On Modification of Properties of P-N-Junctions During Overgrowth

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

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
t
z
y
x
C
D
z
y
t
z
y
x
C
D
y
x
t
z
y
x
C
D
x
t
t
z
y
x
C ,
,
,
,
,
,
,
,
,
,
,
,
(1)

37
( ) ( ) ( ) ( ) 





∫
∇
∂
∂
Ω
+






∫
∇
∂
∂
Ω
+
−
−
z
z L
t
v
S
S
L
t
v
S
S
W
d
t
W
y
x
C
t
z
y
x
T
k
D
y
W
d
t
W
y
x
C
t
z
y
x
T
k
D
x
,
,
,
,
,
,
,
,
,
,
,
, µ
µ .
Boundary and initial conditions for our case could be written as
( ) 0
,
,
,
0
=
∂
∂
=
x
x
t
z
y
x
C
,
( ) 0
,
,
,
=
∂
∂
= x
L
x
x
t
z
y
x
C
,
( ) 0
,
,
,
0
=
∂
∂
=
y
y
t
z
y
x
C
,
( ) 0
,
,
,
=
∂
∂
= y
L
x
y
t
z
y
x
C
,
( ) 0
,
,
,
=
∂
∂
−
= t
v
z
z
t
z
y
x
C
,
( ) 0
,
,
,
=
∂
∂
= z
L
x
z
t
z
y
x
C
, C(x,y,z,0)=fC(x,y,z).
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 ( )
∫
−
z
L
t
v
z
d
t
z
y
x
C ,
,
,
(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
t
z
y
x
V
V
t
z
y
x
V
T
z
y
x
P
t
z
y
x
C
T
z
y
x
D
D S
L
S
S ς
ς
ξ 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]
( ) ( ) ( ) ( ) ( ) ( )×
−






+






= T
z
y
x
k
y
t
z
y
x
I
T
z
y
x
D
y
x
t
z
y
x
I
T
z
y
x
D
x
t
t
z
y
x
I
I
I
I
I ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( ) ( ) ( ) ( )
t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
I
T
z
y
x
D
z
t
z
y
x
I V
I
I ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, ,
2
−






+
×
∂
∂
∂
∂
(3)

38
( ) ( ) ( ) ( ) 





∫
∇
∂
∂
Ω
+






∫
∇
∂
∂
Ω
+
−
−
z
z L
t
v
S
S
I
L
t
v
S
S
I
W
d
t
W
y
x
I
t
z
y
x
T
k
D
y
W
d
t
W
y
x
I
t
z
y
x
T
k
D
x
,
,
,
,
,
,
,
,
,
,
,
, µ
µ
( ) ( ) ( ) ( ) ( ) ( )×
−






+






= T
z
y
x
k
y
t
z
y
x
V
T
z
y
x
D
y
x
t
z
y
x
V
T
z
y
x
D
x
t
t
z
y
x
V
V
V
V
V ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( ) ( ) ( ) ( )+
−






+
× t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
V
T
z
y
x
D
z
t
z
y
x
V V
I
V ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, ,
2
∂
∂
∂
∂
( ) ( ) ( ) ( ) 





∫
∇
∂
∂
Ω
+






∫
∇
∂
∂
Ω
+
−
−
z
z L
t
v
S
VS
L
t
v
S
VS
W
d
t
W
y
x
V
t
z
y
x
T
k
D
y
W
d
t
W
y
x
V
t
z
y
x
T
k
D
x
,
,
,
,
,
,
,
,
,
,
,
, µ
µ .
Boundary and initial conditions for these equations could be written as
( ) 0
,
,
,
0
=
=
x
x
t
z
y
x
I
∂
∂
,
( ) 0
,
,
,
=
= x
L
x
x
t
z
y
x
I
∂
∂
,
( ) 0
,
,
,
0
=
=
y
y
t
z
y
x
I
∂
∂
,
( ) 0
,
,
,
=
= y
L
y
y
t
z
y
x
I
∂
∂
,
( ) 0
,
,
,
=
−
= t
v
z
z
t
z
y
x
I
∂
∂
,
( ) 0
,
,
,
=
= z
L
z
z
t
z
y
x
I
∂
∂
,
( ) 0
,
,
,
0
=
=
x
x
t
z
y
x
V
∂
∂
,
( ) 0
,
,
,
=
= x
L
x
x
t
z
y
x
V
∂
∂
,
( ) 0
,
,
,
0
=
=
y
y
t
z
y
x
V
∂
∂
,
( ) 0
,
,
,
=
= y
L
y
y
t
z
y
x
V
∂
∂
,
( ) 0
,
,
,
=
−
= t
v
z
z
t
z
y
x
V
∂
∂
,
( ) 0
,
,
,
=
= z
L
z
z
t
z
y
x
V
∂
∂
,
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
Φ
Φ T
z
y
x
k
y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x
I
I
I
I
I
I
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( ) ( ) ( ) +






∫Φ
∇
∂
∂
Ω
+





 Φ
+
×
−
Φ
Φ
z
I
I
L
t
v
I
S
S
I
W
d
t
W
y
x
t
z
y
x
T
k
D
x
z
t
z
y
x
T
z
y
x
D
z
t
z
y
x
I ,
,
,
,
,
,
,
,
,
,
,
,
,
,
, µ
∂
∂
∂
∂
( ) ( ) ( ) ( )
t
z
y
x
I
T
z
y
x
k
W
d
t
W
y
x
t
z
y
x
T
k
D
y
I
I
L
t
v
I
S
IS
z
,
,
,
,
,
,
,
,
,
,
,
, 2
,
+






∫Φ
∇
∂
∂
Ω
+
−
Φ
µ (5)
( )
( )
( )
( )
( )
( )×
+





 Φ
+





 Φ
=
Φ
Φ
Φ T
z
y
x
k
y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x
V
V
V
V
V
V
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( )
( )
( ) ( ) +






∫Φ
∇
∂
∂
Ω
+





 Φ
+
×
−
Φ
Φ
z
V
V
L
t
v
V
S
S
V
W
d
t
W
y
x
t
z
y
x
T
k
D
x
z
t
z
y
x
T
z
y
x
D
z
t
z
y
x
V ,
,
,
,
,
,
,
,
,
,
,
,
,
,
, µ
∂
∂
∂
∂

39
( ) ( ) ( ) ( )
t
z
y
x
V
T
z
y
x
k
W
d
t
W
y
x
t
z
y
x
T
k
D
y
V
V
L
t
v
V
S
S z
V
,
,
,
,
,
,
,
,
,
,
,
, 2
,
+






∫Φ
∇
∂
∂
Ω
+
−
Φ
µ .
Boundary and initial conditions could be written as
( ) 0
,
,
,
0
=
Φ
=
x
I
x
t
z
y
x
∂
∂
,
( ) 0
,
,
,
=
Φ
= x
L
x
I
x
t
z
y
x
∂
∂
,
( ) 0
,
,
,
0
=
Φ
=
y
I
y
t
z
y
x
∂
∂
,
( ) 0
,
,
,
=
Φ
= y
L
y
I
y
t
z
y
x
∂
∂
,
( ) 0
,
,
,
=
Φ
−
= t
v
z
I
z
t
z
y
x
∂
∂
,
( ) 0
,
,
,
=
= z
L
z
z
t
z
y
x
I
∂
∂
,
( ) 0
,
,
,
0
=
Φ
=
x
V
x
t
z
y
x
∂
∂
,
( ) 0
,
,
,
=
= x
L
x
x
t
z
y
x
V
∂
∂
,
( ) 0
,
,
,
0
=
Φ
=
y
V
y
t
z
y
x
∂
∂
,
( ) 0
,
,
,
=
Φ
= y
L
y
V
y
t
z
y
x
∂
∂
,
( ) 0
,
,
,
=
Φ
−
= t
v
z
V
z
t
z
y
x
∂
∂
,
( ) 0
,
,
,
=
Φ
= z
L
z
V
z
t
z
y
x
∂
∂
,
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 ( )
t
z
y
x
u ,
,
,
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
t
z
y
x
u
x
t
z
y
x
u
x
t
z
y
x
u
x
t
z
y
x
u
z
E
t
z
y
x δ
ε
µ 0
,
,
,
,
,
,
2
1
,
,
,
,
,
,
2
,
,
,
( )
( )
( ) ( ) ( ) ( )
[ ]





