Using Porosity of Epitaxial Layer to Decrease Quantity of Radiation Defects Generated During...

Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016

DOI:10.5121/msej.2016.3204 43

USING POROSITY OF EPITAXIAL LAYER TO DECREASE

QUANTITY OF RADIATION DEFECTS GENERATED DURING

RADIATION PROCESSING OF A MULTILAYER STRUCTURE

E.L. Pankratov1, 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 redistribution of radiation defects, which were generated during radiation

processing, in a multilayer structure with porous epitaxial layer. It has been shown, that porosity of epitax-

ial layer gives a possibility to decrease quantity of radiation defects.

KEYWORDS

Multilayer structure; porous epitaxial layer; radiation processing; decreasing of quantity of radiation de-

fects.

1. INTRODUCTION

One of actual questions of solid state electronics is increasing of radiation resistance. Several me-

thods are using to increase the radiation resistance of devices of solid state electronics [1-5]. One

way to decrease quantity of radiation defects, generated during radiation processing of materials,

we present in our paper. Framework the approach we consider a heterostructure, which consist of

a substrate and porous epitaxial layer (see Fig. 1). We assume, that the substrate was under influ-

ence of radiation processing (ion implantation, effects of cosmic radiation et al) through the epi-

taxial layer. Radiation processing of materials leads to generation of radiation defects. In this pa-

per we analyzed influence of porosity of material on distribution of concentration of radiation

defects in the considered material.

x

D(x

)

D0

D1

D2

0 a L

C(x,t=0)

P1

P2

x0

Fig. 1. Heterostructure, which includes into itself a substrate and an epitaxial layer. The figure also shows

distribution of concentration of implanted dopant


44

2. METHOD OF SOLUTION

We solve our aim by analysis of distributions of concentrations of radiation defects in space and

time in the considered heterostructure. We determine the above distributions by solving the fol-

lowing system of equations [6-11]

( )( )

( )( )

( )+

+

=

y

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIII

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

( )( )

( ) ( ) ( ) ( ) ×−−

+ TzyxktzyxVtzyxITzyxk

z

tzyxITzyxD

zIIVII

,,,,,,,,,,,,,,,

,,, ,,∂

∂

∂

∂

( )( ) ( ) ( )

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+×

z

tzyx

TkV

D

zy

tzyx

TkV

D

yx

tzyx

TkV

D

xtzyxI

SISISI ,,,,,,,,,,,, 2222 µµµ

( ) ( ) ( ) ( ) ( )+

+

=

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVV

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,, (1)

( )( )

( ) ( ) ( ) ( ) ×−−

+ TzyxktzyxVtzyxITzyxk

z

tzyxVTzyxD

zVVVIV

,,,,,,,,,,,,,,,

,,,,,

∂

∂

∂

∂

( )( ) ( ) ( )

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+×

z

tzyx

TkV

D

zy

tzyx

TkV

D

yx

tzyx

TkV

D

xtzyxV

SVSVSV ,,,,,,,,,,,, 2222 µµµ

with boundary and initial conditions

( )0

,,,

0

==x

x

tzyxI

∂

∂,

( )0

,,,=

= xLxx

tzyxI

∂

∂,

( )0

,,,

0

==y

y

tzyxI

∂

∂,

( )0

,,,=

= yLyy

tzyxI

∂

∂,

( )0

,,,

0

==z

z

tzyxI

∂

∂,

( )0

,,,=

= zLzz

tzyxI

∂

∂,

( )0

,,,

0

==x

x

tzyxV

∂

∂,

( )0

,,,=

= xLxx

tzyxV

∂

∂,

( )0

,,,

0

==y

y

tzyxV

∂

∂,

( )0

,,,=

= yLyy

tzyxV

∂

∂,

( )0

,,,

0

==z

z

tzyxV

∂

∂,

( )0

,,,=

= zLzz

tzyxV

∂

∂,

I (x,y,z,0)=fI (x,y,z), V (x,y,z,0)=fV (x,y,z), ( )

+++=+++

2

1

2

1

2

1

*

111

21,,,

zyxTkVttVztVytVxV

nnn

ωl. (2)

