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Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016

DEPENDENCE OF CHARGE CARRIERS MOBILITY IN THE P-N-HETEROJUNCTIONS ON COMPOSITION OF MULTILAYER STRUCTURE 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 manufacturing a p-n-junctions by dopant diffusion or ion implantation into a multilayer structure. We introduce an approach to increase sharpness of these p-n-junctions and at the same time to increase homogeneity of distributions of dopants in enriched by these dopants areas. We consider influence of the above changing of distribution of dopant on charge carrier mobility. We also consider an approach to decrease value of mismatch-induced stress in the considered multilayer structure by using a buffer layer. The decreasing gives a possibility to increase value of charge carrier mobility.

KEYWORDS: Multilayer structure; charge carriers mobility; p-n-heterojunctions

1. INTRODUCTION In the present time one of the actual questions is increasing of performance of solid state electronic devices (diodes, field-effect and bipolar transistors, ...) [1-6]. To increase the performance it is attracted an interest searching of materials with higher values of the charge carrier mobility [710]. An alternative approach to increase the performance is development of new technological processes and optimization of existing one. Framework this paper we introduce an approach of manufacturing of heterodiodes. The approach gives a possibility to decrease of switching time. At the same using the approach gives a possibility to decrease dimensions of these diodes. With decreasing of switching time and dimensions of these diodes value of mismatch- induced stress also decreases. In this paper we consider a heterostructure on Fig. 1. The heterostructure includes into itself a substrate and an epitaxial layer. A buffer layer has been grown between these substrate and epitaxial layer. We consider doping of the epitaxial layer by diffusion or ion implantation after manufacturing the heterostructure. The doping gives a possibility to produce required type of conductivity (p or n). We assume, that type of conductivity of substrate is known: p or n. After doping of the epitaxial layer optimized annealing of dopant and/or radiation defects has been considered. In this paper we analyzed possibility to increase sharpness of p-n- junction with increasing of homogeneity of concentration of dopant in enriched by the dopant area. Framework the paper we analyzed influence of changing of distribution of concentration of dopant in space and time on charge carrier mobility with account mismatch-induced stress in the considered heterostructure.

DOI:10.5121/msej.2016.3203

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016

2. METHOD OF SOLUTION To analyze influence of changing of distribution of concentration of dopant in space and time on charge carrier mobility we solve the second Fick's law in the following form [1,11-15] ∂ C ( x, y , z , t ) ∂  ∂ C ( x, y , z , t )  ∂ = D + ∂t ∂x ∂x  ∂y

+Ω

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

 ∂ C ( x, y , z , t )  D + ∂z  

(1)

Lz Lz   ∂  DS ∂  DS ∇ S µ1 ( x, y, z, t ) ∫ C ( x, y,W , t ) d W  + Ω ∇ S µ1 (x, y , z , t ) ∫ C (x, y ,W , t ) d W  .   ∂ x kT ∂ y kT 0 0  

Fig. 1. Heterostructure, which consist of a substrate, epitaxial layers and buffer layer

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 ∂x ∂y ∂y x =0 x = Lx y =0 x=Ly ∂ C ( x, y , z , t ) ∂ C ( x, y , z , t ) =0, = 0 , C x,y,z,0)=fC (x,y,z). ∂z ∂z z =0 x = Lz (

The function C(x,y,z,t) describes distribution of concentration of dopant in space and time. Ω is Lz

the atomic volume of dopant; ∇s is the symbol of surficial gradient. The function ∫ C (x, y, z, t ) d z 0

describes distribution of surficial concentration of dopant in space and time on interface between layers of heterostructure (in this situation we assume, that Z-axis is perpendicular to interface between layers of heterostructure). The function µ1(x,y,z,t) describes the chemical potential due to the presence of mismatch-induced stress in space and time. The functions D and DS describe distributions of coefficients of volumetric and surficial diffusions on coordinate and temperature. Values of dopant diffusions coefficients depends on properties of materials of heterostructure, speed of heating and cooling of materials during annealing and spatio-temporal distribution of concentration of dopant. Dependences of dopant diffusions coefficients on parameters could be approximated by the following relations [13-15]

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016

 C γ ( x, y , z , t )   V ( x, y , z , t ) V 2 ( x, y , z , t )  + DC = DL (x, y , z ,T ) 1 + ξ γ ς , 1 + ς 1 2  P (x, y , z ,T )   V* (V * )2  

 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 + ς2 1 + ς 1 .  * P ( x, y , z, T )   V (V * )2  

(2)

Here DL (x,y,z,T) and DLS (x,y,z,T) are the spatial (due to accounting all layers of heterostruicture) and temperature (due to Arrhenius law) dependences of dopant diffusion coefficients; T is the temperature of annealing. Function P (x,y,z,T) describes spatial and temperature dependence of the limit of solubility of dopant. Parameter γ ∈[1,3] with integer values depends on properties of materials [13]. The function V (x,y,z,t) describes distribution of concentration of radiation vacancies in space and time; V* is the equilibrium distribution of vacancies. One can find description of dependence of dopant diffusion coefficient on concentration of dopant in [13]. To determine distributions of concentration of point radiation defects in space and time we solve the following boundary problem [11,12,14,15]

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

+Ω

∂ I ( x, y , z , t )  ∂   D I ( x, y , z , T )  − k I ,V (x, y , z , T ) I (x, y , z, t ) V (x, y , z , t ) + (3) ∂ z ∂ z 

Lz Lz   ∂  D IS ∂  D IS ∇ 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 0 0  

∂ 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 0 0  

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

(4)

The function I (x,y,z,t) describes distribution of concentration of radiation interstitials in space and time with the equilibrium distribution I*. The functions DI(x,y,z,T), DV(x,y,z,T), DIS(x,y,z,T), DVS(x,y,z,T) describe dependences of coefficients of volumetric and surficial diffusions of interstitials and vacancies on coordinate and temperature. Terms V2(x,y,z,t) and I2(x,y,z,t) gives a possibility to take into account generation of divacancies and diinterstitials [15]. The functions 25

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016

kI,V(x,y,z,T), kI,I(x,y, z,T) and kV,V(x,y,z,T) describe dependences of parameters of recombination of point radiation defects and generation of their complexes on coordinate and temperature. We determine spatio-temporal distributions of divacancies ΦV(x,y,z,t) and diinterstitials ΦI(x,y, z,t) by solving the following boundary problem [11,12,14,15]

