An approach to decrease dimensions of drift
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Transcript of An approach to decrease dimensions of drift
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
DOI:10.5121/ijcsa.2014.4503 25
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-
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
26
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 and radiation defects in the considered heterostructure.
Substrate
Dopant 1 Dopant 2 Dopant 3
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]
( ) ( ) ( ) ( )
++
+
=
z
tzyxCD
zy
tzyxCD
yx
tzyxCD
xt
tzyxCCCC
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,,,,,,,,,, (1)
with boundary and initial conditions
( )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)
Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant, T is the temperature
of annealing, DС is the dopant diffusion coefficient. Value of dopant diffusion coefficient depends
on properties of materials 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]
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
27
( ) ( )( )
( ) ( )( )
++
+=
2*
2
2*1
,,,,,,1
,,,
,,,1,,,
V
tzyxV
V
tzyxV
TzyxP
tzyxCTzyxDD
LCςςξ
γ
γ
, (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 γ depends on materials of heterostructure and could be integer in the fol-
lowing interval γ ∈[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. ζ1= ζ2= 0). We de-
termine spatio-temporal distributions of concentrations of point radiation defects by solving of the
following system of equations [21,22]
( ) ( ) ( ) ( ) ( )+
∂
∂
∂
∂+
∂
∂
∂
∂=
∂
∂
y
tzyxITzyxD
yx
tzyxITzyxD
xt
tzyxIII
,,,,,,
,,,,,,
,,,
( ) ( ) ( ) ( ) ( )−−
∂
∂
∂
∂+ tzyxVtzyxITzyxk
z
tzyxITzyxD
zVII
,,,,,,,,,,,,
,,, ,
( ) ( )tzyxITzyxkII
,,,,,, 2
,− (4)
( ) ( ) ( ) ( ) ( )+
∂
∂
∂
∂+
∂
∂
∂
∂=
∂
∂
y
tzyxVTzyxD
yx
tzyxVTzyxD
xt
tzyxVVV
,,,,,,
,,,,,,
,,,
( ) ( ) ( ) ( ) ( )−−
∂
∂
∂
∂+ tzyxVtzyxITzyxk
z
tzyxVTzyxD
zVIV
,,,,,,,,,,,,
,,, ,
( ) ( )tzyxVTzyxkVV
,,,,,, 2
,−
with initial
ρ (x,y,z,0)=fρ (x,y,z) (5a)
and boundary conditions
( )0
,,,
0
=∂
∂
=xx
tzyxρ,
( )0
,,,=
∂
∂
= xLxx
tzyxρ,
( )0
,,,
0
=∂
∂
=yy
tzyxρ,
( )0
,,,=
∂
∂
= yLyy
tzyxρ,
( )0
,,,
0
=∂
∂
=zz
tzyxρ,
( )0
,,,=
∂
∂
= zLzz
tzyxρ. (5b)
Here ρ =I,V; I (x,y,z,t) is the spatio-temporal distribution of concentrations of interstitials; Dρ(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 ΦV (x, y,z,t) and diin-
terstitials ΦI (x,y,z,t) by solving the following system of equations [21,22]
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
28
( )( )
( )( )
( )+
Φ+
Φ=
ΦΦΦ
y
tzyxTzyxD
yx
tzyxTzyxD
xt
tzyxI
I
I
I
I
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,,,,
,,,,,,
,,,
( )( )
( ) ( ) −+
Φ+ Φ tzyxITzyxk
z
tzyxTzyxD
zII
I
I,,,,,,
,,,,,, 2
,∂
∂
∂
∂
( ) ( )tzyxITzyxkI ,,,,,,− (6)
( )( )
( )( )
( )+
Φ+
Φ=
ΦΦΦ
y
tzyxTzyxD
yx
tzyxTzyxD
xt
tzyxV
V
V
V
V
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,,,,
,,,,,,
,,,
( )( )
( ) ( ) −+
Φ+ Φ tzyxVTzyxk
z
tzyxTzyxD
zVV
V
V,,,,,,
,,,,,, 2
,∂
∂
∂
∂
( ) ( )tzyxVTzyxkV ,,,,,,−
with boundary and initial conditions
( )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)
Here DΦI(x,y,z,T) and DΦV(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]
( ) ( ) ( ) ( ) ( ) ( )+
∂
∂
∂
∂+
∂
∂
∂
∂=
∂
∂
y
tzyxTTzyx
yx
tzyxTTzyx
xt
tzyxTTc
,,,,,,
,,,,,,
,,,λλ
( ) ( ) ( )tzyxpz
tzyxTTzyx
z,,,
,,,,,, +
∂
∂
∂
∂+ λ (8)
with boundary and initial conditions
( )0
,,,
0
=∂
∂
=xx
tzyxT,
( )0
,,,=
∂
∂
= xLxx
tzyxT,
( )0
,,,
0
=∂
∂
=yy
tzyxT, (9)
( )0
,,,=
∂
∂
= yLxy
tzyxT,
( )0
,,,
0
=∂
∂
=zz
tzyxT,
( )0
,,,=
∂
∂
= zLxz
tzyxT, 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-η 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]); λ 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 λ(x,y,z,T)=λass(x,y,z) [1+µ
(Td/T(x,y,z,t))ϕ] (see, for example, [23]); p(x,y,z,t) is the volumetric density of heat power, gener-
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
29
ated in heterostructure during annealing; α (x,y,z,T)=λ(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: α ass (x,y,z) =λass(x,y,z)/cas s=α0ass[1+εT
gT(x,y,z)]. Farther we determine solution of Eq.(8) as the following power series
( ) ( )∑ ∑=∞
=
∞
=0 0
,,,,,,i j
ij
ji
TtzyxTtzyxT µε . (10)
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) (i≥1, j≥1).
