Thin Film Growth Model Research Articles

Investigation of structured TiAlN- and TiO2-coatings with moth-eye-like surface morphologies

This paper is to report on the development and investigation of a specific type of growth mode and surface morphology of PVD-coatings, namely a columnar growth mode where the individual columns are terminated by pyramid-shaped tips. TiAlN was chosen as a representative of the cubic crystal system, TiO2 as a representative of the tetragonal crystal system. The coatings were sputter-deposited on silicon and steel substrates by a reactive RF-magnetron process under variation of total pressure, reactive gas flow and RF-power. Investigations of microstructure and surface morphology were carried out with electron microscopy (SEM), atomic force microscopy (AFM) and X-ray diffraction techniques (XRD) and are described in more detail for TiO2-coatings. The individual columns with diameters between 100 and 500 nm were terminated by pyramid-shaped tips where the geometric forms of the pyramids were determined by the crystal system of the coating material. A power law was found for the dependance of the mean column diameter on coating thickness in qualitative agreement with phenomenological models for thin film growth. Both coating materials were deposited on steel tools and hot embossed into the surfaces of PMMA-clear-view screens. The resulting screen's surface morphologies proved to be comparable to the anti-reflective surfaces of moth-eyes and also exhibited a pronounced decrease in reflectivity.

Growth Kinetics of Thin Films Formed by Nucleation during Layer Formation

A kinetic model of thin-film growth on a solid surface is investigated. This model is valid in the case where the layers are formed as a result of two-dimensional nucleation. Under conditions of high supersaturation of a gaseous phase, solutions are obtained for an island-size distribution function at the initial stage of growth, the degree of filling of a substrate by islands at the coalescence stage, the vertical-growth rate of a film, and its surface roughness. These solutions express the structural characteristics of a growing film in terms of physical constants (the interphase energy on the gas-solid interface and the activation barriers of diffusion and desorption) of the system and the growth parameters (the surface temperature and the material-deposition rate). The obtained results make it possible to calculate the growth dynamics for thin films in particular systems.

Feedback control of surface roughness using stochastic PDEs

The surface roughness of thin films deposited from gas phase precursors is an important variable to control, because it strongly affects the quality of such films. The modern integrated circuit technology depends strongly on the uniformity and microstructure of deposited thin films (Granneman, 1993). Due to the increasingly stringent requirements on the quality of such films, on-line estimation and control of thin film deposition becomes important. In a thin film growth process, the film is directly formed from microscopic random processes (for example, particle adsorption, desorption, migration and surface reaction). Therefore, the stochastic nature of thin film growth processes must be fully considered in the modeling and control of the surface roughness of thin films. The desire to understand and control the thin film micro-structure has motivated extensive research on fundamental mathematical models describing the deposition processes, which include (1) kinetic Monte-Carlo methods (for example, Gillespie, 1976; Fichthorn and Weinberg, 1991; Lam and Vlachos, 2001), and (2) stochastic partial-differential equations (PDEs) (for example, Edwards and Wilkinson, 1982; Villain, 1991; Vvedensky et al., 1993). The kinetic Monte-Carlo simulation method can be used to predict average properties of the thin film (which are of interest from a control point of view, for example, surface roughness), by explicitly accounting for the microprocesses that directly shape thin film microstructure. Recently, a methodology for feedback control of thin film growth using kinetic Monte-Carlo models has been developed in (Lou and Christofides, 2003a,b). The methodology leads to the design of (a) real-time roughness estimators by using multiple small lattice kinetic Monte-Carlo simulators, adaptive filters and measurement error compensators, and (b) feedback controllers based on the real-time roughness estimators. The method was successfully applied to control surface roughness in a GaAs deposition process using an experimentally determined kinetic Monte-Carlo process model (Lou and Christofides, 2004b). Other approaches have also been developed to: (a) identify linear models from outputs of kinetic Monte-Carlo simulators and perform controller design by using linear control theory (Siettos et al., 2003), and (b) construct reduced-order approximations of the master equation (Gallivan and Murray, 2003). However, the fact that kinetic Monte-Carlo models are not available in closed-form makes very difficult to perform model-based controller design directly on the basis of kinetic Monte-Carlo models. To achieve better closed-loop performance, it is desirable to design feedback controllers on the basis of deposition process models. This motivates research on feedback control of deposition processes based on stochastic PDE models of thin film growth. Stochastic PDE models have been developed to describe the evolution of the height profile for surfaces in a variety of physical and chemical processes. Examples include deposition processes including adsorption and surface relaxation (Edwards and Wilkinson, 1982; Vvedensky, 2003), crystal growth from atomic-beams with and without desorption (Villain, Correspondence concerning this article should be addressed to P. D. Christofides; e-mail: pdc@seas.ucla.edu.

