![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015
1
VARIATION OF MISMATCH-INDUCED STRESS IN A HE-
TEROSTRUCTURE WITH CHANGING TEMPERATURE OF
GROWTH
E.L. Pankratov1,3
and 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
3
Nizhny Novgorod Academy of the Ministry of Internal Affairs of Russia, 3 Ankudi-
novskoe Shosse, Nizhny Novgorod, 603950, Russia
ABSTRACT
In this paper we analyzed influence of diffusion of material of an epitaxial layer of a heterostructure during
high-temperature growth into next layer (next epitaxial layer or substrate) of the heterostructure on mis-
match-induced stress. It has been shown, that value of mismatch-induced stress in heterostructure depends
on temperature of growth, because the considered diffusion depends on the temperature. We also introduce
an analytical approach to model the diffusion and relaxation of the mismatch-induced stress.
KEYWORDS
Heterostructure; Mismatch-Induced Stress; Temperature Of Growth; Mixing Of Materials Of Layers Of
Heterostructure
1. INTRODUCTION
In the present time large number of solid state electronic devices have been manufactured based
on heterostructures. The widely using of heterostructures leads to necessity to improve of their
properties. It is known, that mismatch-induced stress presents in all heterostructures. The stress in
heterostructures could leads to generation misfit dislocations. One way to decrease mismatch-
induced stress is choosing materials of heterostructure with as small as possible mismatch of lat-
tice constants [1-3]. Another way to decrease mismatch-induced stress is using a buffer layer,
manufactured by using another materials, between layers of heterostructure. Lattice constant of
the buffer layer should be average in comparison with lattice constants of nearest layers of hete-
rostructure [4,5].
In this paper we consider a heterostructure with two layers (see Fig. 1). The layers are a substrate
and an epitaxial layer. At a high temperature of growth (for example, during epitaxy from gas
phase) of heterostructure one can find intensive diffusion of material of the epitaxial layer into the
substrate. Our main aim framework the present paper is analysis of variation of mismatch-
induced stress in the considered heterostructure with diffusion of material of the epitaxial into the
substrate.](https://image.slidesharecdn.com/2215msej01-150708094508-lva1-app6891/85/VARIATION-OF-MISMATCH-INDUCED-STRESS-IN-A-HETEROSTRUCTURE-WITH-CHANGING-TEMPERATURE-OF-GROWTH-1-320.jpg)
![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015
2
2. METHOD OF SOLUTION
Let us to describe diffusion of material of epitaxial layer in the substrate by solution of the fol-
lowing boundary problem [6-11]
( ) ( ) ( ) ( ) +
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC ,,,,,,,,,,,,
( ) ( ) ( ) ( )
∫∇
∂
∂
Ω+
∫∇
∂
∂
Ω+
zz L
S
S
L
S
S
WdtWyxCtzyx
Tk
D
y
WdtWyxCtzyx
Tk
D
x 00
,,,,,,,,,,,, µµ (1)
Fig. 1. Heterostructure, which consist of a substrate and an epitaxial layer
( ) 0
,,,
=
∂
∂
= xLx
x
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=y
y
tzyxC
,
( ) 0
,,,
=
∂
∂
= yLx
y
tzyxC
,
( ) 0
,,,
0
=
∂
∂
=z
z
tzyxC
,
( ) 0
,,,
=
∂
∂
= zLx
z
tzyxC
, C(x,y,z,0)=fC (x,y,z), C(0,y,z,t)=C0.
Here C(x,y,z,t) is the spatio-temporal distribution of concentration of material of epitaxial layer;
atomic volume of the dopant describes by Ω; surficial gradient describes by ∇S; the integral
( )∫
zL
zdtzyxC
0
,,, describes surficial concentration of dopant on interface between materials of he-
terostructure; µ(x,y,z,t) is the chemical potential; D and DS are the coefficients of volumetric and
surficial diffusions (reason of the surficial diffusion is the mismatch-induced stress). Values of
the volumetric and surficial diffusion coefficients are differ in different materials. The diffusion
coefficients will be changed in during heating and cooling of heterostructure. Different levels of
doping also leads to changing of the diffusion coefficients. We approximate dopant diffusion
coefficients by the following functions [7]
( ) ( )
( )
+=
TzyxP
tzyxC
TzyxDD LC
,,,
,,,
1,,, γ
γ
ξ , ( ) ( )
( )
+=
TzyxP
tzyxC
TzyxDD SLSS
,,,
,,,
1,,, γ
γ
ξ . (2)](https://image.slidesharecdn.com/2215msej01-150708094508-lva1-app6891/85/VARIATION-OF-MISMATCH-INDUCED-STRESS-IN-A-HETEROSTRUCTURE-WITH-CHANGING-TEMPERATURE-OF-GROWTH-2-320.jpg)
![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015
3
The functions DL(x,y,z,T) and DLS(x,y,z,T) describes spatial and temperature dependences of diffu-
sion coefficients. The functions could be obtained by accounting all layers of heterostruicture and
Arrhenius law. T is the temperature of grown. P (x,y,z,T) is the limit of solubility of material of
the epitaxial layer in the substrate. Parameter γ depends on properties of materials and will be
integer in the following interval γ∈[1,3] (the dependence described in details in [7]).
Chemical potential µ in the Eq.(1) depends on properties of materials of heterostructure and could
be approximated by the following relation [10]
µ=E(z)Ωσij [uij(x,y,z,t)+uji(x,y,z,t)]/2. (3)
Function E(z) describes spatial dependence of Young modulus. Tensor σij describes the stress
tensor;
∂
∂
+
∂
∂
=
i
j
j
i
ij
x
u
x
u
u
2
1
is the deformation tensor; ui, uj are the components ux(x,y,z,t),
uy(x,y,z,t) and uz(x,y,z,t) of the displacement vector ( )tzyxu ,,,
r
; xi, xj are the coordinate x, y, z.
The Eq. (3) could be written in the following equivalent form
( ) ( )
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
∂
∂
+
∂
∂Ω
=
i
j
j
i
i
j
j
i
x
tzyxu
x
tzyxu
x
tzyxu
x
tzyxu
zEtzyx
,,,,,,
2
1,,,,,,
2
,,,µ
( )
( )
( )
( ) ( ) ( )[ ]
−−
−
∂
∂
−
+− ij
k
kij
ij TtzyxTzzK
x
tzyxu
z
z
δβε
σ
δσ
δε 000 ,,,3
,,,
21
.
Here σ is Poisson coefficient; the normalized difference ε0 = (as-aEL)/aEL describes the mismatch
parameter; parameters as, aEL in the above difference are lattice distances of the substrate and the
epitaxial layer; K is the modulus of uniform compression; β is the coefficient of thermal expan-
sion; Tr is the equilibrium temperature, which coincide (in our case) with room temperature.
