an approach to decrease dimentions of logical elements based on bipolar transistor

8/20/2019 AN APPROACH TO DECREASE DIMENTIONS OF LOGICAL ELEMENTS BASED ON BIPOLAR TRANSISTOR

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International Journal on Computational Science & Applications (IJCSA) Vol.5, No.4,August 2015

DOI:10.5121/ijcsa.2015.5401 1

A N APPROACH TO DECREASE DIMENTIONS OF LOG-

ICAL ELEMENTS BASED ON BIPOLAR TRANSISTOR

E.L.Pankratov1,3

and E.A.Bulaeva1,2

1 Nizhny Novgorod State University,23 Gagarin avenue ,Nizhny Novgorod,603950,Russia2

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

street , Nizhny Novgorod , 603950,Russia3

Nizhny Novgorod Academy of the Ministry of Internal Affairs of Russia ,3 Ankudinovs-

koe Shosse , Nizhny Novgorod , 603950, Russia

A BSTRACT

In this paper we consider manufacturing logical elements with function AND-NOT based on bipolar tran-

sistors. Based on recently considered approach to decrease dimensions of solid state electronic devices

with the same time increasing of their performance we introduce an approach to decrease dimensions of

transistors and p-n-junctions, which became a part of the logical element. Framework the approach a hete-

rostructure with required configuration should be manufactured. After the manufacture required areas of

the heterostructures should be doped by diffusion or ion implantation. The doping should be finished by

optimized annealing of dopant and/or radiation defects.

K EYWORDS

Transistor-transistor logic; optimization of manufacturing; decreasing of dimensions of transistor; analyti-

cal approach for modelling

1.INTRODUCTION

In the present time solid state electronic is intensively elaborating. Framework the elaborating onecan find increasing of integrating rate of elements of integrated circuits ( p-n-junctions, their sys-

tems et al) with decreasing of their dimensions [1-7]. To ways of decreasing of dimensions ofelements of integrated circuits are using laser and microwave types of annealing of dopantsand/or radiation defects [8-16]. Another way to decrease the dimensions is using inhomogeneity

of doped by diffusion or ion implantation heterostructure [1-3,17]. Using the second way leads to

necessity to optimize annealing of dopants and/or radiation defects [17]. It is known, that radia-tion processing of materials leads to changing of distributions of concentrations of dopants [18].

In this situation radiation processing attracted an interest to decrease of dimensions of elements ofintegrated circuits [17,18].

In the next paragraph we introduce an analytical approach to analyze mass transport during tech-

nological process. The approach gives a possibility to generalize recently considered in literatureseveral limiting cases. Framework the approach we consider mass transport in a multilayer struc-

tures without crosslinking solutions on the interfaces between the layers. The approach also gives

a possibility to optimize technological processes.


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International Journal on Computational Science & Applications (IJCSA) Vol.5, No.4, August 2015

2

Fig. 1a. Structure of logical element based on bipolar transistor. View from top. Black marked transistors

and p-n-junction manufactured by using doping of appropriate sections of the epitaxial layer. Dimensions

of these devices are decreased. Transistor 1 is a multiemitter transistor. Emitters have been marked by using

letter E. The index indicates their number in the multiemitter transistor. D1 and D2 mean dopants of p and n

types in p-n-junction. Red marked resistors ( Ri) and wires have no decreasing of their dimensions

Substrate

Epitaxial layer

Sections

Fig. 1b. Heterostructure with a substrate and an epitaxial layer. Some sections have been manufactured by

using another materials in the epitaxial layer

2.METHOD OF SOLUTION

In this paper we consider a heterostructure with a substrate and an epitaxial layer (see Figs. 1).Epitaxial layer includes into itself several sections manufactured by using another materials (see

Figs. 1). The sections have been doped by diffusion or by ion implantation to generate requiredtype of conductivity ( p or n) during manufacturing transistors and p-n-junction. After that anneal-

ing of dopant and/or radiation defects should be considered. Main aim of our paper is determina-

tion of conditions, which correspond to decreasing of the considered elements and increasingswitching time of p-n-junctions by analysis of redistribution of dopant and radiation defects dur-

ing their annealing.

Let us determine spatio-temporal distributions of concentrations of dopants to solve our aim. We

assume, that main part of transport of dopant during annealing is diffusion. Interaction between

atoms of dopant and defects is small. In this situation we determine spatio-temporal distributionof concentration of dopant as solution the second Fick's law in the following form [1,3, 18]


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3

( ) ( ) ( ) ( )

+

+

=

z

t z y xC D

z y

t z y xC D

y x

t z y xC D

xt

t z y xC C C C

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,,,,,,,. (1)

Boundary and initial conditions for the equations are

( )0

,,,

0

=∂

∂

= x x

t z y xC ,

( )0

,,,=

∂

∂

= x L x x

t z y xC ,

( )0

,,,

0

=∂

∂

= y y

t z y xC ,

( )0

,,,=

∂

∂

= y L x y

t z y xC ,

( )0

,,,

0

=∂

∂

= z z

t z y xC ,

( )0

,,,=

∂

∂

= z L x z

t z y xC , C ( x, y, z,0)=f ( x, y, z). (2)

Boundary conditions correspond to absents of flow of dopant through external boundaries of theconsidered heterostructure. Initial distribution depends on type of doping: diffusion doping or ion

doping. In the Eqs.(1) and (2) function C ( x, y, z,t ) describes the spatio-temporal distribution ofconcentration of dopant; T is the temperature of annealing; DС is the dopant diffusion coefficient.

Value of dopant diffusion coefficient could be changed by heating and cooling of heterostructure

(with account Arrhenius law), radiation processing of materials of heterostructure and high dop-ing of materials. Approximation of dopant diffusion coefficient on parameters could be written as[19-21]

( ) ( )

( )( ) ( )

( )

++

+=

2*

2

2*1

,,,,,,1

,,,

,,,1,,,

V

t z y xV

V

t z y xV

T z y xP

t z y xC T z y x D D LC ς ς ξ γ

γ

. (3)

Here function D L ( x, y, z,T ) takes into account the spatial (due to presents several layers in hetero-structure) and temperature (due to Arrhenius law) dependences of dopant diffusion coefficient.

