Henini M. Handbook of Self Assembled Semiconductor Nanostructures for Novel devices in Photonics and Electronics

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Self-organized Quantum Dot Multilayer Structures 29

different symbols and plotted on top of each other. For the elastically isotropic case ( Fig. 1.18a ),

the dots are always vertically aligned on top of each other but there is obviously not any lateral

ordering tendency. This is revealed by the Fourier transform power spectrum of the ﬁ nal dot posi-

tions shown in the insert. The clear discrepancy to the isotropic 1  1D growth simulations by

Tersoff et al. [17] is due to the fact that a uniform dot spacing in one dimension is not sufﬁ cient to

produce an ordered structure in two dimensions if there is no additional in-plane anisotropy.

For the SiGe/Si (100) growth simulation ( Fig. 1.18b ), because the assumed denuded zone R

of 50 nm is signiﬁ cantly larger than the lateral splitting of the energy minima induced by the

elastic anisotropy, the dots are nearly perfectly aligned in the growth direction. In addition, how-

ever, there is also a weak lateral ordering tendency along the in-plane  010  surface directions

as is indicated by the arrows in the 2D FFT power spectrum of the dot positions shown in the

upper inset of Fig. 1.18b . A detailed analysis shows that this preferred dot alignment is due to

the fact that above each dot four maxima in strain energy appear on the surface along the  011 

directions away from the energy minima. This is shown by the lower inset of Fig. 1.18b , where

the strain energy distribution on the 20 nm Si spacer above a buried Ge dot is depicted over an

extended surface area and where the strain maxima are indicated by the dark colour. These

maxima cause a repulsive interlayer dot interaction and, therefore, the subsequent surface dots

tend to nucleate along the  010  surface directions relative to each other. The same result is also

found in simulations for (100) InAs/GaAs dot multilayers because the symmetry and basic shape

of the strain distribution are similar to those for the SiGe system. Due to the higher elastic anisot-

ropy, however, the observed lateral ordering tendency is even enhanced.

For the third case of (111)-oriented PbSe/PbEuTe dot superlattice growth simulations shown

in Fig. 1.18c , clearly, the lateral ordering is not only much more efﬁ cient but also the expected

ABCABC … dot stacking is well reproduced (compare relative dot positions in the 10th, 11th and

12th layers). Even more, the lateral dot spacing in the ﬁ nal layer of 63 nm is exactly equal to the

spacing of the surface energy minima L 



3 d tan α (with d

 50 nm and α  36 °) calculated

from the point-source model and thus reproduces very well the experimental results [18, 30, 50]

(see also section 1.6.1 for details). Figure 1.19a–d shows the evolution of the PbSe dot arrange-

ment in the simulations as a function of the superlattice period at N  1, 5, 10 and 15, respec-

tively, with the FFT power spectra of the dot positions depicted as inserts. Whereas the dots in the

initial layer are completely disordered, already after ﬁ ve superlattice periods a clear hexagonal

lateral ordering has taken place as signiﬁ ed by the appearance of satellite peaks in the FFT power

500

250

250

500

500 250 0 250 500

500

250

250

500

y (nm) [112]

x (nm) [110]

500 250 0 250 500

x (nm) [110]

(a) 1st layer

(b) 5th layer

dot size (a.u.)

Number of dots

0 200 600400

N  1

N  20

(e) Dot size distributions

(d) 15th layer

150

100

Figure 1.19 Monte Carlo simulation of self-organized (111) PbSe/PbEuTe dot superlattice growth, showing the

island positions (triangles) on the surface after the N  1st, 5th, 10th, and 15th superlattice period from (a) to

(d), respectively. The insets show the 2D Fourier power spectra of the island positions indicating the progressing

2D hexagonal in-plane ordering. Panel (e) shows the evolution of the dot size histograms as a function of number

of superlattice periods N , showing a signiﬁ cant reduction of the relative width of the distributions after about four

superlattice periods. Adapted from Springholz et al. [49] .

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30 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

spectra (see inserts). With increasing number of periods, these satellite peaks become increas-

ingly more well deﬁ ned and additional higher-order satellites appear. Thus, the hexagonal order-

ing rapidly improves.

Figure 1.19e shows the corresponding evolution of the size distribution as a function of the

number of superlattice periods. For the initial dot layer, the density and size are determined

merely by the chosen capture radius and deposited number of adatoms. With increasing number

of periods, obviously, a transition from the initially rather small dots to about a factor of two

larger dots in the multilayer stack occurs, which is caused by the reduced dot density induced by

the interlayer correlations. During the concomitant lateral ordering process, the size distribution

at ﬁ rst signiﬁ cantly broadens for the ﬁ rst few superlattice layers but then rapidly narrows, indi-

cating a progressive size homogenization. After about ten layers, the relative width of the size dis-

tribution normalized to the average dot size saturates at a value of about half of that of the initial

layer. This indicates that the lateral ordering in multilayers can improve the uniformity of self-

assembled quantum dots. The observed transient broadening of the size distribution is caused

by the extensive rearrangement of the dot positions in the ﬁ rst superlattice layers, which gives

rise to a larger variation in the dot size (see Fig. 1.19e ). This transient behaviour is in remarkable

agreement with our experimental observations described in section 1.6. Other growth simula-

tions focusing of various other aspects of ordering and correlation formation such as the inﬂ u-

ence of the growth temperature can be found in [148–151] .

