ANALYSIS OF MANUFACTURING OF VOLTAGE RESTORE TO INCREASE DENSITY OF ELEMENTS OF THE CIRCUIT

International Journal on Organic Electronics (IJOE) Vol.5, No.1/2/3/4, October 2016

DOI: 10.5121/ijoe.2016.5401 1

ANALYSIS OF MANUFACTURING OF VOLTAGE RESTORE

TO INCREASE DENSITY OF ELEMENTS OF THE CIRCUIT

E.L. Pankratov, E.A. Bulaeva

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

Russia

ABSTRACT

We introduce an approach for increasing density of voltage restore elements. The approach based on

manufacturing of a heterostructure, which consist of a substrate and an epitaxial layer with special con-

figuration. Several required sections of the layer should be doped by diffusion or ion implantation. After

that dopants and/or radiation defects should be annealed.

KEYWORDS

Voltage restore; optimization of manufacturing; increasing of density of elements

1. INTRODUCTION

Now several actual questions of solid state electronics could be formulated. One of them is in-

creasing of density of elements of integrated circuits. The increasing of density leads to necessity

of decreasing of dimensions of these elements. Recently are have been elaborated several ap-

proaches for the above decreasing. One of them is manufacturing of solid state electronic devices

in thin film heterostructures [1-4]. Other approach based on doping of required areas of a semi-

conductor sample or a heterostructure. After the doping one should consider laser or microwave

annealing of dopant and/or radiation defects [5-7]. Due to these types of annealing one can find

inhomogenous distribution of temperature in doped area. In this situation one can find inhomo-

geneity of doped structure and consequently to decreasing of dimensions of the considered ele-

ments. An alternative approach of changing of properties of materials before or during doping is

radiation processing [8,9].

In this paper we consider an approach to manufacture of a voltage restore. We illustrate the ap-

proach by considering a heterostructure. The heterostructure consist of a substrate and an epitax-

ial layer. We consider that several sections take a place framework the epitaxial layer. We con-

sider doping these sections by diffusion or ion implantation. The doping gives a possibility to

obtain p or n type of conductivity during process of manufacture of diodes and bipolar transistors

so as it is shown on Fig. 1. After finishing the doping we consider annealing of dopant and/or

radiation defects. Our main aim is analysis of redistribution of dopant and/or radiation defects

during annealing.

2. METHOD OF SOLUTION

We calculate distribution of concentration of dopant in space and time by solving the following

boundary problem [10-12]


2

( ) ( ) ( ) ( )

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂

z

tzyxCD

zy

tzyxCD

yx

tzyxCD

xt

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

( )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)=fC (x,y,z).

The function C(x,y,z,t) describes distribution of concentration of dopant in space and time; T

Fig. 1. Structure of voltage restore.

View from top describes the temperature of annealing; DС describes the dopant diffusion coeffi-

cient. Dopant diffusion coefficient is different in different materials of heterostructure. The diffu-

sion coefficient will be also changed with changing of temperature. We approximate dopant dif-

fusion coefficient by the following function based on Refs. [13-15]

.

(3)

Function DL (x,y,z,T) describes spatial and temperature dependences of diffusion coefficient;

function P (x,y,z,T) describes limit of solubility of dopant; parameter γ is differ in different mate-

rials and assume integer values; function V (x,y,z,t) describes distribution of concentration of va-

cancies with equilibrium distribution V*. One can find detailed concentrational dependence of

dopant diffusion coefficient in Ref. [13]. It should be noted, that using diffusive type of doping

did not leads to radiation damage. In this situation ζ1=ζ2= 0. We determine spatio-temporal dis-

tributions of concentrations of point radiation defects by solution the following system of equa-

tions [14,15]

n

n

p

n

p

( ) ( )( )

( ) ( )( )

++

+=

2*

2

2*1

,,,,,,1

,,,

,,,1,,,

V

tzyxV

V

tzyxV

TzyxP

tzyxCTzyxDD LC ςςξ

γ

γ


3

( ) ( ) ( ) ( ) ( ) ( ) ×−

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂Tzyxk

y

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIIIII ,,,

,,,,,,

,,,,,,

,,,,

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

tzyxITzyxD

ztzyxI

VII,,,,,,,,,

,,,,,,,,,

,

2 −

∂

∂

∂

∂+× (4)

( )( )

( )( )

( )( ) ×−

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂Tzyxk

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVVVV ,,,

,,,,,,

,,,,,,

,,,,

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

tzyxVTzyxD

ztzyxV

VIV,,,,,,,,,

,,,,,,,,,

,

2 −

∂

∂

∂

∂+×

with boundary and initial conditions

( )0

,,,

0

=∂

∂

=xx

tzyxρ,

( )0

,,,=

∂

∂

= xLxx

tzyxρ,

( )0

,,,

0

=∂

∂

=yy

tzyxρ,

( )0

,,,=

∂

∂

= yLyy

tzyxρ,

( )0

,,,

0

=∂

∂

=zz

tzyxρ,

( )0

,,,=

∂

∂

= zLzz

tzyxρ, ρ (x,y,z,0)=fρ (x,y,z). (5)

Here ρ =I,V; function I (x,y,z,t) describes distribution of concentration of radiation interstitials in

space and time; functions Dρ(x,y,z,T) describe spatio-temperature dependences of the diffusion

coefficients of point radiation defects; terms V2(x,y,z,t) and I

2(x,y,z,t) describe generation

divacancies and diinterstitials, respectively; function kI,V(x,y,z,T) describes spatio-temperature

dependence of recombination parameter of point radiation defects; functions kI,I(x,y,z,T) and

kV,V(x,y,z,T) describe spatio-temperature dependence of parameters of generation of simplest

complexes of point radiation defects.

We calculate distributions of concentrations of divacancies ΦV (x,y,z,t) and dinterstitials ΦI

(x,y,z,t) in space and time by solving the following system of equations [14,15]

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxI

I

I

I

I

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

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

tzyxTzyxD

zIII

I

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

,,,,,, 2

,−+

Φ+ Φ

∂

∂

∂

∂ (6)

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxV

V

V

V

V

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

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

tzyxTzyxD

zVVV

V

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

,,,,,, 2

,−+

Φ+ Φ ∂

∂

∂

∂

with boundary and initial conditions

( )0

,,,

0

=∂

Φ∂

=xx

tzyxρ,

( )0

,,,=

∂

Φ∂

= xLxx

tzyxρ,

( )0

,,,

0

=∂

Φ∂

=yy

tzyxρ,

( )0

,,,=

∂

Φ∂

= yLyy

tzyxρ,


4

( )0

,,,

0

=∂

Φ∂

=zz

tzyxρ,

( )0

,,,=

∂

Φ∂

= zLzz

tzyxρ, ΦI (x,y,z,0)=fΦI (x,y,z), ΦV (x,y,z,0)=fΦV (x,y,z). (7)

Function DΦρ(x,y,z,T) describes spatio-temperature dependences of diffusion coefficients of the

considered complexes of radiation defects; function kI(x,y,z,T) and kV (x,y,z,T) describes spatio-

temperature dependences of parameters of decay of these complexes.

