In this paper we introduced an approach to model manufacture process of a multiemitter heterotransistor. The approach gives us also possibility to optimize manufacture process. p-n-junctions, which include into the multiemitter heterotransistor, have higher sharpness and higher homogeneity of dopants distributions in enriched by the dopant area.

1. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
1
AN APPROACH FOR OPTIMIZATION OF
MANUFACTURE MULTIEMITTER
HETEROTRANSISTORS
E.L. Pankratov1
and E.A. Bulaeva2
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 introduced an approach to model manufacture process of a multiemitter heterotransistor.
The approach gives us also possibility to optimize manufacture process. p-n-junctions, which include into
the multiemitter heterotransistor, have higher sharpness and higher homogeneity of dopants distributions
in enriched by the dopant area.
KEYWORDS
Multiemitter heterotransistor, Approach to optimize technological processes.
1. INTRODUCTION
Logical elements often include into itself multiemitter transistors [1-4]. To manufacture bipolar
transistors it could be used dopant diffusion, ion doping or epitaxial layer [1-8]. It is attracted an
interest increasing of sharpness of p-n-junctions, which include into bipolar transistors and
decreasing of dimensions of the transistors, which include into integrated circuits [1-4].
Dimensions of transistors will be decreased by using inhomogenous distribution of temperature
during laser or microwave types of annealing [9,10], using defects of materials (for example,
defects could be generated during radiation processing of materials) [11-14], native
inhomogeneity (multilayer property) of heterostructure [12-14].
Framework this paper we consider a heterostructure, which consist of a substrate and three
epitaxial layers (see Fig. 1). Some dopants are infused in the epitaxial layers to manufacture p and
n types of conductivities during manufacture a multiemitter transistor. Let us consider two cases:
infusion of dopant by diffusion or ion implantation. Farther in the first case we consider annealing
of dopant so long, that after the annealing the dopants should achieve interfaces between epitaxial
layers. In this case one can obtain increasing of sharpness of p-n-junctions [12-14]. The
increasing of sharpness could be obtained under conditions, when values of dopant diffusion
coefficients in last epitaxial layers is larger, than values of dopant diffusion coefficient in average
epitaxial layer. At the same time homogeneity of dopant distributions in doped areas increases. It
is attracted an interest higher doping of last epitaxial layers, than doping of average epitaxial
layer, because the higher doping gives us possibility to increase sharpness of p-n-junctions. It is

2. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
2
also practicably to choose materials of epitaxial layers and substrate so; those values of dopant
diffusion coefficients in the substrate should be smaller, than in the epitaxial layer. In this
situation one can obtain thinner transistor. In the case of ion doping of heterostructure it should be
done annealing of radiation defects. It is practicably to choose so regimes of annealing of
radiation defects, that dopant should achieves interface between epitaxial layers. At the same time
the dopant could not diffuse into another epitaxial layer. If dopant did not achieves interface
between epitaxial layers during annealing of radiation defects, additional annealing of dopant
required.
Main aims of the present paper are analysis of dynamics of redistribution of dopants in considered
heterostructure (Figs. 1) and optimization of annealing times of dopants. In some recent papers
analogous analysis have been done [4,10,12-14]. However we can not find in literature any
similar heterostructures as we consider in the paper.
Substrate
Epitaxial
layer 1
Epitaxial
layer 2
Epitaxial
layer 3
n-type dopant p-type dopant n-type dopant
Fig. 1a. Heterostructure, which consist of a substrate and three epitaxial layers. View from above
Epitaxial layer 1 Epitaxial layer 2 Epitaxial layer 3
n-type dopantp-type dopant
n-type dopant 1
n-type dopant 2
n-type dopant 3
n-type dopant 4
Fig. 1b. Heterostructure, which consist of a substrate and three epitaxial layers. View from one side

3. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
3
2. Method of Analysis
Redistribution of dopant in the considered heterostructure has been described by the second
Fick’s low [1-4]
( ) ( ) ( ) ( )






+





+





=
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC
CCC
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,,,,,,,,,,
(1)
with boundary and initial conditions
( ) 0
,,,
0
=
∂
∂
=x
x
tzyxC
,
( ) 0
,,,
=
∂
∂
= xLx
x
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=y
y
tzyxC
,
( ) 0
,,,
=
∂
∂
= yLy
y
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=z
z
tzyxC
,
( ) 0
,,,
=
∂
∂
= zLz
z
tzyxC
, C (x,y,z,0)=fC (x,y,z). (2)
Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant; x, y, z and t are
current coordinates and times; 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 heterostructure (with account Arrhenius low), spatio-temporal distributions of
concentrations of dopants and radiation defects. Two last dependences of dopant diffusion
coefficient could be approximated by the following relation [15,16]
( ) ( )
( )
( ) ( )
( ) 





++





+= 2*
2
2*1
,,,,,,
1
,,,
,,,
1,,,
V
tzyxV
V
tzyxV
TzyxP
tzyxC
TzyxDD LC
ςςξ γ
γ
. (3)
Here DL (x,y,z,T) is the spatial (due to inhomogeneity (многослойности) of heterostructure) and
temperature (due to Arrhenius low) dependences of dopant diffusion coefficient; P (x,y,z,T) is the
limit of solubility of dopant; parameter γ depends of properties of materials and could be integer
in the interval γ ∈[1,3] [15]; V (x,y,z,t) is the spatio-temporal distribution of concentration of
radiation vacancies; V*
is the equilibrium distribution of vacancies. Concentrational dependence
of dopant diffusion coefficient has been discussed in details in [15]. It should be noted, that
radiation damage absents in the case of diffusion doping of haterostructure. In this situation ζ1=
ζ2= 0. Spatio-temporal distributions of concentrations of point radiation defects could be
determine by solution of the following system of equations [17-19]
( ) ( ) ( ) ( ) ( ) ( )×−





∂
∂
∂
∂
+





∂
∂
∂
∂
=
∂
∂
tzyxI
y
tzyxI
TzyxD
yx
tzyxI
TzyxD
xt
tzyxI
II
,,,
,,,
,,,
,,,
,,,
,,, 2
( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk
z
tzyxI
TzyxD
z
Tzyxk VIIII
,,,,,,,,,
,,,
,,,,,, ,,
−





∂
∂
∂
∂
+× (4)
( ) ( ) ( ) ( ) ( ) ( )×−





∂
∂
∂
∂
+





∂
∂
∂
∂
=
∂
∂
tzyxV
y
tzyxV
TzyxD
yx
tzyxV
TzyxD
xt
tzyxV
VV
,,,
,,,
,,,
,,,
,,,
,,, 2
( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk
z
tzyxV
TzyxD
z
Tzyxk VIVVV
,,,,,,,,,
,,,
,,,,,, ,,
−





∂
∂
∂
∂
+×

4. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
4
with initial
ρ(x,y,z,0)=fρ (x,y,z) (5a)
and boundary conditions
( ) 0
,,,
=
∂
∂
= yLy
y
tzyxρ
,
( ) 0
,,,
0
=
∂
∂
=z
z
tzyxρ
,
( ) 0
,,,
=
∂
∂
= zLz
z
tzyxρ
. (5b)
Here ρ =I,V; I (x,y,z,t) is the spatio-temporal distribution of concentration of interstitials; Dρ(x,y,
z,T) are diffusion coefficients of interstitials and vacancies; terms V2
(x,y, z,t) and I2
(x,y,z,t)
correspond to generation of divacancies and analogous complexes of interstitials; kI,V(x,y,z,T),
kI,I(x,y,z,T) and kV,V(x,y,z,T) parameters of recombination of point radiation defects and generation
of complexes.
Spatio-temporal distributions of concentrations of divacancies ΦV(x,y,z,t) and analogous
complexes of interstitials ΦI(x,y,z,t) will be determined by solving the following systems of
equations [17-19]
( )
( )
( )
( )
( )
+




