![International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016
DOI: 10.5121/ijoe.2016.5401 1
ANALYSIS OF MANUFACTURING OF VOLTAGE RESTORE
TO INCREASE DENSITY OF ELEMENTS OF THE CIRCUIT
E.L. Pankratov, E.A. Bulaeva
Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950,
Russia
ABSTRACT
We introduce an approach for increasing density of voltage restore elements. The approach based on
manufacturing of a heterostructure, which consist of a substrate and an epitaxial layer with special con-
figuration. Several required sections of the layer should be doped by diffusion or ion implantation. After
that dopants and/or radiation defects should be annealed.
KEYWORDS
Voltage restore; optimization of manufacturing; increasing of density of elements
1. INTRODUCTION
Now several actual questions of solid state electronics could be formulated. One of them is in-
creasing of density of elements of integrated circuits. The increasing of density leads to necessity
of decreasing of dimensions of these elements. Recently are have been elaborated several ap-
proaches for the above decreasing. One of them is manufacturing of solid state electronic devices
in thin film heterostructures [1-4]. Other approach based on doping of required areas of a semi-
conductor sample or a heterostructure. After the doping one should consider laser or microwave
annealing of dopant and/or radiation defects [5-7]. Due to these types of annealing one can find
inhomogenous distribution of temperature in doped area. In this situation one can find inhomo-
geneity of doped structure and consequently to decreasing of dimensions of the considered ele-
ments. An alternative approach of changing of properties of materials before or during doping is
radiation processing [8,9].
In this paper we consider an approach to manufacture of a voltage restore. We illustrate the ap-
proach by considering a heterostructure. The heterostructure consist of a substrate and an epitax-
ial layer. We consider that several sections take a place framework the epitaxial layer. We con-
sider doping these sections by diffusion or ion implantation. The doping gives a possibility to
obtain p or n type of conductivity during process of manufacture of diodes and bipolar transistors
so as it is shown on Fig. 1. After finishing the doping we consider annealing of dopant and/or
radiation defects. Our main aim is analysis of redistribution of dopant and/or radiation defects
during annealing.
2. METHOD OF SOLUTION
We calculate distribution of concentration of dopant in space and time by solving the following
boundary problem [10-12]](https://image.slidesharecdn.com/5416ijoe01-161104115443/85/ANALYSIS-OF-MANUFACTURING-OF-VOLTAGE-RESTORE-TO-INCREASE-DENSITY-OF-ELEMENTS-OF-THE-CIRCUIT-1-320.jpg)
![International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016
2
( ) ( ) ( ) ( )
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC ,,,,,,,,,,,,
(1)
( ) 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).
The function C(x,y,z,t) describes distribution of concentration of dopant in space and time; T
Fig. 1. Structure of voltage restore.
View from top describes the temperature of annealing; DС describes the dopant diffusion coeffi-
cient. Dopant diffusion coefficient is different in different materials of heterostructure. The diffu-
sion coefficient will be also changed with changing of temperature. We approximate dopant dif-
fusion coefficient by the following function based on Refs. [13-15]
.
(3)
Function DL (x,y,z,T) describes spatial and temperature dependences of diffusion coefficient;
function P (x,y,z,T) describes limit of solubility of dopant; parameter γ is differ in different mate-
rials and assume integer values; function V (x,y,z,t) describes distribution of concentration of va-
cancies with equilibrium distribution V*
. One can find detailed concentrational dependence of
dopant diffusion coefficient in Ref. [13]. It should be noted, that using diffusive type of doping
did not leads to radiation damage. In this situation ζ1=ζ2= 0. We determine spatio-temporal dis-
tributions of concentrations of point radiation defects by solution the following system of equa-
tions [14,15]
n
n
p
n
p
( ) ( )
( )
( ) ( )
( )
++
+= 2*
2
2*1
,,,,,,
1
,,,
,,,
1,,,
V
tzyxV
V
tzyxV
TzyxP
tzyxC
TzyxDD LC ςςξ γ
γ](https://image.slidesharecdn.com/5416ijoe01-161104115443/85/ANALYSIS-OF-MANUFACTURING-OF-VOLTAGE-RESTORE-TO-INCREASE-DENSITY-OF-ELEMENTS-OF-THE-CIRCUIT-2-320.jpg)
![International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016
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( ) ( ) ( ) ( ) ( ) ( ) ×−
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
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; function I(x,y,z,t) describes distribution of concentration of radiation interstitials in
space and time; functions Dρ(x,y,z,T) describe spatio-temperature dependences of the diffusion
coefficients of point radiation defects; terms V2
(x,y,z,t) and I2
(x,y,z,t) describe generation
divacancies and diinterstitials, respectively; function kI,V(x,y,z,T) describes spatio-temperature
dependence of recombination parameter of point radiation defects; functions kI,I(x,y,z,T) and
kV,V(x,y,z,T) describe spatio-temperature dependence of parameters of generation of simplest
complexes of point radiation defects.
We calculate distributions of concentrations of divacancies ΦV (x,y,z,t) and dinterstitials ΦI
(x,y,z,t) in space and time by solving the following system of equations [14,15]
( ) ( ) ( ) ( ) ( ) +
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)
( ) ( ) ( ) ( ) ( ) +
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
,](https://image.slidesharecdn.com/5416ijoe01-161104115443/85/ANALYSIS-OF-MANUFACTURING-OF-VOLTAGE-RESTORE-TO-INCREASE-DENSITY-OF-ELEMENTS-OF-THE-CIRCUIT-3-320.jpg)
![International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016
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( )
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)
Function DΦρ(x,y,z,T) describes spatio-temperature dependences of diffusion coefficients of the
considered complexes of radiation defects; function kI(x,y,z,T) and kV (x,y,z,T) describes spatio-
temperature dependences of parameters of decay of these complexes.
We used recently elaborated approach [16,17] to calculate distributions of concentrations of
point radiation defects in space and time. To use the approach we shall transform spatio-
temperature dependences of diffusion coefficients to the following form: Dρ(x,y,z,T)=D0ρ [1+ερ
gρ(x,y,z,T)]. Here D0ρ are the average values of diffusion coefficients, 0≤ερ<1, |gρ(x,y,z,T)|≤1, ρ
=I,V. We used the same transformation of approximations oof recombination parameters of point
defects and generation parameters of their complexes: kI,V(x,y,z,T)=k0I,V[1+εI,V gI,V(x,y,z,T)],
kI,I(x,y,z,T)=k0I,I[1+ εI,I gI,I(x,y,z,T)] and kV,V (x,y,z,T)= k0V,V [1+εV,V gV,V(x,y,z,T)], where k0ρ1,ρ2 are
the their average values, 0≤εI,V < 1, 0≤εI,I <1, 0≤εV,V<1, | gI,V(x,y,z,T)|≤1, | gI,I(x,y,z,T)|≤1,
|gV,V(x,y,z,T)|≤1. Let us introduce the following dimensionless variables: χ = x/Lx, η = y /Ly,, φ=
z/Lz, ( ) ( ) *
,,,,,,
~
ItzyxItzyxI = , ( ) ( ) *
,,,,,,
~
VtzyxVtzyxV = , VI DDkL 00,0
2
ρρρ =Ω ,
VIVI DDkL 00,0
2
=ω , 2
00 LtDD VI=ϑ . Now we can transform boundary problem Eqs.(4) and
(5) to the following form
( )
( )[ ] ( ) ( )
×
∂
∂
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
η
ϑφηχ
ηχ
ϑφηχ
φηχε
χϑ
ϑφηχ ,,,
~
,,,
~
,,,1
,,,
~
00
0
00
0 I
DD
DI
Tg
DD
DI
VI
I
II
VI
I
( )[ ]} ( )[ ] ( ) ( ) ×−
∂
∂
+
∂
∂
++× ϑφηχ
φ
ϑφηχ
φηχε
φ
φηχε ,,,
~,,,
~
,,,1,,,1
00
0
I
I
Tg
DD
D
Tg II
VI
I
II
( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχεω ,,,
~
,,,1,,,
~
,,,1 2
,,,, ITgVTg IIIIIVIVI +Ω−+× (8)
( ) ( )[ ] ( ) ( )
×
∂
∂
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
η
ϑφηχ
ηχ
ϑφηχ
φηχε
χϑ
ϑφηχ ,,,
~
,,,
~
,,,1
,,,
~
00
0
00
0 V
DD
DV
Tg
DD
DV
VI
V
VV
VI
V
( )[ ]} ( )[ ] ( ) ( ) ×−
∂
∂
+
∂
∂
++× ϑφηχ
φ
ϑφηχ
φηχε
φ
φηχε ,,,
~,,,
~
,,,1,,,1
00
0
I
V
Tg
DD
D
Tg VV
VI
V
VV
( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχεω ,,,
~
,,,1,,,
~
,,,1 2
,,,, VTgVTg VVVVIVIVI +Ω−+×
( ) 0
,,,~
0
=
∂
∂
=χ
χ
ϑφηχρ
,
( ) 0
,,,~
1
=
∂
∂
=χ
χ
ϑφηχρ
,
( ) 0
,,,~
0
=
∂
∂
=η
η
ϑφηχρ
,
( ) 0
,,,~
1
=
∂
∂
=η
η
ϑφηχρ
,
( ) 0
,,,~
0
=
∂
∂
=φ
φ
ϑφηχρ
,
( ) 0
,,,~
1
=
∂
∂
=φ
φ
ϑφηχρ
, ( )
( )
*
,,,
,,,~
ρ
ϑφηχ
ϑφηχρ ρf
= . (9)](https://image.slidesharecdn.com/5416ijoe01-161104115443/85/ANALYSIS-OF-MANUFACTURING-OF-VOLTAGE-RESTORE-TO-INCREASE-DENSITY-OF-ELEMENTS-OF-THE-CIRCUIT-4-320.jpg)
![International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016
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We calculate solutions of boundary problem Eqs.(8) and (9) by using recently introduced ap-
proach [16,17]. In this situation we used the following power series
( ) ( )∑ ∑ ∑Ω=
∞
=
∞
=
∞
=0 0 0
,,,~,,,~
i j k
ijk
kji
ϑφηχρωεϑφηχρ ρρ . (10)
We substitute the series (10) into the boundary problem Eqs.(8) and (9). After that we obtain
equations for initial-order approximations of point defects concentration ( )ϑφηχ ,,,
~
000I and
( )ϑφηχ ,,,
~
000V and corrections for them ( )ϑφηχ ,,,
~
ijkI and ( )ϑφηχ ,,,
~
ijkV , i≥1, j≥1, k≥1. The equa-
tions are presented in the Appendix. We obtain solution of the above equations by the standard
Fourier approach [18,19]. The solutions are presented in the Appendix.
Farther we calculate distributions of concentrations of simplest complexes of point radiation de-
fects in space and time. To determine the distributions we transform approximations of diffusion
coefficients in the following form: DΦρ(x,y,z,T)=D0Φρ[1+εΦρgΦρ(x,y,z,T)]. Here D0Φρ describe av-
erage values of diffusion coefficients. In this situation the Eqs.(6) could be written as
( )
( )[ ] ( ) ( )
×
Φ
+
Φ
+=
Φ
ΦΦΦΦ
y
tzyx
y
D
x
tzyx
Tzyxg
x
D
t
tzyx I
I
I
III
I
∂
∂
∂
∂
∂
∂
ε
∂
∂
∂
∂ ,,,,,,
,,,1
,,,
00
( )[ ] ( )[ ] ( )
( ) ×−
Φ
++
+× ΦΦΦΦΦ Tzyxk
z
tzyx
Tzyxg
z
DTzyxg I
I
IIIII ,,,
,,,
,,,1,,,1 0
∂
∂
ε
∂
∂
ε
( ) ( ) ( )tzyxITzyxktzyxI II ,,,,,,,,, 2
,+×
( )
( )[ ] ( ) ( )
×
Φ
+
Φ
+=
Φ
ΦΦΦΦ
y
tzyx
y
D
x
tzyx
Tzyxg
x
D
t
tzyx V
V
V
VVV
V
∂
∂
∂
∂
∂
∂
ε
∂
∂
∂
∂ ,,,,,,
,,,1
,,,
00
( )[ ] ( )[ ] ( )
( ) ×+
Φ
++
+× ΦΦΦΦΦ Tzyxk
z
tzyx
Tzyxg
z
DTzyxg II
V
VVVVV ,,,
,,,
,,,1,,,1 ,0
∂
∂
ε
∂
∂
ε
( ) ( ) ( )tzyxITzyxktzyxI I ,,,,,,,,,2
−× .