An important conclusion from the growth simulations is that there actually exist two

different ordering mechanisms in multilayer structures. The ﬁ rst one is based on repulsive strain

ﬁ elds from the subsurface dots, which results in an alignment of the surface dots along certain

surface directions in which the mutual dot repulsion is locally reduced. The second mechanism

is caused by the attractive strain ﬁ elds of buried dots and is operative when the surface strain

energy distribution above each dot splits into several side minima. In the ﬁ rst case, the dots are

still aligned along the vertical growth direction and the lateral ordering tendency is rather weak

and essentially only of short-range type. In the second case, the interlayer dot correlations are

of staggered type and the lateral ordering is more efﬁ cient and of longer range. In both cases,

the anisotropy of the strain ﬁ elds on the surface is a crucial prerequisite for the ordering proc-

ess, without this anisotropy no lateral ordering is found even for idealized growth simulations.

This conclusion also holds for the near-ﬁ eld strain interactions when the dots can no longer be

approximated by point sources. Lateral ordering is counteracted by thermal disorder induced

by random surface diffusion and nucleation at other surface sites. Thus, in general the ordering

process in actual growth experiments will be more gradual and less efﬁ cient than in the above

described growth simulations and it will also depend signiﬁ cantly on the growth conditions.

In addition, it is noted that in the near-ﬁ eld limit, anisotropic surface strain ﬁ elds may also be

introduced by anisotropies in the island shapes. Thus, for higher-indexed growth orientations

and growth conditions that produce highly anisotropic dot shapes, a more efﬁ cient lateral dot

ordering can be expected. This conclusion is also supported by recent growth experiments

[60–65, 142, 144] .

1.6 InGaAs/GaAs multilayers

Self-assembled InGaAs quantum dots have attracted great interest due to the high photolumi-

nescence efﬁ ciency achieved in this material system [1–3]. This allows investigation of the

fundamental electronic properties of single dots or single stacks of dots using high-resolution

optical spectroscopy [12–14]. In addition, efﬁ cient lasers (see, e.g. Refs. [1, 168–170]) and detec-

tors have been realized with this system and there are interesting possibilities for single photon

sources and quantum information processing devices. Therefore, this material system has been

investigated in great detail. In fact, the ﬁ rst interlayer correlations in self-assembled quantum dot

multilayers were observed in InAs/GaAs dot superlattices [42, 43] , which has stimulated detailed

further investigations on this self-organization process.

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Self-organized Quantum Dot Multilayer Structures 31

1.6.1 Pairing probability as a function of spacer thickness

For InAs/GaAs quantum dot superlattices, it was found that the InAs dots are vertically aligned

along the (100) growth direction forming one-dimensional dot columns up to GaAs spacer thick-

nesses of around 40 nm [15, 16, 45] as shown in Fig. 1.20 . For larger spacer thicknesses, this

interlayer correlation is lost, i.e. the dots nucleate independently from those in the previous lay-

ers. This transition has been studied by Xie et al. [15] using cross-sectional transmission electron

microscopy ( Fig. 1.20a–c ) from which the pairing probability of the dots in subsequent layers

was determined as a function of spacer thickness, as shown in Fig. 1.20d . For spacer thicknesses

up to about 15 nm, the interlayer pairing probability was found to be equal to one, but then grad-

ually decreases until at about 55 nm spacer thickness no interlayer correlations were found any

more. Similar results were reported in a cross-sectional scanning tunnelling microscopy study

of Legrand et al. [45] , showing that the pairing probability depends also on the InAs dot size, i.e.

for larger dots the interlayer correlations persist to larger spacer thicknesses. In both studies, the

spacer thickness d

1/2

at which the interlayer pairing probability drops to one half was found to

be in the range of 20 to 30 nm for a growth temperature around 500°C and InAs thicknesses of

2–2.3 monolayers. This yields InAs dots with a diameter of 10 to 20 nm. For samples with dif-

ferent dot size, Legrand et al. [45] found that the crossover spacer thickness d

1/2

scales linearly

with the dot size due to the corresponding increasing strength of the strain interactions (see also

Section 1.3.4).

