Geometrical aspect of the problem of model structure growth

In developing the representation of model growth of crystalline and quasicrystalline structures in the form of a growing graph, the convolution operation is used to describe the formation of the growth polyhedron from a specified system of bonds (direct growth problem). It is shown that the growth polyhedron is formed by a system of intermolecular interactions so that the main bonds, which determine the neighborhood graph of the molecules of this system, are oriented in the direction of the growth polyhedron vertices. It is shown that the system of molecular bonds can be determined on the basis of the growth polyhedron (inverse growth problem).     

A model for the growth of anomalous polytype structures in vapour grown SiC

A number of polytype structures observed in vapour grown SiC crystals have a unit-cell which is an integral multiple of the unit-cell of the basic 6H, 15R or 4H structure. The growth of such anomalous structures cannot be understood in terms of spiral growth round a single screw dislocation in a basic matrix. However many of these polytype crystals display a single growth spiral on their (0001) face indicating that they have resulted from spiral growth round a single screw dislocation. It is shown that this anomaly can be resolved if the basic matrix is assumed to contain stacking faults near the surface at the time of the origin of the screw dislocation ledge. This possibility, overlooked in the earlier deduction of polytype structures, must be taken into consideration since vapour grown SiC crystals frequently contain a high concentration of random stacking faults, producing continuous streaks on their X-ray diffraction photographs. The most probable fault configurations that can occur in 6H, 15R and 4H structures of SiC have been deduced from a calculation of their stacking fault energy. These fault configurations are then considered to lie at different distances from the surface at the time of the origin of a screw dislocation ledge. Such a faulted ledge gives rise to polytype structures during subsequent spiral growth even if the screw dislocation has an integral Burgers vector. The most probable series of polytype structures that can result from such a faulted matrix model are deduced. It is shown that nearly all the polytype structures of SiC hitherto regarded as anomalous (such as 36H, 54H, 66H, 45R, 90R etc.) are among the expected structures and there is no need to postulate a complicated configuration of cooperating dislocations to account for their growth.     

A stability diagram for crystal growth from the vapor – a review

Stability of the solid‐vapor interface is investigated. Surface roughening and evolution of flat faces at the growth interface is considered in terms of relation between gradient of temperature in the crystal and gradient of concentration in the vapor. Stability diagram is proposed, based on experimental data. The diagram summarizes the various forms and structures, which can be obtained using a typical system for growth from the vapor. The critical lines for constitutional supersaturation and appearance of low and high index faces were plotted. This attempt to the problem of stability of the growth interface differs from the former investigations mainly in looking for dependence between temperature field and concentration field rather than between more absolute parameters like temperature and supersaturation. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Three-dimensional thermocapillary and buoyancy convections and interface shape in horizontal Bridgman crystal growth

Computer simulation is conducted to study three-dimensional (3D) thermocapillary and buoyancy convections and their effects on the growth interface for horizontal Bridgman crystal growth. The free-boundary model is based on a finite volume approximation of continuity, momentum, and energy equations on a collocated grid. Crystal growth of GaAs is used as an example. From calculated results, it is observed that the effect of buoyancy convection on the growth interface is significant. With the thermocapillary effect, the 3D flow structures are not changed much, but its effect on the growth interface is not trivial. Due to the convections, the growth interface is always concave, and its deflection is affected significantly by the growth rate and thermal environment. A simple strategy of interface control is illustrated. Furthermore, slight crucible tilting can also affect the 3D flows leading to an asymmetric growth interface.     

A Model for Structures Growth by Sodium Electrodiffusion in Quartz Crystals

Dendritic structures were grown inside the quartz crystal by electrodiffusion of sodium ions from a NaCl layer at the anode. The electric field was applied along the optical axis Z and the structures were grown in the xy plane. An experimental arrangement using a pointed cathode and a graphite plate anode was employed. X ray diffraction analysis shows that the structures' chemical composition is a mixture of sodium silicates. To explain the structures' growth a model is proposed. The model is a bidimensional one. The transport of the particles along the Z axis structural channels by the electric field was taken into account by introducing a probability of generating the particles in the growth plane as a function of the distance to the cathode. The computer generated results and the experimental ones were compared and a good correlation was found.     

