1. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
DOI:10.5121/msej.2016.3301 1
ON DECREASING OF DIMENSIONS OF FIELD-
EFFECT TRANSISTORS WITH SEVERAL
SOURCES
E.L. Pankratov, E.A. Bulaeva
Nizhny Novgorod State University, 2 Gagarin avenue, Nizhny Novgorod, 603950, Russia
ABSTRACT
We analyzed mass and heat transport during manufacturing field-effect heterotransistors with several
sources to decrease their dimensions. Framework the result of manufacturing it is necessary to manufacture
heterostructure with specific configuration. After that it is necessary to dope required areas of the hetero-
structure by diffusion or ion implantation to manufacture the required type of conductivity (p or n). After
the doping it is necessary to do optimize annealing. We introduce an analytical approach to prognosis mass
and heat transport during technological processes. Using the approach leads to take into account nonlineari-
ty of mass and heat transport and variation in space and time (at one time) physical parameters of these
processes
KEYWORDS
Field-effect transistor, transistor with several channels, increasing of compactness of transistors
1. INTRODUCTION
Now several problems of solid state electronic intensively solving. The problems are increasing
of density of elements of integrated circuits and at the same time decreasing of dimensions of
these elements [1-4], increasing performance [5-7] and increasing reliability [8,9]. Now one can
find intensive development of both power electronic devices and logical elements. In this paper
we consider an approach to manufacture more compact field-effect heterotransistor with several
sources. Framework the approach it is necessary to manufacture a heterostructure. The hetero-
structure consist of a substrate and an epitaxial layer (see Fig. 1). Several sections have been
manufactured into the epitaxial layer. These sections manufactured by using other materials (see
Fig. 1). The sections have been doped by diffusion or ion implantation to obtain required type of
conductivity (n or p). After the doping one can manufacture a field-effect transistor framework
the considered heterostructure so as it is shown on the Fig. 1. The doping should be finished by
annealing of dopant and/or radiation defects. The annealing should be optimized. The optimiza-
tion attracted an interest to manufacture more compact distributions of concentrations of dopant.
Framework the paper we formulate conditions to increase compactness and at the same time to
increase homogeneity of distribution of concentration of dopant in enriched by the dopant area.
Fig. 1. Structure of a field-effect heterotransistor. Top side of the structure

2. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
2
2. METHOD OF SOLUTION
To solve our aim we determine distribution of concentration of dopant C(x,y,z,t) in space and
time. To determine the distribution we solve the following boundary problem
( ) ( ) ( ) ( )






∂
∂
∂
∂
+






∂
∂
∂
∂
+






∂
∂
∂
∂
=
∂
∂
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 ,
,
,
,
,
,
,
,
,
,
,
,
(1)
( ) 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)=fC(x,y,z). (2)
Here T is the temperature of annealing; DС is the dopant diffusion coefficient. Dopant diffusion
coefficient takes another value in other materials. Heating and cooling of heterostructure (see
Arrhenius law) also leads to changing of value of diffusion coefficient. We consider following
approximation of concentrational dependences of dopant diffusion coefficient [10-12]
( ) ( )
( )
( ) ( )
( ) 







+
+






+
= 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; function P (x,y,z,T) describes dependences of limit of solubility on coordinate
and temperature; parameter γ ∈[1,3] is integer and depends on properties of materials; function V
(x,y,z,t) describes distribution of concentration of radiation vacancies on space and time with
equilibrium distribution V*
. Dependence of dopant diffusion coefficient on concentration of do-
pant has been investigated and described in details in [10]. Diffusion of dopant gives a possibility
to dope materials without generation radiation defects. In this situation ζ1= ζ2= 0. Ion doping of
dopant leads to generation radiation defects. Distributions of concentrations of point radiation
defects have been determined by solving the following boundary problem [11,12]
( ) ( ) ( ) ( ) ( ) ( ) ×
−






∂
∂
∂
∂
+






∂
∂
∂
∂
=
∂
∂
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
−






∂
∂
∂
∂
+
×
( ) 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)

3. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
3
Here ρ=I,V; distribution of concentration of radiation interstitials in space and time describes by
the function I (x,y,z,t); terms of Eqs.(4) with quadric concentrations of point radiation defects
(V2
(x,y,z,t) and I2
(x,y,z,t)) correspond to generation of simplest complexes of radiation defects
(divacancies and diinterstitials); temperature and spatial dependences of diffusion coefficients of
point radiation defects describe by functions Dρ(x,y,z,T); temperature and spatial dependences of
parameter of recombination of point radiation defects describe by function kI,V(x,y,z,T); functions
kI,I(x,y,z,T) and kV,V(x,y,z,T) describe spatial and temperature dependences of parameters of gener-
ation of simplest complexes of point radiation defects.
Concentrations of divacancies ΦV (x,y,z,t) and dinterstitials ΦI (x,y,z,t) have been calculated by
solution of the following boundary problem [11,12]
( ) ( ) ( ) ( ) ( ) +





 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
∂
∂
∂
∂
( )
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)
Functions DΦρ(x,y,z,T) describe spatial and temperature dependences of diffusion coefficients of
simplest complexes of point radiation defects; functions kI(x,y,z,T) and kV (x,y,z,T) describe spatial
and temperature dependences of parameters of decay of the above complexes.
Now let us consider equivalent integro-differential form of Eq.(1)
( ) ( ) ( ) ( )
( )
×
∫ ∫ ∫








+
+
=
∫ ∫ ∫
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
x
z
y x z y
w
y
u
C
T
w
y
u
D
L
L
z
x
L
L
z
y
d
x
w
v
x
C
T
w
v
x
P
w
v
x
C
0
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
∂
τ
∂
τ
∂
τ
∂
τ
ξ γ
γ
( ) ( )
( )
( )
( )
( )
∫ ∫ ∫ ×
+






+








+
+
×
t x
L
y
L
L
y
x x y
T
z
v
u
D
L
L
y
x
d
T
z
y
x
P
w
y
u
C
V
w
y
u
V
V
w
y
u
V
0
2
*
2
2
*
1 ,
,
,
,
,
,
,
,
,
1
,
,
,
,
,
,
1 τ
τ
ξ
τ
ς
τ
ς γ
γ
( ) ( )
( )
( )
( )
( ) ×
+






+








+
+
×
z
y
x L
L
L
z
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
x y z
u
d
v
d
w
d
w
v
u
f ,
, . (1a)

4. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
4
We used the Bubnov-Galerkin approach [13] to calculate solution of the above equation. To use
the approach we consider 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 n χ/Lχ). Number of terms N of the
above series is finite. The series is almost coincides with solution of Eq.(1) in the linear case (i.e.
with ξ = 0) and average value of dopant diffusion coefficient D0. Framework the approach we
substitute the above series into Eq.(1a). After the substitution we obtain
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
∫ ∫ ∫ ×











∑
+
−
=
∑
=
=
t y
L
z
L
N
n
nC
n
n
n
nC
N
n
nC
n
n
n
C
y z T
w
v
x
P
e
w
c
v
c
x
c
a
t
e
z
s
y
s
x
s
n
a
z
y
x
0 1
1
3
2
,
,
,
1 γ
γ
ξ
τ
π
( ) ( )
( )
( ) ( ) ( ) ( ) ( ) −
∑








