ON OPTIMIZATION OF MANUFACTURING OF AN AMPLIFIER TO INCREASE DENSITY OF BIPOLAR TRANSISTOR FRAMEWORK THE AMPLIFIER
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About This Presentation
In this paper we consider a possibility to increase density of bipolar heterotransistor framework an amplifier due to decreasing of their dimensions. The considered approach based on doping of required areas of
heterostructure with specific configuration by diffusion or ion implantation. The doping ...
Slide Content
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
DOI: 10.5121/ijoe.2017.6201 1
O
N OPTIMIZATION OF MANUFACTURING OF AN
AMPLIFIER TO INCREASE DENSITY OF BIPOLAR
TRANSISTOR FRAMEWORK THE AMPLIFIER
E.L. Pankratov and E.A. Bulaeva
Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950,
Russia
ABSTRACT
In this paper we consider a possibility to increase density of bipolar heterotransistor framework an ampli-
fier due to decreasing of their dimensions. The considered approach based on doping of required areas of
heterostructure with specific configuration by diffusion or ion implantation. The doping finished by opti-
mized annealing of dopant and/or radiation defects. Analysis of redistribution of dopant with account re-
distribution of radiation defects (after implantation of ions of dopant) for optimization of the above an-
nealing have been done by using recently introduced analytical approach. The approach gives a possibil-
ity to analyze mass and heat transports in a heterostructure without crosslinking of solutions on interfaces
between layers of the heterostructure with account nonlinearity of these transports and variation in time of
their parameters.
KEYWORDS
Bipolar heterotransistor; increasing of density of transistor; an amplifier circuit
1. INTRODUCTION
In the present time an actual question is decreasing of dimensions of solid state electronic devic-
es. To decrease the dimensions are could be increased density of elements of integrated circuits
and decreased dimensions of these elements. To date, several methods to decrease dimensions of
elements of integrated circuits have been developed. One of them is growth of thin films struc-
tures [1-5]. The second approach is diffusion or ion doping of required areas of samples or
heterostructures and father laser or microwave annealing of dopant and/or radiation defects [6-8].
Using of the above approaches of annealing leads to generation of inhomogenous distribution of
temperature and consequently to decreasing of dimensions of elements of integrated circuits.
Another approach to change properties of doped materials is radiation processing [9,10].
In the present paper we consider circuits of an amplifier [11] (see Fig. 1). Based on recently for-
mulated recommendations to decrease dimensions of single transistors (both bipolar and field-
effect) [12-17] we formulate recommendations to increase density of bipolar framework the con-
sidered circuits. We assume, that the considered circuits were manufactured framework the con-
sidered in Fig.1 heterostructure. The heterostructure consist of a substrate and an epitaxial layer.
The epitaxial layer includes into itself several sections, manufactured by using other materials.
These sections were doped by diffusion or by ion implantation to generate required types of con-
ductivity (n or p) in the required materials. Framework this paper we analyzed redistribution of
dopants during annealing of these dopants and/or radiation defects to formulate conditions,
which correspond to decreasing of the considered circuits.
DOI: 10.5121/ijoe.2017.6201 1
O
N OPTIMIZATION OF MANUFACTURING OF AN
AMPLIFIER TO INCREASE DENSITY OF BIPOLAR
TRANSISTOR FRAMEWORK THE AMPLIFIER
E.L. Pankratov and E.A. Bulaeva
Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950,
Russia
ABSTRACT
In this paper we consider a possibility to increase density of bipolar heterotransistor framework an ampli-
fier due to decreasing of their dimensions. The considered approach based on doping of required areas of
heterostructure with specific configuration by diffusion or ion implantation. The doping finished by opti-
mized annealing of dopant and/or radiation defects. Analysis of redistribution of dopant with account re-
distribution of radiation defects (after implantation of ions of dopant) for optimization of the above an-
nealing have been done by using recently introduced analytical approach. The approach gives a possibil-
ity to analyze mass and heat transports in a heterostructure without crosslinking of solutions on interfaces
between layers of the heterostructure with account nonlinearity of these transports and variation in time of
their parameters.
KEYWORDS
Bipolar heterotransistor; increasing of density of transistor; an amplifier circuit
1. INTRODUCTION
In the present time an actual question is decreasing of dimensions of solid state electronic devic-
es. To decrease the dimensions are could be increased density of elements of integrated circuits
and decreased dimensions of these elements. To date, several methods to decrease dimensions of
elements of integrated circuits have been developed. One of them is growth of thin films struc-
tures [1-5]. The second approach is diffusion or ion doping of required areas of samples or
heterostructures and father laser or microwave annealing of dopant and/or radiation defects [6-8].
Using of the above approaches of annealing leads to generation of inhomogenous distribution of
temperature and consequently to decreasing of dimensions of elements of integrated circuits.
Another approach to change properties of doped materials is radiation processing [9,10].
In the present paper we consider circuits of an amplifier [11] (see Fig. 1). Based on recently for-
mulated recommendations to decrease dimensions of single transistors (both bipolar and field-
effect) [12-17] we formulate recommendations to increase density of bipolar framework the con-
sidered circuits. We assume, that the considered circuits were manufactured framework the con-
sidered in Fig.1 heterostructure. The heterostructure consist of a substrate and an epitaxial layer.
The epitaxial layer includes into itself several sections, manufactured by using other materials.
These sections were doped by diffusion or by ion implantation to generate required types of con-
ductivity (n or p) in the required materials. Framework this paper we analyzed redistribution of
dopants during annealing of these dopants and/or radiation defects to formulate conditions,
which correspond to decreasing of the considered circuits.
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
2
Fig. 1. Structure of inverter [11]. View from top on epitaxial layer
2. METHOD OF SOLUTION
To solve our aims let us determine spatio-temporal distributions of concentrations of dopants.
The required distributions we determined by solving the second Fick's law [9,10,18,19]
( ) ( ) ( ) ( )
+
+
=
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxCCCC
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
,,,,,,,,,,,,
. (1)
Boundary and initial conditions for the equations are
( )
0
,,,
0
=
∂
∂
=x
x
tzyxC
,
( )
0
,,,
=
∂
∂
=
x
Lx
x
tzyxC
,
( )
0
,,,
0
=
∂
∂
=y
y
tzyxC
,
( )
0
,,,
=
∂
∂
=
y
Lx
y
tzyxC
,
( )
0
,,,
0
=
∂
∂
=z
z
tzyxC
,
( )
0
,,,
=
∂
∂
=
z
Lx
z
tzyxC
, C (x,y,z,0)=f (x,y,z). (2)
In the Eqs. (1) and (2) the function C(x,y,z,t) describes the spatio-temporal distribution of con-
centration of dopant; the parameter D
С is the dopant diffusion coefficient. Dopant diffusion coef-
ficient will be changed with changing of materials of heterostructure, heating and cooling of
heterostructure during annealing of dopant or radiation defects (with account Arrhenius law).
Dependences of dopant diffusion coefficient on coordinate in heterostructure, temperature of an-
nealing and concentrations of dopant and radiation defects could be written as [10,18]
( )
( )
( )
( ) ( )
( )
++
+=
2
*
2
2*1
,,,,,,
1
,,,
,,,
1,,,
V
tzyxV
V
tzyxV
TzyxP
tzyxC
TzyxDD
LC ςςξ
γ
γ
. (3)
2
Fig. 1. Structure of inverter [11]. View from top on epitaxial layer
2. METHOD OF SOLUTION
To solve our aims let us determine spatio-temporal distributions of concentrations of dopants.
The required distributions we determined by solving the second Fick's law [9,10,18,19]
( ) ( ) ( ) ( )
+
+
=
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxCCCC
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
,,,,,,,,,,,,
. (1)
Boundary and initial conditions for the equations are
( )
0
,,,
0
=
∂
∂
=x
x
tzyxC
,
( )
0
,,,
=
∂
∂
=
x
Lx
x
tzyxC
,
( )
0
,,,
0
=
∂
∂
=y
y
tzyxC
,
( )
0
,,,
=
∂
∂
=
y
Lx
y
tzyxC
,
( )
0
,,,
0
=
∂
∂
=z
z
tzyxC
,
( )
0
,,,
=
∂
∂
=
z
Lx
z
tzyxC
, C (x,y,z,0)=f (x,y,z). (2)
In the Eqs. (1) and (2) the function C(x,y,z,t) describes the spatio-temporal distribution of con-
centration of dopant; the parameter D
С is the dopant diffusion coefficient. Dopant diffusion coef-
ficient will be changed with changing of materials of heterostructure, heating and cooling of
heterostructure during annealing of dopant or radiation defects (with account Arrhenius law).
Dependences of dopant diffusion coefficient on coordinate in heterostructure, temperature of an-
nealing and concentrations of dopant and radiation defects could be written as [10,18]
( )
( )
( )
( ) ( )
( )
++
+=
2
*
2
2*1
,,,,,,
1
,,,
,,,
1,,,
V
tzyxV
V
tzyxV
TzyxP
tzyxC
TzyxDD
LC ςςξ
γ
γ
. (3)
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
3
Here function D L (x,y,z,T) describes dependences of dopant diffusion coefficient on coordinate
and temperature of annealing T. Function P
(x,y,z,T) describes the same dependences of the limit
of solubility of dopant. The parameter
γ is integer and usually could be varying in the following
interval
γ ∈[1,3]. The parameter describes quantity of charged defects, which interacting (in av-
erage) with each atom of dopant. More detailed information about concentrational dependence of
dopant diffusion coefficient is presented in [18]. The function V
(x,y,z,t) describes distribution of
concentration of radiation vacancies in space and time. The parameter V
*
describes the equilibri-
um distribution of concentration of vacancies. It should be noted, that diffusion type of doping
gives a possibility to obtain doped materials without radiation defects. In this situation
ζ1= ζ2= 0.
We determine spatio-temporal distributions of concentrations of radiation defects by solving the
following system of equations [10,13]
( )
( )
( )
( )
( )
( ) ×−
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
Tzyxk
y
tzyxI
TzyxD
yx
tzyxI
TzyxD
xt
tzyxI
IIII ,,,
,,,
,,,
,,,
,,,
,,,
,
( )
( )
( )
( ) ( ) ( ) tzyxVtzyxITzyxk
z
tzyxI
TzyxD
z
tzyxI
VII
,,,,,,,,,
,,,
,,,,,,
,
2
−
∂
∂
∂
∂
+× (4)
( )
( )
( )
( )
( )
( ) ×−
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
Tzyxk
y
tzyxV
TzyxD
yx
tzyxV
TzyxD
xt
tzyxV
VVVV ,,,
,,,
,,,
,,,
,,,
,,,
,
( )
( )
( )
( ) ( ) ( ) tzyxVtzyxITzyxk
z
tzyxV
TzyxD
z
tzyxV
VIV ,,,,,,,,,
,,,
,,,,,,
,
2−
∂
∂
∂
∂
+× .
Boundary and initial conditions for these equations are
( )
0
,,,
0
=
∂
∂
=x
x
tzyx
ρ
,
( )
0
,,,
=
∂
∂
=
x
Lx
x
tzyx
ρ
,
( )
0
,,,
0
=
∂
∂
=y
y
tzyx
ρ
,
( )
0
,,,
=
∂
∂
=
y
Ly
y
tzyx
ρ
,
( )
0
,,,
0
=
∂
∂
=z
z
tzyx
ρ
,
( )
0
,,,
=
∂
∂
=
z
Lz
z
tzyx
ρ
, ρ (x,y,z,0)=f ρ (x,y,z). (5)
Here
ρ =I,V. The function I (x,y,z,t) describes the distribution of concentration of radiation inter-
stitials in space and time. The functions D
ρ(x,y,z,T) describe dependences of the diffusion coeffi-
cients of point radiation defects on coordinate and temperature. The quadric on concentrations
terms of Eqs. (4) describes generation divacancies and diinterstitials. The function k
I,V(x,y,z,T)
describes dependence of the parameter of recombination of point radiation defects on coordinate
and temperature. The function k
I,I(x,y,z,T) and k V,V(x,y,z,T) describes dependences of the parame-
ters of generation of simplest complexes of point radiation defects on coordinate and tempera-
ture.
Now let us calculate distributions of concentrations of divacancies
ΦV(x,y,z,t) and diinterstitials
ΦI(x,y,z,t) in space and time by solving the following system of equations [10,19]
( )
( )
( )
( )
( )
+
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)
3
Here function D L (x,y,z,T) describes dependences of dopant diffusion coefficient on coordinate
and temperature of annealing T. Function P
(x,y,z,T) describes the same dependences of the limit
of solubility of dopant. The parameter
γ is integer and usually could be varying in the following
interval
γ ∈[1,3]. The parameter describes quantity of charged defects, which interacting (in av-
erage) with each atom of dopant. More detailed information about concentrational dependence of
dopant diffusion coefficient is presented in [18]. The function V
(x,y,z,t) describes distribution of
concentration of radiation vacancies in space and time. The parameter V
*
describes the equilibri-
um distribution of concentration of vacancies. It should be noted, that diffusion type of doping
gives a possibility to obtain doped materials without radiation defects. In this situation
ζ1= ζ2= 0.
We determine spatio-temporal distributions of concentrations of radiation defects by solving the
following system of equations [10,13]
( )
( )
( )
( )
( )
( ) ×−
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
Tzyxk
y
tzyxI
TzyxD
yx
tzyxI
TzyxD
xt
tzyxI
IIII ,,,
,,,
,,,
,,,
,,,
,,,
,
( )
( )
( )
( ) ( ) ( ) tzyxVtzyxITzyxk
z
tzyxI
TzyxD
z
tzyxI
VII
,,,,,,,,,
,,,
,,,,,,
,
2
−
∂
∂
∂
∂
+× (4)
( )
( )
( )
( )
( )
( ) ×−
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
Tzyxk
y
tzyxV
TzyxD
yx
tzyxV
TzyxD
xt
tzyxV
VVVV ,,,
,,,
,,,
,,,
,,,
,,,
,
( )
( )
( )
( ) ( ) ( ) tzyxVtzyxITzyxk
z
tzyxV
TzyxD
z
tzyxV
VIV ,,,,,,,,,
,,,
,,,,,,
,
2−
∂
∂
∂
∂
+× .
Boundary and initial conditions for these equations are
( )
0
,,,
0
=
∂
∂
=x
x
tzyx
ρ
,
( )
0
,,,
=
∂
∂
=
x
Lx
x
tzyx
ρ
,
( )
0
,,,
0
=
∂
∂
=y
y
tzyx
ρ
,
( )
0
,,,
=
∂
∂
=
y
Ly
y
tzyx
ρ
,
( )
0
,,,
0
=
∂
∂
=z
z
tzyx
ρ
,
( )
0
,,,
=
∂
∂
=
z
Lz
z
tzyx
ρ
, ρ (x,y,z,0)=f ρ (x,y,z). (5)
Here
ρ =I,V. The function I (x,y,z,t) describes the distribution of concentration of radiation inter-
stitials in space and time. The functions D
ρ(x,y,z,T) describe dependences of the diffusion coeffi-
cients of point radiation defects on coordinate and temperature. The quadric on concentrations
terms of Eqs. (4) describes generation divacancies and diinterstitials. The function k
I,V(x,y,z,T)
describes dependence of the parameter of recombination of point radiation defects on coordinate
and temperature. The function k
I,I(x,y,z,T) and k V,V(x,y,z,T) describes dependences of the parame-
ters of generation of simplest complexes of point radiation defects on coordinate and tempera-
ture.