Function P ( x, y, z,T ) describes approximation of the limit of solubility of dopant on coordinates in

heterostructure and temperature. The parameter γ shows averaged quantity of charged defects,which interacting which interacting with atoms of dopant [19]. The function V ( x, y, z,t ) describes

the spatio-temporal distribution of concentration of radiation vacancies; V

*

is the equilibrium dis-tribution of concentration of vacancies. The second term of the last part of Eq.(3) taking into ac-

count generation of radiation vacancies. The third term of the last part of Eq.(3) taking into ac-

count generation of radiation divacancies. Dependence of dopant diffusion coefficient on concen-tration of dopant has been described in details in [19]. It should be noted, that using diffusion type

of doping did not generation radiation defects. In this situation ζ 1= ζ 2= 0. We determine spatio-temporal distributions of concentrations of radiation defects by solving the following system of

equations [20,21]

( )( )

( )( )

( )( ) ×−

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂T z y xk

y

t z y x I T z y x D

y x

t z y x I T z y x D

xt

t z y x I I I I I

,,,,,,

,,,,,,

,,,,,,

,

( ) ( ) ( )

( ) ( ) ( )t z y xV t z y x I T z y xk zt z y x I

T z y x D z

t z y x I V I I ,,,,,,,,,,,,

,,,,,, ,2 −

∂

∂

∂∂+× (4)

( )( )

( )( )

( )( ) ×−

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂T z y xk

y

t z y xV T z y x D

y x

t z y xV T z y x D

xt

t z y xV V V V V

,,,,,,

,,,,,,

,,,,,,

,

( ) ( ) ( )

( ) ( ) ( )t z y xV t z y x I T z y xk z

t z y xV T z y x D

zt z y xV

V I V ,,,,,,,,,

,,,,,,,,,

,

2 −

∂

∂

∂

∂+× .


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4

Terms in the right sides of Eqs.(4) with derivatives on coordinate gives a possibility to take intoaccount diffusion of point radiation defects. Terms with V

2( x, y, z,t ) and I

2( x, y, z,t ) gives a possibil-

ity to take into account divacancies and diinterstitials. Term with I ( x, y, z,t ) V ( x, y, z,t ) gives a possi-bility to take into account recombination of point radiation defects. Boundary and initial condi-

tions for Eqs.(4) are

( )0

,,,

0

=∂

∂

= x x

t z y x ρ ,

( )0

,,,=

∂

∂

= x L x x

t z y x ρ ,

( )0

,,,

0

=∂

∂

= y y

t z y x ρ ,

( )0

,,,=

∂

∂

= y L y y

t z y x ρ ,

( )0

,,,

0

=∂

∂

= z z

t z y x ρ ,

( )0

,,,=

∂

∂

= z L z z

t z y x ρ , ρ ( x, y, z,0)=f ρ ( x, y, z). (5)

Here ρ = I ,V . The function I ( x, y, z,t ) describes the spatio-temporal distribution of concentration ofradiation interstitials. The functions D ρ ( x, y, z,T ) describe the diffusion coefficients of point radia-

tion defects. The function k I ,V ( x, y, z,T ) describes the parameter of recombination of point radiation

defects. The functions k ρ , ρ ( x, y, z,T ) describe the parameters of generation of simplest complexes of

point radiation defects.

Spatio-temporal distributions of concentrations of divacancies Φ ρ ( x, y, z,t ) have been calculated bysolving the following system of equations [20,21]

( )( )

( )( )

( )+

Φ+

Φ=

ΦΦΦ

y

t z y xT z y x D

y x

t z y xT z y x D

xt

t z y x I

I

I

I

I

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

( ) ( )

( ) ( ) ( ) ( )t z y x I T z y xk t z y x I T z y xk z

t z y xT z y x D

z I I I

I

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

,,,,,,

2

, −+

Φ+ Φ

∂

∂

∂

∂ (6)

( )( )

( )( )

( )+

Φ+

Φ=

ΦΦΦ

y

t z y xT z y x D

y x

t z y xT z y x D

xt

t z y xV

V

V

V

V

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

( ) ( )

( ) ( ) ( ) ( )t z y xV T z y xk t z y xV T z y xk z

t z y xT z y x D

z V V V

V

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

,,,,,, 2

, −+

Φ+ Φ

∂

∂

∂

∂ .

Terms in the right sides of Eqs.(6) with derivatives on coordinate gives a possibility to take into

account diffusion of complexes of point radiation defects. Terms with V 2( x, y, z,t ) and I

2( x, y, z,t )

gives a possibility to take into account divacancies and diinterstitials. Term with I ( x, y, z,t ) and V ( x, y, z,t ) gives a possibility to take into account decay of complexes of point radiation defects.

Boundary and initial conditions for Eqs.(4) are

( )0

,,,

0=∂

Φ∂

= x x

t z y x ρ ,

( )0

,,,

=∂

Φ∂

= x L x x

t z y x ρ ,

( )0

,,,

0=∂

Φ∂

= y y

t z y x ρ ,

( )0

,,,

=∂

Φ∂

= y L y y

t z y x ρ ,

( )0

,,,

0

=∂

Φ∂

= z z

t z y x ρ ,

( )0

,,,=

∂

Φ∂

= z L z z

t z y x ρ , Φ I ( x, y, z,0)=f Φ I ( x, y, z), Φ V ( x, y, z,0)=f Φ V ( x, y, z). (7)


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5

Here DΦρ ( x, y, z,T ) are the diffusion coefficients of the above complexes of radiation defects;

k I ( x, y, z,T ) and k V ( x, y, z,T ) are the parameters of decay of these complexes. To determine spatio-

temporal distribution of concentration of dopant we transform the Eq.(1) to the following integro-differential form

( ) ( ) ( )( )

×∫ ∫ ∫

++=∫ ∫ ∫t y

L

z

L z y

x

L

y

L

z

L z y x y z x y z V

wv xV

V

wv xV

L L

z yud vd wd t wvuC

L L L

z y x

02*

2

2*1

,,,,,,1,,, τ

ς τ

ς

( ) ( )

( )( )

( ) ×∫ ∫ ∫+

+×

t x

L

z

L L

z x

L

x z

T w yu D L L

z xd

x

wv xC

T wv xP

wv xC T wv x D

0

,,,,,,

,,,

,,,1,,, τ

∂

τ ∂ τ ξ

γ

γ

( ) ( )

( )( )( )

( )+

+

++× τ

∂

τ ∂ τ ξ

τ ς

τ ς

γ

γ

d y

w yuC

T z y xP

w yuC

V

w yuV

V

w yuV ,,,

,,,

,,,1

,,,,,,1

2*

2

2*1

( ) ( ) ( )

( )( )( )

×∫ ∫ ∫

+

+++

t x

L

y

L L

y x x y T z y xP

zvuC

V

zvuV

V

zvuV T zvu D

L L

y x

02*

2

2*1 ,,,

,,,1

,,,,,,1,,,

γ

γ τ ξ

τ ς

τ ς

( ) ( )∫ ∫ ∫+× x

L

y

L

z

L z y x x y z

ud vd wd wvu f L L L z y xd

z zvuC ,,,,, τ

∂ τ ∂ . (1a)

