modeling of manufacturing of a fieldeffect

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

31

MODELING OF MANUFACTURING OF A FIELD-

EFFECT TRANSISTOR TO DETERMINE CONDITIONS

TO DECREASE LENGTH OF CHANNEL

E.L. Pankratov1, E.A. Bulaeva2

1Nizhny 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 introduce an approach to model technological process of manufacture of a field-effect heterotransistor. The modeling gives us possibility to optimize the technological process to decrease length of channel by using mechanical stress. As accompanying results of the decreasing one can find decreasing of thickness of the heterotransistors and increasing of their density, which were comprised in integrated circuits.

KEYWORDS

Modelling of manufacture of a field-effect heterotransistor, Accounting mechanical stress; Optimization the technological process, Decreasing length of channel

1. INTRODUCTION

One of intensively solved problems of the solid-state electronics is decreasing of elements of in-tegrated circuits and their discrete analogs [1-7]. To solve this problem attracted an interest wide-ly used laser and microwave types of annealing [8-14]. These types of annealing leads to genera-tion of inhomogeneous distribution of temperature. In this situation one can obtain increasing of sharpness of p-n-junctions with increasing of homogeneity of dopant distribution in enriched area [8-14]. The first effects give us possibility to decrease switching time of p-n-junction and value of local overheats during operating of the device or to manufacture more shallow p-n-junction with fixed maximal value of local overheat. An alternative approach to laser and microwave types of annealing one can use native inhomogeneity of heterostructure and optimization of annealing of dopant and/or radiation defects to manufacture diffusive junction and implanted-junction rectifi-ers [12-18]. It is known, that radiation processing of materials is also leads to modification of dis-tribution of dopant concentration [19]. The radiation processing could be also used to increase of sharpness of p-n-junctions with increasing of homogeneity of dopant distribution in enriched area [20, 21]. Distribution of dopant concentration is also depends on mechanical stress in heterostruc-ture [15].

In this paper we consider a field-effect heterotransistor. Manufacturing of the transistor based on combination of main ideas of Refs. [5] and [12-18]. Framework the combination we consider a


32

heterostructure with a substrate and an epitaxial layer with several sections. Some dopants have been infused or implanted into the section to produce required types of conductivity. Farther we consider optimal annealing of dopant and/or radiation defects. Manufacturing of the transistor has been considered in details in Ref. [17]. Main aim of the present paper is analysis of influence of mechanical stress on length of channel of the field-effect transistor.

Fig. 1. Heterostructure with substrate and epitaxial layer with two or three sections

a1

a2-a

1L

x-a

2-a

1

c1 c1

c2

oxidesource drain

gate

c3

Fig. 2. Heterostructure with substrate and epitaxial layer with two or three sections. Sectional view


33

2. METHOD OF SOLUTION

To solve our aim we determine spatio-temporal distribution of concentration of dopant in the con-sidered heterostructure and make the analysis. We determine spatio-temporal distribution of con-centration of dopant with account mechanical stress by solving the second Ficks law [1-4,22,23]

( ) ( ) ( ) ( )+

+

+

=

z

tzyxCD

zytzyxC

Dyx

tzyxCD

xt

tzyxC ,,,,,,,,,,,, (1)

( ) ( ) ( ) ( )

+

+zz L

SS

L

SS WdtWyxCtzyx

TkD

yWdtWyxCtzyx

TkD

x 00,,,,,,,,,,,,

with boundary and initial conditions ( ) 0,,,

0

=

=xx

tzyxC,

( ) 0,,, =

= xLx

x

tzyxC,

( ) 0,,,0

=

=yy

tzyxC,

( ) 0,,, =

= yLx

ytzyxC

,

( ) 0,,,0

=

=zz

tzyxC,

( ) 0,,, =

= zLx

z

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

Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant; is the atomic vo-lume; S is the operators of surficial gradient; ( )zL zdtzyxC

0,,, is the surficial concentration of

dopant on interface between layers of heterostructure; (x,y,z,t) is the chemical potential; D and DS are coefficients of volumetric and surficial (due to mechanical stress) of diffusion. Values of the diffusion coefficients depend on properties of materials of heterostructure, speed of heating and cooling of heterostructure, spatio-temporal distribution of concentration of dopant. Concen-traional dependence of dopant diffusion coefficients have been approximated by the following relations [24]

( ) ( )( )

+=

TzyxPtzyxCTzyxDD L

,,,

,,,1,,,

, ( ) ( )( )

+=

TzyxPtzyxCTzyxDD SLSS

,,,

,,,1,,,

. (2)

Here DL (x,y,z,T) and DLS (x,y,z,T) are spatial (due to inhomogeneity of heterostructure) and tem-perature (due to Arrhenius law) dependences of dopant diffusion coefficients; T is the temperature of annealing; P

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

[1,3] [24]; V

(x,y,z,t) is the spatio-temporal distribution of concentration of radiation of vacancies; V* is the equilibrium distribution of vacancies. Concentrational depen-dence of dopant diffusion coefficients has been discussed in details in [24]. Chemical potential could be determine by the following relation [22]

=E(z)ij[uij(x,y,z,t)+uji(x,y,z,t)]/2, (3)

where E is the Young modulus; ij is the stress tensor;

+

=

i

j

j

iij

x

u

x

uu

21

is the deformation

tensor; ui, uj are the components ux(x,y,z,t), uy(x,y,z,t) and uz(x,y,z,t) of the displacement vector ( )tzyxu ,,,r ; xi, xj are the coordinates x, y, z. Relation (3) could be transform to the following form


34

( ) ( ) ( ) ( ) ( )

+

+

=

i

j

j

i

i

j

j

i

x

tzyxux

tzyxux

tzyxux

tzyxutzyx

,,,,,,

21,,,,,,

,,,

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

+

ijk

kijij TtzyxTzzK

x

tzyxuz

zzE

000 ,,,3,,,

212,

where is the Poisson coefficient; 0=(as-aEL)/aEL is the mismatch strain; as, aEL are the lattice spacings for substrate and epitaxial layer, respectively; K is the modulus of uniform compression; is the coefficient of thermal expansion; Tr is the equilibrium temperature, which coincide (for our case) with room temperature. Components of displacement vector could be obtained by solv-ing of the following equations [23]

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

+

+

=

+

+

=

+

+

=

z

tzyxy

tzyxx

tzyxt

tzyxuz

z

tzyxy

tzyxx

tzyxt

tzyxuz

z

tzyxy

tzyxx

tzyxt

tzyxuz

zzzyzxz

yzyyyxy

xzxyxxx

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

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

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

2

2

2

2

2

2

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

+

+

+

=

k

kij

k

kij

i

j

j

iij

x

tzyxux

tzyxux

tzyxux

tzyxuz

zE ,,,,,,3

,,,,,,

12

( ) ( ) ( ) ( )[ ]r

TtzyxTzKzzK ,,, , (z) is the density of materials, ij is the Kronecker sym-bol. With account of the relation the last system of equation takes the form

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

( )+

++

++=

yxtzyxu

z

zEzK

x

tzyxuz

zEzK

t

tzyxuz

yxx,,,

13,,,

165,,, 2

2

2

2

2

( )( )[ ]

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

+++

+

+

+ zKzx

tzyxuz

zEzK

z

tzyxuy

tzyxuz

zE zzy ,,,13

,,,,,,

12

2

2

2

2

2

( ) ( )x

tzyxTz

,,,

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

+

+

=

ytzyxT

zKzyx

tzyxux

tzyxuz

zEt

tzyxuz x

yy ,,,,,,,,,

12,,,

2

2

2

2

2

( )( )[ ]

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

( )+

++

+

+

+

+ 2

2,,,

1125,,,,,,

12 ytzyxu

zKz

zEy

tzyxuz

tzyxuz

zEz

yzy

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

yxtzyxu

zKzy

tzyxuz

zEzK yy

+

++

,,,,,,

16

22

(4)

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

+

+

+

=

zytzyxu

zx

tzyxuy

tzyxux

tzyxut

tzyxuz

yxzzz,,,,,,,,,,,,,,,

22

2

2

2

2

2

2


35

( )( )[ ] ( )

( ) ( ) ( ) ( )

+

+

++

z

tzyxTz

tzyxuy

tzyxux

tzyxuzK

zz

zE xyx ,,,,,,,,,,,,12

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

+

+z

tzyxuy

tzyxux

tzyxuz

tzyxuz

zEz

zzK zyxz,,,,,,,,,,,,6

161

.

