\chapter{Point defects in silicon}
+\label{chapter:defects}
Given the conversion mechnism of SiC in crystalline silicon introduced in section \ref{section:assumed_prec} the understanding of carbon and silicon interstitial point defects in c-Si is of great interest.
Both types of defects are examined in the following both by classical potential as well as density functional theory calculations.
Periodic boundary conditions in each direction are applied.
All point defects are calculated for the neutral charge state.
-\begin{figure}[th]
-\begin{center}
-\includegraphics[width=9cm]{unit_cell_e.eps}
-\end{center}
-\caption[Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration.]{Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration. The black dots ({\color{black}$\bullet$}) correspond to the silicon atoms and the blue lines ({\color{blue}-}) indicate the covalent bonds of the perfect c-Si structure.}
-\label{fig:defects:ins_pos}
-\end{figure}
-
-The interstitial atom positions are displayed in figure \ref{fig:defects:ins_pos}.
-In seperated simulation runs the silicon or carbon atom is inserted at the
-\begin{itemize}
- \item tetrahedral, $\vec{r}=(0,0,0)$, ({\color{red}$\bullet$})
- \item hexagonal, $\vec{r}=(-1/8,-1/8,1/8)$, ({\color{green}$\bullet$})
- \item nearly \hkl<1 0 0> dumbbell, $\vec{r}=(-1/4,-1/4,-1/8)$, ({\color{yellow}$\bullet$})
- \item nearly \hkl<1 1 0> dumbbell, $\vec{r}=(-1/8,-1/8,-1/4)$, ({\color{magenta}$\bullet$})
- \item bond-centered, $\vec{r}=(-1/8,-1/8,-3/8)$, ({\color{cyan}$\bullet$})
-\end{itemize}
-interstitial position.
-For the dumbbell configurations the nearest silicon atom is displaced by $(0,0,-1/8)$ and $(-1/8,-1/8,0)$ respectively of the unit cell length to avoid too high forces.
-A vacancy or a substitutional atom is realized by removing one silicon atom and switching the type of one silicon atom respectively.
-
-From an energetic point of view the free energy of formation $E_{\text{f}}$ is suitable for the characterization of defect structures.
-For defect configurations consisting of a single atom species the formation energy is defined as
-\begin{equation}
-E_{\text{f}}=\left(E_{\text{coh}}^{\text{defect}}
- -E_{\text{coh}}^{\text{defect-free}}\right)N
-\label{eq:defects:ef1}
-\end{equation}
-where $N$ and $E_{\text{coh}}^{\text{defect}}$ are the number of atoms and the cohesive energy per atom in the defect configuration and $E_{\text{coh}}^{\text{defect-free}}$ is the cohesive energy per atom of the defect-free structure.
-The formation energy of defects consisting of two or more atom species is defined as
-\begin{equation}
-E_{\text{f}}=E-\sum_i N_i\mu_i
-\label{eq:defects:ef2}
-\end{equation}
-where $E$ is the free energy of the interstitial system and $N_i$ and $\mu_i$ are the amount of atoms and the chemical potential of species $i$.
-The chemical potential is determined by the cohesive energy of the structure of the specific type in equilibrium at zero Kelvin.
-For a defect configuration of a single atom species equation \eqref{eq:defects:ef2} is equivalent to equation \eqref{eq:defects:ef1}.
-
\section{Silicon self-interstitials}
Point defects in silicon have been extensively studied, both experimentally and theoretically \cite{fahey89,leung99}.
The silicon dumbbell partner remains the same.
The bond to the face-centered silicon atom at the bottom of the unit cell breaks and a new one is formed to the face-centered atom at the forefront of the unit cell.
-\begin{figure}[t!h!]
