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International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
DOI:10.5121/ijcsa.2014.4503 27
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 higherspeed 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 epitaxiallayers (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 epitaxiallayer the section has been doped by diffusion or ion implantation to produce required type of
conductivity (por n) in the section. Farther we consider annealing of dopant and/or radiation de-fects. After that we consider manufacturing of the second and the third epitaxial layers, which
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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 (por 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, Tis the temperature
of annealing,Dis 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]
( ) ( )
( )( ) ( )
( )
++
+=
2*
2
2*1
,,,,,,1
,,,
,,,1,,,
V
tzyxV
V
tzyxV
TzyxP
tzyxCTzyxDD LC
, (3)
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whereDL(x,y,z,T) is the spatial (due to inhomogeneity of hetero structure) and temperature (due toArrhenius law) dependences of diffusion coefficient; P (x,y,z,T) is the limit of solubility of do-
pant; value of parameter depends on materials of heterostructure and could be integer in the fol-lowing interval [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, thatusing diffusion type of doping and radiation damage is absent in the case (i.e. 1= 2= 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
z VII
,,,,,,,,,,,,
,,,,
( ) ( )tzyxITzyxkII
,,,,,,2
, (4)
( )( )
( )( )
( )+
+
=
y
tzyxVTzyxD
yx
tzyxVTzyxD
xt
tzyxVVV
,,,,,,
,,,,,,
,,,
( ) ( )
( ) ( ) ( )
+ tzyxVtzyxITzyxk
z
tzyxVTzyxD
z VIV
,,,,,,,,,,,,
,,,,
( ) ( )tzyxVTzyxkVV
,,,,,,2
,
with initial
(x,y,z,0)=f(x,y,z) (5a)
and boundary conditions
( ) 0,,,0
=
=x
xtzyx , ( ) 0,,, =
= xLxx
tzyx , ( ) 0,,,0
=
=y
ytzyx ,
( )0
,,,=
= yLyy
tzyx,
( )0
,,,
0
=
=zz
tzyx,
( )0
,,,=
= zLzz
tzyx. (5b)
Here =I,V; I (x,y,z,t) is the spatio-temporal distribution of concentrations of interstitials; D(x,y,z,T) are the diffusion coefficients of interstitials and vacancies; terms V
2(x,y,z,t) and I
2(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 V(x,y,z,t) and diin-terstitials I(x,y,z,t) by solving the following system of equations [21,22]
( )( )
( )( )
( )+
+
=
y
tzyxTzyxD
yx
tzyxTzyxD
xt
tzyx II
I
I
I
,,,,,,
,,,,,,
,,,
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( ) ( )
( ) ( )+
+ tzyxITzyxk
z
tzyxTzyxD
z II
I
I,,,,,,
,,,,,, 2
,
( ) ( )tzyxITzyxkI ,,,,,, (6)
( ) ( ) ( ) ( ) ( ) +
+
=
ytzyxTzyxD
yxtzyxTzyxD
xttzyx V
V
V
V
V
,,,,,,,,,,,,,,,
( ) ( )
( ) ( )+
+ tzyxVTzyxk
z
tzyxTzyxD
z VV
V
V,,,,,,
,,,,,,
2
,
( ) ( )tzyxVTzyxkV ,,,,,,
with boundary and initial conditions
( )0
,,,
0
=
=xx
tzyx,
( )0
,,,=
= xLxx
tzyx,
( )0
,,,
0
=
=yy
tzyx,
( )0
,,,=
= yLyy
tzyx,
( )0
,,,
0
=
=zz
tzyx,
( )0
,,,=
= zLzz
tzyx, I(x,y,z,0)=fI (x,y,z), V(x,y,z,0)=fV (x,y,z). (7)
Here DI(x,y,z,T) and DV(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
,,,,,,
,,,,,,
,,,
( ) ( ) ( )tzyxpz
tzyxTTzyxz
,,,,,,,,, +
+ (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-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]); is theheat conduction coefficient, which depends on properties of materials and current temperature ofannealing; temperature dependence of heat conduction coefficient in the most interesting tem-
perature interval could be approximated by the following function (x,y,z,T)=ass(x,y,z) [1+(Td/T(x,y,z,t))
] (see, for example, [23]); p(x,y,z,t) is the volumetric density of heat power, gener-
ated in heterostructure during annealing; (x,y,z,T)=(x,y,z,T)/c (T) is the heat diffusivity. First ofall 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
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approximation of heat diffusivity to the following form: ass (x,y,z) =ass(x,y,z)/cas s=0ass[1+TgT(x,y,z)]. Farther we determine solution of Eq.(8) as the following power series
( ) ( ) =
=
=0 0
,,,,,,i j
ij
ji
T tzyxTtzyxT . (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) (i1,j1).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) (i0,j0). The conditions are presented in the Appendix. The equations for the func-tions Tij(x,y,z,t) (i0,j0) with account boundary and initial conditions have been solved by usingstandard approaches [27,28] for the second-order approximation of the temperature T (x,y,z,t) on
the parameters and. The solutions are presented in the Appendix. The second- order is usuallyenough 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 exactnessand 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 inthe following form: D(x,y,z,T)=D0[1+g(x,y,z,T)], where D0are the average values of diffu-sion coefficients, 0< 1, |g(x,y,z,T)|1, =I,V. The same transformations have been used forapproximations of parameters of recombination of point radiation defects and generation of their
complexes: kI,V(x,y,z,T)=k0I,V [1+I,V gI,V(x,y,z,T)], kI,I(x,y,z,T)=k0I,I[1+I,I gI,I(x,y,z,T)] andkV,V(x,y,z,T) = k0V,V [1+V,V gV,V(x,y,z,T)], where k01,2are the appropriate average values of theseparameters, 0I,V
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( )( )[ ] ( ) ( )+
,,,
~,,,
~,,,1
,,,~
,, VITgV
VIVI
( )[ ] ( ) ,,,~,,,1 2,,
VTgVVVVI
+
( ) 0,,,~
0
=
=
, ( ) 0,,,~
1
=
=
, ( ) 0,,,~
0
=
=
, ( ) 0,,,~
1
=
=
,
( )0
,,,~
0
=
=
,
( )0
,,,~
1
=
=
, ( )
( )*
,,,,,,~
f= . (12)
We determine solutions of Eqs.(11) as the following power series (see [24-26])
( ) ( ) =
=
=
=0 0 0
,,,~,,,~i j k
ijk
kji . (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 ( ) ,,,~
000I and
( ) ,,,~
000V and corrections for them ( ) ,,,~
ijkI and ( ) ,,,~
ijkV , i
1,j
1, k1. 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 inthe 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:
D(x,y,z,T)=D0[1+g(x,y,z,T)], whereD0are the average values of diffusion coefficients.In this situation Eqs.(6) will be transformed to the following form
( )( )[ ]
( )( )[ ]
++
+=
Tzyxgyx
tzyxTzyxg
xD
t
tzyxII
I
III
I ,,,1,,,
,,,1,,,
0
( )( )[ ]
( )+
++
z
tzyxTzyxg
zDD
y
tzyxI
IIII
I
,,,,,,1
,,,00
( )[ ] ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkTzyxg IIIII ,,,,,,,,,,,,,,,12
, +
+
( )( )[ ]
( )( )[ ]
++
+=
Tzyxgyx
tzyxTzyxg
xD
t
tzyxVV
V
VVV
V ,,,1,,,
,,,1,,,
0
( )( )[ ]
( )+
++
z
tzyxTzyxg
zDD
y
tzyxV
VVVV
V
,,,,,,1
,,,00
( )[ ] ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkTzyxgVVVVV
,,,,,,,,,,,,,,,1 2,
+
+ .