 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
,
( )
0
,,,
0
=
∂
Φ∂
=z
z
tzyxρ
,
( )
0
,,,
=
∂
Φ∂
= zLz
z
tzyxρ
, ΦI(x,y,z,0)=fΦI(x,y,z), ΦV(x,y,z,0)=fΦV (x,y,z). (7)
Here 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 the complexes.
To determine spatio-temporal distributions of concentrations of complexes of radiation defects we
used considered in Refs. [14,19] approach. Framework the approach we transform approxi-
mations of diffusion coefficients of complexes of point radiation defects to the following form:
Dρ (x,y,z,T) = D0ρ [1+ερ gρ (x,y,z,T)] and kI,V (x,y,z,T)=k0I,V[1+ζ h(x,y,z,T)], where D0ρ and k0I,V are
the average values of appropriate parameters, 0≤ερV <1, 0≤ζ<1, | gρ(x,y,z,T)|≤1, |h(x,y,z,T)|≤1. Let
us introduced the following dimensionless variables: ( ) ( )
*
,,,
,,,~
ρ
ρ
ρ
tzyx
tzyx = , χ=x/Lx, η= y/Ly,
ψ = z/Lz,








++= 22200
111
zyx
VVVI
LLL
DDtϑ , ( )
VI
zyxVI
DD
VI
LLLk
00
**
222
,0
++=ω , 2
2
2
2
1
z
x
y
x
x
L
L
L
L
b ++= ,

5. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
5
2
2
2
2
1
z
y
x
y
y
L
L
L
L
b ++= , 2
2
2
2
1
y
z
x
z
z
L
L
L
L
b ++= . Using the above variables leads to transformation Eqs. (4) to
the following form
( ) ( )[ ] ( ) ×+






+=
V
I
y
II
V
I
x
D
D
b
I
Tg
D
D
b
I
0
0
0
0 1,,,
~
,,,1
1,,,
~
χ∂
ϑψηχ∂
ψηχε
χ∂
∂
ϑ∂
ϑψηχ∂
( )[ ] ( ) ( )[ ] ( ) ×








++








+×
ψ∂
ϑψηχ∂
ψηχε
ψ∂
∂
η∂
ϑψηχ∂
ψηχε
η∂
∂ ,,,
~
,,,1
1,,,
~
,,,1
I
Tg
b
I
Tg II
z
II
( )[ ] ( ) ( )ϑψηχϑψηχψηχςω ,,,
~
,,,
~
,,,1*
*
0
0
VITh
I
V
D
D
V
I
+−× (8)
( ) ( )[ ] ( ) ×+






+=
I
V
y
VV
I
V
x
D
D
b
V
Tg
D
D
b
V
0
0
0
0 1,,,
~
,,,1
1,,,
~
χ∂
ϑψηχ∂
ψηχε
χ∂
∂
ϑ∂
ϑψηχ∂
( )[ ] ( ) ( )[ ] ( ) ×






++






+×
ψ∂
ϑψηχ∂
ψηχε
ψ∂
∂
η∂
ϑψηχ∂
ψηχε
η∂
∂ ,,,
~
,,,1
,,,
~
,,,1
V
Tg
V
Tg VVVV
( )[ ] ( ) ( )ϑψηχϑψηχψηχςω ,,,
~
,,,
~
,,,1
1
*
*
0
0
VITh
V
I
D
D
b I
V
z
+−× .
We will determine solution of Eqs. (8) as the following power series
( ) ( )∑ ∑ ∑=
∞
=
∞
=
∞
=0 0 0
~ ,,,~,,,~
i j k
ijk
kji
ϑψηχρςωεϑψηχρ ρ
. (9)
Substitution of the series into Eqs. (8) gives us possibility to obtain system of equations for
initial-order approximations of concentrations of defects ( )ϑψηχρ ,,,~
000 and corrections for them
( )ϑψηχρ ,,,~
ijk
(i≥1, j≥1, k≥1). The equations are presented in Appendix.
Substitution of the series (9) into appropriate boundary and initial conditions gives us possibility
to obtain boundary and initial conditions for the functions ( )ϑψηχρ ,,,~
ijk . The conditions dard
approaches [21,22], are presented in Appendix. and solutions for the functions ( )ϑψηχρ ,,,~
ijk
(i≥0, j≥0, k≥0), which have been obtained by stan
Farther we determine spatio-temporal distributions of concentrations of point radiation defects.
To calculate the distributions we transform approximations of diffusion coefficients to the
following form: DΦI(x,y,z,T)=D0ΦI[1+εΦIgΦI(x,y,z,T)], DΦV(x,y,z,T)= D0ΦV[1+εΦVgΦV (x,y,z,T)]. In
this situation Eqs. (6) will be transform to the following form
( )
( )[ ] ( )
( )[ ]






×++





 Φ
+=
Φ
ΦΦΦΦ
Tzyxg
yx
tzyx
Tzyxg
xt
tzyx
II
I
II
I
,,,1
,,,
,,,1
,,,
ε
∂
∂
∂
∂
ε
∂
∂
∂
∂
( )
( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkD
y
tzyx
IIII
I
,,,,,,,,,,,,
,,, 2
,0
+−


Φ
× Φ
∂
∂

6. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
6
( )
( )[ ] ( )
( )[ ]






×++





 Φ
+=
Φ
ΦΦΦΦ
Tzyxg
yx
tzyx
Tzyxg
xt
tzyx
VV
V
VV
V
,,,1
,,,
,,,1
,,,
ε
∂
∂
∂
∂
ε
∂
∂
∂
∂
( )
( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkD
y
tzyx
VVVV
V
,,,,,,,,,,,,
,,, 2
,0
+−


Φ
× Φ
∂
∂
.
We determine solutions of these equations as the following power series
( ) ( )∑ Φ=Φ
∞
=
Φ
0
,,,,,,
i
i
i
tzyxtzyx ρρρ
ε , (10)
where ρ =I,V. Substitution of the series (10) 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 radiation defects, corrections for them, boundary and initial
conditions for all above functions. The equations, conditions and solutions, which have been
obtained by standard approaches [21,22], are presented in the Appendix.
Analysis of spatio-temporal distributions of concentrations of radiation defects has been done by
using the second-order approximations on parameters, which used in appropriate series, and have
been defined more precisely by using numerical simulation.
Farther we determine solution of the Eq.(1). To calculate the solution we used the approach,
which has been considered in [13,18]. Framework the approach we transform approximation of
dopant diffusion coefficient DL(x,y,z,T) into following form: DL(x,y,z,T) =D0L[1+ε L gL(x,y,z,T)].
We determine solution of the Eq.(1) as the following power series
( ) ( )∑ ∑=
∞
=
∞
=0 0
,,
i j
ij
ji
L
txCtxC ξε . (11)
Substitution of the series into Eq. (1) gives us possibility to obtain system of equations for initial-
order approximation of dopant concentration C00(x,y,z,t) and corrections for them Cij(x,y,z,t) (i≥1,
j≥1). Substitution of the series (11) into appropriate boundary and initial conditions for the
functions Cij(x,y,z,t) (i≥0, j≥0)
( )
0
,,,
0
=
=x
ij
x
tzyxC
∂
∂
;
( )
0
,,,
=
= xLx
ij
x
tzyxC
∂
∂
;
( )
0
,,,
0
=
=y
ij
y
tzyxC
∂
∂
;
( )
0
,,,
=
= yLy
ij
y
tzyxC
∂
∂
;
( )
0
,,,
0
=
=z
ij
z
tzyxC
∂
∂
;
( )
0
,,,
=
= zLz
ij
z
tzyxC
∂
∂
; C00(x,y,z,0)=fC (x,y,z); Cij(x,y,z,0)=0, i≥0, j≥0.
Solutions of equations for the functions Cij(x,y,z,t) (i≥0, j≥0), which has been calculated by
standard approaches [21,22], are presented in the Appendix.
Analysis of spatio-temporal distribution of dopant concentration has been done analytically by
using the second-order approximation on parameters, which has been used in appropriate series
and have been defined more precisely by using numerical simulation.

7. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
7
3. Discussion
In this section we analyzed dynamics of redistribution of dopant with account relaxation of
distributions of concentrations of radiation defects (in the case of ion doping of heterostructure).
The Figs. 2 show distributions of concentrations of infused and implanted dopants in one of last
epitaxial layers in direction, which is perpendicular to interfaces between epitaxial layers, after
annealing of dopant and/or radiation defects, respectively. In this case values of dopant diffusion
coefficient in last epitaxial layers is larger, than value of dopant diffusion coefficient in average
epitaxial layer. The figures show, that interfaces between epitaxial layers under the above
conditions give us possibility to increase sharpness of p-n-junctions and at the same time to
increase homogeneity of distributions of concentrations of dopants in enriched by the dopants
areas.
The Figs. 3 show distributions of concentrations of infused and implanted dopants in one of last
epitaxial layers in direction, which is perpendicular to interfaces between epitaxial layers, after
annealing of dopant and/or radiation defects, respectively. In this case values of dopant diffusion
coefficient in last epitaxial layers is smaller, than value of dopant diffusion coefficient in average
epitaxial layer. The figures show, that presents of last epitaxial layers leads to increasing of
homogeneity of dopant in the average epitaxial layers. However sharpness of p-n-junctions
decreases in comparison with p-n-junctions for Figs. 2. In this case one can obtain accelerated
diffusion in last epitaxial layers. The decreasing of sharpness of p-n-junctions is a reason to use
higher level of doping of last epitaxial layer to use nonlinearity of dopant diffusion. In this case
one obtain n+
-p-n+
and/or p+
-n-p+
structures.
x
0.0
0.5
1.0
1.5
C(x,Θ)
1
2
3
4
5
6
0 L/4 L/2 3L/4 L
Interface
Epitaxial
layer2
Epitaxial
layer1
Fig. 2a. Distributions of concentrations of infused dopant in heterostructure from Fig. 1. Increasing of
number of curves corresponds to increasing of difference between values of dopant diffusion coefficient

8. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
8
x
0.0
0.5
1.0
1.5
2.0
C(x,Θ)
1
2
3
4
0 L/4 L/2 3L/4 L
Interface
Epitaxial
layer2
Epitaxial
layer1
Fig. 2b. Distributions of concentrations of implanted dopant in heterostructure from Fig. 1. Increasing of
number of curves corresponds to increasing of difference between values of dopant diffusion coefficient
x
0.0
0.5
1.0
1.5
C(x,Θ)
0 a=2L/3 L-a1 =-L/3
1
3
2
Fig. 3a. Distributions of concentration of infused in average epitaxial layer dopant in the heterostructure in
Fig. 1 without another epitaxial layers (curve 1) and with them for the case, when dopant diffusion
coefficient in the average epitaxial layer is smaller in comparison with dopant diffusion coefficients in
another layers (curves 2 and 3). Increasing of number of curves corresponds to increasing of difference
between values of dopant diffusion coefficients epitaxial layers of heterostructure
x
0.00001
0.00010
0.00100
0.01000
0.10000
1.00000
C(x,Θ)
fC(x)
L/40 L/2 3L/4 Lx0
1
2

9. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
9
Fig. 3b. Distributions of concentration of implanted in average epitaxial layer dopant in the heterostructure
in Fig. 1 without another epitaxial layers (curve 1) and with them for the case, when dopant diffusion
coefficient in the average epitaxial layer is smaller in comparison with dopant diffusion coefficients in
another layers (curves 2 and 3). Increasing of number of curves corresponds to increasing of difference
between values of dopant diffusion coefficients epitaxial layers of heterostructure
Optimal value of annealing time we determine framework recently introduced criterion [12-14].
Framework the criterion we approximate real distributions of dopant by step-wise function and
minimize the following mean-squared error
( ) ( )[ ]∫ −Θ=
L
xdxxC
L
U
0
2
,
1
ψ (12)
at annealing time Θ. Dependences of the optimal value of annealing time from different
parameters are presented on Figs. 4.
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. 4a. Dependences of dimensionless optimal value of annealing time of infused dopant, which has been
calculated by minimization the mean-squared error (12), on different parameters of heterostructure. Curve 1
is the dependence of annealing time on relation a/L, ξ=γ=0 and equal to each other values of dopant
diffusion coefficient in epitaxial layers. Curve 2 is the dependence of annealing time on parameter ε for
a/L=1/2 and ξ=γ=0. Curve 3 is the dependence of annealing time on parameter ξ for a/L=1/2 and ε=γ=0.
Curve 4 is the dependence of annealing time on 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
ΘD0L
-2
3
2
4
1

10. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
10
Fig. 4b. Dependences of dimensionless optimal value time of additional annealing of implanted dopant,
which has been calculated by minimization the mean-squared error (12), on different parameters of
heterostructure. Curve 1 is the dependence of annealing time on relation a/L, ξ=γ =0 and equal to each
other values of dopant diffusion coefficient in epitaxial layers. Curve 2 is the dependence of annealing time
on parameter ε for a/L=1/2 and ξ=γ=0. Curve 3 is the dependence of annealing time on parameter ξ for
a/L=1/2 and ε=γ=0. Curve 4 is the dependence of annealing time on parameter γ for a/L=1/2 and ε=ξ=0
The figures show, that optimal values of annealing time after ion implantation are smaller, than
optimal values of annealing time after infusion of dopant. Reason of the difference is necessity to
use annealing of radiation defects after ion implantation. After that (if necessary) it could be used
additional annealing of dopant. One can obtain spreading of distribution of concentration during
annealing of radiation defects. Increasing of annealing time with increasing of thickness of
epitaxial layer is the consequence of necessity of higher time for dopant to achieve interfaces
between epitaxial layers of heterostructure. Decreasing of annealing time with increasing of
values of parameters ε and ξ is the consequence of increasing of values of dopant diffusion
coefficient in the area, where main part of dopant has been diffused. Increasing of annealing time
with increasing of the parameter γ is the consequence of decreasing of the term [C (x,y,z,t)/ P
(x,y,z,T)]γ
in approximation of dopant diffusion coefficient (3) with increasing of the parameter γ,
because dopant concentration did not usually achieved value of limit of solubility of dopant and
parameter γ is not smaller, than 1.
4. Conclusion
In this paper we introduce an approach of manufacturing multiemitter heterotransistors.
Framework the approach the considered transistors became more compact and will include into
itself p-n-junctions with higher sharpness.
ACKNOWLEDGEMENTS
This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government,
grant of Scientific School of Russia and educational fellowship for scientific research.
APPENDIX
Equations and conditions for functions ( )ϑψηχρ ,,,~
ijk (i≥0, j≥0, k≥0) could be written as
( ) ( ) ( ) ( )