−
−






−
∂
∂
−
+ ij
k
k
ij
T
t
z
y
x
T
z
z
K
x
t
z
y
x
u
z
z
δ
β
ε
σ
δ
σ
0
0 ,
,
,
3
,
,
,
2
1
,
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
t
z
y
x
y
t
z
y
x
x
t
z
y
x
t
t
z
y
x
u
z xz
xy
xx
x
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
2
2
σ
σ
σ
ρ
( )
( ) ( ) ( ) ( )
z
t
z
y
x
y
t
z
y
x
x
t
z
y
x
t
t
z
y
x
u
z yz
yy
yx
y
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
2
2
σ
σ
σ
ρ
( ) ( ) ( ) ( ) ( )
z
t
z
y
x
y
t
z
y
x
x
t
z
y
x
t
t
z
y
x
u
z zz
zy
zx
z
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
2
2
σ
σ
σ
ρ

40
where
( )
( )
[ ]
( ) ( ) ( ) ( ) ( )−
∂
∂
+








∂
∂
−
∂
∂
+
∂
∂
+
=
k
k
ij
k
k
ij
i
j
j
i
ij
x
t
z
y
x
u
z
K
x
t
z
y
x
u
x
t
z
y
x
u
x
t
z
y
x
u
z
z
E ,
,
,
,
,
,
3
,
,
,
,
,
,
1
2
δ
δ
σ
σ
( ) ( ) ( )
[ ]
r
T
t
z
y
x
T
z
K
z −
− ,
,
,
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
( ) ( ) ( ) ( )
( )
[ ]
( ) ( ) ( )
( )
[ ]
( )
+
∂
∂
∂






+
−
+
∂
∂






+
+
=
∂
∂
y
x
t
z
y
x
u
z
z
E
z
K
x
t
z
y
x
u
z
z
E
z
K
t
t
z
y
x
u
z
y
x
x
,
,
,
1
3
,
,
,
1
6
5
,
,
,
2
2
2
2
2
σ
σ
ρ
( )
( )
[ ]
( ) ( ) ( ) ( )
( )
[ ]
( ) −
∂
∂
∂






+
+
+








∂
∂
+
∂
∂
+
+
z
x
t
z
y
x
u
z
z
E
z
K
z
t
z
y
x
u
y
t
z
y
x
u
z
z
E z
z
y ,
,
,
1
3
,
,
,
,
,
,
1
2
2
2
2
2
2
σ
σ
( )
( ) ( )
( )
[ ]
( ) ( ) ( ) ( ) ( )+
∂
∂
−








∂
∂
∂
+
∂
∂
+
=
∂
∂
y
t
z
y
x
T
z
K
z
y
x
t
z
y
x
u
x
t
z
y
x
u
z
z
E
t
t
z
y
x
u
z x
y
y ,
,
,
,
,
,
,
,
,
1
2
,
,
, 2
2
2
2
2
β
σ
ρ
( )
( )
[ ]
( ) ( ) ( )
( )
[ ]
( )
( )
+
∂
∂






+
+
+
















∂
∂
+
∂
∂
+
∂
∂
+ 2
2
,
,
,
1
12
5
,
,
,
,
,
,
1
2 y
t
z
y
x
u
z
K
z
z
E
y
t
z
y
x
u
z
t
z
y
x
u
z
z
E
z
y
z
y
σ
σ
( ) ( )
( )
[ ]
( )
( )
( )
y
x
t
z
y
x
u
z
K
z
y
t
z
y
x
u
z
z
E
z
K
y
y
∂
∂
∂
+
∂
∂
∂






+
−
+
,
,
,
,
,
,
1
6
2
2
σ
(8)
( ) ( ) ( )
( )
[ ]
( ) ( ) ( ) ( )
+








∂
∂
∂
+
∂
∂
∂
+
∂
∂
+
∂
∂
+
=
∂
∂
z
y
t
z
y
x
u
z
x
t
z
y
x
u
y
t
z
y
x
u
x
t
z
y
x
u
z
z
E
t
t
z
y
x
u
z
y
x
z
z
z
,
,
,
,
,
,
,
,
,
,
,
,
1
2
,
,
,
2
2
2
2
2
2
2
2
σ
ρ
( ) ( ) ( ) ( ) ( ) ( ) ( )+
∂
∂
−
















∂
∂
+
∂
∂
+
∂
∂
∂
∂
+
z
t
z
y
x
T
z
z
K
z
t
z
y
x
u
y
t
z
y
x
u
x
t
z
y
x
u
z
K
z
x
y
x ,
,
,
,
,
,
,
,
,
,
,
,
β
( )
( )
( ) ( ) ( ) ( )
















∂
∂
−
∂
∂
−
∂
∂
−
∂
∂
+
∂
∂
+
z
t
z
y
x
u
y
t
z
y
x
u
x
t
z
y
x
u
z
t
z
y
x
u
z
z
E
z
z
y
x
z ,
,
,
,
,
,
,
,
,
,
,
,
6
1
6
1
σ
.
Systems of conditions for these equations could be written as
( ) 0
,
,
,
0
=
∂
∂
=
x
x
t
z
y
x
u
r
;
( ) 0
,
,
,
=
∂
∂
= x
L
x
x
t
z
y
x
u
r
;
( ) 0
,
,
,
0
=
∂
∂
=
y
y
t
z
y
x
u
r
;
( ) 0
,
,
,
=
∂
∂
= y
L
y
y
t
z
y
x
u
r
;
( ) 0
,
,
,
=
∂
∂
−
= t
v
z
z
t
z
y
x
u
r
;
( ) 0
,
,
,
=
∂
∂
= z
L
z
z
t
z
y
x
u
r
; ( ) 0
0
,
,
, u
z
y
x
u
r
r
= ; ( ) 0
,
,
, u
z
y
x
u
r
r
=
∞ .
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]

41
( )
∫ ∫ ∫ ∫
Θ
=
Θ
−
0 0 0
1
1 ,
,
,
1 x y z
L L L
t
v
z
y
x
t
d
x
d
y
d
z
d
t
z
y
x
L
L
L
ρ
α 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 z
L L L
t
v
C
z
y
x
C t
d
x
d
y
d
z
d
z
y
x
f
L
L
L
α ,
( )
4
3
4
1
2
3
2
4
2
3
1
4
4
4 a
A
a
a
a
L
L
L
B
a
B
a
A
a z
y
x
I
+
−







 Θ
+
Θ
+
−
+
=
α ,
( ) 





Θ
−
−
∫ ∫ ∫ ∫
Θ
=
Θ
−
z
y
x
II
I
L L L
t
v
I
I
IV
V L
L
L
S
t
d
x
d
y
d
z
d
z
y
x
f
S
x y z
00
1
0 0 0
1
00
1 ,
,
1
α
α
α ,
( )
∫ ∫ ∫ ∫
+
Θ
+
Θ
=
Θ
−
Φ
Φ
0 0 0
20
1
1 ,
,
1 x y z
I
I
L L L
t
v
z
y
x
z
y
x
II
z
y
x
I
t
d
x
d
y
d
z
d
z
y
x
f
L
L
L
L
L
L
S
L
L
L
R
α ,
( )
∫ ∫ ∫ ∫
+
Θ
+
Θ
=
Θ
−
Φ
Φ
0 0 0
20
1
1 ,
,
1 x y z
V
V
L L L
t
v
z
y
x
z
y
x
VV
z
y
x
V
t
d
x
d
y
d
z
d
z
y
x
f
L
L
L
L
L
L
S
L
L
L
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
1
2
2 ,
,
,
,
,
,
1 x y z
L L L
z
y
x
t
d
x
d
y
d
z
d
t
z
y
x
t
z
y
x
L
L
L
ρ
ρ
α 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
E
b
b
b
L
L
L
F
a
F
b
E
b z
y
x
V
+
−







 Θ
+
Θ
+
−
+
=
α ,
( )
00
2
01
11
02
10
01
2
00
2
2
2
2
IV
V
IV
V
I
V
V
z
y
x
V
I
VV
V
VV
V
V
I
S
S
S
S
L
L
L
S
S
S
C
α
α
α
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

