![International Journal of Computational Science, Information Technology and Control Engineering
(IJCSITCE) Vol.3, No.3, July 2016
DOI : 10.5121/ijcsitce.2016.3302 13
ON INCREASING OF DENSITY OF ELEMENTS IN A
MULTIVIBRATOR ON BIPOLAR TRANSISTORS
E.L. Pankratov1
and 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 an approach to increase density of elements of a multivibrator on bipolar tran-
sistors. The considered approach based on manufacturing a heterostructure with necessity configuration,
doping by diffusion or ion implantation of required areas to manufacture the required type of conductivity
(p or n) in the areas and optimization of annealing of dopant and/or radiation defects to manufacture more
compact distributions of concentrations of dopants. We also introduce an analytical approach to progno-
sis technological process.
KEYWORDS
Bipolar transistors, multivibrator, increasing of density of elements of multivibrator
1. INTRODUCTION
In the present time actual questions of solid state electronic devices are increasing of density of
elements of integrated circuits (in this situation it is necessary to decrease dimensions of these
elements), performance and reliability [1-9]. In this paper we consider an approach to manufac-
ture more compact multivibrator based on bipolar heterotransistors. Framework the approach we
consider a heterostructure with two layers, which consist of a substrate and an epitaxial layer (see
Fig. 1). The epitaxial layer includes into itself several sections manufactured by using another
materials (see Fig. 1). These sections have been doped by diffusion or ion implantation to obtain
required type of conductivity (n or p). The doping gives a possibility to manufacture bipolar tran-
sistor framework the considered heterostructure with account the Fig. 1. After that we consider
annealing of dopant and/or radiation defects. The annealing should be optimized to manufacture
more compact distributions of concentrations of dopant. Main aim of the present paper is deter-
mination of conditions, which correspond to increasing of compactness and at the same time to
increasing of homogeneity of distribution of concentration of dopant in enriched area.
2. METHOD OF SOLUTION
We determine spatio-temporal distribution of concentration of dopant by solving the following
boundary problem
( ) ( ) ( ) ( )
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC ,,,,,,,,,,,,
(1)
with boundary and initial conditions](https://image.slidesharecdn.com/3316ijcsitce02-160806054713/85/on-increasing-of-density-of-elements-in-a-multivibrator-on-bipolar-transistors-1-320.jpg)
![International Journal of Computational Science, Information Technology and Control Engineering
(IJCSITCE) Vol.3, No.3, July 2016
14
( ) 0
,,,
0
=
∂
∂
=x
x
tzyxC
,
( ) 0
,,,
=
∂
∂
= xLx
x
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=y
y
tzyxC
,
( ) 0
,,,
=
∂
∂
= yLx
y
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=z
z
tzyxC
,
( ) 0
,,,
=
∂
∂
= zLx
z
tzyxC
, C(x,y,z,0)=fC (x,y,z).
Fig. 1a. Structure of a transistor multivibrator. View from top
n p n n p n
Substrate
Epitaxial layer
Fig. 1b. Heterostructure with two layers and sections in the epitaxial layer, which has been used for manu-
facture a transistor multivibrator. View from side
Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant; T is the temperature
of annealing; DС is the dopant diffusion coefficient. Value of dopant diffusion coefficient de-
pends on properties of materials of heterostructure, speed of heating and cooling of heterostruc-
ture (with account Arrhenius law). Dependences of dopant diffusion coefficient on parameters
could be approximated by the following relation [10-12]](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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IJCSITCE Vol. 3 No. 3: Increasing Density of Elements in a Multivibrator
- 1. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 DOI : 10.5121/ijcsitce.2016.3302 13 ON INCREASING OF DENSITY OF ELEMENTS IN A MULTIVIBRATOR ON BIPOLAR TRANSISTORS E.L. Pankratov1 and 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 an approach to increase density of elements of a multivibrator on bipolar tran- sistors. The considered approach based on manufacturing a heterostructure with necessity configuration, doping by diffusion or ion implantation of required areas to manufacture the required type of conductivity (p or n) in the areas and optimization of annealing of dopant and/or radiation defects to manufacture more compact distributions of concentrations of dopants. We also introduce an analytical approach to progno- sis technological process. KEYWORDS Bipolar transistors, multivibrator, increasing of density of elements of multivibrator 1. INTRODUCTION In the present time actual questions of solid state electronic devices are increasing of density of elements of integrated circuits (in this situation it is necessary to decrease dimensions of these elements), performance and reliability [1-9]. In this paper we consider an