ON MODIFICATION OF PROPERTIES OF P-N-JUNCTIONS DURING OVERGROWTH by ijitmcjournal

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

ON MODIFICATION OF PROPERTIES OF P-N-JUNCTIONS DURING OVERGROWTH E.L. Pankratov1, E.A. Bulaeva1,2 1

Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950, Russia 2 Nizhny Novgorod State University of Architecture and Civil Engineering, 65 Il'insky street, Nizhny Novgorod, 603950, Russia

ABSTRACT In this paper we consider influence of overgrowth of doped by diffusion and ion implantation areas of heterostructures on distributions of concentrations of dopants. Several conditions to increase sharpness of p-njunctions (single and framework bipolar transistors), which were manufactured during considered technological process, have been determined. At the same time we analyzed influence of speed of overgrowth of doped areas and mechanical stress in the considered heterostructure on distribution of concentrations of dopants in the structure.

KEYWORDS Diffusion-junction heterorectifier; implanted-junction heterorectifier; overgrowth of doped area; analytical approach for modeling

1. INTRODUCTION In the present time they are several approaches could be used to manufacture p-n-junctions diffusion of dopants in a homogenous sample or an epitaxial layer of heterostructure, implantation of ions of dopants in the same situations or doping during epitaxial growth [1-7]. The same approaches could be used to manufacture systems of p-n-junctions: bipolar transistors and thyristors. The first and the second ways of doping are preferable in comparison with the third one because the approaches give us possibility to dope locally materials during manufacture integrated circuits easily in comparison with epitaxial growth. Using diffusion and ion implantation in homogenous sample to manufacturing p-n-junctions leads to production fluently varying and wide distributions of dopants. One of actual problems is increasing sharpness of p-n- junctions [5,7]. The increasing of sharpness gives us possibility to decrease switching time of p-n- junctions. Increasing of homogeneity of dopant distribution in enriched by the dopant area is also attracted an interest [5]. The increasing of homogeneity gives us possibility to decrease local overheat of the doped materials due to streaming of electrical current during operating of p-n-junction or to decrease depth of p-n-junction for fixed value of local overheats. One way to increase sharpness of p-n-junction based on using near-surficial (laser or microwave) types of annealing [8-15]. Framework the types of annealing one can obtain near-surficial heating of the doped materials. In this situation due to the Arrhenius low one can obtain increasing of dopant diffusion coefficient of near-surficial area in comparison with volumetric dopant diffusion coefficient. The increasing of dopant diffusion coefficient of near-surficial area leads to increasing of sharpness of p-n-junction. The second way to increase sharpness of p-n-junction based on using high doping of materials. In this case contribution of nonlinearity of diffusion process increases [4]. The third way to increase DOI: 10.5121/ijitmc.2016.4203

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

sharpness of p-n-junction is using of inhomogeneity of heterostructure [16,17]. Framework the approach we consider simplest heterostructure, which consist of substrate and epitaxial layer. One can find increasing of sharpness of p-n-junction after annealing with appropriate annealing time in the case, when dopant diffusion coefficient in the substrate is smaller, than in the epitaxial layer. The fourth way to increase sharpness of p-n-junction is radiation processing of materials. The radiation processing leads to radiation-enhanced diffusion [18]. However using radiation processing of materials leads to necessity of annealing of radiation defects. Density of elements of integrated circuits could be increases by using mismatch-induced stress [19]. However one could obtain increased unsoundness of doped material (for example, to generation of dislocation of disagreement) by using the approach [7]. The considered approaches gives a possibility to increase sharpness of p-n-junction with increasing of homogeneity of dopant distribution in enriched by the dopant area. Using combination of the above approaches gives us possibility to increase sharpness of p-n-junction and increasing of homogeneity of dopant distribution in enriched area at one time.

Overlayer fC diff(z) fC impl(z)

Epitaxial layer

D(z), P(z)

D1 P1

P2 Substrate D3

Fig. 1. Substrate, epitaxial and overlayers framework heterostructure

Framework this paper we consider a substrate with known type of conductivity (n or p) and an epitaxial layer, included into a heterostructure. The epitaxial layer have been doped by diffusion or by ion implantation to manufacture another type of conductivity (p or n). Farther we consider overgrowth of the epitaxial layer by an overlayer (see Fig. 1). The overlayer has type of conductivity, which coincide with type of conductivity of the substrate. In this paper we analyzed influence of overgrowth of the doped epitaxial layer on distribution of dopants in the considered heterostructure.

2. METHOD OF SOLUTION In this section we calculate spatio-temporal distribution of concentration of dopant in the considered heterostructure to solve our aim. To calculate the distribution we solved the following boundary problem [1,20-22] ∂ C ( x, y , z , t ) ∂  ∂ C ( x, y , z , t )  ∂  ∂ C ( x, y , z , t )  ∂  ∂ C ( x, y , z , t )  = D + D + D + ∂t ∂x ∂x ∂y ∂z  ∂y  ∂z 

(1)

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

+Ω

Lz Lz   ∂  DS ∂  DS ∇ S µ ( x, y , z , t ) ∫ C ( x, y , W , t ) d W  + Ω ∇ S µ ( x, y , z , t ) ∫ C ( x, y , W , t ) d W  .   ∂ x kT ∂ y kT −v t −v t  

Boundary and initial conditions for our case could be written as ∂ C (x, y, z, t ) ∂ C ( x, y , z , t ) ∂ C ( x, y , z , t ) ∂ C ( x, y , z , t ) =0, = 0, =0, = 0, ∂x ∂y ∂y ∂x x=0 y =0 x = Ly x= L x

∂ C ( x, y , z , t ) ∂ C ( x, y , z , t ) =0, = 0 , C (x,y,z,0)=fC (x,y,z). ∂z ∂z z=− v t x = Lz In the above relations we used the function C (x,y,z,t) as the spatio-temporal distribution of concentration of dopant; Ω is the atomic volume; surface concentration of dopant on interface beLz

tween layers of heterostructure could be determined as the following integral ∫ C ( x, y , z, t ) d z −v t

(we assume, that the interface between layers of heterostructure is perpendicular to the direction Oz); surface gradient we denote as symbol ∇S; µ (x,y,z,t) is the chemical potential (reason of accounting of the chemical potential is mismatch-induced stress); the parameters D and DS are the coefficients of volumetric and surface diffusions. One can find the surface diffusions due to mismatch-induced stress. Diffusion coefficients depends on temperature and speed of heating and cooling of heterostructure, properties of materials of heterostructure, spatio-temporal distributions of concentrations of dopant and radiation defects after ion implantation. Approximations of these dependences could be approximated by the following functions [22,23]  C γ (x, y , z , t )   V ( x, y , z , t ) V 2 ( x, y , z , t )  DS = DS L ( x, y, z , T ) 1 + ξ S γ 1 + ς + ς .  1 2 P (x, y , z , T )   V* (V * )2  

(2)

Functions DL (x,y,z,T) and DLS (x,y,z,T) described dependences of diffusion coefficients on coordinate (due to presents several layers in heterostructure, manufactured by using different materials) and temperature of annealing T (due to Arrhenius law); function P (x,y,z,T) describes dependence of limit of solubility of dopant on coordinate and temperature; parameter γ could be integer and depends on properties of materials of heterostructure [23]; V (x,y,z,t) is the spatio-temporal concentration of radiation vacancies; V* is the equilibrium concentration of vacancies. Concentrational dependence of dopant diffusion coefficients has been described in details in [23]. We determine distributions of concentrations of point defects in space and time as solutions of the following system of equations [1,20-22]

∂ I ( x, y, z, t ) ∂  ∂ I ( x, y , z , t )  ∂  ∂ I ( x, y , z , t )  = D ( x, y , z , T ) +  D I ( x, y , z , T )  − k I , I ( x, y , z , T ) × ∂t ∂ x  I ∂x ∂ y ∂y    × I 2 (x, y, z, t ) +

∂ I (x, y, z, t )  ∂   DI ( x, y, z, T )  − k I ,V ( x, y, z, T ) I ( x, y, z, t ) V ( x, y, z, t ) ∂ z ∂z 

(3)

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

+Ω

Lz Lz   ∂  DI S ∂  DI S ∇ S µ ( x, y , z , t ) ∫ I ( x, y , W , t ) d W  + Ω ∇ S µ ( x, y , z , t ) ∫ I ( x, y , W , t ) d W    ∂ x  kT ∂ y  kT −v t −v t  

∂ V (x, y, z, t ) ∂  ∂ V ( x, y, z, t )  ∂  ∂ V (x, y, z, t )  = DV ( x, y, z, T ) + DV (x, y, z, T )  − kV ,V (x, y, z, T ) × ∂t ∂ x ∂x ∂y  ∂ y  × V 2 ( x, y, z, t ) + +Ω

∂ V ( x, y, z, t )  ∂   DV (x, y, z, T )  − k I ,V ( x, y, z, T ) I ( x, y, z, t )V (x, y, z, t ) + ∂ z ∂z 

Lz Lz   ∂  DVS ∂  DVS ∇ S µ ( x , y , z , t ) ∫ V ( x, y , W , t ) d W  + Ω ∇ S µ ( x , y , z , t ) ∫ V ( x, y , W , t ) d W  .   ∂ x  kT ∂ y  kT −v t −v t  

Boundary and initial conditions for these equations could be written as

∂ I ( x, y , z , t ) ∂ I ( x, y , z , t ) ∂ I (x, y, z, t ) ∂ I ( x, y , z , t ) =0, =0, =0, =0, ∂x ∂x ∂ y ∂ y x =0 x = Lx y =0 y = Ly ∂ I ( x, y , z , t ) ∂ I (x, y, z, t ) ∂ V (x, y, z, t ) ∂ V ( x, y , z , t ) = 0, = 0, =0, =0, ∂ x ∂z ∂ z ∂x x=0 z =− v t z =Lz x = Lx ∂ V (x, y, z, t ) ∂ V (x, y, z, t ) ∂ V (x, y, z, t ) ∂ V (x, y, z, t ) = 0, = 0, =0, = 0, ∂y ∂y ∂z ∂z y =0 y = Ly z =− v t z = Lz I (x,y,z,0)=fI (x,y,z), V (x,y,z,0)=fV (x,y,z).