42
( ) ( ) ( ) ( ) ( )
x
t
z
y
x
T
z
z
K
t
t
z
y
x
u
z x
∂
∂
−
=
∂
∂ ,
,
,
,
,
,
2
1
2
β
ρ ,
( )
( )
( ) ( ) ( )
y
t
z
y
x
T
z
z
K
t
t
z
y
x
u
z y
∂
∂
−
=
∂
∂ ,
,
,
,
,
,
2
1
2
β
ρ ,
( ) ( ) ( ) ( ) ( )
z
t
z
y
x
T
z
z
K
t
t
z
y
x
u
z z
∂
∂
−
=
∂
∂ ,
,
,
,
,
,
2
1
2
β
ρ .
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 u
d
d
z
y
x
T
x
z
z
z
K
d
d
z
y
x
T
x
z
z
z
K
t
z
y
x
u 0
0 0
0 0
1 ,
,
,
,
,
,
,
,
, +
∫ ∫
∂
∂
−
∫ ∫
∂
∂
=
∞ϑ
ϑ
ϑ
τ
τ
ρ
β
ϑ
τ
τ
ρ
β
,
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( ) y
t
y u
d
d
z
y
x
T
y
z
z
z
K
d
d
z
y
x
T
y
z
z
z
K
t
z
y
x
u 0
0 0
0 0
1 ,
,
,
,
,
,
,
,
, +
∫ ∫
∂
∂
−
∫ ∫
∂
∂
=
∞ϑ
ϑ
ϑ
τ
τ
ρ
β
ϑ
τ
τ
ρ
β
,
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( ) z
t
z u
d
d
z
y
x
T
z
z
z
z
K
d
d
z
y
x
T
z
z
z
z
K
t
z
y
x
u 0
0 0
0 0
1 ,
,
,
,
,
,
,
,
, +
∫ ∫
∂
∂
−
∫ ∫
∂
∂
=
∞ϑ
ϑ
ϑ
τ
τ
ρ
β
ϑ
τ
τ
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

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 /4
0 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)

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)

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.
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APPENDIX
Eqs. (1), (3) and (5) with account initial distributions could be written as
( ) ( ) ( ) ( ) ( ) ( )+
+






∂
∂
∂
∂
+






∂
∂
∂
∂
+






∂
∂
∂
∂
=
∂
∂
t
z
y
x
f
z
t
z
y
x
C
D
z
y
t
z
y
x
C
D
y
x
t
z
y
x
C
D
x
t
t
z
y
x
C
C δ
,
,
,
,
,
,
,
,
,
,
,
,
,
,
( ) ( ) ( ) ( ) 





∫
∇
∂
∂
Ω
+






∫
∇
∂
∂
Ω
+
−
−
z
z L
t
v
S
S
L
t
v
S
S
W
d
t
W
y
x
C
t
z
y
x
T
k
D
y
W
d
t
W
y
x
C
t
z
y
x
T
k
D
x
,
,
,
,
,
,
,
,
,
,
,
, µ
µ (1a)

47
( ) ( ) ( ) ( ) ( ) ( ) ( )+
+






+






= t
z
y
x
f
y
t
z
y
x
I
T
z
y
x
D
y
x
t
z
y
x
I
T
z
y
x
D
x
t
t
z
y
x
I
I
I
I δ
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
( ) ( ) ( ) ( ) ( ) ( ) ( )+
−
−






+ t
z
y
x
I
T
z
y
x
k
t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
I
T
z
y
x
D
z
I
I
V
I
I ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
,
,
∂
∂
∂
∂
( ) ( ) ( ) ( ) 





∫
∇
∂
∂
Ω
+






∫
∇
∂
∂
Ω
+
−
−
z
z L
t
v
S
IS
L
t
v
S
IS
W
d
t
W
y
x
I
t
z
y
x
T
k
D
y
W
d
t
W
y
x
I
t
z
y
x
T
k
D
x
,
,
,
,
,
,
,
,
,
,
,
, µ
µ (3a)
( ) ( ) ( ) ( ) ( ) ( ) ( )+
+






+






= t
z
y
x
f
y
t
z
y
x
V
T
z
y
x
D
y
x
t
z
y
x
V
T
z
y
x
D
x
t
t
z
y
x
V
V
V
V δ
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
( ) ( ) ( ) ( ) ( ) ( ) ( )+
−
−






+ t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
V
T
z
y
x
D
z
V
V
V
I
V ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
,
,
∂
∂
∂
∂
( ) ( ) ( ) ( ) 





∫
∇
∂
∂
Ω
+






∫
∇
∂
∂
Ω
+
−
−
z
z L
t
v
S
VS
L
t
v
S
VS
W
d
t
W
y
x
V
t
z
y
x
T
k
D
y
W
d
t
W
y
x
V
t
z
y
x
T
k
D
x
,
,
,
,
,
,
,
,
,
,
,
, µ
µ
( ) ( ) ( ) ( ) ( ) ( )×
+





 Φ
+





 Φ
=
Φ
Φ
Φ t
y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x I
I
I
I
I
δ
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
( ) ( ) ( ) ( ) ( ) +






∫Φ
∇
∂
∂
Ω
+





 Φ
+
×
−
Φ
Φ
Φ
z
I
I
I
L
t
v
I
S
S
I
W
d
t
W
y
x
t
z
y
x
T
k
D
x
z
t
z
y
x
T
z
y
x
D
z
z
y
x
f ,
,
,
,
,
,
,
,
,
,
,
,
,
, µ
∂
∂
∂
∂
( ) ( ) ( ) ( ) ( ) ( )
t
z
y
x
I
T
z
y
x
k
t
z
y
x
I
T
z
y
x
k
W
d
t
W
y
x
t
z
y
x
T
k
D
y
I
I
I
L
t
v
I
S
S
z
I
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
,
+
+






∫Φ
∇
∂
∂
Ω
+
−
Φ
µ
( ) ( ) ( ) ( ) ( ) ( )×
+





 Φ
+





 Φ
=
Φ
Φ
Φ t
y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x V
V
V
V
V
δ
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
(5a)
( ) ( ) ( ) ( ) ( ) +






∫Φ
∇
∂
∂
Ω
+





 Φ
+
×
−
Φ
Φ
Φ
z
V
V
V
L
t
v
V
S
S
V
W
d
t
W
y
x
t
z
y
x
T
k
D
x
z
t
z
y
x
T
z
y
x
D
z
z
y
x
f ,
,
,
,
,
,
,
,
,
,
,
,
,
, µ
∂
∂
∂
∂
( ) ( ) ( ) ( ) ( ) ( )
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
T
z
y
x
k
W
d
t
W
y
x
t
z
y
x
T
k
D
y
V
V
V
L
t
v
V
S
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
=
∂
∂
t
z
y
x
T
k
D
z
y
t
z
y
x
T
k
D
z
x
t
t
z
y
x
C
S
S
C
S
S
C ,
,
,
,
,
,
,
,
,
1
1
1
µ
α
µ
α
( ) ( )
t
z
y
x
fC δ
,
,
+ (1b)
( ) ( ) ( ) ( ) ( )−
+






∇
∂
∂
Ω
+






∇
∂
∂
Ω
= t
z
y
x
f
t
z
y
x
T
k
D
z
y
t
z
y
x
T
k
D
x
z
t
t
z
y
x
I
I
S
IS
I
S
IS
I δ
µ
α
µ
α
∂
∂
,
,
,
,
,
,
,
,
,
,
,
1
1
1
( ) ( )
T
z
y
x
k
T
z
y
x
k V
I
V
I
I
I
I ,
,
,
,
,
, ,
1
1
,
2
1 α
α
α −
− (3b)
( ) ( ) ( ) ( ) ( )−
+






∇
∂
∂
Ω
+






∇
∂
∂
Ω
= t
z
y
x
f
t
z
y
x
T
k
D
z
y
t
z
y
x
T
k
D
x
z
t
t
z
y
x
V
V
S
VS
V
S
VS
V δ
µ
α
µ
α
∂
∂
,
,
,
,
,
,
,
,
,
,
,
1
1
1
( ) ( )
T
z
y
x
k
T
z
y
x
k V
I
V
I
V
V
V ,
,
,
,
,
, ,
1
1
,
2
1 α
α
α −
−

48
( ) ( ) ( ) ( ) ( ) ( ) ( )+
+
+
=
Φ
Φ t
z
y
x
f
t
z
y
x
I
T
z
y
x
k
t
z
y
x
I
T
z
y
x
k
t
t
z
y
x
I
I
I
I
I
δ
∂
∂
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
,
1
( ) ( )





