical difusion method enables photothermal transfer Te length-related elastic nonlocal parameter is repre- (mass difusion). When the Laplace transform domain in sented by ξ � ae /l (l is the external characteristic length cylindrical coordinates is utilized, the analytical solutions of scale, a is the internal characteristic length, and e is non- the basic felds are found. Te numerical techniques provide dimensional material property). analytical solutions in the Laplace domain without any Te mass difusion equation is expressed as follows [35]: presumptive limitations on the real physical values. Finally, with changes in nonlocal parameters and changing thermal z z D β e + D cT (r, t) + 􏼠 + τ 􏼡C(r, t) conductivity, the numerical calculations of the important c 2 nn,ii c ,ii d 2 zt zt (4) physical quantities distribution are graphically shown and discussed. Te numerical fndings presented in the current � D bC (r, t). c ,ii study have applications in solid mechanics, acoustics, ma- terial science, and engineering for earthquakes. Te change in thermal conductivity is K of the nonlocal semiconductor medium and β � (3λ + 2μ)α where α is 2 c c 2. Formulation of the Problem and the coefcient of linear difusion. On the other hand, the Basic Equations transport heat coefcients for the nonlocal medium are independent of N, C and T [36–38]. Te four important variables in this problem, respectively, Te strain-stress combinations are as follows: are u(r, t), T(r, t), N(r, t), and C(r, t) which stand in for the displacement (elastic waves), temperature (thermal or heat zu 2 2 1 − ξ ∇ σ � 2μ + λ e − β T − T − β C +(3λ + 2μ)d N, 􏼐 􏼑 􏼁 rr 1 0 2 n zr 2 2 (5) 􏼐1 − ξ ∇ 􏼑σ � 2μ + λ e − β T − T 􏼁 − β C +(3λ + 2μ)d N, ψψ 1 0 2 n 2 2 􏼐1 − ξ ∇ 􏼑σ � λ e − β T − T 􏼁 − β C +(3λ + 2μ)d N, σ � σ � σ � 0. zz 1 0 2 n rψ zψ rz Advances in Condensed Matter Physics 3 Te nonlocal semiconductor medium’s chemical po- K Θ � K(T)T , ⎫ ⎬ 0 ,i ,i tential equation is (9) K Θ � 􏼐K(T)T 􏼑 . 0 ,ii ,i ,i P � −β e + bC − c T − T 􏼁 . (6) 2 nn 0 Another form of equation (9) when the nonlinear terms where P is the chemical potential per unit mass. are neglected can be obtained as follows: It is possible to choose a material’s variable thermal K Θ � K T + KT � K 1 + K T􏼁 T + KT conductivity K, which may be estimated as a linear function 0 ,ii ,i ,i ,ii 0 1 ,i ,ii ,i (10) of temperature [26]: 2 � K K 􏼐T 􏼑 + KT � KT . 0 1 ,i ,ii ,ii K(T) � K (1 + qT), (7) Te time-diferentiation is done in the same manner to where q is a negative parameter and K is a thermal con- both sides of equation (7), resulting in: ductivity when q � 0 (the nonlocal medium is independent zΘ zT of temperature). (11) K � K(T) . zt zt Te map of temperature can be taken in the following form [27]: Using equation (8) and diferentiating equation (1) by z/zx , yields: (8) Θ � 􏽚 K(R)dR. z 1 κK (12) K N � D N − N + Θ . 0 􏼐 􏼑 0 ,i E ,mm ,i ,i ,i zt τ K Diferentiating both sides of equation (7) relative to x , Te other form of the quantity κK /KΘ with neglected 0 ,i we get the nonlinear term can be represented as follows: κK − 1 2 0 ⎫ ⎪ Θ � κ 1 + K T Θ � κ 1 − K T + K T − . . . Θ � 􏼁 􏼐 􏼁 􏼑 ,i 1 ,i 1 1 ,i K 1 + K T􏼁 0 1 (13) κΘ − κK TΘ + K T􏼁 Θ − . . . � κΘ . ,i 1 ,i 1 ,i ,i z u 2 2 ρ 􏼐1 − ξ ∇ 􏼑 � μu +(μ + λ) u Equation (1) results when equation (13) is applied: i,jj i,jj zt (17) z 1 (14) N � D 􏼐N 􏼑 − N + κΘ . ,i E ,mm ,i ,i ,i − β Θ − β C − δ N . zt τ 1 ,i 2 ,i n ,i Integrating equation (14), yields: Te equation for mass difusion equation (4) may be expressed as follows: zN 1 (15) � D N − N + κΘ. E ,ii 2 zt τ D cK z z c 0 D β e + Θ (r, t) + 􏼠 + τ 􏼡C(r, t) c 2 nn,ii ,ii d K zt zt Under the infuence of mapping, the heat (thermal) (18) difusion equation (2) have the following form: � D bC (r, t). c ,ii 1 zΘ β T zu cT zC i 1 0 ,i 0 i Θ � + + − N . (16) ,ii i Te term D cK /KΘ (r, t) can be represented with k zt K zt K zt K τ c 0 ,ii 0 0 0 neglected nonlinear terms in the following form: Te nonlocal motion equation (3) under the temperature map may be simplifed as follows: D cK D cK − 1 2 c 0 c 0 ⎪ Θ � Θ � D c 1 + K T􏼁 Θ � D c􏼐1 − K T + K T􏼁 − .......􏼑Θ � ,ii ,ii c 1 ,ii c 1 1 ,ii ⎪ K K 1 + K T􏼁 0 1 (19) 2 2 ⎭ D c􏼐1 − K T + K T􏼁 − .......􏼑Θ � D cΘ − D cK TΘ + D c K T􏼁 Θ − ....... � D cΘ , c 1 1 ,ii c ,ii c 1 ,ii c 1 ,ii c ,ii In this case, equation (18) can be rewritten as follows: 4 Advances in Condensed Matter Physics 2 2 z z z 1 z z D β e + D cΘ (r, t) + 􏼠 + τ 􏼡C(r, t) 􏼠 + − q − q 􏼡N + ε Θ � 0, c 2 nn,ii c ,ii d 1 2 3 2 2 zt r zr zt zt zr (20) � D bC (r, t). 2 c ,ii ∇ Θ − Θ + ε e + ε C􏼁 + ε N � 0, 1 4 2 zt Te strain in cylindrical 1D form can be represented as follows: z e 2 2 2 ∇ (e − Θ − C − N) − 1 − ξ ∇ � 0, 􏼐 􏼑 zu u zt e � , e � , e � e � e � e rr ψψ rψ ψz zz rz zr r z z (21) ∇ e + q Θ − q C􏼁 + q 􏼠 + τ 􏼡C � 0, 2 4 3 5 d 1 z(ru) z 1 z zt zt � 0, e � ∇ � + . r zr r zr zr P � −e + q C − q T. 3 4 By doing the analysis in the radial direction (r), the (26) problem will be solved in 1D, with the displacement vector having the form u � (u, 0, 0), u(r, t). Using the linear form of variable thermal conductivity Te main governing equations in 1D (radial) are reduced equation (6) and the mapping equation (7), one arrives at the as follows: following result [26]: zN z N 1 zN N 2 1 q q 1 1 (22) � D 􏼠 + 􏼡 − + κ Θ, 2 Θ � 􏽚 K (1 + qT)dT � T + T � T + − , 􏼠 􏼡 zt r zr τ zr K 2 2 q 2q z 1 z 1 zΘ β T ze cT zC 􏽰������ � 1 0 0 1 􏼠 + 􏼡Θ � + + − N. T � [ 1 + 2qΘ − 1], r zr k zt K zt K zt K τ zr 0 0 0 􏽰������ � (23) qT + 1 � 1 + 2qΘ. Taking the divergence on both sides of equation (17), (27) yields: Te dimensionless equations for stress forces may be z e 2 2 2 2 2 2 simplifed as follows: ρ􏼐1 − ξ ∇ 􏼑 �(2μ + λ)∇ e − β ∇ Θ − β ∇ C − δ ∇ N. 1 2 n 􏽰������ � zt u −1 + 1 + 2qΘ 2 2 􏼐1 − ξ ∇ 􏼑σ � e +(β − 1) − 􏼠 􏼡 − N − C, (24) rr r q 􏽰������ � Te equation for mass difusion may be shortened to u −1 + 1 + 2qΘ 2 2 􏼐1 − ξ ∇ 􏼑σ � βe +(1 − β) − 􏼠 􏼡 − N − C, ψψ z z r q 2 2 2 D β ∇ e + D c∇ Θ + 􏼠 + τ 􏼡C − D b∇ C � 0. c 2 c d c zt 􏽰������ � zt −1 + 1 + 2qΘ 2 2 􏼐1 − ξ ∇ 􏼑σ � β e − 􏼠 􏼡 − C − N, (25) zz For simplicity, the dimensionless variables will be rep- (28) ′ ′ ∗ ′ ′ ′ resented as follows: (r , u ,ξ ) � (r, u,ξ)/C t , (t ,τ ,τ ) � T 0 d ∗ ′ where q � K t /D ρτC , q � K /D ρC , ε � β T ′ 1 0 E e 2 0 E e 1 1 0 (t,τ ,τ )/t , Θ � β Θ/2μ + λ, σ � δ σ /2μ + λ, C � β C/ 0 d 1 ij ij ij 2 ∗2 ∗ ∗ ′ ′ ′ t /K ρ, ε � α E t /d ρτK C , ε � d κt /α ρC D , ε � 0 2 T g n 0 e 3 n T e E 4 2μ + λ, N � δ N/2μ + λ, P � P/β , and T � β n 2 1 2 2 ε c C /β K β , C � 2μ + λ/ρ, δ � (2μ + 3λ)d , 3 T 1 0 2 T n n (T − T )/2μ + λ. ∗ 2 2 2 2 t � K /ρC C , β � λ/2μ + λ, q � bρC /β , q � cρC /β β , 0 e 3 4 1 2 T T 2 T According to dimensionless variables, the governing ∗ 2 and q � (2μ + λ)t C /Dβ . 