an approach for optimization of

Advanced Nanoscience and Technology: An International Journal (ANTJ), Vol. 1, No. 1, June 2015

1

AN APPROACH FOR OPTIMIZATION OF

MANUFACTURE MULTIEMITTER

HETEROTRANSISTORS

E.L. Pankratov1 and E.A. Bulaeva2

1 Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950,

Russia 2 Nizhny Novgorod State University of Architecture and Civil Engineering, 65 Il'insky

street, Nizhny Novgorod, 603950, Russia

ABSTRACT

In this paper we introduced an approach to model manufacture process of a multiemitter heterotransistor. The approach gives us also possibility to optimize manufacture process. p-n-junctions, which include into the multiemitter heterotransistor, have higher sharpness and higher homogeneity of dopants distributions in enriched by the dopant area.

KEYWORDS

Multiemitter heterotransistor, Approach to optimize technological processes.

1. INTRODUCTION

Logical elements often include into itself multiemitter transistors [1-4]. To manufacture bipolar transistors it could be used dopant diffusion, ion doping or epitaxial layer [1-8]. It is attracted an interest increasing of sharpness of p-n-junctions, which include into bipolar transistors and decreasing of dimensions of the transistors, which include into integrated circuits [1-4]. Dimensions of transistors will be decreased by using inhomogenous distribution of temperature during laser or microwave types of annealing [9,10], using defects of materials (for example, defects could be generated during radiation processing of materials) [11-14], native inhomogeneity (multilayer property) of heterostructure [12-14].

Framework this paper we consider a heterostructure, which consist of a substrate and three epitaxial layers (see Fig. 1). Some dopants are infused in the epitaxial layers to manufacture p and n types of conductivities during manufacture a multiemitter transistor. Let us consider two cases: infusion of dopant by diffusion or ion implantation. Farther in the first case we consider annealing of dopant so long, that after the annealing the dopants should achieve interfaces between epitaxial layers. In this case one can obtain increasing of sharpness of p-n-junctions [12-14]. The increasing of sharpness could be obtained under conditions, when values of dopant diffusion coefficients in last epitaxial layers is larger, than values of dopant diffusion coefficient in average epitaxial layer. At the same time homogeneity of dopant distributions in doped areas increases. It is attracted an interest higher doping of last epitaxial layers, than doping of average epitaxial layer, because the higher doping gives us possibility to increase sharpness of p-n-junctions. It is


2

also practicably to choose materials of epitaxial layers and substrate so; those values of dopant diffusion coefficients in the substrate should be smaller, than in the epitaxial layer. In this situation one can obtain thinner transistor. In the case of ion doping of heterostructure it should be done annealing of radiation defects. It is practicably to choose so regimes of annealing of radiation defects, that dopant should achieves interface between epitaxial layers. At the same time the dopant could not diffuse into another epitaxial layer. If dopant did not achieves interface between epitaxial layers during annealing of radiation defects, additional annealing of dopant required.

Main aims of the present paper are analysis of dynamics of redistribution of dopants in considered heterostructure (Figs. 1) and optimization of annealing times of dopants. In some recent papers analogous analysis have been done [4,10,12-14]. However we can not find in literature any similar heterostructures as we consider in the paper.

Substrate

Epitaxial layer 1

Epitaxial layer 2

Epitaxial layer 3

n-type dopant p-type dopant n-type dopant

Fig. 1a. Heterostructure, which consist of a substrate and three epitaxial layers. View from above

Epitaxial layer 1 Epitaxial layer 2 Epitaxial layer 3

n-type dopantp-type dopant

n-type dopant 1

n-type dopant 2

n-type dopant 3

n-type dopant 4

Fig. 1b. Heterostructure, which consist of a substrate and three epitaxial layers. View from one side


3

2. Method of Analysis

Redistribution of dopant in the considered heterostructure has been described by the second Ficks low [1-4]

( ) ( ) ( ) ( )

+

+

=

z

tzyxCDzy

tzyxCDyx

tzyxCDxt

tzyxCCCC

,,,,,,,,,,,,

(1)

with boundary and initial conditions

( ) 0,,,0

=

=xx

tzyxC,

( ) 0,,, =

= xLx

x

tzyxC,

( ) 0,,,0

=

=yy

tzyxC,

( ) 0,,, =

= yLy

ytzyxC

,

( ) 0,,,0

=

=zz

tzyxC,

( ) 0,,, =

= zLz

z

tzyxC, C (x,y,z,0)=fC (x,y,z). (2)

Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant; x, y, z and t are current coordinates and times; D is the dopant diffusion coefficient. Value of dopant diffusion coefficient depends on properties of materials in layers of heterostructure, speed of heating and cooling of heterostructure (with account Arrhenius low), spatio-temporal distributions of concentrations of dopants and radiation defects. Two last dependences of dopant diffusion coefficient could be approximated by the following relation [15,16]

( ) ( )( )( ) ( )

( )

++

+= 2*

2

2*1

,,,,,,1,,,

,,,1,,,V

tzyxVV

tzyxVTzyxPtzyxCTzyxDD LC

. (3)

Here DL (x,y,z,T) is the spatial (due to inhomogeneity () of heterostructure) and temperature (due to Arrhenius low) dependences of dopant diffusion coefficient; P

(x,y,z,T) is the limit of solubility of dopant; parameter depends of properties of materials and could be integer in the interval

[1,3] [15]; V

(x,y,z,t) is the spatio-temporal distribution of concentration of radiation vacancies; V* is the equilibrium distribution of vacancies. Concentrational dependence of dopant diffusion coefficient has been discussed in details in [15]. It should be noted, that radiation damage absents in the case of diffusion doping of haterostructure. In this situation 1= 2= 0. Spatio-temporal distributions of concentrations of point radiation defects could be determine by solution of the following system of equations [17-19]

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

+

=

tzyxIy

tzyxITzyxDyx

tzyxITzyxDxt

tzyxIII ,,,

,,,

,,,

,,,

,,,

,,, 2

( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxkz

tzyxITzyxDz

Tzyxk VIIII ,,,,,,,,,,,,

,,,,,,,,

+ (4)

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

+

=

tzyxVy

tzyxVTzyxDyx

tzyxVTzyxDxt

tzyxVVV ,,,

,,,

,,,

,,,

,,,

,,, 2

( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxkz

tzyxVTzyxD

zTzyxk VIVVV ,,,,,,,,,

,,,

,,,,,,,,

+


4

with initial (x,y,z,0)=f (x,y,z) (5a)

and boundary conditions ( ) 0,,, =

= yLy

ytzyx

,

( ) 0,,,0

=

=zz

tzyx,

( ) 0,,, =

= zLz

z

tzyx. (5b)

Here =I,V; I (x,y,z,t) is the spatio-temporal distribution of concentration of interstitials; D(x,y, z,T) are diffusion coefficients of interstitials and vacancies; terms V2(x,y, z,t) and I2(x,y,z,t) correspond to generation of divacancies and analogous complexes of interstitials; kI,V(x,y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T) parameters of recombination of point radiation defects and generation of complexes.