++= 2
000
2
2
000
2
2
000
2
0
0000
,,,
~
1,,,
~
1,,,
~
1,,,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑψηχ∂ I
b
I
b
I
bD
DI
zyxV
I
;
( ) ( ) ( ) ( )








++= 2
000
2
2
000
2
2
000
2
0
0000
,,,
~
1,,,
~
1,,,
~
1,,,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑψηχ∂ V
b
V
b
V
bD
DV
zyxI
V
;
( ) ( ) ( ) ( )
+








++= 2
00
2
2
00
2
2
00
2
0
000
,,,
~
1,,,
~
1,,,
~
1,,,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑψηχ∂ i
z
i
y
i
xV
Ii
I
b
I
b
I
bD
DI
( )
( )
( )
( )




+








+








+ −−
η∂
ϑψηχ∂
ψηχ
η∂
∂
χ∂
ϑψηχ∂
ψηχ
χ∂
∂ ,,,
~
,,,
1,,,
~
,,,
1 100100
0
0 i
I
y
i
I
xV
I
I
Tg
b
I
Tg
bD
D

11. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
11
( )
( )












+ −
ψ∂
ϑψηχ∂
ψηχ
ψ∂
∂ ,,,
~
,,,
1 100i
I
z
I
Tg
b
, i≥1;
( ) ( ) ( ) ( )
+








++= 2
00
2
2
00
2
2
00
2
0
000
,,,
~
1,,,
~
1,,,
~
1,,,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑψηχ∂ i
z
i
y
i
xI
Vi
V
b
V
b
V
bD
DV
( )
( )
( )
( )




+








+








+ −−
η∂
ϑψηχ∂
ψηχ
η∂
∂
χ∂
ϑψηχ∂
ψηχ
χ∂
∂ ,,,
~
,,,
1,,,
~
,,,
1 100100
0
0 i
V
y
i
V
xI
V
V
Tg
b
V
Tg
bD
D
( ) ( )










+ −
ψ∂
ϑψηχ∂
ψηχ
ψ∂
∂ ,,,
~
,,,
1 100i
V
z
V
Tg
b
, i≥1;
( ) ( ) ( ) ( )
−








++= 2
010
2
2
010
2
2
010
2
0
0010
,,,
~
1,,,
~
1,,,
~
1,,,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑψηχ∂ I
b
I
b
I
bD
DI
zyxV
I
( ) ( )ϑψηχϑψηχ ,,,
~
,,,
~
000000*
*
VI
I
V
− ;
( ) ( ) ( ) ( )
−








++= 2
010
2
2
010
2
2
010
2
0
0010
,,,
~
1,,,
~
1,,,
~
1,,,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑψηχ∂ V
b
V
b
V
bD
DV
zyxI
V
( ) ( )ϑψηχϑψηχ ,,,
~
,,,
~
000000*
*
VI
I
V
− ;
( ) ( ) ( ) ( )
−








++= 2
020
2
2
020
2
2
020
2
0
0020
,,,
~
1,,,
~
1,,,
~
1,,,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑψηχ∂ I
b
I
b
I
bD
DI
zyxV
I
( ) ( ) ( ) ( )ϑψηχϑψηχϑψηχϑψηχ ,,,
~
,,,
~
,,,
~
,,,
~
010000*
*
000010*
*
VI
I
V
VI
I
V
−−
( ) ( ) ( ) ( )
−








++= 2
020
2
2
020
2
2
020
2
0
0020
,,,
~
1,,,
~
1,,,
~
1,,,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑψηχ∂ V
b
V
b
V
bD
DV
zyxI
V
( ) ( ) ( ) ( )ϑψηχϑψηχϑψηχϑψηχ ,,,
~
,,,
~
,,,
~
,,,
~
010000*
*
000010*
*
VI
V
I
VI
V
I
−− ;
( ) ( ) ( ) ( )
+








++= 2
110
2
2
110
2
2
110
2
0
0110
,,,
~
1,,,
~
1,,,
~
1,,,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑψηχ∂ I
b
I
b
I
bD
DI
zyxV
I
( )
( )
( )
( )




+








+








+
η∂
ϑψηχ∂
ψηχ
η∂
∂
χ∂
ϑψηχ∂
ψηχ
χ∂
∂ ,,,
~
,,,
1,,,
~
,,,
1 010010
0
0
I
Tg
b
I
Tg
bD
D
I
y
I
xV
I
( )
( )
( ) ( ) ×−−












+ *
*
000100*
*
010
,,,
~
,,,
~,,,
~
,,,
1
I
V
VI
I
VI
Tg
b
I
z
ϑψηχϑψηχ
ψ∂
ϑψηχ∂
ψηχ
ψ∂
∂
( ) ( )ϑψηχϑψηχ ,,,
~
,,,
~
100000
VI× ;
( ) ( ) ( ) ( )
+








++= 2
110
2
2
110
2
2
110
2
0
0110
,,,
~
1,,,
~
1,,,
~
1,,,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑψηχ∂ V
b
V
b
V
bD
DV
zyxI
V
( )
( )
( )
( )




+








+








+
η∂
ϑψηχ∂
ψηχ
η∂
∂
χ∂
ϑψηχ∂
ψηχ
χ∂
∂ ,,,
~
,,,
1,,,
~
,,,
1 010010
0
0
V
Tg
b
V
Tg
bD
D
V
y
V
xI
V

12. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
12
( )
( )
( ) ( ) ×−−












+ *
*
000100*
*
010
,,,
~
,,,
~,,,
~
,,,
1
I
I
VI
I
IV
Tg
b
V
z
ϑψηχϑψηχ
ψ∂
ϑψηχ∂
ψηχ
ψ∂
∂
( ) ( )ϑψηχϑψηχ ,,,
~
,,,
~
100000
VI× ;
( ) ( ) ( ) ( )








++= 2
00
2
2
00
2
2
00
2
0
000
,,,
~
1,,,
~
1,,,
~
1,,,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑψηχ∂ k
z
k
y
k
xV
Ik
I
b
I
b
I
bD
DI
( ) ( ) ( ) ( )








++= 2
00
2
2
00
2
2
00
2
0
000
,,,
~
1,,,
~
1,,,
~
1,,,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑψηχ∂ k
z
k
y
k
xI
Vk
I
b
I
b
I
bD
DV
;
( ) ( ) ( ) ( ) +








++= 2
101
2
2
101
2
2
101
2
0
0101
,,,
~
1,,,
~
1,,,
~
1,,,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑψηχ∂ I
b
I
b
I
bD
DI
zyxV
I
( )
( )
( )
( )




+








+








+
η∂
ϑψηχ∂
ψηχ
η∂
∂
χ∂
ϑψηχ∂
ψηχ
χ∂
∂ ,,,
~
,,,
1,,,
~
,,,
1 001001
0
0
I
Tg
b
I
Tg
bD
D
I
y
I
xV
I
( )
( )












+
ψ∂
ϑψηχ∂
ψηχ
ψ∂
∂ ,,,
~
,,,
1 001
I
Tg
b
I
z
;
( ) ( ) ( ) ( )
+








++= 2
101
2
2
101
2
2
101
2
0
0101
,,,
~
1,,,
~
1,,,
~
1,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑχ∂ V
b
V
b
V
bD
DV
zyxI
V
( )
( )
( )
( )