∇
∂
∂
Ω
+






∇
∂
∂
Ω
+
Φ
Φ
Φ
Φ t
z
y
x
T
k
D
y
z
t
z
y
x
T
k
D
x
z S
S
S
S I
I
I
I
,
,
,
,
,
, 1
1 µ
α
µ
α (5b)
( ) ( ) ( ) ( ) ( ) ( ) ( )+
+
+
=
Φ
Φ t
z
y
x
f
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
T
z
y
x
k
t
t
z
y
x
V
V
V
V
V
δ
∂
∂
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
,
1
( ) ( )





∇
∂
∂
Ω
+






∇
∂
∂
Ω
+
Φ
Φ
Φ
Φ t
z
y
x
T
k
D
y
z
t
z
y
x
T
k
D
x
z S
S
S
S V
V
V
V
,
,
,
,
,
, 1
1 µ
α
µ
α
The first-order approximations of the considered concentrations in the following form
( ) ( )
( )
( ) ( )
( )
∫ ×








+
+






+
∇
∂
∂
=
t
C
S
S
C
V
z
y
x
V
V
z
y
x
V
T
z
y
x
P
z
y
x
x
t
z
y
x
C
0
2
*
2
2
*
1
1
1
1
1
,
,
,
,
,
,
1
,
,
,
1
,
,
,
,
,
,
τ
ς
τ
ς
α
ξ
τ
µ
α γ
γ
( ) ( ) ( ) ( )
( )
∫ ×








+
+
∇
∂
∂
Ω
+



Ω
×
t
S
C
L
S
V
z
y
x
V
V
z
y
x
V
z
y
x
y
d
T
k
z
T
z
y
x
D
0
2
*
2
2
*
1
1
1
,
,
,
,
,
,
1
,
,
,
,
,
,
τ
ς
τ
ς
τ
µ
α
τ
( )
( )
( )
z
y
x
f
d
T
z
y
x
P
T
k
z
T
z
y
x
D C
C
S
L
S ,
,
,
,
,
1
,
,
, 1
+






+
× τ
α
ξ
γ
γ
(1c)
( ) ( ) ( ) ( )−
+
∫ ∇
∂
∂
Ω
+
∫ ∇
∂
∂
Ω
= z
y
x
f
d
z
y
x
T
k
D
y
z
d
z
y
x
T
k
D
x
z
t
z
y
x
I I
t
S
IS
I
t
S
IS
I ,
,
,
,
,
,
,
,
,
,
,
0
1
0
1
1 τ
τ
µ
α
τ
τ
µ
α
( ) ( )
∫
−
∫
−
t
V
I
V
I
t
I
I
I d
T
z
y
x
k
d
T
z
y
x
k
0
,
1
1
0
,
2
1 ,
,
,
,
,
, τ
α
α
τ
α (3c)
( ) ( ) ( ) ( )−
+
∫ ∇
∂
∂
Ω
+
∫ ∇
∂
∂
Ω
= z
y
x
f
d
z
y
x
T
k
D
y
z
d
z
y
x
T
k
D
x
z
t
z
y
x
V V
t
S
IS
V
t
S
IS
V ,
,
,
,
,
,
,
,
,
,
,
0
1
1
0
1
1
1 τ
τ
µ
α
τ
τ
µ
α
( ) ( )
∫
−
∫
−
t
V
I
V
I
t
V
V
V d
T
z
y
x
k
d
T
z
y
x
k
0
,
1
1
0
,
2
1 ,
,
,
,
,
, τ
α
α
τ
α
( ) ( ) ( ) +
∫ ∇
∂
∂
Ω
+
∫ ∇
∂
∂
Ω
=
Φ
Φ
Φ
Φ
Φ
t
S
S
t
S
S
I d
z
y
x
T
k
D
x
z
d
z
y
x
T
k
D
x
z
t
z
y
x I
I
I
I
0
1
0
1
1 ,
,
,
,
,
,
,
,
, τ
τ
µ
α
τ
τ
µ
α
( ) ( ) ( ) ( ) ( )
∫
+
∫
+
+ Φ
t
I
I
t
I d
z
y
x
I
T
z
y
x
k
d
z
y
x
I
T
z
y
x
k
z
y
x
f I
0
2
,
0
,
,
,
,
,
,
,
,
,
,
,
,
,
, τ
τ
τ
τ (5c)
( ) ( ) ( ) +
∫ ∇
∂
∂
Ω
+
∫ ∇
∂
∂
Ω
=
Φ
Φ
Φ
Φ
Φ
t
S
S
t
S
S
V d
z
y
x
T
k
D
x
z
d
z
y
x
T
k
D
x
z
t
z
y
x V
V
V
V
0
1
0
1
1 ,
,
,
,
,
,
,
,
, τ
τ
µ
α
τ
τ
µ
α
( ) ( ) ( ) ( ) ( )
∫
+
∫
+
+ Φ
t
V
V
t
V d
z
y
x
V
T
z
y
x
k
d
z
y
x
V
T
z
y
x
k
z
y
x
f V
0
2
,
0
,
,
,
,
,
,
,
,
,
,
,
,
,
, τ
τ
τ
τ .
Relations for calculations parameters Sρρij, ai, A, B, q, p could be written as
( ) ( ) ( ) ( )
∫ ∫ ∫ ∫
−
Θ
=
Θ
−
0 0 0
1
1
, ,
,
,
,
,
,
,
,
,
x y z
L L L
t
v
j
i
ij t
d
x
d
y
d
z
d
t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
t
S ρ
ρ
ρρ , ( )×
−
= 00
00
2
00
4 VV
II
IV S
S
S
a

49
00
II
S
× , 00
00
2
00
00
00
3 VV
II
IV
II
IV S
S
S
S
S
a −
+
= , ( ) ×
∫ ∫ ∫ ∫
+
Θ
=
Θ
−
0 0 0
2
2
2
00
2 ,
,
2
x y z
L L L
t
v
I
z
y
x
IV t
d
x
d
y
d
z
d
z
y
x
f
L
L
L
S
a
( ) ( ) ×
Θ
−
∫ ∫ ∫ ∫
−
∫ ∫ ∫ ∫
+
×
Θ
−
Θ
−
2
0 0 0
2
00
0 0 0
2
00
00
00
00 ,
,
,
, x
L L L
t
v
I
IV
L L L
t
v
V
IV
IV
II
VV L
t
d
x
d
y
d
z
d
z
y
x
f
S
t
d
x
d
y
d
z
d
z
y
x
f
S
S
S
S
x y z
x y z
00
2
2
VV
z
y S
L
L
× , ( )
∫ ∫ ∫ ∫
=
Θ
−
0 0 0
00
1 ,
,
x y z
L L L
t
v
I
IV t
d
x
d
y
d
z
d
z
y
x
f
S
a , +
+
+
−
−
+
= 3 3
2
3 3
2
q
p
q
q
p
q
B
4
2
6a
a
Θ
+ , ( )
2
0 0 0
00
0 ,
, 





∫ ∫ ∫ ∫
=
Θ
−
x y z
L L L
t
v
I
VV t
d
x
d
y
d
z
d
z
y
x
f
S
a ,
2
1
4
2
2
4
2
3
2
8
4 







+
Θ
−
Θ
= y
a
a
a
a
A , ×
Θ
= 2
4
2
3
24a
a
q
2
4
2
1
4
2
2
2
3
4
3
2
3
4
2
3
2
2
2
4
0
2
4
3
1
0
8
54
4
8
4
a
a
L
L
L
a
a
a
a
a
a
a
a
a
a
L
L
L
a z
y
x
z
y
x
Θ
−
Θ
−








Θ
−
Θ
Θ
−








Θ
−
× , −
Θ
= 2
4
4
0
2
12
4
a
a
a
p
( ) 2
4
4
2
3
1
2
36
2
3 a
a
a
a
a
L
L
L z
y
x +
Θ
Θ
− , ( ) ( ) ( )
∫ ∫ ∫ ∫
−
Θ
=
Θ
−
0 0 0
1 ,
,
,
,
,
,
x y z
L L L
t
v
i
I
i t
d
x
d
y
d
z
d
t
z
y
x
I
T
z
y
x
k
t
Rρ .
Equations for the second-order approximations of concentrations of dopant and radiation defects
could be written as
( ) ( ) ( )
[ ]
( )
( ) ( )
( )




