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

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In 1969, Leo Esaki (1973 Nobel Laureate) and Ray Tsu from IBM, USA, proposed research on

“ man-made crystals ” using a semiconductor superlattice (a semiconductor structure comprising

several alternating ultra-thin layers of semiconductor materials with different properties). This

invention was perhaps the ﬁ rst proposal to advocate the engineering of a new semiconductor mate-

rial, and triggered a wide spectrum of experimental and theoretical investigations. However, the

study of what are now called low dimensional structures (LDS) began in the late 1970s when sufﬁ -

ciently thin epitaxial layers were ﬁ rst produced following developments in the technology of epitax-

ial growth of semiconductors, mainly pioneered in industrial laboratories for device purposes.

The LDS are materials structures whose dimensions are comparable with interatomic distances

in solids (i.e. nanometre, nm). Their electronic properties are signiﬁ cantly different from the same

material in bulk form. These properties are changed by quantum effects. At the inception of their

investigation it was already clear that such structures were of great scientiﬁ c interest and excitement

and their novel properties caused by quantum effects offered potential for application in new devices.

Moreover, these complex LDS offer device engineers new design opportunities for tailor-made new

generation electronic devices. The LDS could be considered as a new branch of condensed matter

physics because of the large variety of possible structures and the changes in the physical processes.

One of the promising fabrication methods to produce and study structures with a dimension less

than one such as quantum dots, in order to realize novel devices that make use of low-dimensional

conﬁ nement effects, is self-organization. The quantum dots are often referred to as artiﬁ cial atoms

due to their small size (typically ⬃ 10 nm). The self-assembling mechanism allows the forma-

tion of tens of billions of dots per cm

with a high degree of uniformity in a single-step process.

Self-assembled nanostructured materials offer a number of advantages over conventional mate-

rial technologies in a widerange of sectors. The research on self-assembled nanostructures

involves a strong interdisciplinary combination, for example material science, physics, chemistry,

biology, electronics and optoelectronics.

Twenty-eight chapters is not a lot in which to convey the current successes of modern self-

organization research. However, the topics covered will give the reader a comprehensive overview

of the ﬁ eld of self-assembled nanostructures and a better evaluation of future trends.

The two parts of this book deal mainly with two fabrication methods, namely epitaxial and col-

loidal techniques that make feasible the design of artiﬁ cial nanostructures. Each part introduces

subjects on properties, characterization and applications of self-assembled nanostructures. We

trust that this book will provide an excellent introduction to the self-organization of nanostruc-

tures to a large number of researchers and scientists active in the nanostructures science and

technology area, and will ﬁ ll an important gap in the market.

I would like to record my thanks to all the authors who have done so much hard work to

achieve this successful book and acknowledge the invaluable help of the staff at Elsevier. Their

efforts are greatly appreciated.

Finally, I would like to thank my family for their tolerance and understanding during the long

hours and many mood swings accompanying my efforts as editor of this book.

Mohamed Henini

School of Physics and Astronomy,

University of Nottingham, Nottingham NG7 2RD, UK

Preface

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Self-organized Quantum Dot Multilayer Structures

Gunther Springholz

Institut für Halbleiter- und Festkörperphysik, Johannes Kepler

Universität Linz, A-4040 Linz, Austria

1.1 Introduction

Strained-layer heteroepitaxy is a powerful tool for the fabrication of self-assembled semiconduc-

tor nanostructures [1–4] . It is based on the tendency of strained layers to spontaneously form

self-organized corrugated surface structures during growth. This process can be driven by ener-

getic as well as by kinetic effects such as strain-induced coherent islanding, faceting, kinetic

step-bunching or step-meandering. One particularly effective process is the formation of three-

dimensional nanoislands in the Stranski–Krastanow (SK) growth mode [4–7] . This islanding is

driven by the highly efﬁ cient elastic strain relaxation within the islands arising from their lat-

eral elastic expansion or compression in the directions of their free side faces [4–11] . For islands

larger than a critical size, the relaxed elastic energy outweighs the corresponding increase in sur-

face energy, leading to an effective lowering of the total energy of the system [4–10] . Therefore,

this island formation occurs in a large number of high-misﬁ t heteroepitaxial material systems.

