on vertical integration framework element of transistor transistor logic · transistor-transistor...
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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
DOI : 10.5121/ijaceee.2015.3301 1
ON VERTICAL INTEGRATION FRAMEWORK
ELEMENT OF TRANSISTOR-TRANSISTOR LOGIC
E.L. Pankratov1, E.A. Bulaeva
1,2
1 Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950,
Russia 2
Nizhny Novgorod State University of Architecture and Civil Engineering, 65 Il'insky
street, Nizhny Novgorod, 603950, Russia
ABSTRACT
In this paper we introduce an approach to increase vertical integration of elements of transistor-transistor
logic with function AND-NOT. Framework the approach we consider a heterostructure with special confi-
guration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation.
Annealing of dopant and/or radiation defects should be optimized.
KEYWORDS
Transistor-transistor logic; optimization of manufacturing; decreasing of dimensions of transistor; analyti-
cal approach for modelling
1. INTRODUCTION
An actual and intensively solving problems of solid state electronics is increasing of integration
rate of elements of integrated circuits (p-n-junctions, their systems et al) [1-8]. Increasing of the
integration rate leads to necessity to decrease their dimensions. To decrease the dimensions are
using several approaches. They are widely using laser and microwave types of annealing of in-
fused dopants. These types of annealing are also widely using for annealing of radiation defects,
generated during ion implantation [9-17]. Using the approaches gives a possibility to increase
integration rate of elements of integrated circuits through inhomogeneity of technological para-
meters due to generating inhomogenous distribution of temperature. In this situation one can ob-
tain decreasing dimensions of elements of integrated circuits [18] with account Arrhenius law
[1,3]. Another approach to manufacture elements of integrated circuits with smaller dimensions is
doping of heterostructure by diffusion or ion implantation [1-3]. However in this case optimiza-
tion of dopant and/or radiation defects is required [18].
In this paper we consider a heterostructure presented in Figs. 1. The heterostructure consist of a
substrate and several epitaxial layers (see Figs. 1). Some sections have been manufactured in the
epitaxial layers so as it is shown on Figs. 1. Further we consider doping of these sections by dif-
fusion or ion implantation. The doping gives a possibility to manufacture transistors and p-n-
junction so as it is shown on Figs. 1. The manufacturing gives a possibility to prepare element of
transistor-transistor logic on Fig. 1a. After the considered doping dopant and/or radiation defects
should be annealed. Framework the paper we analyzed dynamics of redistribution of dopant
and/or radiation defects during their annealing. Similar logical element has been considered in
[19]. We introduce an approach to decrease dimensions of the element. However it is necessary to
complicate technological process.
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
2
Fig. 1a. Composition element transistor-transistor logic. View from above. Black marked transistors and p-
n-junction manufactured by using doping of appropriate sections of the epitaxial layer. Dimensions of these
devices are decreased. Transistor 1 is a multiemitter transistor. Emitters have been marked by using letter E.
The index indicates their number in the multiemitter transistor. D1 and D2 mean dopants of p and n types in
p-n-junction. Red marked resistors (Ri) and wires have no decreasing of their dimensions
Fig. 1b. Heterostructure, which consist of a substrate and epitaxial layer with sections, manufactured by
using another materials. The figure shows integration of a multiemitter and homoemitter transistors. Dashed
lines are illustrated wires
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
3
Substrate
Base
Collector
Emitter
Base
Collector
Emitter
Fig. 1c. Heterostructure, which consist of a substrate and epitaxial layer with sections, manufactured by
using another materials. The figure shows integration of two homoemitter transistors. Dashed lines are illu-
strated wires
2. METHOD OF SOLUTION
In this section we determine spatio-temporal distributions of concentrations of infused and im-
planted dopants. To determine these distributions we calculate appropriate solutions of the second
Fick's law [1,3,18]
( ) ( ) ( ) ( )
+
+
=
z
tzyxCD
zy
tzyxCD
yx
tzyxCD
xt
tzyxCCCC
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,,,,,,,,,,. (1)
Boundary and initial conditions for the equations are
( )0
,,,
0
=∂
∂
=xx
tzyxC,
( )0
,,,=
∂
∂
= xLxx
tzyxC,
( )0
,,,
0
=∂
∂
=yy
tzyxC,
( )0
,,,=
∂
∂
= yLxy
tzyxC,
( )0
,,,
0
=∂
∂
=zz
tzyxC,
( )0
,,,=
∂
∂
= zLxz
tzyxC, C (x,y,z,0)=f (x,y,z). (2)
The function C(x,y,z,t) describes the spatio-temporal distribution of concentration of dopant; T is
the temperature of annealing; DС is the dopant diffusion coefficient. Value of dopant diffusion
coefficient could be changed with changing materials of heterostructure, with changing tempera-
ture of materials (including annealing), with changing concentrations of dopant and radiation de-
fects. We approximate dependences of dopant diffusion coefficient on parameters by the follow-
ing relation with account results in Refs. [20-22]
( ) ( )( )
( ) ( )
( )
++
+=
2*
2
2*1
,,,,,,1
,,,
,,,1,,,
V
tzyxV
V
tzyxV
TzyxP
tzyxCTzyxDD LC ςςξ
γ
γ
. (3)
Here the function DL (x,y,z,T) describes the spatial (in heterostructure) and temperature (due to
Arrhenius law) dependences of diffusion coefficient of dopant. The function P (x,y,z,T) describes
the limit of solubility of dopant. Parameter γ ∈[1,3] describes average quantity of charged defects
interacted with atom of dopant [20]. The function V (x,y,z,t) describes the spatio-temporal distri-
bution of concentration of radiation vacancies. Parameter V* describes the equilibrium distribution
of concentration of vacancies. The considered concentrational dependence of dopant diffusion
coefficient has been described in details in [20]. It should be noted, that using diffusion type of
doping did not generation 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 [21,22]
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
4
( ) ( ) ( ) ( ) ( ) ( ) ×−
∂
∂
∂
∂+
∂
∂
∂
∂=
∂
∂Tzyxk
y
tzyxITzyxD
yx
tzyxITzyxD
xt
tzyxIIIII
,,,,,,
,,,,,,
,,,,,,
,
( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxkz
tzyxITzyxD
ztzyxI
VII,,,,,,,,,
,,,,,,,,, ,
2 −
∂
∂
∂
∂+× (4)
( ) ( ) ( ) ( ) ( ) ( ) ×−
∂
∂
∂
∂+
∂
∂
∂
∂=
∂
∂Tzyxk
y
tzyxVTzyxD
yx
tzyxVTzyxD
xt
tzyxVVVVV ,,,
,,,,,,
,,,,,,
,,,,
( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxkz
tzyxVTzyxD
ztzyxV
VIV,,,,,,,,,
,,,,,,,,,
,
2 −
∂
∂
∂
∂+× .
Boundary and initial conditions for these equations are
( )0
,,,
0
=∂
∂
=xx
tzyxρ,
( )0
,,,=
∂
∂
= xLxx
tzyxρ,
( )0
,,,
0
=∂
∂
=yy
tzyxρ,
( )0
,,,=
∂
∂
= yLyy
tzyxρ,
( )0
,,,
0
=∂
∂
=zz
tzyxρ,
( )0
,,,=
∂
∂
= zLzz
tzyxρ, ρ (x,y,z,0)=fρ (x,y,z). (5)
Here ρ =I,V. The function I (x,y,z,t) describes the spatio-temporal distribution of concentration of
radiation interstitials; Dρ(x,y,z,T) are the diffusion coefficients of point radiation defects; terms
V2(x,y,z,t) and I
2(x,y,z,t) correspond to generation divacancies and diinterstitials; kI,V(x,y,z,T) is the
parameter of recombination of point radiation defects; kI,I(x,y,z,T) and kV,V(x,y,z,T) are the parame-
ters of generation of simplest complexes of point radiation defects.
