In this paper based on recently introduced approach we formulated some recommendations to optimize manufacture drift bipolar transistor to decrease their dimensions and to decrease local overheats during functioning. The approach based on manufacture a heterostructure, doping required parts of the heterostructure by dopant diffusion or by ion implantation and optimization of annealing of dopant and/or radiation defects. The optimization gives us possibility to increase homogeneity of distributions of concentrations of dopants in emitter and collector and specific inhomogenous of concentration of dopant in base and at the same time to increase sharpness of p-n-junctions, which have been manufactured framework the transistor. We obtain dependences of optimal annealing time on several parameters. We also introduced an analytical approach to model nonlinear physical processes (such as mass- and heat transport) in inhomogenous media with time-varying parameters.

1. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
AN APPROACH TO DECREASE DIMENSIONS OF DRIFT
HETERO-BOPOLAR TRANSISTORS
E.L.Pankratov1,3 and E.A.Bulaeva1,2
1Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950,
Russia
2Nizhny Novgorod State University of Architecture and Civil Engineering, 65 Il'insky
street, Nizhny Novgorod, 603950, Russia
3Nizhny Novgorod Academy of the Ministry of Internal Affairs of Russia, 3 Ankudi-novskoe
Shosse, Nizhny Novgorod, 603950, Russia
ABSTRACT
In this paper based on recently introduced approach we formulated some recommendations to optimize
manufacture drift bipolar transistor to decrease their dimensions and to decrease local overheats during
functioning. The approach based on manufacture a heterostructure, doping required parts of the hetero-structure
by dopant diffusion or by ion implantation and optimization of annealing of dopant and/or radia-tion
defects. The optimization gives us possibility to increase homogeneity of distributions of concentrations
of dopants in emitter and collector and specific inhomogenous of concentration of dopant in base and at the
same time to increase sharpness of p-n-junctions, which have been manufactured framework the transistor.
We obtain dependences of optimal annealing time on several parameters. We also introduced an analytical
approach to model nonlinear physical processes (such as mass- and heat transport) in inhomogenous me-dia
with time-varying parameters.
KEYWORDS
Drift heterobipolar transistor, analytical approach to model technological process, decreasing of dimen-sions
of transistor
1.INTRODUCTION
In the present time performance of elements of integrated circuits (p-n-junctions, field-effect and
bipolar transistors, ...) and their discrete analogs are intensively increasing [1-14]. To solve the
problem they are using several ways. One of them is manufacturing new materials with higher
speed of charge carriers [1-18]. Another way to increase the performance is elaboration of new
technological processes or modification of existing one [1-14,19,20]. In this paper we introduce
one of approaches of modification of technological to increase performance of bipolar transistor.
To solve our aim we consider hetero structure, which consist of a substrate and three epitaxial
layers (see Fig. 1). One section have been manufactured in every epitaxial layer by using another
materials so as it is presented on Fig. 1. After manufacturing of the section in the first epitaxial
layer the section has been doped by diffusion or ion implantation to produce required type of
conductivity (p or n) in the section. Farther we consider annealing of dopant and/or radiation de-
DOI:10.5121/ijcsa.2014.4503 25

2. International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
fects. After that we consider manufacturing of the second and the third epitaxial layers, which
also including into itself one section in each new epitaxial layer. The sections are also been manu-factured
by using another materials. Both new sections have been doped by diffusion or ion im-plantation
to produce required type of conductivity (p or n) in the sections. Farther we consider
microwave annealing of dopant and/or radiation defects. Main aim of the paper is analysis of do-pand
26
and radiation defects in the considered heterostructure.
Dopant 1 Dopant 2 Dopant 3
Substrate
Epitaxial layers
Fig. 1. Heterostructure, which consist of a substrate and three epitaxial layers with sections, manufactured
by using another materials. View from side
2. Method of solution
To solve our aims we determine spatio-temporal distribution of concentration of dopant.
We determine the required distribution by solving the second Fick's law [1,3-5]
( ) ( ) ( ) ( )
¶ , , , , , , , , , , , ,
+ +
+
=
C x y z t
z
D
C x y z t
¶
y z
D
C x y z t
¶
x y
D
C x y z t
¶
t x
¶
C C ¶
¶
C ¶
¶
¶
¶
¶
¶
¶
(1)
with boundary and initial conditions
( )
0
, , ,
0
=
C x y z t
¶
¶
x=
x
,
( )
0
, , ,
=
C x y z t
¶
¶
x=Lx
x
,
( )
0
, , ,
0
=
C x y z t
¶
¶
y=
y
,
( )
0
, , ,
=
C x y z t
¶
¶
x=Ly
y
,
( )
0
, , ,
0
=
C x y z t
¶
¶
z=
z
,
( )
0
, , ,
=
C x y z t
¶
¶
x=Lz
z
, C (x,y,z,0)=f (x,y,z). (2)
Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant, T is the temperature
of annealing, D is the dopant diffusion coefficient. Value of dopant diffusion coefficient depends
on properties of materials of the considered hetero structure, speed of heating and cooling of hete-ro
structure (with account Arrhenius law). Dependences of dopant diffusion coefficient could be
approximated by the following relation [3,21]

3. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
27
( ) ( )
, , , , , ,
, , ,
V x y z t
V x y z t
g
C x y z t
D D x y z T C L x V V g
= + * 2
( )
( ) ( )
( )
+ +
2
1 * 2
1
, , ,
, , , 1
V
V
P x y z T
, (3)
where DL (x,y,z,T) is the spatial (due to inhomogeneity of hetero structure) and temperature (due to
Arrhenius law) dependences of diffusion coefficient; P (x,y,z,T) is the limit of solubility of do-pant;
value of parameter g depends on materials of heterostructure and could be integer in the fol-lowing
interval g Î[1,3] [3]; V (x,y,z,t) is the spatio-temporal distribution of concentration of va-cancies;
V* is the equilibrium distribution of concentration of vacancies. Concentrational depen-dence
of dopant diffusion coefficient has been discussed in details in [3]. It should be noted, that
using diffusion type of doping and radiation damage is absent in the case (i.e. z1= z2= 0). We de-termine
spatio-temporal distributions of concentrations of point radiation defects by solving of the
following system of equations [21,22]
( ) ( ) ( ) ( ) ( )
+
I x y z t
¶
¶
¶
¶
+
I x y z t
¶
¶
¶
¶
=
I x y z t
¶
¶
y
D x y z T
x y
D x y z T
t x
I I
, , ,
, , ,
, , ,
, , ,
, , ,
( ) ( ) − ( ) ( ) ( )−
I x y z t
+ k x y z T I x y z t V x y z t
z I I V , , , , , , , , ,
¶
¶
¶
¶
z
D x y z T
, , ,
, , , ,
k (x y z T )I (x y z t ) I I , , , , , , 2
, − (4)
( ) ( ) ( ) ( ) ( )
+
V x y z t
¶
¶
¶
¶
+
V x y z t
¶
¶
¶
¶
=
V x y z t
¶
¶
y
D x y z T
x y
D x y z T
t x
V V
, , ,
, , ,
, , ,
, , ,
, , ,
( ) ( ) − ( ) ( ) ( )−
V x y z t
+ k x y z T I x y z t V x y z t
z V I V , , , , , , , , ,
¶
¶
¶
¶
z
D x y z T
, , ,
, , , ,
k (x y z T )V (x y z t ) V V , , , , , , 2
, −
with initial
r (x,y,z,0)=fr (x,y,z) (5a)
and boundary conditions
( )
0
r x , y , z ,
t
0
=
¶
¶
x=
x
,
( )
0
r x , y , z ,
t
=
¶
¶
x=Lx
x
,
( )
0
r x , y , z ,
t
0
=
¶
¶
y=
y
,
( )
0
r x , y , z ,
t
=
¶
¶
y=Ly
y
,
( )
0
r x , y , z ,
t
0
=
¶
¶
z=
z
,
( )
0
r x , y , z ,
t
=
¶
¶
z=Lz
z
. (5b)
Here r =I,V; I (x,y,z,t) is the spatio-temporal distribution of concentrations of interstitials; Dr(x,
y,z,T) are the diffusion coefficients of interstitials and vacancies; terms V2(x,y,z,t) and I2(x,y,z,t)
correspond to generation of divacancies and diinterstitials; kI,V(x,y,z,T), kI,I(x,y,z,T) and kV,V(x,y,
z,T) are the parameters of recombination of point radiation defects and generation appropriate
their complexes, respectively.
We determine spatio-temporal distributions of concentrations of divacancies FV (x, y,z,t) and diin-terstitials
FI
(x,y,z,t) by solving the following system of equations [21,22]

4. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
x y z t r ,
+ l (8)
28
( )
F
¶ , , ,
¶
¶
x y z t I
( )
( )
F
( )
( )
+
+
=
F
x y z t
D x y z T
x y z t
F x y
F y
D x y z T
t x
I
I
I
I
¶
¶
¶
¶
¶
¶
¶
, , ,
, , ,
, , ,
, , ,
F
( )
( )
( ) ( ) − +
, , ,
x y z t
¶
, , , I
2
¶
+ F k x y z T I x y z t
I , , , , , ,
¶
z
, D x y z T
z I I
¶
k (x y z T )I (x y z t ) I − , , , , , , (6)
( )
F
¶ , , ,
¶
¶
x y z t V
( )
( )
F
( )
( )
+
+
=
F
x y z t
D x y z T
x y z t
F x y
F y
D x y z T
t x
V
V
V
V
¶
¶
¶
¶
¶
¶
¶
, , ,
, , ,
, , ,
, , ,
F
( )
( )
( ) ( )− +
, , ,
x y z t
¶
, , , V
2
¶
+ F k x y z T V x y z t
V , , , , , ,
¶
z
, D x y z T
z V V
¶
k (x y z T )V (x y z t ) V − , , , , , ,
with boundary and initial conditions
( )
x y z t r 0
,
, , ,
0
=
¶
¶F
x=
x
( )
x y z t r ,
0
, , ,
=
¶
¶F
x=Lx
x
( )
x y z t r 0
,
, , ,
0
=
¶
¶F
y=
y
( )
0
, , ,
=
¶
¶F
y=Ly
y
( )
x y z t r 0
,
, , ,
0
=
¶
¶F
z=
z
( )
I
Fx y z t r 0
, , , ,
=
¶
¶F
z=Lz
z
(x,y,z,0)=fFI (x,y,z), FV (x,y,z,0)=fFV (x,y,z). (7)
Here DFI(x,y,z,T) and DFV(x,y,z,T) are the diffusion coefficients of simplest complexes of radia-tion
defects; kI(x,y,z,T) and kV (x,y,z,T) are the parameters of decay of simplest complexes of radi-ation
defects.
We described distribution of temperature by the second law of Fourier [23]
( ) ( ) ( ) ( ) ( ) ( )
+
T x y z t
¶
¶
¶
¶
+
T x y z t
¶
¶
¶
¶
=
T x y z t
¶
¶
y
x y z T
x y
x y z T
t x
c T
, , ,
, , ,
, , ,
, , ,
, , ,
l l
( ) T ( x y z t
) p(x y z t )
z
x y z T
¶
z
, , ,
, , ,
, , , +
¶
¶
¶
with boundary and initial conditions
( )
0
, , ,
0
=
T x y z t
¶
¶
x=
x
,
( )
0
, , ,
=
T x y z t
¶
¶
x=Lx
x
,
( )
0
, , ,
0
=
T x y z t
¶
¶
y=
y
, (9)
( )
0
, , ,
=
T x y z t
¶
¶
x=Ly
y
,
( )
0
, , ,
0
=
T x y z t
¶
¶
z=
z
,
( )
0
, , ,
=
T x y z t
¶
¶
x=Lz
z
, T (x,y,z,0)=fT (x,y,z),
where T(x,y,z,t) is the spatio-temporal distribution of temperature; c (T)=cass[1-h exp(-T(x,y,z,t)/
Td)] is the heat capacitance (in the most interesting case, when temperature of annealing is ap-proximately
equal or larger, than Debay temperature Td, one can assume c (T)»cass [23]); l is the
heat conduction coefficient, which depends on properties of materials and current temperature of
annealing; temperature dependence of heat conduction coefficient in the most interesting tem-perature
interval could be approximated by the following function l(x,y,z,T)=lass(x,y,z) [1+μ
(Td/T(x,y,z,t))j] (see, for example, [23]); p(x,y,z,t) is the volumetric density of heat power, gener-

5. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
ated in heterostructure during annealing; a (x,y,z,T)=l(x,y,z,T)/c (T) is the heat diffusivity. First of
all we determine spatio-temporal distribution of temperature. To calculate the distribution of tem-perature
we used recently introduced approach [24-26]. Framework the approach we transform
approximation of heat diffusivity to the following form: a ass (x,y,z) =lass(x,y,z)/cas s=a0ass[1+eT
gT(x,y,z)]. Farther we determine solution of Eq.(8) as the following power series
i j
T T x y z t e μ T x y z t . (10)
29
¥
( , , , )= ( , , ,
)
=
¥
0 =0
i j
ij
Substitution of the series into Eq.(8) gives us possibility to obtain system of equations for the ini-tial-
order approximation of temperature T00(x,y,z,t) and corrections for them Tij(x,y,z,t) (i³1, j³1).
The equations are presented in the Appendix. Substitution of the series (9) into boundary and ini-tial
conditions for temperature gives us possibility to obtain the same conditions for all functions
Tij(x,y,z,t) (i³0, j³0). The conditions are presented in the Appendix. The equations for the func-tions
Tij(x,y,z,t) (i³0, j³0) with account boundary and initial conditions have been solved by using
standard approaches [27,28] for the second-order approximation of the temperature T (x,y,z,t) on
the parameters e and μ. The solutions are presented in the Appendix. The second- order is usually
enough good approximation to make qualitative analysis and to obtain some quantitative results
(see, for example, [24-26]). Analytical results give us possibility to make more demonstrative
analysis in comparison with numerical one. To calculate the obtained result with higher exactness
and checking the obtain results by independent approaches we used numerical approaches.
To calculate spatio-temporal distributions of concentrations of point of radiation defects we used
recently introduced approach [24-26] and transform approximations of diffusion coefficients in
the following form: Dr(x,y,z,T)=D0r[1+ergr(x,y,z,T)], where D0r are the average values of diffu-sion
coefficients, 0£er 1, |gr(x,y,z,T)|£1, r =I,V. The same transformations have been used for
approximations of parameters of recombination of point radiation defects and generation of their
complexes: kI,V(x,y,z,T)=k0I,V [1+ eI,V gI,V(x,y,z,T)], kI,I(x,y,z,T)=k0I,I[1+ eI,I gI,I(x,y,z,T)] and
kV,V(x,y,z,T) = k0V,V [1+eV,V gV,V(x,y,z,T)], where k0r1,r2 are the appropriate average values of these
parameters, 0£ eI,V 1, 0£ eI,I 1, 0£eV,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: ( ) ( ) * , , , , , ,
~
I x y z t = I x y z t I , c = x/Lx, h = y/Ly, f
~
V x y z t =V x y z t V , I V I V L k D D0 , 0 0
= z/Lz, ( ) ( ) * , , , , , ,
w = 2 , L 2
k D D0 , 0 I 0
V r r r W = ,
2
0 0 D D t L I V J = . The introduction leads to the following transformation of equations (4) and
conditions (5)
( ) [ ( )] ( )
I
c h f J
I D
0
+ ×

6. ¶
¶
0 , , ,
+
¶
¶
c h f J
I D
=
¶
¶
I V
I I
I
I V
D D
g T
D D
0 0
0 0
~
1 , , ,
, , ,
~
c
e c h f
J c
[ ( )] I ( ) D
¶
0 I
{[ + e ( c h f
)] ×
× g T
g T 1 , , ,
I I I I
h ¶
f
+

7. ¶
¶
+
¶
¶
D D
I V
, , ,
~
1 , , ,
0 0
c h f J
e c h f
h
( c h f J
) )− [ + ( )] ( ) ( −
¶
× w e c h f c h f J c h f J
f
¶
, , ,
~
, , ,
~
1 , , ,
, , ,
~
, , g T I V
I
I V I V
~
[ 1 e (c , h , f , )] 2
(c ,h ,f ,J )
, , g T I I I I I I −W + (11)
( ) [ ( )] ( )
0
V
c h f J
V D
+ ×

8. 0 , , ,
¶
¶
+
¶
¶
c h f J
V D
=
¶
¶
I V
V V
V
I V
D D
g T
D D
0 0
0 0
~
1 , , ,
, , ,
~
c
e c h f
J c

9. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
r c h f J r f
= . (12)
i j k r c h f J e w r c h f J r r . (13)
~
000 I and
30
[ ( )] V ( ) D
¶
0 V
{[ + e ( c h f
)]×
+ g T
g T 1 , , ,
V V V V
h ¶
f
+

10. ¶
¶
+
¶
¶
D D
I V
, , ,
~
1 , , ,
0 0
c h f J
e c h f
h
( c h f J
) )− [ + ( )] ( ) ( −
¶
× w e c h f c h f J c h f J
f
¶
, , ,
~
, , ,
~
1 , , ,
, , ,
~
, , g T I V
V
I V I V
~
[ 1 e (c , h , f , )] 2
(c ,h,f ,J )
, , g T V I V V V V −W +
( )
0
~ , , ,
0
=
r c h f J
¶
¶
c = c
,
( )
0
~ , , ,
1
=
r c h f J
¶
¶
c = c
,
( )
0
~ , , ,
0
=
r c h f J
¶
¶
h = h
,
( )
0
~ , , ,
1
=
r c h f J
¶
¶
h = h
,
( )
0
~ , , ,
0
=
r c h f J
¶
¶
f = f
,
( )
0
~ , , ,
1
=
r c h f J
¶
¶
f = f
, ( )
( c , h , f ,
J
)
*
~ , , ,
r
We determine solutions of Eqs.(11) as the following power series (see [24-26])
¥
~ , , , ~ , , ,
( ) = W ( )
=
¥
=
¥
0 0 =0
i j k
ijk
Substitution of the series (13) into Eqs. (11) and conditions (12) gives us possibility to obtain eq-uations
for initial-order approximations of concentrations of point defects (c,h,f,J)
~
V (c,h,f,J)
and corrections for them (c,h,f,J)
000 ~
ijk I and (c ,h,f,J)
~
ijk V , i ³1, j ³1, k ³1. The equa-tions
and conditions for them are presented in the Appendix. The equations have been solved by
standard approaches (see, for example Fourier approach, [27,28]). The equations are presented in
the Appendix.
Farther we determine spatio-temporal distributions of concentrations of complexes of radiation
defects. First of all we transform approximations of diffusion coefficients into the following form:
DFr(x,y,z,T)=D0Fr[1+eFrgFr(x,y,z,T)], where D0Fr are the average values of diffusion coefficients.
In this situation Eqs.(6) will be transformed to the following form

11. ( ) ¶
[ ( )] ( ) ¶
[ ( )]
I 1 , , ,
+ + ×

12. F
= +
F
x y z t
F F F F F g x y z T
x y
g x y z T
x
D
x y z t
t
I I
I
I I I
, , ,
1 , , ,
, , ,
0 e
¶
¶
¶
e
¶
¶
¶
( )
[ ( )] ( )
+

13. ¶ , , ,
F
¶
x y z t I
+ +
F
× F F F F
x y z t
z
g x y z T
z
D D
y
I I I I
I
¶
¶
e
¶
¶
1 , , ,
, , ,
0 0
[ g (x y z T )] k (x y z T ) I (x y z t ) k (x y z T )I (x y z t ) I I I I I 1 , , , , , , , , , , , , , , , 2
, + −
× + F F e
( )

14. [ )] ( )
( ¶
[ ( )]
¶
V 1 , , ,
+ + ×

15. F
= +
F
x y z t
F F F F F g x y z T
x y
g x y z T
x
D
x y z t
t
V V
V
V V V
, , ,
1 , , ,
, , ,
0 e
¶
¶
¶
e
¶
¶
¶
( )
[ ( )] ( )
+

16. F
¶ , , ,
¶
x y z t V
+ +
F
x y z t
× D D
g x y z T
y
F V F V z
F V F V
z
V
¶
¶
e
¶
¶
1 , , ,
, , ,
0 0
[ g (x y z T )] k (x y z T ) V (x y z t ) k (x y z T )V (x y z t ) V V V V V 1 , , , , , , , , , , , , , , , 2
, + −
× + F F e .

17. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
i x y z t x y z t r r r e . (11)
31
Farther we determine solutions of the above equations as the following power series
¥
F ( , , , ) = F ( , , ,
)
=
F
0
i
i
Substitution of the series (14) into Eqs. (6) and appropriate boundary and initial conditions gives
us possibility to obtain equations for initial-order approximations of concentrations of complexes
of radiation defects Fr0(x,y,z,t) and corrections for them Fri(x,y,z,t) (i ³1) and appropriate condi-tions
for all functions Fri(x,y,z,t) (i ³0). The equations and conditions are presented in the Appen-dix.
The obtained equations have been solved by standard approaches (see, for example, [27,28])
with account boundary and initial conditions. The solutions are presented in the Appendix.
We calculate spatio-temporal distribution of dopant concentration by using the same approach,
which have been used for calculation spatio-temporal distribution of concentrations of radiation
defects. In this situation we transform approximation of dopant diffusion coefficient to the fol-lowing
form: DL(x,y,z,T)=D0L[1+eLgL(x,y,z,T)], where D0L is the average value of dopant diffusion
coefficient, 0£eL 1, |gL(x,y,z,T)|£1. Farther we determine solution of Eq.(1) in the following form
¥
( ) = ( )
C x , y , z , t e i x j
C x , y , z ,
t .
L =
¥
0 =1
i j
ij
Substitution of the series into Eq.(1) and conditions (2) gives us possibility to obtain equation for
initial-order approximation of the concentration of dopant C00(x,y,z,t) and corrections for them
Cij(x,y,z,t) (i ³1, j ³1) and boundary and initial conditions for them. The equations and conditions
are presented in the Appendix. The solutions have been calculated by standard approaches (see,
for example, [27,28]). The solutions are presented in the Appendix.
Analysis of spatio-temporal distributions of concentrations of dopant and radiation defects have
been done analytically by using the second-order approximations on all parameters, which are
used in appropriate series. The approximation is usually enough good approximation to make qu-alitative
analysis and to obtain some quantitative results. Results of analytical calculations have
been checked with comparison with numerical one.
3.Discussion
In this section we analyzed redistribution of dopant and radiation defects by using relations, cal-culated
in the previous section. Typical distributions of concentrations of dopant near interface
between materials of hetero structure are presented on Figs. 2 and 3 for diffusion and ion types of
doping, respectively. The distributions have been calculated for the case, when value of dopant
diffusion coefficient in doped area is larger, than value of dopant diffusion coefficient in nearest
areas. The figures show, that presents of interface between materials gives us possibility to in-crease
sharpness of p-n-junctions, which included into the considered heterobipolar transistor. At
the same time homogeneity of distribution of concentration of dopant increases. Increasing of
sharpness of p-n-junctions gives us possibility to decrease their switching time. Increasing of ho-mogeneity
of distribution of concentration of dopant gives us possibility to decrease value of lo-cal
overheats during functioning of the p-n-junctions or to decrease dimensions of p-n-junctions
with fixed maximal value of the overheats. To accelerate transport of charge carriers it is attracted
an interest inhomogenous distribution of dopant in base. In this case it is electrical field has been
generated in the base. This electrical field gives us possibility to accelerate transport of charge
carriers in base of transistors. To manufacture in homogenous distribution of dopant in base it is
practicably to dope required area (section) of the first (nearest to the substrate) epitaxial layer.
After that it is practicably to anneal dopant and/or radiation defects. Farther they are attracted an

18. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
interest the following steps: (i) manufacturing of the second epitaxial layer with section, manufac-tured
by using another materials; (ii) doping the section of the second epitaxial layer by diffusion
or ion implantation; (iii) manufacturing of the third epitaxial layer with section, manufactured by
using another materials; (iv) doping the section of the third epitaxial layer by diffusion or ion im-plantation.
After that we consider microwave annealing of dopant and/or radiation defects. Ad-vantage
of the approach of annealing is formation of inhomogenous distribution of temperature.
In this situation it is practicably to choose parameters of annealing so, that thickness of scin-layer
became larger, than thickness of the third (external) epitaxial layer and smaller, than total of
thickness of the third and the second epitaxial layers. In this case dopant diffusion in nearest to
the substrate side became slower, than in farther side. This is a reason to inhomogeneity of distri-bution
of concentration of dopant in depth of hetero structure. After finishing of manufacturing of
bipolar transistor the section of the average epitaxial layer with inhomogenous distribution of
concentration of dopant assumes function of base.
Fig. 2. Distributions of concentrations of infused dopant in hetero structure from Fig. 1 in direc-tion,
which is perpendicular to interface between layers of heterostructure. Increasing of number
of curves corresponds to increasing of difference between values of dopant diffusion coefficient
in layers of heterostructure. The curves have been calculated under condition, when dopant diffu-sion
32
coefficient in doped layer is larger, than in nearest layer.
Epitaxial layer Substrate
x
2.0
1.5
1.0
0.5
0.0
C(x,Q)
2
3
4
1
0 L/4 L/2 3L/4 L

19. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
Fig. 3. Spatial distributions of implanted dopant concentration after annealing with continuous Q
= 0.0048(Lx
33
2+Ly
2+Lz
2)/D0 (curves 1 and 3) and Q = 0.0057(Lx
2+Ly
2+Lz
2)/ D0 (curves 2 and 4).
Curves 1 and 2 are calculated distributions of dopant concentration in homogenous structure.
Curves 3 and 4 are calculated distributions of dopant concentration in hetero structure under con-dition,
when dopant diffusion coefficient in doped layer is larger, than in nearest layer.
Using of the considered approach to manufacture of transistors leads to necessity of optimization
of annealing time. To optimize the annealing time we used recently introduced criterion
[24,26,29-33]. Framework the approach we approximate real distributions of concentration of
dopant by step-wise function. Farther we determine the required optimal values of annealing time
by minimization of the following mean- squared error
Lx y z L L
= [ ( Q)− ( )]
x y z
C x y z x y z d z d y d x
L L L
U
0 0 0
, , , , ,
1
y , (15)
where y (x) is the approximation function, Q is the required value of annealing time.
0.0 0.1 0.2 0.3 0.4 0.5
a/L, x, e, g
0.5
0.4
0.3
0.2
0.1
0.0
Q D0 L-2
3
2
4
1
Fig.4. Dependences of dimensionless optimal annealing time for doping by diffusion, which have
been obtained by minimization of mean-squared error, on several parameters. Curve 1 is the de-pendence
of dimensionless optimal annealing time on the relation a/L and x = g = 0 for equal to
each other values of dopant diffusion coefficient in all parts of hetero structure. Curve 2 is the
dependence of dimensionless optimal annealing time on value of parameter e for a/L=1/2 and x =
g = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of parameter x
for a/L=1/2 and e = g = 0. Curve 4 is the dependence of dimensionless optimal annealing time on
value of parameter g for a/L=1/2 and e = x = 0

20. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
34
0.0 0.1 0.2 0.3 0.4 0.5
a/L, x, e, g
0.12
0.08
0.04
0.00
Q D0 L-2
3
2
4
1
Fig.5. Dependences of dimensionless optimal annealing time for doping by ion implantation,
which have been obtained by minimization of mean-squared error, on several parameters. Curve 1
is the dependence of dimensionless optimal annealing time on the relation a/L and x = g = 0 for
equal to each other values of dopant diffusion coefficient in all parts of hetero structure. Curve 2
is the dependence of dimensionless optimal annealing time on value of parameter e for a/L=1/2
and x = g = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of
parameter x for a/L=1/2 and e = g = 0. Curve 4 is the dependence of dimensionless optimal an-nealing
time on value of parameter g for a/L=1/2 and e = x = 0
Dependences of optimal values of annealing time are presented in Fig. 4 for diffusion type of
doping. Using ion implantation leads to necessity of annealing of radiation defects. In the ideal
case after finishing of annealing of radiation defects dopant achieves interface between materials
of hetero structure. If the dopant did not achieves the interface during the annealing, it is practica-bly
to use additional annealing of dopant. Dependences of optimal values of additional annealing
time are presented in Fig. 5. Optimal value of time of additional annealing of implanted dopant is
smaller, than in optimal value of infused dopant. Reason of this difference is necessity of anneal-ing
of radiation defects.
4. CONCLUSIONS
In this paper we introduce an approach to manufacture a heterobipolar transistor with inhomo-genous
doping of base. At the same time the introduced approach to manufacture of bipolar tran-sistors
gives us possibility to increase their compactness and to increase sharpness of p-n-junctions,
which included into the transistor. The approach based on manufacturing of a hetero-structure
with special construction, doping of special areas of the hetero structure and optimiza-tion
of annealing of dopant and/or radiation defects.
ACKNOWLEDGEMENTS
This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government,
grant of Scientific School of Russia, the agreement of August 27, 2013 02..49.21.0003 be-tween
The Ministry of education and science of the Russian Federation and Lobachevsky State
University of Nizhni Novgorod and educational fellowship for scientific research of Nizhny Nov-gorod
State University of Architecture and Civil Engineering.

21. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
35
APPENDIX
Equations for the functions Tij(x,y,z,t) (i³0, j³0) have been obtained by substitution the power se-ries
(10) in the equation (8) and equating terms with equal powers of parameters eT and μ. The
equations could be written as
( , , , )
( , , , ) ( , , , ) ( , , , ) ( , , ,
)
ass
00 ass
+ p x y z t
T x y z t
z
T x y z t
y
T x y z t
x
T x y z t
t
n
a
2
00
2
2
00
2
2
00
2
0
¶
¶
+
¶
¶
+
¶
¶
=
¶
¶
( ) ( ) ( ) ( )
a a
× +
0 , , , , , , , , , , , ,
T x y z t
¶
¶
+
T x y z t
¶
¶
+
T x y z t
¶
¶
=
T x y z t
¶
¶
ass
i i i
ass
i
z
y
x
t
2 0
0
2
2
0
2
2
0
2
0
( ) ( ) ( ) ( )

22. +
2 , , ,
T x y z t
¶
¶
g x y z T i
+
T x y z t
¶
¶
× − −
2
10
2
2
10
, , ,
, , ,
, , ,
y
g x y z T
x
T
i
T
( ) ( )
T x y z t
+ −
¶
¶
¶
¶
z
g x y z T
z
i
T
, , ,
, , , 10 , i ³1
( , , , ) ( , , , ) ( , , , ) ( , , ,
)
j a
T x y z t T
ass d
+ × ( )
T x y z t
¶
¶
+
T x y z t
¶
¶
+
T x y z t
¶
¶
=
¶
¶
T x y z t
z
y
x
t
ass , , ,
00
0
2
01
2
2
01
2
2
01
2
0
01
j
a
( ) ( ) ( )
j ja
2 , , ,
T x y z t T
ass d
( )
( )

23. +
T x y z t
¶
¶
−
T x y z t
¶
¶
+
T x y z t
¶
¶
+
¶
¶
× +
2
00
1
00
0
2
00
2
2
00
2
2
00
, , ,
, , , , , , , , ,
x
T x y z t
z
y
x
j
( ) ( )
T x y z t
¶
¶
+
T x y z t
¶
¶
+
2
00
2
00 , , , , , ,
z
y
( , , , ) ( , , , ) ( , , , ) ( , , ,
)
j a
T x y z t T
ass d
+ × ( )
T x y z t
¶
¶
+
T x y z t
¶
¶
+
T x y z t
¶
¶
=
¶
¶
T x y z t
z
y
x
t
ass , , ,
00
0
2
02
2
2
02
2
2
02
2
0
02
j
a
( ) ( ) ( )
j ja
T x y z t ass d , , , , , ,
, , , , , , , , , T
00 01
( )
( ) ( )
+
T x y z t
¶
¶
T x y z t
¶
¶
−
T x y z t
¶
¶
+
T x y z t
¶
¶
+
¶
¶
× x
y
z
T + x y z t
x
x
, , ,
1
00
0
2
01
2
2
01
2
2
01
2
j
( ) ( ) ( ) ( )
T x, y, z, t , , , , , , , , , 00 01 00 01
T x y z t
¶
¶
T x y z t
¶
¶
+
T x y z t
¶
¶
¶
¶
+
z
z
y
y
( ) ( ) ( )
, , , , , , T x , y , z ,
t
, , , , , ,
11 ( )
( )
( ) ( )

24. ×
T x y z t
¶
¶
+
T x y z t
¶
¶
+
T x y z t
¶
¶
=
T x y z t
¶
¶
2
00
2
2
00
2
01
00
2 0
11
2
0
, , ,
, , ,
y
x
g x y z T
T x y z t
x
t
ass ass T a a
¶
( ) ( )
( )
( )
( )

25. +
00 a
g x y z T T T ass T
T x y z t
¶
¶
¶
¶
+
T x y z t
¶
¶
¶
× +
x
g x y z T
z x
g x y z T
z
, , ,
, , ,
, , ,
, , , , , , 01
0
( )
[ ( )] ( )
[ ( )] ( )
+
2 , , ,
¶
+ 2
T x y z t
¶
¶
+ +
T x y z t
¶
¶
+ +
T x y z t
¶
01
2
2
01
2
2
01
1 , , ,
, , ,
1 , , ,
, , ,
z
g x y z T
y
g x y z T
x
T T
( )
( )
( , , , ) T ( x , y , z ,
t
)
, , ,
10
10
10
Tass d , , ,
( )
( ) ( )
T x y z t
+ ×
T x y z t
T x y z t
T x y z t
( )
0 j j j
¶
¶
+
¶
¶
+ T + x y z t
x
T + x y z t
y
T + x y z t
, , , , , ,
, , ,
, , ,
1
00
2
00
2
1
00
2
00
2
1
00
j a

26. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
36
( )
j a
2 , , , , , , , , ,
¶
T x y z t T
× ass d
2
( )
( ) ( ) ( )
+
T x y z t
¶
¶
+
T x y z t
¶
¶
+
T x y z t
¶
¶
+
¶
10
2
2
10
2
2
10
2
00
0
2
00
, , ,
, , ,
z
y
x
T x y z t
z
j
( )
( )
( )
( )
( )

27. × +
g x y z T T T T , , ,
T x y z t
¶
¶
¶
¶
+
T x y z t
¶
¶
+ g x y z T
z
g x y z T
x z
, , ,
, , ,
, , ,
, , , 00
2
10
2
j a
Tass d , , , , , , , , , , , ,
, , ,
( )
( )
( ) ( ) ( )
×
T x y z t
¶
¶
+
T x y z t
¶
¶
T x y z t
¶
¶
−
T x y z t
¶
¶
×
y
x
x
y
T x y z t
10 00 10
2
10
2
00
0
j
( ) ( ) ( )
¶
T x y z t T
× 00 10 00 0
T
+ T + x y z t
( )
( )
( )
, , ,
× −
T x y z t
¶
¶
T x y z t
¶
¶
+
¶
g x y z T
T
T x y z t
z
z
y
ass d
ass d
, , ,
, , ,
, , , , , , , , ,
1
00
1 0
00
j
j
j
j
a j
ja
( ) ( ) ( )

