on increasing of density of elements in a multivibrator on bipolar transistors

International Journal of Computational Science, Information Technology and Control Engineering

(IJCSITCE) Vol.3, No.3, July 2016

DOI : 10.5121/ijcsitce.2016.3302 13

ON INCREASING OF DENSITY OF ELEMENTS IN A

MULTIVIBRATOR ON BIPOLAR TRANSISTORS

E.L. Pankratov1 and E.A. Bulaeva

1,2

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

Russia 2

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

street, Nizhny Novgorod, 603950, Russia

ABSTRACT

In this paper we consider an approach to increase density of elements of a multivibrator on bipolar tran-

sistors. The considered approach based on manufacturing a heterostructure with necessity configuration,

doping by diffusion or ion implantation of required areas to manufacture the required type of conductivity

(p or n) in the areas and optimization of annealing of dopant and/or radiation defects to manufacture more

compact distributions of concentrations of dopants. We also introduce an analytical approach to progno-

sis technological process.

KEYWORDS

Bipolar transistors, multivibrator, increasing of density of elements of multivibrator

1. INTRODUCTION

In the present time actual questions of solid state electronic devices are increasing of density of

elements of integrated circuits (in this situation it is necessary to decrease dimensions of these

elements), performance and reliability [1-9]. In this paper we consider an approach to manufac-

ture more compact multivibrator based on bipolar heterotransistors. Framework the approach we

consider a heterostructure with two layers, which consist of a substrate and an epitaxial layer (see

Fig. 1). The epitaxial layer includes into itself several sections manufactured by using another

materials (see Fig. 1). These sections have been doped by diffusion or ion implantation to obtain

required type of conductivity (n or p). The doping gives a possibility to manufacture bipolar tran-

sistor framework the considered heterostructure with account the Fig. 1. After that we consider

annealing of dopant and/or radiation defects. The annealing should be optimized to manufacture

more compact distributions of concentrations of dopant. Main aim of the present paper is deter-

mination of conditions, which correspond to increasing of compactness and at the same time to

increasing of homogeneity of distribution of concentration of dopant in enriched area.

2. METHOD OF SOLUTION

We determine spatio-temporal distribution of concentration of dopant by solving the following

boundary problem

( ) ( ) ( ) ( )

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂

z

tzyxCD

zy

tzyxCD

yx

tzyxCD

xt

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

with boundary and initial conditions



14

( )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).

Fig. 1a. Structure of a transistor multivibrator. View from top

n p n n p n

Substrate

Epitaxial layer

Fig. 1b. Heterostructure with two layers and sections in the epitaxial layer, which has been used for manu-

facture a transistor multivibrator. View from side

Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant; T is the temperature

of annealing; DС is the dopant diffusion coefficient. Value of dopant diffusion coefficient de-

pends on properties of materials of heterostructure, speed of heating and cooling of heterostruc-

ture (with account Arrhenius law). Dependences of dopant diffusion coefficient on parameters

could be approximated by the following relation [10-12]



15

( ) ( )( )

( ) ( )( )

++

+=

2*

2

2*1

,,,,,,1

,,,

,,,1,,,

V

tzyxV

V

tzyxV

TzyxP

tzyxCTzyxDD

LCςςξ

γ

γ

, (3)

where DL (x,y,z,T) is the spatial (due to presents several layers in heterostructure) and tempera-

ture (due to Arrhenius law) dependences of dopant diffusion coefficient; P (x, y,z,T) is the limit

of solubility of dopant; parameter γ depends on properties of materials and could be integer in the

following interval γ ∈[1,3] [10]; V (x,y,z,t) is the spatio-temporal distribution of concentration of

radiation vacancies; V* is the equilibrium distribution of concentration of vacancies.

Concentrational dependence of dopant diffusion coefficient has been described in details in [10].

Using diffusion type of doping did not generation radiation defects. In this situation ζ1= ζ2= 0.

We determine the spatio-temporal distributions of concentrations of radiation defects by solving

the following system of equations [11,12]

( ) ( ) ( ) ( ) ( ) ( ) ×−

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂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 −

∂

∂

∂

∂+×


( )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; I (x,y,z,t) is the spatio-temporal distribution of concentration of radiation intersti-

tials; Dρ(x,y,z,T) are the diffusion coefficients of point radiation defects; terms V2(x,y,z,t) and

I2(x,y,z,t) correspond to generation divacancies and diinterstitials; kI,V(x,y,z,T) is the parameter of

recombination of point radiation defects; kI,I(x,y,z,T) and kV,V(x,y,z,T) are the parameters of gener-

ation of simplest complexes of point radiation defects.

We determine the spatio-temporal distributions of concentrations of divacancies ΦV (x,y,z,t) and

dinterstitials ΦI (x,y,z,t) by solving the following system of equations [11,12]

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxI

I

I

I

I

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

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

tzyxTzyxD

zIII

I

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

,,,,,, 2

, −+

Φ+ Φ

∂

∂

∂

∂ (6)



16

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxV

V

V

V

V

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

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

tzyxTzyxD

zVVV

V

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

,,,,,, 2

,−+

Φ+ Φ ∂

∂

∂

∂


( )0

,,,

0

=∂

Φ∂

=xx

tzyxρ,

( )0

,,,=

∂

Φ∂

= xLxx

tzyxρ,

( )0

,,,

0

=∂

Φ∂

=yy

tzyxρ,

( )0

,,,=

∂

Φ∂

= yLyy

tzyxρ,

( )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)

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

kI(x,y,z,T) and kV (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

LL

x

L

y

L

z

Lzyx y zx y z V

wvxV

V

wvxVTwvxDudvdwdtwvuC

LLL

zyx

02*

2

2*1

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

τς

τς

( )( )

( )( )