Effective transition rates for epitaxial growth using fast modulation

Thin-film deposition is an industrially important process that is highly dependent on the processing conditions. Most films are grown under constant conditions, but a few studies show that modified properties may be obtained with periodic inputs. However, assessing the effects of modulation experimentally becomes impractical with increasing material complexity. Here we consider periodic conditions in which the period is short relative to the time scales of growth. We analyze a stochastic model of thin-film growth, computing effective transition rates associated with rapid periodic process parameters. Combinations of effective rates may exist that are not attainable under steady conditions, potentially enabling new film properties. An algorithm is presented to construct the periodic input for a desired set of effective transition rates. These ideas are demonstrated in three simple examples using kinetic Monte Carlo simulations of epitaxial growth.

Modeling and visualization of polycrystalline thin film growth

A novel polycrystalline thin film growth simulator, FACET, has been developed. FACET is a multi-scale model with two major components: an atomic level one-dimensional kinetic lattice Monte Carlo (1D KLMC) model and a real time feature scale two-dimensional facet nucleation and growth model. The 1D KLMC model has been developed to calculate inter-facet diffusion rates. By inputting the diffusion activation energies, the model will calculate the inter-facet atomic flux between {1 0 0}, {1 1 0}, and {1 1 1} facets of FCC materials at any temperature. The results of the 1D KLMC model have been verified by comparison with a full three-dimensional kinetic lattice Monte Carlo (3D KLMC) model. The feature scale polycrystalline thin film nucleation and growth model is based on describing grains in terms of two-dimensional faceted surfaces and grain boundaries. The profile of the nuclei are described by crystallographically appropriate facets. The position and orientation of the nuclei can be randomly selected or preferred textures can be created. Growth rates are determined from different deposition fluxes and surface diffusion effects. Quantitative microstructural characterization tools, including roughness analysis, average grain size analysis, and orientation distribution analysis, were incorporated into the model, which allows the users to design, conduct and analyze the virtual experiments within one integrated graphical user interface. Users can also visualize the nucleation and growth process of the film and obtain the final film microstructure. The effects of thickness, temperature, and deposition flux on thin film microstructures have been studied by FACET.

Three-dimensional simulations of self-assembly of hut-shaped Si–Ge quantum dots

This article presents the results of three-dimensional modeling of heteroepitaxial thin film growth with the objective of understanding recent experiments on the early stages of quantum dot formation in SiGe/Si systems. We use a continuum model, based on the underlying physics of crystallographic surface steps, to study the growth of quantum dots, their spatial ordering and coarsening behavior. Using appropriate parameters, obtained from atomistic calculations, the (100) orientation is found to be unstable under compressive strains. The surface energy now develops a minimum at an orientation that may be interpreted as the (105) facet observed in SiGe/Si systems. This form of the surface energy allows for the growth of quantum dots without any barrier to nucleation—dots are seen to start off via a surface instability as shallow stepped mounds, which steepen continuously to reach their low energy orientations. During the very initial stages of growth, mounds are seen to grow in a dense array with several of them impinging on each other and subsequently coalescing to form larger mounds. This behavior occurs due to the competition between surface energy which seeks to minimize the free-energy by the formation of islands with side-walls at the strain stabilized orientations and repulsive elastic interactions between such closely spaced islands. Using simple analytical calculations, we show the existence of a critical island size for this coalescence behavior. A key result of our analysis is the inverse scaling of this critical size with the misfit strain in the film. While energetic analyses may be used to obtain useful insights, the growth of quantum dots is essentially a nonequilibrium process and requires a fundamental understanding of the kinetics. Numerical studies show that the growth kinetics has a profound effect on surface morphology: arrays of well-separated islands or, alternatively, intersecting ridges are obtained in different kinetic regimes. We also study an alternative model of a stable but nonfacet (100) orientation and point out the inconsistencies of this assumption.

