\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.
\section{Silicon self-interstitials}
-For investigating the \si{} structures a Si atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section \ref{section:basics:defects}.
-The formation energies of \si{} configurations are listed in Table \ref{tab:defects:si_self} for both methods used in this work as well as results obtained by other {\em ab initio} studies~\cite{al-mushadani03,leung99}.
+For investigating the \si{} structures a Si atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section~\ref{section:basics:defects}.
+The formation energies of \si{} configurations are listed in Table~\ref{tab:defects:si_self} for both methods used in this work as well as results obtained by other {\em ab initio} studies~\cite{al-mushadani03,leung99}.
\bibpunct{}{}{,}{n}{}{}
\begin{table}[tp]
\begin{center}
\subsection{Defect structures in a nutshell}
-For investigating the \ci{} structures a C atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section \ref{section:basics:defects}.
-Formation energies of the most common C point defects in crystalline Si are summarized in Table \ref{tab:defects:c_ints}.
+For investigating the \ci{} structures a C atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section~\ref{section:basics:defects}.
+Formation energies of the most common C point defects in crystalline Si are summarized in Table~\ref{tab:defects:c_ints}.
The relaxed configurations are visualized in Fig.~\ref{fig:defects:c_conf}.
Again, the displayed structures are the results obtained by the classical potential calculations.
The type of reservoir of the C impurity to determine the formation energy of the defect is chosen to be SiC.
This is consistent with the methods used in the articles~\cite{tersoff90,dal_pino93}, which the results are compared to in the following.
-Hence, the chemical potential of Si and C is determined by the cohesive energy of Si and SiC as discussed in section \ref{section:basics:defects}.
+Hence, the chemical potential of Si and C is determined by the cohesive energy of Si and SiC as discussed in section~\ref{section:basics:defects}.
\begin{table}[tp]
\begin{center}
\begin{tabular}{l c c c c c c}
Due to the high formation energy of the BC defect resulting in a low probability of occurrence of this defect, the wrong description is not posing a serious limitation of the EA potential.
Tersoff indeed predicts a metastable BC configuration.
However, it is not in the correct order and lower in energy than the \ci{} \hkl<1 1 0> DB.
-Quantum-mechanical results of this configuration are discussed in more detail in section \ref{subsection:bc}.
+Quantum-mechanical results of this configuration are discussed in more detail in section~\ref{subsection:bc}.
In another {\em ab initio} study, Capaz~et~al.~\cite{capaz94} in turn found the BC configuration to be an intermediate saddle point structure of a possible migration path, which is \unit[2.1]{eV} higher than the \ci{} \hkl<1 0 0> DB structure.
This is assumed to be due to the neglecting of the electron spin in these calculations.
Another \textsc{vasp} calculation without fully accounting for the electron spin results in the smearing of a single electron over two non-degenerate states for the BC configuration.
This problem is resolved by spin polarized calculations resulting in a net spin of one accompanied by a reduction of the total energy by \unit[0.3]{eV} and the transformation into a metastable local minimum configuration.
It is worth to note that all other listed configurations are not affected by spin polarization.
However, in calculations performed in this work, which fully account for the spin of the electrons, the BC configuration in fact is a real local minimum and an energy barrier is needed to reach this configuration starting from the \ci{} \hkl<1 0 0> DB configuration.
-This is discussed in more detail in section \ref{subsection:100mig}.
+This is discussed in more detail in section~\ref{subsection:100mig}.
To conclude, discrepancies between the results from classical potential calculations and those obtained from first principles are observed.
Within the classical potentials EA outperforms Tersoff and is, therefore, used for further studies.
As the \ci{} \hkl<1 0 0> DB constitutes the ground-state configuration of a C atom incorporated into otherwise perfect c-Si it is the most probable and, hence, one of the most important interstitial configurations of C in Si.
The structure was initially suspected by IR local vibrational mode absorption~\cite{bean70} and finally verified by electron paramagnetic resonance (EPR)~\cite{watkins76} studies on irradiated Si substrates at low temperatures.
