ON MODIFICATION OF PROPERTIES OF P-N-JUNCTIONS DURING OVERGROWTH
0 likes178 views
This paper examines the effects of overgrowth on the properties of p-n-junctions in heterostructures, focusing on the distribution of dopant concentrations. It outlines various methods to enhance the sharpness of these junctions, such as near-surficial annealing, high doping, and the use of mechanical stress during manufacturing. The study models the spatio-temporal distribution of dopants through mathematical equations, providing insights into optimizing integrated circuit production.
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
DOI: 10.5121/ijitmc.2016.4203 35
ON MODIFICATION OF PROPERTIES OF
P-N-JUNCTIONS DURING OVERGROWTH
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 consider influence of overgrowth of doped by diffusion and ion implantation areas of hete-
rostructures on distributions of concentrations of dopants. Several conditions to increase sharpness of p-n-
junctions (single and framework bipolar transistors), which were manufactured during considered technol-
ogical process, have been determined. At the same time we analyzed influence of speed of overgrowth of
doped areas and mechanical stress in the considered heterostructure on distribution of concentrations of
dopants in the structure.
KEYWORDS
Diffusion-junction heterorectifier; implanted-junction heterorectifier; overgrowth of doped area; analytical
approach for modeling
1. INTRODUCTION
In the present time they are several approaches could be used to manufacture p-n-junctions diffu-
sion of dopants in a homogenous sample or an epitaxial layer of heterostructure, implantation of
ions of dopants in the same situations or doping during epitaxial growth [1-7]. The same ap-
proaches could be used to manufacture systems of p-n-junctions: bipolar transistors and thyris-
tors. The first and the second ways of doping are preferable in comparison with the third one be-
cause the approaches give us possibility to dope locally materials during manufacture integrated
circuits easily in comparison with epitaxial growth. Using diffusion and ion implantation in ho-
mogenous sample to manufacturing p-n-junctions leads to production fluently varying and wide
distributions of dopants. One of actual problems is increasing sharpness of p-n- junctions [5,7].
The increasing of sharpness gives us possibility to decrease switching time of p-n-junctions. In-
creasing of homogeneity of dopant distribution in enriched by the dopant area is also attracted an
interest [5]. The increasing of homogeneity gives us possibility to decrease local overheat of the
doped materials due to streaming of electrical current during operating of p-n-junction or to de-
crease depth of p-n-junction for fixed value of local overheats. One way to increase sharpness of
p-n-junction based on using near-surficial (laser or microwave) types of annealing [8-15].
Framework the types of annealing one can obtain near-surficial heating of the doped materials. In
this situation due to the Arrhenius low one can obtain increasing of dopant diffusion coefficient of
near-surficial area in comparison with volumetric dopant diffusion coefficient. The increasing of
dopant diffusion coefficient of near-surficial area leads to increasing of sharpness of p-n-junction.
The second way to increase sharpness of p-n-junction based on using high doping of materials. In
this case contribution of nonlinearity of diffusion process increases [4]. The third way to increase](https://image.slidesharecdn.com/3-160608092455/85/ON-MODIFICATION-OF-PROPERTIES-OF-P-N-JUNCTIONS-DURING-OVERGROWTH-1-320.jpg)
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
36
sharpness of p-n-junction is using of inhomogeneity of heterostructure [16,17]. Framework the
approach we consider simplest heterostructure, which consist of substrate and epitaxial layer. One
can find increasing of sharpness of p-n-junction after annealing with appropriate annealing time
in the case, when dopant diffusion coefficient in the substrate is smaller, than in the epitaxial
layer. The fourth way to increase sharpness of p-n-junction is radiation processing of materials.
The radiation processing leads to radiation-enhanced diffusion [18]. However using radiation
processing of materials leads to necessity of annealing of radiation defects. Density of elements of
integrated circuits could be increases by using mismatch-induced stress [19]. However one could
obtain increased unsoundness of doped material (for example, to generation of dislocation of dis-
agreement) by using the approach [7].
The considered approaches gives a possibility to increase sharpness of p-n-junction with increas-
ing of homogeneity of dopant distribution in enriched by the dopant area. Using combination of
the above approaches gives us possibility to increase sharpness of p-n-junction and increasing of
homogeneity of dopant distribution in enriched area at one time.
z
D(z),P(z)
D2
D3
0 Lz
a
fC impl(z)
fC diff(z)
P2
P3
Substrate
Epitaxial layer
Overlayer
D1
P1
Fig. 1. Substrate, epitaxial and overlayers framework heterostructure
Framework this paper we consider a substrate with known type of conductivity (n or p) and an
epitaxial layer, included into a heterostructure. The epitaxial layer have been doped by diffusion
or by ion implantation to manufacture another type of conductivity (p or n). Farther we consider
overgrowth of the epitaxial layer by an overlayer (see Fig. 1). The overlayer has type of conduc-
tivity, which coincide with type of conductivity of the substrate. In this paper we analyzed influ-
ence of overgrowth of the doped epitaxial layer on distribution of dopants in the considered hete-
rostructure.
2. METHOD OF SOLUTION
In this section we calculate spatio-temporal distribution of concentration of dopant in the consi-
dered heterostructure to solve our aim. To calculate the distribution we solved the following
boundary problem [1,20-22]
( ) ( ) ( ) ( ) +
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC ,,,,,,,,,,,,
(1)](https://image.slidesharecdn.com/3-160608092455/85/ON-MODIFICATION-OF-PROPERTIES-OF-P-N-JUNCTIONS-DURING-OVERGROWTH-2-320.jpg)
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
37
( ) ( ) ( ) ( )
∫∇
∂
∂
Ω+
∫∇
∂
∂
Ω+
−−
zz L
tv
S
S
L
tv
S
S
WdtWyxCtzyx
Tk
D
y
WdtWyxCtzyx
Tk
D
x
,,,,,,,,,,,, µµ .
Boundary and initial conditions for our case could be written as
( ) 0
,,,
0
=
∂
∂
=x
x
tzyxC
,
( ) 0
,,,
=
∂
∂
= xLx
x
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=y
y
tzyxC
,
( ) 0
,,,
=
∂
∂
= yLx
y
tzyxC
,
( ) 0
,,,
=
∂
∂
−= tvz
z
tzyxC
,
( ) 0
,,,
=
∂
∂
= zLx
z
tzyxC
, C(x,y,z,0)=fC(x,y,z).
In the above relations we used the function C (x,y,z,t) as the spatio-temporal distribution of con-
centration of dopant; Ω is the atomic volume; surface concentration of dopant on interface be-
tween layers of heterostructure could be determined as the following integral ( )∫
−
zL
tv
zdtzyxC ,,,
(we assume, that the interface between layers of heterostructure is perpendicular to the direction
Oz); surface gradient we denote as symbol ∇S; µ(x,y,z,t) is the chemical potential (reason of ac-
counting of the chemical potential is mismatch-induced stress); the parameters D and DS are the
coefficients of volumetric and surface diffusions. One can find the surface diffusions due to mis-
match-induced stress. Diffusion coefficients depends on temperature and speed of heating and
cooling of heterostructure, properties of materials of heterostructure, spatio-temporal distributions
of concentrations of dopant and radiation defects after ion implantation. Approximations of these
dependences could be approximated by the following functions [22,23]
( ) ( )
( )
( ) ( )
( )
++
+= 2*
2
2*1
,,,,,,
1
,,,
,,,
1,,,
V
tzyxV
V
tzyxV
TzyxP
tzyxC
TzyxDD SLSS ςςξ γ
γ
. (2)
Functions DL(x,y,z,T) and DLS(x,y,z,T) described dependences of diffusion coefficients on coordi-
nate (due to presents several layers in heterostructure, manufactured by using different materials)
and temperature of annealing T (due to Arrhenius law); function P(x,y,z,T) describes dependence
of limit of solubility of dopant on coordinate and temperature; parameter γ could be integer and
depends on properties of materials of heterostructure [23]; V(x,y,z,t) is the spatio-temporal con-
centration of radiation vacancies; V*
is the equilibrium concentration of vacancies. Concentration-
al dependence of dopant diffusion coefficients has been described in details in [23].
We determine distributions of concentrations of point defects in space and time as solutions of the
following system of equations [1,20-22]
( ) ( ) ( ) ( ) ( ) ( )×−
+
= Tzyxk
y
tzyxI
TzyxD
yx
tzyxI
TzyxD
xt
tzyxI
IIII ,,,
,,,
,,,
,,,
,,,
,,,
,
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk
z
tzyxI
TzyxD
z
tzyxI VII ,,,,,,,,,
,,,
,,,,,, ,
2
−
+×
∂
∂
∂
∂
(3)](https://image.slidesharecdn.com/3-160608092455/85/ON-MODIFICATION-OF-PROPERTIES-OF-P-N-JUNCTIONS-DURING-OVERGROWTH-3-320.jpg)
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
38
( ) ( ) ( ) ( )
∫∇
∂
∂
Ω+
∫∇
∂
∂
Ω+
−−
zz L
tv
S
SI
L
tv
S
SI
WdtWyxItzyx
Tk
D
y
WdtWyxItzyx
Tk
D
x
,,,,,,,,,,,, µµ
( ) ( ) ( ) ( ) ( ) ( )×−
+
= Tzyxk
y
tzyxV
TzyxD
yx
tzyxV
TzyxD
xt
tzyxV
VVVV ,,,
,,,
,,,
,,,
,,,
,,,
,
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( ) ( ) ( ) ( )+−
+× tzyxVtzyxITzyxk
z
tzyxV
TzyxD
z
tzyxV VIV ,,,,,,,,,
,,,
,,,,,, ,
2
∂
∂
∂
∂
( ) ( ) ( ) ( )
∫∇
∂
∂
Ω+
∫∇
∂
∂
Ω+
−−
zz L
tv
S
VS
L
tv
S
VS
WdtWyxVtzyx
Tk
D
y
WdtWyxVtzyx
Tk
D
x
,,,,,,,,,,,, µµ .
Boundary and initial conditions for these equations could be written as
( ) 0
,,,
0
=
=x
x
tzyxI
∂
∂
,
( ) 0
,,,
=
= xLx
x
tzyxI
∂
∂
,
( ) 0
,,,
0
=
=y
y
tzyxI
∂
∂
,
( ) 0
,,,
=
= yLy
y
tzyxI
∂
∂
,
( ) 0
,,,
=
−= tvz
z
tzyxI
∂
∂
,
( ) 0
,,,
=
= zLz
z
tzyxI
∂
∂
,
( ) 0
,,,
0
=
=x
x
tzyxV
∂
∂
,
( ) 0
,,,
=
= xLx
x
tzyxV
∂
∂
,
( ) 0
,,,
0
=
=y
y
tzyxV
∂
∂
,
( ) 0
,,,
=
= yLy
y
tzyxV
∂
∂
,
( ) 0
,,,
=
−= tvz
z
tzyxV
∂
∂
,
( ) 0
,,,
=
= zLz
z
tzyxV
∂
∂
,
I(x,y,z,0)=fI(x,y,z), V(x,y,z,0)=fV(x,y,z). (4)
Here I(x,y,z,t) is the distribution of concentration of radiation interstitials in space and time; I*
is
the equilibrium concentration interstitials; DI(x,y,z,T), DV(x,y,z,T), DIS(x,y,z,T), DVS(x,y,z,T) are the
coefficients of volumetric and surface diffusion; terms V2
(x,y,z,t) and I2
(x,y,z,t) corresponds to
generation divacancies and analogous complexes of interstitials (see, for example, [22] and ap-
propriate references in this work); the functions kI,V(x,y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T) described
dependences of parameters of recombination of point defects and generation their complexes on
coordinate and temperature; k is the Boltzmann constant.
We calculate spatio-temporal distributions of concentrations of divacancies ΦV (x,y,z,t) and diin-
terstitials ΦI (x,y,z,t) by solution of the equations [20-22]
( ) ( ) ( ) ( ) ( ) ( )×+
Φ
+
Φ
=
Φ
ΦΦ Tzyxk
y
tzyx
TzyxD
yx
tzyx
TzyxD
xt
tzyx
I
III
II
,,,
,,,
,,,
,,,
,,,
,,,
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( ) ( ) ( ) +
∫Φ∇
∂
∂
Ω+
Φ
+×
−
Φ
Φ
z
I
I
L
tv
IS
SI
WdtWyxtzyx
Tk
D
xz
tzyx
TzyxD
z
tzyxI ,,,,,,
,,,
,,,,,, µ
∂
∂
∂
∂
( ) ( ) ( ) ( )tzyxITzyxkWdtWyxtzyx
Tk
D
y
II
L
tv
IS
IS
z
,,,,,,,,,,,, 2
,+
∫Φ∇
∂
∂
Ω+
−
Φ
µ (5)
( )
( )
( )
( )
( )
( )×+
Φ
+
Φ
=
Φ
ΦΦ Tzyxk
y
tzyx
TzyxD
yx
tzyx
TzyxD
xt
tzyx
V
VVV
VV
,,,
,,,
,,,
,,,
,,,
,,,
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( )
( )
( ) ( ) +
∫Φ∇
∂
∂
Ω+
Φ
+×
−
Φ
Φ
z
V
V
L
tv
VS
SV
WdtWyxtzyx
Tk
D
xz
tzyx
TzyxD
z
tzyxV ,,,,,,
,,,
,,,,,, µ
∂
∂
∂
∂](https://image.slidesharecdn.com/3-160608092455/85/ON-MODIFICATION-OF-PROPERTIES-OF-P-N-JUNCTIONS-DURING-OVERGROWTH-4-320.jpg)
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
39
( ) ( ) ( ) ( )tzyxVTzyxkWdtWyxtzyx
Tk
D
y
VV
L
tv
VS
S z
V
,,,,,,,,,,,, 2
,+
∫Φ∇
∂
∂
Ω+
−
Φ
µ .