approach to manufac- ture more compact multivibrator based on bipolar heterotransistors. Framework the approach we consider a heterostructure with two layers, which consist of a substrate and an epitaxial layer (see Fig. 1). The epitaxial layer includes into itself several sections manufactured by using another materials (see Fig. 1). These sections have been doped by diffusion or ion implantation to obtain required type of conductivity (n or p). The doping gives a possibility to manufacture bipolar tran- sistor framework the considered heterostructure with account the Fig. 1. After that we consider annealing of dopant and/or radiation defects. The annealing should be optimized to manufacture more compact distributions of concentrations of dopant. Main aim of the present paper is deter- mination of conditions, which correspond to increasing of compactness and at the same time to increasing of homogeneity of distribution of concentration of dopant in enriched area. 2. METHOD OF SOLUTION We determine spatio-temporal distribution of concentration of dopant by solving the following boundary problem ( ) ( ) ( ) ( ) ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC ,,,,,,,,,,,, (1) with boundary and initial conditions
- 2. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 14 ( ) 0 ,,, 0 = ∂ ∂ =x x tzyxC , ( ) 0 ,,, = ∂ ∂ = xLx x tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =y y tzyxC , ( ) 0 ,,, = ∂ ∂ = yLx y tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =z z tzyxC , ( ) 0 ,,, = ∂ ∂ = zLx z tzyxC , C(x,y,z,0)=fC (x,y,z). Fig. 1a. Structure of a transistor multivibrator. View from top n p n n p n Substrate Epitaxial layer Fig. 1b. Heterostructure with two layers and sections in the epitaxial layer, which has been used for manu- facture a transistor multivibrator. View from side Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant; T is the temperature of annealing; DС is the dopant diffusion coefficient. Value of dopant diffusion coefficient de- pends on properties of materials of heterostructure, speed of heating and cooling of heterostruc- ture (with account Arrhenius law). Dependences of dopant diffusion coefficient on parameters could be approximated by the following relation [10-12]
- 3. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 15 ( ) ( ) ( ) ( ) ( ) ( ) ++ += 2* 2 2*1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD LC ςςξ γ γ , (3) where DL (x,y,z,T) is the spatial (due to presents several layers in heterostructure) and tempera- ture (due to Arrhenius law) dependences of dopant diffusion coefficient; P (x, y,z,T) is the limit of solubility of dopant; parameter γ depends on properties of materials and could be integer in the following interval γ ∈[1,3] [10]; V (x,y,z,t) is the spatio-temporal distribution of concentration of radiation vacancies; V* is the equilibrium distribution of concentration of vacancies. Concentrational dependence of dopant diffusion coefficient has been described in details in [10]. Using diffusion type of doping did not generation radiation defects. In this situation ζ1= ζ2= 0. We determine the spatio-temporal distributions of concentrations of radiation defects by solving the following system of equations [11,12] ( ) ( ) ( ) ( ) ( ) ( ) ×− ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ Tzyxk y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI IIII ,,, ,,, ,,, ,,, ,,, ,,, , ( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk z tzyxI TzyxD z tzyxI VII ,,,,,,,,, ,,, ,,,,,, , 2 − ∂ ∂ ∂ ∂ +× (4) ( ) ( ) ( ) ( ) ( ) ( ) ×− ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ Tzyxk y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VVVV ,,, ,,, ,,, ,,, ,,, ,,, , ( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk z tzyxV TzyxD z tzyxV VIV ,,,,,,,,, ,,, ,,,,,, , 2 − ∂ ∂ ∂ ∂ +× with boundary and initial conditions ( ) 0 ,,, 0 = ∂ ∂ =x x tzyxρ , ( ) 0 ,,, = ∂ ∂ = xLx x tzyxρ , ( ) 0 ,,, 0 = ∂ ∂ =y y tzyxρ , ( ) 0 ,,, = ∂ ∂ = yLy y tzyxρ , ( ) 0 ,,, 0 = ∂ ∂ =z z tzyxρ , ( ) 0 ,,, = ∂ ∂ = zLz z tzyxρ , ρ(x,y,z,0)=fρ (x,y,z). (5) Here ρ =I,V; I (x,y,z,t) is the spatio-temporal distribution of concentration of radiation intersti- tials; Dρ(x,y,z,T) are the diffusion coefficients of point radiation defects; terms V2 (x,y,z,t) and I2 (x,y,z,t) correspond to generation divacancies and diinterstitials; kI,V(x,y,z,T) is the parameter of recombination of point radiation defects; kI,I(x,y,z,T) and kV,V(x,y,z,T) are the parameters of gener- ation of simplest complexes of point radiation defects. We determine the spatio-temporal distributions of concentrations of divacancies ΦV (x,y,z,t) and dinterstitials ΦI(x,y,z,t) by solving the following system of equations [11,12] ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx I I I I I ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxk z tzyx TzyxD z III I I ,,,,,,,,,,,, ,,, ,,, 2 , −+ Φ + Φ ∂ ∂ ∂ ∂ (6)
