optimization of technological process to decrease dimensions of circuits xor, manufectured based on...

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International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.6, No.1, January 2016

DOI:10.5121/ijfcst.2016.6101 1

OPTIMIZATION OF TECHNOLOGICAL PROCESS TO

DECREASE DIMENSIONS OF CIRCUITS XOR, MANU-

FECTURED BASED ON FIELD-EFFECT HETEROTRAN-SISTORS

E.L. Pankratov1, E.A. Bulaeva

1,2

1Nizhny 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, Russia

A BSTRACT

The paper describes an approach of increasing of integration rate of elements of integrated circuits. The

approach has been illustrated by example of manufacturing of a circuit XOR. Framework the approach one

should manufacture a heterostructure with specific configuration. After that several special areas of the

heterostructure should be doped by diffusion and/or ion implantation and optimization of annealing of do-

pant and/or radiation defects. We analyzed redistribution of dopant with account redistribution of radiation

defects to formulate recommendations to decrease dimensions of integrated circuits by using analytical

approaches of modeling of technological process.

K EYWORDS

Circuits XOR; increasing of density of elements; optimization of technological process.

1. INTRODUCTION

One of intensively solving problems of solid state electronics is improvement of frequency cha-racteristics of electronic devices and their reliability. Another intensively solving problem is in-

creasing of integration rate of integrated circuits with decreasing of their dimensions [1-9]. Tosolve these problems they were used searching materials with higher values of charge carriers

motilities, development new and elaboration existing technological approaches [1-14]. In the

present paper we consider circuit XOR from [15]. Based on recently considered approaches [16-23] we consider an approach to decrease dimensions of the circuit. The approach based on manu-

facturing a heterostructure, which consist of a substrate and an epitaxial layer. The epitaxial layer

manufactured with several sections. To manufacture these sections another materials have beenused. These sections have been doped by diffusion or ion implantation. The doping gives a possi-

bility to generate another type of conductivity ( p or n). After finishing of manufacturing of thecircuit XOR these sections will be used by sources, drains and gates (see Fig. 1). Dopant and rad-iation defects should be annealed after finishing the dopant diffusion and the ion implantation.

Our aim framework the paper is analysis of redistribution of dopant and redistribution of radiation

defects to prognosis technological process. The accompanying aim of the present paper is devel-opment of analytical approach for prognosis technological processes with account all requiredinfluenced factors.


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2

2. METHOD OF SOLUTION

We solve our aim by calculation and analysis distribution of concentrations of dopants in spaceand time. The required distribution has been determined by solving the second Fick's law in thefollowing form [24,25]

( ) ( ) ( ) ( )

+

+

=

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

Fig. 1. Structure of epitaxial layer. View from top


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3

( )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)

Here the function C ( x, y, z,t ) describes the distribution of concentration of dopant in space and

time. DС describes distribution the dopant diffusion coefficient in space and as a function of tem-perature of annealing. Dopant diffusion coefficient will be changed with changing of materials of

heterostructure, heating and cooling of heterostructure during annealing of dopant or radiation

defects (with account Arrhenius law). Dependences of dopant diffusion coefficient on coordinatein heterostructure, temperature of annealing and concentrations of dopant and radiation defects

could be written as [26-28]

( ) ( )

( )

( ) ( )

( )

++

+=

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 ) describes dependences of dopant diffusion coefficient on coordinateand temperature of annealing T . Function P ( x, y, z,T ) describes the same dependences of the limit

of solubility of dopant. The parameter γ is integer and usually could be varying in the followinginterval γ ∈[1,3]. The parameter describes quantity of charged defects, which interacting (in aver-age) with each atom of dopant. Ref.[26] describes more detailed information about dependence of

dopant diffusion coefficient on concentration of dopant. Spatio-temporal distribution of concen-tration of radiation vacancies described by the function V ( x, y, z,t ). The equilibrium distribution of

concentration of vacancies has been denoted as V *. It is known, that doping of materials by diffu-

sion did not leads to radiation damage of materials. In this situation ζ 1= ζ 2=0. We determine spa-tio-temporal distributions of concentrations of radiation defects by solving the following system

of equations [27,28]

( )( )

( )( )

( )( ) ×−

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂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 z

t z y x I T z y x D

zt 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−

∂

∂

∂

∂+× .

