On Decreasing of Mismatch-Induced Stress During Growth of Films During Magnetron Sputtering
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The paper discusses the analysis of mass transfer during the growth of epitaxial layers through magnetron sputtering, focusing on methods to reduce mismatch-induced stress in the resulting films. An analytical framework is introduced, allowing for the consideration of nonlinear processes and variation over time and space. The study aims to improve the properties of heterostructures utilized in solid-state electronics.
![International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024
DOI:10.5121/ijcax.2024.11402 25
ON DECREASING OF MISMATCH-INDUCED STRESS
DURING GROWTH OF FILMS DURING
MAGNETRON SPUTTERING
E.L. Pankratov
Department of applied mechanics, physics and higher mathematics, Nizhny Novgorod
State Agrotechnical University, 97 Gagarin avenue, Nizhny Novgorod
ABSTRACT
In this paper we analyzed mass transfer during the growth of epitaxial layers during magnetron sputtering.
During the analysis we obtained modifications of properties of films with variation of several parameters.
We analyzed possibility to decrease mismatch -induced stress on the considered films during their growth.
An analytical approach for analyzing mass transfer was introduced, which makes it possible to take into
account the nonlinearity of processes, as well as changes in parameters in space and time.
KEYWORDS
Mass transfer; magnetrons sputtering; analytical approach for modelling; mismatch-induced stress.
1. INTRODUCTION
Development of solid-state electronics and widespread using of heterostructures for
manufacturing of electronic devices leads to the necessity to improve properties of layers of these
heterostructures. Different methods are used to manufacture of heterostructures: molecular beam
epitaxy, epitaxy from the gas phase, magnetron sputtering. A large number of experimental
works have been devoted to the manufacturing and using of heterostructures due to their
widespread using [1-12]. At the same time, a relatively small number of works are devoted to
predicting the growth of heterostructures [11,12].
In this paper in development of references [13-16] we analyzed processes framework growing
films by magnetron sputtering. Structure of the considered magnetron is shown in Fig. 1. In the
framework of the structure electrons are emitting from the cathode (line 1). After that under the
action of a field between the cathode (line 1) and the anode (line 3) an electron flow is formed
between lines 1 and 2. The main purpose of this paper is analysis of mass transfer in magnetrons
during growth of films in order to improve their properties. An additional aim of this paper is
analysis of possibility to decrease mismatch-induced stress during growth of films. To solve this
aim we consider an analytical approach for analysis of mass transfer, which makes a possibility
to take into account the nonlinearity of mass transfer, as well as the changing of its parameters in
space and time.](https://image.slidesharecdn.com/11424ijcax02-241128170357-fbfebb50/85/On-Decreasing-of-Mismatch-Induced-Stress-During-Growth-of-Films-During-Magnetron-Sputtering-1-320.jpg)
![International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024
26
Fig. 1. Structure of magnetron.
In the framework of the structure ions have been emitted from cathode (line 1). After that under
influence of field between cathode (line 1) and anode (line 3) one can obtained flow of ions
between lines 1 and 2
2. METHOD OF SOLUTION
To solve our aims we determine spatio-temporal distribution of electromagnetic field. We
determine the distribution by solving the following boundary problem [17, 18]
t
t
z
r
D
j
t
z
r
H
rot
,
,
,
,
,
,
,
t
t
z
r
B
t
z
r
E
rot
,
,
,
,
,
,
,
t
z
r
D
div ,
,
,
, 0
,
,
,
t
z
r
B
div
, E
D
0
, H
B
0
. (1)
Here and are the dielectric and magnetic constants; 0=0.88610-11
F/m; 0= 1.25610-6
H/m;
E
and H
are the electric and magnetic strengths; D
and B
are the inductions of electric and
magnetic fields; r, and z are the spatial coordinates; t is the current time. Boundary and initial
conditions for the system of equations could be written as
0
,
,
,
t
z
R
Bz
, 0
,
,
,
t
z
R
Dr
, 0
,
,
,
t
z
R
Ez
, 0
,
,
,
t
z
R
Hr
,
0
0
,
,
, E
z
r
E
, 0
0
,
,
, H
z
r
H
.
Current density j
is proportional to speed of ions v
: v
C
j
, where C is the density of ions.