( )×∫ ∫ ∫+

+×

t x

L

z

LL

zxzy x z y

wyuCTwyuD

LL

zx

LL

zyd

x

wvxC

TwvxP

wvxC

0

,,,,,,

,,,

,,,

,,,1

∂

τ∂τ

∂

τ∂τξ

γ

γ

( ) ( )( )

( )( )

( )∫ ∫ ∫ ×+

+

++×

t x

L

y

LL

yx x y

TzvuDLL

yxd

TzyxP

wyuC

V

wyuV

V

wyuV

02*

2

2*1 ,,,,,,

,,,1

,,,,,,1 τ

τξ

τς

τς

γ

γ

( ) ( )( )

( )( )

( )×+

+

++×

zyxLLL

zyxd

z

zvuC

TzyxP

zvuC

V

zvuV

V

zvuVτ

∂

τ∂τξ

τς

τς

γ

γ ,,,

,,,

,,,1

,,,,,,1

2*

2

2*1

( )∫ ∫ ∫×x

L

y

L

z

Lx y z

udvdwdwvuf ,, . (1a)

We determine solution of the above equation by using Bubnov-Galerkin approach [13]. Frame-

work the approach we determine solution of the Eq.(1a) as the following series

( ) ( ) ( ) ( ) ( )∑==

N

nnCnnnnC

tezcycxcatzyxC0

0 ,,, ,

where ( ) ( )[ ]222

0

22exp −−− ++−=zyxCnC

LLLtDnte π , cn(χ) = cos (π n χ/Lχ). 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 into

Eq.(1a) leads to the following result

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

∑+−=∑

==

t y

L

z

L

N

nnCnnnnC

N

nnCnnn

C

y z TwvxPewcvcxcatezsysxs

n

azyx

0 1132 ,,,

1γ

γ ξτ

π

( ) ( )( )

( ) ( ) ( ) ( ) ( ) −∑

++×

=zy

N

nnCnnnnCL

LL

zydewcvcxsanTwvxD

V

wvxV

V

wvxV

12*

2

2*1 ,,,,,,,,,

1 τττ

ςτ

ς



17

( ) ( )( )

( ) ( ) ( ) ( )∫ ∫ ∫

×

∑+

++−

=

t x

L

z

L

N

mmCmmmmC

zx x z

ewcycucaV

wyuV

V

wyuV

LL

zx

0 12*

2

2*1 1,,,,,,

1γ

ττ

ςτ

ς

( )( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫−∑

×=

t x

L

y

LL

yx

N

nnCnnnnCL

x y

TzvuDLL

yxdewcysucanTwyuD

TwyuP 01

,,,,,,,,,

ττξ

γ

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

( )×

++

∑+×

=2*

2

2*11

,,,,,,1

,,,1

V

zvuV

V

zvuVezcvcuca

TzvuP

N

nnCnnnnC

τς

τςτ

ξ γ

γ

( ) ( ) ( ) ( ) ( )∫ ∫ ∫+∑×=

x

L

y

L

z

Lzyx

N

nnCnnnnC

x y z

udvdwdwvufLLL

zyxdezsvcucan ,,

1

ττ ,

where sn(χ) = sin (π n χ/Lχ). We determine coefficients an by using orthogonality condition of

terms of the considered series framework scale of heterostructure. The condition gives us possi-

bility to obtain 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

nnCnnnnCL

N

nnC

nCzyxx y z

TzyxPezcycxcaTzyxDte

n

aLLL

0 0 0 0 1165

222

,,,1,,,

γ

γ ξτ

π

( ) ( )( )

( ) ( ) ( ) ( )[ ] ×∑

−+

++×

=

N

nn

y

nnn

nCzyyc

n

Lysyycxs

n

a

V

zyxV

V

zyxVLL

12*

2

2*1212

,,,,,,1

2 π

τς

τς

π

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

∑−

−+×=

t L L L N

nnCnnnnCn

z

nnCn

x y z

ezcycxcadxdydzdzcn

Lzszezc

0 0 0 0 1

1γ

ττπ

τ

( )( )

( ) ( )( )

( )

++

++

+×*12*

2

2*1

,,,1

,,,,,,1,,,1

,,, V

zyxV

V

zyxV

V

zyxVTzyxD

TzyxPL

τς

τς

τς

ξγ

( )( )

( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )∑ ×

−+

−+

+

=

N

nnnnn

z

nn

x

n

znC zcysxczcn

Lzszxc

n

Lxsx

n

La

V

zyxV

122*

2

2211

2

,,,

πππ

τς

( ) ( ) ( ) ( ) ( )( )∫ ∫ ∫ ∫ ×

∑+×−×

=

t L L L N

nnCnnnnC

yx

nC

x y z

TzyxPezcycxca

LLdxdydzde

0 0 0 0 12 ,,,

12 γ

γ ξτ

πττ

( ) ( )( )

( ) ( ) ( ) ( )[ ] ×∑

−+

++×

=

N

nn

x

nn

nC

Lxc

n

Lxsxxc

n

a

V

zyxV

V

zyxVTzyxD

1*12*

2

2 1,,,,,,

1,,,π

τς

τς

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

−++

−+×=

N

n

L

n

x

nnCn

y

nnn

x

xcn

Lxsxdxdydzdeyc

n

Lysyzsyc

1 0

11π

ττπ

( ) ( )[ ] ( ) ( )[ ] ( )∫ ∫

−+

−+×y z

L L

n

z

nn

y

nxdydzdzyxfzc

n

Lzszyc

n

Lysy

0 0

,,11ππ

.

