AN APPROACH TO DECREASE DIMENSIONS OF DRIFT HETERO-BOPOLAR TRANSISTORS

8/10/2019 AN APPROACH TO DECREASE DIMENSIONS OF DRIFT HETERO-BOPOLAR TRANSISTORS

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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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28

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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29

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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30

( ) ( )

( ) ( )+

+ 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 ,,,,,,


( )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)


( )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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31

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


12/26


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


13/26


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


14/26


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


15/26


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


16/26


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


17/26


18/26


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


( )[ ] ( ) dudvdwdwvuTwvug ,,,~,,,12

000,,+ ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =

=1 0

1

0

1

0

1

0002

2,,,~n


( ) ( ) ( ) 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,,,

~

,,,


19/26


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,,


20/26


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


21/26


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


22/26


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


23/26


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


24/26


50

( ) ( ) ( )

( )( )

( ) ( )

=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,,,

,,,

,,,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

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,,,

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.


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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.

AN APPROACH TO DECREASE DIMENSIONS OF DRIFT HETERO-BOPOLAR TRANSISTORS

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