The speed of ions correlated with strengths of electric and magnetic field by the second Newton
law](https://image.slidesharecdn.com/11424ijcax02-241128170357-fbfebb50/85/On-Decreasing-of-Mismatch-Induced-Stress-During-Growth-of-Films-During-Magnetron-Sputtering-2-320.jpg)
![International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024
27
B
v
E
e
t
d
v
d
m
. (3)
Scalar form of the Eqs.(1) could be written as
z
t
z
r
H
t
z
r
H
r
t
t
z
r
D
v
C z
r
r
,
,
,
,
,
,
1
,
,
,
,
z
t
z
r
H
t
z
r
H
r
t
t
z
r
D
v
C r
z
,
,
,
,
,
,
1
,
,
,
,
t
z
r
H
r
t
z
r
H
r
t
t
z
r
D
r
v
C r
z
z
,
,
,
,
,
,
,
,
,
,
t
t
z
r
B
t
z
r
E
r
z
t
z
r
E r
z
,
,
,
,
,
,
1
,
,
,
,
t
t
z
r
B
t
z
r
E
r
z
t
z
r
E z
r
,
,
,
,
,
,
1
,
,
,
,
t
t
z
r
B
r
t
z
r
E
r
t
z
r
E
r z
r
,
,
,
,
,
,
,
,
,
,
z
t
z
r
D
t
z
r
D
r
r
t
z
r
D
r
r
z
r
,
,
,
,
,
,
1
,
,
,
1
,
0
,
,
,
,
,
,
1
,
,
,
1
z
t
z
r
B
t
z
r
B
r
r
t
z
r
B
r
r
z
r
,
z
r
r
B
v
E
e
t
d
v
d
m
,
z
r
B
v
E
e
t
d
v
d
m
,
B
v
E
e
t
d
v
d
m r
z
z
.
To solve these equations we use the method of averaging functional corrections [19]. To
determine the first-order approximations of the required functions we replace them in the right-
hand sides of equations (4) by their not yet known average values 1s in the right-hand side of
these equations. As a result of this substitution we obtain the following equations to determine
the first-order approximations of components of strength and induction of the considered fields
0
,
,
,
1
t
t
z
r
D
v
C r
r
,
0
,
,
,
1
t
t
z
r
D
v
C
,
H
z
z
t
t
z
r
D
v
C 1
1 ,
,
,
,
0
,
,
,
1
t
t
z
r
B r
,
0
,
,
,
1
t
t
z
r
B
,
E
z
t
t
z
r
B
1
1 ,
,
,
,
C
r
t
z
r
D
r
r
r
,
,
,
1 1
,
0
,
,
,
1 1
r
t
z
r
B
r
r
r
,](https://image.slidesharecdn.com/11424ijcax02-241128170357-fbfebb50/85/On-Decreasing-of-Mismatch-Induced-Stress-During-Growth-of-Films-During-Magnetron-Sputtering-3-320.jpg)
![International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024
28
z
r B
v
E
r
e
t
d
v
d
m 1
1
1
1
,
z
r B
v
E
e
t
d
v
d
m 1
1
1
1
,
B
v
E
z
r
z
e
t
d
v
d
m 1
1
1
1
.
Further after integration of left and right sides of these equations on considered variables we
obtain the first-order approximations of the considered fields and velocity of ions in the following
form
t
r
r d
v
C
t
z
r
D
0
1
1 ,
,
,
,
t
d
v
C
t
z
r
D
0
1
1 ,
,
,
,
0
0
0
1
1
1 ,
,
, E
d
v
C
t
t
z
r
D
t
z
H
z
, t
e
v
m z
r B
v
E
r 1
1
1
1
,
0
,
,
,
1
t
z
r
B r
, 0
0
1 ,
,
, H
t
z
r
B
, t
t
z
r
B E
z
1
1 ,
,
, ,
0
1
1
1
1
v
m
t
e
v
m z
r B
v
E
, 0
1
1
1
1 z
B
v
E
z v
m
t
e
v
m r
z
.
Calculation of average values 1s by using the following standard relation [19]
0 0 0
2
0
1
2
1 ,
,
,
2
1 L R
q
s t
d
z
d
r
d
d
t
z
r
S
r
LR
q
( is the continuance of growth, L is the length of magnetron) gives a possibility to obtain, that
m
E
C
H
e
v z
z v
z
v 4
4 0
0
2
1
2
0
2
0
1
, 0
1
E , 0
1 H
H
,
2
2 1
0
2
0
0
1 C
H
E z
z v
E
, 0
1
r
v
, 0
1
r
E
, 0
1
v
v . Further we
obtain the second-order approximations of components of strength and induction of electrical and
magnetic fields. To obtain these approximations we replace considered fields in right sides of
Eqs. (4) on the following sums S (r,,z,t)2s+S1(r,,z,t). The replacement leads to
transformation of Eqs. (4) to the following form
z
t
z
r
H
t