5 T 2 equations (22)–(25) and the chemical potential equation To solve this problem in Laplace transform domain, the have the following form (drop the dash): initial conditions should be taken mathematically as follows: 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 zσ (r, t)􏼌 zP(r, t)􏼌 zΘ(r, t)􏼌 rr 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 σ (r, t)􏼌 � � 0 P(r, t)| � � 0, Θ(r, t)| � � 0, 􏼌 􏼌 􏼌 rr t�0 t�0 t�0 􏼌 􏼌 􏼌 zt zt zt t�0 t�0 t�0 (29) 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 zC(r, t)􏼌 ze(r, t)􏼌 zN(r, t)􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 C(r, t)| � � 0, e(r, t)| � � 0, N(r, t)| � � 0. 􏼌 􏼌 􏼌 t�0 t�0 t�0 􏼌 􏼌 􏼌 zt zt zt t�0 t�0 t�0 Advances in Condensed Matter Physics 5 3. The Solution in the Laplace Domain 􏼐∇ − α 􏼑N + ε Θ � 0, (31) 1 3 It is possible to express the Laplace transform with pa- ∇ − s Θ + ε N − ε se − ε q C � 0, (32) 􏼐 􏼑 rameter s as follows: 2 1 4 6 −st 2 2 L(Ψ(r, t)) � Ψ(r, s) � 􏽚 Ψ(r, t)e dt. (30) 􏼐∇ − Ω􏼑e − ϖ∇ (Θ + N + C) � 0, (33) 2 2 2 Using the initial conditions equation (29) and per- 􏼐∇ − q 􏼑C − q ∇ e − q ∇ Θ � 0. (34) 6 7 8 forming the Laplace transform including both sides of all mathematical models after a small amount of modifcation, Te stress components relations equations (28)–(30) can we get be represented as follows: 􏽱������ � −1 + 1 + 2qΘ 2 2 ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ 􏼐1 − ξ ∇ 􏼑σ � e +(β − 1) − − N − C, rr r q 􏽱������ � −1 + 1 + 2qΘ 2 2 ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ (35) 􏼐1 − ξ ∇ 􏼑σ � βe +(1 − β) − − N − C, ψψ r q 􏽱������ � −1 + 1 + 2qΘ 2 2 ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ 􏼐1 − ξ ∇ 􏼑σ � β e − − C − N, zz 8 6 4 2 􏼐∇ − Ε ∇ + Ε ∇ − Ε ∇ + Ε 􏼑􏼈e, Θ, N, C􏼉(r, s) � 0. 1 2 3 4 where α � q + sq , q � q (s + τ s )/q , q � 1/q , 1 1 2 6 5 d 3 7 3 (36) 2 2 q � q /q , Ω � s ϖ, and ϖ � 1/1 + s ξ . 8 4 3 Eliminating the set of equations (30)–(33) yields the However, the main coefcients of equation (36) are [28] following expressions for the physical felds e(r, s), N(r, s), as follows: C(r, s), and Θ(r, s) as follows: − 1 Ε � 􏼈Ω + q + s 1 − q + ε 1 + q 􏼁 􏼁 + ε q q + q 􏼁 + ϖα 1 − q 􏼁 􏼉 1 − q 􏼁 , 1 6 7 1 8 4 8 7 8 1 7 7 2 − 1 Ε � 􏽮Ω􏼐q + ε q 􏼑 + s Ω + q 1 + ε 􏼁 􏼁 + ϖα g + g 􏽯 1 − q 􏼁 , 2 6 4 8 7 1 1 1 2 7 (37) − 1 Ε � 􏼈ε sq 1 + ε 􏼁 − ε ε Ω + q 􏼁 + ϖg 􏼉 1 − q 􏼁 3 1 6 3 2 3 6 3 7 − 1 Ε � Ωq α s − ϖε ε 1 − q , 􏼈 􏼁 􏼉 􏼁 4 6 1 2 3 7 ⎫ ⎪ Θ(r, s) � 􏽘λ (s)I k r where g � 􏼈s + q + s(1 − q ) + ε s(1 + q ) + ε q (q + 􏼁 i 0 i 1 6 7 1 8 4 8 7 i�1 q )}, g � 􏼈ε ε (q − 1) + ε s(ε + ε q )􏼉, g � s (sq + ε ⎪ 8 2 2 3 7 3 1 4 7 3 6 4 2 2 2 q ) + α (s q + q (s + q )) 8 1 6 8 6 e(r, s) � 􏽘λ (s)I k r 􏼁 ⎪ In factorized form, equation (36) takes the following i 0 i ⎪ i�1 form: , (i � 1, 2, 3, 4), (40) 2 2 2 2 2 2 2 2 ⎪ ∇ − k ∇ − k ∇ − k ∇ − k e, Θ, N, C (r, s) � 0, 􏼐 􏼑􏼐 􏼑 􏼐 􏼑􏼐 􏼑􏼈 􏼉 ″ ⎪ 1 2 3 4 N(r, s) � 􏽘λ (s)I k r i 0 i i�1 (38) ⎪ where k (n � 1, 2, 3, 4) represent the roots of the following C(r, s) � 􏽘λ (s)I k r􏼁 0 i ⎭ characteristic equation: i�1 8 6 4 2 where I is the zero-order, frst-kind modifed Bessel k − Ε k + Ε k − Ε k + Ε � 0. (39) 1 2 3 4 ′ ″ function. Te unknown parameters are λ ,λ ,λ and i i i For linearity, the solution to equation (36) when r ⟶ 0 λ (i � 1, 2, 3, 4), which are derived from the prepared cir- may be expressed as follows: cular plate and parameter s. From equations (30)–(33) and 6 Advances in Condensed Matter Physics using equation (40), the relationship between these un- where, g � α + q , g � α q − ε ε , g � ε q + q q q , 4 1 6 5 1 6 2 3 6 1 6 4 6 8 known parameters may be determined as follows: g � ε q q , g � g + q + ε q q , g � g + g q ϖ+ 7 1 6 7 8 4 7 4 6 9 9 5 4 7 ε q q α , g � g + (2α + ε )g + ϖg + Ω, g � (2α g + 6 4 2 4 6 9 1 10 7 1 3 6 8 11 1 7 k − g k + g k − g q 2 2 i 8 i 9 i 5 7 α g ) + ε ϖ(α g + g ) + g + Ωg g � (α + ε α )g + λ (s) � λ (s), 6 3 1 6 7 9 8 12 3 1 7 1 1 i 2 2 i 􏼐k − α 􏼑􏼐g k − g 􏼑 g q + Ωg . i 1 6 i 7 5 7 9 Complete analytical solutions for the other main vari- ables are as follows: λ (s) � − λ (s), i i k − α 􏼐 􏼑 i 1 8 6 4 2 g − 1􏼁 k − g k + g k − g k + Ωg q 6 i 10 i 11 i 12 i 5 7 λ (s) � λ (s), i 2 2 2 2 k 􏼐k − α 􏼑 􏼐g k − g 􏼑 1 6 7 i i i (41) 4 6 4 2 k − g k + g k − g q i 8 i 9 i 5 7 e(r, s) � 􏽘 λ (s)I k r , i 0 i 2 2 k − α g k − g 􏼐 􏼑􏼐 􏼑 i�1 i 1 6 i 7 N(r, s) � − 􏽘 λ (s)I k r 􏼁 , (42) i 0 i 􏼐k − α 􏼑 i�1 1 4 8 6 4 2 g − 1 k − g k + g k − g k + Ωg q 6 i 10 i 11 i 12 i 5 7 C(r, s) � 􏽘 λ (s)I k r􏼁 . i 0 i 2 2 2 k 􏼐k − α 􏼑 􏼐g k − g 􏼑 i�1 1 6 7 i i i Using Laplace transform, the displacement component From equations (27) and (29), which may be written as may be derived from equations (21) and (42) as follows: follows, one can see the components of the radial stress and the chemical potential: 6 4 2 k − g k + g k − g q 8 9 5 7 i i i u(r, s) � 􏽘 λ (s)I k r􏼁 . (43) i 1 i 2 2 􏼐k − α 􏼑􏼐g k − g 􏼑 i�1 1 6 7 i i 􏽱��������������� 8 6 4 2 4 ⎧ ⎪ −1 + 1 + 2qλ (s)I k r 􏼁 􏽨􏼐α k − α k + α k − α k + α 􏼑λ (s)I k r􏼁 􏽩 i 0 i 2 i 3 i 4 i 5 i 6 i 0 i ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎝ ⎠ σ (r, s) � 􏽘 − rr ⎪ 2 2 2 2 2 2 2 2 q􏼐1 − ξ k 􏼑 i�1 k 􏼐1 − ξ k 􏼑􏼐k − α 􏼑 􏼐g k − g 􏼑 i i i 1 6 i 7 6 4 2 (β − 1)􏽮k − g k + g k − g q 􏽯 ⎫ ⎬ ⎫ ⎬ 8 9 5 7 i i i ⎤ ⎥ + λ (s)I k r 􏼁 , i 1 i 2 2 2 2 ⎭ ⎭ r􏼐1 − ξ k 􏼑􏼐g k − g 􏼑􏼐k − α 􏼑 i 6 i 7 i 1 (44) 8 6 4 2 4 ⎧ ⎪ ⎫ ⎪ ⎨ 􏼐α k − α k + α k − α k + α 􏼑 ⎪ 7 i 8 i 9 i 10 i 11 ⎪ P(r, s) � 􏽘 λ (s)I k r − 􏼁 ⎪ i 0 i ⎪ 2 ⎪ 2 2 2 ⎩ ⎪ k 􏼐k − α 􏼑 􏼐g k − g 􏼑 ⎪ i�1 i i 1 6 i 7 ⎪ 􏽱��������������� ⎫ ⎪ ⎪ −1 + 1 + 2qλ (s)I k r􏼁 ⎬ ⎪ i 0 i ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎪ ⎝ ⎠ q , ⎪ ⎭ ⎭ where, α � 2 − g , α � g + ϖg , α � g + ε g + g , Based on equation (28), the temperature in the Laplace 2 6 3 9 10 4 9 3 6 11 α � g g + ε g + ϖg , α � Ωg q , α � q (g − 2), transform domain may be expressed as follows: 5 5 7 3 7 12 6 6 7 7 3 6 􏽱������ � α � q g + g , α � g q + ϖg , α � g q + g q , 8 3 10 8 9 11 3 9 10 12 3 5 7 1 T � [ 1 + 2qΘ − 1]. (45) α � Ωg q q . 