Farther we determine solutions of the above equations as the following power series
( ) ( ) =
=
0
,,,,,,i
i
itzyxtzyx . (11)
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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 0(x,y,z,t) and corrections for them i(x,y,z,t) (i1) and appropriate condi-tions for all functionsi(x,y,z,t) (i0). 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 radiationdefects. In this situation we transform approximation of dopant diffusion coefficient to the fol-
lowing form:DL(x,y,z,T)=D0L[1+LgL(x,y,z,T)], whereD0Lis the average value of dopant diffusioncoefficient, 0L< 1, |gL(x,y,z,T)|1. Farther we determine solution of Eq.(1) in the following form
( ) ( ) =
=
=0 1
,,,,,,i j
ij
ji
L tzyxCtzyxC .
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 conditionsare 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 havebeen 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 havebeen 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 ofdoping, 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 nearestareas. The figures show, that presents of interface between materials gives us possibility to in-
crease sharpness ofp-n-junctions, which included into the considered heterobipolar transistor. Atthe same time homogeneity of distribution of concentration of dopant increases. Increasing of
sharpness ofp-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 chargecarriers 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
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 byusing another materials; (iv) doping the section of the third epitaxial layer by diffusion or ion im-
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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 ofthickness 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 ofbipolar 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 coefficientin layers of heterostructure. The curves have been calculated under condition, when dopant diffu-sion coefficient in doped layer is larger, than in nearest layer.
x0.0
0.5
1.0
1.5
2.0
C(x,
)
23
4
1
0 L/4 L/2 3L/4 L
Epitaxial layer Substrate
Fig. 3. Spatial distributions of implanted dopant concentration after annealing with continuous = 0.0048(Lx
2+Ly
2+Lz
2)/D0 (curves 1 and 3) and = 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.
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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 timeby minimization of the following mean- squared error
( ) ( )[ ] =x y zL L L
zyx
xdydzdzyxzyxCLLL
U0 0 0
,,,,,1
, (15)
where (x) is the approximation function,is the required value of annealing time.
0.0 0.1 0.2 0.3 0.4 0.5
a/L, , ,
0.0
0.1
0.2
0.3
0.4
0.5
D0
L-2
3
2
4
1
Fig.4. Dependences of dimensionless optimal annealing time for doping by diffusion, which havebeen 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/Land = = 0 for equal toeach 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 for a/L=1/2 and =
= 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of parameter for a/L=1/2 and = = 0. Curve 4 is the dependence of dimensionless optimal annealing time onvalue of parameter for a/L=1/2 and = = 0
0.0 0.1 0.2 0.3 0.4 0.5
a/L, , ,
0.00
0.04
0.08
0.12
D0
L-2
3
2
4
1
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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/Land = = 0 forequal 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 for a/L=1/2
and = = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value ofparameter for a/L=1/2 and = = 0. Curve 4 is the dependence of dimensionless optimal an-nealing time on value of parameter for a/L=1/2 and = = 0
Dependences of optimal values of annealing time are presented in Fig. 4 for diffusion type ofdoping. 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 annealingtime 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 StateUniversity of Nizhni Novgorod and educational fellowship for scientific research of Nizhny Nov-
gorod State University of Architecture and Civil Engineering.
APPENDIX
Equations for the functions Tij(x,y,z,t) (i0,j0) have been obtained by substitution the power se-ries (10) in the equation (8) and equating terms with equal powers of parameters Tand . Theequations could be written as
( ) ( ) ( ) ( ) ( )ass
asstzyxp
ztzyxT
ytzyxT
xtzyxT
ttzyxT
,,,,,,,,,,,,,,,
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,,,,,,,,,,,,
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37
( ) ( )
( ) ( )
+
+
2
10
2
2
10
2 ,,,,,,
,,,,,,
y
tzyxTTzyxg
x
tzyxTTzyxg i
T
i
T
( ) ( )
+
z
tzyxTTzyxg
z
i
T
,,,,,, 10 , i1
( ) ( ) ( ) ( )( )
+
+
+
=
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxT
t
tzyxTdass
ass,,,
,,,,,,,,,,,,
00
0
2
01
2
2
01
2
2
01
2
0
01
( ) ( ) ( )( )
( )
+
+
+
+
2
00
1
00
0
2
00
2
2
00
2
2
00
2 ,,,
,,,
,,,,,,,,,
x
tzyxT
tzyxT
T
z
tzyxT
y
tzyxT
x
tzyxTdass
( ) ( )
+
+
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
( ) ( ) ( )
( )
( ) ( )
+
+
+
+ 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
( ) ( ) ( ) ( )
+
+
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
( ) ( ) ( )
( ) ( )
+
+
+
x
tzyxTTzyxg
xz
tzyxTTzyxg
z
TzyxgTassTT
,,,,,,
,,,,,,,,, 01
0
00
( )( )[ ]
( )( )[ ]
( )+
++
++
+
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
( )
( )
( ) ( ) ( )+
+
+
+
2
10
2
2
10
2
2
10
2
00
0
2
00
2 ,,,,,,,,,
,,,
,,,
z
tzyxT
y
tzyxT
x
tzyxT
tzyxT
T
z
tzyxTdass
( ) ( )
( ) ( )
( )
+
+
+ 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
( ) ( ) ( )
( )( )( )
+
++tzyxT
TzyxgT
tzyxT
T
z
tzyxT
z
tzyxT
y
tzyxTT
dass
dass
,,,
,,,
,,,
,,,,,,,,,1
00
01
00
0001000
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38
( ) ( ) ( )
+
+
2
00
2
00
2
00,,,,,,,,,
z
tzyxT
y
tzyxT
x
tzyxT.
