The two-temperature effect on a semiconducting thermoelastic solid cylinder based on the modified Moore – Gibson – Thompson heat transfer Текст научной статьи по специальности «Физика»

Original article UDC 536.251

DOI: https://doi.org/10.18721/JPM.16106

THE TWO-TEMPERATURE EFFECT ON A SEMICONDUCTING THERMOELASTIC SOLID CYLINDER BASED ON THE MODIFIED MOORE - GIBSON - THOMPSON HEAT TRANSFER I. Kaur , K. Singh 2

1 Government College for Girls (Palwal), Kurukshetra, India;

2 Kurukshetra University, Kurukshetra, India

и [email protected]

Abstract. This study introduces a new modified Moore — Gibson — Thompson PhotoThermal (MGTPT) theory with two temperatures for semiconductor material. The photo-ther-moelastic effects have been been investigated in an infinitely constrained semiconducting solid cylinder subjected to variable heat flux in the form of an exponential laser pulse. The Laplace transforms were used for the solution of the mathematical model in the transformed domain. The numerical inversion was applied to obtain the displacement components, the conductive temperature, the carrier density, and the thermal stresses in the physical domain. The impact of different theories of thermoelasticity with two temperatures on the displacement, temperature, thermal stresses and carrier density were represented graphically and discussed using Matlab software.

Keywords: semiconducting media, two-temperature effect, modified Moore — Gibson — Thompson heat transfer

For citation: Kaur I., Singh K., The two-temperature effect on a semiconducting thermo-elastic solid cylinder based on the modified Moore — Gibson — Thompson heat transfer, St. Petersburg State Polytechnical University Journal. Physics and Mathematics. 16 (1) (2023) 65-81. DOI: https://doi.org/10.18721/ JPM.16106

This is an open access article under the CC BY-NC 4.0 license (https://creativecommons. org/licenses/by-nc/4.0/)

Научная статья УДК 536.251

DOI: https://doi.org/10.18721/JPM.16106

ДВУХТЕМПЕРАТУРНЫЙ ПОДХОД К АНАЛИЗУ СВОЙСТВ ПОЛУПРОВОДЯЩЕГО ТЕРМОУПРУГОГО СПЛОШНОГО ЦИЛИНДРА, ОСНОВАННЫЙ НА МОДИФИЦИРОВАННОЙ ТЕОРИИ ТЕПЛОПЕРЕДАЧИ МУРА - ГИБСОНА - ТОМПСОНА И. Каур 1И, К. Сингх 2 1 Государственный колледж для девушек (Палвал), г. Курукшетра, Индия; 2 Университет Курукшетры, г. Курукшетра, Индия и [email protected]

Аннотация. В работе вводится новая модифицированная двухтемпературная фототермическая теория Мура — Гибсона — Томпсона (MGTPT) для полупроводникового материала. Исследованы фототермоупругие эффекты в полупроводниковом твердом круговом цилиндре бесконечной длины, который подвергается переменному тепловому потоку, образованному экспоненциальным лазерным импульсом. Для решения математической модели в преобразованной области используются преобразования Лапласа. Для определения компонент смещения, температуры проводимости, плотно© Kaur I., Singh K., 2023. Published by Peter the Great St. Petersburg Polytechnic University.

сти носителей и тепловых напряжений в физической области применяется численная инверсия. Влияние различных двухтемпературных моделей с термоупругостью на смещение, температуру, тепловые напряжения и плотность носителей представлено графически с помощью программного обеспечения Matlab, и полученные результаты обсуждаются.

Ключевые слова: полупроводниковая среда, двухтемпературная модель, модифицированная модель теплопередачи Мура — Гибсона — Томпсона

Ссылка при цитировании: Каур И., Сингх К. Двухтемпературный подход к анализу свойств полупроводящего термоупругого сплошного цилиндра, основанный на модифицированной теории теплопередачи Мура — Гибсона — Томпсона // Научно-технические ведомости СПбГПУ. Физико-математические науки. 2022. Т. 16. № 1. С. 65—81. DOI: https://doi.org/10.18721/ JPM.16106

Статья открытого доступа, распространяемая по лицензии CC BY-NC 4.0 (https:// creativecommons.org/licenses/by-nc/4.0/)

Introduction

In recent years, a lot of attention has been paid to semiconductor materials due to their physical properties. Unlike copper or aluminum, semiconductor materials are not very good conductors at room temperature. At room temperature, they are also not good insulators too, for example, such as glass. The resistance of semiconductor materials gradually decreases with increasing temperature. As a consequence, they provide great electrical conductivity. Because of this property, semiconductor materials become important in modern very large-scale integration logic (VLSI)/electronics/Electrical industries. Under high temperatures, semiconductor materials deform and their physical properties change internally. Two major changes occur as a result of thermal effects: one is thermoelastic deformation and another is electronic deformation. The thermal effect occurs when semiconductor media are exposed to focused laser beams or sunlight beams and, owing to this effect, they will vibrate. These materials have many uses in renewable energy, especially in the solar cells industry which depends heavily on semiconductor materials.

J. M. Duhamel [1] presented the theory of classical uncoupled thermoelasticity. This theory has two limitations. To begin with, the elastic state of material is not related to temperature. Moreover, the parabolic heat equation predicts that the temperature travels at an infinite speed, which is again contradictory to physical experiments. M. A. Biot [2] set up the hypothesis of coupled thermoelasticity to conquer these impediments. According to his theory, the heat conduction equations and the equations of elasticity are related. However, the shortcoming of this theory is that it only predicts heat waves with an unlimited speed of propagation. C. Cattaneo [3] and P. Vernotte [4, 5] suggested a wider form of the Fourier law for the homogeneous and isotropic medium by introducing thermal relaxation time т to the heat flux vector q to establish a steady state at a point when a temperature gradient V T is abruptly imposed on it, as,

(1 + т0 d/dt)q = -K V T, (1)

where K is the coefficient of thermal conductivity, t is the time.