(4)

Here I (x,y,z,t) is the distribution of concentration of radiation interstitials in space and time; I* is the equilibrium concentration interstitials; DI(x,y,z,T), DV(x,y,z,T), DIS(x,y,z,T), DVS(x,y,z,T) are the coefficients of volumetric and surface diffusion; terms V2(x,y,z,t) and I2(x,y,z,t) corresponds to generation divacancies and analogous complexes of interstitials (see, for example, [22] and appropriate references in this work); the functions kI,V(x,y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T) described dependences of parameters of recombination of point defects and generation their complexes on coordinate and temperature; k is the Boltzmann constant. We calculate spatio-temporal distributions of concentrations of divacancies ΦV (x,y,z,t) and diinterstitials ΦI (x,y,z,t) by solution of the equations [20-22]

∂ Φ I (x, y, z, t ) ∂  ∂ Φ I (x, y, z, t )  ∂  ∂ Φ I (x, y, z, t ) = DΦ (x, y, z, T ) + DΦI (x, y, z, T )  + k I (x, y, z, T ) × ∂t ∂ x  I ∂x ∂ y ∂y    × I (x, y, z, t ) +

Lz  ∂ Φ I (x, y, z, t )  ∂  DΦ I S ∂  ∇ S µ ( x, y, z, t ) ∫ Φ I ( x, y,W , t ) d W  +   DΦ I (x, y, z, T ) +Ω ∂z ∂z ∂ x  kT −v t  

+Ω

Lz  ∂  DΦIS 2 ( ) ∇ µ x , y , z , t ∫ Φ I ( x , y , W , t ) d W  + k I , I ( x, y , z , T ) I ( x, y , z , t ) S  ∂ y  kT −v t 

(5)

∂ ΦV (x, y, z, t ) ∂  ∂ ΦV (x, y, z, t )  ∂  ∂ ΦV (x, y, z, t )  = DΦV (x, y, z, T ) + DΦV (x, y, z, T )  + kV (x, y, z, T ) × ∂t ∂x ∂x ∂y  ∂y  ×V ( x , y , z , t ) +

Lz  ∂ Φ V ( x, y , z , t )  ∂  D ΦV S ∂  ∇ S µ ( x, y , z , t ) ∫ Φ V ( x, y , W , t ) d W  +   DΦ V ( x , y , z , T ) +Ω ∂z ∂z ∂ x  kT −v t  

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

+Ω

Lz  ∂  DΦV S ∇ S µ ( x, y, z, t ) ∫ Φ V (x, y ,W , t ) d W  + kV ,V ( x, y , z , T ) V 2 ( x, y, z, t ) .  ∂ y  kT −v t 

Boundary and initial conditions could be written as

∂ Φ I ( x, y , z , t ) ∂ Φ I (x, y , z , t ) ∂ Φ I ( x, y , z , t ) ∂ Φ I (x, y , z , t ) = 0, =0, = 0, = 0, ∂x ∂x ∂ y ∂ y x =0 x = Lx y =0 y=Ly ∂ I ( x, y , z , t ) ∂ Φ I ( x, y , z , t ) ∂ Φ V ( x, y , z , t ) ∂ V (x , y , z , t ) = 0, = 0, = 0, =0, ∂x ∂z ∂x ∂z x =0 z=− v t x = Lx z = Lz

∂ Φ V ( x, y , z , t ) ∂ Φ V ( x, y , z , t ) ∂ ΦV (x, y, z, t ) ∂ Φ V ( x, y , z , t ) =0, = 0, =0, =0, ∂ y ∂ y ∂z ∂z y =0 y=Ly z =−v t z = Lz ΦI (x,y,z,0)=fΦI (x,y,z), ΦV (x,y,z,0)=fΦV (x,y,z).

(6)

Here DΦI(x,y,z,T), DΦV(x,y,z,T), DΦIS(x,y,z,T) and DΦVS(x,y,z,T) are the coefficients of volumetric and surface diffusion; the functions kI(x,y,z,T) and kV(x,y,z,T) described dependences of parameters of decay of complexes of point defects on coordinate and temperature. One can determine chemical potential µ in the Eq.(1) by the following relation [20]

µ =E(z)Ωσij [uij(x,y,z,t)+uji(x,y,z,t)]/2.

(7)

  is the deformation tensor; σij is the   stress tensor; ui, uj are the components ux(x,y,z,t), uy(x,y,z,t) and uz(x,y,z,t) of the displacement tenr sor u (x, y , z , t ) ; xi, xj are the coordinates x, y, z. The relation (3) could be transformed to the following form Here E is the tension (Young) modulus; uij =

µ ( x, y , z , t ) = E ( z ) +

1  ∂ ui ∂ u j + 2  ∂ x j ∂ xi

Ω  ∂ u i (x, y, z, t ) ∂ u j (x, y, z, t )   1  ∂ ui (x, y, z, t ) ∂ u j (x, y, z, t ) + +     − ε 0δ ij + 2  ∂ xj ∂ xi ∂ xj ∂ xi   2  

 σ ( z )δ ij  ∂ u k (x, y, z, t )  − 3 ε 0  − K ( z ) β ( z )[T ( x, y, z, t ) − T0 ] δ ij  ,  1 − 2 σ (z )  ∂ xk  

where σ is the Poisson coefficient; the parameter ε0=(as-aEL)/aEL describes the displacement parameter with lattice distances of the substrate and the epitaxial layer as, aEL; K is the modulus of uniform compression; the parameter b describes the thermal expansion; we assume, that the equilibrium temperature Tr coincides with the room temperature. Components of the displacement vector could be described by solving the following system of equations [24]

ρ (z )

ρ (z ) ρ (z )

∂ 2 u x ( x, y, z , t ) ∂ σ xx ( x, y, z , t ) ∂ σ xy ( x, y , z , t ) ∂ σ xz ( x, y, z , t ) = + + ∂t2 ∂x ∂y ∂z

∂ 2 u y ( x, y , z , t ) ∂ t2

∂ σ yx ( x, y, z , t ) ∂x

∂ u z (x, y , z , t ) ∂ σ zx (x, y , z , t ) = + ∂t2 ∂x 2

∂ σ yy ( x, y , z , t ) ∂y ∂ σ zy (x, y , z , t ) ∂y

+ +

∂ σ yz ( x, y, z , t ) ∂z ∂ σ zz ( x, y, z , t ) ∂z 39

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

σ ij =

where

∂ uk (x, y, z, t ) E ( z )  ∂ ui ( x, y, z, t ) ∂ u j (x, y, z, t ) δ ij ∂ u k ( x, y, z, t )  + − −   + K ( z )δ ij 2 [1 + σ (z )]  ∂ xj ∂ xi 3 ∂ xk ∂ xk 

− β ( z ) K ( z )[T ( x, y , z , t ) − Tr ] , ρ (z) describes the density of materials of heterostructure. The tensor δij describes the Kronecker symbol. Accounting relation for σij in the previous system of equations last system of equation could be written as

ρ (z )

2 ∂ 2 u x ( x, y , z , t )  E ( z )  ∂ u y ( x, y , z , t ) 5 E ( z )  ∂ 2 u x ( x, y , z , t )  = K ( z ) + + K ( z ) − +     ∂t2 ∂ x2 ∂ x∂ y 6 [1 + σ ( z )] 3 [1 + σ (z )]  

ρ (z ) +

2 E ( z )  ∂ u y ( x, y , z , t ) ∂ 2 u z ( x , y , z , t )   E ( z )  ∂ 2 u z ( x, y , z , t ) ( ) + + K z + −     ∂ y2 ∂ z2 2 [1 + σ ( z )]  3 [1 + σ (z )] ∂ x∂ z  

∂ 2 u y ( x, y , z , t ) ∂t2

2 E ( z )  ∂ u y ( x , y , z , t ) ∂ 2 u x ( x, y , z , t )  ∂ T ( x, y , z , t ) + +   − β (z ) K (z ) 2 [1 + σ ( z )]  ∂ x2 ∂ x∂ y ∂y 

2  ∂ u y ( x, y , z , t ) ∂  E ( z )  ∂ u y ( x, y, z , t ) ∂ u z ( x, y , z , t )    5 E ( z ) ( ) + + + + K z       ∂ z  2[1 + σ (z )]  ∂z ∂y ∂ y2    12[1 + σ ( z )] 2 ∂ 2 u y (x, y , z , t )  E ( z )  ∂ u y ( x, y , z , t ) ( ) ( ) + K z − +K z  6 [1 + σ ( z )] ∂ y∂ z ∂ x∂ y 

ρ (z )

(8)

2 ∂ 2 u z (x, y, z, t ) E (z )  ∂ 2 u z (x, y, z, t ) ∂ 2 u z (x, y, z, t ) ∂ 2 u x (x, y, z, t ) ∂ u y (x, y, z, t )  = + + +  + ∂t2 2 [1 + σ (z )]  ∂ x2 ∂ y2 ∂ x∂ z ∂ y∂ z 

