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--> 2 Tool Scale and Feature Scale Models Introduction Chapter 1, "Industrial Perspectives," outlines the most promising ways to apply plasma modeling and simulation to attack problems that most directly affect plasma tool suppliers and chip manufacturers. The purpose of this chapter is to summarize the current state of plasma modeling and simulation and to identify the most pressing issues that could limit the pace of progress in solving industrially relevant problems. The major industrial problems on which modeling may have an impact involve decreasing the time and cost of developing new plasma tools (shortening the time to market), improving the efficiency of optimizing tool performance to meet changing process objectives, and helping to implement real-time process control into operating tools. Each of these goals implies a somewhat different type of plasma modeling approach. In addition, some plasma models can be developed relatively quickly, while others will take more time. They all share the need to be systematically tested against experimental measurements (validated), and they need to be in a form that can be used conveniently by engineers involved in design, optimization, and control of plasma tools in industry. However, the greatest need for any plasma model is in the area of an improved database for the multitude of elementary processes that collectively determine the physical and chemical dynamics of the system. Tool Scale Models Over the past decade, modeling and simulation of plasma reactors at the tool scale, meaning the scale of the entire reactor (including the entire wafer), have attracted much attention.1 This is due in large part to the recognition that the problems in the effective application of plasma processes, outlined in Chapter 1, might be fruitfully attacked using modeling and simulation. However, glow discharge plasma simulation is a challenging task, primarily because of (1) the nonequilibrium nature of the plasma, (2) the disparate time scales (<< 1 ns to several seconds), and (3) the complexity of the gas phase chemistry and especially the surface chemistry. In an electrical glow discharge, external electromagnetic energy is applied to the system, heating electrons and ions. Since the gases rarely are more than a few percent ionized (at most), electrons experience collisions mostly with much heavier neutral species. The very small mass ratio of the collision partners (electrons/neutral molecules) results in a low efficiency of energy transfer in elastic collisions between electrons and neutral species. Electrons continue to gain energy from the electric fields until their mean energy is sufficiently high to excite inelastic collisions with neutrals. Electron mean energies are typically several electron volts (eV), and the resulting ionization of neutral molecules is the primary sustaining mechanism in a glow discharge. (Note that 1 eV is approximately equivalent in temperature to 11,000 K or 20,000 °F). Because of the ambipolar electric fields in the plasma, positive ions are usually accelerated to the boundaries of the plasma and then accelerated further across the sheath potential to impact the surface. Electrons typically diffuse against the confining ambipolar fields to the walls and recombine there with positive ions. In electronegative gases, negative ions form through electron attachment, and are usually lost through gas phase processes such as positive ion-negative ion recombination or electron detachment. Electrons with several eV mean energy can not only ionize molecules, but also can easily dissociate most molecules into fragments. These fragments are the main chemical precursors for both film deposition and etching. Chemisorbed neutral molecular fragments at surfaces, especially when impacted by energetic positive ions from the discharge, are responsible for the chemical reactions that lead to etching. These processes are taking place in chambers with (often) fully
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--> three-dimensional spatial variation, and they are in general not at steady state. Modeling this system therefore requires a treatment of Kinetics and transport of charged species (electrons, positive and negative ions); Kinetics and transport of neutral species in the gas phase; and Interaction of charged and neutral species with surfaces. The nonequilibrium nature of the electrons and ions means that these species not only do not have the same average thermal energies, but in addition do not generally share the same form for the distribution of their velocities. Neither electrons nor ions in general follow the Maxwell-Boltzmann form, characteristic of species at local thermal equilibrium. This means that for a rigorous treatment of electron and ion transport some kind of kinetic scheme is necessary: solution of the Boltzmann equation, for example, or a particle simulation such as a Monte Carlo scheme. Also, neutral transport is often complicated by the fact that the collisional mean free path for neutrals (for some low-pressure etching equipment) is on the same order as the dimensions of the reaction chamber. This brings into question the use of continuum or fluid descriptions of neutral transport. An additional complication