Conditions for the displacement vector could be written as ( ) 0,,,0 =

x

tzyur

; ( ) 0,,, =

x

tzyLu xr

; ( ) 0,,0, =

y

tzxur

; ( )

0,,,

=

ytzLxu y

r

;

( ) 0,0,, =

z

tyxur

; ( ) 0,,, =

z

tLyxu zr

; ( ) 00,,, uzyxu rr = ; ( ) 0,,, uzyxu rr = .

Farther let us analyze spatio-temporal distribution of temperature during annealing of dopant. We determine spatio-temporal distribution of temperature as solution of the second law of Fourier [25]

( ) ( ) ( ) ( ) ( ) +

+

+

=

z

tzyxTzy

tzyxTyx

tzyxTxt

tzyxTTc ,,,,,,,,,,,,

( )tzyxp ,,,+ (5)

with boundary and initial conditions ( ) 0,,,

0

=

=xx

tzyxT,

( ) 0,,, =

= xLx

x

tzyxT,

( ) 0,,,0

=

=yy

tzyxT,

( ) 0,,, =

= yLx

ytzyxT

,

( ) 0,,,0

=

=zz

tzyxT,

( ) 0,,, =

= zLx

z

tzyxT, T (x,y,z,0)=fT (x,y,z).

Here is the heat conduction coefficient. Value of the coefficient depends on materials of hetero-structure and temperature. Temperature dependence of heat conduction coefficient in most inter-est area 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, [25]). c(T)=cass[1- exp(-T(x,y,z,t)/Td)] is the heat capacitance; Td is the Debye temperature [25]. The temperature T(x,y,z,t) is approximately equal or larger, than Debye temperature Td for most interesting for us temperature interval. In this situation one can write the following approximate relation: c(T)cass. p(x,y,z,t) is the volumetric density of heat, which escapes in the heterostructure. Framework the paper it is attracted an interest microwave annealing of dopant. This approach leads to generation inhomogenous distribution of temperature [12-14,26]. Such distribution of temperature gives us possibility to increase sharpness of p-n-junctions with increasing of homogeneity of dopant distribution in enriched area. In this situation it is practicably to choose frequency of electro-magnetic field. After this choosing thickness of scin-layer should be approximately equal to thickness of epitaxial layer.

First of all we calculate spatio-temporal distribution of temperature. We calculate spatio-temporal distribution of temperature by using recently introduce approach [15, 17,18]. Framework the ap-proach we transform approximation of thermal diffusivity

ass(x,y,z) =ass(x,y,z)/cass =0ass[1+T gT(x,y,z)]. Farther we determine solution of Eqs. (5) as the following power series

( ) ( ) = =

=0 0,,,,,,

i j ijji

T tzyxTtzyxT . (6)


36

Substitution of the series into Eq.(6) 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 (6) in boundary and initial conditions for spatio-temporal distribution of temperature gives us possibility to obtain boundary and initial conditions for the functions Tij(x,y,z,t) (i0, j0). The conditions are presented in the Appendix. Solutions of the equations for the functions Tij(x,y,z,t) (i0, j0) have been obtain as the second-order approximation on the parameters and by standard approaches [28,29] and presented in the Appendix. Recently we obtain, that the second-order approximation is enough good approximation to make qualitative analysis and to obtain some quantitative results (see, for example, [15,17,18]). Analytical results have allowed to identify and to illustrate the main depen-dence. To check our results obtained we used numerical approaches.

Farther let us estimate components of the displacement vector. The components could be obtained by using the same approach as for calculation distribution of temperature. However, relations for components could by calculated in shorter form by using method of averaging of function correc-tions [12-14,16,27]. It is practicably to transform differential equations of system (4) to integro-differential form. The integral equations are presented in the Appendix. We determine the first-order approximations of components of the displacement vector by replacement the required functions u(x,y,z,t) on their average values u1. The average values u1 have been calculated by the following relations

us1=Ms1/4L, (7)

where ( ) = 0 0

,,,

x

x

y

y

zL

L

L

L

L

isis tdxdydzdtzyxuM . The replacement leads to the following results

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

=

t zz

xdwd

x

wyxTwwKtdwd

x

wyxTwwKttzyxu

0 00 01

,,,,,,

,,,

( )] 101 ,,, uxxux twyx + , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

=

t zz

y dwdywyxT

wwKtdwdy

wyxTwwKttzyxu

0 00 01

,,,,,,

,,,

( )] 101 ,,, uyyuy twyx + , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

=

t zz

z dwdw

wyxTwwKtdwd

w

wyxTwwKttzyxu

0 00 01

,,,,,,

,,,

( )] 101 ,,, zzuz twyx + .

Substitution of the above relations into relations (7) gives us possibility to obtain values of the parameters u1. Appropriate relations could be written as

( ) ( )[ ]( ) ( )

=

zL

z

xxz

ux

zdzzLL

L

0

3

201

8 ,

( ) ( )[ ]( ) ( )

=

zL

z

yyzuy

zdzzLL

L

0

3

201

8 ,

( ) ( )[ ]( ) ( )

=

zL

z

zzz

uz

zdzzLL

L

0

3

201

8 ,

where ( ) ( ) ( ) ( ) ( )

+=

0 0

,,,1x

x

y

y

zL

L

L

L

L

z

i

is dxdydzds

zyxTzzKzLt .


37

The second-order approximations of components of the displacement vector by replacement of the required functions u(x,y,z,t) on the following sums us2+us1(x,y,z,t), where us2=(Mus2-Mus1)/ 4L3. Results of calculations of the second-order approximations us2(x,y,z,t) and their average values us2 are presented in the Appendix, because the relations are bulky.

Spatio-temporal distribution of concentration of dopant we obtain by solving the Eq.(1). To solve the equations we used recently introduce approach [15,17,18] and transform approximations of dopant diffusion coefficients DL (x,y,z,T) and DSL (x,y,z, T) to the following form: DL(x,y,z,T)=D0L[1+ L gL(x,y,z,T)] and DSL(x,y,z,T)=D0SL[1+ SLgSL(x,y,z,T)]. We also introduce the following dimensionless parameter: = D0SL/ D0L. In this situation the Eq.(1) takes the form

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

+

++

=

ytzyxC

yxtzyxC

TzPtzyxCTzg

xD

t

tzyxCLLL

,,,,,,

,

,,,1,1,,, 0

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

++

+

++

TzyxPtzyxCTzyxg

zDD

TzyxPtzyxCTzg LLLLLL

,,,

,,,1,,,1,,,

,,,1,1 00

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

++

+

zL

SLSLSL WdtWyxCTzPtzyxCTzg

xD

z

tzyxC0

0 ,,,,

,,,1,1,,,

(8) ( ) ( )

( )( ) ( )

+

+

zL

SSLL

S WdtWyxCTk

tzyxTzP

tzyxCy

DDTk

tzyx0

00 ,,,,,,

,

,,,1,,,

.