Spatio-temporal distributions of concentrations of divacancies V(x,y,z,t) and analogous complexes of interstitials I(x,y,z,t) will be determined by solving the following systems of equations [17-19]

( ) ( ) ( ) ( ) ( ) +

+

=

y

tzyxTzyxD

yxtzyx

TzyxDxt

tzyx II

II

I

,,,

,,,

,,,

,,,

,,,

( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkz

tzyxTzyxD

zIII

II ,,,,,,,,,,,,

,,,

,,,2

,+

+

(6)

( ) ( ) ( ) ( ) ( ) +

+

=

y

tzyxTzyxD

yxtzyx

TzyxDxt

tzyx VV

VV

V

,,,

,,,

,,,

,,,

,,,

( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkz

tzyxTzyxD

zVVV

VV ,,,,,,,,,,,,

,,,

,,,2

,+

+

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 diffusion coefficients of complexes of point radiation defects; kI(x,y,z,T) and kV(x,y,z,T) are parameters of decay of the complexes.

To determine spatio-temporal distributions of concentrations of complexes of radiation defects we used considered in Refs. [14,19] approach. Framework the approach we transform approxi-mations of diffusion coefficients of complexes of point radiation defects to the following form: D (x,y,z,T) = D0 [1+ g (x,y,z,T)] and kI,V (x,y,z,T)=k0I,V[1+ h(x,y,z,T)], where D0 and k0I,V are the average values of appropriate parameters, 0V


5

2

2

2

2

1z

y

x

y

y LL

LL

b ++= , 2

2

2

2

1y

z

x

z

z LL

LLb ++= . Using the above variables leads to transformation Eqs. (4) to

the following form

( ) ( )[ ] ( ) +

+=V

I

y

II

V

I

xDD

bITg

DD

bI

0

0

0

0 1,,,~

,,,11,,,~

( )[ ] ( ) ( )[ ] ( )

++

+

,,,~,,,11,,,

~

,,,1 ITgb

ITg II

z

II

( )[ ] ( ) ( ) ,,,~,,,~,,,1*

*

0

0 VIThIV

DD

V

I + (8)

( ) ( )[ ] ( ) +

+=I

V

y

VV

I

V

xDD

bV

TgDD

bV

0

0

0

0 1,,,~

,,,11,,,~

( )[ ] ( ) ( )[ ] ( )

++

+

,,,~,,,1,,,

~

,,,1 VTgVTg VVVV

( )[ ] ( ) ( ) ,,,~,,,~,,,11*

*

0

0 VIThVI

DD

b IV

z

+ .

We will determine solution of Eqs. (8) as the following power series

( ) ( ) = =

=

=0 0 0~ ,,,

~

,,,~

i j k ijkkji . (9)

Substitution of the series into Eqs. (8) gives us possibility to obtain system of equations for initial-order approximations of concentrations of defects ( ) ,,,~000 and corrections for them

( ) ,,,~ijk (i1, j1, k1). The equations are presented in Appendix.

Substitution of the series (9) into appropriate boundary and initial conditions gives us possibility to obtain boundary and initial conditions for the functions ( ) ,,,~ijk . The conditions dard approaches [21,22], are presented in Appendix. and solutions for the functions ( ) ,,,~ijk (i0, j0, k0), which have been obtained by stan

Farther we determine spatio-temporal distributions of concentrations of point radiation defects. To calculate the distributions we transform approximations of diffusion coefficients to the following form: DI(x,y,z,T)=D0I[1+IgI(x,y,z,T)], DV(x,y,z,T)= D0V[1+VgV (x,y,z,T)]. In this situation Eqs. (6) will be transform to the following form

( ) ( )[ ] ( ) ( )[ ]

++

+=

Tzyxgyxtzyx

Tzyxgxt

tzyxII

III

I,,,1,,,,,,1,,,

( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkDy

tzyxIIII

I,,,,,,,,,,,,

,,, 2,0 +


6

( ) ( )[ ] ( ) ( )[ ]

++

+=

Tzyxgyxtzyx

Tzyxgxt

tzyxVV

VVV

V,,,1

,,,

,,,1,,,

( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkDy

tzyxVVVV

V,,,,,,,,,,,,

,,, 2,0 +

.

We determine solutions of these equations as the following power series

( ) ( ) = =

0

,,,,,,

ii

i tzyxtzyx , (10)

where =I,V. Substitution of the series (10) into Eqs. (6) and appropriate boundary and initial conditions gives us possibility to obtain equations for initial-order approximations of concentrations of complexes of point radiation defects, corrections for them, boundary and initial conditions for all above functions. The equations, conditions and solutions, which have been obtained by standard approaches [21,22], are presented in the Appendix.

Analysis of spatio-temporal distributions of concentrations of radiation defects has been done by using the second-order approximations on parameters, which used in appropriate series, and have been defined more precisely by using numerical simulation.

Farther we determine solution of the Eq.(1). To calculate the solution we used the approach, which has been considered in [13,18]. Framework the approach we transform approximation of dopant diffusion coefficient DL(x,y,z,T) into following form: DL(x,y,z,T) =D0L[1+ L gL(x,y,z,T)]. We determine solution of the Eq.(1) as the following power series

( ) ( ) = =

=0 0,,

i j ijji

L txCtxC . (11)

Substitution of the series into Eq. (1) gives us possibility to obtain system of equations for initial-order approximation of dopant concentration C00(x,y,z,t) and corrections for them Cij(x,y,z,t) (i1, j1). Substitution of the series (11) into appropriate boundary and initial conditions for the functions Cij(x,y,z,t) (i0, j0)

( )0

,,,

0

=

=x

ij

x

tzyxC

;

( )0

,,,

=

= xLx

ij

x

tzyxC

;

( )0

,,,

0

=

=y

ij

ytzyxC

; ( )

0,,,

=

= yLy

ij

ytzyxC

;

( )0

,,,

0

=

=z

ij

z

tzyxC

;

( )0

,,,

=

= zLz

ij

z

tzyxC

; C00(x,y,z,0)=fC (x,y,z); Cij(x,y,z,0)=0, i0, j0.

