INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED- HETEROJUNCTION RECTIFIERS

8/19/2019 INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED- HETEROJUNCTION RECTIFIERS

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International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016

DOI : 10.5121/ijitmc.2016.4101 1

INFLUENCEOFO VERLAYERSONDEPTHOF IM-PLANTED-HETEROJUNCTIONR ECTIFIERS

E.L. Pankratov1, E.A. Bulaeva1,2

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

Russia2Nizhny Novgorod State University of Architecture and Civil Engineering , 65 Il'insky

street , Nizhny Novgorod , 603950, Russia

A BSTRACT

In this paper we compare distributions of concentrations of dopants in an implanted-junction rectifiers in a

heterostructures with an overlayer and without the overlayer. Conditions for decreasing of depth of the

considered p-n-junction have been formulated.

K EYWORDS

Heterostructures; implanted-junction rectifiers: overlayers.

1. INTRODUCTION

In the present time integration rate of elements of integrated circuits intensively increasing. At thesame time dimensions of the above elements decreases. To increase integration rate of elements

of integrated circuits and to increase dimensions of the same elements are have been elaborateddifferent approaches [1-10]. In the present paper we consider a heterostructure with two layers: a

substrate and an epitaxial layer. The heterostructure could include into itself a third layer: an

overlayer (see Figs. 1). We assume, that type of conductivity of the substrate (n or p) is known.

The epitaxial layer has been doped by ion implantation to manufacture required type of conduc-tivity ( p or n). Farther we consider annealing of radiation defects. Main aim of the present paper

is comparison two ways of ion doping of the epitaxial layer: ion doping of epitaxial layer and iondoping of epitaxial layer through the overlayer.

Fig. 1a. Heterostructure, which consist of a substrate and an epitaxial layer with initial distribution of

implanted dopant




2

x

D ( x

) D1

D2

0 La

f C ( x)

D0

Substrate

Epitaxial layer

b

Overlayer

Fig. 1b. Heterostructure, which consist of a substrate, an epitaxial layer and an overlayer with initial distri-

bution of implanted dopant

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 [1]

( ) ( ) ( ) ( )

+

+

=

z

t z y xC D

z y

t z y xC D

y x

t z y xC D

xt

t z y xC C C C

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,,,,,,,. (1)

Boundary and initial conditions for the equations are

( )0

,,,

0

=

∂

∂

= x x

t z y xC ,

( )0

,,,=

∂

∂

= x L x x

t z y xC ,

( )0

,,,

0

=

∂

∂

= y y

t z y xC ,

( )0

,,,=

∂

∂

= y L x y

t z y xC ,

( )0

,,,

0

=∂

∂

= z z

t z y xC ,

( )0

,,,=

∂

∂

= z L x z

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

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

defects (with account Arrhenius law). Dependences of dopant diffusion coefficient on coordinate

in heterostructure, temperature of annealing and concentrations of dopant and radiation defectscould be written as [11-13]

( ) ( )

( )( ) ( )

( )

++

+=

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 following




3

interval γ ∈[1,3]. The parameter describes quantity of charged defects, which interacting (in aver-

age) with each atom of dopant. Ref.[11] describes more detailed information about dependence ofdopant 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 ofconcentration 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 [12,13]

( )( )

( )( )

( )( ) ×−

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂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 ,

( )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 diinterstitials

Φ I ( x, y, z,t ) in space and time by solving the following system of equations [12,13]

( )( )

( )( )

( )+

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∂

∂

∂

∂ .

Boundary and initial conditions for these equations are




4

( )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 [14]. 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 ( ) ( )222

0

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

nnnnC 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

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

∫ ∫ ∫ ×

∑+−×

=

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




5

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ×∑

++×

=

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 ,, ,

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 the

common case the relations could be written as

( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫

×

∑+−=∑−

==

t L L L N

nnC nnnnC L

z y N

nnC

nC z y x x y z

e zc yc xcaT z y x D L L

t e

n

a L L L

0 0 0 01

21

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

( ) ( ) ( ) ( ) ( ) ( )[ ] ×−

−+×

222

122 π

τ π

τ π

y x

n

z

nnC nnn

z x L Ld 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

( ) ( )[ ] ( ) ( ) ( )[ ]∑ ∫ ×

−++

−+×=

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 zcn

L

zs z ycn

L

ys y0 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




6

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

∫ ∫ ∫ ∫ ×

−+

−×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 xP

V

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 xcn

L xs x xced e xd yd zd zc

0 0 0 0

21π

τ τ τ

( ) ( )[ ]( )

( ) ( )

( ) ×

++

+

−+×2*

2

2*1

,,,,,,1

,,,11

V

z y xV

V

z y xV

T z y xP zc

n

L zs z

n

y

n

τ ς

τ ς

ξ

π γ

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

−++×t L L

nnn

x

nnnC L

x y

ys yc xcn

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

0 0 0

1,,,π

τ τ

( )[ ] ( ) ( )( )

( )

( )∫

++

+

−+× z L

Lnn

y

V

z y xV

T z y xPT z y x D zs yc

n

L y

02*

2

2

,,,1

,,,1,,,21

τ ς

ξ

π γ

( ) ( )1

65

222

*1

,,, −

−

+ t e

n

L L Ld xd yd zd

V

z y xV nC

z z z

π τ

τ ς .

