Модель двухфазного сдвига фототермоупругих волн в двумерной полупроводниковой среде Текст научной статьи по специальности «Физика»

УДК 539

Модель двухфазного сдвига фототермоупругих волн в двумерной полупроводниковой среде

A.D. Hobiny1, I.A. Abbas12

1 Университет короля Абдулазиза, Джидда, 21589, Саудовская Аравия 2 Сохагский университет, Сохаг, 82524, Египет

В статье с использованием обобщенной модели фототермической волны и модели двухфазного запаздывания проведены расчеты приращения температуры, компонент смещения, плотности носителей и компонент напряжений в двумерных полупроводниковых средах. С помощью преобразований Фурье и Лапласа в рамках подхода к определению собственных значений получены точные решения для всех физических величин. Исследование полупроводниковой среды проведено на примере кремния. С использованием полученных результатов проиллюстрирована разница между моделями динамической связи, Лорда-Шульмана и двухфазного сдвига.

Ключевые слова: фототермические волны, модель двухфазного запаздывания, метод собственных значений

DOI 10.24411/1683-805X-2019-12011

A dual phase lag model of photo-thermoelastic waves in a two-dimensional

semiconducting medium

A.D. Hobiny1 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

In this article, the generalized model for photothermal wave under dual phase lag model is utilized to compute the increment of temperature, the components of displacement, the carrier density and the stress components in a two-dimension semiconducting media. By using Fourier and Laplace transformations with the eigenvalue techniques methodology, the exact solutions of all physical quantities are obtained. A semiconductor media such as silicon has been studied. Finally, the outcomes are represented graphically to display the difference among the models of classical dynamical coupled, the Lord and Shulman and the dual phase lag.

Keywords: photothermal waves, dual phase lag model, eigenvalues approach

1. Introduction

The study of a semiconductor with band gap energy E is lighted by a laser irradiation with energy E higher than E , and followed by the electrons excitation procedures that will take places. Electrons can be transferred to the levels of energy from the valence bands (E - E , while E is the incident photon energy) above the conductance band edge only if E > Eg. Free photoexcited carriers relax at one of the unfilled levels near the bottom of the conductance band during nonradiative transitions. A recombination process will occur during the formation of electron-hole pairs. There is electron-hole plasma before the recombination process. The difference influences of the thermo-

elastic and electronic deformations in semiconductor media with disregard the coupling between the plasma and the thermoelastic equations have been analyzed by numerous researchers [1-3]. Previously, Todorovic et al. [4-6] have studied the theoretical analysis to describe two phenomena that provide information about the properties of transport and carrier recombinations in the semiconducting medium. The changes in the propagations of thermal and plasma waves go back to the linear coupling between the thermal and the mass transport (i.e., thermodiffusion) has included. The local thermoelastic deformation at the model flat cause to the excitation has been studied by Rosencwaig et al. [7]. Opsal and Rosencwaig [8] have presented the depth profil of the plasma and the heat wave

© Hobiny A.D., Abbas I.A., 2019

propagation in silicon [7]. Abbas et al. [9-11] has investigated the photothermal wave in a one-dimension semiconducting material.

Biot [12] presented the coupled thermoelastic model to overcome the first lack of classical thermoelastic model, that it presented two conflicting phenomenons with the physical observation.

Lord and Shulman [13] established a new theory based on one relaxation time. Tzou [14] has presented the dual phase lag (DPL) model to describe the interactions among photon and electron on the microscopical levels. On the microscopical scale the delaying resources caused some delayed responses. Tzou [15] has supported the applicability and the physical meanings of the dual phase lag models through the experimental outcomes. Abbas and Zenkour

[16] have used dual phase lag model to study the thermo-elastic interaction in a semi-infinite thermoelastic media subject to a ramp-type heating. Furthermore, Song et al.

[17] investigated the vibrations under the generalized ther-moelastic theory due to optically excited semiconducting microcantilevers. They concluded that the reflection of waves in a semiconducting plane under photothermal with generalized thermoelastic theories [18, 19]. In the domain of Laplace, the eigenvalues method gave exact solutions without any supposed restrictions on the factual physical variables. Abbas [20-23] applied the eigenvalues method to investigate the thermoelastic problems under fractional order derivative. Ezzat et al. [24] have studied the thermo-elastic interaction in metal film under fractional ultrafast laser-induced effect.