Farther we determine solutions of above equations as the following power series
( ) ( )∑ Φ=Φ
∞
=
Φ
0
,,,,,,
i
i
i
tzyxtzyx ρρρ ε . (11)
Now we substitute series (11) into Eqs.(6) and conditions for them. The substitution gives a ob-
tain equations for initial-order approximations of concentrations of complexes of defects
Φρ0(x,y,z,t) and corrections for them Φρi(x,y,z,t), i ≥1 and boundary and initial conditions for
them. We present the equations and conditions in Appendix. We obtain solutions of the obtain
equations by using standard Fourier approaches [18,19] and presented in the Appendix.
We calculate distribution of concentration of dopant in space and time by using the same ap-
proach, which was used for calculation distributions of concentrations of radiation defects in](https://image.slidesharecdn.com/5416ijoe01-161104115443/85/ANALYSIS-OF-MANUFACTURING-OF-VOLTAGE-RESTORE-TO-INCREASE-DENSITY-OF-ELEMENTS-OF-THE-CIRCUIT-5-320.jpg)
![International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016
6
space and time. To use the approach we transform approximation of dopant diffusion coefficient
to the following form: DL(x,y,z,T)=D0L[1+εLgL(x,y,z,T)]. Here D0L is the average value of diffu-
sion coefficient of dopant, 0≤εL< 1, |gL(x,y,z,T)|≤1. Now we calculate solution of Eq.(1) as the
following power series
( ) ( )∑ ∑=
∞
=
∞
=0 1
,,,,,,
i j
ij
ji
L tzyxCtzyxC ξε .
Now we substitute the above series into boundary problem Eqs.(1) and (2) leads to equations for
the initial-order approximation of concentration of dopant C00(x,y,z,t), corrections for them
Cij(x,y,z,t) (i≥1, j≥1) and conditions for the above equations. The equations are presented in the
Appendix. Solutions of the equations have been calculated by standard Fourier approaches
[18,19]. The solutions are presented in the Appendix.
We analyze distributions of concentrations of dopant and radiation defects in space and time ana-
lytically by using the second-order approximations of considered series. Usually we obtain
enough good qualitative analysis and quantitative results. We check all analytical results by
comparison with results of numerical simulation.
3. DISCUSSION
Now we analyzed distributions of concentrations of dopants in space and time by using calculat-
ed in previous section relations. Figs. 2 show typical distributions of concentrations of dopants
on coordinate near interface between materials of heterostructures in direction, which is perpen-
dicular to the interface. We calculate these distributions for larger value of dopant diffusion coef-
ficient in doped area in comparison of value of dopant diffusion coefficient in nearest areas. The-
se distributions show that in the case sharpness of p-n-junctions increases. At the same time one
can find increasing of homogeneity of distribution of dopant concentration. One can find both
effects for single p-n-junctions and p-n-junctions framework their systems (transistors,
thyristors).
Fig.2a. Spatial distributions of infused dopant concentration in the considered heterostructure. The consid-
ered direction perpendicular to the interface between epitaxial layer substrate. Difference between values
of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves](https://image.slidesharecdn.com/5416ijoe01-161104115443/85/ANALYSIS-OF-MANUFACTURING-OF-VOLTAGE-RESTORE-TO-INCREASE-DENSITY-OF-ELEMENTS-OF-THE-CIRCUIT-6-320.jpg)
![International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016
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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
Fig.2b. Spatial distributions of infused dopant concentration in the considered heterostructure. Curves 1
and 3 corresponds to annealing 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 the considered heterostructure. Difference between values of dopant diffusion coefficient in
layers of heterostructure increases with increasing of number of curves
Now we consider optimization of annealing of dopant and/or radiation defects. To do the optimi-
zation we used recently introduced criterion [16,17,20,21]. Framework the criterion we approxi-
mate real distribution by step-wise function ψ (x,y,z). We calculate optimal annealing time by
minimization the following mean-squared error
( ) ( )[ ]∫ ∫ ∫ −Θ=
x y zL L L
zyx
xdydzdzyxzyxC
LLL
U
0 0 0
,,,,,
1
ψ . (12)
Optimal annealing time as functions of parameters are presented on Figs. 3. It should be noted,
that in the ideal case after finishing of annealing of radiation defects dopant achieves interface
between layers of heterostructure. If the dopant do not achieves the interface, it is practicably to
use additional annealing of the dopant. The Fig. 3b shows just the dependences of optimal values
of additional annealing time. In this situation optimal annealing of implanted dopant is smaller in
comparison with optimal annealing time of infused dopant.
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
Fig.3a. 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](https://image.slidesharecdn.com/5416ijoe01-161104115443/85/ANALYSIS-OF-MANUFACTURING-OF-VOLTAGE-RESTORE-TO-INCREASE-DENSITY-OF-ELEMENTS-OF-THE-CIRCUIT-7-320.jpg)
![International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016
8
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
Fig.3b. 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
We introduce an approach to increase density of elements of an voltage restore. Several recom-
mendations for optimization technological processes were formulated.
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
and educational fellowship for scientific research of Government of Nizhny Novgorod region of
Russia.
REFERENCES
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[2] A. Kerentsev, V. Lanin. Constructive-technological features of MOSFET-transistors. Power Elec-
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[3] 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. Semiconductors. Vol. 43 (7). P. 897-903 (2009).
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ing and nonrectifying contacts. Semiconductors. Vol. 35 (8). P. 1013-1017 (2001).
[5] K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. Dopant distribution in the
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[6] H.T. Wang, L.S. Tan, E. F. Chor. Pulsed laser annealing of Be-implanted GaN. J. Appl. Phys. 98
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[7] Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov, Yu.N.
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[8] V.V. Kozlivsky. Modification of semiconductors by proton beams (Nauka, Sant-Peterburg, 2003, in
Russian).](https://image.slidesharecdn.com/5416ijoe01-161104115443/85/ANALYSIS-OF-MANUFACTURING-OF-VOLTAGE-RESTORE-TO-INCREASE-DENSITY-OF-ELEMENTS-OF-THE-CIRCUIT-8-320.jpg)
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[9] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev,
1979, in Russian).
[10] V.G. Gusev, Yu.M. Gusev. Electronics. ("Higher school", Moscow, 1991).
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Moscow, 1991).
[12] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001).
[13] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991).
[14] P.M. Fahey, P.B. Griffin, J.D. Plummer. Rev. Point defects and dopant diffusion in silicon. Mod.
Phys. 1989. V. 61. № 2. P. 289-388.
[15] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev,
1979, in Russian).
[16] E.L. Pankratov, E.A. Bulaeva. Increasing of sharpness of diffusion-junction heterorectifier by using
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[17] E.L. Pankratov, E.A. Bulaeva. Doping of materials during manufacture p-n-junctions and bipolar
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[18] A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations (Moscow, Nauka 1972) (in
Russian).
[19] H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids (Oxford University Press, 1964).
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(2015).
Appendix
Equations for the functions ( )ϑφηχ ,,,
~
ijkI and ( )ϑφηχ ,,,
~
ijkV , i ≥0, j ≥0, k ≥0 and conditions for
them
( ) ( ) ( ) ( )
2
000
2
0
0
2
000
2
0
0
2
000
2
0
0000 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ I
D
DI
D
DI
D
DI
V
I
V
I
V
I
( ) ( ) ( ) ( )
2
000
2
0
0
2
000
2
0
0
2
000
2
0
0000 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ V
D
DV
D
DV
D
DV
I
V
I
V
I
V
( ) ( ) ( ) ( ) +
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
00
2
2
00
2
2
00
2
0
000 ,,,
~
,,,
~
,,,
~
,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑχ iii
V
Ii III
D
DI
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+ −−
η
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
,,,
~
,,,
,,,
~
,,, 100
0
0100
0
0 i
I
V
Ii
I
V
I I
Tg
D
DI
Tg
D
D
( )
( )
∂
∂
∂
∂
+ −
φ
ϑφηχ
φηχ
φ
,,,
~
,,, 100
0
0 i
I
V
I I
Tg
D
D
, i≥1,
( ) ( ) ( ) ( ) +
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
00
2
2
00
2
2
00
2
0
000 ,,,
~
,,,
~
,,,
~
,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑχ iii
I
Vi VVV
D
DV
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+ −−
η
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
,,,
~
,,,
,,,
~
,,, 100
0
0100
0
0 i
V
I
Vi
V
I
V V
Tg
D
DV
Tg
D
D](https://image.slidesharecdn.com/5416ijoe01-161104115443/85/ANALYSIS-OF-MANUFACTURING-OF-VOLTAGE-RESTORE-TO-INCREASE-DENSITY-OF-ELEMENTS-OF-THE-CIRCUIT-9-320.jpg)
![International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016
10
( )
( )
∂
∂
∂
∂
+ −
φ
ϑφηχ
φηχ
φ
,,,
~
,,, 100
0
0 i
V
I
V V
Tg
D
D
, i≥1;
( ) ( ) ( ) ( ) −
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
010
2
2
010
2
2
010
2
0
0010 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ III
D
DI
V
I
( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,,
~
,,,
~
,,,1 000000,, VITg VIVI+−
( ) ( ) ( ) ( ) −
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
010
2
2
010
2
2
010
2
0
0010 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ VVV
D
DV
I
V
( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,,
~
,,,
~
,,,1 000000,, VITg VIVI+− ;
( ) ( ) ( ) ( ) −
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
020
2
2
020
2
2
020
2
0
0020 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ III
D
DI
V
I
( )[ ] ( ) ( ) ( ) ( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε ,,,
~
,,,
~
,,,
~
,,,
~
,,,1 010000000010,, VIVITg VIVI ++−
( ) ( ) ( ) ( ) −
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
020
2
2
020
2
2
020
2
0
0020 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ VVV
D
DV
V
I
( )[ ] ( ) ( ) ( ) ( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε ,,,
~
,,,
~
,,,
~
,,,
~
,,,1 010000000010,, VIVITg VIVI ++− ;
( ) ( ) ( ) ( ) −
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
001
2
2
001
2
2
001
2
0
0001 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ III
D
DI
V
I
( )[ ] ( )ϑφηχφηχε ,,,
~
,,,1 2
000,, ITg IIII+−
( ) ( ) ( ) ( ) −
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
001
2
2
001
2
2
001
2
0
0001 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ VVV
D
DV
I
V
( )[ ] ( )ϑφηχφηχε ,,,
~
,,,1 2
000,, VTg IIII+− ;
( ) ( ) ( ) ( ) +
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
110
2
2
110
2
2
110
2
0
0110 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ III
D
DI
V
I
( ) ( ) ( ) ( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
η
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
,,,
~
,,,
,,,
~
,,, 010010
0
0 I
Tg
I
Tg
D
D
II
V
I
( )
( )
( )[ ] ( ) ( )[ ++−
∂
∂
∂
∂
+ ϑφηχϑφηχφηχε
φ
ϑφηχ
φηχ
φ
,,,
~
,,,
~
,,,1
,,,
~
,,, 000100,,
010
VITg
I
Tg IIIII
( ) ( )]ϑφηχϑφηχ ,,,
~
,,,
~
100000 VI+
( ) ( ) ( ) ( ) +
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
110
2
2
110
2
2
110
2
0
0110 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ VVV
D
DV
I
V
( ) ( ) ( ) ( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
η
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
,,,
~
,,,
,,,
~
,,, 010010
0
0 V
Tg
V
Tg
D
D
VV
I
V](https://image.slidesharecdn.com/5416ijoe01-161104115443/85/ANALYSIS-OF-MANUFACTURING-OF-VOLTAGE-RESTORE-TO-INCREASE-DENSITY-OF-ELEMENTS-OF-THE-CIRCUIT-10-320.jpg)
![International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016
11
( )
( )
( )[ ] ( ) ( )[ ++−
∂
∂
∂
∂
+ ϑφηχϑφηχφηχε
φ
ϑφηχ
φηχ
φ
,,,
~
,,,
~
,,,1
,,,
~
,,, 000100,,
010
IVTg
V
Tg VVVVV
( ) ( )]ϑφηχϑφηχ ,,,
~
,,,
~
100000 IV+ ;
( ) ( ) ( ) ( ) −
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
002
2
2
002
2
2
002
2
0
0002 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ III
D
DI
V
I
( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,,
~
,,,
~
,,,1 000001,, IITg IIII+−
( ) ( ) ( ) ( ) −
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
002
2
2
002
2
2
002
2
0
0002 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ VVV
D
DV
I
V
( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,,
~
,,,
~
,,,1 000001,, VVЕg VVVV+− ;
( ) ( ) ( ) ( ) +
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
101
2
2
101
2
2
101
2
0
0101 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ III
D
DI
V
I
( ) ( ) ( ) ( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
η
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
,,,
~
,,,
,,,
~
,,, 001001
0
0 I
Tg
I
Tg
D
D
II
V
I
( ) ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχε
φ
ϑφηχ
φηχ
φ
,,,
~
,,,
~
,,,1
,,,
~
,,, 000100
001
VITg
I
Tg III +−
∂
∂
∂
∂
+
( ) ( ) ( ) ( ) +
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
101
2
2
101
2
2
101
2
0
0101 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ VVV
D
DV
I
V
( ) ( ) ( ) ( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
η
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
,,,
~
,,,
,,,
~
,,, 001001
0
0 V
Tg
V
Tg
D
D
VV
I
V
( ) ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχε
φ
ϑφηχ
φηχ
φ
,,,
~
,,,
~
,,,1
,,,
~
,,, 100000
001
VITg
V
Tg VVV +−
∂
∂
∂
∂
+ ;
( ) ( ) ( ) ( ) −
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
011
2
2
011
2
2
011
2
0
0011 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ III
D
DI
V
I
( )[ ] ( ) ( ) ( )[ ] ×+−+− TgIITg VIVIIIII ,,,1,,,
~
,,,
~
,,,1 ,,010000,, φηχεϑφηχϑφηχφηχε
( ) ( )ϑφηχϑφηχ ,,,
~
,,,
~
000001 VI×
( ) ( ) ( ) ( ) −
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
011
2
2
011
2
2
011
2
0
0011 ,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ VVV
D
DV
I
V
( )[ ] ( ) ( ) ( )[ ]](https://image.slidesharecdn.com/5416ijoe01-161104115443/85/ANALYSIS-OF-MANUFACTURING-OF-VOLTAGE-RESTORE-TO-INCREASE-DENSITY-OF-ELEMENTS-OF-THE-CIRCUIT-11-320.jpg)
![International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016
12
( )
0
,,,~
0
=
∂
∂
=φ
φ
ϑφηχρijk
,
( )
0
,,,~
1
=
∂
∂
=φ
φ
ϑφηχρijk
(i≥0, j≥0, k≥0);
( ) ( ) *
000 ,,0,,,~ ρφηχφηχρ ρf= , ( ) 00,,,~ =φηχρijk (i≥1, j≥1, k≥1).
