Continuous Cellular Automaton Research Articles

Etch and growth rates of GaN for surface orientations in the crystallographic zone: Step flow and terrace erosion/filling via the Continuous Cellular Automaton

Compared to other methods, we present the benefits of the Continuous Cellular Automaton (CCA) to describe in a simple and flexible manner anisotropic wet chemical etching of GaN as a combination of step flow and terrace erosion. In fact, the simplicity of the approach enables accounting for epitaxial growth of GaN based on the equivalent perspective of step flow and terrace filling (or build-up). A key ingredient is the direct removal/deposition of GaN groups of atoms, which directly enables describing etching/growth via the removal/deposition of a small number of different crystallographic cell types, each with a different removal/deposition rate. We focus on the derivation of mathematical expressions for the etch rates of the surface orientations contained in the <0001> crystallographic zone as a function of the removal rates of the various cell types, then describing their use also for growth. A least squares approach is used to determine optimal values for the removal/deposition rates of the various cell types. For the first time, Focused Ion Beam (FIB) etching is used to manufacture a well-known, vertically micromachined wagon-wheel structure for subsequent use in wet etching, thus enabling a comparison between the CCA-derived etch rates and the experimental values. For growth, the comparison is performed against data from the literature. The combination of average step flow and average terrace erosion/filling, as inherently contained in the CCA method, explains well the anisotropy of both etching and growth of GaN in the <0001> crystallographic zone.

Characterization and simulation of sputtering etched profile by focused gallium ion beam on GaN substrate

With the continuous development of manufacturing technology of micro/nano devices based on GaN, focused gallium ion beam has become a momentous method for machining complex GaN structures. Therefore, this study focuses on the characterization and simulation of sputtering etched profile by Ga FIB on GaN substrate. First, the profile characterization experiments are carried out to analyze the influence of ion dose, dwelling time, and scan area, and the results reflect that the sputtering etched regularity on GaN substrate differs from that on Si substrate, especially in redeposition effect, milling rate, and droplet phenomenon. In addition, the forming mechanism of droplet phenomenon is discussed, and the milling experiments are conducted to research the droplets avoidance method, which shows that the appearance of droplets is impacted by ion dose and dwelling time. Finally, the sputtering yield of GaN under focused gallium ion beam is measured and sorted, and the simulation program is developed based on continuous cellular automaton (CCA) model. This work can provide a guidance for the further application of GaN-based materials.

Continuous cellular automaton simulation of focused ion beam-induced deposition for nano structures

With the assistance of a gas injection system, focused ion beam-induced deposition (FIBID) can be realized to fabricate complicated three-dimensional structures in nanoscale. The growth rate of the deposited structure depends on the flux of precursor gas and incident ions. In this study, a Continuous Cellular Automaton (CCA) model is put forward to simulate the FIBID process, with four rules established to initiate and dominate the interface evolution, which can calculate the deposited structure profile under different fabrication conditions. In this model, the sputtering and diffusion effects are taken into account to establish a more accurate relationship between the deposition rate and precursor gas flux based on the continuous model, and the influences imposed by precursor gas and incident ions are clarified. In order to verify the precursor gas distribution and the growth mechanism of the FIBID, experiments are carried out with different local precursor gas flux on the substrate surface with a 7 pA and 30 keV Ga ions, using C9H16Pt as the precursor gas. The experimental and simulation results indicate that the CCA model features a high accuracy in predicting the deposited structure profile, and prove that the proposed model can be utilized in fabrication parameters optimization.

Modeling of fungal mycelium growth by fourth-class continuous stochastic cellular automaton with continuously defined growth conditions

The aim of this work was to simulate the growth and spatial structure of the fungal mycelium using a cellular automaton based on the synthesis of various model approaches. The spatial structure of the mycelium is described in the structural submodel of the cellular automaton, which determines the growth rate in the direction of larger resource amount and the number of branches of the mycelium per area unit. The amount of available substrate determines the probability of unidirectional apical growth. Another, biochemical part of the model allows us to describe the rate of transport of resources into the cell, their transport within the mycelium, and also their excretion, and is intended to describe the vertical and horizontal migration in the soil of two nutrients. The proposed model makes it possible to quantitatively describe such a feature of fungal colony growth as more active absorption of resources by external cells, compared to central ones due to separation of transport resources into active and passive resources. The active transport was described using the Michaelis-Menten kinetics. We were able to simulate the stockpiling of surplus resources and their redistribution over the mycelium after the exhaustion of reserves in the external environment, and also to simulate typical growth patterns of mycelial colonies that were observed in experiments published in the literature.

