Influence of Overlayers on Depth of Implanted-Heterojunction Rectifiers

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.1, February 2016
DOI : 10.5121/ijitmc.2016.4101 1
INFLUENCE OF OVERLAYERS ON DEPTH OF IM-
PLANTED-HETEROJUNCTION RECTIFIERS
E.L. Pankratov1
, E.A. Bulaeva1,2
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 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.
KEYWORDS
Heterostructures; implanted-junction rectifiers: overlayers.
1. INTRODUCTION
In the present time integration rate of elements of integrated circuits intensively increasing. At the
same 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 elaborated
different 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 ion
doping 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 L
a
fC(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
x
C
D
z
y
t
z
y
x
C
D
y
x
t
z
y
x
C
D
x
t
t
z
y
x
C
C
C
C
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
. (1)
Boundary and initial conditions for the equations are
( ) 0
,
,
,
0
=
∂
∂
=
x
x
t
z
y
x
C
,
( ) 0
,
,
,
=
∂
∂
= x
L
x
x
t
z
y
x
C
,
( ) 0
,
,
,
0
=
∂
∂
=
y
y
t
z
y
x
C
,
( ) 0
,
,
,
=
∂
∂
= y
L
x
y
t
z
y
x
C
,
( ) 0
,
,
,
0
=
∂
∂
=
z
z
t
z
y
x
C
,
( ) 0
,
,
,
=
∂
∂
= z
L
x
z
t
z
y
x
C
, 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 of
heterostructure, 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 defects
could be written as [11-13]
( ) ( )
( )
( ) ( )
( ) 







+
+






+
= 2
*
2
2
*
1
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
V
t
z
y
x
V
V
t
z
y
x
V
T
z
y
x
P
t
z
y
x
C
T
z
y
x
D
D L
C ς
ς
ξ γ
γ
. (3)
Here function DL (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 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 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 ζ1=ζ2=0. We determine spa-
tio-temporal distributions of concentrations of radiation defects by solving the following system
of equations [12,13]
( ) ( ) ( ) ( ) ( ) ( ) ×
−






∂
∂
∂
∂
+






∂
∂
∂
∂
=
∂
∂
T
z
y
x
k
y
t
z
y
x
I
T
z
y
x
D
y
x
t
z
y
x
I
T
z
y
x
D
x
t
t
z
y
x
I
I
I
I
I ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
( ) ( ) ( ) ( ) ( ) ( )
t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
I
T
z
y
x
D
z
t
z
y
x
I V
I
I ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, ,
2
−






∂
∂
∂
∂
+
× (4)
( ) ( ) ( ) ( ) ( ) ( )×
−






∂
∂
∂
∂
+






∂
∂
∂
∂
=
∂
∂
T
z
y
x
k
y
t
z
y
x
V
T
z
y
x
D
y
x
t
z
y
x
V
T
z
y
x
D
x
t
t
z
y
x
V
V
V
V
V ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
( ) ( ) ( ) ( ) ( ) ( )
t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
V
T
z
y
x
D
z
t
z
y
x
V 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)=fρ (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 Dρ(x,y,z,T). The quadric on concentrations terms of Eqs. (4)
describes generation divacancies and diinterstitials. Parameter of recombination of point radiation
defects and parameters of generation of simplest complexes of point radiation defects have been
denoted as the following functions kI,V(x,y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T), respectively.
Now let us calculate distributions of concentrations of divacancies ΦV(x,y,z,t) and diinterstitials
ΦI(x,y,z,t) in space and time by solving the following system of equations [12,13]
( ) ( ) ( ) ( ) ( ) +





 Φ
+





 Φ
=
Φ
Φ
Φ
y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x I
I
I
I
I
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
( ) ( ) ( ) ( ) ( ) ( )
t
z
y
x
I
T
z
y
x
k
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
T
z
y
x
D
z
I
I
I
I
I ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
, −
+





 Φ
+ Φ
∂
∂
∂
∂
(6)
( ) ( ) ( ) ( ) ( ) +





 Φ
+





 Φ
=
Φ
Φ
Φ
y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x V
V
V
V
V
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
( ) ( ) ( ) ( ) ( ) ( )
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
T
z
y
x
k
z
t
z
y
x
T
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
, ΦI (x,y,z,0)=fΦI (x,y,z), ΦV (x,y,z,0)=fΦV (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 kI(x,y,z,T) and kV(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) to
the 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
w
v
x
V
V
w
v
x
V
T
w
v
x
D
u
d
v
d
w
d
t
w
v
u
C
L
L
L
z
y
x
0
2
*
2
2
*
1
,
,
,
,
,
,
1
,
,
,
,
,
,
τ
ς
τ
ς
( )
( )
( ) ( ) ( )
( )
×
∫ ∫ ∫ 





+
+






+
×
t x
L
z
L
L
z
y x z T
z
y
x
P
w
y
u
C
T
w
y
u
D
L
L
z
y
d
x
w
v
x
C
T
w
v
x
P
w
v
x
C
0 ,
,
,
,
,
,
1
,
,
,
,
,
,
,
,
,
,
,
,
1 γ
γ
γ
γ
τ
ξ
τ
∂
τ
∂
τ
ξ
( ) ( )
( )
( ) ( ) ×
∫ ∫ ∫
+






+
+
×
t x
L
y
L
L
z
x x y
T
z
v
u
D
L
L
z
x
d
y
w
y
u
C
V
w
y
u
V
V
w
y
u
V
0
2
*
2
2
*
1 ,
,
,
,
,
,
,
,
,
,
,
,
1 τ
∂
τ
∂
τ
ς
τ
ς
( ) ( )
( )
( )
( )
( ) +






+






+
+
×
y
x
L
L
y
x
d
z
z
v
u
C
T
z
y
x
P
z
v
u
C
V
z
v
u
V
V
z
v
u
V
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
u
d
v
d
w
d
w
v
u
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
n
nC
n
n
n
nC t
e
z
c
y
c
x
c
a
t
z
y
x
C
0
0 ,
,
, .
Here ( ) ( )
[ ]
2
2
2
0
2
2
exp −
−
−
+
+
−
= z
y
x
C
nC
L
L
L
t
D
n
t
e π , cn(χ)=cos(π nχ/Lχ). 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
n
nC
n
n
n
nC
z
y
N
n
nC
n
n
n
C
y z
e
w
c
v
c
x
c
a
L
L
z
y
t
e
z
s
y
s
x
s
n
a
z
y
x
0 1
1
3
2
1
γ
τ
π
( )
( ) ( )
( )
( ) ( ) ( )×
∑






+
+



×
=
N
n
n
n
nC
L v
c
x
s
a
T
w
v
x
D
V
w
v
x
V
V
w
v
x
V
T
w
v
x
P 1
2
*
2
2
*
1 ,
,
,
,
,
,
,
,
,
1
,
,
,
τ
ς
τ
ς
ξ
γ
( ) ( ) ( ) ( ) ( ) ( )
( )
∫ ∫ ∫ ×