+
+
×
=
z
y
N
n
nC
n
n
n
nC
L
L
L
z
y
d
e
w
c
v
c
x
s
a
n
T
w
v
x
D
V
w
v
x
V
V
w
v
x
V
1
2
*
2
2
*
1 ,
,
,
,
,
,
,
,
,
1 τ
τ
τ
ς
τ
ς
( ) ( )
( )
( ) ( ) ( ) ( )
∫ ∫ ∫



×





∑
+








+
+
−
=
t x
L
z
L
N
m
mC
m
m
m
mC
z
x x z
e
w
c
y
c
u
c
a
V
w
y
u
V
V
w
y
u
V
L
L
z
x
0 1
2
*
2
2
*
1 1
,
,
,
,
,
,
1
γ
τ
τ
ς
τ
ς
( )
( ) ( ) ( ) ( ) ( ) ( ) ×
∫ ∫ ∫
−
∑



×
=
t x
L
y
L
L
y
x
N
n
nC
n
n
n
nC
L
x y
T
z
v
u
D
L
L
y
x
d
e
w
c
y
s
u
c
a
n
T
w
y
u
D
T
w
y
u
P 0
1
,
,
,
,
,
,
,
,
,
τ
τ
ξ
γ
( )
( ) ( ) ( ) ( ) ( ) ( )
( )
×








+
+











∑
+
×
=
2
*
2
2
*
1
1
,
,
,
,
,
,
1
,
,
,
1
V
z
v
u
V
V
z
v
u
V
e
z
c
v
c
u
c
a
T
z
v
u
P
N
n
nC
n
n
n
nC
τ
ς
τ
ς
τ
ξ γ
γ
( ) ( ) ( ) ( ) ( )
∫ ∫ ∫
+
∑
×
=
x
L
y
L
z
L
z
y
x
N
n
nC
n
n
n
nC
x y z
u
d
v
d
w
d
w
v
u
f
L
L
L
z
y
x
d
e
z
s
v
c
u
c
a
n ,
,
1
τ
τ . (8)
Here sn(χ)=sin(πnχ/Lχ). To determine coefficients an it is necessary to use orthogonality condi-
tion of terms of the above series framework scale of heterostructure. Using the condition leads to
the following equations to calculate of coefficients an for any quantity of terms N
( ) ( ) ( ) ( ) ( ) ( )
( )
∫ ∫ ∫ ∫ ×











∑
+
−
=
∑
−
=
=
t L L L N
n
nC
n
n
n
nC
L
N
n
nC
nC
z
y
x
x y z
T
z
y
x
P
e
z
c
y
c
x
c
a
T
z
y
x
D
t
e
n
a
L
L
L
0 0 0 0 1
1
6
5
2
2
2
,
,
,
1
,
,
, γ
γ
ξ
τ
π
( ) ( )
( )
( ) ( ) ( ) ( )
[ ] ×
∑






−
+








+
+
×
=
N
n
n
y
n
n
n
nC
z
y
y
c
n
L
y
s
y
y
c
x
s
n
a
V
z
y
x
V
V
z
y
x
V
L
L
1
2
*
2
2
*
1
2
1
2
,
,
,
,
,
,
1
2 π
τ
ς
τ
ς
π
( ) ( ) ( ) ( )
[ ] ( ) ( ) ( ) ( ) ×
∫ ∫ ∫ ∫








∑
−






−
+
×
=
t L L L N
n
nC
n
n
n
nC
n
z
n
nC
n
x y z
e
z
c
y
c
x
c
a
d
x
d
y
d
z
d
z
c
n
L
z
s
z
e
z
c
0 0 0 0 1
1
γ
τ
τ
π
τ
( )
( ) ( ) ( )
( )
( )



+
+








+
+



+
× *
1
2
*
2
2
*
1
,
,
,
1
,
,
,
,
,
,
1
,
,
,
1
,
,
, V
z
y
x
V
V
z
y
x
V
V
z
y
x
V
T
z
y
x
D
T
z
y
x
P
L
τ
ς
τ
ς
τ
ς
ξ
γ
( )
( )
( ) ( )
[ ] ( ) ( )
[ ] ( ) ( ) ( )
∑ ×






−
+






−
+




+
=
N
n
n
n
n
n
z
n
n
x
n
z
nC
z
c
y
s
x
c
z
c
n
L
z
s
z
x
c
n
L
x
s
x
n
L
a
V
z
y
x
V
1
2
2
*
2
2 2
1
1
2
,
,
,
π
π
π
τ
ς
( ) ( ) ( ) ( ) ( )
( )
∫ ∫ ∫ ∫ ×











∑
+
−
×
=
t L L L N
n
nC
n
n
n
nC
y
x
nC
x y z
T
z
y
x
P
e
z
c
y
c
x
c
a
L
L
d
x
d
y
d
z
d
e
0 0 0 0 1
2
,
,
,
1
2 γ
γ
ξ
τ
π
τ
τ

5. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
5
( ) ( )
( )
( ) ( ) ( ) ( )
[ ] ×
∑






−
+








+
+
×
=
N
n
n
x
n
n
nC
L x
c
n
L
x
s
x
x
c
n
a
V
z
y
x
V
V
z
y
x
V
T
z
y
x
D
1
*
1
2
*
2
2 1
,
,
,
,
,
,
1
,
,
,
π
τ
ς
τ
ς
( ) ( ) ( ) ( )
[ ] ( ) ( ) ( )
[ ]
∑ ∫ ×






−
+
+






−
+
×
=
N
n
L
n
x
n
nC
n
y
n
n
n
x
x
c
n
L
x
s
x
d
x
d
y
d
z
d
e
y
c
n
L
y
s
y
z
s
y
c
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
. (9)
Now we consider several examples. For γ=0 we obtain
( ) ( )
[ ] ( ) ( )
[ ] ( ) ( )
∫ ∫ ∫



×
+






−
+






−
+
=
x y z
L L L
x
n
n
y
n
n
y
n
nC
n
L
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
π
( )
[ ]} ( ) ( ) ( ) ( ) ( ) ( )
[ ] ( )
{









∫ ∫ ∫ ∫ +






−
+
−
×
t L L L
n
n
y
n
L
n
n
n
n
x y z
y
s
y
z
c
n
L
z
s
z
T
z
y
x
D
z
c
y
c
x
s
n
x
d
x
c
0 0 0 0
1
,
,
,
2
2
1
π
( )
[ ] ( ) ( )
( ) ( )
( ) ×






+








+
+



−
+ x
d
y
d
z
d
z
c
T
z
y
x
P
V
z
y
x
V
V
z
y
x
V
y
c
n
L
n
n
y
,
,
,
1
,
,
,
,
,
,
1
1 2
*
2
2
*
1 γ
ξ
τ
ς
τ
ς
π
( ) ( ) ( ) ( ) ( )
[ ] ( ) ( ) ( ) ( )
[ ] ×
∫ ∫ ∫ ∫