Fluctuations During Anisotropic Etching: Local Recalibration and Application to Si{110}

Based on the previous studies of the etch rate of crystalline silicon in alkaline etchants, we stress the fact that the etch rates can noticeably differ between different research groups. This affects the prediction of the etch front, since the simulators typically use experimental data gathered in one laboratory. Considering the most efficient and accurate simulator currently available for the description of anisotropic etching, namely the continuous cellular automaton (CCA), any such variation in the experimental etch rates requires a time-consuming calibration procedure in order to adjust the atomistic removal rates internally used by the method. Since normally it is possible to directly compare the experimental and simulated etch fronts—without actual knowledge of the variations in the macroscopic etch rates—here we propose a local recalibration procedure by which the atomistic removal rates of a few atoms are modified, thus recovering most features of the experimental fronts in the simulated counterparts. As an application, we evaluate for the first time the ability of the CCA to describe wet etching on Si{110}, focusing on a large collection of wet etched structures including cavities and mesas at different stages of the etching process, obtaining excellent agreement between experiment and simulation. [2015-0254]

Evidence for faster etching at the mask-substrate interface: atomistic simulation of complex cavities at the micron-/submicron-scale by the continuous cellular automaton

We combine experiments and simulations to study the acceleration of anisotropic etching of crystalline silicon at the mask-substrate interface, as a function of the coordination number of the substrate atoms located at the junction between obtuse-angled {1 1 1} facets and the mask layer. Atomistic simulations based on the use of the continuous cellular automaton (CCA) conclude that the interface atoms react faster with the etchant, thus initiating a step flow process that results in increased etch rates for the obtuse facets. By generating a wide range of complex cavities on high-index silicon wafers with a single-side, single-step etching, the comparison of the experimental and simulated results strongly indicates that the CCA method is suitable for accurately describing not only the development of micron-scaled structures but also, for the first time, the formation of submicron shapes. The study also describes the acceleration of obtuse facets formed through double-side etching, obtaining results in good agreement with previous experiments.

Particle swarm optimization-based continuous cellular automaton for the simulation of deep reactive ion etching

We combine the particle swarm optimization (PSO) method and the continuous cellular automaton (CCA) in order to simulate deep reactive ion etching (DRIE), also known as the Bosch process. By considering a generic growth/etch process, the proposed PSO–CCA method provides a general, integrated procedure to optimize the parameter values of any given theoretical model conceived to describe the corresponding experiments, which are simulated by the CCA method. To stress the flexibility of the PSO–CCA method, two different theoretical models of the DRIE process are used, namely, the ballistic transport and reaction (BTR) model, and the reactant concentration (RC) model. DRIE experiments are designed and conducted to compare the simulation results with the experiments on different machines and process conditions. Previously reported experimental data are also considered to further test the flexibility of the proposed method. The agreement between the simulations and experiments strongly indicates that the PSO–CCA method can be used to adjust the theoretical parameters by using a limited amount of experimental data. The proposed method has the potential to be applied on the modeling and optimization of other growth/etch processes.

Anisotropic etching on Si{1 1 0}: experiment and simulation for the formation of microstructures with convex corners

We combine experiment, theory and simulation to design and fabricate 3D structures with protected edges and corners on Si{1 1 0} using anisotropic wet chemical etching in 25 wt% tetramethylammonium hydroxide (TMAH) at 71 °C. In order to protect the convex corners formed by <1 1 2 > and <1 1 0 > directions, two methods are considered, namely, corner compensation and two-step etching. The mask design methodology for corner compensation is explained for various microstructures whose edges are aligned along different directions. The detailed geometry of each compensation pattern is shown to depend on the desired etch depth. The two-step wet etching process is explored in order to realize improved sharp convex corners. Using the same etchant concentration and temperature, the second etching is carried out after mask inversion from silicon nitride (Si3N4) to silicon dioxide (SiO2), obtained by local oxidation of silicon (LOCOS) followed by nitride etching. Based on the use of the continuous cellular automaton (CCA), the simulation results for both corner undercutting and two-step etching show that the CCA is suitable for the analysis and prediction of anisotropic etching on Si{1 1 0} wafers.

