MODIFICATION OF DOPANT CONCENTRATION PROFILE IN A FIELD-EFFECT HETEROTRANSISTOR FOR MODIFICATION ENERGY BAND DIAGRAM

8/18/2019 MODIFICATION OF DOPANT CONCENTRATION PROFILE IN A FIELD-EFFECT HETEROTRANSISTOR FOR MODIFICATION …

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Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 1, March 2016

DOI:10.5121/msej.2016.3101 1

MODIFICATION OF DOPANT CONCENTRA-

TION PROFILE IN A FIELD-EFFECT HETERO-

TRANSISTOR FOR MODIFICATION ENERGY B AND DIAGRAM

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

In this paper we consider an approach of manufacturing more compact field-effect heterotransistors. The

approach based on manufacturing a heterostructure, which consist of a substrate and an epitaxial layer

with specific configuration. After that several areas of the epitaxial layer have been doped by diffusion or

ion implantation with optimized annealing of dopant and /or radiation defects. At the same time we intro-

duce an approach of modification of energy band diagram by additional doping of channel of the transis-

tors. We also consider an analytical approach to model and optimize technological process.

K EYWORDS

Modification of profile of dopant; decreasing of dimension of field-effect transistor; modification of energy

band diagram

1. INTRODUCTION

Development of solid state electronic leads to increasing performance of the appropriate electron-ic devices [1-11]. At the same time one can find increasing integration rate of integrated circuits[1-3,5,7]. In this situation dimensions of elements of integrated circuits decreases. To increase

performance of solid-state electronics devices are now elaborating new technological processes of

manufacturing of solid state electronic devices. Another ways to increase the performance areoptimization of existing technological processes and determination new materials with higher

values of charge carriers mobilities. To decrease dimensions of elements of integrated circuits

they are elaborating new and optimizing existing technological processes. Framework this paperwe introduce an approach to decrease dimensions of field-effect heterotransistors. At the same

time we introduce an approach of modification of energy band diagram for regulation of transportof charge carriers. The approach based on manufacturing a field-effect transistor in the hetero-

structure from Fig. 1. The heterostructure consists of a substrate and an epitaxial layer. They arehave been considered four sections in the epitaxial layer. The sections have been doped by diffu-

sion or ion implantation. Left and right sections will be considered in future as source and drain,respectively. Both average sections became as channel of transistor. Using one section instead

two sections leads to simplification of structure of transistor. However using additional dopedsection framework the channel of the considered transistor gives us possibility to modify energyband diagram in the structure. After finishing of the considered doping annealing of dopant and/or

radiation defects should be done. Several conditions for achievement of decreasing of dimensions

of the field-effect transistor have been formulated.


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2

2. METHOD OF SOLUTION

To solve our aim we determine spatio-temporal distributions of concentrations of dopants. We

calculate the required distributions by solving the second Fick's law in the following form [12,13]

( ) ( ) ( ) ( )

+

+

= 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. Heterostructure, which consist of a substrate and an epitaxial layer.

View from side

( )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 andtime. D 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 [14-16]

( ) ( )

( )( ) ( )

( )

++

+=

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 coordinate

and 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-


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3

age) with each atom of dopant. Ref.[14] 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 systemof equations [15,16]

( )( )

( )( )

( )( ) ×−

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂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 z

t 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 ,

( )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 andtemperature 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 [15,16]

( )( )

( )( )

( )+

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

∂

∂

∂

∂ .


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4

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 determine spatio-temporal distribution of concentration of dopant we transform the Eq.(1) tothe 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

( )( )

( )( )

( )( )

×∫ ∫ ∫

++

+×

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 [17]. 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 0

0,,, .