- 4. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 16 ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx V V V V V ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxk z tzyx TzyxD z VVV V V ,,,,,,,,,,,, ,,, ,,, 2 , −+ Φ + Φ ∂ ∂ ∂ ∂ with boundary and initial conditions ( ) 0 ,,, 0 = ∂ Φ∂ =x x tzyxρ , ( ) 0 ,,, = ∂ Φ∂ = xLx x tzyxρ , ( ) 0 ,,, 0 = ∂ Φ∂ =y y tzyxρ , ( ) 0 ,,, = ∂ Φ∂ = yLy y tzyxρ , ( ) 0 ,,, 0 = ∂ Φ∂ =z z tzyxρ , ( ) 0 ,,, = ∂ Φ∂ = zLz z tzyxρ , ΦI(x,y,z,0)=fΦI (x,y,z), ΦV(x,y,z,0)=fΦV (x,y,z). (7) Here DΦρ(x,y,z,T) are the diffusion coefficients of the above complexes of radiation defects; kI(x,y,z,T) and kV (x,y,z,T) are the parameters of decay of these complexes. To determine spatio-temporal distribution of concentration of dopant we transform the Eq.(1) to the following integro-differential form ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ++=∫ ∫ ∫ t y L z L L x L y L z Lzyx y zx y z V wvxV V wvxV TwvxDudvdwdtwvuC LLL zyx 0 2* 2 2*1 ,,,,,, 1,,,,,, τ ς τ ς ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫+ +× t x L z L L zxzy x z y wyuC TwyuD LL zx LL zy d x wvxC TwvxP wvxC 0 ,,, ,,, ,,, ,,, ,,, 1 ∂ τ∂ τ ∂ τ∂τ ξ γ γ ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ×+ + ++× t x L y L L yx x y TzvuD LL yx d TzyxP wyuC V wyuV V wyuV 0 2* 2 2*1 ,,, ,,, ,,, 1 ,,,,,, 1 τ τ ξ τ ς τ ς γ γ ( ) ( ) ( ) ( ) ( ) ( ) ×+ + ++× zyx LLL zyx d z zvuC TzyxP zvuC V zvuV V zvuV τ ∂ τ∂τ ξ τ ς τ ς γ γ ,,, ,,, ,,, 1 ,,,,,, 1 2* 2 2*1 ( )∫ ∫ ∫× x L y L z Lx y z udvdwdwvuf ,, . (1a) We determine solution of the above equation by using Bubnov-Galerkin approach [13]. Frame- work the approach we determine solution of the Eq.(1a) as the following series ( ) ( ) ( ) ( ) ( )∑= = N n nCnnnnC tezcycxcatzyxC 0 0 ,,, , where ( ) ( )[ ]222 0 22 exp −−− ++−= zyxCnC LLLtDnte π , cn(χ) = cos(π n χ/Lχ). The above series includes into itself finite number of terms N. The considered series is similar with solution of linear Eq.(1) (i.e. with ξ = 0) and averaged dopant diffusion coefficient D0. Substitution of the series into Eq.(1a) leads to the following result ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ × ∑+−=∑ == t y L z L N n nCnnnnC N n nCnnn C y z TwvxP ewcvcxcatezsysxs n azyx 0 11 32 ,,, 1 γ γ ξ τ π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) −∑ ++× = zy N n nCnnnnCL LL zy dewcvcxsanTwvxD V wvxV V wvxV 1 2* 2 2*1 ,,, ,,,,,, 1 ττ τ ς τ ς
- 5. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 17 ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ × ∑+ ++− = t x L z L N m mCmmmmC zx x z ewcycuca V wyuV V wyuV LL zx 0 1 2* 2 2*1 1 ,,,,,, 1 γ τ τ ς τ ς ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫−∑ × = t x L y L L yx N n nCnnnnCL x y TzvuD LL yx dewcysucanTwyuD TwyuP 01 ,,,,,, ,,, ττ ξ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ++ ∑+× = 2* 2 2*1 1 ,,,,,, 1 ,,, 1 V zvuV V zvuV ezcvcuca TzvuP N n nCnnnnC τ ς τ ςτ ξ γ γ ( ) ( ) ( ) ( ) ( )∫ ∫ ∫+∑× = x L y L z Lzyx N n nCnnnnC x y z udvdwdwvuf LLL zyx dezsvcucan ,, 1 ττ , where sn(χ) = sin (π n χ/Lχ). We determine coefficients an by using orthogonality condition of terms of the considered series framework scale of heterostructure. The condition gives us possi- bility to obtain relations for calculation of parameters an for any quantity of terms N. In the common case the relations could be written as ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ × ∑+−=∑− == t L L L N n nCnnnnCL N n nC nCzyx x y z TzyxP ezcycxcaTzyxDte n aLLL 0 0 0 0 11 65 222 ,,, 1,,, γ γ ξ τ π ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ×∑ −+ ++× = N n n y nnn nCzy yc n L ysyycxs n a V zyxV V zyxVLL 1 2* 2 2*12 12 ,,,,,, 1 2 π τ ς τ ς π ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ ∑− −+× = t L L L N n nCnnnnCn z nnCn x y z ezcycxcadxdydzdzc n L zszezc 0 0 0 0 1 1 γ ττ π τ ( ) ( ) ( ) ( ) ( ) ( ) ++ ++ +× *12* 2 2*1 ,,, 1 ,,,,,, 1,,,1 ,,, V zyxV V zyxV V zyxV TzyxD TzyxP L τ ς τ ς τ ς ξ γ ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )∑ × −+ −+ + = N n nnnn z nn x n znC zcysxczc n L zszxc n L xsx n La V zyxV 1 22* 2 2 211 2 ,,, πππ τ ς ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ × ∑+×−× = t L L L N n nCnnnnC yx nC x y z TzyxP ezcycxca LL dxdydzde 0 0 0 0 1 2 ,,, 1 2 γ γ ξ τ π ττ ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ×∑ −+ ++× = N n n x nn nC L xc n L xsxxc n a V zyxV V zyxV TzyxD 1 *12* 2 2 1 ,,,,,, 1,,, π τ ς τ ς ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ]∑ ∫ × −++ −+× = N n L n x nnCn y nnn x xc n L xsxdxdydzdeyc n L ysyzsyc 1 0 11 π ττ π ( ) ( )[ ] ( ) ( )[ ] ( )∫ ∫ −+ −+× y z L L n z nn y n xdydzdzyxfzc n L zszyc n L ysy 0 0 ,,11 ππ . As an example for γ=0 we obtain ( ) ( )[ ] ( ) ( )[ ] ( ) ( )∫ ∫ ∫ ×+ −+ −+= x y zL L L x nn y nn y nnC n L xsxydzdzyxfzc n L zszyc n L ysya 0 0 0 ,,11 πππ ( )[ ]} ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ){ ∫ ∫ ∫ ∫ + −+−× t L L L nn y nLnnnn x y z ysyzc n L zszTzyxDzcycxs n xdxc 0 0 0 0 1,,,2 2 1 π ( )[ ] ( ) ( ) ( ) ( ) ( ) × + ++ −+ xdydzdzc TzyxPV zyxV V zyxV yc n L nn y ,,, 1 ,,,,,, 11 2* 2 2*1 γ ξτ ς τ ς π