The objective of the present paper is to introduce the studying is based on the thermoelastic and plasma waves under dual phase lag theory. The photothermoelastic interaction in a two-dimension semiconductor material are studied. Based on the eigenvalues approach, Laplace and Fourier transforms, the governing equations are processed by using the analytical-numerical methods. The numerical calculations have been done for silicon-like semiconductor medium, and the results are represented graphically to show the difference between DPL, LS and CT models.

2. Basic equations

For an isotropic, elastic and homogeneous semiconductor material, the governing relations of motions, plasma and the heat conduction under dual phase lag model can be introduced by [25]

(X + \i)Ujj + liUi j - ynNj - Yt© i = P

%j/

dt2

dN N r © * i1+tef h=- ^ N+

(1) (2)

d tq d2

1 + tq— + m—--r-

q dt 2 dt2

d© ,,, pCelk +Y tTo

dt

(3)

The stress-strain relations tensor can be expressed by

= V(Ui,j + Uji) + &ukk - YnN - Yt©)8ij. (4)

This model can be reduced to:

(i) DPL point to the dual phase lag model

0 < te< tq, m = 1,

(ii) LS point to Lord and Shulman's model

tq = to > 0, te= m = 0,

(iii) CT point to the classical coupled model

te= tq = m = 0,

where Gy are the stress components, te and tq denote the delay times which called the phase lag of the temperature gradient for the delay time te and the phase lag of the heat flux for the other delay time tq, yn = (3X + 2^)dn, dn is the electronic deformation coefficient, Ce is the specific heat at constant strain, ut are the displacement components, Q is the moving heat source, © = T -T0, T0 is the reference temperature, N = n - n0, n0 is the equilibrium carrier concentration, 8E = E - E , Eg is the semiconductor energy gap, E is the excitation energy, De is the carrier diffusion coefficient, p is the medium density, K is the thermal conductivity, X, ^ are the Lame's constants, Yt = (3X + 2^)at, at is the coefficient of linear thermal expansion, t is the photogenerated carrier lifetime and £ = dn0/d© is the thermal activation coupling parameter [26], i, j, k = 1, 2, 3, r is the vector of position, t is the time. In the case of plane strain state for a 2D problem of a semiconducting plane, the displacement components can be defined by

u = (u, v, 0), u = u(x, y, t), v = v(x, y, t). (5)

Therefore, the Eqs. (1)-(4) take the form:

d 2u .. „ . d 2u .. . d2 v d 2u

p-y = (X + 2^)—2 + (X + - + -

dt dx dxdy dy

dN ' dx

d© dx

(6)

d2 v _ .d2 v . d2 v d2 v

dt dy dxdy dx

- dN - d© dy dy

dN_ dt

(a 2

= De

2*r 1

d 2 N d 2 N dx2 + 3y2

N r©

—+Z—,

T T

(7)

(8)

+

(d2© d2©1

dx dy2

v ' J

= -Eg N +

T

d tq2 d2 1

1 + t— + m——t-q dt 2 dt2

(

d© „ PCe^ + YtT0

( d2u d2v +

dtdx dtdy

(9)

( \ CTx

yy

V ■ 7

_ .du - dv ^

(À + 2^)—+K—-ynN-Yt © ox ay

- du . dv ,T

+ (À + -YnN-Yt © ox ay

( du + dv ^ dy dx

(10)

3. Applications

The problem initial conditions can be expressed by

N ( x,y,0) = NdM = 0, at

u(x, y, 0) = 0,

at

v( x, y, 0) = ^^ = 0, (11)

dt

T(x, y, 0) = = 0.

dt

While at x = 0, the boundary conditions can be considered by

- * fi+teAla©( x, y,t) = α2L

f 9dt J dx 16tp

dN

Vxx = xy = 0, De — - S0N = 0,

H(a- | y |),

(12)

where s0 is the velocity of surface, q0 is a constant and H is the Heaviside unit function. For conveniences, the dimen-sionaless variables can be considered by

(x*, y*, u, v*) = nc(X, y, u, v),

(t , T , Îq , tp ) = nc (t, ^ Îq , tq , tp ^

(aXx, °*yy, aXy ) = ""(axx, °yy, axy ),

N = N, ,

n0 T0

where c2 = (À + 2m)/p, n = pcjK.

