Acknowledgement I would like to express my gratitude to all the persons who made this thesis possible. First of all, I gratefully acknowledge my supervisor, Dr. Chiu Cheng-hsin for his invaluable guidance and support. His enthusiasm and active research interests are the constant source of inspiration to me. During the course of this work, I have learnt from him on how to do research work. I would also like to thank our group members, Huang Zhijun, Wang Hangyao and C-T Poh, for their instruction and discussions in the research, sharing of their research experiences and all the assistance to me. Last but not least, I would like to thank my parents and my friends, for their care and encouragement during all my efforts and for their supporting all of my decisions. i Table of Contents Acknowledgement i Table of Contents ii Summary vi List of Figures viii 1 Introduction 1.1 1.2 1 Experimental Observation . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Island Formation . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Prepyramid to Pyramid Transition . . . . . . . . . . . . . 4 Theoretical Study for Island Formation . . . . . . . . . . . . . . . 5 1.2.1 Boundary Perturbation Method . . . . . . . . . . . . . . . 5 1.2.2 Energy Analysis for Island Formation . . . . . . . . . . . . 8 ii Table of Contents 2 Model and Methodology 2.1 2.2 iii 11 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 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 The Critical Thickness for Spontaneous Island Formation . . . . . 34 3.3.1 Numerical Result . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.2 The First Critical Thickness . . . . . . . . . . . . . . . . . 36 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 3.3 3.4 Table of Contents 4 The Formation of Trancated Pyramid Islands iv 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 Island Formation on a Thick Film via Surface Undulation . . . . . 52 4.3.1 A Typical Numerical Result . . . . . . . . . . . . . . . . . 52 4.3.2 Analytical Results . . . . . . . . . . . . . . . . . . . . . . 55 Island Formation on a Thin Film via Surface Undulation . . . . . 65 4.3 4.4 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 At . . . . . . . . . . . . . . . . . . . . . . . 75 The Total Energy Change ∆E . . . . . . . . . . . . . . . . 76 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 5.2.3 5.3 5.4 The Stability of a Pyramid Island on a Thin Film against Trench Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Table of Contents v 6 Conclusion 87 Bibliography 91 Summary The SiGe on Si(001) system has been extensively studied for understanding the heteroepitaxy of the system and there is a wide interest in employing the “selfassembled quantum dots” grown on the system in the nano-technology. In this thesis, we study the formation of the SiGe nano-islands on the Si substrate from the energy point of view. Our energy analyses are based on a continuum threedimensional model for the SiGe/Si system, and the analyses are carried out by employing the first-order boundary perturbation method to calculate the energy change during the island formation process. Three important issues in the island formation process are investigated. The first one is the critical thickness of the wetting layer below which the formation of islands is completely suppressed. The second 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 thesis focuses 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 SUMMARY be suppressed on a thin film. vii List of Figures 1.1 A schematic diagram of the formation of SiGe islands formation on Si(001) substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.1 A schematic diagram of the formation of a facet island . . . . . . 11 2.2 A schematic diagram of a facet island on an SK film-substrate system 16 2.3 A schematic diagram of the illustration of the trapezoidal area for the integral of the function I . . . . . . . . . . . . . . . . . . . . . 3.1 A schematic diagram of an SK film-substrate system containing a trancated pyramid island on a flat wetting layer . . . . . . . . . . 3.2 23 29 The variation of the function U (η) with the width ratio η for a trancated pyramid island . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 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 . . . 3.6 The variation of the second critical thickness H2 with the width ratio η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 42 45 List of figures 4.1 ix A schematic diagram of an SK film-substrate system containing a trancated pyramid island on a flat wetting layer . . . . . . . . . . 4.2 49 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Numerical results of the island morphology transition . . . . . . . 54 4.4 The variation of the critical island volume for the island morphology transition Vcr with the angle φ0 . . . . . . . . . . . . . . . . . 4.5 62 The contours of the critical island volume for the island morphology transition Vˆcr as a function of the surface energy density ratio γ2 /γ1 and the angle φ0 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . The domain (η, V ) of the total energy change ∆Efacet < 0 for a facet island formation on a thick film . . . . . . . . . . . . . . . . 4.7 63 64 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 The variation of the minimum facet angle φmin with the island volume V for the case where H + l = 4Hcr . . . . . . . . . . . . . 4.9 66 67 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 Vcr with the normalized film thickness (H + l)/Hcr for the formation of island on a thin film . 70 List of figures 5.1 A schematic diagram of the cooperative formation . . . . . . . . . 5.2 The variation of the strain energy function Uc (ηc ) with the width ratio ηc and the derivative Uc (ηc ) as a function of ηc . . . . . . . . 5.3 x 74 77 The contours of the total energy change ∆E for the trench formation as a function of the island volume V and the width ratio ηc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The variation of the critical volume VT and the equilibrium trench shape ηc with G3 /(tan φ)5 for the two trench growth modes . . . . 5.5 81 84 The contours of the driving force F0 = 0 and the contours of the total energy change for a pyramid island ∆Epyramid = 0 . . . . . . 85 Chapter 1 Introduction 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) substrate has been intensely studied for many years, driven by the desire to create SiGe heterojunction superlattices, which would form the basis of optoelectronic devices (Moriarty and Krishnamurthy., 1983; Pearsall et al., 1987). The growth of the SiGe/Si(001) system follows the Stranski-Krastanow(SK) mode, which is one of the three basic growth modes for thin film. In the SK mode, Ge or SiGe 1 Chapter 1: Introduction 2 firstly grows in a layer-by-layer mode for several layers, and the flat layers are also called the “wetting layer” (The wetting layer thickness is about 3 to 4 monolayer in the 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 to reduce the elastic energy accumulated in the strained epilayer, because of the 4.2% lattice mismatch between Ge and Si. 1.1.1 Island Formation 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 the same atomic structure on all four facets. Their study also showed that the clusters’ principal axes are strictly along two orthogonal 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 . Similar pyramid islands were observed by Tomitori et al. (1993) and Hornvon 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 al. (1992) reported the formation of oriented islands, with sidewall angles being in the range between 9 and 16 ◦ when they grew Si1−x Gex on Si(100) with x=0.20-0.26. Chapter 1: Introduction 3 A rich body of relevant works showed that the pyramid islands may undergo some 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 a shape of long wire instead of a compact, symmetric shape if the island exceeded a critical 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 pyramids, while larger islands develop a more complex multi-facet shape, usually called “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 coworkers observed not only {105} facets but also {311} and {518} facets on the surfaces of strained Si1−x Gex 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 smaller nanocrystals 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 degenerate pyramids rotated 45 ◦ with respect to each other and with the sharp apex blunted. Floro et al. (1999) found that the morphological evolution of Si1−x Gex film at low mismatch strains is qualitatively consistent with that of a pure Ge film Chapter 1: Introduction 4 on Si(001). The Si1−x Gex films with strains less than 1% undergo the StranskiKrastanow 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 would undergo several distinct configurations during the growth. The earliest stage of growth 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 the island 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)]. 1.1.2 Prepyramid to Pyramid Transition Prepyramids, first reported in (Chen et al., 1997), refer to the small islands that form 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 to the well studied pyramids. Chen et al.’s results were later confirmed in (Vailionis et al., 2000), which presented clear experimental evidence that two-dimensional islands on the wetting layer first evolve into small rounded three-dimensional islands and then transform into {105}-faceted pyramids with oriented Chapter 1: Introduction 5 rectangular bases. In spite of the inspiring experimental findings, the nature of these islands and their role in the growth process have not been fully understood. Studying the morphological evolution of the Si1−x Gex layers on Si(001) by STM, Rastelli and von K¨anel (2003) postulated that the prepyramid-pyramid transition involved the nucleation of facets in the middle of the prepyramid surface and 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 faceted three-dimensional islands on the Si1−x Gex /Si systems with a low Ge concentration. Instead, the facet islands can form via a barrierless and continuous process involving shallow bumps. The slope of these precursor bumps increases continuously until the bumps transform into faceted islands. The experiments of Tromp et al. also showed that the growth of Si1−x Gex islands does not evolve nucleation when the Ge concentration x is between 0.2 and 0.6. These two groups’ studies both presented that, at least in some range of temperature and alloy composition, islands can evolve continuously from surface ripples. 