×








+
+









 +
+
∂
∂
=
∂
∂
2
*
2
2
*
1
1
2
2 ,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
,
,
,
V
t
z
y
x
V
V
t
z
y
x
V
T
z
y
x
P
t
z
y
x
C
T
z
y
x
D
x
t
t
z
y
x
C C
L ς
ς
α
ξ γ
γ
( ) ( ) ( )
( )
( )
[ ]
( )
( )




×
∂
∂









 +
+








+
+
∂
∂
+




∂
∂
×
y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
V
t
z
y
x
V
V
t
z
y
x
V
y
x
t
z
y
x
C C ,
,
,
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
, 1
1
2
2
*
2
2
*
1
1
γ
γ
α
ξ
ς
ς
( )) ( ) ( )
( )
( )
[ ]
( )




×









 +
+








+
+
∂
∂
+
×
T
z
y
x
P
t
z
y
x
C
V
t
z
y
x
V
V
t
z
y
x
V
z
T
z
y
x
D C
L
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
, 1
2
2
*
2
2
*
1 γ
γ
α
ξ
ς
ς
( ) ( ) ( ) ( )
[ ] +






∫ +
∇
∂
∂
Ω
+




∂
∂
×
−
z
L
t
v
C
S
S
L W
d
t
W
y
x
C
t
z
y
x
T
k
D
x
z
t
z
y
x
C
T
z
y
x
D ,
,
,
,
,
,
,
,
,
,
,
, 2
2
1
α
µ
( ) ( )
[ ] ( ) ( )
t
z
y
x
f
W
d
t
W
y
x
C
t
z
y
x
T
k
D
y
C
L
t
v
C
S
S
z
δ
α
µ ,
,
,
,
,
,
,
, 2
2 +






∫ +
∇
∂
∂
Ω
+
−
(1d)
( ) ( ) ( ) ( ) ( ) ( )×
−






+






= T
z
y
x
k
y
t
z
y
x
I
T
z
y
x
D
y
x
t
z
y
x
I
T
z
y
x
D
x
t
t
z
y
x
I
V
I
I
I ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
1
2
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( )
[ ] ( )
[ ] ( ) ( ) ( )
[ ] ×
+
−






+
+
+
×
2
1
1
1
1
1
1
1 ,
,
,
,
,
,
,
,
,
,
,
,
,
,
, t
z
y
x
I
z
t
z
y
x
I
T
z
y
x
D
z
t
z
y
x
V
t
z
y
x
I I
I
V
I α
∂
∂
∂
∂
α
α
( ) ( ) ( )
[ ] +






∫ +
∇
∂
∂
Ω
+
×
−
z
L
t
v
I
S
IS
I
I W
d
t
W
y
x
I
t
z
y
x
T
k
D
x
T
z
y
x
k ,
,
,
,
,
,
,
,
, 1
2
, α
µ
( ) ( )
[ ]






∫ +
∇
∂
∂
Ω
+
−
z
L
t
v
I
S
IS
W
d
t
W
y
x
I
t
z
y
x
T
k
D
y
,
,
,
,
,
, 1
2
α
µ (3d)
( ) ( ) ( ) ( ) ( ) ( )×
−






+






= T
z
y
x
k
y
t
z
y
x
V
T
z
y
x
D
y
x
t
z
y
x
V
T
z
y
x
D
x
t
t
z
y
x
V
V
I
V
V ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
1
2
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( )
[ ] ( )
[ ] ( ) ( ) ( )
[ ] ×
+
−






+
+
+
×
2
1
1
1
1
1
1
1 ,
,
,
,
,
,
,
,
,
,
,
,
,
,
, t
z
y
x
V
z
t
z
y
x
V
T
z
y
x
D
z
t
z
y
x
V
t
z
y
x
I V
V
V
I α
∂
∂
∂
∂
α
α

50
( ) ( ) ( )
[ ] +






∫ +
∇
∂
∂
Ω
+
×
−
z
L
t
v
V
S
VS
V
V W
d
t
W
y
x
V
t
z
y
x
T
k
D
x
T
z
y
x
k ,
,
,
,
,
,
,
,
, 1
2
, α
µ
( ) ( )
[ ]






∫ +
∇
∂
∂
Ω
+
−
z
L
t
v
V
S
VS
W
d
t
W
y
x
V
t
z
y
x
T
k
D
y
,
,
,
,
,
, 1
2
α
µ
( ) ( ) ( ) ( ) ( ) ( )×
+





 Φ
+





 Φ
=
Φ
Φ
Φ T
z
y
x
k
y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x
I
I
I
I
I
I
I
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
1
2
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( )
[ ] ( ) ( )+
+






∫ Φ
+
∇
∂
∂
Ω
+
× Φ
−
Φ
Φ
t
z
y
x
f
W
d
t
W
y
x
t
z
y
x
T
k
D
x
t
z
y
x
I I
z
I
I
L
t
v
I
S
S
δ
α
µ ,
,
,
,
,
,
,
,
,
,
, 1
2
2
( ) ( )
[ ] ( ) ( ) +





 Φ
+






∫ Φ
+
∇
∂
∂
Ω
+ Φ
−
Φ
Φ
z
t
z
y
x
T
z
y
x
D
z
W
d
t
W
y
x
t
z
y
x
T
k
D
y
I
L
t
v
I
S
S
I
z
I
I
∂
∂
∂
∂
α
µ
,
,
,
,
,
,
,
,
,
,
,
, 1
1
2
( ) ( )
t
z
y
x
I
T
z
y
x
kI ,
,
,
,
,
,
+ (5d)
( )
( )
( )
( )
( )
( ) ×
+





 Φ
+





 Φ
=
Φ
Φ
Φ T
z
y
x
k
y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x
V
V
V
V
V
V
V
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
1
2
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( )
[ ] ( ) ( )+
+






∫ Φ
+
∇
∂
∂
Ω
+
× Φ
−
Φ
Φ
t
z
y
x
f
W
d
t
W
y
x
t
z
y
x
T
k
D
x
t
z
y
x
V V
z
V
V
L
t
v
V
S
S
δ
α
µ ,
,
,
,
,
,
,
,
,
,
, 1
2
2
( ) ( )
[ ] ( ) ( )
+





 Φ
+






∫ Φ
+
∇
∂
∂
Ω
+ Φ
−
Φ
Φ
z
t
z
y
x
T
z
y
x
D
z
W
d
t
W
y
x
t
z
y
x
T
k
D
y
V
L
t
v
V
S
S
V
z
V
V
∂
∂
∂
∂
α
µ
,
,
,
,
,
,
,
,
,
,
,
, 1
1
2
( ) ( )
t
z
y
x
V
T
z
y
x
kV ,
,
,
,
,
,
+ .
The second-order approximations of concentrations of dopant and radiation defects could be writ-
ten as
( ) ( ) ( )
[ ]
( )
( ) ( )
( )
∫ ×








+
+









 +
+
∂
∂
=
t
C
L
V
z
y
x
V
V
z
y
x
V
T
z
y
x
P
z
y
x
C
T
z
y
x
D
x
t
z
y
x
C
0
2
*
2
2
*
1
1
2
2
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
,
,
,
τ
ς
τ
ς
τ
α
ξ γ
γ
( ) ( ) ( )
( )
( )
[ ]
( )
∫ ×





 +
+








+
+
∂
∂
+
∂
∂
×
t
C
T
z
y
x
P
z
y
x
C
V
z
y
x
V
V
z
y
x
V
y
d
x
z
y
x
C
0
1
2
2
*
2
2
*
1
1
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
γ
γ
τ
α
ξ
τ
ς
τ
ς
τ
τ
( ) ( ) ( ) ( ) ( )
( )
∫ ×








+
+
∂
∂
+
∂
∂
×
t
L
L
V
z
y
x
V
V
z
y
x
V
T
z
y
x
D
z
d
y
z
y
x
C
T
z
y
x
D
0
2
*
2
2
*
1
1 ,
,
,
,
,
,
1
,
,
,
,
,
,
,
,
,
τ
ς
τ
ς
τ
τ
( ) ( )
[ ]
( )
( )
[ ] ×
∫ ∫ +
∂
∂
Ω
+