z
r
H
r
t
t
z
r
D
v
C z
r
r
,
,
,
,
,
,
1
,
,
, 1
1
2
2
,
z
t
z
r
H
t
z
r
H
r
t
t
z
r
D
v
C r
z
,
,
,
,
,
,
1
,
,
, 1
1
2
2
,
r
t
z
r
H
r
t
z
r
H
t
t
z
r
D
r
v
C H
z
z
,
,
,
,
,
,
,
,
, 1
1
2
2
2
t
z
r
H r ,
,
,
1
,
t
z
r
E
r
z
t
z
r
E
t
t
z
r
B z
r
,
,
,
1
,
,
,
,
,
, 1
1
2
,](https://image.slidesharecdn.com/11424ijcax02-241128170357-fbfebb50/85/On-Decreasing-of-Mismatch-Induced-Stress-During-Growth-of-Films-During-Magnetron-Sputtering-4-320.jpg)
![International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024
30
r
z
r
r
u
d
t
z
u
D
u
z
u
d
t
z
u
D
r
C
t
z
r
D
r
0
1
0
1
2
2
,
,
,
,
,
,
2
,
,
,
,
r
z
r
r u
d
t
z
u
B
u
z
u
d
t
z
u
B
t
z
r
B
r
0
1
0
1
2 ,
,
,
,
,
,
,
,
,
,
t
z
v
B
v
t
r
E
r d
z
r
B
t
d
z
r
E
t
e
v
m z
r
0
1
2
2
2
0
1
2
2 ,
,
,
,
,
,
t
z
t
B d
z
r
B
z
r
v
d
z
r
v
z
0
1
1
0
1
2 ,
,
,
,
,
,
,
,
,
,
t
z
v
B
v
t
E d
z
r
B
t
d
z
r
E
t
e
v
m r
z
r
0
1
2
2
2
0
1
2
2 ,
,
,
,
,
,
t
z
r
t
r
B d
z
r
B
z
r
v
d
z
r
v
z
0
1
1
0
1
2 ,
,
,
,
,
,
,
,
,
,
t
v
B
v
t
z
E
z d
z
r
B
t
d
z
r
E
t
e
v
m r
r
z
0
1
2
2
2
0
1
2
2 ,
,
,
,
,
,
t
r
t
r
B d
z
r
B
z
r
v
d
z
r
v
0
1
1
0
1
2 ,
,
,
,
,
,
,
,
,
.
Average values of the second-order approximations 2s were calculated by using the following
standard relation [19]
0 0 0
2
0
1
2
2
2 ,
,
,
,
,
,
2
1 L R
q
q
s t
d
z
d
r
d
d
t
z
r
S
t
z
r
S
r
LR
q
. (5)
Substitution of obtained approximations of strength and induction of considered fields and
velocities of movement of ions into relations (5) gives a possibility to obtain relations for the
required average values in the following form
0 0 0
2
0
1
2
2
1
2
2
,
,
,
2
6
L R
H
H
D
t
d
z
d
r
d
d
t
z
r
H
LR
t
z
0 0 0
2
0
1
2
2
2
L R
z
z t
d
z
d
r
d
d
v
v
LR
t
, 0
2
r
D
, 0
2
D , 0
2
z
B
,
0
2
r
B
,
0
0
2 H
B
, 0
2
r
v
, 0
2
v
v
,
0
1
0 0 0
2
0
1
2
2
2
,
,
,
2
z
E
L R
z
v v
m
e
t
d
z
d
r
d
d
t
z
r
E
r
LR
t
m
e z
z
.
In this paper we calculated the second-order approximations of the required strengths and
inductions of the considerate fields, as well as the ion velocities by the method of averaging
functional corrections. The approximation is usually sufficient to obtain qualitative conclusions](https://image.slidesharecdn.com/11424ijcax02-241128170357-fbfebb50/85/On-Decreasing-of-Mismatch-Induced-Stress-During-Growth-of-Films-During-Magnetron-Sputtering-6-320.jpg)
![International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024
31
and to obtain some quantitative results. The obtained analytical results were checked by
comparing them with the results of numerical simulation.
Further let us consider changing of components of displacement vector in the grown multilayer
structure with changing of spatial coordinates and time. Equations, which are describe
components ur(r,,z,t), u(r,,z,t) and uz(r,,z,t) of displacement vector
t
z
r
u ,
,
,
, could be
written as [20,21]
z
t
z
r
t
z
r
r
r
t
z
r
r
r
t
t
z
r
u
С
z
t
z
r
y
t
z
r
r
r
t
z
r
r
r
t
t
z
r
u
С
z
t
z
r
t
z
r
r
r
t
z
r
r
r
t
t
z
r
u
C
zz
zy
zx
z
z
r
rz
r
rr
r
,
,
,
,
,
,
1
,
,
,
1
,
,
,
,
,
,
,
,
,
1
,
,
,
1
,
,
,
,
,
,
,
,
,
1
,
,
,
1
,
,
,
2
2
2
2
2
2
where
k
k
ij
i
j
j
i
ij
x
t
z
r
u
x
t
z
r
u
x
t
z
r
u
z
z
E ,
,
,
3
,
,
,
,
,
,
1
2
r
k
k
ij T
t
z
r
T
z
K
z
x
t
z
r
u
z
K
,
,
,
,
,
,
; ij is the Kronecker symbol.