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

Lz  ∂ Φ I (x, y, z, t ) ∂  DΦI S ∂  ∇S µ1 (x, y, z, t ) ∫ Φ I (x, y,W , t ) d W  + k I (x, y, z, T ) × DΦI (x, y, z, T )  +Ω  ∂z  ∂z ∂ x  kT 0  

× I ( x, y , z , t ) + Ω

Lz  ∂  DΦ I S ∇ S µ1 ( x, y, z, t ) ∫ Φ I (x, y , W , t ) d W  + k I , I ( x, y , z, T ) I 2 (x, y , z , t ) (5)  ∂ y  kT 0 

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

Lz  ∂ ΦV (x, y, z, t ) ∂  DΦV S ∂  ∇S µ1 (x, y, z, t ) ∫ ΦV (x, y, W, t ) d W  + kV (x, y, z, T ) × DΦV (x, y, z, T )  +Ω  ∂z  ∂z ∂ x  kT 0  

× V ( x, y , z , t ) + Ω

Lz  ∂  D ΦV S ∇ S µ1 ( 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 0 

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

∂ Φ 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 =0 z = Lz ΦI (x,y,z,0)=fΦI (x,y,z), ΦV (x,y,z,0)=fΦV (x,y,z).

(6)

The functions DΦI(x,y,z,T), DΦV(x,y,z,T), DΦIS (x,y,z,T) and DΦVS(x,y,z,T) describe dependences of coefficients of volumetric and surficial diffusions of complexes of radiation defects on coordinate and temperature. The functions kI(x,y,z,T) and kV(x,y,z,T) describe dependences of parameters of decay of complexes of radiation defects on coordinate and temperature. Chemical potential µ1 in Eq.(1) could be determine by the following relation [11]

µ1=E(z)Ωσij [uij(x,y,z,t)+uji(x,y,z,t)]/2,

(7)

1  ∂ u ∂ u j  where E(z) is the Young modulus, σij is the stress tensor; u ij =  i + is the deformation 2  ∂ x j ∂ xi  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 vector r u ( x, y, z , t ) ; xi, xj are the coordinate x, y, z. The Eq. (3) could be transform to the following form 26

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016

 ∂ u i ( x, y , z , t )

µ ( x, y , z , t ) = 

∂ xj



− ε 0δ ij +

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

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

where σ is Poisson coefficient; the mismatch parameter could be determined as ε0 = (as-aEL)/aEL, where as is the lattice distances of the substrate, aEL and the epitaxial layer; K is the modulus of uniform compression; parameter β describes thermal expansion coefficient, parameter Tr describes the equilibrium temperature. The equilibrium temperature coincides in our case with room temperature. Components of displacement vector could be obtained by solution of the following equations [12] ∂ 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 ) ρ (z ) = + + ∂t2 ∂x ∂y ∂z

ρ (z ) ρ (z ) where σ ij =

∂ 2 u y ( x, y , z , t ) ∂t

∂ σ yx ( x, y, z , t ) ∂x

∂ σ yy ( x, y , z , t ) ∂y

∂ σ yz ( x, y , z , t ) ∂z

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

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

× K ( z ) − β ( z ) K (z ) [T (x, y, z, t ) − Tr ] , ρ (z) is the density of materials of heterostructure, δij is the Kronecker symbol. With account the relation for σij last system of equation could be written as 2 ∂ 2 u x (x, y, z, t )  E ( z )  ∂ u y ( x, y, z, t ) 5E ( z )  ∂ 2 u x (x, y, z, t )  ( ) ( ) = K z + + K z − +     ∂t2 ∂ x2 6 [1 + σ ( z )] 3 [1 + σ ( z )] ∂ x∂ y  

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 + −     2 [1 + σ ( z )]  3[1 + σ (z )] ∂ y2 ∂ z2 ∂ x∂ z  

− K (z ) β (z )

ρ (z )

∂ 2 u y (x, y , z , t ) ∂t2

∂ T ( x, y , z , t ) ∂x

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

2  ∂  E ( z )  ∂ u y ( x, y, z, t ) ∂ u z ( x, y, z, t )   ∂ u 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 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 

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016

 ∂ 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  ∂z ∂x ∂ y ∂z 6 ∂ z  1 + σ (z )   

Conditions for the system of Eq. (8) could be written in the form r r r r ∂ u (x, L y , z , t ) ∂ u (L x , y , z , t ) ∂ u (0, y, z , t ) ∂ u (x,0, z , t ) = 0; =0; =0; =0; ∂x ∂x ∂y ∂y r r ∂ u (x, y , Lz , t ) r r r r ∂ u (x, y ,0, t ) =0; = 0 ; u (x, y, z ,0 ) = u 0 ; u ( x, y, z , ∞ ) = u 0 . ∂z ∂z

We solve the Eqs.(1), (3) and (5) by using standard method of averaging of function corrections [17]. Previously we transform the Eqs.(1), (3) and (5) to the following form with account initial distributions of the considered concentrations ∂ C (x, y, z, t ) ∂  ∂ C (x, y, z, t )  ∂  ∂ C (x, y, z, t )  ∂  ∂ C (x, y, z, t )  = D + D + D  + (1a) ∂t ∂x ∂x ∂y ∂z  ∂y  ∂z   + f C ( x, y , z )δ (t ) + Ω

Lz  ∂  DS ∇ S µ ( x, y , z , t ) ∫ C ( x, y , W , t ) d W  +  ∂ x kT 0 

+Ω

Lz  ∂  DS ∇ S µ ( x , y , z , t ) ∫ C ( x , y ,W , t ) d W   ∂ y kT 0 

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

Lz  ∂ I (x, y, z, t )  ∂  DIS ∂  ∇ S µ1 ( x, y, z, t ) ∫ I (x, y,W , t ) d W  − k I ,V (x, y, z, T ) ×  DI (x, y, z, T ) +Ω  ∂ z ∂z ∂x kT 0  

× I (x, y, z, t )V (x, y, z, t ) + Ω

Lz  ∂  DIS 2 ∇ µ ( x , y , z , t ) ∫ I ( x, y,W , t ) d W  − k I ,I ( x, y, z, T ) I (x, y, z, t ) (3a) S 1  ∂ y  kT 0 

∂ 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 )  + f V (x, y, z)δ (t ) + ∂t ∂ x ∂x ∂y  ∂ y  +

Lz  ∂ V (x, y, z, t ) ∂  ∂  DVS ∇ S µ1 (x, y, z, t ) ∫ V (x, y, W , t ) d W  − k I ,V ( x, y, z, T ) × DV (x, y, z, T )  +Ω  ∂ z ∂z ∂ x k T 0   