Components of the displacement vector have been described by the following equations [11]
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
z
tzyx
y
tzyx
x
tzyx
t
tzyxu
z
z
tzyx
y
tzyx
x
tzyx
t
tzyxu
z
z
tzyx
y
tzyx
x
tzyx
t
tzyxu
z
zzzyzxz
yzyyyxy
xzxyxxx
,,,,,,,,,,,,
,,,,,,,,,,,,
,,,,,,,,,,,,
2
2
2
2
2
2
σσσ
ρ
σσσ
ρ
σσσ
ρ
The stress tensor σij correlates with components of the displacement vector by the following rela-
tion
( )
( )[ ]
( ) ( ) ( ) ( )
×
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂
+
=
k
k
ij
k
kij
i
j
j
i
ij
x
tzyxu
x
tzyxu
x
tzyxu
x
tzyxu
z
zE ,,,,,,
3
,,,,,,
12
δ
δ
σ
σ
( ) ( ) ( ) ( )[ ]rTtzyxTzKzzK −−× ,,,β .](https://image.slidesharecdn.com/2215msej01-150708094508-lva1-app6891/85/VARIATION-OF-MISMATCH-INDUCED-STRESS-IN-A-HETEROSTRUCTURE-WITH-CHANGING-TEMPERATURE-OF-GROWTH-3-320.jpg)
![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015
4
The function ρ(z) describes the density of materials of heterostructure. The tenzor δij describes
the Kronecker symbol. Accounting relation for σij in the previous system of equations last system
of equation could be written as
( )
( )
( ) ( )
( )[ ]
( )
( ) ( )
( )[ ]
( )
+
∂∂
∂
+
−+
∂
∂
+
+=
∂
∂
yx
tzyxu
z
zE
zK
x
tzyxu
z
zE
zK
t
tzyxu
z
yxx
,,,
13
,,,
16
5,,,
2
2
2
2
2
σσ
ρ
( )
( )[ ]
( ) ( )
( ) ( )
( )[ ]
( )
−
∂∂
∂
+
++
∂
∂
+
∂
∂
+
+
zx
tzyxu
z
zE
zK
z
tzyxu
y
tzyxu
z
zE zzy ,,,
13
,,,,,,
12
2
2
2
2
2
σσ
( ) ( ) ( )
x
tzyxT
zzK
∂
∂
−
,,,
β
( )
( ) ( )
( )[ ]
( ) ( )
( ) ( ) ( )
+
∂
∂
−
∂∂
∂
+
∂
∂
+
=
∂
∂
y
tzyxT
zzK
yx
tzyxu
x
tzyxu
z
zE
t
tzyxu
z xyy ,,,,,,,,,
12
,,, 2
2
2
2
2
β
σ
ρ
( )
( )[ ]
( ) ( ) ( )
( )[ ]
( )
( )
+
∂
∂
+
+
+
∂
∂
+
∂
∂
+∂
∂
+ 2
2
,,,
112
5,,,,,,
12 y
tzyxu
zK
z
zE
y
tzyxu
z
tzyxu
z
zE
z
yzy
σσ
( ) ( )
( )[ ]
( )
( )
( )
yx
tzyxu
zK
zy
tzyxu
z
zE
zK
yy
∂∂
∂
+
∂∂
∂
+
−+
,,,,,,
16
22
σ
(4)
( )
( ) ( ) ( ) ( ) ( ) ( )
×
∂∂
∂
+
∂∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
zy
tzyxu
zx
tzyxu
y
tzyxu
x
tzyxuzE
t
tzyxu
z
yxzzz
,,,,,,,,,,,,
2
,,,
22
2
2
2
2
2
2
ρ
( )
( )
( ) ( ) ( )
( ) ( )
×
∂
∂
−
∂
∂
+
∂
∂
+
∂
∂
∂
∂
+
+
×
z
tzyxT
z
z
tzyxu
y
tzyxu
x
tzyxu
zK
zz
xyx ,,,,,,,,,,,,
1
1
β
σ
( ) ( )
( )
( ) ( ) ( ) ( )
∂
∂
−
∂
∂
−
∂
∂
−
∂
∂
+∂
∂
+×
z
tzyxu
y
tzyxu
x
tzyxu
z
tzyxu
z
zE
z
zK zyxz ,,,,,,,,,,,,
6
16
1
σ
.
Conditions for the system of Eqs.(4) are
( ) 0
,,,0
=
∂
∂
x
tzyu
r
;
( )
0
,,,
=
∂
∂
x
tzyLu x
r
;
( ) 0
,,0,
=
∂
∂
y
tzxu
r
;
( ) 0
,,,
=
∂
∂
y
tzLxu y
r
;
( ) 0
,0,,
=
∂
∂
z
tyxu
r
;
( ) 0
,,,
=
∂
∂
z
tLyxu z
r
; ( ) 00,,, uzyxu
rr
= ; ( ) 0,,, uzyxu
rr
=∞ .
We determine spatio-temporal distributions of concentration of material of epitaxial layer in the
substrate by solving the Eq.(1) framework method of averaging of function corrections in the
standard form [12,13]. Previously we transform the Eq.(1) to the following form with account
initial distributions of the considered concentrations
( ) ( ) ( ) ( ) +
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
z
tzyxC
D
zy
tzyxC
D
yx
tzyxC
D
xt
tzyxC ,,,,,,,,,,,,
( ) ( ) ( ) ( ) +
∫∇
∂
∂
Ω+
∫∇
∂
∂
Ω+
zz L
S
S
L
S
S
WdtWyxCtzyx
Tk
D
y
WdtWyxCtzyx
Tk
D
x 00
,,,,,,,,,,,, µµ
( ) ( )tzyxfC δ,,+ (1a)](https://image.slidesharecdn.com/2215msej01-150708094508-lva1-app6891/85/VARIATION-OF-MISMATCH-INDUCED-STRESS-IN-A-HETEROSTRUCTURE-WITH-CHANGING-TEMPERATURE-OF-GROWTH-4-320.jpg)
![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015
5
After that we replace the required function C (x,y,z,t) in right side of Eq. (1a) on its not yet known
average value α1C. In this situation we obtain equation to determine the first- order approxima-
tions of the considered concentrations in the following form
( ) ( ) ( ) +
∇
∂
∂
Ω+
∇
∂
∂
Ω=
∂
∂
tzyx
Tk
D
z
y
tzyx
Tk
D
z
xt
tzyxC
S
S
CS
S
C ,,,,,,
,,,
11
1
µαµα
( ) ( )tzyxfC δ,,+ . (1b)
After integration of the left and right sides of the Eq.(1b) gives us possibility to obtain relation for
the first-order approximation of concentration of material of the epitaxial layer in the substrate in
the following form
( ) ( ) ( )
( )
+∫
+∇
∂
∂
Ω=
t
CS
SLSC d
TzyxP
zyx
Tk
z
TzyxD
x
tzyxC
0
1
11
,,,
1,,,,,,,,, τ
αξ
τµα γ
γ
( ) ( )
( )
( )zyxfd
TzyxP
zyx
Tk
z
TzyxD
y
C
t
CS
SLSC ,,
,,,
1,,,,,,
0
1
1 +∫
+∇
∂
∂
+ τ
αξ
τµα γ
γ
(1c)
We determine average values of the first-order approximations of considered concentration by
using the following relations [12,13]
( )∫ ∫ ∫ ∫
Θ
=
Θ
0 0 0 0
11 ,,,
1 x y zL L L
zyx
C tdxdydzdtzyxC
LLL
α . (5)
Substitution of the relation (1c) into relation (5) gives us possibility to obtain the following rela-
tions
( )∫ ∫ ∫=
x y zL L L
C
zyx
C xdydzdzyxf
LLL 0 0 0
1 ,,
1
α ,
Farther we used standard iteration procedure of method of averaging of function corrections [12,
13] to obtain the second-order approximation of concentration of material of epitaxial layer in the
substrate. Framework this procedure to calculate n-th-order approximation of concentration of
material of epitaxial layer in the substrate we replace the required concentration C(x,y, z,t) in the
right side of Eq. (1a) on the following sum αnC+Cn-1(x,y,z,t). This substitution gives us possibility
to obtain equation for the second-order approximation of the required concentration
( ) ( ) ( )[ ]
( )
( ) +
∂
∂
+
+
∂
∂
=
∂
∂
x
tzyxC
TzyxP
tzyxC
TzyxD
xt
tzyxC C
L
,,,
,,,
,,,
1,,,
,,, 1122
γ
γ
α
ξ
( ) ( )[ ]
( )
( ) ( )
×
∂
∂
∂
∂
+
∂
∂
+
+
∂
∂
+
z
tzyxC
zy
tzyxC
TzyxP
tzyxC
TzyxD
y
C
L
,,,,,,
,,,
,,,
1,,, 1112
γ
γ
α
ξ
( ) ( )[ ]
( )
( ) ( )
×
∂
∂
Ω++
+
+×
Tk
D
x
tzyxf
TzyxP
tzyxC
TzyxD S
C
C
L δ
α
ξ γ
γ
,,
,,,
,,,
1,,, 12
(1d)
( ) ( ) ( ) ( )
∫∇
∂
∂
Ω+
∫∇×
zz L
S
S
L
S WdtWyxCtzyx
Tk
D
y
WdtWyxCtzyx
0
1
0
1 ,,,,,,,,,,,, µµ .](https://image.slidesharecdn.com/2215msej01-150708094508-lva1-app6891/85/VARIATION-OF-MISMATCH-INDUCED-STRESS-IN-A-HETEROSTRUCTURE-WITH-CHANGING-TEMPERATURE-OF-GROWTH-5-320.jpg)
![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015
6
After integration of the left and right sides of the Eq.(1d) gives us possibility to obtain relation for
the second-order approximation of concentration of material of the epitaxial layer in the substrate
in the following form
( ) ( )
( )[ ]
( )
( )
+∫
∂
∂
+
+
∂
∂
=
t
C
L d
x
zyxC
TzyxP
zyxC
TzyxD
x
tzyxC
0
112
2
,,,
,,,
,,,
1,,,,,, τ
ττα
ξ γ
γ
( )
( )[ ]
( )
( ) ( )
∫ ×
∂
∂
∂
∂
+∫
∂
∂
+
+
∂
∂
+
tt
C
L
z
zyxC
z
d
y
zyxC
TzyxP
zyxC
TzyxD
y 0
1
0
112 ,,,,,,
,,,
,,,
1,,,
τ
τ
ττα
ξ γ
γ
( )
( )[ ]
( )
( ) ( )∫ ×∫∇
∂
∂
Ω+
+
+×
t L
S
C
L
z
WdWyxCzyx
x
d
TzyxP
zyxC
TzyxD
0 0
1
12
,,,,,,
,,,
,,,
1,,, ττµτ
τα
ξ γ
γ
( ) ( ) ( )zyxfdWdWyxCzyx
Tk
D
y
d
Tk
D
C
t L
S
SS
z
,,,,,,,,
0 0
1
+∫ ∫∇
∂
∂
Ω+× τττµτ . (1d)
We determine average value of the second-order approximation of the considered concentration
by using standard relation [12,13].