Solutions of equations for the functions Cij(x,y,z,t) (i0, j0), which has been calculated by standard approaches [21,22], are presented in the Appendix.

Analysis of spatio-temporal distribution of dopant concentration has been done analytically by using the second-order approximation on parameters, which has been used in appropriate series and have been defined more precisely by using numerical simulation.


7

3. Discussion

In this section we analyzed dynamics of redistribution of dopant with account relaxation of distributions of concentrations of radiation defects (in the case of ion doping of heterostructure). The Figs. 2 show distributions of concentrations of infused and implanted dopants in one of last epitaxial layers in direction, which is perpendicular to interfaces between epitaxial layers, after annealing of dopant and/or radiation defects, respectively. In this case values of dopant diffusion coefficient in last epitaxial layers is larger, than value of dopant diffusion coefficient in average epitaxial layer. The figures show, that interfaces between epitaxial layers under the above conditions give us possibility to increase sharpness of p-n-junctions and at the same time to increase homogeneity of distributions of concentrations of dopants in enriched by the dopants areas.

The Figs. 3 show distributions of concentrations of infused and implanted dopants in one of last epitaxial layers in direction, which is perpendicular to interfaces between epitaxial layers, after annealing of dopant and/or radiation defects, respectively. In this case values of dopant diffusion coefficient in last epitaxial layers is smaller, than value of dopant diffusion coefficient in average epitaxial layer. The figures show, that presents of last epitaxial layers leads to increasing of homogeneity of dopant in the average epitaxial layers. However sharpness of p-n-junctions decreases in comparison with p-n-junctions for Figs. 2. In this case one can obtain accelerated diffusion in last epitaxial layers. The decreasing of sharpness of p-n-junctions is a reason to use higher level of doping of last epitaxial layer to use nonlinearity of dopant diffusion. In this case one obtain n+-p-n+ and/or p+-n-p+ structures.

x

0.0

0.5

1.0

1.5

C(x,

)

12

34

56

0 L/4 L/2 3L/4 L

Interface

Epitaxial layer2

Epitaxial layer1

Fig. 2a. Distributions of concentrations of infused dopant in heterostructure from Fig. 1. Increasing of number of curves corresponds to increasing of difference between values of dopant diffusion coefficient


8

x

0.0

0.5

1.0

1.5

2.0

C(x,

)

12

34

0 L/4 L/2 3L/4 L

Interface

Epitaxial layer2

Epitaxial layer1

Fig. 2b. Distributions of concentrations of implanted dopant in heterostructure from Fig. 1. Increasing of number of curves corresponds to increasing of difference between values of dopant diffusion coefficient

x

0.0

0.5

1.0

1.5

C(x,

)

0 a=2L/3 L-a1 =-L/3

1

3

2

Fig. 3a. Distributions of concentration of infused in average epitaxial layer dopant in the heterostructure in Fig. 1 without another epitaxial layers (curve 1) and with them for the case, when dopant diffusion

coefficient in the average epitaxial layer is smaller in comparison with dopant diffusion coefficients in another layers (curves 2 and 3). Increasing of number of curves corresponds to increasing of difference

between values of dopant diffusion coefficients epitaxial layers of heterostructure

x

0.00001

0.00010

0.00100

0.01000

0.10000

1.00000

C(x,

)

fC(x)

L/40 L/2 3L/4 Lx0

1

2


9

Fig. 3b. Distributions of concentration of implanted in average epitaxial layer dopant in the heterostructure in Fig. 1 without another epitaxial layers (curve 1) and with them for the case, when dopant diffusion coefficient in the average epitaxial layer is smaller in comparison with dopant diffusion coefficients in another layers (curves 2 and 3). Increasing of number of curves corresponds to increasing of difference

between values of dopant diffusion coefficients epitaxial layers of heterostructure

Optimal value of annealing time we determine framework recently introduced criterion [12-14]. Framework the criterion we approximate real distributions of dopant by step-wise function and minimize the following mean-squared error

( ) ( )[ ] = L xdxxCLU 02

,

1 (12)

at annealing time . Dependences of the optimal value of annealing time from different parameters are presented on Figs. 4.

0.0 0.1 0.2 0.3 0.4 0.5a/L, , ,

0.0

0.1

0.2

0.3

0.4

0.5

D

0 L-

2

3

2

4

1

Fig. 4a. Dependences of dimensionless optimal value of annealing time of infused dopant, which has been calculated by minimization the mean-squared error (12), on different parameters of heterostructure. Curve 1

is the dependence of annealing time on relation a/L, = = 0 and equal to each other values of dopant diffusion coefficient in epitaxial layers. Curve 2 is the dependence of annealing time on parameter for

a/L=1/2 and = = 0. Curve 3 is the dependence of annealing time on parameter for a/L=1/2 and = = 0. Curve 4 is the dependence of annealing time on parameter for a/L=1/2 and = = 0

0.0 0.1 0.2 0.3 0.4 0.5a/L, , ,

0.00

0.04

0.08

0.12

D

0 L-

2

3

2

4

1


10

Fig. 4b. Dependences of dimensionless optimal value time of additional annealing of implanted dopant, which has been calculated by minimization the mean-squared error (12), on different parameters of

heterostructure. Curve 1 is the dependence of annealing time on relation a/L, = = 0 and equal to each other values of dopant diffusion coefficient in epitaxial layers. Curve 2 is the dependence of annealing time

on parameter for a/L=1/2 and = = 0. Curve 3 is the dependence of annealing time on parameter for a/L=1/2 and = = 0. Curve 4 is the dependence of annealing time on parameter for a/L=1/2 and = = 0

The figures show, that optimal values of annealing time after ion implantation are smaller, than optimal values of annealing time after infusion of dopant. Reason of the difference is necessity to use annealing of radiation defects after ion implantation. After that (if necessary) it could be used additional annealing of dopant. One can obtain spreading of distribution of concentration during annealing of radiation defects. Increasing of annealing time with increasing of thickness of epitaxial layer is the consequence of necessity of higher time for dopant to achieve interfaces between epitaxial layers of heterostructure. Decreasing of annealing time with increasing of values of parameters and is the consequence of increasing of values of dopant diffusion coefficient in the area, where main part of dopant has been diffused. Increasing of annealing time with increasing of the parameter is the consequence of decreasing of the term [C (x,y,z,t)/ P (x,y,z,T)] in approximation of dopant diffusion coefficient (3) with increasing of the parameter , because dopant concentration did not usually achieved value of limit of solubility of dopant and parameter is not smaller, than 1.