Boundary and initial conditions could be written as
( ) 0
,,,
0
=
Φ
=x
I
x
tzyx
∂
∂
,
( ) 0
,,,
=
Φ
= xLx
I
x
tzyx
∂
∂
,
( ) 0
,,,
0
=
Φ
=y
I
y
tzyx
∂
∂
,
( ) 0
,,,
=
Φ
= yLy
I
y
tzyx
∂
∂
,
( ) 0
,,,
=
Φ
−= tvz
I
z
tzyx
∂
∂
,
( ) 0
,,,
=
= zLz
z
tzyxI
∂
∂
,
( ) 0
,,,
0
=
Φ
=x
V
x
tzyx
∂
∂
,
( ) 0
,,,
=
= xLx
x
tzyxV
∂
∂
,
( ) 0
,,,
0
=
Φ
=y
V
y
tzyx
∂
∂
,
( ) 0
,,,
=
Φ
= yLy
V
y
tzyx
∂
∂
,
( ) 0
,,,
=
Φ
−= tvz
V
z
tzyx
∂
∂
,
( ) 0
,,,
=
Φ
= zLz
V
z
tzyx
∂
∂
,
ΦI(x,y,z,0)=fΦI(x,y,z), ΦV(x,y,z,0)=fΦV(x,y,z). (6)
Here DΦI(x,y,z,T), DΦV(x,y,z,T), DΦIS(x,y,z,T) and DΦVS(x,y,z,T) are the coefficients of volumetric
and surface diffusion; the functions kI(x,y,z,T) and kV(x,y,z,T) described dependences of parame-
ters of decay of complexes of point defects on coordinate and temperature. One can determine
chemical potential µ in the Eq.(1) by the following relation [20]
µ=E(z)Ωσij[uij(x,y,z,t)+uji(x,y,z,t)]/2. (7)
Here E is the tension (Young) modulus;
∂
∂
+
∂
∂
=
i
j
j
i
ij
x
u
x
u
u
2
1
is the deformation tensor; σij is the
stress tensor; ui, uj are the components ux(x,y,z,t), uy(x,y,z,t) and uz(x,y,z,t) of the displacement ten-
sor ( )tzyxu ,,,
r
; xi, xj are the coordinates x, y, z. The relation (3) could be transformed to the fol-
lowing form
( ) ( ) ( ) ( ) ( ) ( )
+−
∂
∂
+
∂
∂
∂
∂
+
∂
∂Ω
= ij
i
j
j
i
i
j
j
i
x
tzyxu
x
tzyxu
x
tzyxu
x
tzyxu
zEtzyx δεµ 0
,,,,,,
2
1,,,,,,
2
,,,
( )
( )
( ) ( ) ( ) ( )[ ]
−−
−
∂
∂
−
+ ij
k
kij
TtzyxTzzK
x
tzyxu
z
z
δβε
σ
δσ
00 ,,,3
,,,
21
,
where σ is the Poisson coefficient; the parameter ε0=(as-aEL)/aEL describes the displacement pa-
rameter with lattice distances of the substrate and the epitaxial layer as, aEL; K is the modulus of
uniform compression; the parameter b describes the thermal expansion; we assume, that the equi-
librium temperature Tr coincides with the room temperature. Components of the displacement
vector could be described by solving the following system of equations [24]
( ) ( ) ( ) ( ) ( )
z
tzyx
y
tzyx
x
tzyx
t
tzyxu
z xzxyxxx
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ ,,,,,,,,,,,,
2
2
σσσ
ρ
( )
( ) ( ) ( ) ( )
z
tzyx
y
tzyx
x
tzyx
t
tzyxu
z yzyyyxy
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ ,,,,,,,,,,,,
2
2
σσσ
ρ
( ) ( ) ( ) ( ) ( )
z
tzyx
y
tzyx
x
tzyx
t
tzyxu
z zzzyzxz
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ ,,,,,,,,,,,,
2
2
σσσ
ρ](https://image.slidesharecdn.com/3-160608092455/85/ON-MODIFICATION-OF-PROPERTIES-OF-P-N-JUNCTIONS-DURING-OVERGROWTH-5-320.jpg)
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
40
where
( )
( )[ ]
( ) ( ) ( ) ( ) ( )−
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂
+
=
k
k
ij
k
kij
i
j
j
i
ij
x
tzyxu
zK
x
tzyxu
x
tzyxu
x
tzyxu
z
zE ,,,,,,
3
,,,,,,
12
δ
δ
σ
σ
( ) ( ) ( )[ ]rTtzyxTzKz −− ,,,β , ρ (z) describes the density of materials of heterostructure. The ten-
sor δij describes the Kronecker symbol. Accounting relation for σij in the previous system of equa-
tions last system of equation could be written as
( ) ( ) ( ) ( )
( )[ ]
( ) ( ) ( )
( )[ ]
( )
+
∂∂
∂
+
−+
∂
∂
+
+=
∂
∂
yx
tzyxu
z
zE
zK
x
tzyxu
z
zE
zK
t
tzyxu
z
yxx
,,,
13
,,,
16
5,,,
2
2
2
2
2
σσ
ρ
( )
( )[ ]
( ) ( ) ( ) ( )
( )[ ]
( ) −
∂∂
∂
+
++
∂
∂
+
∂
∂
+
+
zx
tzyxu
z
zE
zK
z
tzyxu
y
tzyxu
z
zE zzy ,,,
13
,,,,,,
12
2
2
2
2
2
σσ
( )
( ) ( )
( )[ ]
( ) ( ) ( ) ( ) ( )+
∂
∂
−
∂∂
∂
+
∂
∂
+
=
∂
∂
y
tzyxT
zKz
yx
tzyxu
x
tzyxu
z
zE
t
tzyxu
z xyy ,,,,,,,,,
12
,,, 2
2
2
2
2
β
σ
ρ
( )
( )[ ]
( ) ( ) ( )
( )[ ]
( )
( )
+
∂
∂
+
+
+
∂
∂
+
∂
∂
+∂
∂
+ 2
2
,,,
112
5,,,,,,
12 y
tzyxu
zK
z
zE
y
tzyxu
z
tzyxu
z
zE
z
yzy
σσ
( ) ( )
( )[ ]
( )
( )
( )
yx
tzyxu
zK
zy
tzyxu
z
zE
zK
yy
∂∂
∂
+
∂∂
∂
+
−+
,,,,,,
16
22
σ
(8)
( ) ( ) ( )
( )[ ]
( ) ( ) ( ) ( )
+
∂∂
∂
+
∂∂
∂
+
∂
∂
+
∂
∂
+
=
∂
∂
zy
tzyxu
zx
tzyxu
y
tzyxu
x
tzyxu
z
zE
t
tzyxu
z
yxzzz
,,,,,,,,,,,,
12
,,,
22
2
2
2
2
2
2
σ
ρ
( ) ( ) ( ) ( ) ( ) ( ) ( )+
∂
∂
−
∂
∂
+
∂
∂
+
∂
∂
∂
∂
+
z
tzyxT
zzK
z
tzyxu
y
tzyxu
x
tzyxu
zK
z
xyx ,,,,,,,,,,,,
β
( )
( )
( ) ( ) ( ) ( )
∂
∂
−
∂
∂
−
∂
∂
−
∂
∂
+∂
∂
+
z
tzyxu
y
tzyxu
x
tzyxu
z
tzyxu
z
zE
z
zyxz ,,,,,,,,,,,,
6
16
1
σ
.
Systems of conditions for these equations could be written as
( ) 0
,,,
0
=
∂
∂
=x
x
tzyxu
r
;
( ) 0
,,,
=
∂
∂
= xLx
x
tzyxu
r
;
( ) 0
,,,
0
=
∂
∂
=y
y
tzyxu
r
;
( ) 0
,,,
=
∂
∂
= yLy
y
tzyxu
r
;
( ) 0
,,,
=
∂
∂
−= tvz
z
tzyxu
r
;
( ) 0
,,,
=
∂
∂
= zLz
z
tzyxu
r
; ( ) 00,,, uzyxu
rr
= ; ( ) 0,,, uzyxu
rr
=∞ .
We calculate distribution of concentration of dopant in space in time by method of averaging of
function corrections [25-30]. To use the method we re-write Eqs. (1), (3) and (5) with account
appropriate initial distributions (see Appendix). In future we replace the required concentrations
in right sides of the obtained equations on their average values α1ρ, which are not yet known. The
equations modified after the replacement and solution of these equations are presented in the Ap-
pendix.
We determined average values of the first-order approximations of the considered concentrations
by using the following standard relations [25-30]](https://image.slidesharecdn.com/3-160608092455/85/ON-MODIFICATION-OF-PROPERTIES-OF-P-N-JUNCTIONS-DURING-OVERGROWTH-6-320.jpg)
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
41
( )∫ ∫ ∫ ∫
Θ
=
Θ
−0 0 0
11 ,,,
1 x y zL L L
tvzyx
tdxdydzdtzyx
LLL
ρα ρ . (9)
Substitution of the solutions of the modified equations into the relation (9) gives a possibility to
obtain appropriate average values in the following form
( )∫ ∫ ∫ ∫=
Θ
−0 0 0
1 ,,
1 x y zL L L
tv
C
zyx
C tdxdydzdzyxf
LLL
α ,
( )
4
3
4
1
2
3
2
4
2
3
1
4
4
4 a
Aa
a
aLLLBa
B
a
Aa zyx
I
+
−
Θ+Θ
+−
+
=α ,
( )
Θ−−∫ ∫ ∫ ∫
Θ
=
Θ
−
zyxIII
L L L
tv
I
IIV
V LLLStdxdydzdzyxf
S
x y z
001
0 0 0100
1 ,,
1
α
α
α ,
( )∫ ∫ ∫ ∫+
Θ
+
Θ
=
Θ
−
ΦΦ
0 0 0
201
1 ,,
1 x y z
II
L L L
tvzyxzyx
II
zyx
I
tdxdydzdzyxf
LLLLLL
S
LLL
R
α ,
( )∫ ∫ ∫ ∫+
Θ
+
Θ
=
Θ
−
ΦΦ
0 0 0
201
1 ,,
1 x y z
VV
L L L
tvzyxzyx
VV
zyx
V
tdxdydzdzyxf
LLLLLL
S
LLL
R
α .
Relations for calculations parameters Sρρij, ai, A, B, q, p are presented in the Appendix.
We used standard iterative procedure of method of averaging of function corrections to calculate
the second- and higher-order approximations of the considered concentrations [25-30]. Frame-
work the procedure to calculate approximations of the n-th order of the above concentrations we
replace the functions C(x,y,z,t), I(x,y,z,t), V(x,y,z,t), ΦI(x,y,z,t) and ΦV(x,y,z,t) in the Eqs. (1), (3),
(5) with account initial distributions (see Eqs. (1a), (3a), (5a) in the Appendix) on the sums of the
not yet known average values of the considered approximations and approximations of the pre-
vious order, i.e. αnρ+ρn-1(x,y,z,t). These obtained equations and their solutions, which calculated
are presented in the Appendix.
We calculate average values of the second-order approximations of required functions by using
the following standard relation [25-30]
( ) ( )[ ]∫ ∫ ∫ ∫ −
Θ
=
Θ
0 0 0 0
122 ,,,,,,
1 x y zL L L
zyx
tdxdydzdtzyxtzyx
LLL
ρρα ρ . (10)
Substitution of the second-order approximations of concentrations of dopant into Eq.(10) leads to
the considered average values α2ρ
α2C=0, α2ΦI =0, α2ΦV =0,
( )
4
3
4
1
2
3
2
4
2
3
2
4
4
4 b
Eb
b
bLLLFa
F
b
Eb zyx
V
+
−
Θ+Θ
+−
+
=α ,
( )
00201
11021001200
2
2
2
2
IVVIV
VIVVzyxVIVVVVVVV
I
SS
SSLLLSSSC
α
αα
α
+
−−Θ++−−
= .
Relations for parameters bi, Cρ, F, r, s are presented in the Appendix.
Further we determine solutions of Eqs.(8). In this situation we determine approximation of dis-
placement vector. To determine the first-order approximations of the considered components
framework method of averaging of function corrections we replace the required values on their
not yet known average values α1i. The replacement leads to the following result](https://image.slidesharecdn.com/3-160608092455/85/ON-MODIFICATION-OF-PROPERTIES-OF-P-N-JUNCTIONS-DURING-OVERGROWTH-7-320.jpg)
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
42
( ) ( ) ( ) ( ) ( )
x
tzyxT
zzK
t
tzyxu
z x
∂
∂
−=
∂
∂ ,,,,,,
2
1
2
βρ ,
( )
( )
( ) ( ) ( )
y
tzyxT
zzK
t
tzyxu
z y
∂
∂
−=
∂
∂ ,,,,,,
2
1
2
βρ ,
( ) ( ) ( ) ( ) ( )
z
tzyxT
zzK
t
tzyxu
z z
∂
∂
−=
∂
∂ ,,,,,,
2
1
2
βρ .
Integration of the left and right sides of the previous relations on time t gives a possibility to ob-
tain the required components to the following result
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( ) x
t
x uddzyxT
xz
z
zKddzyxT
xz
z
zKtzyxu 0
0 00 0
1 ,,,,,,,,, +∫ ∫
∂
∂
−∫ ∫
∂
∂
=
∞ϑϑ
ϑττ
ρ
β
ϑττ
ρ
β
,
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( ) y
t
y uddzyxT
yz
z
zKddzyxT
yz
z
zKtzyxu 0
0 00 0
1 ,,,,,,,,, +∫ ∫
∂
∂
−∫ ∫
∂
∂
=
∞ϑϑ
ϑττ
ρ
β
ϑττ
ρ
β
,
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( ) z
t
z uddzyxT
zz
z
zKddzyxT
zz
z
zKtzyxu 0
0 00 0
1 ,,,,,,,,, +∫ ∫
∂
∂
−∫ ∫
∂
∂
=
∞ϑϑ
ϑττ
ρ
β
ϑττ
ρ
β
.
We calculate the second-order approximations of components of the displacement vector by stan-
dard replacement of the required functions in the right sides of the Eqs.(8) on the standard sums
α1i+ui(x,y,z,t) [19,26]. Equations for components of the displacement vector are presented in the
Appendix. Solutions of these equations are also presented in the Appendix.
Framework this paper all required concentrations (concentrations of dopant and radiation defects)
and components of displacement vector have been calculated as the appropriate second-order ap-
proximations by using the method of averaging of function corrections. The second-order approx-
imation gives usually enough information on quantitative behavior of spatio-temporal distribu-
tions of concentrations of dopant and radiation defects and also several quantitative results. We
check all analytical results by using numerical approaches.
3. DISCUSSION
In this section we analyzed redistribution of dopant (for the ion doping of heterostructure) with
account redistribution of radiation defects and their interaction with another defects. If growth
rate is small (v t<D1/v), than overlayer will be fully doped by dopant, which was implanted in the
epitaxial layer. Framework another limiting case it will be doped near-surface area of the overlay-
er only. If dopant diffusion coefficient in the overlayer and in the substrate are smaller, in com-
parison with the epitaxial layer, and type of conductivity of the overlayer and the substrate is dif-
ferent with type of conductivity of the epitaxial layer, than one can find a bipolar transistor. In
this case one can find higher sharpness of p-n-junctions framework the transistor in comparison
with a bipolar transistor in homogenous sample with averaged diffusion coefficient of dopant.
The increasing gives a possibility to increase switching time of p-n-junctions (both single p-n-
junctions and p-n-junctions framework their systems: bipolar transistors, thyristors et al). At the
same time one can find increasing of homogeneity of concentration of dopant (see Fig. 2). In this
situation one can decrease local overheats in doped areas during functioning of the considered
devices or to decrease dimensions of these devices for fixed tolerance for local overheats. Quali-
tatively similar results could be obtained for diffusion type of doping. One can find smaller
sharpness of left p-n-junctions in the case, when dopant diffusion coefficient of the overlayer is
larger, than in doped epitaxial layer. At the same time homogeneity of concentration of dopant in](https://image.slidesharecdn.com/3-160608092455/85/ON-MODIFICATION-OF-PROPERTIES-OF-P-N-JUNCTIONS-DURING-OVERGROWTH-8-320.jpg)
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
45
of dopant. We obtain during the analysis, that p-n-junctions, manufactured near interface between
layers of heterostructures, have higher sharpness and higher homogeneity of concentration of do-
pant in enriched area. Existing mismatch-induced stress leads to changing of distribution of con-
centration of dopant in directions, which are parallel to the considered interface. For example, for
ε0<0 the above distribution in directions x and y became more compact (see Fig. 4). In this situa-
tion one can obtain increasing of density of elements of integrated circuits in this situation, when
the circuits were fabricated in heterostructures. For ε0>0 one can obtain opposite effect (see Fig.
5). It should be noted, that radiation processing of materials of heterostructure during ion doping
of materials gives a possibility to decrease mismatch-induced stress (see Fig. 6).
Further we analyzed influence of mismatch-induced stress on distribution of concentration of do-
pant. We obtain during the analysis, that p-n-junctions, manufactured near interface between lay-
ers of heterostructures, have higher sharpness and higher homogeneity of concentration of dopant
in enriched area. Existing mismatch-induced stress leads to changing of distribution of concentra-
tion of dopant in directions, which are parallel to the considered interface. For example, for ε0<0
the above distribution in directions x and y became more compact (see Fig. 4). For ε0>0 one can
obtain opposite effect (see Fig. 5). It should be noted, that radiation processing of materials of
heterostructure during ion doping of materials gives a possibility to decrease mismatch-induced
stress (see Fig. 6). In this situation component of displacement vector perpendicular to interface
between materials of heterostructure became smaller after radiation processing in comparison
with analogous component of displacement vector in non processed heterostructure.
4. CONCLUSIONS
In this paper we analyzed influence of overgrowth of doped by diffusion or ion implantation areas
of heterostructures on distributions of concentrations of dopants. We determine conditions to in-
crease sharpness if implanted-junction and diffusion- junction rectifiers (single rectifiers and rec-
tifiers framework bipolar transistors). At the same time we analyzed influence of overgrowth rate
of doped areas and mismatch-induced stress in the considered heterostructure on distributions of
concentrations of dopants.
ACKNOWLEDGEMENTS
This work is supported by the agreement of August 27, 2013 № 02.В.49.21.0003 between The
Ministry of education and science of the Russian Federation and Lobachevsky State University of
Nizhni Novgorod, educational fellowship for scientific research of Government of Russian, edu-
cational fellowship for scientific research of Government of Nizhny Novgorod region of Russia
and educational fellowship for scientific research of Nizhny Novgorod State University of Archi-
tecture and Civil Engineering.
REFERENCES
[1] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001).
[2] N.A. Avaev, Yu.E. Naumov, V.T. Frolkin. Basis of microelectronics (Radio and communication,
Moscow, 1991).
[3] V.G. Gusev, Yu.M. Gusev. Electronics (Higher School, Moscow, 1991).
[4] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow 1991).
[5] I.P. Stepanenko. Basis of microelectronics (Soviet Radio, Moscow 1980).
[6] A.G. Alexenko, I.I. Shagurin. Microcircuitry (Radio and communication, Moscow, 1990).
[7] K.V. Shalimova. Physics of semiconductors. (Energoatomizdat, Moscow, 1985.
[8] Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov, Yu.N.
Drozdov, V.D. Skupov. Diffusion processes in semiconductor structures during microwave annealing.
Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836 (2003).](https://image.slidesharecdn.com/3-160608092455/85/ON-MODIFICATION-OF-PROPERTIES-OF-P-N-JUNCTIONS-DURING-OVERGROWTH-11-320.jpg)
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
46
[9] A.V. Dvurechensky, G.A. Kachurin, E.V. Nidaev, L.S. Smirnov. Impulse annealing of semiconductor
materials (Science, Moscow, 1982).