comes from the fact that neutral species may be in excited vibrational or electronic states. In some cases, then, a rigorous treatment of neutral transport also requires a kinetic approach. In spite of this, schemes that assume some form for distribution functions (i.e. fluid models) have proven useful for conditions under which their limitations are well understood. An increasingly popular approach is to combine fluid and kinetic schemes into a hybrid model, in which, ideally, the strengths of both approaches can be combined, while minimizing their weaknesses. In the last 5 years, there have been impressive developments in plasma modeling algorithms. These algorithms use fluid, kinetic, and hybrid methods to treat plasma and neutral transport and kinetics. Electromagnetic modules have been coupled successfully to the transport and kinetics codes and have been applied to systems of industrial interest. The availability of engineering workstations with high performance at modest cost has made these developments possible. In addition, optimizing compilers, convenient "debuggers," "windowing" capabilities, and excellent graphics are widely available and increase the productivity of modelers. An example of a two-dimensional, axisymmetric calculation for an inductively coupled (radio-frequency) plasma discharge is shown in Figure 2.1. The major missing ingredient in further exploitation of large-scale plasma modeling and simulation is the availability of a physical and chemical database for a large and diverse set of species present in the discharge, interacting through a variety of collisional processes in the gas phase and at surfaces. In Figure 2.1 An example of a tool-scale simulation of an inductively coupled plasma reactor and various discharge spatial profiles: (top) contour plot of two important discharge characteristics, electron density and plasma potential; (bottom) plot of electron temperature and the source terms for creation of electrons. (Courtesy of Mark Kushner, University of Illinois at Urbana-Champaign.)
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--> addition, there must be extensive sets of comparisons between experimental measurements and model predictions in order to validate the models that are proposed. An important feature alluded to in Chapter 1 is the role of the mid-level, between the tool scale (tens to hundreds of centimeters) and the microfeature level (~ 1 µm). This mid-level involves effects near the edge of the wafer, the effects of pattern area loading on the wafer, and similar phenomena. The length scale is typically less than 1 cm (on the order of the size of a single chip) but greater than 10 µm. This is an awkward intermediate scale, since resolving tool scale simulations on a scale of ~ 0.1 cm or less becomes very expensive computationally. Further work is needed to develop strategies to couple models of various length scales. Capabilities Needed for Tool Scale Models Tool scale models need to be capable of predicting the behavior and uniformity of the plasma sufficiently well that the following effects can be modeled: Alternative chemistries, Pressure, Power, Gas flow and composition, Changing geometrical configuration (tool shape, wafer edges, clamps), Chamber wall effects: seasoning and cleaning, and Applied magnetic field. In order that the models be useful, the minimum requirement is that the models capture the qualitative trends. In addition, they must be set up in such a way that they can be coupled conveniently to the feature scale models, discussed below. Barriers to Using Tool Scale Models Some of the barriers to the use of current-generation tool scale models are listed below:2 There is a lack of mature models that include physical accuracy, computational accuracy, and robustness. Physical accuracy refers to the accuracy of the underlying equations and assumptions in the model. Computational accuracy refers to possible problems with actually solving the equations in the model. For example, finite difference or finite element methods' solutions to discretized equations do not always correspond to the solution of the original differential equation. Robustness refers to loss of convergence, "touchy" solutions, and similar problems. Robustness is a nontrivial issue to overcome when highly coupled, nonlinear equations are being solved, as in the case of glow discharge plasmas. There is a lack of integration among different parts of the simulation codes. This means that models of the tool geometry, the grid generators, the visualization tools, the graphical interfaces, and other design tools are not linked together in convenient ways. There is a lack of commercial software suppliers. Commercial software suppliers provide documentation, consistent support, model updates, and continuous development. Generally, these features are lacking in university and national laboratory codes (although exceptions exist). There is a lack of a database. All simulations require information about various processes occurring within the system being modeled. In the case of plasma models, this requirement includes information regarding electron-impact phenomena, ion-neutral collisions (e.g. ion-molecule reactions), neutral-neutral reactions, and the various processes occurring at surfaces bounding the plasma.