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) (i≥0, j≥0). The conditions are presented in the Appendix. The equations for the func-
tions Tij(x,y,z,t) (i≥0, j≥0) 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 ε 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: Dρ(x,y,z,T)=D0ρ[1+ερgρ(x,y,z,T)], where D0ρ are the average values of diffu-
sion coefficients, 0≤ερ< 1, |gρ(x,y,z,T)|≤1, ρ =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+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 appropriate average values of these
parameters, 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, φ
= z/Lz, ( ) ( ) *,,,,,,~
VtzyxVtzyxV = , VIVI
DDkL00,0
2=ω , VI
DDkL00,0
2
ρρρ =Ω ,
2
00LtDD
VI=ϑ . The introduction leads to the following transformation of equations (4) and
conditions (5)
( ) ( )[ ] ( )×+
∂
∂+
∂
∂=
∂
∂
VI
I
II
VI
I
DD
DITg
DD
DI
00
0
00
0 ,,,~
,,,1,,,
~
χ
ϑφηχφηχε
χϑ
ϑφηχ
( )[ ] ( ) ( )[ ] ×+∂
∂+
∂
∂+
∂
∂× Tg
DD
DITg
II
VI
I
II,,,1
,,,~
,,,1
00
0 φηχεφη
ϑφηχφηχε
η
( ) ( )[ ] ( ) ( )−+−
∂
∂× ϑφηχϑφηχφηχεω
φ
ϑφηχ,,,
~,,,
~,,,1
,,,~
,, VITgI
VIVI
( )[ ] ( )ϑφηχφηχε ,,,~
,,,1 2
,,ITg
IIIII+Ω− (11)
( ) ( )[ ] ( )×+
∂
∂+
∂
∂=
∂
∂
VI
V
VV
VI
V
DD
DVTg
DD
DV
00
0
00
0 ,,,~
,,,1,,,
~
χ
ϑφηχφηχε
χϑ
ϑφηχ
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
30
( )[ ] ( ) ( )[ ] ×+∂
∂+
∂
∂+
∂
∂+ Tg
DD
DVTg
VV
VI
V
VV,,,1
,,,~
,,,100
0 φηχεφη
ϑφηχφηχε
η
( ) ( )[ ] ( ) ( )−+−
∂
∂× ϑφηχϑφηχφηχεω
φ
ϑφηχ,,,
~,,,
~,,,1
,,,~
,, VITgV
VIVI
( )[ ] ( )ϑφηχφηχε ,,,~
,,,1 2
,,VTg
VVVVI+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
= . (12)
We determine solutions of Eqs.(11) as the following power series (see [24-26])
( ) ( )∑ ∑ ∑Ω=∞
=
∞
=
∞
=0 0 0
,,,~,,,~i j k
ijk
kji ϑφηχρωεϑφηχρ ρρ . (13)
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 ( )ϑφηχ ,,,~
000I and
( )ϑφηχ ,,,~
000V and corrections for them ( )ϑφηχ ,,,
~ijk
I and ( )ϑφηχ ,,,~
ijkV , i ≥1, j ≥1, k ≥1. 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:
DΦρ(x,y,z,T)=D0Φρ[1+εΦρgΦρ(x,y,z,T)], where D0Φρ are the average values of diffusion coefficients.
In this situation Eqs.(6) will be transformed to the following form
( ) ( )[ ] ( ) ( )[ ]
×++
Φ
+=Φ
ΦΦΦΦΦ Tzyxgyx
tzyxTzyxg
xD
t
tzyxII
I
III
I ,,,1,,,
,,,1,,,
0 ε∂
∂
∂
∂ε
∂
∂
∂
∂
( )( )[ ] ( )
+
Φ
++Φ
× ΦΦΦΦz
tzyxTzyxg
zDD
y
tzyxI
IIII
I
∂
∂ε
∂
∂
∂
∂ ,,,,,,1
,,,00
( )[ ] ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkTzyxgIIIII
,,,,,,,,,,,,,,,1 2
, −+
+× ΦΦε
( )( )[ ] ( )
( )[ ]
×++
Φ
+=Φ
ΦΦΦΦΦ Tzyxgyx
tzyxTzyxg
xD
t
tzyxVV
V
VVV
V ,,,1,,,
,,,1,,,
0 ε∂
∂
∂
∂ε
∂
∂
∂
∂
( )( )[ ] ( )
+
Φ
++Φ
× ΦΦΦΦz
tzyxTzyxg
zDD
y
tzyxV
VVVV
V
∂
∂ε
∂
∂
∂
∂ ,,,,,,1
,,,00
( )[ ] ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkTzyxgVVVVV
,,,,,,,,,,,,,,,1 2
, −+
+× ΦΦε .
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
31
Farther we determine solutions of the above equations as the following power series
( ) ( )∑ Φ=Φ∞
=Φ
0
,,,,,,i
i
itzyxtzyx ρρρ ε . (11)
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 Φρ0(x,y,z,t) and corrections for them Φρi(x,y,z,t) (i ≥1) and appropriate condi-
tions for all functions Φρi(x,y,z,t) (i ≥0). 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+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 determine solution of Eq.(1) in the following form
( ) ( )∑ ∑=∞
=
∞
=0 1
,,,,,,i j
ij
ji
LtzyxCtzyxC ξε .
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 ≥1, j ≥1) 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
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
32
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 coefficient in doped layer is larger, than in nearest layer.
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
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
33
Fig. 3. Spatial distributions of implanted dopant concentration after annealing with continuous Θ
= 0.0048(Lx2+Ly
2+Lz2)/D0 (curves 1 and 3) and Θ = 0.0057(Lx
2+Ly2+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
( ) ( )[ ]∫ ∫ ∫ −Θ=x y zL L L
zyx
xdydzdzyxzyxCLLL
U0 0 0
,,,,,1
ψ , (15)
where ψ (x) is the approximation function, Θ is the required value of annealing time.
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.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 ξ = γ = 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 ε for a/L=1/2 and ξ =
γ = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of parameter ξ
for a/L=1/2 and ε = γ = 0. Curve 4 is the dependence of dimensionless optimal annealing time on
value of parameter γ for a/L=1/2 and ε = ξ = 0
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
34
0.0 0.1 0.2 0.3 0.4 0.5a/L, ξ, ε, γ
0.00
0.04
0.08
0.12
Θ D
0 L
-2
3
2
4
1
Fig.5. Dependences of dimensionless optimal annealing time for doping by ion implantation,
which have been obtained by minimization of mean-squared error, on several parameters. Curve 1
is the dependence of dimensionless optimal annealing time on the relation a/L and ξ = γ = 0 for
equal to each other values of dopant diffusion coefficient in all parts of hetero structure. Curve 2
is the dependence of dimensionless optimal annealing time on value of parameter ε for a/L=1/2
and ξ = γ = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of
parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the dependence of dimensionless optimal an-
nealing time on value of parameter γ for a/L=1/2 and ε = ξ = 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.В.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.