FFT

(e)

(g)

(f)

250 nm

FFT

250 nm

(a)

(b)

(c)

[110]

InAs single layer

30 period SL

5-fold InAs stack

XSTM

(h)

5-fold InAs stack

InAs

AlAs

50 nm

set islands

set

islands

(d)

Correlated

regime

Regime with

reduced

correlation

Uncorrelated

regime

[011] cross section

1.0

0.5

0.2

Pairing probability

10 100 1000

Spacer thickness (ML)

[001]

[110]

Figure 1.20 Left-hand side : Cross-sectional transmission electron microscopy images of InAs/GaAs multilayers

with GaAs spacer thicknesses of (a) 42, (b) 92 and (c) 36 monolayers (ML), where for the latter the InAs dots are

perfectly aligned along the growth direction. Adapted from Xie et al. [15] . (d) Pairing probability (open squares) of

these InAs dots deduced from the transmission electron microscopy images as a function of GaAs spacer thickness.

The ﬁ lled circles show the ﬁ t to the experimental data using a model described in [15] . Centre : Atomic force

microscopy images of a single 3 ML InAs dot layer (e) and of a 30 period InAs/GaAs (001) dot superlattice (f)

with 6.5 nm GaAs spacers. The horizontal axis is along the [110] direction. The 2D Fourier transform (FFT) power

spectra of the topography images, shown as insets, indicate a lateral ordering of the dots along the  100  directions.

Adapted from Solomon et al. [74] . Right-hand side : Cross-sectional scanning tunnelling microscopy images of ﬁ ve-

fold stacks of self-assembled 2.4 ML InAs dots separated by 10 nm GaAs spacer layers showing the strong changes

in dot shape and size as a function of the number of bilayers: (g) 150  1 5 0 n m

topography image and (f) enlarged

55  5 5 n m

current image. Adapted from Bruls et al. [73] .

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32 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

The pairing probability in the vertical dot columns was explained by Xie et al. [15] using a

model in which the lowering of the strain energy above the buried dots leads to an attraction

of mobile surface adatoms [15, 45] . The local nucleation probability was then assumed to be

proportional to the amount of accumulated InAs atoms at the strain minima. For small spacer

thickness and correspondingly deeper energy minima, all deposited adatoms within the surface

diffusion length are attracted, leading to a unity dot pairing probability. With increasing spacer

thickness, the depth of the energy minima and thus the pairing probability diminishes until the

minima become too weak to induce a correlated dot nucleation. The predictions of this model are

represented as full circles in Fig. 1.20d and were found to agree well with the experimental data

(open squares) [15, 45] . Also, a gradual transition from correlated to uncorrelated multilayers

was found in Monte Carlo superlattice growth simulations [149–151] .

1.6.2 Lateral ordering

With respect to the lateral dot ordering in (100) InAs/GaAs superlattices, Solomon et al. [74]

have shown a clear short-range dot ordering for superlattices with small GaAs spacer thicknesses

d  10 nm. As shown in Fig. 1.20f the dots are then preferentially aligned along the lateral  010 

directions, whereas for the single dot reference layer no lateral order was found ( Fig. 1.20e ). This

is evidenced by the corresponding Fourier transform power spectra of the atomic force micros-

copy images shown in the inset. For the superlattice sample, four satellite peaks are seen, indi-

cating a preferred square arrangement of the dots. This is explained by Monte Carlo growth

simulations shown in Fig. 1.18 and arises from strain-induced repulsion between the buried dots

and the second nearest surface layer dots (see Section 1.5). The same type of lateral ordering was

also found by Darhuber et al. [38] for InAs dot superlattices with 20 nm GaAs spacers using high-

resolution X-ray diffraction reciprocal space mapping. Solomon et al. [31] have also reported a

signiﬁ cant size homogenization of the dots, with a decrease of the FWHM of the dot height distri-

bution from  20% for single InAs dot reference layers to  8% in the 20 period dot superlattice.

This was also accompanied by a 25% narrowing of the photoluminescence line widths in the

superlattice to 54 meV [16, 31] . A similar narrowing was reported by Nakata et al. [75] for super-

lattices with 3 nm spacers, observing a line width decrease from 90 meV to 27 meV, and compa-

rable results were obtained by He et al. [152] . Also, for vertically aligned InP/GaInP quantum dot

superlattices a signiﬁ cant photoluminescence narrowing from 41 to 16 meV was found [77] .

For vertically aligned (100) InGaAs/GaAs dot superlattices with In concentration of around

50% and 120 monolayer GaAs spacer thickness, another type of ordering has been recently

reported by Wang et al. [153–156] . In this case, the dots were found to be aligned in one-

dimensional dot chains along the [

011

] surface direction, as is exempliﬁ ed by the atomic force

microscopy image shown in Fig. 1.21a [145] . This has been attributed to the anisotropy of the

surface diffusion and the dot shapes [153–156] . The same group has also found a hexagonal

ordering when the superlattices are grown under As

instead of As

ﬂ ux [157] .