Here I (x,y,z,t) and V (x,y,z,t) are the spatio-temporal distributions of concentrations of interstitials

and vacancies; I* and V

* are the equilibrium distributions of concentrations of interstitials and va-

cancies; diffusion coefficients of interstitials and vacancies have been described by the following

functions DI(x,y,z,T) and DV(x,y,z,T); terms with squares of concentrations of interstitials and va-

cancies (V2(x,y,z,t) and I

2(x,y,z,t), respectively) correspond to generation divacancies and analog-

ous complexes of interstitials (see, for example, [10] and appropriate references in this work);

kI,V(x,y,z,T) is the parameter of recombination of point defects; kI,I(x,y,z,T) and kV,V(x,y,z,T) are the

parameters of generation of complexes of point defects and generation; k, V*, l and a are the

Boltzmann constant, the equilibrium distribution of vacancies, the atomic spacing and the specific

surface energy, respectively. ω = a3. Framework taking into account porosity we assume, that at

initial stage porous are approximately cylindrical with average dimensions 2

1

2

1yxr += and z1

[12]. With time small pores decomposing into vacancies. The vacancies are absorbed by large

pores [8]. The large pores takes spherical form during the absorbtion [8]. We determine distribu-


45

tion of concentration of vacancies, which was formed due to porosity, by summing over all pores,

i.e.

( ) ( )∑ ∑ ∑ +++== = =

l

i

m

j

n

kp

tkzjyixVtzyxV0 0 0

,,,,,, χβα , 222zyxR ++= .

Averaged distances between centers of pores are equal to α, β and χ in x, y and z directions, re-

spectively. Quantities of pores in x, y and z directions are equal to i, j and k.

Distributions of concentrations of divacancies ΦV(x,y,z,t) and diinterstitials ΦI (x,y,z,t) in space

and time have been calculated by solving the following system of equations [9-11]

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxIII

II ∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

( ) ( ) ( ) ( ) ( ) ( )+++

Φ+ Φ tzyxITzyxktzyxITzyxk

z

tzyxTzyxD

zIII

I

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

,,,,,, 2

,∂

∂

∂

∂

( ) ( ) ( )

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

ΦΦΦ

z

tzyx

TkV

D

zy

tzyx

TkV

D

yx

tzyx

TkV

D

x

SSS III ,,,,,,,,, 222 µµµ (3)

( )( )

( )( )

( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxVVV

VV ∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

( )( )

( ) ( ) ( ) ( )+++

Φ+ Φ tzyxVTzyxktzyxVTzyxk

z

tzyxTzyxD

zVVV

V

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

,,,,,, 2

,∂

∂

∂

∂

( ) ( ) ( )

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

ΦΦΦ

z

tzyx

TkV

D

zy

tzyx

TkV

D

yx

tzyx

TkV

D

x

SSS VVV ,,,,,,,,,222

µµµ

with boundary and initial conditions

( )0

,,,

0

=Φ

=x

I

x

tzyx

∂

∂,

( )0

,,,=

= xLxx

tzyxI

∂

∂,

( )0

,,,

0

==y

y

tzyxI

∂

∂,

( )0

,,,=

= yLyy

tzyxI

∂

∂,

( )0

,,,

0

=Φ

=z

I

z

tzyx

∂

∂,

( )0

,,,=

= zLzz

tzyxI

∂

∂,

( )0

,,,

0

=Φ

=x

V

x

tzyx

∂

∂,

( )0

,,,=

= xLxx

tzyxV

∂

∂,

( )0

,,,

0

==y

y

tzyxV

∂

∂,

( )0

,,,=

= yLyy

tzyxV

∂

∂,

( )0

,,,

0

==z

z

tzyxV

∂

∂,

( )0

,,,=

Φ

= zLz

V

z

tzyx

∂

∂,

ΦI (x,y,z,0)=fΦI (x,y,z), ΦV (x,y,z,0)=fΦV (x,y,z). (4)

Functions DΦI(x,y,z,T) and DΦV(x,y,z,T) describe spatial and temperature dependences of the dif-

fusion coefficients of complexes of radiation defects; kI(x,y,z,T) and kV(x,y,z,T) are the parameters

of decay of the above complexes.

We calculate distributions of concentrations of radiation defects in space and time by method of

averaging of function corrections [13]. To use the approach we write the Eqs. (1) and (3) with

account initial distributions of defects, i.e.