As an example for γ = 0 we obtain

( ) ( )[ ] ( ) ( )[ ] ( ) ( )∫ ∫ ∫

×+

−+

−+=x y zL L L

x

nn

y

nn

y

nnCn

Lxsxydzdzyxfzc

n

Lzszyc

n

Lysya

0 0 0

,,11πππ

( )[ ]} ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ){

∫ ∫ ∫ ∫ +

−+−×t L L L

nn

y

nLnnnn

x y z

ysyzcn

LzszTzyxDzcycxs

nxdxc

0 0 0 0

1,,,22

1π

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

( ) ×

+

++

−+ xdydzdzcTzyxPV

zyxV

V

zyxVyc

n

Lnn

y

,,,1

,,,,,,11

2*

2

2*1 γ

ξτς

τς

π



18

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

−+

−++×t L L L

n

y

nnnn

y

nnnCnC

x y z

zcn

Lzszzcysxc

n

Lxsxxcede

0 0 0 0

121ππ

τττ

( )( )

( ) ( )( )

+

++

+× τ

τς

τς

ξγ

dxdydzdV

zyxV

V

zyxV

TzyxPTzyxD

L 2*

2

2*1

,,,,,,1

,,,1,,,

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

−+

−++t L L L

Ln

y

nnn

x

nnnC

x y z

TzyxDycn

Lysyycxc

n

Lxsxxce

0 0 0 0

,,,11ππ

τ

( )( )

( )( )

( )1

65

222

*12*

2

2

,,,,,,1

,,,12

−

−

++

+×

n

LLLdxdydzd

V

zyxV

V

zyxV

TzyxPzs zzz

nπ

ττ

ςτ

ςξ

γ.

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

( ) ( ) ( ) ( )∫ ∫ ∫+±−=x y zL L L

nnnnn

n

n

nCxdydzdzyxfzcycxca

0 0 0

2 ,,42

αβα

β,

where ( ) ( )( )

( )( ) ( )

( )∫ ∫ ∫ ∫ ×

++=

t L L LL

nn

zy

n

x y z

V

zyxV

V

zyxV

TzyxP

TzyxDycxs

n

LL

0 0 0 02*

2

2*12

,,,,,,1

,,,

,,,2

2

τς

τς

π

ξα

( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ×+

−+

−+×n

LLdexdydzdzc

n

Lzszyc

n

Lysyzc zx

nCn

z

nn

y

nn 2211

π

ξττ

ππ

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

( )( ) ( )[ ] ×∫ ∫ ∫ ∫

−−

−+×t L L L

n

z

n

L

nn

x

nnnC

x y z

zcn

Lzsz

TzyxP

TzyxDysxc

n

Lxsxxce

0 0 0 0

1,,,

,,,21

ππτ

( ) ( ) ( )( )

( ) ( ) ( ) ×∫ ∫ ∫+

++×

t L L

nnnC

yx

n

x y

ycxcen

LLdxdydzd

V

zyxV

V

zyxVzc

0 0 022*

2

2*12

,,,,,,1 τ

π

ξτ

τς

τς

( )( )

( )( ) ( )

( )( ) ( )[ ] ( ){∫ ×

−+

++×

zL

nn

x

n

L

nysxc

n

Lxsx

V

zyxV

V

zyxV

TzyxP

TzyxDzs

02*

2

2*1 1,,,,,,

1,,,

,,,2

π

τς

τς

( )[ ] τπ

dxdydzdycn

Ly n

y

−+× 1 , ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∫ ∫=t L L L

LnnnnC

zy

n

x y z

TzyxDzcycxsen

LL

0 0 0 02

,,,22

τπ

β

( ) ( )( )

( ) ( )[ ] ( )[ ] ( ) ×

+−

−+

++× ysyyc

n

Lzdzc

n

Lzsz

V

zyxV

V

zyxVnn

y

n

z

n11

,,,,,,1

2*

2

2*1ππ

τς

τς

( ) ( ) ( ) ( )[ ] ( ) ( ) ( )∫ ∫ ∫ ∫

++

−++×t L L L

nnn

x

nnnC

zxx y z

V

zyxVzcysxc

n

Lxsxxce

n

LLdxdyd

0 0 0 0*12

,,,121

2

τς

πτ

πτ

( )( )

( ) ( ) ( )[ ] ( )∫ ×+

−+

+

t

nC

yx

n

z

nLe

n

LLdxdydzdzc

n

LzszTzyxD

V

zyxV

022*

2

22

1,,,,,,

τπ

τπ

τς

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

∫ ∫ ∫ ×

++

−+

−+×x y zL L L

n

y

nn

x

n

V

zyxV

V

zyxVyc

n

Lysyxc

n

Lxsx

0 0 02*

2

2*1

,,,,,,111

τς

τς

ππ

( ) ( ) ( ) ( ) ( ) 65222,,,2 nteLLLdxdxcydyczdTzyxDzs nCzyxnnLn πτ −× .

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. Advan-



19

tage of the approach is absent of necessity to join dopant concentration on interfaces of hetero-structure.

Equations of the system (4) have been also solved by using Bubnov-Galerkin approach. Pre-viously we transform the differential equations to the following integro- differential form

( ) ( ) ( )×+∫ ∫ ∫

∂

∂=∫ ∫ ∫

zx

t y

L

z

LI

zy

x

L

y

L

z

Lzyx

LL

zxdvdwd

x

wvxITwvxD

LL

zyudvdwdtwvuI

LLL

zyx

y zx y z 0

,,,,,,,,, τ

τ

( ) ( ) ( ) ( )−∫ ∫ ∫

∂

∂+∫ ∫ ∫

∂

∂×

t x

L

y

LI

yx

t x

L

z

LI

x yx z

dudvdz

zvuITzvuD

LL

yxdudwd

x

wyuITwyuD

00

,,,,,,

,,,,,, τ

ττ

τ

( ) ( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫−x

L

y

L

z

LII

zyx

x

L

y

L

z

LVI

zyx x y zx y z

TwvukLLL

zyxudvdwdtwvuVtwvuITwvuk

LLL

zyx,,,,,,,,,,,, ,,

( ) ( )∫ ∫ ∫+×x

L

y

L

z

LI

zyx x y z

udvdwdwvufLLL

zyxudvdwdtwvuI ,,,,,2 (4a)