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
35
APPENDIX
Equations for the functions Tij(x,y,z,t) (i≥0, j≥0) have been obtained by substitution the power se-
ries (10) in the equation (8) and equating terms with equal powers of parameters εT and µ. The
equations could be written as
( ) ( ) ( ) ( ) ( )
ass
ass
tzyxp
z
tzyxT
y
tzyxT
x
tzyxT
t
tzyxT
να
,,,,,,,,,,,,,,,2
00
2
2
00
2
2
00
2
0
00 +
∂
∂+
∂
∂+
∂
∂=
∂
∂
( ) ( ) ( ) ( )×+
∂
∂+
∂
∂+
∂
∂=
∂
∂ass
iii
ass
i
z
tzyxT
y
tzyxT
x
tzyxT
t
tzyxT02
0
2
2
0
2
2
0
2
0
0,,,,,,,,,,,,
αα
( ) ( ) ( ) ( )
+∂
∂+
∂
∂× −−
2
10
2
2
10
2,,,
,,,,,,
,,,y
tzyxTTzyxg
x
tzyxTTzyxg i
T
i
T
( ) ( )
∂
∂
∂
∂+ −
z
tzyxTTzyxg
z
i
T
,,,,,, 10 , i ≥1
( ) ( ) ( ) ( )( )
×+
∂
∂+
∂
∂+
∂
∂=
∂
∂
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxT
t
tzyxTdass
ass,,,
,,,,,,,,,,,,
00
0
2
01
2
2
01
2
2
01
2
0
01
ϕ
ϕαα
( ) ( ) ( )( )
( )
+
∂
∂−
∂
∂+
∂
∂+
∂
∂×
+
2
00
1
00
0
2
00
2
2
00
2
2
00
2 ,,,
,,,
,,,,,,,,,
x
tzyxT
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxTdass
ϕ
ϕαϕ
( ) ( )
∂
∂+
∂
∂+
2
00
2
00,,,,,,
z
tzyxT
y
tzyxT
( ) ( ) ( ) ( )( )
×+
∂
∂+
∂
∂+
∂
∂=
∂
∂
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxT
t
tzyxTdass
ass,,,
,,,,,,,,,,,,
00
0
2
02
2
2
02
2
2
02
2
0
02
ϕ
ϕαα
( ) ( ) ( )( )
( ) ( )
+
∂
∂
∂
∂−
∂
∂+
∂
∂+
∂
∂×
+x
tzyxT
x
tzyxT
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxTdass
,,,,,,
,,,
,,,,,,,,,0100
1
00
0
2
01
2
2
01
2
2
01
2
ϕ
ϕαϕ
( ) ( ) ( ) ( )
∂
∂
∂
∂+
∂
∂
∂
∂+
z
tzyxT
z
tzyxT
y
tzyxT
y
tzyxT ,,,,,,,,,,,,01000100
( ) ( ) ( )( )
( )( ) ( )
×∂
∂+
∂
∂+
∂
∂=
∂
∂2
00
2
2
00
2
00
01
02
11
2
0
11,,,,,,
,,,,,,
,,,,,,,,,
y
tzyxT
x
tzyxTTzyxg
tzyxT
tzyxT
x
tzyxT
t
tzyxTTassass
αα
( ) ( )( )
( )( )
+
∂
∂
∂
∂+
∂
∂
∂
∂+×
x
tzyxTTzyxg
xz
tzyxTTzyxg
zTzyxg
TassTT
,,,,,,
,,,,,,,,, 01
0
00 α
( )( )[ ] ( )
( )[ ] ( )+
∂
∂++
∂
∂++
∂
∂+
2
01
2
2
01
2
2
01
2 ,,,,,,1
,,,,,,1
,,,
z
tzyxTTzyxg
y
tzyxTTzyxg
x
tzyxTTT
( )( )
( ) ( )( )
( ) ( )( )
×+
∂
∂+
∂
∂+
+++tzyxT
tzyxT
y
tzyxT
tzyxT
tzyxT
x
tzyxT
tzyxT
tzyxTT
dass,,,
,,,,,,
,,,
,,,,,,
,,,
,,,1
00
10
2
00
2
1
00
10
2
00
2
1
00
10
0 ϕϕϕ
ϕα
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
36
( )( )
( ) ( ) ( )+
∂
∂+
∂
∂+
∂
∂+
∂
∂×
2
10
2
2
10
2
2
10
2
00
0
2
00
2 ,,,,,,,,,
,,,
,,,
z
tzyxT
y
tzyxT
x
tzyxT
tzyxT
T
z
tzyxTdass
ϕ
ϕα
( )( )
( )( )
( )
×+
∂
∂
∂
∂+
∂
∂+ Tzyxg
z
tzyxTTzyxg
zx
tzyxTTzyxg
TTT,,,
,,,,,,
,,,,,, 00
2
10
2
( )( ) ( ) ( ) ( )
×
∂
∂+
∂
∂
∂
∂−
∂
∂×
y
tzyxT
x
tzyxT
x
tzyxT
y
tzyxT
tzyxT
Tdass
,,,,,,,,,,,,
,,,
100010
2
10
2
00
0
ϕ
ϕα
( ) ( ) ( )( )
( )( )
×−
∂
∂
∂
∂+
∂
∂×
++tzyxT
TzyxgT
tzyxT
T
z
tzyxT
z
tzyxT
y
tzyxTT
dass
dass
,,,
,,,
,,,
,,,,,,,,,1
00
01
00
0001000
ϕ
ϕ
ϕ
ϕ
ϕααϕ
( ) ( ) ( )
∂
∂+
∂
∂+
∂
∂×
2
00
2
00
2
00 ,,,,,,,,,
z
tzyxT
y
tzyxT
x
tzyxT.
Conditions for the functions Tij(x,y,z,t) (i≥0, j≥0) have been obtained by the same procedure as appropriate equations and could be written as
( )0
,,,
0
=∂
∂
=x
ij
x
tzyxT,
( )0
,,,=
∂
∂
= xLx
ij
x
tzyxT,
( )0
,,,
0
=∂
∂
=y
ij
y
tzyxT,
( )0
,,,=
∂
∂
= yLx
ij
y
tzyxT,
( )0
,,,
0
=∂
∂
=z
ij
z
tzyxT,
( )0
,,,=
∂
∂
= zLx
ij
z
tzyxT, T00(x,y,z,0)=fT(x,y,z), Tij(x,y,z,0)=0, i ≥1, j ≥1.