( ) ( )[ ]∫ ∫ ∫ ∫ −
Θ
=
Θ
0 0 0 0
122 ,,,,,,
1 x y zL L L
zyx
C tdxdydzdtzyxCtzyxC
LLL
α . (6)
Substitution of the relation (1e) into the relation (6) gives us possibility to obtain zero value of the
required average value α2C.
Now we determine components of displacement vector by solution of the system of Eqs.(4). To
determine components of displacement vector we used method of averaging of function correc-
tions in standard form. Framework the approach we replace the above components in right sides
of Eqs.(4) on their not yet known average values αi. The substitution leads to the following re-
sults
( ) ( ) ( ) ( ) ( )
x
tzyxT
zzK
t
tzyxu
z x
∂
∂
−=
∂
∂ ,,,,,,
2
1
2
βρ , ( )
( )
( ) ( ) ( )
y
tzyxT
zzK
t
tzyxu
z
y
∂
∂
−=
∂
∂ ,,,,,,
2
1
2
βρ ,
( ) ( ) ( ) ( ) ( )
z
tzyxT
zzK
t
tzyxu
z z
∂
∂
−=
∂
∂ ,,,,,,
2
1
2
βρ .
Integration of the left and right sides of the above equations leads to obtain relation for the
second-order approximation of the first-order approximations of components of displacement
vector. The first-order approximations could be written as
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( )
+∫ ∫
∂
∂
−∫ ∫
∂
∂
=
+∫ ∫
∂
∂
−∫ ∫
∂
∂
=
+∫ ∫
∂
∂
−∫ ∫
∂
∂
=
∞
∞
∞
z
t
z
y
t
y
x
t
x
uddzyxT
zz
z
zKddzyxT
zz
z
zKtzyxu
uddzyxT
yz
z
zKddzyxT
yz
z
zKtzyxu
uddzyxT
xz
z
zKddzyxT
xz
z
zKtzyxu
0
0 00 0
1
0
0 00 0
1
0
0 00 0
1
,,,,,,,,,
,,,,,,,,,
,,,,,,,,,
ϑϑ
ϑϑ
ϑϑ
ϑττ
ρ
β
ϑττ
ρ
β
ϑττ
ρ
β
ϑττ
ρ
β
ϑττ
ρ
β
ϑττ
ρ
β](https://image.slidesharecdn.com/2215msej01-150708094508-lva1-app6891/85/VARIATION-OF-MISMATCH-INDUCED-STRESS-IN-A-HETEROSTRUCTURE-WITH-CHANGING-TEMPERATURE-OF-GROWTH-6-320.jpg)
![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015
7
We obtain the second-order approximations of components of displacement vector could be cal-
culated by replacement of the required components in the Eqs. (4) on the following sums αi+
ui(x,y,z,t) [12-14]. This replacement leads to the following result
( )
( )
( ) ( )
( )[ ]
( ) ( ) ( )
( )[ ]
−
+∂∂
∂
−
∂
∂
+
+=
∂
∂
z
zE
yx
tzyxu
x
tzyxu
z
zE
zK
t
tzyxu
z
yxx
σσ
ρ
13
,,,,,,
16
5,,, 1
2
2
1
2
2
2
2
( ) ( )
( )[ ]
( ) ( )
( ) ( )
( )[ ]
( )
−
∂∂
∂
+
++
∂
∂
+
∂
∂
+
+
−
zx
tzyxu
z
zE
zK
z
tzyxu
y
tzyxu
z
zE
zK zzy ,,,
13
,,,,,,
12
1
2
2
1
2
2
1
2
σσ
( ) ( ) ( )
x
tzyxT
zzK
∂
∂
−×
,,,
β
( )
( ) ( )
( )[ ]
( ) ( )
( ) ( ) ( )
+
∂
∂
−
∂∂
∂
+
∂
∂
+
=
∂
∂
y
tzyxT
zzK
yx
tzyxu
x
tzyxu
z
zE
t
tzyxu
z xyy ,,,,,,,,,
12
,,, 1
2
2
1
2
2
2
2
β
σ
ρ
( ) ( )
( )[ ]
( ) ( )
( )[ ]
( ) ( )
+
∂
∂
+
∂
∂
+∂
∂
+
+
+∂
∂
+
y
tzyxu
z
tzyxu
z
zE
z
zK
z
zE
y
tzyxu zyy ,,,,,,
12112
5,,, 11
2
1
2
σσ
( ) ( )
( )[ ]
( )
( )
( )
yx
tzyxu
zK
zy
tzyxu
z
zE
zK
yy
∂∂
∂
+
∂∂
∂
+
−+
,,,,,,
16
1
2
1
2
σ
( )
( ) ( ) ( ) ( ) ( ) ( )
×
∂∂
∂
+
∂∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
zy
tzyxu
zx
tzyxu
y
tzyxu
x
tzyxuzE
t
tzyxu
z
yxzzz
,,,,,,,,,,,,
2
,,, 1
2
1
2
2
1
2
2
1
2
2
2
2
ρ
( )
( )
( ) ( ) ( ) ( )
( )[ ]
×
+
+
∂
∂
+
∂
∂
+
∂
∂
∂
∂
+
+
×
z
zE
z
tzyxu
y
tzyxu
x
tzyxu
zK
zz
xyx
σσ 16
,,,,,,,,,
1
1 111
( ) ( ) ( ) ( ) ( )
×
∂
∂
−
∂
∂
−
∂
∂
−
∂
∂
−
∂
∂
∂
∂
×
z
tzyxT
z
tzyxu
y
tzyxu
x
tzyxu
z
tzyxu
z
zyxz ,,,,,,,,,,,,,,,
6 1111
( ) ( )zzK β× .