Now let us calculate distributions of concentrations of divacancies
ΦV(x,y,z,t) and diinterstitials
ΦI(x,y,z,t) in space and time by solving the following system of equations [10,19]
( )
( )
( )
( )
( )
+
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)
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
4
( )
( )
( )
( )
( )
+
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
∂
∂
∂
∂
.
Boundary and initial conditions for these equations are
( )
0
,,,
0
=
∂
Φ∂
=x
x
tzyx
ρ
,
( )
0
,,,
=
∂
Φ∂
=
x
Lx
x
tzyx
ρ
,
( )
0
,,,
0
=
∂
Φ∂
=y
y
tzyx
ρ
,
( )
0
,,,
=
∂
Φ∂
=
y
Ly
y
tzyx
ρ
,
( )
0
,,,
0
=
∂
Φ∂
=z
z
tzyx
ρ
,
( )
0
,,,
=
∂
Φ∂
=
zLz
z
tzyx
ρ
, ΦI (x,y,z,0)=f ΦI (x,y,z), ΦV (x,y,z,0)=f ΦV (x,y,z). (7)
The functions D
Φρ(x,y,z,T) describe dependences of the diffusion coefficients of the above com-
plexes of radiation defects on coordinate and temperature. The functions k
I(x,y,z,T) and k V(x,y,z,
T) describe the parameters of decay of these complexes on coordinate and temperature.
We calculate distributions of concentrations of point radiation defects in space and time by using
recently elaborated approach [12,13]. To use the approach let us transform dependences of diffu-
sion coefficients of point defects in space and time to the following form: D
ρ(x,y,z,T) =D0ρ [1+ερ
gρ(x,y,z,T)], where D 0ρ are the average values of diffusion coefficients, 0≤ ερ< 1, |gρ(x,y,z, T)|≤1, ρ
=I,V. Let us also transform dependences of another parameters to the similar form: k I,V(x, y,z,T)=
k
0I,V[1+εI,V gI,V(x,y,z,T)], k I,I(x,y,z,T)=k 0I,I [1+εI,I gI,I(x,y,z,T)] and k V,V (x,y,z,T) = k0V,V [1+εV,V
gV,V(x,y,z,T)], where k 0ρ1,ρ2 are the their average values, 0≤ εI,V <1, 0≤εI,I <1, 0≤ εV,V<1, |
gI,V(x,y,z,T)|≤1, | gI,I(x,y,z,T)|≤1, |g V,V(x,y,z,T)|≤1. Let us introduce the following dimensionless
variables:
( )( )
*
,,,,,,
~
ItzyxItzyxI= , χ = x/Lx, η = y /Ly, φ = z/Lz, ( )( )
*
,,,,,,
~
VtzyxVtzyxV = ,
2
00
LtDD
VI
=ϑ ,
VIVI
DDkL
00,0
2
=ω ,
VIDDkL
00,0
2ρρρ=Ω . The introduction leads to trans-
formation of Eqs.(4) and conditions (5) to the following form
( )
( )[ ]
( )
( )[ ]{ ×+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
Tg
I
Tg
DD
DI
IIII
VI
I
,,,1
,,,
~
,,,1
,,,
~
00
0
φηχε
ηχ
ϑφηχ
φηχε
χϑ
ϑφηχ
( )
( )[ ]
( )
( ) ×−
∂
∂
+
∂
∂
+
∂
∂
×
ϑφηχ
φ
ϑφηχ
φηχε
φη
ϑφηχ,,,
~,,,
~
,,,1
,,,
~
00
0
00
0
I
I
Tg
DD
D
DD
DI
II
VI
I
VI
I
( )[ ]( ) ( )[ ]( )ϑφηχφηχεϑφηχφηχεω,,,
~
,,,1,,,
~
,,,1
2
,,,,
ITgVTg
IIIIIVIVI
+Ω−+× (8)
( )
( )[ ]
( )
( )[ ]{ ×+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
Tg
V
Tg
DD
DV
VVVV
VI
V
,,,1
,,,
~
,,,1
,,,
~
00
0
φηχε
ηχ
ϑφηχ
φηχε
χϑ
ϑφηχ
( )
( )[ ]
( )
( ) ×−
∂
∂
+
∂
∂
+
∂
∂
×ϑφηχ
φ
ϑφηχ
φηχε
φη
ϑφηχ,,,
~,,, ~
,,,1
,,,
~
00
0
00
0
I
V
Tg
DD
D
DD
DV
VV
VI
V
VI
V
( )[ ]( ) ( )[ ]( )ϑφηχφηχεϑφηχφηχεω,,,
~
,,,1,,,
~
,,,1
2
,,,,
VTgVTg
VVVVVVIVI
+Ω−+×
( )
0
,,,
~
0
=
∂
∂
=χ
χ
ϑφηχρ
,
( )
0
,,,
~
1
=
∂
∂
=χ
χ
ϑφηχρ
,
( )
0
,,,
~
0
=
∂
∂
=η
η
ϑφηχρ
,
( )
0
,,,
~
1
=
∂
∂
=η
η
ϑφηχρ
,
4
( )
( )
( )
( )
( )
+
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
∂
∂
∂
∂
.
Boundary and initial conditions for these equations are
( )
0
,,,
0
=
∂
Φ∂
=x
x
tzyx
ρ
,
( )
0
,,,
=
∂
Φ∂
=
x
Lx
x
tzyx
ρ
,
( )
0
,,,
0
=
∂
Φ∂
=y
y
tzyx
ρ
,
( )
0
,,,
=
∂
Φ∂
=
y
Ly
y
tzyx
ρ
,
( )
0
,,,
0
=
∂
Φ∂
=z
z
tzyx
ρ
,
( )
0
,,,
=
∂
Φ∂
=
zLz
z
tzyx
ρ
, ΦI (x,y,z,0)=f ΦI (x,y,z), ΦV (x,y,z,0)=f ΦV (x,y,z). (7)
The functions D
Φρ(x,y,z,T) describe dependences of the diffusion coefficients of the above com-
plexes of radiation defects on coordinate and temperature. The functions k
I(x,y,z,T) and k V(x,y,z,
T) describe the parameters of decay of these complexes on coordinate and temperature.
We calculate distributions of concentrations of point radiation defects in space and time by using
recently elaborated approach [12,13]. To use the approach let us transform dependences of diffu-
sion coefficients of point defects in space and time to the following form: D
ρ(x,y,z,T) =D0ρ [1+ερ
gρ(x,y,z,T)], where D 0ρ are the average values of diffusion coefficients, 0≤ ερ< 1, |gρ(x,y,z, T)|≤1, ρ
=I,V. Let us also transform dependences of another parameters to the similar form: k I,V(x, y,z,T)=
k
0I,V[1+εI,V gI,V(x,y,z,T)], k I,I(x,y,z,T)=k 0I,I [1+εI,I gI,I(x,y,z,T)] and k V,V (x,y,z,T) = k0V,V [1+εV,V
gV,V(x,y,z,T)], where k 0ρ1,ρ2 are the their average values, 0≤ εI,V <1, 0≤εI,I <1, 0≤ εV,V<1, |
gI,V(x,y,z,T)|≤1, | gI,I(x,y,z,T)|≤1, |g V,V(x,y,z,T)|≤1. Let us introduce the following dimensionless
variables:
( )( )
*
,,,,,,
~
ItzyxItzyxI= , χ = x/Lx, η = y /Ly, φ = z/Lz, ( )( )
*
,,,,,,
~
VtzyxVtzyxV = ,
2
00
LtDD
VI
=ϑ ,
VIVI
DDkL
00,0
2
=ω ,
VIDDkL
00,0
2ρρρ=Ω . The introduction leads to trans-
formation of Eqs.(4) and conditions (5) to the following form
( )
( )[ ]
( )
( )[ ]{ ×+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
Tg
I
Tg
DD
DI
IIII
VI
I
,,,1
,,,
~
,,,1
,,,
~
00
0
φηχε
ηχ
ϑφηχ
φηχε
χϑ
ϑφηχ
( )
( )[ ]
( )
( ) ×−
∂
∂
+
∂
∂
+
∂
∂
×
ϑφηχ
φ
ϑφηχ
φηχε
φη
ϑφηχ,,,
~,,,
~
,,,1
,,,
~
00
0
00
0
I
I
Tg
DD
D
DD
DI
II
VI
I
VI
I
( )[ ]( ) ( )[ ]( )ϑφηχφηχεϑφηχφηχεω,,,
~
,,,1,,,
~
,,,1
2
,,,,
ITgVTg
IIIIIVIVI
+Ω−+× (8)
( )
( )[ ]
( )
( )[ ]{ ×+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
Tg
V
Tg
DD
DV
VVVV
VI
V
,,,1
,,,
~
,,,1
,,,
~
00
0
φηχε
ηχ
ϑφηχ
φηχε
χϑ
ϑφηχ
( )
( )[ ]
( )
( ) ×−
∂
∂
+
∂
∂
+
∂
∂
×ϑφηχ
φ
ϑφηχ
φηχε
φη
ϑφηχ,,,
~,,, ~
,,,1
,,,
~
00
0
00
0
I
V
Tg
DD
D
DD
DV
VV
VI
V
VI
V
( )[ ]( ) ( )[ ]( )ϑφηχφηχεϑφηχφηχεω,,,
~
,,,1,,,
~
,,,1
2
,,,,
VTgVTg
VVVVVVIVI
+Ω−+×
( )
0
,,,
~
0
=
∂
∂
=χ
χ
ϑφηχρ
,
( )
0
,,,
~
1
=
∂
∂
=χ
χ
ϑφηχρ
,
( )
0
,,,
~
0
=
∂
∂
=η
η
ϑφηχρ
,
( )
0
,,,
~
1
=
∂
∂
=η
η
ϑφηχρ
,
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
5
( )
0
,,,
~
0
=
∂
∂
=φ
φ
ϑφηχρ
,
( )
0
,,,
~
1
=
∂
∂
=φ
φ
ϑφηχρ
, ( )
( )
*
,,,
,,,
~
ρ
ϑφηχ
ϑφηχρ
ρf
=
. (9)
Let us determine solutions of Eqs.(8) as the following power series [12,13]
( ) ( )∑ ∑ ∑ Ω=
∞
=
∞
=
∞
=0 0 0
,,,
~
,,,
~
i j k
ijk
kji
ϑφηχρωεϑφηχρ
ρρ
. (10)
After substitution of the series (10) into Eqs.(8) and conditions (9) we obtain equations for ini-
tial-order approximations of concentration of point defects
( )ϑφηχ,,,
~
000
I and ( )ϑφηχ,,,
~
000
V , correc-
tions ( )ϑφηχ,,,
~
ijkI and ( )ϑφηχ,,,
~
ijkV and conditions for them for all i ≥1, j ≥1, k ≥1. The equations
are presented in the Appendix. We calculate solutions of the equations by standard Fourier ap-
proach [20,21]. The solutions are presented in the Appendix.
Farther we determine spatio-temporal distributions of concentrations of simplest complexes of
point radiation defects. To determine the distributions we transform approximations of diffusion
coefficients in the following form: D
Φρ(x,y,z,T)=D 0Φρ[1+εΦρgΦρ(x,y,z,T)], where D 0Φρ are the av-
erage values of diffusion coefficients. In this situation the Eqs.(6) could be written as
( )
( )[ ]
( )
( ) ( ) ++
Φ
+=
Φ
ΦΦΦ
tzyxITzyxk
x
tzyx
Tzyxg
x
D
t
tzyx
II
I
III
I
,,,,,,
,,,
,,,1
,,,
2
,0
∂
∂
ε
∂
∂
∂
∂
( )[ ]
( )
( )[ ]
( )
−
Φ
++
Φ
++
ΦΦΦΦΦΦ
z
tzyx
Tzyxg
z
D
y
tzyx
Tzyxg
y
D
I
III
I
III
∂
∂
ε
∂
∂
∂
∂
ε
∂
∂
,,,
,,,1
,,,
,,,1
00
( )( )tzyxITzyxk
I ,,,,,, −
( )
( )[ ]
( )
( ) ( ) ++
Φ
+=
Φ
ΦΦΦ tzyxITzyxk
x
tzyx
Tzyxg
x
D
t
tzyx
II
V
VVV
V ,,,,,,
,,,
,,,1
,,,
2
,0
∂
∂
ε
∂
∂
∂
∂
( )[ ]
( )
( )[ ]
( )
−
Φ
++
Φ
++
ΦΦΦΦΦΦ
z
tzyx
Tzyxg
z
D
y
tzyx
Tzyxg
y
D
V
VVV
V
VVV
∂
∂
ε
∂
∂
∂
∂
ε
∂
∂
,,,
,,,1
,,,
,,,1
00
( )( )tzyxITzyxk
I ,,,,,, − .
We determine spatio-temporal distributions of concentrations of complexes of radiation defects
as the following power series
( ) ( )∑Φ=Φ
∞
=
Φ
0
,,,,,,
i
i
i
tzyxtzyx
ρρρε . (11)
Equations for the functions Φρi(x,y,z,t) and conditions for them could be obtained by substitution
of the series (11) into Eqs.(6) and appropriate boundary and initial conditions. We present the
equations and conditions in the Appendix. Solutions of the equations have been calculated by
standard approaches [20,21] and presented in the Appendix.
Now we calculate distribution of concentration of dopant in space and time by using the same
approach, which was used for calculation the same distributions of another concentrations. To
use the approach we transform spatio-temperature approximation of dopant diffusion coefficient
to the form: D
L(x,y,z,T)=D 0L[1+εLgL(x,y,z,T)], where D 0L is the average value of dopant diffusion
coefficient, 0≤
εL< 1, |gL(x,y,z,T)|≤1. Now we solve the Eq.(1) as the following power series
5
( )
0
,,,
~
0
=
∂
∂
=φ
φ
ϑφηχρ
,
( )
0
,,,
~
1
=
∂
∂
=φ
φ
ϑφηχρ
, ( )
( )
*
,,,
,,,
~
ρ
ϑφηχ
ϑφηχρ
ρf
=
. (9)
Let us determine solutions of Eqs.(8) as the following power series [12,13]
( ) ( )∑ ∑ ∑ Ω=
∞
=
∞
=
∞
=0 0 0
,,,
~
,,,
~
i j k
ijk
kji
ϑφηχρωεϑφηχρ
ρρ
. (10)
After substitution of the series (10) into Eqs.(8) and conditions (9) we obtain equations for ini-
tial-order approximations of concentration of point defects
( )ϑφηχ,,,
~
000
I and ( )ϑφηχ,,,
~
000
V , correc-
tions ( )ϑφηχ,,,
~
ijkI and ( )ϑφηχ,,,
~
ijkV and conditions for them for all i ≥1, j ≥1, k ≥1. The equations
are presented in the Appendix. We calculate solutions of the equations by standard Fourier ap-
proach [20,21]. The solutions are presented in the Appendix.