∑
+
−
×
=
t x
L
z
L
N
m
mC
m
m
m
mC
z
x
nC
n
x z T
w
y
u
P
e
w
c
y
c
u
c
a
L
L
z
x
d
e
w
c
n
0 1 ,
,
,
1 γ
γ
ξ
τ
τ
τ

5
( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ×
∑






+
+
×
=
τ
τ
τ
ς
τ
ς d
e
w
c
y
s
u
c
n
V
w
y
u
V
V
w
y
u
V
T
w
y
u
D
N
n
nC
n
n
n
L
1
2
*
2
2
*
1
,
,
,
,
,
,
1
,
,
,
( )
( )
( ) ( ) ( ) ( ) ×
∫ ∫ ∫











∑
+
−
×
=
t x
L
y
L
N
n
nC
n
n
n
nC
L
y
x
nC
x y
e
z
c
v
c
u
c
a
T
z
v
u
P
T
z
v
u
D
L
L
y
x
a
0 1
,
,
,
1
,
,
,
γ
γ
τ
ξ
( ) ( )
( )
( ) ( ) ( ) ( ) ×
+
∑






+
+
×
=
z
y
x
N
n
nC
n
n
n
nC
L
L
L
z
y
x
d
e
z
s
v
c
u
c
a
n
V
z
v
u
V
V
z
v
u
V
τ
τ
τ
ς
τ
ς
1
2
*
2
2
*
1
,
,
,
,
,
,
1
( )
∫ ∫ ∫
×
x
L
y
L
z
L
x y z
u
d
v
d
w
d
w
v
u
f ,
, ,
where sn(χ)=sin(πnχ/Lχ). We used condition of orthogonality to determine coefficients an in the
considered 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
n
nC
n
n
n
nC
L
z
y
N
n
nC
nC
z
y
x
x y z
e
z
c
y
c
x
c
a
T
z
y
x
D
L
L
t
e
n
a
L
L
L
0 0 0 0 1
2
1
6
5
2
2
2
1
,
,
,
2
γ
τ
π
π
( )
( ) ( )
( )
( ) ( ) ( ) ( )×
∑






+
+



×
=
N
n
nC
n
n
n
nC
e
z
c
y
c
x
s
n
a
V
z
y
x
V
V
z
y
x
V
T
z
y
x
P 1
2
*
2
2
*
1 2
,
,
,
,
,
,
1
,
,
,
τ
τ
ς
τ
ς
ξ
γ
( ) ( )
[ ] ( ) ( )
[ ] ( )×
∫ ∫ ∫ ∫
−






−
+






−
+
×
t L L L
L
n
z
n
n
y
n
x y z
T
z
y
x
D
d
x
d
y
d
z
d
z
c
n
L
z
s
z
y
c
n
L
y
s
y
0 0 0 0
,
,
,
1
1 τ
π
π
( ) ( ) ( ) ( ) ( )
( )
( )



+
+











∑
+
×
=
*
1
1
,
,
,
1
,
,
,
1
,
,
,
V
z
y
x
V
T
z
y
x
P
e
z
c
y
c
x
c
a
T
z
y
x
D
N
n
nC
n
n
n
nC
L
τ
ς
ξ
τ γ
γ
( )
( )
( ) ( )
( )
( ) ( )
[ ]
∑ ×






−
+






+
+



+
=
N
n
nC
n
x
n
n
a
x
c
n
L
x
s
x
V
z
y
x
V
V
z
y
x
V
V
z
y
x
V
1
2
*
2
2
*
1
2
*
2
2
1
,
,
,
,
,
,
1
,
,
,
π
τ
ς
τ
ς
τ
ς
( ) ( ) ( ) ( ) ( ) ( )
[ ] ×
−






−
+
× 2
2
2
1
2
2 π
τ
π
τ
π
y
x
n
z
n
nC
n
n
n
z
x
L
L
d
x
d
y
d
z
d
z
c
n
L
z
s
z
e
z
c
y
s
x
c
L
L
( ) ( ) ( ) ( )
( )
( )
( )
∫ ∫ ∫ ∫ 


+
+











∑
+
×
=
t L L L N
n
nC
n
n
n
nC
x y z
V
z
y
x
V
T
z
y
x
P
e
z
c
y
c
x
c
a
0 0 0 0
2
*
2
2
1
,
,
,
1
,
,
,
1
τ
ς
ξ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ] ×
∑






−
+



+
=
N
n
n
x
n
n
n
n
nC
L x
c
n
L
x
s
x
z
s
y
c
x
c
n
a
T
z
y
x
D
V
z
y
x
V
1
*
1 1
,
,
,
,
,
,
π
τ
ς
( ) ( )
[ ] ( ) ( ) ( )
[ ]
∑ ∫ ×






−
+
+






−
+
×
=
N
n
L
n
x
n
nC
n
y
n
x
x
c
n
L
x
s
x
d
x
d
y
d
z
d
e
y
c
n
L
y
s
y
1 0
1
1
π
τ
τ
π
( ) ( )
[ ] ( ) ( )
[ ] ( )
∫ ∫






−
+






−
+
×
y z
L L
n
z
n
n
y
n x
d
y
d
z
d
z
y
x
f
z
c
n
L
z
s
z
y
c
n
L
y
s
y
0 0
,
,
1
1
π
π
.
As an example for γ=0 we obtain
( ) ( )
[ ] ( ) ( )
[ ] ( ) ( )
{
∫ ∫ ∫ +






−
+






−
+
=
x y z
L L L
n
n
y
n
n
y
n
nC x
s
x
y
d
z
d
z
y
x
f
z
c
n
L
z
s
z
y
c
n
L
y
s
y
a
0 0 0
,
,
1
1
π
π

6
( )
[ ] ( ) ( ) ( ) ( )
[ ] ( )







∫ ∫ ∫ ∫ ×






−
+



−
×
t L L L
L
n
y
n
n
n
x
n
x y z
T
z
y
x
D
y
c
n
L
y
s
y
y
c
x
s
n
x
d
n
L
x
c
0 0 0 0
,
,
,
1
2
2
1
π
π
( ) ( )
[ ] ( ) ( )
( ) ( )
×






+






+
+






−
+
×
T
z
y
x
P
V
z
y
x
V
V
z
y
x
V
z
c
n
L
z
s
z n
y
n
,
,
,
1
,
,
,
,
,
,
1
1 2
*
2
2
*
1 γ
ξ
τ
ς
τ
ς
π
( ) ( ) ( ) ( ) ( ) ( )
[ ] ( ) ( )×
∫ ∫ ∫ ∫