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--> Feature Scale Models It is clear that the evaluation of etching or deposition reactors is eventually governed by the feature profiles they develop, in addition to questions of uniformity that can be addressed by tool scale simulations. All choices of equipment type, reactor conditions, and chemistry are judged by etch profiles and/or deposition conformality. Thus, the ideal simulation, from the process engineer's point of view, would produce this profile as an output. The inputs would include the feature geometry, as well as the reactor operating conditions. It should be recognized that it is expensive and often inconvenient to measure feature profiles directly with cross sectional views through scanning electron microscopy (SEM) images. This is especially true when many different runs must be made to determine the effects of tool operating conditions on profiles. It is therefore an important goal to link tool scale models with feature scale models to satisfy the needs of the process engineer. At the current time, it is not yet clear what the weakest link in this chain is. It is possible that the output from the tool scale model is not sufficiently accurate or detailed to provide the feature scale model with input information that will allow accurate predictions of feature profiles. The feature scale models may not be able to use all of the information concerning species composition and velocities coming from the tool scale model because of a lack of a good model for surface processes. One possible scheme to couple reactor or tool scale input into the profile simulator is shown in Figure 2.2. The reactor scale information is sent to a model of the sheath region. The sheath region is the thin (˜ 102 µm to 5 mm) boundary layer just above the surface through which ions are accelerated by relatively strong electric fields. Depending on the conditions (frequency and magnitude of the sheath voltage, sheath thickness, gas pressure, and so on), the ion composition and velocity distribution will be affected. A model of these effects is therefore required since the feature evolution will be affected by these parameters. Given the transport mechanisms within the feature (e.g. ballistic transport and/or surface diffusion) and a model of the surface reaction kinetics, the feature profile shape can then be predicted. Usually, however, the model needs to be calibrated with specific data under conditions similar to the ones that are of interest in the particular simulation. Obviously, one would hope that this calibration step would eventually be minimized or avoided altogether. In the process shown in Figure 2.2, there are several external inputs to three of the steps. The transport mechanism requires information from a test structure. These are specially designed features such that, when processed in the plasma reactor, the changes in their shape can be conveniently translated into information regarding transport and surface kinetics. In effect, these test structures are used to calibrate the models for other feature shapes. In addition to information from test structure experiments, data for surface reaction kinetics can be obtained from surface science experiments, and from atomistic simulations such as molecular dynamics.3 An example of a feature scale simulation is illustrated in Figure 2.3. In this example, the goal is to use a combination of plasma-enhanced chemical vapor deposition (PECVD) with tetraethoxysilane (TEOS) to deposit a silicon oxide film, and argon sputtering to control the film shape to avoid the formation of a void. The sequence begins with a step (Figure 2.3a) in which the PECVD film has been deposited nearly conformally in the metal trench. The second step (Figure 2.3b) is to sputter (with Ar+ in this case) part of the oxide film to open the trench. The third step (Figure 2.3c) is to deposit the oxide again, resulting in a film that does not have the undesirable void space Figure 2.2 Strategy for feature-scale modeling and characterization. (Courtesy of V. Singh, Intel Corporation.)
