In this paper, we introduce a model for the redistribution and interaction of point radiation defects between themselves, as well as their simplest complexes in a material, taking into account the experimentally non-monotonicity of the distribution of the concentration of radiation defects. To take into account this nonmonotonicity, the previously used model in the literature for the analysis of spatiotemporal distributions of the concentration of radiation defects was supplemented by the concentration dependence of their diffusion coefficient.

1. www.ajms.com 71
ISSN 2581-3463
RESEARCH ARTICLE
On Prognosis of Changing of the Rate of Diffusion of Radiation Defects, Generated
during Ion Implantation, with Increasing of Depth of their Penetration
E. L. Pankratov1,2
1
Department of Mathematical and Natural Sciences, Nizhny Novgorod State University, 23 Gagarin Avenue,
Nizhny Novgorod, 603950, Russia, 2
Department of Higher Mathematics, Nizhny Novgorod State Technical
University, 24 Minin Street, Nizhny Novgorod, 603950, Russia
Received: 01-06-2019; Revised: 28-07-2019; Accepted: 10-08-2019
ABSTRACT
In this paper, we introduce a model for the redistribution and interaction of point radiation defects between
themselves, as well as their simplest complexes in a material, taking into account the experimentally
non-monotonicity of the distribution of the concentration of radiation defects. To take into account
this nonmonotonicity, the previously used model in the literature for the analysis of spatiotemporal
distributions of the concentration of radiation defects was supplemented by the concentration dependence
of their diffusion coefficient.
Key words: Analytical approach for modeling, ion implantation, model of radiation defects
INTRODUCTION
Inthepresenttime,theinfluenceofdifferenttypesofradiationprocessingonsemiconductorsisintensively
analyzing.[1-3]
Based on the analysis, several recommendations to increase the radiation resistance have
been formulated.[4-6]
In this paper, we analyze redistribution and interaction between point radiation defects, as well as their
simplest complexes in materials after ion implantation. A modification of the previously proposed
model[7]
describing the redistribution and interaction of radiation defects between themselves is proposed
with the aim of taking into account the experimentally revealed nonmonotonicity of the distribution of
the concentrations of these defects.[8]
METHOD OF SOLUTION
We determine spatiotemporal distributions of concentrations of point defects by the solution of the
following system of equations.[7,9]
∂ ( )
∂
=
∂
∂
( )
∂ ( )
∂





 − ( ) ( )− ( )
I x t
t x
D I V T
I x t
x
K T I x t K T I x
I I r
,
, ,
,
,
2
,
, ,
,
, ,
,
,
t V x t
V x t
t x
D I V T
V x t
x
K T V x
V V
( ) ( )
∂ ( )
∂
=
∂
∂
( )
∂ ( )
∂





 − ( ) 2
t
t K T I x t V x t
r
( )− ( ) ( ) ( )







, ,
(1)
Here, I (x, t) and V (x, t) are the spatiotemporal distributions of concentrations of interstitials and
vacancies. The first term in the right-hand side of Eqs. (1) describe the diffusion of point defects with the
diffusion coefficient, which depends on temperature and concentration of point defects. The dependence
Address for correspondence:
E. L. Pankratov,
E-mail: elp2004@mail.ru

2. Pankratov: On prognosis of transport of radiation defects
AJMS/Jul-Sep-2019/Vol 3/Issue 3 72
could be approximated by the following relation Dp
(I, V, T) = D0
[1+a×I (x, t)/I *+b×V (x, t)/V *],
p = I,V. The second term on the right-hand side of Eqs. (1) describes the generation of the simplest
complexes of radiation defects (divacancies and analogous complexes of radiation defects).[7]
The third
term on the right-hand side of Eqs. (1) describes the recombination of point defects.
∂ ( )
∂
=
=
I x t
x x
,
0
0 ,
∂ ( )
∂
=
=
V x t
x x
,
0
0,
∂ ( )
∂
=
=
I x t
x x L
,
0 ,
∂ ( )
∂
=
=
V x t
x x L
,
0 ,
I x N
x x
x
I
I
I
, exp
0 0
2
2
( ) = −
−
( )








, V x N
x x
x
V
V
V
, exp
0 0
2
2
( ) = −
−
( )