We determine solution of the above equation by using Bubnov-Galerkin approach [22]. Frame-work the approach we determine solution of the Eq.(1a) as the following series

( ) ( ) ( ) ( ) ( )∑==

N

nnC nnnnC t e zc yc xcat z y xC

00

,,, ,

where ( ) ( )2220

22exp −−− ++−= z y xC nC

L L Lt Dnt e π , cn( χ )=cos(π n χ / L χ ). The above series includes

into itself finite number of terms N . The considered series is similar with solution of linear Eq.(1)

(i.e. with ξ =0) and averaged dopant diffusion coefficient D0. Substitution of the series intoEq.(1a) leads to the following result

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫

×

∑+−=∑

==

t y

L

z

L

N

nnC nnnnC

z y

N

nnC nnn

C

y z

ewcvc xca L L

z yt e zs ys xs

n

a z y x

0 1132

1γ

τ π

( )( ) ( )

( ) ( ) ( ) ( )×∑

++

×=

N

nnnnC L vc xsaT wv x D

V

wv xV

V

wv xV

T wv xP 12*

2

2*1,,,

,,,,,,1

,,,

τ ς

τ ς

ξ γ

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

∫ ∫ ∫ ×

∑+−×

=

t x

L

z

L

N

mmC mmmmC

z x

nC n

x z T w yuPewc ycuca

L L

z xd ewcn

0 1 ,,,1

γ

γ ξ τ τ τ

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ×∑

++×

=τ τ

τ ς

τ ς d ewc ysucn

V

w yuV

V

w yuV T w yu D

N

nnC nnn L

12*

2

2*1

,,,,,,1,,,

( )( )

( ) ( ) ( ) ( ) ×∫ ∫ ∫

∑+−×

=

t x

L

y

L

N

nnC nnnnC L

y x

nC

x y

e zcvcucaT zvuP

T zvu D L L

y xa

0 1,,,1,,,

γ

γ τ

ξ

( ) ( )

( ) ( ) ( ) ( ) ( ) ×+∑

++×

= z y x

N

nnC nnnnC

L L L

z y xd e zsvcucan

V

zvuV

V

zvuV τ τ

τ ς

τ ς

12*

2

2*1

,,,,,,1


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6

( )∫ ∫ ∫× x

L

y

L

z

L x y z

ud vd wd wvu f ,, ,

where sn( χ ) =sin (π n χ / L χ ). We determine coefficients an by using orthogonality condition of termsof the considered series framework scale of heterostructure. The condition gives us possibility toobtain relations for calculation of parameters an for any quantity of terms N . In the common case

the relations could be written as

( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫

×

∑+−=∑−

==

t L L L N

nnC nnnnC L

z y N

nnC

nC z y x x y z

e zc yc xcaT z y x D L L

t en

a L L L

0 0 0 0 12

165

222

1,,,2

γ

τ π π

( )( ) ( )

( ) ( ) ( ) ( ) ( )×∑

++

×=

N

nnC nnn

nC e zc yc xsn

a

V

z y xV

V

z y xV

T z y xP 12*

2

2*12

,,,,,,1

,,,τ

τ ς

τ ς

ξ γ

( ) ( )[ ] ( ) ( )[ ] ( )×∫ ∫ ∫ ∫−

−+

−+×t L L L

Ln

z

nn

y

n

x y z

T z y x Dd xd yd zd zcn

L zs z yc

n

L ys y

0 0 0 0

,,,11 τ π π

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

( )

++

∑+×= *11

,,,1

,,,1,,,

V

z y xV

T z y xPe zc yc xcaT z y x D

N

n nC nnnnC L

τ ς

ξ τ

γ

γ

( )

( )( ) ( )

( ) ( ) ( )[ ]∑ ×

−+

++

+

=

N

n

nC

n

x

nn

a xc

n

L xs x

V

z y xV

V

z y xV

V

z y xV

12*

2

2*12*

2

21

,,,,,,1

,,,

π

τ ς

τ ς

τ ς

( ) ( ) ( ) ( ) ( ) ( )[ ] ×−

−+×22 2

122 π

τ π

τ π

y x

n

z

nnC nnn

z x L L

d xd yd zd zcn

L zs ze zc ys xc

L L

( ) ( ) ( ) ( )( )

( )

( )∫ ∫ ∫ ∫

++

∑+×

=

t L L L N

nnC nnnnC

x y z

V

z y xV

T z y xPe zc yc xca

0 0 0 02*

2

21

,,,1

,,,1

τ ς

ξ τ

γ

γ

( )( ) ( ) ( ) ( ) ( ) ( )[ ] ×∑

−+

+

=

N

n

n

x

nnnn

nC

L xc

n

L xs x zs yc xc

n

aT z y x D

V

z y xV

1

*11,,,

,,,

π

τ ς

( ) ( )[ ] ( ) ( ) ( )[ ]∑ ∫ ×

−++

−+×=

N

n

L

n

x

nnC n

y

n

x

xcn

L xs xd xd yd zd e yc

n

L ys y

1 0

11π

τ τ π

( ) ( )[ ] ( ) ( )[ ] ( )∫ ∫

−+

−+× y z

L L

n

z

nn

y

n xd yd zd z y x f zc

n

L zs z yc

n

L ys y

0 0

,,11π π

.

As an example for γ = 0 we obtain

( ) ( )[ ] ( ) ( )[ ] ( ) ( ){∫ ∫ ∫ +

−+

−+= x y z L L L

nn

y

nn

y

nnC xs x yd zd z y x f zc

n

L zs z yc

n

L ys ya

0 0 0

,,11

π π

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

∫ ∫ ∫ ∫ ×

−+

−×t L L L

Ln

y

nnn

x

n

x y z

T z y x D ycn

L ys y yc xs

n xd

n

L xc

0 0 0 0

,,,122

1π π

( ) ( )[ ] ( ) ( )

( ) ( ) ×

+

++

−+×T z y xPV

z y xV

V

z y xV zc

n

L zs z n

y

n,,,

1,,,,,,

112*

2

2*1 γ

ξ τ ς

τ ς

π


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7

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )×∫ ∫ ∫ ∫

−++×t L L L

nnn

y

nnnC nC n

x y z

zc ys xcn

L xs x xced e xd yd zd zc

0 0 0 0

21π

τ τ τ

( ) ( )[ ]( )

( ) ( )

( ) ×

++

+

−+×2*

2

2*1

,,,,,,1

,,,11

V

z y xV

V

z y xV

T z y xP zc

n

L zs z

n

y

n

τ ς

τ ς

ξ

π γ

( ) ( ) ( ) ( ) ( )[ ] ( ) ( ){∫ ∫ ∫ ×

−++×t L L

nnn

x

nnnC L

x y

ys yc xcn

L xs x xced xd yd zd T z y x D

0 0 0

1,,,π

τ τ

( )[ ] ( ) ( )( )

( )

( )∫

++

+

−+× z L

Lnn

y

V

z y xV

T z y xPT z y x D zs yc

n

L y

02*

2

2

,,,1

,,,1,,,21

τ ς

ξ

π γ

( )( )

1

65

222

*1

,,,−

−

+ t e

n

L L Ld xd yd zd

V

z y xV nC

z z z

π τ

τ ς .

