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.