We determine solution of Eq.(1) as the following power series

( ) ( ) = =

=

=0 0 0,,,,,,

i j k ijkkji

L tzyxCtzyxC . (9)

Substitution of the series into Eq.(8) gives us possibility to obtain systems of equations for initial-order approximation of concentration of dopant C000(x,y,z,t) and corrections for them Cijk(x, y,z,t) (i 1, j 1, k 1).The equations are presented in the Appendix. Substitution of the series into ap-propriate boundary and initial conditions gives us possibility to obtain boundary and initial condi-tions for all functions Cijk(x,y,z,t) (i 0, j 0, k 0). Solutions of equations for the functions Cijk(x,y,z,t) (i 0, j 0, k 0) have been calculated by standard approaches [28,29] and presented in the Appendix.

Analysis of spatio-temporal distributions of concentrations of dopant and radiation defects has been done analytically by using the second-order approximations framework recently introduced power series. The approximation is enough good approximation to make qualitative analysis and to obtain some quantitative results. All obtained analytical results have been checked by compari-son with results, calculated by numerical simulation.

3. DISCUSSION

In this section we analyzed redistribution of dopant under influence of mechanical stress based on relations calculated in previous section. Fig. 3 show spatio-temporal distribution of concentration of dopant in a homogenous sample (curve 1) and in heterostructure with negative and positive value of the mismatch strain 0 (curve 2 and 3, respectively). The figure shows, that manufactur-ing field-effect heterotransistors gives us possibility to make more compact field-effect transistors in direction, which is parallel to interface between layers of heterostructure. As a consequence of


38

the increasing of compactness one can obtain decreasing of length of channel of field- effect tran-sistor and increasing of density of the transistors in integrated circuits. It should be noted, that presents of interface between layers of heterostructure give us possibility to manufacture more thin transistors [17].

x, y

C(x,

), C(y,

)

2

3

1

Fig. 3. Distributions of concentration of dopant in directions, which is perpendicular to interface between layers of heterostructure. Curve 1 is a dopant distribution in a homogenous sample. Curve 2 corresponds to

negative value of the mismatch strain. Curve 3 corresponds to positive value of the mismatch strain

4. CONCLUSION

In this paper we consider possibility to decrease length of channel of field-effect transistor by us-ing mechanical stress in heterostructure. At the same time with decreasing of the length it could be increased density of transistors in integrated circuits.

ACKNOWLEDGEMENTS

This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government, Scientific School of Russia SSR-339.2014.2 and educational fellowship for scientific research.

REFERENCES

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APPENDIX

Equations for functions Tij(x,y,z,t) (i0, j0)

( ) ( ) ( ) ( ) ( )ass

ass

tzyxpz

tzyxTy

tzyxTx

tzyxTt

tzyxT

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

200

2

200

2

200

2

000 +

+

+

=

( ) ( ) ( ) ( )+

+

+

=

20

2

20

2

20

2

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

z

tzyxTy

tzyxTx

tzyxTt

tzyxT iiiass

i

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

+

+

+ z

tzyxTzg

zytzyxT

xgx

tzyxTzg iT

iT

iTass

,,,,,,,,, 102

102

210

2

0 , i 1


40

( ) ( ) ( ) ( )( ) +

+

+

=

tzyxTT

z

tzyxTy

tzyxTx

tzyxTt

tzyxT dassass

,,,

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

00

02

012

201

2

201

2

001

( ) ( ) ( )( )

( )

+

+

+

2

00

00

02

002

200

2

200

2,,,

,,,

,,,,,,,,,

x

tzyxTtzyxT

Tz

tzyxTy

tzyxTx

tzyxT dass

( ) ( )

+

+

2

00

2

00 ,,,,,,

z

tzyxTy

tzyxT

( ) ( ) ( ) ( )( )+

+

+

=

tzyxTT

z

tzyxTy

tzyxTx

tzyxTt

tzyxT dassass

,,,

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

00

02

022

202

2

202

2

002

( ) ( ) ( )( )

( )

+

+

+ x

tzyxTtzyxT

Tz

tzyxTy

tzyxTx

tzyxT dass ,,,,,,

,,,,,,,,, 001

00

02

012

201

2

201

2

( ) ( ) ( ) ( ) ( )

+

+

z

tzyxTz

tzyxTy

tzyxTy

tzyxTx

tzyxT ,,,,,,,,,,,,,,, 0100010001

( ) ( ) ( )( ) ( )

( ) ( )

+

+

=

Tzyxgx

tzyxTTzyxg

tzyxTtzyxT

x

tzyxTt

tzyxTTTass ,,,

,,,

,,,

,,,

,,,,,,,,,

200

2

00

012

112

011

( ) ( ) ( ) ( ) ( )

+

+

+

x

tzyxTTzyxg

xz

tzyxTTzyxg

zytzyxT

TassT

,,,

,,,

,,,

,,,

,,, 010

002

002

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

++

++

+assTT

z

tzyxTTzyxg

ytzyxT

Tzyxgx

tzyxT02

012

201

2

201

2,,,

,,,1,,,

,,,1,,,

( )( )

( ) ( )( )

( ) ( )( )

( )

+

+

++++ 2

002

100

102

002

100

102

002

100

10 ,,,

,,,

,,,,,,

,,,

,,,,,,

,,,

,,,

z

tzyxTtzyxT

tzyxTy

tzyxTtzyxT

tzyxTx

tzyxTtzyxT

tzyxT

( )( ) ( ) ( )

( ) +

+

+

+tzyxT

Tz

tzyxTy

tzyxTx

tzyxTtzyxT

TT dassdassdass,,,

,,,,,,,,,

,,, 00

02

102

210

2

210

2

00

00

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

+

+

z

tzyxTTzyxgzy

tzyxTTzyxgx

tzyxTTzyxg TTT,,,

,,,

,,,

,,,

,,,

,,,00

210

2

210

2

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

+

+

z

tzyxTz

tzyxTy

tzyxTy

tzyxTx

tzyxTx

tzyxT ,,,,,,,,,,,,,,,,,, 001000100010

( )( )( )

( ) ( ) ( )

+

+

++

2

002

002

001

001

00

0 ,,,,,,,,,

,,,

,,,

,,, z

tzyxTy

tzyxTx

tzyxTtzyxT

TzyxgtzyxT

T Tdass

dass T0 . Conditions for the functions Tij(x,y,z,t) (i0, j0) ( )

0,,,

0

=

=x

ij

x

tzyxT,

( )0

,,,

=

= xLx

ij

x

tzyxT,

( )0

,,,

0

=

=y

ij

ytzyxT

,

( )0

,,,

=

= yLx

ij

ytzyxT

,

( )0

,,,

0

=

=z

ij

z

tzyxT,

( )0

,,,

=

= zLx

ij

z

tzyxT, T00(x,y,z,0)=fT(x,y,z), Tij(x,y,z,0)=0, i 1, j 1.