−
+






−
+
+
×
t L L L
n
y
n
n
n
n
y
n
n
nC
nC
x y z
z
c
n
L
z
s
z
z
c
y
s
x
c
n
L
x
s
x
x
c
e
d
e
0 0 0 0
1
2
1
π
π
τ
τ
τ
( )
( )
( ) ( )
( )
+








+
+






+
× τ
τ
ς
τ
ς
ξ
γ
d
x
d
y
d
z
d
V
z
y
x
V
V
z
y
x
V
T
z
y
x
P
T
z
y
x
DL 2
*
2
2
*
1
,
,
,
,
,
,
1
,
,
,
1
,
,
,
( ) ( ) ( ) ( )
[ ] ( ) ( ) ( )
[ ] ( ) ×
∫ ∫ ∫ ∫






−
+






−
+
+
t L L L
L
n
y
n
n
n
x
n
n
nC
x y z
T
z
y
x
D
y
c
n
L
y
s
y
y
c
x
c
n
L
x
s
x
x
c
e
0 0 0 0
,
,
,
1
1
π
π
τ
( )
( )
( )
( )
( )
1
6
5
2
2
2
*
1
2
*
2
2
,
,
,
,
,
,
1
,
,
,
1
2
−




−













+
+






+
×
n
L
L
L
d
x
d
y
d
z
d
V
z
y
x
V
V
z
y
x
V
T
z
y
x
P
z
s z
z
z
n
π
τ
τ
ς
τ
ς
ξ
γ
.(10)
For γ=1 calculation of parameters an leads to the following results
( ) ( ) ( ) ( )
∫ ∫ ∫
+
±
−
=
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
. (11)
Here ( ) ( )
( )
( )
( ) ( )
( )
∫ ∫ ∫ ∫ ×








+
+
=
t L L L
L
n
n
z
y
n
x y z
V
z
y
x
V
V
z
y
x
V
T
z
y
x
P
T
z
y
x
D
y
c
x
s
n
L
L
0 0 0 0
2
*
2
2
*
1
2
,
,
,
,
,
,
1
,
,
,
,
,
,
2
2
τ
ς
τ
ς
π
ξ
α
( ) ( ) ( )
[ ] ( ) ( )
[ ] ( ) ×
+






−
+






−
+
×
n
L
L
d
e
x
d
y
d
z
d
z
c
n
L
z
s
z
y
c
n
L
y
s
y
z
c z
x
nC
n
z
n
n
y
n
n 2
2
1
1
π
ξ
τ
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
y
s
x
c
n
L
x
s
x
x
c
e
0 0 0 0
1
,
,
,
,
,
,
2
1
π
π
τ
( ) ( ) ( )
( )
( ) ( ) ( )×
∫ ∫ ∫
+








+
+
×
t L L
n
n
nC
y
x
n
x y
y
c
x
c
e
n
L
L
d
x
d
y
d
z
d
V
z
y
x
V
V
z
y
x
V
z
c
0 0 0
2
2
*
2
2
*
1
2
,
,
,
,
,
,
1 τ
π
ξ
τ
τ
ς
τ
ς

6. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
6
( )
( )
( )
( ) ( )
( )
( ) ( )
[ ] ( )
{
∫ ×






−
+








+
+
×
z
L
n
n
x
n
L
n y
s
x
c
n
L
x
s
x
V
z
y
x
V
V
z
y
x
V
T
z
y
x
P
T
z
y
x
D
z
s
0
2
*
2
2
*
1 1
,
,
,
,
,
,
1
,
,
,
,
,
,
2
π
τ
ς
τ
ς
( )
[ ] τ
π
d
x
d
y
d
z
d
y
c
n
L
y n
y



−
+
× 1 , ( ) ( ) ( ) ( ) ( ) ×
∫ ∫ ∫ ∫
=
t L L L
L
n
n
n
nC
z
y
n
x y z
T
z
y
x
D
z
c
y
c
x
s
e
n
L
L
0 0 0 0
2
,
,
,
2
2
τ
π
β
( ) ( )
( )
( ) ( )
[ ] ( )
[ ] ( ) ×






+
−






−
+








+
+
× y
s
y
y
c
n
L
z
d
z
c
n
L
z
s
z
V
z
y
x
V
V
z
y
x
V
n
n
y
n
z
n 1
1
,
,
,
,
,
,
1 2
*
2
2
*
1
π
π
τ
ς
τ
ς
( ) ( ) ( ) ( )
[ ] ( ) ( ) ( )
∫ ∫ ∫ ∫ 


+
+






−
+
+
×
t L L L
n
n
n
x
n
n
nC
z
x
x y z
V
z
y
x
V
z
c
y
s
x
c
n
L
x
s
x
x
c
e
n
L
L
d
x
d
y
d
0 0 0 0
*
1
2
,
,
,
1
2
1
2
τ
ς
π
τ
π
τ
( )
( )
( ) ( ) ( )
[ ] ( )
∫ ×
+






−
+




+
t
nC
y
x
n
z
n
L e
n
L
L
d
x
d
y
d
z
d
z
c
n
L
z
s
z
T
z
y
x
D
V
z
y
x
V
0
2
2
*
2
2
2
1
,
,
,
,
,
,
τ
π
τ
π
τ
ς
( ) ( )
[ ] ( ) ( )
[ ] ( ) ( )
( )
∫ ∫ ∫ ×