When the islands are subsequently embedded in a higher energy band gap matrix material, the

electronic carriers are spatially conﬁ ned and thus, self-assembled quantum dots with atomic-like

optical and electronic properties are formed [1–3, 12–14] . Because these epitaxial quantum dots

are essentially defect free, their electronic properties are often superior to those fabricated by con-

ventional lithographic processing and etching techniques.

Owing to the statistical nature of growth, ensembles of self-assembled dots exhibit consider-

able variations in size and sometimes even shape. This results in a large inhomogeneous broad-

ening of the energy levels and optical transitions [1–3, 12] . In addition, there is little control over

the lateral arrangement and position of the nanoislands. Both factors pose considerable limita-

tions for practical device applications. Three-dimensional stacking of self-assembled quantum

dots in multilayer or superlattice structures has turned out to be one of the most effective means

for controlling the vertical and lateral arrangement of the dots. Apart from the possible improve-

ments in uniformity [15–21] , this also increases the total amount of active material in quantum

dot devices and allows the tuning of electronic wave functions by quantum mechanical cou-

pling across the spacer layers [22–24] . In this way, quantum dot molecules [23–26] have been

obtained, which are of particular interest for solid-state quantum computation devices [26, 27] .

Multilayering has been found to yield signiﬁ cant improvements in the uniformity of dots

[17, 18, 28–31] and may even result in the formation of ordered quantum dot superstruc-

tures [18, 28] . Experimentally, different types of vertical and lateral dot correlations have been

observed in multilayer structures, ranging from vertically aligned dot columns in Ge/Si [22,

32–41] or InAs/GaAs [15, 16, 31, 42–45] dot superlattices, to vertical anti-correlations for

CHAPTER 1

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

II–VI [46–48] and III–V superlattices [60–65, 87, 88] , oblique replication on high-indexed sur-

faces [142] as well as fcc -like dot stackings for IV–VI semiconductor multilayers [18, 30, 49–57] .

The actual type of correlation has been found to depend on a large number of parameters such

as the spacer thickness [15, 35, 36, 45, 47, 49–57, 87, 88] , dot size [45, 55, 56] , elastic mate-

rial properties [18, 47, 58, 59] , surface orientation [58] , growth conditions [45, 56] as well as

chemical composition of dots and spacer layers [60–66] . In addition, it has been found that lat-

eral dot ordering is particularly effective for multilayers with staggered interlayer dot correlations

[18, 30, 50, 57, 58, 60] .

In this chapter, the mechanisms for vertical and lateral ordering in quantum dot multilayer

structures and the resulting different dot stackings are described. The particular emphasis is on

the prototype Si/Ge, InAs/GaAs and PbSe/PbEuTe material systems in which extensive studies

have been carried out. The chapter is organized as follows: ﬁ rst, a brief general overview of the

different possible interaction mechanisms and various dot stackings for different material sys-

tems is given in Section 1.2. In Section 1.3, the elastic interactions between buried and surface

dots are treated in detail, introducing the far-ﬁ eld and near-ﬁ eld approximations and analysing

the effect of the elastic anisotropy and growth orientation. It is shown that the elastic anisot-

ropy plays a crucial role in the ordering process and is responsible for the formation of staggered

dot stackings. In Section 1.4, the derived theoretical predictions are compared with experi-

mental results, and growth simulations of the stacking and ordering processes are described in

Section 1.5. Sections 1.6 to 1.8 treat various aspects for the three well-studied InAs/GaAs, Si/

Ge and PbSe/PbEuTe multilayer systems, including the correlation lengths, as well as stacking

and ordering transitions as a function of spacer thickness and growth conditions. In section

1.9, other interaction mechanisms such as surface morphology, surface segregation and alloy

decomposition are discussed, and a brief summary and outlook are given in the ﬁ nal section.