1.2 1.2.1 Theoretical Study for Island Formation Boundary Perturbation Method When studying the formation of nano-crystalline islands on the heteroepitaxial film-substrate systems, one question often encountered is to calculate the strain energy change during the formation process. Typical numerical methods such Chapter 1: Introduction 6 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 any island shape. The limitation of the FEM approach is that the numerical efficiency is not high enough to allow calculating a great many cases within reasonable time, especially in the case of three-dimensional islands. The limitation makes it expensive to use the FEM approach to examine problems that involve more than two variables to describe the island shapes. There are several examples of these types of problems: the trancated pyramids with a rectangular base (Tersoff and Tromp, 1993), the equilibrium shapes of two-dimensional faceted islands (Daruka et al., 1999), and the 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 strain energy density on the island surfaces as well as the strain energy associated with the formation of the islands. Gao (1991) proposed the first-order boundary perturbation method for the elasticity problem of a two-dimensional strained solid with a wavy surface. The method is based on Muskhelishvilli’s complex variable potentials for two-dimensional elasticity 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 curves and rounded islands on a flat surface . The method can also be extended to anisotropic solids (Gao, 1991b) and film-substrate systems with different elastic constants between the substrate and the film. Chapter 1: Introduction 7 Tersoff and Tromp (1993) suggested a different perturbation approach. In their approach, which is in essence identical to Gao’s method, the effects of an island surface on the elasticity solution can be approximated by a distribution of surface traction on the flat surface of a semi-infinite solid. Applications of the approach include elastically dissimilar film-substrate systems with three-dimensional undulating surface (Freund and Jonsdottir, 1993) and two-dimensional smooth islands (Spencer and Tersoff, 1997). Except for the smooth profiles, the surface traction approach can also be employed for the cases of faceted islands, by ignoring the weak singularity at the edges between the islands and the flat surface (Tersoff and Tromp, 1993), and the result is accurate to the first-order of the faceted slope. The approach can be employed 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 threedimensional faceted islands is still straightforward. However, the available formula involves a four-dimensional integral, which can be extremely demanding from the computation point of view. This difficulty limits the application of the surface traction approach to studying the variation of the strain energy with the faceted island shapes. A different formula for estimating the strain energy of a three-dimensional faceted island on an elastically similar film-substrate system is presented in (Chiu and Poh, 2004). The most important improvement of the new formula is that it only needs two-dimensional integral when evaluating the island strain energy, Chapter 1: Introduction 8 which greatly reduces the calculation time to a more acceptable duration. The formula will be employed in this thesis to calculate the strain energy change during the island formation. 1.2.2 Energy Analysis for Island Formation Tersoff and LeGoues (1994) presented a widely used theory for the transition from two-dimensional layer to the faceted three-dimensional islands, which assumed that these islands form via a three-dimensional nucleation process. Threedimensional island nucleation is characterized by a misfit-dependent critical volume above which three-dimensional islands are stable against decay towards a planar film, and the nucleation is an activated process involving activation energy (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 that the surface-energy anisotropy allows all orientations near (001), with the first facet being (105). With this assumption, they predicted that tiny prepyramid islands would form without nucleation barrier and that the islands would be unfaceted. As islands increase in size, the prepyramids would undergo a first-order shape transition. The results suggest that there are two critical volumes, V2 and V3 , in the process. The quantity V2 is the volume at which a first-order shape transition can 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. Chapter 1: Introduction 9 Tersoff et al. (2002) analyzed the energy change of island formation. They found 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 at which point the island becomes meta-stable and in equilibrium it transforms to a faceted shape. However, the island may still grow continuously with unfaceted shape because there is an energy barrier for the first-order transition. The energy barrier decreases with increasing size. The energy barrier decreases to zero and the unfaceted island becomes unstable against shape transition to a facetted one as the size reaches V3 . Tersoff’s experimental and theoretical investigations have made remarkable progress in the understanding of island formation. The picture of island formation is explicit and the transition between different island shapes is clearly illustrated through 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 formation has not been fully understood. In this thesis, we study the formation of the SiGe nano-islands on the Si substrate from the 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 by employing the first-order boundary perturbation method to calculate the total energy change during the island formation process. The total energy includes the strain energy, the surface energy, and the film-substrate interaction Chapter 1: Introduction 10 energy. Three important issues in the island formation process are investigated in this thesis. The first one is the critical thickness of the wetting layer below which the formation of islands is completely suppressed. The second 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 thesis focuses 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 be suppressed on a thin film. The thesis is outlined as follows: Chapter 2 describes the model for the filmsubstrate systems examined in the thesis and the methodology for carrying out the energy analysis. Chapter 3 presents the results of the critical thickness for island formation. Chapter 4 focuses on the formation of islands via the surface undulation mode, followed by discussions of the trench formation in Chapter 5. The thesis is concluded in Chapter 6. Chapter 2 Model and Methodology 2.1 A Continuum Model for the SK Film-Substrate System 2.1.1 The Geometry of the Island 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 the island formation process in this thesis. The substrate is assumed to be a semi11 Chapter 2: Model and Methodology 12 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 ,...φN }. The facet island is thought to develop from the flat wetting layer under a mass-conserved shape transformation process as shown in Fig. 2.1. The total energy change during the island formation process is called the energy of the island. Similarly, the energy of other structures on the film-substrate system, for example 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. 2.1.2 The Total Energy of the SK System 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 by the mismatch strain between the film and the substrate and it is well established that the total strain energy of the system decreases as an island or a wavy surface develops (Gao, 1991; Tersoff and Tromp, 1993). The reduction of the strain energy is the driving force for the island formation on the SK systems. Chapter 2: Model and Methodology 13 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 discussed later in Section 2.2. The Surface Energy The second type of energy involved in the SK system is the surface energy. The film surface energy change due to the island formation can be found to be n ∆ES = γi Ai − γ0 A0 (2.1) i=1 where A0 is the area of the island base, Ai is the area of facet Γi , γ0 is the reference surface energy density, and γi is the surface energy density of surface Γi . Equation (2.1) can be simplified to ∆ES = Gγ0 Ao (2.2) where n G= i=1 γˆi − 1, cos φi γˆi = γi . γ0 (2.3) 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; Chapter 2: Model and Methodology 14 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 interface. According to the model, the interaction energy of the system can be given by EI = g(z)dΓ (2.4) where dΓ denotes area integral over the whole film surface. It follows from Eq. (2.4) that the total interaction energy change during the island formation process shown in Fig. 1.1 can be expressed as ∆EI = g(z)dΓ − A0 g(H) − V ∂Ω dg dz (2.5) z=H where ∂Ω and H denote the surface of the island and the height of the wetting layer respectively, and V is the volume of the island. The three terms in Eq. (2.5) can be understood as follows: The first term represents the contribution from the surface of the facet island; the second term determines the interaction energy of the island base area prior to the island formation; and the third term accounts for the interaction energy change due to the decrease of the wetting layer thickness. The wetting layer thickness decreases by an infinitesimal amount, while the corresponding interaction energy change is finite since the area of the wetting layer is infinitely large. In the case where the interaction is caused by the quantum confinement effect (Suo and Zhang, 1998), the interaction energy density is found to be g(z) = g0 l z+l (2.6) Chapter 2: Model and Methodology 15 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 developed 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 consistent with the scheme suggested by Tersoff and Tromp (1993). The difference between the two methods is that Tersoff and Tromp’s scheme involves a fourdimensional 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- Chapter 2: Model and Methodology 16 sion is followed by the solution procedure of the boundary perturbation method and the implementation of the method. 2.2.1 Description of the System 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. The substrate 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 Ri ; 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 φi , and the Chapter 2: Model and Methodology 17 angle φi of each facet may be different; the largest one, denoted as φmax , gives the characteristic slope S of the island S = tan φmax . (2.7) The characteristic slope S is assumed to be much less than 1. The ratio between the slope of facet Γi , tan φi , and the characteristic value S is called the relative slope m of that facet m= tan φi . S (2.8) 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 T refers to the transpose of vectors and tensors. The notation xΓ denotes a point on the film surface, and x0 Γ = (x, y, H)T is a point on the flat surface of the wetting layer. The two points xΓ and x0 , are related by xΓ = x0 + f (x, y)ez (2.9) where f (x, y) describes the island morphology and ez is a unit vector in the zdirection. 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 Chapter 2: Model and Methodology 18 to a mismatch strain εm 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 the system σ(x) · ∇ = 0, (2.10) the traction-free boundary condition on the film surface xΓ σ(xΓ ) · n = 0, (2.11) and the coherent interface condition on the flat interface between the film and the substrate σ f · n = σ s · n, uf = us + εm xex + εm yey (2.12) where subscribes f and s refer to the film and the substrate respectively, and u is the displacement. 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 σ (0) (Nix,1989)  σ (0)   T 0 0       =  0 T 0 ,     0 0 0 T = 2µ(1 + ν)εm . 1−ν (2.13) The mismatch stress T is related to the two strain energy densities, w0 3D and w0 w0 3D = 1−ν T 2, 2µ(1 + ν) w0 = 1−ν 2 T , 4µ (2.14) Chapter 2: Model and Methodology 19 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 direction. In the case of an islanded film, the stress σ can be expressed as     σ (0) + σ ∗ (x) in the film σ(x) =    σ ∗ (x) in the substrate (2.15) where σ ∗ (x) is the effect of the island on the stress in the system. 