Solutions of the equations for the functions Tij(x,y,z,t) (i0, j0) with account appropriate boun-dary and initial conditions could be written as


41

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = =1 00 0 0

002

,,

1,,,

n

L

nnTnnnzyx

L L L

Tzyx

xx y z

uctezcycxcLLL

udvdwdwvufLLL

tzyxT

( ) ( ) ( ) ( ) + + zyx

t L L L

asszyx

L L

Tnn LLLdudvdwdwvup

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

dudvdwdwvupwcvcucetezcycxc

,

where cn() = cos (pi n/L), ( )

++= 2220

22 111expzyx

assnT LLLtnte pi ;

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

20

0 ,,,2,,,n

t L L L

TnnnnTnTnnnzyx

ass

i

x y z

TwvugwcvcusetezcycxcnLLL

tzyxT pi

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) +

=

1 0 0 02010 2,,,

n

t L L

nnnTnTnnnzyx

assix y

vsucetezcycxcnLLL

dudvdwdu

wvuT

pi

( ) ( ) ( ) ( ) ( ) ( ) ( )+

=

12

0

0

10 2,,,,,,n

nTnnnzyx

assL

inT tezcycxcnLLL

dudvdwdv

wvuTwcTwvug

z pi

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

t L L L

iTnnnnT

x y z

dudvdwdw

wvuTTwvugwsvcuce

0 0 0 0

10 ,,,,,,

, i 1,

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

=

=1 0 0 0 02

002

2001,,,2

,,,

n

t L L L

nnnTnTnnnzyx

dass

x y z

u

wvuTvcusetezcycxcn

LLLT

tzyxT

pi

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) +

=1 0 0 0 020

00

2,,, n

t L L L

nnnnTnTnnnzyx

dassn

x y z

wcvsucetezcycxcnLLL

TwvuT

dudvdwdwc

pi

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) +

=1 0 0 020

002

002 2

,,,

,,,

n

t L L

nnnTnTnnnzyx

dass

x y

vcucetezcycxcnLLL

TwvuT

dudvdwdv

wvuT

pi

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

=1 0 0

0

0 002

002

2,,,

,,,

n

t L

nnTnTnnnzyx

ass

d

L

n

xz

ucetezcycxcLLL

TwvuT

dudvdwdu

wvuTws

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

=

+1 0

0

0 01

00

2

00 2,,,

,,,

n

t

nTnTnnnzyx

dassL L

nnetezcycxc

LLLT

wvuTdudvdwd

u

wvuTwcvc

y z

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

=

+1

0

0 0 01

00

2

00 2,,,

,,,

nnTnnn

zyx

ass

d

L L L

nnntezcycxc

LLLT

wvuTdudvdwd

v

wvuTwcvcuc

x y z

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

+

t L L L

nnnnT

x y z

wvuTdudvdwd

w

wvuTwcvcuce

0 0 0 01

00

2

00

,,,

,,,

;

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

=

=1 0 0 0 02

012

2002,,,2

,,,

n

t L L L

nnnTnTnnnzyx

dass

x y z

u

wvuTvcusetezcycxcn

LLLT

tzyxT

pi

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) +

=1 0 0 0 020

00

2,,, n

t L L L

nnnnTnTnnnzyx

dassn

x y z

wcvsucetezcycxcnLLL

TwvuT

dudvdwdwc

pi

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) +

=1 0 020

002

012 2

,,,

,,,

n

t L

nnTnTnnnzyx

dass

x

ucetezcycxcnLLL

TwvuT

dudvdwdv

wvuT

pi


42

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

=1 02

0

0 0 002

012

2,,,

,,,

n

t

nTnTnnnzyx

ass

L L

nnetezcycxc

LLLwvuTdudvdwd

w

wvuTwsvc

y z

pi

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

=

+1

20

0 0 01

00

0100 2,,,

,,,,,,

nn

zyx

ass

d

L L L

nnnd xcLLLT

wvuTdudvdwd

u

wvuTu

wvuTwcvcucT

x y z pi

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

+

t L L L

nnnnTnTnn

x y z

wvuTdudvdwd

v

wvuTv

wvuTwcvcucetezcyc

0 0 0 01

00

0100

,,,

,,,,,,

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

=

+10 0 0 0

100

01002

0

,,,

,,,,,,2n

t L L L

nnnnTzyx

assdx y z

wvuTdudvdwd

w

wvuTw

wvuTwcvcuce

LLLT

pi

( ) ( ) ( ) ( )tezcycxc nTnnn ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

=

=1 0 0 0 02

002

011

,,,2,,,

n

t L L L

nnnnTnTnnnzyx

assx y z

w

wvuTwcvcucetezcycxc

LLLtzyxT

( ) ( ) ( ) ( ) ( ) ( )( ) +

+

+

dudvdwdwvuTwvuT

w

wvuTwg

wv

wvuTTwvugTwvug TTT,,,

,,,,,,,,,

,,,,,,

00

01002

002

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

+

++

dudvdwd

w

wvuTTwvug

wv

wvuTTwvug

u

wvuTTT

,,,

,,,

,,,

,,,1,,, 01

201

2

201

2

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

+

+

=

+1 0 0 0 0

200

2

100

100 ,,,

,,,

,,,

2n

t L L L

nnnTnTnnnzyx

dassx y z

u

wvuTwvuT

wvuTvcucetezcycxc

LLLT

( )( )

( ) ( )( )

( ) ( ) +

+

++yx

ass

n LLdudvdwdwc

w

wvuTwvuT

wvuTv

wvuTwvuT

wvuT 02

002

100

102

002

100

10 ,,,

,,,

,,,,,,

,,,

,,,

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

+

+

=1 0 0 0 02

102

210

2

210

2,,,,,,,,,

2n

t L L L

nnnnTz

dx y z

w

wvuTv

wvuTu

wvuTwcvcuce

LT

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) +

=1 0 0 0

0

00

22,,, n

t L L

nnnTnTnnnzyx

dassnTnnn

x y

vcucetezcycxcLLLT

tezcycxcwvuT

dudvdwd

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

+

+

zL

TTTn Twvugw

wvuTTwvug

wu

wvuTTwvugwc

0

002

002

,,,

,,,

,,,

,,,

,,,

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

=1 0 0 0

0

002

002

2,,,

,,,

n

t L L

nnnTnTnnnzyx

dassx y

vcucetezcycxcLLLT

wvuTdudvdwd

v

wvuT

( ) ( ) ( ) ( ) ( )

+

+

+

zL

nwvuT

dudvdwdw

wvuTv

wvuTu

wvuTwc

01

00

2

00

2

00

2

00

,,,

,,,,,,,,,

.