× I ( x, y, z, t )V ( x, y, z, t ) + Ω

Lz  ∂  DVS ∇ S µ1 ( x, y, z, t ) ∫ V ( x, y, W , t ) d W  − kV ,V (x, y, z, T )V 2 (x, y, z, t )  ∂ y  kT 0 

∂ Φ I ( x, y , z , t ) ∂  ∂ Φ I ( x, y , z , t )  ∂  ∂ Φ I ( x, y , z , t )  = D (x, y , z, T ) +  DΦ I (x, y , z, T ) + ∂t ∂ x  Φ I ∂x ∂ y ∂y    28

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 Lz  ∂ Φ I ( x, y , z , t )  ∂  DΦ I S ∂  ∇ S µ 1 ( x , y , z , t ) ∫ Φ I ( x, y , W , t ) d W  +   DΦ I ( x, y, z , T ) +Ω ∂ x k T ∂z ∂z 0   

+Ω

Lz  ∂  DΦ I S ∇ S µ 1 ( x, y , z , t ) ∫ Φ I ( x, y , W , t ) d W  + k I ( x, y , z , T ) I ( x, y , z , t ) +  ∂ y  kT 0 

+ k I , I (x, y , z , T ) I 2 ( x, y , z , t ) + f Φ I (x, y , z )δ (t )

(5a)

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

Lz  ∂ Φ V ( x, y , z , t )  ∂  DΦV S ∂  ∇ S µ1 ( x , y , z , t ) ∫ Φ V ( x , y , W , t ) d W  +   DΦV (x, y, z , T ) +Ω ∂ x  kT ∂z ∂z 0  

+Ω

Lz  ∂  DΦ I S ∇ S µ1 (x, y, z, t ) ∫ Φ I ( x, y,W , t ) d W  + k I (x, y, z, T ) I ( x, y, z , t ) +  ∂ y  kT 0 

+ kV ,V (x, y, z , T )V 2 ( x, y, z , t ) + f ΦV (x, y , z )δ (t ) .

Farther we replace concentrations of dopant and radiation defects in right sides of Eqs. (1a), (3a) and (5a) on their average values α1ρ. The average values α1ρ are not yet known. In this situation we obtain equations for the first-order approximations of the required concentrations in the following form   ∂ C1 (x, y, z, t ) ∂  D ∂  D = α1C Ω z S ∇S µ1 (x, y, z, t ) + α1C Ω z S ∇S µ1 (x, y, z, t ) + f C (x, y, z)δ (t ) (1b) ∂t ∂x  kT ∂ y  kT     ∂ I 1 ( x, y , z , t ) ∂  DIS ∂  DIS = α 1I z Ω ∇ S µ ( x, y, z, t ) + α1I Ω  z k T ∇ S µ ( x, y, z, t ) + ∂t ∂ x  k T ∂ y    + f I (x, y, z )δ (t ) − α12I k I , I (x, y, z, T ) − α1I α 1V k I ,V (x, y, z ,T )

(3b)

  ∂ V1 (x, y, z , t ) ∂  DVS ∂  DVS = α 1V z Ω ∇ S µ1 ( x, y, z , t ) + α 1V Ω z ∇ S µ1 (x, y, z, t ) +   ∂t ∂ x  kT ∂ y  kT   + f V ( x, y, z )δ (t ) − α 12V kV ,V (x, y, z , T ) − α 1I α 1V k I ,V (x, y, z , T )   ∂ Φ1I (x, y, z, t ) ∂  DΦ S ∂  DΦ I S = α1Φ I z Ω  I ∇ S µ1 (x, y, z, t ) + α1Φ I z Ω ∇ S µ1 (x, y, z, t ) +  ∂t ∂ x  kT ∂ y  kT   + f Φ I ( x, y , z )δ (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 )

(5b)

  ∂ Φ1V (x, y, z, t ) ∂  DΦ S ∂  DΦV S = α1ΦV z Ω  V ∇ S µ1 (x, y, z, t ) + α1ΦV z Ω ∇ S µ1 ( x, y, z, t ) +  ∂t ∂ x  kT ∂ y  kT   + f ΦV (x, y, z )δ (t ) + kV (x, y, z , T )V (x, y, z , t ) + kV ,V (x, y, z, T )V 2 ( x, y, z, t ) . Integration of the left and right sides of the Eqs. (1b), (3b) and (5b) on time gives us possibility to obtain relations for above approximation in the final form

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016

C1 (x, y, z, t ) = α 1C Ω

z kT

 V ( x, y , z , τ ) V 2 (x, y , z , τ )  ∂ t ς + × ∫ DS L (x, y, z, T ) ∇ S µ1 (x, y, z ,τ ) 1 + ς 1 2 ∂x0 V* (V * )2  

  ξ S α 1γC ∂ t z ∫ D S L ( x, y , z , T ) ∇ S µ 1 ( x, y , z , τ ) 1 + γ  d τ + α 1C Ω ( ) ∂y0 kT  P x, y , z , T 

  ξ S α 1γC 1 + γ × ( )  P x, y , z , T 

 V ( x, y , z , τ ) V 2 ( x, y , z , τ )  × 1 + ς 1 + ς  d τ + f C ( x, y , z ) 2 V* (V * )2   I 1 ( x, y , z , t ) = α 1 I z Ω

(1c)

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

+ f I ( x, y, z ) − 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 ) = α1V z Ω

(3c)

∂ t DIS ∂ t DIS ∇ S µ1 ( x, y, z ,τ ) d τ + α1V z Ω ∇ S µ1 ( x, y, z,τ ) d τ + ∫ ∫ ∂ x 0 kT ∂ y 0 kT t

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

Φ 1I ( x, y , z, t ) = α 1Φ I z Ω

∂ t DΦ I S ∂ t DΦ I S ∇ S µ1 ( x, y, z, τ ) d τ + α 1Φ I z Ω ∇ S µ1 ( x, y , z, τ ) d τ + ∫ ∫ ∂ x 0 kT ∂ x 0 kT t

+ 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

Φ 1V (x, y, z, t ) = α 1ΦV z Ω

(5c)

∂ t DΦV S ∂ t DΦV S ∇ S µ1 (x, y , z, τ ) d τ + α 1ΦV z Ω ∇ S µ 1 ( x, y , z , τ ) d τ + ∫ ∫ ∂ x 0 kT ∂ x 0 kT t

+ 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 .