Farther we determine spatio-temporal distributions of concentrations of simplest complexes of
point radiation defects. To determine the distributions we transform approximations of diffusion
coefficients in the following form: D
Φρ(x,y,z,T)=D 0Φρ[1+εΦρgΦρ(x,y,z,T)], where D 0Φρ are the av-
erage values of diffusion coefficients. In this situation the Eqs.(6) could be written as
( )
( )[ ]
( )
( ) ( ) ++
Φ
+=
Φ
ΦΦΦ
tzyxITzyxk
x
tzyx
Tzyxg
x
D
t
tzyx
II
I
III
I
,,,,,,
,,,
,,,1
,,,
2
,0
∂
∂
ε
∂
∂
∂
∂
( )[ ]
( )
( )[ ]
( )
−
Φ
++
Φ
++
ΦΦΦΦΦΦ
z
tzyx
Tzyxg
z
D
y
tzyx
Tzyxg
y
D
I
III
I
III
∂
∂
ε
∂
∂
∂
∂
ε
∂
∂
,,,
,,,1
,,,
,,,1
00
( )( )tzyxITzyxk
I ,,,,,, −
( )
( )[ ]
( )
( ) ( ) ++
Φ
+=
Φ
ΦΦΦ tzyxITzyxk
x
tzyx
Tzyxg
x
D
t
tzyx
II
V
VVV
V ,,,,,,
,,,
,,,1
,,,
2
,0
∂
∂
ε
∂
∂
∂
∂
( )[ ]
( )
( )[ ]
( )
−
Φ
++
Φ
++
ΦΦΦΦΦΦ
z
tzyx
Tzyxg
z
D
y
tzyx
Tzyxg
y
D
V
VVV
V
VVV
∂
∂
ε
∂
∂
∂
∂
ε
∂
∂
,,,
,,,1
,,,
,,,1
00
( )( )tzyxITzyxk
I ,,,,,, − .
We determine spatio-temporal distributions of concentrations of complexes of radiation defects
as the following power series
( ) ( )∑Φ=Φ
∞
=
Φ
0
,,,,,,
i
i
i
tzyxtzyx
ρρρε . (11)
Equations for the functions Φρi(x,y,z,t) and conditions for them could be obtained by substitution
of the series (11) into Eqs.(6) and appropriate boundary and initial conditions. We present the
equations and conditions in the Appendix. Solutions of the equations have been calculated by
standard approaches [20,21] and presented in the Appendix.
Now we calculate distribution of concentration of dopant in space and time by using the same
approach, which was used for calculation the same distributions of another concentrations. To
use the approach we transform spatio-temperature approximation of dopant diffusion coefficient
to the form: D
L(x,y,z,T)=D 0L[1+εLgL(x,y,z,T)], where D 0L is the average value of dopant diffusion
coefficient, 0≤
εL< 1, |gL(x,y,z,T)|≤1. Now we solve the Eq.(1) as the following power series
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
6
( ) ( )∑ ∑=
∞
=
∞
=
0 1
,,,,,,
i j
ij
ji
L
tzyxCtzyxCξε .
The equations for the functions C
ij(x,y,z,t) and conditions for them have been obtained by substi-
tution of the series into Eq.(1) and conditions (2). We presented the equations and conditions for
them in the Appendix. We solve the equations by standard Fourier approach [20,21]. The solu-
tions have been presented in the Appendix.
We analyzed distributions of concentrations of dopant and radiation defects in space and time
analytically by using the second-order approximations on all parameters, which have been used
in appropriate series. The approximations are usually enough good approximations to make qual-
itative analysis and to obtain quantitative results. We check all results of analytical modeling by
comparison with results of numerical simulation.
3. DISCUSSION
In this section based on recently calculated relations we analyzed redistribution of dopant with
account redistribution of radiation defects. These relations give us possibility to obtain spatial
distributions of concentration of dopant. Typical spatial distributions of concentration of dopant
in directions, which is perpendicular to channel, are presented in Fig. 2. Curve 1 of this paper is a
typical distribution of concentration of dopant in directions of channel. The figure shows, that
presents of interface between layers of heterostructure gives us possibility to obtain more com-
pact and more homogenous distribution of concentration of dopant in direction, which is perpen-
dicular to the interface. However in this situation one shall to optimize annealing time. Reason of
the optimization is following. If annealing time is small, dopant can not achieves the interface. If
annealing time is large, dopant will diffuse into another layers of heterostructure too deep. We
calculate optimal value of annealing time by using recently introduced criterion [12-17]. Frame-
work the criterion we approximate real distribution of concentration of dopant by ideal step-wise
function
ψ (x,y,z). Farther the required optimal value of dopant concentration by minimization of
the following mean-squared error
( ) ( )[ ]∫ ∫ ∫ −Θ=
xyz
LLL
zyx
xdydzdzyxzyxC
LLL
U
0 0 0
,,,,,
1ψ . (12)
Dependences of optimal value of annealing time are presented on Figs. 3. Optimal values of an-
nealing time of implanted dopant should be smaller in comparison with the same annealing time
of infused dopant. Reason of the difference is necessity to anneal radiation defects before anneal-
ing of dopant.
6
( ) ( )∑ ∑=
∞
=
∞
=
0 1
,,,,,,
i j
ij
ji
L
tzyxCtzyxCξε .
The equations for the functions C
ij(x,y,z,t) and conditions for them have been obtained by substi-
tution of the series into Eq.(1) and conditions (2). We presented the equations and conditions for
them in the Appendix. We solve the equations by standard Fourier approach [20,21]. The solu-
tions have been presented in the Appendix.
We analyzed distributions of concentrations of dopant and radiation defects in space and time
analytically by using the second-order approximations on all parameters, which have been used
in appropriate series. The approximations are usually enough good approximations to make qual-
itative analysis and to obtain quantitative results. We check all results of analytical modeling by
comparison with results of numerical simulation.
3. DISCUSSION
In this section based on recently calculated relations we analyzed redistribution of dopant with
account redistribution of radiation defects. These relations give us possibility to obtain spatial
distributions of concentration of dopant. Typical spatial distributions of concentration of dopant
in directions, which is perpendicular to channel, are presented in Fig. 2. Curve 1 of this paper is a
typical distribution of concentration of dopant in directions of channel. The figure shows, that
presents of interface between layers of heterostructure gives us possibility to obtain more com-
pact and more homogenous distribution of concentration of dopant in direction, which is perpen-
dicular to the interface. However in this situation one shall to optimize annealing time. Reason of
the optimization is following. If annealing time is small, dopant can not achieves the interface. If
annealing time is large, dopant will diffuse into another layers of heterostructure too deep. We
calculate optimal value of annealing time by using recently introduced criterion [12-17]. Frame-
work the criterion we approximate real distribution of concentration of dopant by ideal step-wise
function
ψ (x,y,z). Farther the required optimal value of dopant concentration by minimization of
the following mean-squared error
( ) ( )[ ]∫ ∫ ∫ −Θ=
xyz
LLL
zyx
xdydzdzyxzyxC
LLL
U
0 0 0
,,,,,
1ψ . (12)
Dependences of optimal value of annealing time are presented on Figs. 3. Optimal values of an-
nealing time of implanted dopant should be smaller in comparison with the same annealing time
of infused dopant. Reason of the difference is necessity to anneal radiation defects before anneal-
ing of dopant.
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
7
Fig.2. The infused dopant concentration distributions. The considered direction is perpendicular to inter-
face between epitaxial layer and substrate. Increasing of number of distributions corresponds to increasing
of difference between values of dopant diffusion coefficient in layers of heterostructure. The distributions
have been calculated under condition, when value of dopant diffusion coefficient in epitaxial layer is larg-
er, than value of dopant diffusion coefficient in substrate
x
0.0
0.5
1.0
1.5
2.0
C
(
x
,
Θ
)
2
3
4
1
0
L/4 L/2 3 L/4 L
Epitaxial layer Substrate
Fig.3. Fig.2. The implanted dopant concentration distributions. The considered direction is perpendicular
to interface between epitaxial layer and substrate. Curves 1 and 3 corresponds to annealing time Θ =
0.0048 (L
x
2+Ly
2+Lz
2)/D0. Curves 2 and 4 corresponds to annealing time Θ = 0.0057(L x
2+Ly
2+Lz
2)/D0.
Curves 1 and 2 have been calculated for homogenous sample. Curves 3 and 4 have been calculated for
heterostructure under condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than
value of dopant diffusion coefficient in substrate
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
7
Fig.2. The infused dopant concentration distributions. The considered direction is perpendicular to inter-
face between epitaxial layer and substrate. Increasing of number of distributions corresponds to increasing
of difference between values of dopant diffusion coefficient in layers of heterostructure. The distributions
have been calculated under condition, when value of dopant diffusion coefficient in epitaxial layer is larg-
er, than value of dopant diffusion coefficient in substrate
x
0.0
0.5
1.0
1.5
2.0
C
(
x
,
Θ
)
2
3
4
1
0
L/4 L/2 3 L/4 L
Epitaxial layer Substrate
Fig.3. Fig.2. The implanted dopant concentration distributions. The considered direction is perpendicular
to interface between epitaxial layer and substrate. Curves 1 and 3 corresponds to annealing time Θ =
0.0048 (L
x
2+Ly
2+Lz
2)/D0. Curves 2 and 4 corresponds to annealing time Θ = 0.0057(L x
2+Ly
2+Lz
2)/D0.
Curves 1 and 2 have been calculated for homogenous sample. Curves 3 and 4 have been calculated for
heterostructure under condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than
value of dopant diffusion coefficient in substrate
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
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
8
Fig.3a. Optimal annealing time of infused dopant as dependences of several parameters. Curve 1 is the
dependence of the considered annealing time on dimensionless thickness of epitaxial layer a/L and
ξ = γ = 0
for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the
dependence of the considered annealing time on the parameter
ε for a/L=1/2 and ξ = γ = 0. Curve 3 is the
dependence of the considered annealing time on the parameter
ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the
dependence of the considered annealing time on parameter
γ for a/L=1/2 and ε = ξ = 0
4. CONCLUSIONS
In this paper we consider a possibility to increase density of elements in circuit of an amplifier
based on bipolar heterotransistors. Several conditions to increase the density have been formulat-
ed. Analysis of redistribution of dopant with account redistribution of radiation defects (after im-
plantation of ions of dopant) for optimization of the above annealing have been done by using
recently introduced analytical approach. The approach gives a possibility to analyze mass and
heat transports in a heterostructure without crosslinking of solutions on interfaces between layers
of the heterostructure with account nonlinearity of these transports and variation in time of their
parameters.
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.3a. Optimal annealing time of implanted dopant as dependences of several parameters. Curve 1 is the
dependence of the considered annealing time on dimensionless thickness of epitaxial layer a/L and
ξ = γ = 0
for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the
dependence of the considered annealing time on the parameter
ε for a/L=1/2 and ξ = γ = 0. Curve 3 is the
dependence of the considered annealing time on the parameter
ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the
dependence of the considered annealing time on parameter
γ for a/L=1/2 and ε = ξ = 0
ACKNOWLEDGEMENTS
This work is supported by the agreement of August 27, 2013
№ 02.В.49.21.0003 between The
Ministry of education and science of the Russian Federation and Lobachevsky State University
of Nizhni Novgorod.
REFERENCES
1. G. Volovich. Modern chips UM3Ch class D manufactured by firm MPS. Modern Electronics. Is-
sue 2. P. 10-17 (2006).
2. A. Kerentsev, V. Lanin. Constructive-technological features of MOSFET-transistors. Power
Electronics. Issue 1. P. 34-38 (2008).
3. O.A. Ageev, A.E. Belyaev, N.S. Boltovets, V.N. Ivanov, R.V. Konakova, Ya.Ya. Kudryk, P.M.
8
Fig.3a. Optimal annealing time of infused dopant as dependences of several parameters. Curve 1 is the
dependence of the considered annealing time on dimensionless thickness of epitaxial layer a/L and
ξ = γ = 0
for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the
dependence of the considered annealing time on the parameter
ε for a/L=1/2 and ξ = γ = 0. Curve 3 is the
dependence of the considered annealing time on the parameter
ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the
dependence of the considered annealing time on parameter
γ for a/L=1/2 and ε = ξ = 0
4. CONCLUSIONS
In this paper we consider a possibility to increase density of elements in circuit of an amplifier
based on bipolar heterotransistors. Several conditions to increase the density have been formulat-
ed. Analysis of redistribution of dopant with account redistribution of radiation defects (after im-
plantation of ions of dopant) for optimization of the above annealing have been done by using
recently introduced analytical approach. The approach gives a possibility to analyze mass and
heat transports in a heterostructure without crosslinking of solutions on interfaces between layers
of the heterostructure with account nonlinearity of these transports and variation in time of their
parameters.
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.3a. Optimal annealing time of implanted dopant as dependences of several parameters. Curve 1 is the
dependence of the considered annealing time on dimensionless thickness of epitaxial layer a/L and
ξ = γ = 0
for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the
dependence of the considered annealing time on the parameter
ε for a/L=1/2 and ξ = γ = 0. Curve 3 is the
dependence of the considered annealing time on the parameter
ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the
dependence of the considered annealing time on parameter
γ for a/L=1/2 and ε = ξ = 0
ACKNOWLEDGEMENTS
This work is supported by the agreement of August 27, 2013
№ 02.В.49.21.0003 between The
Ministry of education and science of the Russian Federation and Lobachevsky State University
of Nizhni Novgorod.
REFERENCES
1. G. Volovich. Modern chips UM3Ch class D manufactured by firm MPS. Modern Electronics. Is-
sue 2. P. 10-17 (2006).
2. A. Kerentsev, V. Lanin. Constructive-technological features of MOSFET-transistors. Power
Electronics. Issue 1. P. 34-38 (2008).
3. O.A. Ageev, A.E. Belyaev, N.S. Boltovets, V.N. Ivanov, R.V. Konakova, Ya.Ya. Kudryk, P.M.
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
9
Lytvyn, V.V. Milenin, A.V. Sachenko. Influence of displacement of the electron-hole equilibrium
on the process of transition metals diffusion in GaAs. Semiconductors. Vol. 43 (7). P. 897-903
(2009).
4. N.I. Volokobinskaya, I.N. Komarov, T.V. Matioukhina, V.I. Rechetniko, A.A. Rush, I.V. Falina,
A.S. Yastrebov. Investigation of technological processes of manufacturing of the bipolar power
high-voltage transistors with a grid of inclusions in the collector region. Semiconductors. Vol. 35
(8). P. 1013-1017 (2001).
5. A. Subramaniam, K. D. Cantley, E.M. Vogel. Active and Passive Electronic Components. Vol.
2013, ID 525017 (2013).
6. K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. Dopant distribution in the
recrystallization transient at the maximum melt depth induced by laser annealing. Appl. Phys.
Lett. 89 (17), 172111-172114 (2006).
7. H.T. Wang, L.S. Tan, E. F. Chor. Pulsed laser annealing of Be-implanted GaN. J. Appl. Phys. 98
(9), 094901-094905 (2006).
8. Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov,
Yu.N. Drozdov, V.D. Skupov. Diffusion processes in semiconductor structures during microwave
annealing. Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836-843 (2003).
9. V.V. Kozlivsky. Modification of semiconductors by proton beams (Nauka, Sant-Peterburg, 2003,
in Russian).
10. V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev,
1979, in Russian).
11. S. Kargarrazi, L. Lanni, C.-M. Zetterling. Solid-State Electronics. Vol. 116. P. 33-37 (2016).
12. E.L. Pankratov, E.A. Bulaeva. Increasing of sharpness of diffusion-junction heterorectifier by us-
ing radiation processing. Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1--1250028-8 (2012).
13. E.L. Pankratov, E.A. Bulaeva. About some ways to decrease quantity of defects in materials for
solid state electronic devices and diagnostics of their realization. Reviews in Theoretical Science.
Vol. 3 (2). P. 177-215 (2015).
14. E.L. Pankratov, E.A. Bulaeva. An approach to decrease dimentions of logical elements based on
bipolar transistor. Int. J. Comp. Sci. Appl. Vol. 5 (4). P. 1-18 (2015).