Solutions of these equations with account boundary and initial conditions could be written as
( ) ( ) ( ) ( ) ( )∑+=
∞
=1
000
21
,,,~
n
nn ecccF
LL
ϑφηχϑφηχρ ρρ ,
where ( ) ( ) ( ) ( )∫ ∫ ∫=
1
0
1
0
1
0
*
,,coscoscos
1
udvdwdwvufwnvnunF nn ρρ πππ
ρ
, ( ) ( )IVnI DDne 00
22
exp ϑπϑ −= ,
( ) ( )VInV DDne 00
22
exp ϑπϑ −= , cn(χ)=cos(πnχ);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
00
0
00 ,,,2,,,
~
n
InnnInIn
V
I
i Twvugvcuseecccn
D
D
I
ϑ
τϑφηχπϑφηχ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−
∂
∂
×
∞
=
−
1 0
1
00
0100
2
,,,
~
n
nnInIn
V
Ii
n uceecccn
D
D
dudvdwd
u
wvuI
wc
ϑ
τϑφηχτ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑−∫ ∫
∂
∂
×
∞
=
−
1
0
0
1
0
1
0
100
2
,,,
~
,,,
n
n
V
Ii
Inn cccn
D
D
dudvdwd
v
wvuI
Twvugwcvs φηχπτ
τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( )
∫ ∫ ∫ ∫
∂
∂
−× −
ϑ
τ
τ
τϑ
0
1
0
1
0
1
0
100 ,,,
~
,,, dudvdwd
w
wvuI
Twvugwsvcucee i
InnnnInI , i≥1,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
00
0
00 ,,,2,,,
~
n
VnnnInVn
I
V
i Twvugvcuseecccn
D
D
V
ϑ
τϑφηχπϑφηχ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫−−
∂
∂
×
∞
=
−
1 0
1
0
1
00
0100 ,
~
n
nnnInVn
I
Vi
n vsuceecccn
D
D
dudvdwd
u
uV
wc
ϑ
τϑφηχτ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫
∂
∂
×
∞
=
−
1
0
0
1
0
100
2
,
~
,,,2
n
nVn
I
Vi
Vn ecccn
D
D
dudvdwd
v
uV
Twvugwc ϑφηχπτ
τ
π
( ) ( ) ( ) ( ) ( ) ( )
∫ ∫ ∫ ∫
∂
∂
−× −
ϑ
τ
τ
τ
0
1
0
1
0
1
0
100 ,
~
,,, dudvdwd
w
uV
Twvugwsvcuce i
VnnnnI , i≥1,
where sn(χ)=sin (π nχ);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−=
∞
=1 0
1
0
1
0
1
0
010 2,,,~
n
nnnnnnnn wcvcuceeccc
ϑ
ρρ τϑφηχϑφηχρ
( )[ ] ( ) ( ) τττε dudvdwdwvuVwvuITwvug VIVI ,,,
~
,,,
~
,,,1 000000,,+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
010
1
0
1
00
0
020 ,,,
~
2,,,~
n
nnnnnnnn
V
I
wvuIwcvcuceeccc
D
D ϑ
ρρ ττϑφηχϑφηχρ
( ) ( ) ( )] ( )[ ] τετττ dudvdwdTwvugwvuVwvuIwvuV VIVI ,,,1,,,
~
,,,
~
,,,
~
,,010000000 ++× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
0
2
000001 ,,,~2,,,~
n
nnnnnnn wvuvcuceeccc
ϑ
ρρ τρτϑφηχϑφηχρ](https://image.slidesharecdn.com/5416ijoe01-161104115443/85/ANALYSIS-OF-MANUFACTURING-OF-VOLTAGE-RESTORE-TO-INCREASE-DENSITY-OF-ELEMENTS-OF-THE-CIRCUIT-12-320.jpg)
![International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016
13
( )[ ] ( ) τε ρρρρ dudvdwdwcTwvug n,,,1 ,,+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ×∫ ∫−−=
∞
=1 0
1
0
1
0
1
0
000002 ,,,~2,,,~
n
nnnnnnnn wvuwcvcuceeccc
ϑ
ρρ τρτϑφηχϑφηχρ
( )[ ] ( ) ττρε ρρρρ dudvdwdwvuTwvug ,,,~,,,1 001,,+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
00
0
110 2,,,
~
n
nnnnInInnn
V
I
ucvcuseecccn
D
D
I
ϑ
τϑφηχπϑφηχ
( ) ( ) ( ) ( ) ( ) ( ) ×∑−
∂
∂
×
∞
=
−
1
0
0100
2
,,,
~
,,,
n
nInnn
V
Ii
I ecccn
D
D
dudvdwd
u
wvuI
Twvug ϑφηχπτ
τ
( ) ( ) ( ) ( ) ( ) ( ) ×−∫ ∫ ∫ ∫
∂
∂
−× −
V
Ii
InnnnI
D
D
dudvdwd
v
wvuI
Twvugucvsuce
0
0
0
1
0
1
0
1
0
100
2
,,,
~
,,, πτ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫
∂
∂
−×
∞
=
−
1 0
1
0
1
0
1
0
100 ,,,
~
,,,
n
i
InnnnInI dudvdwd
w
wvuI
Twvugusvcuceen
ϑ
τ
τ
τϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[∑ ∫ ∫ ∫ ∫ ×+−−×
∞
=1 0
1
0
1
0
1
0
,12
n
VInnnnInnnInnnn vcvcuceccecccc
ϑ
ετφηϑχφηχ
( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvug VI ,,,
~
,,,
~
,,,
~
,,,
~
,,, 100000000100, +×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
00
0
110 2,,,
~
n
nnnnVnVnnn
I
V
ucvcuseecccn
D
D
V
ϑ
τϑφηχπϑφηχ
( ) ( ) ( ) ( ) ( ) ( )×∑−
∂
∂
×
∞
=
−
1
0
0100
2
,,,
~
,,,
n
nVnnn
I
Vi
V ecccn
D
D
dudvdwd
u
wvuV
Twvug ϑφηχπτ
τ
( ) ( ) ( ) ( ) ( ) ( ) ×−∫ ∫ ∫ ∫
∂
∂
−× −
I
Vi
VnnnnV
D
D
dudvdwd
v
wvuV
Twvugucvsuce
0
0
0
1
0
1
0
1
0
100
2
,,,
~
,,, πτ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫
∂
∂
−×
∞
=
−
1 0
1
0
1
0
1
0
100 ,,,
~
,,,
n
i
VnnnnVnV dudvdwd
w
wvuV
Twvugusvcuceen
ϑ
τ
τ
τϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[∑ ∫ ∫ ∫ ∫ ×+−−×
∞
=1 0
1
0
1
0
1
0
,12
n
VInnnnVnnnInnnn vcvcuceccecccc
ϑ
ετφηϑχφηχ
( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvug VI ,,,
~
,,,
~
,,,
~
,,,
~
,,, 100000000100, +× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
00
0
101 ,,,2,,,
~
n
InnnInInnn
V
I
Twvugvcuseecccn
D
D
I
ϑ
τϑφηχπϑφηχ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−
∂
∂
×
∞
=1 0
1
00
0001
2
,,,
~
n
nnInInnn
V
I
n uceecccn
D
D
dudvdwd
u
wvuI
wc
ϑ
τϑφηχπτ
τ
( ) ( ) ( )
( )
( ) ( ) ( ) ×∑ ∫ −−∫ ∫
∂
∂
×
∞
=1 00
0
1
0
1
0
001
2
,,,
~
,,,
n
nInIn
V
I
Inn eec
D
D
dudvdwd
v
wvuI
Twvugwcvs
ϑ
τϑχπτ
τ
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫
∂
∂
×
∞
=1
1
0
1
0
1
0
001
2
,,,
~
,,,
n
nnnnInnn ccccdudvdwd
w
wvuI
Twvugwsvcucn ηχφητ
τ](https://image.slidesharecdn.com/5416ijoe01-161104115443/85/ANALYSIS-OF-MANUFACTURING-OF-VOLTAGE-RESTORE-TO-INCREASE-DENSITY-OF-ELEMENTS-OF-THE-CIRCUIT-13-320.jpg)
![International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016
14
( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ +−×
ϑ
τττετϑφ
0
1
0
000100,, ,,,
~
,,,
~
,,,1 dudvdwdwvuVwvuITwvuguceec VIVInnInIn
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
00
0
101 ,,,2,,,
~
n
VnnnVnVnnn
I
V
Twvugvcuseecccn
D
D
V
ϑ
τϑφηχπϑφηχ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−
∂
∂
×
∞
=1 0
1
00
0001
2
,,,
~
n
nnVnVnnn
I
V
n uceecccn
D
D
dudvdwd
u
wvuV
wc
ϑ
τϑφηχπτ
τ
( ) ( ) ( )
( )
( ) ( ) ( ) ×∑ ∫ −−∫ ∫
∂
∂
×
∞
=1 00
0
1
0
1
0
001
2
,,,
~
,,,
n
nVnVn
I
V
Vnn eec
D
D
dudvdwd
v
wvuV
Twvugwcvs
ϑ
τϑχπτ
τ
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫
∂
∂
×
∞
=1
1
0
1
0
1
0
001
2
,,,
~
,,,
n
nnnnVnnn ccccdudvdwd
w
wvuV
Twvugwsvcucn ηχφητ
τ
( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ +−×
ϑ
τττετϑφ
0
1
0
100000,, ,,,
~
,,,
~
,,,1 dudvdwdwvuVwvuITwvuguceec VIVInnVnVn
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]{∑ ∫ ∫ ∫ ∫ ×+−−=
∞
=1 0
1
0
1
0
1
0
,,011 ,,,12,,,
~
n
IIIInnnnInInnn TwvugwcvcuceecccI
ϑ
ετϑφηχϑφηχ
( ) ( ) ( )[ ] ( ) ( )} τττεττ dudvdwdwvuVwvuITwvugwvuIwvuI VIVI ,,,
~
,,,
~
,,,1,,,
~
,,,
~
000001,,010000 ++×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]{∑ ∫ ∫ ∫ ∫ ×+−−=
∞
=1 0
1
0
1
0
1
0
,,011 ,,,12,,,
~
n
VVVVnnnnVnVnnn TwvugwcvcuceecccV
ϑ
ετϑφηχϑφηχ
( ) ( ) ( )[ ] ( ) ( )} τττεττ dudvdwdwvuVwvuITwvugwvuVwvuV VIVI ,,,
~
,,,
~
,,,1,,,
~
,,,
~
001000,,010000 ++× .