+








+








+
η∂
ϑψηχ∂
ψηχ
η∂
∂
χ∂
ϑψηχ∂
ψηχ
χ∂
∂ ,,,
~
,,,
1,,,
~
,,,
1 001001
0
0
I
Tg
b
I
Tg
bD
D
V
y
V
xI
V
( )
( )












+
ψ∂
ϑψηχ∂
ψηχ
ψ∂
∂ ,,,
~
,,,
1 001
I
Tg
b
V
z
;
( ) ( ) ( ) ( )
+








++= 2
011
2
2
011
2
2
011
2
0
0011
,,,
~
1,,,
~
1,,,
~
1,,,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑψηχ∂ I
b
I
b
I
bD
DI
zyxV
I
( ) ( ) ( ) ( ) ( )[ −−− ϑψηχϑψηχψηχϑψηχϑψηχ ,,,
~
,,,
~
,,,,,,
~
,,,
~
000000000001*
*
VIThVI
I
V
( ) ( )]ϑψηχϑψηχ ,,,
~
,,,
~
001000
VI−
( ) ( ) ( ) ( )
−








++= 2
011
2
2
011
2
2
011
2
0
0011
,,,
~
1,,,
~
1,,,
~
1,,,
~
ψ∂
ϑψηχ∂
η∂
ϑψηχ∂
χ∂
ϑψηχ∂
ϑ∂
ϑψηχ∂ V
b
V
b
V
bD
DV
zyxI
V
( ) ( ) ( ) ( ) ( )[ −−− ϑψηχϑψηχψηχϑψηχϑψηχ ,,,
~
,,,
~
,,,,,,
~
,,,
~
000000000001*
*
VIThVI
I
V
( ) ( )]ϑψηχϑψηχ ,,,
~
,,,
~
001000
VI− .
( )
0
,,,~
0
=
=χ
χ∂
ϑψηχρ∂ ijk
;
( )
0
,,,~
1
=
=χ
χ∂
ϑψηχρ∂ ijk
;
( )
0
,,,~
0
=
=η
η∂
ϑψηχρ∂ ijk
;
( )
0
,,,~
1
=
=η
η∂
ϑψηχρ∂ ijk
;
( )
0
,,,~
0
=
=ψ
ψ∂
ϑψηχρ∂ ijk
;
( )
0
,,,~
1
=
=ψ
ψ∂
ϑψηχρ∂ ijk
, i≥1, j≥1, k≥1; ( )
( )
*
,,
0,,,~
000
ρ
ψηχ
ψηχρ ρ
f
= ;
( ) 00,,,~ =ψηχρijk
, i≥1, j≥1, k≥1.
Solutions of equations for the functions ( )ϑψηχρ ,,,~
ijk
(i≥0, j≥0, k≥0) could be written as

13. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
13
( ) ( ) ( ) ( ) ( )∑+=
∞
=1
000
21
,,,~
n
nn
ecccF
LL
ϑψηχϑψηχρ ρρ
,
where ( ) ( ) ( ) ( )∫ ∫ ∫=
1
0
1
0
1
0
*
,,
1
udvdwdwvufwcvcucF nn ρρ
ρ
, cn(α) = cos(πn α), ( ) ( )IVnI
DDne 00
22
exp ϑπϑ −= ,
( ) ( )VInV
DDne 00
22
exp ϑπϑ −= ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
0
2
0
00
,,,
*
2
,,,~
n
Vnnnnnnn
zyxx
i
Twvugvcuseecccn
LLLb
D ϑ
ρρ
ρ
τϑψηχ
ρ
π
ϑψηχρ
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ −−
∂
∂
×
∞
=
−
1 0
2
0100
*
2,,,~
n
nnnnn
zyxy
i
n
eecccn
LLLb
D
dudvdwd
u
wvu
wc
ϑ
ρρ
ρ
τϑψηχ
ρ
π
τ
τρ
( ) ( ) ( ) ( )
( )
( ) ×∑−∫ ∫ ∫
∂
∂
×
∞
=
−
1
2
0
1
0
1
0
1
0
100
*
2,,,~
,,,
n
n
zyxz
i
Vnnn
cn
LLLb
D
dudvdwd
v
wvu
Twvugwcvsuc χ
ρ
π
τ
τρ ρ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫
∂
∂
−× −
ϑ
ρρ
τ
τρ
τϑψη
0
1
0
1
0
1
0
100
,,,
,,,~
dudvdwdTwvug
w
wvu
wsvcuceecc V
i
nnnnnnn
, i≥1;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
0
*
*
0
010
*
2
,,,
~
n
nnnInInnnn
zyx
I
wcvcuceeccc
I
V
LLLI
D
I
ϑ
τϑψηχϑψηχ
( ) ( ) τττ dudvdwdwvuVwvuI ,,,
~
,,,
~
000000
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
0
*
*
0
010
*
2
,,,
~
n
nnnVnVnnnn
zyx
I
wcvcuceeccc
V
I
LLLV
D
V
ϑ
τϑψηχϑψηχ
( ) ( ) τττ dudvdwdwvuVwvuI ,,,
~
,,,
~
000000
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
0
*
*
0
020
*
2
,,,
~
n
nnnInInnnn
zyx
I
wcvcuceeccc
I
V
LLLI
D
I
ϑ
τϑψηχϑψηχ
( ) ( ) ( ) ( ) ( ) ( ) ×∑−×
∞
=1
*
*
0
000010
*
2
,,,
~
,,,
~
n
Innnn
zyx
I
eccc
I
V
LLLI
D
dudvdwdwvuVwvuI ϑψηχτττ
( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫−×
ϑ
ττττ
0
1
0
1
0
1
0
010000
,,,
~
,,,
~
dudvdwdwvuVwvuIwcvcuce nnnIn
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
0
*
*
0
020
*
2
,,,
~
n
nnnVnVnnnn
zyx
I
wcvcuceeccc
V
I
LLLV
D
V
ϑ
τϑψηχϑψηχ
( ) ( ) ( ) ( ) ( ) ( )×∑−×
∞
=1
*
*
0
000010
*
2
,,,
~
,,,
~
n
Vnnnn
zyx
I
eccc
V
I
LLLV
D
dudvdwdwvuVwvuI ϑψηχτττ
( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫−×
ϑ
ττττ
0
1
0
1
0
1
0
010000
,,,
~
,,,
~
dudvdwdwvuVwvuIwcvcuce nnnVn
;
( ) ( ) ( ) ( ) 0,,,
~
,,,
~
,,,
~
,,,
~
002001002001
==== ϑψηχϑψηχϑψηχϑψηχ VVII ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−=
∞
=1 0
1
0
1
0
1
0
*
*
2
0
110
*
2
,,,
~
n
nnnInInnnn
zyxx
I
wcvcuseecccn
I
V
LLLIb
D
I
ϑ
τϑψηχ
π
ϑψηχ
( )
( )
( ) ( ) ( ) ( ) ×∑+×
∞
=1
*
*
2
0010
*
2,,,
~
,,,
n
Innnn
zyxy
I
I
eccc
I
V
LLLIb
D
dudvdwd
u
wvuI
Twvug ϑψηχ
π
τ
∂
τ∂
( ) ( ) ( ) ( ) ( ) ( ) ×+∫ ∫ ∫ ∫−× 2
0
0
1
0
1
0
1
0
010
*
2,,,
~
,,,
zyxz
I
InnnIn
LLLIb
D
dudvdwd
v
wvuI
Twvugwcvsucen
π
τ
∂
τ∂
τ
ϑ

14. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
14
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−×
∞
=1 0
1
0
1
0
1
0
010
*
*
,,,
~
,,,
n
InnnInnn
dudvdwd
w
wvuI
Twvugwsvcuceccn
I
V ϑ
τ
∂
τ∂
τηχ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−×
∞
=1 0
1
0
1
0
1
0
100*
*
0
,,,
~
*
2
n
nnInInnnn
zyx
I
Inn
wvuIvcuceeccc
I
V
LLLI
D
ec
ϑ
ττϑψηχϑψ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−×
∞
=1 0
1
0
1
0
1
0
*
*
0
000
*
2
,,,
~
n
nnnInInn
I
n
wcvcuceec
I
V
LI
D
dudvdwdwvuVwc
ϑ
τϑψττ
( ) ( ) ( ) ( )ηχτττ nn
ccdudvdwdwvuVwvuI ,,,
~
,,,
~
100000
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−=
∞
=1 0
1
0
1
0
1
0
*
*
2
0
110
*
2
,,,
~
n
nnnVnVnnnn
zyxx
V
wcvcuseecccn
V
I
LLLIb
D
V
ϑ
τϑψηχ
π
ϑψηχ
( )
( )
( ) ( ) ( ) ( ) ×∑+×
∞
=1
*
*
2
0010
*
2,,,
~
,,,
n
Vnnnn
zyxy
V
V
ecccn
V
I
LLLVb
D
dudvdwd
u
wvuV
Twvug ϑψηχ
π
τ
∂
τ∂
( ) ( ) ( ) ( ) ( )
( )
×+∫ ∫ ∫ ∫−× *
*
2
0
0
1
0
1
0
1
0
010
*
2,,,
~
,,,
V
I
LLLVb
D
dudvdwd
v
wvuV
Twvugwcvsuce
zyxz
V
VnnnVn
π
τ
∂
τ∂
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−×
∞
=1 0
1
0
1
0
1
0
010
,,,
~
,,,
n
VnnnVnnnn
dudvdwd
w
wvuV
Twvugwsvcucecccn
ϑ
τ
∂
τ∂
τψηχ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−×
∞
=1 0
1
0
000100
0
,,,
~
,,,
~
*
2
n
nVnVnn
zyx
V
Vn
dudvdwdwvuVwvuIvceec
LLLV
D
e
ϑ
ττττϑχϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−×
∞
=1 0
1
0
1
0
1
0
000*
*
0
*
*
,,,
~
*
2
n
nnnVnnnn
V
nn
wvuIwcvcuceccc
V
I
LV
D
V
I
cc
ϑ
ττψηχψη
( ) ( )ϑττ Vn
edudvdwdwvuV ,,,
~
100
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−=
∞
=1 0
1
0
1
0
1
0
2101
,,,
*
2
,,,
~
n
InnInInnnn
zyxx
Twvugvcuseecccn
LLLIb
I
ϑ
τϑψηχ
π
ϑψηχ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−+×
∞
=1 0
1
0
2
001
*
2,,,
~
n
nInInnnn
zyxy
n
uceeccc
LLLIb
dudvdwd
u
wvuI
wc
ϑ
τϑψηχ
π
τ
∂
τ∂
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑+∫ ∫×
∞
=1
1
0
1
0
001
*
2,,,
~
,,,
n
Innnn
z
Inn
ecccn
Ib
dudvdwd
v
wvuI
Twvugwcvsn ϑψηχ
π
τ
∂
τ∂
( ) ( ) ( ) ( ) ( )
( )
2
0
1
0
1
0
1
0
001 1,,,
~
,,,
zyx
InnnIn
LLL
dudvdwd
w
wvuI
Twvugwsvcuce∫ ∫ ∫ ∫−×
ϑ
τ
∂
τ∂
τ ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−=
∞
=1 0
1
0
1
0
1
0
2101
,,,
*
2
,,,
~
n
VnnVnVnnnn
zyxx
Twvugvcuseeccc
LLLVb
V
ϑ
τϑψηχ
π
ϑψηχ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ −+×
∞
=1 0
2
001
2,,,
~
n
VnVnnnn
yzyx
n
eecccn
bLLL
dudvdwd
u
wvuV
wcn
ϑ
τϑψηχ
π
τ
∂
τ∂
( ) ( ) ( ) ( ) ( ) ( ) ×∑+∫ ∫ ∫×
∞
=1
2
1
0
1
0
1
0
001
*
2,,,
~
*
,,,
n
n
zyxz
V
nnn
cn
LLLVb
dudvdwd
v
wvuV
V
Twvug
wcvsuc χ
π
τ
∂
τ∂
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
∫ ∫ ∫ ∫−×
ϑ
τ
∂
τ∂
τϑψη
0
1
0
1
0
1
0
001
,,,
~
,,, dudvdwd
w
wvuV
Twvugwsvcuceecc VnnnVnVnnn
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
0
001011
,,,
~
*
2
,,,
~
n
nnInInnnn
zyx
wvuIvcuceeccc
LLLI
I
ϑ
ττϑψηχϑψηχ

15. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
15
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ −−×
∞
=1 0
*
*
*
*
000
*
2
,,,
~
n
InInnnn
zyx
n
eeccc
I
V
LLLII
V
dudvdwdwvuVwc
ϑ
τϑψηχττ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫×
∞
=1
*
*1
0
1
0
1
0
001000
*
2
,,,
~
,,,
~
n
nn
zyx
nnn
cc
I
V
ILLL
dudvdwdwvuVwvuIwcvcuc ηχτττ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫−×
ϑ
ττττϑψ
0
1
0
1
0
1
0
001000
,,,
~
,,,
~
,,, dudvdwdwvuVwvuIwcTwvuhvcuceec nnnInInn
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
0
*
*
011
*
2
,,,
~
n
nnnVnVnnnn
zyx
wcvcuceeccc
V
I
LLLV
V
ϑ
τϑψηχϑψηχ
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ −−×
∞
=1 0
000001
*
2
,,,
~
,,,
~
n
VnVnnnn
zyx
eeccc
LLLV
dudvdwdwvuVwvuI
ϑ
τϑψηχτττ
( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫×
∞
=1
1
0
1
0
1
0
001000*
*
*
2
,,,
~
,,,
~
n
n
zyx
nnn
c
LLLV
dudvdwdwvuVwvuIwcvcuc
V
I
χτττ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫−×
ϑ
ττττ
0
1
0
1
0
1
0
001000*
*
,,,
~
,,,
~
,,, dudvdwdwvuVwvuIwcTwvuhvcuce
V
I
nnnVn
( ) ( ) ( )ϑψη Vnnn
ecc× .
Equations for the functions Φρi(x,y,z,t) are
( ) ( ) ( ) ( ) +




 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
ΦΦΦ 2
2
02
2
02
2
0
,,,,,,,,,,,,
z
tzyx
D
y
tzyx
D
x
tzyx
D
t
tzyx iI
I
iI
I
iI
I
iI
∂
∂
∂
∂
∂
∂
∂
∂
( )
( )
( )
( )
+




 Φ
+




 Φ
+ −
ΦΦ
−
ΦΦ
y
tzyx
Tzyxg
y
D
x
tzyx
Tzyxg
x
D iI
II
iI
II
∂
∂
∂
∂
∂
∂
∂
∂ ,,,
,,,
,,,
,,, 1
0
1
0
( )
( )





