Energy analysis for island formation on stranski krastanow systems

494.2 The contours of the total energy change ∆E as a function of the island volume V and the angle φ for the island formation on a thick film.. 644.7 The contours of the total energy ch

I would like to express my gratitude to all the persons who made this thesispossible.

First of all, I gratefully acknowledge my supervisor, Dr Chiu Cheng-hsin forhis invaluable guidance and support His enthusiasm and active research interestsare the constant source of inspiration to me During the course of this work, Ihave learnt from him on how to do research work

I would also like to thank our group members, Huang Zhijun, Wang Hangyaoand C-T Poh, for their instruction and discussions in the research, sharing of theirresearch experiences and all the assistance to me

Last but not least, I would like to thank my parents and my friends, for theircare and encouragement during all my efforts and for their supporting all of mydecisions

i

Acknowledgement i

1.1 Experimental Observation 1

1.1.1 Island Formation 2

1.1.2 Prepyramid to Pyramid Transition 4

1.2 Theoretical Study for Island Formation 5

1.2.1 Boundary Perturbation Method 5

1.2.2 Energy Analysis for Island Formation 8

ii

2 Model and Methodology 11

2.1 A Continuum Model for the SK Film-Substrate System 11

2.1.1 The Geometry of the Island 11

2.1.2 The Total Energy of the SK System 12

2.2 The First-Order Boundary Perturbation Method 15

2.2.1 Description of the System 16

2.2.2 The First-Order Boundary Perturbation Method 19

2.2.3 The Function Ψ(x0) 22

2.2.4 The Numerical Implementation 23

3 The Critical Thickness of the SK Transition 26 3.1 Introduction 26

3.2 Model 28

3.2.1 The Geometry of the Island 28

3.2.2 The Total Energy Change ∆E 30

3.3 The Critical Thickness for Spontaneous Island Formation 34

3.3.1 Numerical Result 34

3.3.2 The First Critical Thickness 36

3.4 The Critical Thickness for Surface Undulation Model 38

3.4.1 Model 38

3.4.2 The Total Energy Change ∆E 39

3.4.3 The Second Critical Thickness 44

4 The Formation of Trancated Pyramid Islands 47

4.1 Introduction 47

4.2 Model 49

4.2.1 The Geometry of the Island 49

4.2.2 The Continuum Model for the SK System 50

4.3 Island Formation on a Thick Film via Surface Undulation 52

4.3.1 A Typical Numerical Result 52

4.3.2 Analytical Results 55

4.4 Island Formation on a Thin Film via Surface Undulation 65

5 The Cooperative Formation 71 5.1 Introduction 71

5.2 Model 73

5.2.1 The Geometry of the Island 73

5.2.2 The Variation of the Total Energy Change ∆E with the Trench Depth A t 75

5.2.3 The Total Energy Change ∆E 76

5.3 The Cooperative Formation on a Thick Film 79

5.3.1 The Total Energy Change ∆E 79

5.3.2 A Typical Numerical Result 80

5.3.3 Analytical Results 81

5.4 The Stability of a Pyramid Island on a Thin Film against Trench Formation 83

6 Conclusion 87

The SiGe on Si(001) system has been extensively studied for understanding theheteroepitaxy of the system and there is a wide interest in employing the “self-assembled quantum dots” grown on the system in the nano-technology In thisthesis, we study the formation of the SiGe nano-islands on the Si substrate fromthe energy point of view Our energy analyses are based on a continuum three-dimensional model for the SiGe/Si system, and the analyses are carried out byemploying the first-order boundary perturbation method to calculate the energychange during the island formation process Three important issues in the islandformation process are investigated The first one is the critical thickness of thewetting layer below which the formation of islands is completely suppressed Thesecond issue is the shape transition from a shallow bump to a faceted pyramid, and

of particular interest is the dependence of the shape transition on the island size,the island shape and the film thickness The third issue examined in this thesisfocuses on the cooperative formation, which is characterized by the development

of trenches surrounding the pyramid island after the bump-pyramid transition It

is demonstrated that it is always energetically favorable for the trench to develop

on a thick film after the bump-pyramid transition, while the trench formation can

vi

be suppressed on a thin film.