[10] A.F. Corballo Sanchez, G. Gonzalez de la Cruz, Yu.G. Gurevich, G.N. Logvinov. Transient heat
transport by carriers and phonons in semiconductors. Phys. Rev. B. Vol. 59 (16). P. 10630 (1999).
[11] V.I. Mazhukin, V.V. Nosov, U. Semmler. Investigation of heat and thermo-elastic fields in semicon-
ductors at pulse processing. Mathematical modelling. Vol. 12 (2), 75 (2000).
[12] K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. Dopant activation in subamor-
phized silicon upon laser annealing. Appl. Phys. Lett. Vol. 89 (17), 172111 (2006).
[13] J.A. Sharp, N.E. B. Cowern, R.P. Webb, K.J. Kirkby, D. Giubertoni, S. Genarro, M. Bersani, M.A.
Foad, F. Cristiano, P.F. Fazzini. Appl. Phys. Lett. Vol. 89, 192105 (2006).
[14] J.E. Epler, F.A. Ponce, F.J. Endicott, T.L. Paoli. J. Appl. Phys. Vol. 64, 3439 (1988).
[15] S.T. Sisianu, T.S. Sisianu, S.K. Railean. Semiconductors. Vol. 36, 581 (2002).
[16] E.L. Pankratov. Influence of spatial, temporal and concentrational dependence of diffusion coefficient
on dopant dynamics: Optimization of annealing time. Phys. Rev. B. Vol. 72 (7). P. 075201 (2005).
[17] E.L. Pankratov. Dopant diffusion dynamics and optimal diffusion time as influenced by diffusion-
coefficient nonuniformity. Russian Microelectronics. Vol. 36 (1). P. 33 (2007).
[18] V.V. Kozlivsky. Modification of semiconductors by proton beams (Science, Sant- Peterburg, 2003).
[19] E.L. Pankratov. Influence of mechanical stress in a multilayer structure on spatial distribution of
dopants in implanted-junction and diffusion-junction rectifiers. Mod. Phys. Lett. B. Vol. 24 (9). P.
867 (2010).
[20] Y.W. Zhang, A.F. Bower. Numerical simulations of island formation in a coherent strained epitaxial
thin film system. Journal of the Mechanics and Physics of Solids. Vol. 47 (11). P. 2273-2297 (1999).
[21] P.M. Fahey, P.B. Griffin, J.D. Plummer. Point defects and dopant diffusion in silicon. Rev. Mod.
Phys. Vol. 61 (2). P. 289-388 (1989).
[22] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev,
1979, in Russian).
[23] Z.Yu. Gotra, Technology of microelectronic devices (Radio and communication, Moscow, 1991).
[24] L.D. Landau, E.M. Lefshits. Theoretical physics. 7 (Theory of elasticity) (Physmatlit, Moscow, 2001,
in Russian).
[25] Yu.D. Sokolov. About the definition of dynamic forces in the mine lifting. Applied Mechanics. Vol. 1
(1). P. 23-35 (1955).
[26] E.L. Pankratov, E.A. Bulaeva. Doping of materials during manufacture p-n-junctions and bipolar
transistors. Analytical approaches to model technological approaches and ways of optimization of dis-
tributions of dopants. Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013).
[27] E.L. Pankratov, E.A. Bulaeva. Increasing of sharpness of diffusion-junction heterorectifier by using
radiation processing. Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1-1250028-8 (2012).
[28] E.L. Pankratov, E.A. Bulaeva. Optimization of manufacturing of emitter-coupled logic to decrease
surface of chip. International Journal of Modern Physics B. Vol. 29 (5). P. 1550023-1-1550023-12
(2015).
[29] E.L. Pankratov, E.A. Bulaeva. An approach to manufacture of bipolar transistors in thin film struc-
tures. On the method of optimization. Int. J. Micro-Nano Scale Transp. Vol. 4 (1). P. 17-31 (2014).
[30] E.L. Pankratov, E.A. Bulaeva. Decreasing of mechanical stress in a semiconductor heterostructure by
radiation processing. J. Comp. Theor. Nanoscience. Vol. 11 (1). P. 91-101 (2014).
APPENDIX
Eqs. (1), (3) and (5) with account initial distributions could be written as
( ) ( ) ( ) ( ) ( ) ( )++
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
tzyxf
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC
C δ,,
,,,,,,,,,,,,
( ) ( ) ( ) ( )
∫∇
∂
∂
Ω+
∫∇
∂
∂
Ω+
−−
zz L
tv
S
S
L
tv
S
S
WdtWyxCtzyx
Tk
D
y
WdtWyxCtzyx
Tk
D
x
,,,,,,,,,,,, µµ (1a)](https://image.slidesharecdn.com/3-160608092455/85/ON-MODIFICATION-OF-PROPERTIES-OF-P-N-JUNCTIONS-DURING-OVERGROWTH-12-320.jpg)
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
49
00IIS× , 0000
2
0000003 VVIIIVIIIV SSSSSa −+= , ( ) ×∫ ∫ ∫ ∫+Θ=
Θ
−0 0 0
222
002 ,,2
x y zL L L
tv
IzyxIV tdxdydzdzyxfLLLSa
( ) ( ) ×Θ−∫ ∫ ∫ ∫−∫ ∫ ∫ ∫+×
Θ
−
Θ
−
2
0 0 0
2
00
0 0 0
2
00000000 ,,,, x
L L L
tv
IIV
L L L
tv
VIVIVIIVV LtdxdydzdzyxfStdxdydzdzyxfSSSS
x y zx y z
00
22
VVzy SLL× , ( )∫ ∫ ∫ ∫=
Θ
−0 0 0
001 ,,
x y zL L L
tv
IIV tdxdydzdzyxfSa , +++−−+= 3 323 32
qpqqpqB
4
2
6a
aΘ
+ , ( )
2
0 0 0
000 ,,
∫ ∫ ∫ ∫=
Θ
−
x y zL L L
tv
IVV tdxdydzdzyxfSa ,
2
1
4
2
2
4
2
32
84
+Θ−Θ= y
a
a
a
a
A , ×
Θ
= 2
4
2
3
24a
a
q
2
4
2
1
4
222
3
4
3
2
3
4
2
32
22
4
02
4
31
0
854
4
8
4
a
a
LLL
a
a
a
a
a
a
a
a
aa
LLLa zyxzyx
Θ
−
Θ
−
Θ−ΘΘ−
Θ−× , −Θ= 2
4
402
12
4
a
aa
p
( ) 2
44231
2
3623 aaaaaLLL zyx +ΘΘ− , ( ) ( ) ( )∫ ∫ ∫ ∫−Θ=
Θ
−0 0 0
1 ,,,,,,
x y zL L L
tv
i
Ii tdxdydzdtzyxITzyxktRρ .
Equations for the second-order approximations of concentrations of dopant and radiation defects
could be written as
( ) ( ) ( )[ ]
( )
( ) ( )
( )
×
++
+
+
∂
∂
=
∂
∂
2*
2
2*1
122 ,,,,,,
1
,,,
,,,
1,,,
,,,
V
tzyxV
V
tzyxV
TzyxP
tzyxC
TzyxD
xt
tzyxC C
L ςς
α
ξ γ
γ
( ) ( ) ( )
( )
( )[ ]
( )
( )
×
∂
∂
+
+
++
∂
∂
+
∂
∂
×
y
tzyxC
TzyxP
tzyxC
V
tzyxV
V
tzyxV
yx
tzyxC C ,,,
,,,
,,,
1
,,,,,,
1
,,, 112
2*
2
2*1
1
γ
γ
α
ξςς
( )) ( ) ( )
( )
( )[ ]
( )
×
+
+
++
∂
∂
+×
TzyxP
tzyxC
V
tzyxV
V
tzyxV
z
TzyxD C
L
,,,
,,,
1
,,,,,,
1,,, 12
2*
2
2*1 γ
γ
α
ξςς
( ) ( ) ( ) ( )[ ] +
∫ +∇
∂
∂
Ω+
∂
∂
×
−
zL
tv
CS
S
L WdtWyxCtzyx
Tk
D
xz
tzyxC
TzyxD ,,,,,,
,,,
,,, 22
1
αµ
( ) ( )[ ] ( ) ( )tzyxfWdtWyxCtzyx
Tk
D
y
C
L
tv
CS
S
z
δαµ ,,,,,,,, 22 +
∫ +∇
∂
∂
Ω+
−
(1d)
( ) ( ) ( ) ( ) ( ) ( )×−
+
= Tzyxk
y
tzyxI
TzyxD
yx
tzyxI
TzyxD
xt
tzyxI
VIII ,,,
,,,
,,,
,,,
,,,
,,,
,
112
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( )[ ] ( )[ ] ( ) ( ) ( )[ ] ×+−
+++×
2
11
1
1111 ,,,
,,,
,,,,,,,,, tzyxI
z
tzyxI
TzyxD
z
tzyxVtzyxI IIVI α
∂
∂
∂
∂
αα
( ) ( ) ( )[ ] +
∫ +∇
∂
∂
Ω+×
−
zL
tv
IS
IS
II WdtWyxItzyx
Tk
D
x
Tzyxk ,,,,,,,,, 12, αµ
( ) ( )[ ]
∫ +∇
∂
∂
Ω+
−
zL
tv
IS
IS
WdtWyxItzyx
Tk
D
y
,,,,,, 12αµ (3d)
( ) ( ) ( ) ( ) ( ) ( )×−
+
= Tzyxk
y
tzyxV
TzyxD
yx
tzyxV
TzyxD
xt
tzyxV
VIVV ,,,
,,,
,,,
,,,
,,,
,,,
,
112
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( )[ ] ( )[ ] ( ) ( ) ( )[ ] ×+−
+++×
2
11
1
1111 ,,,
,,,
,,,,,,,,, tzyxV
z
tzyxV
TzyxD
z
tzyxVtzyxI VVVI α
∂
∂
∂
∂
αα](https://image.slidesharecdn.com/3-160608092455/85/ON-MODIFICATION-OF-PROPERTIES-OF-P-N-JUNCTIONS-DURING-OVERGROWTH-15-320.jpg)
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
50
( ) ( ) ( )[ ] +
∫ +∇
∂
∂
Ω+×
−
zL
tv
VS
VS
VV WdtWyxVtzyx
Tk
D
x
Tzyxk ,,,,,,,,, 12, αµ
( ) ( )[ ]
∫ +∇
∂
∂
Ω+
−
zL
tv
VS
VS
WdtWyxVtzyx
Tk
D
y
,,,,,, 12αµ
( ) ( ) ( ) ( ) ( ) ( )×+
Φ
+
Φ
=
Φ
ΦΦ Tzyxk
y
tzyx
TzyxD
yx
tzyx
TzyxD
xt
tzyx
II
III
II
,,,
,,,
,,,
,,,
,,,
,,,
,
112
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( )[ ] ( ) ( )++
∫ Φ+∇
∂
∂
Ω+× Φ
−
Φ
Φ
tzyxfWdtWyxtzyx
Tk
D
x
tzyxI I
z
I
I
L
tv
IS
S
δαµ ,,,,,,,,,,, 12
2
( ) ( )[ ] ( ) ( ) +
Φ
+
∫ Φ+∇
∂
∂
Ω+ Φ
−
Φ
Φ
z
tzyx
TzyxD
z
WdtWyxtzyx
Tk
D
y
I
L
tv
IS
S
I
z
I
I
∂
∂
∂
∂
αµ
,,,
,,,,,,,,, 1
12
( ) ( )tzyxITzyxkI ,,,,,,+ (5d)
( )
( )
( )
( )
( )
( ) ×+
Φ
+
Φ
=
Φ
ΦΦ Tzyxk
y
tzyx
TzyxD
yx
tzyx
TzyxD
xt
tzyx
VV
VVV
VV
,,,
,,,
,,,
,,,
,,,
,,,
,
112
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( )[ ] ( ) ( )++
∫ Φ+∇
∂
∂
Ω+× Φ
−
Φ
Φ
tzyxfWdtWyxtzyx
Tk
D
x
tzyxV V
z
V
V
L
tv
VS
S
δαµ ,,,,,,,,,,, 12
2
( ) ( )[ ] ( ) ( )
+
Φ
+
∫ Φ+∇
∂
∂
Ω+ Φ
−
Φ
Φ
z
tzyx
TzyxD
z
WdtWyxtzyx
Tk
D
y
V
L
tv
VS
S
V
z
V
V
∂
∂
∂
∂
αµ
,,,
,,,,,,,,, 1
12
( ) ( )tzyxVTzyxkV ,,,,,,+ .
The second-order approximations of concentrations of dopant and radiation defects could be writ-
ten as
( ) ( ) ( )[ ]
( )
( ) ( )
( )
∫ ×
++
+
+
∂
∂
=
t
C
L
V
zyxV
V
zyxV
TzyxP
zyxC
TzyxD
x
tzyxC
0
2*
2
2*1
12
2
,,,,,,
1
,,,
,,,
1,,,,,,
τ
ς
τ
ς
τα
ξ γ
γ
( ) ( ) ( )
( )
( )[ ]
( )∫ ×
+
+
++
∂
∂
+
∂
∂
×
t
C
TzyxP
zyxC
V
zyxV
V
zyxV
y
d
x
zyxC
0
12
2*
2
2*1
1
,,,
,,,
1
,,,,,,
1
,,,
γ
γ
τα
ξ
τ
ς
τ
ςτ
τ
( ) ( ) ( ) ( ) ( )
( )∫ ×
++
∂
∂
+
∂
∂
×
t
LL
V
zyxV
V
zyxV
TzyxD
z
d
y
zyxC
TzyxD
0
2*
2
2*1
1 ,,,,,,
1,,,
,,,
,,,
τ
ς
τ
ςτ
τ
( ) ( )[ ]
( )
( )[ ] ×∫ ∫ +
∂
∂
Ω+
+
+
∂
∂
×
−
t L
v
C
SC
z
WdWyxC
Tk
D
x
d
TzyxP
zyxC
z
zyxC
0
12
121
,,,
,,,
,,,
1
,,,
τ
γ
γ
ϑατ
τα
ξ
τ
( ) ( ) ( )[ ] +∫ ∫ +∇
∂
∂
Ω+∇×
−
t L
v
CS
S
S ddWdWyxCzyx
Tk
D
y
ddzyx
z
0
12 ,,,,,,,,, τϑϑατµτϑτµ
τ
( )zyxfC ,,+ (1e)
( ) ( ) ( ) ( ) ( ) +∫+∫=
t
I
t
I d
y
zyxI
TzyxD
y
d
x
zyxI
TzyxD
x
tzyxI
0
1
0
1
2
,,,
,,,
,,,
,,,,,, τ
∂
τ∂
∂
∂
τ
∂
τ∂
∂
∂
( ) ( ) ( ) ( )[ ] ( )++∫ +−∫+ zyxfdzyxITzyxkd
z
zyxI
TzyxD
z
I
t
III
t
I ,,,,,,,,
,,,
,,,
0
2
12,
0
1
ττατ
∂
τ∂
∂
∂
( ) ( )[ ] ( )×∫ ∇
∂
∂
Ω+∫ ∫ +∇
∂
∂
Ω+
−
t
S
IS
t L
v
IS
IS
zyx
Tk
D
y
ddWdWyxIzyx
Tk
D
x
z
00
12 ,,,,,,,,, τµτϑϑατµ
τ
( )[ ] ( ) ( )[ ] ( )[ ]∫ ++−∫ +×
−
t
VIVI
L
v
I dzyxVzyxITzyxkddWdWyxI
z
0
1212,12 ,,,,,,,,,,,, ττατατϑϑα
τ
(3e)](https://image.slidesharecdn.com/3-160608092455/85/ON-MODIFICATION-OF-PROPERTIES-OF-P-N-JUNCTIONS-DURING-OVERGROWTH-16-320.jpg)
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
51
( ) ( ) ( ) ( ) ( ) +∫+∫=
t
V
t
V d
y
zyxV
TzyxD
y
d
x
zyxV
TzyxD
x
tzyxV
0
1
0
1
2
,,,
,,,
,,,
,,,,,, τ
∂
τ∂
∂
∂
τ
∂
τ∂
∂
∂
( ) ( ) ( ) ( )[ ] ( )++∫ +−∫+ zyxfdzyxVTzyxkd
z
zyxV
TzyxD
z
V
t
VVV
t
V ,,,,,,,,
,,,
,,,
0
2
12,
0
1
ττατ
∂
τ∂
∂
∂
( ) ( )[ ] ( )×∫ ∇
∂
∂
Ω+∫ ∫ +∇
∂
∂
Ω+
−
t
S
VS
t L
v
VS
VS
zyx
Tk
D
y
ddWdWyxVzyx
Tk
D
x
z
00
12 ,,,,,,,,, τµτϑϑατµ
τ
( )[ ] ( ) ( )[ ] ( )[ ]∫ ++−∫ +×
−
t
VIVI
L
v
I dzyxVzyxITzyxkddWdWyxV
z
0
1212,12 ,,,,,,,,,,,, ττατατϑϑα
τ
( ) ( ) ( ) ( ) ( ) +∫
Φ
+∫
Φ
=Φ ΦΦ
t
I
t
I
I d
y
zyx
TzyxD
y
d
x
zyx
TzyxD
x
tzyx II
0
1
0
1
2
,,,
,,,
,,,
,,,,,, τ
∂
τ∂
∂
∂
τ
∂
τ∂
∂
∂
( ) ( ) ( ) ( )[ ] ×∫ ∫ Φ+∇
∂
∂
Ω+∫
Φ
+
−
ΦΦ
t L
v
IS
t
I
z
II
WdWyxzyx
x
d
z
zyx
TzyxD
z 0
12
0
1
,,,,,,
,,,
,,,
τ
τατµτ
∂
τ∂
∂
∂
( ) ( )[ ] ( )++∫ ∫ Φ+∇
∂
∂
Ω+× Φ
−
Φ
ΦΦ
zyxfddWdWyxzyx
Tk
D
y
d
Tk
D
I
z
I
II
t L
v
IS
SS
,,,,,,,,
0
12 τϑϑατµτ
τ
( ) ( ) ( ) ( )∫+∫+
t
I
t
II dzyxITzyxkdzyxITzyxk
00
2
, ,,,,,,,,,,,, ττττ (5e)
( ) ( ) ( ) ( ) ( ) +∫
Φ
+∫
Φ
=Φ ΦΦ
t
V
t
V
V d
y
zyx
TzyxD
y
d
x
zyx
TzyxD
x
tzyx VV
0
1
0
1
2
,,,
,,,
,,,
,,,,,, τ
∂
τ∂
∂
∂
τ
∂
τ∂
∂
∂
( ) ( ) ( ) ( )[ ] ×∫ ∫ Φ+∇
∂
∂
Ω+∫
Φ
+
−
ΦΦ
t L
v
VS
t
V
z
VV
WdWyxzyx
x
d
z
zyx
TzyxD
z 0
12
0
1
,,,,,,
,,,
,,,
τ
τατµτ
∂
τ∂
∂
∂
( ) ( )[ ] ( )++∫ ∫ Φ+∇
∂
∂
Ω+× Φ
−
Φ
ΦΦ
zyxfddWdWyxzyx
Tk
D
y
d
Tk
D
V
z
V
VV
t L
v
VS
SS
,,,,,,,,
0
12 τϑϑατµτ
τ
( ) ( ) ( ) ( )∫+∫+
t
V
t
VV dzyxVTzyxkdzyxVTzyxk
00
2
, ,,,,,,,,,,,, ττττ .