 Φ
+
−
ΦΦ
z
tzyx
Tzyxg
z
D
iI
II
∂
∂
∂
∂ ,,,
,,,
1
0
, i≥1;
( ) ( ) ( ) ( )
+
Φ
+
Φ
+
Φ
=
Φ
ΦΦΦ 2
2
02
2
02
2
0
,,,,,,,,,,,,
z
tzyx
D
y
tzyx
D
x
tzyx
D
t
tzyx iV
V
iV
V
iV
V
iV
∂
∂
∂
∂
∂
∂
∂
∂
( )
( )
( )
( )
+




 Φ
+




 Φ
+
−
ΦΦ
−
ΦΦ
y
tzyx
Tzyxg
y
D
x
tzyx
Tzyxg
x
D
iV
VV
iV
VV
∂
∂
∂
∂
∂
∂
∂
∂ ,,,
,,,
,,,
,,,
1
0
1
0
( )
( )





 Φ
+
−
ΦΦ
z
tzyx
Tzyxg
z
D
iV
VV
∂
∂
∂
∂ ,,,
,,,
1
0
, 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; ΦI0(x,y,z,0)=fΦI (x,y,z), ΦIi(x,y,z,0)=0,

16. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
16
ΦV0(x,y,z,0)=fΦV (x,y,z), ΦVi(x,y,z,0)=0, i≥1.
Solutions of the equations could be written as
( ) ( ) ( ) ( ) ( )∑+=Φ
∞
=
ΦΦ
1
0
21
,,,
n
nn
zyxzyx
tezcycxcF
LLLLLL
tzyx ρρρ
,
where ( )
















++−= ΦΦ 2220
22 111
exp
zyx
n
LLL
tDnte ρρ
π , ( ) ( ) ( ) ( )∫ ∫ ∫= ΦΦ
xL yL zL
nnnn
xdydzdzyxfzcycxcF
0 0 0
,,ρρ
,
cn(α)=cos(πnα/Lα);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−=Φ
∞
=
ΦΦΦ
1 0 0 0 0
2
.,,
2
,,,
n
t xL yL zL
nnnnnnnn
zyx
i
Twvugwcvcusetezcycxcn
LLL
tzyx ρρρρ
τ
π
( )
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−−
Φ
×
∞
=
ΦΦ
−
1 0 0 0
2
1 2,,,
n
t xL yL
nnnnnnn
zyx
iI
vsucetezcycxcn
LLL
dudvdwd
u
wvu
τ
π
τ
∂
τ∂
ρρ
ρ
( ) ( )
( )
( ) ( ) ( ) ( )×∑−∫
Φ
×
∞
=
Φ
−
Φ
1
2
0
1 2,,,
.,,
n
nnnn
zyx
zL
iI
n
tezcycxcn
LLL
dudvdwd
v
wvu
Twvugwc ρ
ρ
ρ
π
τ
∂
τ∂
( ) ( ) ( ) ( ) ( )
( )
∫ ∫ ∫ ∫
Φ
−×
−
ΦΦ
t xL yL zL
iI
nnnn
dudvdwd
w
wvu
Twvugwsvcuce
0 0 0 0
1
,,,
.,, τ
∂
τ∂
τ ρ
ρρ
, i≥1,
where sn(α)=sin(πnα/Lα).
System of equations for functions Cijk(x,y,z,t) (i≥0, j≥0, k≥0) could be written as
( ) ( ) ( ) ( )






∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
00
2
2
00
2
2
00
2
0
00
,,,,,,,,,,,,
z
tzyxC
y
tzyxC
x
tzyxC
D
t
tzyxC
L
;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
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
( )
( )
( )
( )
+





∂
∂
∂
∂
+





∂
∂
∂
∂
+ −−
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
;
( ) ( ) ( ) ( )
×+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
LLLL
D
z
tzyxC
D
y
tzyxC
D
x
tzyxC
D
t
tzyxC
02
02
2
02
02
2
02
02
2
0
02
,,,,,,,,,,,,
( )
( )
( )
( )
( )
( )
( )
( )




+





∂
∂
∂
∂
+





∂
∂
∂
∂
×
−−
y
tzyxC
TzyxP
tzyxC
tzyxC
yx
tzyxC
TzyxP
tzyxC
tzyxC
x
,,,
,,,
,,,
,,,
,,,
,,,
,,,
,,, 00
1
00
01
00
1
00
01 γ
γ
γ
γ

17. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
17
( )
( )
( )
( ) ( )
( )
( )




+





∂
∂
∂
∂
+










∂
∂
∂
∂
+
−
x
tzyxC
TzyxP
tzyxC
x
D
z
tzyxC
TzyxP
tzyxC
tzyxC
z
L
,,,
,,,
,,,,,,
,,,
,,,
,,, 0100
0
00
1
00
01 γ
γ
γ
γ
( )
( )
( ) ( )
( )
( )









∂
∂
∂
∂
+





∂
∂
∂
∂
+
z
tzyxC
TzyxP
tzyxC
zy
tzyxC
TzyxP
tzyxC
y
,,,
,,,
,,,,,,
,,,
,,, 01000100
γ
γ
γ
γ
;
( ) ( ) ( ) ( )
×+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
LLLL
D
z
tzyxC
D
y
tzyxC
D
x
tzyxC
D
t
tzyxC
02
11
2
02
11
2
02
11
2
0
11
,,,,,,,,,,,,
( )
( )
( )
( )
( )
( )
( )
( )
+










∂
∂
∂
∂
+





∂
∂
∂
∂
×
−−
y
tzyxC
TzyxP
tzyxC
tzyxC
yx
tzyxC
TzyxP
tzyxC
tzyxC
x
,,,
,,,
,,,
,,,
,,,
,,,
,,,
,,, 00
1
00
10
00
1
00
10 γ
γ
γ
γ
( )
( )
( )
( ) ( )
( )
( )




+





∂
∂
∂
∂
+










∂
∂
∂
∂
+
−
x
tzyxC
tzyxP
tzyxC
x
D
z
tzyxC
TzyxP
tzyxC
tzyxC
z
L
,,,
,,,
,,,,,,
,,,
,,,
,,, 1000
0
00
1
00
10 γ
γ
γ
γ
( )
( )
( ) ( )
( )
( )
( )








×
∂
∂
+










∂
∂
∂
∂
+





∂
∂
∂
∂
+ Tzyxg
xz
tzyxC
tzyxP
tzyxC
zy
tzyxC
tzyxP
tzyxC
y
L
,,,
,,,
,,,
,,,,,,
,,,
,,, 10001000
γ
γ
γ
γ
( )
( )
( )
( )
( )
LLL
D
z
tzyxC
Tzyxg
zy
tzyxC
Tzyxg
yx
tzyxC
0
010101
,,,
,,,
,,,
,,,
,,,










∂
∂
∂
∂
+





∂
∂
∂
∂
+


∂
∂
× .
Solutions of equations for functions Cij(x,y,z,t) (i≥0, j≥0) with account appropriate conditions
takes the form
( ) ( ) ( ) ( ) ( )∑+=
∞
=1
00
21
,,,
n
nCnC
zyxzyx
tezcycxcF
LLLLLL
tzyxC ,
where ( )
