 +
+
∂
∂
×
−
t L
v
C
S
C
z
W
d
W
y
x
C
T
k
D
x
d
T
z
y
x
P
z
y
x
C
z
z
y
x
C
0
1
2
1
2
1
,
,
,
,
,
,
,
,
,
1
,
,
,
τ
γ
γ
ϑ
α
τ
τ
α
ξ
τ
( ) ( ) ( )
[ ] +
∫ ∫ +
∇
∂
∂
Ω
+
∇
×
−
t L
v
C
S
S
S d
d
W
d
W
y
x
C
z
y
x
T
k
D
y
d
d
z
y
x
z
0
1
2 ,
,
,
,
,
,
,
,
, τ
ϑ
ϑ
α
τ
µ
τ
ϑ
τ
µ
τ
( )
z
y
x
fC ,
,
+ (1e)
( ) ( ) ( ) ( ) ( ) +
∫
+
∫
=
t
I
t
I d
y
z
y
x
I
T
z
y
x
D
y
d
x
z
y
x
I
T
z
y
x
D
x
t
z
y
x
I
0
1
0
1
2
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, τ
∂
τ
∂
∂
∂
τ
∂
τ
∂
∂
∂
( ) ( ) ( ) ( )
[ ] ( )+
+
∫ +
−
∫
+ z
y
x
f
d
z
y
x
I
T
z
y
x
k
d
z
z
y
x
I
T
z
y
x
D
z
I
t
I
I
I
t
I ,
,
,
,
,
,
,
,
,
,
,
,
,
,
0
2
1
2
,
0
1
τ
τ
α
τ
∂
τ
∂
∂
∂
( ) ( )
[ ] ( )×
∫ ∇
∂
∂
Ω
+
∫ ∫ +
∇
∂
∂
Ω
+
−
t
S
IS
t L
v
I
S
IS
z
y
x
T
k
D
y
d
d
W
d
W
y
x
I
z
y
x
T
k
D
x
z
0
0
1
2 ,
,
,
,
,
,
,
,
, τ
µ
τ
ϑ
ϑ
α
τ
µ
τ
( )
[ ] ( ) ( )
[ ] ( )
[ ]
∫ +
+
−
∫ +
×
−
t
V
I
V
I
L
v
I d
z
y
x
V
z
y
x
I
T
z
y
x
k
d
d
W
d
W
y
x
I
z
0
1
2
1
2
,
1
2 ,
,
,
,
,
,
,
,
,
,
,
, τ
τ
α
τ
α
τ
ϑ
ϑ
α
τ
(3e)

51
( ) ( ) ( ) ( ) ( ) +
∫
+
∫
=
t
V
t
V d
y
z
y
x
V
T
z
y
x
D
y
d
x
z
y
x
V
T
z
y
x
D
x
t
z
y
x
V
0
1
0
1
2
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, τ
∂
τ
∂
∂
∂
τ
∂
τ
∂
∂
∂
( ) ( ) ( ) ( )
[ ] ( )+
+
∫ +
−
∫
+ z
y
x
f
d
z
y
x
V
T
z
y
x
k
d
z
z
y
x
V
T
z
y
x
D
z
V
t
V
V
V
t
V ,
,
,
,
,
,
,
,
,
,
,
,
,
,
0
2
1
2
,
0
1
τ
τ
α
τ
∂
τ
∂
∂
∂
( ) ( )
[ ] ( )×
∫ ∇
∂
∂
Ω
+
∫ ∫ +
∇
∂
∂
Ω
+
−
t
S
VS
t L
v
V
S
VS
z
y
x
T
k
D
y
d
d
W
d
W
y
x
V
z
y
x
T
k
D
x
z
0
0
1
2 ,
,
,
,
,
,
,
,
, τ
µ
τ
ϑ
ϑ
α
τ
µ
τ
( )
[ ] ( ) ( )
[ ] ( )
[ ]
∫ +
+
−
∫ +
×
−
t
V
I
V
I
L
v
I d
z
y
x
V
z
y
x
I
T
z
y
x
k
d
d
W
d
W
y
x
V
z
0
1
2
1
2
,
1
2 ,
,
,
,
,
,
,
,
,
,
,
, τ
τ
α
τ
α
τ
ϑ
ϑ
α
τ
( ) ( ) ( ) ( ) ( ) +
∫
Φ
+
∫
Φ
=
Φ Φ
Φ
t
I
t
I
I d
y
z
y
x
T
z
y
x
D
y
d
x
z
y
x
T
z
y
x
D
x
t
z
y
x I
I
0
1
0
1
2
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, τ
∂
τ
∂
∂
∂
τ
∂
τ
∂
∂
∂
( ) ( ) ( ) ( )
[ ] ×
∫ ∫ Φ
+
∇
∂
∂
Ω
+
∫
Φ
+
−
Φ
Φ
t L
v
I
S
t
I
z
I
I
W
d
W
y
x
z
y
x
x
d
z
z
y
x
T
z
y
x
D
z 0
1
2
0
1
,
,
,
,
,
,
,
,
,
,
,
,
τ
τ
α
τ
µ
τ
∂
τ
∂
∂
∂
( ) ( )
[ ] ( )+
+
∫ ∫ Φ
+
∇
∂
∂
Ω
+
× Φ
−
Φ
Φ
Φ
z
y
x
f
d
d
W
d
W
y
x
z
y
x
T
k
D
y
d
T
k
D
I
z
I
I
I
t L
v
I
S
S
S
,
,
,
,
,
,
,
,
0
1
2 τ
ϑ
ϑ
α
τ
µ
τ
τ
( ) ( ) ( ) ( )
∫
+
∫
+
t
I
t
I
I d
z
y
x
I
T
z
y
x
k
d
z
y
x
I
T
z
y
x
k
0
0
2
, ,
,
,
,
,
,
,
,
,
,
,
, τ
τ
τ
τ (5e)
( ) ( ) ( ) ( ) ( ) +
∫
Φ
+
∫
Φ
=
Φ Φ
Φ
t
V
t
V
V d
y
z
y
x
T
z
y
x
D
y
d
x
z
y
x
T
z
y
x
D
x
t
z
y
x V
V
0
1
0
1
2
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, τ
∂
τ
∂
∂
∂
τ
∂
τ
∂
∂
∂
( ) ( ) ( ) ( )
[ ] ×
∫ ∫ Φ
+
∇
∂
∂
Ω
+
∫
Φ
+
−
Φ
Φ
t L
v
V
S
t
V
z
V
V
W
d
W
y
x
z
y
x
x
d
z
z
y
x
T
z
y
x
D
z 0
1
2
0
1
,
,
,
,
,
,
,
,
,
,
,
,
τ
τ
α
τ
µ
τ
∂
τ
∂
∂
∂
( ) ( )
[ ] ( )+
+
∫ ∫ Φ
+
∇
∂
∂
Ω
+
× Φ
−
Φ
Φ
Φ
z
y
x
f
d
d
W
d
W
y
x
z
y
x
T
k
D
y
d
T
k
D
V
z
V
V
V
t L
v
V
S
S
S
,
,
,
,
,
,
,
,
0
1
2 τ
ϑ
ϑ
α
τ
µ
τ
τ
( ) ( ) ( ) ( )
∫
+
∫
+
t
V
t
V
V d
z
y
x
V
T
z
y
x
k
d
z
y
x
V
T
z
y
x
k
0
0
2
, ,
,
,
,
,
,
,
,
,
,
,
, τ
τ
τ
τ .
Parameters bi, Cρ, F, r, s have been determined by the following relations
00
2
00
00
2
00
4
1
1
II
VV
z
y
x
V
V
IV
z
y
x
S
S
L
L
L
S
S
L
L
L
b
Θ
−
Θ
= , ( ) +
Θ
Θ
+
+
−
=
z
y
x
VV
II
z
y
x
IV
VV
L
L
L
S
S
L
L
L
S
S
b 00
00
10
01
3 2
( ) ( ) 3
3
3
3
10
2
00
10
01
2
00
01
10
01
00
00
2
2
z
y
x
IV
IV
z
y
x
IV
VV
z
y
x
IV
z
y
x
IV
II
IV
z
y
x
VV
IV
L
L
L
S
S
L
L
L
S
S
L
L
L
S
L
L
L
S
S
S
L
L
L
S
S
Θ
−
Θ
+
+
Θ
+
Θ
+
+
+
Θ
+ ,
( ) ( ) ( )×
+
+
Θ
Θ
+
+
−
Θ
−
+
+
Θ
= 01
10
01
2
10
01
11
02
00
00
2 2
2 IV
II
z
y
x
x
IV
IV
VV
z
y
x
V
IV
VV
z
y
x
VV
II
S
S
L
L
L
L
S
S
S
L
L
L
C
S
S
L
L
L
S
S
b
( )( ) ×
Θ
−
+
Θ
+
+
+
+
+
Θ
Θ
+
×
z
y
x
IV
IV
z
y
x
VV
IV
II
IV
z
y
x
z
y
x
IV
z
y
VV
L
L
L
S
S
L
L
L
S
S
S
S
L
L
L
L
L
L
S
L
L
S 2
00
10
01
01
10
01
00
00
2
2
2
( ) 01
00
10
2
2
2
2
2
00
11
02
2
IV
z
y
x
IV
IV
z
y
x
IV
I
IV
VV
V S
L
L
L
S
S
L
L
L
S
C
S
S
C
Θ
−
Θ
+
−
−
× , ( ×
Θ
+
Θ
+
+
= x
IV
z
y
x
V
VV
IV
II L
S
L
L
L
C
S
S
S
b 10
02
11
00
1
) ( ) ×
Θ
−
−
−
Θ
+
+
Θ
+
+
Θ
+
+
×
z
y
x
IV
VV
V
z
y
x
IV
VV
z
y
x
IV
II
z
y
x
IV
VV
z
y
L
L
L
S
S
C
L
L
L
S
S
L
L
L
S
S
L
L
L
S
S
L
L 11
02
10
01
01
10
01
01 2
2
2
( )
z
y
x
IV
IV
IV
IV
I
z
y
x
II
IV
IV
L
L
L
S
S
S
S
C
L
L
L
S
S
S
Θ
−
+
Θ
+
+
×
2
01
10
01
00
10
01
00 2
2
3 ,
( )
×
−
Θ
+
= 01
2
02
00
00
0 IV
z
y
x
VV
IV
II S
L
L
L
S
S
S
b