( ) ( ) ( )×+∫ ∫ ∫

∂

∂=∫ ∫ ∫

zx

t y

L

z

LV

zy

x

L

y

L

z

Lzyx

LL

zxdvdwd

x

wvxVTwvxD

LL

zyudvdwdtwvuV

LLL

zyx

y zx y z 0

,,,,,,,,, τ

τ

( )( )

( )( )

−∫ ∫ ∫∂

∂+∫ ∫ ∫

∂

∂×

t x

L

y

LV

yx

t x

L

z

LV

x yx z

dudvdz

zvuVTzvuD

LL

yxdudwd

x

wyuVTwyuD

00

,,,,,,

,,,,,, τ

ττ

τ

( ) ( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫−x

L

y

L

z

LVV

zyx

x

L

y

L

z

LVI

zyx x y zx y z

TwvukLLL

zyxudvdwdtwvuVtwvuITwvuk

LLL

zyx,,,,,,,,,,,, ,,

( ) ( )∫ ∫ ∫+×x

L

y

L

z

LV

zyx x y z

udvdwdwvufLLL

zyxudvdwdtwvuV ,,,,,2 .

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

( ) ( ) ( ) ( ) ( )∑==

N

nnnnnn

tezcycxcatzyx1

0 ,,, ρρρ ,

where anρ are not yet known coefficients. Substitution of the series into Eqs.(4a) leads to the fol-

lowing results

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

N

n

t y

L

z

LInnnInI

zyx

N

nnInnn

nI

y z

dvdwdTwvxDzcyceaLLL

zytezsysxs

n

azyx

1 0133

,,, ττπ

π

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

N

nnnI

zyx

N

n

t x

L

z

LInnnInnI

zyx

nzsa

LLL

yxdudwdTwyuDzcxceysa

LLL

zxxs

x z11 0

,,,π

ττπ

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

∑−∫ ∫ ∫×

=

x

L

y

L

z

L

N

nnInnnnI

zyx

t x

L

y

LInnnI

x y zx y

tewcvcucaLLL

zyxdudvdTzvuDycxce

2

10

,,, ττ

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

x

L

y

L

z

L

N

n

N

nnnnnVnInnnnI

zyx

II

x y z

wcvcucatewcvcucaLLL

zyxudvdwdTvvuk

1 1, ,,,

( ) ( ) ( )∫ ∫ ∫+×x

L

y

L

z

LI

zyx

VInV

x y z

udvdwdwvufLLL

zyxudvdwdTvvukte ,,,,,

,

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

N

n

t y

L

z

LVnnnVnV

zyx

N

nnVnnn

nV

y z

dvdwdTwvxDzcyceaLLL

zytezsysxs

n

azyx

1 0133

,,, ττπ

π



20

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

N

nnnV

zyx

N

n

t x

L

z

LVnnnVnnV

zyx

nzsa

LLL

yxdudwdTwyuDzcxceysa

LLL

zxxs

x z11 0

,,,π

ττπ

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

∑−∫ ∫ ∫×

=

x

L

y

L

z

L

N

nnVnnnnV

zyx

t x

L

y

LVnnnV

x y zx y

tewcvcucaLLL

zyxdudvdTzvuDycxce

2

10

,,, ττ

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

x

L

y

L

z

L

N

n

N

nnnnnVnInnnnI

zyx

VV

x y z

wcvcucatewcvcucaLLL

zyxudvdwdTvvuk

1 1,

,,,

( ) ( ) ( )∫ ∫ ∫+×x

L

y

L

z

LV

zyx

VInV

x y z

udvdwdwvufLLL

zyxudvdwdTvvukte ,,,,,, .

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

The condition gives us possibility to obtain relations to calculate anρ 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