11 5 3 7 q Advances in Condensed Matter Physics 7 and assuming thermal shock as the thermal load, we obtain 4. Boundary Conditions [39, 40]: To establish the unknown parameters λ , mechanical forces (i) Nonmechanical loads are traction-free loads, which and thermal loads will be applied to the nonlocal semi- can be written as follows: conductor medium’s free surface (where a is the radius of the circular plate), which is initially at rest. Te non- σ (a, s) � 0. (46) rr mechanical loads are thus assumed to be traction-free at the cylinder’s surface. Using Laplace transform on both sides Hence, 􏽱��������������� 8 6 4 2 4 ⎧ ⎪ −1 + 1 + 2qλ (s)I k a􏼁 ⎨ 􏽨􏼐α k − α k + α k − α k + α 􏼑λ (s)I k a􏼁 􏽩 i 0 i 2 i 3 i 4 i 5 i 6 i 0 i ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ 􏽘 − ⎪ 2 2 2 2 2 2 2 2 q􏼐1 − ξ k 􏼑 i�1 k 􏼐1 − ξ k 􏼑􏼐k − α 􏼑 􏼐g k − g 􏼑 i i i 1 6 i 7 (47) 6 4 2 (β − 1)􏽮k − g k + g k − g q 􏽯 ⎫ ⎬ ⎫ ⎬ 8 9 5 7 i i i ⎤ ⎦ + λ (s)I k a􏼁 � 0 2 2 2 2 i 1 i ⎭ ⎭ r􏼐1 − ξ k 􏼑􏼐g k − g 􏼑􏼐k − α 􏼑 i 6 i 7 i 1 (ii) Te thermal state is considered a thermal shock When the chemical potential is provided as a known when: function of time and the carriers’ intensities can be obtained using a recombination process, the surface Θ(a, s) � T L(s). (48) boundary conditions are determined. (iii) Tis is how the chemical potential is written as Terefore, follows: 􏽘λ (s)I k a􏼁 � . (49) P(a, s) � P χ(s), (50) i 0 i 0 i�1 which yields: 􏽱��������������� 8 6 4 2 4 ⎧ ⎪ ⎫ ⎪ −1 + 1 + 2qλ (s)I k a􏼁 ⎨ 􏼐α k − α k + α k − α k + α 􏼑 ⎬ i 0 i P 7 8 9 10 11 i i i i ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎝ ⎠ 􏽘 λ (s)I k a􏼁 − q � . (51) 2 i 0 i 4 ⎪ ⎪ 2 2 2 ⎩ q ⎭ s k 􏼐k − α 􏼑 􏼐g k − g 􏼑 i�1 1 6 7 i i i (iv) Te recombination-restricted possibility of carrier- inversion approach [39], the inverse of any function ϑ(t) in free charge density at the cylinder surface is the Laplace domain may be expressed as follows: expressed as follows: n+i∞ − 1 ϑ(r, t) � L 􏽮ϑ(r, s)􏽯 � 􏽚 exp(st)ϑ(r, s)ds, 2πi n−i∞ N(a, s) � ζ(s), (52) e (54) where s � n + im (n, m ∈ R), in this case, equation (55) can which leads to be represented as follows: λ (s)I k a 􏼁 ƛ i 0 i 􏽘 � − , (53) exp(nt) sε D (55) ϑ(r, t) � 􏽚 exp(imt)ϑ(r, n + im)dm. 􏼐k − α 􏼑 3 e i�1 i 1 2π −∞ Fourier series can be utilized to expand the function where ƛ is a constant. On the other hand, the − nt ′ quantities L(t), ζ(s), and χ(t) represent the e ϑ(r, t) during the closed interval [0, 2t ], yields Heaviside unit step function [34, 35]. nt N e 1 ikπ ⎡ ⎣ ⎤ ⎦ ϑ(r, t) � − Re ϑ(r, n) + Re 􏽘 (− 1) ϑ r, n + , 􏼠 􏼡 t 2 t 5. The Numerical Inversion of the k�1 Laplace Transforms (56) √�� � Using the inversion of the Laplace transform, a full solution where i � −1 and Re is the real part. N is a large fnite integer that can be chosen for free. in the time domain was found. Using the numerical 8 Advances in Condensed Matter Physics 1 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0 0 13 912 15 1 510 15 Radial distance (r) Radial distance (r) q=0.0 q=0.0 q= - 0.5 q= - 0.5 -10 -20 -5 1 5 10 15 1 510 15 Radial distance (r) Radial distance (r) q=0.0 q=0.0 q= - 0.5 q= - 0.5 0.4 2.5 0.2 1.5 0.5 -0.2 -0.5 -0.4 1 510 15 1 5 10 15 Radial distance (r) Radial distance (r) q=0.0 q=0.0 q= - 0.5 q= - 0.5 Figure 1: Te variations of the main physical felds against the redial distance at diferent values variable thermal conductivity according to nonlocal semiconductor medium. characteristics of isotropic nonlocal silicon medium, the variable 6. Numerical Results and Discussions thermal conductivity and difusion relaxation time were in- To show the impact of linearly varying thermal conductivity vestigated as a function of temperature [41–44]: λ � 10 2 10 2 3 (which is dependent on the heat), simulations and theo- 3.64x 10 N/m , μ � 5.46 × 10 N/m , ρ � 2330 kg/m , T � − 5 − 31 3 retical discussions are conducted using silicon (n-type) as an 800 K, a � 1, τ � 5x 10 s, d � −9 x 10 m , D � d n E − 3 2 2.5 × 10 m /s, E � 1.11 eV, ƛ � 2 m/s, elastic nonlocal semiconductor medium. Using the physical Stress (σ ) Temperature (T) rr Strain (e) Carrier density (N) Concentration (C) Chemical potential (P) Advances in Condensed Matter Physics 9 1 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0 0 1369 12 15 1 5 10 15 Radial distance (r) Radial distance (r) Local medium (ξ=0.0) Local medium (ξ=0.0) Non-Local medium (ξ=0.2) Non-Local medium (ξ=0.2) 30 4 -1 -10 -2 -20 1 5 10 15 1 5 10 15 Radial distance (r) Radial distance (r) Local medium (ξ=0.0) Local medium (ξ=0.0) Non-Local medium (ξ=0.2) Non-Local medium (ξ=0.2) 0.5 2.5 0.4 2 0.3 1.5 0.2 1 0.1 0.5 0 0 -0.1 -0.5 1 5 10 15 1 5 10 15 Radial distance (r) Radial distance (r) Local medium (ξ=0.0) Local medium (ξ=0.0) Non-Local medium (ξ=0.2) Non-Local medium (ξ=0.2) Figure 2: Te variations of the main physical felds against the redial distance r at diferent values nonlocal parameters with variable thermal conductivity. Strain (e) Temperature (T) Stress (σ ) rr Carrier density (N) Chemical potential (P) Concentration (C) 10 Advances in Condensed Matter Physics − 6 − 1 − 4 3 4 of the main physical quantities seem to exhibit the same α � 4.14x 10 K , α � 1.98x 10 m /kg, c � 1.2x 10 t c 2 − 4 m/Ks , t � 7 × 10 s b � 0.9, C � 695 J/(kg K), D � pattern for various nonlocal parameters. With increasing e c − 8 3 values, the movements of elastic-thermal-plasma- 0.85 × 10 kg s/m . Te real part of the fundamental physical felds is taken mechanical waves are dampened to achieve chemical into consideration when the wave propagation distributions equilibrium. Tese subfgures illustrate that the nonlocal are represented graphically. parameter has a signifcant efect on each of the investigated Figure 1 (consisting of six subfgures) illustrates the distributions. change of physical quantities in this phenomenon versus radial distance for two cases of thermal conductivity that 7. Conclusion vary with distinct values. Te frst case represented by soiled lines refers to the issue of (heat) temperature independence Te efects of changing thermal conductivity and nonlocal q � 0.0. Te second instance is depicted by dashed lines and parameters on the photothermal excitation process and the represents the condition of temperature dependency chemical activity of elastic semiconductor materials have q � −0.5. In response to the boundary conditions, the dis- been investigated. Te model was constructed in one di- tributions of carrier density (plasma), strain (elastic), mension using the Laplace transform according to cylin- chemical potential, concentration (mass difusion), and drical coordinates. Graphs show the infuence of variable temperature (thermal) began with a positive value at the thermal conductivity and nonlocal parameters. Te nu- surface. But the distribution of redial stress begins at zero, merical fndings indicate that the change in thermal con- indicating that traction is free at this surface r � a � 1. Te ductivity has a signifcant impact on the thermal-elastic- frst subfgure depicts the variation in temperature versus mechanical-plasma behavior of nonlocal semiconductor radius r for various nonlocal parameter values (two cases). It medium during photo-electronic deformations. A small is evident from this subfgure that the temperature increases change in the nonlocal parameter has a great infuence and as the radius increases in the frst range due to the thermal leads to diferences in thermal-elastic-mechanical-plasma efect of light beams to reach the maximum value, and the wave propagation in