In the nondimensional terms of variables in (13), the basic equations with boundary and initial conditions can be introduced by the forms (the dash has been neglected for appropriateness)

d2u , .. d2v d2u

a—y + (a-1)-+ —Y -

dx dxdy dy

0 dN 0 d© d 2u

-ßn ä--ßt — = a

dx

t dx ™

d 2v , ^ a—-+(a-1)

dy2

dt2

d 2u + d2v dxdy dx2

-ß dN-ß d© = ad2v

n dy t dy dt2 d2 N d2 N N n © dN

dx2 dy2

(14)

-Y—+ ß— = y^t~ dt

(15)

(16)

1 +19 — 9dt

( d^© + d^© ^ dx2 dy2

d t2 d2 ^ 1 +t — + m-^—r q dt 2 dt2

N

= -E1 — +

d© dt

( 3 2u 3 2 v ^

dtdx dtdy

(17)

( ^

V ■ y

(a^+(a-2) ^-ßnN-ßt ©^ dx dy

a^+(a-2) -ßnN-ßt© dy ox

du + dv

dy dx

(18)

dV©(x, y, t) t2 e'/'p rr

1 + ^ 1 (d H(fl- |y|)'

dt) dx 16'2

nW„n (19)

CT xx =CT xy = o, — = 0,

where a = (A + 2|a)/^, ßn = «0Y„/^, Y = V(n^e), ßt = = ToYt/ ^, ß = CTo/(nonDe), ei = no/(nKTo), £2 = = Yt/ (Pce) and e= sJ(Denc).

The transformation of Laplace of a function ro(x, y, t) can be introduced by

ö>(x, y, s) = L[o>(x, y, t)] = J o>(x, y, t) e"stdt, s > o, (20) o

where s is the parameter of Laplace transformation. Hence, Eqs. (14)-(19) can be expressed as

d2U . d2 a—2-+(a-1)—-—+ dx dx

d2 v

d 2u -ßn dN -ßt d© dx

w dx

d2 v -ßn dN ßt d© ßt dy

dx2 ' dy ~

d2 N d2 N N „© -- + —— = Y--ß—+ syN ,

T T

dx2 dy2

(1 + St9 )

(

(23)

( 3 2© 3 2©^

dx dy

N

= -£,— +

+ S

d t2 2 ^

1 + stq--+ m^—s

q dt 2

© + e-

(- \

yy

Sxy

V ■ y

( 3u + dv ' dx dy

(adu+(a- 2) ^-ßnN-ßt ©

dx dy

adv + (a-2) ^-ßnN-ßt©

dy ox

du + dv dy dx

\

(24)

d©

-qotp

dx 8(1 + ste )( stp +1)

dN

°xx =°xy = 0 N =

-H(a- | y |),

(26)

4. Solutions in the domain of Fourier transformations

The Fourier transform for ro(x, y, s) takes the form

ro* (x, q, s) = J ro(x, y, t) e~iqydx. (27)

0

Hence, we can obtain the following differential equations system

d2u * , dv* 2_» a—2- + (a- 1)iq--q u -

dx2 dx

dN* n d©*

• = s au

Pn dx P' dx

2 _» , ... du* d2v* q av + (a- 1)iq-+ —2

dx dx2

- PniqN* - Ptiq©* = s2av*

d2 N *

dx (1 + ste)

2r-r» N* „©* rr*

+ q N =Y—-P—+ syN

f d2©*

T T

A

(28)

(29)

(30)

2*

dx2 " q ©

V y

N * <

= -e,--+ s

T

.2 A

q 2

1 + stq + m—s q 2

©* + e2

du * dx

■ + iqv

(31)

f- * A

xx

*

yy

*

^ xy /

du*

a — + iq(a - 2)v - PnN - Pt©* dx

du *

aiqv* + (a- 2)—-PnN* -Pt©* dx

* dv*

iqu +-

dx

(32)

-q°tp

2 sin(qa)

d©* =_,

dx 8(1 + ste )( stp +1)3M n q

_* _* _ dN *

Gxx =Gxy = °> "dx"

-ZN * = 0.