2.2.2 The First-Order Boundary Perturbation Method 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 ˆ (1) (x) σ ∗ (x) = ST σ (2.16) where σ ˆ (1) is the normalized first-order term. The second step is to substitute the expression into the boundary condition given 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 surface traction f on the flat wetting layer surface Γ0 f = −T · S · m · n ˆ (1) (2.17) where n ˆ (1) is a unit vector along the projection of the normal vector n onto Γ0 . The two vectors n ˆ (1) and n are related by ˆ (1) = n n − n · ez . sin φ (2.18) Chapter 2: Model and Methodology 20 Based on the finding, the normalized first-order term σ ˆ (1) can be solved analytically to be 2 (1) σ ˆij (x) (1) =− Γ0 k=1 Σkij (x, p) · n ˆ k (p)m(p)dΓp (2.19) where p is a point on Γ0 , dΓp indicates an integration over the entire surface Γ0 , and k ij (x,p) is defined as the stress σij at x due to a point force acting in the 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 1 w = σT · S · σ 2 (2.20) where σ is the stress vector σ = (σ11 , σ22 , σ33 , σ23 , σ13 , σ12 )T , (2.21) and S is the compliance matrix of the materials    S11 S11 S12    S  12 S11 S12    S  12 S12 S11 S=   S44     S44                       S44 (2.22) Chapter 2: Model and Methodology 21 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 σ = T (1 + S σ ˆ11 , 1 + S σ ˆ22 , 0, S σ ˆ23 , S σ ˆ13 , S σ ˆ12 )T . (2.23) Substituting Eq. (2.23) into (2.20), ignoring the higher-order terms and calculating σ ˆ (1) on the surface Γ0 yield the result of w on the film surface w(xΓ ) = w03D − 2Sw0 Ψ(x0 ) (2.24) where the function Ψ(x0 ) is explained later in Eq. (2.25). The first term of Eq. (2.24) w03D stands for the strain energy density on a flat film surface and is given earlier in Eq. (2.14). The second term represents the effect 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 as 2 (1) Ψ(x0 ) = k=1 R gk (x0 , p)ˆ nk (p)m(p)dΓp (2.25) where g1 (x0 , p) = − x−p 1 , π [(x − p)2 + (y − q)2 ] 23 (2.26) g2 (x0 , p) = − y−q 1 . π [(x − p)2 + (y − q)2 ] 23 (2.27) 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 condition of mass conservation can be determined to be ∆W = −w0 S Ψ(x0 )f (x0 )dΓx0 . Γ0 (2.28) Chapter 2: Model and Methodology 22 The quantity ∆W is loosely called the strain energy of the island in this thesis. Equation (2.28) can be rewritten as ∆W = −w0 Vi U S (2.29) where Vi is the island volume and U is given by U= 1 Vi Ψ(x0 )f (x0 )dΓx0 . (2.30) Γ0 The function U depends on the island shape but is independent of the size and the characteristic slopes S of the island. Equation (2.29) suggests that the island strain energy, accurate to the first order of S, is controlled by the strain energy density w0 and the product Vi U S representing the effect of the volume, the shape, and the slope of the island. 2.2.3 The Function Ψ(x0 ) To determine strain energy change ∆W , the key step is to calculate U by Eq. (2.30) which involves f (x0 ) and Ψ(x0 ). The function f (x0 ) describes the shape of the island and it can be expressed analytically. Evaluating the other function Ψ(x0 ), on the other hand, is challenging and crucial. 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 I1 = R I2 = R (1) (2.31) (1) (2.32) g1 (x0 , p)ˆ n1 (p)m(p)dΓp , g2 (x0 , p)ˆ n2 (p)m(p)dΓp . Chapter 2: Model and Methodology 23 Figure 2.3: (a) The trapezoidal area in which the function g1 (x0 , p) can be calculated exactly; (b) the corresponding trapezoidal area for the function g2 (x0 , p). It is shown by Chiu and Poh (2004) that the two surface integrals can be determined analytically on two types of trapezoidal areas. The finding significantly reduces the computational time needed for calculating the function Ψ(x0 ). 2.2.4 The Numerical Implementation Consider the integral I1 first. The integral can be determined analytically in the trapezoidal 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 = m1 q + d1 and p = m2 q + d2 . By (1) assuming the trapezoidal area is part of a projected area Ri , n ˆ 1 is a constant in this area, and I1 can be expressed as (1) mˆ n1 I1 (x, y) = − π (1) = − mˆ n1 π q2 m2 q+d2 g1 (x0 , p)dpdq q1 q2 q1 m1 q+d1 1 (x − p)2 + (y − q)2 m2 q+d2 dq. m1 q+d1 (2.33) Chapter 2: Model and Methodology 24 The quantity I1 in Eq. (2.33) can be further simplified to (1) mˆ n1 I1 (x, y) = − {Ig [1 + m22 , 2m2 (d2 − x) − 2y, (d2 − x)2 + y 2 , q1 , q2 ] π − Ig [1 + m21 , 2m1 (d1 − x) − 2y, (d1 − x)2 + y 2 , q1 , q2 ]} (2.34) where Ig is given by q2 Ig (a, b, c, q1 , q2 ) = 1 aq 2 + bq + c 1 b + 2aq = ± √ ln ±( √ ) + 2 aq 2 + bq + c a a q1 q2 q1 . (2.35) Equation (2.34) shows that the integral I1 can be determined by evaluating the algebraic function Ig in Eq. (2.35). The integral I2 , on the other hand, can be determined analytically in the trapezoidal 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 = m1 p + d1 and q = m2 p + d2 . The integral I2 in the area can be found to be (1) I2 (x, y) = − mˆ n1 {Ig [1 + m22 , 2m2 (d2 − y) − 2x, (d2 − y)2 + x2 , p1 , p2 ] π − Ig [1 + m21 , 2m1 (d1 − y) − 2x, (d1 − y)2 + x2 , p1 , p2 ]} (2.36) There are two expressions for the function Ig (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 . Chapter 2: Model and Methodology 25 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. Chapter 3 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 and von K¨anel, 2003). More recently it is shown that the onset of the SK transition is affected by whether the island formation mechanism is spontaneous formation or surface undulation. The islands formation via surface undulation corresponds to the 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 spon26 Chapter 3: The Critical Thickness of the SK Transition 27 taneous formation means that the faceted islands form on the top of the wetting layer by nucleation. By either mechanism, there is a critical film thickness below which the islands formation is suppressed. However, because of the differences in the island formation processes, the SK critical thicknesses of the two mechanisms are fundamentally different. In most cases, the critical thickness of surface undulation 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 via spontaneous formation but with surface undulation being totally prohibited. The special thickness range makes it possible to control the size, the shape and the location of the nano-structure on the SK systems by the activated SK transition method (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 spontaneous formation of other shapes of island, for example the trancated pyramid observed in experiments, has been ignored. The effects of the island shapes on the critical thickness of spontaneous formation are examined in this chapter. Another issue in the previous study of the critical thickness of the SK transition is that the critical thicknesses for surface undulation and spontaneous formation were derived by different film surface profiles: a wavy profile for the former and a facet island for the latter. The different surface profiles raise a question about whether the different critical thicknesses of the two mechanisms are caused by the different surface profiles or by the mechanisms. This issue is explored in this chapter by using the same surface profile to determine the critical thickness Chapter 3: The Critical Thickness of the SK Transition 28 for the SK transition of the two mechanisms. The results clearly demonstrate that the critical thicknesses difference between the two mechanisms does not come from the shape of islands but from the kinetic pathways of island formation. This chapter is organized as follows: Section 3.2 illustrates the trancated pyramid model for determining the critical thickness for spontaneous island formation and 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 3.2.1 Model The Geometry of the Island Figure 3.1 plots the film-substrate system considered in this chapter for studying the 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     H (x, y) ∈ R f (x, y) =    H + A · fˆ(x, y) (x, y) ∈ R (3.1) Chapter 3: The Critical Thickness of the SK Transition 29 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 fˆ(x, y) is the normalized island shape function     1       D0 − 2x      D − D1   0 D0 − 2y fˆ(x, y) =  D0 − D1     D0 + 2x     D0 − D1     D + 2y    0 D0 − D1 (x, y) ∈ R0 (x, y) ∈ R1 (x, y) ∈ R2 (3.2) (x, y) ∈ R3 (x, y) ∈ R4 where R0 refers to the top surface of island and R1 - R4 refer to the four facet surfaces of the island respectively. The volume of the island can be derived as tan φ(D03 − D13 ) , V = 6 (3.3) and the height of the island A can be expressed as A= tan φ(D0 − D1 ) . 2 (3.4) Chapter 3: The Critical Thickness of the SK Transition 30 The ratio between the top surface width and the base width is termed the width ratio of the island η= 3.2.2 D1 . D0 (3.5) The Total Energy Change ∆E The total energy of an SK film-substrate system consists of the strain energy, the surface energy and the interaction energy. The general formulae for determining the three types of the energy of an SK system containing a facet island on the wetting layer surface are discussed earlier in Section 2.1. The formulae are applied here 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 a trancated-pyramid island can be expressed as ∆W = −wo V U (η) tan φ (3.6) 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. Chapter 3: The Critical Thickness of the SK Transition 31 2 1.8 1.6 1.4 U(η) 1.2 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 η 0.6 0.7 0.8 0.9 1 Figure 3.2: The variation of the function U (η) with the width ratio η for a trancated pyramid island. The Surface Energy The area of the projection of the four facets onto the island base is (D02 − D12 ); thus, the facet surface area is (D02 − D12 )/ cos φ. The total surface energy change due to the formation of trancated pyramid island can be found to be ∆ES = γ2 D02 − D12 − γ1 (D02 − D12 ) cos φ (3.7) where γ1 is the surface energy density of a flat film surface and γ2 is that of a facet surface. By evoking the definition of η given in Eq. (3.5), the surface energy change can be rewritten as ∆ES = γ0 D02 G(1 − η 2 ) (3.8) Chapter 3: The Critical Thickness of the SK Transition 32 where G= γ2 γ1 − . γ0 cos φ γ0 (3.9) Equation (3.8) indicates the surface energy change is controlled by the base area D02 , the shape of the island represented by (1 − η 2 ), and the quantity G representing the effects of the surface energy densities and the angle φ on ∆Es . The quantity G can be negative if γ2 < γ1 cos φ; in such a case, the surface energy favors the formation of the facet, and there will be no energy barrier to the island formation. This scenario is not considered here since it is not observed in the self-assembly of nano-islands on the hetero-epitaxial systems. The Interaction Energy Section 2.1 gives the general formula for the interaction energy. By adopting Eqs. (2.4) and (2.6), and considering the trancated pyramid island geometry which is given by Eq. (3.1), the interaction energy change due to a trancated pyramid island formation can be determined to be ∆EI = g0 lD02 I (3.10) where I= 2(1 − η) 1 D0 (1 − η 3 ) η2 + B− . tan φ + 2 6(H + l) H +l+A A cos φ H +l (3.11) The quantity A in Eq. (3.11) is the height of the island and the quantity B is given by B = 1+ (1 − η)(H + l) A ln 1 + − (1 − η). A H +l (3.12) Chapter 3: The Critical Thickness of the SK Transition 33 The Total Energy The total energy change of the system due to the formation of one island is given by the sum of the strain energy change, the surface energy change and the interaction energy change which are given by Eqs. (3.6) , (3.8) and (3.10) respectively. By employing these equations, the total energy change can be expressed as ∆E = −w0 (tan φ)2 D03 (1 − η 3 )U + γ0 D02 (1 − η 2 )G + g0 lD02 I. 6 (3.13) To simplify Eq. (3.13), two length scales of the system, namely L = τ0 /w0 and gˆ0 l = g0 l/τ0 , are adopted. The quantity gˆ0 l is the length scale associated with the interaction energy, and L represents the length scale at which the strain energy relaxation is balanced with the surface energy increment as the islands form (Gao, 1991). By adopting L and gˆ0 l, Eq. (3.13) can be rewritten by the following nondimensional form ∆Eˆ = 2 ∆E 6 = −Vˆ tan φU + GVˆ 3 (1 − η 2 ) 2 3 γ0 L (1 − η ) tan φ 2 3 2 + gˆ0 lVˆ 3 6 tan φ 2 3 I (3.14) where Vˆ is the normalized island volume D3 tan φ(1 − η 3 ) V . Vˆ = 3 = 0 L 6L3 (3.15) The three terms in the Eq. (3.14) represent the strain energy change, the surface energy change and the interaction energy change respectively. The strain energy change is proportional to Vˆ , while the surface energy change is proportional to 2 Vˆ 3 . The expression implies that as the island’s volume increases, the strain Chapter 3: The Critical Thickness of the SK Transition 34 energy change dominates the total energy change and consequently decides the morphology of the island. 