+
+






−
+






−
+
×
x y z
L L L
n
y
n
n
x
n
V
z
y
x
V
V
z
y
x
V
y
c
n
L
y
s
y
x
c
n
L
x
s
x
0 0 0
2
*
2
2
*
1
,
,
,
,
,
,
1
1
1
τ
ς
τ
ς
π
π
( ) ( ) ( ) ( ) ( ) 6
5
2
2
2
,
,
,
2 n
t
e
L
L
L
d
x
d
x
c
y
d
y
c
z
d
T
z
y
x
D
z
s nC
z
y
x
n
n
L
n π
τ −
× .
It could be used the same approach to calculate values of parameters an for larger values of the
parameterγ. However the relations became more bulky and will not be present in the paper. The
considered approach gives a possibility to calculate distributions of concentrations of dopant and
radiation defects without joining of the above concentration on interfaces of the considered hete-
rostructure.
We solved equations of the system (4) by using Bubnov-Galerkin approach. To use the approach
we previously transform the differential equations to the following integro-differential form
( ) ( ) ( ) ×
+
∫ ∫ ∫
∂
∂
=
∫ ∫ ∫
z
x
t y
L
z
L
I
z
y
x
L
y
L
z
L
z
y
x L
L
z
x
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
y z
x y z 0
,
,
,
,
,
,
,
,
, τ
τ
( ) ( ) ( ) ( ) −
∫ ∫ ∫
∂
∂
+
∫ ∫ ∫
∂
∂
×
t x
L
y
L
I
y
x
t x
L
z
L
I
x y
x z
d
u
d
v
d
z
z
v
u
I
T
z
v
u
D
L
L
y
x
d
u
d
w
d
x
w
y
u
I
T
w
y
u
D
0
0
,
,
,
,
,
,
,
,
,
,
,
, τ
τ
τ
τ
( ) ( ) ( ) ( ) ×
∫ ∫ ∫
−
∫ ∫ ∫
−
x
L
y
L
z
L
I
I
z
y
x
x
L
y
L
z
L
V
I
z
y
x x y z
x y z
T
w
v
u
k
L
L
L
z
y
x
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
,
,
,
,
,
,
,
,
,
,
,
, ,
,
( ) ( )
∫ ∫ ∫
+
×
x
L
y
L
z
L
I
z
y
x 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 ,
,
,
,
,
2
(4a)
( ) ( ) ( ) ×
+
∫ ∫ ∫
∂
∂
=
∫ ∫ ∫
z
x
t y
L
z
L
V
z
y
x
L
y
L
z
L
z
y
x L
L
z
x
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
y z
x y z 0
,
,
,
,
,
,
,
,
, τ
τ
( ) ( ) ( ) ( ) −
∫ ∫ ∫
∂
∂
+
∫ ∫ ∫
∂
∂
×
t x
L
y
L
V
y
x
t x
L
z
L
V
x y
x z
d
u
d
v
d
z
z
v
u
V
T
z
v
u
D
L
L
y
x
d
u
d
w
d
x
w
y
u
V
T
w
y
u
D
0
0
,
,
,
,
,
,
,
,
,
,
,
, τ
τ
τ
τ
( ) ( ) ( ) ( ) ×
∫ ∫ ∫
−
∫ ∫ ∫
−
x
L
y
L
z
L
V
V
z
y
x
x
L
y
L
z
L
V
I
z
y
x x y z
x y z
T
w
v
u
k
L
L
L
z
y
x
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
,
,
,
,
,
,
,
,
,
,
,
, ,
,
( ) ( )
∫ ∫ ∫
+
×
x
L
y
L
z
L
V
z
y
x 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 ,
,
,
,
,
2
.
Now we consider solutions of the above integro-differential equations as the following series

7. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
7
( ) ( ) ( ) ( ) ( )
∑
=
=
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 .
Here anρ are coefficients, which should be determined. After substitution of the series into Eqs.
(4a) one can obtain
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫
−
=
∑
=
=
N
n
t y
L
z
L
I
n
n
nI
nI
z
y
x
N
n
nI
n
n
n
nI
y z
d
v
d
w
d
T
w
v
x
D
z
c
y
c
e
a
L
L
L
z
y
t
e
z
s
y
s
x
s
n
a
z
y
x
1 0
1
3
3
,
,
, τ
τ
π
π
( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑
−
∑ ∫ ∫ ∫
−
×
=
=
N
n
n
nI
z
y
x
N
n
t x
L
z
L
I
n
n
nI
n
nI
z
y
x
n z
s
a
L
L
L
y
x
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
x z 1
1 0
,
,
,
π
τ
τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∫ ∫ ∫ 




∑
−
∫ ∫ ∫
×
=
x
L
y
L
z
L
N
n
nI
n
n
n
nI
z
y
x
t x
L
y
L
I
n
n
nI
x y z
x y
t
e
w
c
v
c
u
c
a
L
L
L
z
y
x
d
u
d
v
d
T
z
v
u
D
y
c
x
c
e
2
1
0
,
,
, τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∫ ∫ ∫ ∑ ∑
−
×
= =
x
L
y
L
z
L
N
n
N
n
n
n
n
nV
nI
n
n
n
nI
z
y
x
I
I
x y z
w
c
v
c
u
c
a
t
e
w
c
v
c
u
c
a
L
L
L
z
y
x
u
d
v
d
w
d
T
v
v
u
k
1 1
, ,
,
,
( ) ( ) ( )
∫ ∫ ∫
+
×
x
L
y
L
z
L
I
z
y
x
V
I
nV
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
v
v
u
k
t
e ,
,
,
,
,
, (12)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫
−
=
∑
=
=
N
n
t y
L
z
L
V
n
n
nV
nV
z
y
x
N
n
nV
n
n
n
nV
y z
d
v
d
w
d
T
w
v
x
D
z
c
y
c
e
a
L
L
L
z
y
t
e
z
s
y
s
x
s
n
a
z
y
x
1 0
1
3
3
,
,
, τ
τ
π
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑
−
∑ ∫ ∫ ∫
−
×
=
=
N
n
n
nV
z
y
x
N
n
t x
L
z
L
V
n
n
nV
n
nV
z
y
x
n z
s
a
L
L
L
y
x
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
x z 1
1 0
,
,
,
π
τ
τ
π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∫ ∫ ∫ 




∑
−
∫ ∫ ∫
×
=
x
L
y
L
z
L
N
n
nV
n
n
n
nV
z
y
x
t x
L
y
L
V
n
n
nV
x y z
x y
t
e
w
c
v
c
u
c
a
L
L
L
z
y
x
d
u
d
v
d
T
z
v
u
D
y
c
x
c
e
2
1
0
,
,
, τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∫ ∫ ∫ ∑ ∑
−
×
= =
x
L
y
L
z
L
N
n
N
n
n
n
n
nV
nI
n
n
n
nI
z
y
x
V
V
x y z
w
c
v
c
u
c
a
t
e
w
c
v
c
u
c
a
L
L
L
z
y
x
u
d
v
d
w
d
T
v
v
u
k
1 1
, ,
,
,
( ) ( ) ( )
∫ ∫ ∫
+
×
x
L
y
L
z
L
V
z
y
x
V
I
nV
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
v
v
u
k
t
e ,
,
,
,
,
, .
We use orthogonality condition of functions in the above series on scale of the heterostructure to
calculate coefficients anρ. Using the condition gives a possibility to obtain equations for calcula-
tion the above coefficients for any quantity N of terms of considered series
( ) ( ) ( )
[ ] ( ) ( )
[ ] ×
∑ ∫ ∫ ∫






−
+
+
−
−
=
∑
−
=
=
N
n
t L L
n
y
n
y
n
nI
nI
x
N
n
nI
nI
z
y
x
x y
y
c
n
L
y
s
y
L
x
c
e
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
τ
π
π
( ) ( ) ( )
[ ] ( )
[ ]
∑ ∫ ∫



+
+
−
−
∫






−
+
×
=
N
n
t L
x
n
x
nI
y
L
n
z
n
I
x
z
L
x
c
n
L
n
a
L
d
x
d
y
d
z
d
z
c
n
L
z
s
z
T
z
y
x
D
1 0 0
2
0
1
2
2
1
1
2
,
,
,
π
π
τ
π
( )} ( ) ( ) ( )
[ ] ( )
[ ] ( ) −
∫ ∫ −






−
+
+
+
y z
L L
nI
n
n
z
n
z
I
n d
e
x
d
y
d
y
c
z
d
z
c
n
L
z
s
z
L
T
z
y
x
D
x
s
x
0 0
2
1
1
2
2
2
,
,
,
2 τ
τ
π
( ) ( ) ( )
[ ] ( ) ( )
[ ] ×
∑ ∫ ∫ ∫