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 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

n

nnnC L vc xsaT wv x DV

wv xV

V

wv xV

T wv xP 12*

2

2*1,,,

,,,,,,1

,,,

τ ς

τ ς

ξ γ


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5

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

∫ ∫ ∫ ×

∑+−×

=

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

n

nC nnn L

1

2*

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 ,, ,

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

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 zc

n

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

x

nnnn

nC

L xc

n

L xs x zs yc xc

n

aT z y x D

V

z y xV

1*1

1,,,,,,

π

τ ς


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6

( ) ( )[ ] ( ) ( ) ( )[ ]∑ ∫ ×

−++

−+×=

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

,,11

π 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

,,11π 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 zcn

L

zs z n y

n,,,1

,,,,,,11 2*

2

2*1 γ

ξ τ ς

τ ς π

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

−++×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

nnC xd yd zd z y x f zc yc xca

0 0 0

2

,,42 α 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


7/19


7

( )

( ) ( ) ( )[ ] ( ) ( )[ ] ×+

−+

−+×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

zc

n

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*1 22

,,,,,,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 02*

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

( ) ( ) ( )[ ] ( ) ( ) ( ) ×∫ ∫ ∫+

−+×t L L

nnnC

z x

n

z

n L

x y

ys xcen

L Ld xd yd zd zc

n

L zs zT z y x D

0 0 02

22

1,,, τ π

τ π

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

( ) ×∫

++

−+× z L

n Ln

x

n

V

z y xV

V

z y xV zcT z y x D xc

n

L xs x

02*

2

2*1

,,,,,,1,,,1

τ ς

τ ς

π

( ) ( )[ ] ( ) ( ) ( )[ ] ×∫ ∫

−++

−+×t L

n

x

nnC

y x

n

z

n

x

xcn

L xs xe

n

L Ld xd yd zd zc

n

L zs z

0 02

12

1π

τ π

τ π

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

( )∫ ∫ ×

++

−+× y z

L L

Ln

y

nn

V

z y xV

V

z y xV T z y x D yc

n

L ys y xc

0 02*

2

2*1

,,,,,,1,,,1

τ ς

τ ς

π

( ) ( ) ( ) 652222 nt e L L Ld xd yd yc zd zsnC z y xnn

π τ −× .

The same approach could be used for calculation parameters an for different values of parameter

γ . However the relations are bulky and will not be presented in the paper. Advantage of the ap-proach is absent of necessity to join dopant concentration on interfaces of heterostructure.The same Bubnov-Galerkin approach has been used for solution the Eqs.(4). Previously we trans-

form the differential equations to the following integro- differential form

( ) ( ) ( ) +∫ ∫ ∫∂

∂=∫ ∫ ∫t y

L

z

L I

z y

x

L

y

L

z

L z y x y z x y z

d vd wd x

wv x I T wv x D L L z yud vd wd t wvu I

L L L z y x

0

,,,,,,,,, τ τ

( ) ( )

( ) ( )×∫ ∫ ∫−∫ ∫ ∫∂

∂+

x

L

y

L

z

LV I

z y x

t x

L

z

L I

z x x y z x z

t wvu I T wvuk L L L

z y xd ud wd

x

w yu I T w yu D

L L

z x,,,,,,

,,,,,,

,0

τ τ


8/19


8

( ) ( )

( ) ×−∫ ∫ ∫∂

∂+×

z y x

t x

L

y

L I

y x L L L

z y xd ud vd T zvu D

z

zvu I

L L

y xud vd wd t wvuV

x y0

,,,,,,

,,, τ τ

( ) ( ) ( )∫ ∫ ∫+∫ ∫ ∫× x

L

y

L

z

L I

z y x

x

L

y

L

z

L I I

x y z x y z


z y xud vd wd t wvu I T wvuk ,,,,,,,, 2

, (4a)

( ) ( ) ( )

+∫ ∫ ∫∂

∂=∫ ∫ ∫

t y

L

z

LV

z y

x

L

y

L

z

L z y x y z x y z

d vd wd x

wv xV T wv x D

L L

z yud vd wd t wvuV

L L L

z y x

0

,,,,,,,,, τ

τ

( ) ( ) ( )

×∫ ∫ ∫∂

∂+∫ ∫ ∫

∂

∂+

t x

L

y

L y x

t x

L

z

LV

z x x y x z z

zvuV

L L

y xd ud wd

x

w yuV T w yu D

L L

z x

00

,,,,,,,,,

τ τ

τ

( ) ( ) ( ) ( ) −∫ ∫ ∫−× x

L

y

L

z

LV I

z y x

V

x y z

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

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

,τ

( ) ( ) ( )∫ ∫ ∫+∫ ∫ ∫− x

L

y

L

z

LV

z y x

x

L

y

L

z

LV V

z y x x y z x y z

ud vd wd wvu f L L L z y xud vd wd t wvuV T wvuk

L L L z y x ,,,,,,,, 2, .