4. Conclusion

In this paper we introduce an approach of manufacturing multiemitter heterotransistors. Framework the approach the considered transistors became more compact and will include into itself p-n-junctions with higher sharpness.

ACKNOWLEDGEMENTS

This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government, grant of Scientific School of Russia and educational fellowship for scientific research.

APPENDIX

Equations and conditions for functions ( ) ,,,~ijk (i0, j0, k0) could be written as

( ) ( ) ( ) ( )

++=

2000

2

2000

2

2000

2

0

0000 ,,,~1,,,~1,,,~1,,,~

I

bI

bI

bDDI

zyxV

I ;

( ) ( ) ( ) ( )

++=

2000

2

2000

2

2000

2

0

0000 ,,,~1,,,~1,,,~1,,,~

V

bV

bV

bDDV

zyxI

V ;

( ) ( ) ( ) ( )+

++=

200

2

200

2

200

2

0

000 ,,,~1,,,~1,,,~1,,,~

i

z

i

y

i

xV

Ii Ib

Ib

IbD

DI

( ) ( ) ( ) ( )

+

+

+

,,,~

,,,

1,,,~,,,

1 1001000

0 iI

y

iI

xV

I ITgb

ITg

bDD


11

( ) ( )

+

,,,~

,,,

1 100iI

z

ITg

b, i 1;

( ) ( ) ( ) ( )+

++=

200

2

200

2

200

2

0

000 ,,,~1,,,~1,,,~1,,,~

i

z

i

y

i

xI

Vi Vb

Vb

VbD

DV

( ) ( ) ( ) ( )

+

+

+

,,,~

,,,

1,,,~,,,

1 1001000

0 iV

y

iV

xI

V VTgb

VTg

bDD

( ) ( )

+

,,,~

,,,

1 100iV

z

VTgb

, i 1;

( ) ( ) ( ) ( )

++=

2010

2

2010

2

2010

2

0

0010 ,,,~1,,,~1,,,~1,,,~

I

bI

bI

bDDI

zyxV

I

( ) ( ) ,,,~,,,~ 000000**

VIIV

;

( ) ( ) ( ) ( )

++=

2010

2

2010

2

2010

2

0

0010 ,,,~1,,,~1,,,~1,,,~

V

bV

bV

bDDV

zyxI

V

( ) ( ) ,,,~,,,~ 000000**

VIIV

;

( ) ( ) ( ) ( )

++=

2020

2

2020

2

2020

2

0

0020 ,,,~1,,,~1,,,~1,,,~

I

bI

bI

bDDI

zyxV

I

( ) ( ) ( ) ( ) ,,,~,,,~,,,~,,,~ 010000**

000010*

*

VIIVVI

IV

( ) ( ) ( ) ( )

++=

2020

2

2020

2

2020

2

0

0020 ,,,~1,,,~1,,,~1,,,~

V

bV

bV

bDDV

zyxI

V

( ) ( ) ( ) ( ) ,,,~,,,~,,,~,,,~ 010000**

000010*

*

VIVIVI

VI

;

( ) ( ) ( ) ( )+

++=

2110

2

2110

2

2110

2

0

0110 ,,,~1,,,~1,,,~1,,,~

I

bI

bI

bDDI

zyxV

I

( ) ( ) ( ) ( )

+

+

+

,,,~

,,,

1,,,~,,,

1 0100100

0 ITgb

ITg

bDD

I

y

I

xV

I

( ) ( ) ( ) ( )

+

*

*

000100*

*

010,,,

~

,,,

~,,,

~

,,,

1IVVI

IVITg

b Iz

( ) ( ) ,,,~,,,~ 100000 VI ; ( ) ( ) ( ) ( )

+

++=

2110

2

2110

2

2110

2

0

0110 ,,,~1,,,~1,,,~1,,,~

V

bV

bV

bDDV

zyxI

V

( ) ( ) ( ) ( )

+

+

+

,,,~

,,,

1,,,~,,,

1 0100100

0 VTgb

VTg

bDD

V

y

V

xI

V


12

( ) ( ) ( ) ( )

+

*

*

000100*

*

010,,,

~

,,,

~,,,

~

,,,

1IIVI

IIVTg

b Vz

( ) ( ) ,,,~,,,~ 100000 VI ; ( ) ( ) ( ) ( )

++=

200

2

200

2

200

2

0

000 ,,,~1,,,~1,,,~1,,,~

k

z

k

y

k

xV

Ik Ib

Ib

IbD

DI

( ) ( ) ( ) ( )

++=

200

2

200

2

200

2

0

000 ,,,~1,,,~1,,,~1,,,~

k

z

k

y

k

xI

Vk Ib

Ib

IbD

DV;

( ) ( ) ( ) ( )+

++=

2101

2

2101

2

2101

2

0

0101 ,,,~1,,,~1,,,~1,,,~

I

bI

bI

bDDI

zyxV

I

( ) ( ) ( ) ( )

+

+

+

,,,~

,,,

1,,,~,,,

1 0010010

0 ITgb

ITg

bDD

I

y

I

xV

I

( ) ( )

+

,,,~

,,,

1 001ITgb I

z

;

( ) ( ) ( ) ( )+

++=

2101

2

2101

2

2101

2

0

0101 ,,,~1,,,~1,,,~1,~

V

bV

bV

bDDV

zyxI

V

( ) ( ) ( ) ( )

+

+

+

,,,~

,,,

1,,,~,,,

1 0010010

0 ITgb

ITg

bDD

V

y

V

xI

V

( ) ( )

+

,,,~

,,,

1 001ITgb V

z

;

( ) ( ) ( ) ( )+

++=

2

0112

2

0112

2

0112

0

0011 ,,,~1,,,~1,,,~1,,,~

I

bI

bI

bDDI

zyxV

I

( ) ( ) ( ) ( ) ( )[ ,,,~,,,~,,,,,,~,,,~ 000000000001** VIThVIIV ( ) ( )] ,,,~,,,~ 001000 VI

( ) ( ) ( ) ( )

++=

2011

2

2011

2

2011

2

0

0011 ,,,~1,,,~1,,,~1,,,~

V

bV

bV

bDDV

zyxI

V

( ) ( ) ( ) ( ) ( )[ ,,,~,,,~,,,,,,~,,,~ 000000000001** VIThVIIV ( ) ( )] ,,,~,,,~ 001000 VI .