1.2 Mechanisms for interlayer correlation formation

The formation of interlayer correlations in multilayer structures obviously requires some kind of

mechanism through which the dots in the buried layers inﬂ uence the dot growth in the subse-

quent layers. By this mechanism, the information on the dot positions is conveyed from one layer

to another and thus vertical and lateral correlations are formed. Conceptually, one can think

of three kinds of such mechanisms, namely, (i) chemical processes such as surface segregation,

(ii) non-planarized corrugated surface morphologies, or (iii) long-range elastic interactions due

to the strain ﬁ elds emerging from the buried dots. These mechanisms are illustrated schemati-

cally in Fig. 1.1 and may produce different kinds of correlations in dependence of the interaction

process.

Nucleation of Stranski–Krastanow dots is a rather complex process that sensitively depends

on parameters such as surface stress, lattice mismatch, thickness and composition of the wetting

layer, free energies and local curvature of the growth surface, surface step structure as well as sur-

face kinetics during growth. On a planar and chemically uniform substrate, these parameters are

invariant across the surface and thus homogeneous dot nucleation at random surface sites occurs.

In multilayer structures, however, the buried dots beneath the surface induce signiﬁ cant variations

of strain, topography and/or chemical composition of the surface, which may lead to preferential

nucleation at particular surface sites that are linked to the position of the buried dots. This prefer-

ential nucleation can be induced, e.g. by a local enhancement of the growth rate, local changes in

surface diffusivity, as well as by local decreases in the critical wetting layer thickness or energy bar-

rier for island nucleation and results in spatial correlations between surface and subsurface dots.

Apart from this, the existence of interlayer interactions is also manifested by the signiﬁ cant changes

in dot size [19, 28, 29, 31, 36, 40, 43, 51, 67, 68, 69, 70, 71, 72, 73] , density [17, 19, 28, 31, 43,

51, 67] , lateral arrangement [18, 19, 28, 30, 31, 51, 74] , shape [34, 51, 67–72, 73] , and criti-

cal thickness for dot nucleation [39, 40, 75] observed in many experiments. Moreover, different

interlayer correlations may be formed, depending on the details of the interaction mechanisms and

growth conditions, as is illustrated schematically in Fig. 1.1 .

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Self-organized Quantum Dot Multilayer Structures 3

As already mentioned above, interlayer correlation may be caused by three main mechanisms,

namely, (i) elastic lattice deformations around the buried dots due to the dot/substrate lattice

mismatch [15, 17, 18, 29, 58] , (ii) corrugations in surface topography due to incomplete surface

planarization [76] , and (iii) surface segregation or alloy decomposition within the spacer layer

that often occurs in highly strained material systems [62–66] . In the case of the interaction via

the elastic strain ﬁ elds ( Fig. 1.1a and b ), preferential dot nucleation is generally induced at the

strain minima on the wetting layer surface. Depending on the elastic properties of the materials,

these minima may be on top (a) or between the buried dots (b), resulting in a vertical or staggered

dot stacking, respectively. This will be treated in detail in sections 1.3 to 1.5. Similar dot stackings

may also result from surface corrugations of the spacer layer [77] when planarization is incom-

plete during spacer layer growth ( Fig. 1.1c and d ). Depending on whether subsequent dots nucle-

ate preferentially at convex or concave surface areas, again vertical or staggered dot stackings

may be formed as shown schematically in Fig. 1.1c and d , respectively. Finally, surface segrega-

tion of dot material ( Fig. 1.1e ) or alloy decomposition of the spacer layer ( Fig. 1.1f ) [62–66] may

produce a non-uniform chemical composition of the surface above the dots. This can affect the

growth of subsequent dots due to the resulting variations in chemical potential, strain and effec-

tive wetting layer thicknesses. As will be discussed in section 1.9, this again may cause a vertical

alignment or staggered dot stackings in dependence of the details of the experimental conditions.

Generally, all three mechanisms may act simultaneously and thus amplify or counteract each

other. Therefore, it is often not easy to determine which is the main driving force for correlation

formation for a given material system and growth conditions.