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 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

N

n

V 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


9/19


9

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

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

( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ +−∫

−+×=

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

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 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 zcn

L

LT z y xk ycn

L

ys y L xs x 0 0, 122,,,12222 π 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


10/19


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 yc

n

L ys y xc

n

L xs x xd yd

1 0 0 0

,,,11

π 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 0212

2212

22

2

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 zc1 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 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,

12

2

2,,,12

2

2

π 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

.


11/19


11

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 xcn

L

xs x LT z y xk t e 0 0 0, 21222,,,2 π 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 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 yc

n

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

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

× , 224 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

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 space

and 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


12/19


12

( ) ( ) ( )

×∫ ∫ ∫ 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

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 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

yw 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

L nnnI n I n z y x

N

n nI nnn

I 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

( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−×=

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


13/19


13

( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−×=

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 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 ofterms 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 L

t en

a L L L1 0 0 01

65

222

122

2212

1π π π

( ) ( ) ( )[ ] ( ) ( ){∑∫ ∫ +−∫

−+×=

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

12

,,,21122 π π

( )( ) ( )[ ] ( ) ( )[ ]∑ ∫ ∫ ∫

+−+

−+−×=

ΦΦ

Φ

N

n

t L L

n

y

nn

x

n

I n

x y

I n

I n

x y

yc

n

L ys y xc

n

L xs x

n

a

L

d

Ln

ea

1 0 0 022

12

2

21

22

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


14/19


14

( )} ( ) 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

} ( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫ ∫

+−+∫ −+ = ΦΦΦΦ

N

n

t L

n x

V nV 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

01

2

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

( )} ( ) ( ) ( )[ ] ( )[ ]∑ ∫ ∫ ∫

+−

−+−+=

Φ

Φ 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

( )} ( ) ( )[ ] ( ) ( ) ×∑+∫

−++=

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 the present paper we analyzed redistribution of infused and implanted dopants in heterostruc-ture, which have been presented on Fig. 1. The analysis has been done by using relations, calcu-

lated in the previous section. First of all we consider situation, when dopant diffusion coefficient

in doped area is larger, than in nearest areas. It has been shown, that in this case distribution of

concentration of dopant became more compact in comparison with wise versa situation (see Figs.2 and 3 for diffusion and ion types of doping). In the wise versa situation (when dopant diffusion

coefficient in doped area is smaller, than in nearest areas) we obtained spreading of distribution ofconcentration of dopant. In this case outside of doping material one can find higher spreading in

comparison with wise versa situation (see Fig. 4).

It should be noted, that properties of layers of multilayer structure varying in space: varying lay-

ers with larger and smaller values of the diffusion coefficient. In this situation with account re-sults, which shown on Figs. 2-4, layers of the considered heterostructure with smaller value of


15/19


15

dopant should has smaller level of doping in comparison with nearest layers to exclude changing

of type of doping in the nearest layers. For a more complete doping of each section and at thesame to decrease diffusion of dopant into nearest sections it is attracted an interest optimization of

annealing of dopant and /or radiation defects. Let us optimize annealing of dopant and/or radia-

tion defects by using recently introduce criterion [18-23]. Framework the criterion we approx-imate real distribution of concentration of dopant by idealized step-wise distribution of concentra-

tion ψ ( x, y, z) (see Figs. 5 and 6 for diffusive or ion types of doping). Farther we determine optim-al annealing time by minimization of mean-squared error

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

The considered direction perpendicular to the interface between epitaxial layer substrate. Differ-

ence between values of dopant diffusion coefficient in layers of heterostructure increases with

increasing of number of curves

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.3. 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 z