We determine average values of the first-order approximations of concentrations of dopant and radiation defects by the following standard relation [17]

α1ρ =

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

(9)

We obtain required average values by substitution of the relations (1c), (3c) and (5c) into relation (9). The obtained relations could be written as

α1C =

1 Lx Ly Lz ∫ ∫ ∫ f C ( x, y, z ) d z d y d x , α 1I = Lx Ly Lz 0 0 0

α 1V =

(a3 + A)2 4 a 42

 Θ a3 B + Θ 2 L x L y Lz a1  a3 + A − , −4 B +   4 a4 a4  

 1  Θ Lx L y Lz ∫ ∫ ∫ f I ( x, y, z ) d z d y d x − α 1I S II 00 − ΘLx L y Lz  .  S IV 00  α 1I 0 0 0 

Lx L y Lz

0 0 0

Here 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 , a4 =(SIV2 00 − SII 00SVV 00 ) × 30

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 Lx L y L z

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

L x L y Lz

× S IV 00 + 2 S VV 00 S II 00 ∫ ∫ ∫ f I (x, y, z ) d z d y d x − Θ L2x L2y L2z S VV 00 − ∫ ∫ ∫ f I ( x, y, z ) d z d y d x × 0 0 0

0 0 0 2

  × S IV2 00 , a1 = S IV 00 ∫ ∫ ∫ f I ( x, y, z ) d z d y d x , a 0 = S VV 00  ∫ ∫ ∫ f I (x, y, z ) d z d y d x  , 0 0 0 0 0 0  L x L y Lz

L x L y Lz

 aa  q 2 + p 3 + q , q =  4a0 − Θ Lx Ly Lz 1 3  × a4   3 2 3 3 4 2 4 a 0 a 4 − Θ L x L y L z a1 a 3 a  a  Θ a2 Θ a2 Θ a1 × − Θ 2 02  4 Θ a 2 − Θ 2 3  − , p = Θ2 − L2x L2y L2z − 2 3 2 24 a 4 8a 4  a 4  54 a 4 8 a4 12 a 42 A = 8 y + Θ2

Θ a2 3 a32 a + q 2 + p3 − q − 3 − 4Θ 2 , B = 2 a4 a4 6 a4

− Θ a 2 18a4 ,

α 1Φ I =

S II 20 RI 1 1 Lx L y L z + + ∫ ∫ ∫ f Φ ( x, y , z ) d z d y d x ΘLx L y Lz ΘLx L y Lz Lx L y Lz 0 0 0 I

α 1ΦV =

SVV 20 RV 1 1 Lx L y L z + + ∫ ∫ ∫ f Φ (x, y , z ) d z d y d x , ΘLx L y Lz ΘLx L y Lz Lx L y Lz 0 0 0 V

Lx L y Lz

0 0 0

Here Rρ i = ∫ (Θ − t ) ∫ ∫ ∫ k I ( x, y, z, T ) I1i ( x, y, z , t ) d z d y d x d t . The second-order approximations and approximations with higher orders of the considered concentrations have been calculated framework standard iterative procedure of method of averaging of function corrections [17]. The considered approximations could be obtained by replacement the required concentrations in the Eqs. (1c), (3c), (5c) on the following sum αnρ+ρ n-1(x,y,z,t). The replacement leads to following results

[α 2C + C1 (x, y, z, t )]γ  1 + ς V (x, y, z, t ) + ς V 2 (x, y, z, t ) × ∂ C 2 (x, y , z , t ) ∂   = + ξ 1    1 2 ∂t ∂ x   P γ ( x, y , z , T ) V* (V * )2    × D L (x, y, z, T )

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

 V (x, y, z, t ) V 2 (x, y, z, t )  ∂ C1 (x, y, z, t ) + ς × 1 + ς 1  2 V* (V * )2  ∂ y 

 [α + C (x, y, z, t )]γ × D L ( x, y, z, T )1 + ξ 2C γ 1 P ( x, y, z, T ) 

× D L ( x, y , z , T ) +Ω

 ∂   V ( x, y, z, t ) V 2 (x, y, z, t )  + ς ×   +  1 + ς 1 2 V* (V * )2    ∂ z  

[α 2C + C1 (x, y, z, t )]γ   + f (x, y, z )δ (t ) + ∂ C1 (x, y, z, t )  1 + ξ   C ∂z P γ ( x, y, z, T )   

Lz  ∂  DS ∇ S µ1 (x, y , z, t ) ∫ [α 2C + C ( x, y,W , t )] d W  +  ∂ x kT 0 

+Ω

Lz  ∂  DS ∇ S µ1 ( x, y, z, t ) ∫ [α 2C + C (x, y,W , t )] d W  (1d)  ∂ y kT 0 

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016

∂ I 2 ( x, y , z , t ) ∂  ∂ I ( x, y , z , t )  ∂  ∂ I 1 (x, y, z , t )  = D ( x, y , z , T ) 1  + ∂ y  DI ( x , y , z , T ) + ∂t ∂ x  I ∂x ∂y    +

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

× [α 1I + I 1 (x, y , z , t )][α 1V + V1 (x, y , z, t )] + Ω

Lz ∂  ∇ S µ ( x, y, z , t ) ∫ [α 2 I + I 1 (x, y,W , t )] d W × ∂x 0

Lz  DIS  ∂  DIS ∇ S µ ( x, y, z, t ) ∫ [α 2 I + I 1 ( x, y ,W , t )] d W  +Ω  kT  ∂ y  kT 0 

(3d)

∂ V2 (x, y, z, t ) ∂  ∂ V ( x, y , z , t )  ∂  ∂ V ( x, y , z , t )  = DV ( x, y, z, T ) 1 + DV (x, y, z, T ) 1    + ∂t ∂ x ∂x ∂y  ∂ y  +

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

× [α 1I + I 1 ( x, y, z , t )][α 1V + V1 ( x, y , z , t )] + Ω

Lz ∂  ∇ S µ ( x, y , z , t ) ∫ [α 2V + V1 ( x, y ,W , t )] d W × ∂x 0

Lz  DVS  ∂  DVS ∇ S µ ( x, y, z, t ) ∫ [α 2V + V1 ( x, y,W , t )] d W  +Ω  kT  ∂ y  kT 0 

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

Lz  ∂  DΦ I S ∇ S µ ( x , y , z , t ) ∫ α 2 Φ I + Φ 1 I ( x , y , W , t ) d W  + k I , I ( x, y , z , T ) I 2 ( x, y , z , t ) +  ∂ x  kT 0 

[

+Ω

]

Lz  ∂  DΦ I S ∇ S µ ( x , y , z , t ) ∫ α 2 Φ I + Φ 1 I ( x , y ,W , t ) d W  + k I ( x , y , z , T ) I ( x , y , z , t ) +  ∂ y  kT 0 