 ∂ u x ( x, y , z , t ) ∂ u y ( x, y , z , t ) ∂ u x ( x, y , z , t )   ∂ T (x, y , z , t ) ∂  + + +  K (z )    − K (z ) β (z ) ∂ z  ∂x ∂y ∂z ∂z   

1 ∂  E ( z )  ∂ u z (x, y, z, t ) ∂ u x ( x, y , z , t ) ∂ u y ( x, y , z , t ) ∂ u z ( x, y , z , t )   − − −  6  . 6 ∂ z 1 + σ ( z )  ∂z ∂x ∂y ∂z  

Systems of conditions for these equations could be written as r r r r ∂ u (x, y , z , t ) ∂ u (x, y , z , t ) ∂ u ( x, y , z , t ) ∂ u (x, y , z , t ) =0; = 0; = 0; = 0; ∂x ∂x ∂y ∂y x =0 x = Lx y =0 y = Ly r r r r r r ∂ u ( x, y , z , t ) ∂ u (x, y , z , t ) = 0; = 0 ; u ( x , y , z ,0 ) = u 0 ; u ( x , y , z , ∞ ) = u 0 . ∂z ∂ z z =− v t z = Lz

We calculate distribution of concentration of dopant in space in time by method of averaging of function corrections [25-30]. To use the method we re-write Eqs. (1), (3) and (5) with account appropriate initial distributions (see Appendix). In future we replace the required concentrations in right sides of the obtained equations on their average values α1ρ, which are not yet known. The equations modified after the replacement and solution of these equations are presented in the Appendix. We determined average values of the first-order approximations of the considered concentrations by using the following standard relations [25-30]

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

α1ρ =

Θ Lx L y Lz 1 ∫ ∫ ∫ ∫ ρ1 (x, y , z, t ) d z d y d x d t . ΘLx L y Lz 0 0 0 −v t

(9)

Substitution of the solutions of the modified equations into the relation (9) gives a possibility to obtain appropriate average values in the following form

α1C =

 Θ a B + Θ2 Lx Ly Lz a1  a3 + A B + 3 − , − 4   4a 4 a42 a4 4    1  Θ Θ Lx Ly Lz = ∫ ∫ ∫ ∫ f I (x, y, z ) d z d y d x d t − α1I S II 00 − ΘLx Ly Lz  ,  S IV 00  α1I 0 0 0 −v t  S II 20 RI 1 1 Θ Lx L y Lz = + + ∫ ∫ ∫ ∫ f Φ ( x, y , z ) d z d y d x d t , ΘLx L y Lz ΘLx Ly Lz Lx L y Lz 0 0 0 −v t I

1 Θ Lx Ly Lz ∫ ∫ ∫ ∫ f C ( x, y, z ) d z d y d x d t , α1I = Lx Ly Lz 0 0 0 −v t

α1V α1Φ I

α1ΦV =

(a3 + A)2

SVV 20 RV 1 1 Θ Lx L y Lz + + ∫ ∫ ∫ ∫ f Φ ( x, y , z ) d z d y d x d t . ΘLx Ly Lz ΘLx Ly Lz Lx Ly Lz 0 0 0 −v t V

Relations for calculations parameters Sρρij, ai, A, B, q, p are presented in the Appendix. We used standard iterative procedure of method of averaging of function corrections to calculate the second- and higher-order approximations of the considered concentrations [25-30]. Framework the procedure to calculate approximations of the n-th order of the above concentrations we replace the functions C(x,y,z,t), I(x,y,z,t), V(x,y,z,t), ΦI(x,y,z,t) and ΦV(x,y,z,t) in the Eqs. (1), (3), (5) with account initial distributions (see Eqs. (1a), (3a), (5a) in the Appendix) on the sums of the not yet known average values of the considered approximations and approximations of the previous order, i.e. αnρ+ρn-1(x,y,z,t). These obtained equations and their solutions, which calculated are presented in the Appendix. We calculate average values of the second-order approximations of required functions by using the following standard relation [25-30]

α2ρ =

Θ Lx L y Lz 1 ∫ ∫ ∫ ∫ [ρ 2 (x, y , z , t ) − ρ1 ( x, y, z , t )] d z d y d x d t . ΘLx Ly Lz 0 0 0 0

(10)

Substitution of the second-order approximations of concentrations of dopant into Eq.(10) leads to the considered average values α 2ρ

α2C=0, α2ΦI =0, α2ΦV =0, α 2V = α2I =

(b3 + E )2 − 4  F + Θ a3 F + Θ 2 Lx Ly Lz b1  − b3 + E , 4 b42

 

 

CV − α SVV 00 − α 2V (2 SVV 01 + S I V 10 + ΘLx L y Lz ) − SV V 02 − S I V 11 2 2V

S IV 01 + α 2V S IV 00 Relations for parameters bi, Cρ, F, r, s are presented in the Appendix.

4 b4

Further we determine solutions of Eqs.(8). In this situation we determine approximation of displacement vector. To determine the first-order approximations of the considered components framework method of averaging of function corrections we replace the required values on their not yet known average values α1i. The replacement leads to the following result 41

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

ρ (z ) ρ (z ) ρ (z )

∂ 2 u1x ( x, y , z , t ) ∂ T ( x, y , z , t ) , = − K ( z ) β (z ) ∂ t2 ∂x ∂ 2 u1 y (x, y , z , t ) ∂ t2

= − K (z ) β ( z )

∂ T ( x, y , z , t ) , ∂y

∂ 2 u1 z ( x, y , z , t ) ∂ T ( x, y , z , t ) . = − K (z ) β ( z ) 2 ∂t ∂z

Integration of the left and right sides of the previous relations on time t gives a possibility to obtain the required components to the following result

β (z ) ∂ t ϑ β (z ) ∂ ∞ ϑ ∫ ∫ T ( x , y , z ,τ ) d τ d ϑ − K ( z ) ∫ ∫ T ( x , y , z ,τ ) d τ d ϑ + u 0 x , ρ (z ) ∂ x 0 0 ρ (z ) ∂ x 0 0 β (z ) ∂ t ϑ β (z ) ∂ ∞ ϑ u1 y ( x, y , z , t ) = K (z ) ∫ ∫ T ( x , y , z ,τ ) d τ d ϑ − K ( z ) ∫ ∫ T ( x , y , z ,τ ) d τ d ϑ + u 0 y , ρ (z ) ∂ y 0 0 ρ (z ) ∂ y 0 0 β (z ) ∂ t ϑ β (z ) ∂ ∞ ϑ u1 z (x, y , z , t ) = K ( z ) ∫ ∫ T ( x , y , z ,τ ) d τ d ϑ − K ( z ) ∫ ∫ T ( x , y , z ,τ ) d τ d ϑ + u 0 z . ρ (z ) ∂ z 0 0 ρ (z ) ∂ z 0 0

u1 x ( x, y , z , t ) = K (z )

We calculate the second-order approximations of components of the displacement vector by standard replacement of the required functions in the right sides of the Eqs.(8) on the standard sums α1i+ui(x,y,z,t) [19,26]. Equations for components of the displacement vector are presented in the Appendix. Solutions of these equations are also presented in the Appendix. Framework this paper all required concentrations (concentrations of dopant and radiation defects) and components of displacement vector have been calculated as the appropriate second-order approximations by using the method of averaging of function corrections. The second-order approximation gives usually enough information on quantitative behavior of spatio- temporal distributions of concentrations of dopant and radiation defects and also several quantitative results. We check all analytical results by using numerical approaches.

3. DISCUSSION In this section we analyzed redistribution of dopant (for the ion doping of heterostructure) with account redistribution of radiation defects and their interaction with another defects. If growth rate is small (v t<D1/v), than overlayer will be fully doped by dopant, which was implanted in the epitaxial layer. Framework another limiting case it will be doped near-surface area of the overlayer only. If dopant diffusion coefficient in the overlayer and in the substrate are smaller, in comparison with the epitaxial layer, and type of conductivity of the overlayer and the substrate is different with type of conductivity of the epitaxial layer, than one can find a bipolar transistor. In this case one can find higher sharpness of p-n-junctions framework the transistor in comparison with a bipolar transistor in homogenous sample with averaged diffusion coefficient of dopant. The increasing gives a possibility to increase switching time of p-n-junctions (both single p-njunctions and p-n-junctions framework their systems: bipolar transistors, thyristors et al). At the same time one can find increasing of homogeneity of concentration of dopant (see Fig. 2). In this situation one can decrease local overheats in doped areas during functioning of the considered devices or to decrease dimensions of these devices for fixed tolerance for local overheats. Qualitatively similar results could be obtained for diffusion type of doping. One can find smaller sharpness of left p-n-junctions in the case, when dopant diffusion coefficient of the overlayer is larger, than in doped epitaxial layer. At the same time homogeneity of concentration of dopant in 42

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

the overlayer increases (see Fig. 3). Qualitatively similar results could be obtained for diffusion type of doping. Further we analyzed influence of mismatch-induced stress on distribution of concentration 2.0 1

C(x,y,z,Î&#x2DC;)

1.5 2

1.0

0.5

0.0 Lz /4

Lz /2

3Lz /4

Fig. 2. Calculated spatial distributions of concentration of implanted dopant in homogenous sample (curve 1) and in heterostructure from Fig. 1 (curve 2) after annealing with the same continuance. Interfaces between layers of heterostructure are: a1=Lz/4 and a2=3Lz/4

C(x,y,z,0)

1.00000 3

C(x,y,z,Î&#x2DC;)