Equations for initial-order approximations of distributions of concentrations of simplest com-
plexes of radiation defects Φρ0(x,y,z,t) and corrections for them Φρi(x,y, z,t), i ≥1 and boundary
and initial conditions for them takes the form
( ) ( ) ( ) ( ) +
Φ
+
Φ
+
Φ
=
Φ
Φ 2
0
2
2
0
2
2
0
2
0
0 ,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx III
I
I
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxk III ,,,,,,,,,,,, 2
, −+
( ) ( ) ( ) ( ) +
Φ
+
Φ
+
Φ
=
Φ
Φ 2
0
2
2
0
2
2
0
2
0
0 ,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx VVV
V
V
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxk VVV ,,,,,,,,,,,, 2
, −+ ;
( ) ( ) ( ) ( )
+
Φ
+
Φ
+
Φ
=
Φ
Φ 2
2
2
2
2
2
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx iIiIiI
I
iI
∂
∂
∂
∂
∂
∂
∂
∂
( )
( )
( )
( )
+
Φ
+
Φ
+
−
Φ
−
ΦΦ
y
tzyx
Tzyxg
yx
tzyx
Tzyxg
x
D
iI
I
iI
II
∂
∂
∂
∂
∂
∂
∂
∂ ,,,
,,,
,,,
,,,
11
0
( )
( )
Φ
+
−
Φ
z
tzyx
Tzyxg
z
iI
I
∂
∂
∂
∂ ,,,
,,,
1
, i≥1,](https://image.slidesharecdn.com/5416ijoe01-161104115443/85/ANALYSIS-OF-MANUFACTURING-OF-VOLTAGE-RESTORE-TO-INCREASE-DENSITY-OF-ELEMENTS-OF-THE-CIRCUIT-14-320.jpg)
![International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016
15
( ) ( ) ( ) ( )
+
Φ
+
Φ
+
Φ
=
Φ
Φ 2
2
2
2
2
2
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx iViViV
V
iV
∂
∂
∂
∂
∂
∂
∂
∂
( )
( )
( )
( )
+
Φ
+
Φ
+
−
Φ
−
ΦΦ
y
tzyx
Tzyxg
yx
tzyx
Tzyxg
x
D
iV
V
iV
VV
∂
∂
∂
∂
∂
∂
∂
∂ ,,,
,,,
,,,
,,,
11
0
( )
( )
Φ
+
−
Φ
z
tzyx
Tzyxg
z
iV
V
∂
∂
∂
∂ ,,,
,,,
1
, i≥1;
( )
0
,,,
0
=
∂
Φ∂
=x
i
x
tzyxρ
,
( )
0
,,,
=
∂
Φ∂
= xLx
i
x
tzyxρ
,
( )
0
,,,
0
=
∂
Φ∂
=y
i
y
tzyxρ
,
( )
0
,,,
=
∂
Φ∂
= yLy
i
y
tzyxρ
,
( )
0
,,,
0
=
∂
Φ∂
=z
i
z
tzyxρ
,
( )
0
,,,
=
∂
Φ∂
= zLz
i
z
tzyxρ
, i≥0; Φρ0(x,y,z,0)=fΦρ (x,y,z), Φρi(x,y,z,0)=0, i≥1.
Solutions of the above equations could be written as
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑+∑+=Φ
∞
=
∞
=
ΦΦ
11
0
221
,,,
n
nnn
n
nnnnn
zyxzyx
zcycxcn
L
tezcycxcF
LLLLLL
tzyx ρρρ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∫ ∫ ∫ ∫ −−× ΦΦ
t L L L
IIInnnn
x y z
wvuITwvukwvuITwvukvcucete
0 0 0 0
2
, ,,,,,,,,,,,, τττρρ
( ) τdudvdwdwcn× ,
where ( ) ( ) ( ) ( )∫ ∫ ∫= ΦΦ
x y zL L L
nnnn udvdwdwvufwcvcucF
0 0 0
,,ρρ
, ( ) ( )[ ]222
0
22
exp −−−
ΦΦ ++−= zyxn LLLtDnte ρρ
π ,
cn(x)=cos(πnx/Lx);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=Φ
∞
=
ΦΦΦ
1 0 0 0 0
2
,,,
2
,,,
n
t L L L
nnnnnnn
zyx
i
x y z
Twvugvcusetezcycxcn
LLL
tzyx ρρρ
τ
π
ρ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−
Φ
×
∞
=
ΦΦ
−
1 0 0
2
1 2,,,
n
t L
nnnnnn
zyx
iI
n
x
ucetezcycxcn
LLL
dudvdwd
u
wvu
wc τ
π
τ
∂
τ∂
ρρ
ρ
( ) ( ) ( )
( )
( ) ( ) ( ) ×∑−∫ ∫
Φ
×
∞
=
−
Φ
1
2
0 0
1 2,,,
,,,
n
nnn
zyx
L L
iI
nn zcycxcn
LLL
dudvdwd
v
wvu
Twvugwcvs
y z π
τ
∂
τ∂ ρ
ρ
( ) ( ) ( ) ( ) ( )
( )
( )∫ ∫ ∫ ∫
Φ
−× Φ
−
ΦΦ
t L L L
iI
nnnnn
x y z
dudvdwdTwvug
w
wvu
wsvcucete
0 0 0 0
1
,,,
,,,
τ
∂
τ∂
τ ρ
ρ
ρρ
, i≥1,
where sn(x)=sin(πnx/Lx). Equations for calculation functions Cij(x,y,z,t) (i≥0, j≥0), boundary and
initial conditions take the form
( ) ( ) ( ) ( )
2
00
2
02
00
2
02
00
2
0
00 ,,,,,,,,,,,,
z
tzyxC
D
y
tzyxC
D
x
tzyxC
D
t
tzyxC
LLL
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
;
( ) ( ) ( ) ( )+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
0
2
02
0
2
02
0
2
0
0 ,,,,,,,,,,,,
z
tzyxC
D
y
tzyxC
D
x
tzyxC
D
t
tzyxC i
L
i
L
i
L
i](https://image.slidesharecdn.com/5416ijoe01-161104115443/85/ANALYSIS-OF-MANUFACTURING-OF-VOLTAGE-RESTORE-TO-INCREASE-DENSITY-OF-ELEMENTS-OF-THE-CIRCUIT-15-320.jpg)
![International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016
17
( )
0
,,,
0
=
=z
ij
z
tzyxC
∂
∂
,
( )
0
,,,
=
= zLz
ij
z
tzyxC
∂
∂
, i≥0, j≥0; C00(x,y,z,0)=fC (x,y,z), Cij(x,y,z,0)=0,
i≥1, j≥1.
