An Approach to Optimize Regimes of Manufacturing of Complementary Horizontal Field-Effect Transistor

International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
DOI : 10.14810/ijrap.2014.3205 55
AN APPROACH TO OPTIMIZE REGIMES OF
MANUFACTURING OF COMPLEMENTARY
HORIZONTAL FIELD-EFFECT TRANSISTOR
E.L. Pankratov1
and E.A. Bulaeva 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 nonlinear model to describe manufacturing complementary horizontal field-effect
heterotransistor. Based on analytical solution of the considered boundary problems some recommendations
have been formulated to optimize technological processes.
KEYWORDS
Horizontal field-effect transistor, modelling of manufacturing of transistor, recommendations for
optimisation of manufacturing of transistor
1. INTRODUCTION
In the present time it is intensively increasing degree of integration of elements of integrated
circuits [1-8]. At the same time it is obtaining decreasing of dimensions of the elements. To
decrease dimensions of elements of integrated circuits it is traditionally using some approaches.
Two of them are laser and microwave types of annealing of dopants and/or radiation defects
during manufacturing p-n-junctions, field-effect and bipolar transistors, thyristors [9-15]. Another
way to increase degree of integration of elements of integrated circuits is using of inhomogeneity
of heterostructures on the basis of which integrated circuits are manufactured [13-19]. However
in this case it is practicably to optimize annealing of dopant and/or radiation defects. It is known,
that distribution of concentrations of dopants in elements of integrated circuits and their discrete
analogs will be changed under influence of radiation processing (for example, during ion
implantation) [20]. Because of this to decrease dimensions of elements of integrated circuits and
their discrete it is attracted an interest radiation processing of materials [21,22].
In this paper we consider manufacturing of complementary field-effect heterotransistor. Structure
of the heterotransistor is presented on the Fig. 1. The heterostructure consist of a substrate and
epitaxial layer. The epitaxial layer has several sections, which have been manufactured by using
another materials. Some dopants have been infused or implanted in the sections to manufacture
required types of conductivity (p or n). Farther we consider annealing of dopant (for doping by
diffusion) and/or radiation defects (during ion doping). Main aim of the present paper we
analyzed dynamics of redistribution of dopant and radiation defects to formulate conditions,
which correspond to manufacture more thin heterotransistor with smaller dimensions into another
dimensions.

56
Substrate
Drain Source
Source Drain
Gate Gate
p p n n
Fig.1. Heterostructure with a substrate and epitaxial layer with several sections
2. METHOD OF SOLUTION
To solve our aims we determine spatio-temporal distribution of concentration of dopant. We
determine the distributions by solving the second Fick’s law [1,3-5]
( ) ( ) ( ) ( )






+






+






=
z
t
z
y
x
C
D
z
y
t
z
y
x
C
D
y
x
t
z
y
x
C
D
x
t
t
z
y
x
C
C
C
C
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
(1)
with boundary and initial conditions
( ) 0
,
,
,
0
=
∂
∂
=
x
x
t
z
y
x
C
,
( ) 0
,
,
,
=
∂
∂
= x
L
x
x
t
z
y
x
C
,
( ) 0
,
,
,
0
=
∂
∂
=
y
y
t
z
y
x
C
,
( ) 0
,
,
,
=
∂
∂
= y
L
x
y
t
z
y
x
C
, (2)
( ) 0
,
,
,
0
=
∂
∂
=
z
z
t
z
y
x
C
,
( ) 0
,
,
,
=
∂
∂
= z
L
x
z
t
z
y
x
C
, C(x,y,z,0)=f(x,y,z).
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 depends
on properties of materials in layers of heterostructure, speed of heating and cooling of hetero-
structure (with account Arrhenius law). Dependences of dopant diffusion coefficient on
parameters could be approximated by the following relation [23-25]
( ) ( )
( )
( ) ( )
( ) 







+
+






+
= 2
*
2
2
*
1
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
V
t
z
y
x
V
V
t
z
y
x
V
T
z
y
x
P
t
z
y
x
C
T
z
y
x
D
D L
C ς
ς
ξ γ
γ
, (3)
where DL (x,y,z,T) is the spatial (due to inhomogeneity of heterostructure) and temperature (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] [23]; V(x,y,z,t) is the spatio-temporal distribution of concentration of vacancies;
V*
is the equilibrium distribution of concentration of vacancies. Concentrational depen-dence of
dopant diffusion coefficient has been discussed in details in the Ref. [23]. It should be noted that
doping of heterostructure by diffusion did not leads to generation of radiation damage and ζ1=ζ2=

57
0. Spatio-temporal distributions of concentrations of point radiation defects we determine by
solving the following system equations [24,25]
( ) ( ) ( ) ( ) ( ) +






∂
∂
∂
∂
+






∂
∂
∂
∂
=
∂
∂
y
t
z
y
x
I
T
z
y
x
D
y
x
t
z
y
x
I
T
z
y
x
D
x
t
t
z
y
x
I
I
I
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
(4)
( ) ( ) ( ) ( ) ( ) ( ) ( )
t
z
y
x
I
T
z
y
x
k
t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
I
T
z
y
x
D
z
I
I
V
I
I ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
,
, −
−






∂
∂
∂
∂
+
( ) ( ) ( ) ( ) ( ) +






∂
∂
∂
∂
+






∂
∂
∂
∂
=
∂
∂
y
t
z
y
x
V
T
z
y
x
D
y
x
t
z
y
x
V
T
z
y
x
D
x
t
t
z
y
x
V
V
V
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
( ) ( ) ( ) ( ) ( ) ( ) ( )
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
V
T
z
y
x
D
z
V
V
V
I
V ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
,
, −
−






∂
∂
∂
∂
+
with initial
ρ(x,y,z,0)=fρ (x,y,z) (5a)
and boundary conditions
( ) 0
,
,
,
0
=
∂
∂
=
x
x
t
z
y
x
ρ
,
( ) 0
,
,
,
=
∂
∂
= x
L
x
x
t
z
y
x
ρ
,
( ) 0
,
,
,
0
=
∂
∂
=
y
y
t
z
y
x
ρ
,
( ) 0
,
,
,
=
∂
∂
= y
L
y
y
t
z
y
x
ρ
,
( ) 0
,
,
,
0
=
∂
∂
=
z
z
t
z
y
x
ρ
,
( ) 0
,
,
,
=
∂
∂
= z
L
z
z
t
z
y
x
ρ
. (5b)
Here ρ =I,V; I (x,y,z,t) are the spatio-temporal distributions of concentrations of radiation
interstitials and radiation vacancies; Dρ(x,y,z,T) are the diffusion coefficients of the interstitials
and vacancies; terms V2
(x,y,z,t) and I2
(x,y,z,t) correspond to generation of divacancies and
diinterstitials, respectively; kI,V(x,y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T) are parameters of
recombination of point defects and generation of their complexes, respectively.
Spatio-temporal distributions of concentrations of divacansies ΦV (x,y,z,t) and diinterstitials ΦI
(x,y,z,t) we determine by solution the following system of equations [24,25]
( ) ( ) ( ) ( ) ( ) +





 Φ
+





 Φ
=
Φ
Φ
Φ
y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x I
I
I
I
I
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
( ) ( ) ( ) ( ) ( ) ( )
t
z
y
x
I
T
z
y
x
k
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
T
z
y
x
D
z
I
I
I
I
I ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
, −
+





 Φ
+ Φ
∂
∂
∂
∂
(6)
( ) ( ) ( ) ( ) ( ) +





 Φ
+





 Φ
=
Φ
Φ
Φ
y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x V
V
V
V
V
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
( ) ( ) ( ) ( ) ( ) ( )
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
T
z
y
x
k
z
t
z
y
x
T
z
y
x
D
z
V
V
V
V
V ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
, −
+