nynnI

nI

x

N

nnI

nIzyxx y

ycn

LysyLxce

n

a

Lte

n

aLLL

1 0 0 02

165

222

122

2212

1

πτ

ππ

( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫

++−−∫

−+×=

N

n

t L

xn

xnI

y

L

n

z

nI

xz

Lxcn

L

n

a

Ldxdydzdzc

n

LzszTzyxD

1 0 02

0

122

11

2,,,

ππτ

π

( )} ( ) ( ) ( )[ ] ( )[ ] ( ) −∫ ∫ −

−+++y z

L L

nInn

z

nzIndexdydyczdzc

n

LzszLTzyxDxsx

0 0

21122

2,,,2 ττπ

( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++

−++−=

N

n

t L L

n

y

nyn

x

nxnI

nI

z

x y

ycn

LysyLxc

n

LxsxLe

n

a

L 1 0 0 02

122

2122

22

1

ππτ

π

( )[ ] ( ) ( ) ( )[ ] ( ) ×∑ ∫

+−+−∫ −×=

N

n

L

nn

x

xnInI

L

In

xz

xsxxcn

LLteadxdydzdTzyxDzc

1 0

2

0

2122

2,,,21π

τ

( ) ( )[ ] ( ) ( )[ ] ( ) −∫ ∫

+−+

−++×y z

L L

nn

z

zIIn

y

nyxdydzdzszzc

n

LLTzyxkyc

n

LysyL

0 0,

2122

,,,122

2ππ

( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫

−++

−++−=

N

n

L L

n

y

nyn

x

nxnVnInVnI

x y

ycn

LysyLxc

n

LxsxLteteaa

1 0 0

122

2122

2ππ

( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫

−++∫

−++×=

N

n

L

n

x

n

L

n

z

nzVI

xz

xcn

Lxsxxdydzdzc

n

LzszLTzyxk

1 00, 112

22,,,

ππ

( ) ( )[ ] ( ) ( ) ( )[ ]∫ ∫

−++

−+×y z

L L

n

z

nzIn

y

nxdydzdzc

n

LzszLTzyxfyc

n

Lysy

0 0

122

2,,,1ππ

( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−==

N

n

t L L

n

y

nynnV

nV

x

N

nnV

nVzyxx y

ycn

LysyLxce

n

a

Lte

n

aLLL

1 0 0 02

165

222

122

2212

1

πτ

ππ

( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫

++−−∫

−+×=

N

n

t L

xn

xnV

y

L

n

z

nV

xz

Lxcn

L

n

a

Ldxdydzdzc

n

LzszTzyxD

1 0 02

0

122

11

2,,,

ππτ

π

( )} ( ) ( ) ( )[ ] ( )[ ] ( ) −∫ ∫ −

−+++y z

L L

nVnn

z

nzVndexdydyczdzc

n

LzszLTzyxDxsx

0 0

21122

2,,,2 ττπ

( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++

−++−=

N

n

t L L

n

y

nyn

x

nxnV

nV

z

x y

ycn

LysyLxc

n

LxsxLe

n

a

L 1 0 0 02

122

2122

22

1

ππτ

π



21

( )[ ] ( ) ( ) ( )[ ] ( ) ×∑ ∫

+−+−∫ −×=

N

n

L

nn

x

xnVnV

L

Vn

xz

xsxxcn

LLteadxdydzdTzyxDzc

1 0

2

0

2122

2,,,21π

τ

( ) ( )[ ] ( ) ( )[ ] ( ) −∫ ∫

+−+

−++×y z

L L

nn

z

zVVn

y

nyxdydzdzszzc

n

LLTzyxkyc

n

LysyL

0 0,

2122

,,,122

2ππ

( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫

−++

−++−=

N

n

L L

n

y

nyn

x

nxnVnInVnI

x y

ycn

LysyLxc

n

LxsxLteteaa

1 0 0

122

2122

2ππ

( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫

−++∫

−++×=

N

n

L

n

x

n

L

n

z

nzVI

xz

xcn

Lxsxxdydzdzc

n

LzszLTzyxk

1 00,

1122

2,,,ππ

( ) ( )[ ] ( ) ( ) ( )[ ]∫ ∫

−++

−+×y z

L L

n

z

nzVn

y

nxdydzdzc

n

LzszLTzyxfyc

n

Lysy

0 0

122

2,,,1ππ

.

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

( )

−+−

+±

+−=

A

ybyb

Ab

b

Aba nInV

nI

2

3

4

2

3

4

3 444

λγ,

nInI

nInInInInI

nVa

aaa

χ

λδγ ++−=

2

,

where ( ) ( ) ( ) ( )[ ] ( )∫ ∫ ∫

×++

−++=x y zL L L

y

ynn

x

nxnnn

LLysyxc

n

LxsxLTzyxkte

0 0 0,

2212

22,,,2

ππγ ρρρρ

( )[ ]} ( ) ( )[ ] xdydzdzcn

LzszLyc

n

z

nzn

−++−× 122

212π

, ( ) ( ){∫ ∫ ∫ +=t L L

nn

x

n

x y

ysyenL 0 0 0

22

1τ

πδ ρρ

( )[ ] ( ) ( )[ ] ( ) ( )[ ] ×+∫ −

−+

−+y

L

nn

z

nn

y

LdxdxcydzdTzyxDzc

n

Lzszyc

n

L z

πτ

ππρ

2

121,,,1

21

2 0

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

+−−

−++×t L L L

n

z

nn

x

nxn

x y z

zcn

LTzyxDycxc

n

LxsxLe

n 0 0 0 02

122

,,,211222

1

ππτ ρρ

( ) } ( ) ( ) ( )[ ] ( ){∫ ∫ ∫ +

−+++++t L L

nn

x

xnn

z

zn

x y

ysyxcn

LLxsxe

nLdxdydzdLzsz

0 0 02

1222

12

πτ

πτ ρ

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

LLLdxdydzdTzyxDzcyc

n

LL

n

zyxL

nn

y

y

z

ρρπ

τπ 65

222

0

,,,2112

−∫ −

−++ , ( ) ×= tenInIV

χ

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

+−+

−+×x y zL L L

nzVInn

y

yn

x

nnV zszLTzyxkysyycn

LLxc

n

Lxsxte

0 0 0, 2,,,212

21

ππ

( )[ ] xdydzdzcn

Ln

z

−+ 122π

, ( ) ( )[ ] ( ) ( )[ ] {∫ ∫ ∫ ×

−+

−+=x y zL L L

n

y

nn

x

nnzyc

n

Lysyxc

n

Lxsx

0 0 0

11ππ

λ ρ

( ) ( )[ ] ( ) xdydzdTzyxfzcn

Lzs

n

z

n,,,1 ρ

π

−+× , 22

4 nInInInVb χγγγ −= , −−= 2

32

nInInInInVb χδδγγ

nInInVγχδ− , ( ) 22

2 2 nInInVnInInVnInVnInInVb χλλδχδγγλδγ −+−+= , nInInVnInVnI

b λχδδγλ −= 21,

2

2

348 bbyA −+= ,

2

4

2

342

9

3

b

bbbp

−= ,

3

4

2

4132

3

3

54

2792

b

bbbbbq

+−= , −++−−+= 3 323 32 qpqqpqy

43 3bb− .