Parameters bi, Cρ, F, r, s have been determined by the following relations
00
2
0000
2
004
11
IIVV
zyx
VVIV
zyx
SS
LLL
SS
LLL
b
Θ
−
Θ
= , ( ) +
Θ
Θ++−=
zyx
VVII
zyxIVVV
LLL
SS
LLLSSb 0000
10013 2
( ) ( ) 3333
10
2
00
1001
2
00
011001
0000
22
zyx
IVIV
zyxIVVV
zyx
IV
zyxIVIIIV
zyx
VVIV
LLL
SS
LLLSS
LLL
S
LLLSSS
LLL
SS
Θ
−Θ++
Θ
+Θ+++
Θ
+ ,
( ) ( ) ( )×++Θ
Θ
++−Θ−++
Θ
= 0110
012
10011102
0000
2 22 IVIIzyx
x
IV
IVVVzyxVIVVV
zyx
VVII
SSLLL
L
S
SSLLLCSS
LLL
SS
b
( )( ) ×
Θ
−+Θ+++++Θ
Θ
+×
zyx
IV
IVzyxVVIVIIIVzyx
zyx
IV
zy
VV
LLL
S
SLLLSSSSLLL
LLL
S
LL
S 2
00
1001011001
0000
222
( ) 01
0010
2222
2
00
1102
2
IV
zyx
IVIV
zyx
IVI
IVVVV S
LLL
SS
LLL
SC
SSC
Θ
−
Θ
+−−× , ( ×Θ+
Θ
++
= xIV
zyx
VVVIV
II LS
LLL
CSS
Sb 10
0211
001
) ( ) ×
Θ
−−
−Θ++
Θ
++Θ
++×
zyx
IVVVV
zyxIVVV
zyx
IVIIzyx
IVVVzy
LLL
SSC
LLLSS
LLL
SSLLL
SSLL 1102
1001
0110
0101 2
2
2
( )
zyx
IVIV
IVIVIzyxIIIVIV
LLL
SS
SSCLLLSSS
Θ
−+Θ++×
2
0110
0100100100 223 ,
( )
×−
Θ
+
= 01
2
0200
000 IV
zyx
VVIV
II S
LLL
SS
Sb](https://image.slidesharecdn.com/3-160608092455/85/ON-MODIFICATION-OF-PROPERTIES-OF-P-N-JUNCTIONS-DURING-OVERGROWTH-17-320.jpg)
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
52
( ) ( )×++Θ
Θ
−−
−++Θ
Θ
−−
× 0110
1102
0110
1102
22 IVIIzyx
zyx
IVVVV
IVIIzyx
zyx
IVVVV
SSLLL
LLL
SSC
SSLLL
LLL
SSC
( )0110
1102
01
2
0101 22 IVIIzyx
zyx
IVVVV
IVIVIIV SSLLL
LLL
SSC
SSCS ++Θ
Θ
−−
−+× , +
Θ
=
zyx
VI
IVI
LLL
SC 11
00
αα
zyx
IV
zyx
IIII
zyx
III
LLL
S
LLL
SS
LLL
S
Θ
−
Θ
−
Θ
+ 11202000
2
1α
, 110200
2
10011 IVVVVVVIVVIV SSSSC −−+= ααα ,
4
2
2
4
2
32
48
a
a
a
a
yE Θ−Θ+= ,
2
4
2
1
4
222
3
4
3
2
3
4
2
32
22
4
2
0
4
31
02
4
2
3
854
4
8
4
24 b
b
LLL
b
b
b
b
b
b
b
b
bb
LLLb
b
b
r zyxzyx
Θ
−
Θ
−
Θ−Θ
Θ
−
Θ−
Θ
= , ( −
Θ
= 402
4
2
4
12
bb
b
s
) 4231 18bbbbLLL zyx Θ−Θ− , 3 323 32
4
2
6
rsrrsr
a
a
F ++−−++
Θ
= .
Equations for components of the displacement vector could be presented in the form
( ) ( ) ( ) ( )
( )[ ]
( ) ( ) ( )
( )[ ]
( )
+
∂∂
∂
+
−+
∂
∂
+
+=
∂
∂
yx
tzyxu
z
zE
zK
x
tzyxu
z
zE
zK
t
tzyxu
z
yxx
,,,
13
,,,
16
5,,, 1
2
2
1
2
2
2
2
σσ
ρ
( )
( )[ ]
( ) ( ) ( ) ( )
( )[ ]
( )
−
∂∂
∂
+
++
∂
∂
+
∂
∂
+
+
zx
tzyxu
z
zE
zK
z
tzyxu
y
tzyxu
z
zE zzy ,,,
13
,,,,,,
12
1
2
2
1
2
2
1
2
σσ
( ) ( ) ( )
x
tzyxT
zzK
∂
∂
−
,,,
β
( )
( ) ( )
( )[ ]
( ) ( ) ( ) ( ) ( )+
∂
∂
−
∂∂
∂
+
∂
∂
+
=
∂
∂
y
tzyxT
zzK
yx
tzyxu
x
tzyxu
z
zE
t
tzyxu
z xyy ,,,,,,,,,
12
,,, 1
2
2
1
2
2
2
2
β
σ
ρ
( )
( )[ ]
( ) ( ) ( ) ( )
( )[ ]
( ) +
+
+∂
∂
+
∂
∂
+
∂
∂
+∂
∂
+ zK
z
zE
y
tzyxu
y
tzyxu
z
tzyxu
z
zE
z
yzy
σσ 112
5,,,,,,,,,
12 2
1
2
11
( ) ( )
( )[ ]
( )
( )
( )
yx
tzyxu
zK
zy
tzyxu
z
zE
zK
yy
∂∂
∂
+
∂∂
∂
+
−+
,,,,,,
16
1
2
1
2
σ
( ) ( ) ( ) ( ) ( ) ( )
×
∂∂
∂
+
∂∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
zy
tzyxu
zx
tzyxu
y
tzyxu
x
tzyxu
t
tzyxu
z
yxzzz
,,,,,,,,,,,,,,, 1
2
1
2
2
1
2
2
1
2
2
2
2
ρ
( )
( )[ ]
( ) ( ) ( ) ( ) ( )
( )[ ]
×
+∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
∂
∂
+
+
×
z
zE
zz
tzyxu
y
tzyxu
x
tzyxu
zK
zz
zE xyx
σσ 16
,,,,,,,,,
12
111
( ) ( ) ( ) ( ) ( ) ( ) ( )
z
tzyxT
zzK
z
tzyxu
y
tzyxu
x
tzyxu
z
tzyxu zyxz
∂
∂
−
∂
∂
−
∂
∂
−
∂
∂
−
∂
∂
×
,,,,,,,,,,,,,,,
6 1111
β .
Integration of the left and the right sides of the above equations on time t leads to final relations
for components of displacement vector
( )
( )
( ) ( )
( )[ ]
( )
( )
( ) ( )
( )[ ]
×
+
−+∫ ∫
∂
∂
+
+=
z
zE
zK
z
ddzyxu
xz
zE
zK
z
tzyxu
t
xx
σρ
ϑττ
σρ
ϑ
13
1
,,,
16
51
,,,
0 0
12
2
2
( ) ( ) ( ) ×
∫ ∫
∂
∂
+∫ ∫
∂
∂
+∫ ∫
∂∂
∂
×
t
z
t
y
t
y ddzyxu
z
ddzyxu
y
ddzyxu
yx 0 0
12
2
0 0
12
2
0 0
1
2
,,,,,,,,,
ϑϑϑ
ϑττϑττϑττ](https://image.slidesharecdn.com/3-160608092455/85/ON-MODIFICATION-OF-PROPERTIES-OF-P-N-JUNCTIONS-DURING-OVERGROWTH-18-320.jpg)
![International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
53
( )
( ) ( )[ ] ( )
( ) ( ) ( )
( )[ ]
( ) ( )
( )
×−
+
+∫ ∫
∂∂
∂
+
+
×
z
z
zK
z
zE
zKddzyxu
zxzzz
zE t
z
ρ
β
σ
ϑττ
ρσρ
ϑ
13
,,,
1
12 0 0
1
2
( )
( )
( ) ( )
( )[ ]
( )
( )
×−∫ ∫
∂
∂
+
+−∫ ∫
∂
∂
×
∞
z
ddzyxu
xz
zE
zK
z
ddzyxT
x
x
t
ρ
ϑττ
σρ
ϑττ
ϑϑ 1
,,,
16
51
,,,
0 0
12
2
0 0
( ) ( )
( )[ ]
( )
( )
( )
+∫ ∫
∂
∂
−∫ ∫
∂∂
∂
+
−×
∞∞
0 0
12
2
0 0
1
2
,,,
2
1
,,,
13
ϑϑ
ϑττ
ρ
ϑττ
σ
ddzyxu
yz
ddzyxu
yxz
zE
zK yy
( ) ( )
( )
( ) ( )
( )[ ]
( ) ×∫ ∫
∂∂
∂
+
+−
+
∫ ∫
∂
∂
+
∞∞
0 0
1
2
0 0
12
2
,,,
131
,,,
ϑϑ
ϑττ
σσ
ϑττ ddzyxu
zxz
zE
zK
z
zE
ddzyxu
z
zz
( )
( ) ( )
( )
( )∫ ∫
∂
∂
++×
∞
0 0
0 ,,,
1 ϑ
ϑττ
ρ
β
ρ
ddzyxT
xz
z
zKu
z
x
( ) ( )
( ) ( )[ ]
( ) ( ) +
∫ ∫
∂∂
∂
+∫ ∫
∂
∂
+
=
t
x
t
xy ddzyxu
yx
ddzyxu
xzz
zE
tzyxu
0 0
1
2
0 0
12
2
2 ,,,,,,
12
,,,
ϑϑ
ϑττϑττ
σρ
( )
( )
( )
( )
( ) ( ) ( )
( )[ ]
+
+
+∫ ∫
∂
∂
+∫ ∫
∂∂
∂
+
z
zE
zKddzyxu
yz
ddzyxu
yxz
zK t
x
t
y
σ
ϑττ
ρ
ϑττ
ρ
ϑϑ
112
5
,,,
1
,,,
0 0
12
2
0 0
1
2
( )
( )[ ] ( )
( )
( )
( ) ( ) −
∫ ∫
∂
∂
+∫ ∫
∂
∂
+∂
∂
+
+
+
t
z
t
y ddzyxu
y
ddzyxu
zz
zE
zzz
zE
0 0
1
0 0
1 ,,,,,,
12
1
112
5 ϑϑ
ϑττϑττ
σρσ
( ) ( )
( )
( )
( )
( ) ( ) ( )
( )[ ]
−
+
−∫ ∫
∂∂
∂
+∫ ∫−
z
zE
zKddzyxu
zyz
ddzyxT
z
z
zK
t
y
t
σ
ϑττ
ρ
ϑττ
ρ
β ϑϑ
16
,,,
1
,,,
0 0
1
2
0 0
( )
( ) ( )[ ]
( ) ( ) ( ) ( )
( )
×−
∫ ∫
∂∂
∂
+∫ ∫
∂
∂
+
−
∞∞
z
z
zKddzyxu
yx
ddzyxu
xzz
zE
xx
ρ
β
ϑττϑττ
σρ
ϑϑ
0 0
1
2
0 0
12
2
,,,,,,
12
( ) ( )
( )
( )
( )
( ) ×∫ ∫
∂
∂
−∫ ∫
∂∂
∂
−∫ ∫×
∞∞∞
0 0
12
2
0 0
1
2
0 0
,,,
1
,,,,,,
ϑϑϑ
ϑττ
ρ
ϑττ
ρ
ϑττ ddzyxu
yz
ddzyxu
yxz
zK
ddzyxT xy
( ) ( )
( )[ ]
( )
( )
( ) ( ) ×
∫ ∫
∂
∂
+∫ ∫
∂
∂
+∂
∂
−
+
+×
∞∞
0 0
1
0 0
1 ,,,,,,
1112
5 ϑϑ
ϑττϑττ
σσ
ddzyxu
y
ddzyxu
zz
zE
zz
zE
zK zy
( ) ( )
( ) ( )
( )[ ]
( ) yy uddzyxu
zyz
zE
zK
zz
0
0 0
1
2
,,,
16
1
2
1
+∫ ∫
∂∂
∂
+
−−×
∞ϑ
ϑττ
σρρ
( ) ( )
( ) ( )[ ]
( ) ( )
+∫ ∫
∂
∂
+∫ ∫
∂
∂
+
=
∞∞
0 0
12
2
0 0
12
2
,,,,,,
12
,,,
ϑϑ
ϑττϑττ
σρ
ddzyxu
y
ddzyxu
xzz
zE
tzyxu zzz
( ) ( ) ( )
+∫ ∫
∂
∂
∂
∂
+
∫ ∫
∂∂
∂
+∫ ∫
∂∂
∂
+
∞∞∞
0 0
1
0 0
1
2
0 0
1
2
,,,,,,,,,
ϑϑϑ
ϑττϑττϑττ ddzyxu
xz
ddzyxu
zy
ddzyxu
zx
xyx
( ) ( ) ( )
( ) ( )
( )
( )
×
+∂
∂
+
∫ ∫
∂
∂
+∫ ∫
∂
∂
+
∞∞
z
zE
zzz
zKddzyxu
z
ddzyxu
y
xx
σρρ
ϑττϑττ
ϑϑ
16
11
,,,,,,
0 0
1
0 0
1
( ) ( ) ( )
−∫ ∫
∂
∂
−∫ ∫
∂
∂
−∫ ∫
∂
∂
×
∞∞∞
0 0
1
0 0
1
0 0
1 ,,,,,,,,,6
ϑϑϑ
ϑττϑττϑττ ddzyxu
y
ddzyxu
x
ddzyxu
z
yxz
( ) ( ) ( )
( )
( ) zz uddzyxT
zz
z
zKddzyxu
z
0
0 00 0
1 ,,,,,, +∫ ∫
∂
∂
−
∫ ∫
∂
∂
−
∞∞ ϑϑ
ϑττ
ρ
β
ϑττ .](https://image.slidesharecdn.com/3-160608092455/85/ON-MODIFICATION-OF-PROPERTIES-OF-P-N-JUNCTIONS-DURING-OVERGROWTH-19-320.jpg)
ON MODIFICATION OF PROPERTIES OF P-N-JUNCTIONS DURING OVERGROWTH
- 1. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 DOI: 10.5121/ijitmc.2016.4203 35 ON MODIFICATION OF PROPERTIES OF P-N-JUNCTIONS DURING OVERGROWTH 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 consider influence of overgrowth of doped by diffusion and ion implantation areas of hete- rostructures on distributions of concentrations of dopants. Several conditions to increase sharpness of p-n- junctions (single and framework bipolar transistors), which were manufactured during considered technol- ogical process, have been determined. At the same time we analyzed influence of speed of overgrowth of doped areas and mechanical stress in the considered heterostructure on distribution of concentrations of dopants in the structure. KEYWORDS Diffusion-junction heterorectifier; implanted-junction heterorectifier; overgrowth of doped area; analytical approach for modeling 1. INTRODUCTION In the present time they are several approaches could be used to manufacture p-n-junctions diffu- sion of dopants in a homogenous sample or an epitaxial layer of heterostructure, implantation of ions of dopants in the same situations or doping during epitaxial growth [1-7]. The same ap- proaches could be used to manufacture systems of p-n-junctions: bipolar transistors and thyris- tors. The first and the second ways of doping are preferable in comparison with the third one be- cause the approaches give us possibility to dope locally materials during manufacture integrated circuits easily in comparison with epitaxial growth. Using diffusion and ion implantation in ho- mogenous sample to manufacturing p-n-junctions leads to production fluently varying and wide distributions of dopants. One of actual problems is increasing sharpness of p-n- junctions [5,7]. The increasing of sharpness gives us possibility to decrease switching time of p-n-junctions. In- creasing of homogeneity of dopant distribution in enriched by the dopant area is also attracted an interest [5]. The increasing of homogeneity gives us possibility to decrease local overheat of the doped materials due to streaming of electrical current during operating of p-n-junction or to de- crease depth of p-n-junction for fixed value of local overheats. One way to increase sharpness of p-n-junction based on using near-surficial (laser or microwave) types of annealing [8-15]. Framework the types of annealing one can obtain near-surficial heating of the doped materials. In this situation due to the Arrhenius low one can obtain increasing of dopant diffusion coefficient of near-surficial area in comparison with volumetric dopant diffusion coefficient. The increasing of dopant diffusion coefficient of near-surficial area leads to increasing of sharpness of p-n-junction. The second way to increase sharpness of p-n-junction based on using high doping of materials. In this case contribution of nonlinearity of diffusion process increases [4]. The third way to increase
- 2. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 36 sharpness of p-n-junction is using of inhomogeneity of heterostructure [16,17]. Framework the approach we consider simplest heterostructure, which consist of substrate and epitaxial layer. One can find increasing of sharpness of p-n-junction after annealing with appropriate annealing time in the case, when dopant diffusion coefficient in the substrate is smaller, than in the epitaxial layer. The fourth way to increase sharpness of p-n-junction is radiation processing of materials. The radiation processing leads to radiation-enhanced diffusion [18]. However using radiation processing of materials leads to necessity of annealing of radiation defects. Density of elements of integrated circuits could be increases by using mismatch-induced stress [19]. However one could obtain increased unsoundness of doped material (for example, to generation of dislocation of dis- agreement) by using the approach [7]. The considered approaches gives a possibility to increase sharpness of p-n-junction with increas- ing of homogeneity of dopant distribution in enriched by the dopant area. Using combination of the above approaches gives us possibility to increase sharpness of p-n-junction and increasing of homogeneity of dopant distribution in enriched area at one time. z D(z),P(z) D2 D3 0 Lz a fC impl(z) fC diff(z) P2 P3 Substrate Epitaxial layer Overlayer D1 P1 Fig. 1. Substrate, epitaxial and overlayers framework heterostructure Framework this paper we consider a substrate with known type of conductivity (n or p) and an epitaxial layer, included into a heterostructure. The epitaxial layer have been doped by diffusion or by ion implantation to manufacture another type of conductivity (p or n). Farther we consider overgrowth of the epitaxial layer by an overlayer (see Fig. 1). The overlayer has type of conduc- tivity, which coincide with type of conductivity of the substrate. In this paper we analyzed influ- ence of overgrowth of the doped epitaxial layer on distribution of dopants in the considered hete- rostructure. 2. METHOD OF SOLUTION In this section we calculate spatio-temporal distribution of concentration of dopant in the consi- dered heterostructure to solve our aim. To calculate the distribution we solved the following boundary problem [1,20-22] ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC ,,,,,,,,,,,, (1)