For γ = 1 one can obtain the following relation to determine required parameters

( ) ( ) ( ) ( )∫ ∫ ∫+±−= x y z L

L L

nnnnn

n

n

nC xd yd zd z y x f zc yc xca

0 0 0

2 ,,42

α β α

β ,

where ( ) ( ) ( ) ( )

( )

( )

( )∫ ∫ ∫ ∫ ×

++=

t L L L L

nnC

z y

n

x y z

T z y xP

T z y x D

V

z y xV

V

z y xV xse

n

L L

0 0 0 02*

2

2*12,,,

,,,,,,,,,12

2

τ ς

τ ς τ

π

ξ α

( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ×∫+

−+

−+×t

nC

z x

n

z

nn

y

nnn en

L Ld xd yd zd zc

n

L zs z yc

n

L ys y zc yc

022

11 τ π

ξ τ

π π

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

( ) ×∫ ∫ ∫

++

−−

−+× x y z L

L L

n

z

nn

x

n

V

z y xV

V

z y xV zc

n

L zs z xc

n

L xs x

0 0 02*

2

2*1

,,,,,,111 τ

ς τ

ς π π

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )

×∫ ∫ ∫ ∫+×t

L L L L

nnnC

y x L

nnn

x y

z

T z y xPT z y x D yc xce

n L Ld xd yd zd

T z y xPT z y x D zc ys xc

0 0 0 02 ,,,

,,,2,,,

,,,2 τ π

ξ τ

( ) ( )

( ) ( ) ( )[ ] ( ) ( )[ ] ×

−+

−+

++× yd yc

n

L ys y xc

n

L xs x

V

z y xV

V

z y xV n

y

nn

x

n11

,,,,,,1

2*

2

2*1 π π

τ ς

τ ς

( ) τ d xd zd zsn 2× , ( ) ( ) ( ) ( ) ( )

( )∫ ∫ ∫ ∫ ×

++=

t L L L

nnnC

z y

n

x y z

V

z y xV

V

z y xV zc xse

n

L L

0 0 0 02*

2

2*12

,,,,,,12

2

τ ς

τ ς τ

π β

( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ×+

−+

−+×22

11,,,π

τ π π n

L Ld xd yd yc

n

L ys y yc zd zc

n

L zs zT z y x D z xn

y

nnn

z

n L

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

( ) ×∫ ∫ ∫ ∫

++

−+×t L L L

nnn

x

nnnC

x y z

V

z y xV

V

z y xV zc ys xc

n

L xs x xce

0 0 0 02

*

2

2*1

,,,,,,121

τ ς

τ ς

π

τ

( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫

−++

−+×t L

n

x

nnC

y x

n

z

n L

x

xcn

L xs xe

n

L Ld xd yd zd zc

n

L zs zT z y x D

0 02

12

1,,,π

τ π

τ π

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

( )∫ ×∫

++

−+× y z

L L

Lnn

y

n zd

V

z y xV

V

z y xV T z y x D zs yc

n

L ys y

0 02*

2

2*1

,,,,,,1,,,21

τ ς

τ ς

π


8/18


8

( ) ( ) ( ) 65222 nt e L L Ld xd xc yd ycnC z y xnn

π τ −× .

Analogous way could be used to calculate values of parameters an for larger values of parameter

γ . However the relations will not be present in the paper because the relations are bulky.Equationsof the system (4) have been also solved by using Bubnov-Galerkin approach. Previously we trans-

form the differential equations to the following integro- differential form

( ) ( ) ( )

×∫ ∫ ∫∂

∂=∫ ∫ ∫

t y

L

z

L I

x

L

y

L

z

L z y x y z x y z

d vd wd x

wv x I T wv x Dud vd wd t wvu I

L L L

z y x

0

,,,,,,,,, τ

τ

( ) ( )

( ) ×∫ ∫ ∫−∫ ∫ ∫∂

∂+×

x

L

y

L

z

LV I

t x

L

z

L I

z x z y x y z x z

T wvuk d ud wd x

w yu I T w yu D

L L

z x

L L

z y,,,

,,,,,,

,0

τ τ

( ) ( ) ( )

×∫ ∫ ∫∂

∂+×

t x

L

y

L y x z y x x y

z

zvu I

L L

y x

L L L

z y xud vd wd t wvuV t wvu I

0

,,,,,,,,,

τ

( ) ( ) ( ) +∫ ∫ ∫−× x

L

y

L

z

L

I I

z y x

I

x y z

ud vd wd t wvu I T wvuk

L L L

z y xd ud vd T zvu D ,,,,,,,,, 2

,τ

( )∫ ∫ ∫+ x

L

y

L

z

L I

z y x x y z

ud vd wd wvu f L L L

z y x,, (4a)

( ) ( ) ( )

×∫ ∫ ∫∂

∂=∫ ∫ ∫

t y

L

z

LV

x

L

y

L

z

L z y x y z x y z

d vd wd x

wv xV T wv x Dud vd wd t wvuV

L L L

z y x

0

,,,,,,,,, τ

τ

( ) ( )

( )×∫ ∫ ∫+∫ ∫ ∫∂

∂+×

t x

L

y

LV

t x

L

z

LV

z x z y x y x z

T zvu Dd ud wd x

w yuV T w yu D

L L

z x

L L

z y

00

,,,,,,

,,, τ τ

( )( ) ( ) ( ) ×∫ ∫ ∫−

∂

∂×

x

L

y

L

z

LV I

y x x y z

ud vd wd t wvuV t wvu I T wvuk L L

y xd ud vd

z

zvuV ,,,,,,,,,

,,,,

τ τ

( ) ( ) ( )

∫ ∫ ∫−∫ ∫ ∫+× x

L

y

L

z

LV V

x

L

y

L

z

LV

z y x z y x x y z x y z

ud vd wd t wvuV T wvuk ud vd wd wvu f L L L

z y x

L L L

z y x,,,,,,,,

2

, .

We determine solutions of the Eqs.(4a) as the following series

( ) ( ) ( ) ( ) ( )∑==

N

nnnnnn t e zc yc xcat z y x

10 ,,, ρ ρ ρ .

