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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 OFP-N-JUNCTIONS DURING OVERGROWTH
E.L. Pankratov1, E.A. Bulaeva
1,2
1Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950,
Russia2Nizhny 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 manufacturep-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 thedoped materials due to streaming of electrical current during operating of p-n-junction or to de-
crease depth ofp-n-junction for fixed value of local overheats. One way to increase sharpness ofp-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 ofnear-surficial area in comparison with volumetric dopant diffusion coefficient. The increasing ofdopant diffusion coefficient of near-surficial area leads to increasing of sharpness ofp-n-junction.
The second way to increase sharpness ofp-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
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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. Onecan 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 ofintegrated 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].
Theconsideredapproaches 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 ofhomogeneity of dopant distribution in enriched area at one time.
z
D(z
),
P(z
)
D2
D3
0 Lza
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 ( nor p) and anepitaxial layer, included into a heterostructure. The epitaxial layer have been doped by diffusion
or by ion implantation to manufacture another type of conductivity (por n). Farther we considerovergrowth 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
tzyxCD
zy
tzyxCD
yx
tzyxCD
xt
tzyxC ,,,,,,,,,,,, (1)
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37
( ) ( ) ( ) ( )
+
+
zz L
tvS
SL
tvS
S WdtWyxCtzyxTk
D
yWdtWyxCtzyx
Tk
D
x,,,,,,,,,,,, .
Boundary and initial conditions for our case could be written as
( )0
,,,
0
=
=xx
tzyxC,
( )0
,,,=
= xLxx
tzyxC,
( )0
,,,
0
=
=yy
tzyxC,
( )0
,,,=
= yLxy
tzyxC,
( )0
,,,=
= tvzz
tzyxC,
( )0
,,,=
= zLxz
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; 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; (x,y,z,t) is the chemical potential (reason of ac-counting of the chemical potential is mismatch-induced stress); the parameters DandDSare 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 andcooling of heterostructure, properties of materials of heterostructure, spatio-temporal distributions
of concentrations of dopant and radiation defects after ion implantation. Approximations of thesedependences could be approximated by the following functions [22,23]
( ) ( )
( )( ) ( )
( )
++
+=
2*
2
2*1
,,,,,,1
,,,
,,,1,,,
V
tzyxV
V
tzyxV
TzyxP
tzyxCTzyxDD
SLSS
. (2)
FunctionsDL(x,y,z,T) andDLS(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 could be integer anddepends 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 thefollowing system of equations [1,20-22]
( )( )
( )( )
( )( )
+
= Tzyxk
y
tzyxITzyxD
yx
tzyxITzyxD
xt
tzyxIIIII
,,,,,,
,,,,,,
,,,,,,
,
( ) ( ) ( ) ( ) ( ) ( )tzyxVtzyxITzyxkz
tzyxITzyxD
ztzyxI
VII,,,,,,,,,
,,,,,,,,,
,
2
+
(3)
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( ) ( ) ( ) ( )
+
+
zz L
tvS
SIL
tvS
SIWdtWyxItzyx
Tk
D
yWdtWyxItzyx
Tk
D
x,,,,,,,,,,,,
( )( )
( )( )
( )( )
+
= Tzyxk
y
tzyxVTzyxD
yx
tzyxVTzyxD
xt
tzyxVVVVV ,,,
,,,,,,
,,,,,,
,,,,
( ) ( ) ( ) ( ) ( ) ( )+
+ tzyxVtzyxITzyxkz
tzyxVTzyxD
ztzyxV
VIV,,,,,,,,,
,,,,,,,,,
,
2
( ) ( ) ( ) ( )
+
+
zz L
tvS
VSL
tvS
VS WdtWyxVtzyxTk
D
yWdtWyxVtzyx
Tk
D
x,,,,,,,,,,,, .
Boundary and initial conditions for these equations could be written as
( )0
,,,
0
=
=xx
tzyxI
,
( )0
,,,=
= xLxx
tzyxI
,
( )0
,,,
0
=
=yy
tzyxI
,
( )0
,,,=
= yLyy
tzyxI
,
( )
0
,,,=
= tvzz
tzyxI
,
( )
0
,,,=
= zLzz
tzyxI
,
( )
0
,,,
0
=
=xx
tzyxV
,
( )
0
,,,=
= xLxx
tzyxV
,
( )0
,,,
0
=
=yy
tzyxV
,
( )0
,,,=
= yLyy
tzyxV
,
( )0
,,,=
= tvzz
tzyxV
,
( )0
,,,=
= zLzz
tzyxV
,
I(x,y,z,0)=fI(x,y,z), V(x,y,z,0)=fV(x,y,z). (4)
HereI(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 thecoefficients of volumetric and surface diffusion; terms V
2(x,y,z,t) and I
2(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) describeddependences of parameters of recombination of point defects and generation their complexes on
coordinate and temperature; kis the Boltzmann constant.