1.1 A schematic diagram of the formation of SiGe islands formation onSi(001) substrate 1

2.1 A schematic diagram of the formation of a facet island 112.2 A schematic diagram of a facet island on an SK film-substrate system 162.3 A schematic diagram of the illustration of the trapezoidal area forthe integral of the function I 233.1 A schematic diagram of an SK film-substrate system containing atrancated pyramid island on a flat wetting layer 29

3.2 The variation of the function U(η) with the width ratio η for a

trancated pyramid island 313.3 The critical thickness for the trancated pyramid island formation 34

3.4 The variation of the first critical thickness H1 with the width ratio η 37 3.5 The variation of the surface energy density with the angle φ 42

3.6 The variation of the second critical thickness H2 with the width

ratio η 45

viii

4.1 A schematic diagram of an SK film-substrate system containing atrancated pyramid island on a flat wetting layer 49

4.2 The contours of the total energy change ∆E as a function of the island volume V and the angle φ for the island formation on a thick

film 524.3 Numerical results of the island morphology transition 544.4 The variation of the critical island volume for the island morphol-

ogy transition V cr with the angle φ0 624.5 The contours of the critical island volume for the island morphologytransition ˆV cr as a function of the surface energy density ratio γ2/γ1

and the angle φ0 63

4.6 The domain (η, V ) of the total energy change ∆Efacet < 0 for a

facet island formation on a thick film 644.7 The contours of the total energy change for the island formation

on a thin film ∆E as a function of the island volume V and the angle φ 66

4.8 The variation of the minimum facet angle φ min with the island

volume V for the case where H + l = 4H cr 67

4.9 The domain (η, V ) of the total energy change ∆Efacet < 0 for a

facet island on a thin film 68

4.10 The variation of the critical volume V cr with the normalized film

thickness (H + l)/H cr for the formation of island on a thin film 70

5.1 A schematic diagram of the cooperative formation 74

5.2 The variation of the strain energy function U c (η c) with the width

ratio η c and the derivative U 0

c (η c ) as a function of η c 77

5.3 The contours of the total energy change ∆E for the trench mation as a function of the island volume V and the width ratio

for-η c 81

5.4 The variation of the critical volume V T and the equilibrium trench

shape η c with G3/(tan φ)5 for the two trench growth modes 84

5.5 The contours of the driving force F0 = 0 and the contours of the

total energy change for a pyramid island ∆Epyramid= 0 85

1.1 Experimental Observation

Figure 1.1: A schematic diagram of the formation of SiGe islands on Si(001) substrate

The epitaxial growth of a Ge or a SiGe alloy film on the Si(001) substratehas been intensely studied for many years, driven by the desire to create Si-

Ge heterojunction superlattices, which would form the basis of optoelectronicdevices (Moriarty and Krishnamurthy., 1983; Pearsall et al., 1987) The growth

of the SiGe/Si(001) system follows the Stranski-Krastanow(SK) mode, which isone of the three basic growth modes for thin film In the SK mode, Ge or SiGe

1

firstly grows in a layer-by-layer mode for several layers, and the flat layers are alsocalled the “wetting layer” (The wetting layer thickness is about 3 to 4 monolayer inthe case of a Ge film on a Si substrate.) After the formation of the wetting layer,three-dimensional coherent islands form on the surface of the wetting layer (Tsaur

et al., 1981; Asai et al., 1985) The island formation pathway is illustrated in Fig.1.1 It is now well understood that the film develops into islands in order toreduce the elastic energy accumulated in the strained epilayer, because of the4.2% lattice mismatch between Ge and Si

Ge islands were first observed as {105}-faceted rectangular pyramids by Mo et

al (1990) In their work, meta-stable three-dimensional clusters were discovered.The small clusters, which are called “hut” clusters, have a prism shape with thesame atomic structure on all four facets Their study also showed that the clusters’

principal axes are strictly along two orthogonal <100> directions and all of the four facets are determined to be {105} planes The {105} plane can be simply understood as a vicinal (001) surface that tilts by 11.3 ◦ along <100>.