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

( ) ( ) ( ) ( )∫ ∫ ∫+±−= x y z L L L

nnnnn

n

n

nC xd yd zd z y x f zc yc xca

0 0 0

2,,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 xPT 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 x

n

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 02*

2

2*1

,,,,,,1

,,,

,,,2

τ ς

τ ς τ

( ) ( )[ ] ( ) ( )[ ] τ π π d xd yd zd ycn

L

ys y xcn

L

xs x n

y

nn

x

n

−+

−+× 11 ,

( ) ×∫=t

nC

z y

n en

L L

02

2τ

π β




7

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

( )∫ ∫ ∫ ×

++

−+× 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 zs

nC z y xnn π τ −× .

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

( ) ( )

( ) ×−∫ ∫ ∫∂

∂+×

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


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




8

( ) ( ) ( ) ( ) ( )∑==

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 thefollowing 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 xsna 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

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

( ) ( ) ( )[ ] ( ) ( ){∑ ∫ ∫ +−∫

−+×

=

N

n

t L

n

nI

y

L

nI n

z

n I

x z xs x

na

Ld e xd yd zd zc

n L zs zT z y x D

1 0 02

0

22

112

,,,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




9

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++

−++−=

N

n

t L L

n

y

n yn

x

n x

nI

z

x y

ycn

L ys y L xc

n

L xs x L

n

a

L 1 0 0 02

122

2122

22

1

π π π

( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫

+−+−∫ −×=

N

n

L

n

x

xnI nI

L

nI I n

x z

xcn

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

1 0

2

0

122

2,,,21π

τ τ

( )} ( ) ( )[ ] ( ) ( )[ ]∫ ∫

+−+

−+++ y z

L L

n

z

z I I n

y

n yn 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 xc

n

L xs x xd yd

1 0 0 0

,,,11π π

( ) ( )[ ] xd yd zd zcn

L zs z Ln

z

n z

−++× 12

22

π

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++−−=∑−==

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 zcn

L

zs z LT z y x Dd e xd yd 0 1222,,, τ τ π τ τ

( ) ( )[ ] ( ) ( )[ ] ×∑ ∫ ∫ ∫

−++

−++−=

N

n

t L L

n

y

n yn

x

n x

nV

z

x y

ycn

L ys y L xc

n

L xs x L

n

a

L 1 0 0 02

122

2122

22

1

π π π

( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫

+−+−∫ −×=

N

n

L

n

x

xnV nV

L

nV V n

x z

xcn

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

1 0

2

0

122

2,,,21π

τ τ

( )} ( ) ( )[ ] ( ) ( )[ ]∫ ∫

+−+

−+++ y z

L L

n

z

zV V n

y

n yn zc

n

L LT z y xk yc

n

L ys y L xs x

0 0,

122

,,,122

22π π

( )} ( ) ( ) ( ) ( )[ ] {∑ ∫ ∫ +

−++−+=

N

n

L L

yn

x

n xnV nI nV nI n

x y

L xc

n

L xs x Lt et eaa xd yd zd zs z

10 0

12

2

22

π

( ) ( )[ ] ( ) ( ) ( )[ ]∫ ×

−++

−++ 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




10

( ) ( )[ ] 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 02

2

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,

122

2,,,2π

,

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

−+

−+

−+= 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 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 λ λ δ 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

−= ,

( ) 3

4

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

.




11

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

(6) to the following integro-differential form

( ) ( ) ( )

×∫ ∫ ∫ Φ

=∫ ∫ ∫Φ Φ

t 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 Φ

t 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

+×ΦΦ

t x

L

y

L

V

y x

t x

L

z

L

V

V

z x z y x y x z

T zvu D

L L

y xd ud wd

y

w yuT w yu D

L L

z x

L L

z y

00

,,,,,,

,,, τ

∂

τ ∂

( )( ) ( ) −∫ ∫ ∫+

Φ×

x

L

y

L

z

LV V

z y x

V

x y z

ud vd wd wvuV T wvuk L L L

z y xd ud vd

z

zvuτ τ

∂

τ ∂ ,,,,,,

,,, 2

,

( ) ( ) ( )∫ ∫ ∫+∫ ∫ ∫−Φ

x

L

y

L

z

LV

z y x

x

L

y

L

z

LV

z y x x y z x y z


z y xud vd wd wvuV T wvuk

L L L

z y x,,,,,,,, τ .