46

( ) ( ) ( ) ( ) ( )+

+

=

y

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIII

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

( )( )

( ) ( ) ( ) ( ) ( ) ++−

+ tzyxftzyxVtzyxITzyxk

z

tzyxITzyxD

zIVII

δ∂

∂

∂

∂,,,,,,,,,,,

,,,,,,

,

( ) ( ) ( )−

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

z

tzyx

TkV

D

zy

tzyx

TkV

D

yx

tzyx

TkV

D

x

SISISI ,,,,,,,,, 222 µµµ

( ) ( )tzyxITzyxkII

,,,,,, 2

,−

(1a)

( ) ( ) ( ) ( ) ( )+

+

=

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVV

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

( ) ( ) ( ) ( ) ( ) ( ) ( )++−

+ tzyxftzyxVtzyxITxk

z

tzyxVTzyxD

zVVIV

δ∂

∂

∂

∂,,,,,,,,,

,,,,,,

,

( ) ( ) ( )−

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

z

tzyx

TkV

D

zy

tzyx

TkV

D

yx

tzyx

TkV

D

x

SVSVSV ,,,,,,,,, 222 µµµ

( ) ( )tzyxVTzyxkVV

,,,,,, 2

,−

( ) ( ) ( ) ( ) ( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxI

I

I

I

I

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

( ) ( ) ( ) ( ) ( ) ( )+++

Φ+ Φ tzyxITzyxktzyxITzyxk

z

tzyxTzyxD

zIII

I

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

,,,,,, 2

,∂

∂

∂

∂

( ) ( ) ( )+

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

ΦΦΦ

z

tzyx

TkV

D

zy

tzyx

TkV

D

yx

tzyx

TkV

D

x

SSS VVV ,,,,,,,,,222

µµµ

( ) ( )tzyxfI

δ,,Φ+

(3a)

( )( )

( )( )

( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxV

V

V

V

V

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,

( )( )

( ) ( ) ( ) ( )+++


z

tzyxTzyxD

zVVV

V

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

,,, 2

,∂

∂

∂

∂

( ) ( ) ( )+

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

ΦΦΦ

z

tzyx

TkV

D

zy

tzyx

TkV

D

yx

tzyx

TkV

D

x

SSS VVV ,,,,,,,,,222

µµµ

( ) ( )tzyxfI

δ,,Φ+ .

Now we use not yet known average values α1ρ of the required concentrations in right sides of the

Eqs. (1a) and (3a) instead of the concentrations. The replacement gives us possibility to obtain

following equations for determination the first-order approximations of concentrations of radia-

tion defects in the following form

( ) ( ) ( ) ( ) ( )++

∂

∂

∂

∂+

∂

∂

∂

∂= tzyxf

y

tzyx

TkV

D

yx

tzyx

TkV

D

xt

tzyxII

SISI δµµ

∂

∂,,

,,,,,,,,, 221


47

( ) ( ) ( )TzyxkTzyxkz

tzyx

TkV

D

zVIVIIII

SI,,,,,,

,,,,11,

2

1

2 αααµ

−−

∂

∂

∂

∂+

(1b)

( ) ( ) ( ) ( ) ( )++

∂

∂

∂

∂+

∂

∂

∂

∂= tzyxf

y

tzyx

TkV

D

yx

tzyx

TkV

D

xt

tzyxVV

SVSV δµµ

∂

∂,,

,,,,,,,,, 221

( ) ( ) ( )TzyxkTzyxkz

tzyx

TkV

D

zVIVIVVV

SV,,,,,,

,,,,11,

2

1

2 αααµ

−−

∂

∂

∂

∂+

( ) ( ) ( )( ) ( )++

∂

∂

∂

∂+

∂

∂

∂

∂=

ΦΦ

ΦΦtzyxf

y

tzyx

TkV

D

yx

tzyx

TkV

D

xt

tzyxI

II SSI δµµ

∂

∂,,

,,,,,,,,, 221

( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkz

tzyx

TkV

D

zIII

SI ,,,,,,,,,,,,,,, 2

,

2 ++

∂

∂

∂

∂+

Φ µ (3b)

( ) ( ) ( )( ) ( )++

∂

∂

∂

∂+

∂

∂

∂

∂=

ΦΦ

ΦΦtzyxf

y

tzyx

TkV

D

yx

tzyx

TkV

D

xt

tzyxV

VV SSV δµµ

∂

∂,,

,,,,,,,,,221

( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkz

tzyx

TkV

D

zVVV

SV ,,,,,,,,,,,,,,, 2

,

2 ++

∂

∂

∂

∂+

Φ µ.