Solutions of the equations for the functions Tij(x,y,z,t) (i≥0, j≥0) 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
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫+∫ ∫ ∫=∞
=1 00 0 000
2,,
1,,,
n
L
nnTnnn
zyx
L L L
T
zyx
xx y z
uctezcycxcLLL
udvdwdwvufLLL
tzyxT
( ) ( ) ( ) ( )×+∫ ∫ ∫ ∫+∫ ∫×
zyx
t L L L
asszyx
L L
TnnLLL
dudvdwdwvup
LLLudvdwdwvufwcvc
x y zy z 2,,,1,,
0 0 0 00 0
τν
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫−×∞
=1 0 0 0 0
,,,
n
t L L L
ass
nnnnTnTnnn
x y z
dudvdwdwvup
wcvcucetezcycxc τν
ττ ,
where cn(χ) = cos (π nχ/L), ( )
++−=
2220
22 111exp
zyx
assnTLLL
tnte απ ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−=∞
=1 0 0 0 02
0
0,,,2,,,
n
t L L L
TnnnnTnTnnn
zyx
ass
i
x y z
TwvugwcvcusetezcycxcnLLL
tzyxT ταπ
( )( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−+
∂
∂×
∞
=
−
1 0 0 02
010 2,,,
n
t L L
nnnTnTnnn
zyx
assix y
vsucetezcycxcnLLL
dudvdwdu
wvuTτ
απτ
τ
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
37
( ) ( )( )
( ) ( ) ( )×∑+∫∂
∂×
∞
=
−
12
0
0
10 2,,,
,,,n
nnn
zyx
assL
i
Tnzcycxcn
LLLdudvdwd
v
wvuTTwvugwc
z απτ
τ
( ) ( ) ( ) ( ) ( ) ( )( )
∫ ∫ ∫ ∫∂
∂−× −
t L L Li
TnnnnTnT
x y z
dudvdwdw
wvuTTwvugwsvcucete
0 0 0 0
10 ,,,,,, τ
ττ , i ≥1,
where sn(χ) = sin (π n χ/L);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−=∞
=1 0 0 0 02001
2,,,
n
t L L L
nnnnTnTnnn
zyx
d
ass
x y z
wcvcusetezcycxcnLLL
TtzyxT τ
πα
ϕ
( )( )
( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−+∂
∂×
∞
=1 0 020
00
2
00
22
,,,
,,,
n
t L
nnTnTnnn
zyx
d
ass
x
ucetezcycxcLLL
T
wvuT
dudvdwd
u
wvuTτ
πα
τ
ττ ϕ
ϕ
( ) ( ) ( )( )
( ) ( ) ( ) ( )×∑+∫ ∫∂
∂×
∞
=120
0 0 00
2
00
22
,,,
,,,
nnTnnn
zyx
d
ass
L L
nntezcycxc
LLL
T
wvuT
dudvdwd
v
wvuTwcvsn
y zϕ
ϕ
πα
τ
ττ
( ) ( ) ( ) ( ) ( )( )
( )×∑−∫ ∫ ∫ ∫∂
∂−×
∞
=1
0
0 0 0 0 00
2
00
2
2,,,
,,,
nn
zyx
ass
d
t L L L
nnnnTxc
LLLT
wvuT
dudvdwd
u
wvuTwsvcucen
x y z αϕ
τ
τττ ϕ
ϕ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
−∫ ∫ ∫ ∫
∂
∂−×
+
t L L L
nnnnTnTnn
x y z
wvuT
dudvdwd
u
wvuTwcvcucetezcyc
0 0 0 01
00
2
00
,,,
,,,
τ
τττ
ϕ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫
∂
∂−−
∞
=1 0 0 0 0
2
000 ,,,2
n
t L L L
nnnnTnTnnn
zyx
assx y z
v
wvuTwcvcucetezcycxc
LLL
ττ
αϕ
( )( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−−×
∞
=+
1 0 0 0
0
1
00
2,,, n
t L L
nnnTnTnnn
zyx
ass
dd
x y
vcucetezcycxcLLL
TwvuT
dudvdwdT τ
αϕ
τ
τ ϕ
ϕ
ϕ
( ) ( )( )∫
∂
∂×
+
zL
nwvuT
dudvdwd
w
wvuTwc
01
00
2
00
,,,
,,,
τ
ττϕ
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−=∞
=1 0 0 0 02002
2,,,
n
t L L L
nnnnTnTnnn
zyx
d
ass
x y z
wcvcusetezcycxcnLLL
TtzyxT τ
πα
ϕ
( )( )
( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−+∂
∂×
∞
=1 0 020
00
2
01
2 2
,,,
,,,
n
t L
nnTnTnnn
zyx
d
ass
x
ucetezcycxcnLLL
T
wvuT
dudvdwd
u
wvuTτ
πα
τ
ττ ϕ
ϕ
( ) ( ) ( )( )
( ) ( ) ( ) ( )×∑+∫ ∫∂
∂×
∞
=120
0 0 00
2
01
22
,,,
,,,
nnTnnn
zyx
d
ass
L L
nntezcycxcn
LLL
T
wvuT
dudvdwd
v
wvuTwcvs
y zϕ
ϕ
πα
τ
ττ
( ) ( ) ( ) ( ) ( )( )
( ) ×∑−∫ ∫ ∫ ∫∂
∂−×
∞
=12
0
0 0 0 0 00
2
01
2
2,,,
,,,
nn
zyx
ass
d
t L L L
nnnnTxc
LLLT
wvuT
dudvdwd
w
wvuTwsvcuce
x y z απ
τ
τττ ϕ
ϕ
( ) ( ) ( ) ( ) ( ) ( ) ( )( )
×∫ ∫ ∫ ∫∂
∂
∂
∂−×
+
t L L L
nnnnTn
x y z
wvuT
dudvdwd
u
wvuT
u
wvuTwcvcuceyc
0 0 0 01
00
0100
,,,
,,,,,,
τ
ττττ
ϕ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫−−×∞
=1 0 0 0 02
02n
t L L L
nnnnTnTnnn
zyx
ass
dnTn
x y z
wcvcucetezcycxcLLL
Ttezc ταπϕ
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
38
( ) ( )( )
( ) ( ) ( ) ( )×∑−∂
∂
∂
∂×
∞
=+
12
0
1
00
0100 2,,,
,,,,,,
nnTnnn
zyx
ass tezcycxcLLLwvuT
dudvdwd
v
wvuT
v
wvuT απ
τ
τττϕ
( ) ( ) ( ) ( ) ( ) ( )( )∫ ∫ ∫ ∫
∂
∂
∂
∂−×
+
t L L L
nnnnTd
x y z
wvuT
dudvdwd
w
wvuT
w
wvuTwcvcuceT
0 0 0 01
00
0100
,,,
,,,,,,
τ
ττττ
ϕ
ϕ ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−=∞
=1 0 0 0 0
0
11 ,,,2
,,,n
t L L L
TnnnTnTnnn
zyx
assx y z
TwvugvcucetezcycxcLLL
tzyxT τα
( ) ( ) ( ) ( ) ( ) ( )×
∂
∂
∂
∂+
∂
∂+
∂
∂× wc
w
wvuTwg
wv
wvuTTwvug
w
wvuTnTT
τττ ,,,,,,,,,
,,,00
2
00
2
2
00
2
( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−+×∞
=1 0 0 0
0
00
012
,,,
,,,
n
t L L
nnnTnTnnn
zyx
assx y
vcucetezcycxcLLL
dudvdwdwvuT
wvuTτ
ατ
τ
τ
( ) ( )[ ] ( ) ( )[ ] ( )∫
+∂
∂++
∂
∂++
∂
∂×
zL
TTv
wvuTTwvug
u
wvuTTwvug
u
wvuT
02
01
2
2
01
2
2
01
2,,,
,,,1,,,
,,,1,,, τττ
( ) ( ) ( ) ( ) ( ) ( ) ×∑+
∂
∂
∂
∂+
∞
=1
001 2,,,
,,,n
