14.4 Summary

Starting with (14.2) and the measurements from [95] we adjusted a model for the formation and dissolution of silicon self-interstitial clusters, namely

$${\partial C\over \partial t} = K_{\mathrm{fi}} {I^2 \over I_*^2} + K_{\mathrm{fc}} {I \over I_*} (C+\alpha I) - K_r C,$$

with values from Table 14.3. Although different values were also examined, the exponents in the first term were found to be equal to $2$ ( ${\mathrm{ifi}} = 2 = {\mathrm{isfi}}$), because two isolated interstitials can form a new cluster. Good results were achieved with ${\mathrm{cf}}=1$ and ${\mathrm{ifc}}=1={\mathrm{isfc}}$, which means the rate of free interstitials joining already existing clusters depends linearly on the number of excess interstitials (interstitials above the equilibrium concentration) and a linear combination of the number of clusters and interstitials. Finally the exponent ${\mathrm{cr}}$ was found to be $1$. This means that the rate of dissolution depends on the concentration of clustered interstitials and on the factor $K_r$.

The terms responsible for cluster formation in both models don't share a common structure, thus we finally compare the results for the dissolution term. In (14.1) the dissolution term is

$$- {D_0 C\over a^2} {\mathrm{e}}^{-(E_b+E_m)/kT},$$

where the values for $E_b$ and $E_m$ given in [99] are $E_b=1.8\,\mathrm{eV}$ and $E_m=1.77\,\mathrm{eV}$. $E_b+E_m=3.57\,\mathrm{eV}$ agrees very well with the values found for $\mathrm{krE}$ in Table 14.3, namely $3.56997\,\mathrm{eV}$ for the high parameter set and $3.84503\,\mathrm{eV}$ for the low parameter set.

In summary, a refined model for the formation and dissolution of silicon self-interstitial clusters was calibrated to published measurements for two different technologies (corresponding to two sets of material parameters) and very good agreement was achieved.