( )0

,,,~

0

=

= ijk ; ( ) 0,,,~

1

=

= ijk ; ( ) 0,,,~

0

=

= ijk ; ( ) 0,,,~

1

=

= ijk ;

( )0

,,,~

0

=

= ijk ; ( ) 0,,,

~

1

=

= ijk

, i 1, j 1, k 1; ( ) ( )*

,,

0,,,~000 f= ;

( ) 00,,,~ = ijk , i 1, j 1, k 1. Solutions of equations for the functions ( ) ,,,~ijk (i0, j0, k0) could be written as


13

( ) ( ) ( ) ( ) ( )+= =1

000

21,,,

~

nnn

ecccFLL

,

where ( ) ( ) ( ) ( ) = 10

1

0

1

0*

,,

1udvdwdwvufwcvcucF

nn , cn() = cos (pi n ), ( ) ( )IVnI DDne 0022exp pi = , ( ) ( )VInV DDne 0022exp pi = ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = =1 0

1

0

1

0

1

02

000 ,,,

*

2,,,

~

nVnnnnnnn

zyxx

i TwvugvcuseecccnLLLbD

pi

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

=

1 02

0100

*

2,,,~n

nnnnn

zyxy

in

eecccnLLLb

Ddudvdwd

u

wvuwc

pi

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

=

12

01

0

1

0

1

0

100

*

2,,,~,,,

nn

zyxz

iVnnn cnLLLb

Ddudvdwd

v

wvuTwvugwcvsuc

pi

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

0

1

0

1

0

1

0

100,,,

,,,~

dudvdwdTwvugw

wvuwsvcuceecc V

innnnnnn

, i 1;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = =1 0

1

0

1

0

1

0*

*

0010

*

2,,,

~

nnnnInInnnn

zyx

I wcvcuceecccIV

LLLIDI

( ) ( ) dudvdwdwvuVwvuI ,,,~,,,~ 000000 ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =

=1 0

1

0

1

0

1

0*

*

0010

*

2,,,

~

nnnnVnVnnnn

zyx

I wcvcuceecccVI

LLLVD

V

( ) ( ) dudvdwdwvuVwvuI ,,,~,,,~ 000000 ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =

=1 0

1

0

1

0

1

0*

*

0020

*

2,,,

~

nnnnInInnnn

zyx

I wcvcuceecccIV

LLLID

I

( ) ( ) ( ) ( ) ( ) ( ) =1*

*

0000010

*

2,,,

~

,,,

~

nInnnn

zyx

I ecccIV

LLLIDdudvdwdwvuVwvuI

( ) ( ) ( ) ( ) ( ) ( ) 0

1

0

1

0

1

0010000 ,,,

~

,,,

~ dudvdwdwvuVwvuIwcvcucennnIn ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = =1 0

1

0

1

0

1

0*

*

0020

*

2,,,

~

nnnnVnVnnnn

zyx

I wcvcuceecccVI

LLLVDV

( ) ( ) ( ) ( ) ( ) ( ) =1*

*

0000010

*

2,,,

~

,,,

~

nVnnnn

zyx

I ecccVI

LLLVDdudvdwdwvuVwvuI

( ) ( ) ( ) ( ) ( ) ( ) 0

1

0

1

0

1

0010000 ,,,

~

,,,

~ dudvdwdwvuVwvuIwcvcucennnVn ;

( ) ( ) ( ) ( ) 0,,,~,,,~,,,~,,,~ 002001002001 ==== VVII ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =

=1 0

1

0

1

0

1

0*

*

20

110*

2,,,

~

nnnnInInnnn

zyxx

I wcvcuseecccnIV

LLLIbDI

pi

( ) ( ) ( ) ( ) ( ) ( ) + =1*

*

20010

*

2,,,~,,,

nInnnn

zyxy

II ecccI

VLLLIb

Ddudvdwdu

wvuITwvug pi

( ) ( ) ( ) ( ) ( ) ( ) + 200

1

0

1

0

1

0

010

*

2,,,~,,,

zyxz

IInnnIn LLLIb

Ddudvdwdv

wvuITwvugwcvsucen pi


14

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =1 0

1

0

1

0

1

0

010*

*,,,

~

,,,

nInnnInnn dudvdwd

w

wvuITwvugwsvcuceccnIV

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =1 0

1

0

1

0

1

0100*

*

0,,,

~

*

2n

nnInInnnn

zyx

IInn wvuIvcuceecccI

VLLLI

Dec

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =1 0

1

0

1

0

1

0*

*

0000

*

2,,,

~

nnnnInInn

In

wcvcuceecIV

LIDdudvdwdwvuVwc

( ) ( ) ( ) ( )nn

ccdudvdwdwvuVwvuI ,,,~,,,~ 100000 ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = =1 0

1

0

1

0

1

0*

*

20

110*

2,,,

~

nnnnVnVnnnn

zyxx

V wcvcuseecccnVI

LLLIbD

V

pi

( ) ( ) ( ) ( ) ( ) ( ) + =1*

*

2

0010

*

2,,,~,,,

nVnnnn

zyxy

V

V ecccnVI

LLLVbD

dudvdwdu

wvuVTwvug

pi

( ) ( ) ( ) ( ) ( ) ( ) + **

20

0

1

0

1

0

1

0

010

*

2,,,~,,,

VI

LLLVbDdudvdwd

v

wvuVTwvugwcvsuce

zyxz

VVnnnVn

pi

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =1 0

1

0

1

0

1

0

010 ,,,~

,,,

nVnnnVnnnn dudvdwd

w

wvuVTwvugwsvcucecccn

( ) ( ) ( ) ( ) ( ) ( ) ( ) =1 0

1

0000100

0,,,

~

,,,

~

*

2n

nVnVnn

zyx

VVn dudvdwdwvuVwvuIvceecLLLV

De

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =1 0

1

0

1

0

1

0000*

*

0*

*

,,,

~

*

2n

nnnVnnnnV

nnwvuIwcvcuceccc

VI

LVD

VI

cc

( ) ( ) VnedudvdwdwvuV ,,,~100 ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =

=1 0

1

0

1

0

1

02101

,,,

*

2,,,

~

nInnInInnnn

zyxx

TwvugvcuseecccnLLLIb

I

pi

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + =1 0

1

02

001

*

2,,,~

nnInInnnn

zyxy

nuceeccc

LLLIbdudvdwd

u

wvuIwc

pi

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )+ =1

1

0

1

0

001

*

2,,,~,,,

nInnnn

z

Inn ecccnIbdudvdwd

v

wvuITwvugwcvsn pi

( ) ( ) ( ) ( ) ( ) ( )2

0

1

0

1

0

1

0

001 1,,,~

,,,

zyx

InnnIn LLLdudvdwd

w

wvuITwvugwsvcuce

;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = =1 0

1

0

1

0

1

02101

,,,

*

2,,,

~

nVnnVnVnnnn

zyxx

TwvugvcuseecccLLLVb

V

pi

( ) ( ) ( ) ( ) ( ) ( ) ( ) + =1 0

2001 2,,,

~

nVnVnnnn

yzyx

neecccn

bLLLdudvdwd

u

wvuVwcn

pi

( ) ( ) ( ) ( ) ( ) ( ) + =12

1

0

1

0

1

0

001

*

2,,,~

*

,,,

nn

zyxz

Vnnn

cnLLLVb

dudvdwdv

wvuVV

Twvugwcvsuc pi

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

0

1

0

1

0

1

0

001 ,,,~

,,, dudvdwdw

wvuVTwvugwsvcuceecc VnnnVnVnnn ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = =1 0

1

0

1

0

1

0001011 ,,,

~

*

2,,,

~

nnnInInnnn

zyx

wvuIvcuceecccLLLI

I


15

( ) ( ) ( ) ( ) ( ) ( ) ( ) =1 0

*

*

*

*

000*

2,,,

~

nInInnnn

zyx

neeccc

IV

LLLIIVdudvdwdwvuVwc

( ) ( ) ( ) ( ) ( ) ( ) ( ) =1*

*1

0

1

0

1

0001000

*

2,,,

~

,,,

~

nnn

zyx

nnncc

IV

ILLLdudvdwdwvuVwvuIwcvcuc

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0

1

0

1

0

1

0001000 ,,,

~

,,,

~

,,, dudvdwdwvuVwvuIwcTwvuhvcuceecnnnInInn ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = =1 0

1

0

1

0

1

0*

*

011*

2,,,

~

nnnnVnVnnnn

zyx

wcvcuceecccVI

LLLVV

( ) ( ) ( ) ( ) ( ) ( ) ( ) =1 0

000001*

2,,,

~

,,,

~

nVnVnnnn

zyx

eecccLLLV

dudvdwdwvuVwvuI

( ) ( ) ( ) ( ) ( ) ( ) =1

1

0

1

0

1

0001000*

*

*

2,,,

~

,,,

~

nn

zyx

nnnc

LLLVdudvdwdwvuVwvuIwcvcuc

VI

( ) ( ) ( ) ( ) ( ) ( ) ( ) 0

1

0

1

0

1

0001000*

*

,,,

~

,,,

~

,,, dudvdwdwvuVwvuIwcTwvuhvcuceVI

nnnVn

( ) ( ) ( ) Vnnn ecc . Equations for the functions i(x,y,z,t) are

( ) ( ) ( ) ( )+

+

+

=

2

02

20

2

20

2

00 ,,,,,,,,,,,,

z

tzyxy

tzyxx

tzyxDt

tzyx IIII

I

( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxk III ,,,,,,,,,,,, 2, + ; ( ) ( ) ( ) ( )

+

+

+

=

2

02

20

2

20

2

00 ,,,,,,,,,,,,

z

tzyxy

tzyxx

tzyxDt

tzyx VVVV

V

( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxk VVV ,,,,,,,,,,,, 2, + ; ( ) ( ) ( ) ( )

+

+

+

=

2

2

02

2

02

2

0

,,,,,,,,,,,,

z

tzyxD

ytzyx

Dx

tzyxD

t

tzyx iII

iII

iII

iI

( ) ( ) ( ) ( ) +

+

+

ytzyx

Tzyxgy

Dx

tzyxTzyxg

xD iIII

iIII

,,,

,,,

,,,

,,,1

01

0

( ) ( )

+

z

tzyxTzyxg

zD iIII

,,,

,,,1

0 , i1;

( ) ( ) ( ) ( )+

+

+

=

2

2

02

2

02

2

0

,,,,,,,,,,,,

z

tzyxD

ytzyx

Dx

tzyxD

t

tzyx iVV

iVV

iVV

iV

( ) ( ) ( ) ( ) +

+

+

ytzyx

Tzyxgy

Dx

tzyxTzyxg

xD iVVV

iV

VV

,,,

,,,

,,,

,,,1

01

0

( ) ( )

+

z

tzyxTzyxg

zD iVVV

,,,

,,,1

0 , i1;

( )0

,,,

0

=

=x

i

x

tzyx,

( )0

,,,

=

= xLx

i

x

tzyx,

( )0

,,,

0

=

=y

i

ytzyx

,

( )0

,,,

=

= yLy

i

ytzyx

,

( )0

,,,

0

=

=z

i

z

tzyx,

( )0

,,,

=

= zLz

i

z

tzyx, i0; I0(x,y,z,0)=fI (x,y,z), Ii(x,y,z,0)=0,


16

V0(x,y,z,0)=fV (x,y,z), Vi(x,y,z,0)=0, i1. Solutions of the equations could be written as

( ) ( ) ( ) ( ) ( )+= =

1

0

21,,,

nnn

zyxzyx

tezcycxcFLLLLLL

tzyx

,

where ( )

++= 2220

22 111expzyx

n LLLtDnte

pi , ( ) ( ) ( ) ( ) = x

L yL zL

nnnnxdydzdzyxfzcycxcF

0 0 0,,

,

cn() = cos (pi n /L); ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =

=

1 0 0 0 0

2.,,

2,,,

n

t xL yL zL

nnnnnnnn

zyx

i TwvugwcvcusetezcycxcnLLLtzyx

pi

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

1 0 0 02

1 2,,,n

t xL yL

nnnnnnn

zyx

iIvsucetezcycxcn

LLLdudvdwd

u

wvu

pi

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

120

1 2,,,.,,

nnnnn

zyx

zL iI

ntezcycxcn

LLLdudvdwd

v

wvuTwvugwc

pi

( ) ( ) ( ) ( ) ( ) ( ) t xL yL zL iI

nnnndudvdwd

w

wvuTwvugwsvcuce

0 0 0 0

1 ,,,

.,,

, i1,

where sn() = sin (pi n/L).