- 3. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 37 ( ) ( ) ( ) ( ) ∫∇ ∂ ∂ Ω+ ∫∇ ∂ ∂ Ω+ −− zz L tv S S L tv S S WdtWyxCtzyx Tk D y WdtWyxCtzyx Tk D x ,,,,,,,,,,,, µµ . Boundary and initial conditions for our case could be written as ( ) 0 ,,, 0 = ∂ ∂ =x x tzyxC , ( ) 0 ,,, = ∂ ∂ = xLx x tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =y y tzyxC , ( ) 0 ,,, = ∂ ∂ = yLx y tzyxC , ( ) 0 ,,, = ∂ ∂ −= tvz z tzyxC , ( ) 0 ,,, = ∂ ∂ = zLx z tzyxC , C(x,y,z,0)=fC(x,y,z). In the above relations we used the function C (x,y,z,t) as the spatio-temporal distribution of con- centration of dopant; Ω is the atomic volume; surface concentration of dopant on interface be- tween layers of heterostructure could be determined as the following integral ( )∫ − zL tv zdtzyxC ,,, (we assume, that the interface between layers of heterostructure is perpendicular to the direction Oz); surface gradient we denote as symbol ∇S; µ(x,y,z,t) is the chemical potential (reason of ac- counting of the chemical potential is mismatch-induced stress); the parameters D and DS are the coefficients of volumetric and surface diffusions. One can find the surface diffusions due to mis- match-induced stress. Diffusion coefficients depends on temperature and speed of heating and cooling of heterostructure, properties of materials of heterostructure, spatio-temporal distributions of concentrations of dopant and radiation defects after ion implantation. Approximations of these dependences could be approximated by the following functions [22,23] ( ) ( ) ( ) ( ) ( ) ( ) ++ += 2* 2 2*1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD SLSS ςςξ γ γ . (2) Functions DL(x,y,z,T) and DLS(x,y,z,T) described dependences of diffusion coefficients on coordi- nate (due to presents several layers in heterostructure, manufactured by using different materials) and temperature of annealing T (due to Arrhenius law); function P(x,y,z,T) describes dependence of limit of solubility of dopant on coordinate and temperature; parameter γ could be integer and depends on properties of materials of heterostructure [23]; V(x,y,z,t) is the spatio-temporal con- centration of radiation vacancies; V* is the equilibrium concentration of vacancies. Concentration- al dependence of dopant diffusion coefficients has been described in details in [23]. We determine distributions of concentrations of point defects in space and time as solutions of the following system of equations [1,20-22] ( ) ( ) ( ) ( ) ( ) ( )×− + = Tzyxk y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI IIII ,,, ,,, ,,, ,,, ,,, ,,, , ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxk z tzyxI TzyxD z tzyxI VII ,,,,,,,,, ,,, ,,,,,, , 2 − +× ∂ ∂ ∂ ∂ (3)
- 4. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 38 ( ) ( ) ( ) ( ) ∫∇ ∂ ∂ Ω+ ∫∇ ∂ ∂ Ω+ −− zz L tv S SI L tv S SI WdtWyxItzyx Tk D y WdtWyxItzyx Tk D x ,,,,,,,,,,,, µµ ( ) ( ) ( ) ( ) ( ) ( )×− + = Tzyxk y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VVVV ,,, ,,, ,,, ,,, ,,, ,,, , ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( ) ( )+− +× tzyxVtzyxITzyxk z tzyxV TzyxD z tzyxV VIV ,,,,,,,,, ,,, ,,,,,, , 2 ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ∫∇ ∂ ∂ Ω+ ∫∇ ∂ ∂ Ω+ −− zz L tv S VS L tv S VS WdtWyxVtzyx Tk D y WdtWyxVtzyx Tk D x ,,,,,,,,,,,, µµ . Boundary and initial conditions for these equations could be written as ( ) 0 ,,, 0 = =x x tzyxI ∂ ∂ , ( ) 0 ,,, = = xLx x tzyxI ∂ ∂ , ( ) 0 ,,, 0 = =y y tzyxI ∂ ∂ , ( ) 0 ,,, = = yLy y tzyxI ∂ ∂ , ( ) 0 ,,, = −= tvz z tzyxI ∂ ∂ , ( ) 0 ,,, = = zLz z tzyxI ∂ ∂ , ( ) 0 ,,, 0 = =x x tzyxV ∂ ∂ , ( ) 0 ,,, = = xLx x tzyxV ∂ ∂ , ( ) 0 ,,, 0 = =y y tzyxV ∂ ∂ , ( ) 0 ,,, = = yLy y tzyxV ∂ ∂ , ( ) 0 ,,, = −= tvz z tzyxV ∂ ∂ , ( ) 0 ,,, = = zLz z tzyxV ∂ ∂ , I(x,y,z,0)=fI(x,y,z), V(x,y,z,0)=fV(x,y,z). (4) Here I(x,y,z,t) is the distribution of concentration of radiation interstitials in space and time; I* is the equilibrium concentration interstitials; DI(x,y,z,T), DV(x,y,z,T), DIS(x,y,z,T), DVS(x,y,z,T) are the coefficients of volumetric and surface diffusion; terms V2 (x,y,z,t) and I2 (x,y,z,t) corresponds to generation divacancies and analogous complexes of interstitials (see, for example, [22] and ap- propriate references in this work); the functions kI,V(x,y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T) described dependences of parameters of recombination of point defects and generation their complexes on coordinate and temperature; k is the Boltzmann constant. We calculate spatio-temporal distributions of concentrations of divacancies ΦV (x,y,z,t) and diin- terstitials ΦI (x,y,z,t) by solution of the equations [20-22] ( ) ( ) ( ) ( ) ( ) ( )×+ Φ + Φ = Φ ΦΦ Tzyxk y tzyx TzyxD yx tzyx TzyxD xt tzyx I III II ,,, ,,, ,,, ,,, ,,, ,,, ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( ) + ∫Φ∇ ∂ ∂ Ω+ Φ +× − Φ Φ z I I L tv IS SI WdtWyxtzyx Tk D xz tzyx TzyxD z tzyxI ,,,,,, ,,, ,,,,,, µ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( )tzyxITzyxkWdtWyxtzyx Tk D y II L tv IS IS z ,,,,,,,,,,,, 2 ,+ ∫Φ∇ ∂ ∂ Ω+ − Φ µ (5) ( ) ( ) ( ) ( ) ( ) ( )×+ Φ + Φ = Φ ΦΦ Tzyxk y tzyx TzyxD yx tzyx TzyxD xt tzyx V VVV VV ,,, ,,, ,,, ,,, ,,, ,,, ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( ) + ∫Φ∇ ∂ ∂ Ω+ Φ +× − Φ Φ z V V L tv VS SV WdtWyxtzyx Tk D xz tzyx TzyxD z tzyxV ,,,,,, ,,, ,,,,,, µ ∂ ∂ ∂ ∂
- 5. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 39 ( ) ( ) ( ) ( )tzyxVTzyxkWdtWyxtzyx Tk D y VV L tv VS S z V ,,,,,,,,,,,, 2 ,+ ∫Φ∇ ∂ ∂ Ω+ − Φ µ . Boundary and initial conditions could be written as ( ) 0 ,,, 0 = Φ =x I x tzyx ∂ ∂ , ( ) 0 ,,, = Φ = xLx I x tzyx ∂ ∂ , ( ) 0 ,,, 0 = Φ =y I y tzyx ∂ ∂ , ( ) 0 ,,, = Φ = yLy I y tzyx ∂ ∂ , ( ) 0 ,,, = Φ −= tvz I z tzyx ∂ ∂ , ( ) 0 ,,, = = zLz z tzyxI ∂ ∂ , ( ) 0 ,,, 0 = Φ =x V x tzyx ∂ ∂ , ( ) 0 ,,, = = xLx x tzyxV ∂ ∂ , ( ) 0 ,,, 0 = Φ =y V y tzyx ∂ ∂ , ( ) 0 ,,, = Φ = yLy V y tzyx ∂ ∂ , ( ) 0 ,,, = Φ −= tvz V z tzyx ∂ ∂ , ( ) 0 ,,, = Φ = zLz V z tzyx ∂ ∂ , ΦI(x,y,z,0)=fΦI(x,y,z), ΦV(x,y,z,0)=fΦV(x,y,z). (6) Here DΦI(x,y,z,T), DΦV(x,y,z,T), DΦIS(x,y,z,T) and DΦVS(x,y,z,T) are the coefficients of volumetric and surface diffusion; the functions kI(x,y,z,T) and kV(x,y,z,T) described dependences of parame- ters of decay of complexes of point defects on coordinate and temperature. One can determine chemical potential µ in the Eq.(1) by the following relation [20] µ=E(z)Ωσij[uij(x,y,z,t)+uji(x,y,z,t)]/2. (7) Here E is the tension (Young) modulus; ∂ ∂ + ∂ ∂ = i j j i ij x u x u u 2 1 is the deformation tensor; σij is the stress tensor; ui, uj are the components ux(x,y,z,t), uy(x,y,z,t) and uz(x,y,z,t) of the displacement ten- sor ( )tzyxu ,,, r ; xi, xj are the coordinates x, y, z. The relation (3) could be transformed to the fol- lowing form ( ) ( ) ( ) ( ) ( ) ( ) +− ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂Ω = ij i j j i i j j i x tzyxu x tzyxu x tzyxu x tzyxu zEtzyx δεµ 0 ,,,,,, 2 1,,,,,, 2 ,,, ( ) ( ) ( ) ( ) ( ) ( )[ ] −− − ∂ ∂ − + ij k kij TtzyxTzzK x tzyxu z z δβε σ δσ 00 ,,,3 ,,, 21 , where σ is the Poisson coefficient; the parameter ε0=(as-aEL)/aEL describes the displacement pa- rameter with lattice distances of the substrate and the epitaxial layer as, aEL; K is the modulus of uniform compression; the parameter b describes the thermal expansion; we assume, that the equi- librium temperature Tr coincides with the room temperature. Components of the displacement vector could be described by solving the following system of equations [24] ( ) ( ) ( ) ( ) ( ) z tzyx y tzyx x tzyx t tzyxu z xzxyxxx ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ,,,,,,,,,,,, 2 2 σσσ ρ ( ) ( ) ( ) ( ) ( ) z tzyx y tzyx x tzyx t tzyxu z yzyyyxy ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ,,,,,,,,,,,, 2 2 σσσ ρ ( ) ( ) ( ) ( ) ( ) z tzyx y tzyx x tzyx t tzyxu z zzzyzxz ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ,,,,,,,,,,,, 2 2 σσσ ρ
- 6. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 40 where ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )− ∂ ∂ + ∂ ∂ − ∂ ∂ + ∂ ∂ + = k k ij k kij i j j i ij x tzyxu zK x tzyxu x tzyxu x tzyxu z zE ,,,,,, 3 ,,,,,, 12 δ δ σ σ ( ) ( ) ( )[ ]rTtzyxTzKz −− ,,,β , ρ (z) describes the density of materials of heterostructure. The ten- sor δij describes the Kronecker symbol. Accounting relation for σij in the previous system of equa- tions last system of equation could be written as ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) + ∂∂ ∂ + −+ ∂ ∂ + += ∂ ∂ yx tzyxu z zE zK x tzyxu z zE zK t tzyxu z yxx ,,, 13 ,,, 16 5,,, 2 2 2 2 2 σσ ρ ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ) − ∂∂ ∂ + ++ ∂ ∂ + ∂ ∂ + + zx tzyxu z zE zK z tzyxu y tzyxu z zE zzy ,,, 13 ,,,,,, 12 2 2 2 2 2 σσ ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )+ ∂ ∂ − ∂∂ ∂ + ∂ ∂ + = ∂ ∂ y tzyxT zKz yx tzyxu x tzyxu z zE t tzyxu z xyy ,,,,,,,,, 12 ,,, 2 2 2 2 2 β σ ρ ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) + ∂ ∂ + + + ∂ ∂ + ∂ ∂ +∂ ∂ + 2 2 ,,, 112 5,,,,,, 12 y tzyxu zK z zE y tzyxu z tzyxu z zE z yzy σσ ( ) ( ) ( )[ ] ( ) ( ) ( ) yx tzyxu zK zy tzyxu z zE zK yy ∂∂ ∂ + ∂∂ ∂ + −+ ,,,,,, 16 22 σ (8) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) + ∂∂ ∂ + ∂∂ ∂ + ∂ ∂ + ∂ ∂ + = ∂ ∂ zy tzyxu zx tzyxu y tzyxu x tzyxu z zE t tzyxu z yxzzz ,,,,,,,,,,,, 12 ,,, 22 2 2 2 2 2 2 σ ρ ( ) ( ) ( ) ( ) ( ) ( ) ( )+ ∂ ∂ − ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + z tzyxT zzK z tzyxu y tzyxu x tzyxu zK z xyx ,,,,,,,,,,,, β ( ) ( ) ( ) ( ) ( ) ( ) ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ +∂ ∂ + z tzyxu y tzyxu x tzyxu z tzyxu z zE z zyxz ,,,,,,,,,,,, 6 16 1 σ . Systems of conditions for these equations could be written as ( ) 0 ,,, 0 = ∂ ∂ =x x tzyxu r ; ( ) 0 ,,, = ∂ ∂ = xLx x tzyxu r ; ( ) 0 ,,, 0 = ∂ ∂ =y y tzyxu r ; ( ) 0 ,,, = ∂ ∂ = yLy y tzyxu r ; ( ) 0 ,,, = ∂ ∂ −= tvz z tzyxu r ; ( ) 0 ,,, = ∂ ∂ = zLz z tzyxu r ; ( ) 00,,, uzyxu rr = ; ( ) 0,,, uzyxu rr =∞ . We calculate distribution of concentration of dopant in space in time by method of averaging of function corrections [25-30]. To use the method we re-write Eqs. (1), (3) and (5) with account appropriate initial distributions (see Appendix). In future we replace the required concentrations in right sides of the obtained equations on their average values α1ρ, which are not yet known. The equations modified after the replacement and solution of these equations are presented in the Ap- pendix. We determined average values of the first-order approximations of the considered concentrations by using the following standard relations [25-30]