15. E.L. Pankratov, E.A. Bulaeva. On approach to decrease dimensions of element OR manufactured
by using field-effect heterotransistor. Nano Science and Nano Technology: An Indian Journal.
Vol. 9 (4). P. 43-60 (2015).
16. E.L. Pankratov, E.A. Bulaeva. Optimization of manufacturing of emitter-coupled logic to de-
crease surface of chip. Int. J. Mod. Phys. B. Vol. 29 (5). P. 1550023-1-1550023-12 (2015).
17. E.L. Pankratov, E.A. Bulaeva. Decreasing of mechanical stress in a semiconductor
heterostructure by radiation processing. J. Comp. Theor. Nanoscience. Vol. 11 (1). P. 91-101
(2014).
18. Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991).
19. P.M. Fahey, P.B. Griffin, J.D. Plummer. Diffusion and point defects in silicon. Rev. Mod. Phys.
1989. V. 61.
№ 2. P. 289-388.
20. A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations (Moscow, Nauka 1972) (in
Russian).
21. H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids (Oxford University Press, 1964).
Authors:
Pankratov Evgeny Leonidovich is a Full Doctor of Science, an Associate Professor of Nizhny Novgorod
State University. He has 186 published papers in area of his researches.
Bulaeva Elena Alexeevna is a PhD student of Nizhny Novgorod State University. She has 134 published
papers in area of her researches.
APPENDIX
Equations for the functions
( )ϑφηχ,,,
~
ijk
I and ( )ϑφηχ,,,
~
ijk
V , i ≥0, j ≥0, k ≥0 and conditions for
them
9
Lytvyn, V.V. Milenin, A.V. Sachenko. Influence of displacement of the electron-hole equilibrium
on the process of transition metals diffusion in GaAs. Semiconductors. Vol. 43 (7). P. 897-903
(2009).
4. N.I. Volokobinskaya, I.N. Komarov, T.V. Matioukhina, V.I. Rechetniko, A.A. Rush, I.V. Falina,
A.S. Yastrebov. Investigation of technological processes of manufacturing of the bipolar power
high-voltage transistors with a grid of inclusions in the collector region. Semiconductors. Vol. 35
(8). P. 1013-1017 (2001).
5. A. Subramaniam, K. D. Cantley, E.M. Vogel. Active and Passive Electronic Components. Vol.
2013, ID 525017 (2013).
6. K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. Dopant distribution in the
recrystallization transient at the maximum melt depth induced by laser annealing. Appl. Phys.
Lett. 89 (17), 172111-172114 (2006).
7. H.T. Wang, L.S. Tan, E. F. Chor. Pulsed laser annealing of Be-implanted GaN. J. Appl. Phys. 98
(9), 094901-094905 (2006).
8. Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov,
Yu.N. Drozdov, V.D. Skupov. Diffusion processes in semiconductor structures during microwave
annealing. Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836-843 (2003).
9. V.V. Kozlivsky. Modification of semiconductors by proton beams (Nauka, Sant-Peterburg, 2003,
in Russian).
10. V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev,
1979, in Russian).
11. S. Kargarrazi, L. Lanni, C.-M. Zetterling. Solid-State Electronics. Vol. 116. P. 33-37 (2016).
12. E.L. Pankratov, E.A. Bulaeva. Increasing of sharpness of diffusion-junction heterorectifier by us-
ing radiation processing. Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1--1250028-8 (2012).
13. E.L. Pankratov, E.A. Bulaeva. About some ways to decrease quantity of defects in materials for
solid state electronic devices and diagnostics of their realization. Reviews in Theoretical Science.
Vol. 3 (2). P. 177-215 (2015).
14. E.L. Pankratov, E.A. Bulaeva. An approach to decrease dimentions of logical elements based on
bipolar transistor. Int. J. Comp. Sci. Appl. Vol. 5 (4). P. 1-18 (2015).
15. E.L. Pankratov, E.A. Bulaeva. On approach to decrease dimensions of element OR manufactured
by using field-effect heterotransistor. Nano Science and Nano Technology: An Indian Journal.
Vol. 9 (4). P. 43-60 (2015).
16. E.L. Pankratov, E.A. Bulaeva. Optimization of manufacturing of emitter-coupled logic to de-
crease surface of chip. Int. J. Mod. Phys. B. Vol. 29 (5). P. 1550023-1-1550023-12 (2015).
17. E.L. Pankratov, E.A. Bulaeva. Decreasing of mechanical stress in a semiconductor
heterostructure by radiation processing. J. Comp. Theor. Nanoscience. Vol. 11 (1). P. 91-101
(2014).
18. Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991).
19. P.M. Fahey, P.B. Griffin, J.D. Plummer. Diffusion and point defects in silicon. Rev. Mod. Phys.
1989. V. 61.
№ 2. P. 289-388.
20. A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations (Moscow, Nauka 1972) (in
Russian).
21. H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids (Oxford University Press, 1964).
Authors:
Pankratov Evgeny Leonidovich is a Full Doctor of Science, an Associate Professor of Nizhny Novgorod
State University. He has 186 published papers in area of his researches.
Bulaeva Elena Alexeevna is a PhD student of Nizhny Novgorod State University. She has 134 published
papers in area of her researches.
APPENDIX
Equations for the functions
( )ϑφηχ,,,
~
ijk
I and ( )ϑφηχ,,,
~
ijk
V , i ≥0, j ≥0, k ≥0 and conditions for
them
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
10
( ) ( ) ( ) ( )
2
000
2
0
0
2
000
2
0
0
2
000
2
0
0000
,,, ~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
I
D
DI
D
DI
D
DI
V
I
V
I
V
I
( ) ( ) ( ) ( )
2
000
2
0
0
2
000
2
0
0
2
000
2
0
0000
,,, ~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
V
D
DV
D
DV
D
DV
I
V
I
V
I
V
;
() ( ) ( ) ( )
( )
×
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
Tg
III
D
DII
iii
V
Ii
,,,
,,,
~
,,,
~
,,,
~
,
~
2
00
2
2
00
2
2
00
2
0
000
φηχ
χφ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑχ
( )
( )
( )
( )
×
∂
∂
+
∂
∂
∂
∂
+
∂
∂
×
−−
Tg
I
Tg
D
D
D
DI
I
i
I
V
I
V
Ii ,,,
,,,
~
,,,
,,,
~
100
0
0
0
0100
φηχ
φη
ϑφηχ
φηχ
ηχ
ϑφηχ
( )
V
Ii
D
DI
0
0100
,,,
~
∂
∂
×
−
φ
ϑφηχ
, i ≥1,
() ( ) ( ) ( )
( )
×
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
Tg
VVV
D
DV
V
iii
I
Vi
,,,
,,,
~
,,,
~
,,,
~
,
~
2
00
2
2
00
2
2
00
2
0
000
φηχ
χφ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑχ
( )
( )
( )
( )
×
∂
∂
+
∂
∂
∂
∂
+
∂
∂
×
−−
Tg
V
Tg
D
D
D
DV
V
i
V
I
V
I
Vi
,,,
,,,
~
,,,
,,,
~
100
0
0
0
0100
φηχ
φη
ϑφηχ
φηχ
ηχ
ϑφηχ
( )
I
Vi
D
DV
0
0100
,,,
~
∂
∂
×
−
φ
ϑφηχ
, i ≥1,
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
010
2
2
010
2
2
010
2
0
0010
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
III
D
DI
V
I
( )[ ]( )( )ϑφηχϑφηχφηχε,,,
~
,,,
~
,,,1
000000,,
VITg
VIVI
+−
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
010
2
2
010
2
2
010
2
0
0010
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
VVV
D
DV
I
V
( )[ ]( )( )ϑφηχϑφηχφηχε,,,
~
,,,
~
,,,1
000000,,
VITg
VIVI
+− ;
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
020
2
2
020
2
2
020
2
0
0020
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
III
D
DI
V
I
( )[ ]( )( ) ( )( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε,,,
~
,,,
~
,,,
~
,,,
~
,,,1
010000000010,,
VIVITg
VIVI
++−
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
020
2
2
020
2
2
020
2
0
0020
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
VVV
D
DV
V
I
( )[ ]( )( ) ( )( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε,,,
~
,,,
~
,,,
~
,,,
~
,,,1
010000000010,,
VIVITg
VIVI
++− ;
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
001
2
2
001
2
2
001
2
0
0001
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
III
D
DI
V
I
( )[ ]( )ϑφηχφηχε,,,
~
,,,1
2
000,,
ITg
IIII
+−
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
001
2
2
001
2
2
001
2
0
0001
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
VVV
D
DV
I
V
( )[ ]( )ϑφηχφηχε,,,
~
,,,1
2
000,,
VTg
IIII
+− ;
10
( ) ( ) ( ) ( )
2
000
2
0
0
2
000
2
0
0
2
000
2
0
0000
,,, ~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
I
D
DI
D
DI
D
DI
V
I
V
I
V
I
( ) ( ) ( ) ( )
2
000
2
0
0
2
000
2
0
0
2
000
2
0
0000
,,, ~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
V
D
DV
D
DV
D
DV
I
V
I
V
I
V
;
() ( ) ( ) ( )
( )
×
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
Tg
III
D
DII
iii
V
Ii
,,,
,,,
~
,,,
~
,,,
~
,
~
2
00
2
2
00
2
2
00
2
0
000
φηχ
χφ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑχ
( )
( )
( )
( )
×
∂
∂
+
∂
∂
∂
∂
+
∂
∂
×
−−
Tg
I
Tg
D
D
D
DI
I
i
I
V
I
V
Ii ,,,
,,,
~
,,,
,,,
~
100
0
0
0
0100
φηχ
φη
ϑφηχ
φηχ
ηχ
ϑφηχ
( )
V
Ii
D
DI
0
0100
,,,
~
∂
∂
×
−
φ
ϑφηχ
, i ≥1,
() ( ) ( ) ( )
( )
×
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
Tg
VVV
D
DV
V
iii
I
Vi
,,,
,,,
~
,,,
~
,,,
~
,
~
2
00
2
2
00
2
2
00
2
0
000
φηχ
χφ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑχ
( )
( )
( )
( )
×
∂
∂
+
∂
∂
∂
∂
+
∂
∂
×
−−
Tg
V
Tg
D
D
D
DV
V
i
V
I
V
I
Vi
,,,
,,,
~
,,,
,,,
~
100
0
0
0
0100
φηχ
φη
ϑφηχ
φηχ
ηχ
ϑφηχ
( )
I
Vi
D
DV
0
0100
,,,
~
∂
∂
×
−
φ
ϑφηχ
, i ≥1,
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
010
2
2
010
2
2
010
2
0
0010
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
III
D
DI
V
I
( )[ ]( )( )ϑφηχϑφηχφηχε,,,
~
,,,
~
,,,1
000000,,
VITg
VIVI
+−
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
010
2
2
010
2
2
010
2
0
0010
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
VVV
D
DV
I
V
( )[ ]( )( )ϑφηχϑφηχφηχε,,,
~
,,,
~
,,,1
000000,,
VITg
VIVI
+− ;
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
020
2
2
020
2
2
020
2
0
0020
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
III
D
DI
V
I
( )[ ]( )( ) ( )( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε,,,
~
,,,
~
,,,
~
,,,
~
,,,1
010000000010,,
VIVITg
VIVI
++−
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
020
2
2
020
2
2
020
2
0
0020
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
VVV
D
DV
V
I
( )[ ]( )( ) ( )( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε,,,
~
,,,
~
,,,
~
,,,
~
,,,1
010000000010,,
VIVITg
VIVI
++− ;
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
001
2
2
001
2
2
001
2
0
0001
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
III
D
DI
V
I
( )[ ]( )ϑφηχφηχε,,,
~
,,,1
2
000,,
ITg
IIII
+−
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
001
2
2
001
2
2
001
2
0
0001
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
VVV
D
DV
I
V
( )[ ]( )ϑφηχφηχε,,,
~
,,,1
2
000,,
VTg
IIII
+− ;
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
11
( ) ( ) ( ) ( )
×+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
V
I
V
I
D
DIII
D
DI
0
0
2
110
2
2
110
2
2
110
2
0
0110
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
( )
( )
( )
( )
( )[
×
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
× Tg
I
Tg
I
Tg
III ,,,
,,,
~
,,,
,,,
~
,,,
010010
φηχ
φη
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
( )
( ) ( ) ( ) ( )[ ] ×+−
∂
∂
×
ϑφηχϑφηχϑφηχϑφηχ
φ
ϑφηχ,,,
~
,,,
~
,,,
~
,,,
~,,,
~
100000000100
010VIVI
I
( )[ ]Tg
IIII
,,,1
,,
φηχε+×
( ) ( ) ( ) ( )
×+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
I
V
I
V
D
DVVV
D
DV
0
0
2
110
2
2
110
2
2
110
2
0
0110
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
( )
( )
( )
( )
( )[
×
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
× Tg
V
Tg
V
Tg
IVV
,,,
,,,
~
,,,
,,,
~
,,,
010010
φηχ
φη
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
( )
( ) ( ) ( ) ( )[ ] ×+−
∂
∂
×ϑφηχϑφηχϑφηχϑφηχ
φ
ϑφηχ,,,
~
,,,
~
,,,
~
,,,
~,,, ~
100000000100
010
IVIV
V
( )[ ]Tg
VVVV ,,,1
,, φηχε+× ;
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
002
2
2
002
2
2
002
2
0
0002
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
III
D
DI
V
I
( )[ ]( )( )ϑφηχϑφηχφηχε,,,
~
,,,
~
,,,1
000001,,
IITg
IIII
+−
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
002
2
2
002
2
2
002
2
0
0002
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
VVV
D
DV
I
V
( )[ ]( )( )ϑφηχϑφηχφηχε,,,
~
,,,
~
,,,1
000001,,
VVЕg
VVVV
+− ;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
101
2
2
101
2
2
101
2
0
0101
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
III
D
DI
V
I
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+ η
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
,,,
~
,,,
,,,
~
,,,
001001
0
0
I
Tg
I
Tg
D
D
II
V
I
( )
( )
( )[ ] ( ) ( ) ϑφηχϑφηχφηχε
φ
ϑφηχ
φηχ
φ,,,
~
,,,
~
,,,1
,,,
~
,,,
000100
001
VITg
I
Tg
III
+−
∂
∂
∂
∂
+
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
101
2
2
101
2
2
101
2
0
0101
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
VVV
D
DV
I
V
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+ η
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
,,,
~
,,,
,,,
~
,,,
001001
0
0
V
Tg
V
Tg
D
D
VV
I
V
( )
( )
( )[ ] ( ) ( ) ϑφηχϑφηχφηχε
φ
ϑφηχ
φηχ
φ,,,
~
,,,
~
,,,1
,,,
~
,,,
100000
001VITg
V
Tg
VVV+−
∂
∂
∂
∂
+ ;
( ) ( ) ( ) ( )
( ) ×−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
ϑφηχ
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ,,,
~,,,
~
,,,
~
,,,
~
,,,
~
0102
011
2
2
011
2
2
011
2
0
0011I
III
D
DI
V
I
11
( ) ( ) ( ) ( )
×+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
V
I
V
I
D
DIII
D
DI
0
0
2
110
2
2
110
2
2
110
2
0
0110
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
( )
( )
( )
( )
( )[
×
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
× Tg
I
Tg
I
Tg
III ,,,
,,,
~
,,,
,,,
~
,,,
010010
φηχ
φη
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
( )
( ) ( ) ( ) ( )[ ] ×+−
∂
∂
×
ϑφηχϑφηχϑφηχϑφηχ
φ
ϑφηχ,,,
~
,,,
~
,,,
~
,,,
~,,,
~
100000000100
010VIVI
I
( )[ ]Tg
IIII
,,,1
,,
φηχε+×
( ) ( ) ( ) ( )
×+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
I
V
I
V
D
DVVV
D
DV
0
0
2
110
2
2
110
2
2
110
2
0
0110
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
( )
( )
( )
( )
( )[
×
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
× Tg
V
Tg
V
Tg
IVV
,,,
,,,
~
,,,
,,,
~
,,,
010010
φηχ
φη
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
( )
( ) ( ) ( ) ( )[ ] ×+−
∂
∂
×ϑφηχϑφηχϑφηχϑφηχ
φ
ϑφηχ,,,
~
,,,
~
,,,
~
,,,
~,,, ~
100000000100
010
IVIV
V
( )[ ]Tg
VVVV ,,,1
,, φηχε+× ;
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
002
2
2
002
2
2
002
2
0
0002
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
III
D
DI
V
I
( )[ ]( )( )ϑφηχϑφηχφηχε,,,
~
,,,
~
,,,1
000001,,
IITg
IIII
+−
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
002
2
2
002
2
2
002
2
0
0002
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
VVV
D
DV
I
V
( )[ ]( )( )ϑφηχϑφηχφηχε,,,
~
,,,
~
,,,1
000001,,
VVЕg
VVVV
+− ;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
101
2
2
101
2
2
101
2
0
0101
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
III
D
DI
V
I
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+ η
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
,,,
~
,,,
,,,
~
,,,
001001
0
0
I
Tg
I
Tg
D
D
II
V
I
( )
( )
( )[ ] ( ) ( ) ϑφηχϑφηχφηχε
φ
ϑφηχ
φηχ
φ,,,
~
,,,
~
,,,1
,,,
~
,,,
000100
001
VITg
I
Tg
III
+−
∂
∂
∂
∂
+
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
101
2
2
101
2
2
101
2
0
0101
,,,
~
,,,
~
,,,
~
,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
VVV
D
DV
I
V
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+ η
ϑφηχ
φηχ
ηχ
ϑφηχ
φηχ
χ
,,,
~
,,,
,,,
~
,,,
001001
0
0
V
Tg
V
Tg
D
D
VV
I
V
( )
( )
( )[ ] ( ) ( ) ϑφηχϑφηχφηχε
φ
ϑφηχ
φηχ
φ,,,
~
,,,
~
,,,1
,,,
~
,,,
100000
001VITg
V
Tg
VVV+−
∂
∂
∂
∂
+ ;
( ) ( ) ( ) ( )
( ) ×−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
ϑφηχ
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ,,,
~,,,
~
,,,
~
,,,
~
,,,
~
0102
011
2
2
011
2
2
011
2
0
0011I
III
D
DI
V
I
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
12
( )[ ]( ) ( )[ ]( )( )ϑφηχϑφηχφηχεϑφηχφηχε,,,
~
,,,
~
,,,1,,,
~
,,,1
000001,,000,,
VITgITg
VIVIIIII
+−+×
( ) ( ) ( ) ( )
( ) ×−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
ϑφηχ
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ,,,
~,,,
~
,,,
~
,,,
~
,,,
~
0102
011
2
2
011
2
2
011
2
0
0011
V
VVV
D
DV
I
V
( )[ ]( ) ( )[ ]( )( )ϑφηχϑφηχφηχεϑφηχφηχε,,,
~
,,,
~
,,,1,,,
~
,,,1
001000,,000,,
VItgVTg
VIVIVVVV
+−+× ;
( )
0
,,,
~
0
=
∂
∂
=x
ijk
χ
ϑφηχρ
,
( )
0
,,,
~
1
=
∂
∂
=x
ijk
χ
ϑφηχρ
,
( )
0
,,,
~
0
=
∂
∂
=η
η
ϑφηχρ
ijk
,
( )
0
,,,
~
1
=
∂
∂
=η
η
ϑφηχρ
ijk
,
( )
0
,,,
~
0
=
∂
∂
=φ
φ
ϑφηχρ
ijk
,
( )
0
,,,
~
1
=
∂
∂
=φ
φ
ϑφηχρ
ijk
(i ≥0, j ≥0, k ≥0);
( )( )
*
000
,,0,,,
~
ρφηχφηχρ
ρ
f= , ( )00,,,
~
=φηχρ
ijk (i ≥1, j ≥1, k ≥1).