-\begin{center}
-\begin{minipage}{6cm}
-\underline{Original}\\
-\includegraphics[width=6cm]{crt_orig.eps}
-\end{minipage}
-\begin{minipage}{1cm}
-\hfill
-\end{minipage}
-\begin{minipage}{6cm}
-\underline{Modified}\\
-\includegraphics[width=6cm]{crt_mod.eps}
-\end{minipage}
-\end{center}
-\caption{Schematic of the constrained relaxation technique (CRT) (left) and of the modified version (right) used to obtain migration pathways and corresponding activation energies.}
-\label{fig:defects:crt}
-\end{figure}
-Since the starting and final structure, which are both local minima of the potential energy surface, are known, the aim is to find the minimum energy path from one local minimum to the other one.
-One method to find a minimum energy path is to move the diffusing atom stepwise from the starting to the final position and only allow relaxation in the plane perpendicular to the direction of the vector connecting its starting and final position.
-This is called the constrained relaxation technique (CRT), which is schematically displayed in the left part of figure \ref{fig:defects:crt}.
-No constraints are applied to the remaining atoms in order to allow relaxation of the surrounding lattice.
-To prevent the remaining lattice to migrate according to the displacement of the defect an atom far away from the defect region is fixed in all three coordinate directions.
-However, it turned out, that this method tremendously failed applying it to the present migration pathways and structures.
-Abrupt changes in structure and free energy occured among relaxed structures of two successive displacement steps.
-For some structures even the expected final configurations were never obtained.
-Thus, the method mentioned above was adjusted adding further constraints in order to obtain smooth transitions, either in energy as well as structure is concerned.
-In this new method all atoms are stepwise displaced towards their final positions.
-Relaxation of each individual atom is only allowed in the plane perpendicular to the last individual displacement vector, as displayed in the right part of figure \ref{fig:defects:crt}.
-The modifications used to add this feature to the VASP code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
-Due to these constraints obtained activation energies can effectively be higher.
-{\color{red}Todo: To refine the migration barrier one has to find the saddle point structure and recalculate the free energy of this configuration with a reduced set of constraints.}
-
\subsection{Migration barriers obtained by quantum-mechanical calculations}
In the following migration barriers are investigated using quantum-mechanical calculations.
The method without updating the constraints but still applying them to all atoms shows a delayed crossing of the saddle point.
This is understandable since the update results in a more aggressive advance towards the final configuration.
In any case the barrier obtained is slightly higher, which means that it does not constitute an energetically more favorable pathway.
-The method in which the constraints are only applied to the diffusing C atom and two Si atoms, ... {\color{red}in progress} ...
+The method in which the constraints are only applied to the diffusing C atom and two Si atoms, ... {\color{red}Todo: does not work!} ...
\subsection{Migration barriers obtained by classical potential calculations}
\label{subsection:defects:mig_classical}
In the first migration path it is the configuration resulting from further relaxation of the rather unstable bond-centered configuration, which is fixed to be a transition point in the migration calculations.
The last two pathways show configurations almost identical to the \hkl<1 1 0> configuration, which constitute a local minimum within the pathway.
Thus, migration pathways with the \hkl<1 1 0> C-Si dumbbell interstitial configuration as a starting or final configuration are further investigated.
-\begin{figure}[ht!]
+\begin{figure}[!ht]
\begin{center}
\includegraphics[width=13cm]{110_mig.ps}
\end{center}
As mentioned earlier the bond-centered configuration itself constitutes a saddle point configuration relaxing into the energetically more favorable \hkl<1 1 0> configuration.
An activation energy of 2.2 eV is necessary to reorientate the \hkl<0 0 -1> dumbbell configuration into the \hkl<1 1 0> configuration, which is 1.3 eV higher in energy.
Residing in this state another 0.9 eV is enough to make the C atom form a \hkl<0 0 -1> dumbbell configuration with the Si atom of the neighboured lattice site.
-In contrast to quantum-mechanical calculations, in which the direct transition is the energetically most favorable transition and the transition composed of the intermmediate migration steps is very unlikely to occur the just presented pathway is much more supposable in classical potential simulations, since the energetically most favorable transition found so far is also composed of two migration steps with activation energies of 2.2 eV and 0.5 eV.