We calculate spatio-temporal distributions of concentrations of divacancies V(x,y,z,t) and diin-terstitials I(x,y,z,t) by solution of the equations [20-22]
( )( )
( )( )
( )( ) +
+
=
Tzyxky
tzyxTzyxD
yx
tzyxTzyxD
xt
tzyxI
III
II,,,
,,,,,,
,,,,,,
,,,
( ) ( ) ( )
( ) ( ) +
+
+
zI
I
L
tvIS
SI WdtWyxtzyxTk
D
xz
tzyxTzyxD
ztzyxI ,,,,,,
,,,,,,,,,
( ) ( ) ( ) ( )tzyxITzyxkWdtWyxtzyxTk
D
y II
L
tvIS
ISz
,,,,,,,,,,,, 2,
+
+
(5)
( )( )
( )( )
( )( ) +
+
=
Tzyxky
tzyxTzyxD
yx
tzyxTzyxD
xt
tzyxV
VVV
VV,,,
,,,,,,
,,,,,,
,,,
( ) ( ) ( )
( ) ( ) +
+
+
zV
V
L
tvVS
SV WdtWyxtzyxTk
D
xz
tzyxTzyxD
ztzyxV ,,,,,,
,,,,,,,,,
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39
( ) ( ) ( ) ( )tzyxVTzyxkWdtWyxtzyxTk
D
y VV
L
tvVS
S z
V ,,,,,,,,,,,, 2,
+
+
.
Boundary and initial conditions could be written as
( )0
,,,
0
=
=x
I
x
tzyx
,
( )0
,,,=
= xLx
I
x
tzyx
,
( )0
,,,
0
=
=y
I
y
tzyx
,
( )0
,,,=
= yLy
I
y
tzyx
,
( )0
,,,=
= tvz
I
z
tzyx
,
( )0
,,,=
= zLzz
tzyxI
,
( )0
,,,
0
=
=x
V
x
tzyx
,
( )0
,,,=
= xLxx
tzyxV
,
( )0
,,,
0
=
=y
V
y
tzyx
,
( )0
,,,=
= yLy
V
y
tzyx
,
( )0
,,,=
= tvz
V
z
tzyx
,
( )0
,,,=
= zLz
V
z
tzyx
,
I(x,y,z,0)=fI(x,y,z), V(x,y,z,0)=fV(x,y,z). (6)
Here DI(x,y,z,T), DV(x,y,z,T), DIS(x,y,z,T) and DVS(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 potentialin the Eq.(1) by the following relation [20]
=E(z)ij[uij(x,y,z,t)+uji(x,y,z,t)]/2. (7)
HereEis the tension (Young) modulus;
+
=
i
j
j
i
ijx
u
x
uu
2
1is the deformation tensor; ijis the
stress tensor; ui, ujare 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,xjare the coordinates x,y,z. The relation (3) could be transformed to the fol-
lowing form
( ) ( ) ( ) ( ) ( ) ( )
+
+
+
=
ij
i
j
j
i
i
j
j
i
x
tzyxu
x
tzyxu
x
tzyxu
x
tzyxuzEtzyx 0
,,,,,,
2
1,,,,,,
2,,,
( )
( )
( )( ) ( ) ( )[ ]
+ ij
k
kij TtzyxTzzKx
tzyxu
z
z
00
,,,3,,,
21,
where is the Poisson coefficient; the parameter 0=(as-aEL)/aELdescribes the displacement pa-rameter with lattice distances of the substrate and the epitaxial layer as, aEL; K is the modulus of
uniform compression; the parameterbdescribes 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
tzyxuz xz
xyxxx
+
+
=
,,,,,,,,,,,,2
2
( ) ( ) ( ) ( ) ( )
z
tzyx
y
tzyx
x
tzyx
t
tzyxuz
yzyyyxy
+
+
=
,,,,,,,,,,,,2
2
( ) ( ) ( ) ( ) ( )
z
tzyx
y
tzyx
x
tzyx
t
tzyxuz zz
zyzxz
+
+
=
,,,,,,,,,,,,2
2
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where( )
( )[ ]( ) ( ) ( )
( ) ( )
+
+
+=
k
k
ij
k
kij
i
j
j
i
ijx
tzyxuzK
x
tzyxu
x
tzyxu
x
tzyxu
z
zE ,,,,,,
3
,,,,,,
12
( ) ( ) ( )[ ]rTtzyxTzKz ,,, , (z) describes the density of materials of heterostructure. The ten-sor ijdescribes the Kronecker symbol. Accounting relation for ijin the previous system of equa-
tions last system of equation could be written as
( ) ( )
( ) ( )
( )[ ]( )
( ) ( )
( )[ ]
( )+
++
++=
yx
tzyxu
z
zEzK
x
tzyxu
z
zEzK
t
tzyxuz
yxx,,,
13
,,,
16
5,,,2
2
2
2
2
( )( )[ ]
( ) ( )( )
( )( )[ ]
( )
+++
+
++
zx
tzyxu
z
zEzK
z
tzyxu
y
tzyxu
z
zE zzy ,,,
13
,,,,,,
12
2
2
2
2
2
( ) ( ) ( )
( )[ ]
( ) ( )( ) ( )
( )+
+
+=
y
tzyxTzKz
yx
tzyxu
x
tzyxu
z
zE
t
tzyxuz x
yy ,,,,,,,,,
12
,,, 2
2
2
2
2
( )( )[ ]
( ) ( ) ( )( )[ ]
( ) ( )
+
++
+
+
+
+
2
2 ,,,
112
5,,,,,,
12 y
tzyxuzK
z
zE
y
tzyxu
z
tzyxu
z
zE
z
yzy
( ) ( )
( )[ ]
( )( )
( )
yx
tzyxuzK
zy
tzyxu
z
zEzK
yy
+
++
,,,,,,
16
22
(8)
( ) ( ) ( )
( )[ ]( ) ( ) ( ) ( )
+
+
+
+
+=
zy
tzyxu
zx
tzyxu
y
tzyxu
x
tzyxu
z
zE
t
tzyxuz
yxzzz,,,,,,,,,,,,
12
,,,22
2
2
2
2
2
2
( ) ( ) ( ) ( )
( ) ( ) ( )
+
+
+
+
z
tzyxTzzK
z
tzyxu
y
tzyxu
x
tzyxuzK
z
xyx ,,,,,,,,,,,,
( )( )
( ) ( ) ( ) ( )
+
+
z
tzyxu
y
tzyxu
x
tzyxu
z
tzyxu
z
zE
z
zyxz,,,,,,,,,,,,
616
1
.