Integro-differential form of equations of systems (4) could be written as

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

+=t z

zyxxx

wx

wyxuyxwyxu

x

wyxuttzyxutzyxu

0 0

22

2

2,,,,,,,,,5,,,,,,

( )( )[ ]

( ) ( ) ( ) ( )( ) +

+

0 0

22

2

2

1,,,,,,,,,5

616z

zyx dw

wdwEwx

wyxuyxwyxu

x

wyxutw

dwdwE

( ) ( ) ( ) ( ) ( )

+

+

+

+

t zx

zzyx

yxwyxudwd

wx

wyxuyxwyxu

x

wyxuwKt

0 0

2

0 0

22

2

2,,,

21,,,,,,,,,


43

( ) ( )( ) ( ) ( )

( ) ( )

+

+

+

+t z yxy

w

wyxuwx

wyxutdt

w

wdwEy

wyxu0 0

2

22

2

2,,,,,,

21

1,,,

( )( )

( ) ( ) ( )( )

( )

+

+

+

+

0 0

2

0 02

22,,,

1,,,,,,

21z

xz

yx

wx

wyxudw

wdwEy

wyxuyxwyxutd

w

wdwE

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

( ) ( ) +

+

+

0 00 02

2,,,

21,,, zt zy

wKtdwdx

wyxTwKwttd

w

wdwEw

wyxu

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

+

tzyxdwdwyxuwtdwd

x

wyxTw x

t zx

,,,

,,,,,,

10 0

2

2

( ) ( )

0 02

2,,,z

x dwdwyxuwt

,

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

+

+

+=

21,,,,,,

21

,,,,,,

0 0

2

2

2td

w

wdwEyxwyxu

x

wyxuttzyxutzyxu

t zxy

yy

( ) ( ) ( )( )

( ) ( )

+ +

+

t zxy

zxy

yxwyxu

ywyxu

dw

wdwEyxwyxu

x

wyxu0 0

2

2

2

0 0

2

2

2,,,,,,5

61

1,,,,,,

( ) ( )( ) ( )

( ) ( ) ( )

+

0 0

22

2

22,,,,,,,,,5

1,,, z zxyz

wywyxu

yxwyxu

ywyxu

dtw

wdwEwywyxu

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) +

+

+

0 00 0

,,,,,,

61zt z

dwdy

wyxTwKwtdwd

ywyxT

wwKttdw

wdwE

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

+

+

+

0 00 0

2

2

22,,,,,,,,, zt z yyy

wKtdwdwywyxu

ywyxu

yxwyxu

wKt

( ) ( ) ( ) ( ) ( )

+

+

+

+

t

yyyy

z

zyxutdwd

wywyxu

ywyxu

yxwyxu

0

2

2

22,,,,,,,,,,,,

( ) ( )( )[ ]

( )( )[ ]

( ) ( )+

+

+

+

+

0

,,,,,,

1212,,,

dy

zyxuz

zyxuz

zEtz

zEdy

zyxu zyz

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

+

tzyxdwdwyxu

wtdwdwyxu

w yt z yz y

,,,

,,,,,,

10 0

2

2

0 02

2

,

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

+

+

+=

21,,,,,,

21

,,,,,,

0 0

2

2

2 tdw

wdwEwx

wyxux

wyxuttzyxutzyxu

t zxz

zz

( ) ( ) ( )( )

( ) ( )

+

+

+

+

t zyz

zxz

wywyxu

ywyxud

w

wdwEwx

wyxux

wyxu0 0

2

2

2

0 0

2

2

2,,,,,,

1,,,,,,

( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) +

+

+

t zzyz wtd

w

wdwEwywyxu

ywyxutdt

w

wdwE0 00 0

2

2

2

1,,,,,,

221

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

+

+

tz

z

z

zyxutdwd

w

wyxTwwKtdwd

w

wyxTwK

00 0

,,,5,,,,,, ( ) ( ) ( )

( )[ ]( )

( )( ) ( )

+

+

0

,,,,,,5116

,,,,,,

x

zyxuz

zyxuz

zEz

zEdy

zyxux

zyxuxzyx


44

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

+

+

+

tzyxy d

z

zyxuy

zyxux

zyxutzKtd

yzyxu

0

,,,,,,,,,

6,,,

( ) ( ) ( ) ( ) ( )

+

+

tzyxdz

zyxuy

zyxux

zyxuzKt zz

yx,,,

,,,,,,,,,

10

,

where ( ) ( ) ( )[ ]= z iis wdtwyxuwtzyx s0

,,,,,, , s =x,y,z, =L/2E0, E0 is the average value of Young modulus.

The second-order approximations of components of displacement vector could be written as

( ) ( ) ( ) ( ) ( )

++=t z

yxxuxx yx

wyxux

wyxuttzyxutzyxu

0 0

12

21

2

1122

,,,,,,561

,,,,,,

( ) ( )( )

( ) ( ) ( )

+

0 0

12

12

21

21

2,,,,,,,,,5

1,,, z zyxz

wx

wyxuyx

wyxux

wyxudw

wdwEwx

wyxu

( )( ) ( ) ( )

( ) ( ) ( )

+

+

++

t z

zyx dwdwx

wyxuyxwyxu

x

wyxuwKttd

w

wdwE0 0

12

12

21

2,,,,,,,,,

61

( ) ( ) ( ) ( ) ( )

+

+

+

+

t zx

zzyx

yxwyxudwd

wx

wyxuyxwyxu

x

wyxuwKt

0 0

12

0 0

12

12

21

2,,,,,,,,,,,,

( ) ( )( )

( ) ( ) ( ) ( )( ) +

+

+

+

+t z

yxy

w

wdwEy

wyxuyxwyxudt

w

wdwEy

wyxu0 0

21

21

2

21

2

1,,,,,,

21

21,,,

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

( ) + +

+

+

0 00 02

12

12

121,,,,,,

21 zt z yx

w

wEtdw

wdwEw

wyxuwx

wyxutdt

( ) ( ) ( ) ( )

+

+

0 02

12

12

21

21

2,,,,,,

2,,,,,,

zyxyx

w

wyxuwx

wyxutdwdy

wyxuyxwyxu

( )( ) ( ) ( )

( ) ( ) ( ) ( ) (

++

t zz

twdx

wyxTwwKdwd

x

wyxTwKwtd

w

wdwE0 00 0

,,,,,,

1

) ( ) ( ) ( ) ( ) ( )

+

0 02

12

0 02

12

,,,,,,z

xt z

x dwdwyxuwtdwdwyxuwtd

( ) ( )}tzyxtzyxxxux

,,,,,, 102 ,

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

+

+

++=t z

xyyuyy d

w

wdwEyxwyxu

x

wyxuttzyxutzyxu

0 0

12

21

2

122 1,,,,,,

2,,,,,,

( ) ( ) ( )( )

( ) ( )

+ +

+

t zxy

zxy

yxwyxu

ywyxu

dw

wdwEyxwyxu

x

wyxut0 0

12

21

2

0 0

12

21

2,,,,,,5

1,,,,,,

2

( ) ( )( )

( ) ( ) ( )

+

0 0

12

12

21

21

2,,,,,,,,,5

61,,,

zzxyz

wywyxu

yxwyxu

ywyxu

dtw

wdwEwy

wyxu

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

+

+

0 00 0

,,,,,,

61zt z

dwdy

wyxTwwKtdwd

ywyxT

wKwttdw

wdwE


45

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

+

+

0 00 0

12

21

21

2,,,,,,,,, zt z yyy

wKtdwdwy

wyxuy

wyxuyxwyxu

wKt

( ) ( ) ( ) ( ) ( )

+

+

+

+

t yyyy

z

zyxutdwd

wywyxu

ywyxu

yxwyxu

0

112

21

21

2,,,

21,,,,,,,,,

( ) ( )( )

( )( )[ ]

( ) ( )

+

+

+

+

20

111 ,,,,,,

121,,,

uyzyz d

yzyxu

z

zyxuz

zEtz

zEdy

zyxu

( ) ( )} 110 ,,,,,, tzyxtzyx yy , ( ) ( ) ( ) ( ) ( ) ( )

+

+

++=

t zxz

zzzw

wdwEwx

wyxux

wyxutzyxutzyxu

0 0

12

21

2

1122 1,,,,,,

21

,,,,,,

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

+

+ +

+

t zz

zxz

ywyxu

tdw

wdwEwx

wyxux

wyxutdt0 0

21

2

0 0

12

21

2,,,

21

1,,,,,,

2

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

( ) ( )

+

+

+

0 02

12

0 0

12

,,,,,,

1,,, z

zt z

y

ywyxudwd

w

wyxTwwKtd

w

wdwEwy

wyxu

( ) ( )( ) ( ) ( )

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

+

+

+ tzy t

z

zEdwdw

wyxTwwKttd

w

wdwEwy

wyxu00 0

12

16,,,

21,,,

( ) ( ) ( ) ( ) ( )

0

11111 ,,,,,,56

,,,,,,,,,5x

zyxuz

zyxutdy

zyxux

zyxuz

zyxu xzyxz

( ) ( )( )[ ] ( )

( ) ( ) ( )

+

+

++

tzyxy d

z

zyxuy

zyxux

zyxut

z

zEtdy

zyxu0

1111 ,,,,,,,,,

16,,,

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

+

+

tzyxdz

zyxuy

zyxux

zyxuzKtzK zz

zyx,,,

,,,,,,,,,

020

111

( )}tzyxz ,,,1 .