52
( ) ( )×
+
+
Θ
Θ
−
−
−
+
+
Θ
Θ
−
−
× 01
10
11
02
01
10
11
02
2
2 IV
II
z
y
x
z
y
x
IV
VV
V
IV
II
z
y
x
z
y
x
IV
VV
V
S
S
L
L
L
L
L
L
S
S
C
S
S
L
L
L
L
L
L
S
S
C
( )
01
10
11
02
01
2
01
01 2
2 IV
II
z
y
x
z
y
x
IV
VV
V
IV
IV
I
IV S
S
L
L
L
L
L
L
S
S
C
S
S
C
S +
+
Θ
Θ
−
−
−
+
× , +
Θ
=
z
y
x
V
I
IV
I
L
L
L
S
C 1
1
00
α
α
z
y
x
IV
z
y
x
II
II
z
y
x
II
I
L
L
L
S
L
L
L
S
S
L
L
L
S
Θ
−
Θ
−
Θ
+ 11
20
20
00
2
1
α
, 11
02
00
2
1
00
1
1 IV
VV
VV
V
IV
V
I
V S
S
S
S
C −
−
+
= α
α
α ,
4
2
2
4
2
3
2
4
8
a
a
a
a
y
E Θ
−
Θ
+
= ,
2
4
2
1
4
2
2
2
3
4
3
2
3
4
2
3
2
2
2
4
2
0
4
3
1
0
2
4
2
3
8
54
4
8
4
24 b
b
L
L
L
b
b
b
b
b
b
b
b
b
b
L
L
L
b
b
b
r z
y
x
z
y
x
Θ
−
Θ
−








Θ
−
Θ
Θ
−








Θ
−
Θ
= , ( −
Θ
= 4
0
2
4
2
4
12
b
b
b
s
) 4
2
3
1 18b
b
b
b
L
L
L z
y
x Θ
−
Θ
− , 3 3
2
3 3
2
4
2
6
r
s
r
r
s
r
a
a
F +
+
−
−
+
+
Θ
= .
Equations for components of the displacement vector could be presented in the form
( ) ( ) ( ) ( )
( )
[ ]
( ) ( ) ( )
( )
[ ]
( )
+
∂
∂
∂






+
−
+
∂
∂






+
+
=
∂
∂
y
x
t
z
y
x
u
z
z
E
z
K
x
t
z
y
x
u
z
z
E
z
K
t
t
z
y
x
u
z
y
x
x
,
,
,
1
3
,
,
,
1
6
5
,
,
, 1
2
2
1
2
2
2
2
σ
σ
ρ
( )
( )
[ ]
( ) ( ) ( ) ( )
( )
[ ]
( )
−
∂
∂
∂






+
+
+








∂
∂
+
∂
∂
+
+
z
x
t
z
y
x
u
z
z
E
z
K
z
t
z
y
x
u
y
t
z
y
x
u
z
z
E z
z
y ,
,
,
1
3
,
,
,
,
,
,
1
2
1
2
2
1
2
2
1
2
σ
σ
( ) ( ) ( )
x
t
z
y
x
T
z
z
K
∂
∂
−
,
,
,
β
( )
( ) ( )
( )
[ ]
( ) ( ) ( ) ( ) ( )+
∂
∂
−








∂
∂
∂
+
∂
∂
+
=
∂
∂
y
t
z
y
x
T
z
z
K
y
x
t
z
y
x
u
x
t
z
y
x
u
z
z
E
t
t
z
y
x
u
z x
y
y ,
,
,
,
,
,
,
,
,
1
2
,
,
, 1
2
2
1
2
2
2
2
β
σ
ρ
( )
( )
[ ]
( ) ( ) ( ) ( )
( )
[ ]
( ) +






+
+
∂
∂
+
















∂
∂
+
∂
∂
+
∂
∂
+ z
K
z
z
E
y
t
z
y
x
u
y
t
z
y
x
u
z
t
z
y
x
u
z
z
E
z
y
z
y
σ
σ 1
12
5
,
,
,
,
,
,
,
,
,
1
2 2
1
2
1
1
( ) ( )
( )
[ ]
( )
( )
( )
y
x
t
z
y
x
u
z
K
z
y
t
z
y
x
u
z
z
E
z
K
y
y
∂
∂
∂
+
∂
∂
∂






+
−
+
,
,
,
,
,
,
1
6
1
2
1
2
σ
( ) ( ) ( ) ( ) ( ) ( )
×








∂
∂
∂
+
∂
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
z
y
t
z
y
x
u
z
x
t
z
y
x
u
y
t
z
y
x
u
x
t
z
y
x
u
t
t
z
y
x
u
z
y
x
z
z
z
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 1
2
1
2
2
1
2
2
1
2
2
2
2
ρ
( )
( )
[ ]
( ) ( ) ( ) ( ) ( )
( )
[ ]



×
+
∂
∂
+
















∂
∂
+
∂
∂
+
∂
∂
∂
∂
+
+
×
z
z
E
z
z
t
z
y
x
u
y
t
z
y
x
u
x
t
z
y
x
u
z
K
z
z
z
E x
y
x
σ
σ 1
6
,
,
,
,
,
,
,
,
,
1
2
1
1
1
( ) ( ) ( ) ( ) ( ) ( ) ( )
z
t
z
y
x
T
z
z
K
z
t
z
y
x
u
y
t
z
y
x
u
x
t
z
y
x
u
z
t
z
y
x
u z
y
x
z
∂
∂
−











∂
∂
−
∂
∂
−
∂
∂
−
∂
∂
×
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
6 1
1
1
1
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
z
E
z
K
z
d
d
z
y
x
u
x
z
z
E
z
K
z
t
z
y
x
u
t
x
x
σ
ρ
ϑ
τ
τ
σ
ρ
ϑ
1
3
1
,
,
,
1
6
5
1
,
,
,
0 0
1
2
2
2
( ) ( ) ( ) ×






∫ ∫
∂
∂
+
∫ ∫
∂
∂
+
∫ ∫
∂
∂
∂
×
t
z
t
y
t
y d
d
z
y
x
u
z
d
d
z
y
x
u
y
d
d
z
y
x
u
y
x 0 0
1
2
2
0 0
1
2
2
0 0
1
2
,
,
,
,
,
,
,
,
,
ϑ
ϑ
ϑ
ϑ
τ
τ
ϑ
τ
τ
ϑ
τ
τ