0.10000 0.01000

4 2

0.00100

0.00010 0.00001 0

Lz /4

Lz /2 x

3Lz /4

Fig. 3. Curves 1 and 2 are the calculated spatial distributions of concentration of implanted dopant in the system of two layers: overlayer and epitaxial layer. Curves 3 and 4 are the calculated spatial distributions of concentration of implanted dopant in the epitaxial layer only. Increasing of number of curves corresponds to increasing of value of relation D1/D2. Coordinates of interfaces between layers of heterostructure are: a1=Lz/4 (between overlayer and epitaxial layer) and a2=3Lz/4 (between epitaxial layer and substrate)

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 2.0 1 2

C(x,y,z,Θ)

1.0

0.0

Fig. 4. Spatial distributions of concentration of dopant in diffusion-junction rectifier after annealing with equal continuance. Curve 1 corresponds to ε0<0. Curve 2 corresponds to ε0=0. Curve 3 corresponds to ε0>0 2.0 1 2

C(x,y,z,Θ)

1.0

0.0 -Lz

Fig. 5. Spatial distributions of concentration of dopant in implanted-junction rectifier after annealing with equal continuance. Curve 1 corresponds to ε0<0. Curve 2 corresponds to ε0=0. Curve 3 corresponds to ε0>0 1.0

0.8

0.6

1 0.4

0.2

0.0 0.0

Fig.6. Normalized dependences of component uz of displacement vector on coordinate z for epitaxial layers before radiation processing (curve 1) and after radiation processing (curve 2) 44

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

of dopant. We obtain during the analysis, that p-n-junctions, manufactured near interface between layers of heterostructures, have higher sharpness and higher homogeneity of concentration of dopant in enriched area. Existing mismatch-induced stress leads to changing of distribution of concentration of dopant in directions, which are parallel to the considered interface. For example, for ε0<0 the above distribution in directions x and y became more compact (see Fig. 4). In this situation one can obtain increasing of density of elements of integrated circuits in this situation, when the circuits were fabricated in heterostructures. For ε0>0 one can obtain opposite effect (see Fig. 5). It should be noted, that radiation processing of materials of heterostructure during ion doping of materials gives a possibility to decrease mismatch-induced stress (see Fig. 6). Further we analyzed influence of mismatch-induced stress on distribution of concentration of dopant. We obtain during the analysis, that p-n-junctions, manufactured near interface between layers of heterostructures, have higher sharpness and higher homogeneity of concentration of dopant in enriched area. Existing mismatch-induced stress leads to changing of distribution of concentration of dopant in directions, which are parallel to the considered interface. For example, for ε0<0 the above distribution in directions x and y became more compact (see Fig. 4). For ε0>0 one can obtain opposite effect (see Fig. 5). It should be noted, that radiation processing of materials of heterostructure during ion doping of materials gives a possibility to decrease mismatch-induced stress (see Fig. 6). In this situation component of displacement vector perpendicular to interface between materials of heterostructure became smaller after radiation processing in comparison with analogous component of displacement vector in non processed heterostructure.

4. CONCLUSIONS In this paper we analyzed influence of overgrowth of doped by diffusion or ion implantation areas of heterostructures on distributions of concentrations of dopants. We determine conditions to increase sharpness if implanted-junction and diffusion- junction rectifiers (single rectifiers and rectifiers framework bipolar transistors). At the same time we analyzed influence of overgrowth rate of doped areas and mismatch-induced stress in the considered heterostructure on distributions of concentrations of dopants.

ACKNOWLEDGEMENTS This work is supported by the agreement of August 27, 2013 № 02.В.49.21.0003 between The Ministry of education and science of the Russian Federation and Lobachevsky State University of Nizhni Novgorod, educational fellowship for scientific research of Government of Russian, educational fellowship for scientific research of Government of Nizhny Novgorod region of Russia and educational fellowship for scientific research of Nizhny Novgorod State University of Architecture and Civil Engineering.

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V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001). N.A. Avaev, Yu.E. Naumov, V.T. Frolkin. Basis of microelectronics (Radio and communication, Moscow, 1991). V.G. Gusev, Yu.M. Gusev. Electronics (Higher School, Moscow, 1991). Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow 1991). I.P. Stepanenko. Basis of microelectronics (Soviet Radio, Moscow 1980). A.G. Alexenko, I.I. Shagurin. Microcircuitry (Radio and communication, Moscow, 1990). K.V. Shalimova. Physics of semiconductors. (Energoatomizdat, Moscow, 1985. Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov, Yu.N. Drozdov, V.D. Skupov. Diffusion processes in semiconductor structures during microwave annealing. Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836 (2003). 45

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[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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A.V. Dvurechensky, G.A. Kachurin, E.V. Nidaev, L.S. Smirnov. Impulse annealing of semiconductor materials (Science, Moscow, 1982). A.F. Corballo Sanchez, G. Gonzalez de la Cruz, Yu.G. Gurevich, G.N. Logvinov. Transient heat transport by carriers and phonons in semiconductors. Phys. Rev. B. Vol. 59 (16). P. 10630 (1999). V.I. Mazhukin, V.V. Nosov, U. Semmler. Investigation of heat and thermo-elastic fields in semiconductors at pulse processing. Mathematical modelling. Vol. 12 (2), 75 (2000). K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. Dopant activation in subamorphized silicon upon laser annealing. Appl. Phys. Lett. Vol. 89 (17), 172111 (2006). J.A. Sharp, N.E. B. Cowern, R.P. Webb, K.J. Kirkby, D. Giubertoni, S. Genarro, M. Bersani, M.A. Foad, F. Cristiano, P.F. Fazzini. Appl. Phys. Lett. Vol. 89, 192105 (2006). J.E. Epler, F.A. Ponce, F.J. Endicott, T.L. Paoli. J. Appl. Phys. Vol. 64, 3439 (1988). S.T. Sisianu, T.S. Sisianu, S.K. Railean. Semiconductors. Vol. 36, 581 (2002). E.L. Pankratov. Influence of spatial, temporal and concentrational dependence of diffusion coefficient on dopant dynamics: Optimization of annealing time. Phys. Rev. B. Vol. 72 (7). P. 075201 (2005). E.L. Pankratov. Dopant diffusion dynamics and optimal diffusion time as influenced by diffusioncoefficient nonuniformity. Russian Microelectronics. Vol. 36 (1). P. 33 (2007). V.V. Kozlivsky. Modification of semiconductors by proton beams (Science, Sant- Peterburg, 2003). E.L. Pankratov. Influence of mechanical stress in a multilayer structure on spatial distribution of dopants in implanted-junction and diffusion-junction rectifiers. Mod. Phys. Lett. B. Vol. 24 (9). P. 867 (2010). Y.W. Zhang, A.F. Bower. Numerical simulations of island formation in a coherent strained epitaxial thin film system. Journal of the Mechanics and Physics of Solids. Vol. 47 (11). P. 2273-2297 (1999). P.M. Fahey, P.B. Griffin, J.D. Plummer. Point defects and dopant diffusion in silicon. Rev. Mod. Phys. Vol. 61 (2). P. 289-388 (1989). V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev, 1979, in Russian). Z.Yu. Gotra, Technology of microelectronic devices (Radio and communication, Moscow, 1991). L.D. Landau, E.M. Lefshits. Theoretical physics. 7 (Theory of elasticity) (Physmatlit, Moscow, 2001, in Russian). Yu.D. Sokolov. About the definition of dynamic forces in the mine lifting. Applied Mechanics. Vol. 1 (1). P. 23-35 (1955). E.L. Pankratov, E.A. Bulaeva. Doping of materials during manufacture p-n-junctions and bipolar transistors. Analytical approaches to model technological approaches and ways of optimization of distributions of dopants. Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013). E.L. Pankratov, E.A. Bulaeva. Increasing of sharpness of diffusion-junction heterorectifier by using radiation processing. Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1-1250028-8 (2012). E.L. Pankratov, E.A. Bulaeva. Optimization of manufacturing of emitter-coupled logic to decrease surface of chip. International Journal of Modern Physics B. Vol. 29 (5). P. 1550023-1-1550023-12 (2015). E.L. Pankratov, E.A. Bulaeva. An approach to manufacture of bipolar transistors in thin film structures. On the method of optimization. Int. J. Micro-Nano Scale Transp. Vol. 4 (1). P. 17-31 (2014). E.L. Pankratov, E.A. Bulaeva. Decreasing of mechanical stress in a semiconductor heterostructure by radiation processing. J. Comp. Theor. Nanoscience. Vol. 11 (1). P. 91-101 (2014).