Concerning the ordering of (100) InGaAs/GaAs multilayers with staggered dot stacking [87,

88] occurring at larger spacer thicknesses as described in Section 1.4.3, no detailed studies on

the lateral dot ordering have been reported yet. From the growth simulations and strain calcula-

tions described in the previous sections, however, it is expected that the ordering is much more

efﬁ cient than for superlattices with vertically aligned dots. First experimental evidence that this

is really the case has been recently reported by Gutierrez et al. [88] , where a square ordering of

InGaAs dots within the growth plane was found for a staggered InGaAs/GaAs (100) multilayer

using plan-view transmission electron microscopy. A particular difference of the lateral ordering

between vertically aligned and staggered InGaAs dot superlattices is that, according to the strain

calculations and Monte Carlo growth simulations, the ordering is of short range type for the

former and of long range for the latter, as is also indicated by the experimentals [74, 88]. This

results from the fact that for vertically aligned superlattices lateral ordering is due to repulsion of

next-nearest surface dots due to the surface strain maxima induced by the buried dots along the

 010  surface directions (see, e.g., inset of Fig. 1.18b ), whereas for the staggered dot stacking,

lateral ordering is caused by the square arrangement of the surface strain minima aligned along

the  010  surface directions (see, e.g., Fig. 1.9c and d ).

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Self-organized Quantum Dot Multilayer Structures 33

An even more efﬁ cient 2D lateral ordering as well as an inclined interlayer dot correlation was

found by the group of Salamo et al. for InGaAs/GaAs superlattices grown on high-indexed ( n 11)B

substrates [142, 144, 145] . This is evidenced by the corresponding series of atomic force micro-

scopy images for 16.5 period superlattices on ( n 11)B substrates with n  9, 7, 5, 4 and 3 shown

in Fig. 1.21b to f , respectively, reported by Lytvyn et al. [145] . Clearly, the dots are preferentially

aligned along the in-plane

〈〉10n directions, resulting in an efﬁ cient square ordering in the super-

lattice stack, although a faint ordering was also found for single dot layers on the high indexed

substrates [145] . The ordering was found to be best for the superlattice grown on the (411)B

orientation and was attributed to the high in-plane anisotropy of the Young’s modulus within

the growth plane with a strong minimum along the

〈〉10n

direction and pronounced maxima

along the

〈〉011

and

〈〉2nn

directions [145] . Lateral ordering of InGaAs/GaAs dots on (311)B

surfaces was also studied by Lee et al. [158] , who found only the formation of one-dimensional

dot chains on the surface. In this case, however, metal organic vapour deposition was used and

the GaAs spacer thickness was only 12 nm. On (311)B InGaAs/GaAs dot superlattices a peculiar

short-range ordering was also recently reported by Lippen et al. [159, 160] . As a consequence,

the exact ordering mechanisms are not yet resolved and remain to be investigated in detail.

1.6.3 Sizes, shapes and critical wetting layer thickness

One general feature of vertically aligned InAs dot superlattices is the signiﬁ cant increase of the

dot size and broadening of shape along the vertical dot columns. This is illustrated in Fig. 1.20g

and h by cross-sectional scanning tunnelling images of ﬁ ve-fold stacked 2.4 ML InAs dots sepa-

rated by 10 nm GaAs spacer layers by Bruls et al. [72, 73] . Similar changes were also found in

numerous cross-sectional scanning tunnelling [67–72] and transmission electron microscopy

[15, 16, 44] studies. From a statistical evaluation of 20 period InAs/GaAs dot superlattices,

500 nm

(a)

100

(b) 911B

(c)

[011]

[2nn]

711B

311B

(f)

411B(e)511B(d)

Figure 1.21 Atomic force microscopy images of 16.5 period In

0.4

0.6

As/GaAs superlattices with ten monolayer

InGaAs alternating with 120 monolayer GaAs grown on (100), (911)B, (711)B, (511)B, (411)B and (311)B

substrates from (a) to (f), respectively. For the different samples on ( n 11) substrates, the horizontal direction of

the atomic force microscopy images is along [011

–

] and the vertical direction along [2

–

nn] . The insets show the 2D

autocorrelation images calculated from the atomic force microscopy images, showing the good nearly square lateral

ordering of the dots along the in-plane  1

–

n0 surface directions rotated by 45° to the horizontal [011

–

] direction.

Adapted from Lytvyn et al. [145] . The oblique interlayer correlation of dots in these samples is shown in Fig. 1.17 .

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34 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

Solomon et al. [31] have found a 50% increase in dot height as well as 25% increase in dot width

as compared to that of a single InAs dot reference layer. Similar changes were reported by Nakata

et al. [75] , who found in addition a signiﬁ cant decrease of the critical wetting layer thickness for

dot nucleation from h

 1.6 monolayers for the ﬁ rst to 1.1 monolayers for the subsequent layers

in the multilayer stack. The latter seems to be the main reason for the substantial increase in dot

size because correspondingly more material is available for dot formation.