We used recently elaborated approach [16,17] to calculate distributions of concentrations of

point radiation defects in space and time. To use the approach we shall transform spatio-

temperature dependences of diffusion coefficients to the following form: Dρ(x,y,z,T)=D0ρ [1+ερ

gρ(x,y,z,T)]. Here D0ρ are the average values of diffusion coefficients, 0≤ερ<1, |gρ(x,y,z,T)|≤1, ρ

=I,V. We used the same transformation of approximations oof recombination parameters of point

defects and generation parameters of their complexes: kI,V(x,y,z,T)=k0I,V[1+εI,V gI,V(x,y,z,T)],

kI,I(x,y,z,T)= k0I,I[1 + εI,I gI,I(x,y,z,T)] and kV,V (x,y,z,T) = k0V,V [1+εV,V gV,V(x,y,z,T)], where k0ρ1,ρ2 are

the their average values, 0≤εI,V < 1, 0≤εI,I <1, 0≤εV,V<1, | gI,V(x,y,z,T)|≤1, | gI,I(x,y,z,T)|≤1,

|gV,V(x,y,z,T)|≤1. Let us introduce the following dimensionless variables: χ = x/Lx, η = y /Ly,, φ =

z/Lz, ( ) ( ) *,,,,,,~

ItzyxItzyxI = , ( ) ( ) *,,,,,,~

VtzyxVtzyxV = , VI

DDkL00,0

2

ρρρ =Ω ,

VIVIDDkL

00,0

2=ω , 2

00LtDD

VI=ϑ . Now we can transform boundary problem Eqs.(4) and

(5) to the following form

( )( )[ ] ( ) ( )

×

∂

∂

∂

∂+

∂

∂+

∂

∂=

∂

∂

η

ϑφηχ

ηχ

ϑφηχφηχε

χϑ

ϑφηχ ,,,~

,,,~

,,,1,,,

~

00

0

00

0 I

DD

DITg

DD

DI

VI

I

II

VI

I

( )[ ] ( )[ ] ( ) ( ) ×−

∂

∂+

∂

∂++× ϑφηχ

φ

ϑφηχφηχε

φφηχε ,,,

~,,,~

,,,1,,,1

00

0 II

TgDD

DTg

II

VI

I

II

( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχεω ,,,~

,,,1,,,~

,,,1 2

,,,,ITgVTg

IIIIIVIVI+Ω−+× (8)

( )( )[ ] ( ) ( )

×

∂

∂

∂

∂+

∂

∂+

∂

∂=

∂

∂

η

ϑφηχ

ηχ

ϑφηχφηχε

χϑ

ϑφηχ ,,,~

,,,~

,,,1,,,

~

00

0

00

0 V

DD

DVTg

DD

DV

VI

V

VV

VI

V

( )[ ] ( )[ ] ( ) ( ) ×−

∂

∂+

∂

∂++× ϑφηχ

φ

ϑφηχφηχε

φφηχε ,,,

~,,,~

,,,1,,,1

00

0 IV

TgDD

DTg

VV

VI

V

VV

( )[ ] ( ) ( )[ ] ( )ϑφηχφηχεϑφηχφηχεω ,,,~

,,,1,,,~

,,,1 2

,,,,VTgVTg

VVVVIVIVI+Ω−+×

( )0

,,,~

0

=∂

∂

=χχ

ϑφηχρ,

( )0

,,,~

1

=∂

∂

=χχ

ϑφηχρ,

( )0

,,,~

0

=∂

∂

=ηη

ϑφηχρ,

( )0

,,,~

1

=∂

∂

=ηη

ϑφηχρ,

( )0

,,,~

0

=∂

∂

=φφ

ϑφηχρ,

( )0

,,,~

1

=∂

∂

=φφ

ϑφηχρ, ( )

( )*

,,,,,,~

ρ

ϑφηχϑφηχρ ρf

= . (9)


5

We calculate solutions of boundary problem Eqs.(8) and (9) by using recently introduced ap-

proach [16,17]. In this situation we used the following power series

( ) ( )∑ ∑ ∑Ω=∞

=

∞

=

∞

=0 0 0

,,,~,,,~i j k

ijk

kji ϑφηχρωεϑφηχρ ρρ . (10)

We substitute the series (10) into the boundary problem Eqs.(8) and (9). After that we obtain

equations for initial-order approximations of point defects concentration ( )ϑφηχ ,,,~

000I and

( )ϑφηχ ,,,~

000V and corrections for them ( )ϑφηχ ,,,

~ijk

I and ( )ϑφηχ ,,,~

ijkV , i ≥1, j ≥1, k ≥1. The equa-

tions are presented in the Appendix. We obtain solution of the above equations by the standard

Fourier approach [18,19]. The solutions are presented in the Appendix.

Farther we calculate distributions of concentrations of simplest complexes of point radiation de-

fects in space and time. To determine the distributions we transform approximations of diffusion

coefficients in the following form: DΦρ(x,y,z,T)=D0Φρ[1+ εΦρgΦρ(x,y,z,T)]. Here D0Φρ describe av-

erage values of diffusion coefficients. In this situation the Eqs.(6) could be written as

( )( )[ ] ( ) ( )

×Φ

+

Φ

+=Φ

ΦΦΦΦy

tzyx

yD

x

tzyxTzyxg

xD

t

tzyxI

I

I

III

I

∂

∂

∂

∂

∂

∂ε

∂

∂

∂

∂ ,,,,,,,,,1

,,,00

( )[ ] ( )[ ] ( )( ) ×−

Φ

++

+× ΦΦΦΦΦ Tzyxkz

tzyxTzyxg

zDTzyxg

I

I

IIIII,,,

,,,,,,1,,,1 0

∂

∂ε

∂

∂ε

( ) ( ) ( )tzyxITzyxktzyxIII

,,,,,,,,, 2

,+×

( )( )[ ] ( ) ( )

×Φ

+

Φ

+=Φ

ΦΦΦΦy

tzyx

yD

x

tzyxTzyxg

xD

t

tzyxV

V

V

VVV

V

∂

∂

∂

∂

∂

∂ε

∂

∂

∂

∂ ,,,,,,,,,1

,,,00

( )[ ] ( )[ ] ( )( ) ×+

Φ

++

+× ΦΦΦΦΦ Tzyxkz

tzyxTzyxg

zDTzyxg II

V

VVVVV ,,,,,,

,,,1,,,1 ,0∂

∂ε

∂

∂ε

( ) ( ) ( )tzyxITzyxktzyxI I ,,,,,,,,,2 −× .

Farther we determine solutions of above equations as the following power series

( ) ( )∑ Φ=Φ∞

=Φ

0

,,,,,,i

i

itzyxtzyx ρρρ ε . (11)

Now we substitute series (11) into Eqs.(6) and conditions for them. The substitution gives a ob-

tain equations for initial-order approximations of concentrations of complexes of defects

Φρ0(x,y,z,t) and corrections for them Φρi(x,y,z,t), i ≥1 and boundary and initial conditions for

them. We present the equations and conditions in Appendix. We obtain solutions of the obtain

equations by using standard Fourier approaches [18,19] and presented in the Appendix.

We calculate distribution of concentration of dopant in space and time by using the same ap-

proach, which was used for calculation distributions of concentrations of radiation defects in


6

space and time. To use the approach we transform approximation of dopant diffusion coefficient

to the following form: DL(x,y,z,T)=D0L[1+εLgL(x,y,z,T)]. Here D0L is the average value of diffu-

sion coefficient of dopant, 0≤εL< 1, |gL(x,y,z,T)|≤1. Now we calculate solution of Eq.(1) as the

following power series

( ) ( )∑ ∑=∞

=

∞

=0 1

,,,,,,i j

ij

ji

LtzyxCtzyxC ξε .