Similar pyramid islands were observed by Tomitori et al (1993) and von Hoegen et al (1993) when growing Ge on Si In addition to the Ge film,the pyramid islands were also found on the SiGe film For example, Pidduck et

Horn-al (1992) reported the formation of <100> oriented islands, with sidewall angles

being in the range between 9 and 16◦ when they grew Si1−xGex on Si(100) withx=0.20-0.26

A rich body of relevant works showed that the pyramid islands may undergosome shape transitions as the island size increases Tersoff and Tromp (1993)studied islands formation of Ag on Si(001) and found that islands would adopt ashape of long wire instead of a compact, symmetric shape if the island exceeded acritical size, and that the aspect ratio of the wire could be greater than 50:1 The

“quantum wires” were observed later in several systems, including GaAs on Si(Ponce and Hetherington, 1989), Au on Mo(111) and Au on Si(111) (Mundschau

et al., 1989), and Au on Ag(110) (Rousset et al., 1992)

Subsequent studies showed that, in equilibrium, small islands are square mids, while larger islands develop a more complex multi-facet shape, usually called

pyra-“dome” shape (Lutz et al., 1994; Floro et al., 1999; Ross et al., 1999)

Instead of finding elongated huts (Tersoff and Tromp, 1993), Lutz and

co-workers observed not only {105} facets but also {311} and {518} facets on the

surfaces of strained Si1−xGex films with x=0.15-0.35 (Lutz et al., 1994) The

growth sequence begins with the shallow {105} facets, followed by the appearance

of steeper facets, oriented on {311} and {518} crystal facets.

Ribeiro and co-workers(1998) determined that in the Ge/Si system the smallernanocrystals were mainly square-based pyramids and the larger nanocrystals were

multifaceted domes The major dome facets they observed are {113} and {102}

planes The shape of the dome can be described as a combination of two nearly generate pyramids rotated 45◦ with respect to each other and with the sharp apexblunted Floro et al (1999) found that the morphological evolution of Si1−xGexfilm at low mismatch strains is qualitatively consistent with that of a pure Ge film

de-on Si(001) The Si1−xGex films with strains less than 1% undergo the

Stranski-Krastanow transition to form islands, followed by the emergence of the {105}

faceted hut clusters and the transformation of the huts into domes characterized

by the {311} and {201} facets.

Further work by Ross and co-workers showed clearly that each island wouldundergo several distinct configurations during the growth The earliest stage ofgrowth is a surface roughening process during which the slope of the surface

increases until the {105} facets appear (Ross et al., 1999) After a dense array of

pyramids form, the islands coarsen, and eventually transform into domes when theisland size is sufficiently large The transition from pyramids to domes involves

two stages: Firstly, the {311} facets appear at the four corners of the pyramids.

The islands, called the transitional pyramid, are characterized by an octagonal

base The second stage is the appearance of the {15 3 23} facets[which is close

to the {518} facets reported in (Lutz et al., 1994)].

Prepyramids, first reported in (Chen et al., 1997), refer to the small islands thatform prior to the emergence of the pyramids The prepyramids are characterized

by a smooth shape without a specific facet, and they appear to be precursors tothe well studied pyramids Chen et al.’s results were later confirmed in (Vailionis

et al., 2000), which presented clear experimental evidence that two-dimensionalislands on the wetting layer first evolve into small rounded three-dimensional

islands and then transform into {105}-faceted pyramids with <100> oriented

rectangular bases In spite of the inspiring experimental findings, the nature ofthese islands and their role in the growth process have not been fully understood.Studying the morphological evolution of the Si1−xGex layers on Si(001) bySTM, Rastelli and von K¨anel (2003) postulated that the prepyramid-pyramidtransition involved the nucleation of facets in the middle of the prepyramid surfaceand the subsequent growth of the facets over the whole prepyramid surface.

In contrast to Rastelli and Von Kanel’s result, Sutter and Lagally (2000)demonstrated that nucleation is not involved in the formation of the facetedthree-dimensional islands on the Si1−xGex/Si systems with a low Ge concentra-tion Instead, the facet islands can form via a barrierless and continuous processinvolving shallow bumps The slope of these precursor bumps increases continu-ously until the bumps transform into faceted islands The experiments of Tromp

et al also showed that the growth of Si1−xGex islands does not evolve nucleationwhen the Ge concentration x is between 0.2 and 0.6 These two groups’ studiesboth presented that, at least in some range of temperature and alloy composition,islands can evolve continuously from surface ripples