Solutions of the above equations with account boundary and initial conditions could be written as
( ) ( ) ( ) ( ) ( )∑+=
∞
=1
00
21
,,,
n
nCnnnnC
zyxzyx
tezcycxcF
LLLLLL
tzyxC ,
where ( ) ( )[ ]222
0
22
exp −−−
++−= zyxCnC LLLtDnte π , ( ) ( ) ( ) ( )∫ ∫ ∫=
x y zL L L
nCnnnC udvdwdwcwvufvcucF
0 0 0
,, ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
20 ,,,
2
,,,
n
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ANALYSIS OF MANUFACTURING OF VOLTAGE RESTORE TO INCREASE DENSITY OF ELEMENTS OF THE CIRCUIT
- 1. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 DOI: 10.5121/ijoe.2016.5401 1 ANALYSIS OF MANUFACTURING OF VOLTAGE RESTORE TO INCREASE DENSITY OF ELEMENTS OF THE CIRCUIT E.L. Pankratov, E.A. Bulaeva Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950, Russia ABSTRACT We introduce an approach for increasing density of voltage restore elements. The approach based on manufacturing of a heterostructure, which consist of a substrate and an epitaxial layer with special con- figuration. Several required sections of the layer should be doped by diffusion or ion implantation. After that dopants and/or radiation defects should be annealed. KEYWORDS Voltage restore; optimization of manufacturing; increasing of density of elements 1. INTRODUCTION Now several actual questions of solid state electronics could be formulated. One of them is in- creasing of density of elements of integrated circuits. The increasing of density leads to necessity of decreasing of dimensions of these elements. Recently are have been elaborated several ap- proaches for the above decreasing. One of them is manufacturing of solid state electronic devices in thin film heterostructures [1-4]. Other approach based on doping of required areas of a semi- conductor sample or a heterostructure. After the doping one should consider laser or microwave annealing of dopant and/or radiation defects [5-7]. Due to these types of annealing one can find inhomogenous distribution of temperature in doped area. In this situation one can find inhomo- geneity of doped structure and consequently to decreasing of dimensions of the considered ele- ments. An alternative approach of changing of properties of materials before or during doping is radiation processing [8,9]. In this paper we consider an approach to manufacture of a voltage restore. We illustrate the ap- proach by considering a heterostructure. The heterostructure consist of a substrate and an epitax- ial layer. We consider that several sections take a place framework the epitaxial layer. We con- sider doping these sections by diffusion or ion implantation. The doping gives a possibility to obtain p or n type of conductivity during process of manufacture of diodes and bipolar transistors so as it is shown on Fig. 1. After finishing the doping we consider annealing of dopant and/or radiation defects. Our main aim is analysis of redistribution of dopant and/or radiation defects during annealing. 2. METHOD OF SOLUTION We calculate distribution of concentration of dopant in space and time by solving the following boundary problem [10-12]
- 2. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 2 ( ) ( ) ( ) ( ) ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC ,,,,,,,,,,,, (1) ( ) 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). The function C(x,y,z,t) describes distribution of concentration of dopant in space and time; T Fig. 1. Structure of voltage restore. View from top describes the temperature of annealing; DС describes the dopant diffusion coeffi- cient. Dopant diffusion coefficient is different in different materials of heterostructure. The diffu- sion coefficient will be also changed with changing of temperature. We approximate dopant dif- fusion coefficient by the following function based on Refs. [13-15] . (3) Function DL (x,y,z,T) describes spatial and temperature dependences of diffusion coefficient; function P (x,y,z,T) describes limit of solubility of dopant; parameter γ is differ in different mate- rials and assume integer values; function V (x,y,z,t) describes distribution of concentration of va- cancies with equilibrium distribution V* . One can find detailed concentrational dependence of dopant diffusion coefficient in Ref. [13]. It should be noted, that using diffusive type of doping did not leads to radiation damage. In this situation ζ1=ζ2= 0. We determine spatio-temporal dis- tributions of concentrations of point radiation defects by solution the following system of equa- tions [14,15] n n p n p ( ) ( ) ( ) ( ) ( ) ( ) ++ += 2* 2 2*1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD LC ςςξ γ γ
- 3. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 3 ( ) ( ) ( ) ( ) ( ) ( ) ×− ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ 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; function I(x,y,z,t) describes distribution of concentration of radiation interstitials in space and time; functions Dρ(x,y,z,T) describe spatio-temperature dependences of the diffusion coefficients of point radiation defects; terms V2 (x,y,z,t) and I2 (x,y,z,t) describe generation divacancies and diinterstitials, respectively; function kI,V(x,y,z,T) describes spatio-temperature dependence of recombination parameter of point radiation defects; functions kI,I(x,y,z,T) and kV,V(x,y,z,T) describe spatio-temperature dependence of parameters of generation of simplest complexes of point radiation defects. We calculate distributions of concentrations of divacancies ΦV (x,y,z,t) and dinterstitials ΦI (x,y,z,t) in space and time by solving the following system of equations [14,15] ( ) ( ) ( ) ( ) ( ) + 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) ( ) ( ) ( ) ( ) ( ) + 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 ,
- 4. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 4 ( ) 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) Function DΦρ(x,y,z,T) describes spatio-temperature dependences of diffusion coefficients of the considered complexes of radiation defects; function kI(x,y,z,T) and kV (x,y,z,T) describes spatio- temperature dependences of parameters of decay of these complexes. We used recently elaborated approach [16,17] to calculate distributions of concentrations of point radiation defects in space and time. To use the approach we shall transform spatio- temperature dependences of diffusion coefficients to the following form: Dρ(x,y,z,T)=D0ρ [1+ερ gρ(x,y,z,T)]. Here D0ρ are the average values of diffusion coefficients, 0≤ερ<1, |gρ(x,y,z,T)|≤1, ρ =I,V. We used the same transformation of approximations oof recombination parameters of point defects and generation parameters of their complexes: kI,V(x,y,z,T)=k0I,V[1+εI,V gI,V(x,y,z,T)], kI,I(x,y,z,T)=k0I,I[1+ εI,I gI,I(x,y,z,T)] and kV,V (x,y,z,T)= k0V,V [1+εV,V gV,V(x,y,z,T)], where k0ρ1,ρ2 are the their average values, 0≤εI,V < 1, 0≤εI,I <1, 0≤εV,V<1, | gI,V(x,y,z,T)|≤1, | gI,I(x,y,z,T)|≤1, |gV,V(x,y,z,T)|≤1. Let us introduce the following dimensionless variables: χ = x/Lx, η = y /Ly,, φ= z/Lz, ( ) ( ) * ,,,,,, ~ ItzyxItzyxI = , ( ) ( ) * ,,,,,, ~ VtzyxVtzyxV = , VI DDkL 00,0 2 ρρρ =Ω , VIVI DDkL 00,0 2 =ω , 2 00 LtDD VI=ϑ . Now we can transform boundary problem Eqs.(4) and (5) to the following form ( ) ( )[ ] ( ) ( ) × ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ η ϑφηχ ηχ ϑφηχ φηχε χϑ ϑφηχ ,,, ~ ,,, ~ ,,,1 ,,, ~ 00 0 00 0 I DD DI Tg DD DI VI I II VI I ( )[ ]} ( )[ ] ( ) ( ) ×− ∂ ∂ + ∂ ∂ ++× ϑφηχ φ ϑφηχ φηχε φ φηχε ,,, ~,,, ~ ,,,1,,,1 00 0 I I Tg DD D Tg II VI I II ( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχεω ,,, ~ ,,,1,,, ~ ,,,1 2 ,,,, ITgVTg IIIIIVIVI +Ω−+× (8) ( ) ( )[ ] ( ) ( ) × ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ η ϑφηχ ηχ ϑφηχ φηχε χϑ ϑφηχ ,,, ~ ,,, ~ ,,,1 ,,, ~ 00 0 00 0 V DD DV Tg DD DV VI V VV VI V ( )[ ]} ( )[ ] ( ) ( ) ×− ∂ ∂ + ∂ ∂ ++× ϑφηχ φ ϑφηχ φηχε φ φηχε ,,, ~,,, ~ ,,,1,,,1 00 0 I V Tg DD D Tg VV VI V VV ( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχεω ,,, ~ ,,,1,,, ~ ,,,1 2 ,,,, VTgVTg VVVVIVIVI +Ω−+× ( ) 0 ,,,~ 0 = ∂ ∂ =χ χ ϑφηχρ , ( ) 0 ,,,~ 1 = ∂ ∂ =χ χ ϑφηχρ , ( ) 0 ,,,~ 0 = ∂ ∂ =η η ϑφηχρ , ( ) 0 ,,,~ 1 = ∂ ∂ =η η ϑφηχρ , ( ) 0 ,,,~ 0 = ∂ ∂ =φ φ ϑφηχρ , ( ) 0 ,,,~ 1 = ∂ ∂ =φ φ ϑφηχρ , ( ) ( ) * ,,, ,,,~ ρ ϑφηχ ϑφηχρ ρf = . (9)
- 5. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 5 We calculate solutions of boundary problem Eqs.(8) and (9) by using recently introduced ap- proach [16,17]. In this situation we used the following power series ( ) ( )∑ ∑ ∑Ω= ∞ = ∞ = ∞ =0 0 0 ,,,~,,,~ i j k ijk kji ϑφηχρωεϑφηχρ ρρ . (10) We substitute the series (10) into the boundary problem Eqs.(8) and (9). After that we obtain equations for initial-order approximations of point defects concentration ( )ϑφηχ ,,, ~ 000I and ( )ϑφηχ ,,, ~ 000V and corrections for them ( )ϑφηχ ,,, ~ ijkI and ( )ϑφηχ ,,, ~ ijkV , i≥1, j≥1, k≥1. The equa- tions are presented in the Appendix. We obtain solution of the above equations by the standard Fourier approach [18,19]. The solutions are presented in the Appendix. Farther we calculate distributions of concentrations of simplest complexes of point radiation de- fects in space and time. To determine the distributions we transform approximations of diffusion coefficients in the following form: DΦρ(x,y,z,T)=D0Φρ[1+εΦρgΦρ(x,y,z,T)]. Here D0Φρ describe av- erage values of diffusion coefficients. In this situation the Eqs.(6) could be written as ( ) ( )[ ] ( ) ( ) × Φ + Φ += Φ ΦΦΦΦ y tzyx y D x tzyx Tzyxg x D t tzyx I I I III I ∂ ∂ ∂ ∂ ∂ ∂ ε ∂ ∂ ∂ ∂ ,,,,,, ,,,1 ,,, 00 ( )[ ] ( )[ ] ( ) ( ) ×− Φ ++ +× ΦΦΦΦΦ Tzyxk z tzyx Tzyxg z DTzyxg I I IIIII ,,, ,,, ,,,1,,,1 0 ∂ ∂ ε ∂ ∂ ε ( ) ( ) ( )tzyxITzyxktzyxI II ,,,,,,,,, 2 ,+× ( ) ( )[ ] ( ) ( ) × Φ + Φ += Φ ΦΦΦΦ y tzyx y D x tzyx Tzyxg x D t tzyx V V V VVV V ∂ ∂ ∂ ∂ ∂ ∂ ε ∂ ∂ ∂ ∂ ,,,,,, ,,,1 ,,, 00 ( )[ ] ( )[ ] ( ) ( ) ×+ Φ ++ +× ΦΦΦΦΦ Tzyxk z tzyx Tzyxg z DTzyxg II V VVVVV ,,, ,,, ,,,1,,,1 ,0 ∂ ∂ ε ∂ ∂ ε ( ) ( ) ( )tzyxITzyxktzyxI I ,,,,,,,,,2 −× . Farther we determine solutions of above equations as the following power series ( ) ( )∑ Φ=Φ ∞ = Φ 0 ,,,,,, i i i tzyxtzyx ρρρ ε . (11) Now we substitute series (11) into Eqs.(6) and conditions for them. The substitution gives a ob- tain equations for initial-order approximations of concentrations of complexes of defects Φρ0(x,y,z,t) and corrections for them Φρi(x,y,z,t), i ≥1 and boundary and initial conditions for them. We present the equations and conditions in Appendix. We obtain solutions of the obtain equations by using standard Fourier approaches [18,19] and presented in the Appendix. We calculate distribution of concentration of dopant in space and time by using the same ap- proach, which was used for calculation distributions of concentrations of radiation defects in
- 6. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 6 space and time. To use the approach we transform approximation of dopant diffusion coefficient to the following form: DL(x,y,z,T)=D0L[1+εLgL(x,y,z,T)]. Here D0L is the average value of diffu- sion coefficient of dopant, 0≤εL< 1, |gL(x,y,z,T)|≤1. Now we calculate solution of Eq.(1) as the following power series ( ) ( )∑ ∑= ∞ = ∞ =0 1 ,,,,,, i j ij ji L tzyxCtzyxC ξε . Now we substitute the above series into boundary problem Eqs.(1) and (2) leads to equations for the initial-order approximation of concentration of dopant C00(x,y,z,t), corrections for them Cij(x,y,z,t) (i≥1, j≥1) and conditions for the above equations. The equations are presented in the Appendix. Solutions of the equations have been calculated by standard Fourier approaches [18,19]. The solutions are presented in the Appendix. We analyze distributions of concentrations of dopant and radiation defects in space and time ana- lytically by using the second-order approximations of considered series. Usually we obtain enough good qualitative analysis and quantitative results. We check all analytical results by comparison with results of numerical simulation. 3. DISCUSSION Now we analyzed distributions of concentrations of dopants in space and time by using calculat- ed in previous section relations. Figs. 2 show typical distributions of concentrations of dopants on coordinate near interface between materials of heterostructures in direction, which is perpen- dicular to the interface. We calculate these distributions for larger value of dopant diffusion coef- ficient in doped area in comparison of value of dopant diffusion coefficient in nearest areas. The- se distributions show that in the case sharpness of p-n-junctions increases. At the same time one can find increasing of homogeneity of distribution of dopant concentration. One can find both effects for single p-n-junctions and p-n-junctions framework their systems (transistors, thyristors). Fig.2a. Spatial distributions of infused dopant concentration in the considered heterostructure. The consid- ered direction perpendicular to the interface between epitaxial layer substrate. Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves
- 7. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 7 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 Fig.2b. Spatial distributions of infused dopant concentration in the considered heterostructure. Curves 1 and 3 corresponds to annealing 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 the considered heterostructure. Difference between values of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves Now we consider optimization of annealing of dopant and/or radiation defects. To do the optimi- zation we used recently introduced criterion [16,17,20,21]. Framework the criterion we approxi- mate real distribution by step-wise function ψ (x,y,z). We calculate optimal annealing time by minimization the following mean-squared error ( ) ( )[ ]∫ ∫ ∫ −Θ= x y zL L L zyx xdydzdzyxzyxC LLL U 0 0 0 ,,,,, 1 ψ . (12) Optimal annealing time as functions of parameters are presented on Figs. 3. It should be noted, that in the ideal case after finishing of annealing of radiation defects dopant achieves interface between layers of heterostructure. If the dopant do not achieves the interface, it is practicably to use additional annealing of the dopant. The Fig. 3b shows just the dependences of optimal values of additional annealing time. In this situation optimal annealing of implanted dopant is smaller in comparison with optimal annealing time of infused dopant. 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 Fig.3a. 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
- 8. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 8 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 Fig.3b. 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 We introduce an approach to increase density of elements of an voltage restore. Several recom- mendations for optimization technological processes were formulated. 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 and educational fellowship for scientific research of Government of Nizhny Novgorod region of Russia. REFERENCES [1] G. Volovich. Modern chips UM3Ch class D manufactured by firm MPS. Modern Electronics. Issue 2. P. 10-17 (2006). [2] A. Kerentsev, V. Lanin. Constructive-technological features of MOSFET-transistors. Power Elec- tronics. Issue 1. P. 34 (2008). [3] 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. Semiconductors. Vol. 43 (7). P. 897-903 (2009). [4] N.I. Volokobinskaya, I.N. Komarov, T.V. Matioukhina, V.I. Rechetniko, A.A. Rush, I.V. Falina, A.S. Yastrebov. Au–TiBx-n-6H-SiC Schottky barrier diodes: the features of current flow in rectify- ing and nonrectifying contacts. Semiconductors. Vol. 35 (8). P. 1013-1017 (2001). [5] K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. Dopant distribution in the recrystallization transient at the maximum melt depth induced by laser annealing. Appl. Phys. Lett. 89 (17), 172111-172114 (2006). [6] H.T. Wang, L.S. Tan, E. F. Chor. Pulsed laser annealing of Be-implanted GaN. J. Appl. Phys. 98 (9), 094901-094905 (2006). [7] Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov, Yu.N. Drozdov, V.D. Skupov. Diffusion processes in semiconductor structures during microwave anneal- ing. Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836-843 (2003). [8] V.V. Kozlivsky. Modification of semiconductors by proton beams (Nauka, Sant-Peterburg, 2003, in Russian).
