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УДК 539.3
Анализ фототермоупругого поведения полупроводниковой среды под воздействием движущегося источника тепла
F.S. Alzahrani1, I.A. Abbas12
1 Университет короля Абдулазиза, Джидда, 21589, Саудовская Аравия 2 Сохагский университет, Сохаг, 82524, Египет
Предложена математическая модель фототермической теории Грина-Нагди для изучения распространения волн, индуцированных движущимся источником тепла, в двумерном полупроводниковом материале. С использованием преобразований Фурье и Лапласа на основе метода собственных значений аналитически получены физические величины. Первоначально предполагается, что среда находится в состоянии покоя и подвергается воздействию источника тепла, движущегося с постоянной скоростью в условиях отсутствия трения. Поведение полупроводниковой среды рассмотрено на примере кремния. Численные расчеты с использованием предлагаемого метода проведены для полупроводниковой среды в упрощенной геометрии. Изучено влияние различных значений скорости движущегося источника тепла для всех физических величин.
Ключевые слова: модель Грина-Нагди, преобразования Лапласа-Фурье, фототермические волны, метод собственных значений
DOI 10.24411/1683-805X-2019-15011
Analysis of photo-thermo-elastic response in a semiconductor media due to moving heat source
F.S. Alzahrani1 and I.A. Abbas12
1 Nonlinear Analysis and Applied Mathematics Research Group, Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Sohag University, Sohag, 82524, Egypt
A mathematical model of Green-Naghdi photothermal theory is given to study the wave propagation in a two-dimensional semiconducting material due to moving heat source. By using the Fourier and Laplace transformations with the eigenvalues method, the physical quantities are obtained analytically. Initially, it is assumed that the medium is at rest and it is subject to a heat source in motion with a constant velocity, which is free of traction. A semiconductor media such as silicon has been studied. The derived method is evaluated with numerical results which are applied to the semiconductor medium in simplified geometry. The influences of the different values of moving heat source speed are discussed for all physical quantities.
Keywords: Green-Naghdi model, Laplace-Fourier transforms, photothermal waves, eigenvalues approach
1. Introduction
The most previous studies considering the thermal and elastic properties of the semiconducting elastic medium are isotropic and homogeneous. Whereas the equations of plasma, thermal and elastic waves are partially coupled and also the coupling between them was neglected. Solving the system with coupling the plasma, the thermal and elastic equation is very complex. But, the analysis with partially coupled is enough in most experimental studies. In those work, the coupling between the plasma, thermal, and elastic waves was neglected. As an especially cases the problem of thermoelastic and electronic distortion were taken into account. The effect of coupling was studied in terms of approximately quantitative analysis. Studying the exci-
tation of short elastic pulses by photothermal means is important for engineers and physicists because it is applied in several areas, such as the monitoring of laser drilling, the determination of the parameters of the thermoelastic material, the photoacoustic microscope, the formation of images by thermal wave, laser annealing and fusion phenomena. The difference influences of thermoelastic and electronic deformations in semiconductor media with disregard the coupling between the plasma and thermoelastic equations have been analyzed. Todorovic et al. [1-3] performed the theoretical analysis to describe two phenomena that provide information about the transport properties and carrier recombination in the semiconductor material. The changes in the propagations of photothermal waves go back to the
© Alzahrani F.S., Abbas I.A., 2019
linear coupling between the heating and mass transports (i.e., thermodiffusions) has inclusive. In the materials science, the variable thermal conductivity that depends on temperature is very important and has many applications in nature. Recent studies of the thermal conductivity dependence of semiconductors on temperature showed that physical properties, especially deformation and thermomecha-nical behavior, are strongly affected by any change in material temperature. Rosencwaig et al. [4] studied the local thermoelastic deformations at the model flat cause to the excitation. Green and Naghdi [5, 6] proposed a new generalized thermoelasticity theorem by consists of the thermal-displacement gradient among the independent constitutive variables. Othman and Marin [7] studied the ther-moelastic interactions on porous material under Green and Naghdi theory due to laser pulse. Abbas and Abbas et al. [8-17] applied the generalized thermoelastic theories to get the numerical and analytical solutions of physical quantities. Marin and Ochsner [18] have presented the effects of a dipolar structure under Green-Naghdi thermoelasticity. Moreover, Song et al. [19] presented the vibration by the generalized theory of thermoelasticity subject to optically excited semiconducting microconductors. Lotfy and Lotfy et al. [20-23] have solved some problems by applied various fields in semiconductors materials. The eigenvalue method gave an analytical solution without any supposed restrictions on the factual physical variables in the Laplace domain.