1.2 Theoretical Study for Island Formation

When studying the formation of nano-crystalline islands on the heteroepitaxialfilm-substrate systems, one question often encountered is to calculate the strainenergy change during the formation process Typical numerical methods such

as the finite element method(FEM) (Freund and Jonsdottir, 1993; Floro et al.,

1998, 1999) for linear elasticity problems can be employed to solve the question;the results are accurate and there is no fundamental difficulty in calculating anyisland shape

The limitation of the FEM approach is that the numerical efficiency is not highenough to allow calculating a great many cases within reasonable time, especially

in the case of three-dimensional islands The limitation makes it expensive to usethe FEM approach to examine problems that involve more than two variables todescribe the island shapes There are several examples of these types of problems:the trancated pyramids with a rectangular base (Tersoff and Tromp, 1993), theequilibrium shapes of two-dimensional faceted islands (Daruka et al., 1999), andthe pyramid islands with a round top during the bump-pyramid transition (Tersoff

et al., 2002) It is therefore desirable to develop methods to estimate the strainenergy density on the island surfaces as well as the strain energy associated withthe formation of the islands

Gao (1991) proposed the first-order boundary perturbation method for theelasticity problem of a two-dimensional strained solid with a wavy surface Themethod is based on Muskhelishvilli’s complex variable potentials for two-dimensionalelasticity solutions and it is accurate to the first-order of the slope of the surface.The method can be applied to any smooth island profile, including cosine curvesand rounded islands on a flat surface The method can also be extended toanisotropic solids (Gao, 1991b) and film-substrate systems with different elasticconstants between the substrate and the film

Tersoff and Tromp (1993) suggested a different perturbation approach Intheir approach, which is in essence identical to Gao’s method, the effects of anisland surface on the elasticity solution can be approximated by a distribution ofsurface traction on the flat surface of a semi-infinite solid Applications of the ap-proach include elastically dissimilar film-substrate systems with three-dimensionalundulating surface (Freund and Jonsdottir, 1993) and two-dimensional smooth is-lands (Spencer and Tersoff, 1997).

Except for the smooth profiles, the surface traction approach can also beemployed for the cases of faceted islands, by ignoring the weak singularity at theedges between the islands and the flat surface (Tersoff and Tromp, 1993), andthe result is accurate to the first-order of the faceted slope The approach can beemployed not only for two-dimensional faceted islands (Daruka et al., 1999; Tersoff

et al., 2002), but also for three-dimensional ones (Tersoff and Tromp, 1993) Even

in the case of complicated shapes, the calculation of the strain energy of dimensional faceted islands is still straightforward However, the available formulainvolves a four-dimensional integral, which can be extremely demanding from thecomputation point of view This difficulty limits the application of the surfacetraction approach to studying the variation of the strain energy with the facetedisland shapes

three-A different formula for estimating the strain energy of a three-dimensionalfaceted island on an elastically similar film-substrate system is presented in (Chiuand Poh, 2004) The most important improvement of the new formula is that

it only needs two-dimensional integral when evaluating the island strain energy,

which greatly reduces the calculation time to a more acceptable duration Theformula will be employed in this thesis to calculate the strain energy change duringthe island formation.

Tersoff and LeGoues (1994) presented a widely used theory for the transitionfrom two-dimensional layer to the faceted three-dimensional islands, which as-sumed that these islands form via a three-dimensional nucleation process Three-dimensional island nucleation is characterized by a misfit-dependent critical vol-ume above which three-dimensional islands are stable against decay towards aplanar film, and the nucleation is an activated process involving activation en-ergy (Tersoff and LeGoues, 1994)

Tersoff et al (2002) proposed barrierless formation of tiny prepyramid islands

by using a simple two-dimensional model A simple assumption was employed thatthe surface-energy anisotropy allows all orientations near (001), with the first facetbeing (105) With this assumption, they predicted that tiny prepyramid islandswould form without nucleation barrier and that the islands would be unfaceted

As islands increase in size, the prepyramids would undergo a first-order shapetransition The results suggest that there are two critical volumes, V2 and V3, inthe process The quantity V2 is the volume at which a first-order shape transitioncan happen (from small smooth islands to faceted islands) and V3 denotes the

volume at which the slope of the island reaches the stability limit When V>V3,the only stable shape is a faceted island with a rounded top