Solutions of the above equations could be written as
( )
( ) ( ) ( ) ( )∑+=
∞
=1
00021
,,,
~
n
nn
ecccF LL
ϑφηχϑφηχρ
ρρ
,
where ( ) ( ) ( ) ( )∫ ∫ ∫=
1
0
1
0
1
0
*
,,coscoscos
1
udvdwdwvufwnvnunF
nnρρ
πππ
ρ ,
() ( )
IVnIDDne
00
22expϑπϑ−= ,
c
n(χ) = cos (π n χ),
() ( )
VInV
DDne
00
22
exp ϑπϑ−= ;
( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
×∑ ∫ ∫ ∫ ∫
∂
∂
−−=
∞
=
−
1 0
1
0
1
0
1
0
100
0
0
00
,,, ~
2,,,
~
n
i
nnnInIn
V
I
i u
wvuI
vcuseecccn
D
D
I
ϑ τ
τϑφηχπϑφηχ
( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−×
∞
=
1 0
1
0
1
0
0
0
2,,,
n
nnnInIn
V
I
In
vsuceecccn
D
D
dudvdwdTwvugwc
ϑ
τϑφηχπτ
( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ×∑ ∫−−∫
∂
∂
×
∞
=
−
1 0
0
0
1
0
100
2
,,,
~
,,,
n
nInIn
V
Ii
In
eecccn
D
D
dudvdwd
v
wvuI
Twvugwc
ϑ
τϑφηχπτ
τ
( ) ( ) ( ) ( )
( )
∫ ∫ ∫
∂
∂
×
−
1
0
1
0
1
0
100 ,,,
~
,,, τ
τdudvdwd
w
wvuI
Twvugwsvcuc
i
Innn
, i ≥1,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫ ∫−−=
∞
=
1 0
1
0
1
0
1
0
0
0
00
,,,2,,,
~
n
VnnnInVn
I
V
i
Twvugvcuseecccn
D
D
V
ϑ
τϑφηχπϑφηχ
( )
()
( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫−−
∂
∂
×
∞
=
−
1 0
1
0
1
0
0
0100,
~
n
nnnInVn
I
Vi
n
vsuceecccn
D
D
dudvdwd
u
uV
wc
ϑ
τϑφηχτ
τ
( ) ( )
()
( ) ( ) ( ) ( )∑ ×−∫
∂
∂
×
∞
=
−
1
0
0
1
0
100
2
,
~
,,,2
n
nVn
I
Vi
Vn
ecccn
D
D
dudvdwd
v
uV
Twvugwcϑφηχπτ
τ
π
( ) ( ) ( ) ( ) ( ) ()
∫ ∫ ∫ ∫
∂
∂
−×
−
ϑ
τ
τ
τ
0
1
0
1
0
1
0
100
,
~
,,, dudvdwd
w
uV
Twvugwsvcuce
i
VnnnnI
, i ≥1,
where s
n(χ) = sin (π n χ);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−=
∞
=1 0
1
0
1
0
1
0
010
2,,,
~
n
nnnnnnnn
wcvcuceeccc
ϑ
ρρ
τϑφηχϑφηχρ
( )[ ]( )( ) τττεdudvdwdwvuVwvuITwvug
VIVI
,,,
~
,,,
~
,,,1
000000,,
+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ×∑ ∫ ∫ ∫ ∫ +−−=
∞
=1 0
1
0
1
0
1
0
,
0
0
020
12,,,
~
n
VInnnnnnnn
V
I
wcvcuceeccc
D
D
ϑ
ρρ
ετϑφηχϑφηχρ
( )]( )( ) ( )( )[ ] τττττdudvdwdwvuVwvuIwvuVwvuITwvug
VI
,,,
~
,,,
~
,,,
~
,,,
~
,,,
010000000010,
+× ;
12
( )[ ]( ) ( )[ ]( )( )ϑφηχϑφηχφηχεϑφηχφηχε,,,
~
,,,
~
,,,1,,,
~
,,,1
000001,,000,,
VITgITg
VIVIIIII
+−+×
( ) ( ) ( ) ( )
( ) ×−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
ϑφηχ
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ,,,
~,,,
~
,,,
~
,,,
~
,,,
~
0102
011
2
2
011
2
2
011
2
0
0011
V
VVV
D
DV
I
V
( )[ ]( ) ( )[ ]( )( )ϑφηχϑφηχφηχεϑφηχφηχε,,,
~
,,,
~
,,,1,,,
~
,,,1
001000,,000,,
VItgVTg
VIVIVVVV
+−+× ;
( )
0
,,,
~
0
=
∂
∂
=x
ijk
χ
ϑφηχρ
,
( )
0
,,,
~
1
=
∂
∂
=x
ijk
χ
ϑφηχρ
,
( )
0
,,,
~
0
=
∂
∂
=η
η
ϑφηχρ
ijk
,
( )
0
,,,
~
1
=
∂
∂
=η
η
ϑφηχρ
ijk
,
( )
0
,,,
~
0
=
∂
∂
=φ
φ
ϑφηχρ
ijk
,
( )
0
,,,
~
1
=
∂
∂
=φ
φ
ϑφηχρ
ijk
(i ≥0, j ≥0, k ≥0);
( )( )
*
000
,,0,,,
~
ρφηχφηχρ
ρ
f= , ( )00,,,
~
=φηχρ
ijk (i ≥1, j ≥1, k ≥1).
Solutions of the above equations could be written as
( )
( ) ( ) ( ) ( )∑+=
∞
=1
00021
,,,
~
n
nn
ecccF LL
ϑφηχϑφηχρ
ρρ
,
where ( ) ( ) ( ) ( )∫ ∫ ∫=
1
0
1
0
1
0
*
,,coscoscos
1
udvdwdwvufwnvnunF
nnρρ
πππ
ρ ,
() ( )
IVnIDDne
00
22expϑπϑ−= ,
c
n(χ) = cos (π n χ),
() ( )
VInV
DDne
00
22
exp ϑπϑ−= ;
( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
×∑ ∫ ∫ ∫ ∫
∂
∂
−−=
∞
=
−
1 0
1
0
1
0
1
0
100
0
0
00
,,, ~
2,,,
~
n
i
nnnInIn
V
I
i u
wvuI
vcuseecccn
D
D
I
ϑ τ
τϑφηχπϑφηχ
( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−×
∞
=
1 0
1
0
1
0
0
0
2,,,
n
nnnInIn
V
I
In
vsuceecccn
D
D
dudvdwdTwvugwc
ϑ
τϑφηχπτ
( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ×∑ ∫−−∫
∂
∂
×
∞
=
−
1 0
0
0
1
0
100
2
,,,
~
,,,
n
nInIn
V
Ii
In
eecccn
D
D
dudvdwd
v
wvuI
Twvugwc
ϑ
τϑφηχπτ
τ
( ) ( ) ( ) ( )
( )
∫ ∫ ∫
∂
∂
×
−
1
0
1
0
1
0
100 ,,,
~
,,, τ
τdudvdwd
w
wvuI
Twvugwsvcuc
i
Innn
, i ≥1,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫ ∫−−=
∞
=
1 0
1
0
1
0
1
0
0
0
00
,,,2,,,
~
n
VnnnInVn
I
V
i
Twvugvcuseecccn
D
D
V
ϑ
τϑφηχπϑφηχ
( )
()
( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫−−
∂
∂
×
∞
=
−
1 0
1
0
1
0
0
0100,
~
n
nnnInVn
I
Vi
n
vsuceecccn
D
D
dudvdwd
u
uV
wc
ϑ
τϑφηχτ
τ
( ) ( )
()
( ) ( ) ( ) ( )∑ ×−∫
∂
∂
×
∞
=
−
1
0
0
1
0
100
2
,
~
,,,2
n
nVn
I
Vi
Vn
ecccn
D
D
dudvdwd
v
uV
Twvugwcϑφηχπτ
τ
π
( ) ( ) ( ) ( ) ( ) ()
∫ ∫ ∫ ∫
∂
∂
−×
−
ϑ
τ
τ
τ
0
1
0
1
0
1
0
100
,
~
,,, dudvdwd
w
uV
Twvugwsvcuce
i
VnnnnI
, i ≥1,
where s
n(χ) = sin (π n χ);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−=
∞
=1 0
1
0
1
0
1
0
010
2,,,
~
n
nnnnnnnn
wcvcuceeccc
ϑ
ρρ
τϑφηχϑφηχρ
( )[ ]( )( ) τττεdudvdwdwvuVwvuITwvug
VIVI
,,,
~
,,,
~
,,,1
000000,,
+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ×∑ ∫ ∫ ∫ ∫ +−−=
∞
=1 0
1
0
1
0
1
0
,
0
0
020
12,,,
~
n
VInnnnnnnn
V
I
wcvcuceeccc
D
D
ϑ
ρρ
ετϑφηχϑφηχρ
( )]( )( ) ( )( )[ ] τττττdudvdwdwvuVwvuIwvuVwvuITwvug
VI
,,,
~
,,,
~
,,,
~
,,,
~
,,,
010000000010,
+× ;
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
13
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
0
001
2,,,
~
n
nnnnnnnn
wcvcuceeccc
ϑ
ρρ
τϑφηχϑφηχρ
( )[ ]( ) ττρε
ρρρρ
dudvdwdwvuTwvug ,,,
~
,,,1
2
000,,
+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ×∫ ∫−−=
∞
=1 0
1
0
1
0
1
0
002
2,,,
~
n
nnnnnnnn
wcvcuceeccc
ϑ
ρρ
τϑφηχϑφηχρ ( )[ ]( )( ) ττρτρε
ρρρρ
dudvdwdwvuwvuTwvug ,,,
~
,,,
~
,,,1
000001,,
+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ −−=
∞
=1 0
1
0
1
0
1
0
0
0
110
2,,,
~
n
nnnnInInnn
V
I
ucvcuseecccn
D
D
I
ϑ
τϑφηχπϑφηχ
( ) ( )
( ) ( ) ( ) ( )×∑−
∂
∂
×
∞
=
−
1
0
0100
2
,,,
~
,,,
n
nInnn
V
Ii
I
ecccn
D
D
dudvdwd
u
wvuI
Twvugϑφηχπτ
τ
( ) ( ) ( ) ( ) ( )
( )
×−∫ ∫ ∫ ∫
∂
∂
−×
−
V
Ii
InnnnI
D
D
dudvdwd
v
wvuI
Twvugucvsuce
0
0
0
1
0
1
0
1
0
100
2
,,,
~
,,, πτ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( )
( )
×∑ ∫ ∫ ∫ ∫
∂
∂
−×
∞
=
−
1 0
1
0
1
0
1
0
100 ,,,
~
,,,
n
i
InnnnInI
dudvdwd
w
wvuI
Twvugusvcuceen
ϑ
τ
τ
τϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[∑ ∫ ∫ ∫ ∫ ×+−−×
∞
=1 0
1
0
1
0
1
0
,
12
n
VInnnnInnnInnnn
vcvcuceccecccc
ϑ
ετφηϑχφηχ
( )]( )( ) ( )( )[ ] τττττdudvdwdwvuVwvuIwvuVwvuITwvug
VI
,,,
~
,,,
~
,,,
~
,,,
~
,,,
100000000100,
+×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ −−=
∞
=1 0
1
0
1
0
1
0
0
0
110
2,,,
~
n
nnnnVnVnnn
I
V
ucvcuseecccn
D
D
V
ϑ
τϑφηχπϑφηχ
( ) ( )
( ) ( ) ( ) ( )×∑−
∂
∂
×
∞
=
−
1
0
0100
2
,,,
~
,,,
n
nVnnn
I
Vi
V
ecccn
D
D
dudvdwd
u
wvuV
Twvugϑφηχπτ
τ
( ) ( ) ( ) ( ) ( )
( )
×−∫ ∫ ∫ ∫
∂
∂
−×
−
I
Vi
VnnnnV
D
D
dudvdwd
v
wvuV
Twvugucvsuce
0
0
0
1
0
1
0
1
0
100
2
,,,
~
,,, πτ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( )
( )
×∑ ∫ ∫ ∫ ∫
∂
∂
−×
∞
=
−
1 0
1
0
1
0
1
0
100 ,,,
~
,,,
n
i
VnnnnVnV
dudvdwd
w
wvuV
Twvugusvcuceen
ϑ
τ
τ
τϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]∑ ∫ ∫ ∫ ∫ ×+−−×
∞
=1 0
1
0
1
0
1
0
,,
,,,12
n
VIVInnnVnnnInnnn
Twvugvcuceccecccc
ϑ
ετφηϑχφηχ
()( )( ) ( )( )[ ] τττττdudvdwdwvuVwvuIwvuVwvuIwc
n ,,,
~
,,,
~
,,,
~
,,,
~
100000000100+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ −−=
∞
=1 0
1
0
1
0
1
0
0
0
101
,,,2,,,
~
n
InnnInInnn
V
I
Twvugvcuseecccn
D
D
I
ϑ
τϑφηχπϑφηχ
( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ −−
∂
∂
×
∞
=1 0
1
0
0
0001
2
,,,
~
n
nnInInnn
V
I
n
uceecccn
D
D
dudvdwd
u
wvuI
wc
ϑ
τϑφηχπτ
τ
( ) ( ) ( )
( )
( ) ( ) ( ) ( )×∑−∫ ∫
∂
∂
×
∞
=1
0
0
1
0
1
0
001
2
,,,
~
,,,
n
nnnnI
V
I