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--> that would have occurred if the sputtering step had not been used. The simulation both allows an understanding of the causes of the void formation and provides a methodology to optimize conditions in such a way as to avoid the problem in practice. General Assessment of Modeling State of the Art and Vision of Future Capability and Implied Needs In summary, the current state of modeling can be described as well established in some regards, but immature in others. For tool scale models, the treatments of neutral gas mass transfer, fluid flow, and heat transfer, while complicated by relatively low pressures and associated problems with transition-regime and sometimes free molecular flow, are well developed fields. Treatments of electric discharge physics, including electron and ion kinetics, and electromagnetic phenomena in gas discharges, are less well developed, but the general strategies for attacking these problems axe fairly well established. If one were interested in modeling rare-gas discharges, one might conclude that most of the major complications had been identified and solution strategies developed, although of course not all problems have been solved. Similarly for feature scale models, profile surface advancement algorithms have been well developed, especially for two-dimensional or axisymmetric features. Progress is being made to extend these techniques to three-dimensional features. Two major issues challenge current capabilities in modeling low-pressure gas discharge plasmas for semiconductor manufacturing. The first is chemical complexity. This issue arises from the use of complex mixtures of halogen-containing gases, as well as other gases that are relatively little studied. The chemical scheme includes reaction products from chamber walls, photoresist reaction products, etching products, and slowly changing wall reactivities as films form on walls. Including these effects increases the difficulty of modeling by orders of magnitude. The second issue has to do with length scale disparities. The tool or wafer scale is tens of centimeters, but the microfeatures that one must control are on the order of fractions of microns. It is very difficult to measure (while the wafer is in the plasma tool) the quantities that one is most interested in controlling, such as the feature linewidth, the sidewall angle, the composition of trace species on the surface, or the introduction of chemical or mechanical imperfections that lead to electrical device failure or reliability problems. There is, in addition the difficult problem of the mid-scale between the feature scale and the tool scale. Little progress has been made to resolve mid-scale model formulation or solution. The need to deal with the first problem, chemical complexity, is the major motivation for the present report, because the way chemical complexity manifests itself in modeling is through the need for parameters to characterize the way the various species in the plasma interact with each other and with walls. This is true for models at all length and time scales. Any vision of future capabilities revolves around the central question of how one represents the chemical species in the plasma. In the immediate future, perhaps the next 5 years, it is not likely that models of industrial plasma processes will be developed that will seriously attempt to include all chemical species present, including all kinetic information (i.e. velocity and internal energy distributions). Furthermore, the proper characterization of the state of surfaces exposed to plasmas is sufficiently far from being fully understood that models of surface processes will necessarily be approximate. Fortunately, it is generally not necessary to make models so comprehensive. In order to make the problem tractable, one must choose some subset of all chemical species present and some approximate treatment of the interactions these species have with bounding surfaces. These choices, along with choices concerning dimensionality, how much of the surrounding environment and materials to include in the model, and how one treats the kinetics of the charged species and electromagnetic phenomena, all constitute the ''model.'' It is axiomatic in modeling that what should be included in a model can properly be judged only by what one wishes to do with the model. There is a wide range of possibilities, from well-mixed reactors that seek to
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--> include a great deal of chemical complexity to three-dimensional, time-dependent models with minimal chemistry that aim to capture only physical phenomena like gas flow or heat transfer. Models that include chemistry make assumptions about which of the chemical species (several tens, probably, for common industrial chemistries) to include in the reaction mechanism. Each of these species is transported through the system via some selected set of transport mechanisms, but more importantly, is Figure 2.3 Optimized oxide deposition profile using sequential deposition and argon ion sputter etching. (a) Initial (approximately conformal) profile of PECVD oxide in a metal trench. (b) Subsequent sputtering and redeposition of the oxide film. (c) Final deposition of oxide to achieve the desired void-free film. (Courtesy of V. Singh, Intel Corporation.)