System of equations for functions Cijk(x,y,z,t) (i0, j0, k0) could be written as

( ) ( ) ( ) ( )

+

+

=

200

2

200

2

200

2

000 ,,,,,,,,,,,,

z

tzyxCy

tzyxCx

tzyxCDt

tzyxCL ;

( ) ( ) ( ) ( )+

+

+

=

20

2

020

2

020

2

00 ,,,,,,,,,,,,

z

tzyxCD

ytzyxC

Dx

tzyxCD

t

tzyxC iL

iL

iL

i

( ) ( ) ( ) ( ) +

+

+ y

tzyxCTzyxg

yD

x

tzyxCTzyxg

xD iLL

iLL

,,,

,,,

,,,

,,,10

010

0

( ) ( )

+ z

tzyxCTzyxg

zD iLL

,,,

,,,10

0 , i 1;

( ) ( ) ( ) ( )+

+

+

=

201

2

0201

2

0201

2

001 ,,,,,,,,,,,,

z

tzyxCD

ytzyxC

Dx

tzyxCD

t

tzyxCLLL

( )( )

( ) ( )( )

( )+

+

+y

tzyxCTzyxPtzyxC

yD

x

tzyxCTzyxPtzyxC

xD LL

,,,

,,,

,,,,,,

,,,

,,, 00000

00000

( )( )

( )

+z

tzyxCTzyxPtzyxC

zD L

,,,

,,,

,,, 00000

;

( ) ( ) ( ) ( )+

+

+

=

LLLL Dz

tzyxCD

ytzyxC

Dx

tzyxCD

t

tzyxC02

022

0202

2

0202

2

002 ,,,,,,,,,,,,

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

( )

+

+

ytzyxC

TzyxPtzyxC

tzyxCyx

tzyxCTzyxPtzyxC

tzyxCx

,,,

,,,

,,,

,,,

,,,

,,,

,,,

,,,00

100

0100

100

01


17

( ) ( )( )( ) ( )

( )( )

+

+

+

x

tzyxCTzyxPtzyxC

xD

z

tzyxCTzyxPtzyxC

tzyxCz

L

,,,

,,,

,,,,,,

,,,

,,,

,,,0100

000

100

01

( )( )

( ) ( )( )

( )

+

+z

tzyxCTzyxPtzyxC

zytzyxC

TzyxPtzyxC

y,,,

,,,

,,,,,,

,,,

,,, 01000100

;

( ) ( ) ( ) ( )+

+

+

=

LLLL Dz

tzyxCD

ytzyxC

Dx

tzyxCD

t

tzyxC02

112

0211

2

0211

2

011 ,,,,,,,,,,,,

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

( )+

+

ytzyxC

TzyxPtzyxC

tzyxCyx

tzyxCTzyxPtzyxC

tzyxCx

,,,

,,,

,,,

,,,

,,,

,,,

,,,

,,,00

100

1000

100

10

( ) ( )( )( ) ( )

( )( )

+

+

+

x

tzyxCtzyxPtzyxC

xD

z

tzyxCTzyxPtzyxC

tzyxCz

L

,,,

,,,

,,,,,,

,,,

,,,

,,,1000

000

100

10

( )( )

( ) ( )( )

( ) ( )

+

+

+ Tzyxgxz

tzyxCtzyxPtzyxC

zytzyxC

tzyxPtzyxC

y L,,,

,,,

,,,

,,,,,,

,,,

,,, 10001000

( ) ( ) ( ) ( ) ( ) LLL Dz

tzyxCTzyxg

zytzyxC

Tzyxgyx

tzyxC0

010101 ,,,,,,

,,,

,,,

,,,

+

+

.

Solutions of equations for functions Cij(x,y,z,t) (i0, j0) with account appropriate conditions takes the form

( ) ( ) ( ) ( ) ( )+= =1

00

21,,,

nnCnC

zyxzyx

tezcycxcFLLLLLL

tzyxC ,

where ( )

++=

222022 111exp

zyx

CnC LLLtDnte pi , ( ) ( ) ( ) ( ) = xL y

L zL

CnnnCn xdydzdzyxfzcycxcF0 0 0

,, ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = =1 0 0 0 0

20,,,

2,,,

n

t xL yL zL

LnnnnCnCnnnnC

zyx

i TwvugwcvcusetezcycxcFLLLtzyxC pi

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

=

1 0 0 02

10 2,,,n

t xL yL

nnnCnCnnnnC

zyx

i vsucetezcycxcFnLLL

ndudvdwdu

wvuC

pi

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

=

1 02

0

10 2,,,,,,

n

t

nCnCnnn

zyx

zLi

Ln etezcycxcLLLdudvdwd

v

wvuCTwvugwc pi

( ) ( ) ( ) ( ) ( )

xL yL zL

iLnnnnC dudvdwd

w

wvuCTwvugwsvcucFn

0 0 0

10 ,,,,,,

, i 1;

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

=1 0 0 0 0

00201

,,,

,,,2,,,

n

t xL yL zL

nnnCnCnnnnC

zyx TwvuPwvuC

vcusetezcycxcFnLLL

tzyxC

pi

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

=1 0 02

00 2,,,n

t xL

nnCnCnnnnC

zyx

nucetezcycxcFn

LLLdudvdwd

u

wvuCwc

pi

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

=120 0

0000 2,,,,,,

,,,

nnnnnC

zyx

yL zL

nnzcycxcFn

LLLdudvdwd

v

wvuCTwvuP

wvuCwcvs

pi

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

t xL yL zL

nnnnCnC dudvdwdw

wvuCTwvuP

wvuCwsvcucete

0 0 0 0

0000 ,,,

,,,

,,,

;


18

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

=

1 0 0 0

100

0202

,,,

,,,2,,,

n

t xL yL zL

nnnnCnCnnnnC

zyx TwvuPwvuC

wcvcusetezcycxcFnLLL

tzyxC

pi

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

=1 0 0 02

00 2,,,n

t xL yL

nnnCnCnnnnC

zyx

vsucetezcycxcFnLLL

dudvdwdu

wvuC

pi

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

=

120

001

00 2,,,,,,

,,,

nnCnnnnC

zyx

zL

ntezcycxcFn

LLLdudvdwd

v

wvuCTwvuP

wvuCwc

pi

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

=

120 0 0 0

001

00 2,,,,,,

,,,

nn

zyx

t xL yL zL

nnnnC xcnLLLdudvdwd

w

wvuCTwvuP

wvuCwsvcuce

pi

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

t xL yL zL

nnnnCnCnnnC dudvdwdu

wvuCTwvuP

wvuCwcvcusetezcycF

0 0 0 0

0100 ,,,

,,,

,,,

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

=1 0 0 0 0

01002

,,,

,,,

,,,2n

t xL yL zL

nnnCnCnnnnC

zyx v

wvuCTwvuP

wvuCvsucetezcycxcFn

LLL

pi

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =1 0 0 0 0

2

2n

t xL yL zL

nnnnCnCnnnnC

zyx

nwsvcucetezcycxcFn

LLLdudvdwdwc pi

( )( )