nnn
zyx
dass
nTzcycxc
LLL
Tdudvdwdwc
w
wvuTTwvug
w
ϕατ
τ
( ) ( ) ( ) ( ) ( ) ( )( )
( ) ( )( )∫ ∫ ×∫ ∫
+
∂
∂−×
++
t L L L
nnnnTnT
x y z
wvuT
wvuT
u
wvuT
wvuT
wvuTwcvcucete
0 0 0 01
00
10
2
00
2
1
00
10
,,,
,,,,,,
,,,
,,,
τ
ττ
τ
ττ
ϕϕ
( ) ( )( )
( ) ( ) ( ) ( )×∑+
∂
∂+
∂
∂×
∞
=+
12
00
2
1
00
10
2
00
2
2,,,
,,,
,,,,,,
nnnn
zcycxcdudvdwdw
wvuT
wvuT
wvuT
v
wvuTτ
τ
τ
ττϕ
( ) ( ) ( ) ( ) ( ) ( ) ( )×∫ ∫ ∫ ∫
∂
∂+
∂
∂+
∂
∂−×
t L L L
nnnnT
x y z
w
wvuT
v
wvuT
u
wvuTwcvcuce
0 0 0 02
10
2
2
10
2
2
10
2 ,,,,,,,,, ττττ
( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫−+×∞
=1 0 0 0
0
00
0 2,,, n
t L L
nnnTnTnnn
zyx
dass
zyx
dass
nT
x y
vcucetezcycxcLLL
T
wvuT
dudvdwd
LLL
Tte τ
α
τ
τα ϕ
ϕ
ϕ
( ) ( ) ( ) ( ) ( ) ( )∫
×+
∂
∂
∂
∂+
∂
∂×
zL
TTTnTwvug
w
wvuTTwvug
wu
wvuTTwvugwc
0
00
2
00
2
,,,,,,
,,,,,,
,,,ττ
( )( )
( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−
∂
∂×
∞
=1 0 0
0
00
2
00
2
2,,,
,,,
n
t L
nnTnTnnn
zyx
dassx
ucetezcycxcLLL
T
wvuT
dudvdwd
v
wvuTτ
αϕ
τ
ττ ϕ
ϕ
( ) ( ) ( ) ( ) ( ) ( )∫ ∫
×
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂×
y zL L
nw
wvuT
v
wvuT
v
wvuT
u
wvuT
u
wvuTvc
0 0
1000100010,,,,,,,,,,,,,,, τττττ
( )( )
( )( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫−−
∂
∂×
∞
=+
1 0 0
0
1
00
00 2,,,
,,,
n
t L
nnTnTnnn
zyx
dass
n
x
ucetezcycxcLLL
T
wvuT
dudvdwdwc
w
wvuTτ
αϕ
τ
ττ ϕ
ϕ
( ) ( ) ( ) ( ) ( )( )∫ ∫
∂
∂+
∂
∂+
∂
∂×
+
y zL L
nnwvuT
dudvdwd
w
wvuT
v
wvuT
u
wvuTwcvc
0 01
00
2
00
2
00
2
00
,,,
,,,,,,,,,
τ
ττττϕ
.
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
39
Equations for the functions ( )ϑφηχ ,,,~
ijkI and ( )ϑφηχ ,,,
~ijk
V , i ≥0, j ≥0, k ≥0 and conditions for
them have been obtain by the same procedure as for the functions Tij(x,y,z,t)
( ) ( ) ( ) ( )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 ;
( ) ( ) ( ) ( )+
∂
∂+
∂
∂+
∂
∂=
∂
∂2
00
2
2
00
2
2
00
2
0
000,,,
~,,,
~,,,
~,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑχiii
V
IiIII
D
DI
( )( )
( )( )
+
∂
∂
∂
∂+
∂
∂
∂
∂+ −−
η
ϑφηχφηχ
ηχ
ϑφηχφηχ
χ
,,,~
,,,,,,
~
,,, 100
0
0100
0
0 i
I
V
Ii
I
V
II
TgD
DITg
D
D
( )( )
∂
∂
∂
∂+ −
φ
ϑφηχφηχ
φ
,,,~
,,, 100
0
0 i
I
V
II
TgD
D, i ≥1,
( ) ( ) ( ) ( )+
∂
∂+
∂
∂+
∂
∂=
∂
∂2
00
2
2
00
2
2
00
2
0
000 ,,,~
,,,~
,,,~
,~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑχiii
I
ViVVV
D
DV
( )( )
( )( )
+
∂
∂
∂
∂+
∂
∂
∂
∂+ −−
η
ϑφηχφηχ
ηχ
ϑφηχφηχ
χ
,,,~
,,,,,,
~
,,, 100
0
0100
0
0 i
V
I
Vi
V
I
VV
TgD
DVTg
D
D
( ) ( )
∂
∂
∂
∂+ −
φ
ϑφηχφηχ
φ
,,,~
,,, 100
0
0 i
V
I
VV
TgD
D, i ≥1;
( ) ( ) ( ) ( )−
∂
∂+
∂
∂+
∂
∂=
∂
∂2
010
2
2
010
2
2
010
2
0
0010,,,
~,,,
~,,,
~,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ III
D
DI
V
I
( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,,~
,,,~
,,,1000000,,
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
++− ;
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
40
( ) ( ) ( ) ( )−
∂
∂+
∂
∂+
∂
∂=
∂
∂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+− ;
( ) ( ) ( ) ( )+
∂
∂+
∂
∂+
∂
∂=
∂
∂2
110
2
2
110
2
2
110
2
0
0110,,,
~,,,
~,,,
~,,,
~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ III
D
DI
V
I
( ) ( ) ( ) ( )
+
∂
∂
∂
∂+
∂
∂
∂
∂+
η
ϑφηχφηχ
ηχ
ϑφηχφηχ
χ
,,,~
,,,,,,
~
,,, 010010
0
0I
TgI
TgD
DII
V
I
( )( )
( )[ ] ( ) ( )[ ++−
∂
∂
∂
∂+ ϑφηχϑφηχφηχε
φ
ϑφηχφηχ
φ,,,
~,,,
~,,,1
,,,~
,,,000100,,
010 VITgI
TgIIIII
( ) ( )]ϑφηχϑφηχ ,,,~
,,,~
100000 VI+
( ) ( ) ( ) ( )+
∂
∂+
∂
∂+
∂
∂=
∂
∂2
110
2
2
110
2
2
110
2
0
0110 ,,,~
,,,~
,,,~
,,,~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ VVV
D
DV
I
V
( ) ( ) ( ) ( )
+
∂
∂
∂
∂+
∂
∂
∂
∂+
η
ϑφηχφηχ
ηχ
ϑφηχφηχ
χ
,,,~
,,,,,,
~
,,, 010010
0
0V
TgV
TgD
DVV
I
V
( ) ( ) ( )[ ] ( )[ ×+−
∂
∂
∂
∂+ ϑφηχφηχε
φ
ϑφηχφηχ
φ,,,
~,,,1
,,,~
,,, 100,,
010 VTgV
TgVVVVV
( ) ( ) ( )]ϑφηχϑφηχϑφηχ ,,,~
,,,~
,,,~
100000000 IVI +× ;
( ) ( ) ( ) ( )−
∂
∂+
∂
∂+
∂
∂=
∂
∂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
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
41
( ) ( ) ( )[ ] ( ) ( )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
+−
∂
∂
∂
∂+ ;
( ) ( ) ( ) ( )−
∂
∂+
∂
∂+
∂
∂=
∂
∂2
011
2
2
011
2
2
011
2
0
0011 ,,,~
,,,~
,,,~
,,,~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ III
D
DI
V
I
( )[ ] ( ) ( ) ( )[ ] ×+−+− TgIITgVIVIIIII
,,,1,,,~
,,,~
,,,1,,010000,,
φηχεϑφηχϑφηχφηχε
( ) ( )ϑφηχϑφηχ ,,,~
,,,~
000001VI×
( ) ( ) ( ) ( )−
∂
∂+
∂
∂+
∂
∂=
∂
∂2
011