Growth of quantum wire structures by selective area chemical beam epitaxy

Selective area growth on silicon dioxide masked gallium arsenide substrates by chemical beam epitaxy is used to fabricate inverted V-shaped mesas. Indium gallium arsenide quantum wells grown on top of these mesas form quantum wire structures. The faceted mesa sidewalls are described as a function of substrate temperature and V/III ratio in terms of a simple geometric model. The photoluminescence spectra show that the wire structure peak is shifted to longer wavelength compared to unpatterned substrates, for all growth temperatures. This shift is explained by the migration of indium. For low temperature growth, a second peak due to sidewall quantum wells is observed.     

Coordination sequences and coordination waves in matter

A possible way of partitioning a space into polycubes (n-dimensional modifications of Golomb polyominoes, which are generally nonconvex) is used as a basic model of ordered matter structure. It is suggested that layer-by-layer growth of a structure, occurring along the geodetics of the digraph of a net defined by the local rules of bonding of polycubes, justifies the phenomenological laws of shaping (self-similarity during the growth, independence of the polyhedron shape on the “seed,” the symmetry of the growth polyhedron, etc.). Specific results of the analysis of number sequences of the increase in coordination circles for planar periodic partitions of model and real crystal structures, as well as the preliminary results of investigation of standing coordination topological waves, revealed for the first time in computer experiments, are reported.     

Generation mechanism of CuAu-I type ordered structures in InGaAs crystals grown on (110) InP substrates by molecular beam epitaxy

The generation mechanism for CuAu-I type ordered structures in InGaAs grown on (110) InP substrates by molecular beam epitaxy is discussed. On the basis of previous work together with considerations of the atomic arrangement of the ordered structure and surface reconstruction on the (110) plane, we propose four possible models for the ordering. Through precise evaluation of these models, two models are selected as the most probable ones: these involve formation of the ordered structures on surfaces dominated by two monolayer steps. This model have been experimentally proven by the analyses of electron diffraction patterns from different crystals grown on different growth surfaces.     

Mathematical model and numerical simulation of faceted crystal growth

A new mathematical macroscopic model is proposed to describe the nonstationary process of faceted crystal growth by the methods of directional crystallization with a slow change in external thermal conditions and low pulling rate of a cell through the growth system. The facet-growth rate is determined by the Stefan condition, integral over the face. Two boundary conditions are set for temperature: the continuity condition and the relation between the heat-flux jump and the supercooling at the facet points. The supercooling is determined by solving the entire heat problem. A facet is selected as a planar part of the phase boundary. The kinetic coefficient at the facet may depend on the supercooling. The energy conservation law is valid within the model developed. Examples of calculations of some model problems are presented.     

Solute trapping and the effects of anti-trapping currents on phase-field models of coupled thermo-solutal solidification

We explore how the inclusion of an anti-trapping current within a phase-field model of coupled thermo-solutal growth formulated in the thin interface limit actually affects the observed levels of solute trapping during dendritic growth. The problem is made computationally tractable by the use of advanced numerical techniques including local mesh adaptivity, implicit temporal discretization and a multigrid solver. Contrary to published results for pure solutal models we find that the inclusion of such an anti-trapping current does not lead to the recovery of the equilibrium partition coefficient, except in the limit of vanishing growth velocity. At higher growth velocities non-vanishing amounts of solute trapping are observed.     

High-velocity banding structure in the laser-resolidified hypoperitectic Ti47Al53 alloy

Stability analysis of a growing solid/liquid interface is the fundamental concept of modern solidification theory. Here, serial laser rapid solidification experiments were performed on a hypoperitectic Ti₄₇Al₅₃ alloy to explore the dendritic growth behavior near the limit of high-velocity absolute stability. SEM and TEM techniques were carried out to investigate the microstructure and identify the phase composition. By adopting an improved sampling method of TEM, the growth morphology evolution of the laser-resolidified layer was observed directly and high-velocity banding structure was firstly detected in Ti–Al peritectic alloys. The high-velocity banding structures are parallel to the solid/liquid interface (normal to the growth direction) and made of the oscillation structures grown alternatively in modes of cell and plane morphologies. In light bands with cellular growth mode, all dislocation assembles are parallel to the growth direction and forms the cell boundaries, while all dislocation distributes randomly in dark bands. The determined growth velocity range for the appearance of high-velocity banding structures is about 0.51.1 m s⁻¹ according to the rapid solidification experiments, and the origin of the banding agrees well with the prediction of the CGZK phenomenological model (Acta Metal. Mater. 40 (1992) 983).     