Integration of left and right sides of the above equations on time t leads to the following results
( )
( )
( ) ( )
( )[ ]
( ) ( ) ( )
( )[ ]
×
+
−+∫ ∫
∂
∂
+
+=
z
zE
zKddzyxu
xz
zE
zK
z
tzyxu
t
xx
σ
ϑττ
σρ
ϑ
13
,,,
16
51
,,,
0 0
12
2
2
( )
( ) ( ) ( ) ×
∫ ∫
∂
∂
+∫ ∫
∂
∂
+∫ ∫
∂∂
∂
×
t
z
t
y
t
y ddzyxu
z
ddzyxu
y
ddzyxu
yxz 0 0
12
2
0 0
12
2
0 0
1
2
,,,,,,,,,
1 ϑϑϑ
ϑττϑττϑττ
ρ
( )
( ) ( )[ ] ( )
( ) ( ) ( )
( )[ ]
( ) ( )
( )
×−
+
+∫ ∫
∂∂
∂
+
+
×
z
z
zK
z
zE
zKddzyxu
zxzzz
zE t
z
ρ
β
σ
ϑττ
ρσρ
ϑ
13
,,,
1
12 0 0
1
2
( )
( )
( ) ( )
( )[ ]
( )
( )
×−∫ ∫
∂
∂
+
+−∫ ∫
∂
∂
×
∞
z
ddzyxu
xz
zE
zK
z
ddzyxT
x
x
t
ρ
ϑττ
σρ
ϑττ
ϑϑ 1
,,,
16
51
,,,
0 0
12
2
0 0
( ) ( )
( )[ ]
( ) ( )
+∫ ∫
∂
∂
−∫ ∫
∂∂
∂
+
−×
∞∞
0 0
12
2
0 0
1
2
,,,,,,
13
ϑϑ
ϑττϑττ
σ
ddzyxu
y
ddzyxu
yxz
zE
zK yy](https://image.slidesharecdn.com/2215msej01-150708094508-lva1-app6891/85/VARIATION-OF-MISMATCH-INDUCED-STRESS-IN-A-HETEROSTRUCTURE-WITH-CHANGING-TEMPERATURE-OF-GROWTH-7-320.jpg)
![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015
8
( ) ( )
( ) ( )[ ]
( ) ( )
( )
( ) −∫ ∫
∂
∂
+
+
∫ ∫
∂
∂
+
∞∞
0 00 0
12
2
,,,
12
,,,
ϑϑ
ϑττ
ρ
β
σρ
ϑττ ddzyxT
xz
z
zK
zz
zE
ddzyxu
z
z
( )
( ) ( ) ( )
( )[ ] xz u
z
zE
zKddzyxu
zxz
0
0 0
1
2
13
,,,
1
+
+
+∫ ∫
∂∂
∂
−
∞
σ
ϑττ
ρ
ϑ
( ) ( )
( ) ( )[ ]
( ) ( ) +
∫ ∫
∂∂
∂
+∫ ∫
∂
∂
+
=
t
x
t
xy ddzyxu
yx
ddzyxu
xzz
zE
tzyxu
0 0
1
2
0 0
12
2
2 ,,,,,,
12
,,,
ϑϑ
ϑττϑττ
σρ
( )
( )
( )
( )
( ) ( )
( )[ ]
( ) +
+
+
∫ ∫
∂
∂
+∫ ∫
∂∂
∂
+ zK
z
zE
ddzyxu
yz
ddzyxu
yxz
zK t
x
t
y
σ
ϑττ
ρ
ϑττ
ρ
ϑϑ
112
5
,,,
1
,,,
0 0
12
2
0 0
1
2
( )
( )
( )
( ) ( ) ( ) ( )
( )
×−
∫ ∫
∂
∂
+∫ ∫
∂
∂
+∂
∂
+
z
z
zKddzyxu
y
ddzyxu
zz
zE
zz
t
z
t
y
ρ
β
ϑττϑττ
σρ
ϑϑ
0 0
1
0 0
1 ,,,,,,
12
1
( )
( )
( ) ( )
( )[ ]
( ) ( )
( )
×−∫ ∫
∂∂
∂
+
−+∫ ∫×
z
zE
ddzyxu
zyz
zE
zK
z
ddzyxT
t
y
t
ρ
ϑττ
σρ
ϑττ
ϑϑ
2
,,,
16
1
,,,
0 0
1
2
0 0
( )
( ) ( ) ( ) ×∫ ∫−
∫ ∫
∂∂
∂
+∫ ∫
∂
∂
+
×
∞∞∞
0 00 0
1
2
0 0
12
2
,,,,,,,,,
1
1 ϑϑϑ
ϑττϑττϑττ
σ
ddzyxTddzyxu
yx
ddzyxu
xz
xx
( ) ( )
( )
( )
( )
( )
( )
( ) ( ){ +∫ ∫
∂
∂
−∫ ∫
∂∂
∂
−×
∞∞
zKddzyxu
yz
ddzyxu
yxz
zK
z
z
zK xy
0 0
12
2
0 0
1
2
,,,
1
,,,
ϑϑ
ϑττ
ρ
ϑττ
ρρ
β
( )
( )[ ] ( )
( ) ( )
( )[ ]
( )
( )
×−+∫ ∫
∂∂
∂
+
−−
+
+
∞
z
uddzyxu
zyz
zE
zK
zz
zE
yy
ρ
ϑττ
σρσ
ϑ
2
1
,,,
16
1
112
5
0
0 0
1
2
( )
( )
( ) ( )
∫ ∫
∂
∂
+∫ ∫
∂
∂
+∂
∂
×
∞∞
0 0
1
0 0
1 ,,,,,,
1
ϑϑ
ϑττϑττ
σ
ddzyxu
y
ddzyxu
zz
zE
z
zy
( ) ( ) ( ) ( )
+∫ ∫
∂∂
∂
+∫ ∫
∂
∂
+∫ ∫
∂
∂
=
∞∞∞
0 0
1
2
0 0
12
2
0 0
12
2
2 ,,,,,,,,,,,,
ϑϑϑ
ϑττϑττϑττ ddzyxu
zx
ddzyxu
y
ddzyxu
x
tzyxu xzzz
( ) ( )
( ) ( )[ ] ( )
( ) ( )
+∫ ∫
∂
∂
∂
∂
+
+
∫ ∫
∂∂
∂
+
∞∞
0 0
1
0 0
1
2
,,,
1
12
,,,
ϑϑ
ϑττ
ρσρ
ϑττ ddzyxu
x
zK
zzzz
zE
ddzyxu
zy
xy
( ) ( )
( ) ( )[ ] ( )
( ) ( )
+∫ ∫
∂
∂
∂
∂
+
+
∫ ∫
∂∂
∂
+
∞∞
0 0
1
0 0
1
2
,,,
1
12
,,,
ϑϑ
ϑττ
ρσρ
ϑττ ddzyxu
x
zK
zzzz
zE
ddzyxu
zy
xy
( ) ( ) ( )
−∫ ∫
∂
∂
∂
∂
+
∫ ∫
∂
∂
+∫ ∫
∂
∂
+
∞∞∞
0 0
1
0 0
1
0 0
1 ,,,6,,,,,,
ϑϑϑ
ϑττϑττϑττ ddzyxu
zz
ddzyxu
z
ddzyxu
y
zxx
( ) ( ) ( ) ( )
( )
×
+
∫ ∫
∂
∂
−∫ ∫
∂
∂
−∫ ∫
∂
∂
−
∞∞∞
z
zE
ddzyxu
z
ddzyxu
y
ddzyxu
x
zyx
σ
ϑττϑττϑττ
ϑϑϑ
1
,,,,,,,,,
0 0
1
0 0
1
0 0
1
( )
( ) ( )
( )
( ) zuddzyxT
zz
z
zK
z
0
0 0
,,,
6
1
+∫∫
∂
∂
−×
∞ϑ
ϑττ
ρ
β
ρ
.
In this paper we calculate the second-order approximations of concentration of material of epitax-
ial layer and components of displacement vector by using the method of averaging of function
corrections. Recently we obtain, that the second-order approximation of a considered solution is
usually enough good approximation to make qualitative analysis and to obtain some quantitative
results. We compare all analytical results by numerical simulations.](https://image.slidesharecdn.com/2215msej01-150708094508-lva1-app6891/85/VARIATION-OF-MISMATCH-INDUCED-STRESS-IN-A-HETEROSTRUCTURE-WITH-CHANGING-TEMPERATURE-OF-GROWTH-8-320.jpg)
![Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015
9
3. DISCUSSION
In this section we used relations obtained in the previous section for analysis influence of diffu-
sion of material of the epitaxial layer of heterostructure from Fig. 1 into the substrate on relaxa-
tion of mismatch-induced stress. Increasing of temperature of growth stimulates acceleration of
the diffusion. The diffusion leads to spreading of interface between layers of heterostructure. At
the same time presents of smooth interface gives us possibility to decrease value of mismatch-
induced stress in neighborhood of the interface between layers of heterostructure. The qualitative-
ly same results have been experimentally obtained in [14]. The decreasing one can obtain due to
more gradual changing of properties of the heterostructure in direction, which is perpendicular to
the interface of the heterostructure. The Fig. 2 shows dependences of components of displace-
ment vector on coordinate deep into the heterostructure for sharp and smooth interfaces. Probably
spreading of interface, which one can obtain during high-temperature growth, is a reason of de-
creasing of mismatch-induced stress. Decreasing of sharpness of the interface could be partially
compensated by nonlinearity of diffusion of material of epitaxial layer in the substrate for the
large quantity of diffusing material. Several spatial distributions of material of epitaxial layer in
the substrate are presented in Fig. 3.
z
0.0
0.2
0.4
0.6
0.8
1.0
Uz
1
2
0.0 a
Fig.2. Normalized dependences of components of displacement vector uz on coordinate z for sharp (curve
1) and smooth (curve 2) interfaces between layers of heterostructure
4. CONCLUSIONS
In this paper we discuss possibility to decrease value of mismatch-induced stress due to increas-
ing of temperature of growth. At the same time one can find decreasing of sharpness of interface
between layers of heterostructure. It has been discussed possibility at least partial compensation
of decreasing of the sharpness.