Now we substitute the above series into boundary problem Eqs.(1) and (2) leads to equations for

the initial-order approximation of concentration of dopant C00(x,y,z,t), corrections for them

Cij(x,y,z,t) (i ≥1, j ≥1) and conditions for the above equations. The equations are presented in the

Appendix. Solutions of the equations have been calculated by standard Fourier approaches

[18,19]. The solutions are presented in the Appendix.

We analyze distributions of concentrations of dopant and radiation defects in space and time ana-

lytically by using the second-order approximations of considered series. Usually we obtain

enough good qualitative analysis and quantitative results. We check all analytical results by

comparison with results of numerical simulation.

3. DISCUSSION

Now we analyzed distributions of concentrations of dopants in space and time by using calculat-

ed in previous section relations. Figs. 2 show typical distributions of concentrations of dopants

on coordinate near interface between materials of heterostructures in direction, which is perpen-

dicular to the interface. We calculate these distributions for larger value of dopant diffusion coef-

ficient in doped area in comparison of value of dopant diffusion coefficient in nearest areas. The-

se distributions show that in the case sharpness of p-n-junctions increases. At the same time one

can find increasing of homogeneity of distribution of dopant concentration. One can find both

effects for single p-n-junctions and p-n-junctions framework their systems (transistors,

thyristors).

Fig.2a. Spatial distributions of infused dopant concentration in the considered heterostructure. The consid-

ered direction perpendicular to the interface between epitaxial layer substrate. Difference between values

of dopant diffusion coefficient in layers of heterostructure increases with increasing of number of curves


7

x

0.0

0.5

1.0

1.5

2.0

C(x

,Θ)

23

4

1

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

Epitaxial layer Substrate

Fig.2b. Spatial distributions of infused dopant concentration in the considered heterostructure. Curves 1

and 3 corresponds to annealing time Θ = 0.0048(Lx2+Ly

2+Lz2)/D0. Curves 2 and 4 corresponds to annealing

time Θ = 0.0057(Lx2+ Ly

2+Lz2)/D0. Curves 1 and 2 corresponds to homogenous sample. Curves 3 and 4

corresponds to the considered heterostructure. Difference between values of dopant diffusion coefficient in

layers of heterostructure increases with increasing of number of curves

Now we consider optimization of annealing of dopant and/or radiation defects. To do the optimi-

zation we used recently introduced criterion [16,17,20,21]. Framework the criterion we approxi-

mate real distribution by step-wise function ψ (x,y,z). We calculate optimal annealing time by

minimization the following mean-squared error

( ) ( )[ ]∫ ∫ ∫ −Θ=x y zL L L

zyx

xdydzdzyxzyxCLLL

U0 0 0

,,,,,1

ψ . (12)

Optimal annealing time as functions of parameters are presented on Figs. 3. It should be noted,

that in the ideal case after finishing of annealing of radiation defects dopant achieves interface

between layers of heterostructure. If the dopant do not achieves the interface, it is practicably to

use additional annealing of the dopant. The Fig. 3b shows just the dependences of optimal values

of additional annealing time. In this situation optimal annealing of implanted dopant is smaller in

comparison with optimal annealing time of infused dopant.

0.0 0.1 0.2 0.3 0.4 0.5a/L, ξ, ε, γ

0.0

0.1

0.2

0.3

0.4

0.5

Θ D

0 L

-2

3

2

4

1

Fig.3a. Optimal annealing time of infused dopant as dependences of several parameters. Curve 1 is the

dependence of the considered annealing time on dimensionless thickness of epitaxial layer a/L and ξ = γ = 0

for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the

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

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

dependence of the considered annealing time on parameter γ for a/L=1/2 and ε = ξ = 0


8

0.0 0.1 0.2 0.3 0.4 0.5a/L, ξ, ε, γ

0.00

0.04

0.08

0.12

Θ D

0 L

-2

3

2

4

1

Fig.3b. Optimal annealing time of implanted dopant as dependences of several parameters. Curve 1 is the

dependence of the considered annealing time on dimensionless thickness of epitaxial layer a/L and ξ = γ = 0

for equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the

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

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

dependence of the considered annealing time on parameter γ for a/L=1/2 and ε = ξ = 0

4. CONCLUSIONS

We introduce an approach to increase density of elements of an voltage restore. Several recom-

mendations for optimization technological processes were formulated.

ACKNOWLEDGEMENTS

This work is supported by the agreement of August 27, 2013 02.В.49.21.0003 between The

Ministry of education and science of the Russian Federation and Lobachevsky State University

of Nizhni Novgorod, educational fellowship for scientific research of Government of Russian

and educational fellowship for scientific research of Government of Nizhny Novgorod region of

Russia.

REFERENCES

[1] G. Volovich. Modern chips UM3Ch class D manufactured by firm MPS. Modern Electronics. Issue

2. P. 10-17 (2006).

[2] A. Kerentsev, V. Lanin. Constructive-technological features of MOSFET-transistors. Power Elec-

tronics. Issue 1. P. 34 (2008).

[3] A.O. Ageev, A.E. Belyaev, N.S. Boltovets, V.N. Ivanov, R.V. Konakova, Ya.Ya. Kudrik, P.M.

Litvin, V.V. Milenin, A.V. Sachenko. Semiconductors. Vol. 43 (7). P. 897-903 (2009).

[4] N.I. Volokobinskaya, I.N. Komarov, T.V. Matioukhina, V.I. Rechetniko, A.A. Rush, I.V. Falina,

A.S. Yastrebov. Au–TiBx-n-6H-SiC Schottky barrier diodes: the features of current flow in rectify-

ing and nonrectifying contacts. Semiconductors. Vol. 35 (8). P. 1013-1017 (2001).

[5] 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.

89 (17), 172111-172114 (2006).

[6] H.T. Wang, L.S. Tan, E. F. Chor. Pulsed laser annealing of Be-implanted GaN. J. Appl. Phys. 98

(9), 094901-094905 (2006).

[7] 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 anneal-

ing. Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836-843 (2003).

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

Russian).


9

[9] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev,

1979, in Russian).

[10] V.G. Gusev, Yu.M. Gusev. Electronics. ("Higher school", Moscow, 1991).

[11] N.A. Avaev, Yu.E. Naumov, V.T. Frolkin. Basis of microelectronics (Radio and communication,

Moscow, 1991).

[12] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001).

[13] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991).

[14] P.M. Fahey, P.B. Griffin, J.D. Plummer. Rev. Point defects and dopant diffusion in silicon. Mod.

Phys. 1989. V. 61. 2. P. 289-388.

[15] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev,

1979, in Russian).

[16] E.L. Pankratov, E.A. Bulaeva. Increasing of sharpness of diffusion-junction heterorectifier by using

radiation processing. Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1--1250028-8 (2012).

[17] E.L. Pankratov, E.A. Bulaeva. Doping of materials during manufacture p-n-junctions and bipolar

transistors. Analytical approaches to model technological approaches and ways of optimization of

distributions of dopants. Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013).

[18] A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations (Moscow, Nauka 1972) (in

Russian).

[19] H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids (Oxford University Press, 1964).

[20] E.L. Pankratov, E.A. Bulaeva. An approach to decrease dimentions of logical elements based on

bipolar transistor. Int. J. Comp. Sci. Appl. Vol. 5 (4). P. 1-18 (2015).