. (2)
We transform Eqs. (1) to the following integral form,
I x t
x
D T
I x
x
d K T I x d
K T
I
t
I
t
r
,
,
,
( ) =
∂
∂
( )
∂ ( )
∂





 − ( ) ( ) −
−
∫ ∫

  
0
2
0
(
( ) ( ) ( ) + ( )
( ) =
∂
∂
( )
∂ ( )
∂


∫
∫
I x V x d I x
V x t
x
D T
V x
x
d
t
V
t
, , ,
,
,
  


0
0
0




 − ( ) ( ) −
− ( ) ( ) ( ) + ( )




∫
∫
K T V x d
K T I x V x d V x
V
t
r
t
2
0
0
0
,
, , ,
 
  










(1a)
Now, we will solve Eqs. (1a) by the method of averaging of function corrections,[10]
with decreasing
quantity of iteration steps.[11]
We used solutions of Eqs. (1) without nonlinear terms and averaged
diffusion coefficients D0IV
as initial-order approximations of solutions of Eqs. (1a). These initial-order
approximations could be solved by standard Fourier approach[10,11]
and could be written as:
( )
2 2
0
0 0 0 2
1
, , cos exp


∞
=
 
   
=
+ −
     
     
∑ I
I nI
n
n D t
r n z
I r z t F F J
R L L
,
( )
2 2
0
0 0 0 2
1
, , cos exp


∞
=
 
   
=
+ −
     
     
∑ V
V nV
n
n D t
r n z
V r z t F F J
R L L
.
Here ( )
0
2
,0 cos

 
=  
 
∫
L
nI
nu
F I u d u
L L
, ( )
0
2
,0 cos

 
=  
 
∫
L
nV
nu
F V u d u
L L
. Substitution of the above
series into Eqs. (1a) gives us a possibility to obtain the first-order approximations of concentrations of
point radiation defects in the following form:
I x t
L
n F
n x
L
D T
n D
L
d
nI I
I
t
1
2 2
0
2
0
, sin exp
( ) = −





 ( ) −






∫
π π π τ
τ






 − ( ) 








=
∞
=
∞
∑ ∑
∫
n
I
n
t
K T
n x
L
1 1
0
cos
π
× −





 +


 − ( ) +


F
n D
L
F
d K T
F
F
n x
L
nI
I I
r
I
nI
exp cos
π τ
τ
π
2 2
0
2
0
2
0
2 2




 −












=
∞
∑
∫ exp
π τ
2 2
0
2
1
0
n D
L
I
n
t
× +





 −











 +
=
∞
∑
F
F
n x
L
n D
L
d I x
V
nV
V
n
0
2 2
0
2
1
2
cos exp
π π τ
τ ,
,0
( ) (1a)

3. Pankratov: On prognosis of transport of radiation defects
AJMS/Jul-Sep-2019/Vol 3/Issue 3 73
V x t
L
n F
n x
L
D T
n D
L
d
nV V
V
t
1
2 2
0
2
0
, sin exp
( ) = −





 ( ) −






∫
π π π τ
τ






 − ( ) 








=
∞
=
∞
∑ ∑
∫
n
V
n
t
K T
n x
L
1 1
0
cos
π
× −





 +


 − ( ) +


F
n D
L
F
d K T
F
F
n x
L
nV
V V
r
I
nI
exp cos
π τ
τ
π
2 2
0
2
0
2
0
2 2




 −












=
∞
∑
∫ exp
π τ
2 2
0
2
1
0
n D
L
I
n
t
× +





 −











 +
=
∞
∑
F
F
n x
L
n D
L
d V x
V
nV
V
n
0
2 2
0
2
1
2
cos exp
π π τ
τ ,
,0
( )
We determine the second and highest orders of approximations of concentrations of point radiation
defects framework standard iterative procedure of method of averaging of function corrections.[10]
In
this case, nth-order approximations of concentrations of defects will be determined by the following
replacement I (x, t)→αnI
+In-1
(x, t), V (x, t)→α nV
+Vn-1
(x, t) in the right sides of Eqs. (1a), where αnI
and αnV
are the average values of the considered approximations. In this situation, the second-order
approximations of concentrations of point defects could be written as:
( ) ( )
( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( )
( ) ( ) ( )
1
2 2 1 2 1
0 0
2
2 1
0
1
2 2 1 2 1
0 0
2
2 1
0
,
, , ,
, ,0
,
, , ,
, ,0
t t
I r I V
t
I I
t t
V r I V
t
V V
I x
I x t D T d K T I x V x d
x x
K T I x d I x
V x
V x t D T d K T I x V x d
x x
K T V x d V x