ACKNOWLEDGEMENTS
This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government,
grant of Scientific School of Russia, the agreement of August 27, 2013 № 02.В.49.21.0003 be-
tween The Ministry of education and science of the Russian Federation and Lobachevsky State
University of Nizhni Novgorod and educational fellowship for scientific research of Nizhny Nov-
gorod State University of Architecture and Civil Engineering.](https://image.slidesharecdn.com/2215msej01-150708094508-lva1-app6891/85/VARIATION-OF-MISMATCH-INDUCED-STRESS-IN-A-HETEROSTRUCTURE-WITH-CHANGING-TEMPERATURE-OF-GROWTH-9-320.jpg)
VARIATION OF MISMATCH-INDUCED STRESS IN A HETEROSTRUCTURE WITH CHANGING TEMPERATURE OF GROWTH
- 1. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015 1 VARIATION OF MISMATCH-INDUCED STRESS IN A HE- TEROSTRUCTURE WITH CHANGING TEMPERATURE OF GROWTH E.L. Pankratov1,3 and 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 3 Nizhny Novgorod Academy of the Ministry of Internal Affairs of Russia, 3 Ankudi- novskoe Shosse, Nizhny Novgorod, 603950, Russia ABSTRACT In this paper we analyzed influence of diffusion of material of an epitaxial layer of a heterostructure during high-temperature growth into next layer (next epitaxial layer or substrate) of the heterostructure on mis- match-induced stress. It has been shown, that value of mismatch-induced stress in heterostructure depends on temperature of growth, because the considered diffusion depends on the temperature. We also introduce an analytical approach to model the diffusion and relaxation of the mismatch-induced stress. KEYWORDS Heterostructure; Mismatch-Induced Stress; Temperature Of Growth; Mixing Of Materials Of Layers Of Heterostructure 1. INTRODUCTION In the present time large number of solid state electronic devices have been manufactured based on heterostructures. The widely using of heterostructures leads to necessity to improve of their properties. It is known, that mismatch-induced stress presents in all heterostructures. The stress in heterostructures could leads to generation misfit dislocations. One way to decrease mismatch- induced stress is choosing materials of heterostructure with as small as possible mismatch of lat- tice constants [1-3]. Another way to decrease mismatch-induced stress is using a buffer layer, manufactured by using another materials, between layers of heterostructure. Lattice constant of the buffer layer should be average in comparison with lattice constants of nearest layers of hete- rostructure [4,5]. In this paper we consider a heterostructure with two layers (see Fig. 1). The layers are a substrate and an epitaxial layer. At a high temperature of growth (for example, during epitaxy from gas phase) of heterostructure one can find intensive diffusion of material of the epitaxial layer into the substrate. Our main aim framework the present paper is analysis of variation of mismatch- induced stress in the considered heterostructure with diffusion of material of the epitaxial into the substrate.
- 2. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015 2 2. METHOD OF SOLUTION Let us to describe diffusion of material of epitaxial layer in the substrate by solution of the fol- lowing boundary problem [6-11] ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC ,,,,,,,,,,,, ( ) ( ) ( ) ( ) ∫∇ ∂ ∂ Ω+ ∫∇ ∂ ∂ Ω+ zz L S S L S S WdtWyxCtzyx Tk D y WdtWyxCtzyx Tk D x 00 ,,,,,,,,,,,, µµ (1) Fig. 1. Heterostructure, which consist of a substrate and an epitaxial layer ( ) 0 ,,, = ∂ ∂ = xLx x tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =y y tzyxC , ( ) 0 ,,, = ∂ ∂ = yLx y tzyxC , ( ) 0 ,,, 0 = ∂ ∂ =z z tzyxC , ( ) 0 ,,, = ∂ ∂ = zLx z tzyxC , C(x,y,z,0)=fC (x,y,z), C(0,y,z,t)=C0. Here C(x,y,z,t) is the spatio-temporal distribution of concentration of material of epitaxial layer; atomic volume of the dopant describes by Ω; surficial gradient describes by ∇S; the integral ( )∫ zL zdtzyxC 0 ,,, describes surficial concentration of dopant on interface between materials of he- terostructure; µ(x,y,z,t) is the chemical potential; D and DS are the coefficients of volumetric and surficial diffusions (reason of the surficial diffusion is the mismatch-induced stress). Values of the volumetric and surficial diffusion coefficients are differ in different materials. The diffusion coefficients will be changed in during heating and cooling of heterostructure. Different levels of doping also leads to changing of the diffusion coefficients. We approximate dopant diffusion coefficients by the following functions [7] ( ) ( ) ( ) += TzyxP tzyxC TzyxDD LC ,,, ,,, 1,,, γ γ ξ , ( ) ( ) ( ) += TzyxP tzyxC TzyxDD SLSS ,,, ,,, 1,,, γ γ ξ . (2)
- 3. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015 3 The functions DL(x,y,z,T) and DLS(x,y,z,T) describes spatial and temperature dependences of diffu- sion coefficients. The functions could be obtained by accounting all layers of heterostruicture and Arrhenius law. T is the temperature of grown. P (x,y,z,T) is the limit of solubility of material of the epitaxial layer in the substrate. Parameter γ depends on properties of materials and will be integer in the following interval γ∈[1,3] (the dependence described in details in [7]). Chemical potential µ in the Eq.(1) depends on properties of materials of heterostructure and could be approximated by the following relation [10] µ=E(z)Ωσij [uij(x,y,z,t)+uji(x,y,z,t)]/2. (3) Function E(z) describes spatial dependence of Young modulus. Tensor σij describes the stress tensor; ∂ ∂ + ∂ ∂ = i j j i ij x u x u u 2 1 is the deformation tensor; ui, uj are the components ux(x,y,z,t), uy(x,y,z,t) and uz(x,y,z,t) of the displacement vector ( )tzyxu ,,, r ; xi, xj are the coordinate x, y, z. The Eq. (3) could be written in the following equivalent form ( ) ( ) ( ) ( ) ( ) ( ) − ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂Ω = i j j i i j j i x tzyxu x tzyxu x tzyxu x tzyxu zEtzyx ,,,,,, 2 1,,,,,, 2 ,,,µ ( ) ( ) ( ) ( ) ( ) ( )[ ] −− − ∂ ∂ − +− ij k kij ij TtzyxTzzK x tzyxu z z δβε σ δσ δε 000 ,,,3 ,,, 21 . Here σ is Poisson coefficient; the normalized difference ε0 = (as-aEL)/aEL describes the mismatch parameter; parameters as, aEL in the above difference are lattice distances of the substrate and the epitaxial layer; K is the modulus of uniform compression; β is the coefficient of thermal expan- sion; Tr is the equilibrium temperature, which coincide (in our case) with room temperature. Components of the displacement vector have been described by the following equations [11] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ z tzyx y tzyx x tzyx t tzyxu z z tzyx y tzyx x tzyx t tzyxu z z tzyx y tzyx x tzyx t tzyxu z zzzyzxz yzyyyxy xzxyxxx ,,,,,,,,,,,, ,,,,,,,,,,,, ,,,,,,,,,,,, 2 2 2 2 2 2 σσσ ρ σσσ ρ σσσ ρ The stress tensor σij correlates with components of the displacement vector by the following rela- tion ( ) ( )[ ] ( ) ( ) ( ) ( ) × ∂ ∂ + ∂ ∂ − ∂ ∂ + ∂ ∂ + = k k ij k kij i j j i ij x tzyxu x tzyxu x tzyxu x tzyxu z zE ,,,,,, 3 ,,,,,, 12 δ δ σ σ ( ) ( ) ( ) ( )[ ]rTtzyxTzKzzK −−× ,,,β .
