14.2 Modeling Silicon Self-Interstitial Cluster Formation and Dissolution

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

$${\partial C\over\partial t} = 4\pi\alpha a D_I I C - {C D_I\over a^2} {\mathrm{e}}^{-E_b/kT},$$ (14.1)

where $D_I=D_0 {\mathrm{e}}^{-E_m/kT}$ is the interstitial diffusivity, $a$ is the average interatomic spacing, $\alpha$ is the capture radius expressed in units of $a$, $C(t,x)$ is the concentration of interstitials trapped in clusters, $I(t,x)$ is the concentration of free interstitials and $T$ is the annealing temperature.

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

$${\partial C\over \partial t} = K_{\mathrm{fi}} {I^{\mathrm{ifi}} ... ...}} \over I_*^{\mathrm{isfc}}} (C+\alpha I)^{\mathrm{cf}} - K_r C^{\mathrm{cr}},$$ (14.2)

where $C(t,x)$ denotes the concentration of clustered interstitials, $t$ time, $I(t,x)$ the concentration of unclustered interstitials, and $I_*(t,x)$ the equilibrium concentration of interstitials (which can be found by solving $\partial C(t,x) / \partial t=0$). There is a number of parameters to be adjusted: $K_{\mathrm{fi}}$, $K_{\mathrm{fc}}$, $K_{\mathrm{r}}$ (the reaction constants); the exponents $I^{\mathrm{ifi}}$, $I^{\mathrm{isfi}}$, $I^{\mathrm{ifc}}$, and $I^{\mathrm{isfc}}$, ${\mathrm{cf}}$, and ${\mathrm{cr}}$; and finally $\alpha$.

The reaction constants have the form

$$K_{\mathrm{fi}} = {\mathrm{kfi0}} \cdot {\mathrm{e}}^{-{\mathrm{kfiE}}/kT},$$

$$K_{\mathrm{fc}} = {\mathrm{kfc0}} \cdot {\mathrm{e}}^{-{\mathrm{kfcE}}/kT},$$

$$K_{\mathrm{r}} = {\mathrm{kr0}} \cdot {\mathrm{e}}^{-{\mathrm{krE}}/kT},$$

with ${\mathrm{kfi0}}>0$, ${\mathrm{kfc0}}>0$, and ${\mathrm{kr0}}>0$. Here $T$ is the temperature (in Kelvin) and $k=8.617\cdot10^{-5}\,\mathrm{eV}\cdot\mathrm{K}^{-1}$ the Boltzmann constant. Since the coefficients ${\mathrm{kfi0}}$, ${\mathrm{kfc0}}$, and ${\mathrm{kr0}}$ 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 $C$ and $I$ remains constant and the initial value of $C$ is $10^9\,\mathrm{cm}^{-3}$.

The ratio $I/I_*$ 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 $I^{\mathrm{ifi}} / I_*^{\mathrm{isfi}}$ and $I^{\mathrm{ifc}} / I_*^{\mathrm{isfc}}$.

Figure 14.2: Result for the high parameter set corresponding to parameter values shown in Table 14.3. The logarithm (base $10$) of the simulated and measured concentration $[\mathrm{cm}^{-3}]$ of interstitial clusters is shown depending on time $[\mathrm{s}]$.

Figure 14.3: Result for the low parameter set corresponding to parameter values shown in Table 14.3. The logarithm (base $10$) of the simulated and measured concentration $[\mathrm{cm}^{-3}]$ of interstitial clusters is shown depending on time $[\mathrm{s}]$.

The first term $K_{\mathrm{fi}} (I^{\mathrm{ifi}} / I_*^{\mathrm{isfi}})$ describes the joining of two clusters and thus the expected values for the exponents are ${\mathrm{ifi}} = 2 = {\mathrm{isfi}}$. The second growth term $K_{\mathrm{fc}} (I^{\mathrm{ifc}} / I_*^{\mathrm{isfc}}) (C+\alpha I)^{\mathrm{cf}}$ 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 $C$ and $I$ with an exponent.

Comparing (14.1) and (14.2), the growth term of (14.1), basically being a reaction constant times $IC$, 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 $(C+\alpha I)^{\mathrm{cf}}$. In the dissolution term an exponent which was later found to be $1$ is added.