The aim of the present article is to introduce a unified mathematical Green-Naghdi model for photothermoelastic case. By using the eigenvalue approach and Fourier-Laplace transformations based on an analytical-numerical method, the governing equations are processed. For the considered variables, the numerical results are obtained and presented graphically.
2. Mathematical model
Consider an isotropic, homogeneous and elastic semiconducting media, the basic equation of plasma, thermal conduction and motion based on Green and Naghdi model can be expressed by [2, 24]
(X + ]\)Uj j + jj - Y„NJ -yt®i -P—f
dt2
= 0,
N s © dN _
JJ T T dt
+ ^ N -
T
(1) (2)
* + * at I©;
d! dt
d© „ PCe~di + yT°
BU;
dt
- Q
= 0.
(3)
The stress-displacement relations can be expressed as = VtVij + Uji) + ßukk - YnN - Yt©)Sj, (4)
where Q is the moving heat source, p is the material density, oij are the components of stress, N = n - n0, n0 is
the equilibrium carrier concentration, © = T - T°, T° is the reference temperature, X, ^ are the Lame's constants, ui are the components of displacement, De is the coefficient of carrier diffusion, Yt = (3X + 2^)at, at is the linear thermal expansion coefficient, yn = (3X + 2^)dn, dn is the electronic deformation coefficient, 8E = E-Eg, E is the excitation energy, Eg is the semiconducting energy gap, K is the thermal conductivity, * is the material constant characteristic of the theory, ce is the specific heat at constant strain, r is the position vector, t is the time, 8 = dn°/d© is the thermal activation coupling parameter [25], t is the photogenerated carrier lifetime, i, j, k = 1, 2, 3. Taking into account the stress state of the plane in a two-dimensional semiconducting problem, the components of the variables are defined by
© = ©(x, y, t), N = N(x, y, t), u = (u, v, 0), u = u(x, y, t), v = v(x, y, t). Subsequently, the equations (1)-(4) can be given by d 2u . d2v d 2u
(5)
(X + 2^) + (X + ^) dx
dxdy dy
dN _
dx
d2 v
d©
dx
d 2u
P^T = 0, dt2
(6)
(X + 2^)-^ + (X + ^) dy
De
dN - d0
Yn ^ ^t
dy dy
f d 2N d 2N ^
d 2u dxdy
d 2v
d^v dx
2
dx dy
"it?=o-- K+^M = 0,
T T dt
(7)
(8)
* *+* f(#*
dt J| dx2 dy2
Eg df d©
+ N-—\ pc— + T dt I e dt
+ YT°
fd 2
u d v
+
2„ ^
-Q = °,
dtdx dtdy
y J
.. _ .du . dv ,T ,,
° xx = (X + 2^) + Y nN-Y A
dx dy
f
du + dv dy dx
\
(9)
(10)
The problem initial and boundary conditions can be defined by
T ( x, y,°) =
N ( x, y, °) =
dT(x, y,°) dt
dN (x, y,°)
u( x, y, °) =
v( x, y, °) =
dt
du(x, y, °) dt
dv(x, y, °) dt
=
=
=
=
0(0, y, t) = 0, De MdZiO = N(0, y, t),
dx (12) CTxx(0, y, t) = ctxy (0, y, t) = 0,
where s0 is the surface recombination speed. For convenience, the dimensionless variables can be taken as (t', T) = Zc2(t, t), (x', y, u , v) = Zc(x, y, u, v),
Q- 0=0 2 ' T '
T0 (13)
-=-, Q = 2 2
«0 KT0Z2c2
1
(CT'xx , ) = T xx > XV )'
M.
xx xV
where c = ^(A + 2^)/P, Z = Pce/k.