     
  

     
  
  
∂
∂
   
= − + +
      
∂ ∂
 


  
− + +
 


 
∂
∂

   
= − + +
     
 ∂ ∂
 


 
− + +
 

∫ ∫
∫
∫ ∫
∫

Average values of the second-order approximations of the considered concentrations could be determined
by the following standard relations:
2 2 1
0
0
1
I
L
L
I x t I x t d x d t
= ( )− ( )

 

∫
∫
Θ
Θ
, , , 2 2 1
0
0
1
V
L
L
V x t V x t d x d t
= ( )− ( )

 

∫
∫
Θ
Θ
, , .
Where Θ is the continuance of observation of the change in the concentration of defects with time.
Substitution of the appropriate approximations of concentrations of point defects into above relations
leads to the following relations:

 

2
2
2
000 2 010 101 010 111 120
2
2 1
V
I I I I r I r I
I
A A A A A A
A
= −
+ + +
( )+ + +
( )
r
r r
A
100 110
+
( )
,
2
2
4
3
2
4
2
3
3
3
4
3
2 3
4
2
0
4
3
1
27 3 27 6 2
I
B
B
B
B
B
B
B B
B
B
B
= −





 − + −
− −





 + − +
1
27 3 27 6 2
2
4
3
2
4
2
3
3
3
4
3
2 3
4
2
0
4
3
B
B
B
B
B
B
B B
B
B
B
.
Here, A
L
t K T I x t V x t d xd t
ijk
i j k
L
 
= −
( ) ( ) ( ) ( )
∫
∫
1
1 1
0
0
Θ
Θ
Θ
, , , B A A A A
I I V r
4 000 000 000 100
2
= −
( ), B AI
3 000
=

4. Pankratov: On prognosis of transport of radiation defects
AJMS/Jul-Sep-2019/Vol 3/Issue 3 74
× + +
( )− + +
( ) +
2 2 1 2 1
000 010 101 100
2
010 101 110 100
A A A A A A A A A
V I r r I r r r r
r V r
A A
100 010 101
2 1
+ +
( )

 

{
+ }
A A
r r
100
2
110 , B A A A A A A A A
V I r I I r I V
2 000 010 101
2
000 010 111 120
2 1 2
= + +
( ) + + +
( )




− 0
010 101 1
+ +
( )
Ar
× − + +
( ) + + +
A A A A A A A A A
I r I r r r r V r
000 110 010 101 110 100 100 010 101
2 1 2 1
(
( )

 
 − + +
( )
A A A
V r V
010 111 120
× − + +
( )−
A A A A A A A A
r r r I r I r r
100
2
100 100 010 111 120 110
2
100
2 , B A A A A
V I r I
1 000 010 101 010
2 1
= + +
( )(
+ + )− + +
( ) + +
( )−
A A A A A A A A
r I r I r V r r
111 120 110 010 101 010 101 100
2 1 2 1 2 A
A A A
V r V
010 111 120
+ +
( )
× − − + +
( ) +
A A A A A A A A A A
r r r I r I r r r V
110 110 110
2
010 111 120 110 100 100 0
2 1
10 101 1
+ +
( )

 

Ar , B AV
0 000
=
× + +
( ) − + +
( ) +
A A A A A A A A A
I r I r I r I V r
010 111 120
2
110 010 111 120 010 101
2 +
+
( )− +
(
1 010 111
A A
V r
+ )
A A
V r
120 110
2
.
Equationsforconcentrationsofsimplestcomplexesofpointdefects(divacanciesΦV
(x,t)anddiinterstitials
ΦI
(x, t)) could be written as:
∂ ( )
∂
=
∂
∂
( )
∂ ( )
∂