−
+
+
×
t L L L
n
n
n
y
n
n
nC
nC
n
x y z
z
c
y
s
x
c
n
L
x
s
x
x
c
e
d
e
x
d
y
d
z
d
z
c
0 0 0 0
2
1
π
τ
τ
τ
( ) ( )
[ ]
( )
( ) ( )
( )
×






+
+






+






−
+
× 2
*
2
2
*
1
,
,
,
,
,
,
1
,
,
,
1
1
V
z
y
x
V
V
z
y
x
V
T
z
y
x
P
z
c
n
L
z
s
z n
y
n
τ
ς
τ
ς
ξ
π γ
( ) ( ) ( ) ( ) ( )
[ ] ( ) ( )
{
∫ ∫ ∫ ×






−
+
+
×
t L L
n
n
n
x
n
n
nC
L
x y
y
s
y
c
x
c
n
L
x
s
x
x
c
e
d
x
d
y
d
z
d
T
z
y
x
D
0 0 0
1
,
,
,
π
τ
τ
( )
[ ] ( ) ( )
( )
( )
( )
∫ 


+
+






+



−
+
×
z
L
L
n
n
y
V
z
y
x
V
T
z
y
x
P
T
z
y
x
D
z
s
y
c
n
L
y
0
2
*
2
2
,
,
,
1
,
,
,
1
,
,
,
2
1
τ
ς
ξ
π γ
( ) ( )
1
6
5
2
2
2
*
1
,
,
,
−




−






+ t
e
n
L
L
L
d
x
d
y
d
z
d
V
z
y
x
V
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
n
n
n
n
n
n
n
nC
x
d
y
d
z
d
z
y
x
f
z
c
y
c
x
c
a
0 0 0
2
,
,
4
2
α
β
α
β
,
where ( ) ( ) ( ) ( ) ( ) ( )
( )
∫ ∫ ∫ ∫ ×






+
+
=
t L L L
n
n
n
nC
z
y
n
x y z
V
z
y
x
V
V
z
y
x
V
z
c
y
c
x
s
e
n
L
L
0 0 0 0
2
*
2
2
*
1
2
,
,
,
,
,
,
1
2
2
τ
ς
τ
ς
τ
π
ξ
α
( )
( )
( ) ( )
[ ] ( ) ( )
[ ] ×
+






−
+






−
+
×
n
L
L
d
x
d
y
d
z
d
z
c
n
L
z
s
z
y
c
n
L
y
s
y
T
z
y
x
P
T
z
y
x
D z
x
n
z
n
n
y
n
L
2
2
1
1
,
,
,
,
,
,
π
ξ
τ
π
π
( ) ( ) ( ) ( )
[ ] ( ) ( )
( )
( ) ( )
[ ] ×
∫ ∫ ∫ ∫






−
−






−
+
×
t L L L
n
z
n
L
n
n
x
n
n
nC
x y z
z
c
n
L
z
s
z
T
z
y
x
P
T
z
y
x
D
z
c
x
c
n
L
x
s
x
x
c
e
0 0 0 0
1
,
,
,
,
,
,
1
π
π
τ
( )
( )
( ) ( )
( )
( ) ×
+






+
+
×
n
L
L
d
x
d
y
d
y
s
z
d
V
z
y
x
V
V
z
y
x
V
T
z
y
x
P
T
z
y
x
D y
x
n
L
2
2
*
2
2
*
1
2
2
,
,
,
,
,
,
1
,
,
,
,
,
,
π
ξ
τ
τ
ς
τ
ς
( ) ( ) ( ) ( ) ( )
( )
( ) ( )
( )
×
∫ ∫ ∫ ∫ 





+
+
×
t L L L
L
n
n
n
nC
x y z
V
z
y
x
V
V
z
y
x
V
T
z
y
x
P
T
z
y
x
D
z
s
y
c
x
c
e
0 0 0 0
2
*
2
2
*
1
,
,
,
,
,
,
1
,
,
,
,
,
,
2
τ
ς
τ
ς
τ
( ) ( )
[ ] ( ) ( )
[ ] τ
π
π
d
x
d
y
d
z
d
y
c
n
L
y
s
y
x
c
n
L
x
s
x n
y
n
n
x
n






−
+






−
+
× 1
1 ,
( )×
∫
=
t
nC
z
y
n e
n
L
L
0
2
2
τ
π
β

7
( ) ( ) ( ) ( )
[ ] ( ) ( ) ( )
( )
∫ ∫ ∫ ×






+
+






−
+
×
x y z
L L L
n
n
y
n
n
n
V
z
y
x
V
V
z
y
x
V
z
c
y
c
n
L
y
s
y
y
c
x
s
0 0 0
2
*
2
2
*
1
,
,
,
,
,
,
1
1
2
τ
ς
τ
ς
π
( ) ( ) ( )
[ ] ( ) ( ) ( )×
∫ ∫ ∫
+






−
+
×
t L L
n
n
nC
z
x
n
z
n
L
x y
y
s
x
c
e
n
L
L
d
x
d
y
d
z
d
z
c
n
L
z
s
z
T
z
y
x
D
0 0 0
2
2
2
1
,
,
, τ
π
τ
π
( ) ( )
[ ] ( ) ( ) ( ) ( )
( )
×
∫ 





+
+






−
+
×
z
L
n
L
n
x
n
V
z
y
x
V
V
z
y
x
V
z
c
T
z
y
x
D
x
c
n
L
x
s
x
0
2
*
2
2
*
1
,
,
,
,
,
,
1
,
,
,
1
τ
ς
τ
ς
π
( ) ( )
[ ] ( ) ( ) ( )
[ ] ×
∫ ∫






−
+
+






−
+
×
t L
n
x
n
nC
y
x
n
z
n
x
x
c
n
L
x
s
x
e
n
L
L
d
x
d
y
d
z
d
z
c
n
L
z
s
z
0 0
2
1
2
1
π
τ
π
τ
π
( ) ( ) ( )
[ ] ( ) ( ) ( )
( )
∫ ∫ ×






