![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016
DOI:10.5121/msej.2016.3204 43
USING POROSITY OF EPITAXIAL LAYER TO DECREASE
QUANTITY OF RADIATION DEFECTS GENERATED DURING
RADIATION PROCESSING OF A MULTILAYER STRUCTURE
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 redistribution of radiation defects, which were generated during radiation
processing, in a multilayer structure with porous epitaxial layer. It has been shown, that porosity of epitax-
ial layer gives a possibility to decrease quantity of radiation defects.
KEYWORDS
Multilayer structure; porous epitaxial layer; radiation processing; decreasing of quantity of radiation de-
fects.
1. INTRODUCTION
One of actual questions of solid state electronics is increasing of radiation resistance. Several me-
thods are using to increase the radiation resistance of devices of solid state electronics [1-5]. One
way to decrease quantity of radiation defects, generated during radiation processing of materials,
we present in our paper. Framework the approach we consider a heterostructure, which consist of
a substrate and porous epitaxial layer (see Fig. 1). We assume, that the substrate was under influ-
ence of radiation processing (ion implantation, effects of cosmic radiation et al) through the epi-
taxial layer. Radiation processing of materials leads to generation of radiation defects. In this pa-
per we analyzed influence of porosity of material on distribution of concentration of radiation
defects in the considered material.
x
D(x)
D0
D1
D2
0 a L
C(x,t=0)
P1
P2
x0
Fig. 1. Heterostructure, which includes into itself a substrate and an epitaxial layer. The figure also shows
distribution of concentration of implanted dopant](https://image.slidesharecdn.com/3216msej04-200826091757/85/USING-POROSITY-OF-EPITAXIAL-LAYER-TO-DECREASE-QUANTITY-OF-RADIATION-DEFECTS-GENERATED-DURING-RADIATION-PROCESSING-OF-A-MULTILAYER-STRUCTURE-1-320.jpg)
![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016
44
2. METHOD OF SOLUTION
We solve our aim by analysis of distributions of concentrations of radiation defects in space and
time in the considered heterostructure. We determine the above distributions by solving the fol-
lowing system of equations [6-11]
( ) ( ) ( ) ( ) ( ) +
+
=
y
tzyxI
TzyxD
yx
tzyxI
TzyxD
xt
tzyxI
II
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,
,,,
,,,
,,,
,,,
( ) ( ) ( ) ( ) ( ) ( ) ×−−
+ TzyxktzyxVtzyxITzyxk
z
tzyxI
TzyxD
z
IIVII ,,,,,,,,,,,,
,,,
,,, ,,
∂
∂
∂
∂
( )
( ) ( ) ( )
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+×
z
tzyx
TkV
D
zy
tzyx
TkV
D
yx
tzyx
TkV
D
x
tzyxI
SISISI ,,,,,,,,,
,,, 2222 µµµ
( ) ( ) ( ) ( ) ( ) +
+
=
y
tzyxV
TzyxD
yx
tzyxV
TzyxD
xt
tzyxV
VV
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,,,
,,,
,,,
,,,
,,,
(1)
( ) ( ) ( ) ( ) ( ) ( ) ×−−
+ TzyxktzyxVtzyxITzyxk
z
tzyxV
TzyxD
z
VVVIV ,,,,,,,,,,,,
,,,
,,, ,,
∂
∂
∂
∂
( )
( ) ( ) ( )
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+×
z
tzyx
TkV
D
zy
tzyx
TkV
D
yx
tzyx
TkV
D
x
tzyxV
SVSVSV ,,,,,,,,,
,,, 2222 µµµ
with boundary and initial conditions
( ) 0
,,,
0
=
=x
x
tzyxI
∂
∂
,
( ) 0
,,,
=
= xLx
x
tzyxI
∂
∂
,
( ) 0
,,,
0
=
=y
y
tzyxI
∂
∂
,
( ) 0
,,,
=
= yLy
y
tzyxI
∂
∂
,
( ) 0
,,,
0
=
=z
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
,,,
0
=
=z
z
tzyxV
∂
∂
,
( ) 0
,,,
=
= zLz
z
tzyxV
∂
∂
,
I(x,y,z,0)=fI (x,y,z), V(x,y,z,0)=fV (x,y,z), ( )
++
+=+++
2
1
2
1
2
1
*
111
2
1,,,
zyxTk
VttVztVytVxV nnn
ωl
. (2)
Here I(x,y,z,t) and V(x,y,z,t) are the spatio-temporal distributions of concentrations of interstitials
and vacancies; I*
and V*
are the equilibrium distributions of concentrations of interstitials and va-
cancies; diffusion coefficients of interstitials and vacancies have been described by the following
functions DI(x,y,z,T) and DV(x,y,z,T); terms with squares of concentrations of interstitials and va-
cancies (V2
(x,y,z,t) and I2
(x,y,z,t), respectively) correspond to generation divacancies and analog-