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--> created and destroyed through a set of chosen reactions. The tens of chemical species present interact through perhaps hundreds of elementary gas phase and surface reactions. Parameters characterizing the rates of creation and loss of the chemical species in the model must be measured, computed, or estimated. It is likely that interactions between many species are synergistic: no single species is responsible for selectivity, for example. Interactions will depend on temperature, competing physical and chemical processes, and probably other factors that are unknown at present. Species present in low concentration may play a key role in the overall chemical balances and cannot be ignored simply because they are not one of the major species. In addition, surface chemical processes can be affected by small concentrations of species in the gas phase, because there are relatively few surface-active sites in comparison to typical fluxes from the gas phase. An important point in modeling is that in many cases a reduced set of chemical species and reactions will serve as a good approximation to the complete set. The goal is then to identify what this reduced set should comprise and how to determine the kinetic parameters that govern their interactions. The most promising way to develop the set of chemical species, their interacting mechanisms, and the parameters in the rate expressions (the "database") is to propose a model, and then critically compare model predictions to measurements. Iteration on the original mechanism will then improve the chosen set. Sensitivity analyses will help identify, within a given model, which of the processes are the most important. In some cases, critical comparisons between model predictions and diagnostics can help to determine the database. Other modelers, starting with a different initial guess of species and mechanisms, may iterate to a different final set. It might be helpful to provide a specific example of how this interaction might work in practice. Let us consider the case of fluorocarbon plasma etching of dielectric films, for example silicon dioxide. This is the largest single etching application in the microelectronics industry, but it is poorly understood and difficult to control. It has proven necessary to use a chemistry that results in polymeric film deposition in order to maintain a high selectivity between silicon dioxide and silicon. The fluorocarbon molecules tend to reduce the etching rate of silicon because the carbon remains involatile in the absence of oxygen at the surface. For a SiO2 surface, this problem does not exist and the fluorocarbon molecules are volatilized along with the silicon and oxygen, resulting in etching. It is thought that this polymeric film forms and is etched by a complex combination of neutral and ionic species interacting at surfaces. This mechanism suggests that developing a model for the system will be challenging, since it is likely that many fluorocarbon radicals and ions participate in the plasma-assisted polymer etching and deposition kinetics. It is important to understand that relatively little progress has been made in developing and testing mechanistic models of plasma chemistry under typical industrial conditions. In a typical high-density plasma oxide etching tool, there will be, in addition to the exposed oxide to be etched, photoresist, various wall materials such as anodized aluminum, other ceramic-like materials, and the deposited polymeric film mentioned above. The inlet gases might include C2F6, O2, and Ar, for example. Consider first the neutral species that might be involved in this plasma: all of the inlet gases, and all combinations of their dissociation products (e.g. CxFy, COFx, CO, CO2, O, Ar, C, F); all products of reactions between these species and the materials to be etched (e.g. SiFx, SiOxFy); all products of reactions between these species and the photoresist (hydrocarbon-based, with other species added for lithographic or other purposes); and, finally, products resulting from reactions between the chemically active species in the plasma and the reactor wall materials and/or the deposited fluorocarbon film. In addition to these neutral species (several tens), there will be many and various positive and negative ions formed from neutral species. Collision cross sections between the neutral species and electrons must be obtained. Ion-molecule rate coefficients are necessary to predict ion composition. Some species will no doubt be in excited states (vibrational and/or electronic). Ultraviolet photons released in the plasma could conceivably play a role in both gas phase and surface chemistry. Each of the proposed species has no doubt many individual reactions, both in the gas phase and especially at surfaces. Surface chemical reactions are difficult to model in the presence of the plasma because of the complicating role of energetic ions in modifying surface
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--> composition and structure. Wall temperatures in the chamber often heat sufficiently to significantly affect reaction rates there. Even if we knew the 10 major chemical species in the plasma (in general we do not), it would be a major task to assemble the data necessary to model even the most important of the chemical reaction rate processes in the gas phase and at the surface. It seems premature at this stage to attempt to identify individual collisional processes (e.g., the cross section for electron impact dissociative excitation of CF2 into CF and excited F) as being especially important and worthy of attention above all the many other similar processes. In order to develop a database to model this system, it will be necessary initially to identify the key chemical species that participate in the main gas phase and surface reactions. One way to begin this process is to make spectroscopic measurements of as many of the radicals and stable species in a fluorocarbon plasma as possible. This might include sampling for a mass spectrometer, optical emission spectroscopy, laser-reduced fluorescence, ultraviolet absorption, and infrared (IR) absorption (with a Fourier transform infrared (FTIR) spectrometer