In these nondimensional terms of the physical quantities in Eq. (13), the above Eqs. (5)-(12) can be expressed as in the following forms (the dash has been dropped for convenience)
d2u d2v —— + mi-
dx2 dxdy
+ m0
d 2u
dV2 '
-ß« d-_ßt =o,
« dx dx dt2
d^v
dv
d 2u
d 2 v
+ m2 d 2 dx
2 1 dxdy lu
_ß d- _ß dQ d2v « dy t dy dt2
- „©
= 0,
d 2 - 3 2 -
dx2 + dy2
d-
_as—+ß--as—- = 0,
dt
Y + dt
3 )f 3 2© d^©) dy 2
3_
dt
dx2
V d©
-
+ ^2--
T
dt
+ni
f 3 2u 3 2 v ) +
dtdx dtdy
_ Q
= o,
du dv „ AT „ ^
CTxx = ^ + m3 ^_ßn-_ßA
ox oy
xy = m2
f du + dv ^ dy dx
0(0, y, t) = o, -^ = <-(o, y, t) = o,
ox
(14)
(15)
(16)
(17)
(18)
(19)
CTxx (0, y, t) = CTxy (0, y, t) = 0, where
m1 = (A + M-)/(A + 2|x), m2 = (A + 2|x), m3 = A/ (A + 2|x), ni =Yt / (PCe), Y = K*/(kpcec2), n2 = ^g «<>/№0), 9 = sJ(Denc), P« = «0 Yn/ (A + 2^), Pt = T Yt/ (A + 2(x), as = 1/ (n De), P = ZT0l(mDe).
Now, we consider that the plane is induced by moving thermal source along x axis with a constant velocity ro which
is assumed in the following nondimensional form
Q = Q0§(x-rot)H(b -|y|), (20)
where 8 is the delta function, Q0 is constant and H is the step function of the Heaviside unit.
3. The Laplace-Fourier transforms
The Laplace transforms for any function Z(x,y, t), can be expressed by
Z(x, y, p) = J Z(x, y, t) e~ptdt, p > 0, (21)
0
while the Fourier transforms for any function can be defined by
Z*(x, q, p) = J Z(x, y, p) e 'qydx.
(22)
Thus, the governing equations with initial and boundary conditions can be written to obtain the following ordinary differential equations system
d2u
dx
/ 2 2x-* ■ dv 2 = (p + m2q )u _ m^q— +
dx
„ d- „ d©' + ß„ — + ßt-
d2 v
dx2
dx dx
p2 + q2 =» + ßniq
(23)
v+
- +
+ ßtiq © » m1iq du
m2 dx
d2 -
dx
2 =|p2as + q2 + -_ß 0»
(24)
(25)
d20*
dx
— = nq
f
2p q +-
Y + p
2 )
-W - +
Y + p
^ » p du _ _
© +n1---fe
Y+ p dx
du
axx =-+ iqm3v _ßnN _ßt©
dx
xy = m2
dv )
iqu +
dx
d-
© --=CTxx = CTxv =°'
(26)
(27)
(28)
dx ' xx xy
where m = p/ro and f = pQ0/ro^j2/n sin(qb^q. The vector-matrix differential equation of Eqs. (23)-(26) can be written by
dV
— = AV + Fe~mx,
dx
where
(29)
V =
u v - ©
du dv d- d©»
dx dx dx dx F = [ooooooo _ f ]T, while the matrix A = [atj ] = o, i, j =1, ..., 8, excepting
T
a15 = a26 = a37 = a48 = ^ a51 = P2 + miq2> a56 =-mliq, a57 = P„ , a58 =$t>
a62 =(P2 + q2Vm2, a63 = P„iq/m2, a64 = $tiqlm2, a65 = -m1iq/m2,
a73 = P«s + q2 + a JT, a74 = -^/t ,
a82 =niiqp2l(l + p), a83 = -^2/(^7 + p)),
a84 = q2 + pV(Y + P), a85 =ni p2/(Y + P). The general solution K of the nonhomogeneous system (29) are the sum of the complementary solution Vc of the homogeneous equation and a particular solution Vp of the nonhomogeneous system. By using the eigenvalues method which proposed in Ref. [25], the exact solutions of homogeneous system can be obtained. Then, the matrix A has the characteristic equation which can be given by
t8 - fg + fg + f£2 + f4 = 0, where
f = a73 + a84 + a51 + a62 + a56a65 + a58a85, f2 = a51a84 + a62 a84 + a56 a65 a84 + a73 a84 +
(30)
f3 = a64a73a82 + a58a65a73a82 - a63a74a82 -
- a57a65a74a82 - a51a62a73 + a51a64a82 -
- a51a62a84 - a51a73a84 - a62a73a84 -
- a56a65a73a84 - a58a62a73a85 + a53a64a73a85 +