- 6. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 18 ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ×∫ ∫ ∫ ∫ −+ −++× t L L L n y nnnn y nnnCnC x y z zc n L zszzcysxc n L xsxxcede 0 0 0 0 121 ππ τττ ( ) ( ) ( ) ( ) ( ) + ++ +× τ τ ς τ ς ξ γ dxdydzd V zyxV V zyxV TzyxP TzyxDL 2* 2 2*1 ,,,,,, 1 ,,, 1,,, ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ×∫ ∫ ∫ ∫ −+ −++ t L L L Ln y nnn x nnnC x y z TzyxDyc n L ysyycxc n L xsxxce 0 0 0 0 ,,,11 ππ τ ( ) ( ) ( ) ( ) ( ) 1 65 222 *12* 2 2 ,,,,,, 1 ,,, 12 − − ++ +× n LLL dxdydzd V zyxV V zyxV TzyxP zs zzz n π τ τ ς τ ς ξ γ . For γ=1 one can obtain the following relation to determine required parameters ( ) ( ) ( ) ( )∫ ∫ ∫+±−= x y zL L L nnnnn n n nC xdydzdzyxfzcycxca 0 0 0 2 ,,4 2 αβ α β , where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫ × ++= t L L L L nn zy n x y z V zyxV V zyxV TzyxP TzyxD ycxs n LL 0 0 0 0 2* 2 2*12 ,,,,,, 1 ,,, ,,, 2 2 τ ς τ ς π ξ α ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ×+ −+ −+× n LL dexdydzdzc n L zszyc n L ysyzc zx nCn z nn y nn 2 2 11 π ξ ττ ππ ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ×∫ ∫ ∫ ∫ −− −+× t L L L n z n L nn x nnnC x y z zc n L zsz TzyxP TzyxD ysxc n L xsxxce 0 0 0 0 1 ,,, ,,, 21 ππ τ ( ) ( ) ( ) ( ) ( ) ( ) ( )×∫ ∫ ∫+ ++× t L L nnnC yx n x y ycxce n LL dxdydzd V zyxV V zyxV zc 0 0 0 22* 2 2*1 2 ,,,,,, 1 τ π ξ τ τ ς τ ς ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ){∫ × −+ ++× zL nn x n L n ysxc n L xsx V zyxV V zyxV TzyxP TzyxD zs 0 2* 2 2*1 1 ,,,,,, 1 ,,, ,,, 2 π τ ς τ ς ( )[ ] τ π dxdydzdyc n L y n y −+× 1 , ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫= t L L L LnnnnC zy n x y z TzyxDzcycxse n LL 0 0 0 0 2 ,,,2 2 τ π β ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( ) × +− −+ ++× ysyyc n L zdzc n L zsz V zyxV V zyxV nn y n z n 11 ,,,,,, 1 2* 2 2*1 ππ τ ς τ ς ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ∫ ∫ ∫ ∫ ++ −++× t L L L nnn x nnnC zx x y z V zyxV zcysxc n L xsxxce n LL dxdyd 0 0 0 0 *12 ,,, 121 2 τ ς π τ π τ ( ) ( ) ( ) ( ) ( )[ ] ( )∫ ×+ −+ + t nC yx n z nL e n LL dxdydzdzc n L zszTzyxD V zyxV 0 22* 2 2 2 1,,, ,,, τ π τ π τ ς ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ∫ ∫ ∫ × ++ −+ −+× x y zL L L n y nn x n V zyxV V zyxV yc n L ysyxc n L xsx 0 0 0 2* 2 2*1 ,,,,,, 111 τ ς τ ς ππ ( ) ( ) ( ) ( ) ( ) 65222 ,,,2 nteLLLdxdxcydyczdTzyxDzs nCzyxnnLn πτ −× . Analogous way could be used to calculate values of parameters an for larger values of parameter γ. However the relations will not be present in the paper because the relations are bulky. Advan-
- 7. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 19 tage of the approach is absent of necessity to join dopant concentration on interfaces of hetero- structure. Equations of the system (4) have been also solved by using Bubnov-Galerkin approach. Pre- viously we transform the differential equations to the following integro- differential form ( ) ( ) ( ) ×+∫ ∫ ∫ ∂ ∂ =∫ ∫ ∫ zx t y L z L I zy x L y L z Lzyx LL zx dvdwd x wvxI TwvxD LL zy udvdwdtwvuI LLL zyx y zx y z 0 ,,, ,,,,,, τ τ ( ) ( ) ( ) ( ) −∫ ∫ ∫ ∂ ∂ +∫ ∫ ∫ ∂ ∂ × t x L y L I yx t x L z L I x yx z dudvd z zvuI TzvuD LL yx dudwd x wyuI TwyuD 00 ,,, ,,, ,,, ,,, τ τ τ τ ( ) ( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫− x L y L z L II zyx x L y L z L VI zyx x y zx y z Twvuk LLL zyx udvdwdtwvuVtwvuITwvuk LLL zyx ,,,,,,,,,,,, ,, ( ) ( )∫ ∫ ∫+× x L y L z L I zyx x y z udvdwdwvuf LLL zyx udvdwdtwvuI ,,,,,2 (4a) ( ) ( ) ( ) ×+∫ ∫ ∫ ∂ ∂ =∫ ∫ ∫ zx t y L z L V zy x L y L z Lzyx LL zx dvdwd x wvxV TwvxD LL zy udvdwdtwvuV LLL zyx y zx y z 0 ,,, ,,,,,, τ τ ( ) ( ) ( ) ( ) −∫ ∫ ∫ ∂ ∂ +∫ ∫ ∫ ∂ ∂ × t x L y L V yx t x L z L V x yx z dudvd z zvuV TzvuD LL yx dudwd x wyuV TwyuD 00 ,,, ,,, ,,, ,,, τ τ τ τ ( ) ( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫− x L y L z L VV zyx x L y L z L VI zyx x y zx y z Twvuk LLL zyx udvdwdtwvuVtwvuITwvuk LLL zyx ,,,,,,,,,,,, ,, ( ) ( )∫ ∫ ∫+× x L y L z L V zyx x y z udvdwdwvuf LLL zyx udvdwdtwvuV ,,,,,2 . Farther we determine solutions of the above equations as the following series ( ) ( ) ( ) ( ) ( )∑= = N n nnnnn tezcycxcatzyx 1 0 ,,, ρρρ , where anρ are not yet known coefficients. Substitution of the series into Eqs.(4a) leads to the fol- lowing results ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−=∑ == N n t y L z L InnnInI zyx N n nInnn nI y z dvdwdTwvxDzcycea LLL zy tezsysxs n azyx 1 01 33 ,,, ττ π π ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑−∑ ∫ ∫ ∫−× == N n nnI zyx N n t x L z L InnnInnI zyx n zsa LLL yx dudwdTwyuDzcxceysa LLL zx xs x z 11 0 ,,, π ττ π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∑−∫ ∫ ∫× = x L y L z L N n nInnnnI zyx t x L y L InnnI x y zx y tewcvcuca LLL zyx dudvdTzvuDycxce 2 10 ,,, ττ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∑ ∑−× = = x L y L z L N n N n nnnnVnInnnnI zyx II x y z wcvcucatewcvcuca LLL zyx udvdwdTvvuk 1 1 , ,,, ( ) ( ) ( )∫ ∫ ∫+× x L y L z L I zyx VInV x y z udvdwdwvuf LLL zyx udvdwdTvvukte ,,,,,, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−=∑ == N n t y L z L VnnnVnV zyx N n nVnnn nV y z dvdwdTwvxDzcycea LLL zy tezsysxs n azyx 1 01 33 ,,, ττ π π