Tersoff et al (2002) analyzed the energy change of island formation Theyfound that there is no energy barrier to the nucleation of an island since the energy

is a monotonically decreasing function of size even for arbitrarily small islands

A growing island will remain stable and unfaceted until the size is up to V2 atwhich point the island becomes meta-stable and in equilibrium it transforms to

a faceted shape However, the island may still grow continuously with unfacetedshape because there is an energy barrier for the first-order transition The energybarrier decreases with increasing size The energy barrier decreases to zero andthe unfaceted island becomes unstable against shape transition to a facetted one

as the size reaches V3

Tersoff’s experimental and theoretical investigations have made remarkableprogress in the understanding of island formation The picture of island formation

is explicit and the transition between different island shapes is clearly illustratedthrough energy analysis However, limitations still exist In the previous work,only a two-dimensional mode was employed to calculate the energy change Also,the development of trenches surrounding the pyramid islands after their formationhas not been fully understood

In this thesis, we study the formation of the SiGe nano-islands on the Sisubstrate from the energy point of view Our energy analyses are based on acontinuum three-dimensional model for the SiGe/Si system, and the analyses arecarried out by employing the first-order boundary perturbation method to calcu-late the total energy change during the island formation process The total energyincludes the strain energy, the surface energy, and the film-substrate interaction

energy Three important issues in the island formation process are investigated inthis thesis The first one is the critical thickness of the wetting layer below whichthe formation of islands is completely suppressed The second issue is the shapetransition from a shallow bump to a faceted pyramid, and of particular interest isthe dependence of the shape transition on the island size, the island shape and thefilm thickness The third issue examined in this thesis focuses on the cooperativeformation, which is characterized by the development of trenches surrounding thepyramid island after the bump-pyramid transition It is demonstrated that it isalways energetically favorable for the trench to develop on a thick film after thebump-pyramid transition, while the trench formation can be suppressed on a thinfilm.

The thesis is outlined as follows: Chapter 2 describes the model for the substrate systems examined in the thesis and the methodology for carrying outthe energy analysis Chapter 3 presents the results of the critical thickness forisland formation Chapter 4 focuses on the formation of islands via the surfaceundulation mode, followed by discussions of the trench formation in Chapter 5.The thesis is concluded in Chapter 6

film-Model and Methodology

2.1 A Continuum Model for the SK Film-Substrate

System

Figure 2.1: A schematic diagram of the formation of a facet island.

Fig 2.1 depicts the film-substrate system that is employed to investigate theisland formation process in this thesis The substrate is assumed to be a semi-

11

infinite solid, while the film consists of a flat wetting layer of thickness H and a

facet island which may contain several types of facets The facets are denoted as

{Γ1,Γ2, ,ΓN } where N is the number of facets of the island The angles between

the facet and the wetting layer surface are called the facet angle and are denoted

as {φ1,φ2, φ N }.

The facet island is thought to develop from the flat wetting layer under amass-conserved shape transformation process as shown in Fig 2.1 The totalenergy change during the island formation process is called the energy of theisland Similarly, the energy of other structures on the film-substrate system, forexample the island with a surrounding trench, refers to the total energy change

as the structure develops from a flat wetting layer by the mass-conserved process

The total energy of an SK system consists of the strain energy, the surface energy,and the film-substrate interaction energy (Tersoff and Tromp, 1993; Chiu and Gao,1995; Suo and Zhang, 1998) The strain energy in the SK system is caused bythe mismatch strain between the film and the substrate and it is well establishedthat the total strain energy of the system decreases as an island or a wavy surfacedevelops (Gao, 1991; Tersoff and Tromp, 1993) The reduction of the strain energy

is the driving force for the island formation on the SK systems

The Strain Energy

The strain energy reduction ∆W depends on the shape and the size of the island,

and it can be calculated by several techniques (Gao, 1991; Tersoff and Tromp,1993; Freund and Jonsdottir, 1993; Chiu and Poh, 2004) The scheme developed

by Chiu and Poh (2004) is adopted in this thesis; the method is briefly discussedlater in Section 2.2

