\chapter{Point defects in silicon}
\label{chapter:defects}
-Regarding the supposed conversion mechanisms of SiC in c-Si as introduced in section~\ref{section:assumed_prec} the understanding of C and Si interstitial point defects in c-Si is of fundamental interest.
+Regarding the supposed conversion mechanisms of SiC in c-Si as introduced in section~\ref{section:assumed_prec}, the understanding of C and Si interstitial point defects in c-Si is of fundamental interest.
During implantation, defects such as vacancies (V), substitutional C (C$_{\text{s}}$), interstitial C (C$_{\text{i}}$) and Si self-interstitials (Si$_{\text{i}}$) are created, which are believed to play a decisive role in the precipitation process.
In the following, these defects are systematically examined by computationally efficient, classical potential as well as highly accurate DFT calculations with the parameters and simulation conditions that are defined in chapter~\ref{chapter:simulation}.
Both methods are used to investigate selected diffusion processes of some of the defect configurations.
It has turned out to be very difficult to capture the results of quantum-mechanical calculations in analytical potential models.
Among the established analytical potentials only the environment-dependent interatomic potential (EDIP)~\cite{bazant97,justo98} and Stillinger-Weber~\cite{stillinger85} potential reproduce the correct order in energy of the defects.
However, these potentials show shortcomings concerning the description of other physical properties and are unable to describe the C-C and C-Si interaction.
-In fact the EA potential calculations favor the tetrahedral defect configuration.
+In fact, the EA potential calculations favor the tetrahedral defect configuration.
This limitation is assumed to arise due to the cut-off.
In the tetrahedral configuration the second neighbors are only slightly more distant than the first neighbors, which creates the particular problem.
Indeed, an increase of the cut-off results in increased values of the formation energies~\cite{albe_sic_pot}, which is most significant for the tetrahedral configuration.
For the vacancy the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site.
No other intrinsic defect configuration, within the ones that are mentioned, is affected by spin polarization.
-In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, i.e.\ if the distance of two atoms is within the cut-off radius $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
+In the case of the classical potential simulations, bonds between atoms are displayed if there is an interaction according to the potential model, i.e.\ if the distance of two atoms is within the cut-off radius $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
For the tetrahedral and the slightly displaced configurations four bonds to the atoms located in the center of the planes of the unit cell exist in addition to the four tetrahedral bonds.
-The length of these bonds are, however, close to the cut-off range and thus are weak interactions not constituting actual chemical bonds.
+The length of these bonds are, however, close to the cut-off range and, thus, are weak interactions not constituting actual chemical bonds.
The same applies to the bonds between the interstitial and the upper two atoms in the \si{} \hkl<1 1 0> DB configuration.
A more detailed description of the chemical bonding is achieved through quantum-mechanical calculations by investigating the accumulation of negative charge between the nuclei.
An experimental value of the formation energy of \cs{} was determined by a fit to solubility data yielding a concentration of $3.5 \times 10^{24} \exp{(-2.3\,\text{eV}/k_{\text{B}}T)} \text{ cm}^{-3}$~\cite{bean71}.
However, there is no particular reason for treating the prefactor as a free parameter in the fit to the experimental data.
It is simply given by the atomic density of pure silicon, which is $5\times 10^{22}\text{ cm}^{-3}$.
-Tersoff~\cite{tersoff90} and Dal~Pino~et~al.~\cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from \unit[1.6--1.89]{eV} an excellent agreement with the experimental solubility data within the entire temperature range of the experiment is obtained.
+Tersoff~\cite{tersoff90} and Dal~Pino~et~al.~\cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from \unit[1.6--1.89]{eV}, an excellent agreement with the experimental solubility data within the entire temperature range of the experiment is obtained.
This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal~Pino~et~al.\ is in good agreement with the results of the quantum-mechanical calculations performed in this work.
-Unfortunately the EA potential undervalues the formation energy roughly by a factor of two, which is a definite drawback of the potential.
+Unfortunately, the EA potential undervalues the formation energy roughly by a factor of two, which is a definite drawback of the potential.
Except for Tersoff's results for the tetrahedral configuration, the \ci{} \hkl<1 0 0> DB is the energetically most favorable interstitial configuration.
As mentioned above, the low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the abrupt cut-off set to \unit[2.5]{\AA} (see Ref. 11 and 13 in~\cite{tersoff90}) and the real formation energy is, thus, supposed to be located between \unit[3--10]{eV}.