Conditions for the functions Tij(x,y,z,t) (i0, j0) have been obtained by the same procedure asappropriate 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, i1,j1.
Solutions of the equations for the functions Tij(x,y,z,t) (i0,j0) with account boundary and initialconditions 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
tL L L
asszyx
L L
TnnLLL
dudvdwdwvup
LLLudvdwdwvufwcvc
x y zy z 2,,,1,,
0 0 0 00 0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
=1 0 0 0 0
,,,
n
t L L L
ass
nnnnTnTnnn
x y z
dudvdwdwvup
wcvcucetezcycxc
,
where cn()=cos(n/L), ( )
++=
2220
22 111expzyx
assnTLLL
tnte ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0 0 0 02
0
0,,,2,,,
n
t L L L
TnnnnTnTnnn
zyx
ass
i
x y z
TwvugwcvcusetezcycxcnLLL
tzyxT
( )( ) ( ) ( ) ( ) ( ) ( ) ( ) +
=
1 0 0 02
010 2,,,
n
t L L
nnnTnTnnn
zyx
assix y
vsucetezcycxcnLLL
dudvdwdu
wvuT
( ) ( ) ( )
( ) ( ) ( )+
=
12
0
0
10 2,,,
,,,n
nnn
zyx
assL
i
Tn zcycxcnLLL
dudvdwdv
wvuTTwvugwc
z
( ) ( ) ( ) ( ) ( ) ( ) ( )
t L L Li
TnnnnTnT
x y z
dudvdwdw
wvuTTwvugwsvcucete
0 0 0 0
10 ,,,,,,
, i1,
where sn() = sin( n/L);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0 0 0 02001
2,,,
n
t L L L
nnnnTnTnnn
zyx
d
ass
x y z
wcvcusetezcycxcnLLL
TtzyxT
( )( )
( ) ( ) ( ) ( ) ( ) ( ) +
=1 0 020
00
2
00
2 2
,,,
,,,
n
t L
nnTnTnnn
zyx
d
ass
x
ucetezcycxcLLL
T
wvuT
dudvdwd
u
wvuT
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39
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )+
=120
0 0 00
2
00
2 2
,,,
,,,
nnTnnn
zyx
d
ass
L L
nn tezcycxcLLL
T
wvuT
dudvdwd
v
wvuTwcvsn
y z
( ) ( ) ( ) ( ) ( )
( ) ( )
=1
0
0 0 0 0 00
2
00
2
2,,,
,,,
nn
zyx
ass
d
t L L L
nnnnT xc
LLLT
wvuT
dudvdwd
u
wvuTwsvcucen
x y z
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
+
t L L L
nnnnTnTnn
x y z
wvuT
dudvdwd
u
wvuTwcvcucetezcyc
0 0 0 01
00
2
00
,,,
,,,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
=1 0 0 0 0
2
000,,,
2n
t L L L
nnnnTnTnnn
zyx
assx y z
v
wvuTwcvcucetezcycxc
LLL
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
=+
1 0 0 0
0
1
00
2,,, n
t L L
nnnTnTnnn
zyx
ass
dd
x y
vcucetezcycxcLLL
TwvuT
dudvdwdT
( ) ( )
( )
+
zL
nwvuT
dudvdwd
w
wvuTwc
01
00
2
00
,,,
,,,
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0 0 0 02002
2,,,
n
t L L L
nnnnTnTnnn
zyx
d
ass
x y z
wcvcusetezcycxcnLLL
TtzyxT
( )( )
( ) ( ) ( ) ( ) ( ) ( ) +
=1 0 020
00
2
01
2 2
,,,
,,,
n
t L
nnTnTnnn
zyx
d
ass
x
ucetezcycxcnLLL
T
wvuT
dudvdwd
u
wvuT
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )+
=120
0 0 00
2
01
2 2
,,,
,,,
nnTnnn
zyx
d
ass
L L
nn tezcycxcnLLL
T
wvuT
dudvdwd
v
wvuTwcvs
y z
( ) ( ) ( ) ( ) ( )
( ) ( )
=12
0
0 0 0 0 00
2
01
2
2,,,
,,,
nn
zyx
ass
d
t L L L
nnnnT xcLLL
TwvuT
dudvdwd
w
wvuTwsvcuce
x y z
( ) ( ) ( ) ( ) ( ) ( ) ( )( )
+
t L L
L
nnnnTn
x y z
wvuTdudvdwd
uwvuT
uwvuTwcvcuceyc
0 0 0 01
00
0100
,,,,,,,,,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
=1 0 0 0 02
02n
t L L L
nnnnTnTnnn
zyx
ass
dnTn
x y z
wcvcucetezcycxcLLL
Ttezc
( ) ( )( )
( ) ( ) ( ) ( )
=+
12
0
1
00
0100 2,,,
,,,,,,
nnTnnn
zyx
ass tezcycxcLLLwvuT
dudvdwd
v
wvuT
v
wvuT
( ) ( ) ( ) ( ) ( ) ( )
( )
+
t L L L
nnnnTd
x y z
wvuT
dudvdwd
w
wvuT
w
wvuTwcvcuceT
0 0 0 01
00
0100
,,,
,,,,,,
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ =
=1 0 0 0 0
0
11,,,
2,,,
n
t L L L
TnnnTnTnnn
zyx
assx y z
Twvugvcucetezcycxc
LLL
tzyxT
( )( )
( )( )
( )( )
+
+
wc
w
wvuTwg
wv
wvuTTwvug
w
wvuTnTT
,,,,,,,,,
,,, 002
00
2
2
00
2
( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) +
=1 0 0 0
0
00
012
,,,
,,,
n
t L L
nnnTnTnnn
zyx
assx y
vcucetezcycxcLLL
dudvdwdwvuT
wvuT
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40
( )( )[ ]
( )( )[ ]
( )
+
++
++
zL
TTv
wvuTTwvug
u
wvuTTwvug
u
wvuT
02
01
2
2
01
2
2
01
2,,,
,,,1,,,
,,,1,,,
( ) ( )
( ) ( ) ( ) ( )+
+
=1
001 2,,,
,,,n
nnn
zyx
dass
nT zcycxc
LLL
Tdudvdwdwc
w
wvuTTwvug
w
( ) ( ) ( ) ( ) ( ) ( )
( )( ) ( )
( )
+
++
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
,,,
,,,,,,
,,,
,,,
( ) ( )( )
( )( ) ( ) ( )+
+
=+
12
00
2
1
00
10
2
00
2
2,,,
,,,
,,,,,,
nnnn zcycxcdudvdwd
w
wvuT
wvuT
wvuT
v
wvuT
( ) ( ) ( ) ( ) ( ) ( ) ( )
+
+
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,,,,,,,,,
( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) +
=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
( ) ( ) ( )
( ) ( )
( )
+
+
zL
TTTn Twvug
w
wvuTTwvug
wu
wvuTTwvugwc
0
00
2
00
2
,,,,,,
,,,,,,
,,,
( )( )
( ) ( ) ( ) ( ) ( ) ( )
=1 0 0
0
00
2
00
2
2,,,
,,,
n
t L
nnTnTnnn
zyx
dassx
ucetezcycxcLLL
T
wvuT
dudvdwd
v
wvuT
( ) ( ) ( ) ( ) ( ) ( )
+
+
y zL L
nw
wvuT
v
wvuT
v
wvuT
u
wvuT
u
wvuTvc
0 0
1000100010 ,,,,,,,,,,,,,,,
( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
=+
1 0 0
0
1
00
00 2,,,
,,,
n
t L
nnTnTnnn
zyx
dass
n
x
ucetezcycxcLLL
T
wvuT
dudvdwdwc
w
wvuT
( ) ( ) ( ) ( ) ( )( )
+
+
+
y zL L
nnwvuT
dudvdwd
w
wvuT
v
wvuT
u
wvuTwcvc
0 01
00
200
200
200
,,,
,,,,,,,,,
.
Equations for the functions ( ) ,,,~
ijkI and ( ) ,,,
~ijk
V , i0, j0, k0 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,,,
~,,,
~,,,
~,,,
~
+
+
=
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,,,
~,,,
~,,,
~,,,
~
+
+
=
V
D
DV
D
DV
D
DV
I
V
I
V
I
V ;
( ) ( ) ( ) ( ) +
+
+
=
2
00
2
2
00
2
2
00
2
0
000 ,,,~
,,,~
,,,~
,~
iii
V
Ii IIIDDI
( ) ( )
( ) ( )
+
+
+
,,,~
,,,,,,
~
,,, 100
0
0100
0
0 i
I
V
Ii
I
V
I I
TgD
DITg
D
D
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41
( ) ( )
+
,,,~
,,, 100
0
0 i
I
V
I ITg
D
D, i 1,
( ) ( ) ( ) ( )+
+
+
=
2
00
2
2
00
2
2
00
2
0
000 ,,,~
,,,~
,,,~
,~
iii
I
Vi VVV
D
DV
( ) ( )
( ) ( )
+
+
+
,,,~
,,,,,,
~
,,, 100
0
0100
0
0 i
V
I
Vi
V
I
V VTgD
DVTg
D
D
( ) ( )
+
,,,~
,,, 100
0
0 i
V
I
V VTgD
D, i 1;
( ) ( ) ( ) ( )
+
+
=
2
010
2
2
010
2
2
010
2
0
0010,,,
~,,,
~,,,
~,,,
~
III
D
DI
V
I
( )[ ] ( ) ( ) ,,,~,,,~,,,1000000,,
VITgVIVI
+
( ) ( ) ( ) ( )
+
+
=
2
0102
2
0102
2
0102
0
0010 ,,,~,,,~,,,~,,,~
VVV
D
DV
I
V
( )[ ] ( ) ( ) ,,,~,,,~,,,1000000,,
VITgVIVI
+ ;
( ) ( ) ( ) ( )
+
+
=
2
020
2
2
020
2
2
020
2
0
0020,,,
~,,,
~,,,
~,,,
~
III
D
DI
V
I
( )[ ] ( ) ( ) ( ) ( ) ,,,~,,,~,,,~,,,~,,,1010000000010,,
VIVITgVIVI
++
( ) ( ) ( ) ( )
+
+
=
2
020
2
2
020
2
2
020
2
0
0020,,,
~,,,
~,,,
~,,,
~
VVV
D
DV
V
I
( )[ ] ( ) ( ) ( ) ( ) ,,,~,,,~,,,~,,,~,,,1010000000010,,
VIVITgVIVI
++ ;
( ) ( ) ( ) ( )
+
+
=
2
001
2
2
001
2
2
001
2
0
0001,,,
~,,,
~,,,
~,,,
~
III
D
DI
V
I
( )[ ] ( ) ,,,~,,,1 2000,,
ITgIIII
+
( ) ( ) ( ) ( )
+
+
=
2
001
2
2
001
2
2
001
2
0
0001,,,
~,,,
~,,,
~,,,
~
VVV
D
DV
I
V
( )[ ] ( ) ,,,~,,,1 2000,,