ous complexes of interstitials (see, for example, [10] and appropriate references in this work);
kI,V(x,y,z,T) is the parameter of recombination of point defects; kI,I(x,y,z,T) and kV,V(x,y,z,T) are the
parameters of generation of complexes of point defects and generation; k, V*
, l and a are the
Boltzmann constant, the equilibrium distribution of vacancies, the atomic spacing and the specific
surface energy, respectively. ω =a3
. Framework taking into account porosity we assume, that at
initial stage porous are approximately cylindrical with average dimensions 2
1
2
1 yxr += and z1
[12]. With time small pores decomposing into vacancies. The vacancies are absorbed by large
pores [8]. The large pores takes spherical form during the absorbtion [8]. We determine distribu-](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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USING POROSITY OF EPITAXIAL LAYER TO DECREASE QUANTITY OF RADIATION DEFECTS GENERATED DURING RADIATION PROCESSING OF A MULTILAYER STRUCTURE
- 1. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 DOI:10.5121/msej.2016.3204 43 USING POROSITY OF EPITAXIAL LAYER TO DECREASE QUANTITY OF RADIATION DEFECTS GENERATED DURING RADIATION PROCESSING OF A MULTILAYER STRUCTURE 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 redistribution of radiation defects, which were generated during radiation processing, in a multilayer structure with porous epitaxial layer. It has been shown, that porosity of epitax- ial layer gives a possibility to decrease quantity of radiation defects. KEYWORDS Multilayer structure; porous epitaxial layer; radiation processing; decreasing of quantity of radiation de- fects. 1. INTRODUCTION One of actual questions of solid state electronics is increasing of radiation resistance. Several me- thods are using to increase the radiation resistance of devices of solid state electronics [1-5]. One way to decrease quantity of radiation defects, generated during radiation processing of materials, we present in our paper. Framework the approach we consider a heterostructure, which consist of a substrate and porous epitaxial layer (see Fig. 1). We assume, that the substrate was under influ- ence of radiation processing (ion implantation, effects of cosmic radiation et al) through the epi- taxial layer. Radiation processing of materials leads to generation of radiation defects. In this pa- per we analyzed influence of porosity of material on distribution of concentration of radiation defects in the considered material. x D(x) D0 D1 D2 0 a L C(x,t=0) P1 P2 x0 Fig. 1. Heterostructure, which includes into itself a substrate and an epitaxial layer. The figure also shows distribution of concentration of implanted dopant
- 2. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 44 2. METHOD OF SOLUTION We solve our aim by analysis of distributions of concentrations of radiation defects in space and time in the considered heterostructure. We determine the above distributions by solving the fol- lowing system of equations [6-11] ( ) ( ) ( ) ( ) ( ) + + = y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI II ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( ) ×−− + TzyxktzyxVtzyxITzyxk z tzyxI TzyxD z IIVII ,,,,,,,,,,,, ,,, ,,, ,, ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ +× z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x tzyxI SISISI ,,,,,,,,, ,,, 2222 µµµ ( ) ( ) ( ) ( ) ( ) + + = y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VV ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, (1) ( ) ( ) ( ) ( ) ( ) ( ) ×−− + TzyxktzyxVtzyxITzyxk z tzyxV TzyxD z VVVIV ,,,,,,,,,,,, ,,, ,,, ,, ∂ ∂ ∂ ∂ ( ) ( ) ( ) ( ) ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ +× z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x tzyxV SVSVSV ,,,,,,,,, ,,, 2222 µµµ with boundary and initial conditions ( ) 0 ,,, 0 = =x