[21] E.L. Pankratov, E.A. Bulaeva. On optimization of manufacturing of field-effect heterotransistors

with several channels. Nano Science and Nano Technology: An Indian Journal. Vol. 9 (4). P. 43-60

(2015).

Appendix

Equations for the functions ( )ϑφηχ ,,,~

ijkI and ( )ϑφηχ ,,,

~ijk

V , i ≥0, j ≥0, k ≥0 and conditions for

them

( ) ( ) ( ) ( )2

000

2

0

0

2

000

2

0

0

2

000

2

0

0000,,,

~,,,

~,,,

~,,,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ

∂

∂+

∂

∂+

∂

∂=

∂

∂ I

D

DI

D

DI

D

DI

V

I

V

I

V

I

( ) ( ) ( ) ( )2

000

2

0

0

2

000

2

0

0

2

000

2

0

0000,,,

~,,,

~,,,

~,,,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ

∂

∂+

∂

∂+

∂

∂=

∂

∂ V

D

DV

D

DV

D

DV

I

V

I

V

I

V

( ) ( ) ( ) ( )+

∂

∂+

∂

∂+

∂

∂=

∂

∂2

00

2

2

00

2

2

00

2

0

000 ,,,~

,,,~

,,,~

,~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑχiii

V

IiIII

D

DI

( )( )

( )( )

+

∂

∂

∂

∂+

∂

∂

∂

∂+ −−

η

ϑφηχφηχ

ηχ

ϑφηχφηχ

χ

,,,~

,,,,,,

~

,,, 100

0

0100

0

0 i

I

V

Ii

I

V

II

TgD

DITg

D

D

( )( )

∂

∂

∂

∂+ −

φ

ϑφηχφηχ

φ

,,,~

,,, 100

0

0 i

I

V

II

TgD

D, i ≥1,

( ) ( ) ( ) ( )+

∂

∂+

∂

∂+

∂

∂=

∂

∂2

00

2

2

00

2

2

00

2

0

000,,,

~,,,

~,,,

~,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑχiii

I

ViVVV

D

DV

( )( )

( )( )

+

∂

∂

∂

∂+

∂

∂

∂

∂+ −−

η

ϑφηχφηχ

ηχ

ϑφηχφηχ

χ

,,,~

,,,,,,

~

,,, 100

0

0100

0

0 i

V

I

Vi

V

I

VV

TgD

DVTg

D

D


10

( )( )

∂

∂

∂

∂+ −

φ

ϑφηχφηχ

φ

,,,~

,,, 100

0

0 i

V

I

VV

TgD

D, i ≥1;

( ) ( ) ( ) ( )−

∂

∂+

∂

∂+

∂

∂=

∂

∂2

010

2

2

010

2

2

010

2

0

0010 ,,,~

,,,~

,,,~

,,,~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ III

D

DI

V

I

( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,,~

,,,~

,,,1000000,,

VITgVIVI

+−

( ) ( ) ( ) ( )−

∂

∂+

∂

∂+

∂

∂=

∂

∂2

010

2

2

010

2

2

010

2

0

0010 ,,,~

,,,~

,,,~

,,,~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ VVV

D

DV

I

V

( )[ ] ( ) ( )ϑφηχϑφηχφηχε ,,,~

,,,~

,,,1 000000,, VITgVIVI

+− ;

( ) ( ) ( ) ( )−

∂

∂+

∂

∂+

∂

∂=

∂

∂2

020

2

2

020

2

2

020

2

0

0020,,,

~,,,

~,,,

~,,,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ III

D

DI

V

I

( )[ ] ( ) ( ) ( ) ( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε ,,,~

,,,~

,,,~

,,,~

,,,1010000000010,,

VIVITgVIVI

++−

( ) ( ) ( ) ( )−

∂

∂+

∂

∂+

∂

∂=

∂

∂2

020

2

2

020

2

2

020

2

0

0020,,,

~,,,

~,,,

~,,,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ VVV

D

DV

V

I

( )[ ] ( ) ( ) ( ) ( )[ ]ϑφηχϑφηχϑφηχϑφηχφηχε ,,,~

,,,~

,,,~

,,,~

,,,1010000000010,,

VIVITgVIVI

++− ;

( ) ( ) ( ) ( )−

∂

∂+

∂

∂+

∂

∂=

∂

∂2

001

2

2

001

2

2

001

2

0

0001,,,

~,,,

~,,,

~,,,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ III

D

DI

V

I

( )[ ] ( )ϑφηχφηχε ,,,~

,,,1 2

000,, ITgIIII

+−

( ) ( ) ( ) ( )−

∂

∂+

∂

∂+

∂

∂=

∂

∂2

001

2

2

001

2

2

001

2

0

0001 ,,,~

,,,~

,,,~

,,,~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ VVV

D

DV

I

V

( )[ ] ( )ϑφηχφηχε ,,,~

,,,1 2

000,,VTg

IIII+− ;

( ) ( ) ( ) ( )+

∂

∂+

∂

∂+

∂

∂=

∂

∂2

110

2

2

110

2

2

110

2

0

0110 ,,,~

,,,~

,,,~

,,,~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ III

D

DI

V

I

( ) ( ) ( ) ( )

+

∂

∂

∂

∂+

∂

∂

∂

∂+

η

ϑφηχφηχ

ηχ

ϑφηχφηχ

χ

,,,~

,,,,,,

~

,,, 010010

0

0I

TgI

TgD

DII

V

I

( )( )

( )[ ] ( ) ( )[ ++−

∂

∂

∂

∂+ ϑφηχϑφηχφηχε

φ

ϑφηχφηχ

φ,,,

~,,,

~,,,1

,,,~

,,, 000100,,

010 VITgI

TgIIIII

( ) ( )]ϑφηχϑφηχ ,,,~

,,,~

100000 VI+

( ) ( ) ( ) ( )+

∂

∂+

∂

∂+

∂

∂=

∂

∂2

110

2

2

110

2

2

110

2

0

0110,,,

~,,,

~,,,

~,,,

~

φ

ϑφηχ

η

ϑφηχ

χ

ϑφηχ

ϑ

ϑφηχ VVV

D

DV

I

V

( ) ( ) ( ) ( )

+

∂

∂

∂

∂+

∂

∂

∂

∂+

η

ϑφηχφηχ

ηχ

ϑφηχφηχ

χ

,,,~

,,,,,,

~

,,, 010010

0

0V

TgV

TgD

DVV

I

V


12

( )0

,,,~

0

=∂

∂

=φφ

ϑφηχρijk

, ( )

0,,,~

1

=∂

∂

=φφ

ϑφηχρijk

(i ≥0, j ≥0, k ≥0);

( ) ( ) *

000,,0,,,~ ρφηχφηχρ ρf= , ( ) 00,,,~ =φηχρ

ijk (i ≥1, j ≥1, k ≥1).