(33)

To obtain the simultaneous equation solutions (28)-(31) in the domains of Fourier-Laplace transformations, write them as follows:

d2w * = s2a + q2 * _ (a - 1)iq dv* +

dx2 a

+ P± N. + P d©*

a dx

a dx a dx

d2v* dx2

= a(s 2 + q 2)v* + PniqN* +

+ Ptiq©* -(a- 1)iq^^, dx

d2 N * f 2 yw P^*

-^ = 1 sy + q2 + ± IN --©*

dx2 i t) t

(36)

d2©* = se2iq(1 + stq + m( t^/2) s2) _,

dx2

1 + st

f

v* -

e

t(1 + ste )

-N * +

2\ A

2 s(1 + st„ + m( t2/2) s 2)

q +---—-

1 + ste

se2(1 + stq + m( t] /2) s2) dw *

1 + ste

dx

©* +

(37)

The forms of matrix-vector of Eqs. (34)-(37) can be ex pressed by [27-31] dO

dx where

O =

= AO,

(38)

u v N ©*

du * dv* dN* d©* dx dx dx dx

A=

f ° 0 0 0

a51 0 0 0

0 0 0 0 0

0 0 0 0 0

a56 a57

a62 a63 a64 a65

0 a.

73 a74

a82 a83 a84 a85

0 A 0

0

1

a58 0 0 0

with

25! = (s2a + q2)/a, a56 = -(a- 1)iq/a, a57 = Pn/a, = Ptla, 262 = a(s2 + q2), 263 = pniq,

= Ptiq, 265 = -(a - 1)iq, 273 = sy+q2 + y/t ,

274 =-ppт, 282 =

se2iq(1 + stq + ms212/2)

283 =

285 =

t(1 + ste )

se2(1 + stq + ms 21212)

1 + ste

2 s(1 + stq + ms21?/2)

284 =q +--

1 + ste

1 + ste

Now, by using the eigenvalue techniques [27-31], the analytical solutions of Eqs. (38) can be obtained. The characteristic equation of matrix A can be given in the form:

- M ^ + Ft,4 + G^2 + R = 0, (39)

where

R = ^^^^ - -

- ^X^2^^ + a51a62a73a84'

M = a5i + a62 + a56a65 + a73 + a84 + a58a85 ,

F = a5ia62 + a51a73 - a62a73 + a56a65a73 -

- a64a82 - a58a65a82 - a74a83 + a51a84 +

+ a62a84 + a56a65a84 + a73a84 + a58a62a85 -

- a56a64a85 + a58a73a85 - a57a74a85,

^ = -a51a62a73 + a51a64a82 + a64a73a82 + + a58a65a73a82 - a63a74a82 - a57a65a74a82 +

a51a62a84 a51a73a84 a62a73a84

51a73a84

a62a73a84

a56a65a73a84 a58a62a73a85 + a56a64a73a85 + + a57a62a74a85 - a56a63a74a85-

The eigenvalue of matrix ^ are the outcomes of Eq. (39) which take the form ±^2' ±^3' ±^4- Thus, the corresponding eigenvectors of eigenvalue ^ can be determined by

N(x, y, s) = * £ J BkNke-^X+qdq, (45)

V2n k=1_^

©(x, y, s) = 1 £ J BkOke~^kx+iqydq, (46)

V2n k=1 _J

CTxx (x, y, S) =

1 4 J

£ J Bk (_at,kuk + iq(a _ 2)Vk _

V2n k=1 _PnN _Pt©k ^+iqydq:

(47)

ctyy (x, y, s) = -7= £ J Bk (iqaVk _ Ik (a _ 2)uk _ V2n k=1 _J

_PnN _Pt©k )e"Çkx+iqydq.