The Surface Energy

The second type of energy involved in the SK system is the surface energy Thefilm surface energy change due to the island formation can be found to be

where A0 is the area of the island base, A i is the area of facet Γi , γ0 is the reference

surface energy density, and γ i is the surface energy density of surface Γi Equation(2.1) can be simplified to

The Film-Substrate Interaction Energy

The third type of energy in the SK system is the film-substrate interaction energy.The film-substrate interaction energy is adopted to account for the SK transition;

it can be modelled as a special type of surface energy of which the density g(z) depends on the distance z between the film surface and the film-substrate inter-

In the case where the interaction is caused by the quantum confinement fect (Suo and Zhang, 1998), the interaction energy density is found to be

ef-g(z) = g0l

where g0 and l are constants depending on the properties of the film and the

substrate

2.2 The First-Order Boundary Perturbation Method

In this section, we summarize the first-order boundary perturbation method

de-veloped by Chiu and Poh (2004) for determining the strain energy change due to

the formation of a facet island on a heteroepitaxial film-substrate system The

method is accurate to the first-order of the slope of the facet island and it is

con-sistent with the scheme suggested by Tersoff and Tromp (1993) The difference

between the two methods is that Tersoff and Tromp’s scheme involves a

four-dimensional integration, while Chiu and Poh’s method only requires one surface

integral when evaluating the island strain energy The method is valid for both

single island and island arrays, and it can be applied to study islands containing

one type of facets as well as the islands involving multiple types of facets Tsao

(1993) demonstrated that when the island aspect ratio is small, the error of the

first-order boundary perturbation method is small; when the island aspect ratio

increases to 0.9, the error of the method is around 5% As in this thesis the island

aspect ratio is much less than 0.9, the results from the calculation is accurate

enough Freund and Suresh (2003) also analyzed the accuracy of the first order

perturbation solution, and their result is same as the result developed by Tsao

(1993)

This section begins with the description of the island geometry The

discus-sion is followed by the solution procedure of the boundary perturbation methodand the implementation of the method.

Figure 2.2: A schematic diagram of a facet island on an SK film-substrate system.

Figure 2.2 depicts the film-substrate system considered in this thesis Thesubstrate of the system is modelled as a semi-infinite solid, while the film consists

of a flat wetting layer and a facet island on top of the flat wetting layer The island

contains N facets, which can be represented by {Γ1,Γ2, ,ΓN } The projected area

of facet Γi onto the wetting layer is denoted as R i; the set containing all of the

project areas is R It is clear that R corresponds to the base of the facet island.

The angle between the facet Γi and the flat wetting layer surface is φ i, and the

angle φ i of each facet may be different; the largest one, denoted as φ max, gives

the characteristic slope S of the island

The characteristic slope S is assumed to be much less than 1 The ratio between

the slope of facet Γi , tan φ i , and the characteristic value S is called the relative slope m of that facet

m = tan φ i

The relative slope m is zero on the flat surface and may be different from one

facet to another one

The film-substrate system is attached by a set of Cartesian coordinate axes

on the flat interface between the substrate and the film The x− and the y− axes lie parallel with the interface, while the z-axis is normal to the interface The notation x = (x, y, z)T represents a point in the system and the superscript Trefers to the transpose of vectors and tensors The notation xΓ denotes a point onthe film surface, and x0Γ = (x, y, H)T is a point on the flat surface of the wettinglayer The two points xΓ and x0, are related by

where f (x, y) describes the island morphology and e z is a unit vector in the

z-direction

The substrate and the film are elastically isotropic materials with the same

Possion’s ratio ν and shear modulus µ However, the two materials are subject

to a mismatch strain ε m between them, and the mismatch stain causes a stress

σ in the system The mismatch stress σ is determined by the balance of force in

If the film surface is perfectly flat, Eq (2.12) requires that there is no

stress in the substrate, σ s =0, and the film experiences a uniform biaxial stress

which, respectively, are the strain energy density in the strained flat film and

that in a solid under the plane strain and subject to the stress T in the lateral

where σ ∗(x) is the effect of the island on the stress in the system

When the characteristic slope S is small, σ ∗(x) can be derived by the boundary

perturbation method The first step is to express σ ∗ as

where ˆσ(1) is the normalized first-order term

The second step is to substitute the expression into the boundary conditiongiven in Eq (2.11) and rewrite the condition by Taylor’s series expansion This

leads to the finding that, accurate to the first order of S, the effects of the island

on the stress in the system can be approximated by a distribution of surfacetraction f on the flat wetting layer surface Γ0

Based on the finding, the normalized first-order term ˆσ(1) can be solved cally to be

k-th direction and at the location p on a flat film surface Substituting Eq (2.19)

into (2.16) and (2.15) and calculating the result at x = x0 give the total stress σ

on the film surface in the presence of the island, accurate to the first-order of the

slope S.