3.3 The Critical Thickness for Spontaneous Island Formation 3.3.1 Numerical Result 80 η=0.001 η=0.5 η=0.9 70 60 V1/3(nm) 50 ∆ E0 0 0 0.5 1 1.5 2 2.5 H(nm) 3 3.5 4 4.5 Figure 3.3: The critical thickness for the trancated pyramid island formation is shown by the regime (V 1/3 , H) of ∆E < 0 in which the trancated pyramid island formation is energetically favorable for the cases where η = 0.001, 0.5, 0.9. Chapter 3: The Critical Thickness of the SK Transition 35 For the spontaneous formation of the trancated pyramid island, the angle φ is fixed. Therefore, the total energy change depends on the island volume V , the film thickness H and the width ratio η. 1 Figure 3.3 plots the regime (V 3 , H) of ∆E < 0 for the case where L = 25 nm, l = 0.1 nm, and gˆ0 = 0.0625. Three different width ratios are considered: 1 η = 0.001, 0.5, and 0.9. The domain (V 3 , H) with ∆E < 0 specifies the regime in which the island formation is energetically favorable. In spite of different width ratios, all the energy analyses show the same characteristics: At large H, the spontaneous island formation is energetically favorable, i.e. ∆E < 0, when the island volume is larger than a critical value Vcr ; the critical island size increases as H decreases; and Vcr approaches infinity as H approaches a critical film thickness, which is called the first critical thickness in this thesis. For comparison, the critical thickness for pyramid island predicted by Chiu et al. (2004) is plotted by the vertical line in Fig. 3.3. The theoretical prediction and our numerical result of η = 0.001 (an almost fully developed pyramid) are consistent with each other. The numerical results indicate that the first critical thickness H1 is larger for islands with a higher width ratio. This mainly comes from the fact that a higher width ratio leads to less strain energy reduction and thus a smaller driving force for the formation of truncated islands. The smaller driving force for truncated islands also explains the finding that, at the same film thickness, the critical island size formation increases with the island width ratio. Measuring the first critical thickness directly in experiments is still a problem because of the difficulty to control the island shape and island size. How to Chapter 3: The Critical Thickness of the SK Transition 36 overcome this issue is a question that needs to be explored in the future. 3.3.2 The First Critical Thickness Figure 3.3 suggests that the first critical thickness H1 is determined by considering ∆E when the island volume V approaches infinity. When the island volume V approaches infinity, the total energy change duo to the formation of a trancated pyramid island can be simplified to ∆E = −w0 tan φV U (η) + g0 l H1 V = V w0 tan φU (η) 1 − 2 (H + l) H +L 2 (3.16) where H1 is given by H + l = H1 = g0 l . w0 U (η) tan φ (3.17) Equation (3.16) indicates the sign of ∆E of a large island is controlled by H. If H + l < H1 , ∆E of large islands is positive, implying island formation of any island size is suppressed. In contrast, if H + l > H1 , ∆E < 0 and the film surface is unstable against the formation of facet islands. The above discussion suggests H + l = H1 is the critical thickness of the SK transition driven by spontaneous formation. Equation (3.17) is reduced to the result derived in Chiu et al. (2004) when η is taken to be 0. Equation (3.17) shows that the first critical thickness is affected by the angle φ. In particular, the critical thickness is large at small angle φ and the critical thickness decreases as the angle increases. Equation (3.17) also reveals that the first critical thickness is controlled by the strain energy density w0 and the interaction energy density g0 l, while the thickness is independent of the surface energy Chapter 3: The Critical Thickness of the SK Transition 37 14 12 H1/H1P 10 8 6 4 2 0 0 0.1 0.2 0.3 0.4 0.5 η 0.6 0.7 0.8 0.9 1 Figure 3.4: The variation of the first critical thickness H1 with the width ratio η where H1 is normalized by the critical thickness for the pyramid island formation H1p . density. The finding suggests the first critical thickness represents the film thickness at which the strain energy reduction and the interaction energy increment due to island formation is balanced. Besides the facet angle φ and the material properties of the SK system, the first critical thickness is also controlled by the width ratio η. The variation of H1 with η is depicted in Fig. 3.4. The result, normalized by the value of the critical thickness for the spontaneous formation of the pyramid island, H1p , confirms our Chapter 3: The Critical Thickness of the SK Transition 38 earlier numerical result in Fig. 3.3 that the first critical thickness increases with the width ratio η. The first critical thickness approaches infinity as η approaches 1. The variation of H1 with η, however, is insignificant at small values of η: H1 /H1p is less than 2 even when top width is as large as 90% of the base. 3.4 The Critical Thickness for Surface Undulation Model 3.4.1 Model The previous section presents the critical thickness of the SK transition driven by the spontaneous island formation. The critical thickness controlled by the surface undulation mechanism is studied in this section. The geometry of the model is same as that depicted in Fig. 3.1, but for simplicity, the width ratio η is fixed to be 0. The major difference in the current case is that the angle φ is allowed to vary to reflect the characteristic of the surface undulation mechanism that islands form by a gradual morphological change. Since φ can take any value, it is necessary to consider the variation of the surface energy density with the surface orientation, and this is addressed in the next section. Chapter 3: The Critical Thickness of the SK Transition 3.4.2 39 The Total Energy Change ∆E The Surface Energy Density The surface energy density γ depends on the surface orientation n, and the dependence plays a significant role in the island shape. In spite of the importance, a general expression for γ(n) is still an open question in the literature. In this section, the following simple model for the surface energy density is adopted 2 2 γ(φ) = γ0 − ∆γ1 e−α1 (φ−φ1 ) − ∆γ2 e−α2 (φ−φ2 ) . (3.18) In this model, γ is a constant, γ0 , except in the vicinity of the orientations where φ = φ1 or φ2 . The deviations of γ from γ0 at the two orientations are given by −∆γ1 and −∆γ2 , respectively. As the value of the island facet angle is small, this simple model is employed to simulate the completely faceted island geometry. If the deviation is positive, the surface energy density is a maximum at that orientation; in contrast, if the deviation is negative, the surface energy density is a minimum. The latter case gives the preferred orientation of the island. The quantity α in Eq. (3.18) controls the range of φ in which γ varies significantly with φ. Thus, a large value of α leads to a sharp minimum or a sharp maximum, depending on the sign of ∆γ. More deviation terms can be added to the model given in Eq. (3.18) to account for variations of γ at different surface orientations, which is important for simulating the morphological evolution of the system. In our current study, however, it is sufficient to consider two terms with φ1 = 0 ◦ and φ2 = 11.3 ◦ . The first term with φ1 = 0 ◦ represents the scenario that the surface energy density Chapter 3: The Critical Thickness of the SK Transition 40 may not be a constant in the vicinity of the vertical direction; the second term with φ = 11.3 ◦ corresponds to the experimental finding that the pyramid islands on the SiGe/Si systems are characterized by the {105} facets. There are some limitations in this model: Firstly, the model only considers a smooth variation of γ with φ; as a consequence, the model cannot generate facet surfaces, which are characterized by perfectly flat plane, in Wulf’s construction. Instead, the model will allow a small variation of orientations on the surface even when a large value of α is taken. Secondly, the model assumes the surface energy density in the vicinity of the minimums can be described by a simple exponential function. The actual situation can be much more complicated. The surface energy density described by Eq. (3.18) does not contain a cusp; thus, the accurate expression for the island morphology should involve non-flat but smooth profiles in some areas and facets in the other areas. This approach was adopted by Tersoff et al. (2002) for the cases of two-dimensional solid. The approach, however, cannot be easily applied to the three dimensional cases because of the high computational costs and the severe challenge to describe a surface containing both flat and non-flat profiles. Instead of using the accurate but expensive expression for the island morphology, we simplify the model for the island shapes. In particular, the smooth non-flat profiles are modelled by shallow flat surfaces with the orientations being allowed to vary. In a sense, the orientations of shallow flat surfaces represent the characteristic slopes of the smooth profiles. The facets are also modelled by flat surfaces but with the orientation being fixed at the facet orientation. The model of the island shape can then be Chapter 3: The Critical Thickness of the SK Transition 41 analyzed from the energy point view by considering the strain energy, the surface energy, and the film-substrate interaction energy. As demonstrated in the thesis, the simple model can capture the essence of the island shape transition from a smooth bump to a faceted pyramid. The result is consistent with the earlier two-dimensional analysis based on the accurate expression and the experimental observations (Tersoff et al., 2002). It is well known that the island geometry is not fully faceted if the surface energy density does not contain a cusp at the facet orientation. This can be simulated and analyzed in the two-dimensional cases, which have been explored by Tersoff et al. (2002). In spite of the simplifications and assumptions, the model can qualitatively capture the effects of the surface energy anisotropy on the island morphology and the critical thickness of the SK transition, as shown later in this chapter. Normalizing the surface energy density γ(φ) by γ0 yields 1 2 2 2 γˆ (φ) = 1 − ∆ˆ γ1 e−α1 (φ−φ0 ) − ∆ˆ γ2 e−α2 (φ−φ0 ) (3.19) where ∆γˆ1 = ∆γ1 , γ0 ∆γˆ2 = ∆γ2 . γ0 (3.20) Figure 3.5 illustrates the variation of the normalized surface energy density with the angle φ for the case where γ contains one minimum at φ = 0 ◦ and another one at φ = 11.3◦ . When investigating the SK island formation via the surface undulation mode, Chapter 3: The Critical Thickness of the SK Transition 42 γ(φ)/γ0 1 0.995 0.99 0 2 4 6 8 10 φ(°) 12 14 16 18 20 Figure 3.5: The variation of the surface energy density with the angle φ. the focus is on the range φ 1 ◦ in which the surface energy density γ(φ) ex- pressed in Eq. (3.18) can be further simplified to γ(φ) = γ0 − ∆γ1 + α1 ∆γ1 φ2 . Substituting Eq. (3.21) into (2.3) and assuming φ (3.21) 1 leads the quantity G to be G= 1 α1 ∆γ1 + φ2 . 