28. T x y z t
¶
¶
+
T x y z t
¶
¶
+
T x y z t
¶
¶
×
2
00
2
00
2
00 , , , , , , , , ,
z
y
x
.
Conditions for the functions Tij(x,y,z,t) (i³0, j³0) have been obtained by the same procedure as
appropriate equations and could be written as
( )
0
, , ,
0
=
T x y z t
¶
¶
x=
ij
x
,
( )
0
, , ,
=
T x y z t
¶
¶
x=Lx
ij
x
,
( )
0
, , ,
0
=
T x y z t
¶
¶
y=
ij
y
,
( )
0
, , ,
=
T x y z t
¶
¶
x=Ly
ij
y
,
( )
0
, , ,
0
=
T x y z t
¶
¶
z=
ij
z
,
( )
0
, , ,
=
T x y z t
¶
¶
x=Lz
ij
z
, T00(x,y,z,0)=fT(x,y,z), Tij(x,y,z,0)=0, i ³1, j ³1.
Solutions of the equations for the functions Tij(x,y,z,t) (i³0, j³0) with account boundary and initial
conditions have been obtained by standard Fourier approach. By using the approach one can ob-tain
the functions Tij(x,y,z,t) in the following form
¥
( ) = ( ) + ( ) ( ) ( ) ( ) ( ) ×
0 0 0 =1 0
00
2
, ,
1
, , ,
n
L
n n n nT n
x y z
L L L
T
x y z
x y z x
c x c y c z e t c u
L L L
f u v w d wd v d u
L L L
T x y z t
( ) ( ) ( ) ( )
y z 1 x y z p u , v , w
, 2
× + + ×
x y z
t L L L
x y z ass
L L
d wd v d u d
, ,
n n T L L L
L L L
c v c w f u v w d wd v d u
0 0 0 0 0 0
t
n
t
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
¥
x y z
× −
=1 0 0 0 0
, , ,
n
t L L L
ass
n n n nT nT n n n
d wd v d u d
p u v w
c x c y c z e t e c u c v c w t
n
t
t ,
where cn(c) = cos (p nc/L), ( )
e t p n a t ;
2 2 1 1 1
= − + + 0 2 2 2
nT ass L L L
exp
x y z
¥
p a
( ) = ( ) ( ) ( ) ( ) (− ) ( ) ( ) ( ) ( )×
T x y z t t
=1 0 0 0 0
2
0
ass
0 , , , 2 , , ,
n
t L L L
n n n nT nT n n n T
x y z
i
x y z
n c x c y c z e t e s u c v c w g u v w T
L L L
( )
+ ( ) ( ) ( ) ( ) (− ) ( ) ( )×
T u v w
10 0 2
i ass
¶
¶
×
¥
=
−
1 0 0 0
2
, , ,
n
t L L
n n n nT nT n n
x y z
x y
n c x c y c z e t e c u s v
L L L
d wd v d u d
u
t
p a
t
t

29. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
37
z p a
( ) ( )
( , , ,
)
T u v w
+ ( ) ( ) ( )×
¶
¶
×
¥
=
−
1
ass
2
0
0
10 2
, , ,
n
n n n
x y z
L
i
n T n c x c y c z
L L L
d wd v d u d
v
c w g u v w T
t
t
t L L L
x y z
( ) ( ) ( ) ( ) ( ) ( )
( )
t , i ³1,
T u v w
¶
¶
× − i
−
nT nT n n n T
d wd v d u d
w
e t e c u c v s w g u v w T
0 0 0 0
t
10 , , ,
, , , t
where sn(c) = sin (p n c/L);
¥
p
2
T
( ) = ( ) ( ) ( ) ( ) (− ) ( ) ( ) ( )×
T x y z t t
=1 0 0 0 0
, , ,
01 0 2
n
t L L L
n n n nT nT n n n
d
x y z
ass
x y z
nc x c y c z e t e s u c v c w
L L L
a
j
( )
t t j
2 2
d wd v d u d
( )
T
+ ( ) ( ) ( ) ( ) (− ) ( )×
T u v w
¶
¶
×
¥
=1 0 0
0 2
00
2
00
, , ,
, , ,
n
t L
n n n nT nT n
d
x y z
ass
x
c x c y c z e t e c u
L L L
T u v w
u
t
p
a
t
j
( ) ( ) ( )
2 2
T
t d wd v d u d
t
T u v w
y z
+ ( ) ( ) ( ) ( )×
( )
¶
¶
×
¥
=1
d
0 2
2
00
0 0 00
, , ,
, , ,
n
n n n nT
x y z
ass
L L
n n c x c y c z e t
L L L
T u v w
v
n s v c w
j
j
p
a
t
2
( ) ( ) ( ) ( ) ( )
x y z ja
t d wd v d u d
t
T u v w
t − j
( )×
( )
¶
¶
× −
¥
=1
0
2
00
0 0 0 0 00
2
, , ,
, , ,
n
n
ass
x y z
d
t L L L
nT n n n c x
L L L
T
T u v w
u
n e c u c v s w
t
j
t L L L
( ) ( ) ( ) ( ) x ( ) y ( ) z
( ) ( )
t d wd v d u d
t
T u v w
− ( )
¶
¶
× − n n nT nT n n n
+
T u v w
u
c y c z e t e c u c v c w
0 0 0 0
1
00
2
00
, , ,
, , ,
t
t j
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
¥
T u v w
×
¶
¶
ass
− −
=1 0 0 0 0
2
0 00 , , ,
2
n
t L L L
n n n nT nT n n n
x y z
x y z
v
c x c y c z e t e c u c v c w
L L L
t
t
ja
d wd v d u d
t j
× T − ( ) ( ) ( ) ( ) (− t
) ( ) ( )×
( )
¥
=
+
1 0 0 0
0
1
00
2
ass
, , , n
t L L
n n n nT nT n n
x y z
d d
x y
c x c y c z e t e c u c v
L L L
T
T u v w
ja
t
j
j
( ) ( )
t d wd v d u d
t
j ;
( )
Lz
T u v w
¶
¶
× +
n w
T u v w
c w
0
1
00
2
00
, , ,
, , ,
t
¥
p
2
T
( ) = ( ) ( ) ( ) ( ) (− ) ( ) ( ) ( )×
T x y z t t
=1 0 0 0 0
, , ,
02 0 2
n
t L L L
n n n nT nT n n n
d
x y z
ass
x y z
nc x c y c z e t e s u c v c w
L L L
a
j
( )
t t j
2 2
d wd vd u d
( )
T
+ ( ) ( ) ( ) ( ) (− ) ( )×
T u v w
¶
¶
×
¥
=1 0 0
0 2
00
2
01
, , ,
, , ,
n
t L
n n n nT nT n
d
x y z
ass
x
nc x c y c z e t e c u
L L L
T u v w
u
t
p
a
t
j
( ) ( ) ( )
2 2
T
t d wd v d u d
t
T u v w
y z
+ ( ) ( ) ( ) ( )×
( )
¶
¶
×
¥
=1
d
0 2
2
01
0 0 00
, , ,
, , ,
n
n n n nT
x y z
ass
L L
n n nc x c y c z e t
L L L
T u v w
v
s v c w
j
j
p
a
t
2
( ) ( ) ( ) ( ) ( )
x y z p a
t d wd v d u d
t
T u v w
t − j
( ) ×
( )
¶
¶
× −
¥
=1
2
0
2
01
0 0 0 0 00
2
, , ,
, , ,
n
n
ass
x y z
d
t L L L
nT n n n c x
L L L
T
T u v w
w
e c u c v s w
t
j
t L L L
( ) ( ) x ( ) y ( ) z
( ) ( ) ( )
t T u v w
t d wd v d u d
t
T u v w
×
( )
¶
¶
¶
¶
× − n nT n n n
+
T u v w
u
u
c y e c u c v c w
0 0 0 0
1
00
00 01
, , ,
, , , , , ,
t
t j
¥
p a j
× c ( z ) e ( t )− T ( ) ( ) ( ) ( ) (− t
) ( ) ( ) ( )×
=1 0 0 0 0
0 2
2
n
t L L L
n n n nT nT n n n
ass
x y z
n nT d
x y z
c x c y c z e t e c u c v c w
L L L

30. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
38
( ) ( )
T u v w p a
t T u v w
t t
00 01 2
( )
− ( ) ( ) ( ) ( )×
¶
¶
¶
¶
×
¥
ass c x c y c z e t
=
+
1
2
0
1
T u v w L L L
00
, , ,
, , , , , ,
n
n n n nT
x y z
d wd v d u d
v
v
t
j
( ) ( ) ( ) ( ) ( ) ( )
t t d wd v d u d
t
j ;
( )
t L L L
x y z
T u v w
¶
¶
T u v w
¶
¶
× − d nT n n n
+
T u v w
w
w
T e c u c v c w
0 0 0 0
1
00
00 01
, , ,
, , , , , ,
t
t j
¥
a
2
( ) = ( ) ( ) ( ) ( ) (− ) ( ) ( ) { ( ) ×
T x y z t t
=1 0 0 0 0
0
ass
, , ,
, , ,
11 n
t L L L
n n n nT nT n n T
x y z
x y z
c x c y c z e t e c u c v g u v w T
L L L
( ) ( ) ( )
( ) ( ) ( )×
¶
× c w
t , , ,t , , ,t
, , , 00
T u v w
¶
¶
¶
¶
+
T u v w
¶
¶
+
T u v w
¶
w
g w
v w
, , ,
g u v w T
w
T T n
2
00
2
2
00
2
( )
( )
¥
t
, , ,
2
01 T u v w
× + ( ) ( ) ( ) ( ) (− ) ( ) ( )×
=1 0 0 0
0
00
, , ,
n
t L L
n n n nT nT n n
ass
x y z
x y
c x c y c z e t e c u c v
L L L
d wd vd u d
T u v w
t
a
t
t
( ) [ ( )] ( ) [ ( )] ( )

31. +
2 , , ,
, , ,t t T u v w
t
¶
¶
+ +
T u v w
¶
¶
+ +
T u v w
¶
¶
×
Lz
g u v w T
T u
T v
g u v w T
u
0
2
01
2
2
01
2
2
01
1 , , ,
, , ,
1 , , ,
( ) T ( u v w
)
( ) T
+ ( ) ( ) ( ) ×
T n c x c y c z
, , ,
¶
¶
¶
¶
+
¥
=1
01 0 2
, , ,
n
n n n
ass d
L L L
x y z
c w d wd v d u d
w
g u v w T
w
j a
t
t
( ) ( ) ( ) ( ) ( ) T ( u , v , w
,
t
)
t j j
( )
( ) ( )
t L L L
x y z
t t
, , , T u , v , w
,
T u v w
10
10
+
× ¶
( )
¶
× − nT nT n n n
+ +
T u v w
u
T u v w
e t e c u c v c w
0 0 0 0
1
00
2
00
2
1
00
, , ,
, , ,
t
t
( , , , t ) T ( u , v , w
,
t
)
( )
( )
+ ( ) ( ) ( )× T u v w
¶
¶
+
T u v w
¶
¶
×
¥
n n n d wd v d u d c x c y c z
=
10
+
1
2
00
2
1
00
2
00
2
2
, , ,
, , ,
n
w
T u v w
v
t
t
t
j
( ) ( ) ( ) ( ) ( ) ( ) ( )
t L L L
x y z
2 T u , v , w
,t T u , v , w
,t T u , v , w
,t
×
¶
¶
+
¶
¶
+
¶
¶
× −
nT n n n
w
v
u
e c u c v c w
0 0 0 0
2
10
2
2
10
2
2
10
t
a T
d wd v d u d
t T
j
× e ( t )
0 + 2
( ) ( ) ( ) ( ) (− t
) ( ) ( )×
( )
¥
=1 0 0 0
0
00
ass d
, , , n
t L L
n n n nT nT n n
x y z
ass d
x y z
nT
x y
c x c y c z e t e c u c v
L L L
T u v w
L L L
a
t
j
j

32. ( ) ( )
( ) ( ) ( ) ( )
× +
T u v w
¶
¶
¶
¶
+
T u v w
¶
¶
×
Lz
n T T T g u v w T
w
g u v w T
u w
c w g u v w T
0
00
2
00
2
, , ,
, , ,
, , ,
, , ,
, , ,
t t
( )
t d wd v d u d
t j
( )
T
− ( ) ( ) ( ) ( ) (− ) ( )×
T u v w
¶
¶
×
¥
=1 0 0
0
00
2
00
2
2
, , ,
, , ,
n
t L
n n n nT nT n
ass d
x y z
x
c x c y c z e t e c u
L L L
T u v w
v
t
a
j
t
j
( ) ( ) ( ) ( ) ( ) ( )
L L
×
T u v w
¶
¶
+
T u v w
¶
¶
T u v w
¶
¶
+
T u v w
¶
¶
T u v w
¶
¶
×
y z
n u
u
v
v
w
c v
0 0
10 00 10 00 10 , , ,t , , ,t , , ,t , , ,t , , ,t
( )
( ) ( ) ( ) ( ) ( ) ( )× − −
t t j
( )
d wd v d u d
T u v w
00 2
( )
¶
¶
×
¥
=
+
1 0 0
0
1
00
, , ,
, , ,
n
t L
n n n nT nT n
T
ass d
x y z
n
x
c x c y c z e t e c u
L L L
T u v w
c w
w
t
a
j
t
j
( ) ( ) ( ) ( ) ( )

33. L L
y z
t T u v w
t T u v w
t d wd v d u d
t
T u v w
( )
¶
¶
+
¶
¶
+
¶
¶
× +
n n u
v
w
T u v w
c v c w
0 0
1
00
2
00
2
00
2
00
, , ,
, , , , , , , , ,
t
j
.

34. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
39
~
ijk I and (c ,h,f ,J )
Equations for the functions (c ,h,f ,J )
~
ijk V , i ³0, j ³0, k ³0 and conditions for
them have been obtain by the same procedure as for the functions Tij(x,y,z,t)
( ) ( ) ( ) ( c h f J
)
¶ c h f J
c h f J
c h f J
I
000 0 , , ,
2
000
2
0
I D
0
2
000
2
0
I D
0
2
000
2
I D
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
J
¶
¶
+
¶
¶
+
¶
¶
=
¶
D
D
D
I
V
I
V
I
V
( ) ( ) ( ) ( c h f J
)
¶ c h f J
c h f J
c h f J
V
000 0 , , ,
V ;
2
000
2
0
V
0
V D
2
000
2
0
V
0
V D
2
000
2
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
V D
J
¶
¶
+
¶
¶
+
¶
¶
=
¶
D
D
D
I
I
I
( ) ( ) ( ) ( )
+
c h f J
c J c h f J
c h f J
i i i
00 0 , , ,
I D
i I I I I
¶
¶
+
¶
¶
+
¶
¶
=
¶
¶
2
00
2
2
00
2
2
00
2
0
~
, , ,
~
, , ,
~
,
~
f
h
c
J
V
D
I I
( )
( )
( )
( )
+
c h f J
I D
D
+ 0 − i
−
c h f J
¶
¶
¶
¶
100 0
i I
+
¶
¶
¶
¶
h
c h f
c h
c h f
c
, , ,
~
, , ,
, , ,
~
, , , 100
0
0
I
V
I
V
g T
D
g T
D
I I
( )
( )
D
+ 0 i
−
c h f J
¶
¶
¶
¶
f
c h f
f
, , ,
~
, , , 100
0
I
V
g T
D
, i ³1,
( ) ( ) ( ) ( )
+
c h f J
c J c h f J
c h f J
i i i
00 0 , , ,
V D
i V V V V
¶
¶
+
¶
¶
+
¶
¶
=
¶
¶
2
00
2
2
00
2
2
00
2
0
~
, , ,
~
, , ,
~
,
~
f
h
c
J
I
D
V V
( )
( )
( )
( )
+
c h f J
V D
D
+ 0 − i
−
c h f J
¶
¶
¶
¶
100 0
i V
+
¶
¶
¶
¶
h
c h f
c h
c h f
c
, , ,
~
, , ,
, , ,
~
, , , 100
0
0
V
I
V
I
g T
D
g T
D
( ) ( )
V V
D
+ 0 i
−
c h f J
¶
¶
¶
¶
f
c h f
f
, , ,
~
, , , 100
0
V
I
g T
D
, i ³1;
( ) ( ) ( ) ( )
−
c h f J
c h f J I c h f J
I c h f J
I
010 0 , , ,
¶
¶
+
¶
¶
+
¶
¶
I D
=
¶
¶
2
010
2
2
010
2
2
010
2
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
J
D
I
V
~
[ e (c h f )] (c , h , f ,
J ) (c ,h,f ,J )
~
1 , , , , , 000 000 g T I V I V I V − +
( ) ( ) ( ) ( )
−
c h f J
c h f J V c h f J
V c h f J
V
010 0 , , ,
¶
¶
+
¶
¶
+
¶
¶
V D
=
¶
¶
2
010
2
2
010
2
2
010
2
V
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
J
D
I
~
[ e (c h f )] (c , h , f ,
J ) (c ,h,f ,J )
~
1 , , , , , 000 000 g T I V I V I V − + ;
( ) ( ) ( ) ( )
−
c h f J
c h f J I c h f J
I c h f J
I
020 0 , , ,
¶
¶
+
¶
¶
+
¶
¶
I D
=
¶
¶
2
020
2
2
020
2
2
020
2
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
J
D
I
V
[ ()] [ ~
~
~
~
e c h f (c , h , f ,
J ) (c , h , f ,
J ) (c , h , f ,
J ) (c ,h,f ,J )]
1 , , , , , 010 000 000 010 g T I V I V I V I V − + +
( ) ( ) ( ) ( )
−
c h f J
c h f J V c h f J
V c h f J
V
020 0 , , ,
¶
¶
+
¶
¶
+
¶
¶
V D
=
¶
¶
2
020
2
2
020
2
2
020
2
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
J
D
I
V
[ ()][ ~
~
~
~
e c h f (c , h , f ,
J ) (c , h , f ,
J ) (c , h , f ,
J ) (c ,h,f ,J )]
1 , , , , , 010 000 000 010 g T I V I V I V I V − + + ;

35. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
40
( ) ( ) ( ) ( )
−
c h f J
c h f J I c h f J
I c h f J
I
001 0 , , ,
¶
¶
+
¶
¶
+
¶
¶
I D
=
¶
¶
2
001
2
2
001
2
2
001
2
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
J
D
I
V
~
[ 1 e (c , h , f , )] 2
(c ,h,f ,J )
, , 000 g T I I I I I − +
( ) ( ) ( ) ( )
−
c h f J
c h f J V c h f J
V c h f J
V
001 0 , , ,
¶
¶
+
¶
¶
+
¶
¶
V D
=
¶
¶
2
001
2
2
001
2
2
001
2
V
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
J
D
I
~
[ 1 e (c , h , f , )] 2
(c ,h,f ,J )
, , 000 g T V I I I I − + ;
( ) ( ) ( ) ( )
+
c h f J
c h f J I c h f J
I c h f J
I
110 0 , , ,
¶
¶
+
¶
¶
+
¶
¶
I D
=
¶
¶
2
110
2
2
110
2
2
110
2
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
J
D
I
V
( ) ( ) ( ) ( )

36. +
D
0 I
c h f J
¶
¶
¶
¶
+
c h f J
¶
¶
¶
¶
+
h
c h f
c h
c h f
c
, , ,
~
, , ,
, , ,
~
, , , 010 010
0
g T
I
g T
D
I I
I
V
( )
( )
− [ + ( )][ ( ) ( ) +
c h f J
010 g T I V
I
+ e c h f c h f J c h f J
¶
¶
¶
¶
f
c h f
f
, , ,
~
, , ,
~
1 , , ,
, , ,
~
, , , , , 100 000
g T I I I I I
(c h f J ) (c ,h ,f ,J )]
~
~
+ I , , ,
V
000 100
( ) ( ) ( ) ( )
+
c h f J
c h f J V c h f J
V c h f J
V
110 0 , , ,
¶
¶
+
¶
¶
+
¶
¶
V D
=
¶
¶
2
110
2
2
110
2
2
110
2
V
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
J
D
I
( ) ( ) ( ) ( )

37. +
0 V
c h f J
¶
¶
¶
¶
+
c h f J
¶
¶
¶
¶
+
h
c h f
c h
c h f
c
, , ,
~
, , ,
, , ,
~
, , , 010 010
V
0
g T
V
g T
D
D
V V
I
( ) V
( c h f J
) 010 − [ + g ( T )][ V
( ) ×
+ e c h f c h f J
¶
¶
¶
¶
f
c h f
f
, , ,
~
1 , , ,
, , ,
~
, , , , , 100
g T V V V V V
(c h f J ) (c h f J ) (c ,h,f ,J )]
~
~
~
× I , , ,
+V , , ,
I ;
000 000 100
( ) ( ) ( ) ( )
−
c h f J
c h f J I c h f J
I c h f J
I
002 0 , , ,
¶
¶
+
¶
¶
+
¶
¶
I D
=
¶
¶
2
002
2
2
002
2
2
002
2
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
J
D
I
V
~
[ e (c h f )] (c , h , f ,
J ) (c ,h,f ,J )
~
1 , , , , , 001 000 g T I I I I I I − +
( ) ( ) ( ) ( )
−
c h f J
c h f J V c h f J
V c h f J
V
002 0 , , ,
¶
¶
+
¶
¶
+
¶
¶
V D
=
¶
¶
2
002
2
2
002
2
2
002
2
V
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
J
D
I
~
[ e (c h f )] (c , h , f ,
J ) (c ,h,f ,J )
~
1 , , , , , 001 000 g V V V V V V − + ;
( ) ( ) ( ) ( )
+
c h f J
c h f J I c h f J
I c h f J
I
101 0 , , ,
¶
¶
+
¶
¶
+
¶
¶
I D
=
¶
¶
2
101
2
2
101
2
2
101
2
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
J
D
I
V
( ) ( ) ( ) ( )

38. +
D
0 I
c h f J
¶
¶
¶
¶
+
c h f J
¶
¶
¶
¶
+
h
c h f
c h
c h f
c
, , ,
~
, , ,
, , ,
~
, , , 001 001
0
g T
I
g T
D
I I
I
V

39. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
J f h c rijk ,
41
( ) ( c h f J
) [ e (c h f )] (c h f J ) (c h f J )
001 g T I V
g T I I I − +
f
c h f
¶
f
, , ,
~
, , ,
~
1 , , ,
, , ,
~
I
, , , 100 000
¶
¶
¶
+
( ) ( ) ( ) ( )
+
c h f J
c h f J V c h f J
V c h f J
V
101 0 , , ,
¶
¶
+
¶
¶
+
¶
¶
V D
=
¶
¶
2
101
2
2
101
2
2
101
2
V
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
J
D
I
( ) ( ) ( ) ( )

40. +
0 V
c h f J
¶
¶
¶
¶
+
c h f J
¶
¶
¶
¶
+
h
c h f
c h
c h f
c
, , ,
~
, , ,
, , ,
~
, , , 001 001
V
0
g T
V
g T
D
D
V V
I
( )
( )
[ e (c h f )] (c h f J ) (c h f J )
c h f J
V
001 g T I V
+ ;
g T V V V − +
f
c h f
¶
f
, , ,
~
, , ,
~
1 , , ,
, , ,
~
, , , 000 100
¶
¶
¶
( ) ( ) ( ) ( )
−
c h f J
c h f J I c h f J
I c h f J
I
011 0 , , ,
¶
¶
+
¶
¶
+
¶
¶
I D
=
¶
¶
2
011
2
2
011
2
2
011
2
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
J
D
I
V
~
−[ + g ( T )] I ( , , ,
)I ( , , , )− [ 1 + g ( , , ,
T )] × I I I I I V I V ~
1 , , , , , 000 010 , , e c h f c h f J c h f J e c h f
~
~
× I V
001 000 (c , h , f ,
J ) (c ,h,f ,J )
( ) ( ) ( ) ( )
−
c h f J
c h f J V c h f J
V c h f J
V
011 0 , , ,
¶
¶
+
¶
¶
+
¶
¶
V D
=
¶
¶
2
011
2
2
011
2
2
011
2
V
0
~
, , ,
~
, , ,
~
, , ,
~
f
h
c
J
D
I
~
−[ + g ( T )]V ( , , ,
)V ( , , , )− [ 1 + g ( , , ,
t )]× V V V V I V I V ~
1 , , , , , 000 010 , , e c h f c h f J c h f J e c h f
~
~
× I (c , h , f ,
J ) V (c ,h,f ,J )
;
000 001 ( )
0
~ , , ,
0
=
r c h f J
¶
¶
x=
ijk
c
,
( )
0
~ , , ,
1
=
r c h f J
¶
¶
x=
ijk
c
,
( )
i
c h f J jk 0
,
r~ , , ,
0
=
¶
¶
h =
h
( )
0
~ , , ,
1
=
¶
¶
h=
h
( )
J f h c ri
jk ,
0
~ , , ,
0
=
¶
¶
f = f
( )
J f h c ri
jk (i ³0, j ³0, k ³0);
0
~ , , ,
1
=
¶
¶
f =
f
i
r~ ( c ,h,f ,0 ) = f ( c ,h,f ) r *
, ~ ( c , h , f , 0 ) = 0 000 r rjk (i ³1, j ³1, k ³1).
Solutions of the above equations have been obtained by standard Fourier approach and could be
written as
¥
( ) = + ( ) ( ) ( ) ( )
r c h f J c h f J r r ,
=1
000
1 2
~ , , ,
n
n n F c c c e
L L
1
where = ( ) ( ) ( ) ( )
* cos cos cos , ,
0
1
0
1
0
1
F nu nv nw f u v w d wd vd u nr nr p p p
r
, cn(c) = cos (p n c),
( ) ( ) nI V I e n D D0 0
2 2 J = exp −p J , ( ) ( ) nV I V e n D D0 0
2 2 J = exp −p J ;
¥
D
( ) = − I
( ) ( ) ( ) ( ) (− ) ( ) ( ) ( ) ×
00 , , , 2 , , ,
i nc c c e e s u c v g u v w T
=1 0
1
0
1
0
1
0
0 0
~
n
n nI nI n n I
V
D
I
J
c h f J p c h f J t
( ) I ( u v w
) D
i 100 − 2
0 I
( ) ( ) ( ) ( ) (− ) ( ) ×
n nc c c e e c u
¶
¶
×
¥
=
−
1 0
1
0 0
, , ,
~
n
n nI nI n
V
D
d wd v d u d
u
c w
J
t c h f J t
t

41. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
42
( ) ( ) ( ) I ( u v w
) D
s v c w g u v w T i t − p I
( c ) ( h ) ( f
) ×
n n I nc c c
¶
¶
×
¥
=
−
1
0
0
1
0
1
0
, , ,
100 2
~
, , ,
n
n
V
D
d wd v d u d
v
t
p
J
( ) ( ) ( ) ( ) ( ) ( ) ( )
e e c u c v
s w g u v w T i
, i ³1,
nI nI n n n I I u v w
¶
¶
× − −
t
t
J t
0
1
0
1
0
1
0
100 , , ,
~
, , , d wd v d u d
w
¥
D
( ) = − V
( ) ( ) ( ) ( ) (− ) ( ) ( ) ( ) ×
00 , , , 2 , , ,
i nc c c e e s u c v g u v w T
=1 0
1
0
1
0
1
0
0 0
~
n
n nV nI n n V
I
D
V
J
c h f J p c h f J t
( ) V ( u
,
) D
i 100 − 0 V
( ) ( ) ( ) ( ) (− ) ( ) ( ) ×
n nc c c e e c u s v
¶
¶
×
¥
=
−
1 0
1
0
1
0 0
~
n
n nV nI n n
I
D
d wd v d u d
u
c w
J
t c h f J t
t
V ( u
) D
c ( w ) g ( u v w T ) i t − p V
( c ) ( h ) ( f ) ( J
) ×
n V nc c c e
¶
¶
×
¥
=
−
1
0
0
1
0
,
100 2
~
2 , , ,
n
n nV
I
D
d wd v d u d
v
t
p
( ) ( ) ( ) ( ) ( ) ( )
J
e c u c v s w g u v w T i
nI n n n V , i ³1,
V u
¶
¶
× − −
t
t
t
0
1
0
1
0
1
0
100 ,
~
, , , d wd v d u d
w
where sn(c) = sin (p n c);
¥
J
( )= − ( ) ( ) ( ) ( ) (− ) ( ) ( ) ( )×
r r r c h f J c h f J t
=1 0
1
0
1
0
1
0
010 ~ , , , 2
n
n n n n n n n n c c c e e c u c v c w
~
[ e g (u v w T )] I (u , v , w,
t )V (u , v , w,
t )d wd vd u dt I V I V ~
1 , , , , , 000 000 × + ;
D ¥
J
( ) = − I c ( ) c ( ) c ( ) e ( ) e (− ) c ( u ) c ( v ) c ( w
) ×
r r r c h f J c h f J t
=1 0
1
0
1
0
1
0
0 0
020 ~ , , , 2
n
n n n n n n n n
V
D
[ ~
( ~
~
~
× I u , v , w ,
t ) V ( u , v , w ,
t )+ I ( u , v , w ,
t ) V ( u , v , w
,t )]×
010 000 000 010 [ e g (u v w T )] d wd v d u dt I V I V 1 , , , , , × + ;
¥
J
( )= − ( ) ( ) ( ) ( ) (− ) ( ) ( ) ( ) ×
r r r c h f J c h f J t
=1 0
1
0
1
0
1
0
001 ~ , , , 2
n
n n n n n n n n c c c e e c u c v c w
[ e ( )]r ( t ) t r r r r 1 g u,v,w,T ~ u,v,w, d wd vd u d 2
, , 000 × + ;
¥
J
( )= − ( ) ( ) ( ) ( ) (− ) ( ) ( ) ( ) ×
r r r c h f J c h f J t
=1 0
1
0
1
0
1
0
002 ~ , , , 2
n
n n n n n n n n c c c e e c u c v c w
[ e ( )]r ( t )r ( t ) t r r r r 1 g u,v,w,T ~ u,v,w, ~ u,v,w, d wd vd u d , , 001 000 × + ;
¥
D
( ) = − I nc ( ) c ( ) c ( ) e ( ) e (− ) s ( u ) c ( v ) c ( u
) ×
=1 0
1
0
1
0
1
0
0 0
~
110 , , , 2
n
n n n nI nI n n n
V
D
I
J
c h f J p c h f J t
( ) I ( u v w
) D
i 100 − 2
0 I
( ) ( ) ( ) ( ) ×
g u v w T t p c h f J
I nc c c e
¶
¶
×
¥
=
−
1
0
, , ,
~
, , ,
n
n n n nI
V
D
d wd vd u d
u
t
( ) ( ) ( ) ( ) ( ) ( )
D
t
, , ,
I u v w
i − 0
I
×
¶
¶
× − −
V
d wd v d u d
nI n n n I v
D
e c u s v c u g u v w T
0
0
1
0
1
0
1
0
100 2
~
, , , t p
t
J
( ) ( ) ( ) ( ) ( ) ( ) ( )
¥
I u v w
J
×
¶
¶
× −
=
−
1 0
1
0
1
0
1
0
100 , , ,
~
, , ,
n
i
nI nI n n n I d wd v d u d
w
n e e c u c v s u g u v w T
t
t
J t

42. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
43
¥
( ) ( × ) ( )− ( ) ( ) ( ) ( ) (− ) ( ) ( ) ( )[ + ×
, 2 1
n n n n nI n n nI n n n I V c c c c e c c e c u c v c v
=1 0
1
0
1
0
1
0
n
J
c h f c J h f t e
~
~
~
~
g (u v w T )][I (u , v , w,
t )V (u , v , w,
t ) I (u , v , w,
t )V (u , v , w,
t )]d wd v d u dt I V , , , , 100 000 000 100 × +
¥
D
( ) = − V nc ( ) c ( ) c ( ) e ( ) e (− ) s ( u ) c ( v ) c ( u
) ×
=1 0
1
0
1
0
1
0
0 0
~
110 , , , 2
n
n n n nV nV n n n
I
D
V
J
c h f J p c h f J t
( ) V ( u v w
) D
i 100 − 2
0 V
( ) ( ) ( ) ( ) ×
g u v w T t p c h f J
V nc c c e
¶
¶
×
¥
=
−
1
0
, , ,
~
, , ,
n
n n n nV
I
D
d wd vd u d
u
t
( ) ( ) ( ) ( ) ( ) ( )
D
t
, , ,
V u v w
i − 0
V
×
¶
¶
× − −
I
d wd v d u d
nV n n n V v
D
e c u s v c u g u v w T
0
0
1
0
1
0
1
0
100 2
~
, , , t p
t
J
( ) ( ) ( ) ( ) ( ) ( ) ( )
¥
V u v w
J
×
¶
¶
× −
=
−
1 0
1
0
1
0
1
0
100 , , ,
~
, , ,
n
i
nV nV n n n V d wd v d u d
w
ne e c u c v s u g u v w T
t
t
J t
¥
× ( ) ( ) ( )− ( ) ( ) ( ) ( ) (− ) ( ) ( ) ( )[ + ×
, 2 1
n n n n nI n n nV n n n I V c c c c e c c e c u c v c v
=1 0
1
0
1
0
1
0
n
J
c h f c J h f t e
~
~
~
~
g (u v w T)][I (u , v , w,
t )V (u , v , w,
t ) I (u , v , w,
t )V (u , v , w,
t )]d wd vd u dt I V , , , , 100 000 000 100 × + ;
¥
D
( ) = − I nc ( ) c ( ) c ( ) e ( ) e (− ) s ( u ) c ( v ) c ( w
) ×
=1 0
1
0
1
0
1
0
0 0
~
101 , , , 2
n
n n n nI nI n n n
V
D
I
J
c h f J p c h f J t
( ) I ( u v w
) D
001 − 2
0 ( ) ( ) ( ) ( ) ×
g u v w T t p c h f J
I nc c c e
¶
¶
×
¥
=1
0
, , ,
~
, , ,
n
n n n nI
I
V
D
d wd v d u d
u
t
~
( ) ( ) ( ) ( ) ( ) ( )
D
t
, , ,
I u v w
− ×
¶
¶
× −
I
0
V
d wd vd u d
nI n n n I v
D
e c u s v c w g u v w T
0
0
1
0
1
0
1
0
001 2
, , , t p
t
J
~
( ) ( ) ( ) ( ) ( ) ( ) ( )
¥
I u v w
J
×
¶
¶
× −
=1 0
1
0
1
0
1
0
001 , , ,
, , ,
n
nI nI n n n I d wd v d u d
w
ne e c u c v s w g u v w T
t
t
J t
¥
× ( ) ( ) ( )− ( ) ( ) ( ) ( ) (− ) [ + ( )] ×
, , 2 1 , , ,
n n n n n n nI nI I V I V c c c c c c e e g u v w T
=1 0
1
0
n
J
c h f c h f J t e
~
~
×
100 000 c (u)I (u , v , w,
t )V (u , v , w,
t )d wd v d u dt n ¥
D
( ) = − V nc ( ) c ( ) c ( ) e ( ) e (− ) s ( u ) c ( v ) c ( w
) ×
=1 0
1
0
1
0
1
0
0 0
~
101 , , , 2
n
n n n nV nV n n n
I
D
V
J
c h f J p c h f J t
( ) V ( u v w
) D
001 − 2
0 ( ) ( ) ( ) ( ) ×
g u v w T t p c h f J
V nc c c e
¶
¶
×
¥
=1
V
0
, , ,
~
, , ,
n
n n n nV
I
D
d wd v d u d
u
t
~
( ) ( ) ( ) ( ) ( ) ( )
D
t
, , ,
V u v w
− ×
¶
¶
× −
0
V
I
d wd vd u d
nV n n n V v
D
e c u s v c w g u v w T
0
0
1
0
1
0
1
0
001 2
, , , t p
t
J
~
( ) ( ) ( ) ( ) ( ) ( ) ( )
¥
V u v w
J
×
¶
¶
× −
=1 0
1
0
1
0
1
0
001 , , ,
, , ,
n
nV nV n n n V d wd v d u d
w
ne e c u c v s w g u v w T
t
t
J t

43. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
44
¥
( ) ( × ) ( )− ( ) ( ) ( ) ( ) (− ) ( ) ( ) ( )[ + ×
, 2 1
n n n n n n nV nV n n n I V c c c c c c e e c u c v c w
=1 0
1
0
1
0
1
0
n
J
c h f c h f J t e
~
g (u v w T )] I (u , v , w,
t )V (u , v , w,
t )d wd v d u dt I V ~
, , , , 000 100 × ;
¥
( ) = − ( ) ( ) ( ) ( ) (− ) ( ) ( ) ( ) ×
n n n nI nI n n n I c c c e e c u c v c w
=1 0
1
0
1
0
1
0
~
011 , , , 2
n
J
c h f J c h f J t
×{[ ()]~
~
+ g u v w T I (u , v , w ,
)I (u , v , w , )+ [ 1 + g (u , v , w ,
T )]× I I I I I V I V 1 , , , , , 000 010 , , e t t e
I (u v w t )V (u, v,w,t )} d wd v d u dt
~
~
×
, , ,
001 000 ¥
( ) = − ( ) ( ) ( ) ( ) (− ) ( ) ( ) ( ) ×
n n n nV nV n n n V c c c e e c u c v c w
=1 0
1
0
1
0
1
0
~
011 , , , 2
n
J
c h f J c h f J t
×{[ ~
~
1 + e g (u , v , w , T )]V (u , v , w ,
t )V (u , v , w , t )+ [ 1 + e
g (u , v , w ,
T )]× V , V V , V 000 010 I , V I , V I (u v wt )V (u,v,w,t )}d wd v d u dt
~
~
× , , ,
.
000 001 Equations for initial-order approximations of distributions of concentrations of simplest complex-es
of radiation defects Fr0(x,y,z,t) and corrections for them Fri(x,y, z,t), i ³1 and boundary and
initial conditions for them have been obtained as the functions Tij(x,y,z,t) and takes the form
( ) ( ) ( ) ( )
+
F
x , y , z , t , , , , , , , , ,
0 I I I
+
F
+
F
=
F
0
2
x y z t
x y z t
x y z t
F 2
2
0
2
2
0
2
0
z
y
x
D
t
I
I
¶
¶
¶
¶
¶
¶
¶
¶
k (x y z T )I (x y z t ) k (x y z T )I (x y z t ) I I I , , , , , , , , , , , , 2
, + −
( ) ( ) ( ) ( )
+
F
x , y , z , t , , , , , , , , ,
0 V V V
+
F
+
F
=
F
0
2
x y z t
x y z t
x y z t
F 2
2
0
2
2
0
2
0
z
y
x
D
t
V
V
¶
¶
¶
¶
¶
¶
¶
¶
k (x y z T )V (x y z t ) k (x y z T )V (x y z t ) V V V , , , , , , , , , , , , 2
, + − ;
( ) ( ) ( ) ( )
+
F
, , , , , , , , , , , ,
x y z t I i I i I i
+
F
+
F
=
F
2
x y z t
x y z t
x y z t
F 2
2
2
2
2
0
z
y
x
D
t
I
I i
¶
¶
¶
¶
¶
¶
¶
¶
F
¶ , , ,
( )
( )
F
( )
( )

44. +
x y z t
¶
¶
+ D I i
−
+
g x y z T
F
−
x y z t
F F y
x y
g x y z T
x
I
I i
I I ¶
¶
¶
¶
¶
, , ,
, , ,
, , , 1 1
0
¶ , , ,
( )
( )
F
¶
+ −
x y z t
g x y z T
F z
z
I i
I ¶
¶
, , , 1 , i³1,
( ) ( ) ( ) ( )
+
F
, , , , , , , , , , , ,
x y z t V i V i V i
+
F
+
F
=
F
2
x y z t
x y z t
x y z t
F 2
2
2
2
2
0
z
y
x
D
t
V
V i
¶
¶
¶
¶
¶
¶
¶
¶
F
¶ , , ,
( )
( )
F
( )
( )

45. +
x y z t
¶
¶
+ D V i
−
+
g x y z T
F
−
x y z t
F F x y
y
g x y z T
x
V
V i
V V ¶
¶
¶
¶
¶
, , ,
, , ,
, , , 1 1
0
F
¶ , , ,
( )
( )
¶
+ −
x y z t
g x y z T
F z
z
V i
V ¶
¶
, , , 1 , i³1;

46. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
p ,
45
( )
x y z t r 0
,
, , ,
0
=
¶
¶F
x=
i
x
( )
x y z t r 0
,
, , ,
=
¶
¶F
x=Lx
i
x
( )
x y z t r 0
,
, , ,
0
=
¶
¶F
y=
i
y
( )
x y z t r 0
,
, , ,
=
¶
¶F
y=Ly
i
y
( )
x y z t r 0
,
, , ,
0
=
¶
¶F
z=
i
z
( )
x y z t r , i³0;
0
, , ,
=
¶
¶F
z=Lz
i
z
Fr0(x,y,z,0)=fFr (x,y,z), Fri(x,y,z,0)=0, i³1.
Solutions of the above equations could be written as
¥
F ( ) = + ( ) ( ) ( ) ( )+ ( ) ( ) ( ) ×
=
¥
=
F F
1 1
0
1 2 2
, , ,
n
n n n
n
n n n n n
x y z x y z
nc x c y c z
L
F c x c y c z e t
L L L L L L
x y z t
r r r
t L L L
( )x y z
× e t e (− ) c ( u ) c ( v ) c ( w )[ k ( u v w T ) I ( u v w )− k ( u v w T
) × F n F
n n n n I I I
0 0 0 0
2
, t , , , , , ,t , , ,
r r
× I (u, v,w,t )]d wd v d u dt ,
Lx y z L L
where = ( ) ( ) ( ) ( ) F F
n n n n F c u c v c w f u v w dwd vdu
0 0 0
, ,
r r
, ( )
2 2 1 1 1
= − + + F 0F 2 2 2
n L L L
exp
x y z
e t n D t
r r
cn(x) = cos (p n x/Lx);
¥
p
F ( x y z t ) = − ( ) ( ) ( ) ( ) (− t
) ( ) ( ) ( ) ×
r r r
=
F F
1 0 0 0 0
2
2
, , ,
n
t L L L
n n n n n n n n
x y z
i
x y z
nc x c y c z e t e s u c v c w
L L L
( )
( )
− ( ) ( ) ( ) ( ) ×
¶ F
t
×
¥
=
F
−
F
1
2
1 , , , 2
, , ,
n
n n n n
x y z
I i
nc x c y c z e t
L L L
d wd v d u d
u v w
u
g u v w T
r
r
r
p
t
¶
¶ t
x y z p
( ) ( ) ( ) ( ) ( )
( )
F
× − −
u v w
− ×
F F
x y
t L L L
I i
d wd vd u d
n n n n v
L L
e c u s v c w g u v w T
t
¶
t r
r r
, , , 2
, , ,
0 0 0 0
1
¶ t
1 , , ,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
F
¥
u v w
t r
×
× −
=
−
F F
1 0 0 0 0
1
2
n
t L L L
I i
n n n n n n n n
z
x y z
w
nc x c y c z e t e c u c v s w
L ¶
r r
( ) t
r
g u,v,w,T d wd v d u d F × , i³1,
where sn(x) = sin (p n x/Lx).
Equations for initial-order approximation of dopant concentration C00(x,y,z,t), corrections for
them Cij(x,y,z,t) (i ³1, j ³1) and boundary and initial conditions take the form
( ) ( ) ( ) ¶
( )
00 , , , , , , , , , , , ,
C x y z t
2
00
2
C x y z t
+
2 0
00
2
C x y z t
2 0
00
2
0
z
D
y
D
x
D
C x y z t
t
L L ¶
L ¶
¶
+
¶
¶
=
¶
¶
;
( ) ( ) ( ) ( )
+
0 , , , , , , , , , , , ,
C x y z t i
C x y z t
¶
¶
+
C x y z t
¶
¶
+
C x y z t
¶
¶
=
¶
¶
2
0
2
2 0
0
2
2 0
0
2
0
z
D
y
D
x
D
t
L
i
L
i
L
i
( ) ( ) ( ) ( )
+
C x y z t
C x y z t
+ D − i
−
¶
¶
¶
¶
+
¶
¶
¶
¶
y
g x y z T
y
D
x
g x y z T
x
L L
i
L L
, , ,
, , ,
, , ,
, , , 10
0
10
0

47. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
46
( ) ( )
C x y z t
+ D i
−
¶
¶
¶
L L
¶
z
g x y z T
z
, , ,
, , , 10
0 , i ³1;
( ) ( ) ( ) ( )
+
01 , , , , , , , , , , , ,
C x y z t
¶
¶
+
C x y z t
¶
¶
+
C x y z t
¶
¶
=
C x y z t
¶
¶
2
01
2
2 0
01
2
2 0
01
2
0
z
D
y
D
x
D
t
L L L
( )
( )
( ) g
( )
, , , , , , C x , y , z ,
t
00 00
( )
( )
+
C x y z t
¶
¶
¶
¶
+
g
D L L
C x y z t
¶
¶
¶
¶
+
y
P x y z T
y
D
x
C x y z t
P x y z T
x
, , ,
, , ,
, , ,
0
00 00
0 g
g
( )
( )
( )
g
, , , 00 00
C x y z t
¶
¶
¶
¶
+
z
C x y z t
P x y z T
z
D L
, , ,
, , ,
0 g
;
( ) ( ) ( ) ( )
+
02 , , , , , , , , , , , ,
C x y z t
¶
¶
+
C x y z t
¶
¶
+
C x y z t
¶
¶
=
C x y z t
¶
¶
2
02
2
2 0
02
2
2 0
02
2
0
z
D
y
D
x
D
t
L L L
g
( ) C ( x , y , z ,
t
)
( )
( )

48. ( )
( ) ( )
×
¶
¶
+
, , ,
C x y z t
¶
¶
¶
¶
+
− −
, , ,
g
C x y z t
P x y z T
, , ,
C x y z t
x y
P x y z T
C x y z t
x , , ,
, , ,
, , ,
1
00
01
00
1
00
01 g
g
( ) g
( ) ( )
C x , y , z ,
t
, , ,
00 ( )
( )
+
C x y z t
¶
¶
¶
¶
+
C x y z t
¶
¶
×
−
L D
z
P x y z T
C x y z t
y z
0
00
1
00
01
, , ,
, , ,
, , ,
g
( )
( )
( ) ( )
g
, , , , , , C x , y , z ,
t
00 01 00 01
( )
( )

49. +
C x y z t
¶
¶
¶
0 g
¶
+
C x y z t
¶
¶
¶
¶
+
y
P x y z T
x y
C x y z t
P x y z T
x
D L
, , ,
, , ,
, , ,
g
g
( )
( )
( )
, , , 00 01
g
C x y z t
¶
¶
¶
¶
+
z
C x y z t
P x y z T
z
, , ,
, , ,
g
;
( ) ( ) ( ) ( )
+
11 , , , , , , , , , , , ,
C x y z t
¶
¶
+
C x y z t
¶
¶
+
C x y z t
¶
¶
=
C x y z t
¶
¶
2
11
2
2 0
11
2
2 0
11
2
0
z
D
y
D
x
D
t
L L L
g
( ) C ( x , y , z ,
t
)
( )
( )

50. ( )
( ) ( )
×
¶
¶
+
, , ,
C x y z t
¶
¶
¶
¶
+
− −
, , ,
g
C x y z t
P x y z T
, , ,
C x y z t
x y
P x y z T
C x y z t
x , , ,
, , ,
, , ,
1
00
10
00
1
00
10 g
g
( ) g
( ) ( )
C x , y , z ,
t
, , ,
00 ( )
( )
+
C x y z t
¶
¶
¶
¶
+
C x y z t
¶
¶
×
−
L D
z
P x y z T
C x y z t
y z
0
00
1
00
10
, , ,
, , ,
, , ,
g
( )
( )
( ) ( )
g
, , , , , , C x , y , z ,
t
00 10 00 10
( )
( )

51. +
C x y z t
¶
¶
¶
0 g
¶
+
C x y z t
¶
¶
¶
¶
+
y
P x y z T
x y
C x y z t
P x y z T
x
D L
, , ,
, , ,
, , ,
g
g
( )
( )

52. ( ) ( ) ( )
+
, , , 01
C x y z t
¶
¶
¶
g
z L L
¶
+
C x y z t
¶
¶
¶
¶
+
x
g x y z T
x
D
z
C x y z t
P x y z T
, , ,
, , ,
, , ,
, , ,
0
00 10
g
( ) ( ) ( ) ( )
C x y z t
¶
¶
¶
¶
+
C x y z t
¶
¶
¶
¶
+
z
g x y z T
y z
g x y z T
y L L
, , ,
, , ,
, , ,
, , , 01 01 ;
( )
0
, , ,
0
=
x=
C x y z t
ij
¶
x
¶
,
( )
0
, , ,
=
x=Lx
C x y z t
ij
¶
x
¶
,
( )
0
, , ,
0
=
y=
C x y z t
ij
y
¶
¶
,

53. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
47
( )
0
, , ,
=
y=Ly
C x y z t
ij
y
¶
¶
,
( )
0
, , ,
0
=
z=
C x y z t
ij
¶
z
¶
,
( )
0
, , ,
=
z=Lz
C x y z t
ij
¶
z
¶
, i ³0, j ³0;
C00(x,y,z,0)=fC (x,y,z), Cij(x,y,z,0)=0, i ³1, j ³1.
Solutions of the above equations with account boundary and initial conditions could be written as
¥
( ) = + ( ) ( ) ( ) ( )
C x y z t ,
=1
00
1 2
, , ,
n
nC n n n nC
x y z x y z
F c x c y c z e t
L L L L L L
Lx y z L L
where = ( ) ( ) ( ) ( ) F
nC n n n F c u c v c w f u v w dwdvdu
0 0 0
, ,
r
, ( )
p ;
2 2 1 1 1
= − + + 0F 2 2 2
nC L L L
exp
x y z
e t n D t
r
¥
p
( )= − ( ) ( ) ( ) ( ) (− ) ( ) ( ) ( ) ×
C x y z t t
0 2 , , ,
=1 0 0 0 0
2
, , ,
n
t L L L
nC n n n nC nC n n L
x y z
i
x y z
n F c x c y c z e t e s u c v g u v w T
L L L
( )
( )
− ( ) ( ) ( ) ( ) (− ) ( ) ×
C u v w
c v t
¶
¶
×
¥
=
−
1 0 0
2
10 , , , 2
n
t L
nC n n n nC nC n
x y z
i
n
x
n F c x c y c z e t e c u
L L L
d wd v d u d
u
p
t
t
y z p
( ) ( ) ( )
( )
C u v w
− ( ) ( ) ( ) ×
¶
¶
×
¥
=
−
1
2
0 0
10 , , , 2
, , ,
n
nC n n n
x y z
L L
i
n n L n F c x c y c z
L L L
d wd v d u d
v
s v c v g u v w T
t
t
t L L L
x y z
( ) ( ) ( ) ( ) ( ) ( )
( )
t , i ³1;
C u v w
¶
¶
× − i
−
nC nC n n n L
d wd v d u d
w
e t e c u c v s v g u v w T
0 0 0 0
t
10 , , ,
, , , t
2 ¥
, , ,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
C u v w
= − − ×
C x y z t g
=1 0 0 0 0
00
, , ,
01 2 n
, , ,
t L L L
nC n n n nC nC n n
x y z
x y z
P u v w T
n F c x c y c z e t e s u c v
L L L
g t
t
p
( )
( )
− ( ) ( ) ( ) ( ) (− ) ×
C u v w
c w t
n n F c x c y c z e t e
¶
¶
×
¥
=1 0
2
00 , , , 2
n
t
nC n n n nC nC
L L L
x y z
d wd v d u d
u
p
t
t
g
x y z p
( ) ( ) ( )
( )
( )
( )
, , ,
C u v w
C u v w
− ( )×
¶
¶
×
¥
=1
2
0 0 0
00 00 , , , 2
, , ,
n
nC nC
x y z
L L L
n n n n F e t
L L L
d wd v d u d
v
P u v w T
c u s v c w
t
t t
g
t L L L
x y z
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
C u v w
¶
¶
× −
n n n nC n n n
d wd v d u d
w
, , ,
g
C u v w
P u v w T
c x c y c z e c u c v s w
0 0 0 0
00 00 , , ,
, , ,
t
t t
t g
;
¥
p
( ) = − ( ) ( ) ( ) ( ) (− ) ( ) ( ) ( ) ×
C x y z t t
=1 0 0 0 0
02 2
2
, , ,
n
t L L L
nC n n n nC nC n n n
x y z
x y z
nF c x c y c z e t e s u c v c w
L L L
C g
( ) ( u , v , w
,
)
( )
( ) ( ) ( )× −
¶
¶
×
¥
=
−
1
2
C u v w
00
1
00
01
, , , 2
, , ,
, , ,
n
nC n n
x y z
F c x c y
L L L
d wd v d u d
u
P u v w T
C u v w
p
t
t t
t g
t L L L C g
−
( ) ( ) ( ) ( ) ( ) ( ) ( u , v , w
,
)
( )
( )
x y z
C u v w
×
¶
¶
× −
n nC nC n n
v
P u v w T
nc z e t e c u s v C u v w
0 0 0 0
00
1
00
01
, , ,
, , ,
, , ,
t t
t t g
¥
p
× c ( w ) d wd v d u d − ( ) ( ) ( ) ( ) (− t
) ( ) ( ) ×
=1 0 0 0
2
2
n
t L L
nC n n n nC nC n n
x y z
n
x y
n F c x c y c z e t e c u c v
L L L
t

54. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
48
g
z ( ) ( ) C ( u , v , w
,
)
( ) p
− ( ) ×
( )
C u v w
¶
¶
×
¥
=
−
1
2
0
00
1
00
01
, , , 2
, , ,
, , ,
n
n
x y z
L
n n c x
L L L
d wd v d u d
w
P u v w T
s w C u v w
t
t t
t g
t L L x y L
( ) ( ) ( ) ( ) ( ) ( ) z
( ) ( ) C ( u v w
)
×
¶
¶
× −
nC n n nC nC n n n
u
F c y c z e t e s u c v c w C u v w
0 0 0 0
00
01
, , ,
, , ,
t
t t
( )
( )
¥
, , ,
g
C u v w
× − ( ) ( ) ( ) ( ) (− ) ( ) ×
=
−
1 0 0
2
1
00 2
, , ,
n
t L
nC n n n nC nC n
x y z
x
n F c x c y c z e t e c u
L L L
d wd v d u d
P u v w T
t
p
t
t
g
g
( ) ( ) ( ) ( )
y z C u , v , w
,
p
( )
( )
C u v w
− ×
¶
¶
×
¥
=
−
1
2
0 0
00
1
00
01
, , , 2
, , ,
, , ,
n
x y z
L L
n n n
L L L
d wd v d u d
v
P u v w T
s v c w C u v w
t
t t
t g
L L g −
( ) ( ) ( ) ( ) t ( ) x ( ) y L ( ) z
( ) ( ) C ( u , v , w
,
t
)
× − ×
( )
t t
nC n n n nC nC n n n
, , , g
P u v w T
F c x c y c z e t e c u c v s w C u v w
0 0 0 0
1
00
01 , , ,
( )
00 , , , 2
− ( ) ( ) ( ) ( ) (− ) ( )×
C u v w
¶
¶
×
¥
=1 0 0
2
n
t L
nC n n n nC nC n
x y z
x
F c x c y c z e t e s u
L L L
d wd vd u d
w
t
p
t
t
g
y ( ) z ( ) C ( u , v , w
,
)
( ) p
− ( ) ( ) ×
( )
C u v w
¶
¶
×
¥
=1
2
0 0
00 01 , , , 2
, , ,
n
n nC
x y z
L L
n n c x e t
L L L
d wd v d u d
u
P u v w T
n c v c w
t
t t
g
g
( ) ( ) ( ) ( ) ( ) C ( u , v , w
,
)
( )
( )
t L L L
x y z
C u v w
×
¶
¶
× −
nC n nC n n n
d wd v d u d
v
P u v w T
F c y e c u s v c w
0 0 0 0
00 01 , , ,
, , ,
t
t t
t g
¥
p
× n c ( z )− ( ) ( ) ( ) ( ) (− t
) ( ) ( ) ( ) ×
=1 0 0 0 0
2
2
n
t L L L
nC n n n nC nC n n n
x y z
n
x y z
nF c x c y c z e t e c u c v s w
L L L
( )
( )
( )
, , , 00 01 ;
t
t t
g
C u v w
g
d wd v d u d
C u v w
w
P u v w T
¶
¶
×
, , ,
, , ,
¥
p
( ) = − ( ) ( ) ( ) ( ) (− ) ( ) ( ) ( ) ×
C x y z t t
=1 0 0 0 0
11 2
2
, , ,
n
t L L L
nC n n n nC nC n n n
x y z
x y z
nF c x c y c z e t e s u c v c w
L L L
( )
( )
− ( ) ( ) ( ) ( ) ×
C u v w
L nF c x c y c z e t
¶
¶
×
¥
=1
2
01 , , , 2
, , ,
n
nC n n n nC
L L L
x y z
d wd v d u d
u
g u v w T
p
t
t
( ) ( ) ( ) ( ) ( ) ( )
x y z C u v w
p
− ×
¶
¶
× − 2
0 0 0 0
01 , , , 2
, , ,
x y z
t L L L
d wd v d u d
nC n n n L v
L L L
e c u s v c w g u v w T
t
t
t
( ) ( ) ( ) ( ) ( ) ( ) ( )
¥
x y z
C u v w
n e t e c u c v s w g u v w T t
×
¶
¶
× −
=1 0 0 0 0
01 , , ,
, , ,
n
t L L L
nC nC n n n L
d wd v d u d
w
t
t
¥
p
× F c ( x ) c ( y ) c ( z )− ( ) ( ) ( ) ( ) (− t
) ( ) ( ) ×
=1 0 0 0
2
2
n
t L L
nC n n n nC nC n n
x y z
nC n n n
x y
F c x c y c z e t e s u c v
L L L
g
z ( ) C ( u , v , w
,
)
( ) p
−
( ) ( )× ( )
C u v w
¶
¶
×
¥
=1
2
0
00 10 , , , 2
, , ,
n
nC n n
x y z
L
n nF c x c y
L L L
d wd v d u d
u
P u v w T
n c w
t
t t
g
g
( ) ( ) ( ) ( ) ( ) ( ) C ( u , v , w
,
)
( )
( )
t L L L
x y z
C u v w
−
¶
¶
× −
n nC nC n n n
d wd v d u d
v
P u v w T
c z e t e c u s v c w
0 0 0 0
00 10 , , ,
, , ,
t
t t
t g

55. International Journal on Computational Sciences Applications (IJCSA) Vol.4, No.5, October 2014
49
g ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( t
)
p
2 C u , v , w
,
− − ×
( )
¥
t
=1 0 0 0 0
00
2 n
, , ,
t L L L
nC n n n nC nC n n n
x y z
x y z
P u v w T
nF c x c y c z e t e c u c v s w
L L L g
( ) ( ) ( ) ( ) ( ) ( ) ( ) × − −
¶
¶
×
¥
=1 0 0
10 , , , 2
2
n
t L
nC n n n nC nC n
x y z
x
n F c x c y c z e t e s u
L L L
d wd v d u d
C u v w
w
t
p
t
t
g
( ) ( ) ( ) ( )
y z C u , v , w
,
p
( )
( )
C u v w
− ×
¶
¶
×
¥
=
−
1
2
0 0
00
1
00
10
, , , 2
, , ,
, , ,
n
x y z
L L
n n n
L L L
d wd v d u d
u
P u v w T
c v c w C u v w
t
t t
t g
L L L −
( ) ( ) ( ) ( ) t ( ) ( ) ( ) ( ) ( )
, , ,t t
g
C u v w
( )
( )
x y z
C u v w
×
¶
¶
× −
nC n n n nC nC n n n
v
P u v w T
F c x c y c z e t e c u s v c w
0 0 0 0
00
1
00 , , ,
, , ,
t g
¥
p
2
× C ( u v w ) d wd v d u d − ( ) ( ) ( ) ( ) (− t
) ( ) ×
=1 0 0
, , ,
10 2
n
t L
nC n n n nC nC n
x y z
x
n F c x c y c z e t e c u
L L L
t t
g
( ) ( ) ( ) C ( u , v , w
,
)
, , , t
( )
( )
C u v w
¶
¶
×
− y z
L L
n n d wd vd u d
w
P u v w T
c v s w C u v w
0 0
00
1
00
10
, , ,
, , ,
t t
t g
.
REFERENCES
[1] I.P. Stepanenko (1980). Basis of Microelectronics, Soviet Radio, Moscow.
[2] A.G. Alexenko, I.I. Shagurin (1990). Microcircuitry, Radio and communication, Moscow.
[3] Z.Yu. Gotra (1991). Technology of microelectronic devices, Radio and communication, Moscow.
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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 102 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 50 published papers in area of her researches.