Accounting of the last relation leads to transformation of the above system of equations to the
following form
r
t
z
r
u
r
r
z
z
E
z
K
r
t
t
z
r
u
C r
r
,
,
,
1
6
5
1
,
,
,
2
2
2
2
2
2
2
2
2
,
,
,
,
,
,
1
,
,
,
1
3
1
z
t
z
r
u
t
z
r
u
r
r
t
z
r
u
r
z
z
E
z
K
r
z
r
t
z
r
T
r
r
z
z
K
z
z
E
z
K
z
r
t
z
r
u
z
z
E z
,
,
,
1
1
3
,
,
,
1
2
2
r
t
z
r
u
r
r
r
t
z
r
u
r
r
r
z
z
E
t
t
z
r
u
C r
y ,
,
,
,
,
,
1
1
2
,
,
,
2
2
t
z
r
T
r
r
z
z
K
t
z
r
u
r
z
t
z
r
u
z
z
E
z
z ,
,
,
,
,
,
1
,
,
,
1
2
r
t
z
r
u
r
r
z
K
t
z
r
u
z
K
z
z
E
r
,
,
,
,
,
,
1
12
5
1
2
2
2
2](https://image.slidesharecdn.com/11424ijcax02-241128170357-fbfebb50/85/On-Decreasing-of-Mismatch-Induced-Stress-During-Growth-of-Films-During-Magnetron-Sputtering-7-320.jpg)
![International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024
33
z
t
z
r
T
z
z
K
t
t
z
r
u
C z
,
,
,
,
,
,
2
1
2
.
Integration of the right and the left sides of the above relations on time t leads to the following
result
t
r d
d
z
r
T
r
r
C
r
z
z
K
t
z
r
u
0 0
1 ,
,
,
,
,
,
r
t
u
d
d
z
r
T
r
r
C
r
z
z
K 0
0 0
,
,
,
,
t
d
d
z
r
T
C
r
z
z
K
t
z
r
u
0 0
1 ,
,
,
,
,
,
0
0 0
,
,
, u
d
d
z
r
T
C
r
z
z
K
,
t
z d
d
z
r
T
z
C
z
z
K
t
z
r
u
0 0
1 ,
,
,
,
,
,
z
u
d
d
z
r
T
z
C
z
z
K 0
0 0
,
,
,
.
Approximations with the second and higher orders of components of displacement vector could
be obtained in the framework of the standard replacement of the required functions in the right
sides of equations (6) on the following sum i+ui(r,,z,t) [16, 19]. The replacement leads to the
following result
z
z
E
r
t
z
r
u
r
r
z
z
E
z
K
r
t
t
z
r
u
C r
r
1
2
,
,
,
1
6
5
1
,
,
, 1
2
2
2
r
t
z
r
u
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1](https://image.slidesharecdn.com/11424ijcax02-241128170357-fbfebb50/85/On-Decreasing-of-Mismatch-Induced-Stress-During-Growth-of-Films-During-Magnetron-Sputtering-9-320.jpg)
![International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024
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some analytical approaches to model physical processes in reactors for epitaxy from gas phase
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[17] Vorob’ev, G.S., Sokolov, S.V., Pisarenko, L.D. & Zhurba, V.O. (2010) Theory of electromagnetic
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PHYSMATLIT.](https://image.slidesharecdn.com/11424ijcax02-241128170357-fbfebb50/85/On-Decreasing-of-Mismatch-Induced-Stress-During-Growth-of-Films-During-Magnetron-Sputtering-15-320.jpg)
On Decreasing of Mismatch-Induced Stress During Growth of Films During Magnetron Sputtering
- 1. International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024 DOI:10.5121/ijcax.2024.11402 25 ON DECREASING OF MISMATCH-INDUCED STRESS DURING GROWTH OF FILMS DURING MAGNETRON SPUTTERING E.L. Pankratov Department of applied mechanics, physics and higher mathematics, Nizhny Novgorod State Agrotechnical University, 97 Gagarin avenue, Nizhny Novgorod ABSTRACT In this paper we analyzed mass transfer during the growth of epitaxial layers during magnetron sputtering. During the analysis we obtained modifications of properties of films with variation of several parameters. We analyzed possibility to decrease mismatch -induced stress on the considered films during their growth. An analytical approach for analyzing mass transfer was introduced, which makes it possible to take into account the nonlinearity of processes, as well as changes in parameters in space and time. KEYWORDS Mass transfer; magnetrons sputtering; analytical approach for modelling; mismatch-induced stress. 1. INTRODUCTION Development of solid-state electronics and widespread using of heterostructures for manufacturing of electronic devices leads to the necessity to improve properties of layers of these heterostructures. Different methods are used to manufacture of heterostructures: molecular beam epitaxy, epitaxy from the gas phase, magnetron sputtering. A large number of experimental works have been devoted to the manufacturing and using of heterostructures due to their widespread using [1-12]. At the same time, a relatively small number of works are devoted to predicting the growth of heterostructures [11,12]. In this paper in development of references [13-16] we analyzed processes framework growing films by magnetron sputtering. Structure of the considered magnetron is shown in Fig. 1. In the framework of the structure electrons are emitting from the cathode (line 1). After that under the action of a field between the cathode (line 1) and the anode (line 3) an electron flow is formed between lines 1 and 2. The main purpose of this paper is analysis of mass transfer in magnetrons during growth of films in order to improve their properties. An additional aim of this paper is analysis of possibility to decrease mismatch-induced stress during growth of films. To solve this aim we consider an analytical approach for analysis of mass transfer, which makes a possibility to take into account the nonlinearity of mass transfer, as well as the changing of its parameters in space and time.