Level set implementation for the simulation of anisotropic etching: application to complex MEMS micromachining

The use of atomistic methods, such as the continuous cellular automaton (CCA), is currently regarded as an accurate and efficient approach for the simulation of anisotropic etching in the development of micro-electro-mechanical systems (MEMS). However, whenever the targeted etching condition is modified (e.g. by changing the substrate material, etchant type, concentration and/or temperature) this approach requires performing a time-consuming recalibration of the full set of internal atomistic rates defined within the method. Based on the level set (LS) approach as an alternative and using the experimental data directly as input, we present a fully operational simulator that exhibits similar accuracy to the latest CCA models. The proposed simulator is tested by describing a wide range of silicon and quartz MEMS structures obtained in different etchants through complex processes, including double-sided etching as well as different mask patterns during different etching steps and/or simultaneous masking materials on different regions of the substrate. The results demonstrate that the LS method is able to simulate anisotropic etching for complex MEMS processes with similar computational times and accuracy as the atomistic models.

Implementation and evaluation of the Level Set method: Towards efficient and accurate simulation of wet etching for microengineering applications

The use of atomistic methods, such as the Continuous Cellular Automaton (CCA), is currently regarded as a computationally efficient and experimentally accurate approach for the simulation of anisotropic etching of various substrates in the manufacture of Micro-electro-mechanical Systems (MEMS). However, when the features of the chemical process are modified, a time-consuming calibration process needs to be used to transform the new macroscopic etch rates into a corresponding set of atomistic rates. Furthermore, changing the substrate requires a labor-intensive effort to reclassify most atomistic neighborhoods. In this context, the Level Set (LS) method provides an alternative approach where the macroscopic forces affecting the front evolution are directly applied at the discrete level, thus avoiding the need for reclassification and/or calibration. Correspondingly, we present a fully-operational Sparse Field Method (SFM) implementation of the LS approach, discussing in detail the algorithm and providing a thorough characterization of the computational cost and simulation accuracy, including a comparison to the performance by the most recent CCA model. We conclude that the SFM implementation achieves similar accuracy as the CCA method with less fluctuations in the etch front and requiring roughly 4 times less memory. Although SFM can be up to 2 times slower than CCA for the simulation of anisotropic etchants, it can also be up to 10 times faster than CCA for isotropic etchants. In addition, we present a parallel, GPU-based implementation (gSFM) and compare it to an optimized, multicore CPU version (cSFM), demonstrating that the SFM algorithm can be successfully parallelized and the simulation times consequently reduced, while keeping the accuracy of the simulations. Although modern multicore CPUs provide an acceptable option, the massively parallel architecture of modern GPUs is more suitable, as reflected by computational times for gSFM up to 7.4 times faster than for cSFM.

Step flow model in continuous cellular automata method for simulation of anisotropic etching of silicon

A continuous cellular automaton (CCA) is presented for the sim- ulation of anisotropic wet chemical etching in the fabrication of microstruc- tures. Based on the step-flow model, the surface atoms are divided into categories according to the atom configurations on different crystal planes. An analytical solution for the dependence of the etch rate on orientation is derived, and the CCA approach makes a direct conversion of experimental macroscopic rates into calibrated microscopic parameters for realistic and reliable simulations. The presented model has been extended to a simu- lation system based on a CCA method. Linear search and variable time stepping are used to simulate a silicon wafer in various etching condition and mask shapes. The simulation results agree well with the experiments. This improved CCA makes possible for the realization of accurate simu- lations of anisotropic etching in engineering applications. © 2013 Society of

Evolutionary continuous cellular automaton for the simulation of wet etching of quartz

Anisotropic wet chemical etching of quartz is a bulk micromachining process for the fabrication of micro-electro-mechanical systems (MEMS), such as resonators and temperature sensors. Despite the success of the continuous cellular automaton for the simulation of wet etching of silicon, the simulation of the same process for quartz has received little attention—especially from an atomistic perspective—resulting in a lack of accurate modeling tools. This paper analyzes the crystallographic structure of the main surface orientations of quartz and proposes a novel classification of the surface atoms as well as an evolutionary algorithm to determine suitable values for the corresponding atomistic removal rates. Not only does the presented evolutionary continuous cellular automaton reproduce the correct macroscopic etch rate distribution for quartz hemispheres, but it is also capable of performing fast and accurate 3D simulations of MEMS structures. This is shown by several comparisons between simulated and experimental results and, in particular, by a detailed, quantitative comparison for an extensive collection of trench profiles.