x tzyxI ∂ ∂ , ( ) 0 ,,, = = xLx x tzyxI ∂ ∂ , ( ) 0 ,,, 0 = =y y tzyxI ∂ ∂ , ( ) 0 ,,, = = yLy y tzyxI ∂ ∂ , ( ) 0 ,,, 0 = =z 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 ,,, 0 = =z z tzyxV ∂ ∂ , ( ) 0 ,,, = = zLz z tzyxV ∂ ∂ , I(x,y,z,0)=fI (x,y,z), V(x,y,z,0)=fV (x,y,z), ( ) ++ +=+++ 2 1 2 1 2 1 * 111 2 1,,, zyxTk VttVztVytVxV nnn ωl . (2) Here I(x,y,z,t) and V(x,y,z,t) are the spatio-temporal distributions of concentrations of interstitials and vacancies; I* and V* are the equilibrium distributions of concentrations of interstitials and va- cancies; diffusion coefficients of interstitials and vacancies have been described by the following functions DI(x,y,z,T) and DV(x,y,z,T); terms with squares of concentrations of interstitials and va- cancies (V2 (x,y,z,t) and I2 (x,y,z,t), respectively) correspond to generation divacancies and analog- ous complexes of interstitials (see, for example, [10] and appropriate references in this work); kI,V(x,y,z,T) is the parameter of recombination of point defects; kI,I(x,y,z,T) and kV,V(x,y,z,T) are the parameters of generation of complexes of point defects and generation; k, V* , l and a are the Boltzmann constant, the equilibrium distribution of vacancies, the atomic spacing and the specific surface energy, respectively. ω =a3 . Framework taking into account porosity we assume, that at initial stage porous are approximately cylindrical with average dimensions 2 1 2 1 yxr += and z1 [12]. With time small pores decomposing into vacancies. The vacancies are absorbed by large pores [8]. The large pores takes spherical form during the absorbtion [8]. We determine distribu-
- 3. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 45 tion of concentration of vacancies, which was formed due to porosity, by summing over all pores, i.e. ( ) ( )∑ ∑ ∑ +++= = = = l i m j n k p tkzjyixVtzyxV 0 0 0 ,,,,,, χβα , 222 zyxR ++= . Averaged distances between centers of pores are equal to α, β and χ in x, y and z directions, re- spectively. Quantities of pores in x, y and z directions are equal to i, j and k. Distributions of concentrations of divacancies ΦV(x,y,z,t) and diinterstitials ΦI (x,y,z,t) in space and time have been calculated by solving the following system of equations [9-11] ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx III II ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( )+++ Φ + Φ tzyxITzyxktzyxITzyxk z tzyx TzyxD z III I I ,,,,,,,,,,,, ,,, ,,, 2 , ∂ ∂ ∂ ∂ ( ) ( ) ( ) ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ΦΦΦ z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SSS III ,,,,,,,,, 222 µµµ (3) ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx VVV VV ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( )+++ Φ + Φ tzyxVTzyxktzyxVTzyxk z tzyx TzyxD z VVV V V ,,,,,,,,,,,, ,,, ,,, 2 , ∂ ∂ ∂ ∂ ( ) ( ) ( ) ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ΦΦΦ z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SSS VVV ,,,,,,,,, 222 µµµ with boundary and initial conditions ( ) 0 ,,, 0 = Φ =x I x tzyx ∂ ∂ , ( ) 0 ,,, = = xLx x tzyxI ∂ ∂ , ( ) 0 ,,, 0 = =y y tzyxI ∂ ∂ , ( ) 0 ,,, = = yLy y tzyxI ∂ ∂ , ( ) 0 ,,, 0 = Φ =z I z tzyx ∂ ∂ , ( ) 0 ,,, = = zLz z tzyxI ∂ ∂ , ( ) 0 ,,, 0 = Φ =x V x tzyx ∂ ∂ , ( ) 0 ,,, = = xLx x tzyxV ∂ ∂ , ( ) 0 ,,, 0 = =y y tzyxV ∂ ∂ , ( ) 0 ,,, = = yLy y tzyxV ∂ ∂ , ( ) 0 ,,, 0 = =z z tzyxV ∂ ∂ , ( ) 0 ,,, = Φ = zLz V z tzyx ∂ ∂ , ΦI(x,y,z,0)=fΦI (x,y,z), ΦV(x,y,z,0)=fΦV(x,y,z). (4) Functions DΦI(x,y,z,T) and DΦV(x,y,z,T) describe spatial and temperature dependences of the dif- fusion coefficients of complexes of radiation defects; kI(x,y,z,T) and kV(x,y,z,T) are the parameters of decay of the above complexes. We calculate distributions of concentrations of radiation defects in space and time by method of averaging of function corrections [13]. To use the approach we write the Eqs. (1) and (3) with account initial distributions of defects, i.e.