Systems of conditions for these equations could be written as
( )0
,,,
0
=
=xx
tzyxur
;( )
0,,,
=
= xLxx
tzyxur
;( )
0,,,
0
=
=yy
tzyxur
;( )
0,,,
=
= yLyy
tzyxur
;
( )0
,,,=
= tvzz
tzyxur
;( )
0,,,
=
= zLzz
tzyxur
; ( )0
0,,, 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 accountappropriate initial distributions (see Appendix). In future we replace the required concentrations
in right sides of the obtained equations on their average values 1, which are not yet known. Theequations 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]
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41
( )
=
0 0 011
,,,1 x y zL L L
tvzyx
tdxdydzdtzyxLLL
. (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
( ) =
0 0 01
,,1 x y zL L L
tvC
zyx
C tdxdydzdzyxf
LLL ,
( )
4
3
4
1
2
3
2
4
2
3
14
44 a
Aa
a
aLLLBaB
a
Aa zyxI
+
++
+= ,
( )
=
zyxIII
L L L
tvI
IIV
V LLLStdxdydzdzyxf
S
x y z
0010 0 0100
1,,
1
,
( ) +
+
=
0 0 0
2011
,,1 x y z
II
L L L
tvzyxzyx
II
zyx
I tdxdydzdzyxfLLLLLL
S
LLL
R ,
( ) +
+
=
0 0 0
201
1,,
1 x y z
VV
L L L
tvzyxzyx
VV
zyx
V tdxdydzdzyxfLLLLLL
S
LLL
R .
Relations for calculations parameters Sij, ai,A,B, q,pare presented in the Appendix.
Weusedstandard iterative procedure of method of averaging of function corrections to calculatethe 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), I(x,y,z,t) and V(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 thenot yet known average values of the considered approximations and approximations of the pre-
vious order, i.e. n+n-1(x,y,z,t). These obtained equations and their solutions, which calculatedare presented in the Appendix.
We calculate average values of the second-order approximations of required functions by using
the following standard relation [25-30]
( ) ( )[ ]
=
0 0 0 0122
,,,,,,1 x y zL L L
zyx
tdxdydzdtzyxtzyxLLL
. (10)
Substitution of the second-order approximations of concentrations of dopant into Eq.(10) leads to
the considered average values 2
2C=0, 2I =0, 2V =0,( )
4
3
4
1
2
3
2
4
2
3
24
44 b
Eb
b
bLLLFaF
b
Eb zyxV
+
++
+= ,
( )
00201
11021001200
2
22
2
IVVIV
VIVVzyxVIVVVVVVVI
SS
SSLLLSSSC
+
++
= .
Relations for parameters bi, C, F, r, sare 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 1i. The replacement leads to the following result
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International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.2, May 2016
42
( ) ( )
( ) ( ) ( )
x
tzyxTzzK
t
tzyxuz x
=
,,,,,,2
1
2
,
( ) ( )
( ) ( ) ( )
y
tzyxTzzK
t
tzyxuz
y
=
,,,,,,2
1
2
,
( ) ( ) ( ) ( ) ( )z
tzyxTzzKt
tzyxuz z
=
,,,,,,
2
1
2
.
Integration of the left and right sides of the previous relations on time tgives a possibility to ob-tain the required components to the following result
( ) ( ) ( )
( ) ( ) ( )
( )( )
( ) x
t
x uddzyxTxz
zzKddzyxT
xz
zzKtzyxu
00 00 0
1,,,,,,,,, +
=
,
( ) ( ) ( )
( ) ( ) ( )
( )( )
( ) y
t
y uddzyxTyz
zzKddzyxT
yz
zzKtzyxu
00 00 0
1,,,,,,,,, +
=
,
( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( )
z
t
z uddzyxT
zz
zzKddzyxT
zz
zzKtzyxu
00 00 0
1,,,,,,,,, +
=
.
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
1i+ui(x,y,z,t) [19,26]. Equations for components of the displacement vector are presented in theAppendix. Solutions of these equations are also presented in the Appendix.
Frameworkthispaperallrequiredconcentrations(concentrationsofdopantandradiation 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) withaccount redistribution of radiation defects and their interaction with another defects. If growth
rate is small (v t
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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,
)
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,
)
C(x,y,z,0)
Lz/40 Lz/2 3Lz/4 Lzx0
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 relationD1/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)
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44
0.0
1.0
2.0
C(x,y,z
,)
0 Lz
12
3
Fig. 4. Spatial distributions of concentration of dopant in diffusion-junction rectifier after annealing with
equal continuance. Curve 1 corresponds to 00
0.0
1.0
2.0
C(x,y,z,
)
Lz-Lz 0
12
3
Fig. 5. Spatial distributions of concentration of dopant in implanted-junction rectifier after annealing with
equal continuance. Curve 1 corresponds to 00
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 uzof displacement vector on coordinatezfor epitaxial layers
before radiation processing (curve 1) and after radiation processing (curve 2)
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45
of dopant. We obtain during the analysis, thatp-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
00 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 dopantin 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 00 one canobtain 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 interfacebetween 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 areasof 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 rateof doped areas and mismatch-induced stress in the considered heterostructure on distributions ofconcentrations of dopants.