软轴型单晶炉提拉系统的振动建模与分析

当软轴型单晶炉提拉系统工作在某一转速附近时,其重锤摆动量会显著增大,严重影响单晶生长质量.为解决此问题,需要建立该系统的数学模型来研究提拉系统的动力学特性.本文在对提拉系统工作原理分析的基础上,应用Lagrange第二类方程建立了考虑该系统面内、面外振动的四自由度非线性振动微分方程.导出了该非线性模型的近似线性模型,对非线性模型响应与线性模型响应做了比较.进一步分析了系统的稳定性,应用Campbell图得到了考虑回转惯性效应后系统的临界转速.通过数值仿真定量说明了减小对中误差和增大阻尼可以减小重锤系统最大摆动幅值.     

Catalytic epitaxy of ZnO whiskers via the vapor–crystal mechanism

A model of oriented growth of (0001) ZnO whiskers on sapphire substrates via the vapor–crystal mechanism using the catalytic properties of gold islands is proposed. The morphological transition from the primary pyramidal ZnO structures to hexagonal ZnO whiskers is described in terms of the minimization of the free energy density of three-dimensional heteroepitaxial islands.     

Theoretical model of crystal growth shaping process

A theoretical investigation of the crystal growth shaping process is carried out on the basis of the dynamic stability concept. The capillary dynamic stability of shaped crystal growth processes for various forms of the liquid menisci is analyzed using the mathematical model of the phenomena in the axisymmetric case. The catching boundary condition of the capillary boundary problem is considered and the limits of its application for shaped crystal growth modeling are discussed.

The static stability of a liquid free surface is taken into account by means of the Jacobi equation analysis. The result is that a large number of menisci having drop-like shapes are statically unstable. A few new non-traditional liquid meniscus shapes (e.g., bubbles and related shapes) are proposed for the case of a catching boundary condition.     

基于数据驱动的晶体直径模型辨识方法研究

直拉法硅单晶生长是一个多场多相耦合、物理变化复杂,且具有大滞后和非线性现象的过程,基于单晶硅生长系统内部机理所构建的机理模型由于存在诸多假设而无法应用于工程实际。因此,本文以现有CL120-97单晶炉拉晶车间的长期、海量晶体生长数据为基础,忽略炉内复杂的晶体生长环境,对影响晶体直径的拉晶参数进行关联性分析及特征量化,探寻拉晶数据中所蕴藏的规律信息,进而建立基于数据驱动的BP神经网络晶体直径预测模型,并针对现有BP神经网络易陷入局部极小值的问题,采用遗传算法对BP神经网络的阈值和权值进行优化,以提高晶体直径预测的准确性。通过实际拉晶数据对模型预测结果进行验证,结果表明,对任意选取的8组拉晶数据进行直径预测,预测的平均相对百分比误差为0.095 71%,证明该模型对于等径阶段晶体直径的预测是可行的。     

Efficient solution of a batch crystallization model with fines dissolution

In this paper, an efficient and accurate numerical method is proposed for solving a batch crystallization model with fines dissolution. The dissolution of small crystals (fines dissolution) is useful for improving the quality of a product. This effectively shifts the crystal size distribution (CSD) towards larger crystal sizes and often makes the distribution narrower. The growth rate can be size-dependent and a time-delay in the dissolution unit is also incorporated in the model. The proposed method has two parts. In the first part, a coupled system of ordinary differential equations (ODEs) for moments and solute mass is numerically solved in the time domain of interest. These discrete values are used to get growth and nucleation rates in the same time domain. In the second part, the discrete growth and nucleation rates along with the initial CSD are used to construct the final CSD. The analytical expression for CSD is obtained by applying the method of characteristics and Duhamel's principle on the given population balance model (PBM). A Gaussian quadrature method, based on orthogonal polynomials, is used for approximating integrals in the ODE-system of moments and solute mass. The efficiency and accuracy of the proposed numerical method is validated by a numerical test problem.     