Inn
cccen
D
D
dudvdwd
v
wvuI
Twvugwcvsφηχϑπτ
τ
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( )×∑−∫ ∫ ∫ ∫
∂
∂
−×
∞
=10
1
0
1
0
1
0
001
2
,,,~
,,,
n
nnnInnnnI
cccdudvdwd
w
wvuI
Twvugwsvcuceφηχτ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( )
[ ]( ) ( )∫ ∫ ∫ ∫ +−×
ϑ
τττετϑ
0
1
0
1
0
1
0
000100,,
,,,
~
,,,
~
,,,1 dudvdwdwvuVwvuITwvugwcvcucee
VIVInnnnInI
13
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0
1
0
1
0
1
0
001
2,,,
~
n
nnnnnnnn
wcvcuceeccc
ϑ
ρρ
τϑφηχϑφηχρ
( )[ ]( ) ττρε
ρρρρ
dudvdwdwvuTwvug ,,,
~
,,,1
2
000,,
+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ×∫ ∫−−=
∞
=1 0
1
0
1
0
1
0
002
2,,,
~
n
nnnnnnnn
wcvcuceeccc
ϑ
ρρ
τϑφηχϑφηχρ ( )[ ]( )( ) ττρτρε
ρρρρ
dudvdwdwvuwvuTwvug ,,,
~
,,,
~
,,,1
000001,,
+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ −−=
∞
=1 0
1
0
1
0
1
0
0
0
110
2,,,
~
n
nnnnInInnn
V
I
ucvcuseecccn
D
D
I
ϑ
τϑφηχπϑφηχ
( ) ( )
( ) ( ) ( ) ( )×∑−
∂
∂
×
∞
=
−
1
0
0100
2
,,,
~
,,,
n
nInnn
V
Ii
I
ecccn
D
D
dudvdwd
u
wvuI
Twvugϑφηχπτ
τ
( ) ( ) ( ) ( ) ( )
( )
×−∫ ∫ ∫ ∫
∂
∂
−×
−
V
Ii
InnnnI
D
D
dudvdwd
v
wvuI
Twvugucvsuce
0
0
0
1
0
1
0
1
0
100
2
,,,
~
,,, πτ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( )
( )
×∑ ∫ ∫ ∫ ∫
∂
∂
−×
∞
=
−
1 0
1
0
1
0
1
0
100 ,,,
~
,,,
n
i
InnnnInI
dudvdwd
w
wvuI
Twvugusvcuceen
ϑ
τ
τ
τϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[∑ ∫ ∫ ∫ ∫ ×+−−×
∞
=1 0
1
0
1
0
1
0
,
12
n
VInnnnInnnInnnn
vcvcuceccecccc
ϑ
ετφηϑχφηχ
( )]( )( ) ( )( )[ ] τττττdudvdwdwvuVwvuIwvuVwvuITwvug
VI
,,,
~
,,,
~
,,,
~
,,,
~
,,,
100000000100,
+×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ −−=
∞
=1 0
1
0
1
0
1
0
0
0
110
2,,,
~
n
nnnnVnVnnn
I
V
ucvcuseecccn
D
D
V
ϑ
τϑφηχπϑφηχ
( ) ( )
( ) ( ) ( ) ( )×∑−
∂
∂
×
∞
=
−
1
0
0100
2
,,,
~
,,,
n
nVnnn
I
Vi
V
ecccn
D
D
dudvdwd
u
wvuV
Twvugϑφηχπτ
τ
( ) ( ) ( ) ( ) ( )
( )
×−∫ ∫ ∫ ∫
∂
∂
−×
−
I
Vi
VnnnnV
D
D
dudvdwd
v
wvuV
Twvugucvsuce
0
0
0
1
0
1
0
1
0
100
2
,,,
~
,,, πτ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( )
( )
×∑ ∫ ∫ ∫ ∫
∂
∂
−×
∞
=
−
1 0
1
0
1
0
1
0
100 ,,,
~
,,,
n
i
VnnnnVnV
dudvdwd
w
wvuV
Twvugusvcuceen
ϑ
τ
τ
τϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]∑ ∫ ∫ ∫ ∫ ×+−−×
∞
=1 0
1
0
1
0
1
0
,,
,,,12
n
VIVInnnVnnnInnnn
Twvugvcuceccecccc
ϑ
ετφηϑχφηχ
()( )( ) ( )( )[ ] τττττdudvdwdwvuVwvuIwvuVwvuIwc
n ,,,
~
,,,
~
,,,
~
,,,
~
100000000100+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ −−=
∞
=1 0
1
0
1
0
1
0
0
0
101
,,,2,,,
~
n
InnnInInnn
V
I
Twvugvcuseecccn
D
D
I
ϑ
τϑφηχπϑφηχ
( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ −−
∂
∂
×
∞
=1 0
1
0
0
0001
2
,,,
~
n
nnInInnn
V
I
n
uceecccn
D
D
dudvdwd
u
wvuI
wc
ϑ
τϑφηχπτ
τ
( ) ( ) ( )
( )
( ) ( ) ( ) ( )×∑−∫ ∫
∂
∂
×
∞
=1
0
0
1
0
1
0
001
2
,,,
~
,,,
n
nnnnI
V
I
Inn
cccen
D
D
dudvdwd
v
wvuI
Twvugwcvsφηχϑπτ
τ
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( )×∑−∫ ∫ ∫ ∫
∂
∂
−×
∞
=10
1
0
1
0
1
0
001
2
,,,~
,,,
n
nnnInnnnI
cccdudvdwd
w
wvuI
Twvugwsvcuceφηχτ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( )
[ ]( ) ( )∫ ∫ ∫ ∫ +−×
ϑ
τττετϑ
0
1
0
1
0
1
0
000100,,
,,,
~
,,,
~
,,,1 dudvdwdwvuVwvuITwvugwcvcucee
VIVInnnnInI
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
14
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ −−=
∞
=1 0
1
0
1
0
1
0
0
0
101
,,,2,,,
~
n
VnnnVnVnnn
I
V
Twvugvcuseecccn
D
D
V
ϑ
τϑφηχπϑφηχ
( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ −−
∂
∂
×
∞
=1 0
1
0
0
0001
2
,,,
~
n
nnVnInnn
I
V
n
uceecccn
D
D
dudvdwd
u
wvuV
wc
ϑ
τϑφηχπτ
τ
( ) ( ) ( )
( )
( ) ( ) ( ) ( )×∑−∫ ∫
∂
∂
×
∞
=1
0
0
1
0
1
0
001
2
,,,
~
,,,
n
nnnnV
I
V
Vnn
cccen
D
D
dudvdwd
v
wvuV
Twvugwcvsφηχϑπτ
τ
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( )×∑−∫ ∫ ∫ ∫
∂
∂
−×
∞
=10
1
0
1
0
1
0
001
2
,,,
~
,,,
n
nnnVnnnnV
cccdudvdwd
w
wvuV
Twvugwsvcuceφηχτ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( )
[ ]( ) ( )∫ ∫ ∫ ∫ +−×
ϑ
τττετϑ
0
1
0
1
0
1
0
000100,,
,,,
~
,,,
~
,,,1 dudvdwdwvuVwvuITwvugwcvcucee
VIVInnnnVnV
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
{∑ ∫ ∫ ∫ ∫ ×−−=
∞
=1 0
1
0
1
0
1
0
000011
,,,
~
2,,,
~
n
nnnnInInnn
wvuIwcvcuceecccI
ϑ
ττϑφηχϑφηχ
( )[ ]( ) ( )[ ]( )( )} τττετεdudvdwdwvuVwvuITwvugwvuITwvug
VIVIIIII ,,,
~
,,,
~
,,,1,,,
~
,,,1
000001,,010,,+++×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) {∑ ∫ ∫ ∫ ∫ ×−−=
∞
=1 0
1
0
1
0
1
0
000011
,,,
~
2,,,
~
n
nnnnVnVnnn
wvuIwcvcuceecccV
ϑ
ττϑφηχϑφηχ
( )[ ]( ) ( )[ ]( )( )} τττετεdudvdwdwvuVwvuITwvugwvuITwvug
VIVIVVVV ,,,
~
,,,
~
,,,1,,,
~
,,,1
000001,,010,,+++× .
Equations for the functions
Φρi(x,y,z,t), boundary and initial conditions for them could be written
as ( ) ( ) ( ) ( )
+
Φ
+
Φ
+
Φ
=
Φ
Φ 2
0
2
2
0
2
2
0
2
0
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx
III
I
I
∂
∂
∂
∂
∂
∂
∂
∂
( )( )( )( )tzyxITzyxktzyxITzyxk
III
,,,,,,,,,,,,
2
,
−+
( ) ( ) ( ) ( )
+
Φ
+
Φ
+
Φ
=
Φ
Φ 2
0
2
2
0
2
2
0
2
0
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx
VVV
V
V
∂
∂
∂
∂
∂
∂
∂
∂
( )( )( )( )tzyxVTzyxktzyxVTzyxk
VVV
,,,,,,,,,,,,
2
,
−+ ;
( ) ( ) ( ) ( )
+
Φ
+
Φ
+
Φ
=
Φ
Φ 2
2
2
2
2
2
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx
iIiIiI
I
iI
∂
∂
∂
∂
∂
∂
∂
∂
( )
( )
( )
( )
+
Φ
+
Φ
+
−
Φ
−
ΦΦ
y
tzyx
Tzyxg
yx
tzyx
Tzyxg
x
D
iI
I
iI
II
∂
∂
∂
∂
∂
∂
∂
∂
,,,
,,,
,,,
,,,
11
0
( )
( )
Φ
+
−
Φ
z
tzyx
Tzyxg
z
iI
I
∂
∂
∂
∂
,,,
,,,
1
, i≥1,
( ) ( ) ( ) ( )
+
Φ
+
Φ
+
Φ
=
Φ
Φ 2
2
2
2
2
2
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx
iViViV
V
iV
∂
∂
∂
∂
∂
∂
∂
∂
( )
( )
( )
( )
+
Φ
+
Φ
+
−
Φ
−
ΦΦ
y
tzyx
Tzyxg
yx
tzyx
Tzyxg
x
D
iV
V
iV
VV
∂
∂
∂
∂
∂
∂
∂
∂
,,,
,,,
,,,
,,,
11
0
( )
( )
Φ
+
−
Φ
z
tzyx
Tzyxg
z
iV
V
∂
∂
∂
∂
,,,
,,,
1
, i≥1;
( )
0
,,,
0
=
∂
Φ∂
=x
i
x
tzyx
ρ
,
( )
0
,,,
=
∂
Φ∂
=
x
Lx
i
x
tzyx
ρ
,
( )
0
,,,
0
=
∂
Φ∂
=y
i
y
tzyx
ρ
,
( )
0
,,,
=
∂
Φ∂
=
y
Ly
i
y
tzyx
ρ
,
14
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ −−=
∞
=1 0
1
0
1
0
1
0
0
0
101
,,,2,,,
~
n
VnnnVnVnnn
I
V
Twvugvcuseecccn
D
D
V
ϑ
τϑφηχπϑφηχ
( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ −−
∂
∂
×
∞
=1 0
1
0
0
0001
2
,,,
~
n
nnVnInnn
I
V
n
uceecccn
D
D
dudvdwd
u
wvuV
wc
ϑ
τϑφηχπτ
τ
( ) ( ) ( )
( )
( ) ( ) ( ) ( )×∑−∫ ∫
∂
∂
×
∞
=1
0
0
1
0
1
0
001
2
,,,
~
,,,
n
nnnnV
I
V
Vnn
cccen
D
D
dudvdwd
v
wvuV
Twvugwcvsφηχϑπτ
τ
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( )×∑−∫ ∫ ∫ ∫
∂
∂
−×
∞
=10
1
0
1
0
1
0
001
2
,,,
~
,,,
n
nnnVnnnnV
cccdudvdwd
w
wvuV
Twvugwsvcuceφηχτ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( )
[ ]( ) ( )∫ ∫ ∫ ∫ +−×
ϑ
τττετϑ
0
1
0
1
0
1
0
000100,,
,,,
~
,,,
~
,,,1 dudvdwdwvuVwvuITwvugwcvcucee
VIVInnnnVnV
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
{∑ ∫ ∫ ∫ ∫ ×−−=
∞
=1 0
1
0
1
0
1
0
000011
,,,
~
2,,,
~
n
nnnnInInnn
wvuIwcvcuceecccI
ϑ
ττϑφηχϑφηχ
( )[ ]( ) ( )[ ]( )( )} τττετεdudvdwdwvuVwvuITwvugwvuITwvug
VIVIIIII ,,,
~
,,,
~
,,,1,,,
~
,,,1
000001,,010,,+++×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) {∑ ∫ ∫ ∫ ∫ ×−−=
∞
=1 0
1
0
1
0
1
0
000011
,,,
~
2,,,
~
n
nnnnVnVnnn
wvuIwcvcuceecccV
ϑ
ττϑφηχϑφηχ
( )[ ]( ) ( )[ ]( )( )} τττετεdudvdwdwvuVwvuITwvugwvuITwvug
VIVIVVVV ,,,
~
,,,
~
,,,1,,,
~
,,,1
000001,,010,,+++× .