 Φ
+ Φ
∂
∂
∂
∂
with boundary and initial conditions

58
( )
0
,
,
,
0
=
∂
Φ
∂
=
x
x
t
z
y
x
ρ
,
( )
0
,
,
,
=
∂
Φ
∂
= x
L
x
x
t
z
y
x
ρ
,
( )
0
,
,
,
0
=
∂
Φ
∂
=
y
y
t
z
y
x
ρ
,
( )
0
,
,
,
=
∂
Φ
∂
= y
L
y
y
t
z
y
x
ρ
,
( )
0
,
,
,
0
=
∂
Φ
∂
=
z
z
t
z
y
x
ρ
,
( )
0
,
,
,
=
∂
Φ
∂
= z
L
z
z
t
z
y
x
ρ
, ΦI(x,y,z,0)=fΦI (x,y,z), ΦV(x,y,z,0)=fΦV (x,y,z). (7)
Here DΦI(x,y,z,T) and DΦV(x,y,z,T) are diffusion coefficients of complexes of point radiation
defects; kI(x,y,z,T) and kV (x,y,z,T) are parameters of decay of complexes of point radiation
defects.
To determine spatio-temporal distributions of concentrations of point radiation defects we used
recently elaborated approach [16,19,22]. Framework the approach we transform approximations
of diffusion coefficients of point radiation defects to the following form:
Dρ(x,y,z,T)=D0ρ[1+ερgρ(x,y,z,T)], where D0ρ are the average values of the diffusion coefficients,
0≤ερ< 1, |gρ(x,y,z,T)|≤1, ρ =I,V. We used the same transformation for approximations of
parameters of recombination of point radiation defects and generation 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)] и kV,V(x,y,z,T) = k0V,V [1+
εV,V gV,V(x,y,z,T)], where k0ρ1,ρ2 are the appropriate 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, ( ) ( ) *
,
,
,
,
,
,
~
I
t
z
y
x
I
t
z
y
x
I = , ( ) ( ) *
,
,
,
,
,
,
~
V
t
z
y
x
V
t
z
y
x
V = ,
2
0
0 L
t
D
D V
I
=
ϑ , V
I
V
I D
D
k
L 0
0
,
0
2
=
ω , V
I D
D
k
L 0
0
,
0
2
ρ
ρ
ρ =
Ω . The introduction leads to
modification of Eqs.(4) and conditions (5)
( ) ( )
[ ] ( ) ( )
[ ]





×
+
∂
∂
+






∂
∂
+
∂
∂
=
∂
∂
T
g
I
T
g
D
D
D
I
I
I
I
I
V
I
I
,
,
,
1
,
,
,
~
,
,
,
1
,
,
,
~
0
0
0
φ
η
χ
ε
η
χ
ϑ
φ
η
χ
φ
η
χ
ε
χ
ϑ
ϑ
φ
η
χ
( ) ( )
[ ] ( ) ( ) ×
−






∂
∂
+
∂
∂
+



∂
∂
× ϑ
φ
η
χ
η
ϑ
φ
η
χ
φ
η
χ
ε
φ
φ
ϑ
φ
η
χ
,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
0
0
0
0
0
0
I
I
T
g
D
D
D
D
D
D
I
I
I
V
I
I
V
I
I
( )
[ ] ( ) ( ) ( )
[ ]
T
g
I
V
T
g I
I
I
I
I
V
I
V
I ,
,
,
1
,
,
,
~
,
,
,
~
,
,
,
1 ,
,
2
,
, φ
η
χ
ε
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε
ω +
Ω
−
+
× (8)
( )
( )
[ ] ( )
( )
[ ]





×
+
∂
∂
+










∂
∂
+
∂
∂
=
∂
∂
T
g
V
T
g
D
D
D
V
V
V
V
V
V
I
V
,
,
,
1
,
,
,
~
,
,
,
1
,
,
,
~
0
0
0
φ
η
χ
ε
η
χ
ϑ
φ
η
χ
φ
η
χ
ε
χ
ϑ
ϑ
φ
η
χ
( ) ( )
[ ] ( ) ( ) ×
−










∂
∂
+
∂
∂
+





∂
∂
× ϑ
φ
η
χ
η
ϑ
φ
η
χ
φ
η
χ
ε
φ
φ
ϑ
φ
η
χ
,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
0
0
0
0
0
0
I
V
T
g
D
D
D
D
D
D
V
V
V
V
I
V
V
I
V
( )
[ ] ( ) ( ) ( )
[ ]
T
g
V
V
T
g V
V
V
V
V
V
I
V
I ,
,
,
1
,
,
,
~
,
,
,
~
,
,
,
1 ,
,
2
,
, φ
η
χ
ε
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε
ω +
Ω
−
+
×
( ) 0
,
,
,
~
0
=
∂
∂
=
χ
χ
ϑ
φ
η
χ
ρ
,
( ) 0
,
,
,
~
1
=
∂
∂
=
χ
χ
ϑ
φ
η
χ
ρ
,
( ) 0
,
,
,
~
0
=
∂
∂
=
η
η
ϑ
φ
η
χ
ρ
,
( ) 0
,
,
,
~
1
=
∂
∂
=
η
η
ϑ
φ
η
χ
ρ
,
( ) 0
,
,
,
~
0
=
∂
∂
=
φ
φ
ϑ
φ
η
χ
ρ
,
( ) 0
,
,
,
~
1
=
∂
∂
=
φ
φ
ϑ
φ
η
χ
ρ
, ( )
( )
*
,
,
,
,
,
,
~
ρ
ϑ
φ
η
χ
ϑ
φ
η
χ
ρ ρ
f
= . (9)
We determine solution of Eqs.(8) and conditions (9) by approach from Refs. [16,19,22], i.e. as the
following power series
( ) ( )
∑ ∑ ∑Ω
=
∞
=
∞
=
∞
=
0 0 0
,
,
,
~
,
,
,
~
i j k
ijk
k
j
i
ϑ
φ
η
χ
ρ
ω
ε
ϑ
φ
η
χ
ρ ρ
ρ . (10)

59
Substitution of the series (10) into Eqs.(8) and conditions (9) gives us possibility to obtain
equations for initial-order approximations of concentrations of point radiation defects
( )
ϑ
φ
η
χ ,
,
,
~
000
I and ( )
ϑ
φ
η
χ ,
,
,
~
000
V and corrections for them ( )
ϑ
φ
η
χ ,
,
,
~
ijk
I and ( )
ϑ
φ
η
χ ,
,
,
~
ijk
V , i ≥1,
j ≥1, k ≥1. The equations and conditions for them are presented in the Appendix. Solutions of
them have been obtained by standard approaches (see, for example, [26,27]). The solutions have
been obtained in the Appendix.
Farther we determine spatio-temporal distributions of concentrations of complexes of point
radiation defects. To obtain the concentrations we transform approximations of diffusion
coefficients to the following form: DΦρ(x,y,z,T)=D0Φρ[1+εΦρgΦρ(x,y,z,T)], where D0Φρ are the
average values of diffusion coefficients. After this transformation the Eqs.(6) takes the form
( ) ( )
[ ] ( )
( )
[ ] ( ) ( )
[ ]
( ) ( ) ( ) ( ) ( )
( ) ( )
[ ] ( )
( )
[ ] ( ) ( )
[ ]
( ) ( ) ( ) ( ) ( )





















−
+



Φ
×



×
+
+





 Φ
+
×
×
+





 Φ
+
=
Φ
−
+



Φ
×



×
+
+





 Φ
+
×
×
+





 Φ
+
=
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
T
z
y
x
k
D
z
t
z
y
x
T
z
y
x
g
z
y
t
z
y
x
T
z
y
x
g
y
D
x
t
z
y
x
T
z
y
x
g
x
D
t
t
z
y
x
t
z
y
x
I
T
z
y
x
k
t
z
y
x
I
T
z
y
x
k
D
z
t
z
y
x
T
z
y
x
g
z
y
t
z
y
x
T
z
y
x
g
y
D
x
t
z
y
x
T
z
y
x
g
x
D
t
t
z
y
x
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
2
,
0
0
0
2
,
0
0
0
∂
∂
ε
∂
∂
∂
∂
ε
∂
∂
∂
∂
ε
∂
∂
∂
∂
∂
∂
ε
∂
∂
∂
∂
ε
∂
∂
∂
∂
ε
∂
∂
∂
∂
We determine solutions of the above equations as the following power series
( ) ( )
∑ Φ
=
Φ
∞
=
Φ
0
,
,
,
,
,
,
i
i
i
t
z
y
x
t
z
y
x ρ
ρ
ρ ε . (11)
Substitution of the series (11) into Eqs.(6) and appropriate boundary and initial conditions gives
us possibility to obtain equations for initial-order approximations of concentrations of complexes
of point of radiation defects Φρ0(x,y,z,t), corrections for them Φρi(x,y,z,t), i ≥1, boundary and
initial conditions for all functions Φρi(x,y, z,t), i≥0. The equations and conditions are presented in
the Appendix. Solutions of the equations have been solved by standard approaches [26,27] and
presented in the Appendix.
Spatio-temporal distribution of dopant concentration we determine framework the same approach
as for determination of concentrations of radiation defects. Framework the approach we transform
approximation of dopant diffusion coefficient in the following form: DL(x,y,z,T)=
D0L[1+εLgL(x,y,z,T)], D0L is the average value of dopant diffusion coefficient, 0≤εL< 1, |gL(x,y,z,
T)|≤1. We determine solution of Eq.(1) as the following power series