- 2. International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024 26 Fig. 1. Structure of magnetron. In the framework of the structure ions have been emitted from cathode (line 1). After that under influence of field between cathode (line 1) and anode (line 3) one can obtained flow of ions between lines 1 and 2 2. METHOD OF SOLUTION To solve our aims we determine spatio-temporal distribution of electromagnetic field. We determine the distribution by solving the following boundary problem [17, 18] t t z r D j t z r H rot , , , , , , , t t z r B t z r E rot , , , , , , , t z r D div , , , , 0 , , , t z r B div , E D 0 , H B 0 . (1) Here and are the dielectric and magnetic constants; 0=0.88610-11 F/m; 0= 1.25610-6 H/m; E and H are the electric and magnetic strengths; D and B are the inductions of electric and magnetic fields; r, and z are the spatial coordinates; t is the current time. Boundary and initial conditions for the system of equations could be written as 0 , , , t z R Bz , 0 , , , t z R Dr , 0 , , , t z R Ez , 0 , , , t z R Hr , 0 0 , , , E z r E , 0 0 , , , H z r H . Current density j is proportional to speed of ions v : v C j , where C is the density of ions. The speed of ions correlated with strengths of electric and magnetic field by the second Newton law
- 3. International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024 27 B v E e t d v d m . (3) Scalar form of the Eqs.(1) could be written as z t z r H t z r H r t t z r D v C z r r , , , , , , 1 , , , , z t z r H t z r H r t t z r D v C r z , , , , , , 1 , , , , t z r H r t z r H r t t z r D r v C r z z , , , , , , , , , , t t z r B t z r E r z t z r E r z , , , , , , 1 , , , , t t z r B t z r E r z t z r E z r , , , , , , 1 , , , , t t z r B r t z r E r t z r E r z r , , , , , , , , , , z t z r D t z r D r r t z r D r r z r , , , , , , 1 , , , 1 , 0 , , , , , , 1 , , , 1 z t z r B t z r B r r t z r B r r z r , z r r B v E e t d v d m , z r B v E e t d v d m , B v E e t d v d m r z z . To solve these equations we use the method of averaging functional corrections [19]. To determine the first-order approximations of the required functions we replace them in the right- hand sides of equations (4) by their not yet known average values 1s in the right-hand side of these equations. As a result of this substitution we obtain the following equations to determine the first-order approximations of components of strength and induction of the considered fields 0 , , , 1 t t z r D v C r r , 0 , , , 1 t t z r D v C , H z z t t z r D v C 1 1 , , , , 0 , , , 1 t t z r B r , 0 , , , 1 t t z r B , E z t t z r B 1 1 , , , , C r t z r D r r r , , , 1 1 , 0 , , , 1 1 r t z r B r r r ,
- 4. International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024 28 z r B v E r e t d v d m 1 1 1 1 , z r B v E e t d v d m 1 1 1 1 , B v E z r z e t d v d m 1 1 1 1 . Further after integration of left and right sides of these equations on considered variables we obtain the first-order approximations of the considered fields and velocity of ions in the following form t r r d v C t z r D 0 1 1 , , , , t d v C t z r D 0 1 1 , , , , 0 0 0 1 1 1 , , , E d v C t t z r D t z H z , t e v m z r B v E r 1 1 1 1 , 0 , , , 1 t z r B r , 0 0 1 , , , H t z r B , t t z r B E z 1 1 , , , , 0 1 1 1 1 v m t e v m z r B v E , 0 1 1 1 1 z B v E z v m t e v m r z . Calculation of average values 1s by using the following standard relation [19] 0 0 0 2 0 1 2 1 , , , 2 1 L R q s t d z d r d d t z r S r LR q ( is the continuance of growth, L is the length of magnetron) gives a possibility to obtain, that m E C H e v z z v z v 4 4 0 0 2 1 2 0 2 0 1 , 0 1 E , 0 1 H H , 2 2 1 0 2 0 0 1 C H E z z v E , 0 1 r v , 0 1 r E , 0 1 v v . Further we obtain the second-order approximations of components of strength and induction of electrical and magnetic fields. To obtain these approximations we replace considered fields in right sides of Eqs. (4) on the following sums S (r,,z,t)2s+S1(r,,z,t). The replacement leads to transformation of Eqs. (4) to the following form z t z r H t z r H r t t z r D v C z r r , , , , , , 1 , , , 1 1 2 2 , z t z r H t z r H r t t z r D v C r z , , , , , , 1 , , , 1 1 2 2 , r t z r H r t z r H t t z r D r v C H z z , , , , , , , , , 1 1 2 2 2 t z r H r , , , 1 , t z r E r z t z r E t t z r B z r , , , 1 , , , , , , 1 1 2 ,
