A Point Defect Based Two-Dimensional Model of the Evolution of Dislocation Loops in Silicon during Oxidation

A point defect based model is developed in two dimensions for the evolution of a group of dislocation loops induced by high dose ion implantation in silicon. Assuming an asymmetric triangular density distribution of periodically oriented circular dislocation loops provides an efficient model reflecting the nonuniform morphology of the loops as observed in transmission electron microscopy (TEM) experiments. The effective pressure from the ensemble of dislocation loops is numerically calculated on the basis of the established formulation of pressure from a single circular loop. The pressure field from the layer of dislocation loops is fundamental to the modeling, as it largely affects equilibrium point defect concentrations and boundary conditions governing emission and absorption of the point defects. Solving the pressure-dependent point defect diffusion equations in association with the simplified loop distribution and geometry makes it possible to model the loop growth and shrinkage incorporating effectively the statistical processes such as loop coalescence and dissolution during oxidation. The simulation with the model shows reduced interstitial supersaturation during the oxidation and correctly predicts the variation of the number of captured silicon atoms and the radii and densities of the dislocation loops in agreement with the TEM experiments. High dose ion implantation is an essential technique for obtaining heavily doped regions in silicon such as the source and drain of metal oxide semiconductor field effect transistors (MOSFETs) and dynamic random access memory (DRAM) cells. The high energy bombardment of incident ions inevitably creates damage in the crystal. The ion implantat ion damage governs the subsequent dopant diffusion during the thermal annealing cycle required for substrate recrystallization and dopant activation. Particularly, the implants of common dopants at a dose above a certain ion mass-dependent threshold amorphize the surface region in silicon substrate, 1 simultaneously producing a large amount of point defects. The subsequent annealing leads to solid-phase epitaxia] regrowth of the amorphous region, and extended defects such as dislocation loops are formed below the amorphous-crystalline interface. The dislocation loops are inherently accompanied by a stress field in the crystal, interacting with the point defects. It is generally accepted ~-7 that the end-of-range dislocation loops affect the distribution of point defects by absorption of interstitials or by emission of vacancies at their core boundary during growth, and by the reverse processes during shrinkage. There has been a lot of effort to model the dopant diffusion by investigating the interaction of dopant and point defects without any extended defects under low dose implant damage and oxidat ion/nitr idation conditions. This work is focused on modeling the evolution of dislocation loops and its effects on the point defect diffusion, which eventually influences the dopant redistribution. Previous work "-1~ established theoretical models for a single circular dislocation loop and its interaction with point defects. Bullough et al. 9 studied the migration of an interstitial impurity atom around a single dislocation loop on the basis of the stress field from the loop solved by Bastecka and Kroupa. 8 Borucki ~1 proposed a model for the growth and shrinkage of a single dislocation loop due to the capture and emission of point defects, and simulated the point defect variation from an assumed initial high supersaturation around a periodic array of the loops in a threedimensional numerical solver of diffusion equations. However, it is necessary to model the effects from the group of dislocation loops formed in the substrate as observed through TEM pictures. TEM measurements 12-~4 show that the variations in distribution and size of the actual dislocation loops during oxidation or annealing are generally not so homogeneous and simple as in the case of one single loop. The dislocation loops usually form a network by * Electrochemical Society Student Member. ** Electrochemical Society Active Member. merging with each other during oxidation. Coalescence and dissolution of dislocation loops are a statistically complicated process, which also depends on the implanted ion species. 1~-14 Since the TEM measurements can only give statistical data on density and size of the dislocation loops, a useful and correct model for the ensemble of dislocation loops should be made in a way to reflect the statistical data from the TEM pictures. Modeling of the Dislocation Loops as a Group The aim of this work is to build up a point defect based model for the evolution of the group of dislocation loops with both accuracy and efficiency, so it can be implemented in two-dimensional process simulators. It involves two-dimensional conversion of Borucki's model for the interaction of a single loop and point defects in three dimensions, and it is extended to the ensemble of the loops. In addition, we developed an efficient model for the network formation of the dislocation loops via a statistical density distribution function of loop radius. Finally, the evolution of the loops and the redistribution of point defects during oxidation are simulated where the surface injection, recombination, and diffusion of point defects have been implemented through a point defect parameter set consistent with the current available data on oxidation enhanced diffusion (OED). 1~ In this work, the model for the group of dislocation loops is developed on the basis of thermodynamics and linear elasticity which govern a single dislocation loop. Major assumptions in the model are: (i) dislocation loops are all circular and evenly distributed on a plane interconnecting their centers; (it) orientation of the loops is periodic in two perpendicular directions, approximating the morphology shown in cross section TEM and plan-view TEM pictures; (iii) radius and density of the loops follow an asymmetric triangular distribution function; (iv) formulation of a pressure field from a single dislocation loop in linear elastic material can be used as a basis in calculating the effective pressure from the ensemble of dislocation loops; (v) local equilibrium is attained around the dislocation loops through their reaction with point defects. The Effective Pressure from the Loop Distribution The pressure from a single circular prismatic dislocation loop has been formulated by Bastecka and Kroupa ~ from the diagonal components of the stress tensor in linear elasticity theory. Figure 1 shows the geometry used in calculating the pressure from a single loop in terms of location of an impurity atom in cylindrical coordinates. The loop has the Burgers vector b in the axial direction. The dilatation factor e of an atom near the loop measures the space for an J. Electrochem. Soc., Vol. 141, No. 3, March 1994 9 The Electrochemical Society, Inc. 759 Downloaded 04 Apr 2011 to 128.227.135.101. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp 760 J. Electrochem. Soc., Vol. 141, No. 3, March 1994 9 The Electrochemical Society, Inc. elastic inclusion of the atom. The pressure is expressed in terms of the loop radius R, the radial distance r, and the height h from the loop s [ R 2 r 2 h 2 ] bP~[, r+R, 2+h2]-u2( ) E(~) K(a) [1] P=-~ " ~ :~y2+h2 + where a = {4rR/[(r + R) 2 + h2]} ~ E(~) and K(~) are the complete elliptic integrals of the first and the second kind, b is the magnitude of the Burgers vector, t~ is the shear modulus, and ~/= 3(1 v)/(1 + v), where v is the Poisson's ratio. Figure 2 shows an example of the calculated pressure at different positions of the atom as a function of the radial distance and the height. In general, the pressure is positive inside the cylindrical region of the loop radius R, and negative outside. Its magnitude decreases approximately in the inverse proportion of the cube of the distance in the outside region, if the distance from the center exceeds the loop diameter. 9 In two-dimensional process simulators, the cross section where dopant and point defect concentrations are calculated is perpendicular to the layer of dislocation loops which is formed inside the substrate in parallel with the surface. Therefore, it is necessary to obtain an effective pressure from the group of dislocation loops at one depth position by considering a certain configuration of the loop ensemble. The configuration of the ensemble of equal-sized loops assumed in this model is shown in Fig. 3 which is viewed from the top of the substrate. We assume that the loops of one radius are evenly distributed on one plane interconnecting their centers. The orientation of the loops are assumed to be periodic in two perpendicular directions, with their Burgers vectors lying on the center plane. This is a good approximation of the loop morphology as observed in plan-view TEM and cross-sectional TEM pictures, where Calculation of pressure from one circular dislocation loop