- 4. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015 4 The function ρ(z) describes the density of materials of heterostructure. The tenzor δij describes the Kronecker symbol. Accounting relation for σij in the previous system of equations last system of equation could be written as ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) + ∂∂ ∂ + −+ ∂ ∂ + += ∂ ∂ yx tzyxu z zE zK x tzyxu z zE zK t tzyxu z yxx ,,, 13 ,,, 16 5,,, 2 2 2 2 2 σσ ρ ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ) − ∂∂ ∂ + ++ ∂ ∂ + ∂ ∂ + + zx tzyxu z zE zK z tzyxu y tzyxu z zE zzy ,,, 13 ,,,,,, 12 2 2 2 2 2 σσ ( ) ( ) ( ) x tzyxT zzK ∂ ∂ − ,,, β ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) + ∂ ∂ − ∂∂ ∂ + ∂ ∂ + = ∂ ∂ y tzyxT zzK yx tzyxu x tzyxu z zE t tzyxu z xyy ,,,,,,,,, 12 ,,, 2 2 2 2 2 β σ ρ ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) + ∂ ∂ + + + ∂ ∂ + ∂ ∂ +∂ ∂ + 2 2 ,,, 112 5,,,,,, 12 y tzyxu zK z zE y tzyxu z tzyxu z zE z yzy σσ ( ) ( ) ( )[ ] ( ) ( ) ( ) yx tzyxu zK zy tzyxu z zE zK yy ∂∂ ∂ + ∂∂ ∂ + −+ ,,,,,, 16 22 σ (4) ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∂∂ ∂ + ∂∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ zy tzyxu zx tzyxu y tzyxu x tzyxuzE t tzyxu z yxzzz ,,,,,,,,,,,, 2 ,,, 22 2 2 2 2 2 2 ρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∂ ∂ − ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + + × z tzyxT z z tzyxu y tzyxu x tzyxu zK zz xyx ,,,,,,,,,,,, 1 1 β σ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ +∂ ∂ +× z tzyxu y tzyxu x tzyxu z tzyxu z zE z zK zyxz ,,,,,,,,,,,, 6 16 1 σ . Conditions for the system of Eqs.(4) are ( ) 0 ,,,0 = ∂ ∂ x tzyu r ; ( ) 0 ,,, = ∂ ∂ x tzyLu x r ; ( ) 0 ,,0, = ∂ ∂ y tzxu r ; ( ) 0 ,,, = ∂ ∂ y tzLxu y r ; ( ) 0 ,0,, = ∂ ∂ z tyxu r ; ( ) 0 ,,, = ∂ ∂ z tLyxu z r ; ( ) 00,,, uzyxu rr = ; ( ) 0,,, uzyxu rr =∞ . We determine spatio-temporal distributions of concentration of material of epitaxial layer in the substrate by solving the Eq.(1) framework method of averaging of function corrections in the standard form [12,13]. Previously we transform the Eq.(1) to the following form with account initial distributions of the considered concentrations ( ) ( ) ( ) ( ) + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC ,,,,,,,,,,,, ( ) ( ) ( ) ( ) + ∫∇ ∂ ∂ Ω+ ∫∇ ∂ ∂ Ω+ zz L S S L S S WdtWyxCtzyx Tk D y WdtWyxCtzyx Tk D x 00 ,,,,,,,,,,,, µµ ( ) ( )tzyxfC δ,,+ (1a)
- 5. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015 5 After that we replace the required function C (x,y,z,t) in right side of Eq. (1a) on its not yet known average value α1C. In this situation we obtain equation to determine the first- order approxima- tions of the considered concentrations in the following form ( ) ( ) ( ) + ∇ ∂ ∂ Ω+ ∇ ∂ ∂ Ω= ∂ ∂ tzyx Tk D z y tzyx Tk D z xt tzyxC S S CS S C ,,,,,, ,,, 11 1 µαµα ( ) ( )tzyxfC δ,,+ . (1b) After integration of the left and right sides of the Eq.(1b) gives us possibility to obtain relation for the first-order approximation of concentration of material of the epitaxial layer in the substrate in the following form ( ) ( ) ( ) ( ) +∫ +∇ ∂ ∂ Ω= t CS SLSC d TzyxP zyx Tk z TzyxD x tzyxC 0 1 11 ,,, 1,,,,,,,,, τ αξ τµα γ γ ( ) ( ) ( ) ( )zyxfd TzyxP zyx Tk z TzyxD y C t CS SLSC ,, ,,, 1,,,,,, 0 1 1 +∫ +∇ ∂ ∂ + τ αξ τµα γ γ (1c) We determine average values of the first-order approximations of considered concentration by using the following relations [12,13] ( )∫ ∫ ∫ ∫ Θ = Θ 0 0 0 0 11 ,,, 1 x y zL L L zyx C tdxdydzdtzyxC LLL α . (5) Substitution of the relation (1c) into relation (5) gives us possibility to obtain the following rela- tions ( )∫ ∫ ∫= x y zL L L C zyx C xdydzdzyxf LLL 0 0 0 1 ,, 1 α , Farther we used standard iteration procedure of method of averaging of function corrections [12, 13] to obtain the second-order approximation of concentration of material of epitaxial layer in the substrate. Framework this procedure to calculate n-th-order approximation of concentration of material of epitaxial layer in the substrate we replace the required concentration C(x,y, z,t) in the right side of Eq. (1a) on the following sum αnC+Cn-1(x,y,z,t). This substitution gives us possibility to obtain equation for the second-order approximation of the required concentration ( ) ( ) ( )[ ] ( ) ( ) + ∂ ∂ + + ∂ ∂ = ∂ ∂ x tzyxC TzyxP tzyxC TzyxD xt tzyxC C L ,,, ,,, ,,, 1,,, ,,, 1122 γ γ α ξ ( ) ( )[ ] ( ) ( ) ( ) × ∂ ∂ ∂ ∂ + ∂ ∂ + + ∂ ∂ + z tzyxC zy tzyxC TzyxP tzyxC TzyxD y C L ,,,,,, ,,, ,,, 1,,, 1112 γ γ α ξ ( ) ( )[ ] ( ) ( ) ( ) × ∂ ∂ Ω++ + +× Tk D x tzyxf TzyxP tzyxC TzyxD S C C L δ α ξ γ γ ,, ,,, ,,, 1,,, 12 (1d) ( ) ( ) ( ) ( ) ∫∇ ∂ ∂ Ω+ ∫∇× zz L S S L S WdtWyxCtzyx Tk D y WdtWyxCtzyx 0 1 0 1 ,,,,,,,,,,,, µµ .