- 4. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 46 ( ) ( ) ( ) ( ) ( ) + + = y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI II ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( ) ( )++− + tzyxftzyxVtzyxITzyxk z tzyxI TzyxD z IVII δ ∂ ∂ ∂ ∂ ,,,,,,,,,,, ,,, ,,, , ( ) ( ) ( ) − ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SISISI ,,,,,,,,, 222 µµµ ( ) ( )tzyxITzyxk II ,,,,,, 2 ,− (1a) ( ) ( ) ( ) ( ) ( ) + + = y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VV ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( ) ( )++− + tzyxftzyxVtzyxITxk z tzyxV TzyxD z VVIV δ ∂ ∂ ∂ ∂ ,,,,,,,,, ,,, ,,, , ( ) ( ) ( ) − ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SVSVSV ,,,,,,,,, 222 µµµ ( ) ( )tzyxVTzyxk VV ,,,,,, 2 ,− ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx I I I I I ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( )+++ Φ + Φ tzyxITzyxktzyxITzyxk z tzyx TzyxD z III I I ,,,,,,,,,,,, ,,, ,,, 2 , ∂ ∂ ∂ ∂ ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ΦΦΦ z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SSS VVV ,,,,,,,,, 222 µµµ ( ) ( )tzyxf I δ,,Φ+ (3a) ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx V V V V V ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, ( ) ( ) ( ) ( ) ( ) ( )+++ Φ + Φ tzyxVTzyxktzyxVTzyxk z tzyx TzyxD z VVV V V ,,,,,,,,,,,, ,,, ,,, 2 , ∂ ∂ ∂ ∂ ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ΦΦΦ z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SSS VVV ,,,,,,,,, 222 µµµ ( ) ( )tzyxf I δ,,Φ+ . Now we use not yet known average values α1ρ of the required concentrations in right sides of the Eqs. (1a) and (3a) instead of the concentrations. The replacement gives us possibility to obtain following equations for determination the first-order approximations of concentrations of radia- tion defects in the following form ( ) ( ) ( ) ( ) ( )++ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = tzyxf y tzyx TkV D yx tzyx TkV D xt tzyxI I SISI δ µµ ∂ ∂ ,, ,,,,,,,,, 221
- 5. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 47 ( ) ( ) ( )TzyxkTzyxk z tzyx TkV D z VIVIIII SI ,,,,,, ,,, ,11, 2 1 2 ααα µ −− ∂ ∂ ∂ ∂ + (1b) ( ) ( ) ( ) ( ) ( )++ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = tzyxf y tzyx TkV D yx tzyx TkV D xt tzyxV V SVSV δ µµ ∂ ∂ ,, ,,,,,,,,, 221 ( ) ( ) ( )TzyxkTzyxk z tzyx TkV D z VIVIVVV SV ,,,,,, ,,, ,11, 2 1 2 ααα µ −− ∂ ∂ ∂ ∂ + ( ) ( ) ( ) ( ) ( )++ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = Φ Φ ΦΦ tzyxf y tzyx TkV D yx tzyx TkV D xt tzyx I II SSI δ µµ ∂ ∂ ,, ,,,,,,,,, 221 ( ) ( ) ( ) ( ) ( )tzyxITzyxktzyxITzyxk z tzyx TkV D z III SI ,,,,,,,,,,,, ,,, 2 , 2 ++ ∂ ∂ ∂ ∂ + Φ µ (3b) ( ) ( ) ( ) ( ) ( )++ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = Φ Φ ΦΦ tzyxf y tzyx TkV D yx tzyx TkV D xt tzyx V VV SSV δ µµ ∂ ∂ ,, ,,,,,,,,, 221 ( ) ( ) ( ) ( ) ( )tzyxVTzyxktzyxVTzyxk z tzyx TkV D z VVV SV ,,,,,,,,,,,, ,,, 2 , 2 ++ ∂ ∂ ∂ ∂ + Φ µ . Integration of the both sides of Eqs. (1b) and (3b) on time gives a possibility to obtain first-order approximations of concentrations of radiation defects in the final form ( ) ( ) ( ) ( )++∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ = zyxfd y tzyx TkV D y d x tzyx TkV D x tzyxI I t SI t SI ,, ,,,,,, ,,, 0 2 0 2 1 τ µ τ µ ( ) ( ) ( )∫−∫−∫ ∂ ∂ ∂ ∂ + t VIVI t III t SI dTzyxkdTzyxkd z tzyx TkV D z 0 ,11 0 , 2 1 0 2 ,,,,,, ,,, ταατατ µ (1c) ( ) ( ) ( ) ( )++∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ = zyxfd y tzyx TkV D y d x tzyx TkV D x tzyxV V t SV t SV ,, ,,,,,, ,,, 0 2 0 2 1 τ µ τ µ ( ) ( ) ( )∫−∫−∫ ∂ ∂ ∂ ∂ + t VIVI t VVV t SV dTzyxkdTzyxkd z tzyx TkV D z 0 ,11 0 , 2 1 0 2 ,,,,,, ,,, ταατατ µ ( ) ( ) ( ) ( ) ( ) ( ) +∫+∫+=Φ Φ t II t II dzyxITzyxkdzyxITzyxkzyxftzyx I 0 2 , 0 1 ,,,,,,,,,,,,,,,,, ττττ ( ) ( ) ( ) ∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ + ΦΦΦ t S t S t S d z zyx TkV D z d y zyx TkV D y d x zyx TkV D x III 0 2 0 2 0 2 ,,,,,,,,, τ τµ τ τµ τ τµ ( ) ( ) ( ) ∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ + ΦΦΦ t S t S t S d z zyx TkV D z d y zyx TkV D y d x zyx TkV D x III 0 2 0 2 0 2 ,,,,,,,,, τ τµ τ τµ τ τµ (3c) ( ) ( ) ( ) ( ) ( ) ( ) +∫+∫+=Φ Φ t VV t VV dzyxVTzyxkdzyxVTzyxkzyxftzyx V 0 2 , 0 1 ,,,,,,,,,,,,,,,,, ττττ ( ) ( ) ( ) ∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ + ΦΦΦ t S t S t S d z zyx TkV D z d y zyx TkV D y d x zyx TkV D x VVV 0 2 0 2 0 2 ,,,,,,,,, τ τµ τ τµ τ τµ . We calculate average values of the required approximations by the following relation [13]
- 6. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 48 ( )∫ ∫ ∫ ∫ Θ = Θ 0 0 0 0 11 ,,, 1 x y zL L L zyx tdxdydzdtzyx LLL ρα ρ . (5) Substitution of the relations (1c) and (3c) into the relation (5) gives a possibility to calculate the required average values in the following form ( ) 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 , ( ) ×∫ ∫ ∫= x y zL L L IV xdydzdzyxf 0 0 0 1 ,,α 00 001 1 1 IV zyxIII I S LLLS Θ−− Θ × α α , ( ) +∫ ∫ ∫+ Θ = ΦΦ x y z II L L L zyxzyx I xdydzdzyxf LLLLLL R 0 0 0 1 1 ,, 1 α zyx II LLL S Θ + 20 , ( ) zyx VVV L L L zyx LLL SR xdydzdzyxf LLL x y z VV Θ + +∫ ∫ ∫= ΦΦ 201 0 0 0 1 ,, 1 α . Here ( ) ( ) ( ) ( )∫ ∫ ∫ ∫−Θ= Θ 0 0 0 0 11, ,,,,,,,,, x y zL L L ji ij tdxdydzdtzyxVtzyxITzyxktS ρρρρ , ( )×−= 0000 2 004 VVIIIV SSSa 00IIS× , 0000 2 0000003 VVIIIVIIIV SSSSSa −+= , ( ) ×+∫ ∫ ∫= 00 0 0 0 2 00002 2,, VV L L L VIVIV SxdydzdzyxfSSa x y z ( ) ( ) ×−Θ+∫ ∫ ∫−∫ ∫ ∫× 222222 00 0 0 0 2 00 0 0 0 00 ,,,, zyxzyxIV L L L IIV L L L III LLLLLLSxdydzdzyxfSxdydzdzyxfS x y zx y z 00VVSΘ× , ( )∫ ∫ ∫= x y zL L L IIV xdydzdzyxfSa 0 0 0 001 ,, , ( ) 2 0 0 0 000 ,, ∫ ∫ ∫= x y zL L L IVV xdydzdzyxfSa , × 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−× , 4 2 2 4 2 32 48 a a a a yA Θ−Θ+= , ( ) ( ) ( )∫ ∫ ∫ ∫−Θ= Θ 0 0 0 0 1 ,,,,,, x y zL L L i Ii tdxdydzdtzyxITzyxktRρ , 2 4 31 4 2 2 4 402 121812 4 a aa LLL a a a aa p zyx Θ − Θ −Θ= , 3 323 32 4 2 6 qpqqpq a a B ++−−++ Θ = . Approximations of concentrations of radiations defects with the second and higher orders have been calculated framework standard iterative procedure of method of averaging of function cor- rections [13]. Framework the procedure we determine the approximation of the n-th order by re- placement of the concentrations of radiation defects I (x,y,z, t), V (x,y,z,t), ΦI (x,y,z,t) and ΦV (x,y,z,t) in right sides of the Eqs.(1b) and (3b) on the following sums αnρ+ρ n-1(x,y,z,t). The re- placement gives a possibility to obtain the second-order approximations of concentrations of radi- ation defects ( ) ( ) ( ) ( ) ( ) + + = y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI II ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, 112 ( ) ( ) ( ) ( )[ ] ( )[ ]+++− + tzyxVtzyxITzyxk z tzyxI TzyxD z VIVII ,,,,,,,,, ,,, ,,, 1111, 1 αα ∂ ∂ ∂ ∂ ( ) ( ) ( ) −∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ + t SI t SI t SI d z tzyx TkV D z d y tzyx TkV D y d x tzyx TkV D x 0 2 0 2 0 2 ,,,,,,,,, τ µ τ µ τ µ ( ) ( )[ ]2 11, ,,,,,, tzyxITzyxk III +− α (1d)
- 7. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 49 ( ) ( ) ( ) ( ) ( ) + + = y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VV ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, 112 ( ) ( ) ( ) ( )[ ] ( )[ ]+++− + tzyxVtzyxITzyxk z tzyxV TzyxD z VIVIV ,,,,,,,,, ,,, ,,, 1111, 1 αα ∂ ∂ ∂ ∂ ( ) ( ) ( ) −∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ + t SI t SI t SI d z tzyx TkV D z d y tzyx TkV D y d x tzyx TkV D x 0 2 0 2 0 2 ,,,,,,,,, τ µ τ µ τ µ ( ) ( )[ ]2 11, ,,,,,, tzyxVTzyxk IVV +− α ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx III II ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, 112 