Stationary Solutions for a Model of Amorphous Thin-Film Growth

We consider a class of stochastic partial differential equations arising as a model for amorphous thin film growth. Using a spectral Galerkin method, we verify the existence of stationary mild solutions, although the specific nature of the nonlinearity prevents us from showing the uniqueness of the solutions as well as their boundedness (in time).

A computer model for the growth of thin films in a structure zone model

A simple two-dimensional model of thin film growth based on the ballistic aggregation ofhard discs is proposed. In this model the impinging hard discs travel in a straight line andinitially ‘stick’ to the first previously deposited hard disc they contact. Previously (Dirksand Leamy 1977 Thin Solid Films 47 219) the criterion for relaxation has been migration ofthe impinging disc to the nearest pocket formed by two additional discs. In the modelproposed here, however, relaxation is dependent on the surrounding geometry of thepreviously deposited atoms. The deposited disc may ‘relax’ into a position where it thenmakes contact with three, two or one previously deposited disc(s). The influences ofsubstrate temperature, deposition rate, angle of incidence, self-shadowing effect andsubstrate roughness, including amorphous substrates are investigated. Results arequalitatively in good agreement with the structure zone model (SZM) predictions for thinmetallic films.

A fourth-order parabolic equation modeling epitaxial thin film growth

We study the continuum model for epitaxial thin film growth from Phys. D 132 (1999) 520–542, which is known to simulate experimentally observed dynamics very well. We show existence, uniqueness and regularity of solutions in an appropriate function space, and we characterize the existence of nontrivial equilibria in terms of the size of the underlying domain. In an investigation of asymptotical behavior, we give a weak assumption under which the ω-limit set of the dynamical system consists only of steady states. In the one-dimensional setting we can characterize the set of steady states and determine its unique asymptotically stable element. The article closes with some illustrative numerical examples.

Computer simulation of thin film growth with defect formation

In this paper, we describe a new model for thin film growth to study the surface structure as well as the defect structure in the film. In the new model, which we call Solid-by-Solid model, the usual Solid-on-Solid criterion is not assumed. This extends to make vacancies to be able to be created in the film during the growth, which enables us to study the surface structure and the void structure at the same time. The growth is controlled by the chemical potential μ, which corresponds to the overpotential in electrodeposition. Spatial and size distributions of voids and the correlation between void structure and the surface roughness are studied for various deposition conditions. The simulation of filling grooves is also performed to study the void formation mechanism during the deposition.

Visualization modeling of thin film growth in photodeposition processes

A computer visualization technique, which analyzes and predicts the spatio-temporal evolution of thin film deposition and growth processes is given. It relies on microscopy sampled or computer generated synthetic micrographs of particles. These are then simulated for deposition, aggregation and coagulation during thin film growth by frequency domain transform techniques. Particle sources and diffusion operators on surfaces are used to predict with high temporal resolution, unattained by real world microscopy the surface structure evolution as time samples and time movies. The simulation program was used to investigate deposition and diffusive profiles in photodeposition experiments, starting from initial synthetic micrographs based on real world scanning electron microscopy (SEM) images. The surface microstructure time ‘tracking’ scheme described here relies on transforming the original image of the deposited particles into a Fourier spatial frequency domain image. The physical models used are that of a material random deposition source and subsequent surface redistribution due to diffusion and other coalescence material surface flow mechanisms. The 2-D inverse Fourier transform (IFT) is finally used to obtain back the real space-time images representing the surface spatio-temporal films morphology changes. False color representation of the images allows for a better discrimination of the films growing details especially during the fast pre-compact thin film layer formation on the substrate.