Solutions of these equations with account boundary and initial conditions could be written as

( ) ( ) ( ) ( ) ( )∑+=∞

=1000

21,,,~

nnn

ecccFLL

ϑφηχϑφηχρ ρρ ,

where ( ) ( ) ( ) ( )∫ ∫ ∫=1

0

1

0

1

0*

,,coscoscos1

udvdwdwvufwnvnunFnn ρρ πππ

ρ, ( ) ( )

IVnIDDne 00

22exp ϑπϑ −= ,

( ) ( )VInV

DDne00

22exp ϑπϑ −= , cn(χ) = cos (π n χ);

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0

1

0

1

0

1

00

0

00 ,,,2,,,~

nInnnInIn

V

I

iTwvugvcuseecccn

D

DI

ϑ

τϑφηχπϑφηχ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂

∂×

∞

=

−

1 0

1

00

0100 2,,,

~

nnnInIn

V

Ii

nuceecccn

D

Ddudvdwd

u

wvuIwc

ϑ

τϑφηχττ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑−∫ ∫∂

∂×

∞

=

−

10

01

0

1

0

100 2,,,

~

,,,n

n

V

Ii

Inn cccnD

Ddudvdwd

v

wvuITwvugwcvs φηχπτ

τπ

( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫

∂

∂−× −

ϑ

ττ

τϑ0

1

0

1

0

1

0

100 ,,,~

,,, dudvdwdw

wvuITwvugwsvcucee i

InnnnInI, i ≥1,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫ ∫−−=∞

=1 0

1

0

1

0

1

00

0

00,,,2,,,

~

nVnnnInVn

I

V

iTwvugvcuseecccn

D

DV

ϑ


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×∫ ∫ ∫−−∂

∂×

∞

=

−

1 0

1

0

1

00

0100,

~

nnnnInVn

I

Vi

nvsuceecccn

D

Ddudvdwd

u

uVwc

ϑ

τϑφηχττ

( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ×−∫∂

∂×

∞

=

−

10

01

0

100 2,

~

,,,2n

nVn

I

Vi

Vnecccn

D

Ddudvdwd

v

uVTwvugwc ϑφηχπτ

τπ

( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫

∂

∂−× −

ϑ

ττ

τ0

1

0

1

0

1

0

100,

~

,,, dudvdwdw

uVTwvugwsvcuce i

VnnnnI, i ≥1,

where sn(χ) = sin (π n χ);

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫ ×−−=∞

=1 0

1

0

1

0

1

0010

2,,,~n

nnnnnnnnwcvcuceeccc

ϑ

ρρ τϑφηχϑφηχρ

( )[ ] ( ) ( ) τττε dudvdwdwvuVwvuITwvugVIVI

,,,~

,,,~

,,,1000000,,

+× ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0

1

0010

1

0

1

00

0

020,,,

~2,,,~

nnnnnnnnn

V

I wvuIwcvcuceecccD

D ϑ

ρρ ττϑφηχϑφηχρ

( ) ( ) ( )] ( )[ ] τετττ dudvdwdTwvugwvuVwvuIwvuVVIVI

,,,1,,,~

,,,~

,,,~

,,010000000++× ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0

1

0

1

0

1

0

2

000001,,,~2,,,~

nnnnnnnn

wvuvcuceecccϑ

ρρ τρτϑφηχϑφηχρ


13

( )[ ] ( ) τε ρρρρ dudvdwdwcTwvugn

,,,1,,

+× ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ×∫ ∫−−=∞

=1 0

1

0

1

0

1

0000002

,,,~2,,,~n

nnnnnnnnwvuwcvcuceeccc

ϑ

ρρ τρτϑφηχϑφηχρ

( )[ ] ( ) ττρε ρρρρ dudvdwdwvuTwvug ,,,~,,,1 001,,+× ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0

1

0

1

0

1

00

0

1102,,,

~

nnnnnInInnn

V

I ucvcuseecccnD

DI

ϑ


( ) ( ) ( ) ( ) ( ) ( ) ×∑−∂

∂×

∞

=

−

10

0100 2,,,

~

,,,n

nInnn

V

Ii

Iecccn

D

Ddudvdwd

u

wvuITwvug ϑφηχπτ

τ

( ) ( ) ( ) ( ) ( ) ( )×−∫ ∫ ∫ ∫

∂

∂−× −

V

Ii

InnnnID

Ddudvdwd

v

wvuITwvugucvsuce

0

0

0

1

0

1

0

1

0

100 2,,,

~

,,, πττ

τϑ

( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫

∂

∂−×

∞

=

−

1 0

1

0

1

0

1

0

100,,,

~

,,,n

i

InnnnInIdudvdwd

w

wvuITwvugusvcuceen

ϑ

ττ

τϑ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[∑ ∫ ∫ ∫ ∫ ×+−−×∞

=1 0

1

0

1

0

1

0,

12n

VInnnnInnnInnnnvcvcuceccecccc

ϑ

ετφηϑχφηχ

( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvugVI

,,,~

,,,~

,,,~

,,,~

,,,100000000100,

+×

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0

1

0

1

0

1

00

0

1102,,,

~

nnnnnVnVnnn

I

V ucvcuseecccnD

DV

ϑ


( ) ( ) ( ) ( ) ( ) ( ) ×∑−∂

∂×

∞

=

−

10

0100 2,,,

~

,,,n

nVnnn

I

Vi

Vecccn

D

Ddudvdwd

u

wvuVTwvug ϑφηχπτ

τ

( ) ( ) ( ) ( ) ( ) ( )×−∫ ∫ ∫ ∫

∂

∂−× −

I

Vi

VnnnnVD

Ddudvdwd

v

wvuVTwvugucvsuce

0

0

0

1

0

1

0

1

0

100 2,,,

~

,,, πττ

τϑ

( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫

∂

∂−×

∞

=

−

1 0

1

0

1

0

1

0

100,,,

~

,,,n

i

VnnnnVnVdudvdwd

w

wvuVTwvugusvcuceen

ϑ

ττ

τϑ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[∑ ∫ ∫ ∫ ∫ ×+−−×∞

=1 0

1

0

1

0

1

0,

12n

VInnnnVnnnInnnnvcvcuceccecccc

ϑ

ετφηϑχφηχ

( )] ( ) ( ) ( ) ( )[ ] τττττ dudvdwdwvuVwvuIwvuVwvuITwvug VI ,,,~

,,,~

,,,~

,,,~

,,, 100000000100, +× ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0

1

0

1

0

1

00

0

101,,,2,,,

~

nInnnInInnn

V

I TwvugvcuseecccnD

DI

ϑ


( )( )

( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂

∂×

∞

=1 0

1

00

0001 2,,,

~

nnnInInnn

V

I

nuceecccn

D

Ddudvdwd

u

wvuIwc

ϑ

τϑφηχπττ

( ) ( ) ( )( )

( ) ( ) ( ) ×∑ ∫ −−∫ ∫∂

∂×

∞

=1 00

01

0

1

0

001 2,,,

~

,,,n

nInIn

V

I

Inneec

D

Ddudvdwd

v

wvuITwvugwcvs

ϑ

τϑχπττ

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫∂

∂×

∞

=1

1

0

1

0

1

0

001 2,,,

~

,,,n

nnnnInnnccccdudvdwd

w

wvuITwvugwsvcucn ηχφητ

τ


14

( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ +−×ϑ

τττετϑφ0

1

0000100,,

,,,~

,,,~

,,,1 dudvdwdwvuVwvuITwvuguceecVIVInnInIn

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0

1

0

1

0

1

00

0

101,,,2,,,

~

nVnnnVnVnnn

I

V TwvugvcuseecccnD

DV

ϑ


( )( )

( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−∂

∂×

∞

=1 0

1

00

0001 2,,,

~

nnnVnVnnn

I

V

nuceecccn

D

Ddudvdwd

u

wvuVwc

ϑ

τϑφηχπττ

( ) ( ) ( )( )