-Fig.~\ref{fig:defects:100db_cmp} schematically shows the \ci{} \hkl<1 0 0> DB structure and Table \ref{tab:defects:100db_cmp} lists the details of the atomic displacements, distances and bond angles obtained by classical potential and quantum-mechanical calculations.
+Fig.~\ref{fig:defects:100db_cmp} schematically shows the \ci{} \hkl<1 0 0> DB structure and Table~\ref{tab:defects:100db_cmp} lists the details of the atomic displacements, distances and bond angles obtained by classical potential and quantum-mechanical calculations.
For comparison, the obtained structures for both methods are visualized in Fig.~\ref{fig:defects:100db_vis_cmp}.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=12cm]{100-c-si-db_cmp.eps}
\end{center}
-\caption[Sketch of the \ci{} \hkl<1 0 0> dumbbell structure.]{Sketch of the \ci{} \hkl<1 0 0> dumbbell structure. Atomic displacements, distances and bond angles are listed in Table \ref{tab:defects:100db_cmp}.}
+\caption[Sketch of the \ci{} \hkl<1 0 0> dumbbell structure.]{Sketch of the \ci{} \hkl<1 0 0> dumbbell structure. Atomic displacements, distances and bond angles are listed in Table~\ref{tab:defects:100db_cmp}.}
\label{fig:defects:100db_cmp}
\end{figure}%
\begin{table}[tp]
This is in agreement with results of the EA potential simulations, which reveal this configuration to be unstable relaxing into the \ci{} \hkl<1 1 0> configuration.
However, this fact could not be reproduced by spin polarized \textsc{vasp} calculations performed in this work.
Present results suggest this configuration to correspond to a real local minimum.
-In fact, an additional barrier has to be passed to reach this configuration starting from the \ci{} \hkl<1 0 0> interstitial configuration, which is investigated in section \ref{subsection:100mig}.
+In fact, an additional barrier has to be passed to reach this configuration starting from the \ci{} \hkl<1 0 0> interstitial configuration, which is investigated in section~\ref{subsection:100mig}.
After slightly displacing the C atom along the \hkl[1 0 0] (equivalent to a displacement along \hkl[0 1 0]), \hkl[0 0 1], \hkl[0 0 -1] and \hkl[1 -1 0] direction the distorted structures relax back into the BC configuration.
As will be shown in subsequent migration simulations the same would happen to structures where the C atom is displaced along the migration direction, which approximately is the \hkl[1 1 0] direction.
These relaxations indicate that the BC configuration is a real local minimum instead of an assumed saddle point configuration.
%These modifications to the usual procedure are applied to avoid abrupt changes in structure and free energy on the one hand and to make sure the expected final configuration is reached on the other hand.
%Due to applying updated constraints on all atoms the obtained migration barriers and pathes might be overestimated and misguided.
%To reinforce the applicability of the employed technique the obtained activation energies and migration pathes for the \hkl<0 0 -1> to \hkl<0 -1 0> transition are compared to two further migration calculations, which do not update the constrainted direction and which only apply updated constraints on three selected atoms, that is the diffusing C atom and the Si dumbbell pair in the initial and final configuration.
-%Results are presented in figure \ref{fig:defects:00-1_0-10_cmp}.
+%Results are presented in figure~\ref{fig:defects:00-1_0-10_cmp}.
%\begin{figure}[tp]
%\begin{center}
%\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_cmp.ps}
\caption{Reorientation barrier of the \ci{} \hkl[0 0 -1] to \hkl[0 -1 0] DB transition in place using the classical EA potential.}
\label{fig:defects:cp_00-1_ip0-10_mig}
\end{figure}
-Figures \ref{fig:defects:cp_00-1_0-10_mig} and \ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of the \ci{} \hkl[0 0 -1] to \hkl[0 -1 0] DB transition.