Calculation of the average values u2 leads to the following results

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

+

=

0 0 0 0

12

12

21

22

2 1,,,,,,,,,5

121 x y zL L L

zzyx

zux Lz

zdzEzx

tzyxuyx

tzyxux

tzyxutL

) ( ) ( ) ( ) ( ) ( )( ) +

0 0 0 0

12

12

21

22

1,,,,,,,,,5

12x y zL L L

zzyx

z

zLzx

tzyxuyx

tzyxux

tzyxuzEtdxdydz

( ) ( ) ( ) ( ) ( )

+

+

+

0 0 0 0

12

12

21

22 ,,,,,,,,,

21 x y zL L L zyx

zzx

tzyxuyx

tzyxux

tzyxuzLttdxdydzd

( ) ( ) ( ) ( ) ( )

+

+

0 0 0 0

12

12

21

22,,,,,,,,,

2x y zL L L

zyxz

zx

tzyxuyx

tzyxux

tzyxuzLtdxdydzdzK

( ) ( ) ( ) ( ) ( ) ( ) +

+

+

0 0 0 02

12

12

2

1,,,,,,

21 x y zL L L zyx ydzd

z

zLy

tzyxuyx

tzyxuzEttdxdydzdzK


46

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

+

+

+

41,,,,,,

21 2

0 0 0 02

12

12

2 xy zL L L yx

z tdxdydz

zdzEz

tzyxuzx

tzyxuzLttdxd

( ) ( )( )( ) ( ) ( ) ( ) +

+

+

+

0 0 0 0

2

0 0 0 02

12

12

141,,,,,,

1x y zx y z L L L

zL L L

yxz

z

zLttdxdydzd

ytzyxu

yxtzyxu

z

zLzE

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

+

+

0 0 0 02

12

21

21

2,,,

2,,,,,, x

y zL L Lx

z

yx

t

tzyxuzzLtdxdydzdzE

z

tzyxuzx

tzyxu

( ) ( ) ( ) ( ) ( ) + 0 0 0 0

2

0 0 0 01

2

2,,,

x y zx y z L L L

z

L L L

xz zLtdxdydzdtzyxuzzLttdxdydzd

( ) ( ) ( ) ( ) ( )

+

zL

zxxx zdzzLLtdxdydzd

t

tzyxu0

20

2

21

2

421

2,,, ,

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

+

+

=

2,,,,,,

121 2

0 0 0 0

12

21

22

2

x y zL L Lxyz

zuy tdxdydzdyxtzyxu

x

tzyxuz

zLzEL

( ) ( )( )( ) ( ) ( ) ( ) +

+

+

+

0 0 0 00 0 0 0

12

21

2

1121,,,,,,

1x y zx y z L L L

zL L L

xyz

z

zLzEtdxdydzd

yxtzyxu

x

tzyxuz

zLzE

( ) ( ) ( ) ( ) ( )

0 0 0 0

221

21

2

21

2

12,,,,,,,,,5

x y zL L Lzxy zEtdxdydzd

zytzyxu

yxtzyxu

ytzyxu

( )( ) ( ) ( ) ( ) ( ) +

+

0 0 0 0

212

12

21

2

2,,,,,,,,,5

1x y zL L L

zxyz zKttdxdydzdzy

tzyxuyx

tzyxuy

tzyxuz

zL

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

+

+

0 0 0 0

12

21

21

2,,,,,,,,, x y zL L L

z

yyyz zLzKtdxdydzd

zytzyxu

ytzyxu

yxtzyxu

zL

( ) ( ) ( ) ( )

+

+

+

+

0 0 0 0

112

21

21

22,,,

21,,,,,,,,,

2x y zL L L yyyy

z

tzyxutdxdydzd

zytzyxu

ytzyxu

yxtzyxu

( ) ( )( ) ( )

( ) ( ) ( )( ) +

+

+

+

0 0 0 0

112

21

1,,,,,,

21,,, x

y zL L Lzyz

z

zdzEy

tzyxuz

tzyxutdxdyd

z

zdzEy

tzyxu

( ) ( ) ( ) ( )

+

0 0 0 02

122

0 0 0 02

12

2 ,,,

2,,,

21 x y zx y z L L L yL L L y

t

tzyxutdxdydzd

t

tzyxuzttdxdyd

( ) ( ) ( ) ( ) ( ) ( ) +

22,,,

022

0 0 0 01

yyL L L

yz

x y z

tdxdydzdtzyxuzzLtdxdydzdz

( ) ( )1

04

zL

z zdzzLL ,

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

+

+

=

2,,,,,,

121 2

0 0 0 0

12

21

22

2

x y zL L Lxzz

zz tdxdydzdzx

tzyxux

tzyxuz

zLzEtL

( ) ( )( )( ) ( ) ( ) ( ) +

+

+

0 0 0 0

2

0 0 0 0

12

21

2

21,,,,,,

1x y zx y z L L LL L L

xzz zEttdxdydzdzx

tzyxux

tzyxuz

zLzE

( ) ( ) ( ) ( )

+

+

0 0 0 0

12

21

21

2

21

2,,,,,,,,,,,, x

y zL L L yzyz

zytzyxu

ytzyxu

tdxdydzdzy

tzyxuy

tzyxu


47

( ) ( )( )[ ]

( ) ( ) ( )

++

0 0 0 0

1112,,,,,,,,,5

121

12x y zL L L yxzz

ytzyxu

x

tzyxuz

tzyxutdxdydzd

z

zLzE

( )( ) ( )

( ) ( ) ( )

+

0 0 0 0

1112

2 ,,,,,,,,,5121

x y zL L L yxz

ytzyxu

x

tzyxuz

tzyxutdtxdyd

z

zdzE

( )( ) ( )

( ) ( ) ( ) ( )

+

+

++

0 0 0 0

1112 ,,,,,,,,,

21x y zL L L

zyx

z

tzyxuy

tzyxux

tzyxuzKttdxdyd

z

zdzE

( ) ( ) ( ) ( ) ( )

+

+

0 0 0 0

1112

2 ,,,,,,,,,

22x y zL L L

zyxz

z

tzyxuy

tzyxux

tzyxuzKtdxdydzd

( ) ( ) ( ) ( ) ( )

+

zx y z L

z

zL L L

zz zLtdxdydzdztzyxuzLtdxdydzd0

02

0 0 0 01 42

,,,

( ) } 1 Lzdz .