Further we determine distributions in space and time of concentrations of divacancies ΦV(x,y,z,t)
and diinterstitials ΦI(x,y,z,t) by solving the following system of equations [21,22]
( ) ( ) ( ) ( ) ( )+
Φ+
Φ=
ΦΦΦ
y
tzyxTzyxD
yx
tzyxTzyxD
xt
tzyxI
I
I
I
I
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,,,,
,,,,,,
,,,
( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkz
tzyxTzyxD
zIII
I
I,,,,,,,,,,,,
,,,,,, 2
,−+
Φ+ Φ
∂
∂
∂
∂ (6)
( ) ( ) ( ) ( ) ( )+
Φ+
Φ=
ΦΦΦ
y
tzyxTzyxD
yx
tzyxTzyxD
xt
tzyxV
V
V
V
V
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,,,,
,,,,,,
,,,
( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkz
tzyxTzyxD
zVVV
V
V,,,,,,,,,,,,
,,,,,, 2
,−+
Φ+ Φ ∂
∂
∂
∂.
Boundary and initial conditions for these equations are
( )0
,,,
0
=∂
Φ∂
=xx
tzyxρ,
( )0
,,,=
∂
Φ∂
= xLxx
tzyxρ,
( )0
,,,
0
=∂
Φ∂
=yy
tzyxρ,
( )0
,,,=
∂
Φ∂
= yLyy
tzyxρ,
( )0
,,,
0
=∂
Φ∂
=zz
tzyxρ,
( )0
,,,=
∂
Φ∂
= zLzz
tzyxρ, ΦI (x,y,z,0)=fΦI (x,y,z), ΦV (x,y,z,0)=fΦV (x,y,z). (7)
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
5
Here DΦρ(x,y,z,T) are the diffusion coefficients of the above complexes of radiation defects;
kI(x,y,z,T) and kV (x,y,z,T) are the parameters of decay of these complexes.
We calculate distributions of concentrations of point radiation defects in space and time by re-
cently elaborated approach [18]. The approach based on transformation of approximations of dif-
fusion coefficients in the following form: Dρ(x,y,z,T)=D0ρ[1+ερ gρ(x,y,z,T)], where D0ρ are the av-
erage values of diffusion coefficients, 0≤ερ<1, |gρ(x,y,z,T)|≤1, ρ =I,V. We also used analogous
transformation of approximations of parameters of recombination of point defects and parameters
of generation of their complexes: kI,V(x,y,z,T)=k0I,V[1+εI,V gI,V(x,y,z,T)], kI,I(x,y,z,T)=k0I,I [1+εI,I
gI,I(x,y,z,T)] and kV,V (x,y,z,T) = k0V,V [1+εV,V gV,V(x,y,z,T)], where k0ρ1,ρ2 are the their average values,
0≤εI,V <1, 0≤εI,I <1, 0≤εV,V<1, | gI,V(x,y,z,T)|≤1, | gI,I(x,y,z,T)|≤1, |gV,V(x,y,z,T)|≤1. Let us introduce
the following dimensionless variables: ( ) ( ) *,,,,,,~
ItzyxItzyxI = , χ = x/Lx, η = y /Ly,
( ) ( ) *,,,,,,~
VtzyxVtzyxV = , 2
00LtDD
VI=ϑ ,
VIVIDDkL
00,0
2=ω , VI
DDkL 00,0
2
ρρρ =Ω , φ =
z/Lz. The introduction leads to transformation of Eqs.(4) and conditions (5) to the following form
( ) ( )[ ] ( ) ( )[ ] ×+∂
∂+
∂
∂+
∂
∂=
∂
∂Tg
ITg
DD
DIIIII
VI
I ,,,1,,,
~
,,,1,,,
~
00
0 φηχεηχ
ϑφηχφηχε
χϑ
ϑφηχ
( ) ( )[ ] ( ) ( ) ×−
∂
∂+
∂
∂+
∂
∂× ϑφηχ
φ
ϑφηχφηχε
φη
ϑφηχ,,,
~,,,~
,,,1,,,
~
00
0
00
0 II
TgDD
D
DD
DIII
VI
I
VI
I
( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχεω ,,,~
,,,1,,,~
,,,1 2
,,,,ITgVTg
IIIIIVIVI+Ω−+× (8)
( ) ( )[ ] ( ) ( )[ ] ×+∂
∂+
∂
∂+
∂
∂=
∂
∂Tg
VTg
DD
DVVVVV
VI
V,,,1
,,,~
,,,1,,,
~
00
0φηχε
ηχ
ϑφηχφηχε
χϑ
ϑφηχ
( ) ( )[ ] ( ) ( ) ×−
∂
∂+
∂
∂+
∂
∂× ϑφηχ
φ
ϑφηχφηχε
φη
ϑφηχ,,,
~,,,~
,,,1,,,
~
00
0
00
0 IV
TgDD
D
DD
DVVV
VI
V
VI
V
( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχεω ,,,~
,,,1,,,~
,,,1 2
,,,,VTgVTg
VVVVVVIVI+Ω−+×
( )0
,,,~
0
=∂
∂
=χχ
ϑφηχρ,
( )0
,,,~
1
=∂
∂
=χχ
ϑφηχρ,
( )0
,,,~
0
=∂
∂
=ηη
ϑφηχρ,
( )0
,,,~
1
=∂
∂
=ηη
ϑφηχρ,
( )0
,,,~
0
=∂
∂
=φφ
ϑφηχρ,
( )0
,,,~
1
=∂
∂
=φφ
ϑφηχρ, ( )
( )*
,,,,,,~
ρ
ϑφηχϑφηχρ ρf
= . (9)
We determine solutions of Eqs.(8) with conditions (9) framework recently introduced approach
[18], i.e. as the power series
( ) ( )∑ ∑ ∑Ω=∞
=
∞
=
∞
=0 0 0
,,,~,,,~i j k
ijk
kji ϑφηχρωεϑφηχρ ρρ . (10)
Substitution of the series (10) into Eqs.(8) and conditions (9) gives us possibility to obtain equa-
tions for initial-order approximations of concentration of point defects ( )ϑφηχ ,,,~
000I and
( )ϑφηχ ,,,~
000V and corrections for them ( )ϑφηχ ,,,
~ijk
I and ( )ϑφηχ ,,,~
ijkV , i ≥1, j ≥1, k ≥1. The equa-
tions are presented in the Appendix. Solutions of the equations could be obtained by standard
Fourier approach [24,25]. The solutions are presented in the Appendix.
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
6
Now we calculate distributions of concentrations of simplest complexes of point radiation defects
in space and time. To determine the distributions we transform approximations of diffusion coef-
ficients in the following form: DΦρ(x,y,z,T)=D0Φρ[1+εΦρgΦρ(x,y,z,T)], where D0Φρ are the average
values of diffusion coefficients. In this situation the Eqs.(6) could be written as
( )( )[ ] ( )
( ) ( )++
Φ
+=Φ
ΦΦΦ tzyxITzyxkx
tzyxTzyxg
xD
t
tzyxII
I
III
I ,,,,,,,,,
,,,1,,, 2
,0∂
∂ε
∂
∂
∂
∂
( )[ ] ( )( )[ ] ( )
−
Φ
++
Φ
++ ΦΦΦΦΦΦz
tzyxTzyxg
zD
y
tzyxTzyxg
yD I
III
I
III∂
∂ε
∂
∂
∂
∂ε
∂
∂ ,,,,,,1
,,,,,,1 00
( ) ( )tzyxITzyxkI
,,,,,,−
( )( )[ ]
( )( ) ( ) ++
Φ
+=Φ
ΦΦΦ tzyxITzyxkx
tzyxTzyxg
xD
t
tzyxII
V
VVV
V ,,,,,,,,,
,,,1,,, 2
,0∂
∂ε
∂
∂
∂
∂
( )[ ] ( ) ( )[ ] ( )−
Φ
++
Φ
++ ΦΦΦΦΦΦz
tzyxTzyxg
zD
y
tzyxTzyxg
yD V
VVV
V
VVV∂
∂ε
∂
∂
∂
∂ε
∂
∂ ,,,,,,1
,,,,,,1
00
( ) ( )tzyxITzyxkI
,,,,,,− .