Equations for the functions
Φρi(x,y,z,t), boundary and initial conditions for them could be written
as ( ) ( ) ( ) ( )
+
Φ
+
Φ
+
Φ
=
Φ
Φ 2
0
2
2
0
2
2
0
2
0
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx
III
I
I
∂
∂
∂
∂
∂
∂
∂
∂
( )( )( )( )tzyxITzyxktzyxITzyxk
III
,,,,,,,,,,,,
2
,
−+
( ) ( ) ( ) ( )
+
Φ
+
Φ
+
Φ
=
Φ
Φ 2
0
2
2
0
2
2
0
2
0
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx
VVV
V
V
∂
∂
∂
∂
∂
∂
∂
∂
( )( )( )( )tzyxVTzyxktzyxVTzyxk
VVV
,,,,,,,,,,,,
2
,
−+ ;
( ) ( ) ( ) ( )
+
Φ
+
Φ
+
Φ
=
Φ
Φ 2
2
2
2
2
2
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx
iIiIiI
I
iI
∂
∂
∂
∂
∂
∂
∂
∂
( )
( )
( )
( )
+
Φ
+
Φ
+
−
Φ
−
ΦΦ
y
tzyx
Tzyxg
yx
tzyx
Tzyxg
x
D
iI
I
iI
II
∂
∂
∂
∂
∂
∂
∂
∂
,,,
,,,
,,,
,,,
11
0
( )
( )
Φ
+
−
Φ
z
tzyx
Tzyxg
z
iI
I
∂
∂
∂
∂
,,,
,,,
1
, i≥1,
( ) ( ) ( ) ( )
+
Φ
+
Φ
+
Φ
=
Φ
Φ 2
2
2
2
2
2
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
D
t
tzyx
iViViV
V
iV
∂
∂
∂
∂
∂
∂
∂
∂
( )
( )
( )
( )
+
Φ
+
Φ
+
−
Φ
−
ΦΦ
y
tzyx
Tzyxg
yx
tzyx
Tzyxg
x
D
iV
V
iV
VV
∂
∂
∂
∂
∂
∂
∂
∂
,,,
,,,
,,,
,,,
11
0
( )
( )
Φ
+
−
Φ
z
tzyx
Tzyxg
z
iV
V
∂
∂
∂
∂
,,,
,,,
1
, i≥1;
( )
0
,,,
0
=
∂
Φ∂
=x
i
x
tzyx
ρ
,
( )
0
,,,
=
∂
Φ∂
=
x
Lx
i
x
tzyx
ρ
,
( )
0
,,,
0
=
∂
Φ∂
=y
i
y
tzyx
ρ
,
( )
0
,,,
=
∂
Φ∂
=
y
Ly
i
y
tzyx
ρ
,
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
15
( )
0
,,,
0
=
∂
Φ∂
=z
i
z
tzyx
ρ
,
( )
0
,,,
=
∂
Φ∂
=
z
Lz
i
z
tzyx
ρ
, i≥0; Φρ0(x,y,z,0)=f Φρ (x,y,z), Φρi(x,y,z,0)=0, i≥1.
Solutions of the above equations could be written as
( )
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑+∑+=Φ
∞
=
∞
=
ΦΦ
11
0221
,,,
n
nnn
n
nnnnn
zyxzyx
zcycxcn
L
tezcycxcF
LLLLLL
tzyx
ρρ
ρ
( ) ( ) ( ) ( ) ( ) ( ) ( )[∫ ∫ ∫ ∫ −−×ΦΦ
t L L L
IInnnnn x y z
wvuITwvukwcvcucete
0 0 0 0
2
,
,,,,,,ττ
ρρ
( )( )] ττ dudvdwdwvuITwvuk
I ,,,,,, − ,
where
( ) ( ) ( ) ( )∫ ∫ ∫= ΦΦ
x y z
L L L
nnnn
udvdwdwvufwcvcucF
0 0 0
,,
ρρ
, ( )
++−=
ΦΦ 2220
22
111
exp
zyx
nLLL
tDnte
ρρ
π ,
c
n(x) = cos (π n x/L x);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=Φ
∞
=
ΦΦΦ
1 0 0 0 0
2
,,,
2
,,,
n
t L L L
nnnnnnn
zyx
i
x y z
Twvugvcusetezcycxcn
LLL
tzyx
ρρρ
τ
π
ρ
( )
( )
( ) ( ) ( ) ( ) ( ) ×∑ ∫−−
Φ
×
∞
=
ΦΦ
−
1 0
2
12,,,
n
t
nnnnn
zyx
iI
n
etezcycxcn
LLL
dudvdwd
u
wvu
wcτ
π
τ
∂
τ∂
ρρ
ρ
( ) ( ) ( ) ( ) ( )
( )
×∑−∫ ∫ ∫ ∫
Φ
−×
∞
=
−
ΦΦ
1
2
0 0 0 0
12,,,
,,,
n
zyx
t L L L
iI
nnnn
n
LLL
dudvdwd
v
wvu
Twvugwcvsuce
x y z π
τ
∂
τ∂
τ
ρ
ρρ
( ) ( ) ( ) ( ) ( )
( )
( ) ×∫ ∫ ∫ ∫
Φ
−×
Φ
−
ΦΦ
t L L L
iI
nnnnn
x y z
dudvdwdTwvug
w
wvu
wsvcucete
0 0 0 0
1
,,,
,,, τ
∂
τ∂
τ
ρ
ρ
ρρ
()()()zcycxc
nnn × , i≥1,
where s
n(x) = sin (π n x/L x).
Equations for the functions C ij(x,y,z,t), boundary and initial conditions for them could be written as ( ) ( ) ( ) ( )
2
00
2
02
00
2
02
00
2
0
00
,,,,,,,,,,,,
z
tzyxC
D
y
tzyxC
D
x
tzyxC
D
t
tzyxC
LLL
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
0
2
2
0
2
2
0
2
0
0
,,,,,,,,,,,,
z
tzyxC
y
tzyxC
x
tzyxC
D
t
tzyxC
iii
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
DL
,,,
,,,
,,,
0000
0
γ
γ
;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
02
2
02
02
2
02
02
2
0
02
,,,,,,,,,,,,
z
tzyxC
D
y
tzyxC
D
x
tzyxC
D
t
tzyxC
LLL
15
( )
0
,,,
0
=
∂
Φ∂
=z
i
z
tzyx
ρ
,
( )
0
,,,
=
∂
Φ∂
=
z
Lz
i
z
tzyx
ρ
, i≥0; Φρ0(x,y,z,0)=f Φρ (x,y,z), Φρi(x,y,z,0)=0, i≥1.
Solutions of the above equations could be written as
( )
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑+∑+=Φ
∞
=
∞
=
ΦΦ
11
0221
,,,
n
nnn
n
nnnnn
zyxzyx
zcycxcn
L
tezcycxcF
LLLLLL
tzyx
ρρ
ρ
( ) ( ) ( ) ( ) ( ) ( ) ( )[∫ ∫ ∫ ∫ −−×ΦΦ
t L L L
IInnnnn x y z
wvuITwvukwcvcucete
0 0 0 0
2
,
,,,,,,ττ
ρρ
( )( )] ττ dudvdwdwvuITwvuk
I ,,,,,, − ,
where
( ) ( ) ( ) ( )∫ ∫ ∫= ΦΦ
x y z
L L L
nnnn
udvdwdwvufwcvcucF
0 0 0
,,
ρρ
, ( )
++−=
ΦΦ 2220
22
111
exp
zyx
nLLL
tDnte
ρρ
π ,
c
n(x) = cos (π n x/L x);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=Φ
∞
=
ΦΦΦ
1 0 0 0 0
2
,,,
2
,,,
n
t L L L
nnnnnnn
zyx
i
x y z
Twvugvcusetezcycxcn
LLL
tzyx
ρρρ
τ
π
ρ
( )
( )
( ) ( ) ( ) ( ) ( ) ×∑ ∫−−
Φ
×
∞
=
ΦΦ
−
1 0
2
12,,,
n
t
nnnnn
zyx
iI
n
etezcycxcn
LLL
dudvdwd
u
wvu
wcτ
π
τ
∂
τ∂
ρρ
ρ
( ) ( ) ( ) ( ) ( )
( )
×∑−∫ ∫ ∫ ∫
Φ
−×
∞
=
−
ΦΦ
1
2
0 0 0 0
12,,,
,,,
n
zyx
t L L L
iI
nnnn
n
LLL
dudvdwd
v
wvu
Twvugwcvsuce
x y z π
τ
∂
τ∂
τ
ρ
ρρ
( ) ( ) ( ) ( ) ( )
( )
( ) ×∫ ∫ ∫ ∫
Φ
−×
Φ
−
ΦΦ
t L L L
iI
nnnnn
x y z
dudvdwdTwvug
w
wvu
wsvcucete
0 0 0 0
1
,,,
,,, τ
∂
τ∂
τ
ρ
ρ
ρρ
()()()zcycxc
nnn × , i≥1,
where s
n(x) = sin (π n x/L x).
Equations for the functions C ij(x,y,z,t), boundary and initial conditions for them could be written as ( ) ( ) ( ) ( )
2
00
2
02
00
2
02
00
2
0
00
,,,,,,,,,,,,
z
tzyxC
D
y
tzyxC
D
x
tzyxC
D
t
tzyxC
LLL
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
0
2
2
0
2
2
0
2
0
0
,,,,,,,,,,,,
z
tzyxC
y
tzyxC
x
tzyxC
D
t
tzyxC
iii
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
DL
,,,
,,,
,,,
0000
0
γ
γ
;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
02
2
02
02
2
02
02
2
0
02
,,,,,,,,,,,,
z
tzyxC
D
y
tzyxC
D
x
tzyxC
D
t
tzyxC
LLL
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
16
( )
( )
( )
( )
( )
( )
( )
×
∂
∂
+
∂
∂
∂
∂
+
−−
TzyxP
tzyxC
tzyxC
yx
tzyxC
TzyxP
tzyxC
tzyxC
x
D
L
,,,
,,,
,,,
,,,
,,,
,,,
,,,
1
00
01
00
1
00
010
γ
γ
γ
γ
( )
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
×
−
z
tzyxC
TzyxP
tzyxC
tzyxC
zy
tzyxC ,,,
,,,
,,,
,,,
,,,
00
1
00
01
00
γ
γ
( )
( )
( )
( )
( ) ( )
( )
×
∂
∂
+
∂
∂
∂
∂
+
∂
∂
×
−
TzyxP
tzyxC
x
D
z
tzyxC
TzyxP
tzyxC
tzyxC
zy
tzyxC
L
,,,
,,,,,,
,,,
,,,
,,,
,,,
00
0
00
1
00
01
00
γ
γ
γ
γ
( ) ( )
( )
( ) ( )
( )
( )
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
×
z
tzyxC
TzyxP
tzyxC
zy
tzyxC
TzyxP
tzyxC
yx
tzyxC ,,,
,,,
,,,,,,
,,,
,,,,,,
0100010001
γ
γ
γ
γ
;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
11
2
02
11
2
02
11
2
0
11
,,,,,,,,,,,,
z
tzyxC
D
y
tzyxC
D
x
tzyxC
D
t
tzyxC
LLL
( )
( )
( )
( )
( )
( )
( )
×
∂
∂
+
∂
∂
∂
∂
+
−−
TzyxP
tzyxC
tzyxC
yx
tzyxC
TzyxP
tzyxC
tzyxC
x ,,,
,,,
,,,
,,,
,,,
,,,
,,,
1
00
10
00
1
00
10
γ
γ
γ
γ
( )
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
×
−
L
D
z
tzyxC
TzyxP
tzyxC
tzyxC
zy
tzyxC
0
00
1
00
10
00
,,,
,,,
,,,
,,,
,,,
γ
γ
( )
( )
( ) ( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
y
tzyxC
TzyxP
tzyxC
yx
tzyxC
TzyxP
tzyxC
x
D
L
,,,
,,,
,,,,,,
,,,
,,,
10001000
0
γ
γ
γ
γ
( )
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
x
tzyxC
Tzyxg
x
D
z
tzyxC
TzyxP
tzyxC
z
LL
,,,
,,,
,,,
,,,
,,,
01
0
1000
γ
γ
( )
( )
( )
( )
∂
∂
∂
∂
+
∂
∂
∂
∂
+
z
tzyxC
Tzyxg
zy
tzyxC
Tzyxg
y
LL
,,,
,,,
,,,
,,,
0101
;
( )
0
,,,
0
=
=x
ij
x
tzyxC∂
∂
,
( )
0
,,,
=
=
x
Lx
ij
x
tzyxC∂
∂
,
( )
0
,,,
0
=
=y
ij
y
tzyxC∂
∂
,
( )
0
,,,
=
=
y
Ly
ij
y
tzyxC∂
∂
,
( )
0
,,,
0
=
=z
ij
z
tzyxC∂
∂
,
( )
0
,,,
=
=
z
Lz
ij
z
tzyxC∂
∂
, i ≥0, j ≥0;
C
00(x,y,z,0)=f C (x,y,z), C ij(x,y,z,0)=0, i ≥1, j ≥1.
Solutions of the above equations with account boundary and initial have been calculated by Fou-
rier approach. The result of calculation could be written as
( )
( ) ( ) ( ) ( )∑+=
∞
=1
0021
,,,
n
nCnnnnC
zyxzyx
tezcycxcF
LLLLLL
tzyxC ,
where
( )
++−=
2220
22
111
exp
zyx
CnCLLL
tDnte
π , ( ) ( ) ( ) ( )∫ ∫ ∫=
x y z
L L L
nCnnnC
udvdwdwcwvufvcucF
0 0 0
,, ;
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
20
,,,
2
,,,
n
t L L L
LnnnCnCnnnnC
zyx
i
x y z
TwvugvcusetezcycxcFn
LLL
tzyxCτ
π
( )
( )
( ) ( ) ( ) ( ) ( ) ×∑ ∫−−
∂
∂
×
∞
=
−
1 0
2
102,,,
n
t
nCnCnnnnC
zyx
i
n
etezcycxcFn
LLL
dudvdwd
u
wvuC
wcτ
π
τ
τ
16
( )
( )
( )
( )
( )
( )
( )
×
∂
∂
+
∂
∂
∂
∂
+
−−
TzyxP
tzyxC
tzyxC
yx
tzyxC
TzyxP
tzyxC
tzyxC
x
D
L
,,,
,,,
,,,
,,,
,,,
,,,
,,,
1
00
01
00
1
00
010
γ
γ
γ
γ
( )
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
×
−
z
tzyxC
TzyxP
tzyxC
tzyxC
zy
tzyxC ,,,
,,,
,,,
,,,
,,,
00
1
00
01
00
γ
γ
( )
( )
( )
( )
( ) ( )
( )
×
∂
∂
+
∂
∂
∂
∂
+
∂
∂
×
−
TzyxP
tzyxC
x
D
z
tzyxC
TzyxP
tzyxC
tzyxC
zy
tzyxC
L
,,,
,,,,,,
,,,
,,,
,,,
,,,
00
0
00
1
00
01
00
γ
γ
γ
γ
( ) ( )
( )
( ) ( )
( )
( )
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
×
z
tzyxC
TzyxP
tzyxC
zy
tzyxC
TzyxP
tzyxC
yx
tzyxC ,,,
,,,
,,,,,,
,,,
,,,,,,
0100010001
γ
γ
γ
γ
;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
11
2
02
11
2
02
11
2
0
11
,,,,,,,,,,,,
z
tzyxC
D
y
tzyxC
D
x
tzyxC
D
t
tzyxC
LLL
( )
( )
( )
( )
( )
( )
( )
×
∂
∂
+
∂
∂
∂
∂
+
−−
TzyxP
tzyxC
tzyxC
yx
tzyxC
TzyxP
tzyxC
tzyxC
x ,,,
,,,
,,,
,,,
,,,
,,,
,,,
1
00
10
00
1
00
10
γ
γ
γ
γ
( )
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
×
−
L
D
z
tzyxC
TzyxP
tzyxC
tzyxC
zy
tzyxC
0
00
1
00
10
00
,,,
,,,
,,,
,,,
,,,
γ
γ
( )
( )
( ) ( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
y
tzyxC
TzyxP
tzyxC
yx
tzyxC
TzyxP
tzyxC
x
D
L
,,,
,,,
,,,,,,
,,,
,,,
10001000
0
γ
γ
γ
γ
( )
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
x
tzyxC
Tzyxg
x
D
z
tzyxC
TzyxP
tzyxC
z
LL
,,,
,,,
,,,
,,,
,,,
01
0
1000
γ
γ
( )
( )
( )
( )
∂
∂
∂
∂
+
∂
∂
∂
∂
+
z
tzyxC
Tzyxg
zy
tzyxC
Tzyxg
y
LL
,,,
,,,
,,,
,,,
0101
;
( )
0
,,,
0
=
=x
ij
x
tzyxC∂
∂
,
( )
0
,,,
=
=
x
Lx
ij
x
tzyxC∂
∂
,
( )
0
,,,
0
=
=y
ij
y
tzyxC∂
∂
,
( )
0
,,,
=
=
y
Ly
ij
y
tzyxC∂
∂
,
( )
0
,,,
0
=
=z
ij
z
tzyxC∂
∂
,
( )
0
,,,
=
=
z
Lz
ij
z
tzyxC∂
∂
, i ≥0, j ≥0;
C
00(x,y,z,0)=f C (x,y,z), C ij(x,y,z,0)=0, i ≥1, j ≥1.
