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

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Introduction
Nanotechnology is currently and in the future will be one of the most crucial cornerstones of human existence.Tis signifcant technology is expanding quickly, and several scientists are engaged in this fascinating sector.Several of the physical characteristics of elastic materials may vary depending on the temperature.Many difculties arise in researching elastic materials without taking varying heat conductivity into account.When thermal conductivity varies, particularly in response to temperature, it becomes essential.Termo-difusion is the relationship between mass difusion and changing thermal conductivity.Termodifusion happens when particles move from an area of greater concentration to an area of lower concentration as a result of a temperature change.Modern engineering has several uses for the study of thermal conductivity in the presence of mass difusion, particularly in the aerospace, electronics, and integrated circuit industries.Highperformance nanostructures, such as nanotubes, nanoflms, and nanowires, have been extensively used as resonators, probes, sensors, transistors, actuators, etc. with the fast development of nanomechanical electromechanical systems (NEMS) technologies.It is crucial to comprehend the precise characterizations of these nanostructures' thermal and mechanical characteristics.
Semiconductor materials (such as silicon) are an excellent research subjects for this phenomenon, particularly when subjected to laser or falling light beams.On the surface, the excited electrons will produce a charge known as free carriers (plasma waves).According to the quantity of light descending, the plasma density is employed to regulate the difusion [1][2][3].Numerous publications [4][5][6] failed to take into account the coupling between thermal-elastic waves and plasma waves during the deformation process in semiconductor materials.Recently, several authors employed photoacoustic spectroscopy to detect photothermal events when a laser beam struck a semiconductor [7,8].Semiconductors' temperature, carrier intensity, and thermal difusion are measured using the photothermal phenomena [9][10][11][12][13].When thermal waves propagate, generating elastic oscillation, and plasma waves are formed by photo-excited free carriers, directly creating a periodic elastic deformation as well [14][15][16], the interaction between the elastic-thermal-plasma waves occurs.Without considering the impact of changing thermal conductivity, several issues in generalized thermoelasticity have been explored [17][18][19][20][21][22][23][24][25].Later, a lot of writers studied generalized thermoelasticity in many areas using variable thermal conductivity.Te thermal-mechanical behavior of the medium may be afected by the deformation of elastic media depending on temperature [26][27][28].Abbas [29][30][31][32][33] studied many problems of the fber-reinforced anisotropic thermoelastic medium in two dimensions with fractional transient heating according to many mathematical methods.
Te nonlocal thermoelastic model with variable thermal conductivity (which may be considered as a linear function of temperature) is utilized in the current study using a theoretical method.Te process of photo-thermaldifusion interactions in semiconductor nanoscale media is investigated.Te variation in temperature caused by the light beam impacting the nonlocal semiconductor medium is the basis for the variable thermal conductivity.Te chemical difusion method enables photothermal transfer (mass difusion).When the Laplace transform domain in cylindrical coordinates is utilized, the analytical solutions of the basic felds are found.Te numerical techniques provide analytical solutions in the Laplace domain without any presumptive limitations on the real physical values.Finally, with changes in nonlocal parameters and changing thermal conductivity, the numerical calculations of the important physical quantities distribution are graphically shown and discussed.Te numerical fndings presented in the current study have applications in solid mechanics, acoustics, material science, and engineering for earthquakes.