2 γ0 − ∆γ1 (3.22) Chapter 3: The Critical Thickness of the SK Transition 43 The Interaction Energy For the surface undulation growth model, the angle φ is small; then, the interaction energy ∆EI between the substrate and the film can be derived as g0 l D02 φ2 D03 φ3 D04 φ2 ∆EI = . + + H +l 2 12(H + l) 48(H + l)2 (3.23) The second term in Eq. (3.23) can be neglected since the order of φ. Therefore, the interaction energy ∆EI is simplified to ∆EI = D04 φ2 g0 l D02 φ2 + . H +l 2 48(H + l)2 (3.24) The Total Energy The total energy of the system under the surface undulation mechanism can still be expressed by Eq. (3.14) except that the quantity G is given by Eq. (3.22) and the angle φ is allowed to vary. The angle φ is assumed to be infinitesimal, here, i.e. φ 1, since the surface undulation process is at its initial stage when the system is just after the onset of the SK transition. Evoking Eq. (3.14) and the condition φ 1, the total energy given in φ 1, the total energy change due to the formation of the island for the surface undulation model can be reduced to 2 ∆E = φ D02 3 1 α1 ∆γ1 g0 l g0 l 2 w0 (1 − η )U D0 + r0 (1−η 2 )( + )+ D0 − 3 48(H + l) 6 2 γ0 − ∆γ1 H + l (3.25) Equation (3.25) indicates the total energy change is controlled by three variables: the island size D0 , the facet angle φ, and the film thickness H. . Chapter 3: The Critical Thickness of the SK Transition 3.4.3 44 The Second Critical Thickness Equation (3.25) can be rewritten as ∆E = φ2 D02 C2 D02 − C1 D0 + C0 (3.26) where 1 α1 ∆γ1 g0 l C0 = r0 (1 − η 2 )( + )+ , 2 γ0 − ∆γ1 H +l (3.27) C1 = w0 (1 − η 3 )U , 6 (3.28) C2 = g0 l . 48(H + l)3 (3.29) Requiring that ∆E > 0 for any value of D0 gives the second critical thickness: 3αˆ g0 lL2 (1 + η) H + l = H2 = 2U 2 (η)(1 − η 3 )(1 + η + η 2 ) 1 3 (3.30) where α=1+ 2α1 ∆γ1 . γ0 − ∆γ1 (3.31) When H + l < H2 , ∆E of shallow island is positive, implying island formation of any island size is suppressed. On the contrary, if H + l > H2 , ∆E < 0 for some range of D0 , and the film surface is unstable against the formation of shallow island with the base width D0 in that range. The above discussion shows that H + l = H2 is the critical thickness of the SK transition driven by surface undulation. Eq. (3.30) indicates that the critical thickness for the island formation by surface undulation depends on the materials properties of the system as well as Chapter 3: The Critical Thickness of the SK Transition 45 the width ratio η which determines the island shape. The quantities α, gˆ0 , and L depend on the materials properties and different composition of the substrate and the film leads to various critical thickness. 6 5 log(H2/H2P) 4 3 2 1 0 0 0.1 0.2 0.3 0.4 0.5 η 0.6 0.7 0.8 0.9 1 Figure 3.6: The variation of the second critical thickness H2 with η where H2 is normalized by the critical thickness for the formation of pyramid island H2p . Figure 3.6 plots the variation of H2 with η where H2 is normalized by the value of the critical thickness for the formation of the pyramid island by surface undulation, H2P . The figure clearly shows that the critical thickness increases with width ratio η. When η is close to 1, which means that the island is like a flat disk, the critical thickness is much larger than that of the island with η close to 0, when the island is almost a pyramid. In contrast with the result based on Chapter 3: The Critical Thickness of the SK Transition 46 wavy surface (Chiu et al., 2004), our result is consistent. Both results indicate that the second critical thickness depends on the material properties of the SK system through the term 3 αˆ g0 lL2 . The two results are not identical, while the relative difference between the two results is less than 10%. Chapter 4 The Formation of Trancated Pyramid Islands 4.1 Introduction After the film exceeds the critical thickness for surface undulation, the film first develops a slightly wavy profile and shallow bumps. The bumps can then evolve into facet islands. The formation process was first observed by Chen et al. (1997) on the Si50 Ge50 /Si system, later by Vailionis et al. (2000) on the Ge/Si system, and more recently by Tersoff et al. (2002) on a similar system. Those experimental results clearly show that surface undulation can play a significant role on the growth of the facet hut islands on the SiGe/Si systems with a wide range of Ge concentration in the film. In addition to the experimental studies, the formation of island by surface undulation has also been extensively explored from the theoretical point of view. 47 Chapter 4: The Formation of Trancated Pyramid Islands 48 The focus of earlier investigations was on the morphological stability of a thick stress film against surface undulation (Asaro and Tiller, 1972; Srolovitz, D.J. , 1989; Gao, 1991; Spencer et al., 1991; Freund and Jonsdottir, 1993) and it is now well understood that it is energetically favorable to form shallow bumps on the flat film surface when the film is sufficiently thick. More recently, the formation of shallow bumps via the surface undulation process was further examined by Tersoff et al. (2002) from the energy point of view. They showed that the smooth bump can transform into a facet island gradually without experiencing an abrupt morphological change during the process, and there is no energy barrier during the process. The results by Tersoff et al. (2002) provide a good model for understanding how the islands formation can be effected by the surface undulation mechanism. However, there are two severe limitations in (Tersoff et al., 2002): Firstly, the structure considered in the study was an infinitely long wire instead of an island. The more realistic cases of three-dimensional islands have not been examined in the literature. Secondly, Tersoff et al. assumed that the system was dominated by the strain energy and the surface energy, overlooking the significant role of the film-substrate interaction energy in the SK systems. Because of the ignorance of the third type of energy, the results are valid for the case of thick films only. The two limitations in (Tersoff et al., 2002) are resolved in this chapter by employing a simple three-dimensional model to study the formation of pyramid islands via the surface undulation mechanism. The formation process, consisting of the bump formation and the bump-pyramid transformation, is investigated Chapter 4: The Formation of Trancated Pyramid Islands 49 Figure 4.1: A schematic diagram of an SK film-substrate system containing a trancated pyramid island on a flat wetting layer. from the energetic point of view, and all of the three types of energy controlling the SK systems are taken into account in the current analysis. The inclusion of the three types of energy allows us to examine the effect of the film thickness on the island formation process driven by surface undulation. The chapter is organized as follows: Section 4.2 illustrates the three-dimensional model for analyzing the formation of facet islands by surface undulation. Sections 4.3 and 4.4 present the result for the cases of thick and thin film, respectively. 4.2 4.2.1 Model The Geometry of the Island Figure 4.1 plots the film-substrate system adopted in this chapter for studying the formation of facet islands by surface undulation on the SK system. The film- Chapter 4: The Formation of Trancated Pyramid Islands 50 substrate system consists of two parts: a thick substrate and a thin film. The film contains a flat wetting layer of thickness H and a trancated pyramid island. The trancated pyramid is characterized by four identical facet surfaces and the angle between the facet surface 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. In the three-dimensional model, the value of the angle φ is used to represent the types of the island. When the angle is small, the island corresponds to a shallow bump; when the angle is in the vicinity of the facet angle, φ ≈ φ0 , the island corresponds to a facet island. The actual value of φ is determined by minimizing the total energy of the SK system. It becomes clear later that the total energy is a minimum at small values of φ when the island is small. This explains that surface undulation occurs prior to the formation of facet islands. As the island volume increases, the angle φ of minimum total energy undergoes a sharp transition to reach the value of the facet angle. The sharp transition of φ symbolizes the formation of facet islands via the bump-pyramid shape transition during the surface undulation process. 4.2.2 The Continuum Model for the SK System The continuum model for the SK system has been discussed in Chapters 2 and 3. In the model, the film morphology of the SK systems is dictated by three types of energy: the strain energy caused by the mismatch strain between the film and the substrate, the surface energy depending on the surface orientation, and the interaction energy between the film and the substrate. Chapter 4: The Formation of Trancated Pyramid Islands 51 By employing Eqs. (3.6) , (3.8) and (3.10), the total energy change for the island formation can be found to be ∆E = −wo (tan φ)V U + γ0 D02 (1 − η 2 )G + g0 lD02 I (4.1) where V is the island volume and η is the ration between the top surface width and the base width; I is defined in Eq. (3.11), and G is expressed as G= γ(φ) γ(0) − . γ0 cos φ γ0 (4.2) The surface energy density γ(φ) follows that expressed in Eq. (3.18). Equation (4.1) can be normalized by the characteristic length L = γ0 /w0 which represents the length scale at which the strain energy relaxation is balanced with the surface energy increment as the island forms ∆Eˆ = 2 ∆E 6 = −Vˆ tan φU + GVˆ 3 (1 − η 2 ) 2 3 γ0 L (1 − η ) tan φ 2 3 2 + gˆ0 lVˆ 3 6 tan φ 2 3 I (4.3) where V Vˆ = 3 . L (4.4) The total energy depends on the island volume V , the angel φ, and the width ratio η; in other words, the total energy change can be expressed by ˆ Vˆ , φ, η). ∆Eˆ = ∆E( (4.5) Chapter 4: The Formation of Trancated Pyramid Islands 4.3 52 Island Formation on a Thick Film via Surface Undulation 4.3.1 A Typical Numerical Result 14 12 + −0.015 φ(°) 10 −0.01 + −0.005 + 8 0 + 6 4 2 0.01 0.02 0.03 0.04 0.05 V/L3 0.06 0.07 0.08 0.09 0.1 Figure 4.2: The contours of the total energy change ∆E as a function of the island volume V and the angle φ for the case where ∆ˆ γ1 = 0.99, ∆ˆ γ2 = 0.999, gˆ0 = 0.0625, and l = 0.1nm. Figure 4.2 plots a typical numerical example of the contours of the total energy change ∆E for the formation of one island on a thick film as a function of the island volume V and the angle φ. In the calculation, ∆ˆ γ1 and ∆ˆ γ2 , defined in Eq. (3.20), are taken to be 0.99 and 0.999 respectively, and gˆ0 and l are fixed at 0.0625 Chapter 4: The Formation of Trancated Pyramid Islands 53 and 0.1 nm, respectively. The width ratio η is determined to be the value that minimizes the total energy change ∆E for the given V and φ. The result indicates that when the island volume V is small, the total energy change is positive except in the regime where φ is small. When the island volume increases and exceeds a critical value, the formation of a facet island with φ ≈ 11.3 ◦ becomes energetically favorable. Nevertheless, the energy of the facet island is still slightly higher than that shallow bumps. As V further increases, the energy change of the facet island becomes lower than that of the shallow bumps and the facet island becomes the most favorable island shape in the system. The result implies that in the initial stage of the surface