+
+






−
+
×
y z
L L
L
n
y
n
n
V
z
y
x
V
V
z
y
x
V
T
z
y
x
D
y
c
n
L
y
s
y
x
c
0 0
2
*
2
2
*
1
,
,
,
,
,
,
1
,
,
,
1
τ
ς
τ
ς
π
( ) ( ) ( ) 6
5
2
2
2
2 n
t
e
L
L
L
d
x
d
y
d
y
c
z
d
z
s nC
z
y
x
n
n 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
v
d
w
d
x
w
v
x
I
T
w
v
x
D
L
L
z
y
u
d
v
d
w
d
t
w
v
u
I
L
L
L
z
y
x
0
,
,
,
,
,
,
,
,
, τ
τ
( ) ( ) ( ) ( )×
∫ ∫ ∫
−
∫ ∫ ∫
∂
∂
+
x
L
y
L
z
L
V
I
z
y
x
t x
L
z
L
I
z
x x y z
x z
t
w
v
u
I
T
w
v
u
k
L
L
L
z
y
x
d
u
d
w
d
x
w
y
u
I
T
w
y
u
D
L
L
z
x
,
,
,
,
,
,
,
,
,
,
,
, ,
0
τ
τ
( ) ( ) ( ) ×
−
∫ ∫ ∫
∂
∂
+
×
z
y
x
t x
L
y
L
I
y
x
L
L
L
z
y
x
d
u
d
v
d
T
z
v
u
D
z
z
v
u
I
L
L
y
x
u
d
v
d
w
d
t
w
v
u
V
x y
0
,
,
,
,
,
,
,
,
, 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
u
d
v
d
w
d
w
v
u
f
L
L
L
z
y
x
u
d
v
d
w
d
t
w
v
u
I
T
w
v
u
k ,
,
,
,
,
,
,
, 2
, (4a)
( ) ( ) ( ) +
∫ ∫ ∫
∂
∂
=
∫ ∫ ∫
t y
L
z
L
V
z
y
x
L
y
L
z
L
z
y
x y z
x y z
d
v
d
w
d
x
w
v
x
V
T
w
v
x
D
L
L
z
y
u
d
v
d
w
d
t
w
v
u
V
L
L
L
z
y
x
0
,
,
,
,
,
,
,
,
, τ
τ
( ) ( ) ( ) ×
∫ ∫ ∫
∂
∂
+
∫ ∫ ∫
∂
∂
+
t x
L
y
L
y
x
t x
L
z
L
V
z
x x y
x z z
z
v
u
V
L
L
y
x
d
u
d
w
d
x
w
y
u
V
T
w
y
u
D
L
L
z
x
0
0
,
,
,
,
,
,
,
,
,
τ
τ
τ
( ) ( ) ( ) ( ) −
∫ ∫ ∫
−
×
x
L
y
L
z
L
V
I
z
y
x
V
x y z
u
d
v
d
w
d
t
w
v
u
V
t
w
v
u
I
T
w
v
u
k
L
L
L
z
y
x
d
u
d
v
d
T
z
v
u
D ,
,
,
,
,
,
,
,
,
,
,
, ,
τ
( ) ( ) ( )
∫ ∫ ∫
+
∫ ∫ ∫
−
x
L
y
L
z
L
V
z
y
x
x
L
y
L
z
L
V
V
z
y
x x y z
x y z
u
d
v
d
w
d
w
v
u
f
L
L
L
z
y
x
u
d
v
d
w
d
t
w
v
u
V
T
w
v
u
k
L
L
L
z
y
x
,
,
,
,
,
,
,
, 2
, .
We determine spatio-temporal distributions of concentrations of point defects as the same series

8
( ) ( ) ( ) ( ) ( )
∑
=
=
N
n
n
n
n
n
n t
e
z
c
y
c
x
c
a
t
z
y
x
1
0 ,
,
, ρ
ρ
ρ .
Parameters anρ 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
n
n
nI
z
y
x
N
n
nI
n
n
n
nI
y z
v
d
w
d
T
w
v
x
D
z
c
y
c
a
L
L
L
z
y
t
e
z
s
y
s
x
s
n
a
z
y
x
1 0
1
3
3
,
,
,
π
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) −
∑ ∫ ∫ ∫
−
×
=
N
n
t x
L
z
L
I
n
n
nI
n
nI
z
y
x
n
nI
x z
d
u
d
w
d
T
w
y
u
D
z
c
x
c
e
y
s
a
L
L
L
z
x
x
s
d
e
1 0
,
,
, τ
τ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( )×
∫ ∫ ∫
−
∑ ∫ ∫ ∫
−
=
x
L
y
L
z
L
I
I
N
n
t x
L
y
L
I
n
n
nI
n
nI
z
y
x x y z
x y
T
v
v
u
k
d
u
d
v
d
T
z
v
u
D
y
c
x
c
e
z
s
a
L
L
L
y
x
,
,
,
,
,
, ,
1 0
τ
τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( )×
∫ ∫ ∫ ∑
−





∑
×
=
=
x
L
y
L
z
L
N
n
n
n
n
nI
z
y
x
z
y
x
N
n
nI
n
n
n
nI
x y z
w
c
v
c
u
c
a
L
L
L
z
y
x
L
L
L
z
y
x
u
d
v
d
w
d
t
e
w
c
v
c
u
c
a
1
2
1
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∫ ∫ ∫
+
∑
×
=
x
L
y
L
z
L
I
N
n
V
I
nV
n
n
n
nV
nI
x y z
u
d
v
d
w
d
w
v
u
f
u
d
v
d
w
d
T
v
v
u
k
t
e
w
c
v
c
u
c
a
t
e ,
,
,
,
,
1
,
z
y
x L
L
L
z
y
x
×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫
−
=
∑
=
=
N
n
t y
L
z
L
V
n
n
nV
z
y
x
N
n
nV
n
n
n
nV
y z
v
d
w
d
T
w
v
x
D
z
c
y
c
a
L
L
L
z
y
t
e
z
s
y
s
x
s
n
a
z
y
x
1 0
1
3
3
,
,
,
π
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) −
∑ ∫ ∫ ∫
−
×
=
N
n
t x
L
z
L
V
n
n
nV
n
nV
z
y
x
n
nV
x z
d
u
d
w
d
T
w
y
u
D
z
c
x
c
e
y
s
a
L
L
L
z
x
x
s
d
e
1 0
,
,
, τ
τ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( )×
∫ ∫ ∫
−
∑ ∫ ∫ ∫
−
=
x
L
y
L
z
L
V
V
N
n
t x
L
y
L
V
n
n
nV
n
nV
z
y
x x y z
x y
T
v
v
u
k
d
u
d
v
d
T
z
v
u
D
y
c
x
c
e
z
s
a
L
L
L
y
x
,
,
,
,
,
, ,
1 0
τ
τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( )×
∫ ∫ ∫ ∑
−





∑
×
=
=
x
L
y
L
z
L
N
n
n
n
n
nI
z
y
x
z
y
x
N
n
nI
n
n
n
nV
x y z
w
c
v
c
u
c
a
L
L
L
z
y
x
L
L
L
z
y
x
u
d
v
d
w
d
t
e
w
c
v
c
u
c
a
1
2
1
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∫ ∫ ∫
+
∑
×
=
x
L
y
L
z
L
V
N
n
V
I
nV
n
n
n
nV
nI
x y z
u
d
v
d
w
d
w
v
u
f
u
d
v
d
w
d
T
v
v
u
k
t
e
w
c
v
c
u
c
a
t
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 anρ. 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
y
n
nI
x
N
n
nI
nI
z
y
x
x y
y
c
n
L
y
s
y
L
x
c
n
a
L
t
e
n
a
L
L
L
1 0 0 0
2
1
6
5
2
2
2
1
2
2
2
2
1
2
1
π
π
π
( ) ( ) ( )
[ ] ( ) ( )
{
∑ ∫ ∫ +
−
∫