- 5. International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024 29 t z r E r z t z r E t t z r B z r , , , 1 , , , , , , 1 1 2 , t z r E r t z r E r t z r E t t z r B r r E z , , , , , , , , , , , , 1 1 1 2 2 , z t z r D t z r D r r t z r D r r z r , , , , , , 1 , , , 1 1 1 2 , z t z r B t z r B r r t z r B r r z r , , , , , , 1 , , , 1 1 1 2 , t z r B t z r v t z r E e t d v d m z B v r E r z r , , , , , , , , , 1 2 1 2 1 2 2 , t z r B t z r v t z r E e t d v d m z B r v E z r , , , , , , , , , 1 2 1 2 1 2 2 , t z r B t z r v t z r E e t d v d m B r v z E z r z , , , , , , , , , 1 2 1 2 1 2 2 . Integration of left and right sides of the above equations on considered variations gives a possibility to obtain the second-order approximations of the considered fields in the following forms t r t t z r d v C d z r H z d z r H r t z r D 0 2 0 1 0 1 2 , , , , , , 1 , , , , t t r t z d v C d z r H z d z r H r t z r D 0 2 0 1 0 1 2 , , , , , , 1 , , , , 0 0 0 1 0 1 2 2 , , , , , , , , , E r d z r H r r d z r H t t z r D r t t H z 0 0 0 2 0 1 , , , E r d v C d z r H t z t r , t t E z d z r E r r d z r E t t z r B r 0 1 0 1 2 2 , , , , , , , , , t r d z r E 0 1 , , , , t z t r d z r E r d z r E z t z r B 0 1 0 1 2 , , , 1 , , , , , , , 0 0 0 1 0 1 2 , , , 1 , , , , , , H d z r E r d z r E z t z r B t z t r ,
- 6. International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024 30 r z r r u d t z u D u z u d t z u D r C t z r D r 0 1 0 1 2 2 , , , , , , 2 , , , , r z r r u d t z u B u z u d t z u B t z r B r 0 1 0 1 2 , , , , , , , , , , t z v B v t r E r d z r B t d z r E t e v m z r 0 1 2 2 2 0 1 2 2 , , , , , , t z t B d z r B z r v d z r v z 0 1 1 0 1 2 , , , , , , , , , , t z v B v t E d z r B t d z r E t e v m r z r 0 1 2 2 2 0 1 2 2 , , , , , , t z r t r B d z r B z r v d z r v z 0 1 1 0 1 2 , , , , , , , , , , t v B v t z E z d z r B t d z r E t e v m r r z 0 1 2 2 2 0 1 2 2 , , , , , , t r t r B d z r B z r v d z r v 0 1 1 0 1 2 , , , , , , , , , . Average values of the second-order approximations 2s were calculated by using the following standard relation [19] 0 0 0 2 0 1 2 2 2 , , , , , , 2 1 L R q q s t d z d r d d t z r S t z r S r LR q . (5) Substitution of obtained approximations of strength and induction of considered fields and velocities of movement of ions into relations (5) gives a possibility to obtain relations for the required average values in the following form 0 0 0 2 0 1 2 2 1 2 2 , , , 2 6 L R H H D t d z d r d d t z r H LR t z 0 0 0 2 0 1 2 2 2 L R z z t d z d r d d v v LR t , 0 2 r D , 0 2 D , 0 2 z B , 0 2 r B , 0 0 2 H B , 0 2 r v , 0 2 v v , 0 1 0 0 0 2 0 1 2 2 2 , , , 2 z E L R z v v m e t d z d r d d t z r E r LR t m e z z . In this paper we calculated the second-order approximations of the required strengths and inductions of the considerate fields, as well as the ion velocities by the method of averaging functional corrections. The approximation is usually sufficient to obtain qualitative conclusions
- 7. International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024 31 and to obtain some quantitative results. The obtained analytical results were checked by comparing them with the results of numerical simulation. Further let us consider changing of components of displacement vector in the grown multilayer structure with changing of spatial coordinates and time. Equations, which are describe components ur(r,,z,t), u(r,,z,t) and uz(r,,z,t) of displacement vector t z r u , , , , could be written as [20,21] z t z r t z r r r t z r r r t t z r u С z t z r y t z r r r t z r r r t t z r u С z t z r t z r r r t z r r r t t z r u C zz zy zx z z r rz r rr r , , , , , , 1 , , , 1 , , , , , , , , , 1 , , , 1 , , , , , , , , , 1 , , , 1 , , , 2 2 2 2 2 2 where k k ij i j j i ij x t z r u x t z r u x t z r u z z E , , , 3 , , , , , , 1 2 r k k ij T t z r T z K z x t z r u z K , , , , , , ; ij is the Kronecker symbol. Accounting of the last relation leads to transformation of the above system of equations to the following form r t z r u r r z z E z K r t t z r u C r r , , , 1 6 5 1 , , , 2 2 2 2 2 2 2 2 2 , , , , , , 1 , , , 1 3 1 z t z r u t z r u r r t z r u r z z E z K r z r t z r T r r z z K z z E z K z r t z r u z z E z , , , 1 1 3 , , , 1 2 2 r t z r u r r r t z r u r r r z z E t t z r u C r y , , , , , , 1 1 2 , , , 2 2 t z r T r r z z K t z r u r z t z r u z z E z z , , , , , , 1 , , , 1 2 r t z r u r r z K t z r u z K z z E r , , , , , , 1 12 5 1 2 2 2 2