−
+
+






−
+
+
−
=
N
n
t L L
n
y
n
y
n
x
n
x
nI
nI
z
x y
y
c
n
L
y
s
y
L
x
c
n
L
x
s
x
L
e
n
a
L 1 0 0 0
2
1
2
2
2
1
2
2
2
2
1
π
π
τ
π

8. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
8
( )
[ ] ( ) ( ) ( )
[ ] ( ) ×
∑ ∫






+
−
+
−
∫ −
×
=
N
n
L
n
n
x
x
nI
nI
L
I
n
x
z
x
s
x
x
c
n
L
L
t
e
a
d
x
d
y
d
z
d
T
z
y
x
D
z
c
1 0
2
0
2
1
2
2
2
,
,
,
2
1
π
τ
( ) ( )
[ ] ( ) ( )
[ ] ( ) −
∫ ∫






+
−
+






−
+
+
×
y z
L L
n
n
z
z
I
I
n
y
n
y x
d
y
d
z
d
z
s
z
z
c
n
L
L
T
z
y
x
k
y
c
n
L
y
s
y
L
0 0
, 2
1
2
2
,
,
,
1
2
2
2
π
π
( ) ( ) ( ) ( )
[ ] ( ) ( )
[ ] ×
∑ ∫ ∫






−
+
+






−
+
+
−
=
N
n
L L
n
y
n
y
n
x
n
x
nV
nI
nV
nI
x y
y
c
n
L
y
s
y
L
x
c
n
L
x
s
x
L
t
e
t
e
a
a
1 0 0
1
2
2
2
1
2
2
2
π
π
( ) ( ) ( )
[ ] ( ) ( )
[ ] ×
∑ ∫






−
+
+
∫






−
+
+
×
=
N
n
L
n
x
n
L
n
z
n
z
V
I
x
z
x
c
n
L
x
s
x
x
d
y
d
z
d
z
c
n
L
z
s
z
L
T
z
y
x
k
1 0
0
, 1
1
2
2
2
,
,
,
π
π
( ) ( )
[ ] ( ) ( ) ( )
[ ]
∫ ∫






−
+
+






−
+
×
y z
L L
n
z
n
z
I
n
y
n x
d
y
d
z
d
z
c
n
L
z
s
z
L
T
z
y
x
f
y
c
n
L
y
s
y
0 0
1
2
2
2
,
,
,
1
π
π
(13)
( ) ( ) ( )
[ ] ( ) ( )
[ ] ×
∑ ∫ ∫ ∫






−
+
+
−
−
=
∑
−
=
=
N
n
t L L
n
y
n
y
n
nV
nV
x
N
n
nV
nV
z
y
x
x y
y
c
n
L
y
s
y
L
x
c
e
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
τ
π
π
( ) ( ) ( )
[ ] ( )
[ ]
∑ ∫ ∫



+
+
−
−
∫






−
+
×
=
N
n
t L
x
n
x
nV
y
L
n
z
n
V
x
z
L
x
c
n
L
n
a
L
d
x
d
y
d
z
d
z
c
n
L
z
s
z
T
z
y
x
D
1 0 0
2
0
1
2
2
1
1
2
,
,
,
π
π
τ
π
( )} ( ) ( ) ( )
[ ] ( )
[ ] ( ) −
∫ ∫ −






−
+
+
+
y z
L L
nV
n
n
z
n
z
V
n d
e
x
d
y
d
y
c
z
d
z
c
n
L
z
s
z
L
T
z
y
x
D
x
s
x
0 0
2
1
1
2
2
2
,
,
,
2 τ
τ
π
( ) ( ) ( )
[ ] ( ) ( )
[ ] ×
∑ ∫ ∫ ∫






−
+
+






−
+
+
−
=
N
n
t L L
n
y
n
y
n
x
n
x
nV
nV
z
x y
y
c
n
L
y
s
y
L
x
c
n
L
x
s
x
L
e
n
a
L 1 0 0 0
2
1
2
2
2
1
2
2
2
2
1
π
π
τ
π
( )
[ ] ( ) ( ) ( )
[ ] ( ) ×
∑ ∫






+
−
+
−
∫ −
×
=
N
n
L
n
n
x
x
nV
nV
L
V
n
x
z
x
s
x
x
c
n
L
L
t
e
a
d
x
d
y
d
z
d
T
z
y
x
D
z
c
1 0
2
0
2
1
2
2
2
,
,
,
2
1
π
τ
( ) ( )
[ ] ( ) ( )
[ ] ( ) −
∫ ∫






+
−
+






−
+
+
×
y z
L L
n
n
z
z
V
V
n
y
n
y x
d
y
d
z
d
z
s
z
z
c
n
L
L
T
z
y
x
k
y
c
n
L
y
s
y
L
0 0
, 2
1
2
2
,
,
,
1
2
2
2
π
π
( ) ( ) ( ) ( )
[ ] ( ) ( )
[ ] ×
∑ ∫ ∫






−
+
+






−
+
+
−
=
N
n
L L
n
y
n
y
n
x
n
x
nV
nI
nV
nI
x y
y
c
n
L
y
s
y
L
x
c
n
L
x
s
x
L
t
e
t
e
a
a
1 0 0
1
2
2
2
1
2
2
2
π
π
( ) ( ) ( )
[ ] ( ) ( )
[ ] ×
∑ ∫






−
+
+
∫






−
+
+
×
=
N
n
L
n
x
n
L
n
z
n
z
V
I
x
z
x
c
n
L
x
s
x
x
d
y
d
z
d
z
c
n
L
z
s
z
L
T
z
y
x
k
1 0
0
, 1
1
2
2
2
,
,
,
π
π
( ) ( )
[ ] ( ) ( ) ( )
[ ]
∫ ∫






−
+
+






−
+
×
y z
L L
n
z
n
z
V
n
y
n x
d
y
d
z
d
z
c
n
L
z
s
z
L
T
z
y
x
f
y
c
n
L
y
s
y
0 0
1
2
2
2
,
,
,
1
π
π
.
Final relations for the above parameters takes the form
( )







 −
+
−
+
±
+
−
=
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
. (14)
Here ( ) ( ) ( ) ( )
[ ] ( )
∫ ∫ ∫



×
+
+






−
+
+
=
x y z
L L L
y
y
n
n
x
n
x
n
n
n
L
L
y
s
y
x
c
n
L
x
s
x
L
T
z
y
x
k
t
e
0 0 0
,
2
2
1
2
2
2
,
,
,
2
π
π
γ ρ
ρ
ρ
ρ

9. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
9
( )
[ ]} ( ) ( )
[ ] x
d
y
d
z
d
z
c
n
L
z
s
z
L
y
c n
z
n
z
n






−
+
+
−
× 1
2
2
2
1
2
π
, ( ) ( )
{
∫ ∫ ∫ +
=
t L L
n
n
x
n
x y
y
s
y
e
n
L 0 0 0
2
2
1
τ
π
δ ρ
ρ
( )
[ ] ( ) ( )
[ ] ( ) ( )
[ ] ×
+
∫ −