−
+
×
=
N
n
t L
n
nI
y
L
nI
n
z
n
I
x
z
x
s
x
n
a
L
d
e
x
d
y
d
z
d
z
c
n
L
z
s
z
T
z
y
x
D
1 0 0
2
0
2
2
1
1
2
,
,
,
π
τ
τ
π
( )
[ ] ( ) ( ) ( )
[ ] ( )
[ ]×
∫ ∫ −






−
+
+



−
+
+
y z
L L
n
n
z
n
z
I
n
x
x
y
c
z
d
z
c
n
L
z
s
z
L
T
z
y
x
D
x
c
n
L
L
0 0
2
1
1
2
2
2
,
,
,
1
2
π
π
( ) ( ) ( ) ( )
[ ] ( ) −
∫






−
+
+
×
z
L
nI
n
z
n
z
I
nI d
e
x
d
y
d
z
d
z
c
n
L
z
s
z
L
T
z
y
x
D
d
e
x
d
y
d
0
1
2
2
2
,
,
, τ
τ
π
τ
τ

9
( ) ( )
[ ] ( ) ( )
[ ] ×
∑ ∫ ∫ ∫






−
+
+






−
+
+
−
=
N
n
t L L
n
y
n
y
n
x
n
x
nI
z
x y
y
c
n
L
y
s
y
L
x
c
n
L
x
s
x
L
n
a
L 1 0 0 0
2
1
2
2
2
1
2
2
2
2
1
π
π
π
( )
[ ] ( ) ( ) ( ) ( )
[ ]
∑ ∫



+
−
+
−
∫ −
×
=
N
n
L
n
x
x
nI
nI
L
nI
I
n
x
z
x
c
n
L
L
t
e
a
d
e
x
d
y
d
z
d
T
z
y
x
D
z
c
1 0
2
0
1
2
2
2
,
,
,
2
1
π
τ
τ
( )} ( ) ( )
[ ] ( ) ( )
[ ]
∫ ∫



+
−
+






−
+
+
+
y z
L L
n
z
z
I
I
n
y
n
y
n z
c
n
L
L
T
z
y
x
k
y
c
n
L
y
s
y
L
x
s
x
0 0
, 1
2
2
,
,
,
1
2
2
2
2
π
π
( )} ( ) ( ) ( ) ( )
[ ] {
∑ ∫ ∫ +






−
+
+
−
+
=
N
n
L L
y
n
x
n
x
nV
nI
nV
nI
n
x y
L
x
c
n
L
x
s
x
L
t
e
t
e
a
a
x
d
y
d
z
d
z
s
z
1 0 0
1
2
2
2
2
π
( ) ( )
[ ] ( ) ( ) ( )
[ ]
∫ ×






−
+
+



−
+
+
z
L
n
z
n
z
V
I
n
y
n z
d
z
c
n
L
z
s
z
L
T
z
y
x
k
y
c
n
L
y
s
y
0
, 1
2
2
2
,
,
,
1
2
2
2
π
π
( ) ( )
[ ] ( ) ( )
[ ] ( )×
∑ ∫ ∫ ∫






−
+






−
+
+
×
=
N
n
L L L
I
n
y
n
n
x
n
x y z
T
z
y
x
f
y
c
n
L
y
s
y
x
c
n
L
x
s
x
x
d
y
d
1 0 0 0
,
,
,
1
1
π
π
( ) ( )
[ ] x
d
y
d
z
d
z
c
n
L
z
s
z
L n
z
n
z






−
+
+
× 1
2
2
2
π
( ) ( )
[ ] ( ) ( )
[ ] ×
∑ ∫ ∫ ∫






−
+
+
−
−
=
∑
−
=
=
N
n
t L L
n
y
n
y
n
nV
x
N
n
nV
nV
z
y
x
x y
y
c
n
L
y
s
y
L
x
c
n
a
L
t
e
n
a
L
L
L
1 0 0 0
2
1
6
5
2
2
2
1
2
2
2
2
1
2
1
π
π
π
( ) ( ) ( )
[ ] ( ) ( )
{
∑ ∫ ∫ +
−
∫






−
+
×
=
N
n
t L
n
nV
y
L
nV
n
z
n
V
x
z
x
s
x
n
a
L
d
e
x
d
y
d
z
d
z
c
n
L
z
s
z
T
z
y
x
D
1 0 0
2
0
2
2
1
1
2
,
,
,
π
τ
τ
π
( )
[ ] ( ) ( ) ( )
[ ] ( )
[ ]×
∫ ∫ −






−
+
+



−
+
+
y z
L L
n
n
z
n
z
V
n
x
x
y
c
z
d
z
c
n
L
z
s
z
L
T
z
y
x
D
x
c
n
L
L
0 0
2
1
1
2
2
2
,
,
,
1
2
π
π
( ) ( ) ( ) ( )
[ ] ( ) −
∫






−
+
+
×
z
L
nV
n
z
n
z
V
nV
d
e
x
d
y
d
z
d
z
c
n
L
z
s
z
L
T
z
y
x
D
d
e
x
d
y
d
0
1
2
2
2
,
,
, τ
τ
π
τ
τ
( ) ( )
[ ] ( ) ( )
[ ] ×
∑ ∫ ∫ ∫






−
+
+






−
+
+
−
=
N
n
t L L
n
y
n
y
n
x
n
x
nV
z
x y
y
c
n
L
y
s
y
L
x
c
n
L
x
s
x
L
n
a
L 1 0 0 0
2
1
2
2
2
1
2
2
2
2
1
π
π
π
( )
[ ] ( ) ( ) ( ) ( )
[ ]
∑ ∫



+
−
+
−
∫ −
×
=
N
n
L
n
x
x
nV
nV
L
nV
V
n
x
z
x
c
n
L
L
t
e
a
d
e
x
d
y
d
z
d
T
z
y
x
D
z
c
1 0
2
0
1
2
2
2
,
,
,
2
1
π
τ
τ
( )} ( ) ( )
[ ] ( ) ( )
[ ]
∫ ∫



+
−
+






−
+
+
+
y z
L L
n
z
z
V
V
n
y
n
y
n z
c
n
L
L
T
z
y
x
k
y
c
n
L
y
s
y
L
x
s
x
0 0
, 1
2
2
,
,
,
1
2
2
2
2
π
π
( )} ( ) ( ) ( ) ( )
[ ] {
∑ ∫ ∫ +