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

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

Φ=

Φ N

n

t y

L

z

LnnnI n I n

z y x

N

nnI nnn

I n

y z

wcvct e xsan L L L

z yt e zs ys xs

n

a z y x

1 0133

π

π

( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−×=

ΦΦΦ

N

n

t x

L

z

L I nn I n

z y x

I

x z

d ud wd T wvu Dwcuca L L L

z xd vd wd T wv x D1 0

,,,,,, τ π τ

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

ΦΦΦΦ

N

n

t x

L

y

L I nn I nn I n

z y x

I nn

x y

d ud vd T zvu Dvcuct e zsan L L L

y xt e ysn

1 0

,,, τ π

( ) ( ) ( ) ×∫ ∫ ∫+∫ ∫ ∫+ Φ

x

L

y

L

z

L I

x

L

y

L

z

L I I

z y x x y z x y z

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

z y x,,,,,,,,

2

, τ

( ) ( )∫ ∫ ∫−× x

L

y

L

z

L I

z y x z y x x y z

ud vd wd wvu I T wvuk L L L

z y x

L L L

z y xτ ,,,,,,

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

=

Φ

=

Φ N

n

t y

L

z

L

nnnV nV n

z y x

N

n

nV nnn

V n

y z

wcvct e xsan

L L L

z yt e zs ys xs

n

a z y x

1 01

33

π

π

( ) ( ) ( ) ( ) ×∑ ∫ ∫ ∫−×=

ΦΦ

N

n

t x

L

z

LV nn

z y x

V

x z

d ud wd T wvu Dwcucn L L L

z xd vd wd T wv x D

1 0

,,,,,, τ π

τ

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

ΦΦΦΦ

N

n

t x

L

y

LV nnV nn

z y x

V nnV n

x y

d ud vd T zvu Dvcuct e zsn L L L

y xt e ysa

1 0

,,, τ π




12

( ) ( ) ( ) ×∫ ∫ ∫+∫ ∫ ∫+×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

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

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 zc

n

L zs zT z y x D

n

a

1 0 00

22

2

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




13

} ( )[ ] ( ) ( ) ( ) ( )[ ]∑ ∫ ∫

+−+∫ −+=

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

( )} ( ) ( )[ ] ( ) ( ) ( )[ ]∫

+−∫

−++ 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 this section we compare spatial distributions of concentrations of dopant in a heterostructurewith overlayer and without the overlayer. Fig. 2 shows result of the above comparison. One can

find from the figure, that using the overlayer gives a possibility to manufacture more shallow p-n-

junction in comparison with p-n-junction in the heterostructure without the overlayer. At the sametime analysis of redistribution of radiation defects shows, that implantation of ions of dopant

through the overlayer gives a possibility to decrease quantity of radiation defects in the epitaxiallayer.

Fig.2. Distributions of concentration of implanted dopant in the considered heterostructure.

Curves 1 and 3 describe distributions of concentration of implanted dopant in the heterostructurewith overlayer. Curve 1 describes distribution of concentration of implanted dopant for the case,

when dopant diffusion coefficient in the overlayer is larger, than in the epitaxial layer. Curve 2describes distribution of concentration of implanted dopant for the case, when dopant diffusion

coefficient in the overlayer is smaller, than in the epitaxial layer. Curve 2 describes distribution of

concentration of implanted dopant in the heterostructure without overlayer.




14

It should be noted, that manufacturing the considered implanted-junction rectifier near interface

between layers of heterostructure gives a possibility to increase sharpness of the p-n-junction andat the same time to increase homogeneity of concentration of the dopant in the enriched area [15-

19]. In this situation it is practicably to choose appropriate thickness of epitaxial layer.

4. CONCLUSIONS

In this paper we analyzed distributions of concentration of implanted dopant in heterostructure

with overlayer and without overlayer during manufacture an implanted -heterojunction rectifier.We determine conditions, which correspond to decrease depth of the p-n-junction.

ACKNOWLEDGEMENTS

This work is supported by the agreement of August 27, 2013 № 02.В.49.21.0003 between TheMinistry 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 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 Radiophysical

Department of Nizhny Novgorod State University. Since 2015 E.L. Pankratov is an Associate Professor ofNizhny 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 ofvillage 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 90 published papers in area of her researches.

Documents

INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED- HETEROJUNCTION RECTIFIERS