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×+++ Φ + ΦΦ TzyxktzyxftzyxITzyxk z tzyx TzyxD z III I II ,,,,,,,,,,, ,,, ,,, 2 , 1 δ ∂ ∂ ∂ ∂ ( ) ( ) ( ) ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ΦΦΦ z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SSS III ,,,,,,,,, 222 µµµ (3d) ( ) ( ) ( ) ( ) ( ) + Φ + Φ = Φ ΦΦ y tzyx TzyxD yx tzyx TzyxD xt tzyx VVV VV ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,,, ,,, ,,, ,,, ,,, 112 ( ) ( ) ( ) ( ) ( ) ( )+++ Φ + Φ tzyxVTzyxktzyxVTzyxk z tzyx TzyxD z VVV I V ,,,,,,,,,,,, ,,, ,,, 2 , 1 ∂ ∂ ∂ ∂ ( ) ( ) ( ) ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ΦΦΦ z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SSS VVV ,,,,,,,,, 222 µµµ . Integration of the both sides of Eqs. (1d) and (3d) gives a possibility to obtain relations for the second-order approximations of the required concentrations of radiation defects in the following form ( ) ( ) ( ) ( ) ( ) +∫+∫= t I t I d y zyxI TzyxD y d x zyxI TzyxD x tzyxI 0 1 0 1 2 ,,, ,,, ,,, ,,,,,, τ ∂ τ∂ ∂ ∂ τ ∂ τ∂ ∂ ∂ ( ) ( ) ( ) ( )[ ] −∫ +−∫+ t III t I dzyxITzyxkd z zyxI TzyxD z 0 2 12, 0 1 ,,,,,, ,,, ,,, ττατ ∂ τ∂ ∂ ∂ ( ) ( )[ ] ( )[ ] ( )++∫ ++− zyxfdzyxVzyxITzyxk I t VIVI ,,,,,,,,,,, 0 1212, ττατα ( ) ( ) ( ) + + + z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SISISI ∂ µ∂ ∂ ∂ ∂ µ∂ ∂ ∂ ∂ µ∂ ∂ ∂ ,,,,,,,,, 222 ( ) ( ) ( ) + + + z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SISISI ∂ µ∂ ∂ ∂ ∂ µ∂ ∂ ∂ ∂ µ∂ ∂ ∂ ,,,,,,,,, 222 (1e) ( ) ( ) ( ) ( ) ( ) +∫+∫= t V t V d y zyxV TzyxD y d x zyxV TzyxD x tzyxV 0 1 0 1 2 ,,, ,,, ,,, ,,,,,, τ ∂ τ∂ ∂ ∂ τ ∂ τ∂ ∂ ∂ ( ) ( ) ( ) ( )[ ] −∫ +−∫+ t VVV t V dzyxVTzyxkd z zyxV TzyxD z 0 2 12, 0 1 ,,,,,, ,,, ,,, ττατ ∂ τ∂ ∂ ∂ ( ) ( )[ ] ( )[ ] ( )++∫ ++− zyxfdzyxVzyxITzyxk V t VVIVI ,,,,,,,,,,, 0 121, ττατα
- 8. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 50 ( ) ( ) ( ) + + + z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SVSVSV ∂ µ∂ ∂ ∂ ∂ µ∂ ∂ ∂ ∂ µ∂ ∂ ∂ ,,,,,,,,, 222 ( ) ( ) ( ) ( ) ( ) +∫ 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 + ΦΦ zyxfdzyxITzyxkd z zyx TzyxD z II t II t I ,,,,,,,, ,,, ,,, 0 2 , 0 1 τττ ∂ τ∂ ∂ ∂ ( ) ( ) ( ) +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ + ΦΦΦ t S t S t S d z zyx TkV D z d y zyx TkV D y d x zyx TkV D x III 0 2 0 2 0 2 ,,,,,,,,, τ τµ τ τµ τ τµ ( ) ( )∫+ t I dzyxITzyxk 0 ,,,,,, ττ (3e) ( ) ( ) ( ) ( ) ( )+∫++∫ Φ =Φ ΦΦΦ tt V V TzyxD y zyxfd x zyx TzyxD x tzyx VVV 00 1 2 ,,,,, ,,, ,,,,,, ∂ ∂ τ ∂ τ∂ ∂ ∂ ( ) ( ) ( ) ( ) ( ) +∫++∫ Φ + ΦΦ t V t V dzyxVTzyxkzyxfd z zyx TzyxD z VV 00 1 ,,,,,,,, ,,, ,,, τττ ∂ τ∂ ∂ ∂ ( ) ( ) ( ) +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ +∫ ∂ ∂ ∂ ∂ + ΦΦΦ t S t S t S d z zyx TkV D z d y zyx TkV D y d x zyx TkV D x VVV 0 2 0 2 0 2 ,,,,,,,,, τ τµ τ τµ τ τµ ( ) ( )∫+ t VV dzyxVTzyxk 0 2 , ,,,,,, ττ . We determine average values of the second-order approximations by the standard relation [13] ( ) ( )[ ]∫ ∫ ∫ ∫ − Θ = Θ 0 0 0 0 122 ,,,,,, 1 x y zL L L zyx tdxdydzdtzyxtzyx LLL ρρα ρ . (6) Substitution of the relations (1e) and (3e) in the relation (6) gives a possibility to obtain relations for the required 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++−− = , where 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 2 00 1001011001 0000 22 zyx IVIV zyx IV zyxIVVVzyxIVIIIV zyx VVIV LLL SS LLL S LLLSSLLLSSS LLL SS Θ − Θ Θ+++Θ+++ Θ + , ( ) ( ) ( )×++Θ++−Θ−++ Θ = 0110 2 10011102 0000 2 22 IVIIzyxIVVVzyxVIVVV zyx VVII SSLLLSSLLLCSS LLL SS b ( )( ) ×−++Θ+++Θ Θ + Θ × 2 001001011001 000001 222 IVIVVVzyxIVIIIVzyx zyx IV zyx VVIV SSSLLLSSSLLL LLL S LLL SS zyx IVIV IV zyx IVI zyx IVVVV LLL SS S LLL SC LLL SSC Θ − Θ + Θ −− × 0100 102222 2 001102 2 , ( )×Θ++= zyxIVVVII LLLSSSb 1001001 2