Formulation of the Problem and Basic Equations
Te four important variables in this problem, respectively, 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 waves), carrier density (plasma waves), and difusive material concentration (mass difusion).When the thermal activation coupling value κ for the nonlocal medium is nonzero, the photothermal difusion transport process takes place.It makes use of cylindrical coordinates (r, ψ, z).When a very thin circular plate is taken into account, all quantities are independent of ψ and z because of the symmetry of the axis z.Elastic-plasma-thermal-difusion wave overlapping processes' governing equations are presented as [34,35], the photo-electronic equation is as follows: Equations for thermal difusion in the photothermal difusion process transport are as follows: ( If there is no body force, the equations of motion for nonlocal medium may be expressed as follows [34]: ( Te length-related elastic nonlocal parameter is represented by ξ � ae 0 /l (l is the external characteristic length scale, a is the internal characteristic length, and e 0 is nondimensional material property).
Te mass difusion equation is expressed as follows [35]: Te change in thermal conductivity is K of the nonlocal semiconductor medium and β 2 � (3λ + 2μ)α c where α c is the coefcient of linear difusion.On the other hand, the transport heat coefcients for the nonlocal medium are independent of N, C and T [36][37][38].
Te strain-stress combinations are as follows: 2 Advances in Condensed Matter Physics Te nonlocal semiconductor medium's chemical potential equation is where P is the chemical potential per unit mass.It is possible to choose a material's variable thermal conductivity K, which may be estimated as a linear function of temperature [26]: where q is a negative parameter and K 0 is a thermal conductivity when q � 0 (the nonlocal medium is independent of temperature).Te map of temperature can be taken in the following form [27]: Diferentiating both sides of equation ( 7) relative to x i , we get Another form of equation ( 9) when the nonlinear terms are neglected can be obtained as follows: Te time-diferentiation is done in the same manner to both sides of equation (7), resulting in: Using equation ( 8) and diferentiating equation (1) by z/zx i , yields: Te other form of the quantity κK 0 /KΘ ,i with neglected the nonlinear term can be represented as follows: Equation ( 1) results when equation ( 13) is applied: Integrating equation ( 14), yields: Under the infuence of mapping, the heat (thermal) difusion equation ( 2) have the following form: Te nonlocal motion equation ( 3) under the temperature map may be simplifed as follows: Te equation for mass difusion equation ( 4) may be expressed as follows: Te term D c cK 0 /KΘ ,ii (r, t) can be represented with neglected nonlinear terms in the following form: In this case, equation ( 18) can be rewritten as follows: Advances in Condensed Matter Physics Te strain in cylindrical 1D form can be represented as follows: By doing the analysis in the radial direction (r), the problem will be solved in 1D, with the displacement vector having the form u → � (u, 0, 0), u(r, t).
Te main governing equations in 1D (radial) are reduced as follows: Taking the divergence on both sides of equation ( 17), yields: Te equation for mass difusion may be shortened to For simplicity, the dimensionless variables will be represented as follows: According to dimensionless variables, the governing equations ( 22)-( 25) and the chemical potential equation have the following form (drop the dash): Using the linear form of variable thermal conductivity equation ( 6) and the mapping equation (7), one arrives at the following result [26]: Te dimensionless equations for stress forces may be simplifed as follows: where To solve this problem in Laplace transform domain, the initial conditions should be taken mathematically as follows:
Based on equation ( 28), the temperature in the Laplace transform domain may be expressed as follows:

Boundary Conditions
To establish the unknown parameters λ i , mechanical forces and thermal loads will be applied to the nonlocal semiconductor medium's free surface (where a is the radius of the circular plate), which is initially at rest.Te nonmechanical loads are thus assumed to be traction-free at the cylinder's surface.Using Laplace transform on both sides and assuming thermal shock as the thermal load, we obtain [39,40]: (i) Nonmechanical loads are traction-free loads, which can be written as follows: Hence, (ii) Te thermal state is considered a thermal shock when: Terefore, When the chemical potential is provided as a known function of time and the carriers' intensities can be obtained using a recombination process, the surface boundary conditions are determined.(iii) Tis is how the chemical potential is written as follows: which yields: (iv) Te recombination-restricted possibility of carrierfree charge density at the cylinder surface is expressed as follows: which leads to where ƛ is a constant.On the other hand, the quantities L(t), ζ(s), and χ(t) represent the Heaviside unit step function [34,35].

The Numerical Inversion of the Laplace Transforms
Using the inversion of the Laplace transform, a full solution in the time domain was found.Using the numerical inversion approach [39], the inverse of any function ϑ(t) in the Laplace domain may be expressed as follows: where s � n + im (n, m ∈ R), in this case, equation (55) can be represented as follows: Fourier series can be utilized to expand the function e − nt ϑ(r, t) during the closed interval [0, 2t ′ ], yields and Re is the real part.N is a large fnite integer that can be chosen for free.
Advances in Condensed Matter Physics