- 8. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 20 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∑ ∫ ∫ ∫−× == N n nnV zyx N n t x L z L VnnnVnnV zyx n zsa LLL yx dudwdTwyuDzcxceysa LLL zx xs x z 11 0 ,,, π ττ π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∑−∫ ∫ ∫× = x L y L z L N n nVnnnnV zyx t x L y L VnnnV x y zx y tewcvcuca LLL zyx dudvdTzvuDycxce 2 10 ,,, ττ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∑ ∑−× = = x L y L z L N n N n nnnnVnInnnnI zyx VV x y z wcvcucatewcvcuca LLL zyx udvdwdTvvuk 1 1 , ,,, ( ) ( ) ( )∫ ∫ ∫+× x L y L z L V zyx VInV x y z udvdwdwvuf LLL zyx udvdwdTvvukte ,,,,,, . We determine coefficients anρ by using orthogonality condition on the scale of heterostructure. The condition gives us possibility to obtain relations to calculate anρ for any quantity N of terms of considered series. In the common case equations for the required coefficients could be written as ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++−−=∑− == N n t L L n y nynnI nI x N n nI nIzyx x y yc n L ysyLxce n a L te n aLLL 1 0 0 0 2 1 65 222 12 2 221 2 1 π τ ππ ( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫ ++−−∫ −+× = N n t L xn xnI y L n z nI xz Lxc n L n a L dxdydzdzc n L zszTzyxD 1 0 0 2 0 12 2 1 1 2 ,,, ππ τ π ( )} ( ) ( ) ( )[ ] ( )[ ] ( ) −∫ ∫ − −+++ y z L L nInn z nzIn dexdydyczdzc n L zszLTzyxDxsx 0 0 2112 2 2,,,2 ττ π ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++ −++− = N n t L L n y nyn x nxnI nI z x y yc n L ysyLxc n L xsxLe n a L 1 0 0 0 2 12 2 212 2 2 2 1 ππ τ π ( )[ ] ( ) ( ) ( )[ ] ( ) ×∑ ∫ +−+−∫ −× = N n L nn x xnInI L In xz xsxxc n L LteadxdydzdTzyxDzc 1 0 2 0 212 2 2,,,21 π τ ( ) ( )[ ] ( ) ( )[ ] ( ) −∫ ∫ +−+ −++× y z L L nn z zIIn y ny xdydzdzszzc n L LTzyxkyc n L ysyL 0 0 , 212 2 ,,,12 2 2 ππ ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ −++ −++− = N n L L n y nyn x nxnVnInVnI x y yc n L ysyLxc n L xsxLteteaa 1 0 0 12 2 212 2 2 ππ ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ −++∫ −++× = N n L n x n L n z nzVI xz xc n L xsxxdydzdzc n L zszLTzyxk 1 00 , 112 2 2,,, ππ ( ) ( )[ ] ( ) ( ) ( )[ ]∫ ∫ −++ −+× y z L L n z nzIn y n xdydzdzc n L zszLTzyxfyc n L ysy 0 0 12 2 2,,,1 ππ ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++−−=∑− == N n t L L n y nynnV nV x N n nV nVzyx x y yc n L ysyLxce n a L te n aLLL 1 0 0 0 2 1 65 222 12 2 221 2 1 π τ ππ ( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫ ++−−∫ −+× = N n t L xn xnV y L n z nV xz Lxc n L n a L dxdydzdzc n L zszTzyxD 1 0 0 2 0 12 2 1 1 2 ,,, ππ τ π ( )} ( ) ( ) ( )[ ] ( )[ ] ( ) −∫ ∫ − −+++ y z L L nVnn z nzVn dexdydyczdzc n L zszLTzyxDxsx 0 0 2112 2 2,,,2 ττ π ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++ −++− = N n t L L n y nyn x nxnV nV z x y yc n L ysyLxc n L xsxLe n a L 1 0 0 0 2 12 2 212 2 2 2 1 ππ τ π
- 9. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 21 ( )[ ] ( ) ( ) ( )[ ] ( ) ×∑ ∫ +−+−∫ −× = N n L nn x xnVnV L Vn xz xsxxc n L LteadxdydzdTzyxDzc 1 0 2 0 212 2 2,,,21 π τ ( ) ( )[ ] ( ) ( )[ ] ( ) −∫ ∫ +−+ −++× y z L L nn z zVVn y ny xdydzdzszzc n L LTzyxkyc n L ysyL 0 0 , 212 2 ,,,12 2 2 ππ ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ −++ −++− = N n L L n y nyn x nxnVnInVnI x y yc n L ysyLxc n L xsxLteteaa 1 0 0 12 2 212 2 2 ππ ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ −++∫ −++× = N n L n x n L n z nzVI xz xc n L xsxxdydzdzc n L zszLTzyxk 1 00 , 112 2 2,,, ππ ( ) ( )[ ] ( ) ( ) ( )[ ]∫ ∫ −++ −+× y z L L n z nzVn y n xdydzdzc n L zszLTzyxfyc n L ysy 0 0 12 2 2,,,1 ππ . In the final form relations for required parameters could be written as ( ) − +− + ± + −= A yb yb Ab b Ab a nInV nI 2 3 4 2 3 4 3 4 44 λγ , nInI nInInInInI nV a aa a χ λδγ ++ −= 2 , where ( ) ( ) ( ) ( )[ ] ( )∫ ∫ ∫ ×++ −++= x y zL L L y ynn x nxnn n L Lysyxc n L xsxLTzyxkte 0 0 0 , 2 212 2 2,,,2 ππ γ ρρρρ ( )[ ]} ( ) ( )[ ] xdydzdzc n L zszLyc n z nzn −++−× 12 2 212 π , ( ) ( ){∫ ∫ ∫ += t L L nn x n x y ysye nL 0 0 0 2 2 1 τ π δ ρρ ( )[ ] ( ) ( )[ ] ( ) ( )[ ] ×+∫ − −+ −+ y L nn z nn y L dxdxcydzdTzyxDzc n L zszyc n L z π τ ππ ρ 2 1 21,,,1 2 1 2 0 ( ) ( ) ( )[ ] ( )[ ] ( ) ( )[ ]∫ ∫ ∫ ∫ +−− −++× t L L L n z nn x nxn x y z zc n L TzyxDycxc n L xsxLe n 0 0 0 0 2 12 2 ,,,21122 2 1 ππ τ ρρ ( ) } ( ) ( ) ( )[ ] ( ){∫ ∫ ∫ + −+++++ t L L nn x xnn z zn x y ysyxc n L Lxsxe nL dxdydzdLzsz 0 0 0 2 122 2 1 2 π τ π τ ρ ( )[ ] ( )[ ] ( ) ( )te n LLL dxdydzdTzyxDzcyc n L L n zyx L nn y y z ρρ π τ π 65 222 0 ,,,211 2 −∫ − −++ , ( ) ×= tenInIVχ ( ) ( ) ( )[ ] ( )[ ] ( ) ( ) ( ){∫ ∫ ∫ ++ +−+ −+× x y zL L L nzVInn y yn x nnV zszLTzyxkysyyc n L Lxc n L xsxte 0 0 0 , 2,,,212 2 1 ππ ( )[ ] xdydzdzc n L n z −+ 12 2π , ( ) ( )[ ] ( ) ( )[ ] {∫ ∫ ∫ × −+ −+= x y zL L L n y nn x nn zyc n L ysyxc n L xsx 0 0 0 11 ππ λ ρ ( ) ( )[ ] ( ) xdydzdTzyxfzc n L zs n z n ,,,1 ρ π −+× , 22 4 nInInInVb χγγγ −= , −−= 2 3 2 nInInInInVb χδδγγ nInInV γχδ− , ( ) 22 2 2 nInInVnInInVnInVnInInVb χλλδχδγγλδγ −+−+= , nInInVnInVnIb λχδδγλ −= 21 , 2 2 3 48 bbyA −+= , 2 4 2 342 9 3 b bbb p − = , 3 4 2 4132 3 3 54 2792 b bbbbb q +− = , −++−−+= 3 323 32 qpqqpqy 43 3bb− .