22

We determine spatio-temporal distributions of concentrations of complexes of radiation defects

in the following form

( ) ( ) ( ) ( ) ( )∑=Φ=

Φ

N

nnnnnn

tezcycxcatzyx1

0 ,,, ρρρ ,

where anΦρ are not yet known coefficients. Let us previously transform the Eqs.(6) to the follow-

ing integro-differential form

( ) ( )( )

+∫ ∫ ∫Φ

=∫ ∫ ∫Φ Φ

t y

L

z

L

I

I

zy

x

L

y

L

z

LI

zyx y zx y z

dvdwdx

wvxTwvxD

LL

zyudvdwdtwvu

LLL

zyx

0

,,,,,,,,, τ

∂

τ∂

( )( )

( )( )

∫ ∫ ∫ ×Φ

+∫ ∫ ∫Φ

+ ΦΦ

t x

L

y

L

I

I

t x

L

z

L

I

I

zx x yx z

dudvdz

zvuTzvuDdudwd

y

wyuTwyuD

LL

zx

00

,,,,,,

,,,,,, τ

∂

τ∂τ

∂

τ∂

( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫+×x

L

y

L

z

LI

zyx

x

L

y

L

z

LII

zyxyx x y zx y z

TwvukLLL

zyxudvdwdwvuITwvuk

LLL

zyx

LL

yx,,,,,,,,, 2

,τ

( ) ( )∫ ∫ ∫+× Φ

x

L

y

L

z

LI

zyx x y z

udvdwdwvufLLL

zyxudvdwdwvuI ,,,,, τ (6a)

( ) ( )( )

+∫ ∫ ∫Φ

=∫ ∫ ∫Φ Φ

t y

L

z

L

V

V

zy

x

L

y

L

z

LV

zyx y zx y z

dvdwdx

wvxTwvxD

LL

zyudvdwdtwvu

LLL

zyx

0

,,,,,,,,, τ

∂

τ∂

( )( )

( )( )

∫ ∫ ∫ ×Φ

+∫ ∫ ∫Φ

+ ΦΦ

t x

L

y

L

V

V

t x

L

z

L

V

V

zx x yx z

dudvdz

zvuTzvuDdudwd

y

wyuTwyuD

LL

zx

00

,,,,,,

,,,,,, τ

∂

τ∂τ

∂

τ∂

( ) ( ) ( ) ×∫ ∫ ∫−∫ ∫ ∫+×x

L

y

L

z

LV

zyx

x

L

y

L

z

LVV

zyxyx x y zx y z

TwvukLLL

zyxudvdwdwvuVTwvuk

LLL

zyx

LL

yx,,,,,,,,, 2

,τ

( ) ( )∫ ∫ ∫+× Φ

x

L

y

L

z

LV

zyx x y z

udvdwdwvufLLL

zyxudvdwdwvuV ,,,,, τ .

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

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

ΦΦ=

ΦN

n

t y

L

z

LInnInIn

zyx

N

nnInnn

In

y z

TwvxDvctexsanLLL

zytezsysxs

n

azyx

1 0133

,,,π

π

( ) ( ) ( ) ( ) ( ) ( ) −∑ ∫ ∫ ∫−×=

ΦΦΦ

N

n

t x

L

z

LInnInnIn

zyx

n

x z

dudwdTwvuDwcucteysnaLLL

zxdvdwdwc

1 0

,,, τπ

τ

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

ΦΦΦ

x

L

y

L

z

LII

N

n

t x

L

y

LInnInnIn

zyx x y zx y

TwvukdudvdTzvuDvcuctezsanLLL

yx,,,,,,

,1 0

τπ

( ) ( ) ( ) ×+∫ ∫ ∫−×zyx

x

L

y

L

z

LI

zyxzyxLLL

zyxudvdwdwvuITwvuk

LLL

zyx

LLL

zyxudvdwdwvuI

x y z

ττ ,,,,,,,,,2

( )∫ ∫ ∫× Φ

x

L

y

L

z

LI

x y z

udvdwdwvuf ,,

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

ΦΦ=

ΦN

n

t y

L

z

LVnnVnVn

zyx

N

nnVnnn

Vn

y z

TwvxDvctexsanLLL

zytezsysxs

n

azyx

1 0133

,,,π

π

( ) ( ) ( ) ( ) ( ) ( ) −∑ ∫ ∫ ∫−×=

ΦΦΦ

N

n

t x

L

z

LVnnVnnVn

zyx

n

x z

dudwdTwvuDwcucteysnaLLL

zxdvdwdwc

1 0

,,, τπ

τ



23

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

ΦΦΦ

x

L

y

L

z

LVV

N

n

t x

L

y

LVnnVnnVn

zyx x y zx y

TwvukdudvdTzvuDvcuctezsanLLL

yx,,,,,, ,

1 0

τπ

( ) ( ) ( ) ×+∫ ∫ ∫−×zyx

x

L

y

L

z

LV

zyxzyxLLL

zyxudvdwdwvuVTwvuk

LLL

zyx

LLL

zyxudvdwdwvuV

x y z

ττ ,,,,,,,,,2

( )∫ ∫ ∫× Φ

x

L

y

L

z

LV

x y z

udvdwdwvuf ,, .