ACKNOWLEDGEMENTS
This work is supported by the agreement of August 27, 2013 02..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
( ) ( ) ( ) ( ) ( ) ( )++
+
+
=
tzyxf
z
tzyxCD
zy
tzyxCD
yx
tzyxCD
xt
tzyxCC ,,
,,,,,,,,,,,,
( ) ( ) ( ) ( )
+
+
zz L
tvS
SL
tvS
S WdtWyxCtzyxTk
D
yWdtWyxCtzyx
Tk
D
x,,,,,,,,,,,, (1a)
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47
( )( )
( )( )
( )( ) ( )++
+
= tzyxf
y
tzyxITzyxD
yx
tzyxITzyxD
xt
tzyxIIII
,,
,,,,,,
,,,,,,
,,,
( ) ( )
( ) ( ) ( ) ( ) ( )+
+ tzyxITzyxktzyxVtzyxITzyxk
z
tzyxITzyxD
z IIVII
,,,,,,,,,,,,,,,,,,
,,, 2,,
( ) ( ) ( ) ( )
+
+
zz L
tvS
ISL
tvS
IS WdtWyxItzyxTk
D
yWdtWyxItzyx
Tk
D
x,,,,,,,,,,,, (3a)
( )( )
( )( )
( )( ) ( )++
+
= tzyxf
y
tzyxVTzyxD
yx
tzyxVTzyxD
xt
tzyxVVVV
,,
,,,,,,
,,,,,,
,,,
( ) ( )
( ) ( ) ( ) ( ) ( )+
+ tzyxVTzyxktzyxVtzyxITzyxk
z
tzyxVTzyxD
z VVVIV
,,,,,,,,,,,,,,,,,,
,,, 2,,
( ) ( ) ( ) ( )
+
+
zz L
tvS
VSL
tvS
VS WdtWyxVtzyxTk
D
yWdtWyxVtzyx
Tk
D
x,,,,,,,,,,,,
( )( )
( )( )
( )( ) +
+
=
t
y
tzyxTzyxD
yx
tzyxTzyxD
xt
tzyxIII
II
,,,,,,
,,,,,,
,,,
( ) ( ) ( )
( ) ( ) +
+
+
zI
II
L
tvIS
SI WdtWyxtzyxTk
D
xz
tzyxTzyxD
zzyxf ,,,,,,
,,,,,,,,
( ) ( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxkWdtWyxtzyxTk
D
y III
L
tvIS
S z
I ,,,,,,,,,,,,,,,,,, 2,
++
+
( )( )
( )( )
( )( )+
+
=
ty
tzyxTzyxD
yx
tzyxTzyxD
xt
tzyxVVV
VV
,,,,,,
,,,,,,
,,,(5a)
( ) ( ) ( )
( ) ( ) +
+
+
zV
VV
L
tvVS
SV WdtWyxtzyxTk
D
xz
tzyxTzyxD
zzyxf ,,,,,,
,,,,,,,,
( ) ( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxkWdtWyxtzyxTk
D
y VVV
L
tvVS
S z
V ,,,,,,,,,,,,,,,,,, 2,
++
+
.
The first-order approximations of concentrations of dopant and radiation defects could by calcu-
lated by solution of the following equations
( )( ) ( ) +
+
=
tzyx
Tk
Dz
ytzyx
Tk
Dz
xt
tzyxCS
S
CS
S
C,,,,,,
,,,11
1
( ) ( )tzyxfC ,,+ (1b)
( ) ( ) ( ) ( ) ( )+
+
= tzyxftzyx
TkDz
ytzyx
TkD
xz
ttzyxI ISISISISI
,,,,,,,,,,, 111
( ) ( )TzyxkTzyxkVIVIIII
,,,,,,,11,
2
1 (3b)
( )( ) ( ) ( ) ( )+
+
= tzyxftzyx
Tk
Dz
ytzyx
Tk
D
xz
t
tzyxVVS
VS
VS
VS
V
,,,,,,,,
,,,11
1
( ) ( )TzyxkTzyxk VIVIVVV ,,,,,, ,11,2
1
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( )( ) ( ) ( ) ( ) ( ) ( )+++=
tzyxftzyxITzyxktzyxITzyxkt
tzyxIIII
I
,,,,,,,,,,,,,,
,,, 2,
1
( ) ( )
+
+
tzyx
Tk
D
yztzyx
Tk
D
xz S
S
S
S I
I
I
I,,,,,, 11 (5b)
( )( ) ( ) ( ) ( ) ( ) ( )+++=
tzyxftzyxVTzyxktzyxVTzyxkt
tzyxVVVV
V
,,,,,,,,,,,,,,
,,, 2,
1
( ) ( )
+
+
tzyx
Tk
D
yztzyx
Tk
D
xz S
S
S
S V
V
V
V,,,,,, 11
The first-order approximations of the considered concentrations in the following form
( ) ( )( )
( ) ( )
( )
++
+
=
tCS
SC
V
zyxV
V
zyxV
TzyxPzyx
xtzyxC
02*
2
2*1
1
111
,,,,,,1
,,,1,,,,,,
( ) ( ) ( ) ( )( )
++
+
t
SCLS
VzyxV
VzyxVzyx
yd
TkzTzyxD
02*
2
2*111,,,,,,1,,,,,,
( )( )