[

]

∂ Φ 1I (x, y , z , t )  ∂  DΦ ( x, y, z, T )  + f Φ I ( x, y, z )δ (t ) ∂ z  I ∂z 

(5d)

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

+Ω

Lz  ∂  DΦV S ∇ S µ ( x, y, z, t ) ∫ α 2 ΦV + Φ 1V ( x, y,W , t ) d W  + kV ,V ( x, y, z, T )V 2 (x, y, z , t ) +  ∂ x  kT 0 

[

]

Lz  ∂  DΦV S ∇ S µ (x, y, z, t ) ∫ α 2 ΦV + Φ1V (x, y,W , t ) d W  + kV (x, y, z, T )V (x, y, z, t ) +  ∂ y  kT 0 

[

]

∂ Φ1V ( x, y , z, t )  ∂   DΦV ( x, y, z , T )  + f ΦV ( x, y, z )δ (t ) . ∂ z ∂z 

Final relations for the second-order approximations could be obtained by integration of the left and the right sides of Eqs. (1d), (3d) and (5d). The final relations could be written as

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016

C2 (x, y, z, t ) =

 [α 2C + C1 (x, y, z,τ )]γ ∂ t ∫ D L ( x, y, z, T ) 1 + ξ ∂x0 P γ (x, y, z, T ) 

 ∂ C 1 ( x, y , z , τ ) ∂ t V ( x, y , z , τ ) V 2 ( x, y , z , τ )  ∂ C 1 ( x, y , z , τ ) +ς2 × dτ +  ∫ D L (x, y, z , T ) 1 + ς 1 * ∂x ∂y0 ∂y V (V * )2  

 [α + C (x, y, z, t )]γ × 1 + ξ 2C γ 1 P (x, y, z, T ) 

 ∂ t  V ( x, y, z, τ ) V 2 ( x, y, z, τ )  ∂ C1 ( x, y, z, τ ) + 1 + + × ς ς   ∫  1 2 ∂z V*  ∂ z 0  (V * )2 

 [α + C (x, y, z,τ )] × D L ( x, y, z, T ) 1 + ξ 2C γ 1 P (x, y , z , T ) 

Lz  ∂ t ∫ ∇ S µ ( x, y, z, τ ) ∫ [α 2C + C1 ( x, y, W ,τ )] × dτ + ∂x0 0 

DS D S Lz ∂ t dW dτ + Ω ∫ ∇ S µ ( x, y , z , τ ) ∫ [α 2 C + C1 ( x, y , W , τ )] d W d τ + f C ( x, y , z ) (1e) kT ∂y0 kT 0

I 2 ( x, y, z, t ) = +

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

∂ I1 (x, y, z,τ ) ∂ I1 (x, y, z,τ ) ∂ t ∂ t dτ + dτ + ∫ DI ( x, y, z, T ) ∫ DI ( x, y, z, T ) ∂x ∂y ∂ x0 ∂ y0

t ∂ I 1 ( x, y , z , τ ) ∂ t 2 d τ + f I (x, y , z ) − ∫ k I , I ( x, y, z , T ) [α 2 I + I 1 (x, y , z ,τ )] d τ − ∫ D I (x, y , z , T ) ∂ z0 ∂z 0 t

− ∫ 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 + Ω 0

× ∫ [α 2 I + I 1 (x, y, W ,τ )] d W d τ + Ω 0

V2 ( x, y, z, t ) = +

∂ t D IS ∇ S µ ( x, y , z , τ ) × ∫ ∂ x 0 kT

DIS Lz ∂ t ∫ ∇ S µ ( x, y , z , τ ) ∫ [α 2 I + I 1 (x, y, W ,τ )] d W d τ (3e) ∂y0 kT 0

∂ V1 (x, y, z,τ ) ∂ V1 (x, y, z,τ ) ∂ t ∂ t dτ + dτ + ∫ DV ( x, y, z, T ) ∫ DV ( x, y, z, T ) ∂ x0 ∂ y0 ∂x ∂y

t ∂ V1 ( x, y, z ,τ ) ∂ t 2 d τ + f V ( x, y , z ) − ∫ k V ,V (x, y , z, T ) [α 2V + V1 (x, y , z,τ )] d τ − ∫ DV (x, y , z , T ) ∂z ∂ z0 0 t

− ∫ 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 + Ω 0 Lz

× ∫ [α 2 I + V1 ( x, y, W , τ )] d W d τ + Ω 0

Φ 2 I ( x, y , z , t ) = × DΦ I ( x, y , z , T ) d τ + Lz

DVS L z ∂ t ∫ ∇ S µ ( x, y , z , τ ) ∫ [α 2 I + V1 (x, y, W , τ )] d W d τ ∂y0 kT 0

∂ t ∂ Φ1I (x, y, z ,τ ) ∂ t ∂ Φ1I (x, y , z ,τ ) dτ + × ∫ DΦ I ( x, y , z, T ) ∫ ∂ x0 ∂x ∂ y0 ∂ y

DΦ I S ∂ Φ 1 I ( x, y , z , τ ) ∂ t ∂ t dτ + Ω × ∫ D Φ I ( x, y , z , T ) ∫ ∇ S µ ( x, y , z , τ ) ∂ z0 ∂ z ∂x0 kT

[

]

× ∫ α 2 Φ I + Φ 1 I (x, y , W , τ ) d W d τ + Ω 0

DΦ I S kT

∂ t DVS ∇ S µ ( x, y , z , τ ) × ∫ ∂ x 0 kT

Lz ∂ t ∫ ∇ S µ ( x, y , z , τ ) ∫ α 2 Φ I + Φ 1 I ( x, y , W , τ ) d W × ∂y0 0

[

]

d 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 + f Φ I ( x, y, z ) (5e)

Φ 2 V ( x, y , z , t ) =

∂ Φ1V (x, y , z ,τ ) ∂ t ∂ t ∂ Φ1V ( x, y , z,τ ) dτ + × ∫ DΦV ( x, y , z , T ) ∫ ∂x ∂ x0 ∂ y0 ∂ y 33

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016

× D ΦV ( x , y , z , T ) d τ + Lz

[

D ΦV S ∂ Φ 1V ( x, y , z,τ ) ∂ t ∂ t dτ + Ω × ∫ D ΦV ( x , y , z , T ) ∫ ∇ S µ (x, y , z , τ ) ∂ z0 ∂ z ∂x0 kT

]