Boundary and initial conditions for these equations 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 ,


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4

( )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 . We denote spatio-temporal distribution of concentration of radiation interstitials as I ( x, y, z,t ). Dependences of the diffusion coefficients of point radiation defects on coordinate and

temperature have been denoted as D ρ ( x, y, z,T ). The quadric on concentrations terms of Eqs. (4)

describes generation divacancies and diinterstitials. Parameter of recombination of point radiationdefects and parameters of generation of simplest complexes of point radiation defects have been

denoted as the following functions k I ,V ( x, y, z,T ), k I , I ( x, y, z,T ) and k V ,V ( x, y, z,T ), respectively.

Now let us calculate distributions of concentrations of divacancies Φ V ( x, y, z,t ) and diinterstitialsΦ I ( x, y, z,t ) in space and time by solving the following system of equations [27,28]

( )( )

( )( )

( )+

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 zt z y x

T 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

∂

∂

∂

∂ .

Boundary and initial conditions for these equations 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)

The functions DΦρ ( x, y, z,T ) describe dependences of the diffusion coefficients of the above com-plexes of radiation defects on coordinate and temperature. The functions k I ( x, y, z,T ) and k V ( x, y, z,

T ) describe the parameters of decay of these complexes on coordinate and temperature.

To describe physical processes they are usually solving nonlinear equations with space and time

varying coefficients. In this situation only several limiting cases have been analyzed [29-32]. Oneway to solve the problem is solving the Eqs. (1), (4), (6) by the Bubnov-Galerkin approach [33]

after appropriate transformation of these transformation. To determine the spatio-temporal distri-bution of concentration of dopant we transform the Eq.(1) to the following integro- differential

form

( ) ( ) ( ) ( )

( ) ×∫ ∫ ∫

++=∫ ∫ ∫

t y

L

z

L L

x

L

y

L

z

L z y x y z x y z V

wv xV

V

wv xV T wv x Dud vd wd t wvuC

L L L

z y x

02*

2

2*1

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

τ ς

τ ς


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5

( )( )

( )( )

( )( )

×∫ ∫ ∫

++

+×

t x

L

z

L L

z y x z T z y xP

w yuC T w yu D

L L

z yd

x

wv xC

T wv xP

wv xC

0 ,,,

,,,1,,,

,,,

,,,

,,,1

γ

γ

γ

γ τ ξ τ

∂

τ ∂ τ ξ

( ) ( )

( )

( )( ) ×∫ ∫ ∫+

++×

t x

L

y

L L

z x x y

T zvu D L L

z xd

y

w yuC

V

w yuV

V

w yuV

02

*

2

2*

1,,,

,,,,,,,,,1 τ

∂

τ ∂ τ ς

τ ς

( ) ( )

( )( )( )

( )+

+

++×

y x L L

y xd

z

zvuC

T z y xP

zvuC

V

zvuV

V

zvuV τ

∂

τ ∂ τ ξ

τ ς

τ ς

γ

γ ,,,

,,,

,,,1

,,,,,,1

2*

2

2*1

( )∫ ∫ ∫+ x

L

y

L

z

L z y x x y z

ud vd wd wvu f L L L

z y x,, . (1a)

Now let us determine solution of Eq.(1a) by Bubnov-Galerkin approach [33]. To use the ap-

proach we consider solution of the Eq.(1a) as the following series

( ) ( ) ( ) ( ) ( )∑= =

N

nnC nnnnC t e zc yc xcat z y xC 00 ,,, .

Here ( ) ( )2220

22exp −−−

++−= z y xC nC

L L Lt Dnt e π , cn( χ ) =cos (π n χ / L χ ). Number of terms N in the

series is finite. The above series is almost the same with solution of linear Eq.(1) (i.e. for ξ =0)and averaged dopant diffusion coefficient D0. Substitution of the series into Eq.(1a) leads to thefollowing 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 xP1

2*

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 ysucnV

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

( )∫ ∫ ∫× x

L

y

L

z

L x y z

ud vd wd wvu f ,, ,


6/20


6

where sn( χ ) = sin (π n χ / L χ ). We used condition of orthogonality to determine coefficients an in theconsidered series. The coefficients an could be calculated for any quantity of terms N . In thecommon case the relations could be written as

( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫

×

∑+−=∑−==

t L L L N

n nC nnnnC L

z y N

n nC

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 0 121 65

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

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

( )

++

∑+×

=*1

1

,,,1

,,,1,,,

V

z y xV

T z y xPe zc yc xcaT z y x D

N

nnC 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

212

2 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

nn

xnnnn

nC L xcn

L xs x zs yc xcn

aT z y x DV

z y xV

1*1 1,,,

,,,

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 zcn

L zs z ycn

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


7/20


7

( ) ( )[ ] ( ) ( )