Experimentally, indeed different dot stackings have been found for different multilayer struc-

tures. Figure 1.2 shows representative examples as revealed by cross-sectional transmission elec-

tron microscopy (TEM) studies. The most common case of vertical dot alignment is illustrated in

Fig. 1.2a for a self-assembled InAs/GaAs quantum dot superlattice [44] , in which case the verti-

cal dot alignment was found to persist up to 50 nm GaAs spacer layer thicknesses [15, 16, 42–

45, 78] . A similar vertical alignment is also found for SiGe/Si dot superlattices up to 70 nm thick

spacers [32–40] , as well as for InP/GaInP [70] and GaN/AlN [79–86] multilayers as shown in

detail in Sections 1.4–1.7. Staggered dot stackings have been found for IV–VI [18, 49, 50, 57] ,

III–V [87, 88] as well as II–VI semiconductors [46–48] and examples are depicted in Fig. 1.2b

and e . Staggered stackings were also observed for self-assembled InAs/AlInAs quantum wire

superlattices [60–65] as exempliﬁ ed by Fig. 1.2d . For a given material system even transitions

Elastically isotropic spacer

Strain

Morphology

Nucleation on mounds Surface segregation

Composition

Alloy decompositionNucleation at troughsAnisotropic spacer and soft direction

Strain

(a) (c)

(b) (d) (f)

(e)

Figure 1.1 Possible mechanisms for formation of interlayer correlations in self-assembled quantum dot

multilayers. Left-hand side : Interlayer correlations caused by the elastic strain ﬁ elds emerging from subsurface dots

and subsequent dot nucleation at the minima of strain on the epilayer surface. Depending on the elastic properties

of the spacer layer as well as surface orientation, these minima may be localized (a) above or (b) between the

buried dots, which will give rise to either a vertical dot alignment or a staggered dot stacking, respectively. Centre :

Interlayer correlations caused by non-planar surface morphologies resulting from incomplete surface planarization

during dot overgrowth. Depending on the dominant mechanism of capillary or stress-driven surface mass transport,

subsequent dots may nucleate either on top of the mounds (c) or in the troughs in between (d), again giving rise to

different types of interlayer correlations. Right-hand side : Correlated nucleation induced by non-uniformities in the

chemical composition of the spacer layer due to (e) surface segregation or (f) alloy decomposition. Clearly, all three

mechanisms may induce vertical dot alignments or staggered dot stackings in multilayer structures.

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

(a) (b) (c) (d)

(e)

InAs/GaAs PbSe/PbEuTe

100 nm

Ge/Si

CdTe/ZnTe

100 nm

InAs/AllnAs

Figure 1.2 Examples of different types of interlayer dot stackings in self-assembled quantum dot multilayers

observed by cross-sectional transmission electron microscopy: (a) vertically aligned (001) InAs quantum dot

superlattice with 20 nm GaAs spacer layers. Adapted from Darhuber et al. [38] . (b) Fcc -like ABCABC … dot

stacking in a (111)-oriented superlattice of 5 ML PbSe dots alternating with 45 nm PbEuTe spacers. Inset: Plan-

view transmission electron microscopy micrograph showing the 2D hexagonal dot ordering within the growth plane.

After Springholz et al. [18, 49, 51] . (c) Inclined dot correlations in a 1.2 nm Ge/40 nm Si (001) dot superlattice.

Adapted from Sutter et al. [69] . (d) Vertically anticorrelated InAs/AlInAs quantum wire superlattice on InP

(001) with 3ML/10 nm thicknesses, respectively. Adapted from Brault et al. [64, 65] . (e) Anticorrelated 2 ML

CdTe islands intercalated with 15 ML ZnTe spacers. The white contour lines indicate iso lattice parameters, i.e. the

chemical composition extracted from the atomically resolved transmission electron microscopy image. Adapted from

Mackowski et al. [48] .

between different interlayer correlations have been observed as exempliﬁ ed by Fig. 1.2c which

shows a Ge/Si dot superlattice where the initial vertical alignment switches to an oblique dot

replication due to an instability in the planarization process [76] . Transitions between different

dot stackings as a function of spacer layer thickness were also found for PbSe/PbEuTe [50, 57] ,

CdSe/ZnSe [47] and InGaAs/GaAs dot superlattices [87, 88] . This will be discussed in detail in

Sections 1.4, 1.6 and 1.8.