Integration of the both sides of Eqs. (1b) and (3b) on time gives a possibility to obtain first- order

approximations of concentrations of radiation defects in the final form

( ) ( ) ( ) ( )++∫∂

∂

∂

∂+∫

∂

∂

∂

∂= zyxfd

y

tzyx

TkV

D

yd

x

tzyx

TkV

D

xtzyxI

I

tSI

tSI

,,,,,,,,

,,,0

2

0

2

1 τµ

τµ

( )( ) ( )∫−∫−∫

∂

∂

∂

∂+

t

VIVI

t

III

tSI

dTzyxkdTzyxkdz

tzyx

TkV

D

z 0,11

0,

2

10

2 ,,,,,,,,,

ταατατµ

(1c)

( ) ( ) ( ) ( )++∫∂

∂

∂

∂+∫

∂

∂

∂

∂= zyxfd

y

tzyx

TkV

D

yd

x

tzyx

TkV

D

xtzyxV

V

tSV

tSV

,,,,,,,,

,,,0

2

0

2

1 τµ

τµ

( )( ) ( )∫−∫−∫

∂

∂

∂

∂+

t

VIVI

t

VVV

tSV

dTzyxkdTzyxkdz

tzyx

TkV

D

z 0,11

0,

2

10

2 ,,,,,,,,,

ταατατµ

( ) ( ) ( ) ( ) ( ) ( ) +∫+∫+=Φ Φ

t

II

t

IIdzyxITzyxkdzyxITzyxkzyxftzyx

I0

2

,0

1,,,,,,,,,,,,,,,,, ττττ

( ) ( ) ( )∫

∂

∂

∂

∂+∫

∂

∂

∂

∂+∫

∂

∂

∂

∂+

ΦΦΦt

St

St

Sd

z

zyx

TkV

D

zd

y

zyx

TkV

D

yd

x

zyx

TkV

D

x

III

0

2

0

2

0

2,,,,,,,,,

ττµ

ττµ

ττµ

( ) ( ) ( )∫

∂

∂

∂

∂+∫

∂

∂

∂

∂+∫

∂

∂

∂

∂+

ΦΦΦt

St

St

Sd

z

zyx

TkV

D

zd

y

zyx

TkV

D

yd

x

zyx

TkV

D

x

III

0

2

0

2

0

2,,,,,,,,,

ττµ

ττµ

ττµ

(3c)

( ) ( ) ( ) ( ) ( ) ( ) +∫+∫+=Φ Φ

t

VV

t

VVdzyxVTzyxkdzyxVTzyxkzyxftzyx

V0

2

,0

1,,,,,,,,,,,,,,,,, ττττ

( ) ( ) ( )∫

∂

∂

∂

∂+∫

∂

∂

∂

∂+∫

∂

∂

∂

∂+

ΦΦΦt

St

St

Sd

z

zyx

TkV

D

zd

y

zyx

TkV

D

yd

x

zyx

TkV

D

x

VVV

0

2

0

2

0

2,,,,,,,,,

ττµ

ττµ

ττµ

.

We calculate average values of the required approximations by the following relation [13]


48

( )∫ ∫ ∫ ∫Θ

=Θ

0 0 0 011 ,,,

1 x y zL L L

zyx

tdxdydzdtzyxLLL

ρα ρ . (5)

Substitution of the relations (1c) and (3c) into the relation (5) gives a possibility to calculate the required average values in the following form

( )

4

3

4

1

2

3

2

4

2

3

14

44 a

Aa

a

aLLLBaB

a

Aa zyx

I

+−

Θ+Θ+−

+=α , ( )

×∫ ∫ ∫=

x y zL L L

IVxdydzdzyxf

0 0 01

,,α

00

001

1

1

IV

zyxIII

I SLLLS

Θ−−

Θ× α

α, ( ) +∫ ∫ ∫+

Θ= ΦΦ

x y z

II

L L L

zyxzyx

I xdydzdzyxfLLLLLL

R

0 0 0

1

1 ,,1

α

zyx

II

LLL

S

Θ+

20, ( )

zyx

VVVL L L

zyxLLL

SRxdydzdzyxf

LLL

x y z

VV Θ

++∫ ∫ ∫= ΦΦ

201

0 0 01 ,,

1α .