APPENDIX Eqs. (1), (3) and (5) with account initial distributions could be written as

∂ C(x, y, z, t ) ∂  ∂ C ( x, y, z, t )  ∂  ∂ C ( x, y, z, t )  ∂  ∂ C (x, y, z, t )  = D + D  + D  + f C (x, y, z )δ (t ) + ∂t ∂x ∂x ∂y ∂z  ∂y  ∂z   +Ω

Lz Lz   ∂  DS ∂  DS ∇ S µ (x, y, z , t ) ∫ C ( x, y,W , t ) d W  + Ω ∇ S µ (x, y, z , t ) ∫ C ( x, y,W , t ) d W  (1a)   ∂ x kT ∂ y k T −v t −v t  

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

∂ I ( x, y , z , t ) ∂  ∂ I ( x, y , z , t )  ∂  ∂ I ( x, y, z, t )  = D ( x, y , z , T ) +  DI (x, y, z, T )  + f I ( x, y, z )δ (t ) + ∂t ∂ x  I ∂x ∂ ∂y y    +

∂  ∂ I (x, y, z, t ) 2 DI (x, y, z, T )  − k I ,V (x, y, z, T ) I ( x, y, z, t )V (x, y, z, t ) − k I ,I (x, y, z, T ) I (x, y, z, t ) + ∂ z ∂z  +Ω

Lz Lz   ∂  DIS ∂  DIS ∇ µ ( x , y , z , t ) I ( x , y , W , t ) d W + Ω ∇ µ ( x , y , z , t ) ∫ ∫ I (x, y,W , t ) d W  (3a) S S    ∂ x  kT ∂ y  kT −v t −v t  

∂ V ( x, y, z, t ) ∂  ∂ V ( x, y, z, t )  ∂  ∂ V ( x, y, z, t )  = D ( x, y, z, T ) + DV ( x, y, z, T )  + fV ( x, y, z )δ (t ) + ∂t ∂ x  V ∂x ∂ ∂y y    +

∂  ∂ V (x, y, z, t ) 2 DV ( x, y, z, T )  − k I ,V (x, y, z, T ) I ( x, y, z, t )V (x, y, z, t ) − kV ,V (x, y, z, T ) V (x, y, z, t ) + ∂ z ∂z 

+Ω

Lz Lz   ∂  DVS ∂  DVS ∇ S µ ( x, y , z , t ) ∫ V ( x , y , W , t ) d W  + Ω ∇ S µ ( x, y , z , t ) ∫ V ( x, y , W , t ) d W    ∂ x  kT ∂ y  kT −v t −v t  

∂ Φ I (x, y, z, t ) ∂  ∂ Φ I (x, y, z, t )  ∂  ∂ Φ I ( x, y , z , t )  =  DΦ I (x, y, z, T ) +  DΦ I ( x, y, z, T )  + δ (t ) × ∂t ∂x ∂x ∂y  ∂y  × f Φ I ( x, y , z ) +

+Ω

Lz  ∂ Φ I ( x, y , z , t )  ∂  DΦ I S ∂  ∇ S µ ( x, y , z , t ) ∫ Φ I ( x, y , W , t ) d W  +   D Φ I ( x, y , z , T ) +Ω ∂z ∂z ∂ x  kT −v t  

Lz  ∂  DΦ I S ∇ S µ (x, y, z, t ) ∫ Φ I (x, y,W , t ) d W  + k I ( x, y, z, T ) I (x, y, z, t ) + k I , I ( x, y, z, T ) I 2 (x, y, z, t )  ∂ y  kT −v t 

∂ ΦV ( x, y, z, t ) ∂  ∂ ΦV (x, y, z, t )  ∂  ∂ ΦV ( x, y, z, t )  = D ( x, y, z, T ) +  DΦV ( x, y, z, T )  + δ (t ) × (5a) ∂t ∂ x  ΦV ∂x ∂ ∂y y    × f ΦV ( x, y, z ) +

+Ω

Lz  ∂ Φ V ( x, y, z, t )  ∂  DΦV S ∂  ∇ S µ ( x, y , z , t ) ∫ Φ V ( x, y , W , t ) d W  +   DΦV ( x, y, z, T ) +Ω ∂ x  kT ∂z ∂z −v t  

Lz  ∂  DΦV S ∇S µ (x, y, z, t ) ∫ ΦV (x, y,W , t ) d W  + kV (x, y, z, T )V (x, y, z, t ) + kV ,V (x, y, z,T )V 2 (x, y, z, t ) .  ∂ y  kT −v t 

The first-order approximations of concentrations of dopant and radiation defects could by calculated by solution of the following equations   ∂ C1 ( x, y, z , t ) ∂  DS ∂  DS = α 1C Ω ∇ S µ ( x, y, z , t ) + α 1C Ω ∇ S µ ( x, y , z , t ) + z z ∂t ∂ x  kT ∂ y k T    + f C ( x, y, z )δ (t ) (1b)

  ∂ I 1 ( x, y , z , t ) ∂  DIS ∂  DIS = α1I z Ω ∇ S µ (x, y, z, t ) + α1I Ω ∇ S µ ( x, y, z, t ) + f I ( x, y, z )δ (t ) −  z ∂t ∂ x kT ∂ y k T   

− α 12I k I , I ( x, y , z , T ) − α 1I α 1V k I ,V ( x, y , z , T )

(3b)

  ∂ V1 (x, y, z, t ) ∂  DVS ∂  DVS = α1V z Ω ∇ S µ (x, y, z, t ) + α1V Ω ∇ S µ (x, y, z, t ) + fV (x, y, z )δ (t ) −  z ∂t ∂ x  kT ∂ y  kT  

− α 12V kV ,V ( x, y , z, T ) − α 1I α 1V k I ,V ( x, y, z, T ) 47

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

∂ Φ 1 I ( x, y , z , t ) = k I (x, y, z , T ) I (x, y , z , t ) + k I , I (x, y , z , T ) I 2 (x, y, z , t ) + f Φ I (x, y, z )δ (t ) + ∂t + α 1Φ I z Ω

  ∂  DΦ I S ∂  DΦ I S ∇ S µ (x, y , z , t ) + α 1Φ I z Ω ∇ S µ (x, y , z , t )   ∂ x  kT ∂ y  kT  

(5b)

∂ Φ 1V ( x, y , z , t ) = kV ( x, y, z , T )V ( x, y, z , t ) + kV ,V ( x, y , z , T )V 2 ( x, y, z , t ) + f ΦV ( x, y, z )δ (t ) + ∂t + α 1ΦV z Ω

  ∂  DΦV S ∂  DΦV S ∇ S µ (x, y , z , t ) + α 1ΦV z Ω ∇ S µ ( x, y , z , t )   y k T ∂ x  kT ∂   

The first-order approximations of the considered concentrations in the following form C1 (x, y, z , t ) = α 1C

  ξ S α 1γC ∂ t V ( x , y , z ,τ ) V 2 ( x , y , z ,τ )  ∇ ( x , y , z , ) 1 + 1 + + µ τ ς ς × ∫ S 1   1 2 γ ∂x0 V* (V * )2   P ( x, y, z , T )  

× Ω DS L ( x, y , z , T )

  z ∂ t V ( x , y , z ,τ ) V 2 ( x , y , z ,τ )  ( ) µ τ ς ς d τ  + α 1C Ω x y z ∇ , , , 1 + +  × ∫ S 1 1 2 kT V* ∂y0 (V * )2    × DS L (x, y, z, T )

I 1 ( x, y , z , t ) = α 1 I z Ω

ξ S α1γC  z  1 + γ  d τ + f C (x, y, z ) k T  P (x, y, z, T ) 

∂ t DIS ∂ t DIS ∇ S µ ( x , y , z ,τ ) d τ + α 1 I z Ω ∇ S µ ( x , y , z ,τ ) d τ + f I ( x , y , z ) − ∫ ∫ ∂ x 0 kT ∂ y 0 kT t

− α 12I ∫ k I , I ( x, y, z, T ) d τ − α 1I α 1V ∫ k I ,V ( x, y , z , T ) d τ V1 (x, y, z , t ) = α 1V z Ω

(1c)

(3c)

∂ t D IS ∂ t D IS ∇ S µ1 (x, y , z ,τ ) d τ + α 1V z Ω ∇ S µ1 ( x, y , z , τ ) d τ + f V ( x, y , z ) − ∫ ∫ ∂ x 0 kT ∂ y 0 kT t

− α 12V ∫ kV ,V ( x, y , z , T ) d τ − α 1I α 1V ∫ k I ,V ( x, y , z , T ) d τ 0

Φ 1I ( x, y, z , t ) = α 1Φ I z Ω

∂ ∂ t DΦ I S ∇ S µ ( x, y, z ,τ ) d τ + α 1Φ I z Ω ∇ S µ ( x , y , z ,τ ) d τ + ∫ ∫ ∂ x 0 kT ∂ x 0 kT t

DΦ I S

+ f Φ I ( x , y , z ) + ∫ k I ( x , y , z , T ) I ( x , y , z ,τ ) d τ + ∫ k I , I ( x , y , z , T ) I 2 ( x , y , z ,τ ) d τ

Φ1V ( x, y, z, t ) = α1ΦV z Ω

(5c)

∂ t DΦV S ∂ t DΦ S ∇S µ ( x, y, z,τ ) d τ + α1ΦV z Ω ∫ V ∇S µ ( x, y, z,τ ) d τ + ∫ ∂ x 0 kT ∂ x 0 kT

+ f ΦV (x, y, z ) + ∫ kV (x, y, z , T )V (x, y, z ,τ ) d τ + ∫ kV ,V (x, y, z, T )V 2 (x, y, z ,τ ) d τ .

Relations for calculations parameters Sρρij, ai, A, B, q, p could be written as Θ

Lx L y Lz

0 0 −v t

S ρρ ij = ∫ (Θ − t ) ∫ ∫ ∫ k ρ ,ρ (x, y, z, T ) I1i (x, y, z, t )V1 j (x, y, z, t ) d z d y d x d t , a 4 = (S IV2 00 − S II 00 SVV 00 ) × 48

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 Θ Lx L y Lz

× S II 00 , a3 = S IV 00 S II 00 + S IV2 00 − S II 00 SVV 00 , a 2 = S IV 00 ΘL2x L2y L2z + 2 ∫ ∫ ∫ ∫ f I (x, y , z ) d z d y d x d t × 0 0 0 −v t Θ Lx L y Lz

Θ Lx L y L z

0 0 0 −v t

× SVV 00 S II 00 + S IV 00 S IV2 00 ∫ ∫ ∫ ∫ f V (x, y, z ) d z d y d x d t − S IV2 00 ∫ ∫ ∫ ∫ f I ( x, y, z ) d z d y d x d t − Θ L2x × Θ Lx L y L z

× L2y L2z SVV 00 , a1 = S IV 00 ∫ ∫ ∫ ∫ f I ( x, y , z ) d z d y d x d t , B = 3

q2 + p3 − q − 3

q 2 + p3 + q +

0 0 0 −v t

 a2 2 a Θ 3 a2 Θ a2 Θ L x L y L z  , a 0 = S VV 00  ∫ ∫ ∫ ∫ f I (x, y , z ) d z d y d x d t  , A =  Θ2 32 − 4 Θ 2 + 8 y  , q = × + a4 6 a4 24 a 42  0 0 0 −v t   a4  4 2  aa  a  a 2  Θ 3 a 23 4a a 2 2 2 Θ a1 ×  4a 0 − Θ Lx L y Lz 1 3  − Θ 2 02  4 Θ a 2 − Θ 2 3  − − L L L , p = Θ 2 0 24 − x y z 3 2 12 a4 a 8 a a 54 a 8 a 4  4  4  4 4  Θ

L x L y Lz

0 0 −v t

− Θ (3Θ 2 Lx Ly Lz a1a3 + 2a2 a4 ) 36a42 , Rρ i = ∫ (Θ − t ) ∫ ∫ ∫ k I ( x, y , z , T ) I1i ( x, y , z , t ) d z d y d x d t .