Simulating anisotropic etching of silicon in any etchant: evolutionary algorithm for the calibration of the continuous cellular automaton

An evolutionary algorithm is presented for the automated calibration of the continuous cellular automaton for the simulation of isotropic and anisotropic wet chemical etching of silicon in as many as 31 widely different and technologically relevant etchants, including KOH, KOH+IPA, TMAH and TMAH+Triton, in various concentrations and temperatures. Based on state-of-the-art evolutionary operators, we implement a robust algorithm for the simultaneous optimization of roughly 150 microscopic removal rates based on the minimization of a cost function with four quantitative error measures, including (i) the error between simulated and experimental macroscopic etch rates for numerous surface orientations all over the unit sphere, (ii) the error due to underetching asymmetries and floor corrugation features observed in simulated silicon samples masked using a circular pattern, (iii) the error associated with departures from a step-flow-based hierarchy in the values of the microscopic removal rates, and (iv) the error associated with deviations from a step-flow-based clustering of the microscopic removal rates. For the first time, we present the calibration and successful simulation of two technologically relevant CMOS compatible etchants, namely TMAH and, especially, TMAH+Triton, providing several comparisons between simulated and experimental MEMS structures based on multi-step etching in these etchants.

Faster and exact implementation of the continuous cellular automaton for anisotropic etching simulations

The current success of the continuous cellular automata for the simulation of anisotropic wet chemical etching of silicon in microengineering applications is based on a relatively fast, approximate, constant time stepping implementation (CTS), whose accuracy against the exact algorithm—a computationally slow, variable time stepping implementation (VTS)—has not been previously analyzed in detail. In this study we show that the CTS implementation can generate moderately wrong etch rates and overall etching fronts, thus justifying the presentation of a novel, exact reformulation of the VTS implementation based on a new state variable, referred to as the predicted removal time (PRT), and the use of a self-balanced binary search tree that enables storage and efficient access to the PRT values in each time step in order to quickly remove the corresponding surface atom/s. The proposed PRT method reduces the simulation cost of the exact implementation from to without introducing any model simplifications. This enables more precise simulations (only limited by numerical precision errors) with affordable computational times that are similar to the less precise CTS implementation and even faster for low reactivity systems.

Octree-based, GPU implementation of a continuous cellular automaton for the simulation of complex, evolving surfaces

Presently, dynamic surface-based models are required to contain increasingly larger numbers of points and to propagate them over longer time periods. For large numbers of surface points, the octree data structure can be used as a balance between low memory occupation and relatively rapid access to the stored data. For evolution rules that depend on neighborhood states, extended simulation periods can be obtained by using simplified atomistic propagation models, such as the Cellular Automata (CA). This method, however, has an intrinsic parallel updating nature and the corresponding simulations are highly inefficient when performed on classical Central Processing Units (CPUs), which are designed for the sequential execution of tasks. In this paper, a series of guidelines is presented for the efficient adaptation of octree-based, CA simulations of complex, evolving surfaces into massively parallel computing hardware. A Graphics Processing Unit (GPU) is used as a cost-efficient example of the parallel architectures. For the actual simulations, we consider the surface propagation during anisotropic wet chemical etching of silicon as a computationally challenging process with a wide-spread use in microengineering applications. A continuous CA model that is intrinsically parallel in nature is used for the time evolution. Our study strongly indicates that parallel computations of dynamically evolving surfaces simulated using CA methods are significantly benefited by the incorporation of octrees as support data structures, substantially decreasing the overall computational time and memory usage.

Orientation-dependent surface morphology of crystalline silicon during anisotropic etching using a continuous cellular automaton

The formation of morphological protrusions on the surface during anisotropic wet etching is addressed computationally by modifying a step-flow-based continuous cellular automaton (CCA) in order to incorporate the stochastic adsorption and desorption of impurity particles on the surface. The CCA method is characterized by the possibility of calibrating the site-specific atomistic removal rates of the step-flow model in order to describe the macroscopic orientation dependence of the etch rate obtained in an experiment. Interestingly, the combination of the deposition model with previously calibrated site-specific atomistic removal rates leads to a realistic description of the experimental surface morphologies for a wide range of crystalline orientations. This is shown by considering the wet etching of crystalline silicon in two very different etchants, namely pure KOH and KOH with isopropanol (IPA). The modified CCA makes possible the realization of accurate and realistic simulations of anisotropic etching in realistic engineering applications.