- 8. International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024 32 z t z r u z z E z K r y , , , 1 6 1 2 (6) 2 2 2 2 2 , , , 1 , , , 1 1 2 , , , t z r u r r t z r u r r r z z E t t z r u C z z z r t z r u r z K z z t z r u r z r t z r u r r r , , , 1 , , , 1 , , , 2 2 z t z r u z z E z z t z r u t z r u r z r , , , 6 1 6 1 , , , , , , 1 z t z r u y t z r u r x t z r u r r z x , , , , , , 1 , , , 1 z t z r T z z K , , , . where E is the Young modulus; is the coefficient of thermal expansion; K is the modulus of uniform compression; is the Poisson coefficient; 0=(as-aEL)/aEL is the mismatch parameter; as, aEL are the lattice distances of the substrate and the epitaxial layer; T (r,,z,t) is the temperature of growth with equilibrium value Tr. Conditions for the system of the above equations can be written in the form 0 , , , 0 r r t z r u ; 0 , , , R r r t z r u ; 0 , , , 0 z z t z r u ; 0 , , , L z z t z r u ; t z r u t z r u , , 2 , , , 0 , ; 0 0 , , , u z r u ; 0 , , , u z r u . Now let us determine solutions of system of equations (6), which describe components of displacement vector. To calculate the first-order approximations of the required components in the framework the method of averaging of function corrections one shall substitute their not yet known average values i in the right sides of the equations (6). The substitution leads to the following result r t z r T r r z z K t t z r u C r , , , , , , 2 1 2 , t z r T r z z K t t z r u C , , , , , , 2 1 2 ,
- 9. International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024 33 z t z r T z z K t t z r u C z , , , , , , 2 1 2 . Integration of the right and the left sides of the above relations on time t leads to the following result t r d d z r T r r C r z z K t z r u 0 0 1 , , , , , , r t u d d z r T r r C r z z K 0 0 0 , , , , t d d z r T C r z z K t z r u 0 0 1 , , , , , , 0 0 0 , , , u d d z r T C r z z K , t z d d z r T z C z z K t z r u 0 0 1 , , , , , , z u d d z r T z C z z K 0 0 0 , , , . Approximations with the second and higher orders of components of displacement vector could be obtained in the framework of the standard replacement of the required functions in the right sides of equations (6) on the following sum i+ui(r,,z,t) [16, 19]. The replacement leads to the following result z z E r t z r u r r z z E z K r t t z r u C r r 1 2 , , , 1 6 5 1 , , , 1 2 2 2 r t z r u r r z z E z K z t z r u t z r u r z y , , , 1 1 3 , , , , , , 1 1 2 2 2 1 2 2 1 2 2 2 1 2 2 1 2 2 1 2 2 , , , , , , 1 1 2 , , , 1 z t z r u t z r u r z z E r t z r u r r z y z r t z r u r z z E z K r r t z r T r z z K z z , , , 1 3 1 , , , 1 2 1 r t z r u r r t z r u r r r z z E t t z r u C r , , , , , , 1 1 2 , , , 1 2 1 2 2 2 t z r u r z t z r u z z E z y t z r T z z K z , , , 1 , , , 1 2 , , , 1 1
- 10. International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024 34 z z E z K r t z r u z K z z E r 1 6 1 , , , 1 12 5 1 2 1 2 2 r t z r u r z K z t z r u , , , , , , 1 2 1 2 2 1 2 2 1 2 2 2 , , , 1 , , , 1 1 2 , , , t z r u r r t z r u r r r z z E t t z r u C z z z r t z r u r r z z t z r u r z r t z r u r r r , , , 1 , , , 1 , , , 1 1 2 1 2 t z r u r z t z r u z z t z r u z t z r u y z r r , , , 1 , , , 6 6 1 , , , , , , 1 1 1 1 z t z r T z z K z z E z t z r u r t z r u r r z r , , , 1 , , , , , , 1 1 1 . Integration of the right and the left sides of the above relations on time t leads to the following result z z E d d z r u r r r z z E z K C t z r u t r r 1 3 , , , 1 6 5 1 , , , 0 0 1 2 t t d d z r u d d z r u r r C z K 0 0 1 2 2 0 0 1 2 , , , , , , 1 t z t z d d z r u r z r C z C r z E d d z r u z 0 0 1 2 2 0 0 1 2 2 , , , 1 1 2 , , , 0 0 1 2 2 , , , 1 6 5 1 1 3 d d z r u r r r z z E z K C z z E z K r z K d d z r u r r C d d z r T r r C z z K y t 0 0 1 2 0 0 , , , 1 , , , 0 0 1 2 2 0 0 1 2 2 2 , , , , , , 1 1 3 d d z r u z d d z r u r z z E z 0 0 1 2 , , , 1 3 1 1 2 d d z r u r z r z z E z K C z C z E z 0 0 0 , , , d d z r T r r C z z K u r