Here ( ) ( ) ( ) ( )∫ ∫ ∫ ∫−Θ=Θ

0 0 0 011, ,,,,,,,,,

x y zL L Lji

ijtdxdydzdtzyxVtzyxITzyxktS ρρρρ , ( )×−=

0000

2

004 VVIIIVSSSa

00IIS× , 0000

2

0000003 VVIIIVIIIVSSSSSa −+= , ( ) ×+∫ ∫ ∫= 00

0 0 0

2

00002 2,,VV

L L L

VIVIVSxdydzdzyxfSSa

x y z

( ) ( ) ×−Θ+∫ ∫ ∫−∫ ∫ ∫× 222222

000 0 0

2

000 0 0

00 ,,,,zyxzyxIV

L L L

IIV

L L L

IIILLLLLLSxdydzdzyxfSxdydzdzyxfS

x y zx y z

00VVSΘ× , ( )∫ ∫ ∫=x y zL L L

IIVxdydzdzyxfSa

0 0 0001

,, , ( )2

0 0 0000

,,

∫ ∫ ∫=x y zL L L

IVVxdydzdzyxfSa , ×

Θ=

2

4

2

3

24 a

aq

2

4

2

1

4

222

3

4

3

2

3

4

2

32

22

4

02

4

31

0854

48

4a

aLLL

a

a

a

aa

a

a

a

aaLLLa zyxzyx

Θ−

Θ−

Θ−ΘΘ−

Θ−× ,

4

2

2

4

2

3248

a

a

a

ayA Θ−Θ+= ,

( ) ( ) ( )∫ ∫ ∫ ∫−Θ=Θ

0 0 0 01 ,,,,,,

x y zL L Li

IitdxdydzdtzyxITzyxktRρ ,

2

4

31

4

2

2

4

402

121812

4

a

aaLLL

a

a

a

aap

zyx

Θ−

Θ−Θ= ,

3 323 32

4

2

6qpqqpq

a

aB ++−−++

Θ= .

Approximations of concentrations of radiations defects with the second and higher orders have

been calculated framework standard iterative procedure of method of averaging of function cor-

rections [13]. Framework the procedure we determine the approximation of the n-th order by re-

placement of the concentrations of radiation defects I (x,y,z, t), V (x,y,z,t), ΦI (x,y,z,t) and ΦV

(x,y,z,t) in right sides of the Eqs.(1b) and (3b) on the following sums αnρ+ρ n-1(x,y,z,t). The re-

placement gives a possibility to obtain the second-order approximations of concentrations of radi-

ation defects

( ) ( ) ( ) ( ) ( )+

+

=

y

tzyxITzyxD

yx

tzyxITzyxD

xt

tzyxIII

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,, 112

( ) ( ) ( ) ( )[ ] ( )[ ]+++−

+ tzyxVtzyxITzyxk

z

tzyxITzyxD

zVIVII

,,,,,,,,,,,,

,,, 1111,

1 αα∂

∂

∂

∂

( ) ( ) ( )−∫

∂

∂

∂

∂+∫

∂

∂

∂

∂+∫

∂

∂

∂

∂+

tSI

tSI

tSI

dz

tzyx

TkV

D

zd

y

tzyx

TkV

D

yd

x

tzyx

TkV

D

x 0

2

0

2

0

2,,,,,,,,,

τµ

τµ

τµ

( ) ( )[ ]2

11,,,,,,, tzyxITzyxk

III+− α (1d)


49

( ) ( ) ( ) ( ) ( )+

+

=

y

tzyxVTzyxD

yx

tzyxVTzyxD

xt

tzyxVVV

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,, 112

( ) ( ) ( ) ( )[ ] ( )[ ]+++−

+ tzyxVtzyxITzyxk

z

tzyxVTzyxD

zVIVIV

,,,,,,,,,,,,

,,,1111,

1 αα∂

∂

∂

∂

( ) ( ) ( )−∫

∂

∂

∂

∂+∫

∂

∂

∂

∂+∫

∂

∂

∂

∂+

tSI

tSI

tSI

dz

tzyx

TkV

D

zd

y

tzyx

TkV

D

yd

x

tzyx

TkV

D

x 0

2

0

2

0

2 ,,,,,,,,,τ

µτ

µτ

µ

( ) ( )[ ]2

11,,,,,,, tzyxVTzyxk

IVV+− α

( )( )

