Self-organized Quantum Dot Multilayer Structures 21
At the critical thickness d
l
dots
/2, and thus α
corr
is as large as α
max
50° for the given InAs
dot array (see arrow in Fig. 1.12 ), compared to α
ps
23° expected for a single buried dot. This
increase in interlayer correlation angle is in good agreement with the experimental values
reported by Wang et al. for InGaAs/GaAs dots at a comparable spacer thickness [87] . According
to Eq. 1.11 , for thicker spacer layers, the dot replication angle should decrease and approach
the value of 23° of the single point-source model. In fact, such difference in correlation angle
has been recently reported by Gutierrez et al. [88] for InGaAs/GaAs multilayers, where for well-
separated isolated dots the interlayer correlation angle was found to be around 23° close to the
single dot model, whereas for densely spaced dots with large strain fi eld overlap a much larger
value of around 50 ° was observed. Finally, it is noted that according to strain calculations, when
L becomes as large as 3
l
dots
/2, the staggered dot stacking of periodic arrays should switch
back to a vertical dot alignment (see Shchukin et al. [59] ). However, since at such large spacer
thicknesses the strain interaction is very weak, there is little chance that this transition can be
actually obtained by experiments.
1.4 Comparison with experimental results
Experimentally, interlayer correlations in multilayer structures have been studied mainly for
low-indexed surface orientations of material systems such as InAs/GaAs (100) [15, 16, 42–45] ,
Si/Ge (100) [19, 22, 32, 35–38] and PbSe/PbEuTe (111) [18, 30, 49–57] . Since in these cases,
the growth direction is parallel to the elastically soft direction, staggered dot stackings are expected
in the far-fi eld limit, whereas the dots should be vertically aligned for near-surface dots. For high-
indexed surfaces, recent experimental studies have been reported for InGaAs/GaAs multilayers
grown on ( n 11)B surfaces, and the results will be compared to theoretical predictions as well.
1.4.1 Vertically aligned dots
Vertically aligned dots are generally expected under three conditions, namely, (i) the materials
exhibit a small elastic anisotropy, (ii) the growth axis is along the elastically hard direction, or
(iii) the spacer layer thickness is small, i.e. less than about 2–3 times the dot height. In all three
cases, the surface strain minima produced by the subsurface dots will be directly above the bur-
ied dots, i.e. the dots will replicate in the vertical growth direction. Experimentally, for multilay-
ers with small spacer thicknesses, a vertical dot alignment has been found in experiments. This
applies not only to (100) InAs/GaAs [15, 16, 31, 42–45, 78] , InGaP/InP [77] , SiGe/Si [19, 22,
32, 34–38] or GaN/AlN [80–85] multilayers, but also for (111) PbSe/PbTe superlattices [50–57] .
Various examples are compiled in Fig. 1.13 for (a) InAs dot-in-a-well superlattices with 30 nm
GaAs spacers (Gutierez et al. [78] ), (b) to (d) InP/Ga
0.52
In
0.48
P dot multilayers on GaAs (100)
with three different spacer layer thicknesses of 16, 4, and 2 nm, respectively (see Zundel et al.
[77] ), as well as (e) an 80 period GaN/AlN dot superlattice on a 6H–SiC substrate consisting of
six monolayers GaN quantum dots alternating with 10 nm thick AlN barriers (see Sarigiannidou
et al. [84] ). Another example was also shown in Fig. 1.2a for an InAs/GaAs dot superlattice with
20 nm spacers (Darhuber et al. [44] ). Clearly, in all cases a well-defi ned vertical dot alignment is
formed and additional results are described in Sections 1.6 and 1.7 for the InAs/GaAs (100) and
SiGe/Si (100) systems. The predominant vertical dot alignment observed for the large majority of
experimental studies is due to the fact that in most works spacer thicknesses smaller than 40 nm
were used, in which case the elastic strain fi elds and thus the strain minima are vertically aligned
due to free surface relaxation. Moreover, for the SiGe/Si (100) system as well as the wurzite GaN/
AlN system, the elastic anisotropy is not very large and, therefore, vertically aligned dots are
observed for all spacer thicknesses up to which interlayer correlations persist. To our knowledge,
other materials with low elastic anisotropy have not been studied yet.
Concerning the growth along the elastically hard direction, i.e. (111) for SiGe, III–V and
II–VI compounds or (100) for IV–VI compounds, it is noted that these growth orientations are less
commonly used in these material systems and therefore little work has been carried out in this
respect. It also turns out, that for these surface orientations, under the usual growth conditions
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