1.3 Strain-ﬁ eld interactions in multilayer structures

Strain is the major driving force for self-assembled Stranski–Krastanow quantum dot forma-

tion [4–10] . Therefore, the elastic strain ﬁ elds produced by buried quantum dots play a crucial

role for the formation of interlayer correlations in multilayer structures. The strain ﬁ elds of the

buried dots are caused by the strong elastic lattice deformation induced by the large lattice mis-

match between the dots and the surrounding matrix material. These strain ﬁ elds extend up to

the epilayer surface and, during subsequent dot layer growth, impose a bias on the diffusion cur-

rent of deposited adatoms due to the concomitant gradients in the surface chemical potential.

At the surface strain minima therefore a local enhancement of growth and preferential island

nucleation may occur. This may also be enforced by a strain-induced local reduction of the island

nucleation barrier.

The strain ﬁ elds created by the buried dots depend on a large variety of parameters such as size,

shape, depth and chemical composition of the buried dots, the spacer material as well as the elas-

tic properties and the crystallographic orientation of the growth surface. In order to understand

the different interlayer correlations and stacking types, obviously, all details of the elastic strain

ﬁ elds must be taken into account. For this purpose, it is useful to distinguish between two limiting

cases, namely, (i) the far-ﬁ eld limit , where the dot depth is large as compared to the dot dimensions,

and (ii) the near-ﬁ eld limit , where the buried dots are very near to the growth surface. In the far-

ﬁ eld limit, the internal structure as well as the actual size and shape of the dots can be ignored, i.e.

the dots can be treated as simple point stress sources. This drastically simpliﬁ es the calculations

and it turns out that the surface strain distribution produced by each buried dot is then solely

determined by the elastic properties of the matrix material and the surface orientation. Thus, the

far-ﬁ eld limit is particularly instructive to reveal the general trends of the elastic dot interactions.

According to calculations and experimental observations, the far-ﬁ eld limit applies well when

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Self-organized Quantum Dot Multilayer Structures 5

the dot depth, i.e. the spacer thickness, exceeds about two times the size of the dots [18, 48] .

For the near-ﬁ eld case, the actual geometry of the dots as well as their depth strongly inﬂ uences the

strain distributions, which therefore have to be evaluated separately for each material system.

1.3.1 The isotropic point-source model

1.3.1.1 Surface strain distribution

For the simplest case of an elastically isotropic matrix material, the far-ﬁ eld stresses created by an

individual point-like buried dot can be derived analytically. As shown by several works [89, 90] ,

the buried dot in this case induces a radially symmetric strain distribution

εεε

() /( )r

xx yy

12

o n

the surface that is independent of surface orientation. This strain distribution can be written as:

()







2252

⋅

⎡

⎣

⎢

⎤

⎦

⎥

(1.1)

In this expression, d is the dot depth underneath the surface, r  ( x

 y

)

1/2

is the radial dis-

tance from the centre above the dot, P is the strength of the point stress source given by

P  ε

( 1  ν )/ π [90, 90] , where ε

is the dot/matrix lattice mismatch, V

is the dot volume

and ν the Poisson’s ratio. If we consider the case of a surface wetting layer of the same mate-

rial as the dots as is illustrated schematically in Fig. 1.3a , then Eq. 1.1 represents the amount by

which the global layer–substrate lattice mismatch strain ε

is locally reduced on the wetting layer

due to the presence of the buried dot. This is accounted for by the negative prefactor of Eq. 1.1 .

3 2 1012

10

8

6

4

2

1.0 0.5 0.0 0.5 1.0

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Normalized strain ε

/ε

(%)

Normalized radial distance r/d

Vertical distance z/d

Normalized radial distance r/d

Dot

spacer

(a) (b)

2(1)



Minimum

Maximum

 d/3

 d

min

h  d/3

Figure 1.3 (a) Schematic illustration of a buried pyramidal Stranski–Krastanow dot located at a depth d below the

surface with a height of h  d /3, a square base of b  d and a volume of V

 d

/9. (b) Corresponding calculated

normalized surface strain energy distribution ε

⎢⎢

/ ε

plotted as a function of normalized radial distance r / d from the

centre above the dot calculated within the point-source approximation (Eq. 1.1) for an elastically isotropic medium

with ν  0.3. The arrows indicate the minima and maxima of the strain distribution.