−
+



−
+
y
L
n
n
z
n
n
y
L
d
x
d
x
c
y
d
z
d
T
z
y
x
D
z
c
n
L
z
s
z
y
c
n
L z
π
τ
π
π
ρ
2
1
2
1
,
,
,
1
2
1
2 0
( ) ( ) ( )
[ ] ( )
[ ] ( ) ( )
[ ]
∫ ∫ ∫ ∫



+
−
−






−
+
+
×
t L L L
n
z
n
n
x
n
x
n
x y z
z
c
n
L
T
z
y
x
D
y
c
x
c
n
L
x
s
x
L
e
n 0 0 0 0
2
1
2
2
,
,
,
2
1
1
2
2
2
1
π
π
τ ρ
ρ
( ) } ( ) ( ) ( )
[ ] ( )
{
∫ ∫ ∫ +






−
+
+
+
+
+
t L L
n
n
x
x
n
n
z
z
n
x y
y
s
y
x
c
n
L
L
x
s
x
e
n
L
d
x
d
y
d
z
d
L
z
s
z
0 0 0
2
1
2
2
2
1
2
π
τ
π
τ ρ
( )
[ ] ( )
[ ] ( ) ( )
t
e
n
L
L
L
d
x
d
y
d
z
d
T
z
y
x
D
z
c
y
c
n
L
L n
z
y
x
L
n
n
y
y
z
ρ
ρ
π
τ
π 6
5
2
2
2
0
,
,
,
2
1
1
2
−
∫ −



−
+
+ , ( ) ×
= t
enI
nIV
χ
( ) ( ) ( )
[ ] ( )
[ ] ( ) ( ) ( )
{
∫ ∫ ∫ +
+






+
−
+






−
+
×
x y z
L L L
n
z
V
I
n
n
y
y
n
x
n
nV z
s
z
L
T
z
y
x
k
y
s
y
y
c
n
L
L
x
c
n
L
x
s
x
t
e
0 0 0
, 2
,
,
,
2
1
2
2
1
π
π
( )
[ ] x
d
y
d
z
d
z
c
n
L
n
z



−
+ 1
2
2π
, ( ) ( )
[ ] ( ) ( )
[ ] {
∫ ∫ ∫ ×






−
+






−
+
=
x y z
L L L
n
y
n
n
x
n
n z
y
c
n
L
y
s
y
x
c
n
L
x
s
x
0 0 0
1
1
π
π
λ ρ
( ) ( )
[ ] ( ) x
d
y
d
z
d
T
z
y
x
f
z
c
n
L
z
s n
z
n ,
,
,
1 ρ
π 


−
+
× , 2
2
4 nI
nI
nI
nV
b χ
γ
γ
γ −
= , −
−
= 2
3 2 nI
nI
nI
nI
nV
b χ
δ
δ
γ
γ
nI
nI
nV γ
χ
δ
− , ( ) 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
nI
nV
nI
nV
nI
b λ
χ
δ
δ
γ
λ −
= 2
1 ,
2
2
3 4
8 b
b
y
A −
+
= , 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
+
−
= , −
+
+
−
−
+
= 3 3
2
3 3
2
q
p
q
q
p
q
y
4
3 3b
b
− .
Now we will calculate distributions of concentrations of simplest complexes of radiation defects
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 (15)
with not yet known coefficients anΦρ. To calculate these coefficient we transform the Eqs.(6) to
the following integro-differential form
( ) ( )
( )
+
∫ ∫ ∫
Φ
=
∫ ∫ ∫ Φ Φ
t y
L
z
L
I
I
z
y
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
L
L
z
y
u
d
v
d
w
d
t
w
v
u
L
L
L
z
y
x
0
,
,
,
,
,
,
,
,
, τ
∂
τ
∂
( )
( )
( )
( )
∫ ∫ ∫ ×
Φ
+
∫ ∫ ∫
Φ
+ Φ
Φ
t x
L
y
L
I
I
t x
L
z
L
I
I
z
x x y
x z
d
u
d
v
d
z
z
v
u
T
z
v
u
D
d
u
d
w
d
y
w
y
u
T
w
y
u
D
L
L
z
x
0
0
,
,
,
,
,
,
,
,
,
,
,
, τ
∂
τ
∂
τ
∂
τ
∂
( ) ( ) ( ) ×
∫ ∫ ∫
−
∫ ∫ ∫
+
×
x
L
y
L
z
L
I
z
y
x
x
L
y
L
z
L
I
I
z
y
x
y
x x y z
x y z
T
w
v
u
k
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
L
L
y
x
,
,
,
,
,
,
,
,
, 2
, τ
( ) ( )
∫ ∫ ∫
+
× Φ
x
L
y
L
z
L
I
z
y
x 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 ,
,
,
,
, τ (6a)
( ) ( )
( )
+
∫ ∫ ∫
Φ
=
∫ ∫ ∫ Φ Φ
t y
L
z
L
V
V
z
y
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
L
L
z
y
u
d
v
d
w
d
t
w
v
u
L
L
L
z
y
x
0
,
,
,
,
,
,
,
,
, τ
∂
τ
∂
( )
( )
( )
( )
∫ ∫ ∫ ×
Φ
+
∫ ∫ ∫
Φ
+ Φ
Φ
t x
L
y
L
V
V
t x
L
z
L
V
V
z
x x y
x z
d
u
d
v
d
z
z
v
u
T
z
v
u
D
d
u
d
w
d
y
w
y
u
T
w
y
u
D
L
L
z
x
0
0
,
,
,
,
,
,
,
,
,
,
,
, τ
∂
τ
∂
τ
∂
τ
∂

10. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
10
( ) ( ) ( ) ×
∫ ∫ ∫
−
∫ ∫ ∫
+
×
x
L
y
L
z
L
V
z
y
x
x
L
y
L
z
L
V
V
z
y
x
y
x x y z
x y z
T
w
v
u
k
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
L
L
y
x
,
,
,
,
,
,
,
,
, 2
, τ
( ) ( )
∫ ∫ ∫
+
× Φ
x
L
y
L
z
L
V
z
y
x 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 ,
,
,
,
, τ .
Further we substitute the previously considered series in the Eqs.(6a). In this situation we obtain
the following equation
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ ∫ ∫
−
=
∑
−
=
Φ
Φ
=
Φ
N
n
t y
L
z
L
I
n
nI
n
I
n
z
y
x
N
n
nI
n
n
n
I
n
y z
T
w
v
x
D
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
n
I
n
z
y
x
n
x z
d
u
d
w
d
T
w
v
u
D
w
c
u
c
t
e
y
s
n
a
L
L
L
z
x
d
v
d
w
d
w
c
1 0
,
,
, τ
π
τ
( ) ( ) ( ) ( ) ( ) ( ) ×
∫ ∫ ∫
+
∑ ∫ ∫ ∫
−
=
Φ
Φ
Φ
x
L
y
L
z
L
I
I
N
n
t x
L
y
L
I
n
n
I
n
n
I
n
z
y
x x y z
x y
T
w
v
u
k
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
,
,
,
,
,
, ,
1 0
τ
π
( ) ( ) ( ) ×
+
∫ ∫ ∫
−
×
z
y
x
x
L
y
L
z
L
I
z
y
x
z
y
x 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
L
L
L
z
y
x
u
d
v
d
w
d
w
v
u
I
x y z
τ
τ ,
,
,
,
,
,
,
,
,
2
( )
∫ ∫ ∫
× Φ
x
L
y
L
z
L
I
x y z
u
d
v
d
w
d
w
v
u
f ,
, (16)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫
−
=
∑
−
=
Φ
Φ
=
Φ
N
n
t y
L
z
L
V
n
nV
n
V
n
z
y
x
N
n
nV
n
n
n
V
n
y z
T
w
v
x
D
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
V
n
n
V
n
n
V
n
z
y
x
n
x z
d
u
d
w
d
T
w
v
u
D
w
c
u
c
t
e
y
s
n
a
L
L
L
z
x
d
v
d
w
d
w
c
1 0
,
,
, τ
π
τ
( ) ( ) ( ) ( ) ( ) ( ) ×
∫ ∫ ∫
+
∑ ∫ ∫ ∫
−
=
Φ
Φ
Φ
x
L
y
L
z
L
V
V
N
n
t x
L
y
L
V
n
n
V
n
n
V
n
z
y
x x y z
x y
T
w
v
u
k
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
,
,
,
,
,
, ,
1 0
τ
π
( ) ( ) ( ) ×
+
∫ ∫ ∫
−
×
z
y
x
x
L
y
L
z
L
V
z
y
x
z
y
x 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
L
L
L
z
y
x
u
d
v
d
w
d
w
v
u
V
x y z
τ
τ ,
,
,
,
,
,
,
,
,
2
( )
∫ ∫ ∫
× Φ
x
L
y
L
z
L
V
x y z
u
d
v
d
w
d
w
v
u
f ,
, .
We use orthogonality condition of functions in the above series on scale of the heterostructure to
calculate coefficients anΦρ. Using the condition gives a possibility to obtain equations for calcula-
tion the above coefficients for any quantity N of terms of considered series
( ) ( )
[ ] ( ) ( )
[ ] ×
∑ ∫ ∫ ∫






−
+
+
−
−
=
∑
−
=
Φ
=
Φ
Φ
N
n
t L L
n
y
n
y
n
I
n
x
N
n
I
n
I
n
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
( ) ( ) ( )
[ ] ( ) ( )
{
∑ ∫ ∫ +
+
−
∫






−
+
×
=
Φ
Φ
Φ
N
n
t L
x
n
y
I
n
L
I
n
n
z
n
I
x
z
L
x
s
x
L
n
a
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
( ) ( )
[ ] ( ) ( ) ( )
[ ] ( ) ×
−
∫ ∫






−
+
−



−
+ Φ
Φ
x
L L
I
n
n
z
n
I
n
n
x
L
d
e
x
d
y
d
z
d
z
c
n
L
z
s
z
T
z
y
x
D
y
c
n
x
c
L
y z
π
τ
τ
π
π
1
1
2
,
,
,
2
1
2
1
2
0 0

11. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
11
( ) ( ) ( )
[ ] ( ) ( )
[ ] ( ) ×
∑ ∫ ∫ ∫ ∫






+
−
+






−
+
×
=
Φ
Φ
Φ
N
n
t L L L
I
y
n
y
n
n
x
n
I
n
I
n
x y z
T
z
y
x
D
L
y
c
n
L
y
s
y
x
c
n
L
x
s
x
e
n
a
1 0 0 0 0
2
,
,
,
1
2
2
2
1
2
2 π
π
τ
( )
[ ] ( ) ( )
[ ] ( ) ( ) ( )
[ ] ×
∑∫ ∫ ∫






−
+






+
−
+
−
×
=
Φ
N
n
t L L
n
y
n
n
n
x
I
n
n
x y
y
c
n
L
y
s
y
x
s
x
x
c
n
L
e
d
x
d
y
d
z
d
y
c
10 0 0
1
2
1
2
2
1
π
π
τ
τ
( ) ( ) ( )
[ ] ( ) ( ) ( )
{
∑∫ ∫ +
−
∫






+
−
×
=
Φ
Φ
N
n
t L
n
I
n
L
n
n
z
I
I
I
n
x
z
x
s
x
e
x
d
y
d
z
d
z
s
z
z
c
n
L
T
z
y
x
k
t
z
y
x
I
n
a
10 0
0
,
2
3
3
1
2
,
,
,
,
,
, τ
π
π
( )
[ ] ( )
[ ] ( ) ( ) ( ) ( )
[ ]
∫ ∫



+
−






+
−



−
+ Φ
y z
L L
n
z
I
n
n
y
I
n
n
x
z
c
n
L
t
z
y
x
I
T
z
y
x
k
y
s
y
y
c
n
L
n
a
x
c
n
L
0 0
3
3
1
2
,
,
,
,
,
,
1
2
1
2 π
π
π
π
( )} ( ) ( ) ( )
[ ] ( ) ( )
[ ] ×
∑ ∫ ∫ ∫






−
+






−
+
+
+
=
Φ
Φ
N
n
t L L
n
y
n
n
x
n
I
n
I
n
n
x y
y
c
n
L
y
s
y
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 π
π
τ
π
( )
[ ] ( ) ( )
∫






+
−
× Φ
z
L
I
n
n
z
x
d
y
d
z
d
z
y
x
f
z
s
z
z
c
n
L
0
,
,
1
2π
(17)
( ) ( )
[ ] ( ) ( )
[ ] ×
∑ ∫ ∫ ∫






−
+
+
−
−
=
∑
−
=
Φ
=
Φ
Φ
N
n
t L L
n
y
n
y
n
V
n
x
N
n
V
n
V
n
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
( ) ( ) ( )
[ ] ( ) ( )
{
∑ ∫ ∫ +
+
−
∫






−
+
×
=
Φ
Φ
Φ
N
n
t L
x
n
y
V
n
L
V
n
n
z
n
V
x
z
L
x
s
x
L
n
a
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
( )
( )
[ ] ( ) ( ) ( )
[ ] ( ) ×
−
∫ ∫






−
+
−



−
+ Φ
Φ
x
L L
V
n
n
z
n
V
n
n
x
L
d
e
x
d
y
d
z
d
z
c
n
L
z
s
z
T
z
y
x
D
y
c
n
x
c
L
y z
π
τ
τ
π
π
1
1
2
,
,
,
2
1
2
1
2
0 0
( ) ( ) ( )
[ ] ( ) ( )
[ ] ( ) ×
∑ ∫ ∫ ∫ ∫






+
−
+






−
+
×
=
Φ
Φ
Φ
N
n
t L L L
V
y
n
y
n
n
x
n
V
n
V
n
x y z
T
z
y
x
D
L
y
c
n
L
y
s
y
x
c
n
L
x
s
x
e
n
a
1 0 0 0 0
2
,
,
,
1
2
2
2
1
2
2 π
π
τ
( )
[ ] ( ) ( )
[ ] ( ) ( ) ( )
[ ] ×
∑∫ ∫ ∫