Farther we determine solutions of above equations as the following power series
( ) ( )∑ Φ=Φ∞
=Φ
0
,,,,,,i
i
itzyxtzyx ρρρ ε . (11)
Now we used the series (11) into Eqs.(6) and appropriate boundary and initial conditions. The
using gives the possibility to obtain equations for initial-order approximations of concentrations
of complexes of defects Φρ0(x,y,z,t), corrections for them Φρi(x,y,z,t) (for them i ≥1) and boundary
and initial conditions for them. We remove equations and conditions to the Appendix. Solutions
of the equations have been calculated by standard approaches [24,25] and presented in the Ap-
pendix.
Now we calculate distribution of concentration of dopant in space and time by using the ap-
proach, which was used for analysis of radiation defects. To use the approach we consider follow-
ing transformation of approximation of dopant diffusion coefficient: DL(x,y,z,T)=D0L[1+
εLgL(x,y,z,T)], where D0L is the average value of dopant diffusion coefficient, 0≤εL< 1,
|gL(x,y,z,T)|≤1. Farther we consider solution of Eq.(1) as the following series:
( ) ( )∑ ∑=∞
=
∞
=0 1
,,,,,,i j
ij
ji
LtzyxCtzyxC ξε .
Using the relation into Eq.(1) and conditions (2) leads to obtaining equations for the functions
Cij(x,y,z,t) (i ≥1, j ≥1), boundary and initial conditions for them. The equations are presented in
the Appendix. Solutions of the equations have been calculated by standard approaches (see, for
example, [24,25]). The solutions are 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. Usually the second-order approximations are enough good approximations to
make qualitative analysis and to obtain quantitative results. All analytical results have been
checked by numerical simulation.
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
7
3. DISCUSSION
In this section we analyzed spatio-temporal distributions of concentrations of dopants. Figs. 2
shows typical spatial distributions of concentrations of dopants in neighborhood of interfaces of
heterostructures. We calculate these distributions of concentrations of dopants under the follow-
ing condition: value of dopant diffusion coefficient in doped area is larger, than value of dopant
diffusion coefficient in nearest areas. In this situation one can find increasing of sharpness of p-n-
junctions with increasing of homogeneity of distribution of concentration of dopant at one time.
These both effects could be obtained in both situations, when p-n-junctions are single and frame-
work their systems (transistors, thyristors). Changing relation between values of dopant diffusion
coefficients leads to opposite result (see Figs. 3).
Fig. 2a. Dependences of concentration of dopant, infused in heterostructure from Figs. 1, on coordinate in
direction, which is perpendicular to interface between epitaxial layer substrate. Difference between values
of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves.
Value of dopant diffusion coefficient in the epitaxial layer is larger, than value of dopant diffusion coeffi-
cient in the substrate
x
0.0
0.5
1.0
1.5
2.0
C(x
,Θ)
23
4
1
0 L/4 L/2 3L/4 L
Epitaxial layer Substrate
Fig. 2b. Dependences of concentration of dopant, implanted in heterostructure from Figs. 1, on coordinate
in direction, which is perpendicular to interface between epitaxial layer substrate. Difference between val-
ues of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of
curves. Value of dopant diffusion coefficient in the epitaxial layer is larger, than value of dopant diffusion
coefficient in the substrate. Curve 1 corresponds to homogenous sample and annealing time Θ = 0.0048
(Lx2+Ly
2+Lz
2)/D0. Curve 2 corresponds to homogenous sample and annealing time Θ = 0.0057 (Lx
2+Ly
2+
Lz2)/D0. Curves 3 and 4 correspond to heterostructure from Figs. 1; annealing times Θ = 0.0048 (Lx
2+Ly
2+
Lz2)/D0 and Θ = 0.0057 (Lx
2+Ly2+ Lz
2)/D0, respectively
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
8
Fig.3a. Distributions of concentration of dopant, infused in average section of epitaxial layer of heterostruc-
ture from Figs. 1 in direction parallel to interface between epitaxial layer and substrate of heterostructure.
Difference between values of dopant diffusion coefficients increases with increasing of number of curves.
Value of dopant diffusion coefficient in this section is smaller, than value of dopant diffusion coefficient in
nearest sections
x
0.00000
0.00001
0.00010
0.00100
0.01000
0.10000
1.00000
C(x
, Θ)
fC(x)
L/40 L/2 3L/4 Lx0
1
2
Substrate
Epitaxial layer 1
Epitaxial layer 2
Fig.3b. Calculated distributions of implanted dopant in epitaxial layers of heterostructure. Solid lines are
spatial distributions of implanted dopant in system of two epitaxial layers. Dushed lines are spatial distribu-
tions of implanted dopant in one epitaxial layer. Annealing time increases with increasing of number of
curves
It should be noted, that framework the considered approach one shall optimize annealing of do-
pant and/or radiation defects. To do the optimization we used recently introduced criterion [26-
34]. The optimization based on approximation real distribution by step-wise function ψ (x,y, z)
(see Figs. 4). Farther the required values of optimal annealing time have been calculated by mi-
nimization the following mean-squared error
( ) ( )[ ]∫ ∫ ∫ −Θ=x y zL L L
zyx
xdydzdzyxzyxCLLL
U0 0 0
,,,,,1
ψ . (12)
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
9
C(x
,Θ)
0 Lx
2
13
4
Fig.4a. Distributions of concentration of infused dopant in depth of heterostructure from Fig. 1 for different
values of annealing time (curves 2-4) and idealized step-wise approximation (curve 1). Increasing of num-
ber of curve corresponds to increasing of annealing time
x
C(x
,Θ)
1
23
4
0 L
Fig.4b. Distributions of concentration of implanted dopant in depth of heterostructure from Fig. 1 for dif-
ferent values of annealing time (curves 2-4) and idealized step-wise approximation (curve 1). Increasing of
number of curve corresponds to increasing of annealing time
We show optimal values of annealing time as functions of parameters on Figs. 5. It is known, that
standard step of manufactured ion-doped structures is annealing of radiation defects. In the ideal
case after finishing the annealing dopant achieves interface between layers of heterostructure. If
the dopant has no enough time to achieve the interface, it is practicably to anneal the dopant addi-
tionally. The Fig. 5b shows the described dependences of optimal values of additional annealing
time for the same parameters as for Fig. 5a. Necessity to anneal radiation defects leads to smaller
values of optimal annealing of implanted dopant in comparison with optimal annealing time of
infused dopant.
0.0 0.1 0.2 0.3 0.4 0.5a/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
Fig.5a. Dimensionless optimal annealing time of infused dopant as a function of several parameters. Curve
1 describes the dependence of the annealing time on the relation a/L and ξ = γ = 0 for equal to each other
values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 describes the dependence of
the annealing time on value of parameter ε for a/L=1/2 and ξ = γ = 0. Curve 3 describes the dependence of
the annealing time on value of parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 describes the dependence of
the annealing time on value of parameter γ for a/L=1/2 and ε = ξ = 0
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
10
0.0 0.1 0.2 0.3 0.4 0.5a/L, ξ, ε, γ
0.00
0.04
0.08
0.12
Θ D
0 L
-2
3
2
4
1
Fig.5b. Dimensionless optimal annealing time of implanted dopant as a function of several parameters.
Curve 1 describes the dependence of the annealing time on the relation a/L and ξ = γ = 0 for equal to each
other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 describes the dependence
of the annealing time on value of parameter ε for a/L=1/2 and ξ = γ = 0. Curve 3 describes the dependence
of the annealing time on value of parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 describes the dependence of
the annealing time on value of parameter γ for a/L=1/2 and ε = ξ = 0
4. CONCLUSIONS
In this paper we introduce an approach of vertical integration framework element of transistor-
transistor logic. The approach gives us possibility to decrease area of the elements with smaller
increasing of the element’s thickness.