Then the generalized theory of thermoelasticity with one relaxation time was put forward by H. W. Lord and V. Shulman [6] for the particular case of an isotropic body. According to this theory, the heat equation being hyperbolic has a finite speed of propagation for temperature. After that a more precise version of thermoelasticity was presented by the two-temperature theory of thermoelasticity introduced by P. J. Chen and M. E. Gurtin [7]. This model includes the conductive temperature ф (the outcome of thermodynamic processes) and the thermodynamic temperature T (the result of mechanical processes). A. E. Green and K. A. Lindsay [8] showed that the linear heat conduction tensor is symmetric. R. S. Dhaliwal and H. H. Sherief [9] gave the equations of generalized thermoelasticity for an anisotropic medium. However, A. E. Green and P. M. Naghdi [10 - 12] further contributed to the thermoelastic theories with and without energy dissipation and further developed the Fourier law as

© Каур И., Сингх К., 2023. Издатель: Санкт-Петербургский политехнический университет Петра Великого.

q = -K, VT- K* V0, e = T, T = 9 - a, 9,, (2)

where K..* is the materialistic constant; a., is the two-temperature parameter.

Now the law can be formulated as follows:

Based on entropy equality, they proposed three new thermoelastic theories. Their theories are known as the thermoelasticity theory of type I, the thermoelasticity theory of type II (i.e., thermo-elasticity without energy dissipation), and the thermoelasticity theory of type III (i.e., thermoelas-ticity with energy dissipation). On linearization, type I becomes the classical heat equation whereas on linearization of type-II, as well as type-III theories, give the finite speed of thermal wave propagation.

Recently, there has been a myriad of academic articles that study and interpret the Moore -Gibson - Thompson (MGT) equation. I. Lasiecka and X. Wang [13] founded their theory on a 3rd-order differential equation, important to various fluid dynamics. Using the MGT equation with 2T, R. Quintanilla [14, 15] devised a novel heat conduction model. MGT equation of the modified Fourier law is, as follows:

(1 + t0 d/dt)q = -K.. VT- K * V0, where e = T. (3)

J. R. Fernandez and R. Quintanilla [16] discussed linear thermoelastic deformations of dielectrics. N. Bazarra et al. [17] studied a thermoelastic problem using MGT thermoelastic equation. Suppose that the semiconductor elastic media is exposed to external laser beams and which create carrier-free charge density due to excited free electrons with semiconductor gap energy E . In response to absorbed optical energy, there is a change in the electronic deformation and the elastic vibration. In this case, heat conductivity equations will be affected by thermal-elastic-plasma waves. The revised Fourier law for semiconductor materials with plasma impact in a generalized form can be written as

(1 + t0 d/dt)q = -K .. VT - K.. * V0 - J (ENh)dx, where e = T; (4)

here N is the carrier density.

The photoexcitation effect is represented by the final term in Eq. (4). When the above equation is differentiated with respect to x, the result is

(1 + T0 a/at) V •q = - V •(K .. VT - K..* V0) - EN/t, where e = T. (5)

Some other researchers also worked on similar research on Hall current effect and semiconductor medium such as M. Marin [18], P. Lata et al. [19], A. M. S. Mahdy et al. [20], I. Kaur and K. Singh [21, 22], M. Marin et al. [23, 24], I. Kaur et al. [25, 26], M. M. Bhatti et al. [27, 28], M. Conti et al. [29], M. I. A. Othman and M. Marin [30], J. A. Conejero et al. [31], M. Marin et al. [32], E. M. Craciun et al. [33], M. E. Nasr et al. [34], A. E. Abouelregal et al. [35]. However, from the literature review, it has been observed that no work has been carried out in the transient study of semiconductor cylinders exposed to ultrashort pulsed laser heating and photogenerated plasma with two temperatures. The importance of this issue is the main motivation for its in-depth review in this study.

This study aimed to explore the photo-thermoelastic interactions in an infinite semiconducting solid cylinder acted upon by the variable heat flux in the form of an exponential laser pulse along the boundary surface.

The governing equations are expressed using a new generalized photo-thermoelastic model as MGTPT heat transfer for semiconducting medium with two temperatures. The Laplace transforms are used for the solution of the mathematical model in the transformed domain. And the numerical inversion is applied to obtain the displacement components, the conductive temperature, the carrier density and the thermal stresses in the physical domain. The impact of different theories of thermoelasticity with two temperatures on the displacement, temperature, thermal stresses, and the carrier density are represented graphically using Matlab software.

Basic equations

Following the work of A. E. Abouelregal and D. Atta [36], we present below the constitutive relations, the equation of motion, the plasma diffusion equation governing the plasma transportation process in semiconductor nanostructure medium, the MGTPT heat conduction equation with thermal-plasma-elastic interaction. The constitutive relations:

^ = ( Ч* -вТ — SnN)Sj + ц (uuу + Uj,,),

в = (3Х-2ц)ат Sn = (3A, + 2ц) dn, T = p—a. ф.,

where о.., N/m2, are the stress tensor components; X, ц, Pa, are the Lame's elastic constants; u,., u , m, are the displacement tensor components; dn is the coefficient of electronic deformation; ' 5.. is the Kronecker delta; aT, K-1, is the linear thermal expansion coefficient. The equation of motion:

°y,j + F, =Pu,, (7)

where F,, N, is the body force; p, kg/m3, is the medium density; iii, m/s2, are components of the body acceleration.

The plasma diffusion equation:

^ = devN-K + k(I-а^)ф (8)

8t т

where DE is the carrier diffusion coefficient; т, s, is the photo-generated carrier lifetime; a is the measure of thermoelastic diffusion effect; к is the coupling parameter for thermal activation, к = (T/т) dN/дТ (N is the carrier concentration at equilibrium position).