undulation process when the island volume V is small, it is energetically favorable to form shallow bumps first. As the surface undulation process proceeds, the shallow bumps grow, and when the volume is sufficiently large, the bumps are unstable against the shape transition to form facet islands. The shape transition can be demonstrated by studying the effects of the island volume on the angle φmin (V ) that minimizes the total energy of the system for the given island volume V . The result, depicted in Fig. 4.3(a), indicates that the angle φmin is small at small values of V , and the value of φmin increases linearly with V . When V exceeds a critical value, φmin increases sharply and then maintains in the vicinity of the facet angle φmin = 11.3 ◦ . The result confirms that during the surface undulation process, shallow bumps form first and later the bumps transform into facet islands. Figure 4.3(b) plots the variation of the minimum total energy ∆Emin with the island volume V . The result indicates that the total energy decreases with in- Chapter 4: The Formation of Trancated Pyramid Islands (a) 12 (b) 0 0.05 0 8 ∆ Emin φmin(°) 10 6 −5 −10 4 −15 2 0 x 10 −3 5 54 0 0.05 −20 0.1 V/L3 V/L3 (c) 1 0.1 0.8 η 0.6 0.4 0.2 0 0 0.05 V/L3 0.1 Figure 4.3: Numerical results of the island morphology transition: (a) The variation of the angle φmin with the island volume V . (b) The variation of the minimum energy change ∆Emin with the island volume V . (c) The variation of the equilibrium width ratio ηmin with the island volume V , where V is normalized by L3 . creasing V monotonically, which means that the island can grow gradually during the surface undulation process without experiencing any energy barrier, in contrast with the nucleation process. Another important finding in Fig. 4.3(b) is that the total energy reduction rate with respect to the island volume, |d∆Emin /dV |, changes significantly during the island formation process. The reduction rate |d∆Emin /dV | is extremely small when the shallow bump is the “equilibrium” structures, and the rate |d∆Emin /dV | is much higher when V is large and the facet island is more energetically favorable than the shallow bumps. Chapter 4: The Formation of Trancated Pyramid Islands 55 Figure 4.3(c) depicts how the “equilibrium” width ratio ηmin varies with the island volume V . The result shows that ηmin is a constant at small values of V , it increases sharply to about 0.2 during the bump-pyramid transition, and then it decreases as V further increases. The result ηmin = 0.2 at the onset of the shape transition implies that the facet island after the bump-pyramid transition is a trancated pyramid. This is consistent with the experimental findings and the theoretical prediction made by Tersoff et al. (2002). The value ηmin = 0.2, however, is smaller than the theoretical result in (Tersoff et al., 2002) mainly because our analysis is based on a three-dimensional islands and theirs is on infinitely long wires. The width ratio during the bump-pyramid transition is further discussed later. 4.3.2 Analytical Results Introduction Figures 4.2 and 4.3 in the previous section illustrate the island formation pathway from the energy point of view: In the early stage of the surface undulation process when the island volume V is small, the angle φmin is small and the magnitude of the energy reduction ∆Emin is also small; when V is larger than the critical value for the shape transition, the angle φmin is in the vicinity of 11.3 ◦ and the magnitude of the energy reduction is significantly larger. The numerical results suggests that the islands can be categorized into two types: shallow bumps and facet islands. For the shallow bump, which is favored Chapter 4: The Formation of Trancated Pyramid Islands at small V , the angle φ can vary but the value of φ is small, φ 56 1. In contrast, the angle φ is fixed at 11.3 ◦ for facet islands which are favored at large V . By employing this model for describing the two types of islands, it is possible to simplify the expression for the total energy of the SK system and this allows us to derive the analytical results for the formation of islands during the surface undulation process. The derivation and the analysis of the analytical results are the focus of this section. In particular, we first study the equilibrium shape of the shallow bumps, followed by an analysis for the critical volume Vcr for the bumppyramid transition and discussions of the dependence of the critical volume Vcr on the materials properties. The Equilibrium Shape of Shallow Bumps It is assumed that φ 1 in the shallow bump; therefore, the quantity defined in Eq. (3.9) G can be rewritten as G= α1 ∆γ1 1 + φ2 . 2 γ0 − ∆γ1 (4.6) Substituting Eq. (4.6) into Eq. (4.3) simplifies the total energy change due to the formation of one island on a thick film to ∆ˆ e= 4 1 α1 ∆γ1 ∆Eˆ 1 + S(η) = −φU + Vˆ − 3 φ 3 2 γ0 − ∆γ1 Vˆ (4.7) where S(η) is a function of η S(η) = (1 − η 2 ) 6 (1 − η 3 ) 2 3 . (4.8) Equation (4.7) shows that for a given value of Vˆ , ∆ˆ e only depends on φ and η. Chapter 4: The Formation of Trancated Pyramid Islands 57 The equilibrium shape of the shallow bumps are determined by the following two conditions ∂ˆ e = 0, ∂φ (4.9) ∂ˆ e = 0. ∂η (4.10) Equations (4.9) and (4.10) can be simplified to the following forms φ Vˆ 1/3 φ Vˆ 1/3 = 3U 1 α1 ∆γ1 + 4S(η) 2 γ0 − ∆γ1 −1 = U (η) 1 α1 ∆γ1 + S (η) 2 γ0 − ∆γ1 −1 , (4.11) . (4.12) Solving Eqs. (4.11) and (4.12) yields the equilibrium shape of the shallow bumps 3U (η) U (η) = . 4S(η) S (η) (4.13) The solution, denoted as η0 , is found to be 0.126; this is the only solution to Eq. (4.13). The result η = η0 for the equilibrium shape of shallow bumps is unaffected by the material properties of the systems and the volume of the island. This, however, does not mean the equilibrium island shape is self-similar during the surface undulation process since the angle φ will increase. This issue is further examined in the following section. The angle φ of the equilibrium bump shape can be derived by substituting η = η0 into Eq. (4.13) φmin = C Vˆ (4.14) Chapter 4: The Formation of Trancated Pyramid Islands 58 where C is a constant given by 3U (η0 ) 1 α1 ∆γ1 C= + 4S(η0 ) 2 γ0 − ∆γ1 −1 3 . (4.15) Equation (4.14) suggests that the angle φmin , or equivalently the slope of the equilibrium bump shape, increases linearly with the island volume Vˆ in the early stage of the surface undulation process. The result is consistent with Fig. 4.3(a), which is obtained by numerical calculation. By adopting Eqs. (3.3) and (4.14), the base width of the shallow bumps can be derived to be D0 = 6 (1 − η03 )C 1 3 L. (4.16) Equation (4.16) shows that the width of the shallow island base is a constant prior to the bump-pyramid transition. This result was first observed experimentally by Vailionis et al. (2000), who investigated the island morphological transition of Ge island on a Si substrate by using scanning tunnelling microscopy. They observed that the island size of the shallow bumps, called “prepyramid”, remains approximately constant before transforming into pyramids. The island size is about 15 to 20 nm. Turn to the theoretical result in Eq. (4.16). For the Ge/Si system, L in Eq. (4.16) can be approximated to be 6.25 nm (Chiu and Gao, 1995). The other quantity C in Eq. (4.16) depends on ∆γ1 and α, representing the effect of the surface energy anisotropy on the size of the bump. When ∆γ1 = 0, D0 can be ˚ This corresponds to the case where the surface energy calculated to be 123A. density is not a minimum or a maximum at the (001) direction. As ∆γ1 and α Chapter 4: The Formation of Trancated Pyramid Islands 59 increases, meaning γ is a minimum at (001), the value of C decreases, resulting in ˚ when ∆γ1 = 0.02 a larger island size. For example, the size D0 increases to 172A and ∆γ1 γ0 . The result is consistent with the width range observed by Vailionis et al. (2000). By substituting Eqs. (4.14) and (4.15) into Eq. (4.7), the total energy change of the equilibrium shallow bumps can be expressed as ∆ˆ e = −0.045944Vˆ 1 α1 ∆γ1 + 2 γ0 − ∆γ1 −3 . (4.17) Equation (4.17) indicates the total energy change decreases monotonically as the island size increases, which means that there is no energy barrier for the shallow bumps to form. It becomes clear later that the magnitude of the energy reduction is small when compared with that of the facet island formation. The Critical Volume for the Bump-Pyramid Transition When the volume of the shallow bump increases to a critical value, the shallow bump undergoes a morphological transition from a shallow island to a facet one. The critical volume Vcr can be determined approximately by two conditions ∂E(V, η)/∂η = 0 (4.18) ∆E(V, η) = 0 (4.19) and where E(V, η) refers to the total energy of the facet island 2 2 ∆E(V, η) = −(tan 11.3 ◦ )U Vˆ + GVˆ 3 S(η)(tan 11.3 ◦ )− 3 (4.20) Chapter 4: The Formation of Trancated Pyramid Islands 60 where the quantity G is given by G= γ0 − ∆γ2 − 1. (γ0 − ∆γ1 ) cos φ0 (4.21) The first condition ∂E/∂η = 0 in Eq. (4.18) means that the total energy is a minimum for given island volume V and the second condition ∆E = 0 in Eq. (4.19) is simplified from the actual condition ∆Efacet = ∆Ebump by approximating the total energy change of a shallow bump to be zero, ∆Ebump ≈ 0. The approximation is valid according to the numerical result shown in Fig. 4.3(b), and it is further verified later. The two conditions for the bump-pyramid transition, Eqs. (4.18) and (4.19), can be reduced to the following two equations U (η) = S (η)R (4.22) U (η) = S(η)R (4.23) and where R is a quantity depending on the island volume V G R= 3 (tan φ0 )5 Vˆ . (4.24) The variable φ0 in Eq. (4.24) denotes the facet angle of the pyramid island. In the SiGe/Si (001) system, φ0 is given by 11.3 ◦ . Equations (4.22) and (4.23) can be solved numerically to determine the critical values of η and R at the onset of the bump-pyramid transition η = ηcr ≈ 0.2, R = Rcr ≈ 0.6. (4.25) Chapter 4: The Formation of Trancated Pyramid Islands 61 The solutions ηcr and Rcr are constants and are both independent of the materials properties of the SK system. According to Eq. (4.24), the critical island volume beyond which the facet islands are energetically more favorable than the shallow bumps can be expressed as Vcr = G 3 L3 . 