1.6.4 Photoluminescence

As a result of the increase in dot size, a signiﬁ cant red shift of the photoluminescence emission of

vertically aligned InAs dots superlattice by around 100 meV has been found with respect to the

photoluminescence of single InAs dot layers [16, 31, 74, 75, 152] . The size increase should also

broaden the overall photoluminescence emission in spite of the narrowing of the size distribu-

tion in the ﬁ nal dot layer. Therefore, contradictory results have been reported by various groups.

While on the one hand some groups have reported a signiﬁ cant narrowing of the photolumi-

nescence emission of InAs multilayers, as mentioned above [16, 31, 75, 152, 161] , others have

found little changes [162, 163] or even a broadening of the photoluminescence emission [164–

166] . This difference may be explained also by taking the electronic coupling between the dots

into account. For very small spacer thicknesses (as applies to most experimental studies), there is

a strong overlap of the electronic wave functions of the dots in adjacent layers and thus the car-

riers are actually delocalized over several stacked dots. As a result, a certain averaging over the

vertical inhomogeneities in the dot sizes occurs. By tunnelling, the excited carriers may be trans-

ferred to the lowest energy eigenstates along the dot columns and thus a narrower photolumi-

nescence is obtained. For intermediate spacer thicknesses around 20 nm, the electronic coupling

becomes small and thus the differently red-shifted photoluminescence emission of successive dot

layers is superimposed. As a result, the emission may be signiﬁ cantly broader as compared to that

of a single dot layer [167] , an effect that should be also strongly temperature dependent. For even

larger spacer thicknesses, the structural interaction between the dots also becomes negligible and

thus the multilayer emission will eventually approach that of a several times repeated single dot

layer [162, 163] . In spite of these complications, in most studies a notable increase in photo-

luminescence efﬁ ciency was achieved with stacked InAs dots [16, 31, 74, 75, 152, 162, 163] .

Therefore, InAs multilayer dot structures have been successfully utilized to improve the perform-

ance of quantum dot lasers (see, e.g., [1, 4, 168–170] for reviews).

A special technique has been developed by Fafard et al. [164] to reduce the variations in dot

sizes in vertically aligned InAs multilayers. It is based on an indium ﬂ ush step after each InAs

layer after partial capping of the dots with a thin GaAs layer. This indium ﬂ ush step consists of

a short growth interruption during which the excess InAs that sticks out of the GaAs layer is

desorbed by a short annealing step at 610°C before continuing with superlattice growth at the

520°C growth temperature. This reduces the height of the InAs dots to the thickness of the GaAs

capping layer and the layer-to-layer dot height uniformity is signiﬁ cantly improved [164] . As a

result, very narrow photoluminescence emission was obtained from these multilayers such that

even the s , p , and d shell structure of the electronic transitions could be observed [164] .

1.7 Ordering in SiGe/Si dot superlattices

For SiGe/Si (100) quantum dot multilayers, the general trends are quite similar as for InAs/GaAs

superlattices. From cross-sectional transmission electron microscopy [19, 32–40] as well as high

resolution X-ray diffraction [36, 38] studies, a predominant vertical alignment of SiGe dots was

found for Si spacer layers up to 50–70 nm. This is demonstrated in Fig. 1.22a by the transmis-

sion electron microscopy micrograph of a vertically aligned SiGe/Si dot superlattice with 20 nm

Si spacers grown by MBE at a temperature of 620°C [147]. The degree of interlayer correlation,

i.e. the interlayer pairing probability of Ge/Si multilayers as a function of spacer thickness, has

been studied by Kienzle et al. [35] and Stangl et al. [36] for a constant Ge thickness of 6.5 ML.

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Self-organized Quantum Dot Multilayer Structures 35

The results are shown in Fig. 1.22b , where the probability of vertical dot alignment, deﬁ ned as

the degree of base area overlap of subsequent dots in the vertical directions as deﬁ ned by the

inset in Fig. 1.22b , is plotted versus spacer layer thickness. For Si spacers up to 30 nm, a perfect

vertical alignment with 100% pairing probability is found. For thicker spacers the pairing proba-

bility gradually decreases until at 100 nm no interlayer correlations are formed. At 70 nm spacer

thickness, a pairing probability of 50% was found. Thus, interlayer correlations persist up to sig-

niﬁ cantly larger spacer thickness than for the InAs/GaAs case, which is attributed to the fact that

the Ge islands in these studies were about a factor of 5–10 times larger in diameter (85 nm) [34]

than the typical InAs dots of InAs/GaAs structures [15, 45] . In the same study, it was also found

that along the vertical Ge dot columns the dot size increases linearly with layer index number

[35] , which is most pronounced for superlattices with very thin spacers.

Apart from the changes in dot size, several studies have shown that in Si/Ge multilayers the

critical thickness for island formation decreases with increasing number of superlattice periods.