The modified Moore — Gibson —Thompson heat conduction equation with two temperature quantities:

(K jф.), + (Кф, ), + = [1 + To I) [PCe (l — aV2) cp + - PC^} (9)

where C is the specific heat at constant strain; eare the strain tensor components; Q is the

source of heat; К. = К. 5 , К * = К *S , i is not summed.

. . . .

The subscripts marked by commas mean the partial derivatives with respect to the spatial coordinates, whereas one or two dots on top of the notation mean the first or second derivative with respect to the time variable t.

Formulation and solution of the problem

A one-dimensional (1D) symmetrical, thermally homogenous, semiconductor solid cylinder of radius r0 was considered (Fig. 1). An external laser pulse heating system was used to irradiate the external surface of the solid cylinder. A cylindrical coordinate system (r, 0, z) with the z-axis arranged along the cylinder axis was taken. Initially, it was believed the cylinder to maintain constant and uniform temperature T0.

Furthermore, all examined fields were assumed to be finite within the medium for the regularity condition. Due to symmetry, and for the 1D problem, all the functions considered depended on the radial distance r and the time t.

For the 1D problem, displacement components of u and the displacement-strain relations are given by

u = (u,0,0) (r, t), e = u/r, = du/dr, e Q = e = e = e = 0, (10)

5 00 5 Г0 rz 0z zz 5 V '

where eare the strain tensor components.

Fig. 1. The illustration to the problem formulation:

a semiconductor solid cylinder of radius ro and an external laser heat

The stress-strain-temperature-carrier relations (6) using Eqs. (10) will be the form

rs

■■2+ -[p(l-aV2 )^+8„JV],

u r

azz -[(l-a V2 )ç+8„tf ],

= •

2 + A,e -[-a V2 )9+8kN ]

(11) (12) (13)

V 2 =-d 2

1 d

dr r dr

where ekk is the cubical dilatation, ekk = e = (1/r) d(ru) / dr. Hence, the dynamic motion equation becomes

cte 1 , .

-T-- + -(°rr-CTee) = P drr

(du2 02 ^ —^ - 0 u dt2

(14)

Using Eqs. (11) — (13), in Eqs. (14) and (8), (9), the governing equations for the considered semiconducting medium are:

1 d( ru)

r dr

d vu „2\ i „ dN f 52u ^

(£ [(I )9]-8, £ = p

d r

St2 ,

dN = D. (V22N I-^MV22 )<p,

„ d ^2 .gN L d"

pC. [(l-aV2 )9] + prc

de 1

O r

(15)

(16) (17)

Pre-operating both sides of Eq. (15) by (1/r + d/dr), we get:

(À + 2^) V2e - (V2 (l - aV2 ) 9 - 8„V2N -

Kdt2,

(18)

In order to obtain the above equations in dimensionless form, the dimensionless quantities are given by the following:

(ru') = v0n(r,u), (T ,Na.,9') = -^(pT,5„N,a.,9),

Pvo

C I- (19)

(t0,t',t') = v2n(t 0,t,f), n =

pv2 =À + Y =.

£ u + Here, the magnetic parameter M (also known as the Hartmann number) measures the strength

of the magnetic field. Using Eq. (19) in Eqs. (16) — (18) and after suppressing the primes, yields:

V2e-V2 (l-aV2)^-V2N

8l (V2 N) -82 N + 83 (l - aV2) 9

d

Kdt2,

dN dt

—V> + S4V> + S5 N = J1 + t0 —

dt

dt

4 (l-aV2 )9 + S6 ^ dt2K ; d*2

dt2

(20) (21) (22)

where 5„ are the electronic deformation coefficients that follow the expressions

K *

S1 = DEn, S2 = 1 S3 = KS„

t ß

^ S4 =

S5 =

E„

. S6 =■

(X + 2|l) Ce' ß2To

(23)

S „Ce + 6 pCe (X + 2H)"

Making use of dimensionless quantities defined by Eq. (19) in Eqs. (10) — (12) and after suppressing the primes, yields:

2 y22 + (1 -2 y2) e-[(l-a V2)9 + N ] a00 = 2y2 U + (l-2y2 ) e-[(l-aV2 + N (l-2y2 ) e-[(l-aV2 + N

= 11 -

The initial conditions of the problem are taken as

u ( r ,0) = 0 = f ( r ,0),

<p( r ,0) = 0 = ( r ,0),

N ( r ,0) = 0 = f ( r ,0).

(24)

(25)

(26)

(27)

(28) (29)

The Laplace transform of a function f with respect to the time variable t, with 5 as a Laplace transform variable, is defined as

L (f (t )) = f (, ) = J f (t >-5tdt.

0

Using Laplace transforms on Eq. (30) to Eqs. (20) — (22) we obtain:

(V2 -52)e -V2 (l-aV2)9-V2N = 0, [81V2 - (82 +5)] N + 83 (l - aV2) 9 = 0, (1 + t0 5 )86 52 e +[-(5 + 84 )V2 +(1 + t0 5) 52 (l-aV2)] 9 - 85 N = 0. Using Laplace transforms on Eq. (30) to Eqs. (24) — (26), we obtain:

2y2 — + (l-2y2 ) e-[(l-aV2 )9 + N'

°rr =

(30)

(31)

(32)

(33)

(34)

arr =

c00 = 2y2 r + (1 -2y2 ) e-[(1 -aV2 )9 + N ôzz =(1 -2y2 ) e-[(1 -aV2 )9 + N '. When Eqs. (31) to (33) are decoupled, we get

( V6 - BV4 + CV2 - D ) ( e, 9, N) = 0,

where

A

B = -[a83855 + (-A87 -8l8l0 -8l89 +888ll -a8889 )]/ A, C = (-83855 + a838587 + 8389 -8l878l0 + 88878ll + 888l0 + 8889) / A, D = (8387855 - 88878l0) / A,

8 7 =(-5 2 ) , 8 8 = 82 +5, 89 =(l + T0 5 )865 8l0 = (l + T)5) 52, 8n =-( 5 + 84 )-(l + T05) 52 a.

