# 14.2 Modeling Silicon Self-Interstitial Cluster Formation and Dissolution

In [99] the following equation describing interstitial cluster kinetics is given:

 (14.1)

where is the interstitial diffusivity, is the average interatomic spacing, is the capture radius expressed in units of , is the concentration of interstitials trapped in clusters, is the concentration of free interstitials and is the annealing temperature.

Here the main formula of the model for the change of the concentration of clustered interstitials is

 (14.2)

where denotes the concentration of clustered interstitials, time, the concentration of unclustered interstitials, and the equilibrium concentration of interstitials (which can be found by solving ). There is a number of parameters to be adjusted: , , (the reaction constants); the exponents , , , and , , and ; and finally .

The reaction constants have the form

with , , and . Here is the temperature (in Kelvin) and the Boltzmann constant. Since the coefficients , , and are positive, the first two terms in (14.2) are responsible for the formation of clusters and the last term for the dissolution. The sum of interstitials counted in  and  remains constant and the initial value of  is .

The ratio of the concentration of the unclustered interstitials and its equilibrium concentration is often called the interstitial supersaturation. Here additional exponents modify the interstitial supersaturation which appears in the form and .

The first term describes the joining of two clusters and thus the expected values for the exponents are . The second growth term governs the case when an unclustered interstitials joins an interstitial cluster. Here we can expect the exponents to be unity. The second factor is a linear combination of  and  with an exponent.

Comparing (14.1) and (14.2), the growth term of (14.1), basically being a reaction constant times , is split into two parts providing greater flexibility: one depending on a modified interstitial supersaturation term and one depending on a modified interstitial supersaturation term times . In the dissolution term an exponent which was later found to be  is added.

Clemens Heitzinger 2003-05-08