−
+
+
−
+
=
N
n
L L
y
n
x
n
x
nV
nI
nV
nI
n
x y
L
x
c
n
L
x
s
x
L
t
e
t
e
a
a
x
d
y
d
z
d
z
s
z
1 0 0
1
2
2
2
2
π
( ) ( )
[ ] ( ) ( ) ( )
[ ]
∫ ×






−
+
+



−
+
+
z
L
n
z
n
z
V
I
n
y
n z
d
z
c
n
L
z
s
z
L
T
z
y
x
k
y
c
n
L
y
s
y
0
, 1
2
2
2
,
,
,
1
2
2
2
π
π
( ) ( )
[ ] ( ) ( )
[ ] ( )×
∑ ∫ ∫ ∫






−
+






−
+
+
×
=
N
n
L L L
V
n
y
n
n
x
n
x y z
T
z
y
x
f
y
c
n
L
y
s
y
x
c
n
L
x
s
x
x
d
y
d
1 0 0 0
,
,
,
1
1
π
π

10
( ) ( )
[ ] x
d
y
d
z
d
z
c
n
L
z
s
z
L n
z
n
z






−
+
+
× 1
2
2
2
π
.
In the final form relations for required parameters could be written as
( )







 −
+
−
+
±
+
−
=
A
y
b
y
b
A
b
b
A
b
a nI
nV
nI
2
3
4
2
3
4
3
4
4
4
λ
γ
,
nI
nI
nI
nI
nI
nI
nI
nV
a
a
a
a
χ
λ
δ
γ +
+
−
=
2
,
where ( ) ( ) ( ) ( )
[ ] ( )
{
∫ ∫ ∫ +
+






−
+
+
=
x y z
L L L
y
n
n
x
n
x
n
n L
y
s
y
x
c
n
L
x
s
x
L
T
z
y
x
k
t
e
0 0 0
, 2
1
2
2
2
,
,
,
2
π
γ ρ
ρ
ρ
ρ
( )
[ ] ( ) ( )
[ ] x
d
y
d
z
d
z
c
n
L
z
s
z
L
y
c
n
L
n
z
n
z
n
y






−
+
+



−
+ 1
2
2
2
1
2
2 π
π
, ( )×
∫
=
t
n
x
n e
n
L 0
2
2
1
τ
π
δ ρ
ρ
( ) ( )
[ ] ( ) ( )
[ ] ( ) [
∫ ∫ ∫ −






−
+






−
+
×
x y z
L L L
n
z
n
n
y
n y
d
z
d
T
z
y
x
D
z
c
n
L
z
s
z
y
c
n
L
y
s
y
0 0 0
1
,
,
,
1
2
1
2
ρ
π
π
( )] ( ) ( ) ( )
[ ] ( )
[ ] {
∫ ∫ ∫ ∫ +
−






−
+
+
+
−
t L L L
z
n
n
x
n
x
n
y
n
x y z
L
y
c
x
c
n
L
x
s
x
L
e
n
L
d
x
d
x
c
0 0 0 0
2
2
1
1
2
2
2
1
2
π
τ
π
τ ρ
( ) ( )
[ ] ( ) ( ) ( )
{
∫ ∫ +
+



−
+
+
t L
n
n
z
n
z
n
x
x
s
x
e
n
L
d
x
d
y
d
z
d
T
z
y
x
D
z
c
n
L
z
s
z
0 0
2
2
2
1
,
,
,
1
2
2
2 τ
π
τ
π
ρ
ρ
( )
[ ] ( ) ( )
[ ] ( )
[ ] ( ) ×
∫ ∫ −






−
+
+



−
+
+
y z
L L
n
n
y
n
y
n
x
x z
d
T
z
y
x
D
z
c
y
c
n
L
y
s
y
L
x
c
n
L
L
0 0
,
,
,
2
1
1
2
1
2 ρ
π
π
( )
t
e
n
L
L
L
d
x
d
y
d n
z
y
x
ρ
π
τ 6
5
2
2
2
−
× , ( ) ( )
[ ] ( )
[ ]
∫ ∫



+
−
+






−
+
=
x y
L L
n
y
y
n
x
n
nIV y
c
n
L
L
x
c
n
L
x
s
x
0 0
1
2
2
1
π
π
χ
( )} ( ) ( ) ( )
[ ] ( ) ( )
t
e
t
e
x
d
y
d
z
d
z
c
n
L
z
s
z
L
T
z
y
x
k
y
s
y nV
nI
L
n
z
n
z
V
I
n
z
∫