VTgIIII
+ ;
( ) ( ) ( ) ( ) +
+
+
=
2
110
2
2
110
2
2
110
2
0
0110 ,,,
~
,,,
~
,,,
~
,,,
~
III
DDI
V
I
( ) ( )
( ) ( )
+
+
+
,,,~
,,,,,,
~
,,, 010010
0
0 I
TgI
TgD
DII
V
I
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42
( ) ( )
( )[ ] ( ) ( )[ ++
+
,,,
~,,,
~,,,1
,,,~
,,,000100,,
010 VITgI
TgIIIII
( ) ( ) ,,,~
,,,~
100000 VI+
( ) ( ) ( ) ( ) +
+
+
=
2
1102
2
1102
2
1102
0
0110 ,,,~,,,~,,,~,,,~
VVV
D
DV
I
V
( ) ( )
( ) ( )
+
+
+
,,,~
,,,,,,
~
,,, 010010
0
0 V
TgV
TgD
DVV
I
V
( ) ( )
( )[ ] ( )[ +
+
,,,
~,,,1
,,,~
,,,100,,
010 VTgV
Tg VVVVV
( ) ( ) ( ) ,,,~
,,,~
,,,~
100000000 IVI + ;
( ) ( ) ( ) ( )
+
+
=
2
002
2
2
002
2
2
002
2
0
0002,,,
~,,,
~,,,
~,,,
~
III
D
DI
V
I
( )[ ] ( ) ( ) ,,,~,,,~,,,1000001,,
IITgIIII
+
( ) ( ) ( ) ( )
+
+
=
2
002
2
2
002
2
2
002
2
0
0002,,,
~,,,
~,,,
~,,,
~
VVV
D
DV
I
V
( )[ ] ( ) ( ) ,,,~,,,~,,,1000001,,
VVgVVVV
+ ;
( ) ( ) ( ) ( )+
+
+
=
2
101
2
2
101
2
2
101
2
0
0101,,,
~,,,
~,,,
~,,,
~
III
D
DI
V
I
( ) ( ) ( ) ( ) +
+
+
,,,
~
,,,,,,
~
,,, 001001
0
0 ITgITgDD
II
V
I
( ) ( )
( )[ ] ( ) ( )
,,,
~,,,
~,,,1
,,,~
,,,000100
001 VITgI
TgIII
+
+
( ) ( ) ( ) ( )+
+
+
=
2
101
2
2
101
2
2
101
2
0
0101 ,,,~
,,,~
,,,~
,,,~
VVV
D
DV
I
V
( ) ( )
( ) ( )
+
+
+
,,,~
,,,,,,
~
,,, 001001
0
0 V
TgV
TgD
DVV
I
V
( ) ( ) ( )[ ] ( ) ( )
,,,
~,,,
~,,,1
,,,~,,, 100000
001 VITgV
TgVVV
+
+ ;
( ) ( ) ( ) ( )
+
+
=
2
011
2
2
011
2
2
011
2
0
0011,,,
~,,,
~,,,
~,,,
~
III
D
DI
V
I
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44
( ) ( ) ( ) ( ) ( ) ( )
0
1
0
1
0
1
0
100,
~
,,, dudvdwdw
uVTwvugwsvcuce iVnnnnI , i 1,
where sn() = sin(n);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0
1
0
1
0
1
0010 2,,,
~n
nnnnnnnn wcvcuceeccc
( )[ ] ( ) ( ) dudvdwdwvuVwvuITwvugVIVI
,,,~
,,,~
,,,1000000,,
+ ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0
1
0
1
0
1
00
0
0202,,,~
nnnnnnnnn
V
I wcvcuceecccD
D
( ) ( ) ( ) ( ) + ,,,~
,,,~
,,,~
,,,~
010000000010 wvuVwvuIwvuVwvuI
( ) dudvdwdTwvugVIVI
,,,1,,
+ ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0
1
0
1
0
1
0001
2,,,~n
nnnnnnnn wcvcuceeccc
( )[ ] ( ) dudvdwdwvuTwvug ,,,~,,,12
000,,+ ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0
1
0
1
0
1
0002
2,,,~n
nnnnnnnn wcvcuceeccc
( ) ( ) ( ) dudvdwdwvuwvuTwvug ,,,~,,,~,,,1
000001,,+ ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0
1
0
1
0
1
00
0
1102,,,
~
nnnnnInInnn
V
I ucvcuseecccnD
DI
( ) ( )
( ) ( ) ( ) ( )
=
10
0100 2,,,
~
,,,n
nInnn
V
Ii
I ecccnD
Ddudvdwd
u
wvuITwvug
( ) ( ) ( ) ( ) ( ) ( )
V
IiInnnnI
D
Ddudvdwd
v
wvuITwvugucvsuce
0
0
0
1
0
1
0
1
0
100 2,,,
~
,,,
( ) ( ) ( ) ( ) ( ) ( ) ( )
=
1 0
1
0
1
0
1
0
100 ,,,~
,,,n
i
InnnnInI dudvdwd
w
wvuITwvugusvcuceen
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ +
=1 0
1
0
1
0
1
0,
12n
VInnnnInnnInnnn vcvcuceccecccc
( )] ( ) ( ) ( ) ( ) dudvdwdwvuVwvuIwvuVwvuITwvugVI
,,,~
,,,~
,,,~
,,,~
,,,100000000100,
+
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0
1
0
1
0
1
00
0
1102,,,
~
nnnnnVnVnnn
I
V ucvcuseecccnD
DV
( ) ( ) ( ) ( ) ( ) ( )
=
10
0100 2,,,~
,,,n
nVnnn
I
Vi
V ecccnD
Ddudvdwd
u
wvuVTwvug
( ) ( ) ( ) ( ) ( ) ( )
I
Vi
VnnnnVD
Ddudvdwd
v
wvuVTwvugucvsuce
0
0
0
1
0
1
0
1
0
100 2,,,
~
,,,
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45
( ) ( ) ( ) ( ) ( ) ( ) ( )