- 9. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 51 ( ) × Θ −Θ++ Θ ++Θ + Θ ++ × zyx IV zyxIVVV zyx IVIIzyx IV zyx VVVIV LLL S LLLSS LLL SSLLL S LLL CSS 00 1001 0110 01 0211 2 2 ( )( ) zyx IVIV IVIVIIVVVVzyxIIIV LLL SS SSCSSCLLLSS Θ −+−−Θ++× 2 0110 010011021001 223 , × Θ = zyx II LLL S b 00 0 ( ) ( ) × Θ −− −++Θ Θ −− −+× 01 1102 0110 1102 01 2 0200 2 IV zyx IVVVV IVIIzyx zyx IVVVV IVVVIV S LLL SSC SSLLL LLL SSC SSS ( ) 2 010110 22 IVIIVIIzyx SCSSLLL +++Θ× , zyx IV zyx II zyx III IV zyx VI I LLL S LLL S LLL S S LLL C Θ − Θ − Θ + Θ = 112000 2 1 00 11 ααα , 110200 2 10011 IVVVVVVIVVIV SSSSC −−+= ααα , 3 323 32 4 2 6 rsrrsr a a F ++−−++ Θ = , × Θ = 2 4 2 3 24b b r 2 4 2 1 4 222 3 4 3 2 3 4 2 32 22 4 2 0 4 31 0 854 4 8 4 b b LLL b b b b b b b b bb LLLb zyxzyx Θ − Θ − Θ−Θ Θ − Θ−× , 4 2 2 4 2 32 48 a a a a yE Θ−Θ+= , ( ) 4 2 31402 4 2 18 4 12 b b bbLLLbb b s zyx Θ −Θ− Θ = . Framework this paper the required spatio-temporal distributions of concentrations of radiations defects have been determined by using the second-order approximations by using method of av- eraging of function corrections. The approximations give enough good qualitative and some quantitative results. We check all analytical results by comparison with numerical one. 3. DISCUSSION In the previous section we analytically take into account porosity of materials in comparison with cited similar works. In this situation we obtain decreasing quantity of radiation defects (one can find decreasing both types of accounted defects: point defects and their simplest complexes) in comparison with nonporous materials. Probably this effect could be obtained due to drain of these defects to pores. x 0.0 0.5 1.0 1.5 ρ(x,Θ) 0 L/4 L/2 3L/4 L 1 2 Fig. 2. Distributions of concentrations of point radiation defects for fixed value of annealing time. Curve 1 corresponds to implantation of ions of dopant through nonporous epitaxial layer. Curve 2 corresponds to implantation of ions of dopant through porous epitaxial layer Typical distributions of concentrations of point radiation defects and their simplest complexes are presented on Figs. 2 and 3, respectively. In this situation using overlayer over device area gives a
- 10. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 52 possibility to increase radiation resistance of the devices during radiation processing. Using por- ous overlayer gives a possibility to obtain larger increasing of radiation resistance. The Figs. 2 and 3 also shows, that quantity of point defects is larger, than quantity of simplest complexes of point defects. This effect could be find because only part of point defects could generate their complexes. It should be also noted, that we analyzed relaxation of concentrations of radiation de- fects analytically and in more common case in comparison with similar results in literature. The figures show, that porosity of materials of epitaxial layer gives a possibility to decrease quantity of radiation defects. In this situation porous overlayer over device area gives a possibility to in- crease safety during radiation processing. Analogous situation could be find during using nonpor- ous overlayer over device area. However pores probably became drains of radiation defects. In this situation porosity of overlayer leads to higher safety of device area. Fig. 3. Distributions of concentrations of simplest complexes of point radiation defects for fixed value of annealing time. Curve 1 corresponds to implantation of ions of dopant through nonporous epitaxial layer. Curve 2 corresponds to implantation of ions of dopant through porous epitaxial layer 4. CONCLUSIONS In the present paper we analyzed redistributions of radiations defects in material with porous and nonporous overlayer after radiation processing. We show, that presents of porous overlayer gives a possibility to decrease quantity of radiation defects. 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.