( ) ( ) ( ) ×∑ ∫ −−∫ ∫∂

∂×

∞

=1 00

01

0

1

0

001 2,,,

~

,,,n

nVnVn

I

V

Vnneec

D

Ddudvdwd

v

wvuVTwvugwcvs

ϑ

τϑχπττ

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ×∑−∫ ∫ ∫∂

∂×

∞

=1

1

0

1

0

1

0

001 2,,,

~

,,,n

nnnnVnnnccccdudvdwd

w

wvuVTwvugwsvcucn ηχφητ

τ

( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∫ ∫ +−×ϑ

τττετϑφ0

1

0100000,, ,,,

~,,,

~,,,1 dudvdwdwvuVwvuITwvuguceec

VIVInnVnVn

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∑ ∫ ∫ ∫ ∫ ×+−−=∞

=1 0

1

0

1

0

1

0,,011 ,,,12,,,

~

nIIIInnnnInInnn

TwvugwcvcuceecccIϑ

ετϑφηχϑφηχ

( ) ( ) ( )[ ] ( ) ( ) τττεττ dudvdwdwvuVwvuITwvugwvuIwvuIVIVI

,,,~

,,,~

,,,1,,,~

,,,~

000001,,010000 ++×

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∑ ∫ ∫ ∫ ∫ ×+−−=∞

=1 0

1

0

1

0

1

0,,011 ,,,12,,,

~

nVVVVnnnnVnVnnn

TwvugwcvcuceecccVϑ

ετϑφηχϑφηχ

( ) ( ) ( )[ ] ( ) ( ) τττεττ dudvdwdwvuVwvuITwvugwvuVwvuVVIVI

,,,~

,,,~

,,,1,,,~

,,,~

001000,,010000 ++× .

Equations for initial-order approximations of distributions of concentrations of simplest com-

plexes of radiation defects Φρ0(x,y,z,t) and corrections for them Φρi(x,y, z,t), i ≥1 and boundary

and initial conditions for them takes the form

( ) ( ) ( ) ( )+

Φ+

Φ+

Φ=

ΦΦ 2

0

2

2

0

2

2

0

2

0

0,,,,,,,,,,,,

z

tzyx

y

tzyx

x

tzyxD

t

tzyxIII

I

I

∂

∂

∂

∂

∂

∂

∂

∂

( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkIII

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

,−+

( ) ( ) ( ) ( )+

Φ+

Φ+

Φ=

ΦΦ 2

0

2

2

0

2

2

0

2

0

0 ,,,,,,,,,,,,

z

tzyx

y

tzyx

x

tzyxD

t

tzyxVVV

V

V

∂

∂

∂

∂

∂

∂

∂

∂

( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkVVV

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

,−+ ;

( ) ( ) ( ) ( )+

Φ+

Φ+

Φ=

ΦΦ 2

2

2

2

2

2

0

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

z

tzyx

y

tzyx

x

tzyxD

t

tzyxiIiIiI

I

iI

∂

∂

∂

∂

∂

∂

∂

∂

( )( )

( )( )

+

Φ+

Φ+

−

Φ

−

ΦΦy

tzyxTzyxg

yx

tzyxTzyxg

xD

iI

I

iI

II∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

11

0

( )( )

Φ+

−

Φz

tzyxTzyxg

z

iI

I∂

∂

∂

∂ ,,,,,,

1, i ≥1,


15

( ) ( ) ( ) ( )+

Φ+

Φ+

Φ=

ΦΦ 2

2

2

2

2

2

0

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

z

tzyx

y

tzyx

x

tzyxD

t

tzyxiViViV

V

iV

∂

∂

∂

∂

∂

∂

∂

∂

( )( )

( )( )

+

Φ+

Φ+

−

Φ

−

ΦΦy

tzyxTzyxg

yx

tzyxTzyxg

xD

iV

V

iV

VV∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

11

0

( )( )

Φ+

−

Φz

tzyxTzyxg

z

iV

V∂

∂

∂

∂ ,,,,,,

1, i≥1;

( )0

,,,

0

=∂

Φ∂

=x

i

x

tzyxρ,

( )0

,,,=

∂

Φ∂

= xLx

i

x

tzyxρ,

( )0

,,,

0

=∂

Φ∂

=y

i

y

tzyxρ,

( )0

,,,=

∂

Φ∂

= yLy

i

y

tzyxρ,

( )0

,,,

0

=∂

Φ∂

=z

i

z

tzyxρ,

( )0

,,,=

∂

Φ∂

= zLz

i

z

tzyxρ, i≥0; Φρ0(x,y,z,0)=fΦρ (x,y,z), Φρi(x,y,z,0)=0, i≥1.

Solutions of the above equations could be written as

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑+∑+=Φ∞

=

∞

=ΦΦ

110

221,,,

nnnn

nnnnnn

zyxzyx

zcycxcnL

tezcycxcFLLLLLL

tzyxρρρ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∫ ∫ ∫ ∫ −−× ΦΦ

t L L L

IIInnnn

x y z

wvuITwvukwvuITwvukvcucete0 0 0 0

2

, ,,,,,,,,,,,, τττρρ

( ) τdudvdwdwcn

× ,

where ( ) ( ) ( ) ( )∫ ∫ ∫= ΦΦ

x y zL L L

nnnnudvdwdwvufwcvcucF

0 0 0

,,ρρ

, ( ) ( )[ ]222

0

22exp −−−

ΦΦ ++−=zyxn

LLLtDnteρρ

π ,

cn(x) = cos (π n x/Lx);

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=Φ∞

=ΦΦΦ

1 0 0 0 02

,,,2

,,,n

t L L L

nnnnnnn

zyx

i

x y z

TwvugvcusetezcycxcnLLL

tzyxρρρ

τπ

ρ

( )( )

( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−Φ

×∞

=ΦΦ

−

1 0 02

1 2,,,

n

t L

nnnnnn

zyx

iI

n

x

ucetezcycxcnLLL

dudvdwdu

wvuwc τ

πτ

∂

τ∂ρρ

ρ

( ) ( ) ( )( )

( ) ( ) ( ) ×∑−∫ ∫Φ

×∞

=

−

Φ1

20 0

1 2,,,,,,

nnnn

zyx

L LiI

nnzcycxcn

LLLdudvdwd

v

wvuTwvugwcvs

y z πτ

∂

τ∂ρ

ρ

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

( )∫ ∫ ∫ ∫Φ

−× Φ

−

ΦΦ

t L L LiI

nnnnn

x y z

dudvdwdTwvugw

wvuwsvcucete

0 0 0 0

1

,,,,,,

τ∂

τ∂τ

ρ

ρ

ρρ, i≥1,

where sn(x) = sin (π n x/Lx). Equations for calculation functions Cij(x,y,z,t) (i ≥0, j ≥0), boundary and

initial conditions take the form

( ) ( ) ( ) ( )2

00

2

02

00

2

02

00

2

0

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

z

tzyxCD

y

tzyxCD

x

tzyxCD

t

tzyxCLLL

∂

∂+

∂

∂+

∂

∂=

∂

∂;