Solutions of the above equations with account boundary and initial have been calculated by Fou-
rier approach. The result of calculation could be written as
( )
( ) ( ) ( ) ( )∑+=
∞
=1
0021
,,,
n
nCnnnnC
zyxzyx
tezcycxcF
LLLLLL
tzyxC ,
where
( )
++−=
2220
22
111
exp
zyx
CnCLLL
tDnte
π , ( ) ( ) ( ) ( )∫ ∫ ∫=
x y z
L L L
nCnnnC
udvdwdwcwvufvcucF
0 0 0
,, ;
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
20
,,,
2
,,,
n
t L L L
LnnnCnCnnnnC
zyx
i
x y z
TwvugvcusetezcycxcFn
LLL
tzyxCτ
π
( )
( )
( ) ( ) ( ) ( ) ( ) ×∑ ∫−−
∂
∂
×
∞
=
−
1 0
2
102,,,
n
t
nCnCnnnnC
zyx
i
n
etezcycxcFn
LLL
dudvdwd
u
wvuC
wcτ
π
τ
τ
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
17
( ) ( ) ( ) ( )
( )
( )∑ ×−∫ ∫ ∫
∂
∂
×
∞
=
−
1
2
0 0 0
102,,,
,,,
n
nCnC
zyx
L L L
i
Lnnn
teFn
LLL
dudvdwd
v
wvuC
Twvugvcvsuc
x y z π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
∫ ∫ ∫ ∫
∂
∂
−×
−
t L L L
i
LnnnnCnnn
x y z
dudvdwd
w
wvuC
Twvugvsvcucezcycxc
0 0 0 0
10
,,,
,,,
τ
τ
τ, i ≥1;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
2012
,,,
n
t L L L
nnnnCnCnnnnC
zyx
x y z
wcvcusetezcycxcFn
LLL
tzyxCτ
π
( )
( )
( )
( ) ( ) ( ) ( )×∑−
∂
∂
×
∞
=1
2
00002,,,
,,,
,,,
n
nCnnnnC
zyx
tezcycxcFn
LLL
dudvdwd
u
wvuC
TwvuP
wvuC
π
τ
ττ
γ
γ
( ) ( ) ( ) ( )
( )
( )
( )
( )×∑−∫ ∫ ∫ ∫
∂
∂
−×
∞
=1
2
0 0 0 0
00002,,,
,,,
,,,
n
nC
zyx
t L L L
nnnnC
ten
LLL
dudvdwd
v
wvuC
TwvuP
wvuC
wcvsuce
x y z π
τ
ττ
τ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
∫ ∫ ∫ ∫
∂
∂
−×
t L L L
nnnnCnnnnC x y z
dudvdwd
w
wvuC
TwvuP
wvuC
wsvcucezcycxcF
0 0 0 0
0000
,,,
,,,
,,,
τ
ττ
τ
γ
γ
;
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=1 0 0 0 0
2022
,,,
n
t L L L
nnnnCnCnnnnC
zyx
x y z
wcvcusetezcycxcFn
LLL
tzyxCτ
π
( )
( )
( )
( )
( ) ( )×∑−
∂
∂
×
∞
=
−
1
2
00
1
00
012,,,
,,,
,,,
,,,
n
nnnC
zyx
ycxcF
LLL
dudvdwd
u
wvuC
TwvuP
wvuC
wvuC
π
τ
ττ
τ
γ
γ
( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
×∫ ∫ ∫ ∫
∂
∂
−×
−
t L L L
nnnCnCn
x y z
v
wvuC
TwvuP
wvuC
wvuCvsucetezcn0 0 0 0
00
1
00
01
,,,
,,,
,,,
,,,ττ
ττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−×
∞
=1 0 0 0
22
n
t L L
nnnCnCnnnnC
zyx
n
x y
vcucetezcycxcFn
LLL
dudvdwdwcτ
π
τ
( ) ( )
( )
( )
( )
( )×∑−∫
∂
∂
×
∞
=
−
1
2
0
00
1
00
012,,,
,,,
,,,
,,,
n
n
zyx
L
n
xcn
LLL
dudvdwd
w
wvuC
TwvuP
wvuC
wvuCws
z π
τ
ττ
τ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
×∫ ∫ ∫ ∫
∂
∂
−×
t L L L
nnnnCnCnnnC x y z
u
wvuC
wvuCwcvcusetezcycF0 0 0 0
00
01
,,,
,,,τ
ττ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−×
∞
=
−
1 0 0
2
1
002
,,,
,,,
n
t L
nnCnCnnnnC
zyx
x
ucetezcycxcFn
LLL
dudvdwd
TwvuP
wvuCτ
π
τ
τ
γ
γ
( ) ( ) ( )
( )
( )
( )
×∑−∫ ∫
∂
∂
×
∞
=
−
1
2
0 0
00
1
00
012,,,
,,,
,,,
,,,
n
zyx
L L
nn
n
LLL
dudvdwd
v
wvuC
TwvuP
wvuC
wvuCwcvs
y z
π
τ
ττ
τ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
×∫ ∫ ∫ ∫−×
−
t L L L
nnnnCnCnnnnC x y z
TwvuP
wvuC
wvuCwsvcucetezcycxcF0 0 0 0
1
00
01 ,,,
,,,
,,,
γ
γ
τ
ττ
( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ −−
∂
∂
×
∞
=1 0 0
2
002,,,
n
t L
nnCnCnnnnC
zyx
x
usetezcycxcF
LLL
dudvdwd
w
wvuCτ
π
τ
τ
( ) ( )
( )
( )
( )
( ) ( )∑ ×−∫ ∫
∂
∂
×
∞
=
1
2
0 0
0100
2,,,
,,,
,,,
n
nCn
zyx
L L
nn
texc
LLL
dudvdwd
u
wvuC
TwvuP
wvuC
wcvcn
y z
π
τ
ττ
γ
γ
( ) ( ) ( ) ( ) ( )
( )
( )
( )
×∫ ∫ ∫ ∫
∂
∂
−×
t L L L
nnnnCnnC x y z
dudvdwd
v
wvuC
TwvuP
wvuC
wcvsuceycF
0 0 0 0
0100
,,,
,,,
,,,
τ
ττ
τ
γ
γ
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ −−×
∞
=1 0 0 0 0
22
n
t L L L
nnnnCnCnnnnC
zyx
n
x y z
wsvcucetezcycxcFn
LLL
zcnτ
π
17
( ) ( ) ( ) ( )
( )
( )∑ ×−∫ ∫ ∫
∂
∂
×
∞
=
−
1
2
0 0 0
102,,,
,,,
n
nCnC
zyx
L L L
i
Lnnn
teFn
LLL
dudvdwd
v
wvuC
Twvugvcvsuc
x y z π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
∫ ∫ ∫ ∫
∂
∂
−×
−
t L L L
i
LnnnnCnnn
x y z
dudvdwd
w
wvuC
Twvugvsvcucezcycxc
0 0 0 0
10
,,,
,,,
τ
τ
τ, i ≥1;
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∞
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2012
,,,
n
t L L L
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π
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( )
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2
00002,,,
,,,
,,,
n
nCnnnnC
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n
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t L L L
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;
( )
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n
nnnC
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τ
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γ
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t L L L
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v
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00
1
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γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−×
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22
n
t L L
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zyx
n
x y
vcucetezcycxcFn
LLL
dudvdwdwcτ
π
τ
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×
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0
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1
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n
n
zyx
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n
xcn
LLL
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z π
τ
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τ
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γ
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( )
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t L L L
nnnnCnCnnnC x y z
u
wvuC
wvuCwcvcusetezcycF0 0 0 0
00
01
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,,,τ
ττ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−×
∞
=
−
1 0 0
2
1
002
,,,
,,,
n
t L
nnCnCnnnnC
zyx
x
ucetezcycxcFn
LLL
dudvdwd
TwvuP
wvuCτ
π
τ
τ
γ
γ
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( )
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1
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0 0
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1
00
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,,,
n
zyx
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nn
n
LLL
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v
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τ
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−
t L L L
nnnnCnCnnnnC x y z
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wvuCwsvcucetezcycxcF0 0 0 0
1
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∞
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2
002,,,
n
t L
nnCnCnnnnC
zyx
x
usetezcycxcF
LLL
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w
wvuCτ
π
τ
τ
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=
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0 0
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,,,
n
nCn
zyx
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nn
texc
LLL
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wcvcn
y z
π
τ
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( )
( )
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∂
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t L L L
nnnnCnnC x y z
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v
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0 0 0 0
0100
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τ
ττ
τ
γ
γ
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫ −−×
∞
=1 0 0 0 0
22
n
t L L L
nnnnCnCnnnnC
zyx
n
x y z
wsvcucetezcycxcFn
LLL
zcnτ
π
International Journal on Organic Electronics (IJOE) Vol.6, No.2, April 2017
18
( )
( )
( )
τ
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γ
γ
dudvdwd
w
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∂
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0100
;
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
∞
=
1 0 0 0 0
211
2
,,,
n
t L L L
nnnnCnCnnnnC
zyx
x y z
wcvcusetezcycxcFn
LLL
tzyxCτ
π
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( )
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∂
∂
×
∞
=1
2
012,,,
,,,
n
nCnnnnC
zyx
L
tezcycxcFn
LLL
dudvdwd
u
wvuC
Twvug
π
τ
τ
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( )
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2
0 0 0 0
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t L L L
LnnnnCLLL
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τ
τ
τ
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∂
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01
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n
t L L L
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τ
τ
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∞
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22
n
t L L
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( )
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( )
( ) ( )×∑−∫
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=
1
2
0
1000
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,,,
,,,
n
nnnC
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L
n
ycxcFn
LLL
dudvdwd
u
wvuC
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z π
τ
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γ
γ
( ) ( ) ( ) ( ) ( ) ( )
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t L L L
nnnnCnCn x y z
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v
wvuC
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wcvsucetezc
0 0 0 0
1000
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,,,
τ
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τ
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γ
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( )
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1 0 0 0 0
00
2 ,,,
,,,2n
t L L L
nnnnCnCnnnnC
zyx
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wsvcucetezcycxcFn
LLL
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τ
τ
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∞
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2
102,,,
n
t L
nnCnCnnnnC
zyx
x
usetezcycxcFn
LLL
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w
wvuCτ
π
τ
τ
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( )
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∂
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−
1
2
0 0
00
1
00
10
2,,,
,,,
,,,
,,,
n
zyx
L L
nn
n
LLL
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wvuCwcvc
y z
π
τ
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τ
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γ
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( )
( )
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×∫ ∫ ∫ ∫
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−
t L L L
nnnnCnCnnnnC
x y z
v
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TwvuP
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wcvsucetezcycxcF0 0 0 0
00
1
00
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τ
γ
γ
( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ −−×
∞
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2102
,,,
n
t L
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zyx
x
ucetezcycxcFn
LLL
dudvdwdwvuCτ
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( )
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( )
∫ ∫
∂
∂
×
−
y z
L L
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dudvdwd
w
wvuC
TwvuP
wvuC
wvuCwsvc
0 0
00
1
00
10
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,,,
,,,
,,,
τ
ττ
τ
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.
18
( )
( )
( )
τ
ττ
γ
γ
dudvdwd
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∂
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,,,
,,,
,,,
0100
;
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=
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=
1 0 0 0 0
211
2
,,,
n
t L L L
nnnnCnCnnnnC
zyx
x y z
wcvcusetezcycxcFn
LLL
tzyxCτ
π
( )
( )
( ) ( ) ( ) ( )×∑−
∂
∂
×
∞
=1
2
012,,,
,,,
n
nCnnnnC
zyx
L
tezcycxcFn
LLL
dudvdwd
u
wvuC
Twvug
π
τ
τ
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( )
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∂
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2
0 0 0 0
01
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LnnnnCLLL
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v
wvuC
Twvugwcvsuce
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τ
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n
t L L L
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22
n
t L L
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LLL
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( ) ( )×∑−∫
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∂
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1
2
0
1000
2,,,
,,,
,,,
n
nnnC
zyx
L
n
ycxcFn
LLL
dudvdwd
u
wvuC
TwvuP
wvuC
wcn
z π
τ
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γ
γ
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( )
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t L L L
nnnnCnCn x y z
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v
wvuC
TwvuP
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wcvsucetezc
0 0 0 0
1000
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,,,
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τ
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τ
γ
γ
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( )
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1 0 0 0 0
00
2 ,,,
,,,2n
t L L L
nnnnCnCnnnnC
zyx
x y z
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wsvcucetezcycxcFn
LLL
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γ
τ
τ
π
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∂
×
∞
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2
102,,,
n
t L
nnCnCnnnnC
zyx
x
usetezcycxcFn
LLL
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w
wvuCτ
π
τ
τ
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∂
×
∞
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−
1
2
0 0
00
1
00
10
2,,,
,,,
,,,
,,,
n
zyx
L L
nn
n
LLL
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u
wvuC
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wvuC
wvuCwcvc
y z
π
τ
ττ
τ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
×∫ ∫ ∫ ∫
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∂
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t L L L
nnnnCnCnnnnC
x y z
v
wvuC
TwvuP
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wcvsucetezcycxcF0 0 0 0
00
1
00
,,,
,,,
,,,ττ
τ
γ
γ
( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ −−×
∞
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2102
,,,
n
t L
nnCnCnnnnC
zyx
x
ucetezcycxcFn
LLL
dudvdwdwvuCτ
π
ττ
( ) ( ) ( )
( )
( )
( )
∫ ∫
∂
∂
×
−
y z
L L
nn
dudvdwd
w
wvuC
TwvuP
wvuC
wvuCwsvc
0 0
00
1
00
10
,,,
,,,
,,,
,,,
τ
ττ
τ
γ
γ
.
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