++−= 2220
22 111
exp
zyx
CnC
LLL
tDnte π , ( ) ( ) ( ) ( )∫ ∫ ∫=
xL yL zL
CnnnCn
xdydzdzyxfzcycxcF
0 0 0
,, ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
20
,,,
2
,,,
n
t xL yL zL
LnnnnCnCnnnnC
zyx
i
TwvugwcvcusetezcycxcF
LLL
tzyxC τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫−−
∂
∂
×
∞
=
−
1 0 0 0
2
10
2,,,
n
t xL yL
nnnCnCnnnnC
zyx
i
vsucetezcycxcFn
LLL
ndudvdwd
u
wvuC
τ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ −−∫
∂
∂
×
∞
=
−
1 0
2
0
10
2,,,
,,,
n
t
nCnCnnn
zyx
zL
i
Ln
etezcycxc
LLL
dudvdwd
v
wvuC
Twvugwc τ
π
τ
τ
( ) ( ) ( ) ( )
( )
∫ ∫ ∫
∂
∂
× −
xL yL zL
i
LnnnnC
dudvdwd
w
wvuC
TwvugwsvcucFn
0 0 0
10
,,,
,,, τ
τ
, i≥1;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
∑ ×∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
00
201
,,,
,,,2
,,,
n
t xL yL zL
nnnCnCnnnnC
zyx
TwvuP
wvuC
vcusetezcycxcFn
LLL
tzyxC γ
γ
τ
τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫−−
∂
∂
×
∞
=1 0 0
2
00
2,,,
n
t xL
nnCnCnnnnC
zyx
n
ucetezcycxcFn
LLL
dudvdwd
u
wvuC
wc τ
π
τ
τ
( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ×∑−∫ ∫
∂
∂
×
∞
=1
2
0 0
0000
2,,,
,,,
,,,
n
nnnnC
zyx
yL zL
nn
zcycxcFn
LLL
dudvdwd
v
wvuC
TwvuP
wvuC
wcvs
π
τ
ττ
γ
γ
( ) ( ) ( ) ( ) ( )
( )
( )
( )
∫ ∫ ∫ ∫
∂
∂
−×
t xL yL zL
nnnnCnC
dudvdwd
w
wvuC
TwvuP
wvuC
wsvcucete
0 0 0 0
0000
,,,
,,,
,,,
τ
ττ
τ γ
γ
;

18. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
18
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
∑ ×∫ ∫ ∫ ∫−−=
∞
=
−
1 0 0 0
1
00
0
202
,,,
,,,2
,,,
n
t xL yL zL
nnnnCnCnnnnC
zyx
TwvuP
wvuC
wcvcusetezcycxcFn
LLL
tzyxC γ
γ
τ
τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫−−
∂
∂
×
∞
=1 0 0 0
2
00
2,,,
n
t xL yL
nnnCnCnnnnC
zyx
vsucetezcycxcFn
LLL
dudvdwd
u
wvuC
τ
π
τ
τ
( ) ( )
( )
( ) ( ) ( ) ( ) ( )×∑−∫
∂
∂
×
∞
=
−
1
2
0
00
1
00
2,,,
,,,
,,,
n
nCnnnnC
zyx
zL
n
tezcycxcFn
LLL
dudvdwd
v
wvuC
TwvuP
wvuC
wc
π
τ
ττ
γ
γ
( ) ( ) ( ) ( ) ( )
( )
( ) ( )×∑−∫ ∫ ∫ ∫
∂
∂
−×
∞
=
−
1
2
0 0 0 0
00
1
00
2,,,
,,,
,,,
n
n
zyx
t xL yL zL
nnnnC
xcn
LLL
dudvdwd
w
wvuC
TwvuP
wvuC
wsvcuce
π
τ
ττ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( ) −∫ ∫ ∫ ∫
∂
∂
−×
t xL yL zL
nnnnCnCnnnC
dudvdwd
u
wvuC
TwvuP
wvuC
wcvcusetezcycF
0 0 0 0
0100
,,,
,,,
,,,
τ
ττ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ×∑ ∫ ∫ ∫ ∫
∂
∂
−−
∞
=1 0 0 0 0
0100
2
,,,
,,,
,,,2
n
t xL yL zL
nnnCnCnnnnC
zyx
v
wvuC
TwvuP
wvuC
vsucetezcycxcFn
LLL
ττ
τ
π
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−×
∞
=1 0 0 0 0
2
2
n
t xL yL zL
nnnnCnCnnnnC
zyx
n
wsvcucetezcycxcFn
LLL
dudvdwdwc τ
π
τ
( )
( )
( )
τ
ττ
γ
γ
dudvdwd
w
wvuC
TwvuP
wvuC
∂
∂
×
,,,
,,,
,,, 0100
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
211
,,,
2
,,,
n
t xL yL zL
LnnnnCnCnnnnC
zyx
TwvugwcvcusetezcycxcFn
LLL
tzyxC τ
π
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫ ∫−−
∂
∂
×
∞
=1 0 0 0 0
2
01 2,,,
n
t xL yL zL
nnnnCnCnnn
zyx
wcvsucetezcycxc
LLL
dudvdwd
u
wvuC
τ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ −−
∂
∂
×
∞
=1 0
2
01
2,,,
,,,
n
t
nCnCnnn
zyx
LnC
etezcycxc
LLL
dudvdwd
v
wvuC
TwvugFn τ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( )×∑−∫ ∫ ∫
∂
∂
×
∞
=1
2
0 0 0
01
2,,,
,,,
n
nnC
zyx
xL yL zL
LnnnnC
xcFn
LLL
dudvdwd
w
wvuC
TwvugwsvcucFn
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( ) −∫ ∫ ∫ ∫
∂
∂
−×
t xL yL zL
nnnnCnCnn
dudvdwd
u
wvuC
TwvuP
wvuC
wcvcusetezcyc
0 0 0 0
1000
,,,
,,,
,,,
τ
ττ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ×∑ ∫ ∫ ∫ ∫
∂
∂
−−
∞
=1 0 0 0 0
1000
2
,,,
,,,
,,,2
n
t xL yL zL
nnnnCn
zyx
dudvdwd
v
wvuC
TwvuP
wvuC
wcvsucexcn
LLL
τ
ττ
τ
π
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫ ∫−−×
∞
=1 0 0 0 0
2
2
n
t xL yL zL
nnnnCnCnnnnC
zyx
nCnnnC
wsvcucetezcycxcFn
LLL
tezcycF τ
π
( )
( )
( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ −−
∂
∂
×
∞
=1 0
2
1000
2,,,
,,,
,,,
n
t
nCnCnnnnC
zyx
etezcycxcFn
LLL
dudvdwd
w
wvuC
TwvuP
wvuC
τ
π
τ
ττ
γ
γ
( ) ( ) ( )
( )
( )
( )
( ) ×∑−∫ ∫ ∫
∂
∂
×
∞
=
−
1
2
0 0 0
00
1
00 2,,,
,,,
,,,
n
nnC
zyx
xL yL zL
nnn
xcFn
LLL
dudvdwd
u
wvuC
TwvuP
wvuC
wcvcus
π
τ
ττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
−∫ ∫ ∫ ∫
∂
∂
−×
−
t xL yL zL
nnnnCnCnn
dudvdwd
v
wvuC
TwvuP
wvuC
wcvsucetezcyc
0 0 0 0
00
1
00
,,,
,,,
,,,
τ
ττ
τ γ
γ
( ) ( ) ( ) ( )
( )
( )
( )
∑ ×
∂
∂
∫ ∫ ∫ ∫−−
∞
=
−
1
00
0 0 0 0
1
00
2
,,,
,,,
,,,2
n
t xL yL zL
nnnnC
zyx
dudvdwd
w
wvuC
TwvuP
wvuC
wsvcucen
LLL
τ
ττ
τ
π
γ
γ
( ) ( ) ( ) ( )tezcycxcF nCnnnnC
× .

19. Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015
19
REFERENCES
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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] V.I. Lachin, N.S. Savelov. Electronics,Phoenix, Rostov-na-Donu, 2001.
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Author
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.

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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.