× ∫ α 2 ΦV + Φ 1V ( x, y , W , τ ) d W d τ + Ω 0

D ΦV S kT

Lz ∂ t ∫ ∇ S µ ( x, y, z ,τ ) ∫ α 2 ΦV + Φ 1V (x, y , W ,τ ) d W × ∂y0 0

[

]

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

Average values of the second-order approximations of required approximations by using the following standard relation [17] Θ L x L y Lz 1 ∫ ∫ ∫ ∫ [ρ 2 ( x, y, z, t ) − ρ1 ( x, y , z, t )] d z d y d x d t . ΘLx L y Lz 0 0 0 0

α 2ρ =

(10)

Substitution of the relations (1e), (3e), (5e) into relation (10) gives us possibility to obtain relations for required average values α 2ρ

α2C=0, α2ΦI =0, α2ΦV =0, α 2V = α 2I = where b4 = +

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

 

 

4 b4

CV − α 22V SVV 00 − α 2V (2 SVV 01 + S I V 10 + ΘLx L y Lz ) − SV V 02 − S I V 11 S IV 01 + α 2V S IV 00

S II 00 SVV 00 1 1 2 (2SVV 01 + S IV10 + ΘLx Ly Lz ) + S IV2 00 SV V 00 − SVV 00 S II 00 , b3 = − ΘLx Ly Lz ΘLx Ly Lz ΘLx Ly Lz

2 2 S IV 00 SVV 00 (S IV 01 + 2S II10 + S IV 01 + ΘLx Ly Lz ) + S IV 00 (2SVV 01 + S IV10 + ΘLx Ly Lz ) − S IV3 003 S 3IV103 , ΘLx L y Lz ΘLx L y Lz Θ Lx L y Lz

b2 =

S II 00 S VV 00 (SVV 02 + S IV 11 + CV ) − (S IV 10 − 2SVV 01 + Θ L x L y Lz )2 + (ΘL x L y L z + 2S II10 + S IV 01 ) × ΘL x L y L z

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

S IV 00 (S IV 01 + 2S II 10 + 2S IV 01 + ΘL x L y L z )(2SVV 01 + ΘL x L y Lz + S IV 10 ) − (CV − SVV 02 − S IV 11 ) × ΘL x L y L z

S IV2 00 C S2 S S S + S VV 02 + CV (2 SVV 01 + S IV 10 + + 2 I 2 IV200 2 − 2 S IV 01 IV 10 IV 00 , b1 = S II 00 IV 11 ΘL x L y L z Θ L x L y L z ΘL x L y L z ΘL x L y L z

+ Θ Lx L y Lz ) + −

2 S IV 01 ( Θ L x L y L z + 2S II 10 + S IV 01 )(2SVV 01 + S IV 10 + ΘLL y L z ) − S IV 10 S IV 01 − ΘL x L y L z ΘL x L y L z

S IV 00 (3S IV 01 + 2S II 10 + ΘL x L y L z )(CV − SVV 02 − S IV 11 ) + 2 C I S IV 00 S IV 01 , b0 = S II 00 × ΘL x L y L z ΘL x L y L z

× (S IV 00 + SVV 02 ) − 2

S IV 01 (ΘLx Ly Lz + 2 S II10 + S IV 01 )(CV − SVV 02 − S IV11 ) − S IV 01 CV − SVV 02 − S IV11 × Lx L y Lz Θ ΘLx L y Lz

× S IV 01 (ΘL x L y L z + 2 S II 10 + S IV 01 ) + 2C I S IV2 01 , C I =

−

α 1I α 1V ΘL x L y L z

S IV 00 +

α 12I S II 00 ΘL x L y L z

−

S II 20 S II 20 −, ΘL x L y L z

S IV 11 a2 Θ a2 a , CV = α 1I α1V S IV 00 + α 12V SVV 00 − SVV 02 − S IV 11 , E = 8 y + Θ2 32 − 4 Θ 2 , F = + ΘL x L y L z a4 a4 6 a4 34

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 b b  Θ 3 b23  b2   4b0 − Θ Lx L y L z 1 3  − −  4 Θ b2 − Θ 2 3  × 3 b4  54 b4  b4   4 2 2 4b b − Θ Lx L y Lz b1b3 Θ b2 Θ b1 Θ , s = Θ2 0 4 . × b0 2 − L2x L2y L2z − 2 8b4 8 b4 12 b42 18 b4 +3

r 2 + s3 − r − 3

r 2 + s3 + r , r =

Θ 3 b2 24 b42

Farther we determine solutions of Eqs.(8), i.e. components of displacement vector. To determine the first-order approximations of the considered components framework method of averaging of function corrections we replace the required functions in the right sides of the equations by their not yet known average values αi. The substitution leads to the following result

ρ (z )

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

ρ (z )

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

Integration of the left and the right sides of the above relations on time t leads to the following result u1x (x, y, z, t ) = u 0 x + K ( z )

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 , ρ (z ) ∂ x 0 0 ρ (z ) ∂ x 0 0

u1 y ( x, y, z, t ) = u 0 y + K ( z )

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 , ρ (z ) ∂ y 0 0 ρ (z ) ∂ y 0 0

u1z (x, y, z, t ) = u 0 z + K (z )

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 . ρ (z ) ∂ z 0 0 ρ (z ) ∂ z 0 0

Approximations of the second and higher orders of components of displacement vector could be determined by using standard replacement of the required components on the following sums αi+ui(x,y,z,t) [17]. The replacement leads to the following result 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 ) − + ρ (z )   ∂t2 6 [1 + σ (z )] ∂ x2 3 [1 + σ ( z )] ∂ x∂ y  

∂ 2 u1 y ( x, y , z , t ) ∂ x∂ y

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

 E ( z )  ∂ 2 u1 z ( x, y , z , t ) + K (z ) +  3 [ 1 + σ ( z )] ∂ x∂ z 

ρ (z )

∂ 2 u 2 y (x, y, z, t ) ∂t2

2 E (z )  ∂ u1 y (x, y, z, t ) ∂ 2 u1x (x, y, z, t )  ∂ T (x, y, z, t ) + +   − K ( z ) β (z ) ∂ x2 ∂ x∂ y ∂y 2[1 + σ (z )]  

2  ∂  E ( z )  ∂ u1 y (x, y , z , t ) ∂ u1z ( x, y, z, t )   ∂ u1 y (x, y , z , t )  5E ( 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 

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ρ (z )

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

 ∂ 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    6 [1 + σ (z )]

∂  ∂ u1z ( x, y , z, t ) ∂ u1x (x, y , z , t ) ∂ u1 y ( x, y , z, t ) ∂ u1z ( x, y , z, t )   ∂ T ( x, y, z , t ) − − − × 6  − ∂z  ∂z ∂x ∂y ∂z ∂z   × K (z ) β ( z ) .