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 terms

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

as

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−=

Φ

=Φ

ΦN

n

t L L

n

y

nyn

In

x

N

nIn

Inzyxx y

ycn

LysyLxc

n

a

Lte

n

aLLL

1 0 0 02

165

222

122

2212

1

πππ

( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ ++−∫

−+×=

Φ

ΦΦ

N

n

t L

xn

y

InL

Inn

z

nI

xz

LxsxLn

adexdydzdzc

n

LzszTzyxD

1 0 02

0

22

11

2,,,

πττ

π

( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ×−∫ ∫

−+−−

+ ΦΦ

x

L L

Inn

z

nIn

n

xL

dexdydzdzcn

LzszTzyxDyc

n

xcL

y z

πττ

ππ

11

2,,,21

2

12

0 0

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

+−+

−+×=

ΦΦΦ

N

n

t L L L

Iyn

y

nn

x

nIn

Inx y z

TzyxDLycn

Lysyxc

n

Lxsxe

n

a

1 0 0 0 02

,,,122

2122 ππ

τ

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

−+

+−+−×=

Φ

N

n

t L L

n

y

nnn

x

Inn

x y

ycn

Lysyxsxxc

n

Ledxdydzdyc

10 0 0

12

12

21ππ

ττ

( ) ( ) ( )[ ] ( ) ( ) ( ){∑ ∫ ∫ +−∫

+−×=

ΦΦ

N

n

t L

nIn

L

nn

z

II

Inxz

xsxexdydzdzszzcn

LTzyxktzyxI

n

a

10 00,

2

331

2,,,,,, τ

ππ

( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ]∫ ∫

+−

+−

−+ Φy z

L L

n

z

Inn

yIn

n

x zcn

LtzyxITzyxkysyyc

n

L

n

axc

n

L

0 033

12

,,,,,,12

12 ππππ

( )} ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−+

−+++=

Φ

ΦN

n

t L L

n

y

nn

x

nIn

In

n

x y

ycn

Lysyxc

n

Lxsxe

n

axdydzdzsz

1 0 0 033

12

12 ππ

τπ

( )[ ] ( ) ( )∫

+−× Φ

zL

Inn

z xdydzdzyxfzszzcn

L

0

,,12π

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−=

Φ

=Φ

ΦN

n

t L L

n

y

nyn

Vn

x

N

nVn

Vnzyxx y

ycn

LysyLxc

n

a

Lte

n

aLLL

1 0 0 02

165

222

122

2212

1

πππ

( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ ++−∫

−+×=

Φ

ΦΦ

N

n

t L

xn

y

VnL

Vnn

z

nV

xz

LxsxLn

adexdydzdzc

n

LzszTzyxD

1 0 02

0

22

11

2,,,

πττ

π

( )( )[ ] ( ) ( ) ( )[ ] ( ) ×−∫ ∫

−+−−

+ ΦΦ

x

L L

Vnn

z

nVn

n

xL

dexdydzdzcn

LzszTzyxDyc

n

xcL

y z

πττ

ππ

11

2,,,21

2

12

0 0

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

+−+

−+×=

ΦΦ

ΦN

n

t L L L

Vyn

y

nn

x

nVn

Vnx y z

TzyxDLycn

Lysyxc

n

Lxsxe

n

a

1 0 0 0 02

,,,122

2122 ππ

τ



24

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

−+

+−+−×=

Φ

N

n

t L L

n

y

nnn

x

Vnn

x y

ycn

Lysyxsxxc

n

Ledxdydzdyc

1 0 0 0

12

12

21ππ

ττ

( ) ( ) ( )[ ] ( ) ( ) ( ){∑ ∫ ∫ +−∫

+−×=

ΦΦ

N

n

t L

nVn

L

nn

z

VV

Vnxz

xsxexdydzdzszzcn

LTzyxktzyxV

n

a

10 00,

2

331

2,,,,,, τ

ππ

( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ]∫ ∫

+−

+−

−+ Φy z

L L

n

z

Vnn

yVn

n

x zcn

LtzyxVTzyxkysyyc

n

L

n

axc

n

L

0 033

12

,,,,,,12

12 ππππ

( )} ( ) ( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−+

−+++=

Φ

ΦN

n

t L L

n

y

nn

x

nVn

Vn

n

x y

ycn

Lysyxc

n

Lxsxe

n

axdydzdzsz

1 0 0 033

12

12 ππ

τπ

( )[ ] ( ) ( )∫

+−× Φ

zL

Vnn

z xdydzdzyxfzszzcn

L

0

,,12π

.

3. DISCUSSION

In this section we analyzed distribution of concentration of dopant, infused (see Fig. 2a) or im-

planted (see Fig. 2b) into epitaxial layer. Annealing time is the same for all curves of these fig-

ures. Increasing of number of curves corresponds to increasing of difference between values of

dopant diffusion coefficients in layers of hetero structure. The figures show, that presents of in-

terface between layers of hetero structure gives a possibility to increase absolute value of gra-

dient of concentration of dopant in direction, which is perpendicular to the considered interface.

A consequence of this result is decreasing of the dimensions of transistors included in the multi-

vibrator. At the same time with increasing of the considered gradient homogeneity of concentra-

tion of dopant in enriched area.

Fig.2a. Distributions of concentration of infused dopant in heterostructure from Figs. 1 in direction, which

is perpendicular to interface between epitaxial layer substrate. Increasing of number of curve corresponds

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 is larger, than value of dopant dif-

fusion coefficient in substrate

To estimate annealing time it is necessary to estimate decreasing of absolute value of gradient of

concentration of dopant near interface between substrate and epitaxial layer with increasing of

annealing time. Decreasing of annealing time leads to manufacturing more inhomogenous distri-

bution of concentration of dopant (see Fig. 3a for diffusion type of doping and Fig. 3b for ion

type of doping). We determine compromise value of annealing time framework recently intro-

duced criterion [14-20]. Frame-work the criterion we approximate real distribution of concentra-

tion of dopant by idealized step-wise function ψ (x,y,z). Further we determine the required an-

nealing time by minimization of mean-square error



25

( ) ( )[ ]∫ ∫ ∫ −Θ=x y zL L L

zyx

xdydzdzyxzyxCLLL

U0 0 0

,,,,,1

ψ . (8)

Dependences of optimal annealing time are shown on Figs. 4. It should be noted, that radiation

defects, generated during ion implantation, should be annealed. In the ideal case after the anneal-

ing the implanted dopant should achieve the interface between layers of heterostructure. If the

dopant has no enough time to achieve the interface, it is attracted an interest to make additional

annealing. The Fig. 4b shows dependences of additional annealing time. Necessity of annealing

of radiation defects leads to smaller value of optimal annealing time of dopant in comparison

with optimal annealing time of dopant during diffusion type of doping. Diffusion type of doping

did not leads to radiation damage of materials. Ion type of doping gives us possibility to decrease

mismatch-induced stress in heterostructure [20].