( )

dudvdwdw

wvuCTwvuP

wvuC

,,,

,,,

,,, 0100 ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = =1 0 0 0 0

211,,,

2,,,

n

t xL yL zL

LnnnnCnCnnnnC

zyx

TwvugwcvcusetezcycxcFnLLL

tzyxC pi

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

=1 0 0 0 02

01 2,,,n

t xL yL zL

nnnnCnCnnn

zyx

wcvsucetezcycxcLLL

dudvdwdu

wvuC

pi

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

=1 02

01 2,,,,,,

n

t

nCnCnnn

zyx

LnC etezcycxcLLLdudvdwd

v

wvuCTwvugFn pi

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

=120 0 0

01 2,,,,,,

nnnC

zyx

xL yL zL

LnnnnC xcFnLLLdudvdwd

w

wvuCTwvugwsvcucFn pi

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

t xL yL zL

nnnnCnCnn dudvdwdu

wvuCTwvuP

wvuCwcvcusetezcyc

0 0 0 0

1000 ,,,

,,,

,,,

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

=1 0 0 0 0

10002

,,,

,,,

,,,2n

t xL yL zL

nnnnCn

zyx

dudvdwdv

wvuCTwvuP

wvuCwcvsucexcn

LLL

pi

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =1 0 0 0 0

2

2n

t xL yL zL

nnnnCnCnnnnC

zyx

nCnnnC wsvcucetezcycxcFnLLLtezcycF pi

( )( )

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

=1 02

1000 2,,,,,,

,,,

n

t

nCnCnnnnC

zyx

etezcycxcFnLLL

dudvdwdw

wvuCTwvuP

wvuC

pi

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

=

120 0 0

001

00 2,,,,,,

,,,

nnnC

zyx

xL yL zL

nnnxcFn

LLLdudvdwd

u

wvuCTwvuP

wvuCwcvcus

pi

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

t xL yL zL

nnnnCnCnn dudvdwdv

wvuCTwvuP

wvuCwcvsucetezcyc

0 0 0 0

001

00 ,,,

,,,

,,,

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

=

1

00

0 0 0 0

100

2

,,,

,,,

,,,2n

t xL yL zL

nnnnC

zyx

dudvdwdw

wvuCTwvuP

wvuCwsvcucen

LLL

pi

( ) ( ) ( ) ( )tezcycxcFnCnnnnC .


19

REFERENCES

[1] I.P. Stepanenko. Basis of Microelectronics, Soviet Radio, Moscow, 1980. [2] A.G. Alexenko, I.I. Shagurin. Microcircuitry Radio and Communication, Moscow, 1990. [3] V.G. Gusev, Yu.M. Gusev. Electronics, Moscow: Vysshaya shkola, 1991, in Russian. [4] V.I. Lachin, N.S. Savelov. Electronics,Phoenix, Rostov-na-Donu, 2001. [5] Jung-Hui Tsai, Shao-Yen Chiu, Wen-Shiung Lour, Der-Feng Guo. High-performance InGaP/GaAs

pnp delta-doped heterojunction bipolar transistor, Semiconductors. Vol. 43 (7). P. 971-974 (2009). [6] T. Y. Peng, S. Y. Chen, L. C. Hsieh, C. K. Lo, Y. W. Huang, W. C. Chien, Y. D. Yao. Impedance

behavior of spin-valve transistor, J. Appl. Phys. Vol. 99 (8). P. 08H710-08H712 (2006). [7] S. Maximenko, S. Soloviev, D. Cherednichenko, T. Sudarshan. Observation of dislocations in

diffused 4HSiC p-i-n diodes by electron-beam induced current, J. Appl. Phys. Vol. 97 (1). P. 013533-013538 (2005).

[8] I.E. Tyschenko, V.A. Volodin. Quantum confinement effect in silicon-on-insulator films implanted with high dose hydrogen ions, Semiconductors. Vol.46 (10). P.1309-1313 (2012).

[9] K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. Dopant distribution in the recrystallization transient at the maximum melt depth induced by laser annealing, Appl. Phys. Lett. Vol. 89 (17). P. 172111 (2006).

[10] Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov, Yu.N. Drozdov, V.D. Skupov. Diffusion processes in semiconductor structures during microwave annealing, Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836-843 (2003).

[11] V.V. Kozlivsky. Modification of semiconductors by proton beams, Sant-Peterburg, Nauka, 2003, in Russian.

[12] E.L. Pankratov. Application of porous layers and optimization of annealing of dopant and radiation defects to increase sharpness of p-n-junctions in a bipolar heterotransistors, J. Optoel. Nanoel. Vol. 6 (2). P. 188 (2011).

[13] E.L. Pankratov. Increasing of homogeneity of dopant distribution in a p-n-junction by using an overlayers, J. Comp. Theor. Nanoscience. Vol. 8 (2). P. 207 (2011).

[14] E.L. Pankratov, E.A. Bulaeva. Increasing of sharpness of diffusion-junction heterorectifier by using radiation pro-cessing, Int. J. Nanoscience. Vol. 11 (5). P. 1250028 (2012).

[15] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991). [16] E.I. Zorin, P.V. Pavlov and D.I. Tetelbaum. Ion doping of semiconductors, Energy, Moscow, 1975. [17] P.M. Fahey, P.B. Griffin, J.D. Plummer. Point defects and dopant diffusion in silicon, Rev. Mod.

Phys. 1989. V. 61. 2. P. 289-388. [18] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. (Naukova Dumka, Kiev, 1979,

in Russian). [19] E.L. Pankratov. Analysis of redistribution of radiation defects with account diffusion and several

secondary processes, Mod. Phys. Lett. B. 2008. Vol. 22 (28). P. 2779-2791. [20] K.V. Shalimova. Physics of semiconductors, Moscow: Energoatomizdat, 1985, in Russian. [21] A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations, Moscow, Nauka 1972) (in

Russian. [22] H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids, Oxford University Press, 1964.

Author

Pankratov Evgeny Leonidovich was born at 1977. From 1985 to 1995 he was educated in a secondary school in Nizhny Novgorod. From 1995 to 2004 he was educated in Nizhny Novgorod State University: from 1995 to 1999 it was bachelor course in Radiophysics, from 1999 to 2001 it was master course in Radiophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in Radiophysics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of Microstructures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny Novgorod 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 96 published papers in area of his researches.


20

Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school of village Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school Center for gifted children. From 2009 she is a student of Nizhny Novgorod State University of Architecture 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 29 published papers in area of her researches.

an approach for optimization of

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