Coefficients an ρ are not yet known. Substitution of the series into Eqs.(4a) leads to the followingresults

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

N

n

t y

L

z

L

I nnnI

z y x

N

n

nI nnn

nI

y z

vd wd T wv x D zc yca

L L L

z yt e zs ys xs

n

a z y x

1 01

33,,,

π

π

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

N

n

t x

L

z

L I nnnI nnI

z y x

nnI

x z

d ud wd T w yu D zc xce ysa L L L

z x xsd e

1 0

,,, τ τ π

τ τ

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

x

L

y

L

z

L I I

N

n

t x

L

y

L I nnnI nnI

z y x x y z x y

T vvuk d ud vd T zvu D yc xce zsa L L L

y x,,,,,,

,1 0

τ τ π


9/18


9

( ) ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∑−

∑×

==

x

L

y

L

z

L

N

nnnnnI

z y x z y x

N

nnI nnnnI

x y z

wcvcuca L L L

z y x

L L L

z y xud vd wd t ewcvcuca

1

2

1

( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫+∑×=

x

L

y

L

z

L I

z y x

N

nV I nV nnnnV nI

x y z


z y xud vd wd T vvuk t ewcvcucat e ,,,,,

1,

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

N

n

t y

L

z

LV nnnV

z y x

N

nnV nnn

nV

y z

vd wd T wv x D zc yca L L L

z yt e zs ys xs

n

a z y x

1 0133

,,,π

π

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

N

n

t x

L

z

LV nnnV nnV

z y x

nnV

x z

d ud wd T w yu D zc xce ysa L L L

z x xsd e

1 0

,,, τ τ π

τ τ

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

x

L

y

L

z

LV V

N

n

t x

L

y

LV nnnV nnV

z y x x y z x y

T vvuk d ud vd T zvu D yc xce zsa L L L

y x,,,,,,

,1 0

τ τ π

( ) ( ) ( ) ( ) ( ) ( ) ( )×∫ ∫ ∫ ∑−

∑×

==

x

L

y

L

z

L

N

nnnnnI

z y x z y x

N

nnI nnnnV

x y z

wcvcuca L L L

z y x

L L L

z y xud vd wd t ewcvcuca

1

2

1

( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫+∑× =

x

L

y

L

z

LV

z y x

N

nV I nV nnnnV nI

x y zud vd wd wvu f L L L

z y x

ud vd wd T vvuk t ewcvcucat e ,,,,,1, .

We determine coefficients an ρ by using orthogonality condition on the scale of heterostructure.

The condition gives us possibility to obtain relations to calculate an ρ for any quantity N of terms

of considered series. In the common case equations for the required coefficients could be written

as

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−==

N

n

t L L

n

y

n yn

nI

x

N

nnI

nI z y x x y

ycn

L ys y L xc

n

a

Lt e

n

a L L L

1 0 0 02

165

222

122

2212

1

π π π

( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ +−∫

−+×=

N

n

t L

n

nI

y

L

nI n

z

n I

x z

xs xn

a

Ld e xd yd zd zc

n

L zs zT z y x D

1 0 020

22

11

2,,,

π τ τ

π

( )[ ] ( ) ( ) ( )[ ] ( )[ ]×∫ ∫ −

−++

−++ y z

L L

nn

z

n z I n

x

x yc zd zc

n

L zs z LT z y x D xc

n

L L

0 0

21122

2,,,12π π

( ) ( ) ( ) ( )[ ] ( ) −∫

−++× z L

nI n

z

n z I nI d e xd yd zd zc

n

L zs z LT z y x Dd e xd yd

0

122

2,,, τ τ π

τ τ

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++

−++−=

N

n

t L L

n

y

n yn

x

n x

nI

z

x y

ycn

L ys y L xc

n

L xs x L

n

a

L 1 0 0 02

122

2122

22

1

π π π

( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫

+−+−∫ −×=

N

n

L

n

x

xnI nI

L

nI I n

x z

xcn

L Lt ead e xd yd zd T z y x D zc

1 0

2

0

122

2,,,21π

τ τ

( )} ( ) ( )[ ] ( ) ( )[ ]∫ ∫

+−+

−+++ y z

L L

n

z

z I I n

y

n yn zcn

L LT z y xk yc

n

L ys y L xs x

0 0,

122

,,,122

22π π

( )} ( ) ( ) ( ) ( )[ ] {∑ ∫ ∫ +

−++−+=

N

n

L L

yn

x

n xnV nI nV nI n

x y

L xcn

L xs x Lt et eaa xd yd zd zs z

1 0 0

122

22π


10/18


10

( ) ( )[ ] ( ) ( ) ( )[ ]∫ ×

−++

−++ z L

n

z

n zV I n

y

n zd zc

n

L zs z LT z y xk yc

n

L ys y

0,

122

2,,,122

2π π

( ) ( )[ ] ( ) ( )[ ] ( )×∑ ∫ ∫ ∫

−+

−++×=

N

n

L L L

I n

y

nn

x

n

x y z

T z y x f ycn

L ys y xc

n

L xs x xd yd

1 0 0 0

,,,11π π

( ) ( )[ ] xd yd zd zcn

L zs z L

n

z

n z

−++× 122

2π

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−==

N

n

t L L

n

y

n yn

nV

x

N

nnV

nV z y x x y

ycn

L ys y L xc

n

a

Lt e

n

a L L L

1 0 0 02

165

222

122

2212

1

π π π

( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ +−∫

−+×=

N

n

t L

n

nV

y

L

nV n

z

nV

x z

xs xn

a

Ld e xd yd zd zc

n

L zs zT z y x D

1 0 02

0

22

11

2,,,

π τ τ

π

( )[ ] ( ) ( ) ( )[ ] ( )[ ] ×∫ ∫ −

−++

−++ y z

L L

nn

z

n zV n

x

x yc zd zc

n

L zs z LT z y x D xc

n

L L

0 0

21122

2,,,12π π

( ) ( ) ( ) ( )[ ] ( ) −∫

−++×

z L

nV n

z

n zV nV d e xd yd zd zc

n

L zs z LT z y x Dd e xd yd

0

122

2,,, τ τ π

τ τ

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++

−++−=

N

n

t L L

n

y

n yn

x

n x

nV

z

x y

ycn

L ys y L xc

n

L xs x L

n

a

L 1 0 0 02

122

2122

22

1

π π π

( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫

+−+−∫ −×=

N

n

L

n

x

xnV nV

L

nV V n

x z

xcn

L Lt ead e xd yd zd T z y x D zc

1 0

2

0

122

2,,,21π

τ τ

( )} ( ) ( )[ ] ( ) ( )[ ]∫ ∫

+−+

−+++ y z

L L

n

z

zV V n

y

n yn zc

n

L LT z y xk yc

n

L ys y L xs x

0 0,

122

,,,122

22π π

( )} ( ) ( ) ( ) ( )[ ] {∑ ∫ ∫ +

−++−+

=

N

n

L L

yn x

n xnV nI nV nI n

x y

L xcn

L xs x Lt et eaa xd yd zd zs z1 0 0

122

22π

( ) ( )[ ] ( ) ( ) ( )[ ]∫ ×

−++

−++ z L

n

z

n zV I n

y

n zd zc

n

L zs z LT z y xk yc

n

L ys y

0,

122

2,,,122

2π π

( ) ( )[ ] ( ) ( )[ ] ( )×∑ ∫ ∫ ∫

−+

−++×=

N

n

L L L

V n

y

nn

x

n

x y z

T z y x f ycn

L ys y xc

n

L xs x xd yd

1 0 0 0

,,,11π π

( ) ( )[ ] xd yd zd zcn

L zs z L

n

z

n z

−++× 122

2π

.