the elastic medium. Tus, the non- exponential decreases until it agrees with the zero line. Tis local parameter’s ability to conduct and transfer thermal subfgure indicates that the variable thermal conductivity energy may serve as an additional identifer. Various uses of infuences the temperature change. Te second subfgure the variable thermal conductivity of nonlocal semiconductor shows the propagation of plasma waves with increasing elastic media in current physics via photo-elastic-thermal- radial distance for two diferent values of the variable difusion excitation processes are applied in many industries. thermal conductivity. It is clear that the carrier density In particular, mass and heat transfer mechanisms are im- distribution starts with a positive value increases slightly to portant in photovoltaic cells, display technologies, opto- reach the maximum value, and then decreases exponentially electronic applications, and photoconductor devices. until it reaches equilibrium by difusion within the nonlocal semiconductor material, following the zero line. From the Nomenclature frst and second subfgures, it is clear that the theoretical numerical results obtained in this work are consistent with λ,μ: Lame’s parameters the experimental results [45]. Te third and fourth sub- N : Equilibrium carrier concentration fgures were produced to study the nonlocal strain and δ : Te diference in deformation potential chemical potential variation against the radius r for varying θ � T − T : Termodynamical temperature thermal conductivity. As shown in the fourth subfgure, the T: Absolute temperature nonlocal chemical potential begins at a positive value at the T : Reference temperature and boundary plane for all boundary-satisfying situations. |(T − T )/T | < 1 0 0 However, the distribution of radial nonlocal stress (ffth β � (3λ + 2μ)α : Te volume thermal expansion 1 T subfgure) begins at zero, indicating that traction is free near σ : Components of the stress tensor ij the surface, and then begins to rise to its maximum value ρ: Te density of the medium before decreasing quickly and convergently to zero as the α : Te coefcient of linear thermal distance increases to reach the equilibrium state. Te con- expansion centration distribution begins with a positive value at the e: Cubical dilatation beginning and then drops gradually with exponential be- τ : Te difusion relaxation time havior to reach the zero-state line. A slight variation in C : Specifc heat at a constant strain of the linearly variable thermal conductivity has a signifcant efect solid plate on the wave propagation behavior, as shown by these ρC /K � 1/k: Te thermal viscosity e 0 subfgures. D : Te carrier difusion coefcient Figure 2 depicts the variation of the principal variables τ: Te photogenerated carrier lifetime (distributions of the carrier density (plasma waves), the E : Te energy gap of the semiconductor concentration (difusion), the strain (elastic waves), the κ � zN /zTT/τ: Te thermal activation coupling temperature (thermal waves), and the radial stress (me- parameter chanical waves)) as a function of radial distance r for varying c: Measure the efect of thermoelastic nonlocal parameter values. We observe that the distributions difusion Advances in Condensed Matter Physics 11 [10] A. C. 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Photo Thermal Diffusion of Excited Nonlocal Semiconductor Circular Plate Medium with Variable Thermal Conductivity

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Abstract

Hindawi Advances in Condensed Matter Physics Volume 2023, Article ID 1106568, 12 pages https://doi.org/10.1155/2023/1106568 Review Article Photo Thermal Diffusion of Excited Nonlocal Semiconductor Circular Plate Medium with Variable Thermal Conductivity 1 2,3 4,5 2,3 Shreen El-Sapa, Khaled Lotfy , Alaa A. El-Bary, and M. H. Ahmed Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt Department of Mathematics, Faculty of Science, Taibah University, Madinah, Saudi Arabia Arab Academy for Science, Technology and Maritime Transport, P.O. Box 1029, Alexandria, Egypt Council of Future Studies and Risk Management, Academy of Scientifc Research and Technology, Cairo, Egypt Correspondence should be addressed to Khaled Lotfy; khlotfy_1@yahoo.com Received 18 November 2022; Revised 12 January 2023; Accepted 17 April 2023; Published 26 April 2023 Academic Editor: Sergio Ulloa Copyright © 2023 Shreen El-Sapa et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. To examine the efects of the nonlocal thermoelastic parameters in a nanoscale semiconductor material, a novel nonlocal model with variable thermal conductivity is provided in this study. Te photothermal difusion (PTD) processes in a chemical action are utilized in the framework of the governing equations. When elastic, thermal, and plasma waves interact, the nonlocal continuum theory is used to create this model. For the main formulations to get the analytical solutions of the thermal stress, displacement, carrier density, and temperature during the nanoscale thermo-photo-electric medium, the Laplace transformation approach in one dimension (1D) of a thin circular plate is utilized. To create the physical felds, mechanical forces and thermal loads are applied to the semiconductor’s free surface. To acquire the full solutions of the research areas in the time-space domains, the inverse of the Laplace transform is applied with several numerical approximation techniques. Under the impact of nonlocal factors, the principal physical felds are visually depicted and theoretically explained. presence of mass difusion, particularly in the aerospace, 1. Introduction electronics, and integrated circuit industries. High- Nanotechnology is currently and in the future will be one of performance nanostructures, such as nanotubes, nano- the most crucial cornerstones of human existence. Tis flms, and nanowires, have been extensively used as reso- signifcant technology is expanding quickly, and several nators, probes, sensors, transistors, actuators, etc. with the scientists are engaged in this fascinating sector. Several of the fast development of nanomechanical electromechanical physical characteristics of elastic materials may vary systems (NEMS) technologies. It is crucial to comprehend the precise characterizations of these nanostructures’ ther- depending on the temperature. Many difculties arise in researching elastic materials without taking varying heat mal and mechanical characteristics. conductivity into account. When thermal conductivity Semiconductor materials (such as silicon) are an ex- varies, particularly in response to temperature, it becomes cellent research subjects for this phenomenon, particularly