Equations for the second-order approximations of concentrations of dopant and radiation defects could be written as  [α + C (x, y, z, t )]γ ∂ C2 (x, y, z, t ) ∂  = D L (x, y, z, T )1 + ξ 2C γ 1 P (x, y, z, T ) ∂t ∂ x   ×

∂ C1 ( x, y, z, t )  ∂   + ∂x  ∂ y 

× D L ( x, y , z , T ) ) +

× D L ( x, y , z , T ) +Ω

  V (x, y, z, t ) V 2 (x, y, z, t ) +ς2 ×  1 + ς 1 * V   (V * )2 

 [α 2C + C1 (x, y, z, t )]γ V (x, y, z, t ) V 2 ( x, y, z, t )   1 + + ς ξ 1 + ς 1   2 V* Pγ (x, y, z, T ) (V * )2   

 ∂ C1 ( x, y, z, t ) ×  ∂y 

[α 2C + C1 (x, y, z, t )]γ ∂   V ( x, y , z , t ) V 2 ( x, y , z , t )   ς ξ 1+ ς1 + 1 +   2 ∂ z   V* P γ ( x, y , z , T ) (V * )2  

 × 

Lz  ∂ C1 ( x, y, z, t )  ∂  DS  + Ω ∇ S µ 2 (x, y , z , t ) ∫ [α 2C + C ( x, y ,W , t )] d W  +  ∂z ∂ x kT −v t  

Lz  ∂  DS ∇ S µ 2 ( x, y , z , t ) ∫ [α 2C + C (x, y ,W , t )] d W  + f C ( x, y , z )δ (t )  ∂ y k T −v t 

(1d)

∂ I 2 ( x, y , z , t ) ∂  ∂ I ( x, y , z , t )  ∂  ∂ I 1 ( x, y , z , t )  = D I ( x, y , z , T ) 1 +  DI ( x, y, z, T )  − k I ,V ( x, y, z, T ) × ∂t ∂ x  ∂x ∂ y ∂y    × [α 1I + I 1 ( x, y , z , t )][α 1V + V1 (x, y , z , t )] +

× k I , I ( x, y , z , T ) + Ω

∂ I ( x, y , z , t )  ∂  2 D I ( x, y , z , T ) 1  − [α 1I + I 1 ( x, y , z , t )] × ∂ z  ∂ z 

Lz  ∂  D IS ∇ S µ ( x, y , z, t ) ∫ [α 2 I + I 1 (x, y, W , t )] d W  +  ∂ x kT −v t 

+Ω

Lz  ∂  DIS ∇ S µ ( x, y , z , t ) ∫ [α 2 I + I1 ( x, y, W , t )] d W  (3d)  ∂ y kT −v t 

∂ V2 (x, y, z, t ) ∂  ∂ V1 (x, y, z, t )  ∂  ∂ V1 (x, y, z, t )  =  DV (x, y, z, T ) +  DV (x, y, z, T )  − k I ,V (x, y, z, T ) × ∂t ∂ x ∂x ∂y  ∂ y  × [α 1I + I 1 ( x, y , z , t )][α 1V + V1 ( x, y , z , t )] +

∂ V1 ( x, y , z , t )  ∂  2  DV ( x, y , z , T )  − [α 1V + V1 ( x, y , z , t )] × ∂ z ∂z  49

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

× kV ,V ( x, y, z, T ) + Ω

Lz  ∂  DVS ∇ S µ ( x, y, z, t ) ∫ [α 2V + V1 ( x, y,W , t )] d W  +  ∂ x  kT −v t 

+Ω

Lz  ∂  DVS ∇ S µ ( x, y , z, t ) ∫ [α 2V + V1 ( x, y ,W , t )] d W   ∂ y  kT −v t 

∂ Φ2I (x, y, z, t ) ∂  ∂ Φ ( x, y, z, t )  ∂  ∂ Φ1I (x, y, z, t )  = DΦI (x, y, z,T ) 1I  + DΦI (x, y, z,T )  + kI ,I ( x, y, z,T ) × ∂t ∂x ∂x ∂y  ∂y  × I 2 ( x, y , z , t ) + Ω

Lz  ∂  DΦ I S ∇ S µ (x, y , z , t ) ∫ α 2 Φ I + Φ 1I ( x, y,W , t ) d W  + f Φ I ( x, y , z )δ (t ) +  ∂ x  kT −v t 

[

]

Lz  ∂  ∂ Φ1I (x, y, z, t ) ∂  DΦ I S ∇ S µ (x, y, z, t ) ∫ α 2Φ I + Φ1I (x, y,W , t ) d W  +   DΦ I (x, y, z, T ) + ∂ y  kT ∂z −v t   ∂ z

[

+Ω

]

+ k I ( x, y , z , T ) I ( x, y , z , t )

(5d)

∂ Φ 2V ( x, y, z , t ) ∂  ∂ Φ 1V ( x, y, z , t )  ∂  ∂ Φ 1V ( x, y, z , t )  = D Φ ( x, y , z , T )  + kV ,V (x, y, z , T ) × +  D Φ V ( x, y , z , T ) ∂t ∂ x  V ∂x ∂ y ∂y   

× V 2 ( x, y , z , t ) + Ω +Ω

Lz  ∂  DΦ V S ∇ S µ (x, y , z , t ) ∫ α 2 ΦV + Φ 1V ( x, y ,W , t ) d W  + f ΦV ( x, y , z )δ (t ) +  ∂ x  kT −v t 

[

]

Lz  ∂  ∂ Φ 1V ( x, y , z , t )  ∂  DΦV S ∇ S µ (x, y, z , t ) ∫ α 2 ΦV + Φ 1V ( x, y,W , t ) d W  +   DΦ V ( x , y , z , T ) + ∂ y  kT ∂ z ∂z −v t   

[

]

+ kV ( x, y , z , T ) V ( x, y , z , t ) . The second-order approximations of concentrations of dopant and radiation defects could be written as [α 2C + C1 (x, y, z,τ )]γ  1 + ς V (x, y, z,τ ) + ς V 2 (x, y, z,τ ) × ∂ t  C2 (x, y, z, t ) =  ∫ D L (x, y, z, T ) 1 + ξ  1 2 P γ (x, y, z, T ) V* ∂x0    (V * )2  [α 2C + C1 (x, y, z,τ )]γ  × ∂ C ( x , y , z ,τ ) ∂ t V ( x , y , z ,τ ) V 2 ( x , y , z ,τ )   ς ξ × 1 dτ + + 1 +  ∫ 1 + ς 1   2 ∂x ∂ y 0  V* P γ ( x, y , z , T ) (V * )2   

× D L ( x, y , z , T )

 ∂ C1 (x, y, z,τ ) ∂ t V ( x , y , z ,τ ) V 2 ( x , y , z ,τ )  dτ + +ς2  × ∫ D L (x, y , z , T ) 1 + ς 1 * ∂y ∂z 0 V (V * )2  

[α 2C + C1 (x, y, z,τ )]γ  d τ + Ω ∂ t DS Lz[α + C (x, y,W ,ϑ )] d W × ∂ C1 ( x, y , z ,τ )  1 + ξ ∫ ∫ 2C   1 ∂z P γ ( x, y , z , T ) ∂ x 0 k T −v τ   Lz ∂ t DS × ∇ S µ ( x , y , z ,τ ) d ϑ d τ + Ω ∇ S µ ( x, y, z ,τ ) ∫ [α 2 C + C1 (x, y ,W ,ϑ )] d W d ϑ d τ + ∫ ∂ y 0 kT −v τ (1e) + f C ( x, y , z ) t t ∂ ∂ I 1 ( x, y , z ,τ ) ∂ ∂ I 1 ( x, y , z ,τ ) I 2 ( x, y , z , t ) = dτ + dτ + ∫ D ( x, y , z , T ) ∫ D ( x, y , z , T ) ∂ y0 I ∂y ∂ x0 I ∂x t ∂ I 1 ( x , y , z ,τ ) ∂ t 2 + d τ − ∫ k I , I ( x, y , z , T ) [α 2 I + I 1 ( x, y , z ,τ )] d τ + f I (x, y, z ) + ∫ D I ( x, y , z , T ) ∂ z0 ∂z 0 Lz ∂ t DIS ∂ t DIS +Ω ∇ S µ ( x, y, z ,τ ) ∫ [α 2 I + I 1 ( x, y, W ,ϑ )] d W d ϑ d τ + Ω ∇ S µ ( x , y , z ,τ ) × ∫ ∫ ∂ x 0 kT ∂ y 0 kT −v τ ×