( ) ( ) ( ) ( )+

∂

∂+

∂

∂+

∂

∂=

∂

∂2

0

2

02

0

2

02

0

2

0

0,,,,,,,,,,,,

z

tzyxCD

y

tzyxCD

x

tzyxCD

t

tzyxCi

L

i

L

i

L

i


16

( ) ( ) ( ) ( )+

∂

∂

∂

∂+

∂

∂

∂

∂+ −−

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

yD

x

tzyxC

TzyxP

tzyxC

xD

LL

,,,

,,,

,,,,,,

,,,

,,,0000

0

0000

0 γ

γ

γ

γ

( )( )

( )

∂

∂

∂

∂+

z

tzyxC

TzyxP

tzyxC

zD

L

,,,

,,,

,,, 0000

0 γ

γ

;

( ) ( ) ( ) ( )+

∂

∂+

∂

∂+

∂

∂=

∂

∂2

02

2

02

02

2

02

02

2

0

02 ,,,,,,,,,,,,

z

tzyxCD

y

tzyxCD

x

tzyxCD

t

tzyxCLLL

( )( )

( )( )

( )( )

( )

×

∂

∂+

∂

∂

∂

∂+

−−

TzyxP

tzyxCtzyxC

yx

tzyxC

TzyxP

tzyxCtzyxC

xD

L,,,

,,,,,,

,,,

,,,

,,,,,,

1

00

01

00

1

00

010 γ

γ

γ

γ

( )( )

( )( )

( ) ( )( )

×

∂

∂+

∂

∂

∂

∂+

∂

∂×

−

TzyxP

tzyxC

xD

z

tzyxC

TzyxP

tzyxCtzyxC

zy

tzyxCL

,,,

,,,,,,

,,,

,,,,,,

,,,00

0

00

1

00

01

00

γ

γ

γ

γ

( ) ( )( )

( ) ( )( )

( )

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂×

z

tzyxC

TzyxP

tzyxC

zy

tzyxC

TzyxP

tzyxC

yx

tzyxC ,,,

,,,

,,,,,,

,,,

,,,,,, 0100010001

γ

γ

γ

γ

;

( ) ( ) ( ) ( )+

∂

∂+

∂

∂+

∂

∂=

∂

∂2

11

2

02

11

2

02

11

2

0

11 ,,,,,,,,,,,,

z

tzyxCD

y

tzyxCD

x

tzyxCD

t

tzyxCLLL

( ) ( )( )

( ) ( ) ( )( )

×

∂

∂+

∂

∂

∂

∂+

−−

TzyxP

tzyxCtzyxC

yx

tzyxC

TzyxP

tzyxCtzyxC

x ,,,

,,,,,,

,,,

,,,

,,,,,,

1

00

10

00

1

00

10 γ

γ

γ

γ

( ) ( ) ( )( )

( )+

∂

∂

∂

∂+

∂

∂×

−

LD

z

tzyxC

TzyxP

tzyxCtzyxC

zy

tzyxC0

00

1

00

10

00 ,,,

,,,

,,,,,,

,,,γ

γ

( )( )

( ) ( )( )

( )

+

∂

∂

∂

∂+

∂

∂

∂

∂+

y

tzyxC

TzyxP

tzyxC

yx

tzyxC

TzyxP

tzyxC

xD

L

,,,

,,,

,,,,,,

,,,

,,,10001000

0 γ

γ

γ

γ

( )( )

( ) ( ) ( )

+

∂

∂

∂

∂+

∂

∂

∂

∂+

x

tzyxCTzyxg

xD

z

tzyxC

TzyxP

tzyxC

zLL

,,,,,,

,,,

,,,

,,, 01

0

1000

γ

γ

( ) ( ) ( ) ( )

∂

∂

∂

∂+

∂

∂

∂

∂+

z

tzyxCTzyxg

zy

tzyxCTzyxg

yLL

,,,,,,

,,,,,, 0101 ;

( )0

,,,

0

==x

ij

x

tzyxC

∂

∂,

( )0

,,,=

= xLx

ij

x

tzyxC

∂

∂,

( )0

,,,

0

==y

ij

y

tzyxC

∂

∂,

( )0

,,,=

= yLy

ij

y

tzyxC

∂

∂,


17

( )0

,,,

0

==z

ij

z

tzyxC

∂

∂,

( )0

,,,=

= zLz

ij

z

tzyxC

∂

∂, i ≥0, j ≥0; C00(x,y,z,0)=fC (x,y,z), Cij(x,y,z,0)=0,

i ≥1, j ≥1.

Solutions of the above equations with account boundary and initial conditions could be written as

( ) ( ) ( ) ( ) ( )∑+=∞

=100

21,,,

nnCnnnnC

zyxzyx

tezcycxcFLLLLLL

tzyxC ,

where ( ) ( )[ ]222

0

22exp

−−− ++−=zyxCnC

LLLtDnte π , ( ) ( ) ( ) ( )∫ ∫ ∫=x y zL L L

nCnnnCudvdwdwcwvufvcucF

0 0 0

,, ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0 0 0 020 ,,,2

,,,n

t L L L

LnnnnCnCnnnnC

zyx

i

x y z

TwvugvcvcusetezcycxcFnLLL

tzyxC τπ

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−

∂

∂×

∞

=

−

1 0 0 02

10 2,,,

n

t L L

nnnCnCnnnnC

zyx

ix y

vsucetezcycxcFnLLL

dudvdwdu

wvuCτ

πτ

τ

( ) ( )( )

( ) ( ) ( ) ( ) ×∑−∫∂

∂×

∞

=

−

12

0

10 2,,,,,,

nnCnnnnC

zyx

Li

LntezcycxcFn

LLLdudvdwd

v

wvuCTwvugvc

z πτ

τ

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

∫ ∫ ∫ ∫∂

∂−× −

t L L Li

LnnnnC

x y z

dudvdwdw

wvuCTwvugvsvcuce

0 0 0 0

10 ,,,,,, τ

ττ , i ≥1;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0 0 0 0201

2,,,

n

t L L L

nnnnCnCnnnnC

zyx

x y z

wcvcusetezcycxcFnLLL

tzyxC τπ

( )( )

( )( ) ( ) ( ) ( ) ( ) ×∑ ∫ −−

∂

∂×

∞

=1 02

0000 2,,,

,,,

,,,

n

t

nCnCnnnnC

zyx

etezcycxcFnLLL

dudvdwdu

wvuC

TwvuP

wvuCτ

πτ

ττγ

γ

( ) ( ) ( )( )( )

( )( ) ×∑−∫ ∫ ∫

∂

∂×

∞

=12

0 0 0

0000 2,,,

,,,

,,,

nnC

zyx

L L L

nnnten

LLLdudvdwd

v

wvuC

TwvuP

wvuCwcvsuc

x y z πτ

ττγ

γ

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

( )∫ ∫ ∫ ∫

∂

∂−×

t L L L

nnnnCnnnnC

x y z

dudvdwdw

wvuC

TwvuP

wvuCwsvcucezcycxcF

0 0 0 0

0000,,,

,,,

,,,τ

τττ

γ

γ

;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0 0 0 0202

2,,,

n

t L L L

nnnnCnCnnnnC

zyx

x y z


tzyxC τπ

( ) ( )( )