- 7. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 41 ( )∫ ∫ ∫ ∫ Θ = Θ −0 0 0 11 ,,, 1 x y zL L L tvzyx tdxdydzdtzyx LLL ρα ρ . (9) Substitution of the solutions of the modified equations into the relation (9) gives a possibility to obtain appropriate average values in the following form ( )∫ ∫ ∫ ∫= Θ −0 0 0 1 ,, 1 x y zL L L tv C zyx C tdxdydzdzyxf LLL α , ( ) 4 3 4 1 2 3 2 4 2 3 1 4 4 4 a Aa a aLLLBa B a Aa zyx I + − Θ+Θ +− + =α , ( ) Θ−−∫ ∫ ∫ ∫ Θ = Θ − zyxIII L L L tv I IIV V LLLStdxdydzdzyxf S x y z 001 0 0 0100 1 ,, 1 α α α , ( )∫ ∫ ∫ ∫+ Θ + Θ = Θ − ΦΦ 0 0 0 201 1 ,, 1 x y z II L L L tvzyxzyx II zyx I tdxdydzdzyxf LLLLLL S LLL R α , ( )∫ ∫ ∫ ∫+ Θ + Θ = Θ − ΦΦ 0 0 0 201 1 ,, 1 x y z VV L L L tvzyxzyx VV zyx V tdxdydzdzyxf LLLLLL S LLL R α . Relations for calculations parameters Sρρij, ai, A, B, q, p are presented in the Appendix. We used standard iterative procedure of method of averaging of function corrections to calculate the second- and higher-order approximations of the considered concentrations [25-30]. Frame- work the procedure to calculate approximations of the n-th order of the above concentrations we replace the functions C(x,y,z,t), I(x,y,z,t), V(x,y,z,t), ΦI(x,y,z,t) and ΦV(x,y,z,t) in the Eqs. (1), (3), (5) with account initial distributions (see Eqs. (1a), (3a), (5a) in the Appendix) on the sums of the not yet known average values of the considered approximations and approximations of the pre- vious order, i.e. αnρ+ρn-1(x,y,z,t). These obtained equations and their solutions, which calculated are presented in the Appendix. We calculate average values of the second-order approximations of required functions by using the following standard relation [25-30] ( ) ( )[ ]∫ ∫ ∫ ∫ − Θ = Θ 0 0 0 0 122 ,,,,,, 1 x y zL L L zyx tdxdydzdtzyxtzyx LLL ρρα ρ . (10) Substitution of the second-order approximations of concentrations of dopant into Eq.(10) leads to the considered average values α2ρ α2C=0, α2ΦI =0, α2ΦV =0, ( ) 4 3 4 1 2 3 2 4 2 3 2 4 4 4 b Eb b bLLLFa F b Eb zyx V + − Θ+Θ +− + =α , ( ) 00201 11021001200 2 2 2 2 IVVIV VIVVzyxVIVVVVVVV I SS SSLLLSSSC α αα α + −−Θ++−− = . Relations for parameters bi, Cρ, F, r, s are presented in the Appendix. Further we determine solutions of Eqs.(8). In this situation we determine approximation of dis- placement vector. To determine the first-order approximations of the considered components framework method of averaging of function corrections we replace the required values on their not yet known average values α1i. The replacement leads to the following result
- 8. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 42 ( ) ( ) ( ) ( ) ( ) x tzyxT zzK t tzyxu z x ∂ ∂ −= ∂ ∂ ,,,,,, 2 1 2 βρ , ( ) ( ) ( ) ( ) ( ) y tzyxT zzK t tzyxu z y ∂ ∂ −= ∂ ∂ ,,,,,, 2 1 2 βρ , ( ) ( ) ( ) ( ) ( ) z tzyxT zzK t tzyxu z z ∂ ∂ −= ∂ ∂ ,,,,,, 2 1 2 βρ . Integration of the left and right sides of the previous relations on time t gives a possibility to ob- tain the required components to the following result ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x t x uddzyxT xz z zKddzyxT xz z zKtzyxu 0 0 00 0 1 ,,,,,,,,, +∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ = ∞ϑϑ ϑττ ρ β ϑττ ρ β , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) y t y uddzyxT yz z zKddzyxT yz z zKtzyxu 0 0 00 0 1 ,,,,,,,,, +∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ = ∞ϑϑ ϑττ ρ β ϑττ ρ β , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) z t z uddzyxT zz z zKddzyxT zz z zKtzyxu 0 0 00 0 1 ,,,,,,,,, +∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ = ∞ϑϑ ϑττ ρ β ϑττ ρ β . We calculate the second-order approximations of components of the displacement vector by stan- dard replacement of the required functions in the right sides of the Eqs.(8) on the standard sums α1i+ui(x,y,z,t) [19,26]. Equations for components of the displacement vector are presented in the Appendix. Solutions of these equations are also presented in the Appendix. Framework this paper all required concentrations (concentrations of dopant and radiation defects) and components of displacement vector have been calculated as the appropriate second-order ap- proximations by using the method of averaging of function corrections. The second-order approx- imation gives usually enough information on quantitative behavior of spatio-temporal distribu- tions of concentrations of dopant and radiation defects and also several quantitative results. We check all analytical results by using numerical approaches. 3. DISCUSSION In this section we analyzed redistribution of dopant (for the ion doping of heterostructure) with account redistribution of radiation defects and their interaction with another defects. If growth rate is small (v t<D1/v), than overlayer will be fully doped by dopant, which was implanted in the epitaxial layer. Framework another limiting case it will be doped near-surface area of the overlay- er only. If dopant diffusion coefficient in the overlayer and in the substrate are smaller, in com- parison with the epitaxial layer, and type of conductivity of the overlayer and the substrate is dif- ferent with type of conductivity of the epitaxial layer, than one can find a bipolar transistor. In this case one can find higher sharpness of p-n-junctions framework the transistor in comparison with a bipolar transistor in homogenous sample with averaged diffusion coefficient of dopant. The increasing gives a possibility to increase switching time of p-n-junctions (both single p-n- junctions and p-n-junctions framework their systems: bipolar transistors, thyristors et al). At the same time one can find increasing of homogeneity of concentration of dopant (see Fig. 2). In this situation one can decrease local overheats in doped areas during functioning of the considered devices or to decrease dimensions of these devices for fixed tolerance for local overheats. Quali- tatively similar results could be obtained for diffusion type of doping. One can find smaller sharpness of left p-n-junctions in the case, when dopant diffusion coefficient of the overlayer is larger, than in doped epitaxial layer. At the same time homogeneity of concentration of dopant in
- 9. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 43 the overlayer increases (see Fig. 3). Qualitatively similar results could be obtained for diffusion type of doping. Further we analyzed influence of mismatch-induced stress on distribution of con- centration z 0.0 0.5 1.0 1.5 2.0 C(x,y,z,Θ) 0 Lz /4 Lz /2 3Lz /4 Lz 1 2 Fig. 2. Calculated spatial distributions of concentration of implanted dopant in homogenous sample (curve 1) and in heterostructure from Fig. 1 (curve 2) after annealing with the same continuance. Interfaces be- tween layers of heterostructure are: a1=Lz/4 and a2=3Lz/4 z 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 C(x,y,z,Θ) C(x,y,z,0) Lz /40 Lz /2 3Lz /4 Lz x0 1 2 3 4 Fig. 3. Curves 1 and 2 are the calculated spatial distributions of concentration of implanted dopant in the system of two layers: overlayer and epitaxial layer. Curves 3 and 4 are the calculated spatial distributions of concentration of implanted dopant in the epitaxial layer only. Increasing of number of curves corresponds to increasing of value of relation D1/D2. Coordinates of interfaces between layers of heterostructure are: a1=Lz/4 (between overlayer and epitaxial layer) and a2=3Lz/4 (between epitaxial layer and substrate)
- 10. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 44 0.0 1.0 2.0 C(x,y,z,Θ) 0 Lz 1 2 3 Fig. 4. Spatial distributions of concentration of dopant in diffusion-junction rectifier after annealing with equal continuance. Curve 1 corresponds to ε0<0. Curve 2 corresponds to ε0=0. Curve 3 corresponds to ε0>0 0.0 1.0 2.0 C(x,y,z,Θ) Lz-Lz 0 1 2 3 Fig. 5. Spatial distributions of concentration of dopant in implanted-junction rectifier after annealing with equal continuance. Curve 1 corresponds to ε0<0. Curve 2 corresponds to ε0=0. Curve 3 corresponds to ε0>0 z 0.0 0.2 0.4 0.6 0.8 1.0 Uz 1 2 0.0 a Fig.6. Normalized dependences of component uz of displacement vector on coordinate z for epitaxial layers before radiation processing (curve 1) and after radiation processing (curve 2)
- 11. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 45 of dopant. We obtain during the analysis, that p-n-junctions, manufactured near interface between layers of heterostructures, have higher sharpness and higher homogeneity of concentration of do- pant in enriched area. Existing mismatch-induced stress leads to changing of distribution of con- centration of dopant in directions, which are parallel to the considered interface. For example, for ε0<0 the above distribution in directions x and y became more compact (see Fig. 4). In this situa- tion one can obtain increasing of density of elements of integrated circuits in this situation, when the circuits were fabricated in heterostructures. For ε0>0 one can obtain opposite effect (see Fig. 5). It should be noted, that radiation processing of materials of heterostructure during ion doping of materials gives a possibility to decrease mismatch-induced stress (see Fig. 6). Further we analyzed influence of mismatch-induced stress on distribution of concentration of do- pant. We obtain during the analysis, that p-n-junctions, manufactured near interface between lay- ers of heterostructures, have higher sharpness and higher homogeneity of concentration of dopant in enriched area. Existing mismatch-induced stress leads to changing of distribution of concentra- tion of dopant in directions, which are parallel to the considered interface. For example, for ε0<0 the above distribution in directions x and y became more compact (see Fig. 4). For ε0>0 one can obtain opposite effect (see Fig. 5). It should be noted, that radiation processing of materials of heterostructure during ion doping of materials gives a possibility to decrease mismatch-induced stress (see Fig. 6). In this situation component of displacement vector perpendicular to interface between materials of heterostructure became smaller after radiation processing in comparison with analogous component of displacement vector in non processed heterostructure. 4. CONCLUSIONS In this paper we analyzed influence of overgrowth of doped by diffusion or ion implantation areas of heterostructures on distributions of concentrations of dopants. We determine conditions to in- crease sharpness if implanted-junction and diffusion- junction rectifiers (single rectifiers and rec- tifiers framework bipolar transistors). At the same time we analyzed influence of overgrowth rate of doped areas and mismatch-induced stress in the considered heterostructure on distributions of concentrations of dopants. ACKNOWLEDGEMENTS This work is supported by the agreement of August 27, 2013 № 02.В.49.21.0003 between The Ministry of education and science of the Russian Federation and Lobachevsky State University of Nizhni Novgorod, educational fellowship for scientific research of Government of Russian, edu- cational fellowship for scientific research of Government of Nizhny Novgorod region of Russia and educational fellowship for scientific research of Nizhny Novgorod State University of Archi- tecture and Civil Engineering. REFERENCES [1] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001). [2] N.A. Avaev, Yu.E. Naumov, V.T. Frolkin. Basis of microelectronics (Radio and communication, Moscow, 1991). [3] V.G. Gusev, Yu.M. Gusev. Electronics (Higher School, Moscow, 1991). [4] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow 1991). [5] I.P. Stepanenko. Basis of microelectronics (Soviet Radio, Moscow 1980). [6] A.G. Alexenko, I.I. Shagurin. Microcircuitry (Radio and communication, Moscow, 1990). [7] K.V. Shalimova. Physics of semiconductors. (Energoatomizdat, Moscow, 1985. [8] Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov, Yu.N. Drozdov, V.D. Skupov. Diffusion processes in semiconductor structures during microwave annealing. Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836 (2003).