or a diode laser) for the gas phase species. Surface species might be identified with reflectance-absorbance IR measurements, or with total internal reflection methods, or possibly some form of spectroscopic ellipsometry. Laser-induced desorption of surface species followed by one of the gas phase techniques can be a powerful surface diagnostic. Each of these measurement techniques could contribute information about plasma and surface composition, and changes in these compositions under varying conditions could give hints about possible mechanisms for interspecies conversions. Pulsing the plasma and observing the decay or growth of various species could provide further information about kinetics. Ideally, these measurements would be made in a plasma configuration that is easily modeled, and the identified species could be included in the model, along with electrons and the relevant positive and negative ions. Estimates of electron-neutral and ion-neutral collision cross sections, coupled with preliminary values for kinetic rate coefficients for gas phase and surface reactions, would be included in the model. The model would predict spatial profiles of various chemical species, and these profiles would be tested by direct comparison with measurements. The inevitable initial disagreements between model and measurements provide the basis for iterative improvement of the "first cut" database. As key species and processes are identified, this information would be made available to specialists in measuring specific collision cross sections (between, for example, electrons and radical intermediates) through specialized experimental techniques such as crossed electron-radical beam or electron swarm methods. Ion drift tubes have been used for many years to study the kinetics of selected ion-molecule reactions. Special vacuum chambers designed to measure radical-surface and ion-surface interactions can be used for plasma-surface interaction studies. In some cases, ultrahigh-vacuum beam-surface studies might be useful. These techniques are described in greater detail in subsequent chapters of the report. This vision of plasma model development implies several types of interactions within the plasma processing research community. Modelers will work directly with plasma diagnostic experimentalists to test their mechanisms and kinetic databases. Teams of modelers and plasma diagnosticians will also work directly with database experts: researchers measuring or computing electron-neutral cross sections, ion-molecule rate coefficients, radical-molecule reactions, radical-wall reactions, and ion-wall reactions. Since it is difficult and time-consuming to make accurate measurements or accurate computations of data, empirical methods to estimate or extend existing database parameters will be developed. In some cases, model results can be made consistent with measurements if collisional data have values in restricted ranges. In this way, models can be used to help determine parts of an improved database. Researchers will interact through the normal scientific channels of collaborative interactions, reporting through the scientific and technical literature and at scientific meetings, conferences, and workshops. Industrial research and development teams will participate in this process by playing important roles in helping to direct and influence academic and government laboratory researchers through feedback about what processes, chemistries, and problems are most important. Companies that are best able to utilize the
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--> flow of information in the research community will be able to gain the most commercial advantage. Individual companies will no doubt develop proprietary databases and proprietary models of their tools and processes. However, this practice should not restrict the public flow of information to and from research teams. Toward this end, it seems desirable to establish at least one data center, in which database information, including collisional process characterization data and mechanism data, would be archived, evaluated, and disseminated. Findings Models of low-temperature, nonequilibrium plasmas, especially for the description of physical phenomena, have developed rapidly in the last 5 years. Computing power per unit cost continues to increase rapidly. However, few of the currently available plasma models can be used easily by process engineers. Although attempts have been made to model plasmas with realistic chemistries, the parameter space that can be addressed is limited. Only a handful of studies have been made that attempt to validate models of plasma processes with industrially relevant chemistries. Models that attempt to link the relevant length scales (from tool scale to feature scale to atomic scale) are just now emerging. Simulations can be no more accurate than the data and assumptions on which they are based. The lack of fundamental data for the most important chemical species is the single largest factor limiting the successful application of models to problems of industrial interest. Conclusions The main roadblock to development of plasma models that will have industrially important uses is the lack of fundamental data on collisional, reactive processes occurring in the plasma and on walls bounding the plasma. Among the most important missing data are the identities of key chemical species and the dominant kinetic pathways that determine the concentrations and reactivities of these key species. The lack of a central location to collect, analyze, and disseminate the data that are currently available, or that will be available in the future, is limiting progress in the field. References 1. See, for example, "Special Issue on Modeling Collisional Low-Temperature Plasmas," eds. M.J. Kushner and D.B. Graves, IEEE Trans. Plasma Sci . 19(2):61 (1991), and "Special Issue on Modeling Collisional Low-Temperature Plasmas," eds. J. Wu, M. Meyyappan, and D. Economou, IEEE Trans. Plasma Sci. (August 1995). 2. A. Krishnan, Workshop on Database Needs in Plasma Processing, Washington, D.C., April 1-2, 1995. 3. M.E. Barone and D.B. Graves, J. Appl. Phys. 77:12-65 (1995).
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