53
( )
( ) ( )
[ ] ( )
( ) ( ) ( )
( )
[ ]
( ) ( )
( )
×
−






+
+
∫ ∫
∂
∂
∂
+
+
×
z
z
z
K
z
z
E
z
K
d
d
z
y
x
u
z
x
z
z
z
z
E t
z
ρ
β
σ
ϑ
τ
τ
ρ
σ
ρ
ϑ
1
3
,
,
,
1
1
2 0 0
1
2
( )
( )
( ) ( )
( )
[ ]
( )
( )
×
−
∫ ∫
∂
∂






+
+
−
∫ ∫
∂
∂
×
∞
z
d
d
z
y
x
u
x
z
z
E
z
K
z
d
d
z
y
x
T
x
x
t
ρ
ϑ
τ
τ
σ
ρ
ϑ
τ
τ
ϑ
ϑ 1
,
,
,
1
6
5
1
,
,
,
0 0
1
2
2
0 0
( ) ( )
( )
[ ]
( )
( )
( )



+
∫ ∫
∂
∂
−
∫ ∫
∂
∂
∂






+
−
×
∞
∞
0 0
1
2
2
0 0
1
2
,
,
,
2
1
,
,
,
1
3
ϑ
ϑ
ϑ
τ
τ
ρ
ϑ
τ
τ
σ
d
d
z
y
x
u
y
z
d
d
z
y
x
u
y
x
z
z
E
z
K y
y
( ) ( )
( )
( ) ( )
( )
[ ]
( ) ×
∫ ∫
∂
∂
∂






+
+
−
+



∫ ∫
∂
∂
+
∞
∞
0 0
1
2
0 0
1
2
2
,
,
,
1
3
1
,
,
,
ϑ
ϑ
ϑ
τ
τ
σ
σ
ϑ
τ
τ d
d
z
y
x
u
z
x
z
z
E
z
K
z
z
E
d
d
z
y
x
u
z
z
z
( )
( ) ( )
( )
( )
∫ ∫
∂
∂
+
+
×
∞
0 0
0 ,
,
,
1 ϑ
ϑ
τ
τ
ρ
β
ρ
d
d
z
y
x
T
x
z
z
z
K
u
z
x
( ) ( )
( ) ( )
[ ]
( ) ( ) +






∫ ∫
∂
∂
∂
+
∫ ∫
∂
∂
+
=
t
x
t
x
y d
d
z
y
x
u
y
x
d
d
z
y
x
u
x
z
z
z
E
t
z
y
x
u
0 0
1
2
0 0
1
2
2
2 ,
,
,
,
,
,
1
2
,
,
,
ϑ
ϑ
ϑ
τ
τ
ϑ
τ
τ
σ
ρ
( )
( )
( )
( )
( ) ( ) ( )
( )
[ ]
+






+
+
∫ ∫
∂
∂
+
∫ ∫
∂
∂
∂
+
z
z
E
z
K
d
d
z
y
x
u
y
z
d
d
z
y
x
u
y
x
z
z
K t
x
t
y
σ
ϑ
τ
τ
ρ
ϑ
τ
τ
ρ
ϑ
ϑ
1
12
5
,
,
,
1
,
,
,
0 0
1
2
2
0 0
1
2
( )
( )
[ ] ( )
( )
( )
( ) ( ) −












∫ ∫
∂
∂
+
∫ ∫
∂
∂
+
∂
∂
+



+
+
t
z
t
y d
d
z
y
x
u
y
d
d
z
y
x
u
z
z
z
E
z
z
z
z
E
0 0
1
0 0
1 ,
,
,
,
,
,
1
2
1
1
12
5 ϑ
ϑ
ϑ
τ
τ
ϑ
τ
τ
σ
ρ
σ
( ) ( )
( )
( )
( )
( ) ( ) ( )
( )
[ ]
−






+
−
∫ ∫
∂
∂
∂
+
∫ ∫
−
z
z
E
z
K
d
d
z
y
x
u
z
y
z
d
d
z
y
x
T
z
z
z
K
t
y
t
σ
ϑ
τ
τ
ρ
ϑ
τ
τ
ρ
β ϑ
ϑ
1
6
,
,
,
1
,
,
,
0 0
1
2
0 0
( )
( ) ( )
[ ]
( ) ( ) ( ) ( )
( )
×
−






∫ ∫
∂
∂
∂
+
∫ ∫
∂
∂
+
−
∞
∞
z
z
z
K
d
d
z
y
x
u
y
x
d
d
z
y
x
u
x
z
z
z
E
x
x
ρ
β
ϑ
τ
τ
ϑ
τ
τ
σ
ρ
ϑ
ϑ
0 0
1
2
0 0
1
2
2
,
,
,
,
,
,
1
2
( ) ( )
( )
( )
( )
( ) ×
∫ ∫
∂
∂
−
∫ ∫
∂
∂
∂
−
∫ ∫
×
∞
∞
∞
0 0
1
2
2
0 0
1
2
0 0
,
,
,
1
,
,
,
,
,
,
ϑ
ϑ
ϑ
ϑ
τ
τ
ρ
ϑ
τ
τ
ρ
ϑ
τ
τ d
d
z
y
x
u
y
z
d
d
z
y
x
u
y
x
z
z
K
d
d
z
y
x
T x
y
( ) ( )
( )
[ ]
( )
( )
( ) ( ) ×












∫ ∫
∂
∂
+
∫ ∫
∂
∂
+
∂
∂
−






+
+
×
∞
∞
0 0
1
0 0
1 ,
,
,
,
,
,
1
1
12
5 ϑ
ϑ
ϑ
τ
τ
ϑ
τ
τ
σ
σ
d
d
z
y
x
u
y
d
d
z
y
x
u
z
z
z
E
z
z
z
E
z
K z
y
( ) ( )
( ) ( )
( )
[ ]
( ) y
y u
d
d
z
y
x
u
z
y
z
z
E
z
K
z
z
0
0 0
1
2
,
,
,
1
6
1
2
1
+
∫ ∫
∂
∂
∂






+
−
−
×
∞ϑ
ϑ
τ
τ
σ
ρ
ρ
( ) ( )
( ) ( )
[ ]
( ) ( )



+
∫ ∫
∂
∂
+
∫ ∫
∂
∂
+
=
∞
∞
0 0
1
2
2
0 0
1
2
2
,
,
,
,
,
,
1
2
,
,
,
ϑ
ϑ
ϑ
τ
τ
ϑ
τ
τ
σ
ρ
d
d
z
y
x
u
y
d
d
z
y
x
u
x
z
z
z
E
t
z
y
x
u z
z
z
( ) ( ) ( )






+
∫ ∫
∂
∂
∂
∂
+



∫ ∫
∂
∂
∂
+
∫ ∫
∂
∂
∂
+
∞
∞
∞
0 0
1
0 0
1
2
0 0
1
2
,
,
,
,
,
,
,
,
,
ϑ
ϑ
ϑ
ϑ
τ
τ
ϑ
τ
τ
ϑ
τ
τ d
d
z
y
x
u
x
z
d
d
z
y
x
u
z
y
d
d
z
y
x
u
z
x
x
y
x
( ) ( ) ( )
( ) ( )
( )
( )



×
+
∂
∂
+






∫ ∫
∂
∂
+
∫ ∫
∂
∂
+
∞
∞
z
z
E
z
z
z
z
K
d
d
z
y
x
u
z
d
d
z
y
x
u
y
x
x
σ
ρ
ρ
ϑ
τ
τ
ϑ
τ
τ
ϑ
ϑ
1
6
1
1
,
,
,
,
,
,
0 0
1
0 0
1
( ) ( ) ( )



−
∫ ∫
∂
∂
−
∫ ∫
∂
∂
−
∫ ∫
∂
∂
×
∞
∞
∞
0 0
1
0 0
1
0 0
1 ,
,
,
,
,
,
,
,
,
6
ϑ
ϑ
ϑ
ϑ
τ
τ
ϑ
τ
τ
ϑ
τ
τ d
d
z
y
x
u
y
d
d
z
y
x
u
x
d
d
z
y
x
u
z
y
x
z
( ) ( ) ( )
( )
( ) z
z u
d
d
z
y
x
T
z
z
z
z
K
d
d
z
y
x
u
z
0
0 0
0 0
1 ,
,
,
,
,
, +
∫ ∫
∂
∂
−






∫ ∫
∂
∂
−
∞
∞ ϑ
ϑ
ϑ
τ
τ
ρ
β
ϑ
τ
τ .

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.

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