( )zyxfdTzyxPTk
zTzyxD
C
CS
LS,,
,,,1,,, 1 +
+
(1c)
( ) ( ) ( ) ( )+
+
= zyxfdzyx
Tk
D
yzdzyx
Tk
D
xztzyxI
I
t
S
IS
I
t
S
IS
I,,,,,,,,,,,
01
011
( ) ( )t
VIVI
t
III dTzyxkdTzyxk0
,110
,
2
1,,,,,, (3c)
( ) ( ) ( ) ( )+
+
= zyxfdzyxTk
D
yzdzyxTk
D
xztzyxVV
t
SISV
t
SISV ,,,,,,,,,,,0
110
111
( ) ( )t
VIVI
t
VVV dTzyxkdTzyxk
0,11
0,
2
1,,,,,,
( ) ( ) ( ) +
+
=
t
S
St
S
S
I dzyx
Tk
D
xzdzyx
Tk
D
xztzyx I
I
I
I0
10
11,,,,,,,,,
( ) ( ) ( ) ( ) ( )+++ t
II
t
I dzyxITzyxkdzyxITzyxkzyxf
I0
2
,0
,,,,,,,,,,,,,, (5c)
( ) ( ) ( ) +
+
=
t
S
St
S
S
V dzyxTk
D
xzdzyx
Tk
D
xztzyx V
V
V
V0
10
11,,,,,,,,,
( ) ( ) ( ) ( ) ( )+++ t
VV
t
V dzyxVTzyxkdzyxVTzyxkzyxf
V0
2,
0
,,,,,,,,,,,,,, .
Relations for calculations parameters Sij, ai,A,B, q,pcould be written as
( ) ( ) ( ) ( ) =
0 0 011,
,,,,,,,,,x y zL L L
tv
ji
ij tdxdydzdtzyxVtzyxITzyxktS , ( )= 0000
2
004 VVIIIV SSSa
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49
00IIS ,
0000
2
0000003 VVIIIVIIIV SSSSSa += , ( ) +=
0 0 0
222
002,,2
x y zL L L
tvIzyxIV tdxdydzdzyxfLLLSa
( ) ( ) +
2
0 0 0
2
000 0 0
2
00000000,,,, x
L L L
tvIIV
L L L
tvVIVIVIIVV
LtdxdydzdzyxfStdxdydzdzyxfSSSSx y zx y z
0022
VVzy SLL , ( ) =
0 0 0001 ,,
x y zL L L
tvIIV tdxdydzdzyxfSa , ++++= 3 323 32 qpqqpqB
4
2
6a
a+ , ( )
2
0 0 0000
,,
=
x y zL L L
tvIVV tdxdydzdzyxfSa ,
2
1
4
2
2
4
2
32 84
+= y
a
a
a
aA ,
=
2
4
2
3
24a
aq
2
4
2
1
4
222
3
4
3
2
3
4
2
32
22
4
02
4
31
0854
48
4a
aLLL
a
a
a
aa
a
a
a
aaLLLa
zyxzyx
, =
2
4
402
12
4
a
aap
( ) 244231
2 3623 aaaaaLLL zyx + , ( ) ( ) ( ) =
0 0 01
,,,,,,x y zL L L
tv
i
Ii tdxdydzdtzyxITzyxktR .
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
tzyxCTzyxD
xt
tzyxC CL
( ) ( ) ( )
( )( )[ ]
( )( )
++
++
+
y
tzyxC
TzyxP
tzyxC
V
tzyxV
V
tzyxV
yx
tzyxC C ,,,
,,,
,,,1
,,,,,,1
,,,112
2*
2
2*1
1
( )) ( ) ( )( )
( )[ ]( )
++
++
+
TzyxP
tzyxC
V
tzyxV
V
tzyxV
zTzyxD C
L,,,
,,,1
,,,,,,1,,, 12
2*
2
2*1
( ) ( )
( ) ( )[ ] +
+
+
zL
tv
CS
S
L WdtWyxCtzyx
Tk
D
xz
tzyxCTzyxD ,,,,,,
,,,,,,
22
1
( ) ( )[ ] ( ) ( )tzyxfWdtWyxCtzyxTk
D
y C
L
tvCS
Sz
,,,,,,,,22
+
+
+
(1d)
( )( )
( )( )
( )( )
+
= Tzyxk
y
tzyxITzyxD
yx
tzyxITzyxD
xt
tzyxIVIII
,,,,,,
,,,,,,
,,,,,,
,
112
( )[ ] ( )[ ] ( ) ( )
( )[ ] +
+++
2
11
1
1111,,,
,,,,,,,,,,,, tzyxI
z
tzyxITzyxD
ztzyxVtzyxI
IIVI
( ) ( ) ( )[ ] +
+
+
zL
tvIS
IS
II WdtWyxItzyx
Tk
D
xTzyxk ,,,,,,,,,
12,
( ) ( )[ ]
+
+
zL
tvIS
IS WdtWyxItzyxTk
D
y,,,,,,
12 (3d)
( )( )
( )( )
( )( )
+
= Tzyxk
y
tzyxVTzyxD
yx
tzyxVTzyxD
xt
tzyxVVIVV
,,,,,,
,,,,,,
,,,,,,
,
112
( )[ ] ( )[ ] ( ) ( )
( )[ ] +
+++
2
11
1
1111,,,
,,,,,,,,,,,, tzyxV
z
tzyxVTzyxD
ztzyxVtzyxI
VVVI
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50
( ) ( ) ( )[ ] +
+
+
zL
tvVS
VS
VV WdtWyxVtzyx
Tk
D
xTzyxk ,,,,,,,,,
12,
( ) ( )[ ]
+
+
zL
tvVS
