Module Defects

Fig. 1. Crystal and Melt Weight Calculation

Fig. 2. Automatic Reconstruction of the Crystal Positions

Module Defects is a part of the CGSim package specially designed for analysis of:

• thermal elastic stress distribution;
• behavior of initial defects such as self-interstitials and vacancies in silicon crystals;
• cluster distribution in silicon crystals (voids and oxygen precipitates).

Elastic Stress analysis

The elastic stress analysis is performed in the module in 2D axisymmetrical approximation. The numerical algorithm used in the module operates with the displacement vector u_i. The axisymmetric computational domain representing the crystal is meshed using cylindrical coordinates { r, j, z } with the temperature distribution found from the heat transfer modeling on the corresponding crystal position. The thermoelastic problem is solved using the Finite Volume Method. The following boundary conditions are used to solve the problem formulated above:

• zero pressure is assigned along the crystal external boundary;
• radial component of deformation vector u_r is equal to zero along the symmetry axis.

Fig. 3. Module Defects Window

Several parameters can be used by the user to characterize strain-stress state in the crystal:

• maximum s_max and minimum s_min principal stresses;
• maximum shear stress s^sh_max which is critical for the gliding dislocation multiplication;
• Von-Mises stress s_VM governing the change of crystal shape.

Simulation of initial defect incorporation

Initial lattice defects (single vacancies and self-interstitial atoms) are the sources for point defect clusterization during a crystal thermal treatment. Their concentrations determine the crystal quality to a large extent. The governing equation of initial defect incorporation into the crystal and their subsequent recombination in a hot region in vicinity of the crystallization front could be presented as
   
   ,
where C^eq_x is the equilibrium defect concentration and C_x is the actual defect concentration, D_x is the defect diffusivity, V is the crystal pulling rate, a_r is the recombination capture radius, and DG is the recombination free energy barrier (x = i, n  for self-interstitials and vacancies, respectively).

Model of point defect clusterization

Certain characteristics of point defects formed during the crystal growth and subsequent wafer annealing are required for further making the integrated circuits (IC). Voids and oxygen precipitates near wafer surface can damage precise sub-micron IC elements. The model incorporated into the module accounts for simultaneous formation and evolution of voids and oxygen precipitates, allowing prediction of point defect concentrations and size distributions.