=
1 0
1
0
1
0
1
0
100,,,
~
,,,n
i
VnnnnVnV dudvdwdw
wvuVTwvugusvcuceen
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ +
=1 0
1
0
1
0
1
0,
12n
VInnnnVnnnInnnn vcvcuceccecccc
( )] ( ) ( ) ( ) ( ) dudvdwdwvuVwvuIwvuVwvuITwvugVI
,,,~,,,~,,,~,,,~,,,100000000100,
+ ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0
1
0
1
0
1
00
0
1012,,,
~
nnnnnInInnn
V
I wcvcuseecccnD
DI
( ) ( )
( ) ( ) ( ) ( )
=10
0001 2,,,
~
,,,n
nInnn
V
I
I ecccnD
Ddudvdwd
u
wvuITwvug
( ) ( ) ( ) ( ) ( ) ( )
V
I
InnnnID
Ddudvdwd
v
wvuITwvugwcvsuce
0
0
0
1
0
1
0
1
0
001 2,,,
~
,,,
( ) ( ) ( ) ( ) ( ) ( )
( )
=1 0
1
0
1
0
1
0
001,,,
~
,,,n InnnnInI dudvdwdw
wvuI
Twvugwsvcuceen
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] +
=1 0
1
0,,
,,,12n
VIVInInInnnnnn Twvugeecccccc
( ) ( ) ( ) dudvdwdwvuVwvuIucn ,,,~
,,,~
000100
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0
1
0
1
0
1
00
0
1012,,,
~
nnnnnVnVnnn
I
V wcvcuseecccnD
DV
( ) ( )
( ) ( ) ( ) ( )
=10
0001 2,,,
~
,,,n
nVnnn
I
V
V ecccnD
Ddudvdwd
u
wvuVTwvug
( ) ( ) ( ) ( ) ( ) ( )
I
V
VnnnnVDDdudvdwd
vwvuVTwvugwcvsuce
0
0
0
1
0
1
0
1
0
001 2,,,~
,,,
( ) ( ) ( ) ( ) ( ) ( ) ( )
=1 0
1
0
1
0
1
0
001,,,
~
,,,n
VnnnnVnV dudvdwdw
wvuVTwvugwsvcuceen
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ +
=1 0
1
0
1
0
1
0,
12n
VInnnnVnVnnnnnn wcvcuceecccccc
( )] ( ) ( ) dudvdwdwvuVwvuITwvugVI
,,,~
,,,~
,,,100000,
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0
1
0
1
0
1
0011
2,,,~
nnnnnInInnn
wcvcuceecccI
( )[ ] ( ) ( ) ( )[ ]{ +++ TwvugwvuIwvuITwvug VIVIIIII ,,,1,,,~
,,,~
,,,1 ,,010000,,
( ) ( ) dudvdwdwvuVwvuI ,,,~
,,,~
000001
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0
1
0
1
0
1
0011
2,,,~
nnnnnVnVnnn
wcvcuceecccV
( )[ ] ( ) ( ) ( )[ ]{ +++ TwvugwvuVwvuVTwvugVIVIVVVV
,,,1,,,~
,,,~
,,,1,,010000,,
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46
( ) ( ) dudvdwdwvuVwvuI ,,,~
,,,~
001000 .
Equations for initial-order approximations of distributions of concentrations of simplest complex-
es of radiation defects 0(x,y,z,t) and corrections for them i(x,y, z,t), i1 and boundary and
initial conditions for them have been obtained as the functions Tij(x,y,z,t) and takes the form
( ) ( ) ( ) ( )+
+
+
=
2
0
2
2
0
2
2
0
2
0
0,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyxD
t
tzyxIII
I
I
( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkIII
,,,,,,,,,,,, 2,
+
( ) ( ) ( ) ( )+
+
+
=
2
0
2
2
0
2
2
0
2
0
0,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyxD
t
tzyxVVV
V
V
( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkVVV
,,,,,,,,,,,, 2,
+ ;
( ) ( ) ( ) ( )
+
+
+
=
2
2
2
2
2
2
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyx
Dt
tzyxiIiIiI
I
iI
( ) ( )
( ) ( )
+
+
+
y
tzyxTzyxg
yx
tzyxTzyxg
xD
iI
I
iI
II
,,,,,,
,,,,,,
11
0
( ) ( )
+
z
tzyxTzyxg
z
iI
I
,,,,,,
1, i1,
( ) ( ) ( ) ( )+
+
+
=
2
2
2
2
2
2
0
,,,,,,,,,,,,
z
tzyx
y
tzyx
x
tzyxD
t
tzyxiViViV
V
iV
( )
( )
( )
( )
+
+
+
y
tzyx
Tzyxgyx
tzyx
TzyxgxD iV
V
iV
VV
,,,
,,,
,,,
,,,11
0
( ) ( )
+
z
tzyxTzyxg
z
iV
V
,,,,,,
1, i1;
( )0
,,,
0
=
=x
i
x
tzyx,
( )0
,,,=
= xLx
i
x
tzyx,
( )0
,,,
0
=
=y
i
y
tzyx,
( )0
,,,=
= yLy
i
y
tzyx,
( )0
,,,
0
=
=z
i
z
tzyx,
( )0
,,,=
= zLz
i
z
tzyx, i0;
0(x,y,z,0)=f(x,y,z), i(x,y,z,0)=0, i1.