essential. Termo-difusion is the relationship between mass when subjected to laser or falling light beams. On the difusion and changing thermal conductivity. Termo- surface, the excited electrons will produce a charge known as difusion happens when particles move from an area of free carriers (plasma waves). According to the quantity of greater concentration to an area of lower concentration as light descending, the plasma density is employed to regulate a result of a temperature change. Modern engineering has the difusion [1–3]. Numerous publications [4–6] failed to several uses for the study of thermal conductivity in the take into account the coupling between thermal-elastic 2 Advances in Condensed Matter Physics waves and plasma waves during the deformation process in waves), carrier density (plasma waves), and difusive ma- semiconductor materials. Recently, several authors terial concentration (mass difusion). When the thermal employed photoacoustic spectroscopy to detect photo- activation coupling value κ for the nonlocal medium is thermal events when a laser beam struck a semiconductor nonzero, the photothermal difusion transport process takes [7, 8]. Semiconductors’ temperature, carrier intensity, and place. It makes use of cylindrical coordinates (r,ψ, z). When thermal difusion are measured using the photothermal a very thin circular plate is taken into account, all quantities phenomena [9–13]. When thermal waves propagate, gen- are independent of ψ and z because of the symmetry of the erating elastic oscillation, and plasma waves are formed by axis z. Elastic-plasma-thermal-difusion wave overlapping photo-excited free carriers, directly creating a periodic processes’ governing equations are presented as [34, 35], the elastic deformation as well [14–16], the interaction between photo-electronic equation is as follows: the elastic-thermal-plasma waves occurs. Without consid- zN(r, t) N(r, t) (1) ering the impact of changing thermal conductivity, several � D N (r, t) − + κ T(r, t). E ,ii zt τ issues in generalized thermoelasticity have been explored [17–25]. Later, a lot of writers studied generalized ther- Equations for thermal difusion in the photothermal moelasticity in many areas using variable thermal con- difusion process transport are as follows: ductivity. Te thermal-mechanical behavior of the medium z K may be afected by the deformation of elastic media 􏼐KT (r, t)􏼑 � 􏼒 T(r, t) + β T u (r, t) + cT C􏼓 ,i 1 0 ,i 0 ,i zt k depending on temperature [26–28]. Abbas [29–33] studied (2) many problems of the fber-reinforced anisotropic ther- moelastic medium in two dimensions with fractional − N(r, t). transient heating according to many mathematical methods. Te nonlocal thermoelastic model with variable thermal If there is no body force, the equations of motion for conductivity (which may be considered as a linear function nonlocal medium may be expressed as follows[34]: of temperature) is utilized in the current study using z u 2 2 a theoretical method. Te process of photo-thermal- ρ 􏼐1 − ξ ∇ 􏼑 � μu (r, t) +(μ + λ) u (r, t) i,jj i,jj zt difusion interactions in semiconductor nanoscale media is investigated. Te variation in temperature caused by the − β T (r, t) − β C (r, t) − δ N (r, t). 1 ,i 2 ,i n ,i light beam impacting the nonlocal semiconductor medium is the basis for the variable thermal conductivity. Te (3) chemical difusion method enables photothermal transfer Te length-related elastic nonlocal parameter is repre- (mass difusion). When the Laplace transform domain in sented by ξ � ae /l (l is the external characteristic length cylindrical coordinates is utilized, the analytical solutions of scale, a is the internal characteristic length, and e is non- the basic felds are found. Te numerical techniques provide dimensional material property). analytical solutions in the Laplace domain without any Te mass difusion equation is expressed as follows [35]: presumptive limitations on the real physical values. Finally, with changes in nonlocal parameters and changing thermal z z D β e + D cT (r, t) + 􏼠 + τ 􏼡C(r, t) conductivity, the numerical calculations of the important c 2 nn,ii c ,ii d 2 zt zt (4) physical quantities distribution are graphically shown and discussed. Te numerical fndings presented in the current � D bC (r, t). c ,ii study have applications in solid mechanics, acoustics, ma- terial science, and engineering for earthquakes. Te change in thermal conductivity is K of the nonlocal semiconductor medium and β � (3λ + 2μ)α where α is 2 c c 2. Formulation of the Problem and the coefcient of linear difusion. On the other hand, the Basic Equations transport heat coefcients for the nonlocal medium are independent of N, C and T [36–38]. Te four important variables in this problem, respectively, Te strain-stress combinations are as follows: are u(r, t), T(r, t), N(r, t), and C(r, t) which stand in for the displacement (elastic waves), temperature (thermal or heat zu 2 2 1 − ξ ∇ σ � 2μ + λ e − β T − T − β C +(3λ + 2μ)d N, 􏼐 􏼑 􏼁 rr 1 0 2 n zr 2 2 (5) 􏼐1 − ξ ∇ 􏼑σ � 2μ + λ e − β T − T 􏼁 − β C +(3λ + 2μ)d N, ψψ 1 0 2 n 2 2 􏼐1 − ξ ∇ 􏼑σ � λ e − β T − T 􏼁 − β C +(3λ + 2μ)d N, σ � σ � σ � 0. zz 1 0 2 n rψ zψ rz Advances in Condensed Matter Physics 3 Te nonlocal semiconductor medium’s chemical po- K Θ � K(T)T , ⎫ ⎬ 0 ,i ,i tential equation is (9) K Θ � 􏼐K(T)T 􏼑 . 0 ,ii ,i ,i P � −β e + bC − c T − T 􏼁 . (6) 2 nn 0 Another form of equation (9) when the nonlinear terms where P is the chemical potential per unit mass. are neglected can be obtained as follows: It is possible to choose a material’s variable thermal K Θ � K T + KT � K 1 + K T􏼁 T + KT conductivity K, which may be estimated as a linear function 0 ,ii ,i ,i ,ii 0 1 ,i ,ii ,i (10) of temperature [26]: 2 � K K 􏼐T 􏼑 + KT � KT . 0 1 ,i ,ii ,ii K(T) � K (1 + qT), (7) Te time-diferentiation is done in the same manner to where q is a negative parameter and K is a thermal con- both sides of equation (7), resulting in: ductivity when q � 0 (the nonlocal medium is independent zΘ zT of temperature). (11) K � K(T) . zt zt Te map of temperature can be taken in the following form [27]: Using equation (8) and diferentiating equation (1) by z/zx , yields: (8) Θ � 􏽚 K(R)dR. z 1 κK (12) K N � D N − N + Θ . 