Equilibrium morphologies of epitaxially strained islands

An analytical expression of the free energy consisting of the strain energy, surface energy and interfacial energy for the coherent island/substrate system, as well as the evolving relations of aspect ratio against volume of the island, and misfit of the system, which provides a broad perspective on island behavior, is obtained, and used to study the equilibrium shapes of the systems. A two-dimensional model assuming linear elastic behavior is used to analyze an isolated island with elastic properties similar to those of the substrate. The results show that in order to minimize the total free energy, a coherent island will adopt a particular shape and height-to-width aspect ratio that are a function of only the island volume. The effect of a misfit dislocation on the equilibrium shape of an island is in passing examined. These can serve as a basis for interpretation of experiments.     

Ge/Si半导体量子点的应变分布与平衡形态

基于各向异性弹性理论的有限元方法，研究了金字塔形自组织Ge/Si半导体量子点应变能随高宽比变化的规律：系统的应变能随着高宽比的增大而逐渐减小.并通过自由能(应变能与表面能之和)讨论了量子点的平衡形态.结果表明，对于固定体积的量子点，存在一个高宽比值，称之为平衡高宽比，使得系统的自由能最低.同时，还给出了量子点的应力、应变、流体静应变及双轴应变分布.这些可以作为阐明应变自组织量子点实验的理论基础. 关键词： 量子点 应变分布 自由能 平衡形态     

The size dependence of equilibrium elastic strain in finite epitaxial islands

The equilibrium elastic strain in epitaxial islands of small size is shown to have a sawtooth dependence on island width by minimizing the systems energy as calculated from a periodic interaction potential between substrate and overgrowth. Such a sawtooth variation is consistent with Vincent's observations and idea that a given misfit between island and substrate may not be entirely accommodated by an integral number of identical misfit dislocations, and hence, the remaining misfit, which depends in part on island width, will be accommodated by residual elastic strain. The present calculations of mechanical equilibrium support these ideas and further reveal that in some cases misfit dislocations may over compensate for the misfit, thereby introducing an elastic strain of opposite sign to that normally expected. Other results of the calculations are in agreement with earlier theoretical and experimental observations.     

The influence of size on the composition of nano-precipitates in coherent precipitation

Various experiments show that the solute content in small coherent precipitates in their early stages of growth-coarsening is significantly lower than its equilibrium value and gradually increases with the particle size until equilibrium composition is reached. In this paper, we investigate the thermodynamic stability of a mono-dispersed assembly of coherent precipitates in a finite matrix by minimizing the total free energy of the system to account for this size effect observed for nano-phases. Both interfacial energy and misfit elastic energy were taken into account and simple regular solid solutions were considered for both phases. It is found that for small sizes, precipitates with a low solute content are favoured energetically. The solute content in precipitates decreases when increasing either interfacial energy or coherency misfit. The solute concentration in nano-particles is observed to increase with size. The asymptotic composition remains under the equilibrium concentration even for small misfits. The model was confronted to atom probe tomography experiments performed in FeCr system. Predictions exhibit a composition trend with precipitate size that is in good agreement with experiments.     

Bistability in the shape transition of strained islands

The equilibrium shape of a monatomic strained island on a substrate depends on the step free energies and the difference in surface stress between the island and the substrate. For small island sizes the step free energies dominate, resulting in compact islands. Beyond a critical island size, however, the strain energy becomes dominant and the island maximizes its perimeter, resulting in elongated islands. Here we show that for strained islands with force monopoles pointing in opposing directions at neighboring steps, a regime exists near the critical island size where both compact and elongated shapes can coexist.     

Kinetic Monte Carlo simulations of heteroepitaxial growth with an atomistic model of elasticity

We have implemented Kinetic Monte Carlo (KMC) simulations of growth of heteroepitaxial thin films. A simple cubic Solid-on-Solid (SOS) model is used to describe the atomic configurations and nearest neighbor bonds are used to describe the energetics. Elastic effects are modeled using harmonic springs between atoms displaced from their lattice positions. The misfit strain is a consequence of different equilibrium spring lengths for the substrate and film. The consistency of this elastic model with continuum theories for strained surfaces has been shown by performing elastic energy calculations for various morphologies. KMC simulations for submonolayer deposition show scaling behavior in the island size distribution. The resulting island shapes are predominantly square and do not show any shape transitions in the physically relevant range of conditions. This method gives a detailed understanding of elastic interactions and their interplay with surface diffusion in heteroepitaxial systems.     

Phase-field study for the splitting behaviour of precipitate under applied stress

This paper employs the phase-field method to study the splitting behaviour of a single coherent particle under an applied uniaxiai stress. It finds that the splitting behaviour is greatly influenced by the initial shape of precipitates, the bulk free energy condition, the degree of supersaturation and the ratio of the interfacial energy to the elastic strain energy, etc. The simulated results demonstrate that the aspect ratio of the particle determines whether the splitting can occur and how many split plates can be obtained. The splitting of particle is sensitive to the interracial energy, i.e. the splitting becomes more and more difficult with increasing the ratio of the interracial energy to the elastic strain energy. And increasing the magnitude of the applied stress is favourable to the splitting. The splitting process is also explained from the point of view of the corresponding diffusion potential.     