+Figures~\ref{fig:defects:cp_00-1_0-10_mig} and~\ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of the \ci{} \hkl[0 0 -1] to \hkl[0 -1 0] DB transition.
In the first case, the transition involves a change in the lattice site of the C atom whereas in the second case, a reorientation at the same lattice site takes place.
In the first case, the pathways for the two different time constants look similar.
A local minimum exists in between two peaks of the graph.
The probability of already rare diffusion events is further decreased for this reason.
However, agglomeration of C and diffusion of Si self-interstitials are an important part of the proposed SiC precipitation mechanism.
Thus, a serious limitation that has to be taken into account for appropriately modeling the C/Si system using the otherwise quite promising EA potential is revealed.
-Possible workarounds are discussed in more detail in section \ref{section:md:limit}.
+Possible workarounds are discussed in more detail in section~\ref{section:md:limit}.
\section{Combination of point defects and related diffusion processes}
\end{figure}
Fig.~\ref{fig:defects:combos} schematically displays the initial \ci{} \hkl[0 0 -1] DB structure (Fig.~\ref{fig:defects:combos_ci}) as well as the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (Fig.~\ref{fig:defects:combos_si}) and various positions for the second defect (1--5) that are used for investigating defect pairs.
The color of the number denotes the amount of possible atoms for the second defect resulting in equivalent configurations.
-Binding energies of the defect pair are determined by equation \ref{eq:basics:e_bind}.
+Binding energies of the defect pair are determined by equation~\ref{eq:basics:e_bind}.
Next to formation and binding energies, migration barriers are investigated, which allow to draw conclusions on the probability of the formation of such defect complexes by thermally activated diffusion processes.
\subsection[Pairs of \ci{} \hkl<1 0 0>-type interstitials]{\boldmath Pairs of \ci{} \hkl<1 0 0>-type interstitials}
The bond of Si atoms 1 and 2 does not persist.
Instead, the Si atom forms a bond with the initial \ci{} and the second C atom forms a bond with Si atom 1 forming four bonds in total.
The C atoms are spaced by \unit[3.14]{\AA}, which is very close to the expected C-C next neighbor distance of \unit[3.08]{\AA} in SiC.
-Figure \ref{fig:defects:205} displays the results of a \hkl[0 0 1] DB inserted at position 3.
+Figure~\ref{fig:defects:205} displays the results of a \hkl[0 0 1] DB inserted at position 3.
The binding energy is \unit[-2.05]{eV}.
Both DBs are tilted along the same direction remaining aligned in parallel and the second DB is pushed downwards in such a way, that the four DB atoms form a rhomboid.
Both C atoms form tetrahedral bonds to four Si atoms.
\end{figure}
Energetically beneficial configurations of defect combinations are observed for interstitials of all orientations placed at position 5, a position two bonds away from the initial interstitial along the \hkl[1 1 0] direction.
Relaxed structures of these combinations are displayed in Fig.~\ref{fig:defects:comb_db_03}.
-Fig.~\ref{fig:defects:153} and \ref{fig:defects:166} show the relaxed structures of \hkl[0 0 1] and \hkl[0 0 -1] DBs.
+Fig.~\ref{fig:defects:153} and~\ref{fig:defects:166} show the relaxed structures of \hkl[0 0 1] and \hkl[0 0 -1] DBs.
The upper DB atoms are pushed towards each other forming fourfold coordinated bonds.
While the displacements of the Si atoms in case (b) are symmetric to the \hkl(1 1 0) plane, in case (a) the Si atom of the initial DB is pushed a little further in the direction of the C atom of the second DB than the C atom is pushed towards the Si atom.
The bottom atoms of the DBs remain in threefold coordination.
The symmetric configuration is energetically more favorable ($E_{\text{b}}=-1.66\,\text{eV}$) since the displacements of the atoms is less than in the antiparallel case ($E_{\text{b}}=-1.53\,\text{eV}$).