 + ( ) ( )− ( )
Φ Φ
Φ
I I
I I I
x t
t x
D T
x t
x
k T I x t k T I
I
, ,
,
,
2
x
x t
x t
t x
D T
x t
x
k T V x t
V V
V V
V
,
, ,
,
,
( )
∂ ( )
∂
=
∂
∂
( )
∂ ( )
∂





 + ( ) ( )−
Φ Φ
Φ
2
k
k T V x t
V ( ) ( )







,
(3)
With boundary and initial conditions,
∂ ( )
∂
=
=
ΦI
x
x t
x
,
0
0 ,
∂ ( )
∂
=
=
ΦV
x
x t
x
,
0
0,
∂ ( )
∂
=
=
ΦI
x L
x t
x
,
0 ,
∂ ( )
∂
=
=
ΦV
x L
x t
x
,
0, (4)
Φ Φ
Φ
Φ
I x N
x x
x
I
I
I
, exp
0
0
2
2
( ) = −
−
( )








, Φ Φ
Φ
Φ
V x N
x x
x
V
V
V
, exp
0
0
2
2
( ) = −
−
( )








.
Here, DΦr
(T) are the diffusion coefficients of the above complexes of radiation defects; kr
(T) are
the parameters of decay of the above complexes. To simplify the solution of the above equations, we
transform them to the following integral form:
Φ
Φ
Φ
Φ
I
I
t
I
t
I
x t
x
D T
x
x
d k T I x d x
I
,
,
, ,
( ) =
∂
∂
( )
∂ ( )
∂
− ( ) ( ) + ( )+
∫ ∫

  
0 0
0
+
+ ( ) ( )
( ) =
∂
∂
( )
∂ ( )
∂
− (
∫
∫
k T I x d
x t
x
D T
x
x
d k T
I I
t
V
V
t
V
V
, ,
,
,
2
0
0
 


Φ
Φ
Φ )
) ( ) + ( )+
+ ( ) ( )













∫
∫
V x d x
k T V x d
t
V
V V
t
, ,
,
,
 
 
0
2
0
0
Φ
(3a)

5. Pankratov: On prognosis of transport of radiation defects
AJMS/Jul-Sep-2019/Vol 3/Issue 3 75
Now, we solve systems of Eqs. (3a) by the method of averaging of function corrections with decreasing
quantity of iteration steps. We used solutions of Eqs. (3) without nonlinear terms and averaged
diffusion coefficients D0IV
as initial-order approximations of solutions of Eqs. (3a). These initial-order
approximations could be solved by standard Fourier approach and could be written as:
( )
2 2
0 0
0 2
1
, cos exp
2


∞
Φ Φ
Φ
=
 
 
Φ = + −
   
   
∑
I I
I
I n
n
F n D t
n x
x t F
L L
,
( )
2 2
0 0
0 2
1
, cos exp
2


∞
Φ Φ
Φ
=
 
 
Φ = + −
   
   
∑
V V
V
V n
n
F n D t
n x
x t F
L L
.
Here, ( )
0
2
,0 cos

Φ
 
= Φ  
 
∫
I
L
n I
nv
F v d v
L L
, ( )
0
2
,0 cos

Φ
 
= Φ  
 
∫
I
L
n I
nv
F v d v
L L
. Substitution of the above
series into Eqs.(3a) gives us a possibility to obtain the first-order approximations of concentrations of
point radiation defects in the following form:
Φ Φ Φ
Φ
1
1
2 2
0
2
I n
n
x t
L
F
n x
L
D T
n D
L
I I
I
, sin exp
( ) = −





 ( ) −



=
∞
∑
π π π τ 


 − ( ) +



∫ ∑
∫ =
∞
d k T
F
F
t
I n
n
t
I
I
τ
0
0
1
0
2
Φ
Φ
×





 −










+ ( )
cos exp cos
,
π π τ
τ
π
n x
L
n D
L
d k T F
n
I
I
I I n
2 2
0
2
Φ
Φ
x
x
L
n D
L
I
n
t





 −









 =
∞
∑
∫ exp
π τ
2 2
0
2
1
0
Φ
+



1
2
0
2
F d
I
Φ  .
Φ Φ Φ
Φ
1
1
2 2
0
2
V n
n
x t
L
F
n x
L
D T
n D
L
V V
V
, sin exp
( ) = −