- 9. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 9 [9] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev, 1979, in Russian). [10] V.G. Gusev, Yu.M. Gusev. Electronics. ("Higher school", Moscow, 1991). [11] N.A. Avaev, Yu.E. Naumov, V.T. Frolkin. Basis of microelectronics (Radio and communication, Moscow, 1991). [12] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001). [13] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991). [14] P.M. Fahey, P.B. Griffin, J.D. Plummer. Rev. Point defects and dopant diffusion in silicon. Mod. Phys. 1989. V. 61. № 2. P. 289-388. [15] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev, 1979, in Russian). [16] E.L. Pankratov, E.A. Bulaeva. Increasing of sharpness of diffusion-junction heterorectifier by using radiation processing. Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1--1250028-8 (2012). [17] E.L. Pankratov, E.A. Bulaeva. Doping of materials during manufacture p-n-junctions and bipolar transistors. Analytical approaches to model technological approaches and ways of optimization of distributions of dopants. Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013). [18] A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations (Moscow, Nauka 1972) (in Russian). [19] H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids (Oxford University Press, 1964). [20] E.L. Pankratov, E.A. Bulaeva. An approach to decrease dimentions of logical elements based on bipolar transistor. Int. J. Comp. Sci. Appl. Vol. 5 (4). P. 1-18 (2015). [21] E.L. Pankratov, E.A. Bulaeva. On optimization of manufacturing of field-effect heterotransistors with several channels. Nano Science and Nano Technology: An Indian Journal. Vol. 9 (4). P. 43-60 (2015). Appendix Equations for the functions ( )ϑφηχ ,,, ~ ijkI and ( )ϑφηχ ,,, ~ ijkV , i ≥0, j ≥0, k ≥0 and conditions for them ( ) ( ) ( ) ( ) 2 000 2 0 0 2 000 2 0 0 2 000 2 0 0000 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ I D DI D DI D DI V I V I V I ( ) ( ) ( ) ( ) 2 000 2 0 0 2 000 2 0 0 2 000 2 0 0000 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ V D DV D DV D DV I V I V I V ( ) ( ) ( ) ( ) + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 00 2 2 00 2 2 00 2 0 000 ,,, ~ ,,, ~ ,,, ~ , ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑχ iii V Ii III D DI ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + −− η ϑφηχ φηχ ηχ ϑφηχ φηχ χ ,,, ~ ,,, ,,, ~ ,,, 100 0 0100 0 0 i I V Ii I V I I Tg D DI Tg D D ( ) ( ) ∂ ∂ ∂ ∂ + − φ ϑφηχ φηχ φ ,,, ~ ,,, 100 0 0 i I V I I Tg D D , i≥1, ( ) ( ) ( ) ( ) + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 00 2 2 00 2 2 00 2 0 000 ,,, ~ ,,, ~ ,,, ~ , ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑχ iii I Vi VVV D DV ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + −− η ϑφηχ φηχ ηχ ϑφηχ φηχ χ ,,, ~ ,,, ,,, ~ ,,, 100 0 0100 0 0 i V I Vi V I V V Tg D DV Tg D D
- 10. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 10 ( ) ( ) ∂ ∂ ∂ ∂ + − φ ϑφηχ φηχ φ ,,, ~ ,,, 100 0 0 i V I V V Tg D D , i≥1; ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 010 2 2 010 2 2 010 2 0 0010 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ III D DI V I ( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,, ~ ,,, ~ ,,,1 000000,, VITg VIVI+− ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 010 2 2 010 2 2 010 2 0 0010 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV I V ( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,, ~ ,,, ~ ,,,1 000000,, VITg VIVI+− ; ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 020 2 2 020 2 2 020 2 0 0020 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ III D DI V I ( )[ ] ( ) ( ) ( ) ( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε ,,, ~ ,,, ~ ,,, ~ ,,, ~ ,,,1 010000000010,, VIVITg VIVI ++− ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 020 2 2 020 2 2 020 2 0 0020 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV V I ( )[ ] ( ) ( ) ( ) ( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε ,,, ~ ,,, ~ ,,, ~ ,,, ~ ,,,1 010000000010,, VIVITg VIVI ++− ; ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 001 2 2 001 2 2 001 2 0 0001 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ III D DI V I ( )[ ] ( )ϑφηχφηχε ,,, ~ ,,,1 2 000,, ITg IIII+− ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 001 2 2 001 2 2 001 2 0 0001 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV I V ( )[ ] ( )ϑφηχφηχε ,,, ~ ,,,1 2 000,, VTg IIII+− ; ( ) ( ) ( ) ( ) + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 110 2 2 110 2 2 110 2 0 0110 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ III D DI V I ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + η ϑφηχ φηχ ηχ ϑφηχ φηχ χ ,,, ~ ,,, ,,, ~ ,,, 010010 0 0 I Tg I Tg D D II V I ( ) ( ) ( )[ ] ( ) ( )[ ++− ∂ ∂ ∂ ∂ + ϑφηχϑφηχφηχε φ ϑφηχ φηχ φ ,,, ~ ,,, ~ ,,,1 ,,, ~ ,,, 000100,, 010 VITg I Tg IIIII ( ) ( )]ϑφηχϑφηχ ,,, ~ ,,, ~ 100000 VI+ ( ) ( ) ( ) ( ) + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 110 2 2 110 2 2 110 2 0 0110 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV I V ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + η ϑφηχ φηχ ηχ ϑφηχ φηχ χ ,,, ~ ,,, ,,, ~ ,,, 010010 0 0 V Tg V Tg D D VV I V
- 11. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 11 ( ) ( ) ( )[ ] ( ) ( )[ ++− ∂ ∂ ∂ ∂ + ϑφηχϑφηχφηχε φ ϑφηχ φηχ φ ,,, ~ ,,, ~ ,,,1 ,,, ~ ,,, 000100,, 010 IVTg V Tg VVVVV ( ) ( )]ϑφηχϑφηχ ,,, ~ ,,, ~ 100000 IV+ ; ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 002 2 2 002 2 2 002 2 0 0002 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ III D DI V I ( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,, ~ ,,, ~ ,,,1 000001,, IITg IIII+− ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 002 2 2 002 2 2 002 2 0 0002 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV I V ( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,, ~ ,,, ~ ,,,1 000001,, VVЕg VVVV+− ; ( ) ( ) ( ) ( ) + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 101 2 2 101 2 2 101 2 0 0101 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ III D DI V I ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + η ϑφηχ φηχ ηχ ϑφηχ φηχ χ ,,, ~ ,,, ,,, ~ ,,, 001001 0 0 I Tg I Tg D D II V I ( ) ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχε φ ϑφηχ φηχ φ ,,, ~ ,,, ~ ,,,1 ,,, ~ ,,, 000100 001 VITg I Tg III +− ∂ ∂ ∂ ∂ + ( ) ( ) ( ) ( ) + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 101 2 2 101 2 2 101 2 0 0101 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV I V ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + η ϑφηχ φηχ ηχ ϑφηχ φηχ χ ,,, ~ ,,, ,,, ~ ,,, 001001 0 0 V Tg V Tg D D VV I V ( ) ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχε φ ϑφηχ φηχ φ ,,, ~ ,,, ~ ,,,1 ,,, ~ ,,, 100000 001 VITg V Tg VVV +− ∂ ∂ ∂ ∂ + ; ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 011 2 2 011 2 2 011 2 0 0011 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ III D DI V I ( )[ ] ( ) ( ) ( )[ ] ×+−+− TgIITg VIVIIIII ,,,1,,, ~ ,,, ~ ,,,1 ,,010000,, φηχεϑφηχϑφηχφηχε ( ) ( )ϑφηχϑφηχ ,,, ~ ,,, ~ 000001 VI× ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 011 2 2 011 2 2 011 2 0 0011 ,,, ~ ,,, ~ ,,, ~ ,,, ~ φ ϑφηχ η ϑφηχ χ ϑφηχ ϑ ϑφηχ VVV D DV I V ( )[ ] ( ) ( ) ( )[ ]