CTxy (x, y, S) = 1 4

(48)

-\/2n k =1 _,

£ J BkKkVk + iquk )e"^kx+iqydq, (49)

X =

(_(a58 _ a62) + a56a64 )(^2 _ a73 ) _ (a57 _ a62 ) + a56a63 )a74

2 2 2 2 2 _^58(((H, _a51>a64 + E, a58a65)(^ _a73> + ((E, _a51>a63 + E, a57a63)a74>

£a58 (-(^2 _ a51)(^2 _ a62) + ^a56a65 )a74

_^a58 ((^2 _ a51)(^2 _ a62) _ ^a56a65 )(^2 _ a73 )

£(^a58 (_(a58 (^2 _ a62) + a56a64 _ a73) _ (a57 _ a62) + a56a63 )a74 ))

^(_^a58(((^2 _a51)a64 + £2a58a65)(£2 _a73) + ((£2 _a51)a63 + £2a57a65)a74))

£(£a58 (-(^2 _ a51)(^2 _ a62) + £2a56a65 )a74 ) ^(_^a58((^2 _ a51)(^ _ a62) _^2a56a65)(^2 _ a73 ))

(40)

The solutions of Eq. (38) can be expressed by

0(X, q, s) = £ BXe-^, (41)

i=1

where the terms containing exponentials of growing nature in the space variable x are discarded due to the regularity condition of the solution at infinity, B1, B2, B3 and B4 are constants to be determined from the boundary condition of the problem.

5. Double transforms inversion

The domain of Fourier transform expressions for functions $(x, q, s) can be determined by

0(x, y, s) = J 0(x, q, s)eiqydq.

V2n _J

(42)

Thus, the solutions of physical quantities which depend on s, x and y can be given by

1 4

r(x, y, s) = ' £ J Bkuke-^qydq, V2n k=1 _J

;(x, y, s) = * £ J Bkvke~^qydq, V2n k=1 _J

(43)

(44)

where uk, vk, Nk and ©k are the corresponding eigenvector components. To obtain the solutions at the time t for the carrier density, the temperature increment, the components of displacement and the stress components with respect to the spaces x andy, we use the Stehfest [32] method of numerical inverse. The inverse œ(x, y, t) of the Laplace transformations ffl(x, y, s) can be approximated by

ln2 n ( ln2N

ro(x, y, s) =-£ F,œl x, y,-I, (50)

(51)

t n=1 ^ t

where Vn is given by the following equation: Fn = (_1)N 2+1 x

PN 2+1(2 p )!

min( n, N 2)

x £ —, ,

p^t+D/2 p!(N/2_ p)!(2n_ 1)!(n_ p)!

where N is the number of terms used in the summation in Eq. (50) and should be optimized by trial and error.

6. Numerical results and discussion

To understand the theoretical outcomes shown in the previous section, the numerical values of the physical parameters are presented. In this problem, the parameters can

+ a51a74a83 + a62a74a83 + a56a65a74a83

N 15010050-

-CT

---LS

К ----DPL

л

\\

0

1.00.60.2-0.2

xV — CT LA ---LS ----DPL

0

1

Fig. 2. The variation of carrier density N (a) and temperature © (b) with the distance x when y = 0.2 for three different theories

Fig. 1. Contour plots of the temperature (a) and carrier density distribution (b) for the DPL model at t = 1.5 (color online)

be considered in SI unit [33]

X = 3.64• 1010 N• m-2, p = 2330 kg- m-3, ц = 5.46 -1010 N - m-2, E = 2.33 eV, Eg = 1.11 eV,

s0 = 2m-s-1, De = 2.5- 10_J mz -s

,-3

at

- 3-10-6 K-1, ce

695 J - kg-1K-1, T0 = 300 K,

tQ = 0.3, n0 = 1020 m-3, dn = -9• 10-31 m3, t = 5-10-5 s,tq =0.35.