( )( )

( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxIII

II ∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,, 112

( )( )

( ) ( ) ( ) ( ) ( ) ×+++

Φ+ ΦΦ TzyxktzyxftzyxITzyxk

z

tzyxTzyxD

zIII

I

II,,,,,,,,,,,

,,,,,, 2

,

1 δ∂

∂

∂

∂

( ) ( ) ( )

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

ΦΦΦ

z

tzyx

TkV

D

zy

tzyx

TkV

D

yx

tzyx

TkV

D

x

SSS III ,,,,,,,,,222

µµµ (3d)

( )( )

( )( )

( )+

Φ+

Φ=

ΦΦΦ

y

tzyxTzyxD

yx

tzyxTzyxD

xt

tzyxVVV

VV ∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ,,,,,,

,,,,,,

,,,112

( )( )

( ) ( ) ( ) ( )+++


z

tzyxTzyxD

zVVV

I

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

,,,,,, 2

,

1

∂

∂

∂

∂

( ) ( ) ( )

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

ΦΦΦ

z

tzyx

TkV

D

zy

tzyx

TkV

D

yx

tzyx

TkV

D

x

SSS VVV ,,,,,,,,,222

µµµ.

Integration of the both sides of Eqs. (1d) and (3d) gives a possibility to obtain relations for the second-order approximations of the required concentrations of radiation defects in the following

form

( ) ( ) ( ) ( ) ( )+∫+∫=

t

I

t

Id

y

zyxITzyxD

yd

x

zyxITzyxD

xtzyxI

0

1

0

1

2

,,,,,,

,,,,,,,,, τ

∂

τ∂

∂

∂τ

∂

τ∂

∂

∂

( ) ( ) ( ) ( )[ ] −∫ +−∫+t

III

t

IdzyxITzyxkd

z

zyxITzyxD

z 0

2

12,0

1 ,,,,,,,,,

,,, ττατ∂

τ∂

∂

∂

( ) ( )[ ] ( )[ ] ( ) ++∫ ++− zyxfdzyxVzyxITzyxkI

t

VIVI,,,,,,,,,,,

01212,

ττατα

( ) ( ) ( )

+

+

+

z

tzyx

TkV

D

zy

tzyx

TkV

D

yx

tzyx

TkV

D

x

SISISI

∂

µ∂

∂

∂

∂

µ∂

∂

∂

∂

µ∂

∂

∂ ,,,,,,,,,222

( ) ( ) ( )

+

+

+

z

tzyx

TkV

D

zy

tzyx

TkV

D

yx

tzyx

TkV

D

x

SISISI

∂

µ∂

∂

∂

∂

µ∂

∂

∂

∂

µ∂

∂

∂ ,,,,,,,,,222 (1e)

( ) ( ) ( ) ( ) ( )+∫+∫=

t

V

t

Vd

y

zyxVTzyxD

yd

x

zyxVTzyxD

xtzyxV

0

1

0

1

2

,,,,,,

,,,,,,,,, τ

∂

τ∂

∂

∂τ

∂

τ∂

∂

∂

( ) ( ) ( ) ( )[ ] −∫ +−∫+t

VVV

t

VdzyxVTzyxkd

z

zyxVTzyxD

z 0

2

12,0

1 ,,,,,,,,,

,,, ττατ∂

τ∂

∂

∂

( ) ( )[ ] ( )[ ] ( )++∫ ++− zyxfdzyxVzyxITzyxkV

t

VVIVI,,,,,,,,,,,

0121,

ττατα


50

( ) ( ) ( )

+

+

+

z

tzyx

TkV

D

zy

tzyx

TkV

D

yx

tzyx

TkV

D

x

SVSVSV

∂

µ∂

∂

∂

∂

µ∂

∂

∂

∂

µ∂

∂

∂ ,,,,,,,,,222

( ) ( )( )

( )( )

+∫Φ

+∫Φ

=Φ ΦΦ

tI

tI

I dy

zyxTzyxD

yd

x

zyxTzyxD

xtzyx

II0

1

0

1

2

,,,,,,

,,,,,,,,, τ

∂

τ∂

∂

∂τ

∂

τ∂

∂

∂

( )( )