−
+
+
+
0
, 1
2
2
2
,
,
,
2
π
,
( ) ( )
[ ] ( ) ( )
[ ] ( ) ( )
[ ]
∫ ∫ ∫ ×






−
+






−
+






−
+
=
x y z
L L L
n
z
n
n
y
n
n
x
n
n z
c
n
L
z
s
z
y
c
n
L
y
s
y
x
c
n
L
x
s
x
0 0 0
1
1
1
π
π
π
λ ρ
( ) x
d
y
d
z
d
T
z
y
x
f ,
,
,
ρ
× , 2
2
4 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 −
−
= 2
3
2 ,
2
2
3
4
8 b
b
y
A −
+
= , ( ) 2
2
2
2 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
3
3 3
2
3 3
2
3b
b
q
p
q
q
p
q
y −
+
+
−
−
+
= , 2
4
2
3
4
2
9
3
b
b
b
b
p
−
= ,
( ) 3
4
2
4
1
3
2
3
3
54
27
9
2 b
b
b
b
b
b
q +
−
= .
We determine distributions of concentrations of simplest complexes of radiation defects in space
and time as the following functional series
( ) ( ) ( ) ( ) ( )
∑
=
Φ
=
Φ
N
n
n
n
n
n
n t
e
z
c
y
c
x
c
a
t
z
y
x
1
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
v
d
w
d
x
w
v
x
T
w
v
x
D
u
d
v
d
w
d
t
w
v
u
L
L
L
z
y
x
0
,
,
,
,
,
,
,
,
, τ
∂
τ
∂
( ) ( ) ( )
∫ ∫ ∫ ×
+
∫ ∫ ∫
Φ
+
× Φ
Φ
t x
L
y
L
I
y
x
t x
L
z
L
I
I
z
x
z
y x y
x z
T
z
v
u
D
L
L
y
x
d
u
d
w
d
y
w
y
u
T
w
y
u
D
L
L
z
x
L
L
z
y
0
0
,
,
,
,
,
,
,
,
, τ
∂
τ
∂
( ) ( ) ( ) −
∫ ∫ ∫
+
Φ
×
x
L
y
L
z
L
I
I
z
y
x
I
x y z
u
d
v
d
w
d
w
v
u
I
T
w
v
u
k
L
L
L
z
y
x
d
u
d
v
d
z
z
v
u
τ
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
u
d
v
d
w
d
w
v
u
f
L
L
L
z
y
x
u
d
v
d
w
d
w
v
u
I
T
w
v
u
k
L
L
L
z
y
x
,
,
,
,
,
,
,
, τ
( ) ( ) ( ) ×
∫ ∫ ∫
Φ
=
∫ ∫ ∫Φ Φ
t y
L
z
L
V
V
x
L
y
L
z
L
V
z
y
x y z
x y z
d
v
d
w
d
x
w
v
x
T
w
v
x
D
u
d
v
d
w
d
t
w
v
u
L
L
L
z
y
x
0
,
,
,
,
,
,
,
,
, τ
∂
τ
∂
( ) ( ) ( )
∫ ∫ ∫ ×
+
∫ ∫ ∫
Φ
+
× Φ
Φ
t x
L
y
L
V
y
x
t x
L
z
L
V
V
z
x
z
y x y
x z
T
z
v
u
D
L
L
y
x
d
u
d
w
d
y
w
y
u
T
w
y
u
D
L
L
z
x
L
L
z
y
0
0
,
,
,
,
,
,
,
,
, τ
∂
τ
∂
( ) ( ) ( ) −
∫ ∫ ∫
+
Φ
×
x
L
y
L
z
L
V
V
z
y
x
V
x y z
u
d
v
d
w
d
w
v
u
V
T
w
v
u
k
L
L
L
z
y
x
d
u
d
v
d
z
z
v
u
τ
τ
∂
τ
∂
,
,
,
,
,
,
,
,
, 2
,
( ) ( ) ( )
∫ ∫ ∫
+
∫ ∫ ∫
− Φ
x
L
y
L
z
L
V
z
y
x
x
L
y
L
z
L
V
z
y
x x y z
x y z
u
d
v
d
w
d
w
v
u
f
L
L
L
z
y
x
u
d
v
d
w
d
w
v
u
V
T
w
v
u
k
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
n
n
nI
n
I
n
z
y
x
N
n
nI
n
n
n
I
n
y z
w
c
v
c
t
e
x
s
a
n
L
L
L
z
y
t
e
z
s
y
s
x
s
n
a
z
y
x
1 0
1
3
3
π
π
( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫
−
×
=
Φ
Φ
Φ
N
n
t x
L
z
L
I
n
n
I
n
z
y
x
I
x z
d
u
d
w
d
T
w
v
u
D
w
c
u
c
a
L
L
L
z
x
d
v
d
w
d
T
w
v
x
D
1 0
,
,
,
,
,
, τ
π
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) +
∑ ∫ ∫ ∫
−
×
=
Φ
Φ
Φ
Φ
N
n
t x
L
y
L
I
n
n
I
n
n
I
n
z
y
x
I
n
n
x y
d
u
d
v
d
T
z
v
u
D
v
c
u
c
t
e
z
s
a
n
L
L
L
y
x
t
e
y
s
n
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
u
d
v
d
w
d
w
v
u
f
u
d
v
d
w
d
w
v
u
I
T
w
v
u
k
L
L
L
z
y
x
,
,
,
,
,
,
,
, 2
, τ
( ) ( )
∫ ∫ ∫
−
×
x
L
y
L
z
L
I
z
y
x
z
y
x x y z
u
d
v
d
w
d
w
v
u
I
T
w
v
u
k
L
L
L
z
y
x
L
L
L
z
y
x
τ
,
,
,
,
,
,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ ∫ ∫
−
=
∑
−
=
Φ
=
Φ
N
n
t y
L
z
L
n
n
nV
n
V
n
z
y
x
N
n
nV
n
n
n
V
n
y z
w
c
v
c
t
e
x
s
a
n
L
L
L
z
y
t
e
z
s
y
s
x
s
n
a
z
y
x
1 0
1
3
3
π
π
( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫
−
×
=
Φ
Φ
N
n
t x
L
z
L
V
n
n
z
y
x
V
x z
d
u
d
w
d
T
w
v
u
D
w
c
u
c
n
L
L
L
z
x
d
v
d
w
d
T
w
v
x
D
1 0
,
,
,
,
,
, τ
π
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫
−
×
=
Φ
Φ
Φ
Φ
N
n
t x
L
y
L
V
n
n
V
n
n
z
y
x
V
n
n
V
n
x y
d
u
d
v
d
T
z
v
u
D
v
c
u
c
t
e
z
s
n
L
L
L
y
x
t
e
y
s
a
1 0
,
,
, τ
π

12
( ) ( ) ( ) ×
∫ ∫ ∫
+
∫ ∫ ∫
+
× Φ
Φ
x
L
y
L
z
L
V
x
L
y
L
z
L
V
V
z
y
x
V
n
x y z
x y z
u
d
v
d
w
d
w
v
u
f
u
d
v
d
w
d
w
v
u
V
T
w
v
u
k
L
L
L
z
y
x
a ,
,
,
,
,
,
,
, 2
,
τ
( ) ( )
∫ ∫ ∫
−
×
x
L
y
L
z
L
V
z
y
x
z
y
x x y z
u
d
v
d
w
d
w
v
u
V
T
w
v
u
k
L
L
L
z
y
x
L
L
L
z
y
x
τ
,
,
,
,
,
, .
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 of
terms N. In the common case equations for the required coefficients could be written as
( ) ( )
[ ] ( ) ( )
[ ] ×
∑∫ ∫ ∫






−
+
+
−
−
=
∑
−
=
=
Φ
Φ
N
n
t L L
n
y
n
y
n
x
N
n
I
n
I
n
z
y
x
x y
y
c
n
L
y
s
y
L
x
c
L
t
e
n
a
L
L
L
1 0 0 0
1
6
5
2
2
2
1
2
2
2
2
1
2
1
π
π
π
( ) ( ) ( )
[ ] ( ) ( )
{
∑∫ ∫ +
−
∫






−
+
×
=
Φ
Φ
Φ
N
n
t L
n
L
I
n
n
z
n
I
I
n
x
z
x
s
x
d
e
x
d
y
d
z
d
z
c
n
L
z
s
z
T
z
y
x
D
n
a
1 0 0
0
2
2
2
1
1
2
,
,
,
π
τ
τ
π
( )
[ ] ( )
[ ] ( ) ( ) ( )
[ ] ×
∫ ∫






−
+
−



−
+
+ Φ
y z
L L
n
z
n
I
n
n
x
x x
d
y
d
z
d
z
c
n
L
z
s
z
T
z
y
x
D
y
c
x
c
n
L
L
0 0
1
2
,
,
,
2
1
1
2
2 π
π
( ) ( ) ( )
[ ] ( ) ( )
[ ]
∑ ∫ ∫ ∫



+
−
+






−
+
−
×
=
Φ
Φ
Φ
N
n
t L L
n
y
n
n
x
n
I
n
x
y
I
n
I
n
x y
y
c
n
L
y
s
y
x
c
n
L
x
s
x
n
a
L
d
L
n
e
a
1 0 0 0
2
2
1
2
2
2
1
2
2
1
π
π
π
τ
τ
} ( )
[ ] ( ) ( ) ( ) ( )
[ ]
∑ ∫ ∫