Solutions of the above equations could be written as
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ++=
=
=
110
221,,,
nnnn
nnnnnn
zyxzyx
zcycxcnL
tezcycxcFLLLLLL
tzyx
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47
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ t L L L
IIInnnnn
x y z
TwvukwvuITwvukwcvcucete0 0 0 0
2
,,,,,,,,,,
( )] dudvdwdwvuI ,,, ,
where ( ) ( ) ( ) ( ) = x y zL L L
nnnn udvdwdwvufwcvcucF0 0 0
,, , ( )
++= 222022 111exp
zyx
nLLL
tDnte ,
cn(x)=cos(nx/Lx);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=
1 0 0 0 02
2,,,
n
t L L L
nnnnnnnn
zyx
i
x y z
wcvcusetezcycxcnLLL
tzyx
( ) ( )
( ) ( ) ( ) ( )
=
1
2
1 2,,,,,,
nnnnn
zyx
iI
tezcycxcnLLL
dudvdwdu
wvuTwvug
( ) ( ) ( ) ( ) ( ) ( )
yx
t L L LiI
nnnnLL
dudvdwdv
wvuTwvugwcvsuce
x y z
2,,,,,,
0 0 0 0
1
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
=
1 0 0 0 0
1
2
,,,1
n
t L L L iI
nnnnnnnn
z
x y z
w
wvuwsvcucetezcycxcnL
( )
dudvdwdTwvug ,,, , i1,
where sn(x)=sin(nx/Lx).Equations for initial-order approximation of dopant concentration C00(x,y,z,t), corrections for
them Cij(x,y,z,t) (i1,j1) 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
tzyxCDy
tzyxCDx
tzyxCDt
tzyxCi
L
i
L
i
L
i
( ) ( )
( ) ( )
+
+
+
y
tzyxCTzyxg
yD
x
tzyxCTzyxg
xD i
LL
i
LL
,,,,,,
,,,,,, 10
0
10
0
( ) ( )
+
z
tzyxCTzyxg
zD i
LL
,,,,,, 10
0, i 1;
( ) ( ) ( ) ( )+
+
+
=
2
01
2
02
01
2
02
01
2
0
01,,,,,,,,,,,,
z
tzyxCD
y
tzyxCD
x
tzyxCD
t
tzyxCLLL
( )
( )
( ) ( )
( )
( )+
+
+
y
tzyxC
TzyxP
tzyxC
y
D
x
tzyxC
TzyxP
tzyxC
x
DLL
,,,
,,,
,,,,,,
,,,
,,,0000
0
0000
0
( )( )
( )
+
z
tzyxC
TzyxP
tzyxC
zD L
,,,
,,,
,,,0000
0
;
( ) ( ) ( ) ( )+
+
+
=
2
02
2
02
02
2
02
02
2
0
02,,,,,,,,,,,,
z
tzyxCD
y
tzyxCD
x
tzyxCD
t
tzyxCLLL
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48
( ) ( )
( )( )
( ) ( )
( )
+
+
TzyxP
tzyxCtzyxC
yx
tzyxC
TzyxP
tzyxCtzyxC
x ,,,
,,,,,,
,,,
,,,
,,,,,,
1
00
01
00
1
00
01
( )( )
( )
( )
( )+
+
LD
z
tzyxC
TzyxP
tzyxCtzyxC
zy
tzyxC0
00
1
00
01
00 ,,,
,,,
,,,,,,
,,,
( )( )
( ) ( )( )
( )
+
+
+
y
tzyxC
TzyxP
tzyxC
yx
tzyxC
TzyxP
tzyxC
xD
L
,,,
,,,
,,,,,,
,,,
,,,01000100
0
( )( )
( )
+
z
tzyxC
TzyxP
tzyxC
z
,,,
,,,
,,,0100
;
( ) ( ) ( ) ( )+
+
+
=
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
( )( )
( )( )
( )+
+
LD
z
tzyxC
TzyxP
tzyxCtzyxC
zy
tzyxC0
00
1
00
10
00,,,
,,,
,,,,,,
,,,
( )( )
( ) ( )( )
( )
+
+
+
y
tzyxC
TzyxP
tzyxC
yx
tzyxC
TzyxP
tzyxC
xD
L
,,,
,,,
,,,,,,
,,,
,,,10001000
0
( )( )
( )( )
( )
+
+
+
x
tzyxCTzyxg
xD
z
tzyxC
TzyxP
tzyxC
z LL
,,,,,,
,,,
,,,
,,,01
0
1000
( ) ( )
( ) ( )
+
+
z
tzyxCTzyxg
zy
tzyxCTzyxg
y
LL
,,,,,,
,,,,,, 0101 ;
( )0
,,,
0
==x
ij
x
tzyxC
,
( )0
,,,=
= xLx
ij
x
tzyxC
,
( )0
,,,
0
==y
ij
y
tzyxC
,
( )0
,,,=
= yLy
ij
y
tzyxC
,
( )0
,,,
0
==z
ij
z
tzyxC
,
( )0
,,,=
= zLz
ij
z
tzyxC
, i0,j0;
C00(x,y,z,0)=fC (x,y,z), Cij(x,y,z,0)=0, i1,j1.