In the final form relations for required parameters could be written as

( )

−+−

+±

+−=

A

yb yb

Ab

b

Aba nI nV

nI

2

3

4

2

3

4

3 444

λ γ ,

nI nI

nI nI nI nI nI

nV a

aaa

χ

λ δ γ ++−=

2

,

where ( ) ( ) ( ) ( )[ ] ( ){∫ ∫ ∫ ++

−++= x y z L L L

ynn

x

n xnn L ys y xc

n

L xs x LT z y xk t e

0 0 0,

2122

2,,,2π

γ ρ ρ ρ ρ


11/18


11

( )[ ] ( ) ( )[ ] xd yd zd zcn

L zs z L yc

n

Ln

z

n zn

y

−++

−+ 122

2122 π π

, ( )×∫=t

n

x

n en L 0

22

1τ

π δ ρ ρ

( ) ( )[ ] ( ) ( )[ ] ( ) [∫ ∫ ∫ −

−+

−+× x y z L L L

n

z

nn

y

n yd zd T z y x D zc

n

L zs z yc

n

L ys y

0 0 0

1,,,12

12

ρ π π

( )] ( ) ( ) ( )[ ] ( )[ ] {∫ ∫ ∫ ∫ +−

−+++−

t L L L

znn

x

n xn

y

n

x y z

L yc xcn

L xs x Le

n Ld xd xc

0 0 0 02

211222

12

π τ

π τ ρ

( ) ( )[ ] ( ) ( ) ( ){∫ ∫ ++

−++t L

nn

z

n

z

n

x

xs xen L

d xd yd zd T z y x D zcn

L zs z

0 02

22

1,,,12

22 τ

π τ

π ρ ρ

( )[ ] ( ) ( )[ ] ( )[ ] ( ) ×∫ ∫ −

−++

−++ y z

L L

nn

y

n yn

x

x zd T z y x D zc ycn

L ys y L xc

n

L L

0 0

,,,2112

12 ρ

π π

( )t en

L L Ld xd yd

n

z y x

ρ π

τ 65

222

−× , ( ) ( )[ ] ( )[ ]∫ ∫

+−+

−+= x y L L

n

y

yn

x

nnIV ycn

L L xc

n

L xs x

0 0

122

1π π

χ

( )} ( ) ( ) ( )[ ] ( ) ( )t et e xd yd zd zcn

L zs z LT z y xk ys y

nV nI

L

n

z

n zV I n

z

∫

−+++0

,12

2

2,,,2

π

,

( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ]∫ ∫ ∫ ×

−+

−+

−+= x y z L

L L

n

z

nn

y

nn

x

nn zc

n

L zs z yc

n

L ys y xc

n

L xs x

0 0 0

111π π π

λ ρ

( ) xd yd zd T z y x f ,,, ρ × ,22

4 nI nI nI nV b χ γ γ γ −= ,

nI nI nV nI nI nI nI nV b γ χ δ χ δ δ γ γ −−= 23 2 ,

2

2

348 bb y A −+= , ( ) 22

2 2 nI nI nV nI nI nV nI nV nI nI nV b χ λ λ δ χ δ γ γ λ δ γ −+−+= , ×= nI b λ 21

nI nI nV nI nV λ δ δ −× ,4

33 323 32

3b

bq pqq pq y −++−−+= ,

2

4

2

342

9

3

b

bbb p

−= ,

( ) 34

2

4132

3

3542792 bbbbbbq +−= .

We determine solutions of the Eqs.(4a) as the following series

( ) ( ) ( ) ( ) ( )∑=Φ=

Φ

N

nnnnnn t e zc yc xcat z y x

10 ,,, ρ ρ ρ .

Coefficients anΦρ are not yet known. Let us previously transform the Eqs.(6) to the following in-

tegro-differential form

( ) ( ) ( )

×∫ ∫ ∫ Φ

=∫ ∫ ∫Φ Φt y

L

z

L

I

I

x

L

y

L

z

L I

z y x y z x y z

d vd wd x

wv xT wv x Dud vd wd t wvu

L L L

z y x

0

,,,,,,,,, τ

∂

τ ∂

( ) ( )

( )∫ ∫ ∫ ×+∫ ∫ ∫ Φ

+× ΦΦt x

L

y

L I

y x

t x

L

z

L

I

I

z x z y x y x z

T zvu D L L

y xd ud wd

y

w yuT w yu D

L L

z x

L L

z y

00

,,,,,,

,,, τ ∂

τ ∂

( ) ( ) ( ) −∫ ∫ ∫+Φ× x

L

y

L

z

L I I

z y x

I

x y z

ud vd wd wvu I T wvuk L L L z y xd ud vd

z zvu τ τ

∂ τ ∂ ,,,,,,,,, 2

, (6a)

( ) ( ) ( )∫ ∫ ∫+∫ ∫ ∫− Φ x

L

y

L

z

L I

z y x

x

L

y

L

z

L I

z y x x y z x y z


z y xud vd wd wvu I T wvuk

L L L

z y x,,,,,,,, τ


12/18


12

( ) ( ) ( )

×∫ ∫ ∫ Φ

=∫ ∫ ∫Φ Φt y

L

z

L

V

V

x

L

y

L

z

LV

z y x y z x y z

d vd wd x

wv xT wv x Dud vd wd t wvu

L L L

z y x

0

,,,,,,,,, τ

∂

τ ∂

( ) ( )

( )∫ ∫ ∫ ×+∫ ∫ ∫ Φ

+× ΦΦt x

L

y

LV

y x

t x

L

z

L

V

V

z x z y x y x z

T zvu D L L

y xd ud wd

y

w yuT w yu D

L L

z x

L L

z y

00

,,,,,,

,,, τ ∂

τ ∂

( ) ( ) ( ) −∫ ∫ ∫+Φ

× x

L

y

L

z

LV V

z y x

V

x y z

ud vd wd wvuV T wvuk L L L

z y xd ud vd

z

zvuτ τ

∂

τ ∂ ,,,,,,

,,, 2,

( ) ( ) ( )∫ ∫ ∫+∫ ∫ ∫− Φ x

L

y

L

z

LV

z y x

x

L

y

L

z

LV

z y x x y z x y z


z y xud vd wd wvuV T wvuk

L L L

z y x,,,,,,,, τ .