Plastic relaxation of mixed dislocation in axial nanowire heterostructures using Peach–Koehler approach

In this Letter, the plastic relaxation introduced by typical 60° mixed dislocation in zinc‐blende axial nanowire (NW) heterostructures is evaluated by numerical Peach–Koehler approach which reflects the interaction between coherent strain field and misfit dislocation. Cylindrical NW epilayers grown on NW and bulk substrate are separately modeled. We reveal that straight 60° dislocation with Burgers vector (a /2) 〈101〉 would always generate at NW center orienting [111], while aspect ratio dependent off‐center equilibrium position should be expected for NW epilayer along [001]. Moreover, the critical diameters, below which coherency is maintained, are predicted based on energy balance criterion for axial NW heterostructures along predominating growth direction [111]. The results are compared with existing theory and experimental observations. Our work provides a necessary quantitative complement to the understanding of plastic relaxation and coherency limit in axial NW heteroepitaxy. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

晶体相场法模拟异质外延过程中界面形态演化与晶向倾侧

采用晶体相场模型研究了异质外延过程中失配应变与应力弛豫对外延层界面形态演化的影响, 并对由衬底倾角引起的外延层晶向倾侧进行了分析.研究结果表明: 在有一定倾角的衬底晶体上进行外延生长时,若衬底和外延层之间失配度较大 (ε>0.08),外延层中弹性畸变能会以失配位错的形式释放, 最终薄膜以稳定的流动台阶形式生长且外延层的晶向倾角与衬底倾角呈近似线性关系. 而当衬底和外延层之间失配度较小(ε<0.04)不足以形成失配位错时, 外延层中弹性畸变能会以表面能的形式释放,最终使薄膜以岛状形态生长. 在高过冷度条件下,衬底倾角和失配度较大时,衬底和外延层之间会形成由大量位错规则排列而成的小角度晶界从而显著改变外延层的生长位向.     

The strain relaxation of InAs/GaAs self-organized quantum dot

This paper presents a detailed analysis of the dependence of degree of strain relaxation of the self-organized InAs/GaAs quantum dot on the geometrical parameters. Differently shaped quantum dots arranged with different transverse periods are simulated in this analysis. It investigates the total residual strain energy that stored in the quantum dot and the substrate for all kinds of quantum dots with the same volume, as well as the dependence on both the aspect ratio and transverse period. The calculated results show that when the transverse period is larger than two times the base of the quantum dots, the influence of transverse periods can be ignored. The larger aspect ratio will lead more efficient strain relaxation. The larger angle between the faces and the substrate will lead more efficient strain relaxation. The obtained results can help to understand the shape transition mechanism during the epitaxial growth from the viewpoint of energy, because the strain relaxation is the main driving force of the quantum dot's self-organization.     

Formation of strained ring-shaped islands around square notches

The location and morphology of a two-dimensional island has been studied theoretically as a function of the misfit stress in the neighbourhood of a square notch present on the free surface of an epitaxially stressed film deposited on a substrate. From a static energy calculation, it has been shown that the notches can drive the motion of the islands towards the notches. It was then found that, depending on the side length and depth of the notch, self-organized formation at constant volume of a two-dimensional ring-shaped island can be favoured along the periphery of the pre-existing notch with respect to the notch shrinking.     

Edge misfit dislocation formation at the interface of a nanopore and infinite substrate with surface/interface effects

The model of an edge misfit dislocation at the interface of the hollow nanopore and the infinite substrate with surface/interface stress is investigated. Using the complex variable method, analytical solutions for complex potentials of a film due to an edge misfit dislocation located in the film with surface/interface effect are derived, and the stress fields of the film and the edge misfit dislocation formation energy can be obtained. The critical conditions for edge misfit dislocation formation are given at which the generation of an edge misfit dislocation is energetically favourable. The influence of the ratio of the shear modulus between the film and the infinite substrate, the misfit strain, the radius of the nanopore and the surface/interface stress on the critical thickness of the film is discussed.     

Au/Cu(001)异质外延岛演化的分子动力学研究

利用分子动力学弛豫方法模拟了Au/Cu(001)异质外延生长初期Au异质外延岛的形貌演化，分析了Au外延岛演化过程中的局域应力及与基体结合能随表面岛尺寸的变化. 研究结果表明：当异质外延岛小于7×7时，外延岛原子分布呈现赝Cu点阵形貌；当外延岛达到8×8后，外延岛内开始出现失配位错，失配位错数量随外延岛尺寸的增加而增加. 局域压力分析指出，外延岛上原子之间的近邻环境不同导致了所受应力的差异，而外延岛的形变则是由外延岛原子的应力分布所决定. 研究还发现，失配位错的产生导致错位原子与基体原子之间的结合强度减弱，但相对增加了非错位原子与基体原子之间的结合强度. 关键词： 异质外延 表面形貌 局域压力 分子动力学模拟     

Bistability of bilayer islands under anisotropic misfit stress

The equilibrium shapes of bilayer islands under anisotropic misfit stress have been theoretically characterized from a thermodynamics point of view. Based on an energy variation calculation, it is found that when the ratio σ between the components of the misfit stress is such that σ = ? 1, both square and rectangular shaped bilayer islands may coexist depending on the island volume and stress level. When σ ≠ ? 1 and σ < 0, the possibility of formation rectangular islands along both perpendicular directions is also discussed.     