- 12. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 12 ( ) 0 ,,,~ 0 = ∂ ∂ =φ φ ϑφηχρijk , ( ) 0 ,,,~ 1 = ∂ ∂ =φ φ ϑφηχρijk (i≥0, j≥0, k≥0); ( ) ( ) * 000 ,,0,,,~ ρφηχφηχρ ρf= , ( ) 00,,,~ =φηχρijk (i≥1, j≥1, k≥1). Solutions of these equations with account boundary and initial conditions could be written as ( ) ( ) ( ) ( ) ( )∑+= ∞ =1 000 21 ,,,~ n nn ecccF LL ϑφηχϑφηχρ ρρ , where ( ) ( ) ( ) ( )∫ ∫ ∫= 1 0 1 0 1 0 * ,,coscoscos 1 udvdwdwvufwnvnunF nn ρρ πππ ρ , ( ) ( )IVnI DDne 00 22 exp ϑπϑ −= , ( ) ( )VInV DDne 00 22 exp ϑπϑ −= , cn(χ)=cos(πnχ); ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 1 0 1 0 1 00 0 00 ,,,2,,, ~ n InnnInIn V I i Twvugvcuseecccn D D I ϑ τϑφηχπϑφηχ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−− ∂ ∂ × ∞ = − 1 0 1 00 0100 2 ,,, ~ n nnInIn V Ii n uceecccn D D dudvdwd u wvuI wc ϑ τϑφηχτ τ ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑−∫ ∫ ∂ ∂ × ∞ = − 1 0 0 1 0 1 0 100 2 ,,, ~ ,,, n n V Ii Inn cccn D D dudvdwd v wvuI Twvugwcvs φηχπτ τ π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫ ∂ ∂ −× − ϑ τ τ τϑ 0 1 0 1 0 1 0 100 ,,, ~ ,,, dudvdwd w wvuI Twvugwsvcucee i InnnnInI , i≥1, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫ ∫−−= ∞ =1 0 1 0 1 0 1 00 0 00 ,,,2,,, ~ n VnnnInVn I V i Twvugvcuseecccn D D V ϑ τϑφηχπϑφηχ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫−− ∂ ∂ × ∞ = − 1 0 1 0 1 00 0100 , ~ n nnnInVn I Vi n vsuceecccn D D dudvdwd u uV wc ϑ τϑφηχτ τ ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫ ∂ ∂ × ∞ = − 1 0 0 1 0 100 2 , ~ ,,,2 n nVn I Vi Vn ecccn D D dudvdwd v uV Twvugwc ϑφηχπτ τ π ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫ ∂ ∂ −× − ϑ τ τ τ 0 1 0 1 0 1 0 100 , ~ ,,, dudvdwd w uV Twvugwsvcuce i VnnnnI , i≥1, where sn(χ)=sin (π nχ); ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−= ∞ =1 0 1 0 1 0 1 0 010 2,,,~ n nnnnnnnn wcvcuceeccc ϑ ρρ τϑφηχϑφηχρ ( )[ ] ( ) ( ) τττε dudvdwdwvuVwvuITwvug VIVI ,,, ~ ,,, ~ ,,,1 000000,,+× ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 1 0 010 1 0 1 00 0 020 ,,, ~ 2,,,~ n nnnnnnnn V I wvuIwcvcuceeccc D D ϑ ρρ ττϑφηχϑφηχρ ( ) ( ) ( )] ( )[ ] τετττ dudvdwdTwvugwvuVwvuIwvuV VIVI ,,,1,,, ~ ,,, ~ ,,, ~ ,,010000000 ++× ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 1 0 1 0 1 0 2 000001 ,,,~2,,,~ n nnnnnnn wvuvcuceeccc ϑ ρρ τρτϑφηχϑφηχρ
- 13. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 13 ( )[ ] ( ) τε ρρρρ dudvdwdwcTwvug n,,,1 ,,+× ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ×∫ ∫−−= ∞ =1 0 1 0 1 0 1 0 000002 ,,,~2,,,~ n nnnnnnnn wvuwcvcuceeccc ϑ ρρ τρτϑφηχϑφηχρ ( )[ ] ( ) ττρε ρρρρ dudvdwdwvuTwvug ,,,~,,,1 001,,+× ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 1 0 1 0 1 00 0 110 2,,, ~ n nnnnInInnn V I ucvcuseecccn D D I ϑ τϑφηχπϑφηχ ( ) ( ) ( ) ( ) ( ) ( ) ×∑− ∂ ∂ × ∞ = − 1 0 0100 2 ,,, ~ ,,, n nInnn V Ii I ecccn D D dudvdwd u wvuI Twvug ϑφηχπτ τ ( ) ( ) ( ) ( ) ( ) ( ) ×−∫ ∫ ∫ ∫ ∂ ∂ −× − V Ii InnnnI D D dudvdwd v wvuI Twvugucvsuce 0 0 0 1 0 1 0 1 0 100 2 ,,, ~ ,,, πτ τ τ ϑ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ ∂ ∂ −× ∞ = − 1 0 1 0 1 0 1 0 100 ,,, ~ ,,, n i InnnnInI dudvdwd w wvuI Twvugusvcuceen ϑ τ τ τϑ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[∑ ∫ ∫ ∫ ∫ ×+−−× ∞ =1 0 1 0 1 0 1 0 ,12 n VInnnnInnnInnnn vcvcuceccecccc ϑ ετφηϑχφηχ ( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvug VI ,,, ~ ,,, ~ ,,, ~ ,,, ~ ,,, 100000000100, +× ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 1 0 1 0 1 00 0 110 2,,, ~ n nnnnVnVnnn I V ucvcuseecccn D D V ϑ τϑφηχπϑφηχ ( ) ( ) ( ) ( ) ( ) ( )×∑− ∂ ∂ × ∞ = − 1 0 0100 2 ,,, ~ ,,, n nVnnn I Vi V ecccn D D dudvdwd u wvuV Twvug ϑφηχπτ τ ( ) ( ) ( ) ( ) ( ) ( ) ×−∫ ∫ ∫ ∫ ∂ ∂ −× − I Vi VnnnnV D D dudvdwd v wvuV Twvugucvsuce 0 0 0 1 0 1 0 1 0 100 2 ,,, ~ ,,, πτ τ τ ϑ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ ∂ ∂ −× ∞ = − 1 0 1 0 1 0 1 0 100 ,,, ~ ,,, n i VnnnnVnV dudvdwd w wvuV Twvugusvcuceen ϑ τ τ τϑ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[∑ ∫ ∫ ∫ ∫ ×+−−× ∞ =1 0 1 0 1 0 1 0 ,12 n VInnnnVnnnInnnn vcvcuceccecccc ϑ ετφηϑχφηχ ( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvug VI ,,, ~ ,,, ~ ,,, ~ ,,, ~ ,,, 100000000100, +× ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 1 0 1 0 1 00 0 101 ,,,2,,, ~ n InnnInInnn V I Twvugvcuseecccn D D I ϑ τϑφηχπϑφηχ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−− ∂ ∂ × ∞ =1 0 1 00 0001 2 ,,, ~ n nnInInnn V I n uceecccn D D dudvdwd u wvuI wc ϑ τϑφηχπτ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−∫ ∫ ∂ ∂ × ∞ =1 00 0 1 0 1 0 001 2 ,,, ~ ,,, n nInIn V I Inn eec D D dudvdwd v wvuI Twvugwcvs ϑ τϑχπτ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫ ∂ ∂ × ∞ =1 1 0 1 0 1 0 001 2 ,,, ~ ,,, n nnnnInnn ccccdudvdwd w wvuI Twvugwsvcucn ηχφητ τ
- 14. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 14 ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ +−× ϑ τττετϑφ 0 1 0 000100,, ,,, ~ ,,, ~ ,,,1 dudvdwdwvuVwvuITwvuguceec VIVInnInIn ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 1 0 1 0 1 00 0 101 ,,,2,,, ~ n VnnnVnVnnn I V Twvugvcuseecccn D D V ϑ τϑφηχπϑφηχ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−− ∂ ∂ × ∞ =1 0 1 00 0001 2 ,,, ~ n nnVnVnnn I V n uceecccn D D dudvdwd u wvuV wc ϑ τϑφηχπτ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−∫ ∫ ∂ ∂ × ∞ =1 00 0 1 0 1 0 001 2 ,,, ~ ,,, n nVnVn I V Vnn eec D D dudvdwd v wvuV Twvugwcvs ϑ τϑχπτ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫ ∂ ∂ × ∞ =1 1 0 1 0 1 0 001 2 ,,, ~ ,,, n nnnnVnnn ccccdudvdwd w wvuV Twvugwsvcucn ηχφητ τ ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ +−× ϑ τττετϑφ 0 1 0 100000,, ,,, ~ ,,, ~ ,,,1 dudvdwdwvuVwvuITwvuguceec VIVInnVnVn ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]{∑ ∫ ∫ ∫ ∫ ×+−−= ∞ =1 0 1 0 1 0 1 0 ,,011 ,,,12,,, ~ n IIIInnnnInInnn TwvugwcvcuceecccI ϑ ετϑφηχϑφηχ ( ) ( ) ( )[ ] ( ) ( )} τττεττ dudvdwdwvuVwvuITwvugwvuIwvuI VIVI ,,, ~ ,,, ~ ,,,1,,, ~ ,,, ~ 000001,,010000 ++× ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]{∑ ∫ ∫ ∫ ∫ ×+−−= ∞ =1 0 1 0 1 0 1 0 ,,011 ,,,12,,, ~ n VVVVnnnnVnVnnn TwvugwcvcuceecccV ϑ ετϑφηχϑφηχ ( ) ( ) ( )[ ] ( ) ( )} τττεττ dudvdwdwvuVwvuITwvugwvuVwvuV VIVI ,,, ~ ,,, ~ ,,,1,,, ~ ,,, ~ 001000,,010000 ++× . Equations for initial-order approximations of distributions of concentrations of simplest com- plexes of radiation defects Φρ0(x,y,z,t) and corrections for them Φρi(x,y, z,t), i ≥1 and boundary and initial conditions for them takes the form ( ) ( ) ( ) ( ) + Φ + Φ + Φ = Φ Φ 2 0 2 2 0 2 2 0 2 0 0 ,,,,,,,,,,,, z tzyx y tzyx x tzyx D t tzyx III I I ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxk III ,,,,,,,,,,,, 2 , −+ ( ) ( ) ( ) ( ) + Φ + Φ + Φ = Φ Φ 2 0 2 2 0 2 2 0 2 0 0 ,,,,,,,,,,,, z tzyx y tzyx x tzyx D t tzyx VVV V V ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxk VVV ,,,,,,,,,,,, 2 , −+ ; ( ) ( ) ( ) ( ) + Φ + Φ + Φ = Φ Φ 2 2 2 2 2 2 0 ,,,,,,,,,,,, z tzyx y tzyx x tzyx D t tzyx iIiIiI I iI ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) + Φ + Φ + − Φ − ΦΦ y tzyx Tzyxg yx tzyx Tzyxg x D iI I iI II ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, 11 0 ( ) ( ) Φ + − Φ z tzyx Tzyxg z iI I ∂ ∂ ∂ ∂ ,,, ,,, 1 , i≥1,
- 15. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 15 ( ) ( ) ( ) ( ) + Φ + Φ + Φ = Φ Φ 2 2 2 2 2 2 0 ,,,,,,,,,,,, z tzyx y tzyx x tzyx D t tzyx iViViV V iV ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) + Φ + Φ + − Φ − ΦΦ y tzyx Tzyxg yx tzyx Tzyxg x D iV V iV VV ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, 11 0 ( ) ( ) Φ + − Φ z tzyx Tzyxg z iV V ∂ ∂ ∂ ∂ ,,, ,,, 1 , i≥1; ( ) 0 ,,, 0 = ∂ Φ∂ =x i x tzyxρ , ( ) 0 ,,, = ∂ Φ∂ = xLx i x tzyxρ , ( ) 0 ,,, 0 = ∂ Φ∂ =y i y tzyxρ , ( ) 0 ,,, = ∂ Φ∂ = yLy i y tzyxρ , ( ) 0 ,,, 0 = ∂ Φ∂ =z i z tzyxρ , ( ) 0 ,,, = ∂ Φ∂ = zLz i z tzyxρ , i≥0; Φρ0(x,y,z,0)=fΦρ (x,y,z), Φρi(x,y,z,0)=0, i≥1. Solutions of the above equations could be written as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑+∑+=Φ ∞ = ∞ = ΦΦ 11 0 221 ,,, n nnn n nnnnn zyxzyx zcycxcn L tezcycxcF LLLLLL tzyx ρρρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∫ ∫ ∫ ∫ −−× ΦΦ t L L L IIInnnn x y z wvuITwvukwvuITwvukvcucete 0 0 0 0 2 , ,,,,,,,,,,,, τττρρ ( ) τdudvdwdwcn× , where ( ) ( ) ( ) ( )∫ ∫ ∫= ΦΦ x y zL L L nnnn udvdwdwvufwcvcucF 0 0 0 ,,ρρ , ( ) ( )[ ]222 0 22 exp −−− ΦΦ ++−= zyxn LLLtDnte ρρ π , cn(x)=cos(πnx/Lx); ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=Φ ∞ = ΦΦΦ 1 0 0 0 0 2 ,,, 2 ,,, n t L L L nnnnnnn zyx i x y z Twvugvcusetezcycxcn LLL tzyx ρρρ τ π ρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−− Φ × ∞ = ΦΦ − 1 0 0 2 1 2,,, n t L nnnnnn zyx iI n x ucetezcycxcn LLL dudvdwd u wvu wc τ π τ ∂ τ∂ ρρ ρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ Φ × ∞ = − Φ 1 2 0 0 1 2,,, ,,, n nnn zyx L L iI nn zcycxcn LLL dudvdwd v wvu Twvugwcvs y z π τ ∂ τ∂ ρ ρ ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ Φ −× Φ − ΦΦ t L L L iI nnnnn x y z dudvdwdTwvug w wvu wsvcucete 0 0 0 0 1 ,,, ,,, τ ∂ τ∂ τ ρ ρ ρρ , i≥1, where sn(x)=sin(πnx/Lx). Equations for calculation functions Cij(x,y,z,t) (i≥0, j≥0), boundary and initial conditions take the form ( ) ( ) ( ) ( ) 2 00 2 02 00 2 02 00 2 0 00 ,,,,,,,,,,,, z tzyxC D y tzyxC D x tzyxC D t tzyxC LLL ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ; ( ) ( ) ( ) ( )+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 0 2 02 0 2 02 0 2 0 0 ,,,,,,,,,,,, z tzyxC D y tzyxC D x tzyxC D t tzyxC i L i L i L i