−v τ

× ∫ [α 2 I + I 1 (x, y,W ,ϑ )] d W d ϑ d τ − ∫ k I ,V (x, y, z, T )[α 2 I + I1 (x, y, z,τ )][α 2V + V1 (x, y, z,τ )] d τ (3e) 50

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

∂ V1 ( x, y , z,τ ) ∂ V1 ( x, y , z ,τ ) ∂ t ∂ t dτ + dτ + ∫ DV ( x, y , z , T ) ∫ DV (x, y , z , T ) ∂x ∂ x0 ∂ y0 ∂y t ∂ V1 (x, y , z ,τ ) ∂ t 2 + d τ − ∫ kV ,V ( x, y , z , T ) [α 2V + V1 ( x, y , z ,τ )] d τ + f V ( x, y , z ) + ∫ DV ( x, y, z , T ) ∂ z0 ∂z 0 V2 (x, y , z , t ) =

+Ω

Lz ∂ t DVS ∂ t DVS ∇ S µ (x, y , z ,τ ) ∫ [α 2V + V1 ( x, y, W , ϑ )] d W d ϑ d τ + Ω ∇ S µ ( x , y , z ,τ ) × ∫ ∫ ∂ x 0 kT ∂ y 0 kT −v τ

× ∫ [α 2 I + V1 ( x, y,W ,ϑ )] d W d ϑ d τ − ∫ k I ,V ( x, y, z, T )[α 2 I + I 1 ( x, y, z ,τ )][α 2V + V1 ( x, y, z,τ )] d τ −v τ

∂ Φ 1 I ( x , y , z ,τ ) ∂ Φ 1 I ( x , y , z ,τ ) ∂ ∂ t dτ + dτ + ∫ DΦ I ( x , y , z , T ) ∫ DΦ I ( x , y , z , T ) ∂ x ∂ x0 ∂ y0 ∂ y Lz ∂ Φ 1 I ( x , y , z ,τ ) ∂ t ∂ t + dτ + Ω ∫ DΦ I ( x , y , z , T ) ∫ ∇ S µ ( x, y , z ,τ ) ∫ [α 2 Φ I + Φ 1 I (x, y ,W ,τ )] d W × ∂ z0 ∂x0 ∂ z −v τ t

Φ 2 I ( x, y , z , t ) =

DΦ I S kT

dτ + Ω

Lz ∂ t DΦ I S ∇ S µ ( x , y , z ,τ ) ∫ α 2 Φ I + Φ 1 I ( x , y , W , ϑ ) d W d ϑ d τ + f Φ I ( x , y , z ) + ∫ ∂ y 0 kT −v τ

[

]

+ ∫ k I , I ( x , y , z , T ) I 2 ( x , y , z ,τ ) d τ + ∫ k I ( x , y , z , T ) I ( x , y , z ,τ ) d τ

(5e)

∂ Φ 1V (x, y, z ,τ ) ∂ Φ 1V (x, y, z ,τ ) ∂ t ∂ t dτ + dτ + ∫ DΦ V ( x , y , z , T ) ∫ DΦ ( x , y , z , T ) ∂ x0 ∂ y0 V ∂ x ∂ y Lz ∂ Φ 1V (x, y, z,τ ) ∂ t ∂ t + dτ + Ω ∫ DΦV ( x, y, z, T ) ∫ ∇ S µ (x, y, z,τ ) ∫ α 2ΦV + Φ1V ( x, y,W ,τ ) d W × ∂ z0 ∂x0 ∂ z −v τ L z DΦ S ∂ t DΦ V S × V dτ + Ω ∇ S µ ( x, y , z,τ ) ∫ [α 2 Φ V + Φ 1V ( x, y,W ,ϑ )] d W d ϑ d τ + f Φ V ( x, y , z ) + ∫ kT ∂ y 0 kT −v τ

Φ 2V (x, y, z , t ) =

[

]

+ ∫ kV ,V ( x, y , z , T ) V 2 ( x, y , z ,τ ) d τ + ∫ kV ( x, y, z, T ) V ( x, y , z,τ ) d τ .

Parameters bi, Cρ, F, r, s have been determined by the following relations b4 = +

2 2 S IV 00 SVV 00 (S IV 01 + 2S II10 + S IV 01 + ΘLx Ly Lz ) + S IV 00 (2SVV 01 + S IV 10 + ΘLx Ly Lz ) − S IV3 003 S3IV 103 , ΘLx L y Lz ΘLx L y Lz Θ L x L y Lz

b2 =

S II 00 SVV 00 1 1 2 S IV2 00 SV V 00 − SVV + 00 S II 00 , b3 = −(2 S VV 01 + S IV 10 + ΘL x L y L z ) ΘLx Ly Lz ΘLx L y Lz ΘLx L y Lz

S II 00 SVV 00 (SVV 02 + S IV 11 + CV ) − (ΘLx L y Lz − 2SVV 01 + S IV 10 )2 + S IV 01 (ΘLx L y Lz + 2S II 10 + S IV 01 ) × ΘLx L y Lz Θ Lx

2 SVV 00 S IV 00 (ΘLx L y Lz + S IV 01 + 2S II 10 + 2S IV 01 ) (2SVV 01 + +ΘLx L y Lz + S IV 10 ) − S IV 00 × + L y L z ΘLx L y Lz ΘL x L y L z

× (CV − SVV 02 − S IV 11 ) +

C I S IV2 00 2S S S + SVV 02 + CV (S IV 10 + ΘLx × − IV 10 IV 00 S IV 01 , b1 = S II 00 IV 11 2 2 2 2 Θ Lx L y Lz ΘLx L y Lz ΘLx L y Lz

× L y Lz + 2 SVV 01 ) + S IV 01

ΘL x L y L z + 2S II 10 + S IV 01 ΘL x L y L z

(2S

× S IV 00 (3S IV 01 + 2S II10 + ΘLx Ly Lz ) + 2 CI S IV 00 S IV 01 −

VV 01

+ S IV 10 + ΘLx L y Lz ) −

CV − SVV 02 − S IV 11 × ΘLx L y Lz

(S + SVV 02 ) S IV 10 S IV2 01 , b0 = S II 00 IV 00 − S IV 01 × ΘLx Ly Lz ΘL x L y L z 2

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

CV − SVV 02 − S IV 11 (ΘLx L y Lz + 2 S II10 + S IV 01 ) − CV − SVV 02 − S IV 11 (ΘLx L y Lz + 2 S II 10 + S IV 01 ) × ΘLx L y Lz Θ Lx L y Lz

× S IV 01 + 2C I S IV2 01 − S IV 01

α12I S II 00 ΘLx Ly Lz

−

CV − SVV 02 − S IV 11 (ΘLx L y Lz + 2 S II10 + S IV 01 ) , C I = S IV 00 α 1I α 1V + Θ Lx L y L z ΘLx L y Lz

S II 20S II 20 S IV11 a2 a , CV = α1I α1V S IV 00 + α12V SVV 00 − SVV 02 − S IV 11 , E = 8 y + Θ2 32 − 4Θ 2 , − a4 a4 ΘLx Ly Lz ΘLx Ly Lz

 bb  b 2  Θ3b23 Θ 4 b12 Θ2 Θ2   4b0 − Θ Lx Ly Lz 1 3  − b0 2  4 Θ b2 − Θ 2 3  − ,s = (4b0 b4 − − L2x L2y L2z 3 2 b4  8b4  b4  54b4 8 b4 12 b42  Θ a2 3 − Θ Lx L y L z b1b3 ) − Θ b2 18b4 , F = + r 2 + s3 − r − 3 r 2 + s3 + r . 6 a4

Θ3b2 24b42

Equations for components of the displacement vector could be presented in the form

ρ (z )

2 ∂ 2 u 2 x ( x, y , z , t )  5E (z )  ∂ 2 u1x (x, y, z , t )  E (z )  ∂ u1 y (x, y, z , t ) ( ) ( ) = K z + + K z − +     6 [1 + σ (z )] 3 [1 + σ (z )] ∂t2 ∂ x2 ∂ x∂ y  

2 E ( z )  ∂ u1 y ( x, y , z , t ) ∂ 2 u1z ( x, y , z, t )   E ( z )  ∂ 2 u1z ( x, y , z, t ) + + −   + K (z ) +  2 2 2 [1 + σ ( z )]  ∂y ∂z 3 [1 + σ ( z )] ∂ x∂ z  

− K (z ) β (z )

ρ (z )

∂ 2 u 2 y (x, y , z , t ) ∂t2

∂ T (x, y , z , t ) ∂x

2 ∂ T ( x, y , z , t ) E ( z )  ∂ u1 y ( x, y, z, t ) ∂ 2 u1x ( x, y, z, t )  + +   − K ( z ) β (z ) 2 2[1 + σ (z )]  ∂x ∂ x∂ y ∂y 

2  ∂  E ( z )  ∂ u1 y ( x, y, z , t ) ∂ u1 z ( x, y, z , t )   ∂ u1 y (x, y , z , t )  5 E ( z ) + + + K ( z ) +     + 2 ( ) ( ) ∂ z  2 [1 + σ z ]  ∂z ∂y ∂y 12 [1 + σ z ]   

2 ∂ 2 u1 y ( x, y , z, t )  E (z )  ∂ u1 y ( x, y , z , t ) + K (z ) − + K (z )  6 [1 + σ (z )] ∂ y∂ z ∂ x∂ y  2 ∂ 2 u 2 z ( x, y , z , t )  ∂ 2 u1 z (x, y , z , t ) ∂ 2 u1 z ( x, y, z , t ) ∂ 2 u1x ( x, y, z , t ) ∂ u1 y (x, y, z , t )  ρ (z ) = + + +  × ∂t2 ∂ x2 ∂ y2 ∂ x∂ z ∂ y∂ z  