0 􏼐 􏼑 0 ,i E ,mm ,i ,i ,i zt τ K Diferentiating both sides of equation (7) relative to x , Te other form of the quantity κK /KΘ with neglected 0 ,i we get the nonlinear term can be represented as follows: κK − 1 2 0 ⎫ ⎪ Θ � κ 1 + K T Θ � κ 1 − K T + K T − . . . Θ � 􏼁 􏼐 􏼁 􏼑 ,i 1 ,i 1 1 ,i K 1 + K T􏼁 0 1 (13) κΘ − κK TΘ + K T􏼁 Θ − . . . � κΘ . ,i 1 ,i 1 ,i ,i z u 2 2 ρ 􏼐1 − ξ ∇ 􏼑 � μu +(μ + λ) u Equation (1) results when equation (13) is applied: i,jj i,jj zt (17) z 1 (14) N � D 􏼐N 􏼑 − N + κΘ . ,i E ,mm ,i ,i ,i − β Θ − β C − δ N . zt τ 1 ,i 2 ,i n ,i Integrating equation (14), yields: Te equation for mass difusion equation (4) may be expressed as follows: zN 1 (15) � D N − N + κΘ. E ,ii 2 zt τ D cK z z c 0 D β e + Θ (r, t) + 􏼠 + τ 􏼡C(r, t) c 2 nn,ii ,ii d K zt zt Under the infuence of mapping, the heat (thermal) (18) difusion equation (2) have the following form: � D bC (r, t). c ,ii 1 zΘ β T zu cT zC i 1 0 ,i 0 i Θ � + + − N . (16) ,ii i Te term D cK /KΘ (r, t) can be represented with k zt K zt K zt K τ c 0 ,ii 0 0 0 neglected nonlinear terms in the following form: Te nonlocal motion equation (3) under the temperature map may be simplifed as follows: D cK D cK − 1 2 c 0 c 0 ⎪ Θ � Θ � D c 1 + K T􏼁 Θ � D c􏼐1 − K T + K T􏼁 − .......􏼑Θ � ,ii ,ii c 1 ,ii c 1 1 ,ii ⎪ K K 1 + K T􏼁 0 1 (19) 2 2 ⎭ D c􏼐1 − K T + K T􏼁 − .......􏼑Θ � D cΘ − D cK TΘ + D c K T􏼁 Θ − ....... � D cΘ , c 1 1 ,ii c ,ii c 1 ,ii c 1 ,ii c ,ii In this case, equation (18) can be rewritten as follows: 4 Advances in Condensed Matter Physics 2 2 z z z 1 z z D β e + D cΘ (r, t) + 􏼠 + τ 􏼡C(r, t) 􏼠 + − q − q 􏼡N + ε Θ � 0, c 2 nn,ii c ,ii d 1 2 3 2 2 zt r zr zt zt zr (20) � D bC (r, t). 2 c ,ii ∇ Θ − Θ + ε e + ε C􏼁 + ε N � 0, 1 4 2 zt Te strain in cylindrical 1D form can be represented as follows: z e 2 2 2 ∇ (e − Θ − C − N) − 1 − ξ ∇ � 0, 􏼐 􏼑 zu u zt e � , e � , e � e � e � e rr ψψ rψ ψz zz rz zr r z z (21) ∇ e + q Θ − q C􏼁 + q 􏼠 + τ 􏼡C � 0, 2 4 3 5 d 1 z(ru) z 1 z zt zt � 0, e � ∇ � + . r zr r zr zr P � −e + q C − q T. 3 4 By doing the analysis in the radial direction (r), the (26) problem will be solved in 1D, with the displacement vector having the form u � (u, 0, 0), u(r, t). Using the linear form of variable thermal conductivity Te main governing equations in 1D (radial) are reduced equation (6) and the mapping equation (7), one arrives at the as follows: following result [26]: zN z N 1 zN N 2 1 q q 1 1 (22) � D 􏼠 + 􏼡 − + κ Θ, 2 Θ � 􏽚 K (1 + qT)dT � T + T � T + − , 􏼠 􏼡 zt r zr τ zr K 2 2 q 2q z 1 z 1 zΘ β T ze cT zC 􏽰������ � 1 0 0 1 􏼠 + 􏼡Θ � + + − N. T � [ 1 + 2qΘ − 1], r zr k zt K zt K zt K τ zr 0 0 0 􏽰������ � (23) qT + 1 � 1 + 2qΘ. Taking the divergence on both sides of equation (17), (27) yields: Te dimensionless equations for stress forces may be z e 2 2 2 2 2 2 simplifed as follows: ρ􏼐1 − ξ ∇ 􏼑 �(2μ + λ)∇ e − β ∇ Θ − β ∇ C − δ ∇ N. 1 2 n 􏽰������ � zt u −1 + 1 + 2qΘ 2 2 􏼐1 − ξ ∇ 􏼑σ � e +(β − 1) − 􏼠 􏼡 − N − C, (24) rr r q 􏽰������ � Te equation for mass difusion may be shortened to u −1 + 1 + 2qΘ 2 2 􏼐1 − ξ ∇ 􏼑σ � βe +(1 − β) − 􏼠 􏼡 − N − C, ψψ z z r q 2 2 2 D β ∇ e + D c∇ Θ + 􏼠 + τ 􏼡C − D b∇ C � 0. c 2 c d c zt 􏽰������ � zt −1 + 1 + 2qΘ 2 2 􏼐1 − ξ ∇ 􏼑σ � β e − 􏼠 􏼡 − C − N, (25) zz For simplicity, the dimensionless variables will be rep- (28) ′ ′ ∗ ′ ′ ′ resented as follows: (r , u ,ξ ) � (r, u,ξ)/C t , (t ,τ ,τ ) � T 0 d ∗ ′ where q � K t /D ρτC , q � K /D ρC , ε � β T ′ 1 0 E e 2 0 E e 1 1 0 (t,τ ,τ )/t , Θ � β Θ/2μ + λ, σ � δ σ /2μ + λ, C � β C/ 0 d 1 ij ij ij 2 ∗2 ∗ ∗ ′ ′ ′ t /K ρ, ε � α E t /d ρτK C , ε � d κt /α ρC D , ε � 0 2 T g n 0 e 3 n T e E 4 2μ + λ, N � δ N/2μ + λ, P � P/β , and T � β n 2 1 2 2 ε c C /β K β , C � 2μ + λ/ρ, δ � (2μ + 3λ)d , 3 T 1 0 2 T n n (T − T )/2μ + λ. ∗ 2 2 2 2 t � K /ρC C , β � λ/2μ + λ, q � bρC /β , q � cρC /β β , 0 e 3 4 1 2 T T 2 T According to dimensionless variables, the governing ∗ 2 and q � (2μ + λ)t C /Dβ . 5 T 2 equations (22)–(25) and the chemical potential equation To solve this problem in Laplace transform domain, the have the following form (drop the dash): initial conditions should be taken mathematically as follows: 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 zσ (r, t)􏼌 zP(r, t)􏼌 zΘ(r, t)􏼌 rr 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 σ (r, t)􏼌 � � 0 P(r, t)| � � 0, Θ(r, t)| � � 0, 􏼌 􏼌 􏼌 rr t�0 t�0 t�0 􏼌 􏼌 􏼌 zt zt zt t�0 t�0 t�0 (29) 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 zC(r, t)􏼌 ze(r, t)􏼌 zN(r, t)􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 C(r, t)| � � 0, e(r, t)| � � 0, N(r, t)| � � 0. 􏼌 􏼌 􏼌 t�0 t�0 t�0 􏼌 􏼌 􏼌 zt zt zt t�0 t�0 t�0 Advances in Condensed Matter Physics 5 3. The Solution in the Laplace Domain 􏼐∇ − α 􏼑N + ε Θ � 0, (31) 1 3 It is possible to express the Laplace transform with pa- ∇ − s Θ + ε N − ε se − ε q C � 0, (32) 􏼐 􏼑 rameter s as follows: 2 1 4 6 −st 2 2 L(Ψ(r, t)) � Ψ(r, s) � 􏽚 Ψ(r, t)e dt. (30) 􏼐∇ − Ω􏼑e − ϖ∇ (Θ + N + C) � 0, (33) 2 2 2 Using the initial conditions equation (29) and per- 􏼐∇ − q 􏼑C − q ∇ e − q ∇ Θ � 0. (34) 6 7 8 forming the Laplace transform including both sides of all mathematical models after a small amount of modifcation, Te stress components relations equations (28)–(30) can we get be represented as follows: 􏽱������ � −1 + 1 + 2qΘ 2 2 ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ 􏼐1 − ξ ∇ 􏼑σ � e +(β − 1) − − N − C, rr r q 􏽱������ � −1 + 1 + 2qΘ 2 2 ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ (35) 􏼐1 − ξ ∇ 􏼑σ � βe +(1 − β) − − N − C, ψψ r q 􏽱������ � −1 + 1 + 2qΘ 2 2 ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ 􏼐1 − ξ ∇ 􏼑σ � β e − − C − N, zz 8 6 4 2 􏼐∇ − Ε ∇ + Ε ∇ − Ε ∇ + Ε 􏼑􏼈e, Θ, N, C􏼉(r, s) � 0. 1 2 3 4 where α � q + sq , q � q (s + τ s )/q , q � 1/q , 1 1 2 6 5 d 3 7 3 (36) 2 2 q � q /q , Ω � s ϖ, and ϖ � 1/1 + s ξ . 8 4 3 Eliminating the set of equations (30)–(33) yields the However, the main coefcients of equation (36) are [28] following expressions for the physical felds e(r, s), N(r, s), as follows: C(r, s), and Θ(r, s) as follows: − 1 Ε � 􏼈Ω + q + s 1 − q + ε 1 + q 􏼁 􏼁 + ε q q + q 􏼁 + ϖα 1 − q 􏼁 􏼉 1 − q 􏼁 , 1 6 7 1 8 4 8 7 8 1 7 7 2 − 1 Ε � 􏽮Ω􏼐q + ε q 􏼑 + s Ω + q 1 + ε 􏼁 􏼁 + ϖα g + g 􏽯 1 − q 􏼁 , 2 6 4 8 7 1 1 1 2 7 (37) − 1 Ε � 􏼈ε sq 1 + ε 􏼁 − ε ε Ω + q 􏼁 + ϖg 􏼉 1 − q 􏼁 3 1 6 3 2 3 6 3 7 − 1 Ε � Ωq α s − ϖε ε 1 − q , 􏼈 􏼁 􏼉 􏼁 4 6 1 2 3 7 ⎫ ⎪ Θ(r, s) � 􏽘λ (s)I k r where g � 􏼈s + q + s(1 − q ) + ε s(1 + q ) + ε q (q + 􏼁 i 0 i 1 6 7 1 8 4 8 7 i�1 q )}, g � 􏼈ε ε (q − 1) + ε s(ε + ε q )􏼉, g � s (sq + ε ⎪ 8 2 2 3 7 3 1 4 7 3 6 4 2 2 2 q ) + α (s q + q (s + q )) 8 1 6 8 6 e(r, s) � 􏽘λ (s)I k r 􏼁 ⎪ In factorized form, equation (36) takes the following i 0 i ⎪ i�1 form: , (i � 1, 2, 3, 4), (40) 2 2 2 2 2 2 2 2 ⎪ ∇ − k ∇ − k ∇ − k ∇ − k e, Θ, N, C (r, s) � 0, 􏼐 􏼑􏼐 􏼑 􏼐 􏼑􏼐 􏼑􏼈 􏼉 ″ ⎪ 1 2 3 4 N(r, s) � 􏽘λ (s)I k r i 0 i i�1 (38) ⎪ where k (n � 1, 2, 3, 4) represent the roots of the following C(r, s) � 􏽘λ (s)I k r􏼁 0 i ⎭ characteristic equation: i�1 8 6 4 2 where I is the zero-order, frst-kind modifed Bessel k − Ε k + Ε k − Ε k + Ε � 0. (39) 1 2 3 4 ′ ″ function. Te unknown parameters are λ ,λ ,λ and i i i For linearity, the solution to equation (36) when r ⟶ 0 λ (i � 1, 2, 3, 4), which are derived from the prepared cir- may be expressed as follows: cular plate and parameter s. From