2
2
011
2
2
011
2
0
0011 ,,,~
,,,~
,,,~
,,,~
φ
ϑφηχ
η
ϑφηχ
χ
ϑφηχ
ϑ
ϑφηχ VVV
D
DV
I
V
( )[ ] ( ) ( ) ( )[ ]×+−+− tgVVTgVIVIVVVV
,,,1,,,~
,,,~
,,,1,,010000,,
φηχεϑφηχϑφηχφηχε
( ) ( )ϑφηχϑφηχ ,,,~
,,,~
001000VI× ;
( )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 have been obtained by standard Fourier approach and could be
written as
( ) ( ) ( ) ( ) ( )∑+=∞
=1000
21,,,~
nnn ecccF
LLϑφηχϑφηχρ ρρ ,
where ( ) ( ) ( ) ( )∫ ∫ ∫=1
0
1
0
1
0*
,,coscoscos1
udvdwdwvufwnvnunFnn ρρ πππ
ρ, cn(χ) = cos (π n χ),
( ) ( )IVnI
DDne00
22exp ϑπϑ −= , ( ) ( )VInV
DDne00
22exp ϑπϑ −= ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0
1
0
1
0
1
00
0
00 ,,,2,,,~
nInnnInIn
V
I
iTwvugvcuseecccn
D
DI
ϑ
τϑφηχπϑφηχ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂
∂×
∞
=
−
1 0
1
00
0100 2,,,
~
nnnInIn
V
Ii
nuceecccn
D
Ddudvdwd
u
wvuIwc
ϑ
τϑφηχττ
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
42
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫∂
∂×
∞
=
−
10
01
0
1
0
100 2,,,
~
,,,n
n
V
Ii
Inncccn
D
Ddudvdwd
v
wvuITwvugwcvs φηχπτ
τπ
( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫
∂
∂−× −
ϑ
ττ
τϑ0
1
0
1
0
1
0
100,,,
~
,,, dudvdwdw
wvuITwvugwsvcucee i
InnnnInI, 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
( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫∂
∂×
∞
=
−
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
00
0
0202,,,~
nnnnnnnnn
V
I wcvcuceecccD
D ϑ
ρρ τϑφηχϑφηχρ
( ) ( ) ( ) ( )[ ]×+× ττττ ,,,~
,,,~
,,,~
,,,~
010000000010wvuVwvuIwvuVwvuI
( )[ ] τε dudvdwdTwvugVIVI
,,,1,,
+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0
1
0
1
0
1
0001
2,,,~n
nnnnnnnnwcvcuceeccc
ϑ
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 ,,,~,,,~,,,1 000001,,+× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=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
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
43
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[∑ ∫ ∫ ∫ ∫ ×+−−×∞
=1 0
1
0
1
0
1
0,
12n
VInnnnInnnInnnnvcvcuceccecccc
ϑ
ετφηϑχφηχ
( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvugVI
,,,~
,,,~
,,,~
,,,~
,,,100000000100,
+×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=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
VInnnnVnnnInnnnvcvcuceccecccc
ϑ
ετφηϑχφηχ
( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvug VI ,,,~
,,,~
,,,~
,,,~
,,, 100000000100, +× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0
1
0
1
0
1
00
0
1012,,,
~
nnnnnInInnn
V
I wcvcuseecccnD
DI
ϑ
τϑφηχπϑφηχ
( ) ( ) ( ) ( ) ( ) ( ) ×∑−∂
∂×
∞
=10
0001 2,,,
~
,,,n
nInnn
V
I
Iecccn
D
Ddudvdwd
u
wvuITwvug ϑφηχπτ
τ
( ) ( ) ( ) ( ) ( ) ( )×−∫ ∫ ∫ ∫
∂
∂−×
V
I
InnnnID
Ddudvdwd
v
wvuITwvugwcvsuce
0
0
0
1
0
1
0
1
0
001 2,,,
~
,,, πττ
τϑ
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫
∂
∂−×
∞
=1 0
1
0
1
0
1
0
001,,,
~
,,,n
InnnnInIdudvdwd
w
wvuITwvugwsvcuceen
ϑ
ττ
τϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ×∑ ∫ ∫ +−−×∞
=1 0
1
0,,
,,,12n
VIVInInInnnnnnTwvugeecccccc
ϑ
ετϑφηχφηχ
( ) ( ) ( ) τττ dudvdwdwvuVwvuIucn
,,,~
,,,~
000100×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0
1
0
1
0
1
00
0
1012,,,
~
nnnnnVnVnnn
I
V wcvcuseecccnD
DV
ϑ
τϑφηχπϑφηχ
( ) ( ) ( ) ( ) ( ) ( ) ×∑−∂
∂×
∞
=10
0001 2,,,
~
,,,n
nVnnn
I
V
Vecccn
D
Ddudvdwd
u
wvuVTwvug ϑφηχπτ
τ
( ) ( ) ( ) ( ) ( ) ( )×−∫ ∫ ∫ ∫
∂
∂−×
I
V
VnnnnVD
Ddudvdwd
v
wvuVTwvugwcvsuce
0
0
0
1
0
1
0
1
0
001 2,,,
~
,,, πττ
τϑ
( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫
∂
∂−×
∞
=1 0
1
0
1
0
1
0
001,,,
~
,,,n
VnnnnVnVdudvdwd
w
wvuVTwvugwsvcuceen
ϑ
ττ
τϑ
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
44
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[∑ ∫ ∫ ∫ ∫ ×+−−×∞
=1 0
1
0
1
0
1
0,
12n
VInnnnVnVnnnnnnwcvcuceecccccc
ϑ
ετϑφηχφηχ
( )] ( ) ( ) τττ dudvdwdwvuVwvuITwvugVI
,,,~
,,,~
,,,100000,
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−=∞
=1 0
1
0
1
0
1
0011
2,,,~
nnnnnInInnn
wcvcuceecccIϑ
τϑφηχϑφηχ
( )[ ] ( ) ( ) ( )[ ] ×+++× TwvugwvuIwvuITwvugVIVIIIII
,,,1,,,~
,,,~
,,,1,,010000,,
εττε
( ) ( ) τττ dudvdwdwvuVwvuI ,,,~
,,,~
000001×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−=∞
=1 0
1
0
1
0
1
0011
2,,,~
nnnnnVnVnnn
wcvcuceecccVϑ
τϑφηχϑφηχ
( )[ ] ( ) ( ) ( )[ ] ×+++× TwvugwvuVwvuVTwvugVIVIVVVV
,,,1,,,~
,,,~
,,,1,,010000,,
εττε
( ) ( ) τττ dudvdwdwvuVwvuI ,,,~
,,,~
001000× .