 ∂ u1x ( x, y , z , t ) ∂ u1 y ( x, y, z, t ) ∂ u1x ( x, y , z, t )   ∂  E ( z ) E (z ) ∂  + + + ×  K (z )    + 2 [1 + σ (z )] ∂ z  ∂x ∂y ∂z    ∂ z  6 [1 + σ ( z )]

 ∂ u ( x, y , z , t ) ∂ u1 x ( x, y , z , t ) ∂ u1 y (x, y , z , t ) ∂ u1 z ( x, y , z , t )   ∂ T ( x, y , z , t ) × 6 1 z − − − .   − K (z ) β (z ) ∂z ∂x ∂y ∂z ∂z   

Integration of the left and the right sides of the above equations on time t leads to final relations for components of displacement vector 5E (z )  ∂ 2 t ϑ E (z )  1  1  ∫ ∫ u1x (x, y , z,τ ) d τ d ϑ + K (z ) +   K (z ) − × ρ (z )  6 [1 + σ ( z )] ∂ x 2 0 0 ρ (z )  3 [1 + σ (z )]  ∂2 t ϑ  ∂2 t ϑ ∂2 t ϑ × ∫ ∫ u1 y ( x, y , z ,τ ) d τ d ϑ +  2 ∫ ∫ u1 y ( x, y , z ,τ ) d τ d ϑ + 2 ∫ ∫ u1z ( x, y, z,τ ) d τ d ϑ  × ∂ x∂ y 0 0 ∂z 00 ∂ y 0 0 

u 2 x ( x, y , z , t ) =

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

 β (z ) E (z ) 1 ∂2 t ϑ E (z )  + × ∫ ∫ u1z ( x, y , z ,τ ) d τ d ϑ  K ( z ) +  − K (z ) ( ) [ ( ) ] ( ) [ ( ) ] 2 ρ z 1+σ z ρ z ∂ x∂ z 0 0 3 1+σ z  ρ (z ) 

5E (z )  ∂ 2 ∞ ϑ ∂ tϑ 1  1 × ∫ ∫ T ( x , y , z ,τ ) d τ d ϑ − K (z ) +  2 ∫ ∫ u1 x ( x , y , z , τ ) d τ d ϑ − ∂x 00 ρ (z )  6 [1 + σ ( z )] ∂ x 0 0 ρ (z )  E (z )  ∂ 2 ∞ ϑ 1  ∂ 2 ∞ϑ × K (z ) − ∫ ∫ u1 y ( x, y, z,τ ) d τ d ϑ − ∫ ∫ u1 y ( x, y , z ,τ ) d τ d ϑ +   3 [1 + σ ( z )] ∂ x ∂ y 0 0 2 ρ (z )  ∂ y 2 0 0 

 E (z )  E (z )  ∂ 2 ∞ ϑ ∂ 2 ∞ϑ ( ) ( ) u x , y , z , τ d τ d ϑ − K z + ∫ ∫ 1z ∫ ∫ u1 z (x, y , z ,τ ) d τ d ϑ ×    ∂ z2 0 0 3 [1 + σ ( z )] ∂ x ∂ z 0 0  1 + σ (z ) 

β (z ) ∂ ∞ ϑ 1 + u 0 x + K (z ) ∫ ∫ T ( x , y , z ,τ ) d τ d ϑ ρ (z ) ρ (z ) ∂ x 0 0

 ∂2 t ϑ  E (z ) ∂2 t ϑ ( ) u x , y , z , τ d τ d ϑ + ∫ ∫ u1x ( x, y , z ,τ ) d τ d ϑ  +  2 ∫ ∫ 1x ∂ x∂ y 0 0 2 ρ ( z )[1 + σ ( z )]  ∂ x 0 0 

u 2 y ( x, y , z , t ) =

 K (z ) ∂ 2 t ϑ 5 E (z )  1 ∂2 t ϑ ∫ ∫ u1 y ( x , y , z , τ ) d τ d ϑ + ∫ ∫ u1 x ( x, y , z ,τ ) d τ d ϑ  K (z ) + + [ ρ (z ) ∂ x ∂ y 0 0 ρ (z ) ∂ y 2 0 0 12 1 + σ ( z )] 

 5E (z )  1 ∂  E (z )  ∂ t ϑ ∂ tϑ ∫ ∫ u1 y (x, y, z,τ ) d τ d ϑ + ∫ ∫ u1z (x, y , z ,τ ) d τ d ϑ   − +   ∂y 00 12 [1 + σ (z )] 2 ρ ( z ) ∂ z 1 + σ (z )  ∂ z 0 0 

− K (z )

−

 E (z )  β (z ) t ϑ 1 ∂2 t ϑ ( ) τ τ ϑ , , , T x y z d d + ∫∫ ∫ ∫ u1 y ( x , y , z , τ ) d τ d ϑ  K ( z ) − − [ ρ (z ) 0 0 ρ (z ) ∂ y ∂ z 0 0 6 1 + σ ( z )] 

 ∂ 2 ∞ϑ  β (z ) E (z ) ∂ 2 ∞ϑ × ∫ ∫ u1x ( x, y , z,τ ) d τ d ϑ  − K ( z )  2 ∫ ∫ u1x ( x, y , z ,τ ) d τ d ϑ + ( ) [ ( ) ] 2 ρ z 1+σ z ∂ x 0 0 ∂ x∂ y 0 0 ρ (z ) 

∞ϑ

× ∫ ∫ T ( x , y , z ,τ ) d τ d ϑ − 00

K (z ) ∂ 2 ∞ ϑ 1 ∂ 2 ∞ϑ ∫ ∫ u1 y (x, y, z,τ ) d τ d ϑ − ∫ ∫ u1x (x, y, z,τ ) d τ d ϑ × ρ (z ) ∂ x ∂ y 0 0 ρ (z ) ∂ y 2 0 0

  5E (z )  ∂  E (z )  ∂ ∞ ϑ ∂ ∞ϑ × K (z ) + ∫ ∫ u1 y ( x , y , z ,τ ) d τ d ϑ + ∫ ∫ u1 z ( x , y , z ,τ ) d τ d ϑ   × −   [ ( ) ] ( ) σ σ 12 1 + z ∂ z 1 + z ∂ z ∂ y 0 0 0 0     

1 2 ρ (z )

u z ( x, y , z , t ) =

−

E (z )  ∂ 2 ∞ ϑ 1  ∫ ∫ u1 y ( x , y , z , τ ) d τ d ϑ + u 0 y  K (z ) −  6 [1 + σ ( z )] ∂ y ∂ z 0 0 ρ (z ) 

 ∂ 2 ∞ϑ E (z ) ∂ 2 ∞ϑ ( ) u x , y , z , τ d τ d ϑ + ∫ ∫ 1z ∫ ∫ u1z ( x, y, z,τ ) d τ d ϑ +  2 ρ (z )[1 + σ ( z )]  ∂ x 2 0 0 ∂ y2 0 0

 ∂   ∂ ∞ϑ ∂ 2 ∞ϑ ∂ 2 ∞ϑ ∫ ∫ u1x (x, y, z,τ ) d τ d ϑ + ∫ ∫ u1 y (x, y, z,τ ) d τ d ϑ  +   ∫ ∫ u1x (x, y, z,τ ) d τ d ϑ + ∂ x∂ z 0 0 ∂ y∂ z 0 0  ∂ z  ∂ x 0 0

 1  ∂ ∞ϑ ∂ ∞ϑ ∂  E (z ) 1 + × ∫ ∫ u1x ( x, y, z ,τ ) d τ d ϑ + ∫ ∫ u1x ( x, y, z ,τ ) d τ d ϑ  K ( z )   ( ) ( ) ∂y 00 ∂z 00 ρ z 6 ρ z ∂ z 1 + σ ( z )    ∂ ∞ϑ ∂ ∞ϑ ∂ ∞ϑ × 6 ∫ ∫ u1 z ( x , y , z , τ ) d τ d ϑ − ∫ ∫ u1 x ( x, y, z,τ ) d τ d ϑ − ∫ ∫ u1 y ( x , y , z , τ ) d τ d ϑ − ∂x 00 ∂y 00  ∂z 00  β (z ) ∂ ∞ ϑ ∂ ∞ϑ − ∫ ∫ u1z ( x, y, z ,τ ) d τ d ϑ   − K ( z ) ∫ ∫ T ( x , y , z ,τ ) d τ d ϑ + u 0 z . ρ (z ) ∂ z 0 0 ∂z 00  53

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016

AUTHORS Pankratov Evgeny Leonidovich was born at 1977. From 1985 to 1995 he was educated in a secondary school in Nizhny Novgorod. From 1995 to 2004 he was educated in Nizhny Novgorod State University: from 1995 to 1999 it was bachelor course in Radiophysics, from 1999 to 2001 it was master course in Radiophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in Radiophysics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of Microstructures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny Novgorod State University of Architecture and Civil Engineering. 2012-2015 Full Doctor course in Radiophysical Department of Nizhny Novgorod State University. Since 2015 E.L. Pankratov is an Associate Professor of Nizhny Novgorod State University. He has 155 published papers in area of his researches. Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school of village Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school “Center for gifted children”. From 2009 she is a student of Nizhny Novgorod State University of Architecture and Civil Engineering (spatiality “Assessment and management of real estate”). At the same time she is a student of courses “Translator in the field of professional communication” and “Design (interior art)” in the University. Since 2014 E.A. Bulaeva is in a PhD program in Radiophysical Department of Nizhny Novgorod State University. She has 105 published papers in area of her researches.