- 10. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 22 We determine spatio-temporal distributions of concentrations of complexes of radiation defects in the following form ( ) ( ) ( ) ( ) ( )∑=Φ = Φ N n nnnnn tezcycxcatzyx 1 0 ,,, ρρρ , where anΦρ are not yet known coefficients. Let us previously transform the Eqs.(6) to the follow- ing integro-differential form ( ) ( ) ( ) +∫ ∫ ∫ Φ =∫ ∫ ∫ Φ Φ t y L z L I I zy x L y L z L I zyx y zx y z dvdwd x wvx TwvxD LL zy udvdwdtwvu LLL zyx 0 ,,, ,,,,,, τ ∂ τ∂ ( ) ( ) ( ) ( ) ∫ ∫ ∫ × Φ +∫ ∫ ∫ Φ + ΦΦ t x L y L I I t x L z L I I zx x yx z dudvd z zvu TzvuDdudwd y wyu TwyuD LL zx 00 ,,, ,,, ,,, ,,, τ ∂ τ∂ τ ∂ τ∂ ( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫+× x L y L z L I zyx x L y L z L II zyxyx x y zx y z Twvuk LLL zyx udvdwdwvuITwvuk LLL zyx LL yx ,,,,,,,,, 2 , τ ( ) ( )∫ ∫ ∫+× Φ x L y L z L I zyx x y z udvdwdwvuf LLL zyx udvdwdwvuI ,,,,, τ (6a) ( ) ( ) ( ) +∫ ∫ ∫ Φ =∫ ∫ ∫ Φ Φ t y L z L V V zy x L y L z L V zyx y zx y z dvdwd x wvx TwvxD LL zy udvdwdtwvu LLL zyx 0 ,,, ,,,,,, τ ∂ τ∂ ( ) ( ) ( ) ( ) ∫ ∫ ∫ × Φ +∫ ∫ ∫ Φ + ΦΦ t x L y L V V t x L z L V V zx x yx z dudvd z zvu TzvuDdudwd y wyu TwyuD LL zx 00 ,,, ,,, ,,, ,,, τ ∂ τ∂ τ ∂ τ∂ ( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫+× x L y L z L V zyx x L y L z L VV zyxyx x y zx y z Twvuk LLL zyx udvdwdwvuVTwvuk LLL zyx LL yx ,,,,,,,,, 2 , τ ( ) ( )∫ ∫ ∫+× Φ x L y L z L V zyx x y z udvdwdwvuf LLL zyx udvdwdwvuV ,,,,, τ . Substitution of the previously considered series in the Eqs.(6a) leads to the following form ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−=∑− = ΦΦ = Φ N n t y L z L InnInIn zyx N n nInnn In y z TwvxDvctexsan LLL zy tezsysxs n a zyx 1 01 33 ,,, π π ( ) ( ) ( ) ( ) ( ) ( ) −∑ ∫ ∫ ∫−× = ΦΦΦ N n t x L z L InnInnIn zyx n x z dudwdTwvuDwcucteysna LLL zx dvdwdwc 1 0 ,,, τ π τ ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫+∑ ∫ ∫ ∫− = ΦΦΦ x L y L z L II N n t x L y L InnInnIn zyx x y zx y TwvukdudvdTzvuDvcuctezsan LLL yx ,,,,,, , 1 0 τ π ( ) ( ) ( ) ×+∫ ∫ ∫−× zyx x L y L z L I zyxzyx LLL zyx udvdwdwvuITwvuk LLL zyx LLL zyx udvdwdwvuI x y z ττ ,,,,,,,,,2 ( )∫ ∫ ∫× Φ x L y L z L I x y z udvdwdwvuf ,, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−=∑− = ΦΦ = Φ N n t y L z L VnnVnVn zyx N n nVnnn Vn y z TwvxDvctexsan LLL zy tezsysxs n a zyx 1 01 33 ,,, π π ( ) ( ) ( ) ( ) ( ) ( ) −∑ ∫ ∫ ∫−× = ΦΦΦ N n t x L z L VnnVnnVn zyx n x z dudwdTwvuDwcucteysna LLL zx dvdwdwc 1 0 ,,, τ π τ
- 11. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 23 ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫+∑ ∫ ∫ ∫− = ΦΦΦ x L y L z L VV N n t x L y L VnnVnnVn zyx x y zx y TwvukdudvdTzvuDvcuctezsan LLL yx ,,,,,, , 1 0 τ π ( ) ( ) ( ) ×+∫ ∫ ∫−× zyx x L y L z L V zyxzyx LLL zyx udvdwdwvuVTwvuk LLL zyx LLL zyx udvdwdwvuV x y z ττ ,,,,,,,,,2 ( )∫ ∫ ∫× Φ x L y L z L V x y z udvdwdwvuf ,, . We determine coefficients anΦρ by using orthogonality condition on the scale of heterostructure. The condition gives us possibility to obtain relations to calculate anΦρ for any quantity N of terms of considered series. In the common case equations for the required coefficients could be written as ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++−−=∑− = Φ = Φ Φ N n t L L n y nyn In x N n In Inzyx x y yc n L ysyLxc n a L te n aLLL 1 0 0 0 2 1 65 222 12 2 221 2 1 πππ ( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ ++−∫ −+× = Φ ΦΦ N n t L xn y In L Inn z nI xz Lxsx Ln a dexdydzdzc n L zszTzyxD 1 0 0 2 0 2 2 1 1 2 ,,, π ττ π ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ×−∫ ∫ −+− − + ΦΦ x L L Inn z nIn n x L dexdydzdzc n L zszTzyxDyc n xc L y z π ττ ππ 1 1 2 ,,,21 2 12 0 0 ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ×∑ ∫ ∫ ∫ ∫ +−+ −+× = ΦΦ Φ N n t L L L Iyn y nn x nIn In x y z TzyxDLyc n L ysyxc n L xsxe n a 1 0 0 0 0 2 ,,,12 2 21 22 ππ τ ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ] ×∑∫ ∫ ∫ −+ +−+−× = Φ N n t L L n y nnn x Inn x y yc n L ysyxsxxc n L edxdydzdyc 10 0 0 1 2 1 2 21 ππ ττ ( ) ( ) ( )[ ] ( ) ( ) ( ){∑∫ ∫ +−∫ +−× = Φ Φ N n t L nIn L nn z II In xz xsxexdydzdzszzc n L TzyxktzyxI n a 10 00 , 2 33 1 2 ,,,,,, τ ππ ( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ]∫ ∫ +− +− −+ Φ y z L L n z Inn yIn n x zc n L tzyxITzyxkysyyc n L n a xc n L 0 0 33 1 2 ,,,,,,1 2 1 2 ππππ ( )} ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −+ −+++ = Φ Φ N n t L L n y nn x nIn In n x y yc n L ysyxc n L xsxe n a xdydzdzsz 1 0 0 0 33 1 2 1 2 ππ τ π ( )[ ] ( ) ( )∫ +−× Φ zL Inn z xdydzdzyxfzszzc n L 0 ,,1 2π ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −++−−=∑− = Φ = Φ Φ N n t L L n y nyn Vn x N n Vn Vnzyx x y yc n L ysyLxc n a L te n aLLL 1 0 0 0 2 1 65 222 12 2 221 2 1 πππ ( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ ++−∫ −+× = Φ ΦΦ N n t L xn y Vn L Vnn z nV xz Lxsx Ln a dexdydzdzc n L zszTzyxD 1 0 0 2 0 2 2 1 1 2 ,,, π ττ π ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ×−∫ ∫ −+− − + ΦΦ x L L Vnn z nVn n x L dexdydzdzc n L zszTzyxDyc n xc L y z π ττ ππ 1 1 2 ,,,21 2 12 0 0 ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ×∑ ∫ ∫ ∫ ∫ +−+ −+× = ΦΦ Φ N n t L L L Vyn y nn x nVn Vn x y z TzyxDLyc n L ysyxc n L xsxe n a 1 0 0 0 0 2 ,,,12 2 21 22 ππ τ