Mathematical modeling of the multi‐run process of crystal pulling from the melt by EFG (Stepanov) technique in dependence on the angle of inclination of the working edges of the dies

A mathematical model for the determination of melt hydrodynamics, impurity concentrations and thermal stresses in the multi‐run process of the growth of sapphire ribbons by EFG (Stepanov) technique with inclinated working surfaces of the dies is considered. The mathematical model deals with thermal conductivity equation, Navier‐Stokes equation, diffusion equation, capillary Laplace equation. This problem has been solved by the finite‐element method. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Growth of EuTe islands on SnTe by molecular beam epitaxy

Semiconductor magnetic quantum dots are very promising structures, with novel properties that find multiple applications in spintronic devices. EuTe is a wide gap semiconductor with NaCl structure, and strong magnetic moments S=7/2 at the half filled 4f⁷ electronic levels. On the other hand, SnTe is a narrow gap semiconductor with the same crystal structure and 4% lattice mismatch with EuTe. In this work, we investigate the molecular beam epitaxial growth of EuTe on SnTe after the critical thickness for island formation is surpassed, as a previous step to the growth of organized magnetic quantum dots. The topology and strain state of EuTe islands were studied as a function of growth temperature and EuTe nominal layer thickness. Reflection high energy electron diffraction (RHEED) was used in-situ to monitor surface morphology and strain state. RHEED results were complemented and enriched with atomic force microscopy and grazing incidence X-ray diffraction measurements made at the XRD2 beamline of the Brazilian Synchrotron. EuTe islands of increasing height and diameter are obtained when the EuTe nominal thickness increases, with higher aspect ratio for the islands grown at lower temperatures. As the islands grow, a relaxation toward the EuTe bulk lattice parameter was observed. The relaxation process was partially reverted by the growth of the SnTe cap layer, vital to protect the EuTe islands from oxidation. A simple model is outlined to describe the distortions caused by the EuTe islands on the SnTe buffer and cap layers. The SnTe cap layers formed interesting plateau structures with easily controlled wall height, that could find applications as a template for future nanostructures growth.     

Growth and etching of silicon in chemical vapour deposition systems; The influence of thermal diffusion and temperature gradient

In the growth of epitaxial silicon using SiCl₄ as source material it is known, that the growth rate at first increases with increasing p_SiCl4, but at higher p_SiCl4 decreases and finally becomes negative. The exact shape of the growth curve and especi ally the point of zero growth, where the etch rate counterbalances the growth rate, differs considerably amongst various investigators. This phenomenon was subject to theoretical considerations. Three models were evaluated: (i) a simple equilibrium model in which gas transport limited kinetics are used with one single diffusion coefficient for all species; (ii) an extended model using individual diffusion coefficients; (iii) a model as in (ii) but including the effect of thermal diffusion. It is shown that models (i) and (ii) do not give a general solution of the problem. The best results are obtained with model (iii). The growth rate appears to be very sensitive to the temperature gradient ▽T. As ▽T depends on the kind of apparatus used in the deposition process, it provides a reasonable explanation for the observed differences.     

In situ optical monitoring of AlGaN thickness and composition during MOVPE growth of AlGaN/GaN microwave HFETs

Real-time spectral reflectometry has been implemented to monitor the MOVPE growth of AlGaN/GaN microwave HFET structures. The aim is to monitor and control the thickness and composition of the thin AlGaN layer during growth. In order to extract useful information from the in situ spectra the optical constants of AlGaN as a function of alloy composition are required at the growth temperature (1050°C). As the first step to obtaining the high temperature optical constants, a room temperature spectroscopic ellipsometry study (energy range 1.65–4.95 eV) has been carried out on thin AlGaN films of various thickness (30 and 100 nm) and aluminium content (0.15 and 0.25). The multilayer model of each sample from the ellipsometry study is used to generate a reflectance spectrum which is compared with the in situ spectral reflectometry spectrum of the same sample acquired at room temperature to verify the technique. Further work is in progress to model the bandgap and optical constants of GaN and AlGaN at growth temperature.