( ) ( ) ( )×∑−∂

∂×

∞

=

−

12

00

1

00

01

2,,,

,,,

,,,,,,

nnnnC

zyx

ycxcFLLL

dudvdwdu

wvuC

TwvuP

wvuCwvuC

πτ

τττ

γ

γ

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

( )×∫ ∫ ∫ ∫

∂

∂−×

−t L L L

nnnCnCn

x y z

v

wvuC

TwvuP

wvuCwvuCvsucetezcn

0 0 0 0

00

1

00

01

,,,

,,,

,,,,,,

ττττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−×∞

=1 0 0 02

2

n

t L L

nnnCnCnnnnC

zyx

n

x y

vcucetezcycxcFnLLL

dudvdwdwc τπ

τ

( ) ( ) ( )( )

( ) ( ) ×∑−∫∂

∂×

∞

=

−

12

0

00

1

00

01

2,,,

,,,

,,,,,,

nn

zyx

L

nxcn

LLLdudvdwd

w

wvuC

TwvuP

wvuCwvuCws

z πτ

τττ

γ

γ


18

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×∫ ∫ ∫ ∫

∂

∂−×

t L L L

nnnnCnCnnnC

x y z

u

wvuCwvuCwcvcusetezcycF

0 0 0 0

00

01

,,,,,,

τττ

( )( )

( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−×∞

=

−

1 0 02

1

00 2

,,,

,,,

n

t L

nnCnCnnnnC

zyx

x

ucetezcycxcFnLLL

dudvdwdTwvuP

wvuCτ

πτ

τγ

γ

( ) ( ) ( ) ( )( )

( )×∑−∫ ∫

∂

∂×

∞

=

−

12

0 0

00

1

00

01

2,,,

,,,

,,,,,,

nzyx

L L

nnn

LLLdudvdwd

v

wvuC

TwvuP

wvuCwvuCwcvs

y z πτ

τττ

γ

γ

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

×∫ ∫ ∫ ∫−×−t L L L

nnnnCnCnnnnC

x y z

TwvuP

wvuCwvuCwsvcucetezcycxcF

0 0 0 0

1

00

01,,,

,,,,,,

γ

γ τττ

( )( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−

∂

∂×

∞

=1 0 02

00 2,,,

n

t L

nnCnCnnnnC

zyx

x

usetezcycxcFLLL

dudvdwdw

wvuCτ

πτ

τ

( ) ( ) ( )( )

( ) ( ) ( )∑ ×−∫ ∫∂

∂×

∞

=12

0 0

0100 2,,,

,,,

,,,

nnCn

zyx

L L

nntexc

LLLdudvdwd

u

wvuC

TwvuP

wvuCwcvcn

y z πτ

ττγ

γ

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

( )×∫ ∫ ∫ ∫

∂

∂−×

t L L L

nnnnCnnC

x y z

dudvdwdv

wvuC

TwvuP

wvuCwcvsuceycF

0 0 0 0

0100 ,,,

,,,

,,,τ

τττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−×∞

=1 0 0 0 02

2

n

t L L L

nnnnCnCnnnnC

zyx

n

x y z

wsvcucetezcycxcFnLLL

zcn τπ

( )( )

( )τ

ττγ

γ

dudvdwdw

wvuC

TwvuP

wvuC

∂

∂×

,,,

,,,

,,, 0100 ;

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫ ∫−−=∞

=1 0 0 0 0211

2,,,

n

t L L L

nnnnCnCnnnnC

zyx

x y z


tzyxC τπ

( )( )

( ) ( ) ( ) ( ) ×∑−∂

∂×

∞

=12

01 2,,,,,,

nnCnnnnC

zyx

LtezcycxcFn

LLLdudvdwd

u

wvuCTwvug

πτ

τ

( ) ( ) ( ) ( ) ( ) ( )×−∫ ∫ ∫ ∫

∂

∂−×

20 0 0 0

01 2,,,,,,

zyx

t L L L

LnnnnCLLL

dudvdwdv

wvuCTwvugwcvsuce

x y z πτ

ττ

( ) ( ) ( ) ( ) ( ) ( ) ( )×∑ ∫ ∫ ∫ ∫

∂

∂−×

∞

=1 0 0 0 0

01 ,,,,,,

n

t L L L

LnnnnCnC

x y z

dudvdwdw

wvuCTwvugwsvcuceten τ

ττ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−−×∞

=1 0 0 02

2

n

t L L

nnnCnCnnnnC

zyx

nnnnC

x y

vcusetezcycxcFLLL

zcycxcF τπ

( )( )( )

( )( ) ( ) ×∑−∫

∂

∂×

∞

=12

0

1000 2,,,

,,,

,,,

nnnnC

zyx

L

nycxcFn

LLLdudvdwd

u

wvuC

TwvuP

wvuCwcn

z πτ

ττγ

γ

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

( )−∫ ∫ ∫ ∫

∂

∂−×

t L L L

nnnnCnCn

x y z

dudvdwdv

wvuC

TwvuP

wvuCwcvsucetezc

0 0 0 0

1000 ,,,

,,,

,,,τ

τττ

γ

γ

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

×∑ ∫ ∫ ∫ ∫−−∞

=1 0 0 0 0

00

2,,,

,,,2

n

t L L L

nnnnCnCnnnnC

zyx

x y z

TwvuP

wvuCwsvcucetezcycxcFn

LLLγ

γ ττ

π

( )( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−

∂

∂×

∞

=1 0 02

10 2,,,

n

t L

nnCnCnnnnC

zyx

x

usetezcycxcFnLLL

dudvdwdw

wvuCτ

πτ

τ


19

( ) ( ) ( ) ( )( )

( )×∑−∫ ∫

∂

∂×

∞

=

−

12

0 0

00

1

00

10

2,,,

,,,

,,,,,,

nzyx

L L

nnn

LLLdudvdwd

u

wvuC

TwvuP

wvuCwvuCwcvc

y z πτ

τττ

γ

γ

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

( )×∫ ∫ ∫ ∫

∂

∂−×

−t L L L

nnnnCnCnnnnC

x y z

v

wvuC

TwvuP

wvuCwcvsucetezcycxcF

0 0 0 0

00

1

00,,,

,,,

,,, τττ

γ

γ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∑ ∫ ∫−−×∞

=1 0 0210

2,,,

n

t L

nnCnCnnnnC

zyx

x

ucetezcycxcFnLLL

dudvdwdwvuC τπ

ττ

( ) ( ) ( ) ( )( )

( )∫ ∫

∂

∂×

−y zL L

nndudvdwd

w

wvuC

TwvuP

wvuCwvuCwsvc

0 0

00

1

00

10

,,,

,,,

,,,,,, τ

τττ

γ

γ

.

Authors:

Pankratov Evgeny Leonidovich was born at 1977. From 1985 to 1995 he was educated in a secondary

school in Nizhny Novgorod. From 1995 to 2004 he was educated in Nizhny Novgorod State University:

from 1995 to 1999 it was bachelor course in Radiophysics, from 1999 to 2001 it was master course in

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. 2012-2015 Full Doctor course in

Radiophysical Department of Nizhny Novgorod State University. Since 2015 E.L. Pankratov is an Associ-

ate Professor of Nizhny Novgorod State University. He has 155 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. Since 2014 E.A. Bulaeva is in a PhD program in Radiophysical Department of Nizhny

Novgorod State University. She has 103 published papers in area of her researches.

ANALYSIS OF MANUFACTURING OF VOLTAGE RESTORE TO INCREASE DENSITY OF ELEMENTS OF THE CIRCUIT

Engineering

Transcript of ANALYSIS OF MANUFACTURING OF VOLTAGE RESTORE TO INCREASE DENSITY OF ELEMENTS OF THE CIRCUIT