60
( ) ( )
∑ ∑
=
∞
=
∞
=
0 1
,
,
,
,
,
,
i j
ij
j
i
L
t
z
y
x
C
t
z
y
x
C ξ
ε .
Substitution of the series into Eq.(1) and conditions (2) gives us possibility to obtain equations for
initial-order approximation of concentration of dopant C00(x,y,z,t), corrections for the
approximation Cij(x,y,z,t) (i≥1, j≥1), boundary and initial conditions for all functions Cij(x,y,z,t) (i
≥0, j ≥0). All these equations and conditions for them are presented in the Appendix. The above
equations have been solved by standard approaches (see, for example, [26,27]). The solutions are
presented in the Appendix.
Analysis of spatio-temporal distributions of concentrations of dopant and radiation defects have
been done analytically by using the second-order approximation by all parameters, which have
been used in considered power series. The second-order approximation is usually enough good
approximation to make qualitative analysis and to obtain some quantitative results. All analytical
results have been checked by numerical simulation.
3. DISCUSSION
In this section based on relations, which have been calculated in previous section, we analyzed
dynamics of redistribution of dopant and radiation defects during the annealing. Figs. 2 and 3
shows distributions of concentrations of dopants (for diffusive and ion types of doping) in
neighborhood of interface between layers of heterostructure under condition, when dopant
diffusion coefficient in the epitaxial layer is larger, than in the substrate. The figure shows, that
the interface gives us possibility to manufacture more thin field- effect transistors. Similar
choosing of properties of sections in the epitaxial layer gives us possibility to manufacture more
compact transistors in other directions.
Fig.2. Distributions of concentration of infused dopant in heterostructure from Figs. 1 and 2 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 diffusion coefficient in substrate

61
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.3. Distributions of concentration of implanted dopant in heterostructure from Figs. 1 and 2 in
direction, which is perpendicular to interface between epitaxial layer substrate. 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 heterostructure under condition, when value of dopant diffusion
coefficient in epitaxial layer is larger, than value of dopant diffusion coefficient in substrate
It should be noted, that interface between layers of heterostructure leads to influence on
distribution of concentration of dopant at appropriate value of annealing time: annealing time of
dopant should be neither much, no small. In this situation it should be done optimization of
annealing. The optimization of annealing has been done framework recently introduce criterion
[13-19,21,22]. By using the optimization we obtain values of optimal annealing time.
Dependences of the values are presented on the Figs. 4 and 5. It should be noted, that after ion
implantation one shall make annealing of radiation defects. One can find spreading of distribution
of concentration of dopant during the annealing. In the ideal case parameters of technological
process should be chosen so, that after finishing the annealing dopant should achieve interface
between layers of heterostructure. If after finishing of the annealing dopant did not achieved the
interface, it is attracted an interest additional annealing of dopant. In this situation additional
optimal annealing time of dopant decreases in the case of ion doping.
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
Θ
D
0
L
-2
3
2
4
1

62
Fig.4. Dependences of dimensionless optimal annealing time for doping by diffusion, which have
been obtained by minimization of mean-squared error, on several parameters. Curve 1 is the
dependence of dimensionless optimal annealing time on the relation 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 dimensionless optimal annealing time on value of parameter ε for a/L=1/2 and ξ =
γ = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of parameter ξ
for a/L=1/2 and ε =γ =0. Curve 4 is the dependence of dimensionless optimal annealing time on
value of parameter γ for a/L=1/2 and ε=ξ=0
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
Θ
D
0
L
-2
3
2
4
1
Fig.5. Dependences of dimensionless optimal annealing time for doping by ion implantation,
which have been obtained by minimization of mean-squared error, on several parameters. Curve 1
is the dependence of dimensionless optimal annealing time on the relation 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 dimensionless optimal annealing time on value of parameter ε for a/L=1/2 and
ξ = γ = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of
parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the dependence of dimensionless optimal
annealing time on value of parameter γ for a/L=1/2 and ε=ξ=0
4. CONCLUSION
In this paper we introduce an approach to manufacture thinner field-effect heterotransistor with
decreasing of their dimensions into another directions.
ACKNOWLEDGEMENTS
This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government,
Scientific School of Russia SSR-339.2014.2 and educational fellowship for scientific research.
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[2] A.G. Alexenko, I.I. Shagurin. Microcircuitry (Radio and communication, Moscow, 1990).
[3] V.G. Gusev, Yu.M. Gusev. Electronics (Moscow: Vysshaya shkola, 1991, in Russian).
[4] N.A. Avaev, Yu.E. Naumov, V.T. Frolkin. Basis of microelectronics (Radio and communication,
Moscow, 1991).

63
[5] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001).
[6] A. Kerentsev, V. Lanin. "Constructive-technological features of MOSFET-transistors" Power
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[7] A.N. Andronov, N.T. Bagraev, L.E. Klyachkin, S.V. Robozerov. "Ultrashallow p+−n junctions in
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144 (1998).
[8] S.T. Shishiyanu, T.S. Shishiyanu, S.K. Railyan. "Shallow p−n junctions in Si prepared by pulse
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[10] 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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[11] J. A. Sharp, N. E. B. Cowern, R. P. Webb, K. J. Kirkby, D. Giubertoni, S. Genarro, M. Bersani, M. A.
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[13] E.L. Pankratov. "Redistribution of dopant during microwave annealing of a multilayer structure for
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[14] E.L. Pankratov. "Optimization of near-surficial annealing for decreasing of depth of p-n-junction in
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[15] E.L. Pankratov. "Decreasing of depth of implanted-junction rectifier in semiconductor heterostructure
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[20] V.V. Kozlivsky. Modification of semiconductors by proton beams (Nauka, Sant-Peterburg, 2003, in
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[21] E.L. Pankratov. "Decreasing of depth of p-n-junction in a semiconductor heterostructure by serial
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[22] E.L. Pankratov, E.A. Bulaeva. "Increasing of sharpness of diffusion-junction heterorectifier by using
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[27] H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids (Oxford University Press, 1964).
APPENDIX
Equations for the functions ( )
ϑ
φ
η
χ ,
,
,
~
ijk
I and ( )
ϑ
φ
η
χ ,
,
,
~
ijk
V , i ≥0, j ≥0, k ≥0 and conditions for
them are

64
( ) ( ) ( ) ( )
2
000
2
0
0
2
000
2
0
0
2
000
2
0
0
000 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ I
D
D
I
D
D
I
D
D
I
V
I
V
I
V
I
( ) ( ) ( ) ( )
2
000
2
0
0
2
000
2
0
0
2
000
2
0
0
000 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ V
D
D
V
D
D
V
D
D
V
I
V
I
V
I
V
;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
00
2
0
0
2
00
2
0
0
2
00
2
0
0
00 ,
,
,
~
,
,
,
~
,
,
,
~
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
χ i
V
I
i
V
I
i
V
I
i I
D
D
I
D
D
I
D
D
I
( )
( )
( )
( )
+








∂
∂
∂
∂
+








∂
∂
∂
∂
+ −
−
η
ϑ
φ
η
χ
φ
η
χ
η
χ
ϑ
φ
η
χ
φ
η
χ
χ
,
,
,
~
,
,
,
,
,
,
~
,
,
, 100
0
0
100
0
0 i
I
V
I
i
I
V
I I
T
g
D
D
I
T
g
D
D
( )
( )








∂
∂
∂
∂
+ −
φ
ϑ
φ
η
χ
φ
η
χ
φ
,
,
,
~
,
,
, 100
0
0 i
I
V
I I
T
g
D
D
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
00
2
0
0
2
00
2
0
0
2
00
2
0
0
00 ,
,
,
~
,
,
,
~
,
,
,
~
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
χ i
I
V
i
I
V
i
I
V
i V
D
D
V
D
D
V
D
D
V
( )
( )
( )
( )
+








∂
∂
∂
∂
+








∂
∂
∂
∂
+ −
−
η
ϑ
φ
η
χ
φ
η
χ
η
χ
ϑ
φ
η
χ
φ
η
χ
χ
,
,
,
~
,
,
,
,
,
,
~
,
,
, 100
0
0
100
0
0 i
V
I
V
i
V
I
V V
T
g
D
D
V
T
g
D
D
( )
( )








∂
∂
∂
∂
+ −
φ
ϑ
φ
η
χ
φ
η
χ
φ
,
,
,
~
,
,
, 100
0
0 i
V
I
V V
T
g
D
D , i≥1;
( ) ( ) ( ) ( )
−








∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
010
2
2
010
2
2
010
2
0
0
010 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ I
I
I
D
D
I
V
I
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 000
000
,
, V
I
T
g V
I
V
I
+
−
( ) ( ) ( ) ( )
+








∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
010
2
2
010
2
2
010
2
0
0
010 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ V
V
V
D
D
V
I
V
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 000
000
,
, V
I
T
g V
I
V
I
+
− ;
( ) ( ) ( ) ( )
−








∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
020
2
2
020
2
2
020
2
0
0
020 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ I
I
I
D
D
I
V
I
( )
[ ] ( ) ( ) ( ) ( )
[ ]
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
1 010
000
000
010
,
, V
I
V
I
T
g V
I
V
I +
+
−
( ) ( ) ( ) ( )
−








∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
020
2
2
020
2
2
020
2
0
0
020 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ V
V
V
D
D
V
V
I
( )
[ ] ( ) ( ) ( ) ( )
[ ]
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
1 010
000
000
010
,
, V
I
V
I
T
g V
I
V
I +
+
− ;
( ) ( ) ( ) ( )
−








∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
001
2
2
001
2
2
001
2
0
0
001 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ I
I
I
D
D
I
V
I
( )
[ ] ( )
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
1 2
000
,
, I
T
g I
I
I
I
+
−

65
( ) ( ) ( ) ( )
−








∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
001
2
2
001
2
2
001
2
0
0
001 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ V
V
V
D
D
V
I
V
( )
[ ] ( )
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
1 2
000
,
, V
T
g I
I
I
I
+
− ;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
110
2
0
0
2
110
2
0
0
2
110
2
0
0
110 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ I
D
D
I
D
D
I
D
D
I
V
I
V
I
V
I
( )
( )
( )
( )
+








∂
∂
∂
∂
+








∂
∂
∂
∂
+
η
ϑ
φ
η
χ
φ
η
χ
η
χ
ϑ
φ
η
χ
φ
η
χ
χ
,
,
,
~
,
,
,
,
,
,
~
,
,
, 010
0
0
010
0
0 I
T
g
D
D
I
T
g
D
D
I
V
I
I
V
I
( )
( )
( )
[ ]×
+
−








∂
∂
∂
∂
+ T
g
I
T
g
D
D
I
I
I
I
I
V
I
,
,
,
1
,
,
,
~
,
,
, ,
,
010
0
0
φ
η
χ
ε
φ
ϑ
φ
η
χ
φ
η
χ
φ
( ) ( ) ( ) ( )
[ ]
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
100
000
000
100 V
I
V
I +
×
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
110
2
0
0
2
110
2
0
0
2
110
2
0
0
110 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ V
D
D
V
D
D
V
D
D
V
I
V
I
V
I
V
( )
( )
( )
( )
+








∂
∂
∂
∂
+








∂
∂
∂
∂
+
η
ϑ
φ
η
χ
φ
η
χ
η
χ
ϑ
φ
η
χ
φ
η
χ
χ
,
,
,
~
,
,
,
,
,
,
~
,
,
, 010
0
0
010
0
0 V
T
g
D
D
V
T
g
D
D
V
I
V
V
I
V
( )
( )
( )
[ ]×
+
−








∂
∂
∂
∂
+ T
g
V
T
g
D
D
V
V
V
V
V
I
V
,
,
,
1
,
,
,
~
,
,
, ,
,
010
0
0
φ
η
χ
ε
φ
ϑ
φ
η
χ
φ
η
χ
φ
( ) ( ) ( ) ( )
[ ]
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
100
000
000
100 I
V
I
V +
× ;
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
002
2
0
0
2
002
2
0
0
2
002
2
0
0
002 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ I
D
D
I
D
D
I
D
D
I
V
I
V
I
V
I
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 000
001
,
, I
I
T
g I
I
I
I
+
−
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
002
2
0
0
2
002
2
0
0
2
002
2
0
0
002 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ V
D
D
V
D
D
V
D
D
V
I
V
I
V
I
V
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 000
001
,
, V
V
Е
g V
V
V
V
+
− ;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
101
2
0
0
2
101
2
0
0
2
101
2
0
0
101 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ I
D
D
I
D
D
I
D
D
I
V
I
V
I
V
I
( )
( )
( )
( )
+








∂
∂
∂
∂
+








∂
∂
∂
∂
+
η
ϑ
φ
η
χ
φ
η
χ
η
χ
ϑ
φ
η
χ
φ
η
χ
χ
,
,
,
~
,
,
,
,
,
,
~
,
,
, 001
0
0
001
0
0 I
T
g
D
D
I
T
g
D
D
I
V
I
I
V
I
( )
( )
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε
φ
ϑ
φ
η
χ
φ
η
χ
φ
,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
, 000
100
001
0
0
V
I
T
g
I
T
g
D
D
I
I
I
V
I
+
−








∂
∂
∂
∂
+
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
101
2
0
0
2
101
2
0
0
2
101
2
0
0
101 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ V
D
D
V
D
D
V
D
D
V
I
V
I
V
I
V

66
( )
( )
( )
( )
+








∂
∂
∂
∂
+








∂
∂
∂
∂
+
η
ϑ
φ
η
χ
φ
η
χ
η
χ
ϑ
φ
η
χ
φ
η
χ
χ
,
,
,
~
,
,
,
,
,
,
~
,
,
, 001
0
0
001
0
0 V
T
g
D
D
V
T
g
D
D
V
I
V
V
I
V
( )
( )
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε
φ
ϑ
φ
η
χ
φ
η
χ
φ
,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
, 100
000
001
0
0
V
I
T
g
V
T
g
D
D
V
V
V
I
V
+
−








∂
∂
∂
∂
+ ;
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
011
2
0
0
2
011
2
0
0
2
011
2
0
0
011 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ I
D
D
I
D
D
I
D
D
I
V
I
V
I
V
I
( )
[ ] ( ) ( )−
+
− ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 010
000
,
, I
I
T
g I
I
I
I
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 000
001
,
, V
I
T
g V
I
V
I
+
−
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
011
2
0
0
2
011
2
0
0
2
011
2
0
0
011 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ V
D
D
V
D
D
V
D
D
V
I
V
I
V
I
V
( )
[ ] ( ) ( )−
+
− ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 010
000
,
, V
V
T
g V
V
V
V
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 001
000
,
, V
I
t
g V
I
V
I
+
− ;
( )
0
,
,
,
~
0
=
∂
∂
=
x
ijk
χ
ϑ
φ
η
χ
ρ
,
( )
0
,
,
,
~
1
=
∂
∂
=
x
ijk
χ
ϑ
φ
η
χ
ρ
,
( )
0
,
,
,
~
0
=
∂
∂
=
η
η
ϑ
φ
η
χ
ρijk
,
( )
0
,
,
,
~
1
=
∂
∂
=
η
η
ϑ
φ
η
χ
ρijk
,
( )
0
,
,
,
~
0
=
∂
∂
=
φ
φ
ϑ
φ
η
χ
ρijk
,
( )
0
,
,
,
~
1
=
∂
∂
=
φ
φ
ϑ
φ
η
χ
ρijk
(i≥0, j≥0, k≥0);
( ) ( ) *
000 ,
,
0
,
,
,
~ ρ
φ
η
χ
φ
η
χ
ρ ρ
f
= , ( ) 0
0
,
,
,
~ =
φ
η
χ
ρijk (i≥1, j≥1, k≥1).
Solutions of these equations with account boundary and initial conditions could be written as
( ) ( ) ( ) ( ) ( )
∑
+
=
∞
=1
000
2
1
,
,
,
~
n
n
n e
c
c
c
F
L
L
ϑ
φ
η
χ
ϑ
φ
η
χ
ρ ρ
ρ ,
where ( ) ( ) ( ) ( )
∫ ∫ ∫
=
1
0
1
0
1
0
*
,
,
cos
cos
cos
1
u
d
v
d
w
d
w
v
u
f
w
n
v
n
u
n
F n
n ρ
ρ π
π
π
ρ
, cn(χ) = cos (π n χ), ( )=
ϑ
nI
e
( )
I
V D
D
n 0
0
2
2
exp ϑ
π
−
= , ( ) ( )
V
I
nV D
D
n
e 0
0
2
2
exp ϑ
π
ϑ −
= ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
∂
∂
−
−
=
∞
=
−
1 0
1
0
1
0
1
0
100
0
0
00
,
,
,
~
2
,
,
,
~
n
i
n
n
nI
nI
n
V
I
i
u
w
v
u
I
v
c
u
s
e
e
c
c
c
n
D
D
I
ϑ τ
τ
ϑ
φ
η
χ
π
ϑ
φ
η
χ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ ∫ ∫
−
−
×
∞
=1 0
1
0
1
0
0
0
2
,
,
,
n
n
n
nI
nI
n
V
I
I
n v
s
u
c
e
e
c
c
c
D
D
d
u
d
v
d
w
d
T
w
v
u
g
w
c
ϑ
τ
ϑ
φ
η
χ
π
τ
( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑
−
∫
∂
∂
×
∞
=
−
1
0
0
1
0
100
2
,
,
,
~
,
,
,
n
nI
n
V
I
i
I
n e
c
c
c
n
D
D
d
u
d
v
d
w
d
v
w
v
u
I
T
w
v
u
g
w
c
n ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( )
∫ ∫ ∫ ∫
∂
∂
−
× −
ϑ
τ
τ
τ
0
1
0
1
0
1
0
100 ,
,
,
~
,
,
, d
u
d
v
d
w
d
w
w
v
u
I
T
w
v
u
g
w
s
v
c
u
c
e i
I
n
n
n
nI