+ a57a62a74a85 - a56a63a74a85 + a51a74a83 +
+ a62a74a83 + a56a65a74a83, /4 = a51a62a73a84 - a51a62a74a83 + a51a63a74a82 -
- a51a64a73a82*
To obtain the solutions of Eq. (29), the eigenvalues and corresponding eigenvectors of matrix A must be calculated. If the eigenvalues take the form ±^2> ±^3> ±^4> the corresponding eigenvectors of eigenvalues t can be considered as
Hence, the complementary solutions of Eq. (29) can be given by
Vc( x, q, p) = i BiYi^'x- (32)
i=1
From Eq. (29), in the inhomogeneous terms, there is the exponential function e~mx, which in the homogeneous equation solution coincides with the exponential function. Thus, the particular solutions hp* should be sought in the form of a quasi-polynomial vector:
Vp = Re~mx, (33)
where R depends on q and p. Thus, the general solutions of the field variables can be written for q, p and t as
4
N*(x, q, p) = 1B1N,e~t-x + De"mx, ¿=1 D
©*( x, q, p) = 1 BTe-^x + D e~mx, 1=1 D
4
u
'( x, q, P) = 1 Bue-^ix + D e-mx, i=1 D
V(x, q, p) = iBv^ix + De-mx, i=1 D
°*xx (x, q, p) =
= 1 B(-tu+1qm3 V-VnN)e~tx + =1
+ i5 e~mx, D
°*xy (x, q, p) =
= 1 Bm2(-tt V + ¿qui )e_t 1 x+^ e~mx,
(34)
(35)
(36)
(37)
=1
D
(38)
(39)
where the terms containing exponentials of growing nature in the space variable x have been discarded due to the regularity condition of the solution at infinity, B1, B2, B3 and B4 are constants which can be calculated by using the problem boundary conditions while ut, vi, Ni and ©i are the components of corresponding eigenvectors with
^ = fa74 (a51(m2 - a62) + m2 (-m2 + a62 + a56a65 X)>
(
Y =
-W
((a58 (a62 + ^2) - a64a56 )(t - a73> - a74 ((t - a62 )a57 + a63a56
2 2 2 2 2 -ta58((a64(t - a51) + t a65a58)(t - a73) + a74(a63(t - a51) + t a63a57))
ta74a58(^2a56a65 - (t2 - a51)(^2 - a62))
ta58 (^2 - a73 )(^2a56a65 + - a51X^ - a62 ))
(a58^2 (a56a64 + a58(^2 - a62 ))(a73 - ^ + - a62 )a57 + a63a56 )a74 №
(-ta58(((t2 -a51)a64 + t2a65a58)(t2 - a73) + (a63(t2 -a51) + t2a65a57)a74))t
(t.a58((t - a51)(a62 -^2) + ^a65a56)a74)t
(ta58 ((t2 - a51 )(a62 - 2) + ^a65a56 X^ - a73 ))t
(31)
+ a51a62 + a51a73 + a62a73 + a56a65a73
a64a82 a58 a65a82 a74a83 + a58 a73a85
s2 = -f (m2 -«7s)(a51(-m2 + a62) +
+ m2(m2 - a56a65 - a62))'
S3 =-fm(a58(m2 -a62)(m2 -a73) + + a56 (a64 (m2 - a73 ) + a63a74) + + a57(m2 - a62)074), s4 = -f (a64(m2 -a62)(m2 -a51) +
+ m2a58a65 (m2 - a73 ) + ((m2 - a51)a63 + + m 2 057065) 074),
^5 = -ms3 + fqm3s4 - p^s1 - 2,
s6 = m2( - ms 4 + iqs3),
D = m8 - fm6 + /2m4 _+/3m2 + /4. Now, for any function Z (x, q, p), its inversion of Fourier transform can be defined by
_ 1 ~ _ .
Z (x, y, p) = f Z *(x, q, p) eiqydq. V2n -L
(40)
Finally, to get the general solutions of temperature, the displacements, carrier density and stresses along distances x, y at any time t, we choose the Stehfest numerical inver-
sion approach [26]. In this approach, the inverse of Laplace transforms for h (x, y, p) can be approximated by
ln2^
In 2 N -
Z ( x, y, t ) =-E VnZ
t n =1
x, y, n-
t
(41)
where Vn is defined by the following relation: Vn = (-1)( N 2+1) X
(2 p )! p( N 2+1)
min( n, N 2)
x E -,
p=(n+1)/2 p!(n- p)!(N2- p)!(2n- 1)!
