A systems-based approach for generating quantitative models of microstructural evolution in silicon materials processing

Abstract

A systematic approach based on multiple model regression to several distinct types of experimental data is used to create a robust modeling framework for defect and impurity evolution during crystalline silicon processing. The three experimental systems considered here are Czochralski single-crystal growth, zinc diffusion, and oxide precipitation during wafer thermal annealing. All three systems are intimately connected at the atomistic level by the thermodynamic and transport properties of single point defects. The present work demonstrates how this microscopic overlap in these macroscopically distinct systems can substantially reduce uncertainty in model regression to experimental data, leading to strict bounds on the transport, reaction, and thermodynamic properties of the various species present. The resulting parameters lead to a solid, quantitative basis for modeling a wide variety of technologically important defect-related phenomena in silicon processing. Several different stochastic global optimization methods are employed to perform the computationally expensive multi-objective function minimization, namely simulated annealing, genetic algorithms, Tabu search and particle swarm optimization. These methods are compared and contrasted as part of the investigation and it is shown that a hybrid genetic algorithm that periodically incorporates a local search is the most robust approach for the types of problems considered in this work.

Introduction

The development of predictive models for defect and dopant chemistry and physics during the various stages of silicon growth and processing remains important despite the success of experimentally based approaches. As the level of microelectronic device integration continues to increase and the minimum feature size decreases, the requirements placed on the perfection of the silicon substrate material continue to tighten. Furthermore, the tendency to continually increase wafer sizes and process complexity implies that processes such as Czochralski (CZ) crystal growth and the various stages of wafer thermochemical annealing must be accordingly modified. Both of these trends require substantial experimentation with increasingly expensive materials and systems, driving the need for predictive process models for microstructural evolution.

The quality of a crystalline silicon substrate is determined by the number, size, and spatial distribution of native (or intrinsic) and impurity related defects. The term native refers to defect structures that are comprised of self-interstitials, vacancies, and their respective aggregates. Equally important are impurity related defects created in the presence of unintentionally incorporated chemical species such as carbon, or metals, such as copper and iron (Gilles, Weber, & Hahn, 1990; Harada, Tanaka, Matsubara, Shimanuki, & Furuya, 1995; Isomae, Ishida, Itoga, & Hozawaa, 2002; Istratov, Hieslmair, & Weber, 2000). Additionally, intentional dopants such as oxygen can, under certain conditions, lead to undesirable defect structures (Yamazaki, Matsushita, Sugamoto, & Tsuchiya, 2000). Defects and impurities also impact a wide range of material properties such as resistivity and mechanical strength.

The standard approach for modeling and predicting the formation and evolution of defects as a function of processing conditions is based on the numerical solution of a system of coupled reaction/diffusion/convection partial differential equations. While these models are usually posed at the continuum scale, they must accurately represent many complex events that take place on the atomistic and meso-scales, thereby requiring the specification of a large number of difficult-to-measure parameters. The input parameters include defect and impurity diffusion coefficients, reaction and/or nucleation rates of mesoscopic clusters, and a host of temperature dependent thermodynamic properties for numerous chemical species. Selected examples of defect and impurity models can be found in Sinno, Brown, von Ammon, and Dornberger (1998), Falster, Voronkov, and Quast (2000a), Brown, Maroudas, and Sinno (1994), Bergholz and Gilles (2000), Habu, Iwasaki, Harada, and Tomiura (1994), Esfandyari, Schmeiser, Senkader, Hobler, and Murphy (1996) and Fahey, Griffin, and Plummer (1989). Such models are not yet in widespread commercial use because they are rarely fully predictive, even though most are able to fit any given experimental data set. The lack of predictive ability arises because there is usually insufficient information in a single data set to robustly parameterize a model containing several fitting parameters. Simply put, most models contain too many adjustable parameters to be commercially useful. An important goal of the present work is to provide an estimate of the information content in each of the systems considered.

There have been numerous efforts aimed at independently calculating some of the thermophysical parameters needed for process models, such as reaction rates, equilibrium properties and diffusion coefficients. A common approach is to employ atomistic simulations, which offer the advantage of allowing a particular phenomenon to be isolated (a brief review of some of these calculations can be found in Sinno (2002)). However, even state-of-the-art electronic density functional theory (DFT) results still exhibit too much uncertainty to provide all parameters completely independently of experiment because of the extreme sensitivity of these models to some of the parameters. In addition, some of the properties required, such as defect aggregate geometries and formation energies, are beyond the scope of these computationally expensive simulations (Staab et al., 2002). That being said, DFT results were used to fix at least some of the required parameters, such as activation energies (see Section 2.4), as well as provide strong constraints on the values of others.

In this paper we describe work aimed at robustly parameterizing models for three defect-related phenomena using quantitative experimental data, with particular emphasis on the properties of self-interstitials and vacancies. A multi-model approach is employed that exploits the microscopic links between three macroscopically distinct systems. The experimental systems considered are (1) zinc in-diffusion in wafers, (2) oxygen precipitation during wafer thermal annealing, and (3) CZ crystal growth from the melt. Each of these systems is sensitively dependent on (at least some) native point defect properties without requiring the introduction of many system-specific fitting parameters. The net result of the parametric overlap is that most of the adjustable parameters in the collective system are exposed to multiple (different) data sets, greatly increasing the probability of finding a unique parameterization; details are given in Section 2.

The multi-model parameterization task is complicated by several features. Firstly, repeated evaluation of an objective function (generally defined as the difference between model predictions and experimental data) for each of the described systems is computationally expensive. Secondly, the PDE models for the systems considered are non-linear and can lead to multiple minima in the objective function landscape. We have therefore chosen to utilize several stochastic optimization approaches including simulated annealing (SA), genetic algorithms (GA), Tabu search (TS), and particle swarm optimization (PSO). Another goal of this paper is to systematically analyze the relative performance of these methods for non-linear PDE model fitting.

The remainder of this paper is organized as follows. In the following section, a physiochemical description is given of each of the model systems. The fitting parameters for each system are also described and the parametric overlap discussed is outlined explicitly. In Section 3, the essential features of each of the proposed optimization algorithms are presented. Our results are presented in Section 4, and both the relative performance of each optimization algorithm and the final model fits are discussed in detail. Finally, conclusions and future outlook are provided in Section 5.