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 and educational fellowship for scientific research of Government of Russian
and of Nizhny Novgorod State University of Architecture and Civil Engineering.
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[28] E.L. Pankratov. Decreasing of depth of implanted-junction rectifier in semiconductor heterostruc-
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[29] E.L. Pankratov, E.A. Bulaeva. Application of native inhomogeneities to increase compactness of
vertical field-effect transistors. J. Comp. Theor. Nanoscience. Vol. 10 (4). P. 888-893 (2013).
[30] E.L. Pankratov, E.A. Bulaeva. Optimization of doping of heterostructure during manufacturing of p-
i-n-diodes. Nanoscience and Nanoengineering. Vol. 1 (1). P. 7-14 (2013).
[31] E.L. Pankratov, E.A. Bulaeva. An approach to decrease dimensions of field-effect transistors. Uni-
versal Journal of Materials Science. Vol. 1 (1). P.6-11 (2013).
[32] E.L. Pankratov, E.A. Bulaeva. An approach to manufacture a heterobipolar transistors in thin film
structures. On the method of optimization. Int. J. Micro-Nano Scale Transp. Vol. 4 (1). P. 17-31
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[33] E.L. Pankratov, E.A. Bulaeva. Application of native inhomogeneities to increase compactness of
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(2012).
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International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
12
Authors
Pankratov Evgeny Leonidovich was born at 1977. From 1985 to 1995 he was educated in a secondary
school in Nizhny Novgorod. From 1995 to 2004 he was educated in Nizhny Novgorod State University:
from 1995 to 1999 it was bachelor course in Radiophysics, from 1999 to 2001 it was master course in Ra-
diophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in Radio-
physics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of Micro-
structures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny Novgo-
rod State University of Architecture and Civil Engineering. Now E.L. Pankratov is in his Full Doctor
course in Radiophysical Department of Nizhny Novgorod State University. He has 110 published papers in
area of his researches.
Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school of
village Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school
“Center for gifted children”. From 2009 she is a student of Nizhny Novgorod State University of Architec-
ture and Civil Engineering (spatiality “Assessment and management of real estate”). At the same time she
is a student of courses “Translator in the field of professional communication” and “Design (interior art)” in
the University. E.A. Bulaeva was a contributor of grant of President of Russia (grant MK-548.2010.2).
She has 74 published papers in area of her researches.
APPENDIX
Equations for the functions ( )ϑφηχ ,,,~
ijkI and ( )ϑφηχ ,,,
~ijk
V , i ≥0, j ≥0, k ≥0 and conditions for
them
( ) ( ) ( ) ( )2
000
2
0
0
2
000
2
0
0
2
000
2
0
0000,,,
~,,,
~,,,
~,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
∂
∂+
∂
∂+
∂
∂=
∂
∂ I
D
DI
D
DI
D
DI
V
I
V
I
V
I
( ) ( ) ( ) ( )2
000
2
0
0
2
000
2
0
0
2
000
2
0
0000,,,
~,,,
~,,,
~,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ
∂
∂+
∂
∂+
∂
∂=
∂
∂ V
D
DV
D
DV
D
DV
I
V
I
V
I
V ;
( ) ( ) ( ) ( )( )
×
∂
∂+
∂
∂+
∂
∂+
∂
∂=
∂
∂Tg
III
D
DII
iii
V
Ii ,,,,,,
~,,,
~,,,
~,
~
2
00
2
2
00
2
2
00
2
0
000 φηχχφ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑχ
( )( )
( )( )
×
∂
∂+
∂
∂
∂
∂+
∂
∂× −− Tg
ITg
D
D
D
DII
i
I
V
I
V
Ii ,,,,,,
~
,,,,,,
~100
0
0
0
0100 φηχφη
ϑφηχφηχ
ηχ
ϑφηχ
( )
V
Ii
D
DI
0
0100,,,
~
∂
∂× −
φ
ϑφηχ, i ≥1,
( ) ( ) ( ) ( )( )
×
∂
∂+
∂
∂+
∂
∂+
∂
∂=
∂
∂Tg
VVV
D
DVV
iii
I
Vi ,,,,,,
~,,,
~,,,
~,
~
2
00
2
2
00
2
2
00
2
0
000 φηχχφ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑχ
( )( )
( )( )
×
∂
∂+
∂
∂
∂
∂+
∂
∂× −− Tg
VTg
D
D
D
DVV
i
V
I
V
I
Vi ,,,,,,
~
,,,,,,
~100
0
0
0
0100 φηχφη
ϑφηχφηχ
ηχ
ϑφηχ
( )
I
Vi
D
DV
0
0100,,,
~
∂
∂× −
φ
ϑφηχ, i ≥1,
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
13
( ) ( ) ( ) ( )−
∂
∂+
∂
∂+
∂
∂=
∂
∂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,, VITgVIVI
+−
( ) ( ) ( ) ( )−
∂
∂+
∂
∂+
∂
∂=
∂
∂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 ,,,~
,,,~
,,,1000000,,
VITgVIVI
+− ;
( ) ( ) ( ) ( )−
∂
∂+
∂
∂+
∂
∂=
∂
∂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 ,,,~
,,,~
,,,~
,,,~
,,,1010000000010,,
VIVITgVIVI
++−
( ) ( ) ( ) ( )−
∂
∂+
∂
∂+
∂
∂=
∂
∂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 ,,,~
,,,~
,,,~
,,,~
,,,1010000000010,,
VIVITgVIVI
++− ;
( ) ( ) ( ) ( )−
∂
∂+
∂
∂+
∂
∂=
∂
∂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+− ;
( ) ( ) ( ) ( )×+
∂
∂+
∂
∂+
∂
∂=
∂
∂
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
ITg
ITg III ,,,
,,,~
,,,,,,
~
,,, 010010 φηχφη
ϑφηχφηχ
ηχ
ϑφηχφηχ
χ
( )( ) ( ) ( ) ( )[ ] ×+−
∂
∂× ϑφηχϑφηχϑφηχϑφηχ
φ
ϑφηχ,,,
~,,,
~,,,
~,,,
~,,,~
100000000100
010 VIVII
( )[ ]Tg IIII ,,,1 ,, φηχε+×
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
14
( ) ( ) ( ) ( )×+
∂
∂+
∂
∂+
∂
∂=
∂
∂
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
VTg
VTg
IVV,,,
,,,~
,,,,,,
~
,,, 010010 φηχφη
ϑφηχφηχ
ηχ
ϑφηχφηχ
χ
( )( ) ( ) ( ) ( )[ ] ×+−
∂
∂× ϑφηχϑφηχϑφηχϑφηχ
φ
ϑφηχ,,,
~,,,
~,,,
~,,,
~,,,~
100000000100
010 IVIVV
( )[ ]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 ,,,~
,,,~
,,,1000001,,
IITgIIII
+−
( ) ( ) ( ) ( )−
∂
∂+
∂
∂+
∂
∂=
∂
∂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 ,,,~
,,,~
,,,1000001,,
VVЕgVVVV
+− ;
( ) ( ) ( ) ( )+
∂
∂+
∂
∂+
∂
∂=
∂
∂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
ϑφηχφηχ
ηχ
ϑφηχφηχ
χ
,,,~
,,,,,,
~
,,, 001001
0
0I
TgI
TgD
DII
V
I
( ) ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχεφ
ϑφηχφηχ
φ,,,
~,,,
~,,,1
,,,~
,,,000100
001 VITgI
TgIII
+−
∂
∂
∂
∂+
( ) ( ) ( ) ( )+
∂
∂+
∂
∂+
∂
∂=
∂
∂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
ϑφηχφηχ
ηχ
ϑφηχφηχ
χ
,,,~
,,,,,,
~
,,, 001001
0
0V
TgV
TgD
DVV
I
V
( ) ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχεφ
ϑφηχφηχ
φ,,,
~,,,
~,,,1
,,,~
,,,100000
001 VITgV
TgVVV
+−
∂
∂
∂
∂+ ;
( ) ( ) ( ) ( )( ) ×−
∂
∂+
∂
∂+
∂
∂=
∂
∂ϑφηχ
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ,,,
~,,,~
,,,~
,,,~
,,,~
0102
011
2
2
011
2
2
011
2
0
0011 IIII
D
DI
V
I
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
15
( )[ ] ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχεϑφηχφηχε ,,,~
,,,~
,,,1,,,~
,,,1000001,,000,,
VITgITgVIVIIIII
+−+×
( ) ( ) ( ) ( )( ) ×−
∂
∂+
∂
∂+
∂
∂=
∂
∂ϑφηχ
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ,,,
~,,,~
,,,~
,,,~
,,,~
0102
011
2
2
011
2
2
011
2
0
0011 VVVV
D
DV
I
V
( )[ ] ( ) ( )[ ] ( ) ( )ϑφηχϑφηχφηχεϑφηχφηχε ,,,~
,,,~
,,,1,,,~
,,,1001000,,000,,
VItgVTgVIVIVVVV
+−+× ;
( )0
,,,~
0
=∂
∂
=x
ijk
χ
ϑφηχρ,
( )0
,,,~
1
=∂
∂
=x
ijk
χ
ϑφηχρ,
( )0
,,,~
0
=∂
∂
=ηη
ϑφηχρijk
, ( )
0,,,~
1
=∂
∂
=ηη
ϑφηχρijk
,
( )0
,,,~
0
=∂
∂
=φφ
ϑφηχρijk
, ( )
0,,,~
1
=∂
∂
=φφ
ϑφηχρijk
(i ≥0, j ≥0, k ≥0);
( ) ( ) *
000,,0,,,~ ρφηχφηχρ ρf= , ( ) 00,,,~ =φηχρ
ijk (i ≥1, j ≥1, k ≥1).