- 12. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 24 ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ] ×∑∫ ∫ ∫ −+ +−+−× = Φ N n t L L n y nnn x Vnn x y yc n L ysyxsxxc n L edxdydzdyc 10 0 0 1 2 1 2 21 ππ ττ ( ) ( ) ( )[ ] ( ) ( ) ( ){∑ ∫ ∫ +−∫ +−× = Φ Φ N n t L nVn L nn z VV Vn xz xsxexdydzdzszzc n L TzyxktzyxV n a 10 00 , 2 33 1 2 ,,,,,, τ ππ ( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ]∫ ∫ +− +− −+ Φ y z L L n z Vnn yVn n x zc n L tzyxVTzyxkysyyc n L n a xc n L 0 0 33 1 2 ,,,,,,1 2 1 2 ππππ ( )} ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫ −+ −+++ = Φ Φ N n t L L n y nn x nVn Vn n x y yc n L ysyxc n L xsxe n a xdydzdzsz 1 0 0 0 33 1 2 1 2 ππ τ π ( )[ ] ( ) ( )∫ +−× Φ zL Vnn z xdydzdzyxfzszzc n L 0 ,,1 2π . 3. DISCUSSION In this section we analyzed distribution of concentration of dopant, infused (see Fig. 2a) or im- planted (see Fig. 2b) into epitaxial layer. Annealing time is the same for all curves of these fig- ures. Increasing of number of curves corresponds to increasing of difference between values of dopant diffusion coefficients in layers of hetero structure. The figures show, that presents of in- terface between layers of hetero structure gives a possibility to increase absolute value of gra- dient of concentration of dopant in direction, which is perpendicular to the considered interface. A consequence of this result is decreasing of the dimensions of transistors included in the multi- vibrator. At the same time with increasing of the considered gradient homogeneity of concentra- tion of dopant in enriched area. Fig.2a. Distributions of concentration of infused dopant in heterostructure from Figs. 1 in direction, which is perpendicular to interface between epitaxial layer substrate. Increasing of number of curve corresponds to increasing of difference between values of dopant diffusion coefficient in layers of heterostructure under condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than value of dopant dif- fusion coefficient in substrate To estimate annealing time it is necessary to estimate decreasing of absolute value of gradient of concentration of dopant near interface between substrate and epitaxial layer with increasing of annealing time. Decreasing of annealing time leads to manufacturing more inhomogenous distri- bution of concentration of dopant (see Fig. 3a for diffusion type of doping and Fig. 3b for ion type of doping). We determine compromise value of annealing time framework recently intro- duced criterion [14-20]. Frame-work the criterion we approximate real distribution of concentra- tion of dopant by idealized step-wise function ψ (x,y,z). Further we determine the required an- nealing time by minimization of mean-square error
- 13. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 25 ( ) ( )[ ]∫ ∫ ∫ −Θ= x y zL L L zyx xdydzdzyxzyxC LLL U 0 0 0 ,,,,, 1 ψ . (8) Dependences of optimal annealing time are shown on Figs. 4. It should be noted, that radiation defects, generated during ion implantation, should be annealed. In the ideal case after the anneal- ing the implanted dopant should achieve the interface between layers of heterostructure. If the dopant has no enough time to achieve the interface, it is attracted an interest to make additional annealing. The Fig. 4b shows dependences of additional annealing time. Necessity of annealing of radiation defects leads to smaller value of optimal annealing time of dopant in comparison with optimal annealing time of dopant during diffusion type of doping. Diffusion type of doping did not leads to radiation damage of materials. Ion type of doping gives us possibility to decrease mismatch-induced stress in heterostructure [20]. Fig.2b. Distributions of concentration of implanted dopant in heterostructure from Figs. 1 in direction, which is perpendicular to interface between epitaxial layer substrate. Curves 1 and 3 corresponds to an- nealing time Θ=0.0048(Lx 2 +Ly 2 +Lz 2 )/D0. Curves 2 and 4 corresponds to annealing time Θ=0.0057(Lx 2 + Ly 2 +Lz 2 )/D0. Curves 1 and 2 corresponds to homogenous sample. Curves 3 and 4 corresponds to hetero- structure under condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than value of dopant diffusion coefficient in substrate Fig. 3a. Spatial distributions of dopant in heterostructure after dopant infusion. Curve 1 is idealized distri- bution of dopant. Curves 2-4 are real distributions of dopant for different values of annealing time. In- creasing of number of curve corresponds to increasing of annealing time x C(x,Θ) 1 2 3 4 0 L x 0.0 0.5 1.0 1.5 2.0 C(x,Θ) 2 3 4 1 0 L/4 L/2 3L/4 L Epitaxial layer Substrate C(x,Θ) 0 Lx 2 1 3 4