Presenting X (i = l, 2, 3) in Eqs. (39) we obtain:

(v2-x? )(v2 -x? )(v2-x? )(e, 9, N ) = 0,

where X? (i = 1, 2, 3) are the roots of the equation

(X6 - BX4 + CX2 - D) = 0,

l

that are given by

with

œ =

Xf =-( 2œ sin ^ + B ), = 1 (-œ sin ^->/3œcos ^ + B ), Xl = 1 (-œ sin £ + >/3œcos ^ + B )

Vb2 -3C, s =1

f 2B3 - 9BC + 27D ^

2œ

The general solution of Eq. (38) can be written in the form

( e, 9, N ) = ]T (1, z, n )gI0 (X;r ),

i=1

where I ( ) indicates the second type of modified Bessel functions of order n. We get the following relations by inserting Eq. (42) into Eqs. (31) — (33):

Zi =

-(X2 +57 )(89X2 -8s )

Ô3Ô5 +(8„X 2 +810 )(8,X 2-8g )'

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

n =

"(\2 +07 )8s

Ô3Ô5 + (8nX? +810)(81X? -8g)■

The displacement u may be represented in the Laplace transform domain, as follows:

_ 3 i

u (V).

i=i Ki

We obtained Eq. (45) with the help of the Bessel function relation

| xI0 ( x) dx = xI1 ( x). Differentiating Eq. (45) in terms of r gives

(44)

(45)

(46)

du = = ^ gi

i=1

10 (V(V)

Thus, the final thermal stress solutions are generated in closed form as follows:

5 rr ^ ' Si i =1

II (V)- ^/1 far) Xr

(47)

(48)

5ee =Z S

i=1

2Y2

/1 (V) + l/o (V)

X/

5zz =Z Sili/0 (V)

(49)

(50)

l =1 -2y2 -[Zi (1 -aX2) + n

Boundary conditions

We presume that the cylinder's outside surface is compelled. Therefore, the mechanical boundary condition can be expressed as

u(r, t) = 0 at r = r

(51)

Also, the boundary condition for variable heat flux (exponentially laser pulsed heat) is applied to the boundary surface:

t2 -f

qp = q016t2e pat r = r0-

(52)

Using dimensionless variables (32) on Eq. (3) yields the equation

i1 + T0 £ ^ qp = "ff + ]f ■

Eqs. (52) and (53) give the following boundary condition:

f t N

a- f1+T0 -1-16t 21 0 dt )dt

12e ''

d

dT

= -l —+ 8, I — at r = r.

dt

dr

(53)

(54)

The carriers can reach the sample surface during the diffusion phase, with a finite probability of recombination.

The boundary condition for the carrier density:

dN - (55)

De-T- = SVN at r = r0,

dr

where DE is the carrier diffusion coefficient, s is the surface recombination velocity.

The boundary conditions (51), (54), and (55) have the following forms after performing the Laplace transform:

U (r>, s) = 0,

d(1 - aV2 )

9

dr

D,

dN

dr

q (1+T0s )s

;(1+stp )3 ( s+84 )

= svN(ro, s).

= -G ( s ),

Eqs. (42) and (45) are substituted into Eqs. (56) — (58), giving

t gt 1I (Xi.r0 ) = 0,

i=1 Xi

t gil! (Xi.r0)ZiXi (1 -aXl) = --MlM^ = -G(s),

i=1

8 (1 + stp ) (s + 84 ) t ngi [ De XI1 (Xi-r, )-sv 10 (V0 )] = 0.

(56)

(57)

(58)

(59)

(60)

(61)

The values of g (i = 1, 2, 3) can be obtained by solving Eqs. (59) — (61) by the Cramer's rule:

gi (s)f,

A = Gj [G5G9 - G8G6 ] - Gi [G4G9 - G6G7 ] + G3 [G4G8 - G5G7 ],

A1 = G(s)[GiG9 -G8G3], A 2 =-G( s)[GG9 - G7G3 ] ,

A3 = G(s)[G1G8 -G2G7], 1

Gi =^ï~9i, Xi

G+3 = 9iZiXi (1 -aXl ),

Gi+6 =n [DExi9i - svWi j, I1 (Xir0 ) = 9i, 10 (Xir0 ) = Wi, Î = 1, 2, 3,

and putting the values of gi(s) into Eqs. (42), (45), (48) — (50), the various components of displacement, temperature distribution, carrier density, and stresses are

U = GA)|[GiG9 -G8G3 - [GG9 -G7G3 jX2 + [G1G8 -G2G7 jX3}, (62)

t = {[G2G9 - GG ]Z A -[g,G9 - GG ]Z 2 s 2 +[GG8 - G2G7 ]Z 3S},

N = ^ {[G2G9 - GsG3 ] niSi - [G1G9 - G7G3 ] ^2S + [GiGs - G2G ]n3^3},

(63)

(64)

=G^{[G22G9-GÄ](i9,-,)-[G,G»-G,G3](,92-,2)+ ^

+ [G1G8 G2G7 ](l3S3 )} >

0„= Gf {[G2G9-G,G3](,+ll9l)-[G,G»-G7G5](,22+l292)+ (66)

[G1G8 - G2G7 ](^3 + l3S3 )} >

where

^ = ^ {[GG - GG ] 'A -iG!G9 - G G3 ] ^2 + [GG - G2G7 ] ^3 } , (67) ^ = I0 far), 0( = Ii (V), ^ = ^Ii (V), i = 1, 2, 3.

V

Inversion of the transforms

In order to obtain the result of the problem in the physical domain, transforms in Eqs. (62) — (67) are inverted using

1 e+™

f (^ t) = — J f (^ s) S'ds. (68)

e~liii

Finally, let us evaluate the integral in Eq. (68) using the Romberg's integration (W. H. Press et al. [37]) with an adaptive step size.