67
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∑ ×
∫ ∫ ∫ ∫
∂
∂
−
−
=
∞
=
−
1 0
1
0
1
0
1
0
100
0
0
00
,
~
2
,
,
,
~
n
i
n
n
n
nI
nV
n
I
V
i
u
u
V
w
c
v
c
u
s
e
e
c
c
c
n
D
D
V
ϑ τ
τ
ϑ
φ
η
χ
π
ϑ
φ
η
χ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∑ ×
∫ ∫ ∫ ∫
−
−
×
∞
=1 0
1
0
1
0
1
0
0
0
2
,
,
,
n
n
n
n
nI
nV
n
I
V
V w
c
v
s
u
c
e
e
c
c
c
D
D
d
u
d
v
d
w
d
T
w
v
u
g
ϑ
τ
ϑ
φ
η
χ
π
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ ∫
−
−
∂
∂
×
∞
=
−
1 0
1
0
0
0
100
2
,
~
,
,
,
n
n
nI
nV
n
I
V
i
V u
c
e
e
c
c
c
D
D
d
u
d
v
d
w
d
v
u
V
T
w
v
u
g
n
ϑ
τ
ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( )
( )
∫ ∫
∂
∂
× −
1
0
1
0
100 ,
~
,
,
, τ
τ
d
u
d
v
d
w
d
w
u
V
T
w
v
u
g
w
s
v
c
n i
V
n
n , i ≥1,
where sn(χ)=sin(πnχ);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∑ ∫ ∫ ∫ ∫ ×
−
−
=
∞
=1 0
1
0
1
0
1
0
010 2
,
,
,
~
n
n
n
n
n
n
n
n
n w
c
v
c
u
c
e
e
c
c
c
ϑ
ρ
ρ τ
ϑ
φ
η
χ
ϑ
φ
η
χ
ρ
( )
[ ] ( ) ( ) τ
τ
τ
ε d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
T
w
v
u
g V
I
V
I ,
,
,
~
,
,
,
~
,
,
,
1 000
000
,
,
+
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]×
∑ ∫ ∫ ∫ ∫ +
−
−
=
∞
=1 0
1
0
1
0
1
0
,
,
0
0
020 ,
,
,
1
2
,
,
,
~
n
V
I
V
I
n
n
n
n
n
V
I
T
w
v
u
g
e
e
c
c
c
D
D ϑ
ρ
ρ ε
τ
ϑ
φ
η
χ
ϑ
φ
η
χ
ρ
( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ] τ
τ
τ
τ
τ d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
w
v
u
V
w
v
u
I
w
c
v
c
u
c n
n
n ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
010
000
000
010 +
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0
1
0
1
0
1
0
001 2
,
,
,
~
n
n
n
n
n
n
n
n
n w
c
v
c
u
c
e
e
c
c
c
ϑ
ρ
ρ τ
ϑ
φ
η
χ
ϑ
φ
η
χ
ρ
( )
[ ] ( ) τ
τ
ρ
ε ρ
ρ
ρ
ρ d
u
d
v
d
w
d
w
v
u
T
w
v
u
g ,
,
,
~
,
,
,
1 2
000
,
,
+
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∑ ∫ ∫ ×
∫ ∫
−
−
=
∞
=1 0
1
0
1
0
1
0
002 2
,
,
,
~
n
n
n
n
n
n
n
n
n w
c
v
c
u
c
e
e
c
c
c
ϑ
ρ
ρ τ
ϑ
φ
η
χ
ϑ
φ
η
χ
ρ
( )
[ ] ( ) ( ) τ
τ
ρ
τ
ρ
ε ρ
ρ
ρ
ρ d
u
d
v
d
w
d
w
v
u
w
v
u
T
w
v
u
g ,
,
,
~
,
,
,
~
,
,
,
1 000
001
,
,
+
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0
1
0
1
0
1
0
0
0
110 ,
,
,
2
,
,
,
~
n
I
n
n
n
nI
nI
n
n
n
V
I
T
w
v
u
g
u
c
v
c
u
s
e
e
c
c
c
D
D
I
ϑ
τ
ϑ
φ
η
χ
π
ϑ
φ
η
χ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ ∫ ∫
−
−
∂
∂
×
∞
=
−
1 0
1
0
1
0
0
0
100
2
,
,
,
~
n
n
n
nI
nI
n
n
n
V
I
i
v
s
u
c
e
e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
w
v
u
I
n
ϑ
τ
ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑
×
−
∫
∂
∂
×
∞
=
−
1
0
0
1
0
100
2
,
,
,
~
,
,
,
n
nI
n
n
n
V
I
i
I
n e
c
c
c
n
D
D
d
u
d
v
d
w
d
v
w
v
u
I
T
w
v
u
g
u
c ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑
−
∫ ∫ ∫ ∫
∂
∂
−
×
∞
=
−
1
0
1
0
1
0
1
0
100
2
,
,
,
~
,
,
,
n
n
n
n
i
I
n
n
n
nI c
c
c
d
u
d
v
d
w
d
w
w
v
u
I
T
w
v
u
g
u
s
v
c
u
c
e φ
η
χ
τ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]
∫ ∫ ∫ ∫ ×
+
−
×
ϑ
τ
τ
τ
τ
τ
ϑ
0
1
0
1
0
1
0
100
000
000
100 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
w
v
u
V
w
v
u
I
w
v
u
V
w
v
u
I
v
c
u
c
e
e n
n
nI
nI
( )
[ ] ( ) τ
ε d
u
d
v
d
w
d
w
c
T
w
v
u
g n
V
I
V
I ,
,
,
1 ,
,
+
×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0
1
0
1
0
1
0
0
0
110 ,
,
,
2
,
,
,
~
n
V
n
n
nV
nV
n
n
n
I
V
T
w
v
u
g
v
c
u
s
e
e
c
c
c
n
D
D
V
ϑ
τ
ϑ
φ
η
χ
π
ϑ
φ
η
χ