The relative strain variation

εε

given by Eq. 1.1 is plotted in Fig. 1.3b as a function of nor-

malized surface coordinates r / d . Obviously, directly above the buried dot, the strain is signiﬁ cantly

reduced by a value of

Δεε νπ ε 

() ()/(/)rVd021

. Considering the case of a buried

dot pyramid with square base b  d and a height of h  d /3, the dot volume is V

 d

/9 and thus

the original misﬁ t strain ε

is reduced by about 10% above the buried dot (see Fig. 1.3b ). Also,

there exist two weak side maxima at a lateral distance of r / d  2 at which the strain is slightly

increased by about 0.2%. Since the shape of the normalized strain distribution is invariant as

a function of normalized surface coordinates r/d , the width (FWHM) of the central strain mini-

mum is inversely proportional to the dot depth, i.e. FWHM  1 / d , and the strain reduction above

the dot decreases inversely proportional to the cube of the dot depth. This is a general property of

the far-ﬁ eld point-source solution that also applies for elastically anisotropic materials.

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

For subsequent dot layer deposition, the variation in the strain energy E

( x, y ) on the wetting

layer surface results in a gradient in the surface chemical potential Δμ  Δ E

across the surface.

This produces a diffusional bias of surface adatoms towards the strain minima on the surface,

where preferred dot nucleation will occur. The strain energy distribution on the surface is cal-

culated using E ( x, y )  1 / 2 c

ijkl

( x, y ) ε

( x, y ) where c

ijkl

are the elastic constants of the wetting

layer, and ε

( x, y ) are the tensor components of the surface strain distribution. The latter is given

by the sum of the misﬁ t strain ε

and the additional inhomogeneous strain ﬁ elds

()r

induced by

the buried dots. In the far-ﬁ eld limit

εε

()r 

and thus the isotropic surface strain energy vari-

ation Δ E

( r )  E

( r )  E

can be approximated by [18, 19] :

ΔEr

() ( )

()

  





2252

⋅

⎡

⎣

⎢

⎤

⎦

⎥

(1.2a)

with:

21 1μν νε()/()

(1.2b)

where E

is the constant background strain energy within the biaxially strained 2D wetting

layer arising from the homogeneous dot/matrix lattice mismatch. The strain energy variation

(Eq. 1.2) shows exactly the same form of the strain energy distribution of Eq. 1.1 with only a dif-

ferent prefactor. Therefore, the strain energy distribution Δ E

( r ) is radially symmetric as well with

a minimum directly above the buried dot with an amplitude of Δ E

,min

  2 ( 1  ν )/ π



/ d

1.3.1.2 Lateral ordering in 1D growth simulations

According to the isotropic strain calculations of Fig. 1.3 , a single buried dot will always produce

one central strain minimum on the surface directly along the growth direction. This would lead

to a vertical dot alignment in multilayer structures. For a whole array of buried dots, however,

there is also a lateral overlap of the strain ﬁ elds of neighbouring dots which modiﬁ es the sur-

face strain distribution. Based on a simple one-dimensional growth model, Tersoff et al. [17] have

shown that this overlap may cause a lateral ordering of the dots as well. The effect of overlapping

strain ﬁ elds is illustrated in Fig. 1.4a [17] , where the surface strain distribution produced by four

(b)(a)

5 10 0 25 50 75

Island position x (d)

N =10

N = 5

BAAC

2

4

Island position x (d)

Surface strain  (P·d

3

)