 ( ) −



=
∞
∑
π π π τ 


 − ( ) +



∫ ∑
∫ =
∞
d k T
F
F
t
V n
n
t
V
V
τ
0
0
1
0
2
Φ
Φ
×





 −










+ ( )
cos exp cos
,
π π τ
τ
π
n x
L
n D
L
d k T F
n
V
V
V V n
2 2
0
2
Φ
Φ
x
x
L
n D
L
V
n
t





 −









 =
∞
∑
∫ exp
π τ
2 2
0
2
1
0
Φ
+



1
2
0
2
F d
V
Φ  .
We determine the second and highest orders of approximations of concentrations of simplest complexes
of point radiation defects framework standard iterative procedure of method of averaging of function
corrections.[10]
In this case, nth-order approximations of concentrations of complexes of defects will
be determined by the following replacement: ΦI
(x,t)→αnΦI
+ΦIn-1
(x,t), ΦV
(x,t)→αnΦV
+ΦVn-1
(x,t) in the
right sides of Eqs. (3a), where αnI
and αnV
are the average values of the considered approximations. In
this situation, the second-order approximations of concentrations of complexes of point defects could be
written as,
Φ
Φ
Φ
2
1
0
2
0
I
I
t
I I
t
I
x t
x
D T
x
x
d k T I x d k T
I
,
,
,
,
( ) =
∂
∂
( )
∂ ( )
∂
+ ( ) ( ) −
∫ ∫

   (
( ) ( ) + ( )
( ) =
∂
∂
( )
∂ ( )
∂
+
∫
∫
I x d x
x t
x
D T
x
x
d k
t
I
V
V
t
V
, ,
,
,
 


0
2
1
0
0
Φ
Φ
Φ
Φ V
V V
t
V
t
V
T V x d k T V x d x
, , , ,
( ) ( ) − ( ) ( ) + ( )






 ∫ ∫
2
0 0
0
    Φ

6. Pankratov: On prognosis of transport of radiation defects
AJMS/Jul-Sep-2019/Vol 3/Issue 3 76
We determine spatiotemporal distribution of temperature, generated during generation of radiation
defects due to radiation processing, by the solution of the following equation:
c T
T x t
t x
T
T x t
x
( )
∂ ( )
∂
=
∂
∂
( )
∂ ( )
∂






, ,
 . (5)
Temperature dependence of heat capacitance c (T) could be approximated by the following function:
c(T)=c0
[1-exp (-T/Td
)], where Td
is the Debye temperature. If current temperature is larger than the Debye
temperature, heat conduction coefficient could be approximated by the following function: λ(T)=λ0
{1+µ
[Td
/T(x,t)]ϕ
}. In the same area of temperatures, one can consider the following relation: c(T) ≈ c0
. The
Eq. (5) is complemented by the following boundary and initial conditions:
∂ ( )
∂
=
=
T x t
x x
,
0
0 ,
∂ ( )
∂
=
=
T x t
x x L
,
0, T(x,0)=Tr
, (6)
Here, Tr
is the room temperature. With account dependence of heat diffusion coefficient on temperature,
one can transform the Eq.(5) to the following differential (6a) or integral (6b) form:
c T x t
t
T T x t
T x t
x
T
T x t
x
d d
0
0
2
2 0
φ
µ λ µφ λ
φ
φ φ
∂ ( )
∂
= ( )
∂ ( )
∂
−
∂ ( )
∂




,
,
, ,



2
(7a)
c
T x t T T x t
T x
x
d T
T x
x
d
t
d
0
0
2
2
0
0
φ
µ λ
τ
τ µφ λ
τ
φ φ φ
, ,
, ,
( ) = ( )
∂ ( )
∂
−
∂ ( )
∂


∫ 




∫
2
0
d
t
τ . (7b)
We solved the Eq. (6b) by method of averaging of function corrections with a decreased quantity
of iterative steps. Framework the approach we used solutions of linear Eqs. (5) with averaged heat
diffusion coefficients λ0
as initial-order approximations of solutions of Eqs. (7b). These initial-order
approximations could be solved by standard Fourier approach and could be written as:
T x t
F
F
n x
L
n t
L c
n
n
0
0
2 2
0
2
0
1
2
, cos exp
( ) = +