VS WdtWyxVtzyxTk
D
y,,,,,,
12
( ) ( ) ( ) ( ) ( ) ( ) +
+
=
Tzyxky
tzyxTzyxD
yx
tzyxTzyxD
xt
tzyxII
III
II,,,
,,,,,,
,,,,,,
,,,,
112
( ) ( ) ( )[ ] ( ) ( )++
+
+
tzyxfWdtWyxtzyxTk
D
xtzyxI
I
z
I
I
L
tvIS
S ,,,,,,,,,,,
12
2
( ) ( )[ ] ( ) ( ) +
+
+
+
z
tzyxTzyxD
zWdtWyxtzyx
Tk
D
y
IL
tvIS
S
I
z
I
I
,,,,,,,,,,,, 112
( ) ( )tzyxITzyxkI
,,,,,,+ (5d)
( )( )
( )( )
( )( )+
+
=
Tzyxky
tzyxTzyxD
yx
tzyxTzyxD
xt
tzyxVV
VVV
VV,,,
,,,,,,
,,,,,,
,,,,
112
( ) ( ) ( )[ ] ( ) ( )++
+
+
tzyxfWdtWyxtzyxTk
D
xtzyxV V
z
V
V
L
tvVS
S ,,,,,,,,,,, 122
( ) ( )[ ] ( ) ( ) +
+
+
+
z
tzyxTzyxD
zWdtWyxtzyx
Tk
D
y
VL
tvVS
S
V
z
V
V
,,,,,,,,,,,, 112
( ) ( )tzyxVTzyxkV ,,,,,,+ .The second-order approximations of concentrations of dopant and radiation defects could be writ-ten as
( ) ( ) ( )[ ]
( )
( ) ( )
( )
++
++
=
tC
L
V
zyxV
V
zyxV
TzyxP
zyxCTzyxD
xtzyxC
02*
2
2*1
12
2
,,,,,,1
,,,
,,,1,,,,,,
( ) ( ) ( )
( )
( )[ ]
( )
++
++
+
tC
TzyxP
zyxC
V
zyxV
V
zyxV
y
d
x
zyxC
0
12
2*
2
2*1
1
,,,
,,,1
,,,,,,1
,,,
( ) ( )
( ) ( ) ( )
( )
++
+
t
LL
V
zyxV
V
zyxVTzyxD
zd
y
zyxCTzyxD
02*
2
2*1
1 ,,,,,,1,,,,,,
,,,
( ) ( )[ ]( )
( )[ ] +
+
++
t L
vC
SCz
WdWyxCTk
D
xd
TzyxP
zyxC
z
zyxC
012
121 ,,,,,,
,,,1
,,,
( ) ( ) ( )[ ] + +
+
t L
vCS
S
S ddWdWyxCzyx
Tk
D
yddzyx
z
012
,,,,,,,,,
( )zyxfC ,,+ (1e)
( ) ( ) ( )
( ) ( )
++=t
I
t
I d
y
zyxITzyxD
yd
x
zyxITzyxD
xtzyxI
0
1
0
12
,,,,,,
,,,,,,,,,
( ) ( )
( ) ( )[ ] ( )++ ++ zyxfdzyxITzyxkdz
zyxITzyxD
z I
t
III
t
I,,,,,,,,
,,,,,,
0
2
12,0
1
( ) ( )[ ] ( )
+ +
+
t
S
ISt L
vIS
IS zyxTk
D
yddWdWyxIzyx
Tk
D
x
z
0012
,,,,,,,,,
( )[ ] ( ) ( )[ ] ( )[ ] ++ +
t
VIVI
L
vI
dzyxVzyxITzyxkddWdWyxIz
01212,12
,,,,,,,,,,,,
(3e)
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51
( ) ( ) ( )
( ) ( )
++=t
V
t
V dy
zyxVTzyxD
yd
x
zyxVTzyxD
xtzyxV
0
1
0
1
2
,,,,,,
,,,,,,,,,
( ) ( )
( ) ( )[ ] ( )++ ++ zyxfdzyxVTzyxkdz
zyxVTzyxD
z V
t
VVV
t
V ,,,,,,,,,,,
,,,0
2
12,0
1
( ) ( )[ ] ( )
+ +
+
t
SVS
t L
vVS
VS zyxTk
Dy
ddWdWyxVzyxTk
Dx
z
0012 ,,,,,,,,,
( )[ ] ( ) ( )[ ] ( )[ ] ++ +
t
VIVI
L
vI dzyxVzyxITzyxkddWdWyxV
z
01212,12
,,,,,,,,,,,,
( ) ( ) ( )
( ) ( )
+
+
=
tI
tI
I dy
zyxTzyxD
yd
x
zyxTzyxD
xtzyx
II0
1
0
1
2
,,,,,,
,,,,,,,,,
( ) ( )
( ) ( )[ ] +
+
+
t L
vIS
tI
z
IIWdWyxzyx
xd
z
zyxTzyxD
z 012
0
1 ,,,,,,,,,
,,,
( ) ( )[ ] ( )++ +
+
zyxfddWdWyxzyxTk
D
yd
Tk
D
I
z
I
IIt L
vIS
SS,,,,,,,,
012
( ) ( ) ( ) ( )++
t
I
t
II dzyxITzyxkdzyxITzyxk00
2
, ,,,,,,,,,,,, (5e)
( ) ( ) ( )
( ) ( )
+
+
=
tV
tV
V dy
zyxTzyxD
yd
x
zyxTzyxD
xtzyx
VV0
1
0
1
2
,,,,,,
,,,,,,,,,
( ) ( )
( ) ( )[ ] +
+
+
t L
vVS
tV
z
VVWdWyxzyx
xd
z
zyxTzyxD
z 012
0
1 ,,,,,,,,,
,,,
( ) ( )[ ] ( )++ +
+
zyxfddWdWyxzyxTk
D
yd
Tk
D
V
z
V
VVt L
vVS
SS,,,,,,,,
012
( ) ( ) ( ) ( )++t
V
t
VV dzyxVTzyxkdzyxVTzyxk
00
2
,,,,,,,,,,,,, .