Fig.2b. Distributions of concentration of implanted dopant in heterostructure from Figs. 1 in direction,

which is perpendicular to interface between epitaxial layer substrate. Curves 1 and 3 corresponds to an-

nealing time Θ = 0.0048(Lx2+Ly

2+Lz

2)/D0. Curves 2 and 4 corresponds to annealing time Θ = 0.0057(Lx

2+

Ly2+Lz

2)/D0. Curves 1 and 2 corresponds to homogenous sample. Curves 3 and 4 corresponds to hetero-

structure under condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than value

of dopant diffusion coefficient in substrate

Fig. 3a. Spatial distributions of dopant in heterostructure after dopant infusion. Curve 1 is idealized distri-

bution of dopant. Curves 2-4 are real distributions of dopant for different values of annealing time. In-

creasing of number of curve corresponds to increasing of annealing time

x

C(x

,Θ)

1

23

4

0 L

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

C(x

,Θ)

0 Lx

2

13

4



26

Fig.3b. Spatial distributions of dopant in heterostructure after ion implantation. Curve 1 is idealized distri-

bution of dopant. Curves 2-4 are real distributions of dopant for different values of annealing time. In-

creasing of number of curve corresponds to increasing of annealing time

Fig.4a. 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

Fig.4b. 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

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

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



27

In this paper prognosis of time varying of spatial distributions of concentrations of infused and

implanted dopants during manufacturing a transistor multivibrator has been done. Several rec-

ommendations to optimize manufacture the heterotransistors have been formulated. Analytical

approach to prognosis diffusion and ion types of doping has been introduced. The approach gives

a possibility to take into account variation in space and time parameters of doped material and at

the same time nonlinearity of mass and heat transport.

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,

educational fellowship for scientific research of Government of Nizhny Novgorod region of Rus-

sia and educational fellowship for scientific research of Nizhny Novgorod State University of

Architecture and Civil Engineering.

REFERENCES

[1] Z. Ramezani, A.A. Orouji. A silicon-on-insulator metal semiconductor field-effect transistor with

an L-shaped buried oxide for high output-power density. Mat. Sci. Sem. Proc. Vol. 19. P. 124-129

(2014).

[2] Ch. Dong, J. Shi, J. Wu, Y. Chen, D. Zhou, Z. Hu, H. Xie, R. Zhan, Zh. Zou. Improvements in

passivation effect of amorphous InGaZnO thin film transistors. Mat. Sci. Sem. Proc. Vol. 20. P. 7-

11 (2014).

[3] D. Fathi, B. Forouzandeh. Accurate analysis of global interconnects in nano-fpgas. Nano. Vol. 4

(3). P. 171-176 (2009).

[4] D. Fathi, B. Forouzandeh, N. Masoumi. New enhanced noise analysis in active mixers in nanoscale

technologies. Nano. Vol. 4 (4). P. 233-238 (2009).

[5] 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. Au-TiBx-n-6H-SiC Schottky barrier diodes: Specific fea-

tures of charge transport in rectifying and nonrectifying contacts. Semiconductors. Vol. 43 (7). P.

897-903 (2009).

[6] A.G. Alexenko, I.I. Shagurin. Microcircuitry (Radio and communication, Moscow, 1990).

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

Moscow, 1991).

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

[9] R. Pal, R. Pandey, N. Pandey, R.Ch. Tiwari. Single CDBA Based Voltage Mode Bistable Multivi-

brator and Its Applications. Circuits and Systems. Vol. 6 (11). P. 237-251 (2015).

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

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

1979, in Russian).

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

Phys. Vol. 61 (2). P. 289-388 (1989).

[13] M.L. Krasnov, A.I. Kiselev, G.I. Makarenko. Integral equations ("Science", Moscow, 1976).

[14] E.L. Pankratov. Dopant diffusion dynamics and optimal diffusion time as influenced by diffusion-

coefficient nonuniformity Russian Microelectronics. 2007. V.36 (1). P. 33-39.

[15] E.L. Pankratov. Redistribution of dopant during annealing of radiative defects in a multilayer

structure by laser scans for production an implanted-junction rectifiers. Int. J. Nanoscience. Vol. 7

(4-5). P. 187–197 (2008).

[16] E.L. Pankratov. Decreasing of depth of implanted-junction rectifier in semiconductor heterostruc-

ture by optimized laser annealing. J. Comp. Theor. Nanoscience. Vol. 7 (1). P. 289-295 (2010).

[17] E.L. Pankratov, E.A. Bulaeva. Optimization of manufacturing of emitter-coupled logic to decrease

surface of chip. International Journal of Modern Physics B. Vol. 29 (5). P. 1550023-1-1550023-12

(2015).

[18] E.L. Pankratov, E.A. Bulaeva. An approach to manufacture of bipolar transistors in thin film struc-

tures. On the method of optimization. Int. J. Micro-Nano Scale Transp. Vol. 4 (1). P. 17-31 (2014).



28

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

ing radiation processing. Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1250035 (2012).

[20] E.L. Pankratov, E.A. Bulaeva. Decreasing of mechanical stress in a semiconductor heterostructure

by radiation processing. J. Comp. Theor. Nanoscience. Vol. 11 (1). P. 91-101 (2014).

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 Radiophysi-

cal Department of Nizhny Novgorod State University. Since 2015 E.L. Pankratov is an Associate Profes-

sor 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.