 −






=
∞
∑
π π λ
.
Here, ( )
0
2
,0 cos

 
=  
 
∫
L
n
nv
F T v d v
L L
. Substitution of the above series into Eqs. (7b) gives us a possibility
to obtain the first-order approximation of temperature in the following form:
c
T x t
L
F
F
n x
L
n
L c
n
0
1
2
2
0
2 2
0
2
0
2
φ
µ
π π π λ τ
φ
, cos exp
( ) = − +





 −






n
n n
t
n
L c
d
=
∞
=
∞
∑ ∑
∫





 −






1
2 2
0
2
0
1
0
exp
π λ τ
τ
×





 −





 −
n F
n x
L
T T
L
n F
n x
L
n
n d d n
2
0
2
2
2
2 2
cos sin exp
π
µφ λ
π π π
φ φ λ
λ τ τ
τ
0
2
0
1
2
0
L c
T x
x
d
n
t






∂ ( )
∂






=
∞
∑
∫
,
.
Wedeterminethesecondandhighestordersofapproximationoftemperatureframeworkstandarditerative
procedure of method of averaging of function corrections.[10]
In this case, nth-order approximation of
temperature will be determined by following replacement: T (x, t)→αnT
+Tn-1
(x,t) in the right sides of
Eqs. (7b), where αnT
is the average value of the considered approximation. In this situation, the second-
order approximation of temperature could be written as:
T x t T
c
T x t
T x
x
d T
d T
t
d
2 0
0
2 1
2
1
2
0
0
2
, ,
,
( ) = + ( )

 

∂ ( )
∂
−
∫
µ λ
φ
α
τ
τ µ λ
φ
φ φ
c
c
T x
x
d
t
0
1
2
0
1
∂ ( )
∂
















∫
,τ
τ
φ
.

7. Pankratov: On prognosis of transport of radiation defects
AJMS/Jul-Sep-2019/Vol 3/Issue 3 77
Not yet known average value α2T
by solution the following equation:
α µ λ
φ
α
τ
τ µ λ
φ
φ φ
2 0
0
2 1
2
1
2
0
0
2
0
T d T
t
d
T
c
T x t
T x
x
d T
c
T
= + ( )

 

∂ ( )
∂
−
∂
∫ ,
, 1
1
2
0
1
1
0
x
x
d T x t d xd t
t
L
,
,
τ
τ
φ
( )
∂
















− ( )










∫
∫
0
0
Θ
∫ .
The value of this average value depends on the parameter Φ, determined by the experimental data.
DISCUSSION
In this paper, we analyzed spatiotemporal distribution of the concentrations of radiation defects in
doped with ion implantation material. Figure 1 shows the distribution of the concentration of point
radiation defects for two doses: 5×1014
cm−2
(curves 1) and 5×1015
cm−2
(curves 2). The solid lines are
the calculated curves, and the dashed lines are the experimental curves.[8]
According to the experimental
data, the concentration of defects increases to the irradiated sample boundary while having a maximum
in the depth of the sample. Apparently, an increase in the concentration of defects with approach to the
irradiated surface is a consequence of a large number of defects in the near-surface region at the initial
stage of doping. Over time, the defects recombine among themselves and diffuse from the near-surface
region. To account for these changes, the concentration dependence of the defect diffusion coefficient
was used.
CONCLUSION
In this paper, we introduce a model for the redistribution and interaction of point radiation defects between
themselves, as well as their simplest complexes in a material, taking into account the experimentally non-
monotonicity of the distribution of the concentration of radiation defects. To take into account this non-
monotonicity, the previously used model in the literature for the analysis of spatiotemporal distributions
of the concentration of radiation defects was supplemented by the concentration dependence of their
diffusion coefficient.
Figure 1: The distribution of the concentration of radiation defects at a dose 5×1014
cm−2
(curve 1) and a dose 5×1015
cm−2
(curve 2). Solid curves are the calculated results. Dotted curves are experimental results

8. Pankratov: On prognosis of transport of radiation defects
AJMS/Jul-Sep-2019/Vol 3/Issue 3 78
CONFLICTS OF INTERESTS
The authors declare that they have no conflicts of interest.
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