Integration of the left and right sides of the above relations on time t leads to the following result u 2 x ( x, y, z, t ) =

 5E ( z )  ∂ 2 t ϑ E (z )  1  ( ) K z +   2 ∫ ∫ u1x ( x, y, z,τ ) d τ d ϑ +  K ( z ) − × ρ (z )  6 [1 + σ (z )] ∂ x 0 0 3 [1 + σ ( z )] 

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

 β (z ) E (z ) E (z )  ∂2 t ϑ 1 + × ∫ ∫ u1 z ( 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 − ∂x00 ρ (z )  6 [1 + σ (z )] ∂ x 0 0 ρ (z )

 E (z )  ∂ 2 ∞ ϑ E (z )  ∂ 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 1 + σ (z )  ∂ y 2 0 0 

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

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

1 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 + ∫ ∫ u1 x ( x, y , z ,τ ) d τ d ϑ  +  2 ∫ ∫ 1x 2 ρ ( z )[1 + σ (z )]  ∂ x 0 0 ∂ x∂ y 0 0 

 5E (z ) K (z ) ∂ 2 t ϑ ∂2 t ϑ u ( x, y , z , τ ) d τ d ϑ  + ∫ ∫ u1 y ( x, y , z , τ ) d τ d ϑ + 2 ∫ ∫ 1x ρ (z ) ∂ x ∂ y 0 0 ∂y 00 12 [1 + σ (z )]

 1   ∂  E (z )  ∂ t ϑ ∂ tϑ + K ( z ) + ∫ ∫ u 1 y ( x, y , z , τ ) d τ d ϑ + ∫ ∫ u1 z ( x, y , z , τ ) d τ d ϑ   ×   ∂y 00    ρ (z ) ∂ z 1 + σ ( z )  ∂ z 0 0 ×

1 2 ρ (z )

− K (z )

 E (z )  ∂2 t ϑ β (z ) t ϑ − K (z ) ∫ ∫ T ( x, y , z , τ ) d τ d ϑ −  ∫ ∫ u 1 y ( x, y , z , τ ) d τ d ϑ × ρ (z ) 0 0  6 [1 + σ ( z )] ∂ y∂ z 00

 ∂ 2 ∞ϑ  E (z ) 1 ∂ 2 ∞ϑ − u ( x , y , z , τ ) d τ d ϑ + ∫ ∫ u1 x ( x , y , z , τ ) d τ d ϑ  − K ( z ) ×  2 ∫ ∫ 1x ∂ x∂ y 0 0 ρ (z ) 2 ρ (z ) [1 + σ ( z )]  ∂ x 0 0 

K (z ) ∂ 2 ∞ ϑ β (z ) ∞ ϑ 1 ∂ 2  5E (z ) + ∫ ∫ T ( x, y , z , τ ) d τ d ϑ − ∫ ∫ u 1 y ( x, y , z , τ ) d τ d ϑ −  ρ (z ) 0 0 ρ (z ) ∂ x ∂ y 0 0 ρ (z ) ∂ y 2  12 [1 + σ (z )] 36

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016

 ∞ ϑ  ∂  ∂ ∞ ϑ ∂ ∞ϑ + K ( z ) ∫ ∫ u1x ( x, y , z , τ ) d τ d ϑ − ∫ ∫ u 1 y ( x, y , z , τ ) d τ d ϑ + ∫ ∫ u 1 z ( x, y , z , τ ) d τ d ϑ  ×  ∂ z  ∂ z 0 0 ∂y 00   0 0

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

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

 ∂2 ∞ϑ E (z ) ∂2 ∞ϑ ( ) u x , y , z , τ d τ d ϑ + ∫ ∫ ∫ ∫ u1 z ( x , y , z , τ ) d τ d ϑ + 1z  2 ρ ( z ) [1 + σ ( z )]  ∂ x 2 0 0 ∂ y2 0 0

 ∂   ∂ ∞ ϑ ∂ 2 ∞ϑ ∂ 2 ∞ϑ ∫ ∫ u 1 x ( x, y , z , τ ) d τ d ϑ + ∫ ∫ u 1 y ( x, y , z , τ ) d τ d ϑ  + ∫ ∫ u 1 x ( x, y , z , τ ) d τ d ϑ +  ∂ x∂ z 0 0 ∂ y∂ z 00  ∂ z   ∂ x 0 0

 1  ∂ ∞ϑ ∂ ∞ϑ 1 ∂  E (z ) + × ∫ ∫ u 1 x ( 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 ∫ ∫ u 1 z ( x, y , z , τ ) d τ d ϑ − ∫ ∫ u 1 x ( x, y , z , τ ) d τ d ϑ − ∫ ∫ u 1 y ( x, y , z , τ ) d τ d ϑ − ∂x 00 ∂y 00  ∂z 00

−

  β (z ) ∂ ∞ ϑ ∂ ∞ϑ ∫ ∫ u1 z ( 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  

In this paper we calculate distributions of concentration of dopant, concentrations of radiation defects and components of displacement vector in space and time by using the second-order approximation framework method of averaging of function corrections. This approximation is usually enough good approximation to make qualitative analysis and to obtain some quantitative results. All obtained results have been checked by comparison with results of numerical simulations.

3. DISCUSSION Now we analyzed changing concentrations of dopant and radiation defects in space and time in a heterostructure with account mismatch-induced stress. Several spatial distributions of dopant concentration are presented on Fig. 2 for diffusion type of doping and on Fig. 3 for ion type of doping. In the both cases dopant diffusion coefficient in the epitaxial layer is larger, than in the substrate. One can find from these figures, that using in homogeneity of heterostructure leads to increasing of sharpness of p-n-junctions. In this situation the accompanying effect is increasing homogeneity of dopant distribution in doped part of heterostructure. Switching time of p-n- junction correlated with the junction sharpness and could be increased with increasing of the sharpness. Increasing homogeneity of dopant distribution leads to decreasing local overheat leads to decreasing local heating during functioning of p-n-junction or to decrease dimensions of the p-njunction for fixed maximal value of local overheat. However framework this approach of manufacturing of p-n-junction and their systems (bipolar transistor and thyristors) it is necessary to optimize annealing of dopant and/or radiation defects. Reason of this optimization is following. If annealing time is small, the dopant did not achieve any interfaces between materials of heterostructure. In this situation one cannot find any modifications of distribution of concentration of dopant. Increasing of annealing time leads to increasing of homogeneity of distribution of concentration of dopant. We optimize annealing time framework recently introduces approach [16,18-24]. To use the criterion one shall to approximate real distribution of concentration of dopant by step-wise function (see Figs. 4 for diffusion doping and 5 for ion doping). The optimal values of annealing time have been calculated by minimization of the following mean-squared error 37