Solutions of the above equations could be written as
( ) ( ) ( ) ( ) ( )∑+=∞
=1000
21,,,~
nnn
ecccFLL
ϑφηχϑφηχρ ρρ ,
where ( ) ( ) ( ) ( )∫ ∫ ∫=1
0
1
0
1
0*
,,coscoscos1
udvdwdwvufwnvnunFnn ρρ πππ
ρ, ( ) ( )
IVnIDDne 00
22exp ϑπϑ −= ,
cn(χ) = cos (π n χ), ( ) ( )VInV
DDne00
22exp ϑπϑ −= ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
×∑ ∫ ∫ ∫ ∫∂
∂−−=
∞
=
−
1 0
1
0
1
0
1
0
100
0
0
00
,,,~
2,,,~
n
i
nnnInIn
V
I
iu
wvuIvcuseecccn
D
DI
ϑ ττϑφηχπϑφηχ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−×∞
=1 0
1
0
1
00
02,,,n
nnnInIn
V
I
Invsuceecccn
D
DdudvdwdTwvugwc
ϑ
τϑφηχπτ
( ) ( )( )
( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−∫∂
∂×
∞
=
−
1 00
01
0
100 2,,,
~
,,,n
nInIn
V
Ii
In eecccnD
Ddudvdwd
v
wvuITwvugwc
ϑ
τϑφηχπττ
( ) ( ) ( ) ( )( )
∫ ∫ ∫∂
∂× −
1
0
1
0
1
0
100,,,
~
,,, ττ
dudvdwdw
wvuITwvugwsvcuc i
Innn, i ≥1,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫ ∫−−=∞
=1 0
1
0
1
0
1
00
0
00,,,2,,,
~
nVnnnInVn
I
V
iTwvugvcuseecccn
D
DV
ϑ
τϑφηχπϑφηχ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫−−∂
∂×
∞
=
−
1 0
1
0
1
00
0100,
~
nnnnInVn
I
Vi
nvsuceecccn
D
Ddudvdwd
u
uVwc
ϑ
τϑφηχττ
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
16
( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫∂
∂×
∞
=
−
10
01
0
100 2,
~
,,,2n
nVn
I
Vi
Vnecccn
D
Ddudvdwd
v
uVTwvugwc ϑφηχπτ
τπ
( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫
∂
∂−× −
ϑ
ττ
τ0
1
0
1
0
1
0
100,
~
,,, dudvdwdw
uVTwvugwsvcuce i
VnnnnI, i ≥1,
where sn(χ) = sin (π n χ);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−=∞
=1 0
1
0
1
0
1
0010
2,,,~n
nnnnnnnnwcvcuceeccc
ϑ
ρρ τϑφηχϑφηχρ
( )[ ] ( ) ( ) τττε dudvdwdwvuVwvuITwvugVIVI
,,,~
,,,~
,,,1000000,,
+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ×∑ ∫ ∫ ∫ ∫ +−−=∞
=1 0
1
0
1
0
1
0,
0
0
02012,,,~
nVInnnnnnnn
V
I wcvcuceecccD
D ϑ
ρρ ετϑφηχϑφηχρ
( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvugVI
,,,~
,,,~
,,,~
,,,~
,,,010000000010,
+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0
1
0
1
0
1
0001 2,,,~
nnnnnnnnn
wcvcuceecccϑ
ρρ τϑφηχϑφηχρ
( )[ ] ( ) ττρε ρρρρ dudvdwdwvuTwvug ,,,~,,,1 2
000,,+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ×∫ ∫−−=∞
=1 0
1
0
1
0
1
0002
2,,,~n
nnnnnnnnwcvcuceeccc
ϑ
ρρ τϑφηχϑφηχρ
( )[ ] ( ) ( ) ττρτρε ρρρρ dudvdwdwvuwvuTwvug ,,,~,,,~,,,1000001,,
+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0
1
0
1
0
1
00
0
1102,,,
~
nnnnnInInnn
V
I ucvcuseecccnD
DI
ϑ
τϑφηχπϑφηχ
( ) ( ) ( ) ( ) ( ) ( ) ×∑−∂
∂×
∞
=
−
10
0100 2,,,
~
,,,n
nInnn
V
Ii
Iecccn
D
Ddudvdwd
u
wvuITwvug ϑφηχπτ
τ
( ) ( ) ( ) ( ) ( ) ( )×−∫ ∫ ∫ ∫
∂
∂−× −
V
Ii
InnnnID
Ddudvdwd
v
wvuITwvugucvsuce
0
0
0
1
0
1
0
1
0
100 2,,,
~
,,, πττ
τϑ
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫
∂
∂−×
∞
=
−
1 0
1
0
1
0
1
0
100,,,
~
,,,n
i
InnnnInIdudvdwd
w
wvuITwvugusvcuceen
ϑ
ττ
τϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[∑ ∫ ∫ ∫ ∫ ×+−−×∞
=1 0
1
0
1
0
1
0,
12n
VInnnnInnnInnnnvcvcuceccecccc
ϑ
ετφηϑχφηχ
( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvugVI
,,,~
,,,~
,,,~
,,,~
,,,100000000100,
+×
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
17
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0
1
0
1
0
1
00
0
1102,,,
~
nnnnnVnVnnn
I
V ucvcuseecccnD
DV
ϑ
τϑφηχπϑφηχ
( ) ( ) ( ) ( ) ( ) ( ) ×∑−∂
∂×
∞
=
−
10
0100 2,,,
~
,,,n
nVnnn
I
Vi
Vecccn
D
Ddudvdwd
u
wvuVTwvug ϑφηχπτ
τ
( ) ( ) ( ) ( ) ( ) ( )×−∫ ∫ ∫ ∫
∂
∂−× −
I
Vi
VnnnnVD
Ddudvdwd
v
wvuVTwvugucvsuce
0
0
0
1
0
1
0
1
0
100 2,,,
~
,,, πττ
τϑ
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫
∂
∂−×
∞
=
−
1 0
1
0
1
0
1
0
100,,,
~
,,,n
i
VnnnnVnVdudvdwd
w
wvuVTwvugusvcuceen
ϑ
ττ
τϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∑ ∫ ∫ ∫ ∫ ×+−−×∞
=1 0
1
0
1
0
1
0,,
,,,12n
VIVInnnVnnnInnnnTwvugvcuceccecccc
ϑ
ετφηϑχφηχ
( ) ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuIwcn
,,,~
,,,~
,,,~
,,,~
100000000100+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0
1
0
1
0
1
00
0
101,,,2,,,
~
nInnnInInnn
V
I TwvugvcuseecccnD
DI
ϑ
τϑφηχπϑφηχ
( )( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂
∂×
∞
=1 0
1
00
0001 2,,,
~
nnnInInnn
V
I
nuceecccn
D
Ddudvdwd
u
wvuIwc
ϑ
τϑφηχπττ
( ) ( ) ( )( )
( ) ( ) ( ) ( ) ×∑−∫ ∫∂
∂×
∞
=10
01
0
1
0