The results of a systematic UHV-CVD growth study of Le Thanh et al. [40] using in situ reﬂ ection

high-energy electron diffraction is shown in Fig. 1.23 where the critical thickness of the 2D–3D

transition is plotted as a function of the number N of deposited Si/Ge bilayers for Si spacer thick-

nesses of 22 and 2.5 nm for a constant 4 ML thickness of the Ge layers. For the thicker Si spacers,

the critical thickness decreases by a factor of two within the ﬁ rst four superlattice periods and

saturates thereafter at a value of about two monolayers compared to four monolayers for the ﬁ rst

Ge layer. For the 2.5 nm thin Si spacers, the reduction is even larger, saturating at a critical thick-

ness of about 1.4 monolayers within the superlattice stack (see Fig. 1.23a ). The dependence of

the critical thickness of the second Ge layer on the Si spacer thickness is depicted in Fig. 1.23c

[40] , showing that for spacer thicknesses lower than 90 nm, the critical thickness of the second

Ge layer decreases linearly with decreasing spacer thickness, whereas for thicker spacer the criti-

cal thickness for the second Ge dot layer approaches the value of the ﬁ rst Ge layer. This indicates

that at this point, no interlayer interactions occur any more, which agrees qualitatively with the

results of Kienzle et al. [35] displayed in Fig. 1.22b . A similar behaviour was found by Schmidt

et al. [39] for MBE grown ﬁ ve-period Si/Ge multilayers with 25 nm thick Si spacers, where the Ge

wetting layer thickness was found to decrease from 3.8 ML for the ﬁ rst layer to 2.5 ML in the ﬁ fth

layer. These changes in the critical wetting layer thickness have also been shown to signiﬁ cantly

inﬂ uence the photoluminescence spectra from multilayer samples as is discussed in more detail,

e.g., in [22, 40] .

(a) SiGe/Si dot SL

Pairing probability

(b)

[010]

[001]

1 ␮m

50 nm

100 nm

Ge/Si SLs

0.0

0.2

0.4

0.6

0.8

1.0

050t

100

150

200

Si spacer thickness (nm)

i1

= 1

i1

= 0

Figure 1.22 (a) Cross-sectional transmission electron micrograph of a self-assembled (100) Si

0.5

/Si dot

superlattice showing the typical vertical alignment of the dots along the growth direction. The SiGe and Si layer

thicknesses are 2.1 and 20 nm, respectively. Adapted from Holy et al. [147]. (b) Pairing probability of 6.5 ML Ge

dots in ﬁ ve-stack Ge/Si multilayers grown at 620°C determined by cross-sectional transmission electron microscopy

as a function of Si spacer thickness. For spacer thicknesses smaller than 40 nm, a nearly perfect dot alignment is

obtained. For larger thicknesses the pairing probability decreases such that above 100 nm, no interlayer correlations

are found. Adapted from Kienzle et al. [35] . (c) Atomic force microscopy image of the last SiGe dot layer of a 20

period SiGe/Si dot superlattice with 5.5 ML SiGe dots alternating with 30 nm Si spacers. Inset: Image of a single

SiGe dot, illustrating the typical pyramidal island shape with { 105 } side facets.

CH001-I046325.indd 35CH001-I046325.indd 35 6/25/2008 12:27:36 PM6/25/2008 12:27:36 PM

36 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

Concerning the lateral dot ordering, one has to distinguish between two different cases accord-

ing to the density and spacing of the dots in the ﬁ rst superlattice layer that strongly depend on

the composition and thickness of the SiGe dot layers and the chosen growth conditions. For the

case of high-density SiGe hut cluster islands deposited at temperatures around 550°C, Teichert

et al. [17, 21, 28] have demonstrated a quite effective lateral ordering of SiGe hut islands along

the  100  directions, with the formation of a preferred square nearest neighbour dot arrange-

ment. This is illustrated by the atomic force microscopy images and corresponding Fourier trans-

form power spectra of Teichert et al. [21, 28] obtained for Si

0.25

0.75

/Si multilayers with N  1 ,

2 and 20 periods shown in Fig. 1.23a to c , respectively. Clearly, the lateral ordering progresses

rapidly with increasing number of deposited periods. This is accompanied by an almost three-fold

increase of the dot size but the uniformity of the dots in the ﬁ nal dot layer signiﬁ cantly improves.

This is demonstrated by the data displayed in Fig. 1.23d–f , where the evolution of the lateral dot

spacings, the lateral aspect ratio as well as root mean square roughness that is proportional to

the dot height are plotted versus N , respectively.