- 11. International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024 35 t r t r d d z r u r r d d z r u r r r t z r u 0 0 1 2 0 0 1 2 , , , , , , , , , C r z K d d z r u z K z z E C r z C r z E t r 0 0 1 2 2 2 , , , 1 12 5 1 1 2 t t y d d z r u z z z E z C d d z r u r r 0 0 1 0 0 1 2 , , , 1 2 1 , , , t t z d d z r u z z z E z K C r d d z r u r 0 0 1 2 0 0 1 , , , 1 6 1 , , , 1 t t d d z r T z C r z E d d z r T C z z K 0 0 0 0 , , , 1 2 , , , z C r z E d d z r u z z z E z K C r t 1 2 , , , 1 6 1 0 0 1 2 C z z K d d z r u r r d d z r u r r r r r 0 0 1 2 0 0 1 , , , , , , z z E z K d d z r u C r d d z r T r 1 12 5 , , , 1 , , , 0 0 1 2 2 2 0 0 0 0 0 1 0 0 1 , , , 1 , , , 1 2 1 u d d z r u r d d z r u z z z E z C z 0 0 1 2 0 0 1 2 , , , , , , d d z r u z d d z r u r r C r z K С r z K z z E 1 1 6 0 0 1 2 2 0 0 1 2 , , , 1 , , , , , , d d z r u r d d z r u r r r t z r u z z z z C r z E d d z r u z d d z r u r z r y r 1 2 , , , , , , 0 0 1 2 0 0 1 2 0 0 1 0 0 1 , , , 1 , , , 1 1 d d z r u r d d z r u r r z K z C r r r 0 0 1 0 0 1 , , , 6 1 6 1 , , , d d z r u z z z E z C d d z r u z z r
- 12. International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024 36 0 0 1 0 0 1 , , , 1 , , , 1 d d z r u r d d z r u r r r r z z u d d z r T z C z z K d d z r u z 0 0 0 0 0 1 , , , , , , . In this paper we calculated the second-order approximations of components of displacement vector by using the method of averaging functional corrections. The approximation is usually sufficient to obtain qualitative conclusions and to obtain some quantitative results. The obtained analytical results were checked by comparing them with the results of numerical simulation. 3. DISCUSSION In this section we analyzed dynamics of mass transport during growth of films by magnetron sputtering to determine conditions to change properties of epitaxial layers. The Fig. 2 shows dependence of concentration of sputtered material on cyclotron frequency c. Increasing of induction of magnetic field B0 leads to increasing of the cyclotron frequency c and to increase homogeneity of epitaxial layer. The Fig. 3 shows dependence of concentration of sputtered material on electric strengths E0. Increasing of the strengths leads to increasing of speed of transport of ions to target and their concentration. In this case the opposite effect was obtained with increasing of the ion mass (see Fig. 3), radius (see Fig. 4) and length (see Fig. 5) of the magnetron. During analysis of components of displacement vector we obtained, that increasing of velocity of ions of growing material leads to decreasing of value of the considered vector (see Fig. 6). 0 5 10 15 c 0.0 0.2 0.4 0.6 0.8 1.0 C/C 0 3 2 1 Fig. 2. Typical dependence of concentration of sputtered material on cyclotron frequency c. Increasing of density of curves correspond to increasing of strength of electrical field
- 13. International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024 37 0 5 10 15 E0 0.0 0.2 0.4 0.6 0.8 1.0 C/C 0 1 2 3 Fig. 3. Typical dependence of concentration of sputtered material on electric strengths E0. Increasing of density of curves correspond to increasing of mass of ions 0 5 10 15 E0 0.0 0.2 0.4 0.6 0.8 C/C 0 1 2 3 Fig. 4. Typical dependence of concentration of sputtered material on electric strengths E0. Increasing of density of curves correspond to increasing of radius of magnetron
- 14. International Journal of Computer- Aided Technologies (IJCAx) Vol.11, No.1/2/3/4, October 2024 38 0 5 10 15 E0 0.0 0.2 0.4 0.6 0.8 C/C 0 1 2 3 Fig. 5. Typical dependence of concentration of sputtered material on electric strengths E0. Increasing of density of curves correspond to increasing of length of magnetron z 0.0 0.2 0.4 0.6 0.8 1.0 Uz 1 2 a L Fig. 6. Normalized dependences of component of displacement vector uz on coordinate z at small velocity of ions (curve 1) and large velocity of ions (curve 2) of growing material 4. CONCLUSIONS In the present paper we analyzed mass transport during magnetron sputtering of materials. Based on results of analysis we formulate recommendations to control of properties of epitaxial layers. We analyzed possibility to decrease mismatch-induced stress in multilayer structure during growth. We also introduce an analytical approach for prognosis of mass transport during growth of layers in magnetron.
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