- 12. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 46 [9] A.V. Dvurechensky, G.A. Kachurin, E.V. Nidaev, L.S. Smirnov. Impulse annealing of semiconductor materials (Science, Moscow, 1982). [10] A.F. Corballo Sanchez, G. Gonzalez de la Cruz, Yu.G. Gurevich, G.N. Logvinov. Transient heat transport by carriers and phonons in semiconductors. Phys. Rev. B. Vol. 59 (16). P. 10630 (1999). [11] V.I. Mazhukin, V.V. Nosov, U. Semmler. Investigation of heat and thermo-elastic fields in semicon- ductors at pulse processing. Mathematical modelling. Vol. 12 (2), 75 (2000). [12] K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. Dopant activation in subamor- phized silicon upon laser annealing. Appl. Phys. Lett. Vol. 89 (17), 172111 (2006). [13] J.A. Sharp, N.E. B. Cowern, R.P. Webb, K.J. Kirkby, D. Giubertoni, S. Genarro, M. Bersani, M.A. Foad, F. Cristiano, P.F. Fazzini. Appl. Phys. Lett. Vol. 89, 192105 (2006). [14] J.E. Epler, F.A. Ponce, F.J. Endicott, T.L. Paoli. J. Appl. Phys. Vol. 64, 3439 (1988). [15] S.T. Sisianu, T.S. Sisianu, S.K. Railean. Semiconductors. Vol. 36, 581 (2002). [16] E.L. Pankratov. Influence of spatial, temporal and concentrational dependence of diffusion coefficient on dopant dynamics: Optimization of annealing time. Phys. Rev. B. Vol. 72 (7). P. 075201 (2005). [17] E.L. Pankratov. Dopant diffusion dynamics and optimal diffusion time as influenced by diffusion- coefficient nonuniformity. Russian Microelectronics. Vol. 36 (1). P. 33 (2007). [18] V.V. Kozlivsky. Modification of semiconductors by proton beams (Science, Sant- Peterburg, 2003). [19] E.L. Pankratov. Influence of mechanical stress in a multilayer structure on spatial distribution of dopants in implanted-junction and diffusion-junction rectifiers. Mod. Phys. Lett. B. Vol. 24 (9). P. 867 (2010). [20] Y.W. Zhang, A.F. Bower. Numerical simulations of island formation in a coherent strained epitaxial thin film system. Journal of the Mechanics and Physics of Solids. Vol. 47 (11). P. 2273-2297 (1999). [21] P.M. Fahey, P.B. Griffin, J.D. Plummer. Point defects and dopant diffusion in silicon. Rev. Mod. Phys. Vol. 61 (2). P. 289-388 (1989). [22] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev, 1979, in Russian). [23] Z.Yu. Gotra, Technology of microelectronic devices (Radio and communication, Moscow, 1991). [24] L.D. Landau, E.M. Lefshits. Theoretical physics. 7 (Theory of elasticity) (Physmatlit, Moscow, 2001, in Russian). [25] Yu.D. Sokolov. About the definition of dynamic forces in the mine lifting. Applied Mechanics. Vol. 1 (1). P. 23-35 (1955). [26] E.L. Pankratov, E.A. Bulaeva. Doping of materials during manufacture p-n-junctions and bipolar transistors. Analytical approaches to model technological approaches and ways of optimization of dis- tributions of dopants. Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013). [27] E.L. Pankratov, E.A. Bulaeva. Increasing of sharpness of diffusion-junction heterorectifier by using radiation processing. Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1-1250028-8 (2012). [28] E.L. Pankratov, E.A. Bulaeva. Optimization of manufacturing of emitter-coupled logic to decrease surface of chip. International Journal of Modern Physics B. Vol. 29 (5). P. 1550023-1-1550023-12 (2015). [29] E.L. Pankratov, E.A. Bulaeva. An approach to manufacture of bipolar transistors in thin film struc- tures. On the method of optimization. Int. J. Micro-Nano Scale Transp. Vol. 4 (1). P. 17-31 (2014). [30] E.L. Pankratov, E.A. Bulaeva. Decreasing of mechanical stress in a semiconductor heterostructure by radiation processing. J. Comp. Theor. Nanoscience. Vol. 11 (1). P. 91-101 (2014). APPENDIX Eqs. (1), (3) and (5) with account initial distributions could be written as ( ) ( ) ( ) ( ) ( ) ( )++ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ tzyxf z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC C δ,, ,,,,,,,,,,,, ( ) ( ) ( ) ( ) ∫∇ ∂ ∂ Ω+ ∫∇ ∂ ∂ Ω+ −− zz L tv S S L tv S S WdtWyxCtzyx Tk D y WdtWyxCtzyx Tk D x ,,,,,,,,,,,, µµ (1a)
- 13. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 47 ( ) ( ) ( ) ( ) ( ) ( ) ( )++ + = tzyxf y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI III δ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,, ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( ) ( )+−− + tzyxITzyxktzyxVtzyxITzyxk z tzyxI TzyxD z IIVII ,,,,,,,,,,,,,,, ,,, ,,, 2 ,, ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ∫∇ ∂ ∂ Ω+ ∫∇ ∂ ∂ Ω+ −− zz L tv S IS L tv S IS WdtWyxItzyx Tk D y WdtWyxItzyx Tk D x ,,,,,,,,,,,, µµ (3a) ( ) ( ) ( ) ( ) ( ) ( ) ( )++ + = tzyxf y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VVV δ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,, ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( ) ( )+−− + tzyxVTzyxktzyxVtzyxITzyxk z tzyxV TzyxD z VVVIV ,,,,,,,,,,,,,,, ,,, ,,, 2 ,, ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ∫∇ ∂ ∂ Ω+ ∫∇ ∂ ∂ Ω+ −− zz L tv S VS L tv S VS WdtWyxVtzyx Tk D y WdtWyxVtzyx Tk D x ,,,,,,,,,,,, µµ ( ) ( ) ( ) ( ) ( ) ( )×+ Φ + Φ = Φ ΦΦ t y tzyx TzyxD yx tzyx TzyxD xt tzyx III II δ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) + ∫Φ∇ ∂ ∂ Ω+ Φ +× − Φ ΦΦ z I II L tv IS SI WdtWyxtzyx Tk D xz tzyx TzyxD z zyxf ,,,,,, ,,, ,,,,, µ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkWdtWyxtzyx Tk D y III L tv IS S z I ,,,,,,,,,,,,,,,,,, 2 ,++ ∫Φ∇ ∂ ∂ Ω+ − Φ µ ( ) ( ) ( ) ( ) ( ) ( )×+ Φ + Φ = Φ ΦΦ t y tzyx TzyxD yx tzyx TzyxD xt tzyx VVV VV δ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, (5a) ( ) ( ) ( ) ( ) ( ) + ∫Φ∇ ∂ ∂ Ω+ Φ +× − Φ ΦΦ z V VV L tv VS SV WdtWyxtzyx Tk D xz tzyx TzyxD z zyxf ,,,,,, ,,, ,,,,, µ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkWdtWyxtzyx Tk D y VVV L tv VS S z V ,,,,,,,,,,,,,,,,,, 2 ,++ ∫Φ∇ ∂ ∂ Ω+ − Φ µ . The first-order approximations of concentrations of dopant and radiation defects could by calcu- lated by solution of the following equations ( ) ( ) ( ) + ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω= ∂ ∂ tzyx Tk D z y tzyx Tk D z xt tzyxC S S CS S C ,,,,,, ,,, 11 1 µαµα ( ) ( )tzyxfC δ,,+ (1b) ( ) ( ) ( ) ( ) ( )−+ ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω= tzyxftzyx Tk D z y tzyx Tk D x z t tzyxI IS IS IS IS I δµαµα ∂ ∂ ,,,,,,,, ,,, 11 1 ( ) ( )TzyxkTzyxk VIVIIII ,,,,,, ,11, 2 1 ααα −− (3b) ( ) ( ) ( ) ( ) ( )−+ ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω= tzyxftzyx Tk D z y tzyx Tk D x z t tzyxV VS VS VS VS V δµαµα ∂ ∂ ,,,,,,,, ,,, 11 1 ( ) ( )TzyxkTzyxk VIVIVVV ,,,,,, ,11, 2 1 ααα −−
- 14. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 48 ( ) ( ) ( ) ( ) ( ) ( ) ( )+++= Φ Φ tzyxftzyxITzyxktzyxITzyxk t tzyx IIII I δ ∂ ∂ ,,,,,,,,,,,,,, ,,, 2 , 1 ( ) ( ) ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω+ Φ Φ Φ Φ tzyx Tk D y ztzyx Tk D x z S S S S I I I I ,,,,,, 11 µαµα (5b) ( ) ( ) ( ) ( ) ( ) ( ) ( )+++= Φ Φ tzyxftzyxVTzyxktzyxVTzyxk t tzyx VVVV V δ ∂ ∂ ,,,,,,,,,,,,,, ,,, 2 , 1 ( ) ( ) ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω+ Φ Φ Φ Φ tzyx Tk D y ztzyx Tk D x z S S S S V V V V ,,,,,, 11 µαµα The first-order approximations of the considered concentrations in the following form ( ) ( ) ( ) ( ) ( ) ( )∫ × ++ +∇ ∂ ∂ = t CS SC V zyxV V zyxV TzyxP zyx x tzyxC 0 2* 2 2*1 1 111 ,,,,,, 1 ,,, 1,,,,,, τ ς τ ς αξ τµα γ γ ( ) ( ) ( ) ( ) ( )∫ × ++∇ ∂ ∂ Ω+ Ω× t SCLS V zyxV V zyxV zyx y d Tk z TzyxD 0 2* 2 2*111 ,,,,,, 1,,,,,, τ ς τ ςτµατ ( ) ( ) ( )zyxfd TzyxPTk z TzyxD C CS LS ,, ,,, 1,,, 1 + +× τ αξ γ γ (1c) ( ) ( ) ( ) ( )−+∫ ∇ ∂ ∂ Ω+∫ ∇ ∂ ∂ Ω= zyxfdzyx Tk D y zdzyx Tk D x ztzyxI I t S IS I t S IS I ,,,,,,,,,,, 0 1 0 11 ττµαττµα ( ) ( )∫−∫− t VIVI t III dTzyxkdTzyxk 0 ,11 0 , 2 1 ,,,,,, ταατα (3c) ( ) ( ) ( ) ( )−+∫ ∇ ∂ ∂ Ω+∫ ∇ ∂ ∂ Ω= zyxfdzyx Tk D y zdzyx Tk D x ztzyxV V t S IS V t S IS V ,,,,,,,,,,, 0 11 0 111 ττµαττµα ( ) ( )∫−∫− t VIVI t VVV dTzyxkdTzyxk 0 ,11 0 , 2 1 ,,,,,, ταατα ( ) ( ) ( ) +∫ ∇ ∂ ∂ Ω+∫ ∇ ∂ ∂ Ω=Φ Φ Φ Φ Φ t S S t S S I dzyx Tk D x zdzyx Tk D x ztzyx I I I I 0 1 0 11 ,,,,,,,,, ττµαττµα ( ) ( ) ( ) ( ) ( )∫+∫++ Φ t II t I dzyxITzyxkdzyxITzyxkzyxf I 0 2 , 0 ,,,,,,,,,,,,,, ττττ (5c) ( ) ( ) ( ) +∫ ∇ ∂ ∂ Ω+∫ ∇ ∂ ∂ Ω=Φ Φ Φ Φ Φ t S S t S S V dzyx Tk D x zdzyx Tk D x ztzyx V V V V 0 1 0 11 ,,,,,,,,, ττµαττµα ( ) ( ) ( ) ( ) ( )∫+∫++ Φ t VV t V dzyxVTzyxkdzyxVTzyxkzyxf V 0 2 , 0 ,,,,,,,,,,,,,, ττττ . Relations for calculations parameters Sρρij, ai, A, B, q, p could be written as ( ) ( ) ( ) ( )∫ ∫ ∫ ∫−Θ= Θ −0 0 0 11, ,,,,,,,,, x y zL L L tv ji ij tdxdydzdtzyxVtzyxITzyxktS ρρρρ , ( )×−= 0000 2 004 VVIIIV SSSa
- 15. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 49 00IIS× , 0000 2 0000003 VVIIIVIIIV SSSSSa −+= , ( ) ×∫ ∫ ∫ ∫+Θ= Θ −0 0 0 222 002 ,,2 x y zL L L tv IzyxIV tdxdydzdzyxfLLLSa ( ) ( ) ×Θ−∫ ∫ ∫ ∫−∫ ∫ ∫ ∫+× Θ − Θ − 2 0 0 0 2 00 0 0 0 2 00000000 ,,,, x L L L tv IIV L L L tv VIVIVIIVV LtdxdydzdzyxfStdxdydzdzyxfSSSS x y zx y z 00 22 VVzy SLL× , ( )∫ ∫ ∫ ∫= Θ −0 0 0 001 ,, x y zL L L tv IIV tdxdydzdzyxfSa , +++−−+= 3 323 32 qpqqpqB 4 2 6a aΘ + , ( ) 2 0 0 0 000 ,, ∫ ∫ ∫ ∫= Θ − x y zL L L tv IVV tdxdydzdzyxfSa , 2 1 4 2 2 4 2 32 84 +Θ−Θ= y a a a a A , × Θ = 2 4 2 3 24a a q 2 4 2 1 4 222 3 4 3 2 3 4 2 32 22 4 02 4 31 0 854 4 8 4 a a LLL a a a a a a a a aa LLLa zyxzyx Θ − Θ − Θ−ΘΘ− Θ−× , −Θ= 2 4 402 12 4 a aa p ( ) 2 44231 2 3623 aaaaaLLL zyx +ΘΘ− , ( ) ( ) ( )∫ ∫ ∫ ∫−Θ= Θ −0 0 0 1 ,,,,,, x y zL L L tv i Ii tdxdydzdtzyxITzyxktRρ . Equations for the second-order approximations of concentrations of dopant and radiation defects could be written as ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) × ++ + + ∂ ∂ = ∂ ∂ 2* 2 2*1 122 ,,,,,, 1 ,,, ,,, 1,,, ,,, V tzyxV V tzyxV TzyxP tzyxC TzyxD xt tzyxC C L ςς α ξ γ γ ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) × ∂ ∂ + + ++ ∂ ∂ + ∂ ∂ × y tzyxC TzyxP tzyxC V tzyxV V tzyxV yx tzyxC C ,,, ,,, ,,, 1 ,,,,,, 1 ,,, 112 2* 2 2*1 1 γ γ α ξςς ( )) ( ) ( ) ( ) ( )[ ] ( ) × + + ++ ∂ ∂ +× TzyxP tzyxC V tzyxV V tzyxV z TzyxD C L ,,, ,,, 1 ,,,,,, 1,,, 12 2* 2 2*1 γ γ α ξςς ( ) ( ) ( ) ( )[ ] + ∫ +∇ ∂ ∂ Ω+ ∂ ∂ × − zL tv CS S L WdtWyxCtzyx Tk D xz tzyxC TzyxD ,,,,,, ,,, ,,, 22 1 αµ ( ) ( )[ ] ( ) ( )tzyxfWdtWyxCtzyx Tk D y C L tv CS S z δαµ ,,,,,,,, 22 + ∫ +∇ ∂ ∂ Ω+ − (1d) ( ) ( ) ( ) ( ) ( ) ( )×− + = Tzyxk y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI VIII ,,, ,,, ,,, ,,, ,,, ,,, , 112 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( )[ ] ( )[ ] ( ) ( ) ( )[ ] ×+− +++× 2 11 1 1111 ,,, ,,, ,,,,,,,,, tzyxI z tzyxI TzyxD z tzyxVtzyxI IIVI α ∂ ∂ ∂ ∂ αα ( ) ( ) ( )[ ] + ∫ +∇ ∂ ∂ Ω+× − zL tv IS IS II WdtWyxItzyx Tk D x Tzyxk ,,,,,,,,, 12, αµ ( ) ( )[ ] ∫ +∇ ∂ ∂ Ω+ − zL tv IS IS WdtWyxItzyx Tk D y ,,,,,, 12αµ (3d) ( ) ( ) ( ) ( ) ( ) ( )×− + = Tzyxk y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VIVV ,,, ,,, ,,, ,,, ,,, ,,, , 112 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( )[ ] ( )[ ] ( ) ( ) ( )[ ] ×+− +++× 2 11 1 1111 ,,, ,,, ,,,,,,,,, tzyxV z tzyxV TzyxD z tzyxVtzyxI VVVI α ∂ ∂ ∂ ∂ αα