Numerical Results and Discussions
To show the impact of linearly varying thermal conductivity (which is dependent on the heat), simulations and theoretical discussions are conducted using silicon (n-type) as an elastic nonlocal semiconductor medium.Using the physical characteristics of isotropic nonlocal silicon medium, the variable thermal conductivity and difusion relaxation time were investigated as a function of temperature [41][42][43][44] Te real part of the fundamental physical felds is taken into consideration when the wave propagation distributions are represented graphically.
Figure 1 (consisting of six subfgures) illustrates the change of physical quantities in this phenomenon versus radial distance for two cases of thermal conductivity that vary with distinct values.Te frst case represented by soiled lines refers to the issue of (heat) temperature independence q � 0.0.Te second instance is depicted by dashed lines and represents the condition of temperature dependency q � −0.5.In response to the boundary conditions, the distributions of carrier density (plasma), strain (elastic), chemical potential, concentration (mass difusion), and temperature (thermal) began with a positive value at the surface.But the distribution of redial stress begins at zero, indicating that traction free at this surface r � a � 1. Te frst subfgure depicts the variation in temperature versus radius r for various nonlocal parameter values (two cases).It is evident from this subfgure that the temperature increases as the radius increases in the frst range due to the thermal efect of light beams to reach the maximum value, and the exponential decreases until it agrees with the zero line.Tis subfgure indicates that the variable thermal conductivity infuences the temperature change.Te second subfgure shows the propagation of plasma waves with increasing radial distance for two diferent values of the variable thermal conductivity.It is clear that the carrier density distribution starts with a positive value increases slightly to reach the maximum value, and then decreases exponentially until it reaches equilibrium by difusion within the nonlocal semiconductor material, following the zero line.From the frst and second subfgures, it is clear that the theoretical numerical results obtained in this work are consistent with the experimental results [45].Te third and fourth subfgures were produced to study the nonlocal strain and chemical potential variation against the radius r for varying thermal conductivity.As shown in the fourth subfgure, the nonlocal chemical potential begins at a positive value at the boundary plane for all boundary-satisfying situations.However, the distribution of radial nonlocal stress (ffth subfgure) begins at zero, indicating that traction is free near the surface, and then begins to rise to its maximum value before decreasing quickly and convergently to zero as the distance increases to reach the equilibrium state.Te concentration distribution begins with a positive value at the beginning and then drops gradually with exponential behavior to reach the zero-state line.A slight variation in linearly variable thermal conductivity has a signifcant efect on the wave propagation behavior, as shown by these subfgures.
Figure 2 depicts the variation of the principal variables (distributions of the carrier density (plasma waves), the concentration (difusion), the strain (elastic waves), the temperature (thermal waves), and the radial stress (mechanical waves)) as a function of radial distance r for varying nonlocal parameter values.We observe that the distributions of the main physical quantities seem to exhibit the same pattern for various nonlocal parameters.With increasing values, the movements of elastic-thermal-plasmamechanical waves are dampened to achieve chemical equilibrium.Tese subfgures illustrate that the nonlocal parameter has a signifcant efect on each of the investigated distributions.

Conclusion
Te efects of changing thermal conductivity and nonlocal parameters on the photothermal excitation process and the chemical activity of elastic semiconductor materials have been investigated.Te model was constructed in one dimension using the Laplace transform according to cylindrical coordinates.Graphs show the infuence of variable thermal conductivity and nonlocal parameters.Te numerical fndings indicate that the change in thermal conductivity has a signifcant impact on the thermal-elasticmechanical-plasma behavior of nonlocal semiconductor medium during photo-electronic deformations.A small change in the nonlocal parameter has a great infuence and leads to diferences in thermal-elastic-mechanical-plasma wave propagation in the elastic medium.Tus, the nonlocal parameter's ability to conduct and transfer thermal energy may serve as an additional identifer.Various uses of the variable thermal conductivity of nonlocal semiconductor elastic media in current physics via photo-elastic-thermaldifusion excitation processes are applied in many industries.In particular, mass and heat transfer mechanisms are important in photovoltaic cells, display technologies, optoelectronic applications, and photoconductor devices.Te coefcient of electronic deformation.

5 Figure 1 : 2 Figure 2 :
Figure 1: Te variations of the main physical felds against the redial distance at diferent values variable thermal conductivity according to nonlocal semiconductor medium.