001 2,,,
~
,,,n
nnnnI
V
I
Inncccen
D
Ddudvdwd
v
wvuITwvugwcvs φηχϑπτ
τ
( ) ( ) ( ) ( ) ( )( )
( ) ( ) ( ) ×∑−∫ ∫ ∫ ∫∂
∂−×
∞
=10
1
0
1
0
1
0
001 2,,,
~
,,,n
nnnInnnnIcccdudvdwd
w
wvuITwvugwsvcuce φηχτ
ττ
ϑ
( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ ∫ ∫ +−×ϑ
τττετϑ0
1
0
1
0
1
0000100,,
,,,~
,,,~
,,,1 dudvdwdwvuVwvuITwvugwcvcuceeVIVInnnnInI
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0
1
0
1
0
1
00
0
101,,,2,,,
~
nVnnnVnVnnn
I
V TwvugvcuseecccnD
DV
ϑ
τϑφηχπϑφηχ
( )( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂
∂×
∞
=1 0
1
00
0001 2,,,
~
nnnVnInnn
I
V
nuceecccn
D
Ddudvdwd
u
wvuVwc
ϑ
τϑφηχπττ
( ) ( ) ( )( )
( ) ( ) ( ) ( ) ×∑−∫ ∫∂
∂×
∞
=10
01
0
1
0
001 2,,,
~
,,,n
nnnnV
I
V
Vnncccen
D
Ddudvdwd
v
wvuVTwvugwcvs φηχϑπτ
τ
( ) ( ) ( ) ( ) ( )( )
( ) ( ) ( ) ×∑−∫ ∫ ∫ ∫∂
∂−×
∞
=10
1
0
1
0
1
0
001 2,,,
~
,,,n
nnnVnnnnVcccdudvdwd
w
wvuVTwvugwsvcuce φηχτ
ττ
ϑ
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
18
( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ ∫ ∫ +−×ϑ
τττετϑ0
1
0
1
0
1
0000100,,
,,,~
,,,~
,,,1 dudvdwdwvuVwvuITwvugwcvcuceeVIVInnnnVnV
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−=∞
=1 0
1
0
1
0
1
0000011
,,,~
2,,,~
nnnnnInInnn
wvuIwcvcuceecccIϑ
ττϑφηχϑφηχ
( )[ ] ( ) ( )[ ] ( ) ( ) τττετε dudvdwdwvuVwvuITwvugwvuITwvugVIVIIIII
,,,~
,,,~
,,,1,,,~
,,,1000001,,010,,
+++×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−=∞
=1 0
1
0
1
0
1
0000011
,,,~
2,,,~
nnnnnVnVnnn
wvuIwcvcuceecccVϑ
ττϑφηχϑφηχ
( )[ ] ( ) ( )[ ] ( ) ( ) τττετε dudvdwdwvuVwvuITwvugwvuITwvugVIVIVVVV
,,,~
,,,~
,,,1,,,~
,,,1000001,,010,,
+++× .
Equations for functions Φρi(x,y,z,t), i ≥0 to describe concentrations of simplest complexes of radi-
ation defects.
( ) ( ) ( ) ( )+
Φ+
Φ+
Φ=
ΦΦ 2
0
2
2
0
2
2
0
2
0
0,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyxD
t
tzyxIII
I
I
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkIII
,,,,,,,,,,,, 2
,−+
( ) ( ) ( ) ( )+
Φ+
Φ+
Φ=
ΦΦ 2
0
2
2
0
2
2
0
2
0
0,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyxD
t
tzyxVVV
V
V
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkVVV
,,,,,,,,,,,, 2
,−+ ;
( ) ( ) ( ) ( )+
Φ+
Φ+
Φ=
ΦΦ 2
2
2
2
2
2
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyxD
t
tzyxiIiIiI
I
iI
∂
∂
∂
∂
∂
∂
∂
∂
( )( )
( )( )
+
Φ+
Φ+
−
Φ
−
ΦΦy
tzyxTzyxg
yx
tzyxTzyxg
xD
iI
I
iI
II∂
∂
∂
∂
∂
∂
∂
∂ ,,,,,,
,,,,,,
11
0
( )( )
Φ+
−
Φz
tzyxTzyxg
z
iI
I∂
∂
∂
∂ ,,,,,,
1, i≥1,
( ) ( ) ( ) ( )+
Φ+
Φ+
Φ=
ΦΦ 2
2
2
2
2
2
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyxD
t
tzyxiViViV
V
iV
∂
∂
∂
∂
∂
∂
∂
∂
( )( )
( )( )
+
Φ+
Φ+
−
Φ
−
ΦΦy
tzyxTzyxg
yx
tzyxTzyxg
xD
iV
V
iV
VV∂
∂
∂
∂
∂
∂
∂
∂ ,,,,,,
,,,,,,
11
0
( )( )
Φ+
−
Φz
tzyxTzyxg
z
iV
V∂
∂
∂
∂ ,,,,,,
1, i≥1;
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
19
Boundary and initial conditions for the functions takes the form
( )0
,,,
0
=∂
Φ∂
=x
i
x
tzyxρ,
( )0
,,,=
∂
Φ∂
= xLx
i
x
tzyxρ,
( )0
,,,
0
=∂
Φ∂
=y
i
y
tzyxρ,
( )0
,,,=
∂
Φ∂
= yLy
i
y
tzyxρ,
( )0
,,,
0
=∂
Φ∂
=z
i
z
tzyxρ,
( )0
,,,=
∂
Φ∂
= zLz
i
z
tzyxρ, i≥0; Φρ0(x,y,z,0)=fΦρ (x,y,z), Φρi(x,y,z,0)=0, i≥1.
Solutions of the above equations could be written as
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑+∑+=Φ∞
=
∞
=ΦΦ
110
221,,,
nnnn
nnnnnn
zyxzyx
zcycxcnL
tezcycxcFLLLLLL
tzyxρρρ
( ) ( ) ( ) ( ) ( ) ( ) ( )[∫ ∫ ∫ ∫ −−× ΦΦ
t L L L
IInnnnn
x y z
wvuITwvukwcvcucete0 0 0 0
2
, ,,,,,, ττρρ
( ) ( )] ττ dudvdwdwvuITwvukI
,,,,,,− ,
where ( ) ( ) ( ) ( )∫ ∫ ∫= ΦΦ
x y zL L L
nnnnudvdwdwvufwcvcucF
0 0 0
,,ρρ
, ( )
++−= ΦΦ 2220
22 111exp
zyx
nLLL
tDnteρρ
π ,
cn(x) = cos (π n x/Lx);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=Φ∞
=ΦΦΦ
1 0 0 0 02
,,,2
,,,n
t L L L
nnnnnnn
zyx
i
x y z
TwvugvcusetezcycxcnLLL
tzyxρρρ
τπ
ρ
( )( )
( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−Φ
×∞
=ΦΦ
−
1 02
1 2,,,
n
t
nnnnn
zyx
iI
netezcycxcn
LLLdudvdwd
u
wvuwc τ
πτ
∂
τ∂ρρ
ρ
( ) ( ) ( ) ( ) ( )( )
×∑−∫ ∫ ∫ ∫Φ
−×∞
=
−
ΦΦ1
20 0 0 0
1 2,,,,,,
nzyx
t L L LiI
nnnnn
LLLdudvdwd
v
wvuTwvugwcvsuce
x y z πτ
∂
τ∂τ ρ
ρρ
( ) ( ) ( ) ( ) ( )( )
( ) ×∫ ∫ ∫ ∫Φ
−× Φ
−
ΦΦ
t L L LiI
nnnnn
x y z
dudvdwdTwvugw
wvuwsvcucete
0 0 0 0
1
,,,,,,
τ∂
τ∂τ
ρ
ρ
ρρ
( ) ( ) ( )zcycxcnnn
× , i ≥1,
where sn(x) = sin (π n x/Lx).