( ) ( ) ×

+

++

−+×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

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

−++×t L L L

nnn

y

nnnC nC n

x y z

zc ys xc

n

L xs x xced e xd yd zd zc

0 0 0 0

21

π

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,,4

2α β

α

β ,

where ( ) ( ) ( ) ( ) ( ) ( )

( )∫ ∫ ∫ ∫ ×

++=

t L L L

nnnnC

z y

n

x y z

V

z y xV

V

z y xV zc yc xse

n

L L

0 0 0 02*

2

2*12

,,,,,,12

2

τ ς

τ ς τ

π

ξ α

( )

( ) ( ) ( )[ ] ( ) ( )[ ] ×+

−+

−+×n

L Ld xd yd zd zc

n

L zs z yc

n

L ys y

T z y xP

T z y x D z x

n

z

nn

y

n

L

2211

,,,

,,,

π

ξ τ

π π

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

( ) ( ) ( )[ ] ×∫ ∫ ∫ ∫

−−

−+×t L L L

n

z

n

L

nn

x

nnnC

x y z

zcn

L zs z

T z y xP

T z y x D zc xc

n

L xs x xce

0 0 0 0

1,,,

,,,1

π π τ

( )

( )

( ) ( )

( ) ( ) ×+

++×

n

L Ld xd yd ys zd

V

z y xV

V

z y xV

T z y xP

T z y x D y xn

L

22*

2

2*12

2,,,,,,

1,,,

,,,

π

ξ τ

τ ς

τ ς

( ) ( ) ( ) ( ) ( )

( )

( ) ( )

( ) ×∫ ∫ ∫ ∫

++×

t L L L L

nnnnC

x y z

V

z y xV

V

z y xV

T z y xP

T z y x D zs yc xce

0 0 0 0 2*

2

2*1

,,,,,,1

,,,

,,,2

τ ς

τ ς τ

( ) ( )[ ] ( ) ( )[ ] τ π π

d xd yd zd ycn

L ys y xc

n

L xs x

n

y

nn

x

n

−+

−+× 11 , ( ) ×∫=t

nC

z y

n en

L L

02

2τ

π β

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

( )∫ ∫ ∫ ×

++

−+× x y z L

L L

nn

y

nnn

V

z y xV

V

z y xV zc yc

n

L ys y yc xs

0 0 02*

2

2*1

,,,,,,112

τ ς

τ ς

π


8/20


9/20


9

We determine spatio-temporal distributions of concentrations of point defects as the same series

( ) ( ) ( ) ( ) ( )∑==

N

nnnnnn

t e zc yc xcat z y x1

0,,, ρ ρ ρ .

Parameters an ρ should be determined in future. Substitution of the series into Eqs.(4a) leads to the

following results

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

N

n

t y

L

z

L I nnnI

z y x

N

nnI 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 0133

,,,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

( ) ( ) ( ) ( ) ( ) ( ) ( )×∫ ∫ ∫ ∑−

∑× ==

x

L

y

L

z

L

N

nnnnnI

z y x z y x

N

nnI nnnnI

x y zwcvcuca L L L

z y x

L L L

z y x

ud vd wd t ewcvcuca 1

2

1

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

x

L

y

L

z

L I

N

nV I nV nnnnV nI

x y z

ud vd wd wvu f ud vd wd T vvuk t ewcvcucat e ,,,,,1

,

z y x L L L z y x×

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

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

N

nV I nV nnnnV nI

x y z

ud vd wd wvu f ud vd wd T vvuk t ewcvcucat e ,,,,,1

,

z y x L L L z y x× .

We used orthogonality condition of functions of the considered series framework the heterostruc-

ture to calculate coefficients an ρ . The coefficients an could be calculated for any quantity of terms N . 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


10/20


10

( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ +−∫

−+×=

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 02

0

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

2112

2

2,,,12

π 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 0212

2212

22

2

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 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 z

1 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

I n y

nn x

n

x y z

T z y x f ycn

L ys y xcn

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 x

n

a

L

d e xd yd zd zc

n

L zs zT z y x D

1 0 02

0

2

2

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


11/20


11

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++

−++−=

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 0212

2212

22

2

1

π π π

( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫

+−+−∫ −×=

N

n

L

n

x

xnV nV

L

nV V n

x z

xc

n

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

1 0

2

0

12

2

2,,,21

π

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 z

1 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

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

L zs z L yc

n

Ln

z

n zn

y

−++

−+ 122

2122 π π

, ( )×∫=t

n

x

n e

n L 022

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 Len 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