- 11. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 3, No. 2, June 2016 53 REFERENCES [1] A.A. Lebedev, A.M. Ivanov, N.B. Strokan. Radiation hardness of SiC and nuclear radiation detectors based on the SiC films. Semiconductors. Vol. 38 (2). P. 129-150 (2004). [2] E.V. Kalinina, V.G. Kossov, R.R. Yafaev, A.M. Strelchuk, G.N. Violina. A high-temperature radiation- resistant rectifier based on p+ -n junctions in 4H-SiC ion-implanted with aluminum. Semiconductors. Vol. 44 (6). P. 778-788 (2010). [3] A.E. Belyaev, N.S. Boltovets, A.V. Bobyl', V.N. Ivanov, L.M. Kapitanchuk, V.P. Klad'ko, R.V. Kona- kova, Ya.Ya. Kudrik, A.A. Korchevoy, O.S. Litvinin, V.V. Milenin, S.V. Novitskiy, V.N. Sheremet. Radiation effects and interphase interactions in ohmic and barrier contacts to indium phosphide as in- duced by rapid thermal annealing and irradiation with γ-ray 60 Co photons. Semiconductors. Vol. 44 (12). P. 1559-1566 (2010). [4] G.P. Gaydar. On the kinetics of electron processes in 60 Co γ-irradiated n-Ge single crystals. Semicon- ductors. Vol. 48 (9). P. 1141-1144 (2014). [5] P.A. Alexandrov, N.E. Belova, K.D. Demakov, S.G. Shemardov. On the generation of charge-carrier recombination centers in the sapphire substrates of silicon-on-sapphire structures. Semiconductors. Vol. 49 (8). P. 1099-1103 (2015). [6] 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). [7] M. Kitayama, T. Narushima, W.C. Carter, R.M. Cannon, A.M. Glaeser. The Wulff shape of alumina: I. Modeling the kinetics of morphological evolution. J. Am. Ceram. Soc. Vol. 83. P. 2561 (2000); M. Ki- tayama, T. Narushima, A.M. Glaeser. The Wulff shape of Alumina: II, experimental measurements of pore shape evolution rates. J. Am. Ceram. Soc. Vol. 83. P. 2572 (2000). [8] P.G. Cheremskoy, V.V. Slesov, V.I. Betekhtin. Pore in solid bodies (Energoatomizdat, Moscow, 1990, in Russian). [9] 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). [10] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev, 1979, in Russian). [11] E.L. Pankratov. Application of porous layers and optimization of annealing of dopant and radiation defects to increase sharpness of p-n-junctions in a bipolar heterotransistors. Journal of Nanoelectronics and Optoelectronics. Vol. 6 (2). P. 188-206 (2011). [12] M.G. Mynbaeva, E.N. Mokhov, A.A. Lavrent'ev, K.D. Mynbaev. About hightemperature diffusion doping of porous SiC. Techn. Phys. Lett. Vol. 34 (17). P. 13 (2008). [13] Yu.D. Sokolov. About the definition of dynamic forces in the mine lifting. Applied Mechanics. Vol.1 (1). P. 23-35 (1955). AUTHORS: Pankratov Evgeny Leonidovich was born at 1977. From 1985 to 1995 he was educated in a secondary school in Nizhny Novgorod. From 1995 to 2004 he was educated in Nizhny Novgorod State University: from 1995 to 1999 it was bachelor course in Radiophysics, from 1999 to 2001 it was master course in Ra- diophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in Radio- physics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of Micro- structures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny Novgo- rod State University of Architecture and Civil Engineering. 2012-2015 Full Doctor course in Radiophysical Department of Nizhny Novgorod State University. Since 2015 E.L. Pankratov is an Associate Professor of Nizhny Novgorod State University. He has 155 published papers in area of his researches. Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school of village Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school “Center for gifted children”. From 2009 she is a student of Nizhny Novgorod State University of Architec- ture and Civil Engineering (spatiality “Assessment and management of real estate”). At the same time she is a student of courses “Translator in the field of professional communication” and “Design (interior art)” in the University. Since 2014 E.A. Bulaeva is in a PhD program in Radiophysical Department of Nizhny Novgorod State University. She has 103 published papers in area of her researches.