(42)
where N is the term numbers.
4. Numerical results and discussion
The polymeric silicone appears in the photovoltaic solar cell of the p-n junction and can be manufactured quickly and economically. The values of thermal properties for silicon (Si) like material have been written as [27]
p = 5.46-1010 N• m-2, p = 2330 kg• m-3,
-2
-3
À = 3.64 -1010 N -m-2, E = 2.33 eV, De = 2.5 -10-3 m2 - s-1,50 = 2 m - s-1,
= 1.11eV,at = 3-10-6 K-1,n
-6 ix-1 „ _ 1020 _-3
m
u, 105- l£
0- 1
- 4
-4- //
/
i
1 2 X
d
\
A3 N
X 2 x \ _ •• \ \ ---^ N .
) 1 7 X
Fig. 1. The variation of temperature 0 (a), carrier density N (b), horizontal u (c) and vertical displacement v (d), stress axx (e) and a^ f) along to distance x when y = 0.5 for different values of heat source velocity to = 0.01 (1), 0.03 (2), 0.05 (3)
Fig. 2. The variation of temperature © (a), carrier density N (b), horizontal u (c) and vertical displacement v (d), stress axx (e) and a f) along to distance y when x = 0.5 for different values of heat source velocity to = 0.01 (1), 0.03 (2), 0.05 (3)
dn =-9-10-31 m3, t = 5-10-5 s, T0 =300 K, ce = 695 J • kg-1K-1, b = 0.4.
The above data have been applied to study the effects of the moving heat source speed in the variations of temperature ©, the variations of carrier density N, the components of displacement u, v and the stresses axx, axy. The media is considered to be an isotropic and homogeneous two-dimension semiconducting material. In addition, the thermal and elastic properties are considered without leaving the conjunction between the waves subjected to the plasma and the thermoelastic conditions.
Figure 1, a predicts the increment of temperature along the distance x. It is noticed that it starts from zeros according to the application of boundary condition and increase with x to have utmost values at x = 0.3 and decreases gradually with increasing x to close to zero beyond a wave front for the generalized photothermal theory. Figure 1, b shows the carrier density variation along x. It is observed that the carrier density increasing with increases x to have maximum values on x = 0.4 and decreases with the increasing x
until attaining zero on x = 3. Figure 1, c displays the variation of vertical displacement along x which have maximums values on x = 0 and decreases with increasing x. Figure 1, d shows the variation of horizontal displacement u along x. It is observed that it attains utmost negative value and progressively increases until it attains peak values at a particular location in close proximity to x = 0.0 and then continuously decreases to zero. Figure 1, e display the stress component variation axx along x. It is clear that the stress magnitude always starts from zero which satisfies the boundary conditions. Figure 1, f predicts the variation of stress component axy along x. The stress magnitude always starts from zero which satisfies the problem boundary conditions.
Figures 2, a and 2, b show the variations of the increment of temperature © and the carrier density N along y and they point that the carrier density and the increment of temperature have ultimate values at the length of thermal surface (|y| < 0.4) and they start to reduce just near the edge (|y| < 0.4) where they smoothly decrease and finally close to zero values. Figure 2, c shows the variations of vertical displacement v along y. We find that it starts raising at the
beginning and ending of the thermal surface (|y| < 0.4), and has smallest values at the middle of the thermal surface, then it starts increasing and come to highest values just near the edge (y = ±0.4), after that it decreases to reach to zero. Figure 2, d displays the variations of horizontal displacement u with respect to x and it indicates that it has ultimate values at the length of the thermal surface (|y| < < 0.4), and it begins to reduce just near the edge (y = ±0.4), and after that reduces to zero value. Respectively, stresses axx and axy with respect to y are shown in Figs. 2, e and 2, f As expected, it can be found that the speed of moving heat source have the great effects on the values of all the studied fields.
Funding
This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. D-197-130-1439. The authors, therefore, gratefully acknowledge the DSR technical and financial support.
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Received July 08, 2019, revised August 21, 2019, accepted August 26, 2019
Сведения об авторах
Faris S. Alzahrani, Assoc. Prof., King Abdulaziz University, Saudi Arabia, falzahrani1@kau.edu.sa
Ibrahim A. Abbas, Prof. Dr., King Abdulaziz University, Saudi Arabia; Prof., Sohag University, Egypt, ibrabbas7@gmail.com