12/20


12

( )[ ] ( ) ( )[ ] ( )[ ] ( ) ×∫ ∫ −

−++

−++ 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 e

n

L L Ld xd yd

n

z y x

ρ

π

τ 65

222

−× , ( ) ( )[ ] ( )[ ]∫ ∫

+−+

−+= x y L L

n

y

yn

x

nnIV yc

n

L L xc

n

L xs x

0 0

12

2

1

π 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

22,,,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 −−= 2

32 ,

2

2

348 bb y A −+= , ( ) 22

22

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 2

1

nI nI nV nI nV λ χ δ δ γ −× ,

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 distributions of concentrations of simplest complexes of radiation defects in spaceand time as the following functional series

( ) ( ) ( ) ( ) ( )∑=Φ=

Φ

N

nnnnnn

t e zc yc xcat z y x1

0,,, ρ ρ ρ .

Here anΦρ are the coefficients, which should be determined. Let us previously transform the Eqs.

(6) to the following integro-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

+×ΦΦ

t 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

zvuτ 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

( ) ( ) ( )

×∫ ∫ ∫

Φ

=∫ ∫ ∫Φ Φt y

L

z

L

V

V

x

L

y

L

z

LV

z y x y z x y zd vd wd x

wv x

T wv x Dud vd wd t wvu L L L

z y x

0

,,,,,,,,, τ ∂

τ ∂

( ) ( )

( )∫ ∫ ∫ ×+∫ ∫ ∫ Φ

+× ΦΦ

t 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 ∂


13/20


13

( )( ) ( ) −∫ ∫ ∫+

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 D

1 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

LnnnV nV n

z y x

N

nnV nnn

V n

y z

wcvct e xsan L L L

z yt e zs ys xs

n

a z y x

1 0133

π

π

( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−× = ΦΦ N

n

t x

L

z

LV nn

z y x

V

x zd 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 used orthogonality condition of functions of the considered series framework the heterostruc-

ture to calculate coefficients anΦρ . The coefficients anΦρ could be calculated for any quantity of

terms N . In the common case equations for the required coefficients could be written as

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−==

Φ

Φ 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

1 0 0 0165

222

122

2212

1

π π π


14/20


14

( ) ( ) ( )[ ] ( ) ( ){∑∫ ∫ +−∫

−+×=

ΦΦ

Φ 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

1 0 002

22

11

2,,,

π τ τ

π

( )[ ] ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫

−+−

−++ Φ

y z L L

n

z

n I nn

x

x xd yd zd zc

n

L zs zT z y x D yc xc

n

L L

0 0

1

2

,,,2112

2 π π

( )( ) ( )[ ] ( ) ( )[ ]∑ ∫ ∫ ∫

+−+

−+−×=

ΦΦ

Φ

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

} ( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫ ∫

+−+∫ −+=

Φ

Φ

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 zc

n

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

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

+−

−+

−+× Φ

t 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

+

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−==

Φ

Φ N

n

t L L

n

y

n yn

x

N

nV n

V n z y x x y

ycn

L ys y L xc

Lt e

n

a L L L

1 0 0 0165

222

122

2212

1

π π π

( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ +−∫

−+×=

ΦΦ

Φ 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

1 0 002

22

11

2,,,

π τ τ

π

( )[ ] ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫

−+−

−++ Φ

y z L L

n

z

nV nn

x

x xd yd zd zc

n

L zs zT z y x D yc xc

n

L L

0 0

12

,,,21122 π π

( )( ) ( )[ ] ( ) ( )[ ]∑ ∫ ∫ ∫

+−+

−+−×=

ΦΦ

Φ

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

} ( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫ ∫

+−+∫ −+=

Φ

Φ

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


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15

( )} ( ) ( )[ ] ( ) ( ) ( )[ ]∫

+−∫

−++ 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

( )} ( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫ ∫

+−

−+−+=

Φ

Φ N

n

t L L

n

y

n

x

nV n

V n

n

x y

yc

n

L xc

n

L xs xe

n

a xd yd zd zs z

1 0 0 033

1

2

1

2

1

π π

τ

π

( )} ( ) ( )[ ] ( ) ( ) ×∑+∫

−++=

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

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

+−

−+

−+× Φ

t 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 zV n

,,Φ

+ .