equations (30)–(33) and 6 Advances in Condensed Matter Physics using equation (40), the relationship between these un- where, g � α + q , g � α q − ε ε , g � ε q + q q q , 4 1 6 5 1 6 2 3 6 1 6 4 6 8 known parameters may be determined as follows: g � ε q q , g � g + q + ε q q , g � g + g q ϖ+ 7 1 6 7 8 4 7 4 6 9 9 5 4 7 ε q q α , g � g + (2α + ε )g + ϖg + Ω, g � (2α g + 6 4 2 4 6 9 1 10 7 1 3 6 8 11 1 7 k − g k + g k − g q 2 2 i 8 i 9 i 5 7 α g ) + ε ϖ(α g + g ) + g + Ωg g � (α + ε α )g + λ (s) � λ (s), 6 3 1 6 7 9 8 12 3 1 7 1 1 i 2 2 i 􏼐k − α 􏼑􏼐g k − g 􏼑 g q + Ωg . i 1 6 i 7 5 7 9 Complete analytical solutions for the other main vari- ables are as follows: λ (s) � − λ (s), i i k − α 􏼐 􏼑 i 1 8 6 4 2 g − 1􏼁 k − g k + g k − g k + Ωg q 6 i 10 i 11 i 12 i 5 7 λ (s) � λ (s), i 2 2 2 2 k 􏼐k − α 􏼑 􏼐g k − g 􏼑 1 6 7 i i i (41) 4 6 4 2 k − g k + g k − g q i 8 i 9 i 5 7 e(r, s) � 􏽘 λ (s)I k r , i 0 i 2 2 k − α g k − g 􏼐 􏼑􏼐 􏼑 i�1 i 1 6 i 7 N(r, s) � − 􏽘 λ (s)I k r 􏼁 , (42) i 0 i 􏼐k − α 􏼑 i�1 1 4 8 6 4 2 g − 1 k − g k + g k − g k + Ωg q 6 i 10 i 11 i 12 i 5 7 C(r, s) � 􏽘 λ (s)I k r􏼁 . i 0 i 2 2 2 k 􏼐k − α 􏼑 􏼐g k − g 􏼑 i�1 1 6 7 i i i Using Laplace transform, the displacement component From equations (27) and (29), which may be written as may be derived from equations (21) and (42) as follows: follows, one can see the components of the radial stress and the chemical potential: 6 4 2 k − g k + g k − g q 8 9 5 7 i i i u(r, s) � 􏽘 λ (s)I k r􏼁 . (43) i 1 i 2 2 􏼐k − α 􏼑􏼐g k − g 􏼑 i�1 1 6 7 i i 􏽱��������������� 8 6 4 2 4 ⎧ ⎪ −1 + 1 + 2qλ (s)I k r 􏼁 􏽨􏼐α k − α k + α k − α k + α 􏼑λ (s)I k r􏼁 􏽩 i 0 i 2 i 3 i 4 i 5 i 6 i 0 i ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎝ ⎠ σ (r, s) � 􏽘 − rr ⎪ 2 2 2 2 2 2 2 2 q􏼐1 − ξ k 􏼑 i�1 k 􏼐1 − ξ k 􏼑􏼐k − α 􏼑 􏼐g k − g 􏼑 i i i 1 6 i 7 6 4 2 (β − 1)􏽮k − g k + g k − g q 􏽯 ⎫ ⎬ ⎫ ⎬ 8 9 5 7 i i i ⎤ ⎥ + λ (s)I k r 􏼁 , i 1 i 2 2 2 2 ⎭ ⎭ r􏼐1 − ξ k 􏼑􏼐g k − g 􏼑􏼐k − α 􏼑 i 6 i 7 i 1 (44) 8 6 4 2 4 ⎧ ⎪ ⎫ ⎪ ⎨ 􏼐α k − α k + α k − α k + α 􏼑 ⎪ 7 i 8 i 9 i 10 i 11 ⎪ P(r, s) � 􏽘 λ (s)I k r − 􏼁 ⎪ i 0 i ⎪ 2 ⎪ 2 2 2 ⎩ ⎪ k 􏼐k − α 􏼑 􏼐g k − g 􏼑 ⎪ i�1 i i 1 6 i 7 ⎪ 􏽱��������������� ⎫ ⎪ ⎪ −1 + 1 + 2qλ (s)I k r􏼁 ⎬ ⎪ i 0 i ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎪ ⎝ ⎠ q , ⎪ ⎭ ⎭ where, α � 2 − g , α � g + ϖg , α � g + ε g + g , Based on equation (28), the temperature in the Laplace 2 6 3 9 10 4 9 3 6 11 α � g g + ε g + ϖg , α � Ωg q , α � q (g − 2), transform domain may be expressed as follows: 5 5 7 3 7 12 6 6 7 7 3 6 􏽱������ � α � q g + g , α � g q + ϖg , α � g q + g q , 8 3 10 8 9 11 3 9 10 12 3 5 7 1 T � [ 1 + 2qΘ − 1]. (45) α � Ωg q q . 11 5 3 7 q Advances in Condensed Matter Physics 7 and assuming thermal shock as the thermal load, we obtain 4. Boundary Conditions [39, 40]: To establish the unknown parameters λ , mechanical forces (i) Nonmechanical loads are traction-free loads, which and thermal loads will be applied to the nonlocal semi- can be written as follows: conductor medium’s free surface (where a is the radius of the circular plate), which is initially at rest. Te non- σ (a, s) � 0. (46) rr mechanical loads are thus assumed to be traction-free at the cylinder’s surface. Using Laplace transform on both sides Hence, 􏽱��������������� 8 6 4 2 4 ⎧ ⎪ −1 + 1 + 2qλ (s)I k a􏼁 ⎨ 􏽨􏼐α k − α k + α k − α k + α 􏼑λ (s)I k a􏼁 􏽩 i 0 i 2 i 3 i 4 i 5 i 6 i 0 i ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ 􏽘 − ⎪ 2 2 2 2 2 2 2 2 q􏼐1 − ξ k 􏼑 i�1 k 􏼐1 − ξ k 􏼑􏼐k − α 􏼑 􏼐g k − g 􏼑 i i i 1 6 i 7 (47) 6 4 2 (β − 1)􏽮k − g k + g k − g q 􏽯 ⎫ ⎬ ⎫ ⎬ 8 9 5 7 i i i ⎤ ⎦ + λ (s)I k a􏼁 � 0 2 2 2 2 i 1 i ⎭ ⎭ r􏼐1 − ξ k 􏼑􏼐g k − g 􏼑􏼐k − α 􏼑 i 6 i 7 i 1 (ii) Te thermal state is considered a thermal shock When the chemical potential is provided as a known when: function of time and the carriers’ intensities can be obtained using a recombination process, the surface Θ(a, s) � T L(s). (48) boundary conditions are determined. (iii) Tis is how the chemical potential is written as Terefore, follows: 􏽘λ (s)I k a􏼁 � . (49) P(a, s) � P χ(s), (50) i 0 i 0 i�1 which yields: 􏽱��������������� 8 6 4 2 4 ⎧ ⎪ ⎫ ⎪ −1 + 1 + 2qλ (s)I k a􏼁 ⎨ 􏼐α k − α k + α k − α k + α 􏼑 ⎬ i 0 i P 7 8 9 10 11 i i i i ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎝ ⎠ 􏽘 λ (s)I k a􏼁 − q � . (51) 2 i 0 i 4 ⎪ ⎪ 2 2 2 ⎩ q ⎭ s k 􏼐k − α 􏼑 􏼐g k − g 􏼑 i�1 1 6 7 i i i (iv) Te recombination-restricted possibility of carrier- inversion approach [39], the inverse of any function ϑ(t) in free charge density at the cylinder surface is the Laplace domain may be expressed as follows: expressed as follows: n+i∞ − 1 ϑ(r, t) � L 􏽮ϑ(r, s)􏽯 � 􏽚 exp(st)ϑ(r, s)ds, 2πi n−i∞ N(a, s) � ζ(s), (52) e (54) where s � n + im (n, m ∈ R), in this case, equation (55) can which leads to be represented as follows: λ (s)I k a 􏼁 ƛ i 0 i 􏽘 � − , (53) exp(nt) sε D (55) ϑ(r, t) � 􏽚 exp(imt)ϑ(r, n + im)dm. 􏼐k − α 􏼑 3 e i�1 i 1 2π −∞ Fourier series can be utilized to expand the function where ƛ is a constant. On the other hand, the − nt ′ quantities L(t), ζ(s), and χ(t) represent the e ϑ(r, t) during the closed interval [0, 2t ], yields Heaviside unit step function [34, 35]. nt N e 1 ikπ ⎡ ⎣ ⎤ ⎦ ϑ(r, t) � − Re ϑ(r, n) + Re 􏽘 (− 1) ϑ r, n + , 􏼠 􏼡 t 2 t 5. The Numerical Inversion of the k�1 Laplace Transforms (56) √�� � Using the inversion of the Laplace transform, a full solution where i � −1 and Re is the real part. N is a large fnite integer that can be chosen for free. in the time domain was found. Using the numerical 8 Advances in Condensed Matter Physics 1 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0 0 13 912 15 1 510 15 Radial distance (r) Radial distance (r) q=0.0 q=0.0 q= - 0.5 q= - 0.5 -10 -20 -5 1 5 10 15 1 510 15 Radial distance (r) Radial distance (r) q=0.0 q=0.0 q= - 0.5 q= - 0.5 0.4 2.5 0.2 1.5 0.5 -0.2 -0.5 -0.4 1 510 15 1 5 10 15 Radial distance (r) Radial distance (r) q=0.0 q=0.0 q= - 0.5 q= - 0.5 Figure 1: Te variations of the main physical felds against the redial distance at diferent values variable thermal conductivity according to nonlocal semiconductor medium. characteristics of isotropic nonlocal silicon medium, the variable 6. Numerical Results and Discussions thermal conductivity and difusion relaxation time were in- To show the impact of linearly varying thermal conductivity vestigated as a function of temperature [41–44]: λ � 10 2 10 2 3 (which is dependent on the heat), simulations and theo- 3.64x 10 N/m , μ � 5.46 × 10 N/m , ρ � 2330 kg/m , T � − 5 − 31 3 retical discussions are conducted using silicon (n-type) as an 800 K, a � 1, τ � 5x 10 s, d � −9 x 10 m , D � d n E − 3 2 2.5 × 10 m /s, E � 1.11 eV, ƛ � 2 m/s, elastic nonlocal semiconductor medium. Using the physical Stress (σ ) Temperature (T) rr Strain (e) Carrier density (N) Concentration (C) Chemical potential (P) Advances