Equilibrium shape and size of supported heteroepitaxial nanoislands

We study the equilibrium shape, shape transitions and optimal size of strained heteroepitaxial nanoislands with a two-dimensional atomistic model using simply adjustable interatomic pair potentials. We map out the global phase diagram as a function of substrate-adsorbate misfit and interaction. This phase diagram reveals all the phases corresponding to different well-known growth modes. In particular, for large enough misfits and attractive substrate there is a Stranski-Krastanow regime, where nano-sized islands grow on top of wetting films. We analyze the various terms contributing to the total island energy in detail, and show how the competition between them leads to the optimal shape and size of the islands. Finally, we also develop an analytic interpolation formula for the various contributions to the total energy of strained nanoislands.     

Melting of embedded anisotropic particles: PbIn inclusions in Al

Small particles embedded in the solid state can assume facetted shapes that indicate the anisotropy in interfacial free energy between the particle and the embedding phase. Previous in situ transmission electron microscopy (TEM) of Al–In has shown that such facetted nanoparticles with cubic symmetry embedded in a solid can produce a series of complex melt configurations. To determine the dependence of melt configuration on solid–solid facet energies, this work numerically calculates the equilibrium melt trajectory for a family of cuboctahedral particles parameterized by the ratio of matrix–particle interfacial free energies for orientations taken with respect to the crystal axes of the matrix, ??=?γ₁₀₀/γ₁₁₁. The calculations assume that equilibrium occurs with the minimization of the total interfacial free energy to determine the stable internal melt configuration for a fixed volume of melt confined within a member of the cuboctahedral family. At particular melt volume fractions, abrupt transitions in shape occur when a stable configuration reaches a touching or necking instability. In situ TEM of a dilute Al alloy containing embedded nanosized PbIn inclusions that gradually melt as the temperature is increased from 230 to 260°C is consistent with the results calculated for ??=?1.245.     

Mesoscopic and microscopic modeling of island formation in strained film epitaxy

The instability of strained films for island formation is examined through an approach incorporating both discrete microscopic details and continuum mechanics. A linear relationship between the island wave number and misfit strain is found for large strains, while only in the small strain limit is a crossover to the continuum elasticity result obtained. A universal scaling relation accommodating all range of misfit strains is identified. Our results indicate that continuum mechanics may break down even at relatively small misfit stress due to the discrete nature of crystalline surfaces.     

Effect of out-of-plane misfit strain on phase diagrams and ferroelectric properties of ferroelectric films in vertical nanocomposite structures

A phenomenological thermodynamic Landau–Devonshire theory is developed to investigate phase diagrams of epitaxial ferroelectric films with out-of-plane misfit strain induced by vertical nanocomposites. The thermodynamic potential of ferroelectric films is obtained based on the boundary conditions of three-dimensional clamping induced by the vertical nanocomposites. Our calculated results indicate that the out-of-plane misfit strain modulates the transition temperature and spontaneous polarization of ferroelectric films in a wide range even the substrate does not provide an effective in-plane misfit strain control. An enhanced critical transition temperature up to 803 °C in BaTiO₃ films under a tensile out-of-plane misfit strain is predicted, which is consistent with the experimental result very well. The polarization properties of BaTiO₃ films can also be effectively modulated by the out-of-plane misfit strain which is controlled by the volume fraction of nanopillars in the vertical nanocomposites. Our method provides a theoretical guide for the out-of-plane strain engineering of ferroelectric films.     

Origin of apparent critical thickness for island formation in heteroepitaxy

We find that a continuum model of heteroepitaxy exhibits a sharp crossover with increasing coverage, from planar growth to island formation. The "critical thickness" at which this Stranski-Krastanov transition occurs depends sensitively on misfit strain, with a dependence strikingly similar to that seen experimentally. The initial planar growth occurs because of intermixing of deposited material with the substrate. While the transition is strictly kinetic in nature, it depends only weakly on growth rate. The role of surface segregation is also discussed.     

Stable unidimensional arrays of coherent strained islands

We investigate the equilibrium properties of arrays of coherent strained islands in heteroepitaxial thin films of bidimensional materials. The model we use takes into account only three essential ingredients: surface energies, elastic energies of the film and of the substrate and interaction energies between islands via the substrate. Using numerical simulations for a simple Lennard-Jones solid, we can assess the validity of the analytical expressions used to describe each of these contributions. A simple analytical expression is obtained for the total energy of the system. Minimizing this energy, we show that arrays of coherent islands can exist as stable configurations. Even in this simple approach, the quantitative results turn out to be very sensitive to some details of the surface energy.