-In Fig.~\ref{fig:defects:188} and \ref{fig:defects:138} the non-parallel orientations, namely the \hkl[0 -1 0] and \hkl[1 0 0] DBs, are shown.
+In Fig.~\ref{fig:defects:188} and~\ref{fig:defects:138} the non-parallel orientations, namely the \hkl[0 -1 0] and \hkl[1 0 0] DBs, are shown.
Binding energies of \unit[-1.88]{eV} and \unit[-1.38]{eV} are obtained for the relaxed structures.
In both cases the Si atom of the initial interstitial is pulled towards the near by atom of the second DB.
Both atoms form fourfold coordinated bonds to their neighbors.
Table~\ref{tab:defects:c-s} lists the energetic results of \cs{} combinations with the initial \ci{} \hkl[0 0 -1] DB.
For \cs{} located at position 1 and 3, the configurations $\alpha$ and A correspond to the naive relaxation of the structure by substituting the Si atom by a C atom in the initial \ci{} \hkl[0 0 -1] DB structure at positions 1 and 3 respectively.
However, small displacements of the involved atoms near the defect result in different stable structures labeled $\beta$ and B respectively.
-Fig.~\ref{fig:093-095} and \ref{fig:026-128} show structures A, B and $\alpha$, $\beta$ together with the barrier of migration for the A to B and $\alpha$ to $\beta$ transition respectively.
+Fig.~\ref{fig:093-095} and~\ref{fig:026-128} show structures A, B and $\alpha$, $\beta$ together with the barrier of migration for the A to B and $\alpha$ to $\beta$ transition respectively.
% A B
%./visualize_contcar -w 640 -h 480 -d results/c_00-1_c3_csub_B -nll -0.20 -0.4 -0.1 -fur 0.9 0.6 0.9 -c 0.5 -1.5 0.375 -L 0.5 0 0.3 -r 0.6 -A -1 2.465
% old c_int - c_substitutional stuff
-%Figures \ref{fig:defects:comb_db_04} and \ref{fig:defects:comb_db_05} show relaxed structures of substitutional carbon in combination with the \hkl<0 0 -1> dumbbell for several positions.
-%In figure \ref{fig:defects:comb_db_04} positions 1 (a)), 3 (b)) and 5 (c)) are displayed.
+%Figures~\ref{fig:defects:comb_db_04} and~\ref{fig:defects:comb_db_05} show relaxed structures of substitutional carbon in combination with the \hkl<0 0 -1> dumbbell for several positions.
+%In figure~\ref{fig:defects:comb_db_04} positions 1 (a)), 3 (b)) and 5 (c)) are displayed.
%A substituted carbon atom at position 5 results in an energetically extremely unfavorable configuration.
%Both carbon atoms, the substitutional and the dumbbell atom, pull silicon atom number 1 towards their own location regarding the \hkl<1 1 0> direction.
%Due to this a large amount of tensile strain energy is needed, which explains the high positive value of 0.49 eV.
%The lowest binding energy is observed for a substitutional carbon atom inserted at position 3.
%The substitutional carbon atom is located above the dumbbell substituting a silicon atom which would usually be bound to and displaced along \hkl<0 0 1> and \hkl<1 1 0> by the silicon dumbbell atom.
%In contrast to the previous configuration strain compensation occurs resulting in a binding energy as low as -0.93 eV.
-%Substitutional carbon at position 2 and 4, visualized in figure \ref{fig:defects:comb_db_05}, is located below the initial dumbbell.
+%Substitutional carbon at position 2 and 4, visualized in figure~\ref{fig:defects:comb_db_05}, is located below the initial dumbbell.
%Silicon atom number 1, which is bound to the interstitial carbon atom is displaced along \hkl<0 0 -1> and \hkl<-1 -1 0>.
%In case a) only the first displacement is compensated by the substitutional carbon atom.
%This results in a somewhat higher binding energy of -0.51 eV.
%The binding energy gets even higher in case b) ($E_{\text{b}}=-0.15\text{ eV}$), in which the substitutional carbon is located further away from the initial dumbbell.