3 (tan φ0 )5 Rcr (4.26) Equation (4.26) describes the dependence of the critical island volume for the bump-pyramid transition on the material properties of the SK film-substrate system: Firstly, the critical volume is proportional to L3 ; therefore, the island size is large at small mismatch strains, while the island size decreases as the mismatch strain increases. This is consistent with earlier theoretical predictions and agrees with the experimental findings. Secondly, the critical volume is proportional to G3 , which is controlled by the ratio γ2 /γ1 and the facet angle, while the critical volume is insensitive to the quantity α representing the width of the minimum/maximum of the surface energy density in the vertical direction. Figure 4.4 plots the variation of Vˆcr with the facet angle φ0 for the case where γ2 /γ1 is fixed at 0.995. The figure shows that when the angle is small (φ0 < 8 ◦ ), the critical island volume for the pyramid formation is small. As the angle increases, the critical volume also increases rapidly. Figure 4.5 plots the contours of Vˆcr as a function of γ2 /γ1 and φ0 . The range of γ2 /γ1 is between 0.99 and 1.01, and that of φ0 is between 5 to 25 degrees. The regime of negative values of Vˆcr corresponds to the situation when the facet island Chapter 4: The Formation of Trancated Pyramid Islands 62 0.05 0.045 0.04 0.035 Vcr/L3 0.03 0.025 0.02 0.015 0.01 0.005 0 5 10 15 φ0 20 25 Figure 4.4: The variation of the critical island volume for the island morphology transition Vcr with the angle φ0 for the case where γ2 /γ1 is fixed at 0.995. can form directly on the substrate without experiencing surface undulation or nucleation. Both γ2 /γ1 and φ0 have critical values beyond which there is always a critical volume for the formation of the pyramid island. The Width Ratio at the Bump-Pyramid Transition The critical value of the width ratio ηcr given by Eq. (4.25) corresponds to the situation when the bump-pyramid transition occurs at Vˆ = Vˆcr . However, in actual situation, because of the kinetic constrains, the bump-pyramid transition may occur at a much larger island volume. As a consequence, the width ratio can be different from ηcr . Since the actual value of η is affected by the kinetic Chapter 4: The Formation of Trancated Pyramid Islands 63 1.01 1.008 1.006 0.04 1.004 γ2/γ1 1.002 1 0.01 0.998 0.02 0.996 0.994 0.992 0.99 0 5 10 15 φ0 20 25 Figure 4.5: The contours of the critical island volume for the island morphology transition Vˆcr as a function of the surface energy density ratio γ2 /γ1 and the angle φ0 . mechanism of the shape transition, it cannot be predicted by an energy analysis. Nevertheless, the energy analysis can still be adopted to determine the admissible range of η for the bump-pyramid transition. The bump-pyramid transition can occur if ∆Efacet < ∆Ebump . Since the magnitude of the total energy reduction by the bump formation is small, the condition can be approximated by ∆Efacet < 0. (4.27) In other words, the domain (η, V ) of ∆Efacet < 0 describes the regime in which the bump-pyramid transition is energetically favorable. Chapter 4: The Formation of Trancated Pyramid Islands 64 0.9 ∆ E>0 0.8 0.7 η 0.6 0.5 ∆ E 0, the cooperative formation will further proceed; if F < 0, the cooperative formation will follow the opposite direction that the trench will shrink; and if F = 0, the cooperative formation will stop and reach an equilibrium state. Chapter 5: The Cooperative Formation 76 The driving force F varies with the trench depth At , and of particular importance is the value at At = 0, denoted as F0 . The quantity F0 determines the stability of the island against the formation of a surrounding trench. 5.2.3 The Total Energy Change ∆E The key task in our analyses is to evaluate the total energy change ∆E during the shape transition shown in Fig 5.1 under the condition that the total volume of the film is fixed. The total energy, similar to our analyses in earlier chapters, consists of the strain energy, the surface energy, and the interaction energy. The Strain Energy Change According Eq. (2.29), the strain energy ∆W0 of the structure depicted in Fig. 5.1(a) can be written as ∆W0 = −wo tan φV0 U0 (5.7) where V0 = D03 tan φ/6 and U0 can be calculated by using Eq. (2.30). Similarly, the strain energy ∆Wc of the structure in Fig. 5.1(b) can be expressed as ∆Wc = −wo tan φV Uc (ηc ) (5.8) where V is defined in Eq. (5.1) and Uc (ηc ) can be calculated by adopting Eq. (2.30). The difference between ∆Wc and ∆W0 yields the strain energy change due to the trench formation ∆W = ∆Wc − ∆W0 . (5.9) Chapter 5: The Cooperative Formation (a) 35 70 30 60 25 50 20 15 40 30 10 20 5 10 0 0 0.2 0.4 ηc 0.6 (b) 80 Uc′(ηc) Uc(ηc) 40 77 0.8 1 0 0 0.2 0.4 ηc 0.6 0.8 1 Figure 5.2: The variation of the strain energy function Uc (ηc ) with the width ratio ηc in (a); the derivative Uc (ηc ) as a function of ηc in (b). Figure 5.2(a) plots the variation of the Uc (ηc ) with ηc . The result indicates that Uc (ηc ) increases monotonically with ηc , meaning the total strain energy is reduced as the trench deepens with the island base D being fixed. The result demonstrates that the strain energy favors the trench formation. Figure 5.2(b) depicts the derivative Uc (ηc ) as a function of ηc . The Surface Energy Change The area of the projection of the facet surface onto the wetting layer is D12 ; accordingly, the facet surface area is D12 / cos φ, and the total surface energy of the structure shown in Fig. 5.1(b) can be determined to be ∆ESc = γ0 D2 (1 + 2ηc )2 G (5.10) where G= γ2 γ1 − . γ0 cos φ γ0 (5.11) Chapter 5: The Cooperative Formation 78 In Eq. (5.11), γ1 is the surface energy density of a flat film surface, and γ2 is that of a facet surface. By the same procedure, the surface energy of the pyramid island depicted in Fig. 5.1(a) can be deduced as ∆ES0 = γ0 D02 G. (5.12) Subtracting ∆ES0 from ∆ESc leads to the surface energy change ∆ES = ∆ESc − ∆ES0 . (5.13) The Film-Substrate Interaction Energy Change The interaction energy change due to the trench formation can be expressed as ∆EI = g0 lD02 P H +l (5.14) where g0 is the interaction energy density, D0 is original pyramid island base width and P is a polynomial. For the fixed island size mode, P is found to be P = 2(H + l) 2η (1 + 2η)2 P3 P1 + 4 (1 + η)2 − P2 − cos φ 2 D0 A + 1 − (1 + 2η)2 (5.15) where A is the height of the island, H is the thickness of the wetting layer and P1 , P2 , P3 are given by P1 = + P2 = 1 D2 − D1 −1 A 2D2 − D1 H +l 1 1+ A A ln(A + H + l) − ln (ac − bd)[ln d − ln(d − c)] + bc , c2 P3 = −1 + 1 + A H +l ln 1 + A H +l D2 − D1 +H +l 2 , (5.16) (5.17) (5.18) Chapter 5: The Cooperative Formation 79 where a = D1 , b = D1 − D2 , c = (D1 − D2 ) tan φ and d = 2(H + l). For the local transportation mode, P can be expressed as 2 1 2 (1 + 2η)2 r 3 2(H + l) 2ηr 3 P3 P = 4 (1 + η)2 r 3 − P1 + P2 − cos φ 2 D0 A 2 D0 tan φ(r − 1) + 1 − (1 + 2η)2 r 3 + 6(H + l) (5.19) where P1 , P2 and P3 are defined in Eqs. (5.16), (5.17), and (5.18), respectively. 5.3 The Cooperative Formation on a Thick Film In this section, we study the cooperative formation on a thick film. This means that the interaction energy can be neglected when evaluating the total energy. 5.3.1 The Total Energy Change ∆E When the wetting layer is thick, the interaction energy can be neglected and the total energy change consists of the strain energy change and the surface energy change only ∆E = ∆W + ∆ES (5.20) where ∆W and ∆ES are the strain energy change and the surface energy change which are given by Eqs. (5.9) and (5.13) respectively. Equation (5.32) can also rewritten as ∆E = ∆Ec − ∆E0 (5.21) Chapter 5: The Cooperative Formation 80 where ∆Ec = ∆Wc + ∆ESc = −wo (tan φ)V Uc + γ0 D2 (1 + 2ηc )2 G, (5.22) ∆E0 = ∆W0 + ∆ES0 = −wo (tan φ)V0 U0 + γ0 D02 G. (5.23) The quantity ∆Ec defined in Eq. (5.22) is the total energy of the pyramid surrounded by a trench shown in Fig. 5.1(a); the quantity ∆E0 defined in Eq. (5.23) is that of the pyramid illustrated in Fig. 5.1(b). For convenience, the two quantities can be normalized by γ0 L2 to be 2 6(1 + 2ηc )3 ∆Eˆc = −(tan φ)Vˆ Uc + Vˆ 3 tan φ 2 3 ∆Eˆ0 = −(tan φ)Vˆ0 U0 + Vˆ0 6 tan φ 2 3 2 3 G, G (5.24) (5.25) where Vˆ and Vˆ0 are the normalized volume for trenched island and pyramid island respectively V Vˆ = 3 , L V0 Vˆ0 = 3 . L (5.26) For the fixed island size mode, V = V0 ; for the local transportation mode, V = [2(1 + ηc )3 − (1 + 2ηc )3 ]V0 . 5.3.2 A Typical Numerical Result Figure 5.3 plots the contours of ∆E as a function of V /Vcr and ηc where Vcr refers to the critical island volume above which the formation of a pyramid island is energetically favorable. In our calculation, φ = 11.3 ◦ , γ1 /γ0 = 0.99, and γ2 /γ0 = 0.999. Figure 5.3(a) depicts the result of the first mode, i.e. the fixed Chapter 5: The Cooperative Formation 81 (b) 3 3 2.5 2.5 V/Vcr V/Vcr (a) ∆ E0 0.05 0.1 0.15 0.2 ηc 0.25 0.3 Figure 5.3: The contours of the total energy change for the trench formation ∆E as a function of the normalized island volume V /Vcr and the width ratio ηc for the cases of (a) the fixed island size mode and (b) the local transportation mode. island size mode, and Fig 5.3(b) depicts that of the second mode, i.e. the local transportation mode. The results show that for both modes, ∆E < 0 when V /Vcr > 1. In other words, the surface undulation always occurs once the pyramid island forms. However, the cooperative formation may proceed first and then be stopped if the island size is below a critical value. Larger than this critical value, the cooperative formation decreases with ∆V for any value of ∆V . 5.3.3 Analytical Results Figure 5.3 indicates that the island on a thick film is unstable against cooperative formation if the island exceeds the critical volume of island formation Vcr . In other words, the driving force for the cooperative formation F > 0 at At = 0 when V > Vcr . As the island’s volume increases and is sufficiently large, it is Chapter 5: The Cooperative Formation 82 energetically favorable for the trench to grow deeper and deeper. In such a case, the trench is unstable against coarsening. These two results are reexamined in this section analytically. The driving force for the cooperative formation F for the two modes can be derived as 2 1 ∂Uc 6 tan φVˆ − 4Vˆ 3 G A ∂ηc tan φ 2 3 6 2Vˆ − 3 G(1 + 2ηc )2 F2 = F1 − 3A tan φ 2 3 F1 = 1 (1 + 2ηc ) , − tan φUc ∂V . A ∂ηc (5.27) (5.28) The value of F at At = 0 is denoted as F0 . When At = 0, ∂V /∂ηc = 0, therefore, F0 is identical for both trench growth models and can be written as F0 = 1 ∂Uc tan φVˆ0 A ∂ηc 2 ηc =0 − 4Vˆ03 G 6 tan φ 2 3 . (5.29) The sign of F0 decides the stability of the island against the trench formation. When F0 < 0, the trench formation results in an energy increase, and the island is stable against the trench formation. On the other hand, if F > 0, the total energy is reduced as the trench forms, and the island is unstable. Equation (5.29) indicates that F0 is controlled by the island volume V0 and the critical island volume VT beyond which F0 > 0 can be determined to be VˆT = 2304 G3 · [∂Uc /∂ηc ]3 (tan φ)5 ηc =0 , (5.30) which is smaller than the critical volume Vcr for a pyramid island formation, which can be determined by the condition ∆E = 0 G3 36 Vˆcr = 3 · Uc (tan φ)5 ηc =0 . (5.31) Chapter 5: The Cooperative Formation 83 The stability of a trench formed around an island of volume V against coarsening is controlled by the condition that F (ηc ) > 0 for any value of ηc . If the condition is not satisfied, on the contrary, the trench will deepen first and then stop at the equilibrium depth. In such a case, the island is stable against trench coarsening. As implied by Eq. (5.30), the critical volume of island stable against trench coarsening is controlled by the parameter G3 /(tan φ)5 . The variation of the critical volume with the parameter G3 /(tan φ)5 can be determined numerically and Fig. 5.4 plots the variation of the critical volume and the equilibrium trench shape with G3 /(tan φ)5 for the two modes. 