68
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫
−
−
∂
∂
×
∞
=
−
1 0
1
0
0
0
100
2
,
,
,
~
n
n
nV
nV
n
n
n
I
V
i
n u
c
e
e
c
c
c
D
D
d
u
d
v
d
w
d
u
w
v
u
V
u
c
ϑ
τ
ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ −
−
∫ ∫
∂
∂
×
∞
=
−
1 0
0
0
1
0
1
0
100
2
,
,
,
~
,
,
,
n
nV
nV
I
V
i
V
n
n e
e
D
D
d
u
d
v
d
w
d
v
w
v
u
V
T
w
v
u
g
u
c
v
s
n
ϑ
τ
ϑ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
∑ ×
−
∫ ∫ ∫
∂
∂
×
∞
=
−
1
1
0
1
0
1
0
100
2
,
,
,
~
,
,
,
n
n
i
V
n
n
n
n
n
n c
d
u
d
v
d
w
d
w
w
v
u
V
T
w
v
u
g
u
s
v
c
u
c
c
c
c
n χ
τ
τ
φ
η
χ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]
∫ ∫ ∫ ∫ ×
+
−
×
ϑ
τ
τ
τ
τ
τ
ϑ
φ
η
0
1
0
1
0
1
0
100
000
000
100 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
w
v
u
V
w
v
u
I
w
v
u
V
w
v
u
I
u
c
e
e
c
c n
nV
nI
n
n
( ) ( ) τ
d
u
d
v
d
v
c
w
d
w
c n
n
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0
1
0
1
0
1
0
0
0
101 ,
,
,
2
,
,
,
~
n
I
n
n
n
nI
nI
n
n
n
V
I
T
w
v
u
g
w
c
v
c
u
s
e
e
c
c
c
n
D
D
I
ϑ
τ
ϑ
φ
η
χ
π
ϑ
φ
η
χ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫
−
−
∂
∂
×
∞
=1 0
1
0
1
0
0
0
001
2
,
,
,
~
n
n
n
nI
nI
n
n
n
V
I
v
s
u
c
e
e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
w
v
u
I ϑ
τ
ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑
−
∫
∂
∂
×
∞
=1
0
0
1
0
001
2
,
,
,
~
,
,
,
n
nI
n
n
n
V
I
I
n e
c
c
c
n
D
D
d
u
d
v
d
w
d
v
w
v
u
I
T
w
v
u
g
w
c ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑
−
∫ ∫ ∫ ∫
∂
∂
−
×
∞
=1
0
1
0
1
0
1
0
001
2
,
,
,
~
,
,
,
n
n
n
I
n
n
n
nI c
c
d
u
d
v
d
w
d
w
w
v
u
I
T
w
v
u
g
w
s
v
c
u
c
e η
χ
τ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( )
[ ] ( ) ( ) ×
∫ ∫ ∫ ∫ +
−
×
ϑ
τ
τ
ε
τ
ϑ
φ
0
1
0
1
0
1
0
000
100
,
, ,
,
,
~
,
,
,
~
,
,
,
1 w
v
u
V
w
v
u
I
T
w
v
u
g
v
c
u
c
e
e
c V
I
V
I
n
n
nI
nI
n
( ) τ
d
u
d
v
d
w
d
w
cn
×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0
1
0
1
0
1
0
0
0
101 ,
,
,
2
,
,
,
~
n
V
n
n
nV
nV
n
n
n
I
V
T
w
v
u
g
v
c
u
s
e
e
c
c
c
n
D
D
V
ϑ
τ
ϑ
φ
η
χ
π
ϑ
φ
η
χ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫
−
−
∂
∂
×
∞
=1 0
1
0
0
0
001
2
,
,
,
~
n
n
nV
nV
n
n
n
I
V
u
c
e
e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
w
v
u
V
w
c
ϑ
τ
ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑
−
∫ ∫
∂
∂
×
∞
=1
0
0
1
0
1
0
001
2
,
,
,
~
,
,
,
n
n
n
n
I
V
V
n
n c
c
c
n
D
D
d
u
d
v
d
w
d
v
w
v
u
V
T
w
v
u
g
w
c
v
s φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑
−
∫ ∫ ∫ ∫
∂
∂
−
×
∞
=1
0
1
0
1
0
1
0
001
2
,
,
,
~
,
,
,
n
n
V
n
n
n
nV
nV c
d
u
d
v
d
w
d
w
w
v
u
V
T
w
v
u
g
w
s
v
c
u
c
e
e χ
τ
τ
τ
ϑ
ϑ
( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ] ( ) ( ) ×
∫ ∫ ∫ ∫ +
−
×
ϑ
τ
τ
ε
τ
ϑ
φ
η
0
1
0
1
0
1
0
100
000
,
, ,
,
,
~
,
,
,
~
,
,
,
1 w
v
u
V
w
v
u
I
T
w
v
u
g
v
c
u
c
e
e
c
c V
I
V
I
n
n
nV
nV
n
n
( ) τ
d
u
d
v
d
w
d
w
cn
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]
{
∑ ∫ ∫ ∫ ∫ ×
+
−
−
=
∞
=1 0
1
0
1
0
1
0
,
,
011 ,
,
,
1
2
,
,
,
~
n
I
I
I
I
n
n
nI
nI
n
n
n T
w
v
u
g
v
c
u
c
e
e
c
c
c
I
ϑ
ε
τ
ϑ
φ
η
χ
ϑ
φ
η
χ
( ) ( ) ( )
[ ] ( ) ( )}×
+
+
× τ
τ
ε
τ
τ ,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
,
~
000
001
,
,
010
000 w
v
u
V
w
v
u
I
T
w
v
u
g
w
v
u
I
w
v
u
I V
I
V
I
( ) τ
d
u
d
v
d
w
d
w
cn
×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]
{
∑ ∫ ∫ ∫ ∫ ×
+
−
−
=
∞
=1 0
1
0
1
0
1
0
,
,
011 ,
,
,
1
2
,
,
,
~
n
V
V
V
V
n
n
nV
nV
n
n
n T
w
v
u
g
v
c
u
c
e
e
c
c
c
V
ϑ
ε
τ
ϑ
φ
η
χ
ϑ
φ
η
χ

69
( ) ( ) ( )
[ ] ( ) ( )}×
+
+
× τ
τ
ε
τ
τ ,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
,
~
001
000
,
,
010
000 w
v
u
V
w
v
u
I
T
w
v
u
g
w
v
u
V
w
v
u
V V
I
V
I
( ) τ
d
u
d
v
d
w
d
w
cn
× .
Inutial-order approximations of distributions of concentrations of complexes of radiation defects
Φρ0(x,y,z,t), corrections for the approximations Φρi(x,y,z,t) i≥1, boundary and initial conditions for
them
( ) ( ) ( ) ( )+
Φ
+
Φ
+
Φ
=
Φ
Φ
Φ
Φ 2
0
2
0
2
0
2
0
2
0
2
0
0 ,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
D
y
t
z
y
x
D
x
t
z
y
x
D
t
t
z
y
x I
I
I
I
I
I
I
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( ) ( )
t
z
y
x
I
T
z
y
x
k
t
z
y
x
I
T
z
y
x
k I
I
I ,
,
,
,
,
,
,
,
,
,
,
, 2
, −
+
( ) ( ) ( ) ( )+
Φ
+
Φ
+
Φ
=
Φ
Φ
Φ
Φ 2
0
2
0
2
0
2
0
2
0
2
0
0 ,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
D
y
t
z
y
x
D
x
t
z
y
x
D
t
t
z
y
x V
V
V
V
V
V
V
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( ) ( )
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
T
z
y
x
k V
V
V ,
,
,
,
,
,
,
,
,
,
,
, 2
, −
+ ;
( ) ( ) ( ) ( )
+
Φ
+
Φ
+
Φ
=
Φ
Φ
Φ
Φ 2
2
0
2
2
0
2
2
0
,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
D
y
t
z
y
x
D
x
t
z
y
x
D
t
t
z
y
x i
I
I
i
I
I
i
I
I
i
I
∂
∂
∂
∂
∂
∂
∂
∂
( )
( )
( )
( )
+





 Φ
+





 Φ
+
−
Φ
Φ
−
Φ
Φ
y
t
z
y
x
T
z
y
x
g
y
D
x
t
z
y
x
T
z
y
x
g
x
D
i
I
I
I
i
I
I
I
∂
∂
∂
∂
∂
∂
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
1
0
1
0
( )
( )





 Φ
+
−
Φ
Φ
z
t
z
y
x
T
z
y
x
g
z
D
i
I
I
I
∂
∂
∂
∂ ,
,
,
,
,
,
1
0
( ) ( ) ( ) ( )
+
Φ
+
Φ
+
Φ
=
Φ
Φ
Φ
Φ 2
2
0
2
2
0
2
2
0
,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
D
y
t
z
y
x
D
x
t
z
y
x
D
t
t
z
y
x i
V
V
i
V
V
i
V
V
i
V
∂
∂
∂
∂
∂
∂
∂
∂
( )
( )
( )
( )
+





 Φ
+





 Φ
+
−
Φ
Φ
−
Φ
Φ
y
t
z
y
x
T
z
y
x
g
y
D
x
t
z
y
x
T
z
y
x
g
x
D
i
V
V
V
i
V
V
V
∂
∂
∂
∂
∂
∂
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
1
0
1
0
( )
( )





 Φ
+
−
Φ
Φ
z
t
z
y
x
T
z
y
x
g
z
D
i
V
V
V
∂
∂
∂
∂ ,
,
,
,
,
,
1
0 , i≥1;
( )
0
,
,
,
0
=
∂
Φ
∂
=
x
i
x
t
z
y
x
ρ
,
( )
0
,
,
,
=
∂
Φ
∂
= x
L
x
i
x
t
z
y
x
ρ
,
( )
0
,
,
,
0
=
∂
Φ
∂
=
y
i
y
t
z
y
x
ρ
,
( )
0
,
,
,
=
∂
Φ
∂
= y
L
y
i
y
t
z
y
x
ρ
,
( )
0
,
,
,
0
=
∂
Φ
∂
=
z
i
z
t
z
y
x
ρ
,
( )
0
,
,
,
=
∂
Φ
∂
= z
L
z
i
z
t
z
y
x
ρ
, 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
∞
=
∞
=
Φ
Φ
1
1
0
2
2
1
,
,
,
n
n
n
n
n
n
n
n
n
n
z
y
x
z
y
x
z
c
y
c
x
c
n
L
t
e
z
c
y
c
x
c
F
L
L
L
L
L
L
t
z
y
x ρ
ρ
ρ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]
∫ ∫ ∫ ∫ ×
−
−
× Φ
Φ
t L L L
I
I
I
n
n
n
n
x y z
w
v
u
I
T
w
v
u
k
w
v
u
I
T
w
v
u
k
v
c
u
c
e
t
e
0 0 0 0
2
, ,
,
,
,
,
,
,
,
,
,
,
, τ
τ
τ
ρ
ρ
( ) τ
d
u
d
v
d
w
d
w
cn
× ,

70
where ( ) ( ) ( ) ( )
∫ ∫ ∫
= Φ
Φ
x y z
L L L
n
n
n
n u
d
v
d
w
d
w
v
u
f
w
c
v
c
u
c
F
0 0 0
,
,
ρ
ρ
, ( )
