- 14. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 26 Fig.3b. Spatial distributions of dopant in heterostructure after ion implantation. Curve 1 is idealized distri- bution of dopant. Curves 2-4 are real distributions of dopant for different values of annealing time. In- creasing of number of curve corresponds to increasing of annealing time Fig.4a. Optimal annealing time of infused dopant as dependences of several parameters. Curve 1 is the dependence of the considered annealing time on dimensionless thickness of epitaxial layer a/L and ξ = γ = 0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of the considered annealing time on the parameter ε for a/L=1/2 and ξ = γ = 0. Curve 3 is the dependence of the considered annealing time on the parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the dependence of the considered annealing time on parameter γ for a/L=1/2 and ε = ξ = 0 Fig.4b. Optimal annealing time of implanted dopant as dependences of several parameters. Curve 1 is the dependence of the considered annealing time on dimensionless thickness of epitaxial layer a/L and ξ=γ=0 for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the dependence of the considered annealing time on the parameter ε for a/L=1/2 and ξ=γ=0. Curve 3 is the dependence of the considered annealing time on the parameter ξ for a/L=1/2 and ε=γ=0. Curve 4 is the dependence of the considered annealing time on parameter γ for a/L=1/2 and ε=ξ=0 4. CONCLUSIONS 0.0 0.1 0.2 0.3 0.4 0.5 a/L, ξ, ε, γ 0.0 0.1 0.2 0.3 0.4 0.5 ΘD0L -2 3 2 4 1 0.0 0.1 0.2 0.3 0.4 0.5 a/L, ξ, ε, γ 0.00 0.04 0.08 0.12 ΘD0L -2 3 2 4 1
- 15. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 27 In this paper prognosis of time varying of spatial distributions of concentrations of infused and implanted dopants during manufacturing a transistor multivibrator has been done. Several rec- ommendations to optimize manufacture the heterotransistors have been formulated. Analytical approach to prognosis diffusion and ion types of doping has been introduced. The approach gives a possibility to take into account variation in space and time parameters of doped material and at the same time nonlinearity of mass and heat transport. 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 Rus- sia and educational fellowship for scientific research of Nizhny Novgorod State University of Architecture and Civil Engineering. REFERENCES [1] Z. Ramezani, A.A. Orouji. A silicon-on-insulator metal semiconductor field-effect transistor with an L-shaped buried oxide for high output-power density. Mat. Sci. Sem. Proc. Vol. 19. P. 124-129 (2014). [2] Ch. Dong, J. Shi, J. Wu, Y. Chen, D. Zhou, Z. Hu, H. Xie, R. Zhan, Zh. Zou. Improvements in passivation effect of amorphous InGaZnO thin film transistors. Mat. Sci. Sem. Proc. Vol. 20. P. 7- 11 (2014). [3] D. Fathi, B. Forouzandeh. Accurate analysis of global interconnects in nano-fpgas. Nano. Vol. 4 (3). P. 171-176 (2009). [4] D. Fathi, B. Forouzandeh, N. Masoumi. New enhanced noise analysis in active mixers in nanoscale technologies. Nano. Vol. 4 (4). P. 233-238 (2009). [5] A.O. Ageev, A.E. Belyaev, N.S. Boltovets, V.N. Ivanov, R.V. Konakova, Ya.Ya. Kudrik, P.M. Litvin, V.V. Milenin, A.V. Sachenko. Au-TiBx-n-6H-SiC Schottky barrier diodes: Specific fea- tures of charge transport in rectifying and nonrectifying contacts. Semiconductors. Vol. 43 (7). P. 897-903 (2009). [6] A.G. Alexenko, I.I. Shagurin. Microcircuitry (Radio and communication, Moscow, 1990). [7] N.A. Avaev, Yu.E. Naumov, V.T. Frolkin. Basis of microelectronics (Radio and communication, Moscow, 1991). [8] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001). [9] R. Pal, R. Pandey, N. Pandey, R.Ch. Tiwari. Single CDBA Based Voltage Mode Bistable Multivi- brator and Its Applications. Circuits and Systems. Vol. 6 (11). P. 237-251 (2015). [10] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991). [11] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev, 1979, in Russian). [12] 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). [13] M.L. Krasnov, A.I. Kiselev, G.I. Makarenko. Integral equations ("Science", Moscow, 1976). [14] E.L. Pankratov. Dopant diffusion dynamics and optimal diffusion time as influenced by diffusion- coefficient nonuniformity Russian Microelectronics. 2007. V.36 (1). P. 33-39. [15] E.L. Pankratov. Redistribution of dopant during annealing of radiative defects in a multilayer structure by laser scans for production an implanted-junction rectifiers. Int. J. Nanoscience. Vol. 7 (4-5). P. 187–197 (2008). [16] E.L. Pankratov. Decreasing of depth of implanted-junction rectifier in semiconductor heterostruc- ture by optimized laser annealing. J. Comp. Theor. Nanoscience. Vol. 7 (1). P. 289-295 (2010). [17] 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). [18] E.L. Pankratov, E.A. Bulaeva. An approach to manufacture of bipolar transistors in thin film struc- tures. On the method of optimization. Int. J. Micro-Nano Scale Transp. Vol. 4 (1). P. 17-31 (2014).
- 16. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.3, No.3, July 2016 28 [19] E.L. Pankratov, E.A. Bulaeva. Increasing of sharpness of diffusion-junction heterorectifier by us- ing radiation processing. Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1250035 (2012). [20] 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). 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 Ra- diophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in Radio- physics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of Micro- structures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny Novgo- rod State University of Architecture and Civil Engineering. 2012-2015 Full Doctor course in Radiophysi- cal Department of Nizhny Novgorod State University. Since 2015 E.L. Pankratov is an Associate Profes- sor 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 Architec- ture 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.