Equations for the functions Cij(x,y,z,t) (i ≥0, j ≥0), boundary and initial conditions could be writ-
ten as
( ) ( ) ( ) ( )2
00
2
02
00
2
02
00
2
0
00 ,,,,,,,,,,,,
z
tzyxCD
y
tzyxCD
x
tzyxCD
t
tzyxCLLL
∂
∂+
∂
∂+
∂
∂=
∂
∂;
( ) ( ) ( ) ( )+
∂
∂+
∂
∂+
∂
∂=
∂
∂2
0
2
2
0
2
2
0
2
0
0,,,,,,,,,,,,
z
tzyxC
y
tzyxC
x
tzyxCD
t
tzyxCiii
L
i
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
20
( )( )
( )( )
+
∂
∂
∂
∂+
∂
∂
∂
∂+ −−
y
tzyxCTzyxg
yD
x
tzyxCTzyxg
xD i
LL
i
LL
,,,,,,
,,,,,, 10
0
10
0
( ) ( )
∂
∂
∂
∂+ −
z
tzyxCTzyxg
zD i
LL
,,,,,, 10
0, i ≥1;
( ) ( ) ( ) ( )+
∂
∂+
∂
∂+
∂
∂=
∂
∂2
01
2
02
01
2
02
01
2
0
01 ,,,,,,,,,,,,
z
tzyxCD
y
tzyxCD
x
tzyxCD
t
tzyxCLLL
( )( )
( ) ( )( )
( )+
∂
∂
∂
∂+
∂
∂
∂
∂+
y
tzyxC
TzyxP
tzyxC
yD
x
tzyxC
TzyxP
tzyxC
xD
LL
,,,
,,,
,,,,,,
,,,
,,,0000
0
0000
0 γ
γ
γ
γ
( )( )
( )
∂
∂
∂
∂+
z
tzyxC
TzyxP
tzyxC
zD
L
,,,
,,,
,,, 0000
0 γ
γ
;
( ) ( ) ( ) ( )+
∂
∂+
∂
∂+
∂
∂=
∂
∂2
02
2
02
02
2
02
02
2
0
02 ,,,,,,,,,,,,
z
tzyxCD
y
tzyxCD
x
tzyxCD
t
tzyxCLLL
( )( )
( )( )
( )( )
( )
×
∂
∂+
∂
∂
∂
∂+
−−
TzyxP
tzyxCtzyxC
yx
tzyxC
TzyxP
tzyxCtzyxC
xD
L,,,
,,,,,,
,,,
,,,
,,,,,,
1
00
01
00
1
00
010 γ
γ
γ
γ
( )( )
( )( )
( )+
∂
∂
∂
∂+
∂
∂×
−
z
tzyxC
TzyxP
tzyxCtzyxC
zy
tzyxC ,,,
,,,
,,,,,,
,,, 00
1
00
01
00
γ
γ
( )( )
( )( )
( ) ( )( )
×
∂
∂+
∂
∂
∂
∂+
∂
∂×
−
TzyxP
tzyxC
xD
z
tzyxC
TzyxP
tzyxCtzyxC
zy
tzyxCL
,,,
,,,,,,
,,,
,,,,,,
,,, 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
tzyxCD
y
tzyxCD
x
tzyxCD
t
tzyxCLLL
( ) ( )( )
( ) ( ) ( )( )
×
∂
∂+
∂
∂
∂
∂+
−−
TzyxP
tzyxCtzyxC
yx
tzyxC
TzyxP
tzyxCtzyxC
x ,,,
,,,,,,
,,,
,,,
,,,,,,
1
00
10
00
1
00
10 γ
γ
γ
γ
( ) ( ) ( )( )
( )+
∂
∂
∂
∂+
∂
∂×
−
LD
z
tzyxC
TzyxP
tzyxCtzyxC
zy
tzyxC0
00
1
00
10
00 ,,,
,,,
,,,,,,
,,,γ
γ
( )( )
( ) ( )( )
( )
+
∂
∂
∂
∂+
∂
∂
∂
∂+
y
tzyxC
TzyxP
tzyxC
yx
tzyxC
TzyxP
tzyxC
xD
L
,,,
,,,
,,,,,,
,,,
,,,10001000
0 γ
γ
γ
γ
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
21
( )( )
( ) ( ) ( )
+
∂
∂
∂
∂+
∂
∂
∂
∂+
x
tzyxCTzyxg
xD
z
tzyxC
TzyxP
tzyxC
zLL
,,,,,,
,,,
,,,
,,,01
0
1000
γ
γ
( ) ( ) ( ) ( )
∂
∂
∂
∂+
∂
∂
∂
∂+
z
tzyxCTzyxg
zy
tzyxCTzyxg
yLL
,,,,,,
,,,,,, 0101 ;
( )0
,,,
0
==x
ij
x
tzyxC
∂
∂,
( )0
,,,=
= xLx
ij
x
tzyxC
∂
∂,
( )0
,,,
0
==y
ij
y
tzyxC
∂
∂,
( )0
,,,=
= yLy
ij
y
tzyxC
∂
∂,
( )0
,,,
0
==z
ij
z
tzyxC
∂
∂,
( )0
,,,=
= zLz
ij
z
tzyxC
∂
∂, i ≥0, j ≥0;
C00(x,y,z,0)=fC (x,y,z), Cij(x,y,z,0)=0, i ≥1, j ≥1.
Functions Cij(x,y,z,t) (i ≥0, j ≥0) could be approximated by the following series during solutions
of the above equations
( ) ( ) ( ) ( ) ( )∑+=∞
=100
21,,,
nnCnnnnC
zyxzyx
tezcycxcFLLLLLL
tzyxC .