−
+






+
−
+
−
×
=
Φ
N
n
t L L
n
y
n
n
n
x
V
n
n
x y
y
c
n
L
y
s
y
x
s
x
x
c
n
L
e
d
x
d
y
d
z
d
y
c
10 0 0
1
2
1
2
2
1
π
π
τ
τ
( ) ( ) ( )
[ ] ( ) ( ) ( )
{
∑∫ ∫ +
−
∫






+
−
×
=
Φ
Φ
N
n
t L
n
V
n
L
n
n
z
V
V
V
n
x
z
x
s
x
e
x
d
y
d
z
d
z
s
z
z
c
n
L
T
z
y
x
k
t
z
y
x
V
n
a
10 0
0
,
2
3
3
1
2
,
,
,
,
,
, τ
π
π
( )
[ ] ( )
[ ] ( ) ( ) ( ) ( )
[ ]
∫ ∫



+
−






+
−



−
+ Φ
y z
L L
n
z
V
n
n
y
V
n
n
x
z
c
n
L
t
z
y
x
V
T
z
y
x
k
y
s
y
y
c
n
L
n
a
x
c
n
L
0 0
3
3
1
2
,
,
,
,
,
,
1
2
1
2 π
π
π
π
( )} ( ) ( ) ( )
[ ] ( ) ( )
[ ] ×
∑ ∫ ∫ ∫






−
+






−
+
+
+
=
Φ
Φ
N
n
t L L
n
y
n
n
x
n
V
n
V
n
n
x y
y
c
n
L
y
s
y
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 π
π
τ
π
( )
[ ] ( ) ( )
∫






+
−
× Φ
z
L
V
n
n
z
x
d
y
d
z
d
z
y
x
f
z
s
z
z
c
n
L
0
,
,
1
2π
.

12. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
12
Fig.2a. Spatial distributions of concentration of dopant after infusion and annealing with the same anneal-
ing time before and after interface between layers of heterostructure. Curve 1 is the distribution of concen-
tration of dopant in homogenous sample with averaged dopant diffusion coefficient D0. Curves 2-6 are the
distribution of concentration of dopant in heterostructure with increasing difference between values of dif-
fusion coefficient. Value of dopant diffusion coefficient in the substrate is smaller, than in epitaxial the
layer
3. DISCUSSION
In this section analysis of distributions of concentration of dopant, infused (see Fig. 2a) or im-
planted (see Fig. 2b) into epitaxial layer have been done. Annealing time is the same for all
curves of these figures. One can find from these figures, that absolute value of gradient of con-
centration of dopant increases due to presents an interface between layers of heterostructure. In
this situation dimensions of the considered transistors decreases. At the same time homogeneity of
concentration of dopant in enriched area increases.
x
0.0
0.5
1.0
1.5
2.0
C(x,
Θ
)
2
3
4
1
0 L/4 L/2 3L/4 L
Epitaxial layer Substrate
Fig.2b. Spatial distributions of concentration of dopant after implantation and annealing with the same an-
nealing time before and after interface between layers of heterostructure. Annealing time for curves 1 and 3
is equal to Θ = 0.0048(Lx
2
+Ly
2
+Lz
2
)/D0. Annealing time for curves 2 and 4 is equal to Θ = 0.0057(Lx
2
+Ly
2
+Lz
2
)/D0. Curves 1 and 2 are the distributions of concentration of dopant in homogenous sample.
Curves 3 and 4 are distributions of concentration of dopant in heterostructure under condition, when value
of dopant diffusion coefficient in the substrate is smaller, than in epitaxial the layer

13. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
13
To estimate optimal annealing time we estimate decreasing of absolute value of gradient of con-
centration of dopant near interface between layers of the heterostructure with increasing of an-
nealing time. Decreasing of annealing time leads to increasing inhomogeneity of distribution of
concentration of dopant. Estimation of the compromise value of annealing time has been done by
using recently introduced criterion [14,15]. To use the criterion we approximate real spatial dis-
tribution of concentration of dopant by idealized step-wise function ψ(x,y,z). After that we esti-
mate the required optimal annealing time by minimization of mean-square error [16-20]
( ) ( )
[ ]
∫ ∫ ∫ −
Θ
=
x y z
L L L
z
y
x
x
d
y
d
z
d
z
y
x
z
y
x
C
L
L
L
U
0 0 0
,
,
,
,
,
1
ψ . (18)
Minimization of the above mean-squared error leads to dependences of optimal annealing time on
parameters, which are presented on Figs. 3. It should be noted, that radiation defects, generated
during ion implantation, should be annealed. After ideal optimization of annealing time the im-
planted dopant should achieve the interface between layers of heterostructure. If annealing time is
smaller, it is attracted an interest to make additional annealing to achieve the interface. The Fig.
3b shows dependences of additional annealing time.
The figures shows, that optimal annealing time of implanted dopant is smaller in comparison with
optimal annealing time of infused dopant. If the considered heterostructure have been doped by
diffusion, any radiation damage of materials of layers is absent. On the other hand radiation
processing of materials of heterostructure (including of ion implantation) leads to decreasing of
mismatch-induced stress in the processed heterostructure [20].
0.0 0.1 0.2 0.3 0.4 0.5
a/L, ξ, ε, γ
0.0
0.1
0.2
0.3
0.4
0.5
Θ
D
0
L
-2
3
2
4
1
Fig. 3a. Dependences of optimal annealing time of infused dopant on several parameters. Dependence of
the optimal annealing time on normalized thickness of the epitaxial layer a/L describes by curve 1 for ξ=γ=
0 for equal to each other values of dopant diffusion coefficient in both parts of heterostructure. Curve 2
describes dependence of the optimal annealing time on the parameter ε for a/L=1/2 and ξ= γ = 0. Curve 3
describes dependence of the optimal annealing time on the parameter ξ for a/L=1/2 and ε= γ = 0. Curve 4
describes dependence of the optimal annealing time on parameter γ for a/L=1/2 and ε=ξ=0

14. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 3, September 2016
14
0.0 0.1 0.2 0.3 0.4 0.5
a/L, ξ, ε, γ
0.00
0.04
0.08
0.12
Θ
D
0
L
-2
3
2
4
1
Fig.3b. Dependences of optimal annealing time of implanted dopant on several parameters. Dependence of
the optimal annealing time on normalized thickness of the epitaxial layer a/L describes by curve 1 for ξ=γ=
0 for equal to each other values of dopant diffusion coefficient in both parts of heterostructure. Curve 2
describes dependence of the optimal annealing time on the parameter ε for a/L=1/2 and ξ= γ = 0. Curve 3
describes dependence of the optimal annealing time on the parameter ξ for a/L=1/2 and ε= γ = 0. Curve 4
describes dependence of the optimal annealing time on parameter γ for a/L=1/2 and ε=ξ=0
4. CONCLUSIONS
In this paper we formulate several recommendations to optimize manufacture heterotransistor
with several source based on prognosis of time varying of spatial distributions of concentrations
of infused and implanted dopants in specific heterostructure. We also introduce analytical ap-
proach to prognosis diffusion and ion types of doping with account variation in space and time
parameters of technological parameters and nonlinearity of mass and heat transport.
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 and
educational fellowship for scientific research of Government of Nizhny Novgorod region of Rus-
sia.
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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.