- 6. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015 6 After integration of the left and right sides of the Eq.(1d) gives us possibility to obtain relation for the second-order approximation of concentration of material of the epitaxial layer in the substrate in the following form ( ) ( ) ( )[ ] ( ) ( ) +∫ ∂ ∂ + + ∂ ∂ = t C L d x zyxC TzyxP zyxC TzyxD x tzyxC 0 112 2 ,,, ,,, ,,, 1,,,,,, τ ττα ξ γ γ ( ) ( )[ ] ( ) ( ) ( ) ∫ × ∂ ∂ ∂ ∂ +∫ ∂ ∂ + + ∂ ∂ + tt C L z zyxC z d y zyxC TzyxP zyxC TzyxD y 0 1 0 112 ,,,,,, ,,, ,,, 1,,, τ τ ττα ξ γ γ ( ) ( )[ ] ( ) ( ) ( )∫ ×∫∇ ∂ ∂ Ω+ + +× t L S C L z WdWyxCzyx x d TzyxP zyxC TzyxD 0 0 1 12 ,,,,,, ,,, ,,, 1,,, ττµτ τα ξ γ γ ( ) ( ) ( )zyxfdWdWyxCzyx Tk D y d Tk D C t L S SS z ,,,,,,,, 0 0 1 +∫ ∫∇ ∂ ∂ Ω+× τττµτ . (1d) We determine average value of the second-order approximation of the considered concentration by using standard relation [12,13]. ( ) ( )[ ]∫ ∫ ∫ ∫ − Θ = Θ 0 0 0 0 122 ,,,,,, 1 x y zL L L zyx C tdxdydzdtzyxCtzyxC LLL α . (6) Substitution of the relation (1e) into the relation (6) gives us possibility to obtain zero value of the required average value α2C. Now we determine components of displacement vector by solution of the system of Eqs.(4). To determine components of displacement vector we used method of averaging of function correc- tions in standard form. Framework the approach we replace the above components in right sides of Eqs.(4) on their not yet known average values αi. The substitution leads to the following re- sults ( ) ( ) ( ) ( ) ( ) x tzyxT zzK t tzyxu z x ∂ ∂ −= ∂ ∂ ,,,,,, 2 1 2 βρ , ( ) ( ) ( ) ( ) ( ) y tzyxT zzK t tzyxu z y ∂ ∂ −= ∂ ∂ ,,,,,, 2 1 2 βρ , ( ) ( ) ( ) ( ) ( ) z tzyxT zzK t tzyxu z z ∂ ∂ −= ∂ ∂ ,,,,,, 2 1 2 βρ . Integration of the left and right sides of the above equations leads to obtain relation for the second-order approximation of the first-order approximations of components of displacement vector. The first-order approximations could be written as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) +∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ = +∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ = +∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ = ∞ ∞ ∞ z t z y t y x t x uddzyxT zz z zKddzyxT zz z zKtzyxu uddzyxT yz z zKddzyxT yz z zKtzyxu uddzyxT xz z zKddzyxT xz z zKtzyxu 0 0 00 0 1 0 0 00 0 1 0 0 00 0 1 ,,,,,,,,, ,,,,,,,,, ,,,,,,,,, ϑϑ ϑϑ ϑϑ ϑττ ρ β ϑττ ρ β ϑττ ρ β ϑττ ρ β ϑττ ρ β ϑττ ρ β
- 7. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015 7 We obtain the second-order approximations of components of displacement vector could be cal- culated by replacement of the required components in the Eqs. (4) on the following sums αi+ ui(x,y,z,t) [12-14]. This replacement leads to the following result ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] − +∂∂ ∂ − ∂ ∂ + += ∂ ∂ z zE yx tzyxu x tzyxu z zE zK t tzyxu z yxx σσ ρ 13 ,,,,,, 16 5,,, 1 2 2 1 2 2 2 2 ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ) − ∂∂ ∂ + ++ ∂ ∂ + ∂ ∂ + + − zx tzyxu z zE zK z tzyxu y tzyxu z zE zK zzy ,,, 13 ,,,,,, 12 1 2 2 1 2 2 1 2 σσ ( ) ( ) ( ) x tzyxT zzK ∂ ∂ −× ,,, β ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) + ∂ ∂ − ∂∂ ∂ + ∂ ∂ + = ∂ ∂ y tzyxT zzK yx tzyxu x tzyxu z zE t tzyxu z xyy ,,,,,,,,, 12 ,,, 1 2 2 1 2 2 2 2 β σ ρ ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( ) + ∂ ∂ + ∂ ∂ +∂ ∂ + + +∂ ∂ + y tzyxu z tzyxu z zE z zK z zE y tzyxu zyy ,,,,,, 12112 5,,, 11 2 1 2 σσ ( ) ( ) ( )[ ] ( ) ( ) ( ) yx tzyxu zK zy tzyxu z zE zK yy ∂∂ ∂ + ∂∂ ∂ + −+ ,,,,,, 16 1 2 1 2 σ ( ) ( ) ( ) ( ) ( ) ( ) ( ) × ∂∂ ∂ + ∂∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ zy tzyxu zx tzyxu y tzyxu x tzyxuzE t tzyxu z yxzzz ,,,,,,,,,,,, 2 ,,, 1 2 1 2 2 1 2 2 1 2 2 2 2 ρ ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] × + + ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + + × z zE z tzyxu y tzyxu x tzyxu zK zz xyx σσ 16 ,,,,,,,,, 1 1 111 ( ) ( ) ( ) ( ) ( ) × ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ ∂ ∂ × z tzyxT z tzyxu y tzyxu x tzyxu z tzyxu z zyxz ,,,,,,,,,,,,,,, 6 1111 ( ) ( )zzK β× . Integration of left and right sides of the above equations on time t leads to the following results ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] × + −+∫ ∫ ∂ ∂ + += z zE zKddzyxu xz zE zK z tzyxu t xx σ ϑττ σρ ϑ 13 ,,, 16 51 ,,, 0 0 12 2 2 ( ) ( ) ( ) ( ) × ∫ ∫ ∂ ∂ +∫ ∫ ∂ ∂ +∫ ∫ ∂∂ ∂ × t z t y t y ddzyxu z ddzyxu y ddzyxu yxz 0 0 12 2 0 0 12 2 0 0 1 2 ,,,,,,,,, 1 ϑϑϑ ϑττϑττϑττ ρ ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ×− + +∫ ∫ ∂∂ ∂ + + × z z zK z zE zKddzyxu zxzzz zE t z ρ β σ ϑττ ρσρ ϑ 13 ,,, 1 12 0 0 1 2 ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ×−∫ ∫ ∂ ∂ + +−∫ ∫ ∂ ∂ × ∞ z ddzyxu xz zE zK z ddzyxT x x t ρ ϑττ σρ ϑττ ϑϑ 1 ,,, 16 51 ,,, 0 0 12 2 0 0 ( ) ( ) ( )[ ] ( ) ( ) +∫ ∫ ∂ ∂ −∫ ∫ ∂∂ ∂ + −× ∞∞ 0 0 12 2 0 0 1 2 ,,,,,, 13 ϑϑ ϑττϑττ σ ddzyxu y ddzyxu yxz zE zK yy
- 8. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015 8 ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) −∫ ∫ ∂ ∂ + + ∫ ∫ ∂ ∂ + ∞∞ 0 00 0 12 2 ,,, 12 ,,, ϑϑ ϑττ ρ β σρ ϑττ ddzyxT xz z zK zz zE ddzyxu z z ( ) ( ) ( ) ( ) ( )[ ] xz u z zE zKddzyxu zxz 0 0 0 1 2 13 ,,, 1 + + +∫ ∫ ∂∂ ∂ − ∞ σ ϑττ ρ ϑ ( ) ( ) ( ) ( )[ ] ( ) ( ) + ∫ ∫ ∂∂ ∂ +∫ ∫ ∂ ∂ + = t x t xy ddzyxu yx ddzyxu xzz zE tzyxu 0 0 1 2 0 0 12 2 2 ,,,,,, 12 ,,, ϑϑ ϑττϑττ σρ ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) + + + ∫ ∫ ∂ ∂ +∫ ∫ ∂∂ ∂ + zK z zE ddzyxu yz ddzyxu yxz zK t x t y σ ϑττ ρ ϑττ ρ ϑϑ 112 5 ,,, 1 ,,, 0 0 12 2 0 0 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×− ∫ ∫ ∂ ∂ +∫ ∫ ∂ ∂ +∂ ∂ + z z zKddzyxu y ddzyxu zz zE zz t z t y ρ β ϑττϑττ σρ ϑϑ 0 0 1 0 0 1 ,,,,,, 12 1 ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ×−∫ ∫ ∂∂ ∂ + −+∫ ∫× z zE ddzyxu zyz zE zK z ddzyxT t y t ρ ϑττ σρ ϑττ ϑϑ 2 ,,, 16 1 ,,, 0 0 1 2 0 0 ( ) ( ) ( ) ( ) ×∫ ∫− ∫ ∫ ∂∂ ∂ +∫ ∫ ∂ ∂ + × ∞∞∞ 0 00 0 1 2 0 0 12 2 ,,,,,,,,, 1 1 ϑϑϑ ϑττϑττϑττ σ ddzyxTddzyxu yx ddzyxu xz xx ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ +∫ ∫ ∂ ∂ −∫ ∫ ∂∂ ∂ −× ∞∞ zKddzyxu yz ddzyxu yxz zK z z zK xy 0 0 12 2 0 0 1 2 ,,, 1 ,,, ϑϑ ϑττ ρ ϑττ ρρ β ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) ×−+∫ ∫ ∂∂ ∂ + −− + + ∞ z uddzyxu zyz zE zK zz zE yy ρ ϑττ σρσ ϑ 2 1 ,,, 16 1 112 5 0 0 0 1 2 ( ) ( ) ( ) ( ) ∫ ∫ ∂ ∂ +∫ ∫ ∂ ∂ +∂ ∂ × ∞∞ 0 0 1 0 0 1 ,,,,,, 1 ϑϑ ϑττϑττ σ ddzyxu y ddzyxu zz zE z zy ( ) ( ) ( ) ( ) +∫ ∫ ∂∂ ∂ +∫ ∫ ∂ ∂ +∫ ∫ ∂ ∂ = ∞∞∞ 0 0 1 2 0 0 12 2 0 0 12 2 2 ,,,,,,,,,,,, ϑϑϑ ϑττϑττϑττ ddzyxu zx ddzyxu y ddzyxu x tzyxu xzzz ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) +∫ ∫ ∂ ∂ ∂ ∂ + + ∫ ∫ ∂∂ ∂ + ∞∞ 0 0 1 0 0 1 2 ,,, 1 12 ,,, ϑϑ ϑττ ρσρ ϑττ ddzyxu x zK zzzz zE ddzyxu zy xy ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) +∫ ∫ ∂ ∂ ∂ ∂ + + ∫ ∫ ∂∂ ∂ + ∞∞ 0 0 1 0 0 1 2 ,,, 1 12 ,,, ϑϑ ϑττ ρσρ ϑττ ddzyxu x zK zzzz zE ddzyxu zy xy ( ) ( ) ( ) −∫ ∫ ∂ ∂ ∂ ∂ + ∫ ∫ ∂ ∂ +∫ ∫ ∂ ∂ + ∞∞∞ 0 0 1 0 0 1 0 0 1 ,,,6,,,,,, ϑϑϑ ϑττϑττϑττ ddzyxu zz ddzyxu z ddzyxu y zxx ( ) ( ) ( ) ( ) ( ) × + ∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ −∫ ∫ ∂ ∂ − ∞∞∞ z zE ddzyxu z ddzyxu y ddzyxu x zyx σ ϑττϑττϑττ ϑϑϑ 1 ,,,,,,,,, 0 0 1 0 0 1 0 0 1 ( ) ( ) ( ) ( ) ( ) zuddzyxT zz z zK z 0 0 0 ,,, 6 1 +∫∫ ∂ ∂ −× ∞ϑ ϑττ ρ β ρ . In this paper we calculate the second-order approximations of concentration of material of epitax- ial layer and components of displacement vector by using the method of averaging of function corrections. Recently we obtain, that the second-order approximation of a considered solution is usually enough good approximation to make qualitative analysis and to obtain some quantitative results. We compare all analytical results by numerical simulations.
- 9. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015 9 3. DISCUSSION In this section we used relations obtained in the previous section for analysis influence of diffu- sion of material of the epitaxial layer of heterostructure from Fig. 1 into the substrate on relaxa- tion of mismatch-induced stress. Increasing of temperature of growth stimulates acceleration of the diffusion. The diffusion leads to spreading of interface between layers of heterostructure. At the same time presents of smooth interface gives us possibility to decrease value of mismatch- induced stress in neighborhood of the interface between layers of heterostructure. The qualitative- ly same results have been experimentally obtained in [14]. The decreasing one can obtain due to more gradual changing of properties of the heterostructure in direction, which is perpendicular to the interface of the heterostructure. The Fig. 2 shows dependences of components of displace- ment vector on coordinate deep into the heterostructure for sharp and smooth interfaces. Probably spreading of interface, which one can obtain during high-temperature growth, is a reason of de- creasing of mismatch-induced stress. Decreasing of sharpness of the interface could be partially compensated by nonlinearity of diffusion of material of epitaxial layer in the substrate for the large quantity of diffusing material. Several spatial distributions of material of epitaxial layer in the substrate are presented in Fig. 3. z 0.0 0.2 0.4 0.6 0.8 1.0 Uz 1 2 0.0 a Fig.2. Normalized dependences of components of displacement vector uz on coordinate z for sharp (curve 1) and smooth (curve 2) interfaces between layers of heterostructure 4. CONCLUSIONS In this paper we discuss possibility to decrease value of mismatch-induced stress due to increas- ing of temperature of growth. At the same time one can find decreasing of sharpness of interface between layers of heterostructure. It has been discussed possibility at least partial compensation of decreasing of the sharpness. ACKNOWLEDGEMENTS This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government, grant of Scientific School of Russia, the agreement of August 27, 2013 № 02.В.49.21.0003 be- tween The Ministry of education and science of the Russian Federation and Lobachevsky State University of Nizhni Novgorod and educational fellowship for scientific research of Nizhny Nov- gorod State University of Architecture and Civil Engineering.
- 10. Advances in Materials Science and Engineering: An International Journal (MSEJ), Vol. 2, No. 2, June 2015 10 REFERENCES 1. A.A. Andronov, N.T. Bagraev, L.E. Klyachkin, A.M. Malyarenko, S.V. Robozerov. Growth and electrophysical properties of heterostructures Si/Ge-on-insulator produced by ion implantation and subsequent hydrogen transferring. Semiconductors. Vol. 33 (1). P. 58-63 (1999). 2. M. Nichterwitz, R. Caballero, C.A. Kaufmann, H.-W. Schock, T. Unold. Generation-dependent charge carrier transport in Cu(In,Ga)Se2/CdS/ZnO thin-film solar-cells. J. Appl. Phys. Vol. 113 (16). P. 044515-044530 (2013). 3. J.Y. Fang, G.Y. Lee, J.I. Chyi, C.P. Hsu, Y.W. Kang, K.C. Fang, W.L. Kao, D.J. Yao, C.H. Hsu, Y.F. Huang, C.C. Chen, S.S. Li, J.A. Yeh, F. Ren, Y.L. Wang. Viscosity-dependent drain current noise of AlGaN/GaN high electron mobility transistor in polar liquids. J. Appl. Phys. Vol. 114 (20). P. 204503-204507 (2013). 4. S.Yu. Davydov. On the effect of SiC substrate spontaneous polarization on buffer layer and quasi- free standing graphene. Semiconductors. Vol. 46 (9). P. 1209-1212 (2012). 5. M.G. Mynbaeva, A.A. Sitnikova, K.D. Mynbaev. Photoelectrical properties of porous GaN/SiC structures. Semiconductors. Vol. 45 (10). P. 1369-1372 (2013). 6. I.P. Stepanenko. Basis of Microelectronics (Soviet Radio, Moscow, 1980). 7. Z.Yu. Gotra, Technology of microelectronic devices (Radio and communication, Moscow, 1991). 8. N.A. Avaev, Yu.E. Naumov, V.T. Frolkin. Basis of microelectronics (Radio and communication, Moscow, 1991). 9. V.I. Lachin, N.S. Savelov. Electronics. Rostov-on-Don: Phoenix, 2001. 10. Y.W. Zhang, A.F. Bower. Numerical simulations of island formation in a coherent strained epitax- ial thin film system. Journal of the Mechanics and Physics of Solids. Vol. 47 (11). P. 2273-2297 (1999). 11. L.D. Landau, E.M. Lefshits. Theoretical physics. 7 (Theory of elasticity) (Physmatlit, Moscow, 2001, in Russian). 12. Yu.D. Sokolov. About the definition of dynamic forces in the mine lifting. Applied Mechanics. Vol.1 (1). P. 23-35 (1955). 13. E.L. Pankratov, E.A. Bulaeva. Doping of materials during manufacture p-n-junctions and bipolar transistors. Analytical approaches to model technological approaches and ways of optimization of distributions of dopants. Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013). 14. E.L. Pankratov, E.A. Bulaeva. On prognozisys of manufacturing double-base heterotransistor and optimization of technological process. Advances in Materials Science and Engineering: An Inter- national Journal (MSEJ). Vol. 2 (1). P. 1-12 (2015). 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. Now E.L. Pankratov is in his Full Doctor course in Radiophysical Department of Nizhny Novgorod State University. He has 110 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. E.A. Bulaeva was a contributor of grant of President of Russia (grant № MK- 548.2010.2). She has 65 published papers in area of her researches.