N = 2

N = 1

Figure 1.4 Self-organized lateral ordering of strained Stranski–Krastanow dots in a multilayer structure predicted

by an isotropic 1  1 dimensional growth model by Tersoff et al. [17] . (a) Strain distribution produced by four sub-

surface dots located at different positions x below the surface as indicated by the upward pointing arrows, calculated

using the isotropic point-source approximation. The downward arrows at the top indicate local strain minima

on the surface above the dots, representing favoured positions for dot nucleation. Properly spaced dots (labelled A)

replicate vertically along the growth direction, whereas for closely spaced dots only one new strain minimum and

thus one new dot in position B will be formed in the subsequent layer. For widely spaced buried dots, a weak local

strain minimum is produced at point C, where an additional new dot is assumed to grow. (b) Deterministic growth

simulation assuming dot nucleation at each local surface strain minimum induced by the buried dots. The vertical

lines indicate the lateral position of the dots in the bilayer N  1, 2, 5 and 10 of the superlattice structure, starting

from a random distribution in the initial dot layer N  1. Clearly, a rapid equalization of the lateral dot separations

due to the lateral strain interactions occurs. The height of each vertical line represents the size of each dot, which

becomes more uniform with increasing N . Adapted from Teichert et al. [21].

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Self-organized Quantum Dot Multilayer Structures 7

buried dots located at different positions x within an interface below the surface is plotted (the

position of the buried dots is indicated by the upward pointing arrows). Obviously, three differ-

ent cases can be distinguished: (i) for widely spaced buried dots (labelled A in Fig. 1.4a ), the sur-

face strain minima are exactly on top of the dots. Thus, in the subsequent layers, these dots will

be replicated along the growth direction. (ii) For closely spaced dots (labelled B in Fig. 1.4a ), the

surface strain minima are merged to one and thus only one new dot will nucleate on the surface

in the middle between the buried dots. (iii) When the spacing of the buried dots is very wide, an

additional weak local strain minimum is produced at point C in between the buried dots, where

subsequently an additional new dot may start to grow. As a result of this strain ﬁ eld overlap, the

dot positions gradually change from one layer to another as multilayer growth proceeds.

To demonstrate the possible lateral ordering induced by this effect, Tersoff et al. [17] have

performed a deterministic one-dimensional growth simulation based on the assumption that at

every local surface strain minimum a new dot is created with a volume that is proportional to the

area of the Voronoi polygons around each surface dot that represents the average capture area

of adatoms for each dot. After each dot layer growth, the dots are covered by a spacer layer with

thickness d and the subsequent surface strain distribution is recalculated. Starting from an initial

random dot layer, this sequence is repeated N times to produce the desired multilayer structure.

The results of this deterministic model are shown in Fig. 1.4 (b), where the lateral dot positions

in the N  1st, 2nd, 5th and 10th dot layers are plotted as vertical lines with a length propor-

tional the volume of the individual dots. Starting with a randomly distributed dot layer with high

density, one can see that the lateral dot spacing as well as dot size becomes progressively more

uniform as the number of periods increases. This demonstrates that in a multilayer structure a

lateral dot ordering may be induced. It is noted, however, that for elastically isotropic materials

this conclusion applies only for a one-dimensional system (see Section1.5 for details). Therefore,

for more realistic growth models, the elastic anisotropy, random surface diffusion as well as the

size and shape of the dots must be taken into account. The 1D simulation also yields a signiﬁ cant

narrowing of the dot size distribution of up to a factor of 5 and the preferred ﬁ nal lateral dot

separation is found to be equal to 3.5 times the spacer thickness [17] . Thus, the spacer layer acts

like a band pass ﬁ lter for a certain lateral dot separation.

1.3.2 The effect of elastic anisotropy

Most semiconductors exhibit a rather high elastic anisotropy. As a result, the surface strain dis-

tribution produced by a buried dot is signiﬁ cantly altered compared to the isotropic model. In

particular, the strain distributions show a strong dependence on surface orientation. To obtain

the resulting strain distributions, the equilibrium stress equations must be solved by taking the

true elastic properties of the matrix material around the dot as well as the boundary condition

of a free surface with vanishing normal surface stresses into account. This can be done using a

Fourier or Green’s function method [18, 58, 59, 91–93] . Alternatively, ﬁ nite element methods

[50, 94–100] or atomistic calculations using semi-empirical atom potentials [100, 103] have

been used, which all yield very similar results for the strain ﬁ elds well outside of the buried dots

[100] .