%
\begin{minipage}{15cm}
\centering
-\framebox{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
+\framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 0 1]}\\
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
\end{minipage}
%
\begin{minipage}{15cm}
\centering
-\framebox{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
+\framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 -1 0]}\\
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
\end{minipage}
%
\begin{minipage}{15cm}
\centering
-\framebox{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
+\framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 -1 0] (in place)}\\
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
\end{minipage}
\label{img:defects:c_mig_path}
\end{figure}
Three different migration paths are accounted in this work, which are displayed in Fig.~\ref{img:defects:c_mig_path}.
-The first migration investigated is a transition of a \hkl<0 0 -1> into a \hkl<0 0 1> DB interstitial configuration.
+The first migration investigated is a transition of a \hkl[0 0 -1] into a \hkl[0 0 1] DB interstitial configuration.
During this migration the C atom is changing its Si DB partner.
-The new partner is the one located at $a_{\text{Si}}/4 \hkl<1 1 -1>$ relative to the initial one, where $a_{\text{Si}}$ is the Si lattice constant.
+The new partner is the one located at $a_{\text{Si}}/4 \hkl[1 1 -1]$ relative to the initial one, where $a_{\text{Si}}$ is the Si lattice constant.
Two of the three bonds to the next neighbored Si atoms are preserved while the breaking of the third bond and the accompanying formation of a new bond is observed.
-The C atom resides in the \hkl(1 1 0) plane.
+The C atom resides in the \hkl(-1 1 0) plane.
This transition involves the intermediate BC configuration.
However, results discussed in the previous section indicate that the BC configuration is a real local minimum.
-Thus, the \hkl<0 0 -1> to \hkl<0 0 1> migration can be thought of a two-step mechanism in which the intermediate BC configuration constitutes a metastable configuration.
-Due to symmetry it is enough to consider the transition from the BC to the \hkl<1 0 0> configuration or vice versa.
+Thus, the \hkl[0 0 -1] to \hkl[0 0 1] migration can be thought of a two-step mechanism, in which the intermediate BC configuration constitutes a metastable configuration.
+Due to symmetry, it is enough to consider the transition from the BC to the \hkl<1 0 0> configuration or vice versa.
In the second path, the C atom is changing its Si partner atom as in path one.
-However, the trajectory of the C atom is no longer proceeding in the \hkl(1 1 0) plane.
-The orientation of the new DB configuration is transformed from \hkl<0 0 -1> to \hkl<0 -1 0>.
+However, the trajectory of the C atom is no longer proceeding in the \hkl(-1 1 0) plane.
+The orientation of the new DB configuration is transformed from \hkl[0 0 -1] to \hkl[0 -1 0].
Again, one bond is broken while another one is formed.
As a last migration path, the defect is only changing its orientation.
Thus, this path is not responsible for long-range migration.
\begin{center}
\includegraphics[width=0.7\textwidth]{im_00-1_nosym_sp_fullct_thesis_vasp_s.ps}
\end{center}
-\caption[Migration barrier and structures of the {\hkl[0 0 -1]} DB to BC transition.]{Migration barrier and structures of the \hkl<0 0 -1> DB (left) to BC (right) transition. Bonds of the C atom are illustrated by blue lines.}
+\caption[Migration barrier and structures of the {\hkl[0 0 -1]} DB to BC transition.]{Migration barrier and structures of the \hkl[0 0 -1] DB (left) to BC (right) transition. Bonds of the C atom are illustrated by blue lines.}
\label{fig:defects:00-1_001_mig}
\end{figure}
In Fig.~\ref{fig:defects:00-1_001_mig} results of the \hkl<0 0 -1> to \hkl<0 0 1> migration fully described by the migration of the \hkl<0 0 -1> to the BC configuration is displayed.
As already known from the migration of the \hkl<0 0 -1> to the BC configuration discussed in Fig.~\ref{fig:defects:00-1_001_mig}, another \unit[0.25]{eV} are needed to turn back from the BC to a \hkl<1 0 0>-type interstitial.
However, due to the fact that this migration consists of three single transitions with the second one having an activation energy slightly higher than observed for the direct transition, this sequence of paths is considered very unlikely to occur.
