Modeling Microdefect Formation in Czochralski Silicon

Abstract

An internally consistent model is presented for the dynamic formation of microdefects in single‐crystal silicon. The model is built on the dynamics of point defects, vacancies and self‐interstitials, and is extended to include the growth of clusters of these point defects into microdefects. A hybrid finite‐element/finite‐difference numerical method is used to solve the coupled system of partial differential equations, which includes sets of discrete rate equations for small clusters and Fokker‐Planck equations for larger ones. As described previously by a point defect dynamics model [J. Electrochem. Soc., 145, 303 (1998)], the oxidation‐induced stacking fault (OSF)‐ring position delineates the vacancy‐rich region inside from the external interstitial‐rich crystal. In Czochralski silicon, the radial position of the OSF‐ring correlates well with the expression V/G false(ROSFfalse)=1.34×10−3 cm2 min−1 K−1 . Simulations are used to explore the formation of voids in the vacancy‐rich region inside the OSF‐ring. Predictions of the total concentration of observable voids and the dependence of this concentration on the cooling rate agree with experiments and point to the importance of the axial temperature profile in the crystal from the melting point (1685 K) down to about 1150 K in setting the number and size of voids. The total number of voids correlates with V 〈G〉 where 〈G〉 is a measure of the temperature gradient in the temperature range 1173 K ≤ T ≤ 1685 K. The appearance of the OSF‐ring is explained qualitatively in terms of the residual vacancy concentration remaining in the crystal after aggregation has ceased. © 1999 The Electrochemical Society. All rights reserved.

Internally consistent approach for modeling solid-state aggregation. II. Mean-field representation of atomistic processes

“…Such models are necessary for extending the scope of atomistic simulations to realistic processing environments such as crystal growth and wafer annealing 1,2,3,29 . The model is first developed using a single reaction pathway in which only monomers are assumed to be mobile and then is extended to the general case of cluster diffusion and reaction.…”

“…An important challenge in the formulation of continuum rate equation-based models for inherently atomistic processes is verification of the physics and chemistry embodied within the model 1,2,3,4,5,6 . Typically in such models, both the assumed mechanisms and the model parameters are uncertain.…”

Internally consistent approach for modeling solid-state aggregation. I. Atomistic calculations of vacancy clustering in silicon

“…A common assumption 50,51 in the process modeling of solid-state aggregation of point defects and impurities in silicon is diffusion-limited growth by monomer addition (and dissociation). In the case of vacancy aggregation, the reaction pathway generally is expressed as a sequence of single-vacancy events…”

On-lattice kinetic Monte Carlo simulations of point defect aggregation in entropically influenced crystalline systems

“…This particular system choice is based on the fact that vacancy aggregation is a technologically important pro-cess that has been characterized in detail with experiments 8,9 and detailed continuum models, [10][11][12] and also because accurate empirical interatomic potentials are readily available. [13][14][15] Vacancy aggregation is also often assumed to be a prototypical "on-lattice system," an assumption that we show here to be invalid, particularly at elevated temperature.…”