In cubic materials, the main elastic anisotropy axes are the crystallographic  100  and  111 

directions, in which Young’s modulus E

hkl

has its extremal values. This is illustrated in Fig. 1.5 ,

where the variation of E

hkl

as a function of crystallographic direction [ hkl ] is plotted for several

semiconductors such as PbTe and PbSe (a) and Si, Ge, GaAs and ZnSe (b). Correspondingly, the

 100  or  111 directions represent the elastically hard or soft directions of cubic materials. The

degree of deviation from the isotropic case is characterized by the dimensionless anisotropy ratio:

Acc c EE2

44 11 12 111 100

/( ) / ≈

(1.3)

which is essentially equal to the ratio of the elastic moduli E

hkl

along the main  111 and  100 

anisotropy axes. For isotropic materials, E

hkl

does not depend on the direction of the applied

stresses and thus A  1. For anisotropic materials, one has to consider two opposite cases,

namely, that A is either larger or smaller than one. In the ﬁ rst case of A  1, the  111 directions

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

are the elastically hard directions and the  001  directions the soft directions, i.e. E

111

/ E

100

 1 .

As shown in Fig. 1.5b , this essentially applies to all group IV, III–V and II–VI semiconductors

with diamond or zinc-blende crystal structures because the chemical bonds are along the  111

directions. The anisotropy is largest for the II–VI compounds, with A  2.04 for ZnTe and 2.53

for ZnS ( Fig. 1.5c ). For C, Si and Ge, A increases from 1.21, 1.56 to 1.64, respectively, and for the

III–V compounds A ranges from 1.83 for GaAs to 2.08 for InAs. In the opposite case of A  1 ,

now  100  are the elastically hard directions and  111  the soft directions (E

111

/ E

100

 1, see Fig.

1.5a) . This applies, e.g., for materials with rock salt crystal structure in which the nearest neigh-

bours are along the  100  directions. In particular, for the narrow gap IV–VI semiconductors the

elastic anisotropy is particularly large, with A  0.18, 0.27 and 0.51 for SnTe, PbTe and PbS,

respectively.

1.3.2.1 Far-ﬁ eld strain distributions

With respect to the elastic strain ﬁ elds, obviously, the strongest changes will occur for materials

with high elastic anisotropy. Thus, when A strongly deviates from one, the strain distributions

will become strongly dependent on surface orientation. Selecting two materials with large ani-

sotropy but opposite directions of the anisotropy axes, the normalized surface strain energy dis-

tributions Δ E

/ E

induced by a buried point-like stress source are shown in Fig. 1.6 for PbTe

(top row) and ZnSe (centre row) for different surface orientations and scaled surface coordinates

r / d . It is evident that not only the energy distributions strongly differ as a function of surface ori-

entation, but that they also show the opposite trend when A is larger (ZnSe) or smaller than one

(PbTe). In particular, it is found that only when the surface orientation is parallel to the elastically

PbSe

PbTe

[110]

GaAs

ZnSe

[110]

[001]

[111]

[110]

[001]

[111]

[110]

[001]

[

111]

[001]

[111]

0.0 0.5 1.0 1.5 2.0 2.5 3.0

SnTe

PbTe

PbS

PbSe

Isotropic

Diamond

GaA

InA

ZnSe

ZnS

anisotropy ratio A

NaCl

IV–VI Compounds IV, III–V and II–VI Compounds

(c)

Rock salt structure

A  1 and [111] soft directions.

Zinc-blende structure

A  1 and [100] soft direction.

hkl

(10

Pa)

(a) (b)

hkl

(10

Pa)

Figure 1.5 Polar plots of the Young’s modulus E

hkl

as a function of [ hkl ] direction within the (1

–

10) plane. (a)

IV–VI compounds PbTe and PbSe with rock salt structure and elastic anisotropy ratio A  1, i.e.  001  as elastically

hard directions with maximum E

hkl

value. (b) Group IV, III–V and II–VI semiconductors (Si, Ge, GaAs and ZnSe)

with zinc-blende structure and  111 as elastically hard axes ( A  1). Bottom: (c) Elastic anisotropy ratio

A  2 c

/( c

 c

) plotted for various cubic semiconductors.

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