After knowing the stress on the film surface, we turn our attention to the

strain energy density w, which is related to σ by

where S11 = 1/2µ(1 + ν), S12 = −ν/2µ(1 + ν), and S44 = 1/µ The first-order solution of the stress vector σ can be expressed as

where the function Ψ(x0) is explained later in Eq (2.25)

The first term of Eq (2.24) w 3D

0 stands for the strain energy density on a flatfilm surface and is given earlier in Eq (2.14) The second term represents theeffect of the island on the surface strain energy density, which is proportional to

w0, the slope S of the island and the function Ψ(x0) illustrating the variation of

w on the film surface due to the island The function Ψ(x0) can be expressed asΨ(x0) =

The function Ψ, independent of the elastic properties of the system, is controlled

by the shape of the island

The strain energy change due to the formation of the island under the tion of mass conservation can be determined to be

condi-∆W = −w0S

Z

Γ

The quantity ∆W is loosely called the strain energy of the island in this thesis.

Equation (2.28) can be rewritten as

and the slope of the island

It follows from Eq (2.25) that the function Ψ(x0) can be expressed as the

sum of two surface integrals, I1 and I2, given by

Figure 2.3: (a) The trapezoidal area in which the function g1 (x 0, p) can be calculated exactly;

(b) the corresponding trapezoidal area for the function g2 (x 0, p).

It is shown by Chiu and Poh (2004) that the two surface integrals can be mined analytically on two types of trapezoidal areas The finding significantlyreduces the computational time needed for calculating the function Ψ(x0)

Consider the integral I1 first The integral can be determined analytically in thetrapezoidal area shown in Fig 2.3(a), which is characterized by the two parallel

lines in the p-direction The two parallel lines of the trapezoid are q = q1 and

q = q2, and the two nonparallel lines are p = m1q + d1 and p = m2q + d2 By

assuming the trapezoidal area is part of a projected area R i, ˆn(1)1 is a constant in

this area, and I1 can be expressed as

I1(x, y) = − mˆ n

(1) 1

π

Z q2

q1

1p

The quantity I1 in Eq (2.33) can be further simplified to

I1(x, y) = − mˆ n

(1) 1

Equation (2.34) shows that the integral I1 can be determined by evaluating the

algebraic function I g in Eq (2.35)

The integral I2, on the other hand, can be determined analytically in thetrapezoidal area shown in Fig 2.3(b), which is characterized by the two parallel

lines in the q-direction The two parallel lines of the trapezoid are p = p1 and

p = p2, and the two nonparallel lines are q = m1p + d1 and q = m2p + d2 The

integral I2 in the area can be found to be

I2(x, y) = − mˆ n

(1) 1

There are two expressions for the function I g (a, b, c, q1, q2) given in Eq (2.35)

The two expressions lead to the same result, and both can be adopted when a > 0 and b2 − 4ac < 0 One of the two expressions cannot be evaluated numerically

when b2−4ac = 0 since the argument of the logarithmic function is zero; however,

this problem would not occur in the other expression The range b2− 4ac > 0 can

be excluded when calculating ∆W

In summary, the numerical procedure for determining ∆W involves several steps: Firstly, divide the island base into smaller triangle for calculating I1(x, y) and I2(x, y) defined in Eqs (2.34) and (2.36) Secondly, summing I1(x, y) and

I2(x, y) in the triangles yields the function Ψ(x0) defined in Eq (2.25) Thirdly,substituting the result Ψ(x0) into Eq (2.24) gives the variation of the strain

energy density w(xΓ) on the film surface Finally, carrying out the surface integral

in Eq (2.30) yields U, which represents the effect of the island shape on the strain energy change ∆W during the formation of the island.