Breakdown of the approximations of small perturbations in continuum modeling of amorphous thin film growth

Morphology evolution of thin film growth can be quantitatively modeled by employing rate equations for two-dimensional, single-valued functions within a treatment of small perturbations. This description can be expected to be valid in an intermediate film thickness regime, where substrate influences and three-dimensional effects are minor. The breakdown of the mathematical description in its upper film thickness limit is systematically investigated by a detailed comparison of experiments and simulation. Possible reasons for the failure are discussed and improvements are suggested.

FACET: a novel model of simulation and visualization of polycrystalline thin film growth

We present a two-dimensional polycrystalline thin film growth model based on describing grains in terms of faceted surfaces and grain boundaries. The profiles of the nuclei are described by crystallographically appropriate facets, with the position and orientation of the nuclei semi-randomly selected. Growth rates are determined from deposition fluxes and a new one-dimensional kinetic lattice Monte Carlo (KLMC) method developed to treat surface diffusion effects. The one-dimensional KLMC was verified by comparison with a full three-dimensional KLMC code. A continuum facet model has been developed to simulate grain growth and interaction. Users can visualize the nucleation and growth process of the film and obtain the final film structure, including the topography as well as the texture. The codes are written in Visual Basic and can be run in a Windows 98/2000/NT environment. An integrated graphical user interface has been implemented to facilitate use. The β version of FACET can be freely downloaded from our web site: http://ceaspub.eas.asu.edu/cms/.

Reduced Order Modeling and Control of Thin Film Growth in an HPCVD Reactor

This paper describes the development of a reduced order model-based feedback control methodology for regulation of the growth of thin films in a high-pressure chemical vapor deposition (HPCVD) reactor. Precise control of the film thickness and composition is highly desirable, making real-time control of the deposition process very important. The source vapor species transport is modeled by the standard gas dynamics partial differential equations, with species decomposition reactions, reduced down to a small number of ordinary differential equations through use of the proper orthogonal decomposition technique. This system is coupled with a reduced order model of the surface reactions involved in the source vapor decomposition and film growth on the substrate. Also modeled is the real-time observation technique used to obtain a partial measurement of the deposition process. The utilization of reduced order models greatly simplifies the mathematical formulation of the physical process so that it can be solved quickly enough to be used for real-time model-based feedback control. This control problem is fairly complicated, however, because the surface reactions render the model nonlinear. To deal with this we use a nonlinear feedback control method based on the state-dependent Riccati equation (SDRE). A second SDRE is contained in a state estimator which uses the nonlinear partial observations of the growth process to obtain an estimated state on which to base the feedback control. These nonlinear control techniques are implemented on the HPCVD model, and the results analyzed as to the effectiveness of the reduced order model and nonlinear control at tracking the desired film growth profile.

Thin film microstructure and thermal transport simulation using 3D-films

Simulations of the flow of heat through porous thin films by the three-dimensional microstructural thin film simulation framework 3D-Films are discussed in this paper. For each simulation, the film structures are generated by the thin film growth model 3D-Films and then used to generate a finite difference based thermal model by the program 3D-Films/Thermal. This program creates a block based data structure using a 3D quadtree mesh and subsequently solves for the steady state heat flow through the film structure. In this paper the film growth and thermal models are used to analyze and suggest optimization of porous thermal barrier coatings produced by glancing angle deposition techniques. The paper also deals with the determination of the accuracy and efficiency of the thermal model. Studies on the effect of reducing the resolution of the simulated film for less memory intensive thermal simulations are presented, indicating that a reduction of the resolution by a factor of 3 and the number of solution variables by as much as a factor of 27 is feasible. The simulations Of ZrO 2 thermal barriers are compared to experimental results with a relatively close match being obtained. Finally, the simulator is used to analyze the effectiveness of a number of potential thermal barrier structures produced by glancing angle deposition techniques. These results suggest that the most effective thermal barrier film microstructures will be porous films consisting of slanted posts or large pitch helices.