( ) ( ) ( ) ++∫+∫Φ

+ ΦΦ zyxfdzyxITzyxkdz

zyxTzyxD

z II

t

II

tI ,,,,,,,,

,,,,,,

0

2

,0

1 τττ∂

τ∂

∂

∂

( ) ( ) ( )+∫

∂

∂

∂

∂+∫

∂

∂

∂

∂+∫

∂

∂

∂

∂+

ΦΦΦt

St

St

Sd

z

zyx

TkV

D

zd

y

zyx

TkV

D

yd

x

zyx

TkV

D

x

III

0

2

0

2

0

2,,,,,,,,,

ττµ

ττµ

ττµ

( ) ( )∫+t

IdzyxITzyxk

0

,,,,,, ττ (3e)

( ) ( )( )

( ) ( )+∫++∫Φ

=Φ ΦΦΦ

ttV

VTzyxD

yzyxfd

x

zyxTzyxD

xtzyx

VVV00

1

2,,,,,

,,,,,,,,,

∂

∂τ

∂

τ∂

∂

∂

( )( )

( ) ( ) ( ) +∫++∫Φ

+ ΦΦ

t

V

tV

dzyxVTzyxkzyxfdz

zyxTzyxD

z VV00

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

,,, τττ∂

τ∂

∂

∂

( ) ( ) ( )+∫

∂

∂

∂

∂+∫

∂

∂

∂

∂+∫

∂

∂

∂

∂+

ΦΦΦt

St

St

Sd

z

zyx

TkV

D

zd

y

zyx

TkV

D

yd

x

zyx

TkV

D

x

VVV

0

2

0

2

0

2,,,,,,,,,

ττµ

ττµ

ττµ

( ) ( )∫+t

VVdzyxVTzyxk

0

2

,,,,,,, ττ .

We determine average values of the second-order approximations by the standard relation [13]

( ) ( )[ ]∫ ∫ ∫ ∫ −Θ

=Θ

0 0 0 0122

,,,,,,1 x y zL L L

zyx

tdxdydzdtzyxtzyxLLL

ρρα ρ . (6)

Substitution of the relations (1e) and (3e) in the relation (6) gives a possibility to obtain relations

for the required values α 2ρ

α2C=0, α2ΦI =0, α2ΦV =0, ( )