2)/ D0. Curves 2 and 4 corres-

ponds to annealing time Θ= 0.0057( L x2+ L y

2+ L z

2)/ D0. Curves 1 and 2 corresponds to homogenous

sample. Curves 3 and 4 corresponds to the considered heterostructure. Difference between values

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


16/19


16

Fig.4. 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

( ) ( )[ ]∫ ∫ ∫ −Θ= x L y

L z L

z y x

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

U 0 0 0

,,,,,1

ψ . (8)

Dependences of optimal annealing time are shown on Figs. 7 and 8 for diffusion and ion types of

doping. It should be noted, that after finishing of ion doping of material it is necessary to anneal

radiation defects. In the ideal case after finishing of the annealing dopant achieves nearest inter-face between materials of heterostructure. If the dopant has no time to achieve the interface, it is

practicably to additionally anneal the dopant. The Fig. 8 shows dependences of the exactly addi-tional annealing time of implanted dopant. Necessity of annealing of radiation defects leads to

smaller values of optimal annealing time for ion doping in comparison with values of optimalannealing time for diffusion type of doping. Using diffusion type of doping did not leads to so

large damage in comparison with damage during ion type of doping.

C ( x ,

Θ )

0 L x

2

13

4

Fig. 5. 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 concentra-

tions of dopant for different values of annealing time for increasing of annealing time with in-creasing of number of curve


17/19


17

x

C ( x ,

Θ )

1

23

4

0 L

Fig. 6. 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 concentra-tions of dopant for different values of annealing time for increasing of annealing time with in-

creasing of number of curve

Dependences of optimal annealing time are shown on Figs. 7 and 8 for diffusion and ion types ofdoping. It should be noted, that after finishing of ion doping of material it is necessary to annealradiation defects. In the ideal case after finishing of the annealing dopant achieves nearest inter-

face between materials of heterostructure. If the dopant has no time to achieve the interface, it is

practicably to additionally anneal the dopant. The Fig. 8 shows dependences of the exactly addi-tional annealing time of implanted dopant. Necessity of annealing of radiation defects leads to

smaller values of optimal annealing time for ion doping in comparison with values of optimal

annealing time for diffusion type of doping. Using diffusion type of doping did not leads to solarge damage in comparison with damage during ion type of doping.

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.7. 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 epitax-

ial layer a/L and ξ = γ = 0 for equal to each other values of dopant diffusion coefficient in all partsof 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 para-meter ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the dependence of the considered annealing time onparameter γ for a/L=1/2 and ε = ξ = 0


18/19


18

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.8. 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 epi-

taxial layer a/L and ξ = γ = 0 for equal to each other values of dopant diffusion coefficient in allparts of heterostructure. Curve 2 is the dependence of the considered annealing time on the para-

meter ε for a/L=1/2 and ξ = γ = 0. Curve 3 is the dependence of the considered annealing time onthe parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the dependence of the considered annealingtime on parameter γ for a/L=1/2 and ε = ξ = 0

4. CONCLUSIONS

In this paper we introduced an approach to manufacture of field-effect of transistors which gives a

possibility to decrease their dimensions. The decreasing based on manufacturing the transistors ina heterostructure with specific configuration, doping of required areas of the heterostructure by

diffusion or ion implantation and optimization of annealing of dopant and/or radiation defects.Framework the approach we introduce an approach of additional doping of channel. The addi-

tional doping gives us possibility to modify energy band diagram. We also consider an analyticalapproach to model and optimize technological process.

ACKNOWLEDGEMENTS

This work is supported by the agreement of August 27, 2013 02..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 Russia, educa-

tional fellowship for scientific research of Government of Nizhny Novgorod region of Russia andeducational fellowship for scientific research of Nizhny Novgorod State University of Architec-

ture 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 Radio physics, from 1999 to 2001 it was master course in Ra-

diophysics with specialization in Statistical Radio-physics, 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 Radio physi-

cal 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-

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 Radio-physical Department of Nizhny

Novgorod State University. She has 90 published papers in area of her researches.

MODIFICATION OF DOPANT CONCENTRATION PROFILE IN A FIELD-EFFECT HETEROTRANSISTOR FOR MODIFICATION ENERGY BAND DIAGRAM

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