130
5 Mechanical Properties
[434–437] can be followed. Both approaches have been shown to be equivalent [438] (if a periodic
array of dislocations is considered). In [439] it was pointed out that the finite speed of plastic flow also
has to be considered to explain experimental data. Temperature affects the observed critical thickness
and a kinetic model is needed. Another way of introducing dislocations is the plastic relaxation at the
edge of coherent strained islands (cmp. Fig. 14.37).
In the following, isotropic materials and identical elastic constants of substrate and thin film are
assumed, following [438]. The interface plane is the (x,y)-plane, the growth direction is z. The energy
E d of a periodic dislocation array with period p and Burgers vector b = (b 1 , b 2 , b 3 ) is
E d =
Y
8π (1 − ν 2 )
β
2
(5.92)
β
2
=
b
2
1 + (1 − ν) b
2
2 + b
2
3
ln
p
1 − exp(−4π h/ p)
2π q
+
b
2
1 − b
2
3
4π h
p
exp(−4π h/ p)
1 − exp(−4π h/ p)
−
1
2
b
2
1 + b
2
3
4π h
p
2
exp(−4π h/ p)
1 − exp(−4π h/ p)
2
+b
2
3
2π h
p
exp(−2π h/ p)
1 − exp(−2π h/ p)
,
where h is the film thickness and q denotes the cutoff length for the dislocation core, taken as q = b.
The misfit strain including the relaxation due to dislocations with Burger’s vectors b and ˆ
b in the
two orthogonal interface 110 directions n and ˆ
n. We chose the coordinate system such that n =
(1, 0, 0) and ˆ
n = (0, 1, 0) (the z direction remains). With respect to these axes the Burger’s vectors are
±
1
2
,
1
2
,
1
√
2
a 0 /
√
2. The misfit strain
m
i j is reduced due to the dislocation formation to the ‘relaxed’
misfit strain
r
i j with
r
i j =
m
i j +
b i n j + b j n i
2 p
+
ˆ
b i ˆ
n j + ˆ
b j ˆ
n i
2 p
,
(5.93)
(a)
(b)
g
002
Fig. 5.40 a Series of cross-sectional TEM images of 100 nm thick Ge x Si 1−x layers on Si(001) with different ternary
compositions x = 0.1, 0.2, 0.5, and 1.0. The growth temperature was 550 ◦ C. The transition from commensurate
to incommensurate growth is obvious. Adapted from [430]. b Plan view 022 TEM bright field image of a 250 nm
Ge 0.15 Si 0.85 layer on Si (001), annealed at about 700 ◦ C. The arrow denotes the position of a dislocation loop. Reprinted
with permission from [431], c
1989 AVS
5 Mechanical Properties
[434–437] can be followed. Both approaches have been shown to be equivalent [438] (if a periodic
array of dislocations is considered). In [439] it was pointed out that the finite speed of plastic flow also
has to be considered to explain experimental data. Temperature affects the observed critical thickness
and a kinetic model is needed. Another way of introducing dislocations is the plastic relaxation at the
edge of coherent strained islands (cmp. Fig. 14.37).
In the following, isotropic materials and identical elastic constants of substrate and thin film are
assumed, following [438]. The interface plane is the (x,y)-plane, the growth direction is z. The energy
E d of a periodic dislocation array with period p and Burgers vector b = (b 1 , b 2 , b 3 ) is
E d =
Y
8π (1 − ν 2 )
β
2
(5.92)
β
2
=
b
2
1 + (1 − ν) b
2
2 + b
2
3
ln
p
1 − exp(−4π h/ p)
2π q
+
b
2
1 − b
2
3
4π h
p
exp(−4π h/ p)
1 − exp(−4π h/ p)
−
1
2
b
2
1 + b
2
3
4π h
p
2
exp(−4π h/ p)
1 − exp(−4π h/ p)
2
+b
2
3
2π h
p
exp(−2π h/ p)
1 − exp(−2π h/ p)
,
where h is the film thickness and q denotes the cutoff length for the dislocation core, taken as q = b.
The misfit strain including the relaxation due to dislocations with Burger’s vectors b and ˆ
b in the
two orthogonal interface 110 directions n and ˆ
n. We chose the coordinate system such that n =
(1, 0, 0) and ˆ
n = (0, 1, 0) (the z direction remains). With respect to these axes the Burger’s vectors are
±
1
2
,
1
2
,
1
√
2
a 0 /
√
2. The misfit strain
m
i j is reduced due to the dislocation formation to the ‘relaxed’
misfit strain
r
i j with
r
i j =
m
i j +
b i n j + b j n i
2 p
+
ˆ
b i ˆ
n j + ˆ
b j ˆ
n i
2 p
,
(5.93)
(a)
(b)
g
002
Fig. 5.40 a Series of cross-sectional TEM images of 100 nm thick Ge x Si 1−x layers on Si(001) with different ternary
compositions x = 0.1, 0.2, 0.5, and 1.0. The growth temperature was 550 ◦ C. The transition from commensurate
to incommensurate growth is obvious. Adapted from [430]. b Plan view 022 TEM bright field image of a 250 nm
Ge 0.15 Si 0.85 layer on Si (001), annealed at about 700 ◦ C. The arrow denotes the position of a dislocation loop. Reprinted
with permission from [431], c
1989 AVS