The Critical Thickness of the SK Transition

3.1 Introduction

The Stranski-Krastanow (SK) transition refers to the morphological change from

a flat film surface to a wavy or an island one when the film thickness exceeds

a critical value The SK transition is commonly observed in the self-assembly

of nano-islands on the heteroepitaxial systems (Jesson et al., 1996; Chen et al.,1997; Vailionis et al., 2000; Tersoff et al., 2002; Rastelli et al., 2003; Rastelli andvon K¨anel, 2003) More recently it is shown that the onset of the SK transition isaffected by whether the island formation mechanism is spontaneous formation orsurface undulation The islands formation via surface undulation corresponds tothe gradual morphological change from a smooth wavy surface, to round mounds,and then to faceted islands (Tersoff et al., 2002) On the other hand, the spon-

26

taneous formation means that the faceted islands form on the top of the wettinglayer by nucleation By either mechanism, there is a critical film thickness belowwhich the islands formation is suppressed However, because of the differences inthe island formation processes, the SK critical thicknesses of the two mechanismsare fundamentally different In most cases, the critical thickness of surface un-dulation is higher than that for spontaneous formation In such a case, there is

a film thickness range within which the film surface can develop into islands viaspontaneous formation but with surface undulation being totally prohibited Thespecial thickness range makes it possible to control the size, the shape and thelocation of the nano-structure on the SK systems by the activated SK transitionmethod (Chiu et al., 2004)

The previous study of the critical thickness of the SK transition has focused

on the case of pyramid islands (Chiu et al., 2004) However, the spontaneousformation of other shapes of island, for example the trancated pyramid observed

in experiments, has been ignored The effects of the island shapes on the criticalthickness of spontaneous formation are examined in this chapter

Another issue in the previous study of the critical thickness of the SK tion is that the critical thicknesses for surface undulation and spontaneous forma-tion were derived by different film surface profiles: a wavy profile for the formerand a facet island for the latter The different surface profiles raise a questionabout whether the different critical thicknesses of the two mechanisms are caused

transi-by the different surface profiles or transi-by the mechanisms This issue is explored inthis chapter by using the same surface profile to determine the critical thickness

for the SK transition of the two mechanisms The results clearly demonstrate thatthe critical thicknesses difference between the two mechanisms does not come fromthe shape of islands but from the kinetic pathways of island formation.

This chapter is organized as follows: Section 3.2 illustrates the trancated

pyra-mid model for determining the critical thickness for spontaneous island formationand surface undulation The critical thickness for spontaneous formation of the

trancated pyramid islands is presented in Section 3.3, and the critical thickness for surface undulation is discussed in Section 3.4.

3.2 Model

Figure 3.1 plots the film-substrate system considered in this chapter for studyingthe critical thickness of the SK transition The film-substrate system consists of

an infinitely thick substrate and a thin film which contains a flat wetting layer of

thickness H and a trancated pyramid island on the top of the wetting layer The

trancated pyramid is characterized by four identical facets and the angle between

the facet and the flat wetting layer is φ Both the base and the top of the island are square, and the width of them are D0 and D1 respectively

The island surface geometry can be described as

Figure 3.1: A schematic diagram of an SK film-substrate system containing a trancated pyramid island on a flat wetting layer.

where R refers to the base area of the island and ˆ f (x, y) is the normalized island

The ratio between the top surface width and the base width is termed the widthratio of the island

η = D1

D0

The total energy of an SK film-substrate system consists of the strain energy, thesurface energy and the interaction energy The general formulae for determiningthe three types of the energy of an SK system containing a facet island on thewetting layer surface are discussed earlier in Section 2.1 The formulae are appliedhere to write down the total energy of the SK system considered in this chapter

The Strain Energy

According to Eq (2.29), the strain energy change due to the formation of atrancated-pyramid island can be expressed as

where V is the volume of the island which can be found in Eq (3.3) and U is given by Eq (2.30), which is a function of the width ratio η The variation of

U(η) with η is plotted by Fig 3.2 The figure indicates that when the width

ratio η approaches 0, which means the island shape is nearly a pyramid, U(η) has

the maximum value On the contrary, when the width ratio approaches 1, which

means the island shape is like a flat disk, U(η) has the minimum value The variation of U(η) with η shows that the strain energy favors the pyramid shape.