5.3 Elasticity
129
(a)
(b)
[010]
=14°
[100]
Fig. 5.39 a Strain energy (in units of the strain energy of the flat pseudomorphic layers) of a scroll of a 4-layer SiGe
structure (Si 0.3 Ge 0.7 , Si 0.6 Ge 0.4 and Si 0.8 Ge 0.2 , each 3 nm thick and a 1 nm Si cap) as a function of radius for winding
directions along 100 and 110. Top (bottom) curves without (with complete) strain relaxation along the cylinder axis.
Vertical lines indicate the positions of the respective energy minima [419]. b SEM image of curled (In, Ga)As/GaAs
nanoscroll rolled φ = 14 ◦ off 100. The stripe from which the film was rolled off is indicated by white dashed lines.
Part (b) from [429]
The strain energy versus bending radius (= κ
−1 ) is shown for a SiGe nanoscroll in Fig. 5.39. First,
the relaxation along the cylinder axis plays a minor role. The smallest strain energy is reached for
scrolling along 100, also yielding the smaller bending radius (larger curvature). Therefore, the film
preferentially scrolls along 100. This explains the observed ‘curl’ behavior of scrolls winding up for
φ = 0 [423, 427] (Fig. 5.39b). The effect of surface strain needs to be included to yield improved
quantitative agreement with experimental values of κ((, d) [428].
5.4 Plasticity
5.4.1 Critical Thickness
Strained epitaxial films are called pseudomorphic when they do not contain defects and the strain
relaxes elastically, e.g. by tetragonal distortion. When the layer thickness increases, however, strain
energy is accumulated that will lead at some point to plastic relaxation via the formation of defects. In
many cases, a grid of misfit dislocations forms at the interface (Figs. 4.18 and 5.40).
In Fig. 5.41 the strain around misfit dislocations at a GaAs/CdTe heterointerface, as calculated from
a TEM image (Fig. 4.14), is shown.
The average distance p of the dislocations is related to the misfit f = (a 1 − a 2 )/a 2 and the edge
component b ⊥ of the Burger’s vector (for a 60
◦ dislocation b ⊥ = a 0 /
√
8)
p =
b ⊥
f
.
(5.91)
Two mechanisms have been proposed for the formation of misfit dislocations (Fig. 5.42), the elongation
of a grown-in threading dislocation [432, 433] and the nucleation and growth of dislocation half-loops
[434]. For the modeling of such systems a mechanical approach based on the forces on dislocations
[432] or an energy consideration based on the minimum strain energy necessary for defect formation
129
(a)
(b)
[010]
=14°
[100]
Fig. 5.39 a Strain energy (in units of the strain energy of the flat pseudomorphic layers) of a scroll of a 4-layer SiGe
structure (Si 0.3 Ge 0.7 , Si 0.6 Ge 0.4 and Si 0.8 Ge 0.2 , each 3 nm thick and a 1 nm Si cap) as a function of radius for winding
directions along 100 and 110. Top (bottom) curves without (with complete) strain relaxation along the cylinder axis.
Vertical lines indicate the positions of the respective energy minima [419]. b SEM image of curled (In, Ga)As/GaAs
nanoscroll rolled φ = 14 ◦ off 100. The stripe from which the film was rolled off is indicated by white dashed lines.
Part (b) from [429]
The strain energy versus bending radius (= κ
−1 ) is shown for a SiGe nanoscroll in Fig. 5.39. First,
the relaxation along the cylinder axis plays a minor role. The smallest strain energy is reached for
scrolling along 100, also yielding the smaller bending radius (larger curvature). Therefore, the film
preferentially scrolls along 100. This explains the observed ‘curl’ behavior of scrolls winding up for
φ = 0 [423, 427] (Fig. 5.39b). The effect of surface strain needs to be included to yield improved
quantitative agreement with experimental values of κ((, d) [428].
5.4 Plasticity
5.4.1 Critical Thickness
Strained epitaxial films are called pseudomorphic when they do not contain defects and the strain
relaxes elastically, e.g. by tetragonal distortion. When the layer thickness increases, however, strain
energy is accumulated that will lead at some point to plastic relaxation via the formation of defects. In
many cases, a grid of misfit dislocations forms at the interface (Figs. 4.18 and 5.40).
In Fig. 5.41 the strain around misfit dislocations at a GaAs/CdTe heterointerface, as calculated from
a TEM image (Fig. 4.14), is shown.
The average distance p of the dislocations is related to the misfit f = (a 1 − a 2 )/a 2 and the edge
component b ⊥ of the Burger’s vector (for a 60
◦ dislocation b ⊥ = a 0 /
√
8)
p =
b ⊥
f
.
(5.91)
Two mechanisms have been proposed for the formation of misfit dislocations (Fig. 5.42), the elongation
of a grown-in threading dislocation [432, 433] and the nucleation and growth of dislocation half-loops
[434]. For the modeling of such systems a mechanical approach based on the forces on dislocations
[432] or an energy consideration based on the minimum strain energy necessary for defect formation