Particular cases:

i. If K* f 0, K f 0 and t0 f 0, a f 0 in Eqs. (62) - (67), the results for the MGTPT + 2TT can be obtained (2TT are the two temperature quantities).

ii. If K f 0, K f 0 and t0 = 0, a f 0 in Eqs. (62) - (67), the results for the Photothermal Green and Naghdi III model (PGN-III) [12] with 2TT can be obtained.

iii. If K f 0 and t0 = 0, a f 0 in Eqs. (62) — (67), the results for the Photothermal Green and Naghdi II model (PGN-II) [11] with 2TT can be obtained.

iv. If K* = 0 and t0 = 0, a f 0 in Eqs. (62) — (67), the results for the Coupled Photothermo-elasticity theory (CPTE) with 2TT are obtained.

v. If K* = 0 and a f 0 in Eqs. (62) — (67), the results for the generalized Lord and Shulman Photothermoelasticity model (PLS) [6] with 2TT are obtained.

vi. If a = 0 in Eqs. (62) — (67), the results for the MGTPT model are obtained.

Numerical results and discussion

In order to demonstrate the theoretical results and to show the effect of Hall current, rotation, and the effect of the modified photothermal heat equation (MGTPT) graphically using Matlab software, the isotropic silicon material (Si) with its physical properties was taken. The used data are listed in Table.

Fig. 2, a depicts the variation in the radial displacement u for MGTPT and PGN-III theories with two temperatures. Notice that the radial displacement is minimum in the absence of two temperature quantities in the MGTPT theory. However, as the value of the two-temperature parameters in the MGTPT theory increases, the radial displacement increases as well. Moreover, PGN-III theories with and without two temperatures show a higher variation in the radial displacement. Furthermore, as the radial distance increases, the radial displacement decreases.

Table

Parameter values of silicon material [36] used in our study

Parameter Notation Unit Value

Lame's elastic constants I GPa 36.4

^ 54.6

Thermal elastic coupling P MPa/(deg) 7.04

Electronic deformation coefficient 5 n m3 -9-10-31

Medium density P kg/m3 2.33403

Specific heat at constant strain CE J/(kg-K) 695

Coefficient of thermal conductivity K W/(m-K) 150

Materialistic constant K* kW-s 1.54

Carrier diffusion coefficient de m2/s 2.5-10-3

Magnetic constant H/m 1.25740-6

Reference temperature (s. t. \T/TQ\ <<1) T0 K 300

Photo-generated carrier lifetime T ^s 50

Carrier concentration at equilibrium position N0 m-3 1020

Electric constant e0 F/m 8.85440-12

Semiconductor gap energy E g eV 1.11

Linear thermal expansion coefficient aT K-1 340-6

Surface recombination velocity s v m/s 2

a)

o.o

0.2

- MGTPT with a = 0.0

- MGTPT with a = 0.2

- MGTPT with a = 0.4 PGN-III with a = 0.0

- PGN-III with a = 0.2

0.4

0.6

0.8

1.0 r

b)

MGTPT with a = 0.0 MGTPT with a = 0.2 MGTPT with a = 0.4 PGN-III with a = 0.0 PGN-III with a = 0.2

Fig. 2. The radial displacement (a) and temperature (b) variations for various models with two temperature quantities (a is the two-temperature parameter)

Fig. 2, b illustrates the variation in the temperature distribution for MGTPT and PGN-III theories with two temperatures. Note that temperature distribution is higher in the inner core of the cylinder as compared to the outer one of the cylinder. In addition, the presence of two temperatures causes a higher temperature distribution.

Fig. 3 shows the variation in the carrier density for MGTPT and PGN-III theories with two temperatures. It should be noted that PGN-III depicts the minimum variation in carrier density than MGTPT theory in the absence of two temperature quantities. At the same time, PGN-III theory results in maximum variation in carrier density. Notice that variation in the carrier density sharply decreases as the radial distance r increases.

Fig. 4 shows the variation in the stress components for MGTPT and PGN-III theories with two temperatures. It has been noticed that in the absence of two temperature quantities in the MGTPT theory, the variation in the stress components are minimum as compared to the PGN-III theory. However, as the value of the

Q 0.0003

ca O

-MGTPT with a = 0.0

- MGTPT with a = 0.2

- MGTPT with a = 0.4 -PGN-III with a = 0.0

PGN-III with a = 0.2

0.0

0.2

0.4

—I— 0.6

0.8

Fig. 3. The variation in the carrier density for various models with two temperature quantities

a)

0.0002 -

b)

ee

0.0008

- MGTPT with a = 0.0

- MGTPT with a - 0.2

- MGTPT with a = 0.4

- PGN-III with a = 0.0 PGN-III with a = 0.2

0.2

0.4

0.6

0.6

1.0 r

c)

0.0005-

0.0

- MGTPT with a - 0.0 -MGTPT with a = 0.2

- MGTPT with a - 0.4 -PGN-III with a = 0.0

PGN-III with a = 0.2

0.6

1.0 r

Fig. 4. The deviations in radial (a), hoop (b) and vertical (c) stresses for various models with two temperatures

a

7 7

a

zz

two- temperature parameter increases, there is a sharp increase in the stress components. Note that the variation in the stress components decreases drastically as the radial distance r increases.

Conclusions

This study lays out several photo-thermoelasticity models that are generalized in the Moore — Gibson — Thompson photo-thermal (MGTPT) model with two temperatures. In addition, this study included the photo-thermoelastic Green — Naghdi type III model. The generalized MGTPT model is used to solve some of the physical consequences of some earlier models. In this study, the infinite semiconducting solid cylinder subjected to the exponential laser pulse on its boundary surface has been studied. The governing equations are expressed with the help of the MGTPT with two temperatures.

Effects of different thermoelastic theories with two temperatures on the components of displacement, temperature field, carrier density, and thermal stresses are represented graphically. As seen in the graphs, all of the domains examined are significantly impacted by the two temperatures. The thermal effect occurs when semiconductor media are exposed to focused laser beams or sunlight ones and, because of the thermal effect, they will vibrate. These materials have many uses in renewable energy, especially in the solar cells industry which depends heavily on semiconductor materials.

