Matthias K. Gobbert

Multiple scale integrated modeling of deposition processes

The ability to predict feature profile evolution across wafers during processing using equipment scale operating conditions is one important goal of process engineers. We present an integrated approach for simulating the multiple length scales needed to address this problem for thermal chemical vapour deposition (CVD) processes. In this approach, continuum models on the reactor scale and mesoscopic scales are coupled tightly with ballistic transport models on the feature scale to predict micro and macro loading effects in a transient environment. As an example of this approach, the transient simulation results for thermal deposition of silicon dioxide from tetraethoxysilane (TEOS) are presented. The efficiency of the approach presented and extensions to more complex systems are briefly discussed.

A Multiscale Simulator for Low Pressure Chemical Vapor Deposition

An integrated simulator for chemical vapor deposition is introduced. In addition to reactor scale and feature scale simulators, it includes a mesoscopic scale simulator with the typical length scale of a die. It is shown that the three-scale integrated simulator used is a proper extension of two-scale deposition simulators that consist of reactor scale and feature scale simulation models. Moreover, it is demonstrated that information is provided on a new length scale, for which no information is available from the two-scale approach, as well as important corrections to the simulation results on the reactor scale. This enables, for instance, studies of microloading. Thermally induced deposition of silicon dioxide from tetraethyoxysilane is chosen as the application example. The deposition chemistry is modeled using six gaseous reacting species involved in four gas-phase and eight surface reactions.

Mesoscopic Scale Modeling of Microloading during Low Pressure Chemical Vapor Deposition

A model designed to deal with pattern dependences of deposition processes is discussed. It is a mesoscopic scale model in the sense that it deals with spatial scales on the order of 10 -3 to 10 -2 m, which is intermediate between reactor scale and feature scale. This model accounts for the effects of the microscopic surface structure via suitable averages obtained by a homogenization technique from asymptotic analysis. Two studies on the low pressure chemical vapor deposition of silicon dioxide from tetraethoxysilane are presented to demonstrate the mesoscopic scale model. The first study shows the effects of microloading in regions of higher feature density. The second study shows the effects of varying operating conditions on loading and introduces a generalized Damkoehler number, which includes information about the surface patterns, for quantifying the degree of transport limitations. Some thoughts on how this model can be used to bridge reactor scale and feature scale models are presented.

Integrated multiscale process simulation

Abstract We summarize two approaches to integrated multiscale process simulation (IMPS), particularly relevant to integrated circuit (IC) fabrication, in which models for equipment (m) and feature (μm) scales are solved simultaneously. The first approach uses regular grids, and is applied to low-pressure chemical vapor deposition (LPCVD) of silicon dioxide from tetraethoxysilane (TEOS). The second approach uses unstructured meshes, and is applied to electrochemical deposition (ECD) of copper. The goal is to develop approaches to estimate “loading” in these processes; i.e., the effects of pattern density and topography on local deposition rates. This is accomplished by resolving pattern (mesoscopic, mm) scales, which are between equipment (0.1–1 m) and feature scales (0.1–1 μ m). In this work, we focus on steady-state simulation results. We close with a few thoughts on extending IMPS to the grain scale, and the conversion of discrete atomistic representations to continuum representations of islands during deposition.

An asymptotic analysis for a model of chemical vapor deposition on a microstructured surface

We consider a model for chemical vapor deposition, the process of adsorption of gas onto a surface together with the associated deposition of a chemical reactant on the surface. The surface has a microscopic structure which, in the context of semiconductor manufacturing, arises from a preprocessing of the semiconductor wafer. Using singular perturbation analysis, a boundary condition for the corresponding diffusion equation is derived, which allows for the replacement of the microstructured surface by a flat boundary. The asymptotic analysis is numerically verified with a simple test example.

Modeling and simulation of atomic layer deposition at the feature scale

We present a transient Boltzmann equation based transport and reaction model for atomic layer deposition (ALD) at the feature scale. The transport model has no adjustable parameters. In this article, we focus on the reaction step and the postreaction purge steps of ALD. The heterogeneous chemistry model consists of reversible adsorption of a reactant on a single site, and irreversible reaction of a second gaseous reactant with the adsorbed reactant. We conduct studies on the effect of the kinetic rate parameter associated with the reaction. We provide results for number densities of gaseous species, fluxes to the surface of the feature, and surface coverage of the adsorbing reactant as functions of time. For reasonable reaction rate parameter values, the time scale for gas transport is much smaller than that for reaction and desorption. For these cases, an analytic expression for the time evolution of the surface coverage of the adsorbing reactant provides a good approximation to the solution obtained fro...