System of equations for the functions Cijk(x,y,z,t) (i 0, j 0, k 0) could be written as

( ) ( ) ( ) ( )

+

+

=

2000

2

2000

2

2000

2

0000 ,,,,,,,,,,,,

z

tzyxCy

tzyxCx

tzyxCDt

tzyxCL ;

( ) ( ) ( ) ( ) ( )

+

+

=

ytzyxC

Tzgyx

tzyxCTzg

xD

t

tzyxC iL

iLL

i ,,,,

,,,

,

,,, 1001000

00

( ) ( ) ( ) ( ) ( )

+

+

+

+ 200

2

200

2

200

2100 ,,,,,,,,,,,,

,

z

tzyxCy

tzyxCx

tzyxCz

tzyxCTzg

ziiii

L , i 1;

( ) ( ) ( ) ( )+

+

+

=

2010

2

2010

2

2010

2

0010 ,,,,,,,,,,,,

z

tzyxCy

tzyxCx

tzyxCD

t

tzyxCL

( )( )

( ) ( )( )

( )

+

+

+y

tzyxCTzyxPtzyxC

yxtzyxC

TzyxPtzyxC

xD L

,,,

,,,

,,,,,,

,,,

,,, 0000000000000

( )( )

( )

+z

tzyxCTzyxPtzyxC

z

,,,

,,,

,,, 000000

;

( ) ( ) ( ) ( )+

+

+

=

LL Dz

tzyxCy

tzyxCx

tzyxCD

t

tzyxC02

0202

2020

2

2020

2

0020 ,,,,,,,,,,,,

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

+

ytzyxC

tzyxCyy

tzyxCTzyxPtzyxC

tzyxCx

,,,

,,,

,,,

,,,

,,,

,,,000

010000

1000

010

( )( ) ( )

( )( )

( ) ( )( )

+

+

TzyxPtzyxC

xD

z

tzyxCTzyxPtzyxC

tzyxCzTzyxP

tzyxCL

,,,

,,,,,,

,,,

,,,

,,,

,,,

,,, 0000

0001

000010

1000

( ) ( )( )

( ) ( )( )

( )

+

+

z

tzyxCTzyxPtzyxC

zytzyxC

TzyxPtzyxC

yxtzyxC ,,,

,,,

,,,,,,

,,,

,,,,,, 010000010000010

;

( ) ( ) ( ) ( )+

+

+

=

2001

2

2001

2

2001

2

0001 ,,,,,,,,,,,,

z

tzyxCy

tzyxCx

tzyxCD

t

tzyxCL


48

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

+

++

+

Tktzyx

WdtWyxCTzyxPtzyxC

Tzyxgx

D SL

SLSLSSL

z ,,,

,,,

,,,

,,,1,,,10

000000

0

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

++

+

Tktzyx

WdtWyxCTzyxPtzyxC

Tzyxgy

D SL

SLSLSSL

z ,,,

,,,

,,,

,,,1,,,10

000000

0

;

( ) ( ) ( ) ( )+

+

+

=

2002

2

2002

2

2002

2

0002 ,,,,,,,,,,,,

z

tzyxCy

tzyxCx

tzyxCDt

tzyxCL

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

+

++

+

Tktzyx

WdtWyxCTzyxPtzyxC

Tzyxgx

D SL

SLSLSSL

z ,,,

,,,

,,,

,,,1,,,10

001000

0

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

+

++

+

Tktzyx

WdtWyxCTzyxPtzyxC

Tzyxgy

D SL

SLSLSSL

z ,,,

,,,

,,,

,,,1,,,10

001000

0

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

++

+

zL

SLSLSSL WdtWyxCTzyxPtzyxC

tzyxCTzyxgx

D0

000

1000

0010 ,,,,,,

,,,

,,,1,,,1

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

+

+

zLS

SS WdtWyxC

Tktzyx

TzyxPtzyxC

tzyxCyTk

tzyx0

000

1000

001 ,,,,,,

,,,

,,,

,,,1,,,

( )[ ]} SLLSLS DTzyxg 0,,,1 + ; ( ) ( ) ( ) ( ) ( )

+

+

+

=

x

tzyxCxz

tzyxCy

tzyxCx

tzyxCDt

tzyxCL

,,,,,,,,,,,,,,, 0102

1102

2110

2

2110

2

0110

( ) ( ) ( ) ( ) ( ) +

+

+

LLLL D

z

tzyxCTzyxg

zytzyxC

Tzyxgy

Tzyxg 0010010 ,,,

,,,

,,,

,,,,,,

( )( )

( ) ( )( )

( ) ( )( )

+

+

+TzyxPtzyxC

zytzyxC

TzyxPtzyxC

yxtzyxC

TzyxPtzyxC

x ,,,

,,,,,,

,,,

,,,,,,

,,,

,,, 000100000100000

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

( )

+

+

TzyxPtzyxC

yxtzyxC

TzyxPtzyxC

tzyxCx

Dz

tzyxCL

,,,

,,,,,,

,,,

,,,

,,,

,,,1

0000001

0001000

100

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

LDz

tzyxCTzyxPtzyxC

tzyxCzy

tzyxCtzyxC 0000

1000

100000

100,,,

,,,

,,,

,,,

,,,

,,,

+

;

( ) ( ) ( ) ( ) ( )

+

+

+

=

ytzyxC

xz

tzyxCy

tzyxCx

tzyxCD

t

tzyxCL

,,,,,,,,,,,,,,, 0002

1002

2100

2

2100

2

0101

( ) ( ) ( ) ( ) ( ) +

+

+

LLLL D

z

tzyxCTzyxgzy

tzyxCTzyxgy

Tzyxg 0000000,,,

,,,

,,,

,,,,,,

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

++

+

zL

SLSLSSL WdtWyxCTzPtzyxC

tzyxCTzyxgx

D0

100

1000

1000 ,,,,

,,,

,,,1,,,1

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

+

+

++

+

zz L

LSLSS

L

SLSLSSLS

WdtWyxCTzyxgxTk

tzyxWdtWyxC

TzPtzyxC

tzyxCTzyxgy

DTk

tzyx

0100

0100

1000

1000

,,,,,,1,,,,,,

,

,,,

,,,1,,,1,,,


49

( ) ( ) ( )[ ] ( )

+

+

Tk

tzyxTzyxgy

DWdtWyxCTk

tzyx SLSLSSL

LS

z ,,,

,,,1,,,,,, 00

100

( ) ( ) SLLL

DWdtWyxCWdtWyxCzz

00

1000

100 ,,,,,,

;

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

+

+

+

=

TzyxPtzyxC

xz

tzyxCy

tzyxCx

tzyxCDt

tzyxCL

,,,

,,,,,,,,,,,,,,, 0002

0112

2011

2

2011

2

0011

( ) ( )( )

( ) ( )( )

( )+

+

+

LDz

tzyxCTzyxPtzyxC

zytzyxC

TzyxPtzyxC

yxtzyxC

0001000001000001 ,,,

,,,

,,,,,,

,,,

,,,,,,

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

+

+

ytzyxC

tzyxCyx

tzyxCTzyxPtzyxC

tzyxCx

D L,,,

,,,

,,,

,,,

,,,

,,,000

001000

1000

0010

( )( ) ( )

( )( )

( ) ( )

+

+

Tktzyx

xz

tzyxCTzyxPtzyxC

tzyxCzTzyxP

tzyxC S ,,,,,,,,,

,,,

,,,

,,,

,,, 0001

000001

1000

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

+

++

Tktzyx

xDWdtWyxC

TzyxPtzyxCTzyxg SSL

L

SLSLS

z ,,,

,,,

,,,

,,,1,,,1 00

010000

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

++

SLSL

L

SLSLS DDWdtWyxCTzyxPtzyxC

tzyxCTzyxgz

000

000

1000

010 ,,,,,,

,,,

,,,1,,,1

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

++

zLS

SLSLS WdtWyxCTktzyx

TzyxPtzyxC

tzyxCTzyxgy 0 000

1000

010 ,,,,,,

,,,

,,,

,,,1,,,1

.