Parameters bi, C, F, r, shave been determined by the following relations
00
2
0000
2
004
11IIVV
zyx
VVIV
zyx
SSLLL
SSLLL
b
= , ( ) +
++=
zyx
VVII
zyxIVVVLLL
SSLLLSSb 0000
100132
( ) ( )3333
10
2
00
1001
2
00
011001
0000 22zyx
IVIV
zyxIVVV
zyx
IV
zyxIVIIIV
zyx
VVIV
LLL
SSLLLSS
LLL
SLLLSSS
LLL
SS
++
++++
+ ,
( ) ( ) ( )++
++++
=0110
012
10011102
0000
2 22 IVIIzyxx
IV
IVVVzyxVIVVV
zyx
VVII SSLLLL
SSSLLLCSS
LLL
SSb
( ) ( )
++++++
+
zyx
IV
IVzyxVVIVIIIVzyx
zyx
IV
zy
VV
LLL
SSLLLSSSSLLL
LLL
S
LL
S 200
1001011001
0000 222
( )01
0010
2222
2
00
1102
2IV
zyx
IVIV
zyx
IVI
IVVVV SLLL
SS
LLL
SCSSC
+ , ( +
++=
xIV
zyx
VVVIV
II LS
LLL
CSSSb
10
0211
001
) ( )
++
++++
zyx
IVVVV
zyxIVVV
zyx
IVIIzyx
IVVVzyLLL
SSCLLLSS
LLL
SSLLLSSLL 1102
1001
0110
01012
22
( )zyx
IVIV
IVIVIzyxIIIVIVLLL
SSSSCLLLSSS
+++
2
0110
0100100100223 ,
( )
+= 01
2
0200
000 IV
zyx
VVIV
II S
LLL
SSSb
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52
( ) ( )++
++
0110
1102
0110
1102 22IVIIzyx
zyx
IVVVV
IVIIzyx
zyx
IVVVV SSLLLLLL
SSCSSLLL
LLL
SSC
( )0110
1102
01
2
010122
IVIIzyx
zyx
IVVVV
IVIVIIV SSLLL
LLL
SSCSSCS ++
+ , +
=
zyx
VI
IVILLL
SC 1100
zyx
IV
zyx
IIII
zyx
III
LLLS
LLLSS
LLLS
+
11202000
2
1 ,110200
2
10011 IVVVVVVIVVIV SSSSC += ,4
2
2
4
2
32 48aa
aayE += ,
2
4
2
1
4
222
3
4
3
2
3
4
2
32
22
4
2
0
4
31
02
4
2
3
8544
84
24 b
bLLL
b
b
b
bb
bb
b
bbLLLb
b
br
zyxzyx
= , (
=
402
4
2
412
bbb
s
423118bbbbLLL
zyx ,
3 323 32
4
2
6rsrrsr
a
aF ++++
= .