Discrete and continuous cellular automata for the simulation of propagating surfaces

We present the geometrical and kinetic features of the Continuous Cellular Automaton (CCA) as an alternative method for the propagation of fronts using a scalar field instead of traditional velocity fields governing the direction and amount of advancement for the interface points. By comparing the features of the discrete and continuous cellular automata we stress the flexibility of the CCA approach, which makes possible the direct conversion of experimental macroscopic rates into calibrated microscopic parameters for realistic and reliable simulations. Using the example of anisotropic etching as a complex application we show that the method also enables the determination of atomistic activation energies for the microscopic process rates, thus simplifying additional calibrations at different temperatures. The experimental etch rate distribution for anisotropically etched spherical samples is correctly described and the orientation-dependence of the etch rate is properly reproduced for two typical, significantly different etchants. In addition, detailed comparisons between simulation and experiment for the time evolution in two different etchants and eight different mask patterns are provided.

Analytical Solution of the Continuous Cellular Automaton for Anisotropic Etching

The fabrication of micro- and nanoelectromechanical systems (MEMS/NEMS) is based on a wide variety of growth and etching technologies sequentially applied throughout process flows which may involve a dozen or more steps, their realistic simulation having become an essential part of the overall design. By focusing in the simulation of anisotropic etching as a complex example of microfabrication, in this paper, we show how to solve analytically the time evolution of the continuous cellular automaton method, thus providing a particularly suitable choice for the realization of realistic simulations for MEMS and NEMS applications. This paper presents a complete theoretical derivation of the analytical solution based on geometrical and kinetic aspects of step flow on any surface, including a new classification of the surface sites based on a mean-field treatment of the propagation of the steps. The results of the corresponding simulations are in good agreement with the experiments. The study can be seen as an example of a general procedure that is applicable to other interface propagation problems.

VisualTAPAS: an example of density functional theory assisted understanding andsimulation of anisotropic etching (abstract only)

VisualTAPAS is a self-contained, user-friendly, graphical user interface based simulator of wetetching and deep reactive ion etching with multi-masking capabilities, built upon an octreerepresentation of the silicon substrate (www.fyslab.hut.fi/∼mag/VisualTAPAS/Home.html).The program allows the use of a wide range of kinetic Monte Carlo (KMC) and cellularautomata time-evolution algorithms, including a fast octree search algorithm for the KMCsimulations. ‘VisualTAPAS’ stands for ‘visual three-dimensional anisotropic processing atall scales’. The use of the term ‘visual’ stresses the interactive visual capabilities of theprogram.Here, a brief history of the evolution of VisualTAPAS as a research tool willbe presented: from the initial efforts, explaining the anisotropy of wet etchingas a result of steric hindrance using a combination of density functional theory(DFT) and KMC simulations, to the most recent implementation, focusing on thepropagation of the etch front for engineering applications by making use of theanalytical solution of the continuous cellular automaton (CCA) method; and inbetween, a recent example of DFT assisted understanding of the effects of metalimpurities on the surface morphology of the etched surfaces will be presented.We try to bridge together three complementary simulation tools, namely, DFT,KMC and CCA methods. Our experience with the use of the three methods forthe simulation of anisotropic etching shows that DFT is very useful and, manytimes, an unavoidable approach. Similarly, the KMC approach is the method ofchoice for understanding the large variety of etched surface morphologies while theCCA is an outstanding tool for the simulation of the process at an engineeringlevel.

A cellular automaton-based simulator for silicon anisotropic etching processes considering high index planes

This paper presents a novel 3D continuous cellular automaton (CA) model with high index planes such as (2 1 1), (3 1 1), (3 3 1) and (4 1 1) efficiently incorporated. A dynamic algorithm has also been developed to speed up the simulation process and reduce the memory usage. A 3D silicon anisotropic etching simulator, SEAES, has been implemented based on the 3D continuous CA model and the dynamic algorithm. The simulation results by SEAES have been found to be in good agreement with the experimental results, and the arbitrarily complex mask shapes and the merging of 3D etching profiles can be both efficiently and accurately handled. This is identified to be useful for the research of anisotropic etching technologies and the development of micro-electro-mechanical systems (MEMS) design.