Equations for initial-order approximations of distributions of concentrations of simplest complex-
es of radiation defects Φρ0(x,y,z,t) and corrections for them Φρi(x,y, z,t), i ≥1 and boundary and
initial conditions for them have been obtained as the functions Tij(x,y,z,t) and takes the form
( ) ( ) ( ) ( )+
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 on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
45
( )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
IIInnnnn
x y z
TwvukwvuITwvukwcvcucete0 0 0 0
2
,,,,,,,,,, ττ
ρρ
( )] ττ dudvdwdwvuI ,,,× ,
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∞
=ΦΦ
1 0 0 0 02
2,,,
n
t L L L
nnnnnnnn
zyx
i
x y z
wcvcusetezcycxcnLLL
tzyx τπ
ρρρ
( )( )
( ) ( ) ( ) ( ) ×∑−Φ
×∞
=Φ
−
Φ1
2
1 2,,,,,,
nnnnn
zyx
iI
tezcycxcnLLL
dudvdwdu
wvuTwvug
ρ
ρ
ρ
πτ
∂
τ∂
( ) ( ) ( ) ( ) ( )( )
×−∫ ∫ ∫ ∫Φ
−×−
ΦΦ
yx
t L L LiI
nnnnLL
dudvdwdv
wvuTwvugwcvsuce
x y z πτ
∂
τ∂τ ρ
ρρ
2,,,,,,
0 0 0 0
1
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
×∑ ∫ ∫ ∫ ∫Φ
−×∞
=
−
ΦΦ1 0 0 0 0
1
2
,,,1
n
t L L LiI
nnnnnnnn
z
x y z
w
wvuwsvcucetezcycxcn
L ∂
τ∂τ ρ
ρρ
( ) τρ
dudvdwdTwvug ,,,Φ× , i≥1,
where sn(x) = sin (π 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 ≥1, j ≥1) and boundary and initial conditions take the form
( ) ( ) ( ) ( )2
00
2
02
00
2
02
00
2
0
00,,,,,,,,,,,,
z
tzyxCD
y
tzyxCD
x
tzyxCD
t
tzyxCLLL
∂
∂+
∂
∂+
∂
∂=
∂
∂;
( ) ( ) ( ) ( )+
∂
∂+
∂
∂+
∂
∂=
∂
∂2
0
2
02
0
2
02
0
2
0
0 ,,,,,,,,,,,,
z
tzyxCD
y
tzyxCD
x
tzyxCD
t
tzyxCi
L
i
L
i
L
i
( ) ( ) ( ) ( )+
∂
∂
∂
∂+
∂
∂
∂
∂+ −−
y
tzyxCTzyxg
yD
x
tzyxCTzyxg
xD i
LL
i
LL
,,,,,,
,,,,,, 10
0
10
0
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
46
( ) ( )
∂
∂
∂
∂+ −
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
x ,,,
,,,,,,
,,,
,,,
,,,,,,
1
00
01
00
1
00
01 γ
γ
γ
γ
( ) ( ) ( )( )
( )+
∂
∂
∂
∂+
∂
∂×
−
LD
z
tzyxC
TzyxP
tzyxCtzyxC
zy
tzyxC0
00
1
00
01
00,,,
,,,
,,,,,,
,,,γ
γ
( )( )
( ) ( )( )
( )
+
∂
∂
∂
∂+
∂
∂
∂
∂+
y
tzyxC
TzyxP
tzyxC
yx
tzyxC
TzyxP
tzyxC
xD
L
,,,
,,,
,,,,,,
,,,
,,, 01000100
0 γ
γ
γ
γ
( )( )
( )
∂
∂
∂
∂+
z
tzyxC
TzyxP
tzyxC
z
,,,
,,,
,,,0100
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
( )( )
( ) ( ) ( )
+
∂
∂
∂
∂+
∂
∂
∂
∂+
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
∂
∂,
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
47
( )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.
Solutions of the above equations with account boundary and initial conditions could be written as
( ) ( ) ( ) ( ) ( )∑+=∞
=100
21,,,
nnCnnnnC
zyxzyx
tezcycxcFLLLLLL
tzyxC ,
where ( ) ( ) ( ) ( )∫ ∫ ∫= Φ
x y zL L L
nnnnCudvdwdwvufwcvcucF
0 0 0
,,ρ
, ( )
++−= Φ 2220
22 111exp
zyx
nCLLL
tDnteρ
π ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=1 0 0 0 020 ,,,2
,,,n
t L L L
LnnnCnCnnnnC
zyx
i
x y z
TwvugvcusetezcycxcFnLLL
tzyxC τπ
( )( )
( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂
∂×
∞
=
−
1 0 02
10 2,,,
n
t L
nnCnCnnnnC
zyx
i
n
x
ucetezcycxcFnLLL
dudvdwdu
wvuCvc τ
πτ
τ
( ) ( ) ( )( )
( ) ( ) ( ) ×∑−∫ ∫∂
∂×
∞
=
−
12
0 0
10 2,,,,,,
nnnnnC
zyx
L Li
LnnzcycxcFn
LLLdudvdwd
v
wvuCTwvugvcvs
y z πτ
τ
( ) ( ) ( ) ( ) ( ) ( )( )
∫ ∫ ∫ ∫∂
∂−× −
t L L Li
LnnnnCnC
x y z
dudvdwdw
wvuCTwvugvsvcucete
0 0 0 0
10,,,
,,, ττ
τ , i ≥1;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( )
×∑ ∫ ∫ ∫ ∫−−=∞
=1 0 0 0 0
00
201,,,
,,,2,,,
n
t L L L
nnnCnCnnnnC
zyx
x y z
TwvuP
wvuCvcusetezcycxcFn
LLLtzyxC
γ
γ ττ
π
( )( )
( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−∂
∂×
∞
=1 02
00 2,,,
n
t
nCnCnnnnC
zyx
netezcycxcFn
LLLdudvdwd
u
wvuCwc τ
πτ
τ
( ) ( ) ( )( )( )
( )( )×∑−∫ ∫ ∫
∂
∂×
∞
=12
0 0 0
0000 2,,,
,,,
,,,
nnCnC
zyx
L L L
nnnteFn
LLLdudvdwd
v
wvuC
TwvuP
wvuCwcvsuc
x y z πτ
ττγ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( )( )( )
( )∫ ∫ ∫ ∫
∂
∂−×
t L L L
nnnnCnnn
x y z
dudvdwdw
wvuC
TwvuP
wvuCwsvcucezcycxc
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
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
48
( ) ( ) ( )( )
( ) ( ) ×∑−∫∂
∂×
∞
=
−
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 ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞
=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
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
49
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
×∑ ∫ ∫ ∫ ∫−−∞
=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
.
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.
[4] N.A. Avaev, Yu.E. Naumov, V.T. Frolkin (1991). Basis of microelectronics, Radio and communica-
tion, Moscow.
[5] V.I. Lachin, N.S. Savelov (2001). Electronics, Phoenix, Rostov-na-Donu.
[6] A. Polishscuk (2004), Ultrashallow p+−n junctions in silicon: electron-beam diagnostics of sub-
surface region. Modern Electronics. 12, P. 8-11.