Photothermal methods are not only simple and sensitive but they can also be used to gain some insight into the process of de-excitation in materials with optical absorption. The ideas presented in this paper will come in handy for physicists, material designers, thermal engineers, and geo-physicists. A wide range of photo-thermoelasticity and thermodynamic problems can be solved using the technique used in the above study.

REFERENCES

1. Duhamel J. M., Memories of the molecular actions developed by changes in temperatures in solids, Mummy Div. Sav. (l'Acad. des Sci., Paris). 5 (1938) 440-498.

2. Biot M. A., Thermoelasticity and irreversible thermodynamics, J. Appl. Phys. 27 (3) (1956) 240-253.

3. Cattaneo C., A form of heat-conduction equations which eliminates the paradox of instantaneous propagation, Comptes Rendus Acad. Sci. Paris. Ser. II. 247 (01 January) (1958) 431-433.

4. Vernotte P., Les paradoxes de la theorie continue de l'equation de lachaleur, Comptes Rendus Acad. Sci. Paris. Ser. II. 246 (1958) 3154-3155.

5. Vernotte P., Some possible complications in the phenomena of thermal conduction, Comptes Rendus Acad. Sci. Paris. Ser. II. 252 (1961) 2190-2191.

6. Lord H. W., Shulman Y., A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids. 15 (5) (1967) 299-309.

7. Chen P. J., Gurtin M. E., On a theory of heat conduction involving two temperatures, Zeitschrift für Angew. Math. Phys. 19 (4) (1968) 614-627.

8. Green A. E., Lindsay K. A., Thermoelasticity, J. Elast. 2 (1) (1972) 1-7.

9. Dhaliwal R. S., Sheriff H. H., Generalized thermoelasticity for anisotropic media, Q. Appl. Math. 38 (1) (1980) 1-8.

10. Green A. E., Naghdi P. M., A re-examination of the basic postulates of thermomechanics, Proc. R. Soc. A: Math. Phys. Eng. Sci. 432 (1885) (1991) 171-194.

11. Green A. E., Naghdi P. M., On undamped heat waves in an elastic solid, J. Therm. Stress. 15 (2) (1992) 253-264.

12. Green A. E., Naghdi P. M., Thermoelasticity without energy dissipation, J. Elast. 31 (3) (1993) 189-208.

13. Lasiecka I., Wang X., Moore - Gibson - Thompson equation with memory, part II: General decay of energy, J. Diff. Eq. 259 (12) (2015) 7610-7635.

14. Quintanilla R., Moore - Gibson - Thompson thermoelasticity, Math. Mech. Solids. 24 (12) (2019) 4020-4031.

15. Quintanilla R., Moore - Gibson - Thompson thermoelasticity with two temperatures, Appl. Eng. Sci. 1 (March) (2020) 100006.

16. Fernández J. R., Quintanilla R., Moore — Gibson — Thompson theory for thermoelastic dielectrics, Appl. Math. Mech. 42 (2) (2021) 309-316.

17. Bazarra N., Fernández J. R., Quintanilla R., Analysis of a Moore -Gibson — Thompson thermoelastic problem, J. Comput. Appl. Math. 382 (15 January) (2021) 113058.

18. Marin M., On weak solutions in elasticity of dipolar bodies with voids, J. Comput. Appl. Math. 82 (1—2) (1997) 291—297.

19. Lata P., Kaur I., Singh K., Propagation of plane wave in transversely isotropic magneto-thermoelastic material with multi-dual-phase lag and two temperature, Coupled Syst. Mech. 9 (5) (2020) 411—432.

20. Mahdy A. M. S., Lotfy K., Ahmed M. H., et al., Electromagnetic Hall current effect and fractional heat order for microtemperature photoexcited semiconductor medium with laser pulses, Res. Phys. 17 (June) (2020) 103161.

21. Kaur I., Singh K., Fiber-reinforced magneto-thermoelastic composite material with hyperbolic two-temperature, fractional-order three-phase lag and new modified couple stress theory, Waves Random Complex Media. 2021 (October) (2021) 1—24.

22. Kaur I., Singh K., Thermoelastic damping in a thin circular transversely isotropic Kirchhoff — Love plate due to GN theory of type III, Arch. Appl. Mech. 91 (5) (2021) 2143—2157.

23. Marin M., Ochsner A., Craciun E. M., A generalization of the Saint-Venant's principle for an elastic body with dipolar structure, Contin. Mech. Thermodyn. 32 (1) (2020) 269—278.

24. Marin M., Ochsner A., Craciun E. M., A generalization of the Gurtin's variational principle in thermoelasticity without energy dissipation of dipolar bodies, Contin. Mech. Thermodyn. 32 (6) (2020) 1685—1694.

25. Kaur I., Lata P., Singh K., Memory-dependent derivative approach on magneto-thermoelastic transversely isotropic medium with two temperature, Int. J. Mech. Mater. Eng. 15 (2020) 10.

26. Kaur I., Lata P., Singh K., Reflection of plane harmonic wave in rotating media with fractional order heat transfer and two temperature, Part. Differ. Eq. Appl. Math. 4 (December) (2021) 100049.

27. Bhatti M. M., Ellahi R., Zeeshan A., et al., Numerical study of heat transfer and Hall current impact on peristaltic propulsion of particle-fluid suspension with compliant wall properties, Mod. Phys. Lett. B. 33 (35) (2019) 1950439.

28. Bhatti M. M., Yousif M. A., Mishra S. R., Shahid A., Simultaneous influence of thermodiffusion and diffusion-thermo on non-Newtonian hyperbolic tangent magnetised nanofluid with Hall current through a nonlinear stretching surface, Pramana. 93 (6) (2019) 88.

29. Conti M., Pata V., Quintanilla R., Thermoelasticity of Moore — Gibson —Thompson type with history dependence in temperature, Asymptot. Anal. 120 (1—2) (2020) 1—21.