The half-space is considered to be two-dimension semiconductor medium which is an isotropic and homogeneous. Moreover, the elastic and thermal properties are considered without drop out the coupling between the waves due to the thermoelastic and plasma conditions. Figure 1 shows the temperature and carrier density contours. We find that the temperature changed zones restricted in a finite area while the temperature does not change out of this area. In addition, there are zones with a temperature gradient much higher than that of another zone. This means that the heat is done at a limited velocity.

Figures 3 to 7 show the carrier density variation N, the increment of temperature ©, the components of displacement u, v and the components of stress axx, axy distributions along the distance x and the distance y at t = 1.5. These figures display the predict curves under various theories of thermoelasticity. The classical dynamically coupled theory (CT) appeared in solid lines, the dashed lines refer to Lord and Shulman's theory, while the DPL solutions are taken as dotted lines. It is noted that the carrier density N has high value at the surface x = 0.0 and reduces with the increase of x to reach to zero when x = 3.5 as in Fig. 2, a.

Figure 2, b displays the variations of temperature along x. It is noticed that the increment of temperature has utmost value at x = 0.0 and decrease gradually with the increasing of x till reach to zero beyond a wave front for the theory of generalized photothermoelasticity. Figure 3, a predicts the variations of horizontal displacement u along x. It is clearly noted that it reaches the highest negative values and increases progressively till it reaches peak values at a particular location near to x = 0, after that decreases to close to zero. The variations of vertical displacement attain maximum values on x = 0 and decreases with the increasing of x as in Fig. 3, b. Figure 4 displays the variations of stress components axx, axy with along x. It is noticed that the stress magnitudes always begin from zero that satisfied the

T

о

-1

5-з-

-5-

f " ^Vs^ a

ff 1

- -У

f

-CT

-7 ---LS

-1-1- ----DPL

10

0 ^

1

2-

-2

л — CT LA ---LS ----DPL

л

0

1

Fig. 3. The variation of horizontal u (a) and vertical displacement v (b) with the distance x when y = 0.2 for three different theories

Fig. 4. The variation of stress a^ (a) and a(b) with the distance x when y = 0.2 for three different theories

boundary conditions and then increases with the increasing of the distance x till x = 1.

Figure 5 displays the carrier density N and the temperature 0 variations along y and they point that the carrier density and the increment of temperature have ultimate values at the length of thermal surface (|y| < 0.5) and they start to reduce just near the edges (y = ±0.5) where they smoothly decrease and finally reach zero value. Figure 6, a displays the variations of horizontal displacement u with respect to x and it indicate that the magnitude of the displacement has ultimate values at the length of the thermal surface (|y| < 0.5), and it begins to reduce just near the edges (y = ±0.5), and after that reduces to zero value. Figure 6, b predicts the variations of vertical displacement v with

Fig. 5. The variation of carrier density N (a) and temperature 0 (b) with the distance y when x = 0.2 for three different theories

theories

respect to y. We find that the vertical displacement starts raising at the beginning and ending of the thermal surface (|y| < 0.5), and has a minimum value at the middle of the thermal surface, then it starts increasing and come to a maximum just near the edges (y = ±0.5), after that it decreases to reach to zero. The stress components axx and axy with respect to the distance y are shown in Figs. 6, b and 7, b respectively. The results predicted the difference between the use of coupled photothermoelastic theory, the generalized photothermoelastic theory with one relaxation time and generalized photothermoelastic theory with the dual phase lag model. In this problem, the most important observed phenomenon where the medium is unbounded is that the solution of any of the considered function for the

lb

l/\

ji \

i 7

V // -CT

W---LS

V ----DPL

Fig. 7. The variation of stress axx (a) and a^ (b) with the distance y when x = 0.2 for three different theories

generalized models vanishes identically outside a bounded region of space.

Funding

This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No (D-061-130-1439). The authors, gratefully acknowledge the DSR technical and financial support.

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Received 22.03.2019, revised 22.03.2019, accepted 29.03.2019

Сведения об авторах

Aatef D. Hobiny, Dr., King Abdulaziz University, Saudi Arabia, ahobany@kau.edu.sa

Ibrahim A. Abbas, Prof. Dr., King Abdulaziz University, Saudi Arabia; Prof., Sohag University, Egypt, ibrabbas7@science.sohag.edu.eg