The migration barrier of the \hkl<1 1 0> to \hkl<0 0 -1> transition, in which the C atom is changing its Si partner and, thus, moving to the neighbored lattice site, corresponds to approximately \unit[1.35]{eV}.
-During this transition the C atom is escaping the \hkl(1 1 0) plane approaching the final configuration on a curved path.
+During this transition the C atom is escaping the \hkl(-1 1 0) plane approaching the final configuration on a curved path.
This barrier is much higher than the ones found previously, which again make this transition very unlikely to occur.
For this reason, the assumption that C diffusion and reorientation is achieved by transitions of the type presented in Fig.~\ref{fig:defects:00-1_0-10_mig} is reinforced.
Fig.~\ref{fig:defects:cp_bc_00-1_mig} shows the evolution of structure and energy along the \ci{} BC to \hkl[0 0 -1] DB transition.
Since the \ci{} BC configuration is unstable relaxing into the \hkl[1 1 0] DB configuration within this potential, the low kinetic energy state is used as a starting configuration.
Two different pathways are obtained for different time constants of the Berendsen thermostat.
-With a time constant of \unit[1]{fs}, the C atom resides in the \hkl(1 1 0) plane
+With a time constant of \unit[1]{fs}, the C atom resides in the \hkl(-1 1 0) plane
resulting in a migration barrier of \unit[2.4]{eV}.
-However, weaker coupling to the heat bath realized by an increase of the time constant to \unit[100]{fs} enables the C atom to move out of the \hkl(1 1 0) plane already at the beginning, which is accompanied by a reduction in energy, approaching the final configuration on a curved path.
-The energy barrier of this path is \unit[0.2]{eV} lower in energy than the direct migration within the \hkl(1 1 0) plane.
+However, weaker coupling to the heat bath realized by an increase of the time constant to \unit[100]{fs} enables the C atom to move out of the \hkl(-1 1 0) plane already at the beginning, which is accompanied by a reduction in energy, approaching the final configuration on a curved path.
+The energy barrier of this path is \unit[0.2]{eV} lower in energy than the direct migration within the \hkl(-1 1 0) plane.
However, the investigated pathways cover an activation energy approximately twice as high as the one obtained by quantum-mechanical calculations.
If the entire transition of the \hkl[0 0 -1] into the \hkl[0 0 1] configuration is considered a two step process passing the intermediate BC configuration, an additional activation energy of \unit[0.5]{eV} is necessary to escape the BC towards the \hkl[0 0 1] configuration.
Assuming equal preexponential factors for both diffusion steps, the total probability of diffusion is given by $\exp\left((2.2\,\text{eV}+0.5\,\text{eV})/k_{\text{B}}T\right)$.
\label{subsection:si-cs}
So far the C-Si \hkl<1 0 0> DB interstitial was found to be the energetically most favorable configuration.
-In fact substitutional C exhibits a configuration more than \unit[3]{eV} lower with respect to the formation energy.
+In fact, substitutional C exhibits a configuration more than \unit[3]{eV} lower with respect to the formation energy.
However, the configuration does not account for the accompanying Si self-interstitial that is generated once a C atom occupies the site of a Si atom.
With regard to the IBS process, in which highly energetic C atoms enter the Si target being able to kick out Si atoms from their lattice sites, such configurations are absolutely conceivable and a significant influence on the precipitation process might be attributed to them.
Thus, combinations of \cs{} and an additional \si{} are examined in the following.
With an activation energy of \unit[0.9]{eV}, the C$_{\text{i}}$ carbon interstitial can be expected to be highly mobile at prevailing temperatures in the process under investigation, i.e.\ IBS.
Since the \ci{} \hkl<1 0 0> DB is the ground-state configuration and highly mobile, possible migration of these DBs to form defect agglomerates, as demanded by the model introduced in section~\ref{section:assumed_prec}, is considered possible.
-Unfortunately the description of the same processes fails if classical potential methods are used.
+Unfortunately, the description of the same processes fails if classical potential methods are used.
Already the geometry of the most stable DB configuration differs considerably from that obtained by first-principles calculations.
The classical approach is unable to reproduce the correct character of bonding due to the deficiency of quantum-mechanical effects in the potential.
Nevertheless, both methods predict the same type of interstitial as the ground-state configuration and also the order in energy of the remaining defects is reproduced fairly well.
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