Here ( )
++−=
2220
22 111exp
zyx
CnCLLL
tDnte π , ( ) ( ) ( ) ( )∫ ∫ ∫=x y zL L L
nCnnnCudvdwdwcwvufvcucF
0 0 0
,, ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0 0 0 020
,,,2
,,,n
t L L L
LnnnCnCnnnnC
zyx
i
x y z
TwvugvcusetezcycxcFnLLL
tzyxC τπ
( )( )
( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−∂
∂×
∞
=
−
1 02
10 2,,,
n
t
nCnCnnnnC
zyx
i
netezcycxcFn
LLLdudvdwd
u
wvuCwc τ
πτ
τ
( ) ( ) ( ) ( )( )
( )∑ ×−∫ ∫ ∫∂
∂×
∞
=
−
12
0 0 0
10 2,,,,,,
nnCnC
zyx
L L Li
LnnnteFn
LLLdudvdwd
v
wvuCTwvugvcvsuc
x y z πτ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
∫ ∫ ∫ ∫∂
∂−× −
t L L Li
LnnnnCnnn
x y z
dudvdwdw
wvuCTwvugvsvcucezcycxc
0 0 0 0
10 ,,,,,, τ
ττ , i ≥1;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−=∞
=1 0 0 0 0201
2,,,
n
t L L L
nnnnCnCnnnnC
zyx
x y z
wcvcusetezcycxcFnLLL
tzyxC τπ
( )( )
( ) ( ) ( ) ( ) ( ) ×∑−∂
∂×
∞
=12
0000 2,,,
,,,
,,,
nnCnnnnC
zyx
tezcycxcFnLLL
dudvdwdu
wvuC
TwvuP
wvuC πτ
ττγ
γ
( ) ( ) ( ) ( )( )( )
( )( )×∑−∫ ∫ ∫ ∫
∂
∂−×
∞
=12
0 0 0 0
0000 2,,,
,,,
,,,
nnC
zyx
t L L L
nnnnCten
LLLdudvdwd
v
wvuC
TwvuP
wvuCwcvsuce
x y z πτ
τττ
γ
γ
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
22
( ) ( ) ( ) ( ) ( ) ( ) ( )( )( )
( )∫ ∫ ∫ ∫
∂
∂−×
t L L L
nnnnCnnnnC
x y z
dudvdwdw
wvuC
TwvuP
wvuCwsvcucezcycxcF
0 0 0 0
0000,,,
,,,
,,,τ
τττ
γ
γ
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0 0 0 0202
2,,,
n
t L L L
nnnnCnCnnnnC
zyx
x y z
wcvcusetezcycxcFnLLL
tzyxC τπ
( ) ( )( )
( ) ( ) ( )×∑−∂
∂×
∞
=
−
12
00
1
00
01
2,,,
,,,
,,,,,,
nnnnC
zyx
ycxcFLLL
dudvdwdu
wvuC
TwvuP
wvuCwvuC
πτ
τττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( )( )
( )×∫ ∫ ∫ ∫
∂
∂−×
−t L L L
nnnCnCn
x y z
v
wvuC
TwvuP
wvuCwvuCvsucetezcn
0 0 0 0
00
1
00
01
,,,
,,,
,,,,,,
ττττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−×∞
=1 0 0 02
2
n
t L L
nnnCnCnnnnC
zyx
n
x y
vcucetezcycxcFnLLL
dudvdwdwc τπ
τ
( ) ( ) ( )( )
( ) ( ) ×∑−∫∂
∂×
∞
=
−
12
0
00
1
00
01
2,,,
,,,
,,,,,,
nn
zyx
L
nxcn
LLLdudvdwd
w
wvuC
TwvuP
wvuCwvuCws
z πτ
τττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∫ ∫ ∫ ∫
∂
∂−×
t L L L
nnnnCnCnnnC
x y z
u
wvuCwvuCwcvcusetezcycF
0 0 0 0
00
01
,,,,,,
τττ
( )( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−×∞
=
−
1 0 02
1
00 2
,,,
,,,
n
t L
nnCnCnnnnC
zyx
x
ucetezcycxcFnLLL
dudvdwdTwvuP
wvuCτ
πτ
τγ
γ
( ) ( ) ( ) ( )( )
( )×∑−∫ ∫
∂
∂×
∞
=
−
12
0 0
00
1
00
01
2,,,
,,,
,,,,,,
nzyx
L L
nnn
LLLdudvdwd
v
wvuC
TwvuP
wvuCwvuCwcvs
y z πτ
τττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
×∫ ∫ ∫ ∫−×−t L L L
nnnnCnCnnnnC
x y z
TwvuP
wvuCwvuCwsvcucetezcycxcF
0 0 0 0
1
00
01,,,
,,,,,,
γ
γ τττ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂
∂×
∞
=1 0 02
00 2,,,
n
t L
nnCnCnnnnC
zyx
x
usetezcycxcFLLL
dudvdwdw
wvuCτ
πτ
τ
( ) ( ) ( )( )
( ) ( ) ( )∑ ×−∫ ∫∂
∂×
∞
=12
0 0
0100 2,,,
,,,
,,,
nnCn
zyx
L L
nntexc
LLLdudvdwd
u
wvuC
TwvuP
wvuCwcvcn
y z πτ
ττγ
γ
( ) ( ) ( ) ( ) ( ) ( )( )
( )×∫ ∫ ∫ ∫
∂
∂−×
t L L L
nnnnCnnC
x y z
dudvdwdv
wvuC
TwvuP
wvuCwcvsuceycF
0 0 0 0
0100 ,,,
,,,
,,,τ
τττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−×∞
=1 0 0 0 02
2
n
t L L L
nnnnCnCnnnnC
zyx
n
x y z
wsvcucetezcycxcFnLLL
zcn τπ
( )( )
( )τ
ττγ
γ
dudvdwdw
wvuC
TwvuP
wvuC
∂
∂×
,,,
,,,
,,,0100 ;
International Journal of Applied Control, Electrical and Electronics Engineering (IJACEEE) Vol 3, No.3, August 2015
23
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0 0 0 0211
2,,,
n
t L L L
nnnnCnCnnnnC
zyx
x y z
wcvcusetezcycxcFnLLL
tzyxC τπ
( )( )
( ) ( ) ( ) ( ) ×∑−∂
∂×
∞
=12
01 2,,,,,,
nnCnnnnC
zyx
LtezcycxcFn
LLLdudvdwd
u
wvuCTwvug
πτ
τ
( ) ( ) ( ) ( ) ( ) ( )×−∫ ∫ ∫ ∫
∂
∂−×
20 0 0 0
01 2,,,,,,
zyx
t L L L
LnnnnCLLL
dudvdwdv
wvuCTwvugwcvsuce
x y z πτ
ττ
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫
∂
∂−×
∞
=1 0 0 0 0
01 ,,,,,,
n
t L L L
LnnnnCnC
x y z
dudvdwdw
wvuCTwvugwsvcuceten τ
ττ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−×∞
=1 0 0 02
2
n
t L L
nnnCnCnnnnC
zyx
nnnnC
x y
vcusetezcycxcFLLL
zcycxcF τπ
( ) ( )( )
( ) ( ) ( )×∑−∫∂
∂×
∞
=12
0
1000 2,,,
,,,
,,,
nnnnC
zyx
L
nycxcFn
LLLdudvdwd
u
wvuC
TwvuP
wvuCwcn
z πτ
ττγ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( )( )
( )−∫ ∫ ∫ ∫
∂
∂−×
t L L L
nnnnCnCn
x y z
dudvdwdv
wvuC
TwvuP
wvuCwcvsucetezc
0 0 0 0
1000 ,,,
,,,
,,,τ
τττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
×∑ ∫ ∫ ∫ ∫−−∞
=1 0 0 0 0
00
2,,,
,,,2
n
t L L L
nnnnCnCnnnnC
zyx
x y z
TwvuP
wvuCwsvcucetezcycxcFn
LLLγ
γ ττ
π
( )( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−
∂
∂×
∞
=1 0 02
10 2,,,
n
t L
nnCnCnnnnC
zyx
x
usetezcycxcFnLLL
dudvdwdw
wvuCτ
πτ
τ
( ) ( ) ( ) ( )( )
( )×∑−∫ ∫
∂
∂×
∞
=
−
12
0 0
00
1
00
10
2,,,
,,,
,,,,,,
nzyx
L L
nnn
LLLdudvdwd
u
wvuC
TwvuP
wvuCwvuCwcvc
y z πτ
τττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
( )×∫ ∫ ∫ ∫
∂
∂−×
−t L L L
nnnnCnCnnnnC
x y z
v
wvuC
TwvuP
wvuCwcvsucetezcycxcF
0 0 0 0
00
1
00,,,
,,,
,,, τττ
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−×∞
=1 0 0210
2,,,
n
t L
nnCnCnnnnC
zyx
x
ucetezcycxcFnLLL
dudvdwdwvuC τπ
ττ
( ) ( ) ( ) ( )( )
( )∫ ∫
∂
∂×
−y zL L
nndudvdwd
w
wvuC
TwvuP
wvuCwvuCwsvc
0 0
00
1
00
10
,,,
,,,
,,,,,, τ
τττ
γ
γ
.