+
+
−
= Φ
Φ 2
2
2
0
2
2 1
1
1
exp
z
y
x
n
L
L
L
t
D
n
t
e ρ
ρ
π ,
cn(x)=cos(πnx/Lx);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ ∫ ∫ ∫
−
−
=
Φ
∞
=
Φ
Φ
Φ
1 0 0 0 0
2
,
,
,
2
,
,
,
n
t L L L
n
n
n
n
n
n
n
z
y
x
i
x y z
T
w
v
u
g
v
c
u
s
e
t
e
z
c
y
c
x
c
n
L
L
L
t
z
y
x ρ
ρ
ρ
τ
π
ρ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫
−
−
Φ
×
∞
=
Φ
Φ
−
1 0 0
2
1 2
,
,
,
n
t L
n
n
n
n
n
n
z
y
x
i
I
n
x
u
c
e
t
e
z
c
y
c
x
c
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
w
c τ
π
τ
∂
τ
∂
ρ
ρ
ρ
( ) ( ) ( )
( )
( ) ( ) ( ) ×
∑
−
∫ ∫
Φ
×
∞
=
−
Φ
1
2
0 0
1 2
,
,
,
,
,
,
n
n
n
n
z
y
x
L L
i
I
n
n z
c
y
c
x
c
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
T
w
v
u
g
w
c
v
s
y z π
τ
∂
τ
∂ ρ
ρ
( ) ( ) ( ) ( ) ( )
( )
( )
∫ ∫ ∫ ∫
Φ
−
× Φ
−
Φ
Φ
t L L L
i
I
n
n
n
n
n
x y z
d
u
d
v
d
w
d
T
w
v
u
g
w
w
v
u
w
s
v
c
u
c
e
t
e
0 0 0 0
1
,
,
,
,
,
,
τ
∂
τ
∂
τ ρ
ρ
ρ
ρ
, i≥1,
where sn(x)=sin(πnx/Lx).
Equation for initial-order approximation of dopant concentration C00(x,t), corrections for the
approximation Cij(x,y,z,t) (i ≥1, j ≥1), boundary and initial conditions of the above functions are
( ) ( ) ( ) ( )
2
00
2
0
2
00
2
0
2
00
2
0
00 ,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
C
D
y
t
z
y
x
C
D
x
t
z
y
x
C
D
t
t
z
y
x
C
L
L
L
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
;
( ) ( ) ( ) ( )
+






∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
0
2
2
0
2
2
0
2
0
0 ,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
C
y
t
z
y
x
C
x
t
z
y
x
C
D
t
t
z
y
x
C i
i
i
L
i
( ) ( ) ( ) ( ) +






∂
∂
∂
∂
+






∂
∂
∂
∂
+ −
−
y
t
z
y
x
C
T
z
y
x
g
y
D
x
t
z
y
x
C
T
z
y
x
g
x
D i
L
L
i
L
L
,
,
,
,
,
,
,
,
,
,
,
, 10
0
10
0
( ) ( )






∂
∂
∂
∂
+ −
z
t
z
y
x
C
T
z
y
x
g
z
D i
L
L
,
,
,
,
,
, 10
0 , i≥1;
( ) ( ) ( ) ( ) ( )
( )






×
∂
∂
+






∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
T
z
y
x
P
t
z
y
x
C
x
z
t
z
y
x
C
y
t
z
y
x
C
x
t
z
y
x
C
D
t
t
z
y
x
C
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 00
2
01
2
2
01
2
2
01
2
0
01
γ
γ
( ) ( )
( )
( ) ( )
( )
( )
L
D
z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
z
y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
y
x
t
z
y
x
C
0
00
00
00
00
00 ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,











∂
∂
∂
∂
+






∂
∂
∂
∂
+



∂
∂
× γ
γ
γ
γ
;
( ) ( ) ( ) ( ) ( )
( )



×
+






∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ −
T
z
y
x
P
t
z
y
x
C
z
t
z
y
x
C
y
t
z
y
x
C
x
t
z
y
x
C
D
t
t
z
y
x
C
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 1
00
2
02
2
2
02
2
2
02
2
0
02
γ
γ
( ) ( ) ( ) ( )
( )
( ) +






∂
∂
∂
∂
+



∂
∂
×
−
y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
y
x
t
z
y
x
C
t
z
y
x
C
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 00
1
00
01
00
01 γ
γ
( ) ( )
( )
( ) ( )
( )
( )





×






∂
∂
∂
∂
+











∂
∂
∂
∂
+
−
x
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
x
D
z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
z
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 01
00
0
00
1
00
01 γ
γ
γ
γ

71
( )
( )
( ) ( )
( )
( )
L
D
z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
z
y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
y
0
01
00
01
00 ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,











∂
∂
∂
∂
+






∂
∂
∂
∂
+ γ
γ
γ
γ
;
( ) ( ) ( ) ( ) ( )
( )








×
∂
∂
+






∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ −
T
z
y
x
P
t
z
y
x
C
x
z
t
z
y
x
C
y
t
z
y
x
C
x
t
z
y
x
C
D
t
t
z
y
x
C
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 1
00
2
11
2
2
11
2
2
11
2
0
11
γ
γ
( )
( )
( )
( )
( )
( )
+






∂
∂
∂
∂
+



∂
∂
×
−
y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
y
x
t
z
y
x
C
t
z
y
x
C
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 00
1
00
10
00
10 γ
γ
( )
( )
( )
( ) ( )








×
∂
∂
∂
∂
+











∂
∂
∂
∂
+
−
x
t
z
y
x
C
x
D
D
z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
z
L
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 10
0
0
00
1
00
10 γ
γ
( )
( )
( )
( )
( ) ( )
( )
( )
+











∂
∂
∂
∂
+






∂
∂
∂
∂
+



×
y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
z
y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
y
T
z
y
x
P
t
z
y
x
C ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 10
00
10
00
00
γ
γ
γ
γ
γ
γ
( )
( )
( )
( )
+











∂
∂
∂
∂
+






∂
∂
∂
∂
+
y
t
z
y
x
C
T
z
y
x
g
y
x
t
z
y
x
C
T
z
y
x
g
x
D L
L
L
,
,
,
,
,
,
,
,
,
,
,
, 01
01
0
( )
( )











∂
∂
∂
∂
+
z
t
z
y
x
C
T
z
y
x
g
z
L
,
,
,
,
,
, 01
;
( )
0
,
,
,
0
=
=
x
ij
x
t
z
y
x
C
∂
∂
,
( )
0
,
,
,
=
= x
L
x
ij
x
t
z
y
x
C
∂
∂
,
( )
0
,
,
,
0
=
=
y
ij
y
t
z
y
x
C
∂
∂
,
( )
0
,
,
,
=
= y
L
y
ij
y
t
z
y
x
C
∂
∂
,
( )
0
,
,
,
0
=
=
z
ij
z
t
z
y
x
C
∂
∂
,
( )
0
,
,
,
=
= z
L
z
ij
z
t
z
y
x
C
∂
∂
, 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
2
1
,
,
,
n
nC
n
n
n
nC
z
y
x
z
y
x
t
e
z
c
y
c
x
c
F
L
L
L
L
L
L
t
z
y
x
C ,
where ( )
















+
+
−
= 2
2
2
0
2
2 1
1
1
exp
z
y
x
C
nC
L
L
L
t
D
n
t
e π , ( ) ( ) ( ) ( )
∫ ∫ ∫
=
x y z
L L L
C
n
n
n
nC u
d
v
d
w
d
w
v
u
f
w
c
v
c
u
c
F
0 0 0
,
, ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0 0 0 0
2
0 ,
,
,
2
,
,
,
n
t L L L
L
n
n
nC
nC
n
n
n
nC
z
y
x
i
x y z
T
w
v
u
g
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
t
z
y
x
C τ
π
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫
−
−
∂
∂
×
∞
=
−
1 0 0
2
10 2
,
,
,
n
t L
n
nC
nC
n
n
n
nC
z
y
x
i
n
x
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
w
c τ
π
τ
τ
( ) ( ) ( )
( )
( ) ( ) ( ) ×
∑
−
∫ ∫
∂
∂
×
∞
=
−
1
2
0 0
10 2
,
,
,
,
,
,
n
n
n
n
nC
z
y
x
L L
i
L
n
n z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
g
v
c
v
s
y z π
τ
τ
( ) ( ) ( ) ( ) ( ) ( )
( )
∫ ∫ ∫ ∫
∂
∂
−
× −
t L L L
i
L
n
n
n
nC
nC
x y z
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
g
v
s
v
c
u
c
e
t
e
0 0 0 0
10 ,
,
,
,
,
, τ
τ
τ , i≥1;

