12 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics
until the dots appear again at the same position along the growth direction. This replication
thus produces an ABCABC … dot stacking sequence that is shown schematically in Fig. 1.8b . The
resulting dot arrangement is similar to the atom stacking in face-centred cubic lattices but in
general the ratio between the lateral dot spacing within the 2D sheet of hexagonally ordered dots
and the vertical spacer thickness will not be equal to 1.155 as in fcc lattices. Therefore, the dot
crystal lattice is expanded or compressed in the (111) direction, i.e. the 3D dot arrangement rep-
resents an overall trigonal lattice of dots. As shown in Fig. 3.4e, also for the (110) growth orienta-
tion two well-defi ned side energy minima may occur on the surface above the dots. Accordingly,
this may result in the formation of vertical sheets of 2D rhombohedrally ordered dots in a multi-
layer structure.
1.3.3 Near-fi eld strain interactions
For multilayers with small spacer layer thicknesses, the buried dots can no longer be approxi-
mated as simple point stress sources but their actual size, shape and gradients in compositions
must be taken into account. These parameters obviously differ strongly from one material to
another and moreover depend on the growth and overgrowth conditions, the dot layer thickness
and utilized growth technique (see, e.g., Stangl et al. [92] for a review). For (001) SiGe/Si dots,
e.g., hut cluster islands with { 105 } facets are formed at low growth temperatures and small cov-
erages [21, 124] , whereas for thicker dot layers and higher temperatures dome-shaped islands
with { 113 } facets are formed [125–127] , with several transitional shapes in between [126,
127] . Even steeper dots with { 111 } side facets have been observed for SiGe dots grown by liquid
phase epitaxy [104] . For (001) InAs/GaAs islands, on the other hand, multifaceted islands com-
posed of { 317 } , { 011 } and { 111 } facets have been reported [128, 129] , whereas for (111) PbSe
islands pyramids with triangular base and { 100 } side facets were observed [18, 130] . On high-
indexed surfaces, asymmetric island shapes have been found, such as for InAs on GaAs (113)A
[105] and Ge on (113) Si surfaces [106] . A further complication arises from the fact that during
overgrowth signifi cant changes in dot shape and composition occur due to intermixing with the
surrounding matrix material [92, 107–113] . This intermixing strongly depends on the growth
conditions [109, 110] as well as the chemical composition of the spacer layer [112, 113] and
changes the chemical composition of the dots as well [91, 110–113] . As a consequence, no gen-
eral solution of the near-fi eld strain interactions can be given but each particular experimental
situation and material system must be considered separately.
To calculate the strain fi elds of near-surface dots several methods have been used [92] . If the
elastic constants of dots and matrix material do not differ much, the strain fi elds can be obtained
by convoluting the point-source solution with the given dot shape [18, 92] . Alternatively, one
can also apply the fi nite element method [50, 110, 114, 118, 142] or atomistic calculations
using semi-empirical atom potentials [119–122] . These methods have been employed exten-
sively for InAs/GaAs [93, 117, 119] and Si/Ge [110, 118–121] dots but also for other materials
such as InN/AlN [84, 123] . As shown by Pryor et al. [119] , all three methods give quite similar
results for the strain fi elds well outside of the buried dots, as applies for the surface strain fi elds
well above the dots relevant for multilayer structures. As a general trend, for near-surface dots
the strain fi elds are focused towards the surface normal direction, i.e. the surface energy minima
are confi ned more closely to the region directly above the buried dots. This arises from the fact
that the free surface allows a very effi cient strain relaxation due to the outward or inward relaxa-
tion of the surface lattice planes.
1.3.3.1 (100) surfaces
For (100) surfaces, the changes of the strain energy distributions in dependence of the dot
depth are demonstrated in Fig. 1.9 for InAs dots embedded in GaAs. In this example, the dots
were assumed as truncated square base InAs pyramids with a fi xed base width of 20 nm and a
height of 7 nm (see insert of Fig. 1.9 ), similar to what has been found in cross-sectional scan-
ning tunnelling microscopy studies [72, 73, 111] . For simplicity, variations in the chemical com-
position within the dots due to intermixing [92, 111] were neglected, i.e. pure InAs dots were
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