Transient Adsorption and Desorption in Micrometer Scale Features

In an ideal atomic layer deposition ~ALD! process, the deposition of solid material on the substrate is accomplished one atomic or molecular layer at a time, in a self-limiting fashion; this property is responsible for the recent interest in ALD. To accomplish this, a representative ALD process consists of repeating a sequence of reactant flows and reactor purges. In the first step of a cycle, a gaseous species ~A! is directed into the reactor, and ~ideally! one monolayer adsorbs onto the surface. After purging the reactor, a second gaseous reactant ~B! is directed through the reactor. A self-limiting chemical reaction forms the next layer of deposited film. By repeating this cycle of reactant flows and purges, a film of ideally uniform and controlled thickness is deposited. 1-4 Although ALD has the potential to deposit films of uniform thickness, to keep average rates at reasonably high levels, high switching frequencies of reactants and purges are desired. A pulse frequency that is too high may lead to nonuniform film deposition on the feature scale, on the wafer scale, or both. As a start toward understanding species transport in features during ALD, we model the adsorption of species A and the desorption that occurs during the following purge in one cycle of a ALD process. We use a Boltzmann equation-based transport model with no adjustable parameters. The chemical reaction model used for the adsorption and desorption rates are those found in typical mechanistic representations of simple reversible Langmuir adsorption. We consider the case in which the flux of reactive species from the source volume to the wafer surface is constant with time, either at zero or at some selected value. Essentially, we idealize the transients at the reactor scale, and focus on modeling feature scale processes in response to these idealized steps in concentration above the wafer surface. More general boundary conditions at the interface between the wafer surface and the reactor volume are possible, but the results presented below provide considerable insight. We also assume that the process is isothermal in time and space.

A Galerkin Method for the Simulation of the Transient 2-D/2-D and 3-D/3-D Linear Boltzmann Equation

Many production steps used in the manufacturing of integrated circuits involve the deposition of material from the gas phase onto wafers. Models for these processes should account for gaseous transport in a range of flow regimes, from continuum flow to free molecular or Knudsen flow, and for chemical reactions at the wafer surface. We develop a kinetic transport and reaction model whose mathematical representation is a system of transient linear Boltzmann equations. In addition to time, a deterministic numerical solution of this system of kinetic equations requires the discretization of both position and velocity spaces, each two-dimensional for 2-D/2-D or each three-dimensional for 3-D/3-D simulations. Discretizing the velocity space by a spectral Galerkin method approximates each Boltzmann equation by a system of transient linear hyperbolic conservation laws. The classical choice of basis functions based on Hermite polynomials leads to dense coefficient matrices in this system. We use a collocation basis instead that directly yields diagonal coefficient matrices, allowing for more convenient simulations in higher dimensions. The systems of conservation laws are solved using the discontinuous Galerkin finite element method. First, we simulate chemical vapor deposition in both two and three dimensions in typical micron scale features as application example. Second, stability and convergence of the numerical method are demonstrated numerically in two and three dimensions. Third, we present parallel performance results which indicate that the implementation of the method possesses very good scalability on a distributed-memory cluster with a high-performance Myrinet interconnect.

Predictive modeling of atomic layer deposition on the feature scale

Abstract A feature scale simulator for atomic layer deposition (ALD) is presented that combines a Boltzmann equation transport model with chemistry models. A simple but instructive chemistry is considered; one reactant species adsorbs onto the surface, and a second reactant reacts with it from the gas phase (Eley–Rideal). This work includes potential desorption of the adsorbed species during purge steps, which may or may not play a role in any given ALD system. Three sets (cases) of rate parameters are chosen to compare chemical rates with transport rates. The duration of the ALD pulses and the geometry of the representative feature are the same for each case. Simulation results are presented for all four steps in one ALD cycle, adsorption, post-adsorption purge, reaction, and post-reaction purge. The results are extended to multiple ALD cycles, and the monolayers per cycle are estimated. We highlight the potential trade-off between pulse durations and deposition rate (wafer throughput); e.g. the time penalty required to increase the amount adsorbed during the adsorption step. The simulation methodology we present can be used to determine the pulse durations that maximize throughput for a given chemistry and chemical rate parameters. One overall observation is that transport is fast relative to chemical reactions, for reasonable kinetic parameters.

A Discontinuous Finite Element Method for Solving a Multiwell Problem

Many physical materials of practical relevance can attain several variants of crystalline microstructure. The appropriate energy functional is necessarily nonconvex, and the minimization of the functional becomes a challenging problem. A new numerical method based on discontinuous finite elements and a scaled energy functional is proposed. It exhibits excellent convergence behavior for the energy (second order) as well as other crucial quantities of interest for general spatial meshes, contrary to standard (non-)conforming methods. Both theoretical analyses and numerical test calculations are presented and contrasted to other current finite element methods for this problem.