Substitution of the previously considered series in the Eqs.(6a) leads to the following form

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

Φ=

Φ N

n

t y

L

z

LnnnI n I n

z y x

N

nnI nnn

I n

y z

wcvct e xsan L L L

z yt e zs ys xs

n

a z y x

1 0133

π

π

( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−×=

ΦΦΦ

N

n

t x

L

z

L I nn I n

z y x

I

x z

d ud wd T wvu Dwcuca L L L

z xd vd wd T wv x D1 0

,,,,,, τ π τ

( ) ( ) ( ) ( ) ( ) ( ) ( ) +∑ ∫ ∫ ∫−×=

ΦΦΦΦ

N

n

t x

L

y

L I nn I nn I n

z y x

I nn

x y

d ud vd T zvu Dvcuct e zsan L L L

y xt e ysn

1 0

,,, τ π

( ) ( ) ( ) ×∫ ∫ ∫+∫ ∫ ∫+ Φ x

L

y

L

z

L I

x

L

y

L

z

L I I

z y x x y z x y z

ud vd wd wvu f ud vd wd wvu I T wvuk L L L

z y x,,,,,,,, 2

, τ

( ) ( )∫ ∫ ∫−× x

L

y

L

z

L I

z y x z y x x y z

ud vd wd wvu I T wvuk L L L

z y x

L L L

z y xτ ,,,,,,

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

Φ

=

Φ N

n

t y

L

z

L

nnnV nV n

z y x

N

n

nV nnn

V n

y z

wcvct e xsan

L L L

z yt e zs ys xs

n

a z y x

1 01

33

π

π

( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−×=

ΦΦ

N

n

t x

L

z

LV nn

z y x

V

x z

d ud wd T wvu Dwcucn L L L

z xd vd wd T wv x D

1 0

,,,,,, τ π

τ

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

ΦΦΦΦ

N

n

t x

L

y

LV nnV nn

z y x

V nnV n

x y

d ud vd T zvu Dvcuct e zsn L L L

y xt e ysa

1 0

,,, τ π

( ) ( ) ( ) ×∫ ∫ ∫+∫ ∫ ∫+× ΦΦ x

L

y

L

z

LV

x

L

y

L

z

LV V

z y x

V n

x y z x y z

ud vd wd wvu f ud vd wd wvuV T wvuk L L L

z y xa ,,,,,,,,

2

, τ

( ) ( )∫ ∫ ∫−× x

L

y

L

z

LV

z y x z y x x y z

ud vd wd wvuV T wvuk L L L

z y x

L L L

z y xτ ,,,,,, .

We determine coefficients anΦρ by using orthogonality condition on the scale of heterostructure.

The condition gives us possibility to obtain relations to calculate anΦρ for any quantity N of termsof considered series. In the common case equations for the required coefficients could be written