Solutions of the above equations with account boundary and initial conditions could be written as
( ) ( ) ( ) ( ) ( )+=
=100
21,,,
nnCnnnnC
zyxzyx
tezcycxcFLLLLLL
tzyxC ,
where ( ) ( ) ( ) ( ) = x y zL L L
nnnnC udvdwdwvufwcvcucF0 0 0
,,
, ( )
++= 2220
22 111expzyx
nCLLL
tDnte
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0 0 0 020
,,,2
,,,n
t L L L
LnnnCnCnnnnC
zyx
i
x y z
TwvugvcusetezcycxcFnLLL
tzyxC
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International Journal on Computational Sciences & Applications (IJCSA) Vol.4, No.5, October 2014
49
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
=
1 0 02
10 2,,,
n
t L
nnCnCnnnnC
zyx
i
n
x
ucetezcycxcFnLLL
dudvdwdu
wvuCvc
( ) ( ) ( ) ( )
( ) ( ) ( )
=
12
0 0
10 2,,,,,,n
nnnnC
zyx
L Li
Lnn zcycxcFn
LLLdudvdwd
v
wvuCTwvugvcvs
y z
( ) ( ) ( ) ( ) ( ) ( ) ( )
t L L Li
LnnnnCnC
x y z
dudvdwdw
wvuCTwvugvsvcucete
0 0 0 0
10 ,,,,,,
, i 1;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) =
=1 0 0 0 0
00
201 ,,,
,,,2,,,
n
t L L L
nnnCnCnnnnC
zyx
x y z
TwvuP
wvuCvcusetezcycxcFn
LLLtzyxC
( ) ( )
( ) ( ) ( ) ( ) ( )
=1 02
00 2,,,
n
t
nCnCnnnnC
zyx
n etezcycxcFnLLL
dudvdwdu
wvuCwc
( ) ( ) ( ) ( )
( )( )
( )
=12
0 0 0
0000 2,,,
,,,
,,,
nnCnC
zyx
L L L
nnn teFn
LLLdudvdwd
v
wvuC
TwvuP
wvuCwcvsuc
x y z
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
( )
t L
L L
nnnnCnnn
x y z
dudvdwdw
wvuCTwvuP
wvuCwsvcucezcycxc0 0 0 0
0000 ,,,,,,,,,
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0 0 0 0202
2,,,
n
t L L L
nnnnCnCnnnnC
zyx
x y z
wcvcusetezcycxcFnLLL
tzyxC
( ) ( )
( )( )
( ) ( )
=
12
00
1
00
01
2,,,
,,,
,,,,,,
nnnnC
zyx
ycxcFLLL
dudvdwdu
wvuC
TwvuP
wvuCwvuC
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )( )
t L L L
nnnCnCn
x y z
v
wvuC
TwvuP
wvuCwvuCvsucetezcn
0 0 0 0
00
1
00
01
,,,
,,,
,,,,,,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
=1 0 0 02
2
n
t L L
nnnCnCnnnnCzyx
n
x y
vcucetezcycxcFnLLL
dudvdwdwc
( ) ( ) ( )
( )( )
( )
=
12
0
00
1
00
01
2,,,
,,,
,,,,,,
nn
zyx
L
n xcn
LLLdudvdwd
w
wvuC
TwvuP
wvuCwvuCws
z
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
t L L L
nnnnCnCnnnC
x y z
u
wvuCwvuCwcvcusetezcycF
0 0 0 0
00
01
,,,,,,
( )( )
( ) ( ) ( ) ( ) ( ) ( )
=
1 0 02
1
00 2
,,,
,,,
n
t L
nnCnCnnnnC
zyx
x
ucetezcycxcFnLLL
dudvdwdTwvuP
wvuC
( ) ( ) ( ) ( )
( )( )
=
12
0 0
00
1
00
01
2,,,
,,,
,,,,,,
nzyx
L L
nn n
LLLdudvdwd
v
wvuC
TwvuP
wvuCwvuCwcvs
y z
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
t L L L
nnnnCnCnnnnC
x y z
TwvuP
wvuCwvuCwsvcucetezcycxcF
0 0 0 0
1
00
01,,,
,,,,,,
( )( ) ( ) ( ) ( ) ( ) ( )
=1 0 02
00 2,,,
n
t L
nnCnCnnnnC
zyx
x
usetezcycxcFLLL
dudvdwdw
wvuC
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( ) ( ) ( )
( )( )
( ) ( )
=12
0 0
0100 2,,,
,,,
,,,
nnCn
zyx
L L
nn texcLLL
dudvdwdu
wvuC
TwvuP
wvuCwcvcn
y z
( ) ( ) ( ) ( ) ( ) ( )
( )( )
t L L L
nnnnCnnC
x y z
dudvdwdv
wvuC
TwvuP
wvuCwcvsuceycF
0 0 0 0
0100,,,
,,,
,,,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
=1 0 0 0 02
2
n
t L L L
nnnnCnCnnnnC
zyx
n
x y z
wsvcucetezcycxcFnLLL
zcn
( )( )
( )
dudvdwdw
wvuC
TwvuP
wvuC
,,,
,,,
,,,0100 ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =
=1 0 0 0 0211
2,,,
n
t L L L
nnnnCnCnnnnC
zyx
x y z
wcvcusetezcycxcFnLLL
tzyxC
( ) ( )
( ) ( ) ( ) ( )
=12
01 2,,,,,,n
nCnnnnC
zyx
L tezcycxcFn
LLLdudvdwd
u
wvuCTwvug
( ) ( ) ( ) ( ) ( ) ( )
2
0 0 0 0
01 2,,,,,,zyx
t L L L
LnnnnCLLL
dudvdwdv
wvuCTwvugwcvsucex y
z
( ) ( ) ( ) ( ) ( ) ( ) ( )
=1 0 0 0 0
01,,,
,,,n
t L L L
LnnnnCnC
x y z
dudvdwdw
wvuCTwvugwsvcuceten
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
=1 0 0 02
2
n
t L L
nnnCnCnnnnC
zyx
nnnnC
x y
vcusetezcycxcFLLL
zcycxcF
( ) ( )
( )( )
( ) ( )
=12
0
1000 2,,,
,,,
,,,
nnnnC
zyx
L
n ycxcFn
LLLdudvdwd
u
wvuC
TwvuP
wvuCwcn
z
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
t L L L
nnnnCnCn
x y z
dudvdwd
v
wvuC
TwvuP
wvuCwcvsucetezc
0 0 0 0
1000,,,
,,,
,,,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
=1 0 0 0 0
00
2 ,,,
,,,2
n
t L L L
nnnnCnCnnnnC
zyx
x y z
TwvuP
wvuCwsvcucetezcycxcFn
LLL
( )( ) ( ) ( ) ( ) ( ) ( )
=1 0 02
10 2,,,
n
t L
nnCnCnnnnC
zyx
x
usetezcycxcFnLLL
dudvdwdw
wvuC
( ) ( ) ( ) ( )
( )( )
=
12
0 0
00
1
00
10
2,,,
,,,
,,,,,,
nzyx
L L
nn n
LLLdudvdwd
u
wvuC
TwvuP
wvuCwvuCwcvc
y z
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )( )
t L L L
nnnnCnCnnnnC
x y z
v
wvuC
TwvuP
wvuCwcvsucetezcycxcF
0 0 0 0
00
1
00,,,
,,,
,,,
( ) ( ) ( ) ( ) ( ) ( ) ( )
=1 0 0210
2,,,
n
t L
nnCnCnnnnC
zyx
x
ucetezcycxcFnLLL
dudvdwdwvuC
( ) ( ) ( ) ( )
( )( )
y zL L
nn dudvdwdw
wvuC
TwvuP
wvuCwvuCwsvc
0 0
00
1
00
10
,,,
,,,
,,,,,,
.
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Authors
Pankratov Evgeny Leonidovichwas 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 Alexeevnawas 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.