5.4 Plasticity
131
Fig. 5.41 Components
xx xz
zx zz
of the strain tensor (with respect to the GaAs lattice constant) of the dislocation array
shown in Fig. 4.14, red/blue: positive/negative value, white: zero. From [321]
(a)
h 1
h 2
h 3
a
b
b
c
c
a
(b)
Fig. 5.42 Schematic formation of misfit dislocations by a elongation of a grown-in threading dislocation and b by the
nucleation and growth of dislocation half-loops. a depicts a threading dislocation. Initially, for thickness h 1 the interface
is coherent ‘a’, for larger thickness h 2 the interface is critical and the force of the interface on the dislocation is equal
to the tension in the dislocation line, ‘b’. For larger thickness, e.g. h 3 , the dislocation line is elongated in the plane of
the interface, ‘c’. In b ‘a’ denotes a subcritical dislocation half-loop, ‘b’ depicts a half-loop being stable under the misfit
stress and for ‘c’ the loop has grown under the misfit stress into a misfit dislocation line along the interface
with an associated stress σ i j . The strain energy E s of the layer due to the relaxed misfit is then
E s =
1
2
h σ i j
r
i j
(5.94)
lim
p→∞
E s = 2h
Y (1 + ν)
1 − ν
f
2
.
(5.95)
The total strain energy E is given by
pE = 2E d + 2E c + p E s
(5.96)
E ∞ = lim
p→∞
E ,
(5.97)
with the core energy E c of the dislocation that needs to be calculated with an atomistic model (not considered further here). This energy is shown in Fig. 5.43a for the material parameters of Ge 0.1 Si 0.9 /Si(001)
(misfit −0.4%) for various layer thicknesses as a function of 1/ p. This plot looks similar to that for
a first-order phase transition (with 1/ p as the order parameter). For a certain critical thickness h c1 the
131
Fig. 5.41 Components
xx xz
zx zz
of the strain tensor (with respect to the GaAs lattice constant) of the dislocation array
shown in Fig. 4.14, red/blue: positive/negative value, white: zero. From [321]
(a)
h 1
h 2
h 3
a
b
b
c
c
a
(b)
Fig. 5.42 Schematic formation of misfit dislocations by a elongation of a grown-in threading dislocation and b by the
nucleation and growth of dislocation half-loops. a depicts a threading dislocation. Initially, for thickness h 1 the interface
is coherent ‘a’, for larger thickness h 2 the interface is critical and the force of the interface on the dislocation is equal
to the tension in the dislocation line, ‘b’. For larger thickness, e.g. h 3 , the dislocation line is elongated in the plane of
the interface, ‘c’. In b ‘a’ denotes a subcritical dislocation half-loop, ‘b’ depicts a half-loop being stable under the misfit
stress and for ‘c’ the loop has grown under the misfit stress into a misfit dislocation line along the interface
with an associated stress σ i j . The strain energy E s of the layer due to the relaxed misfit is then
E s =
1
2
h σ i j
r
i j
(5.94)
lim
p→∞
E s = 2h
Y (1 + ν)
1 − ν
f
2
.
(5.95)
The total strain energy E is given by
pE = 2E d + 2E c + p E s
(5.96)
E ∞ = lim
p→∞
E ,
(5.97)
with the core energy E c of the dislocation that needs to be calculated with an atomistic model (not considered further here). This energy is shown in Fig. 5.43a for the material parameters of Ge 0.1 Si 0.9 /Si(001)
(misfit −0.4%) for various layer thicknesses as a function of 1/ p. This plot looks similar to that for
a first-order phase transition (with 1/ p as the order parameter). For a certain critical thickness h c1 the