30. Othman M. I. A., Marin M., Effect of thermal loading due to laser pulse on thermoelastic porous medium under G-N theory, Res. Phys. 7 (2017) 3863—3872.

31. Conejero J. A., Lizama C., Rodenas F., Ramos J. S., Chaotic behaviour of the solutions of the Moore — Gibson — Thompson equation, Appl. Math. Inf. Sci. 9 (5) (2015) 2233—2238.

32. Marin M., Othman M. I. A., Seadawy A. R., Carstea C., A domain of influence in the Moore — Gibson — Thompson theory of dipolar bodies, J. Taibah Univ. Sci. 14 (1) (2020) 653—660.

33. Craciun E. M., Baesu E., Soós E., General solution in terms of complex potentials for incremental antiplane states in prestressed and prepolarized piezoelectric crystals: Application to Mode III fracture propagation, IMA J. Appl. Math. (The Institute of Mathematics and its Applications). 70 (1) (2005) 39—52.

34. Nasr M. E., Abouelregal A. E., Light absorption process in a semiconductor infinite body with a cylindrical cavity via a novel photo-thermoelastic MGT model, Arch. Appl. Mech. 92 (5) (2022) 1529—1549.

35. Abouelregal A. E., Sedighi H. M., Sofiyev A. H., Modeling photoexcited carrier interactions in a solid sphere of a semiconductor material based on the photothermal Moore — Gibson — Thompson model, Appl. Phys. A. 127 (11) (2021) 845.

36. Abouelregal A. E., Atta D., A rigid cylinder of a thermoelastic magnetic semiconductor material based on the generalized Moore — Gibson — Thompson heat equation model, Appl. Phys. A. 128 (2) (2022) 118.

37. Press W. H., Teukolsky S. A., Flannery B. P., Numerical recipes in Fortran, Cambridge University Press, Cambridge, 1980.

СПИСОК ЛИТЕРАТУРЫ

1. Duhamel J. M. Memories of the molecular actions developed by changes in temperatures in solids // Mummy Div. Sav. (l'Academie des Sciences, Paris). 1938. Vol. 5. Pp. 440-498.

2. Biot M. A. Thermoelasticity and irreversible thermodynamics // Journal of Applied Physics. 1956. Vol. 27. No. 3. Pp. 240-253.

3. Cattaneo C. A form of heat-conduction equations which eliminates the paradox of instantaneous propagation // Comptes Rendus de l'Académie des Sciences de Paris. Ser. II. 1958. Vol. 247. 01 January. Pp. 431-433.

4. Vernotte P. Les paradoxes de la theorie continue de l'equation de lachaleur // Comptes Rendus de l'Acadñmie des Sciences de Paris. Ser. II. 1958. Vol. 246. Pp. 3154-3155.

5. Vernotte P. Some possible complications in the phenomena of thermal conduction, Comptes Rendus de l'Acadñmie des Sciences de Paris. Ser. II. 1961. Vol. 252. Pp. 2190-2191.

6. Lord H. W., Shulman Y. A generalized dynamical theory of thermoelasticity // Journal of the Mechanics and Physics of Solids. 1967. Vol. 15. No. 5. Pp. 299-309.

7. Chen P. J., Gurtin M. E. On a theory of heat conduction involving two temperatures // Zeitschrift für Angewandte Mathematik und Physik. 1968. Vol. 19. No. 4. Pp. 614-627.

8. Green A. E., Lindsay K. A. Thermoelasticity // Journal of Elastisity. 1972. Vol. 2. No. 1. Pp. 1-7.

9. Dhaliwal R. S., Sheriff H. H. Generalized thermoelasticity for anisotropic media // Quarterly of Applied Mathematics. 1980. Vol. 38. No. 1. Pp. 1-8.

10. Green A. E., Naghdi P. M. A re-examination of the basic postulates of thermomechanics // Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 1991. Vol. 432. No. 1885. Pp. 171-194.

11. Green A. E., Naghdi P. M. On undamped heat waves in an elastic solid // Journal of Thermal Stresses. 1992. Vol. 15. No. 2. Pp. 253-264.

12. Green A. E., Naghdi P. M. Thermoelasticity without energy dissipation // Journal of Elastisity. 1993. Vol. 31. No. 3. Pp. 189-208.

13. Lasiecka I., Wang X. Moore - Gibson - Thompson equation with memory, part II: General decay of energy // Journal of Differential Equations. 2015. Vol. 259. No. 12. Pp. 7610-7635.

14. Quintanilla R. Moore - Gibson - Thompson thermoelasticity // Mathematics and Mechanics of Solids. 2019. Vol. 24. No. 12. Pp. 4020-4031.

15. Quintanilla R. Moore - Gibson - Thompson thermoelasticity with two temperatures // Applications in Engineering Science. 2020. Vol. 1. March. P. 100006.

16. Fernández J. R., Quintanilla R. Moore - Gibson - Thompson theory for thermoelastic dielectrics // Applied Mathematics and Mechanics. 2021. Vol. 42. No. 2. Pp. 309-316.

17. Bazarra N., Fernández J. R., Quintanilla R. Analysis of a Moore -Gibson - Thompson thermoelastic problem // Journal of Computational and Applied Mathematics. 2021. Vol. 382. 15 January. P. 113058.

18. Marin M. On weak solutions in elasticity of dipolar bodies with voids // Journal of Computational and Applied Mathematics. 1997. Vol. 82. No. 1-2. Pp. 291-297.

19. Lata P., Kaur I., Singh K. Propagation of plane wave in transversely isotropic magneto-thermo-elastic material with multi-dual-phase lag and two temperature // Coupled System Mechanics. 2020. Vol. 9. No. 5. Pp. 411-432.