[7] G. Volovich, Integration of on-chip field-effect transistor switches with dopantless Si/SiGe quantum
dots for high-throughput testing (2006). Modern Electronics. 2, P. 10-17.
[8] A. Kerentsev, V. Lanin, Design and technological features of MOSFETs (2008). Power Electronics.
Issue 1. P. 34.
[9] A.O. Ageev, A.E. Belyaev, N.S. Boltovets, V.N. Ivanov, R.V. Konakova, Ya.Ya. Kudrik, P.M. Litvin,
V.V. Milenin, A.V. Sachenko, Au–TiBx−n-6H-SiC Schottky barrier diodes: the features of current
flow in rectifying and nonrectifying contacts (2009). Semiconductors. Vol. 43 (7). P. 897-903.
[10] Jung-Hui Tsai, Shao-Yen Chiu, Wen-Shiung Lour, Der-Feng Guo, High-performance InGaP/GaAs
pnp δ-doped heterojunction bipolar transistor (2009). Semiconductors. Vol. 43 (7). P. 971-974.
[11] E.I. Gol’dman, N.F. Kukharskaya, V.G. Naryshkina, G.V. Chuchueva, The manifestation of excessive
centers of the electron-hole pair generation, appeared as a result to field and thermal stresses, and their
subsequent annihilation in the dynamic current-voltage characteristics of Si-MOS-structures with the
ultrathin oxide (2011). Semiconductors. Vol. 45 (7). P. 974-979.
[12] T.Y. Peng, S.Y. Chen, L.C. Hsieh C.K. Lo, Y.W. Huang, W.C. Chien, Y.D. Yao, Impedance behavior
of spin-valve transistor (2006). J. Appl. Phys. Vol. 99 (8). P. 08H710-08H712.
[13] W. Ou-Yang, M. Weis, D. Taguchi, X. Chen, T. Manaka, M. Iwamoto, Modeling of threshold voltage
in pentacene organic field-effect transistors (2010). J. Appl. Phys. Vol. 107 (12). P. 124506-124510.
[14] J. Wang, L. Wang, L. Wang, Z. Hao, Yi Luo, A. Dempewolf, M. M ller, F. Bertram, J rgen, Christen,
An improved carrier rate model to evaluate internal quantum efficiency and analyze efficiency droop
origin of InGaN based light-emitting diodes (2012). J. Appl. Phys. Vol. 112 (2). P. 023107-023112.
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
50
[15] O.V. Alexandrov, A.O. Zakhar’in, N.A. Sobolev, E.I. Shek, M.M. Makoviychuk, E.O. Parshin, Forma-
tion of donor centers after annealing of dysprosium and holmium implanted silicon (1998). Semicon-
ductors. Vol. 32 (9). P. 1029-1032.
[16] I.B. Ermolovich, V.V. Milenin, R.A. Red’ko, S.M. Red’ko, Features of recombination processes in
CdTe films, preparing at different temperature conditions and further annealing (2009). Semiconduc-
tors. Vol. 43 (8). С. 1016-1020.
[17] P. Sinsermsuksakul, K. Hartman, S.B. Kim, J. Heo, L. Sun, H.H. Park, R. Chakraborty, T. Buonassisi,
R.G. Gordon, Enhancing the efficiency of SnS solar cells via band-offset engineering with a zinc oxy-
sulfide buffer layer (2013). Appl. Phys. Lett. Vol. 102 (5). P. 053901-053905.
[18] J.G. Reynolds, C.L. Reynolds, Jr.A. Mohanta, J.F. Muth, J.E. Rowe, H.O. Everitt, D.E. Aspnes, Shal-
low acceptor complexes in p-type ZnO (2013). Appl. Phys. Lett. Vol. 102 (15). P. 152114-152118.
[19] A.E. Zhukov, L.V. Asryan, Yu.M. Shernyakov, M.V. Maksimov, F.I. Zubov, N.V. Kryzhanovskay, K.
Yvind, E.S. Semenova, Effect of asymmetric barrier layers in the waveguide region on the temperature
characteristics of quantum-well lasers (2012). Semiconductors. Vol. 46 (8). P. 1049-1053.
[20] V.P. Konyaev, A.A. Marmalyuk, M.A. Ladugin, T.A. Bagaev, M.V. Zverkov, V.V. Krichevsky, A.A.
Padalitsa, S.M. Sapozhnikov, V.A. Simakov, Laser diodes arrays based on epitaxial integrated hetero-
structures with increased power and pulsed emission bright (2014). Semiconductors. Vol. 48 (1). P.
104-108.
[21] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors (1979). "Naukova Dumka",
Kiev.
[22] P.M. Fahey, P.B. Griffin, J.D. Plummer, point defects and dopant diffusion in silicon (1989). Rev.
Mod. Phys. Vol. 61. 2. P. 289-388.
[23] K.V. Shalimova, Physics of semiconductors (1985). Energoatomizdat, Moscow.
[24] E.L. Pankratov, Dopant Diffusion Dynamics and Optimal Diffusion Time as Influenced by Diffusion-
Coefficient Nonuniformity. Russian Microelectronics (2007). V.36 (1). P. 33-39.
[25] E.L. Pankratov, E.A. Bulaeva. Decreasing of quantity of radiation defects in an implanted-junction
rectifiers by using overlayers, Int. J. Micro-Nano Scale Transp (2012). Vol. 3 (3). P. 119-130.
[26] E.L. Pankratov, E.A. Bulaeva, Doping of materials during manufacture p-n-junctions and bipolar tran-
sistors. Analytical approaches to model technological approaches and ways of optimization of distribu-
tions of dopants. Rev. Theor. Sci. Vol. 1 (1). P. 58-82 (2013).
[27] A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations (1972). Moscow, Nauka (in Rus-
sian).
[28] H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids (1964). Oxford University Press.
[29] E.L. Pankratov, Redistribution of dopant during microwave annealing of a multilayer structure for
production p-n-junction (2008). J. Appl. Phys. Vol. 103 (6). P. 064320-064330.
[30] E.L. Pankratov, Redistribution of dopant during annealing of radiative defects in a multilayer structure
by laser scans for production an implanted-junction rectifiers (2008). Int. J. Nanoscience. Vol. 7 (4-5).
P. 187–197.
[31] E.L. Pankratov, Decreasing of depth of implanted-junction rectifier in semiconductor heterostructure
by optimized laser annealing. J. Comp. Theor. Nanoscience (2010). Vol. 7 (1). P. 289-295.
[32] E.L. Pankratov, E.A. Bulaeva, Application of native inhomogeneities to increase compactness of ver-
tical field-effect transistors. J. Comp. Theor. Nanoscience (2013). Vol. 10 (4). P. 888-893.
[33] E.L. Pankratov, E.A. Bulaeva, Optimization of doping of heterostructure during manufacturing of p-i-
n-diodes. Nanoscience and Nanoengineering (2013). Vol. 1 (1). P. 7-14.
International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
51
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.