Boundary and initial conditions for functions Cijk(x,t) (i 0, j 0, k 0) could be written as ( )

0,,,

0

=

=x

ijk

x

tzyxC

;

( )0

,,,

=

= xLx

ijk

x

tzyxC

;

( )0

,,,

0

=

=y

ijk

ytzyxC

; ( )

0,,,

=

= yLy

ijk

ytzyxC

;

( )0

,,,

0

=

=z

ijk

z

tzyxC

;

( )0

,,,

=

= zLz

ijk

z

tzyxC

; C000(x,y,z,0)=fC (x,y,z); Cijk(x,y,z,0)=0.

Solutions of equations for functions Cijk(x,t) (i 0, j 0, k 0) could be written as

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

0000

2,,,

nnCnnnnC

zyxzyx

C tezcycxcFLLLLLL

FtzyxC ,

where ( )

++= 2220

22 111expzyx

LnC LLLtDnte pi , ( ) ( ) ( ) ( ) = x y zL

L L

CnnnnC udvdwdwvufwcvcucF0 0 0

,, ;

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

20

00 ,,,2

,,,

n

t L L L

LnnnCnCnnnnCzyx

Li

x y z

TwvugvcusetezcycxcFnLLL

DtzyxC

pi

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

=

1 0 02

0100 2,,,n

t L

nnCnCnnnnCzyx

Lin

x

ucetezcycxcFnLLL

Ddudvdwdu

wvuCwc

pi

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

=

12

0

0 0

100 2,,,,,,

nnCnnn

zyx

LL L

iLnn tezcycxcLLL

Ddudvdwdv

wvuCTwvugwcvsy z pi


50

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

t L L L

iLnnnnCnC

x y z

dudvdwdw

wvuCTwvugwsvcuceFn

0 0 0 0

100 ,,,,,,

, i 1;

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

=1 0 0 0 0

0002

0010

,,,

,,,2,,,

n

t L L L

nnnCnCnnnnCzyx

Lx y z

TwvuPwvuC

vcusetezcycxcFnLLL

DtzyxC

pi

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

=1 0 02

0000 2,,,n

t L

nnCnCnnnnCzyx

Ln

x

ucetezcycxcFnLLL

Ddudvdwdu

wvuCwc

pi

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

=12

0

0 0 0

000000 2,,,,,,

,,,

nnn

zyx

LL L L

nnnycxcn

LLLDdudvdwd

v

wvuCTwvuP

wvuCwcwcvs

y z z pi

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

t L L L

nnnnCnCnnC

x y z

dudvdwdw

wvuCTwvuP

wvuCwsvcucetezcF

0 0 0 0

000000 ,,,

,,,

,,,

;

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

01020

020 ,,,2

,,,

n

t L L L

nnnCnCnnnnCzyx

Lx y z

wvuCvcusetezcycxcFnLLL

DtzyxC pi

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

=

12

00001

000 2,,,,,,

,,,

nnCnnnnC

zyx

Ln

tezcycxcFnLLL

Ddudvdwdu

wvuCTwvuP

wvuCwc

pi

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

t L L L

nnnnC

x y z

dudvdwdv

wvuCTwvuP

wvuCwvuCwcvcuse

0 0 0 0

0001

000010

,,,

,,,

,,,

,,,

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

=

1 0 0 0 0

1000

010,,,

,,,

,,,

n

t L L L

nnnnCnCnnnnC

x y z

TwvuPwvuC

wvuCwcvcusetezcycxcFn

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

=1 02

00002

0 2,,,2n

t

nCnCnnnnCzyx

L

zyx

L etezcycxcFnLLL

Ddudvdwdw

wvuCLLL

D

pi

pi

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

=12

0

0 0 0

010000 2,,,,,,

,,,

nnnC

zyx

LL L L

nnnxcFn

LLLDdudvdwd

u

wvuCTwvuP

wvuCwcvcus

x y z pi

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

t L L L

nnnnCnCnn

x y z

dudvdwdv

wvuCTwvuP

wvuCwcvsucetezcyc

0 0 0 0

010000 ,,,

,,,

,,,

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

=1 0 0 0 0

0100002

0 ,,,

,,,

,,,2n

t L L L

nnnnCnCzyx

Lx y z

dudvdwdw

wvuCTwvuP

wvuCwsvcuceFn

LLLD

pi

( ) ( ) ( ) ( )tezcycxcnCnnn ;

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

20

001,,,2

,,,

n

t L L LS

nnCnCnnnnCzyx

Lx y z

Tkwvu

usetezcycxcFnLLL

DtzyxC pi

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

++

dudvdvcwdwcWdWvuCTwvuP

wvuCTwvugnn

L

SLSLS

z

0000

000,,,

,,,

,,,1,,,1

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

00020

,,,

2n

t L L L L

nnnnCnCnnnnCzyx

Lx y z z

WdWvuCwcvsucetezcycxcFnLLL

D

pi

( )[ ] ( )( )( )

dudvdwdTk

wvu

TwPwvuC

Twvug SSLSLS,,,

,

,,,

1,,,1 000

++ ;

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

20

002,,,2

,,,

n

t L L LS

nnnCnCnnnnCzyx

Lx y z

Tkwvu

vcusetezcycxcFnLLL

DtzyxC pi

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

++

zyx

Ln

L

SLSLS LLLDdudvdwdwcWdWvuC

TwvuPwvuCTwvug

z

20

0001

000,,,

,,,

,,,1,,,1 pi


51

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

001 ,,,,,,

n

t L L L LS

nnnnCnCnnnnC

x y z z

WdWvuCTk

wvuwcvsucetezcycxcFn

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

++

=12

0000 2,,,

,,,1,,,12n

nnnnCzyx

LSLSLS zcycxcFLLL

DdudvdwdTwvuP

wvuCTwvug pi

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

+

t L L L L

SnnnnCnC

x y z z

WdWvuCTwvuP

wvuCtzyxCwcvcuseten

0 0 0 0 0000

1000

001 ,,,,,,

,,,

,,,1

( )[ ] ( ) ( ) ( ) ( ) ( ) + =1

202,,,

,,,1n

nCnnnnCzyx

LSLSLS tezcycxcFLLL

DdudvdwdTk

wvuTwvug pi

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

+

t L L L L

SS

nnn

x y z z

WdWvuCTwPwvuC

tzyxCTk

wvuwcvcusn

0 0 0 0 0000

1000

001 ,,,,

,,,

,,,1,,,

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52

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LSLSnnnCnCnnnnCzyx

Lx y z


D

pi

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n

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Lx y z

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53

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++

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x y z

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x y z

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( ) ( )( )( )

dudvdwdTk

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wvuC SS,,,

,

,,,

,,,11

000010

+

.

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 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 96 published papers in area of his researches.

Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school of village Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school Center for gifted children. From 2009 she is a student of Nizhny Novgorod State University of Architec-ture and Civil Engineering (spatiality Assessment and management of real estate). At the same time she is a student of courses Translator in the field of professional communication and Design (interior art) in the University. E.A. Bulaeva was a contributor of grant of President of Russia (grant MK-548.2010.2). She has 29 published papers in area of her researches.

modeling of manufacturing of a fieldeffect

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