5.4 The Stability of a Pyramid Island on a Thin Film against Trench Formation The former section focuses on the cooperative formation on a thick film; in this section, we examine the stability of a pyramid island against the formation of trench around the island for the case of a thin film. The approach is similar to that in Section 5.3, while the total energy includes the film-substrate interaction energy in addition to the stain energy and the surface energy ∆E = ∆W + ∆ES + ∆EI (5.32) The stability of an island against trench formation can be determined by the sign of the driving force F = −∂∆E/∂At , for the cooperative formation at At = 0. Chapter 5: The Cooperative Formation 84 (a) 0.04 (b) 0.25 0.036 0.2 ηc VT 0.038 0.034 0.15 0.032 0.1 0.03 0.05 0.028 3 3.5 0 4 G5/tan3(φ) x 10−3 (c) 0.036 4 0.032 0.2 ηc 0.25 VT 3.5 G5/tan3(φ) x 10−3 (d) 0.034 0.03 0.15 0.028 0.1 0.026 0.05 0.024 3 3 3.5 4 G5/tan3(φ) x 10−3 0 3 3.5 4 G5/tan3(φ) x 10−3 Figure 5.4: The variation of the critical volume VT and the equilibrium trench shape ηc with G3 /(tan φ)5 for the two trench growth modes. Parts (a) and (b) show those for the first growth mode; while parts (c) and (d) show those for the second growth mode. The stability of an island is controlled by F0 = − ∂∆E ∂At . (5.33) At =0 The sign of the driving force F0 decides the stability of the island against the trench formation. If F0 > 0, the island is unstable and the trench will form; If F0 < 0, the island is stable and the trench’s formation will be prohibited. The quantity F0 is controlled by the island volume and the film thickness, Chapter 5: The Cooperative Formation 85 while F0 is independent of the growth mode of the trench, 1 ∂Uc F0 = tan φVˆ0 A ∂ηc 2 − 4Vˆ03 G ηc =0 6 tan φ 2 3 8g0 l sin φ 6Vˆ0 − · (H + l)(1 − cos φ) tan φ 1 3 . (5.34) 60 F =0 0 ∆ Epyramid=0 50 V1/3(nm) 40 30 20 10 0 2 4 6 8 H(nm) 10 12 14 Figure 5.5: The contours of the driving force F0 = 0 and the total energy change for a pyramid island ∆Epyramid = 0. Figure 5.5 plots a typical example of the contours F0 = 0 and ∆Epyramid = 0 as functions of island size and the film thickness H. On the one hand, the contour ∆Epyramid = 0 specifies, for given film thickness, the minimum island volume for the formation of pyramid islands by the bump-pyramid transition during the surface undulation process. The contour F0 = 0, on the other hand, illustrates the maximum volume below which the pyramid island is stable against the trench Chapter 5: The Cooperative Formation 86 formation. The contour ∆Epyramid = 0 is higher than F0 = 0 when the film thickness is small or large, while the former becomes lower than the latter when the value of H is medium. The film thicknesses at the two intersection points of the two contours are denoted as H1a and H1b where H1a < H1b . The two quantities, H1a and H1b , together with the critical film thickness H1 defined by ∆Epyramid > 0 for any value of V divide the island formation process into four characteristic regimes. • [0, H1 ]: The island can become unstable against trench formation if the island is sufficiently large; however, it is energetically unfavorable for any island to form. • [H1 , H1a ]: Extremely large islands can form energetically, and once they form, they are unstable against trench formation. • [H1a , H1b ]: There exists a range of island sizes in which it is energetically favorable to form the islands and the islands are stable against trench formation. The islands with a size higher than this range can induce trench formation, while the islands with a size lower than this range are unfavorable from the energy point of view. • [H1b , ∞]: Above this thickness H1b , the result is similar to the case of an infinitely thick film that all of the pyramid islands can cause trench formation if it is energetically favorable to form the islands. Chapter 6 Conclusion In this thesis, the epitaxial growth of SiGe nano-crystalline islands on Si(001) system which is an empirical classification of Stranski-Krastanow growth mode, has been studied from the energy point of view. Three-dimensional models are employed to investigate the morphological evolution of the SK system. We illustrate the critical thickness of the wetting layer for both the spontaneous island formation and the surface undulation mechanisms. The facet island’s formation on thick and thin film is investigated. The nature of the cooperative formation after the facet island formation is also examined. Model The SK system considered in this thesis consists of a substrate, which is assumed to be a semi-infinite solid, and a thin film which contains a flat wetting layer and a facet island on the top of the wetting layer. The island formation process is studied from the energy point of view. The system’s energy for island formation 87 Chapter 6: Conclusion 88 consists of three kinds of energy: the strain energy caused by the mismatch strain between the film and the substrate, the surface energy, and the film-substrate interaction energy, which can be modelled as a special type of surface energy. Due to the FEM’s low numerical efficiency, the first-order boundary perturbation method is adopted to calculate the strain energy for three-dimensional island formation. The Critical Thickness for Island Formation The critical thickness of wetting layer for island formation has been derived for both the spontaneous island formation and the surface undulation mechanisms. For the spontaneous island formation mechanism, there is a critical thickness H1 below which island formation of any size is forbidden. H1 is called the first critical thickness. As the film thickness increases and the value becomes larger than H1 , islands can form on the top of the wetting layer. For the surface undulation model, there is the second critical thickness H2 for island formation below which any size island’s formation is suppressed. Both critical thicknesses H1 and H2 increase with the island width ratio η. The Trancated Pyramid Island Formation A simple three-dimensional model is employed to study the formation of pyramid islands via the surface undulation mechanism. The formation process, which consists of the bump formation and the bump-pyramid transformation, is investigated from the energetic point of view. The three types of energy controlling Chapter 6: Conclusion 89 the SK systems are taken into account in the current analysis, which allows us to examine the effect of the film thickness on the island formation process driven by surface undulation. Our investigation illustrates the following island formation pathway: Firstly, shallow bumps form on the top of the wetting layer with facet angle φ 1◦ and the island shape does not change; secondly, bumps’ volume increases and the islands transform to trancated pyramid island while the facet angle φ increases to big value; thirdly, as the facet island’s volume increases, the facet angle does not change and the width ratio of the island decreases until the island transforms into a pyramid. The critical volume and the width ratio at the bump-pyramid transition are derived and the result is consistent with the findings in the literature. The film thickness’ effect on the process of island formation is investigated. When the film thickness is sufficiently small, either the bump’s or the facet island’s formation is suppressed for any island volume. As the film thickness increases and becomes larger than the critical value for the spontaneous formation of facet island but smaller than that for the bump formation, the island formation via surface undulation is prohibited and only islands larger than the critical volume can form on the film via the process of nucleation. When the film thickness further increases, the bump formation becomes energetically favorable in the early stage of the island formation and the shallow island forms on the film via the process of surface undulation first followed by the bump-pyramid transition to develop into a pyramid island. Chapter 6: Conclusion 90 The Cooperative Formation After the pyramid island forms on the top of the wetting layer, trenches may form around the island. There are two growth modes for the trench formation and both modes reduce the system’s energy. When the wetting layer is sufficiently thick, the trench can form on the wetting layer if the island volume is larger than a critical value Vt , which is smaller than the critical volume for a pyramid island formation Vc . On the other hand, for the thin wetting layer, the cooperative formation depends on the film thickness and the island volume. The island formation process can be divided into four regimes by three characteristic values of the film thickness. In the first regime, [0, H1 ], the island can become unstable against trench formation if the island is sufficiently large; however, it is energetically unfavorable for any island to form; in the second regime, [H1 , H1a ], extremely large islands can form energetically, and once they form, they are unstable against trench formation; in the third regime, [H1a , H1b ], there exists a range of island size in which it is energetically favorable to form the islands and the islands thus formed are stable against trench formation. 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Lett. 83, 5205-5207. [...]... 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 and von K¨anel, 2003) More recently it is shown that the onset of the SK transition is affected by whether the island formation mechanism is spontaneous formation or surface undulation The islands formation via surface undulation corresponds to... trench formation can be suppressed on a thin film The thesis is outlined as follows: Chapter 2 describes the model for the filmsubstrate systems examined in the thesis and the methodology for carrying out the energy analysis Chapter 3 presents the results of the critical thickness for island formation Chapter 4 focuses on the formation of islands via the surface undulation mode, followed by discussions... that it only needs two-dimensional integral when evaluating the island strain energy, Chapter 1: Introduction 8 which greatly reduces the calculation time to a more acceptable duration The formula will be employed in this thesis to calculate the strain energy change during the island formation 1.2.2 Energy Analysis for Island Formation Tersoff and LeGoues (1994) presented a widely used theory for the... understanding of island formation The picture of island formation is explicit and the transition between different island shapes is clearly illustrated through 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 formation has not been... the formation of the SiGe nano-islands on the Si substrate from the 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 by employing the first-order boundary perturbation method to calculate the total energy change during the island formation process The total energy includes the strain energy, the surface energy, ... concentration x is between 0.2 and 0.6 These two groups’ studies both presented that, at least in some range of temperature and alloy composition, islands can evolve continuously from surface ripples 1.2 1.2.1 Theoretical Study for Island Formation Boundary Perturbation Method When studying the formation of nano-crystalline islands on the heteroepitaxial film-substrate systems, one question often encountered... the energy change of island formation They found 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 at which point the island becomes meta-stable and in equilibrium it transforms to a faceted shape However, the island. .. the island shape on the strain energy change ∆W during the formation of the island Chapter 3 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... shape transformation process as shown in Fig 2.1 The total energy change during the island formation process is called the energy of the island Similarly, the energy of other structures on the film-substrate system, for example 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 2.1.2 The Total Energy. .. 1991; Tersoff and Tromp, 1993) The reduction of the strain energy is the driving force for the island formation on the SK systems Chapter 2: Model and Methodology 13 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 ... total energy change ∆Efacet < for a facet island formation on a thick film 4.7 63 64 The contours of the total energy change for the island formation on a thin film ∆E as a function... Transition Theoretical Study for Island Formation 1.2.1 Boundary Perturbation Method 1.2.2 Energy Analysis for Island Formation ii Table of Contents... strain energy change during the island formation 1.2.2 Energy Analysis for Island Formation Tersoff and LeGoues (1994) presented a widely used theory for the transition from two-dimensional layer