In the second case of low-density dots formed by growth of pure Ge at higher temperatures

as used in most other studies [33–36, 38–40] , the dots of single layers are already much more

uniform and widely spaced. As a result, the Ge dots in these multilayers show only a very faint

lateral ordering tendency (see Fig. 1.22c ) even at small spacer thicknesses, with only a very weak

tendency of a lateral alignment of the dots along the  100  directions, as has been found, e.g., by

X-ray diffraction studies [32, 36, 38] . These ﬁ ndings are attributed to the weak lateral overlap of

the strain ﬁ eld of the dots and the weak ordering tendency is in good agreement with our Monte

Carlo growth simulations [58, 147] (see section 1.5 and Fig. 1.18 ). In these simulations, the lat-

eral dot alignment along  100  as found in the experiments was explained to result from the elas-

tic anisotropy of the Si spacer material. The apparent discrepancy between the work of Teichert

et al. [21, 28] and those of the other groups [33–36, 38–40] is ascribed to the different growth

conditions, which for the former result in closely spaced dots with a large lateral strain ﬁ eld over-

lap, whereas in the other works the lateral dot interaction is rather weak.

1.8 PbSe/PbEuTe dot superlattices

Self-assembled PbSe/PbEuTe quantum dot superlattices represent a particularly interesting sys-

tem for investigation of interlayer correlations in multilayer structures. On the one hand, the

elastic anisotropy is particularly large and, therefore, an exceedingly efﬁ cient vertical and lateral

ordering takes place [18, 30] . On the other hand, different dot stacking types occur in dependence

of spacer thicknesses as well as growth conditions [50–56, 57] . Because in the IV–VI compounds

(a)

Critical thickness d

Ge/Si SLs

Number of deposited layers

2D–3D transition time (sec)

280

240

200

160

120

= 22 nm

= 2.5 nm

Critical thickness (ML)

Si spacer-layer thickness (nm)

Ge critical thickness d

(ML)

0 50 100 150 200

0.66

1.33

2.66

3.33

4.66

Si sub.

[001]

50 nm

Si spacer: 22 nm

(b)

(c)

24681012

layer d

= 4 ML

Ge/Si double layer

Figure 1.23 (a) Change of the critical thickness for Ge islanding as a function of the number of deposited layers

N in a multilayer consisting of ten periods of 4 ML Ge alternating with 22 nm (  ) or 2.5 nm (  ) Si spacer layers.

The 2D–3D transition was determined using in situ reﬂ ection high-energy electron diffraction and the growth was

carried out using UHV-CVD at a temperature of 550°C. (b) Change of the critical thickness d

of the second Ge

layer as a function of the Si spacer thickness at a growth temperature of 600°C. The critical thickness d

 4 M L o f

the ﬁ rst Ge layer is indicated by the dashed horizontal line. Adapted from Le Thanh et al. [40] .

CH001-I046325.indd 36CH001-I046325.indd 36 6/25/2008 12:27:36 PM6/25/2008 12:27:36 PM

Self-organized Quantum Dot Multilayer Structures 37

the (111) direction is the elastically soft direction, fcc -like dot stackings are formed in (111) mul-

tilayer structures. Therefore, this system is particularly well suited for testing the theoretical pre-

dictions of superlattice growth models. In addition, extensive systematic studies on interlayer

correlation formation have been carried out in this material system.

Self-assembled PbSe dots are produced by heteroepitaxial growth on PbEuTe (111) at a criti-

cal coverage of 2.5 monolayers [130] . For the (100) growth orientation, this islanding transi-

tion is suppressed because misﬁ t dislocation formation sets in already at a smaller thickness of

one monolayer [134, 135] . Therefore, self-assembled quantum dots cannot be obtained in this

growth orientation. PbSe dots are under tensile strain because their lattice constant is 5.4%

smaller than that of the underlying PbEuTe buffer layers. This is in contrast to the compres-

sively strained dots present in most other studied material systems, but from an energetic point

of view this does not make any difference because the sign of strain is removed in the elastic

energy calculations. The growth properties of single PbSe dot layers have been studied in detail

by our group [130, 171] , showing that the basic behaviour is quite similar to that of other self-

assembled quantum dot systems. In particular, PbSe dots show a well-deﬁ ned pyramidal shape

with { 100 } side facets and very narrow size distributions [130] . The size and dot density can be

readily controlled by adjusting the growth temperature or dot layer thickness [171] , with the dot

height being typically in the range of 60 to 200 Å, and the density between 1  1 0

 2 0 

c m

 2

. Because of the narrow energy band gaps, PbSe dots are also of interest for fabrica-

tion of mid-infrared quantum lasers as demonstrated in our previous work [172] . Multilayer

structures are always grown on thick strain relaxed PbTe buffer layers as virtual substrates. In

the superlattice structures, the Pb

1 

Te spacer layer composition is typically in the range of

x  4 to 10%. By proper adjustment of the ternary composition, strain symmetrized superlattices

SiGe/Si SLs

(d)

(f)

FFT

N  2

N  1

N  20

1.5

1.4

1.3

1.2

1.1

1.0

0 1020304050

0.0

0.5

1.0

1.5

<1> (nm)

150

100

<a/b>

SiGe terminated

Si terminated

 (nm)