72
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0 0 0 0
00
2
01
,
,
,
,
,
,
2
,
,
,
n
t L L L
n
n
nC
nC
n
n
n
nC
z
y
x
x y z
T
w
v
u
P
w
v
u
C
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
t
z
y
x
C γ
γ
τ
τ
π
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫
−
−
∂
∂
×
∞
=1 0 0
2
00 2
,
,
,
n
t L
n
nC
nC
n
n
n
nC
z
y
x
n
x
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
w
c τ
π
τ
τ
( ) ( )
( )
( )
( )
( ) ( ) ( ) ×
∑
−
∫ ∫
∂
∂
×
∞
=1
2
0 0
00
00 2
,
,
,
,
,
,
,
,
,
n
n
n
n
nC
z
y
x
L L
n
n z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
P
w
v
u
C
w
c
v
s
y z π
τ
τ
τ
γ
γ
( ) ( ) ( ) ( ) ( )
( )
( )
( )
∫ ∫ ∫ ∫
∂
∂
−
×
t L L L
n
n
n
nC
nC
x y z
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
P
w
v
u
C
w
s
v
c
u
c
e
t
e
0 0 0 0
00
00 ,
,
,
,
,
,
,
,
,
τ
τ
τ
τ γ
γ
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0 0 0 0
01
2
02 ,
,
,
2
,
,
,
n
t L L L
n
n
nC
nC
n
n
n
nC
z
y
x
x y z
w
v
u
C
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
t
z
y
x
C τ
τ
π
( )
( )
( )
( )
( ) ( ) ( ) ( ) ×
∑
−
∂
∂
×
∞
=
−
1
2
00
1
00 2
,
,
,
,
,
,
,
,
,
n
nC
n
n
n
nC
z
y
x
n t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
P
w
v
u
C
w
c
π
τ
τ
τ
γ
γ
( ) ( ) ( ) ( ) ( )
( )
( )
( )
−
∫ ∫ ∫ ∫
∂
∂
−
×
−
t L L L
n
n
n
nC
x y z
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
u
C
w
c
v
s
u
c
e
0 0 0 0
00
1
00
01
,
,
,
,
,
,
,
,
,
,
,
, τ
τ
τ
τ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
×
∑ ∫ ∫ ∫ ∫
∂
∂
−
−
∞
=1 0 0 0 0
00
2
,
,
,
2
n
t L L L
n
n
n
nC
nC
n
n
n
nC
z
y
x
x y z
w
w
v
u
C
w
s
v
c
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L
τ
τ
π
( )
( )
( )
( ) ( ) ( ) ( ) ( )×
∑ ∫ −
−
×
∞
=
−
1 0
2
1
00
01
2
,
,
,
,
,
,
,
,
,
n
t
nC
nC
n
n
n
nC
z
y
x
e
t
e
z
c
y
c
x
c
F
L
L
L
d
u
d
v
d
w
d
T
w
v
u
P
w
v
u
C
w
v
u
C τ
π
τ
τ
τ γ
γ
( ) ( ) ( )
( )
( )
( )
×
∑
−
∫ ∫ ∫
∂
∂
×
∞
=
−
1
2
0 0 0
00
1
00
01
2
,
,
,
,
,
,
,
,
,
,
,
,
n
nC
z
y
x
L L L
n
n F
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
u
C
v
c
u
s
n
x y z π
τ
τ
τ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
×
∫ ∫ ∫ ∫
∂
∂
−
×
−
t L L L
n
n
nC
nC
n
n
n
x y z
w
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
u
C
v
c
u
c
e
t
e
z
c
y
c
x
c
n
0 0 0 0
00
1
00
01
,
,
,
,
,
,
,
,
,
,
,
,
τ
τ
τ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
−
−
×
∞
=1 0 0 0 0
2
2
n
t L L L
n
n
n
nC
nC
n
n
n
nC
z
y
x
n
x y z
w
c
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
w
s τ
π
τ
( )
( )
( )
( ) ( ) ( ) ( ) ( ) ×
∑ ∫ −
−
∂
∂
×
∞
=1 0
2
01
00 2
,
,
,
,
,
,
,
,
,
n
t
nC
nC
n
n
n
nC
z
y
x
e
t
e
z
c
y
c
x
c
F
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
P
w
v
u
C
τ
π
τ
τ
τ
γ
γ
( ) ( ) ( )
( )
( )
( )
( ) ×
∑
−
∫ ∫ ∫
∂
∂
×
∞
=1
2
0 0 0
01
00 2
,
,
,
,
,
,
,
,
,
n
n
nC
z
y
x
L L L
n
n
n x
c
F
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
P
w
v
u
C
w
c
v
s
u
c
n
x y z π
τ
τ
τ
γ
γ
| ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
∫ ∫ ∫ ∫
∂
∂
−
×
t L L L
n
n
n
nC
nC
n
n
x y z
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
P
w
v
u
C
w
s
v
c
u
c
e
t
e
z
c
y
c
0 0 0 0
01
00 ,
,
,
,
,
,
,
,
,
τ
τ
τ
τ γ
γ
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0 0 0 0
2
11 ,
,
,
2
,
,
,
n
t L L L
L
n
n
nC
nC
n
n
n
nC
z
y
x
x y z
T
w
v
u
g
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
t
z
y
x
C τ
π
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫
−
−
∂
∂
×
∞
=1 0 0
2
01 2
,
,
,
n
t L
n
nC
nC
n
n
n
nC
z
y
x
n
x
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
w
c τ
π
τ
τ
( ) ( ) ( )
( )
( ) ( ) ( ) ×
∑
−
∫ ∫
∂
∂
×
∞
=1
2
0 0
01 2
,
,
,
,
,
,
n
n
n
n
nC
z
y
x
L L
L
n
n z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
g
w
c
v
s
y z π
τ
τ

73
( ) ( ) ( ) ( ) ( ) ( )
( )
×
−
∫ ∫ ∫ ∫
∂
∂
−
×
z
y
x
t L L L
L
n
n
n
nC
nC
L
L
L
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
g
w
s
v
c
u
c
e
t
e
x y z
2
0 0 0 0
01 2
,
,
,
,
,
,
π
τ
τ
τ
( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
×
∑ ∫ ∫ ∫ ∫
∂
∂
−
×
∞
=1 0 0 0 0
10
00 ,
,
,
,
,
,
,
,
,
n
t L L L
n
n
n
nC
nC
n
nC
x y z
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
P
w
v
u
C
w
c
v
c
u
s
e
t
e
x
c
F
n τ
τ
τ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
×
∑ ∫ ∫ ∫ ∫
−
−
×
∞
=1 0 0 0 0
00
2
,
,
,
,
,
,
2
n
t L L L
n
n
n
nC
nC
n
n
n
z
y
x
n
n
x y z
T
w
v
u
P
w
v
u
C
w
c
v
s
u
c
e
t
e
z
c
y
c
x
c
n
L
L
L
z
c
y
c γ
γ
τ
τ
π
( )
( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫
−
−
∂
∂
×
∞
=1 0 0
2
10 2
,
,
,
n
t L
n
nC
nC
n
n
n
nC
z
y
x
nC
x
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
C
F τ
π
τ
τ
( ) ( )
( )
( )
( )
( ) ( ) ( ) ×
∑
−
∫ ∫
∂
∂
×
∞
=1
2
0 0
10
00 2
,
,
,
,
,
,
,
,
,
n
n
n
n
nC
z
y
x
L L
n
n z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
P
w
v
u
C
w
s
v
c
y z π
τ
τ
τ
γ
γ
( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
−
∫ ∫ ∫ ∫
∂
∂
−
×
−
t L L L
n
n
n
nC
nC
x y z
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
u
C
w
c
v
c
u
s
e
t
e
0 0 0 0
00
1
00
10
,
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, τ
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τ γ
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( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
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( )
×
∑ ∫ ∫ ∫ ∫
∂
∂
−
−
∞
=
−
1 0 0 0 0
00
1
00
2
,
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2
n
t L L L
n
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L
L
L
τ
τ
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π
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( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫
−
−
×
∞
=1 0 0 0
2
10
2
,
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τ
( ) ( )
( )
( )
( )
∫
∂
∂
×
−
z
L
n
nC d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
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C
w
s
F
0
00
1
00
10
,
,
,
,
,
,
,
,
,
,
,
, τ
τ
τ
τ γ
γ
.
Short Biographies:
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. Now E.L. Pankratov is in his Full Doctor
course in Radiophysical Department of Nizhny Novgorod State University. He has 96 published papers in
area of his researches.
Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school
of village Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school
“Center for gifted children”. From 2009 she is a student of Nizhny Novgorod State University of
Architecture and Civil Engineering (spatiality “Assessment and management of real estate”). At the same
time she is a student of courses “Translator in the field of professional communication” and “Design
(interior art)” in the University. E.A. Bulaeva was a contributor of grant of President of Russia (grant №
MK-548.2010.2). She has 29 published papers in area of her researches.

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An Approach to Optimize Regimes of Manufacturing of Complementary Horizontal Field-Effect Transistor