3. DISCUSSION

In this section we analyzed redistribution of dopant with account redistribution of radiation de-fects. The analysis shown, that presents of interface between materials of heterostructure gives a

possibility to increase absolute value of gradient of concentration of dopant outside of enrichedare by the dopant (see Figs. 2 and 3). At the same time homogeneity of concentration of dopant inenriched area increases (see Figs. 2 and 3). The effects could be find, when dopant diffusion coef-

ficient in the doped area is larger, than in the nearest areas. Otherwise absolute value of gradientof concentration of dopant decreases outside enriched by the dopant area (see Fig. 4). However

the decreasing could be partially or fully compensated by using high doping of materials. Thehigh doping leads to significant nonlinearity of diffusion of dopant. To increase compactness of

the considered circuits XOR it is attracted an interest the first relation between values of dopantdiffusion coefficient.

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

dered 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


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16

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. Spatial distributions of infused dopant concentration in the considered heterostructure. 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 z2

)/ D0. Curves 1 and 2 corresponds to homogenous sample. Curves 3 and 4 corres-ponds to the considered heterostructure. Difference between values of dopant diffusion coefficient in layers

of heterostructure increases with increasing of number of curves

x

0.00000

0.00001

0.00010

0.00100

0.01000

0.10000

1.00000

C ( x ,

Θ )

f C ( x)

L /40 L /2 3 L /4 L x0

1

2

Substrate

Epitaxial layer 1

Epitaxial layer 2

Fig.3. Implanted dopant distributions in heterostructure in heterostructure with two epitaxial layers (solid

lines) and with one epitaxial layer (dushed lines) for different values of annealing time. Difference between

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

curves

Increasing of annealing time leads to acceleration of diffusion. In this situation one can find in-

creasing quantity of dopant in materials near doped sections. If annealing time is small, the do-pant can not achieves nearest interface between layers of heterostructure. These effects are shown

by Figs. 5 and 6. We used recently introduced criterion [16-23] to estimate compromise value of

annealing time. Framework the criterion we approximate real distribution of concentration of do-

pant by idealized step-wise distribution ψ ( x, y, z), which would be better to use for minimizationdimensions of elements of the considered circuit XOR [19-26]. Farther the required compromiseannealing time has been calculated by minimization the following mean-squared error


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( ) ( )[ ]∫ ∫ ∫ −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)

C ( x ,

Θ )

0 L x

2

13

4

Fig. 4. Distributions of concentrations of infused dopant in the considered heterostructure. Curve 1 is the

idealized distribution of dopant. Curves 2-4 are the real distributions of concentrations of dopant for differ-

ent values of annealing time for increasing of annealing time with increasing of number of curve

x

C ( x ,

Θ )

1

23

4

0 L

Fig. 5. Distributions of concentrations of implanted dopant in the considered heterostructure. Curve 1 is the

idealized distribution of dopant. Curves 2-4 are the real distributions of concentrations of dopant for differ-

ent values of annealing time for increasing of annealing time with increasing of number of curve

We analyzed optimal value of annealing time. The analysis shows, that optimal value of anneal-ing time for ion type of doping is smaller, than optimal value of annealing time for diffusion type

of doping. It is known, that ion doping of materials leads to radiation damage of doped materials.In this situation radiation defects should be annealed. The annealing leads to the above difference

between optimal values of annealing time of dopant. The annealing of implanted dopant is neces-sary in the case, when the dopant did not achieved nearest interface between layers of heterostruc-ture during annealing of radiation defects.


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18

It should be noted, that using diffusion type of doping did not leads to radiation damage of mate-

rials. However radiation damage of materials during ion doping gives a possibility to decreasemismatch-induced stress in heterostructure [34].

4. CONCLUSIONS

In this paper we introduced an approach to decrease dimensions of a circuit XOR. The approach

based on optimization of manufacturing field-effect heterotransistors, which includes into itselfthe considered circuit.

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 ofNizhni Novgorod, educational fellowship for scientific research of Government of Russian, edu-

cational fellowship for scientific research of Government of Nizhny Novgorod region of Russiaand 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. 2012-2015 Full Doctor course in Radi-

ophysical Department of Nizhny Novgorod State University. Since 2015 E.L. Pankratov is an Associate

Professor of Nizhny Novgorod State University. He has 135 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-


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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 90 published papers in area of her researches.

optimization of technological process to decrease dimensions of circuits xor, manufectured based on...

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