- 16. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 16 ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + −− y tzyxC Tzyxg y D x tzyxC Tzyxg x D i LL i LL ,,, ,,, ,,, ,,, 10 0 10 0 ( ) ( ) ∂ ∂ ∂ ∂ + − z tzyxC Tzyxg z D i LL ,,, ,,, 10 0 , i≥1; ( ) ( ) ( ) ( )+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 01 2 02 01 2 02 01 2 0 01 ,,,,,,,,,,,, z tzyxC D y tzyxC D x tzyxC D t tzyxC LLL ( ) ( ) ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + y tzyxC TzyxP tzyxC y D x tzyxC TzyxP tzyxC x D LL ,,, ,,, ,,,,,, ,,, ,,, 0000 0 0000 0 γ γ γ γ ( ) ( ) ( ) ∂ ∂ ∂ ∂ + z tzyxC TzyxP tzyxC z D L ,,, ,,, ,,, 0000 0 γ γ ; ( ) ( ) ( ) ( ) + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 02 2 02 02 2 02 02 2 0 02 ,,,,,,,,,,,, z tzyxC D y tzyxC D x tzyxC D t tzyxC LLL ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ ∂ ∂ + −− TzyxP tzyxC tzyxC yx tzyxC TzyxP tzyxC tzyxC x D L ,,, ,,, ,,, ,,, ,,, ,,, ,,, 1 00 01 00 1 00 010 γ γ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ × − TzyxP tzyxC x D z tzyxC TzyxP tzyxC tzyxC zy tzyxC L ,,, ,,,,,, ,,, ,,, ,,, ,,, 00 0 00 1 00 01 00 γ γ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ × z tzyxC TzyxP tzyxC zy tzyxC TzyxP tzyxC yx tzyxC ,,, ,,, ,,,,,, ,,, ,,,,,, 0100010001 γ γ γ γ ; ( ) ( ) ( ) ( )+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 11 2 02 11 2 02 11 2 0 11 ,,,,,,,,,,,, z tzyxC D y tzyxC D x tzyxC D t tzyxC LLL ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ ∂ ∂ + −− TzyxP tzyxC tzyxC yx tzyxC TzyxP tzyxC tzyxC x ,,, ,,, ,,, ,,, ,,, ,,, ,,, 1 00 10 00 1 00 10 γ γ γ γ ( ) ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ × − LD z tzyxC TzyxP tzyxC tzyxC zy tzyxC 0 00 1 00 10 00 ,,, ,,, ,,, ,,, ,,, γ γ ( ) ( ) ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + y tzyxC TzyxP tzyxC yx tzyxC TzyxP tzyxC x D L ,,, ,,, ,,,,,, ,,, ,,, 10001000 0 γ γ γ γ ( ) ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + x tzyxC Tzyxg x D z tzyxC TzyxP tzyxC z LL ,,, ,,, ,,, ,,, ,,, 01 0 1000 γ γ ( ) ( ) ( ) ( ) ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + z tzyxC Tzyxg zy tzyxC Tzyxg y LL ,,, ,,, ,,, ,,, 0101 ; ( ) 0 ,,, 0 = =x ij x tzyxC ∂ ∂ , ( ) 0 ,,, = = xLx ij x tzyxC ∂ ∂ , ( ) 0 ,,, 0 = =y ij y tzyxC ∂ ∂ , ( ) 0 ,,, = = yLy ij y tzyxC ∂ ∂ ,
- 17. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 17 ( ) 0 ,,, 0 = =z ij z tzyxC ∂ ∂ , ( ) 0 ,,, = = zLz ij z tzyxC ∂ ∂ , i≥0, j≥0; C00(x,y,z,0)=fC (x,y,z), Cij(x,y,z,0)=0, i≥1, j≥1. Solutions of the above equations with account boundary and initial conditions could be written as ( ) ( ) ( ) ( ) ( )∑+= ∞ =1 00 21 ,,, n nCnnnnC zyxzyx tezcycxcF LLLLLL tzyxC , where ( ) ( )[ ]222 0 22 exp −−− ++−= zyxCnC LLLtDnte π , ( ) ( ) ( ) ( )∫ ∫ ∫= x y zL L L nCnnnC udvdwdwcwvufvcucF 0 0 0 ,, ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 0 0 0 20 ,,, 2 ,,, n t L L L LnnnnCnCnnnnC zyx i x y z TwvugvcvcusetezcycxcFn LLL tzyxC τ π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−− ∂ ∂ × ∞ = − 1 0 0 0 2 10 2,,, n t L L nnnCnCnnnnC zyx i x y vsucetezcycxcFn LLL dudvdwd u wvuC τ π τ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∂ ∂ × ∞ = − 1 2 0 10 2,,, ,,, n nCnnnnC zyx L i Ln tezcycxcFn LLL dudvdwd v wvuC Twvugvc z π τ τ ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫ ∂ ∂ −× − t L L L i LnnnnC x y z dudvdwd w wvuC Twvugvsvcuce 0 0 0 0 10 ,,, ,,, τ τ τ , i≥1; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 0 0 0 201 2 ,,, n t L L L nnnnCnCnnnnC zyx x y z wcvcusetezcycxcFn LLL tzyxC τ π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ −− ∂ ∂ × ∞ =1 0 2 0000 2,,, ,,, ,,, n t nCnCnnnnC zyx etezcycxcFn LLL dudvdwd u wvuC TwvuP wvuC τ π τ ττ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫ ∂ ∂ × ∞ =1 2 0 0 0 0000 2,,, ,,, ,,, n nC zyx L L L nnn ten LLL dudvdwd v wvuC TwvuP wvuC wcvsuc x y z π τ ττ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫ ∂ ∂ −× t L L L nnnnCnnnnC x y z dudvdwd w wvuC TwvuP wvuC wsvcucezcycxcF 0 0 0 0 0000 ,,, ,,, ,,, τ ττ τ γ γ ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 0 0 0 202 2 ,,, n t L L L nnnnCnCnnnnC zyx x y z wcvcusetezcycxcFn LLL tzyxC τ π ( ) ( ) ( ) ( ) ( ) ( )×∑− ∂ ∂ × ∞ = − 1 2 00 1 00 01 2,,, ,,, ,,, ,,, n nnnC zyx ycxcF LLL dudvdwd u wvuC TwvuP wvuC wvuC π τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ ∂ ∂ −× −t L L L nnnCnCn x y z v wvuC TwvuP wvuC wvuCvsucetezcn 0 0 0 0 00 1 00 01 ,,, ,,, ,,, ,,, ττ ττ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−−× ∞ =1 0 0 0 2 2 n t L L nnnCnCnnnnC zyx n x y vcucetezcycxcFn LLL dudvdwdwc τ π τ ( ) ( ) ( ) ( ) ( ) ( )×∑−∫ ∂ ∂ × ∞ = − 1 2 0 00 1 00 01 2,,, ,,, ,,, ,,, n n zyx L n xcn LLL dudvdwd w wvuC TwvuP wvuC wvuCws z π τ ττ τ γ γ
- 18. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 18 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ ∂ ∂ −× t L L L nnnnCnCnnnC x y z u wvuC wvuCwcvcusetezcycF 0 0 0 0 00 01 ,,, ,,, τ ττ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−× ∞ = − 1 0 0 2 1 00 2 ,,, ,,, n t L nnCnCnnnnC zyx x ucetezcycxcFn LLL dudvdwd TwvuP wvuC τ π τ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∂ ∂ × ∞ = − 1 2 0 0 00 1 00 01 2,,, ,,, ,,, ,,, n zyx L L nn n LLL dudvdwd v wvuC TwvuP wvuC wvuCwcvs y z π τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫−× −t L L L nnnnCnCnnnnC x y z TwvuP wvuC wvuCwsvcucetezcycxcF 0 0 0 0 1 00 01 ,,, ,,, ,,, γ γ τ ττ ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−− ∂ ∂ × ∞ =1 0 0 2 00 2,,, n t L nnCnCnnnnC zyx x usetezcycxcF LLL dudvdwd w wvuC τ π τ τ ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫ ∫ ∂ ∂ × ∞ =1 2 0 0 0100 2,,, ,,, ,,, n nCn zyx L L nn texc LLL dudvdwd u wvuC TwvuP wvuC wcvcn y z π τ ττ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ ∂ ∂ −× t L L L nnnnCnnC x y z dudvdwd v wvuC TwvuP wvuC wcvsuceycF 0 0 0 0 0100 ,,, ,,, ,,, τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−× ∞ =1 0 0 0 0 2 2 n t L L L nnnnCnCnnnnC zyx n x y z wsvcucetezcycxcFn LLL zcn τ π ( ) ( ) ( ) τ ττ γ γ dudvdwd w wvuC TwvuP wvuC ∂ ∂ × ,,, ,,, ,,, 0100 ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−= ∞ =1 0 0 0 0 211 2 ,,, n t L L L nnnnCnCnnnnC zyx x y z wcvcusetezcycxcFn LLL tzyxC τ π ( ) ( ) ( ) ( ) ( ) ( ) ×∑− ∂ ∂ × ∞ =1 2 01 2,,, ,,, n nCnnnnC zyx L tezcycxcFn LLL dudvdwd u wvuC Twvug π τ τ ( ) ( ) ( ) ( ) ( ) ( ) ×−∫ ∫ ∫ ∫ ∂ ∂ −× 2 0 0 0 0 01 2,,, ,,, zyx t L L L LnnnnC LLL dudvdwd v wvuC Twvugwcvsuce x y z π τ τ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ ∂ ∂ −× ∞ =1 0 0 0 0 01 ,,, ,,, n t L L L LnnnnCnC x y z dudvdwd w wvuC Twvugwsvcuceten τ τ τ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−× ∞ =1 0 0 0 2 2 n t L L nnnCnCnnnnC zyx nnnnC x y vcusetezcycxcF LLL zcycxcF τ π ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∂ ∂ × ∞ =1 2 0 1000 2,,, ,,, ,,, n nnnC zyx L n ycxcFn LLL dudvdwd u wvuC TwvuP wvuC wcn z π τ ττ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) −∫ ∫ ∫ ∫ ∂ ∂ −× t L L L nnnnCnCn x y z dudvdwd v wvuC TwvuP wvuC wcvsucetezc 0 0 0 0 1000 ,,, ,,, ,,, τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−− ∞ =1 0 0 0 0 00 2 ,,, ,,,2 n t L L L nnnnCnCnnnnC zyx x y z TwvuP wvuC wsvcucetezcycxcFn LLL γ γ τ τ π ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−− ∂ ∂ × ∞ =1 0 0 2 10 2,,, n t L nnCnCnnnnC zyx x usetezcycxcFn LLL dudvdwd w wvuC τ π τ τ
- 19. International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016 19 ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∂ ∂ × ∞ = − 1 2 0 0 00 1 00 10 2,,, ,,, ,,, ,,, n zyx L L nn n LLL dudvdwd u wvuC TwvuP wvuC wvuCwcvc y z π τ ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫ ∂ ∂ −× −t L L L nnnnCnCnnnnC x y z v wvuC TwvuP wvuC wcvsucetezcycxcF 0 0 0 0 00 1 00 ,,, ,,, ,,, ττ τ γ γ ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−× ∞ =1 0 0 210 2 ,,, n t L nnCnCnnnnC zyx x ucetezcycxcFn LLL dudvdwdwvuC τ π ττ ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∂ ∂ × −y z L L nn dudvdwd w wvuC TwvuP wvuC wvuCwsvc 0 0 00 1 00 10 ,,, ,,, ,,, ,,, τ ττ τ γ γ . 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 Associ- ate 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 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.