- 16. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 50 ( ) ( ) ( )[ ] + ∫ +∇ ∂ ∂ Ω+× − zL tv VS VS VV WdtWyxVtzyx Tk D x Tzyxk ,,,,,,,,, 12, αµ ( ) ( )[ ] ∫ +∇ ∂ ∂ Ω+ − zL tv VS VS WdtWyxVtzyx Tk D y ,,,,,, 12αµ ( ) ( ) ( ) ( ) ( ) ( )×+ Φ + Φ = Φ ΦΦ Tzyxk y tzyx TzyxD yx tzyx TzyxD xt tzyx II III II ,,, ,,, ,,, ,,, ,,, ,,, , 112 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( )[ ] ( ) ( )++ ∫ Φ+∇ ∂ ∂ Ω+× Φ − Φ Φ tzyxfWdtWyxtzyx Tk D x tzyxI I z I I L tv IS S δαµ ,,,,,,,,,,, 12 2 ( ) ( )[ ] ( ) ( ) + Φ + ∫ Φ+∇ ∂ ∂ Ω+ Φ − Φ Φ z tzyx TzyxD z WdtWyxtzyx Tk D y I L tv IS S I z I I ∂ ∂ ∂ ∂ αµ ,,, ,,,,,,,,, 1 12 ( ) ( )tzyxITzyxkI ,,,,,,+ (5d) ( ) ( ) ( ) ( ) ( ) ( ) ×+ Φ + Φ = Φ ΦΦ Tzyxk y tzyx TzyxD yx tzyx TzyxD xt tzyx VV VVV VV ,,, ,,, ,,, ,,, ,,, ,,, , 112 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ( ) ( ) ( )[ ] ( ) ( )++ ∫ Φ+∇ ∂ ∂ Ω+× Φ − Φ Φ tzyxfWdtWyxtzyx Tk D x tzyxV V z V V L tv VS S δαµ ,,,,,,,,,,, 12 2 ( ) ( )[ ] ( ) ( ) + Φ + ∫ Φ+∇ ∂ ∂ Ω+ Φ − Φ Φ z tzyx TzyxD z WdtWyxtzyx Tk D y V L tv VS S V z V V ∂ ∂ ∂ ∂ αµ ,,, ,,,,,,,,, 1 12 ( ) ( )tzyxVTzyxkV ,,,,,,+ . The second-order approximations of concentrations of dopant and radiation defects could be writ- ten as ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ∫ × ++ + + ∂ ∂ = t C L V zyxV V zyxV TzyxP zyxC TzyxD x tzyxC 0 2* 2 2*1 12 2 ,,,,,, 1 ,,, ,,, 1,,,,,, τ ς τ ς τα ξ γ γ ( ) ( ) ( ) ( ) ( )[ ] ( )∫ × + + ++ ∂ ∂ + ∂ ∂ × t C TzyxP zyxC V zyxV V zyxV y d x zyxC 0 12 2* 2 2*1 1 ,,, ,,, 1 ,,,,,, 1 ,,, γ γ τα ξ τ ς τ ςτ τ ( ) ( ) ( ) ( ) ( ) ( )∫ × ++ ∂ ∂ + ∂ ∂ × t LL V zyxV V zyxV TzyxD z d y zyxC TzyxD 0 2* 2 2*1 1 ,,,,,, 1,,, ,,, ,,, τ ς τ ςτ τ ( ) ( )[ ] ( ) ( )[ ] ×∫ ∫ + ∂ ∂ Ω+ + + ∂ ∂ × − t L v C SC z WdWyxC Tk D x d TzyxP zyxC z zyxC 0 12 121 ,,, ,,, ,,, 1 ,,, τ γ γ ϑατ τα ξ τ ( ) ( ) ( )[ ] +∫ ∫ +∇ ∂ ∂ Ω+∇× − t L v CS S S ddWdWyxCzyx Tk D y ddzyx z 0 12 ,,,,,,,,, τϑϑατµτϑτµ τ ( )zyxfC ,,+ (1e) ( ) ( ) ( ) ( ) ( ) +∫+∫= t I t I d y zyxI TzyxD y d x zyxI TzyxD x tzyxI 0 1 0 1 2 ,,, ,,, ,,, ,,,,,, τ ∂ τ∂ ∂ ∂ τ ∂ τ∂ ∂ ∂ ( ) ( ) ( ) ( )[ ] ( )++∫ +−∫+ zyxfdzyxITzyxkd z zyxI TzyxD z I t III t I ,,,,,,,, ,,, ,,, 0 2 12, 0 1 ττατ ∂ τ∂ ∂ ∂ ( ) ( )[ ] ( )×∫ ∇ ∂ ∂ Ω+∫ ∫ +∇ ∂ ∂ Ω+ − t S IS t L v IS IS zyx Tk D y ddWdWyxIzyx Tk D x z 00 12 ,,,,,,,,, τµτϑϑατµ τ ( )[ ] ( ) ( )[ ] ( )[ ]∫ ++−∫ +× − t VIVI L v I dzyxVzyxITzyxkddWdWyxI z 0 1212,12 ,,,,,,,,,,,, ττατατϑϑα τ (3e)
- 17. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 51 ( ) ( ) ( ) ( ) ( ) +∫+∫= t V t V d y zyxV TzyxD y d x zyxV TzyxD x tzyxV 0 1 0 1 2 ,,, ,,, ,,, ,,,,,, τ ∂ τ∂ ∂ ∂ τ ∂ τ∂ ∂ ∂ ( ) ( ) ( ) ( )[ ] ( )++∫ +−∫+ zyxfdzyxVTzyxkd z zyxV TzyxD z V t VVV t V ,,,,,,,, ,,, ,,, 0 2 12, 0 1 ττατ ∂ τ∂ ∂ ∂ ( ) ( )[ ] ( )×∫ ∇ ∂ ∂ Ω+∫ ∫ +∇ ∂ ∂ Ω+ − t S VS t L v VS VS zyx Tk D y ddWdWyxVzyx Tk D x z 00 12 ,,,,,,,,, τµτϑϑατµ τ ( )[ ] ( ) ( )[ ] ( )[ ]∫ ++−∫ +× − t VIVI L v I dzyxVzyxITzyxkddWdWyxV z 0 1212,12 ,,,,,,,,,,,, ττατατϑϑα τ ( ) ( ) ( ) ( ) ( ) +∫ Φ +∫ Φ =Φ ΦΦ t I t I I d y zyx TzyxD y d x zyx TzyxD x tzyx II 0 1 0 1 2 ,,, ,,, ,,, ,,,,,, τ ∂ τ∂ ∂ ∂ τ ∂ τ∂ ∂ ∂ ( ) ( ) ( ) ( )[ ] ×∫ ∫ Φ+∇ ∂ ∂ Ω+∫ Φ + − ΦΦ t L v IS t I z II WdWyxzyx x d z zyx TzyxD z 0 12 0 1 ,,,,,, ,,, ,,, τ τατµτ ∂ τ∂ ∂ ∂ ( ) ( )[ ] ( )++∫ ∫ Φ+∇ ∂ ∂ Ω+× Φ − Φ ΦΦ zyxfddWdWyxzyx Tk D y d Tk D I z I II t L v IS SS ,,,,,,,, 0 12 τϑϑατµτ τ ( ) ( ) ( ) ( )∫+∫+ t I t II dzyxITzyxkdzyxITzyxk 00 2 , ,,,,,,,,,,,, ττττ (5e) ( ) ( ) ( ) ( ) ( ) +∫ Φ +∫ Φ =Φ ΦΦ t V t V V d y zyx TzyxD y d x zyx TzyxD x tzyx VV 0 1 0 1 2 ,,, ,,, ,,, ,,,,,, τ ∂ τ∂ ∂ ∂ τ ∂ τ∂ ∂ ∂ ( ) ( ) ( ) ( )[ ] ×∫ ∫ Φ+∇ ∂ ∂ Ω+∫ Φ + − ΦΦ t L v VS t V z VV WdWyxzyx x d z zyx TzyxD z 0 12 0 1 ,,,,,, ,,, ,,, τ τατµτ ∂ τ∂ ∂ ∂ ( ) ( )[ ] ( )++∫ ∫ Φ+∇ ∂ ∂ Ω+× Φ − Φ ΦΦ zyxfddWdWyxzyx Tk D y d Tk D V z V VV t L v VS SS ,,,,,,,, 0 12 τϑϑατµτ τ ( ) ( ) ( ) ( )∫+∫+ t V t VV dzyxVTzyxkdzyxVTzyxk 00 2 , ,,,,,,,,,,,, ττττ . Parameters bi, Cρ, F, r, s have been determined by the following relations 00 2 0000 2 004 11 IIVV zyx VVIV zyx SS LLL SS LLL b Θ − Θ = , ( ) + Θ Θ++−= zyx VVII zyxIVVV LLL SS LLLSSb 0000 10013 2 ( ) ( ) 3333 10 2 00 1001 2 00 011001 0000 22 zyx IVIV zyxIVVV zyx IV zyxIVIIIV zyx VVIV LLL SS LLLSS LLL S LLLSSS LLL SS Θ −Θ++ Θ +Θ+++ Θ + , ( ) ( ) ( )×++Θ Θ ++−Θ−++ Θ = 0110 012 10011102 0000 2 22 IVIIzyx x IV IVVVzyxVIVVV zyx VVII SSLLL L S SSLLLCSS LLL SS b ( )( ) × Θ −+Θ+++++Θ Θ +× zyx IV IVzyxVVIVIIIVzyx zyx IV zy VV LLL S SLLLSSSSLLL LLL S LL S 2 00 1001011001 0000 222 ( ) 01 0010 2222 2 00 1102 2 IV zyx IVIV zyx IVI IVVVV S LLL SS LLL SC SSC Θ − Θ +−−× , ( ×Θ+ Θ ++ = xIV zyx VVVIV II LS LLL CSS Sb 10 0211 001 ) ( ) × Θ −− −Θ++ Θ ++Θ ++× zyx IVVVV zyxIVVV zyx IVIIzyx IVVVzy LLL SSC LLLSS LLL SSLLL SSLL 1102 1001 0110 0101 2 2 2 ( ) zyx IVIV IVIVIzyxIIIVIV LLL SS SSCLLLSSS Θ −+Θ++× 2 0110 0100100100 223 , ( ) ×− Θ + = 01 2 0200 000 IV zyx VVIV II S LLL SS Sb
- 18. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 52 ( ) ( )×++Θ Θ −− −++Θ Θ −− × 0110 1102 0110 1102 22 IVIIzyx zyx IVVVV IVIIzyx zyx IVVVV SSLLL LLL SSC SSLLL LLL SSC ( )0110 1102 01 2 0101 22 IVIIzyx zyx IVVVV IVIVIIV SSLLL LLL SSC SSCS ++Θ Θ −− −+× , + Θ = zyx VI IVI LLL SC 11 00 αα zyx IV zyx IIII zyx III LLL S LLL SS LLL S Θ − Θ − Θ + 11202000 2 1α , 110200 2 10011 IVVVVVVIVVIV SSSSC −−+= ααα , 4 2 2 4 2 32 48 a a a a yE Θ−Θ+= , 2 4 2 1 4 222 3 4 3 2 3 4 2 32 22 4 2 0 4 31 02 4 2 3 854 4 8 4 24 b b LLL b b b b b b b b bb LLLb b b r zyxzyx Θ − Θ − Θ−Θ Θ − Θ− Θ = , ( − Θ = 402 4 2 4 12 bb b s ) 4231 18bbbbLLL zyx Θ−Θ− , 3 323 32 4 2 6 rsrrsr a a F ++−−++ Θ = . Equations for components of the displacement vector could be presented in the form ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) + ∂∂ ∂ + −+ ∂ ∂ + += ∂ ∂ yx tzyxu z zE zK x tzyxu z zE zK t tzyxu z yxx ,,, 13 ,,, 16 5,,, 1 2 2 1 2 2 2 2 σσ ρ ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ) − ∂∂ ∂ + ++ ∂ ∂ + ∂ ∂ + + zx tzyxu z zE zK z tzyxu y tzyxu z zE zzy ,,, 13 ,,,,,, 12 1 2 2 1 2 2 1 2 σσ ( ) ( ) ( ) x tzyxT zzK ∂ ∂ − ,,, β ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )+ ∂ ∂ − ∂∂ ∂ + ∂ ∂ + = ∂ ∂ y tzyxT zzK yx tzyxu x tzyxu z zE t tzyxu z xyy ,,,,,,,,, 12 ,,, 1 2 2 1 2 2 2 2 β σ ρ ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ) + + +∂ ∂ + ∂ ∂ + ∂ ∂ +∂ ∂ + zK z zE y tzyxu y tzyxu z tzyxu z zE z yzy σσ 112 5,,,,,,,,, 12 2 1 2 11 ( ) ( ) ( )[ ] ( ) ( ) ( ) yx tzyxu zK zy tzyxu z zE zK yy ∂∂ ∂ + ∂∂ ∂ + −+ ,,,,,, 16 1 2 1 2 σ ( ) ( ) ( ) ( ) ( ) ( ) × ∂∂ ∂ + ∂∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ zy tzyxu zx tzyxu y tzyxu x tzyxu t tzyxu z yxzzz ,,,,,,,,,,,,,,, 1 2 1 2 2 1 2 2 1 2 2 2 2 ρ ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( )[ ] × +∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + + × z zE zz tzyxu y tzyxu x tzyxu zK zz zE xyx σσ 16 ,,,,,,,,, 12 111 ( ) ( ) ( ) ( ) ( ) ( ) ( ) z tzyxT zzK z tzyxu y tzyxu x tzyxu z tzyxu zyxz ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ × ,,,,,,,,,,,,,,, 6 1111 β . Integration of the left and the right sides of the above equations on time t leads to final relations for components of displacement vector ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] × + −+∫ ∫ ∂ ∂ + += z zE zK z ddzyxu xz zE zK z tzyxu t xx σρ ϑττ σρ ϑ 13 1 ,,, 16 51 ,,, 0 0 12 2 2 ( ) ( ) ( ) × ∫ ∫ ∂ ∂ +∫ ∫ ∂ ∂ +∫ ∫ ∂∂ ∂ × t z t y t y ddzyxu z ddzyxu y ddzyxu yx 0 0 12 2 0 0 12 2 0 0 1 2 ,,,,,,,,, ϑϑϑ ϑττϑττϑττ
- 19. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 53 ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ×− + +∫ ∫ ∂∂ ∂ + + × z z zK z zE zKddzyxu zxzzz zE t z ρ β σ ϑττ ρσρ ϑ 13 ,,, 1 12 0 0 1 2 ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ×−∫ ∫ ∂ ∂ + +−∫ ∫ ∂ ∂ × ∞ z ddzyxu xz zE zK z ddzyxT x x t ρ ϑττ σρ ϑττ ϑϑ 1 ,,, 16 51 ,,, 0 0 12 2 0 0 ( ) ( ) ( )[ ] ( ) ( ) ( ) +∫ ∫ ∂ ∂ −∫ ∫ ∂∂ ∂ + −× ∞∞ 0 0 12 2 0 0 1 2 ,,, 2 1 ,,, 13 ϑϑ ϑττ ρ ϑττ σ ddzyxu yz ddzyxu yxz zE zK yy ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ×∫ ∫ ∂∂ ∂ + +− + ∫ ∫ ∂ ∂ + ∞∞ 0 0 1 2 0 0 12 2 ,,, 131 ,,, ϑϑ ϑττ σσ ϑττ ddzyxu zxz zE zK z zE ddzyxu z zz ( ) ( ) ( ) ( ) ( )∫ ∫ ∂ ∂ ++× ∞ 0 0 0 ,,, 1 ϑ ϑττ ρ β ρ ddzyxT xz z zKu z x ( ) ( ) ( ) ( )[ ] ( ) ( ) + ∫ ∫ ∂∂ ∂ +∫ ∫ ∂ ∂ + = t x t xy ddzyxu yx ddzyxu xzz zE tzyxu 0 0 1 2 0 0 12 2 2 ,,,,,, 12 ,,, ϑϑ ϑττϑττ σρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] + + +∫ ∫ ∂ ∂ +∫ ∫ ∂∂ ∂ + z zE zKddzyxu yz ddzyxu yxz zK t x t y σ ϑττ ρ ϑττ ρ ϑϑ 112 5 ,,, 1 ,,, 0 0 12 2 0 0 1 2 ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) − ∫ ∫ ∂ ∂ +∫ ∫ ∂ ∂ +∂ ∂ + + + t z t y ddzyxu y ddzyxu zz zE zzz zE 0 0 1 0 0 1 ,,,,,, 12 1 112 5 ϑϑ ϑττϑττ σρσ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] − + −∫ ∫ ∂∂ ∂ +∫ ∫− z zE zKddzyxu zyz ddzyxT z z zK t y t σ ϑττ ρ ϑττ ρ β ϑϑ 16 ,,, 1 ,,, 0 0 1 2 0 0 ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ×− ∫ ∫ ∂∂ ∂ +∫ ∫ ∂ ∂ + − ∞∞ z z zKddzyxu yx ddzyxu xzz zE xx ρ β ϑττϑττ σρ ϑϑ 0 0 1 2 0 0 12 2 ,,,,,, 12 ( ) ( ) ( ) ( ) ( ) ( ) ×∫ ∫ ∂ ∂ −∫ ∫ ∂∂ ∂ −∫ ∫× ∞∞∞ 0 0 12 2 0 0 1 2 0 0 ,,, 1 ,,,,,, ϑϑϑ ϑττ ρ ϑττ ρ ϑττ ddzyxu yz ddzyxu yxz zK ddzyxT xy ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) × ∫ ∫ ∂ ∂ +∫ ∫ ∂ ∂ +∂ ∂ − + +× ∞∞ 0 0 1 0 0 1 ,,,,,, 1112 5 ϑϑ ϑττϑττ σσ ddzyxu y ddzyxu zz zE zz zE zK zy ( ) ( ) ( ) ( ) ( )[ ] ( ) yy uddzyxu zyz zE zK zz 0 0 0 1 2 ,,, 16 1 2 1 +∫ ∫ ∂∂ ∂ + −−× ∞ϑ ϑττ σρρ ( ) ( ) ( ) ( )[ ] ( ) ( ) +∫ ∫ ∂ ∂ +∫ ∫ ∂ ∂ + = ∞∞ 0 0 12 2 0 0 12 2 ,,,,,, 12 ,,, ϑϑ ϑττϑττ σρ ddzyxu y ddzyxu xzz zE tzyxu zzz ( ) ( ) ( ) +∫ ∫ ∂ ∂ ∂ ∂ + ∫ ∫ ∂∂ ∂ +∫ ∫ ∂∂ ∂ + ∞∞∞ 0 0 1 0 0 1 2 0 0 1 2 ,,,,,,,,, ϑϑϑ ϑττϑττϑττ ddzyxu xz ddzyxu zy ddzyxu zx xyx ( ) ( ) ( ) ( ) ( ) ( ) ( ) × +∂ ∂ + ∫ ∫ ∂ ∂ +∫ ∫ ∂ ∂ + ∞∞ z zE zzz zKddzyxu z ddzyxu y xx σρρ ϑττϑττ ϑϑ 16 11 ,,,,,, 0 0 1 0 0 1 ( ) ( ) ( ) −∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ × ∞∞∞ 0 0 1 0 0 1 0 0 1 ,,,,,,,,,6 ϑϑϑ ϑττϑττϑττ ddzyxu y ddzyxu x ddzyxu z yxz ( ) ( ) ( ) ( ) ( ) zz uddzyxT zz z zKddzyxu z 0 0 00 0 1 ,,,,,, +∫ ∫ ∂ ∂ − ∫ ∫ ∂ ∂ − ∞∞ ϑϑ ϑττ ρ β ϑττ .
- 20. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016 54 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 105 published papers in area of her researches.