One-dimensional analytical model for oxide thin film growth on Ti metal layers during laser heating in air

This paper presents the theoretical and experimental results for oxide thin film growth on titanium films previously deposited over glass substrate. Ti films of thickness 0.1μm were heated by Nd:YAG laser pulses in air. The oxide tracks were created by moving the samples with a constant speed of 2mm/s, under the laser action. The micro-topographic analysis of the tracks was performed by a microprofiler. The results taken along a straight line perpendicular to the track axis revealed a Gaussian profile that closely matches the laser’s spatial mode profile, indicating the effectiveness of the surface temperature gradient on the film’s growth process. The sample’s micro-Raman spectra showed two strong bands at 447 and 612cm−1 associated with the TiO2 structure. This is a strong indication that thermo-oxidation reactions took place at the Ti film surface that reached an estimated temperature of 1160K just due to the action of the first pulse. The results obtained from the numerical integration of the analytical equation which describes the oxidation rate (Wagner equation) are in agreement with the experimental data for film thickness in the high laser intensity region. This shows the partial accuracy of the one-dimensional model adopted for describing the film growth rate.

CISCuT absorber layers — the present model of thin film growth

Experimental results and numeric calculations that have been contributed to a better understanding of how layer growth proceeds in the CISCuT process are given in this work. The heat transfer resistance between the heater and copper tape has been found to mainly determine the temperature of the copper tape during the CISCuT process. From this, more insight into the strongly temperature-dependent precursor formation has been gathered. The initial layer growth model of CISCuT absorber layer growth will be described and the partial solubility of CuInS 2 in CuIn-melts has been taken into account. Tape-like CISCuT-based solar cells, with a best efficiency of 5.4% achieved so far on an area of 4 cm 2, are presented.

Modeling of metal thin film growth: Linking angstrom-scale molecular dynamics results to micron-scale film topographies

A general method for modeling ionized physical vapor deposition is presented. As an example, the method is applied to growth of an aluminum film in the presence of an ionized argon flux. Molecular dynamics techniques are used to examine the surface adsorption, reflection, and sputter reactions taking place during ionized physical vapor deposition. We predict their relative probabilities and discuss their dependence on energy and incident angle. Subsequently, we combine the information obtained from molecular dynamics with a line of sight transport model in a two-dimensional feature, incorporating all effects of reemission and resputtering. This provides a complete growth rate model that allows inclusion of energy- and angular-dependent reaction rates. Finally, a level-set approach is used to describe the morphology of the growing film. We thus arrive at a computationally highly efficient and accurate scheme to model the growth of thin films. We demonstrate the capabilities of the model predicting the major differences on Al film topographies between conventional and ionized sputter deposition techniques studying thin film growth under ionized physical vapor deposition conditions with different Ar fluxes.

Nucleation and growth of clusters on heterogeneous surfaces

A thin film growth model, with the consideration of the transient nucleation and surface heterogeneity, was proposed to simulate the nucleation and growth rates of clusters in the physical vapor deposition. The process is divided into a sequence of time stages in order to solve the time dependent problem. The concentration of the adatoms is obtained by solving the adatom diffusion equations around the stable clusters. The effect of surface heterogeneity was treated by a suitable distribution of contact angle of the condensate on the substrate. The results show that a surface with higher heterogeneity will lead to a higher nucleation rate and cluster density. However, the clusters grow slowly due to the serious depletion of adatom by the existing clusters and thus, the clusters distribute more narrowly and a fine-grain polycrystal structure is resulted. The importance of the transient stage was also compared between the cases of complete and incomplete condensation. The simulation result shows that the transient stage can not be neglected in the completed condensation.

Simulation of epitaxial growth under ion-beam sputtering

Models of thin-film growth under ion-beam sputtering in a vacuum are reviewed. The direct molecular-dynamics simulation of atom flux deposition, as well as the simulation of the single-atom dynamics on the Ag/Cu(001) surface, are discussed. The component mixing effect at the film-substrate interface due to impact substitution is shown to take place for incident ion energies between 3 and 10 eV. Exchange processes are statistically studied, and their probabilities are estimated. Atomic mechanisms and the energy characteristics of surface diffusion near atomic steps are considered for the Cu/Cu(001) system. The dynamics of surface diffusion is investigated by the method of molecular statics. With step roughness taken into account, the activation barriers lower. Texture formation in a growing film under ion bombardment is analyzed in terms of the kinetic Monte Carlo model. This model allows us to explain the enhanced evolution of the axial texture at low substrate temperatures by introducing a concept of competitive growth of variously oriented grains under ion bombardment