+
−
+
∫ −
+
=
Φ
Φ
Φ
Φ
N
n
t L
n
x
I
n
I
n
L
I
n
I
n
y
x
z
x
c
n
L
e
n
a
d
e
x
d
y
d
z
d
T
z
y
x
D
y
c
L
1 0 0
3
3
0
1
2
1
,
,
,
2
1
π
τ
π
τ
τ
( )} ( ) ( )
[ ] ( ) ( ) ( )
[ ]
∫



+
−
∫






−
+
+
z
y L
n
z
I
I
L
n
y
n
n z
c
n
L
T
z
y
x
k
t
z
y
x
I
y
c
n
L
y
s
y
x
s
x
0
,
2
0
1
2
,
,
,
,
,
,
1
2 π
π
( )} ( ) ( ) ( )
[ ] ( )
[ ]
∑ ∫ ∫ ∫



+
−






−
+
−
+
=
Φ
Φ
N
n
t L L
n
y
n
x
n
I
n
I
n
n
x y
y
c
n
L
x
c
n
L
x
s
x
e
n
a
x
d
y
d
z
d
z
s
z
1 0 0 0
3
3
1
2
1
2
1
π
π
τ
π
( )} ( ) ( )
[ ] ( ) ( ) ×
∑
+
∫






−
+
+
=
Φ
N
n
I
n
L
I
n
z
n
n
n
a
x
d
y
d
z
d
t
z
y
x
I
T
z
y
x
k
z
c
n
L
z
s
z
y
s
y
z
1
3
3
0
1
,
,
,
,
,
,
1
2 π
π
( ) ( ) ( )
[ ] ( ) ( )
[ ] ( )
[ ]
∫ ∫ ∫ ∫



+
−






−
+






−
+
× Φ
t L L L
n
z
n
y
n
n
x
n
I
n
x y z
z
c
n
L
y
c
n
L
y
s
y
x
c
n
L
x
s
x
e
0 0 0 0
1
2
1
2
1
2 π
π
π
τ
( )} ( ) x
d
y
d
z
d
z
y
x
f
z
s
z I
n
,
,
Φ
+
( ) ( )
[ ] ( ) ( )
[ ] ×
∑∫ ∫ ∫






−
+
+
−
−
=
∑
−
=
=
Φ
Φ
N
n
t L L
n
y
n
y
n
x
N
n
V
n
V
n
z
y
x
x y
y
c
n
L
y
s
y
L
x
c
L
t
e
n
a
L
L
L
1 0 0 0
1
6
5
2
2
2
1
2
2
2
2
1
2
1
π
π
π
( ) ( ) ( )
[ ] ( ) ( )
{
∑∫ ∫ +
−
∫






−
+
×
=
Φ
Φ
Φ
N
n
t L
n
L
V
n
n
z
n
V
V
n
x
z
x
s
x
d
e
x
d
y
d
z
d
z
c
n
L
z
s
z
T
z
y
x
D
n
a
1 0 0
0
2
2
2
1
1
2
,
,
,
π
τ
τ
π
( )
[ ] ( )
[ ] ( ) ( ) ( )
[ ] ×
∫ ∫






−
+
−



−
+
+ Φ
y z
L L
n
z
n
V
n
n
x
x x
d
y
d
z
d
z
c
n
L
z
s
z
T
z
y
x
D
y
c
x
c
n
L
L
0 0
1
2
,
,
,
2
1
1
2
2 π
π
( ) ( ) ( )
[ ] ( ) ( )
[ ]
∑ ∫ ∫ ∫



+
−
+






−
+
−
×
=
Φ
Φ
Φ
N
n
t L L
n
y
n
n
x
n
V
n
x
y
V
n
V
n
x y
y
c
n
L
y
s
y
x
c
n
L
x
s
x
n
a
L
d
L
n
e
a
1 0 0 0
2
2
1
2
2
2
1
2
2
1
π
π
π
τ
τ

13
} ( )
[ ] ( ) ( ) ( ) ( )
[ ]
∑ ∫ ∫



+
−
+
∫ −
+
=
Φ
Φ
Φ
Φ
N
n
t L
n
x
V
n
V
n
L
V
n
V
n
y
x
z
x
c
n
L
e
n
a
d
e
x
d
y
d
z
d
T
z
y
x
D
y
c
L
1 0 0
3
3
0
1
2
1
,
,
,
2
1
π
τ
π
τ
τ
( )} ( ) ( )
[ ] ( ) ( ) ( )
[ ]
∫



+
−
∫






−
+
+
z
y L
n
z
V
V
L
n
y
n
n z
c
n
L
T
z
y
x
k
t
z
y
x
V
y
c
n
L
y
s
y
x
s
x
0
,
2
0
1
2
,
,
,
,
,
,
1
2 π
π
( )} ( ) ( ) ( )
[ ] ( )
[ ]
∑ ∫ ∫ ∫



+
−






−
+
−
+
=
Φ
Φ
N
n
t L L
n
y
n
x
n
V
n
V
n
n
x y
y
c
n
L
x
c
n
L
x
s
x
e
n
a
x
d
y
d
z
d
z
s
z
1 0 0 0
3
3
1
2
1
2
1
π
π
τ
π
( )} ( ) ( )
[ ] ( ) ( ) ×
∑
+
∫






−
+
+
=
Φ
N
n
V
n
L
V
n
z
n
n
n
a
x
d
y
d
z
d
t
z
y
x
V
T
z
y
x
k
z
c
n
L
z
s
z
y
s
y
z
1
3
3
0
1
,
,
,
,
,
,
1
2 π
π
( ) ( ) ( )
[ ] ( ) ( )
[ ] ( )
[ ]
∫ ∫ ∫ ∫



+
−






−
+






−
+
× Φ
t L L L
n
z
n
y
n
n
x
n
V
n
x y z
z
c
n
L
y
c
n
L
y
s
y
x
c
n
L
x
s
x
e
0 0 0 0
1
2
1
2
1
2 π
π
π
τ
( )} ( ) x
d
y
d
z
d
z
y
x
f
z
s
z V
n
,
,
Φ
+ .
3. DISCUSSION
In this section we compare spatial distributions of concentrations of dopant in a heterostructure
with 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 same
time 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 epitaxial
layer.
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 heterostructure
with 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 2
describes 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 and
at 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 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 Russia, educa-
tional fellowship for scientific research of Government of Nizhny Novgorod region of Russia and
educational fellowship for scientific research of Nizhny Novgorod State University of Architec-
ture and Civil Engineering.
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15
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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 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 Radiophysical Department of Nizhny
Novgorod State University. She has 90 published papers in area of her researches.

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