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1 Lx L y Lz ∫ ∫ ∫ [C (x, y , z, Θ ) − ψ (x, y , z )] d z d y d x . L x L y Lz 0 0 0

(15)

Fig.2. Distributions of concentration dopant in heterostructure from Fig. 1 for diffusion type of doping. Direction of diffusion is perpendicular to interface between layers of heterostructure. Dopant diffusion coefficient in the epitaxial layer is larger, than in the substrate and increasing with increasing of number of curves. 2.0 1

C(x,Θ)

1.5

2 3

1.0 Epitaxial layer

0.5

Substrate

0.0 L/4

L/2 x

3L/4

Fig.3. Distributions of concentration dopant in heterostructure from Fig. 1 for ion type of doping. Direction of diffusion is perpendicular to interface between layers of heterostructure. Curves 1 and 3 corresponds to annealing time Θ = 0.0048(Lx2+Ly2+Lz2)/D0. Curves 2 and 4 corresponds to annealing time Θ = 0.0057(Lx2+Ly2+Lz2)/D0. Curves 1 and 2 are the distributions of concentrations of dopant in homogenous sample. Curves 3 and 4 are the distributions of concentrations of dopant in the heterostructure. Dopant diffusion coefficient in the epitaxial layer is larger, than in the substrate and increasing with increasing of number of curves.

C(x,Θ)

2 3 1

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Fig. 4. Distributions of concentration dopant in heterostructure from Fig. 1 for diffusion type of doping. Curve 1 is idealized distribution of dopant. Curves 2-4 are the calculated distributions of dopant for several values of annealing time. Annealing time increased with increasing of number of curves. 2 3 4 C(x,Θ)

Fig. 5. Distributions of concentration dopant in heterostructure from Fig. 1 for ion type of doping. Curve 1 is idealized distribution of dopant. Curves 2-4 are the calculated distributions of dopant for several values of annealing time. Annealing time increased with increasing of number of curves. 0.5

2 4

0.3

ΘDL

-2

0.4

0.2

1 0.1 0.0 0.0

0.1

0.2 0.3 a/L, ξ, ε, γ

0.4

0.5

Fig.6. Dimensionless optimal annealing time of infused dopant. Curve 1 describes dependence of the annealing time on normalized thickness of epitaxial layer, ξ = γ = 0 and equal to each other values of dopant diffusion coefficient in all layers of heterostructure. Curve 2 describes dependence of the annealing time on normalized difference between diffusion coefficients ε = (DEL-DS)/ DS, a/L=1/2 and ξ = γ = 0. Curve 3 describes dependence of the annealing time on value of parameter ξ for a/L=1/2 and ε = γ = 0. Curve 3 describes dependence of the annealing time on value of parameter γ for a/L=1/2 and ε = ξ = 0. 0.12 2 0.08 Θ D0 L

-2

0.04

0.00 0.0

0.1

0.2 0.3 a/L, ξ, ε, γ

0.4

0.5

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Fig.7. Dimensionless optimal annealing time of implanted dopant. Curve 1 describes dependence of the annealing time on normalized thickness of epitaxial layer, ξ = γ = 0 and equal to each other values of dopant diffusion coefficient in all layers of heterostructure. Curve 2 describes dependence of the annealing time on normalized difference between diffusion coefficients ε = (DEL-DS)/ DS, a/L=1/2 and ξ = γ = 0. Curve 3 describes dependence of the annealing time on value of parameter ξ for a/L=1/2 and ε = γ = 0. Curve 3 describes dependence of the annealing time on value of parameter γ for a/L=1/2 and ε = ξ = 0. Here ψ (x,y,z) is the step-wise approximation function. We calculate dependences of optimal values of annealing time on parameters. These dependences are presented on Figs. 6 (for diffusion type of doping) and 7 (for ion types of doping). It is known, that radiation defects should be annealed after ion implantation. Distribution of concentration of these defects will be spreads during the annealing. If distribution of dopant concentration achieves nearest interface, than we obtain rare ideal case. If the dopant has not enough time for the achievement, it is practicably to additionally anneal the dopant. In this situation the considered the additional annealing time became smaller, than annealing time for diffusion type of doping. Now we analyzed correlation between relaxations of mismatch-induced stress and distribution of concentration of dopant. If ε0< 0, one obtain compression of distribution of concentration of dopant. Contrary one obtain spreading of distribution of concentration of dopant. Laser annealing gives a possibility to decrease influence of mismatch-induced stress on distribution of concentration of dopant [22]. Using the annealing leads to acceleration of dopant diffusion in the irradiated area. Taking into account mismatch-induced stress leads to changing of optimal values of annealing time. Mismatch-induced stress could be used to increase density of elements of integrated circuits. On the other hand could leads to generation dislocations of the discrepancy. Fig. 8 shows distributions of component of displacement vector in direction, which is perpendicular to epitaxial layer and substrate. 1.0

0.8

0.6

Uz 0.4

2 0.2

0.0 0.0

Fig. 8. Normalized dependences of component uz of displacement vector on coordinate z for nonporous (curve 1) and porous (curve 2) epitaxial layers

Fig. 9. Normalized distributions of charge carrier mobility in the considered heterostructure.

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Curve 1 corresponds to the heterostructure, which has been considered in Fig. 1. Curve 2 correspond to a homogenous material with averaged parameters of heterostructure from Fig. 1 Now we consider the changing of charge carrier mobility with changing of distribution of concentration of dopant. It has been recently shown, that the mobility is a function of concentration of dopant: µ ~ C1/3 [25]. The situation is illustrated by Fig. 9. The figure shows normalized distributions of charge carrier mobility in the considered heterostructure. Curve 1 corresponds to the heterostructure, which has been considered in Fig. 1. Curve 2 correspond to material after averaging parameters of heterostructure from Fig. 1.

4. CONCLUSIONS In this paper we consider a possibility to increase sharpness of diffusion-junction and implantedjunction heterorectifiers. At the same time homogeneity of distribution of concentration of dopant in enriched by the dopant increases. We also consider an approach to decrease mismatch-induced stress in the considered heterostructure. We analyzed influence of changing of concentration of dopant on value of charge carrier mobility.

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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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 103 published papers in area of her researches.