4

3

4

1

2

3

2

4

2

3

24

44 b

Eb

b

bLLLFaF

b

Eb zyx

V

+−

Θ+Θ+−

+=α ,

( )00201

11021001200

2

2

2

2

IVVIV

VIVVzyxVIVVVVVVV

ISS

SSLLLSSSC

α

ααα

+

−−Θ++−−= ,

where 00

2

0000

2

004

11IIVV

zyx

VVIV

zyx

SSLLL

SSLLL

bΘ

−Θ

= , ( ) +Θ

Θ++−=zyx

VVII

zyxIVVVLLL

SSLLLSSb 0000

100132

( ) ( )3333

10

2

00

2

00

1001011001

0000 22zyx

IVIV

zyx

IV

zyxIVVVzyxIVIIIV

zyx

VVIV

LLL

SS

LLL

SLLLSSLLLSSS

LLL

SS

Θ−

ΘΘ+++Θ+++

Θ+ ,

( ) ( ) ( ) ×++Θ++−Θ−++Θ

=0110

2

10011102

0000

222

IVIIzyxIVVVzyxVIVVV

zyx

VVII SSLLLSSLLLCSSLLL

SSb

( )( ) ×−++Θ+++ΘΘ

+Θ

× 2

001001011001

000001 222IVIVVVzyxIVIIIVzyx

zyx

IV

zyx

VVIV SSSLLLSSSLLLLLL

S

LLL

SS

zyx

IVIV

IV

zyx

IVI

zyx

IVVVV

LLL

SSS

LLL

SC

LLL

SSC

Θ−

Θ+

Θ

−−× 0100

102222

2

001102 2 , ( )×Θ++=zyxIVVVII

LLLSSSb1001001

2


51

( ) ×Θ

−Θ++Θ

++Θ+

Θ

++×

zyx

IV

zyxIVVV

zyx

IVIIzyx

IV

zyx

VVVIV

LLL

SLLLSS

LLL

SSLLLS

LLL

CSS 00

1001

0110

01

0211 22

( )( )zyx

IVIV

IVIVIIVVVVzyxIIIVLLL

SSSSCSSCLLLSS

Θ−+−−Θ++×

2

0110

010011021001 223 , ×Θ

=zyx

II

LLL

Sb 00

0

( ) ( ) ×Θ

−−−++Θ

Θ

−−−+× 01

1102

0110

1102

01

2

0200 2IV

zyx

IVVVV

IVIIzyx

zyx

IVVVV

IVVVIVS

LLL

SSCSSLLL

LLL

SSCSSS

( ) 2

01011022

IVIIVIIzyxSCSSLLL +++Θ× ,

zyx

IV

zyx

II

zyx

III

IV

zyx

VI

ILLL

S

LLL

S

LLL

SS

LLLC

Θ−

Θ−

Θ+

Θ= 112000

2

1

00

11 ααα,

110200

2

10011 IVVVVVVIVVIVSSSSC −−+= ααα ,

3 323 32

4

2

6rsrrsr

a

aF ++−−++

Θ= , ×

Θ=

2

4

2

3

24b

br

2

4

2

1

4

222

3

4

3

2

3

4

2

32

22

4

2

0

4

31

0854

48

4b

bLLL

b

b

b

bb

bb

b

bbLLLb

zyxzyx

Θ−

Θ−

Θ−Θ

Θ−

Θ−× ,

4

2

2

4

2

32 48a

a

a

ayE Θ−Θ+= ,

( )4

2

31402

4

2

184

12 b

bbbLLLbb

bs

zyx

Θ−Θ−

Θ= .

Framework this paper the required spatio-temporal distributions of concentrations of radiations

defects have been determined by using the second-order approximations by using method of av-eraging of function corrections. The approximations give enough good qualitative and some

quantitative results. We check all analytical results by comparison with numerical one.

3. DISCUSSION In the previous section we analytically take into account porosity of materials in comparison with

cited similar works. In this situation we obtain decreasing quantity of radiation defects (one can

find decreasing both types of accounted defects: point defects and their simplest complexes) in

comparison with nonporous materials. Probably this effect could be obtained due to drain of these defects to pores.

x

0.0

0.5

1.0

1.5

ρ(x

,Θ)

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

1

2

Fig. 2. Distributions of concentrations of point radiation defects for fixed value of annealing time.

Curve 1 corresponds to implantation of ions of dopant through nonporous epitaxial layer. Curve 2

corresponds to implantation of ions of dopant through porous epitaxial layer

Typical distributions of concentrations of point radiation defects and their simplest complexes are

presented on Figs. 2 and 3, respectively. In this situation using overlayer over device area gives a


52

possibility to increase radiation resistance of the devices during radiation processing. Using por-ous overlayer gives a possibility to obtain larger increasing of radiation resistance. The Figs. 2

and 3 also shows, that quantity of point defects is larger, than quantity of simplest complexes of

point defects. This effect could be find because only part of point defects could generate their complexes. It should be also noted, that we analyzed relaxation of concentrations of radiation de-

fects analytically and in more common case in comparison with similar results in literature. The

figures show, that porosity of materials of epitaxial layer gives a possibility to decrease quantity

of radiation defects. In this situation porous overlayer over device area gives a possibility to in-

crease safety during radiation processing. Analogous situation could be find during using nonpor-

ous overlayer over device area. However pores probably became drains of radiation defects. In

this situation porosity of overlayer leads to higher safety of device area.

Fig. 3. Distributions of concentrations of simplest complexes of point radiation defects for fixed value of

annealing time.

Curve 1 corresponds to implantation of ions of dopant through nonporous epitaxial layer. Curve 2

corresponds to implantation of ions of dopant through porous epitaxial layer

4. CONCLUSIONS

In the present paper we analyzed redistributions of radiations defects in material with porous and nonporous overlayer after radiation processing. We show, that presents of porous overlayer gives

a possibility to decrease quantity of radiation defects.

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, edu-

cational fellowship for scientific research of Government of Nizhny Novgorod region of Russia

and educational fellowship for scientific research of Nizhny Novgorod State University of Archi-

tecture and Civil Engineering.


53

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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 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.

Using Porosity of Epitaxial Layer to Decrease Quantity of Radiation Defects Generated During...

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