%In both cases, silicon atom number 1 is displaced in such a way, that the bond to silicon atom number 5 vanishes.
-%In case of \ref{fig:defects:comb_db_04} a) the carbon atoms form a bond with a distance of 1.5 \AA, which is close to the C-C distance expected in diamond or graphit.
+%In case of~\ref{fig:defects:comb_db_04} a) the carbon atoms form a bond with a distance of 1.5 \AA, which is close to the C-C distance expected in diamond or graphit.
%Both carbon atoms are highly attracted by each other resulting in large displacements and high strain energy in the surrounding.
%A binding energy of 0.26 eV is observed.
%Substitutional carbon at positions 2, 3 and 4 are the energetically most favorable configurations and constitute promising starting points for SiC precipitation.
\caption[Relaxed structures of defect combinations obtained by creating a vacancy at positions 2, 3, 4 and 5.]{Relaxed structures of defect combinations obtained by creating a vacancy at positions 2 (a), 3 (b), 4 (c) and 5 (d).}
\label{fig:defects:comb_db_06}
\end{figure}
-Figure \ref{fig:defects:comb_db_06} shows the associated configurations.
+Figure~\ref{fig:defects:comb_db_06} shows the associated configurations.
All investigated structures are preferred compared to isolated, largely separated defects.
In contrast to C$_{\text{s}}$ this is also valid for positions along \hkl[1 1 0] resulting in an entirely attractive interaction between defects of these types.
Even for the largest possible distance (R) achieved in the calculations of the periodic supercell a binding energy as low as \unit[-0.31]{eV} is observed.
Strain reduced by this huge displacement is partially absorbed by tensile strain on Si atom number 1 originating from attractive forces of the C atom and the vacancy.
A binding energy of \unit[-0.50]{eV} is observed.
-The migration pathways of configuration \ref{fig:defects:314} and \ref{fig:defects:059} into the ground-state configuration, i.e.\ the \cs{} configuration, are shown in Fig.~\ref{fig:314-539} and \ref{fig:059-539} respectively.
+The migration pathways of configuration~\ref{fig:defects:314} and~\ref{fig:defects:059} into the ground-state configuration, i.e.\ the \cs{} configuration, are shown in Fig.~\ref{fig:314-539} and~\ref{fig:059-539} respectively.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{314-539.ps}
This is particularly important since the energy of formation of C$_{\text{s}}$ is drastically underestimated by the EA potential.
A possible occurrence of C$_{\text{s}}$ could then be attributed to a lower energy of formation of the C$_{\text{s}}$-Si$_{\text{i}}$ combination due to the low formation energy of C$_{\text{s}}$, which is obviously wrong.
-Since quantum-mechanical calculations reveal the Si$_{\text{i}}$ \hkl<1 1 0> DB as the ground-state configuration of Si$_{\text{i}}$ in Si, it was assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$ in the calculations carried out in section \ref{subsection:si-cs}.
+Since quantum-mechanical calculations reveal the Si$_{\text{i}}$ \hkl<1 1 0> DB as the ground-state configuration of Si$_{\text{i}}$ in Si, it was assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$ in the calculations carried out in section~\ref{subsection:si-cs}.
Empirical potentials, however, predict Si$_{\text{i}}$ T to be the energetically most favorable configuration.
Thus, investigations of the relative energies of formation of defect pairs need to include combinations of C$_{\text{s}}$ with Si$_{\text{i}}$ T.
Results of {\em ab initio} and classical potential calculations are summarized in Table~\ref{tab:defect_combos}.
A net magnetization of two electrons, which is already clear by simple molecular orbital theory considerations on the bonding of the $sp$ hybridized C atom, is settled.
By investigating the charge density isosurface it turns out that the two resulting spin up electrons are localized in a torus around the C atom.
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.
+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.
Already the geometry of the most stable DB configuration differs considerably from that obtained by first-principles calculations.
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