3.7 Dopant Diffusion in the Presence of Exdendend Crystal Defects

During ion implementation simple point defect aggregate to build extended defects such as $\{311\}$ defects and dislocation loops. Clustering of dopants occurs always when the peak of the implants dose exceeds the solid solubility. Clustered dopants are electrically inactive and immobile and on such way they significantly influence dopant diffusion. Even at small implant dose a build up of the interstitial clusters takes place, but the dynamics of the interstitals clusters significantly increases for an implant dose larger than $5\times{10^{13} cm^{-2}}$ as so-called $\{311\}$ begin to evolve. These defects can be described as rod-like in shape running in $<110>$ directions. If the anneal temperature is larger than $950^{\circ}C$, the $\{311\}$ defects which consist of $2000$ or more interstitials unfalts to form dislocations loops. The clusters of dopants and point defects do not evolve individually, but they also interact with each other. It is possible [36] to model all these defects together with their dynamics in a consistent and fundamental way within a general framework. The so called moment-based approach models the size distribution of dopant and point defect clusters. In this section we will illustrate moment approach for the case of $\{311\}$ extended defects and phosphorus diffusion.
The central role in the moment approach plays the size distribution function $f_n^{311}(x,t)$ of the $\{311\}$ extended defects. This distribution function determines the density per unit volume of extended defects consisting of $n$ interstitials at a given location and time. The behavior of $\{311\}$ defects is described with the following equations which deal with the moments of the distribution $f_n^{311}$, $m_i=\sum_{n=2}^{\infty}n_i f_n^{311}$ [36,30].

$$\frac{\partial m_0}{\partial t}=D \lambda[(C_I^{311})^2-m_0C_{ss}\gamma_0],$$ (3.69)

$$\frac{\partial m_1}{\partial t}=2\frac{\partial m_0}{\partial t}+ D \lambda m_0[C_I^{311}-m_0C_{ss}\gamma_1],$$ (3.70)

$$\frac{\partial C_I^{311}}{\partial t}=-\frac{\partial m_0}{\partial t},$$ (3.71)

with,

$$\gamma_0=\frac{1}{C_{ss}m_0}C_1^{*}f_2, \gamma_1=\frac{1}{C_{ss}m_0}\sum_{n=2}^{\infty}C_n^{*}f_{n+1},$$ (3.72)

where $C_I^{311}$ is the concentration of the interstitials bound in $\{311\}$ extended defects, $C_{ss}$ is the solid solubility and $C_n^{*}$ is the interstitial concentration at which there will be no change in the free energy for precipitates to grow from size $n$ to $n+1$.
This model for $\{311\}$ extended defects can be readily combined with the three stream Mulavaney-Richardson model in order to get an equation set which describes dopant behavior in the presence of interstitial clusters. Since boron diffuses primarily through pairing with interstitals one can omit the vacancy-dopant term from the first and third equation of the model.

$$\frac{\partial C_B}{\partial t}= \nabla\cdot \Bigl[ \frac{f_I D_B}{C_{I}^{eq}}\nabla (C_I C_B) + \frac{f_I D_B}{C_{I}^{eq}}C_I C_B \nabla ( n)\Bigr],$$ (3.73)

$$\frac{\partial C_I}{\partial t}=\nabla\cdot(D_I\nabla C_I) - k_{f} ( C_I C_V - C_{I}^{eq} C_{V}^{eq}) +$$

$$+\nabla\cdot \Bigl [ \frac{f_I D_B}{C_{I}^{eq}}\nabla (C_I C_B) + \frac{f_I D_B}{C_{I}^{eq}}C_I C_B \nabla ( n)\Bigr]-\frac{\partial C_I^{311}}{\partial t},$$ (3.74)

$$\frac{\partial C_V}{\partial t}=\nabla\cdot(D_V\nabla C_V) - k_{f} ( C_I C_V - C_{I}^{eq} C_{V}^{eq}).$$ (3.75)

As shown in [37], this kind of model is suitable to simulate diffusion which controls the junction depth for ultra-low energy boron implants.