Number of bilayers N

(e)

(a)

(b)

(c)

Δl/l

Figure 1.24 Atomic force microscopy images (left column) and corresponding 2D power spectra (right column) of

a SiGe surface layer of Si

0.25

0.75

/Si multilayers with increasing number of multilayer periods N  1, 2 and 20

from (a) to (c), respectively (see schematic drawings on the left). The SiGe layer thickness is 2.5 nm and the Si spacer

thickness 10 nm. Panels (d) to (f) show the evolution of characteristic island parameters as a function of number

of superlattice periods N . (d) Mean island spacing  l  along [100] (solid line) with the error bars indicating the

width of the size distribution, and relative width Δ l / l  of the distribution of island spacings (dashed line) w.

(e) Average aspect ratio of the SiGe island bases  a/b  (see inset). (f) Overall root mean square roughness of the

surface topography. Adapted from Teichert et al. [21, 28] .

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38 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

can be produced [30] in which the number of deposited bilayers is not limited by misﬁ t dislocation

formation.

1.8.1 Stackings as a function of spacer thickness

For PbSe/PbEuTe dot superlattices, the dependence of the vertical and lateral dot correlations on

spacer thickness was studied for spacer thicknesses varying from 100 to 800 Å, while the PbSe

dot layer thickness of ﬁ ve monolayers as well as the growth conditions were kept constant. The

lateral dot ordering is illustrated in Fig. 1.25a–d , where representative atomic force microscopy

images of the ﬁ nal dot layer after 30 superlattice periods are depicted for spacer thicknesses of

160, 320, 465 and 660 Å, respectively. In addition, the average lateral dot spacing formed in the

multilayer structure is plotted in Fig. 1.25e for a large number of samples as a function of verti-

cal superlattice period d

[112]

300

500

700

900

1100

1300

100 200 300 400 500 600 700 800

Vertical superlattice period d

(Å)

PbSe

dot SLs

(II)

Lateral dot spacing L (Å)

(I)

(III)

(d) d  660 Å(c) d  465 Å

20 nm

(g)

d  680 Å

[110]

Vertically

aligned

ABC

stacked

uncor-

(f)

d  320 Å

(e)

(a) d  160 Å

(b) d  320 Å

Figure 1.25 Left-hand side : Atomic force microscopy surface images of 30 period PbSe/PbEuTe dot superlattices

with different spacer layer thickness of (a) 160, (b) 320, (c) 465 and (d) 660 Å, demonstrating the different

dot spacings and ordering tendency in the structures as evidenced by the FFT power spectra shown as insets.

(e) Preferred lateral dot spacing L plotted as a function of vertical superlattice period: squares indicate superlattices

with vertically aligned dots, diamonds those with fcc -like ABCABC … stacking and full circles those of uncorrelated

superlattices. Right-hand side : Cross-sectional transmission electron microscopy images of a vertically aligned

(f) and an uncorrelated (g) PbSe/PbEuTe dot superlattice with spacer thicknesses of 320 and 680 Å, respectively.

See Springholz et al. [50, 51] .

For spacer thicknesses lower than about 400 Å ( Fig. 1.25a and b ), there is a clear hexagonal

ordering tendency of the dots in the lateral directions [51] that is also evidenced by six peaks

in the Fourier transform power spectra of the atomic force microscopy images shown as inserts.

The lateral spacing of the dots is larger than 800 Å and rapidly increases with increasing spacer

thickness (see solid line in region I of Fig. 1.25e ). As demonstrated by the cross-sectional trans-

mission electron microscopy image of a sample with 320 Å spacer thickness shown in Fig. 1.25f ,

the dots in these samples are vertically aligned along the growth direction.

For samples with intermediate spacer thicknesses between 400 and 550 Å, the average lateral

dot spacing abruptly drops by more than a factor of two from 1300 Å to 580 Å and a very efﬁ cient

2D hexagonal lateral ordering takes place (see Fig. 1.25c ) [18, 30, 49] . From detailed high resolu-

tion X-ray diffraction [18–20, 49, 50, 53, 55–57] as well as cross-sectional transmission electron

microscopy [19, 20, 50, 52, 54] studies already presented in Section 1.4.2, for these samples, a

well-deﬁ ned fcc -like interlayer dot stacking (see Fig. 1.14) is formed, corresponding to an overall

trigonal three-dimensional lattice of dots as shown in Fig. 1.8a . In addition, the lateral dot spac-

ing within the superlattice stacks increases linearly with increasing spacer thickness or superlat-

tice period d

, as indicated by the data and solid line depicted in region II in Fig. 1.25e . In this

range of spacer thicknesses, the interlayer correlation angle is also essentially constant at about

39° [18] , which means that the lattice constant of the resulting three-dimensional trigonal dot

lattice can be tuned over a range from 500 to 700 Å just by changes in the spacer thickness [18] .

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