20. Mahdy A. M. S., Lotfy K., Ahmed M. H., El-Bary A., Ismail E. A. Electromagnetic Hall current effect and fractional heat order for microtemperature photo-excited semiconductor medium with laser pulses // Results in Physics. 2020. Vol. 17. June. P. 103161.

21. Kaur I., Singh K. Fiber-reinforced magneto-thermoelastic composite material with hyperbolic two-temperature, fractional-order three-phase lag and new modified couple stress theory // Waves in Random and Complex Media. 2021. October. Pp. 1-24.

22. Kaur I., Singh K. Thermoelastic damping in a thin circular transversely isotropic Kirchhoff -Love plate due to GN theory of type III // Archive of Applied Mechanics. 2021. Vol. 91. No. 5. Pp. 2143-2157.

23. Marin M., Öchsner A., Craciun E. M. A generalization of the Saint-Venant's principle for an elastic body with dipolar structure // Continuum Mechanics and Thermodynamics. 2020. Vol. 32. No. 1. Pp. 269-278.

24. Marin M., Öchsner A., Craciun E. M. A generalization of the Gurtin's variational principle

in thermoelasticity without energy dissipation of dipolar bodies // Continuum Mechanics and Thermodynamics. 2020. Vol. 32. No. 6. Pp. 1685-1694.

25. Kaur I., Lata P., Singh K. Memory-dependent derivative approach on magneto-thermoelastic transversely isotropic medium with two temperatures // International Journal of Mechanical and Materials Engineering. 2020. Vol. 15. Article No. 10.

26. Kaur I., Lata P., Singh K. Reflection of plane harmonic wave in rotating media with fractional order heat transfer and two temperature // Partial Differential Equations in Applied Mathematics.

2021. Vol. 4. December. P. 100049.

27. Bhatti M. M., Ellahi R., Zeeshan A., Marin M., Ijaz N. Numerical study of heat transfer and Hall current impact on peristaltic propulsion of particle-fluid suspension with compliant wall properties // Modern Physics Letters. B. 2019. Vol. 33. No. 35. P. 1950439.

28. Bhatti M. M., Yousif M. A., Mishra S. R., Shahid A. Simultaneous influence of thermodiffusion and diffusion-thermo on non-Newtonian hyperbolic tangent magnetised nanofluid with Hall current through a nonlinear stretching surface // Pramana. 2019. Vol. 93. No. 6. P. 88.

29. Conti M., Pata V., Quintanilla R. Thermoelasticity of Moore — Gibson -Thompson type with history dependence in temperature // Asymptotic Analysis. 2020. Vol. 120. No. 1—2. Pp. 1—21.

30. Othman M. I. A., Marin M. Effect of thermal loading due to laser pulse on thermoelastic porous medium under G-N theory // Results in Physics. 2017. Vol. 7. Pp. 3863—3872.

31. Conejero J. A., Lizama C., Rodenas F., Ramos J. S. Chaotic behaviour of the solutions of the Moore — Gibson — Thompson equation // Applied Mathematics & Information Sciences. 2015. Vol. 9. No. 5. P. 2233—2238.

32. Marin M., Othman M. I. A., Seadawy A. R., Carstea C. A domain of influence in the Moore — Gibson — Thompson theory of dipolar bodies // Journal of Taibah University for Science. 2020. Vol. 14. No. 1. Pp. 653—660.

33. Craciun E. M., Baesu E., Soys E. General solution in terms of complex potentials for incremental antiplane states in prestressed and prepolarized piezoelectric crystals: Application to Mode III fracture propagation // IMA Journal of Applied Mathematics (The Institute of Mathematics and its Applications). 2005. Vol. 70. No. 1. Pp. 39—52.

34. Nasr M. E., Abouelregal A. E. Light absorption process in a semiconductor infinite body with a cylindrical cavity via a novel photo-thermoelastic MGT model // Archive of Applied Mechanics. 2022 Vol. 92. No. 5. Pp. 1529—1549.

35. Abouelregal A. E., Sedighi H. M., Sofiyev A. H. Modeling photoexcited carrier interactions in a solid sphere of a semiconductor material based on the photothermal Moore — Gibson — Thompson model // Applied Physics. A. 2021. Vol. 127. No. 11. P. 845.

36. Abouelregal A. E., Atta D. A rigid cylinder of a thermoelastic magnetic semiconductor material based on the generalized Moore — Gibson — Thompson heat equation model // Applied Physics. A.

2022. Vol. 128. No. 2. P. 118.

37. Пресс У. Х., Теукольски С. А., Фланнери Б. П. Числовые рецепты на Фортране 77. Искусство научных вычислений (2-е изд.). Нью-Йорк: Изд-во Кембриджского университета, 1992.

THE AUTHORS

KAUR Iqbal

Government College for Girls, Department of Mathematics Palwal, Kurukshetra, Haryana, India [email protected] ORCID: 0000-0002-2210-7701

SINGH Kulvinder

Kurukshetra University, University Institute of Engineering & Technology, Department of Computer Science & Engineering

Kurukshetra, Haryana, India-136119

[email protected]

ORCID: 0000-0002-2717-0419

СВЕДЕНИЯ ОБ АВТОРАХ

КАУР Икбал — Ph.D., адъюнкт-профессор математики кафедры математики Государственного колледжа для девушек (городок Палвал), г. Курукшетра, штат Харьяна, Индия. Palwal, Kurukshetra, Haryana, India [email protected] ORCID: 0000-0002-2210-7701

СИНГХ Кулвиндер — Ph.D., доцент кафедры информатики и инженерии Инженерно-технологического института Университета Курукшетры, г. Курукшетра, штат Харьяна, Индия. Kurukshetra, Haryana, 136119, India [email protected] ORCID: 0000-0002-2717-0419

Received 10.05.2022. Approved after reviewing 11.10.2022. Accepted 11.10.2022. Статья поступила в редакцию 10.05.2022. Одобрена после рецензирования 11.10.2022. Принята 11.10.2022.

© Санкт-Петербургский политехнический университет Петра Великого, 2023