as


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13

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−==

ΦΦ

N

n

t L L

n

y

n yn

x

N

n I n

I n z y x x y

ycn

L ys y L xc

Lt e

n

a L L L

10 0 0165

222

122

2212

1

π π π

( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ +−∫

−+×=

ΦΦΦ

N

n

t L

n

L

I nn

z

n I

I n x z

xs xd e xd yd zd zcn

L zs zT z y x D

n

a

10 002

22

11

2,,,

π τ τ

π

( )[ ] ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫

−+−

−++ Φ y z

L L

n

z

n I nn

x

x xd yd zd zcn

L zs zT z y x D yc xc

n

L L

0 0

12

,,,21122 π π

( )( ) ( )[ ] ( ) ( )[ ]∑ ∫ ∫ ∫

+−+

−+−×=

ΦΦΦ

N

n

t L L

n

y

nn

x

n

I n

x y

I n

I n

x y

ycn

L ys y xc

n

L xs x

n

a

Ld

Ln

ea

1 0 0 022

122

2122

1

π π π τ

τ

} ( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫ ∫

+−+∫ −+=

ΦΦ

ΦΦ

N

n

t L

n

x

I n

I n L

I n I n y

x z

xcn

Le

n

ad e xd yd zd T z y x D yc L

1 0 033

0

12

1,,,21

π τ

π τ τ

( )} ( ) ( )[ ] ( ) ( ) ( )[ ]∫

+−∫

−++ z y L

n

z

I I

L

n

y

nn zcn

LT z y xk t z y x I yc

n

L ys y xs x

0,

2

0

12

,,,,,,12 π π

( )} ( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫ ∫ +−

−+−+

=Φ

Φ N

n

t L L

n

y

n

x

n I n

I n

n

x y

ycn

L xc

n

L xs xe

n

a xd yd zd zs z

1 0 0 033

12

12

1

π π τ

π

( )} ( ) ( )[ ] ( ) ( ) ×∑+∫

−++=

Φ N

n

I n L

I n

z

nnn

a xd yd zd t z y x I T z y xk zc

n

L zs z ys y

z

133

0

1,,,,,,1

2 π π

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

+−

−+

−+× Φt L L L

n

z

n

y

nn

x

n I n

x y z

zcn

L yc

n

L ys y xc

n

L xs xe

0 0 0 0

12

12

12 π π π

τ

( )} ( ) xd yd zd z y x f zs z I n ,,Φ+

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−==

ΦΦ

N

n

t L L

n

y

n yn

x

N

nV n

V n z y x x y

yc

n

L ys y L xc

L

t e

n

a L L L

10 0 0165

222

12

2

221

2

1

π π π

( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ +−∫

−+×=

ΦΦΦ

N

n

t L

n

L

V nn

z

nV

V n x z

xs xd e xd yd zd zcn

L zs zT z y x D

n

a

10 002

22

11

2,,,

π τ τ

π

( )[ ] ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫

−+−

−++ Φ y z

L L

n

z

nV nn

x

x xd yd zd zcn

L zs zT z y x D yc xc

n

L L

0 0

12

,,,21122 π π

( )( ) ( )[ ] ( ) ( )[ ]∑ ∫ ∫ ∫

+−+

−+−×=

ΦΦΦ

N

n

t L L

n

y

nn

x

n

V n

x y

V n

V n

x y

ycn

L ys y xc

n

L xs x

n

a

Ld

Ln

ea

1 0 0 022

122

2122

1

π π π τ

τ

} ( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫ ∫

+−+∫ −+=

ΦΦ

ΦΦ

N

n

t L

n

x

V n

V n L

V nV n y

x z

xcn

Le

n

ad e xd yd zd T z y x D yc L

1 0 033

0

12

1,,,21

π τ

π τ τ

( )} ( ) ( )[ ] ( ) ( ) ( )[ ]∫

+−∫

−++ z y L

n

z

V V

L

n

y

nn zc

n

LT z y xk t z y xV yc

n

L ys y xs x

0,

2

0

12

,,,,,,12 π π

( )} ( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫ ∫

+−

−+−+=

ΦΦ

N

n

t L L

n

y

n

x

nV n

V n

n

x y

ycn

L xc

n

L xs xe

n

a xd yd zd zs z

1 0 0 033

12

12

1

π π τ

π


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14

( )} ( ) ( )[ ] ( ) ( ) ×∑+∫

−++=

Φ N

n

V n L

V n

z

nnn

a xd yd zd t z y xV T z y xk zc

n

L zs z ys y

z

133

0

1,,,,,,1

2 π π

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

+−

−+

−+× Φt L L L

n

z

n

y

nn

x

nV n

x y z

zcn

L yc

n

L ys y xc

n

L xs xe

0 0 0 0

12

12

12 π π π

τ

( )} ( ) xd yd zd z y x f zs z V n ,,Φ+ .

It should be noted, that we consider analytical algorithm of calculation of concentrations of do-

pant and radiation defects. We use only two simplifications: (i) we consider approximate using of

Arrhenius law for analytical calculation of appropriate integrals; (ii) we consider so approxima-

tions of initial distributions, which give a possibility to calculate appropriate integrals analytical-ly. Summation of considered series after calculation of integrals could be done by using any soft,

which gives a possibility of the summation.

3.Discussion

In this section we used calculated in the previous sections relations for analysis the spatio-tem-poral distributions of concentrations of infused and implanted dopants. During analysis of spatialdistributions of concentrations of dopants it has been shown, that inhomogeneity of heterostruc-

ture gives us possibility to increase sharpness of p-n-heterojunctions both single and frameworktransistors. Figs. 2 show spatial distributions of concentrations of dopants in direction, which isperpendicular to interface between epitaxial layer substrate. Increasing of number of curve cor-

responds to increasing of difference between values of dopant diffusion coefficient in layers of

heterostructure under condition, when value of dopant diffusion coefficient in epitaxial layer islarger, than value of dopant diffusion coefficient in substrate. We calculate distributions of con-

centrations of dopants for larger value of dopant diffusion coefficient in doped area in comparisonwith undoped one.

Fig.2a. Distributions of concentration of infused dopant in heterostructure from Figs. 1 and 2


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15

x

0.0

0.5

1.0

1.5

2.0

C ( x , Θ

)

23

4

1

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

Epitaxial layer Substrate

Fig.2b. Distributions of concentration of implanted dopant. Curves 1 and 3 corresponds to annealing time Θ

= 0.0048( L x2+ L y

2+ L z2)/ D0. Curves 2 and 4 corresponds to annealing time Θ = 0.0057( L x

2+ L y2+ L z

2)/ D0

One can find from the Figs.2 show, that in homogeneity of heterostructure leads to increasingsharpness of p-n-junctions and homogeneity of distribution of concentration of dopant. However

in this situation one shall optimize annealing of dopant and/or radiation defects during manufac-

turing p-n-junctions and transistors. Reasons of this necessity is following. If annealing time ofinfused dopant is small, dopant have no enough time to achieve interface between layers of hete-

rostructure. In this situation one can not find any changing of distribution of concentration of do-pant with presents of interface between layers of hterostructure. If annealing time is too large,

distribution of concentration of dopant became too homogenous. We determine optimal value of

annealing time framework recently introduced criterion [17,23-31]. To use the criterion one shall

approximate real distribution of concentration of dopant by step-wise function ψ ( x, y, z). After thatwe determine required optimal value of annealing time by minimization the following mean-squared error

( ) ( )[ ]∫ ∫ ∫ −Θ= x y z L L L

z y x

xd yd zd z y x z y xC L L L

U 0 0 0

,,,,,1

ψ . (8)

In this case we obtain optimal value of annealing time, which corresponds to minimal differencebetween real distribution of concentration of dopant and required step-wise approximation of the

concentration.

Dependences of optimal values of annealing time of dopant are presented on Figs. 3. Curve 1 is

the dependence of dimensionless optimal annealing time on the relation a/L and ξ = γ = 0 for equalto each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the

dependence of dimensionless optimal annealing time on value of parameter ε for a/L=1/2 and ξ = γ = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of parameter ξ for a/L =1/2 and ε = γ = 0. Curve 4 is the dependence of dimensionless optimal annealing time onvalue of parameter γ for a/L=1/2 and ε = ξ = 0. It is known, that radiation defects after ion implan-tation should be annealed. In the ideal case annealing time of radiation defects should be so, thatdopant during the annealing should be achieves interface between layers of heterostructure. If the

dopant has not enough time to achieve the interface during the annealing, it is practicably to use

additional annealing of dopant. Dependences of optimal values of the above additional annealing


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16

of dopant are presented on the Fig. 3b. Necessity of annealing of radiation defects leads to smallervalues of optimal annealing time of implanted dopant in comparison with the optimal annealing

time of infused dopant. It is known, that diffusion type of doping did not leads to any radiationdamage of materials of heterostructure. However ion implantation gives us possibility gives us

possibility to decrease mismatch-induced stress in heterostructure [32].

0.0 0.1 0.2 0.3 0.4 0.5

a/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. Dependences of dimensionless optimal annealing time for doping by diffusion

0.0 0.1 0.2 0.3 0.4 0.5

a/L, ξ, ε, γ

0.00

0.04

0.08

0.12

Θ

D 0

L - 2

3

2

4

1

Fig.3b. Dependences of dimensionless optimal annealing time for doping by ion implantation

4.CONCLUSIONS

In this paper we introduce an approach to manufacture more compact logical elements based on

bipolar transistor. The approach based on fabrication a heterostructure with required configura-tion, doping by diffusion or ion implantation of required parts of heterostructure and optimal an-nealing of dopant and/or radiation defects. At the same time we consider an analytical approach

to model technological processes, which gives us possibility to do without crosslinking solutionsat the interfaces between the layers of the heterostructure.


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17

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 and educational fellowship for scientific research of Government of Russian

and of Nizhny Novgorod State University of Architecture and Civil Engineering.

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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 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 105 published papers inarea 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 52 published papers in area of her researches.

an approach to decrease dimentions of logical elements based on bipolar transistor

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