Equations for components of the displacement vector could be presented in the form
( )
( )
( )
( )
( )[ ]
( )
( )
( )
( )[ ]
( )
+
++
++=
yx
tzyxu
z
zE
zKx
tzyxu
z
zE
zKt
tzyxu
z
yxx,,,
13
,,,
16
5,,, 12
2
1
2
2
2
2
( )( )[ ]
( ) ( )( )
( )( )[ ]
( )
+++
+
++
zx
tzyxu
z
zEzK
z
tzyxu
y
tzyxu
z
zE zzy ,,,
13
,,,,,,
12
1
2
2
1
2
2
1
2
( ) ( ) ( )
x
tzyxTzzK
,,,
( ) ( ) ( )
( )[ ]
( ) ( )( ) ( )
( )+
+
+=
y
tzyxTzzK
yx
tzyxu
x
tzyxu
z
zE
t
tzyxuz x
yy ,,,,,,,,,
12
,,,1
2
2
1
2
2
2
2
( )( )[ ]
( ) ( ) ( ) ( )( )[ ]
( ) +
++
+
+
+
+ zK
z
zE
y
tzyxu
y
tzyxu
z
tzyxu
z
zE
z
yzy
112
5,,,,,,,,,
12 21
2
11
( ) ( )( )[ ]
( ) ( ) ( )yx
tzyxuzK
zy
tzyxu
z
zEzK
yy
+
++
,,,,,,
16
12
12
( ) ( ) ( ) ( ) ( ) ( )
+
+
+
=
zy
tzyxu
zx
tzyxu
y
tzyxu
x
tzyxu
t
tzyxuz
yxzzz,,,,,,,,,,,,,,, 1
2
1
2
2
1
2
2
1
2
2
2
2
( )( )[ ]
( ) ( ) ( ) ( ) ( )
( )[ ]
+
+
+
+
+
+
z
zE
zz
tzyxu
y
tzyxu
x
tzyxuzK
zz
zE xyx
16
,,,,,,,,,
12
111
( ) ( ) ( ) ( )( ) ( )
( )
z
tzyxTzzK
z
tzyxu
y
tzyxu
x
tzyxu
z
tzyxuzyxz
,,,,,,,,,,,,,,,6 1
111 .
Integration of the left and the right sides of the above equations on time tleads to final relationsfor components of displacement vector
( )( )
( ) ( )
( )[ ] ( )
( ) ( )
( )( )[ ]
++
++=
z
zEzK
zddzyxu
xz
zEzK
ztzyxu
t
xx
13
1,,,
16
51,,,
0 012
2
2
( ) ( ) ( )
+
+
t
z
t
y
t
y ddzyxu
zddzyxu
yddzyxu
yx 0 012
2
0 012
2
0 01
2
,,,,,,,,,
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53
( )( ) ( )[ ] ( )
( ) ( ) ( )
( )[ ] ( )
( )( )
++
+
+
z
zzK
z
zEzKddzyxu
zxzzz
zE tz
13,,,
1
12 0 01
2
( )( )
( ) ( )
( )[ ] ( )
( )
++
zddzyxu
xz
zEzK
zddzyxT
x x
t
1,,,
16
51,,,
0 012
2
0 0
( ) ( )( )[ ]
( )( )
( )
+
+
0 012
2
0 01
2
,,,2
1,,,
13
ddzyxuyz
ddzyxuyxz
zEzK
yy
( ) ( )
( ) ( )
( )( )[ ]
( )
++
+
+
0 01
2
0 012
2
,,,131
,,,
ddzyxuzxz
zEzK
z
zEddzyxu
z zz
( ) ( )
( )( )
( )
++
0 00
,,,1
ddzyxT
xz
zzKu
z x
( ) ( )
( ) ( )[ ] ( ) ( ) +
+
+=
t
x
t
xy ddzyxu
yxddzyxu
xzz
zEtzyxu
0 01
2
0 012
2
2,,,,,,
12,,,
( )( ) ( ) ( ) ( ) ( )
( )( )[ ] +
++
+
+ z
zE
zKddzyxuyzddzyxuyxz
zK t
x
t
y
112
5
,,,
1
,,,0 0
12
2
0 01
2
( )( )[ ] ( )
( )( )
( ) ( )
+
+
+
++
t
z
t
y ddzyxuy
ddzyxuzz
zE
zzz
zE
0 01
0 01 ,,,,,,
12
1
112
5
( ) ( )
( ) ( )
( ) ( ) ( )
( )( )[ ]
+
+
z
zEzKddzyxu
zyzddzyxT
z
zzK
t
y
t
16,,,
1,,,
0 01
2
0 0
( )( ) ( )[ ]
( ) ( ) ( ) ( )
( )
+
+
z
zzKddzyxu
yxddzyxu
xzz
zExx
0 01
2
0 012
2
,,,,,,12
( ) ( )
( ) ( )
( ) ( )
0 012
2
0 01
2
0 0
,,,1
,,,,,,
ddzyxuyz
ddzyxuyxz
zKddzyxT
xy
( ) ( )( )[ ]
( )( )
( ) ( )
+
+
++
0 01
0 01
,,,,,,1112
5
ddzyxu
yddzyxu
zz
zE
zz
zEzK
zy
( ) ( ) ( )
( )( )[ ]
( ) yy
uddzyxuzyz
zEzK
zz0
0 01
2
,,,16
1
2
1+
+
( ) ( )
( ) ( )[ ] ( ) ( )
+
+
+=
0 012
2
0 012
2
,,,,,,12
,,,
ddzyxuy
ddzyxuxzz
zEtzyxu
zzz
( ) ( ) ( )
+
+
+
+
0 01
0 01
2
0 01
2
,,,,,,,,,
ddzyxuxz
ddzyxuzy
ddzyxuzx
xyx
( ) ( ) ( )( ) ( )
( )( )
+
+
+
+
zzE
zzzzKddzyxu
zddzyxu
y xx
1611,,,,,,
0 01
0 01
( ) ( ) ( )
0 01
0 01
0 01
,,,,,,,,,6
ddzyxuy
ddzyxux
ddzyxuz
yxz
( ) ( ) ( )
( ) ( )
zz uddzyxTzz
zzKddzyxu
z0
0 00 01
,,,,,, +
.
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54
AUTHORS
Pankratov Evgeny Leonidovichwas 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 Alexeevnawas born at 1991. From 1997 to 2007 she was educated in secondary school ofvillage 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.