in Condensed Matter Physics 9 1 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0 0 1369 12 15 1 5 10 15 Radial distance (r) Radial distance (r) Local medium (ξ=0.0) Local medium (ξ=0.0) Non-Local medium (ξ=0.2) Non-Local medium (ξ=0.2) 30 4 -1 -10 -2 -20 1 5 10 15 1 5 10 15 Radial distance (r) Radial distance (r) Local medium (ξ=0.0) Local medium (ξ=0.0) Non-Local medium (ξ=0.2) Non-Local medium (ξ=0.2) 0.5 2.5 0.4 2 0.3 1.5 0.2 1 0.1 0.5 0 0 -0.1 -0.5 1 5 10 15 1 5 10 15 Radial distance (r) Radial distance (r) Local medium (ξ=0.0) Local medium (ξ=0.0) Non-Local medium (ξ=0.2) Non-Local medium (ξ=0.2) Figure 2: Te variations of the main physical felds against the redial distance r at diferent values nonlocal parameters with variable thermal conductivity. Strain (e) Temperature (T) Stress (σ ) rr Carrier density (N) Chemical potential (P) Concentration (C) 10 Advances in Condensed Matter Physics − 6 − 1 − 4 3 4 of the main physical quantities seem to exhibit the same α � 4.14x 10 K , α � 1.98x 10 m /kg, c � 1.2x 10 t c 2 − 4 m/Ks , t � 7 × 10 s b � 0.9, C � 695 J/(kg K), D � pattern for various nonlocal parameters. With increasing e c − 8 3 values, the movements of elastic-thermal-plasma- 0.85 × 10 kg s/m . Te real part of the fundamental physical felds is taken mechanical waves are dampened to achieve chemical into consideration when the wave propagation distributions equilibrium. Tese subfgures illustrate that the nonlocal are represented graphically. parameter has a signifcant efect on each of the investigated Figure 1 (consisting of six subfgures) illustrates the distributions. change of physical quantities in this phenomenon versus radial distance for two cases of thermal conductivity that 7. Conclusion vary with distinct values. Te frst case represented by soiled lines refers to the issue of (heat) temperature independence Te efects of changing thermal conductivity and nonlocal q � 0.0. Te second instance is depicted by dashed lines and parameters on the photothermal excitation process and the represents the condition of temperature dependency chemical activity of elastic semiconductor materials have q � −0.5. In response to the boundary conditions, the dis- been investigated. Te model was constructed in one di- tributions of carrier density (plasma), strain (elastic), mension using the Laplace transform according to cylin- chemical potential, concentration (mass difusion), and drical coordinates. Graphs show the infuence of variable temperature (thermal) began with a positive value at the thermal conductivity and nonlocal parameters. Te nu- surface. But the distribution of redial stress begins at zero, merical fndings indicate that the change in thermal con- indicating that traction is free at this surface r � a � 1. Te ductivity has a signifcant impact on the thermal-elastic- frst subfgure depicts the variation in temperature versus mechanical-plasma behavior of nonlocal semiconductor radius r for various nonlocal parameter values (two cases). It medium during photo-electronic deformations. A small is evident from this subfgure that the temperature increases change in the nonlocal parameter has a great infuence and as the radius increases in the frst range due to the thermal leads to diferences in thermal-elastic-mechanical-plasma efect of light beams to reach the maximum value, and the wave propagation in the elastic medium. Tus, the non- exponential decreases until it agrees with the zero line. Tis local parameter’s ability to conduct and transfer thermal subfgure indicates that the variable thermal conductivity energy may serve as an additional identifer. Various uses of infuences the temperature change. Te second subfgure the variable thermal conductivity of nonlocal semiconductor shows the propagation of plasma waves with increasing elastic media in current physics via photo-elastic-thermal- radial distance for two diferent values of the variable difusion excitation processes are applied in many industries. thermal conductivity. It is clear that the carrier density In particular, mass and heat transfer mechanisms are im- distribution starts with a positive value increases slightly to portant in photovoltaic cells, display technologies, opto- reach the maximum value, and then decreases exponentially electronic applications, and photoconductor devices. until it reaches equilibrium by difusion within the nonlocal semiconductor material, following the zero line. From the Nomenclature frst and second subfgures, it is clear that the theoretical numerical results obtained in this work are consistent with λ,μ: Lame’s parameters the experimental results [45]. Te third and fourth sub- N : Equilibrium carrier concentration fgures were produced to study the nonlocal strain and δ : Te diference in deformation potential chemical potential variation against the radius r for varying θ � T − T : Termodynamical temperature thermal conductivity. As shown in the fourth subfgure, the T: Absolute temperature nonlocal chemical potential begins at a positive value at the T : Reference temperature and boundary plane for all boundary-satisfying situations. |(T − T )/T | < 1 0 0 However, the distribution of radial nonlocal stress (ffth β � (3λ + 2μ)α : Te volume thermal expansion 1 T subfgure) begins at zero, indicating that traction is free near σ : Components of the stress tensor ij the surface, and then begins to rise to its maximum value ρ: Te density of the medium before decreasing quickly and convergently to zero as the α : Te coefcient of linear thermal distance increases to reach the equilibrium state. Te con- expansion centration distribution begins with a positive value at the e: Cubical dilatation beginning and then drops gradually with exponential be- τ : Te difusion relaxation time havior to reach the zero-state line. A slight variation in C : Specifc heat at a constant strain of the linearly variable thermal conductivity has a signifcant efect solid plate on the wave propagation behavior, as shown by these ρC /K � 1/k: Te thermal viscosity e 0 subfgures. D : Te carrier difusion coefcient Figure 2 depicts the variation of the principal variables τ: Te photogenerated carrier lifetime (distributions of the carrier density (plasma waves), the E : Te energy gap of the semiconductor concentration (difusion), the strain (elastic waves), the κ � zN /zTT/τ: Te thermal activation coupling temperature (thermal waves), and the radial stress (me- parameter chanical waves)) as a function of radial distance r for varying c: Measure the efect of thermoelastic nonlocal parameter values. We observe that the distributions difusion Advances in Condensed Matter Physics 11 [10] A. C. 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