-{\color{red}Todo: Stress out that this is actually more probable, since BC conf is unstable!}
+In contrast to quantum-mechanical calculations, in which the direct transition is the energetically most favorable transition and the transition composed of the intermmediate migration steps is very unlikely to occur the just presented pathway is much more supposable in classical potential simulations, since the energetically most favorable transition found so far is also composed of two migration steps with activation energies of 2.2 eV and 0.5 eV, for which the intermediate state is the bond-centered configuration, which is unstable.
+Thus the just proposed migration path involving the \hkl<1 1 0> interstitial configuration becomes even more probable than path 1 involving the unstable bond-centered configuration.
Although classical potential simulations reproduce the order in energy of the \hkl<1 0 0> and \hkl<1 1 0> C-Si dumbbell interstitial configurations as obtained by more accurate quantum-mechanical calculations the obtained migration pathways and resulting activation energies differ to a great extent.
On the one hand the most favorable pathways differ.
As expected by the initialization conditions the two carbon atoms form a bond.
This bond has a length of 1.38 \AA{} close to the nex neighbour distance in diamond or graphite, which is approximately 1.54 \AA.
The minimum of binding energy observed for this configuration suggests prefered C clustering as a competing mechnism to the C-Si dumbbell interstitial agglomeration inevitable for the SiC precipitation.
-{\color{red}Todo: Activation energy to obtain a configuration of separated C atoms again or vice versa to obtain this configuration from separated C confs?}
+{\color{red}Todo: Activation energies to obtain separated C confs FAILED (again?) - could be added in the combined defect migration chapter and mentioned here, too!}
However, for the second most favorable configuration, presented in figure \ref{fig:defects:comb_db_01} a), the amount of possibilities for this configuration is twice as high.
In this configuration the initial Si (I) and C (I) dumbbell atoms are displaced along \hkl<1 0 0> and \hkl<-1 0 0> in such a way that the Si atom is forming tetrahedral bonds with two silicon and two carbon atoms.
The carbon and silicon atom constituting the second defect are as well displaced in such a way, that the carbon atom forms tetrahedral bonds with four silicon neighbours, a configuration expected in silicon carbide.
The ground state of a single Si self-interstitial was found to be the Si \hkl<1 1 0> self-interstitial configuration.
For the follwoing study the same type of self-interstitial is assumed to provide the energetically most favorable configuration in combination with substitutional C.
-\begin{table}[ht!]
+\begin{table}[!ht]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\caption{Equivalent configurations of \hkl<1 1 0>-type Si self-interstitials created at position I of figure \ref{fig:defects:pos_of_comb} and substitutional C created at positions 1 to 5.}
\label{tab:defects:comb_csub_si110}
\end{table}
-\begin{table}[ht!]
+\begin{table}[!ht]
\begin{center}
\begin{tabular}{l c c c c c c c c c c}
\hline
\begin{center}
\includegraphics[width=12cm]{c_sub_si110.ps}
\end{center}
-\caption{Binding energy of combinations of a substitutional C and a Si \hkl<1 1 0> dumbbell self-interstitial with respect to the separation distance.}
+\caption[Binding energy of combinations of a substitutional C and a Si \hkl<1 1 0> dumbbell self-interstitial with respect to the separation distance.]{Binding energy of combinations of a substitutional C and a Si \hkl<1 1 0> dumbbell self-interstitial with respect to the separation distance. The binding energy of the defect pair is well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}
\label{fig:defects:csub_si110}
\end{figure}
According to the formation energies none of the investigated structures is energetically preferred over the C-Si \hkl<1 0 0> dumbbell interstitial, which exhibits a formation energy of 3.88 eV.
Thus, the C-Si \hkl<1 0 0> dumbbell structure remains the ground state configuration of a C interstitial in c-Si with a constant number of Si atoms.
{\color{blue}
-However the binding energy quickly drops to zero with respect of the distance indicating a possibly low interaction capture radius of the defect pair.
+However the binding energy quickly drops to zero with respect to the distance, which is reinforced by the Lennard-Jones fit estimating almost zero interaction energy already at 0.6 nm.
+This indicates a possibly low interaction capture radius of the defect pair.
Highly energetic collisions in the IBS process might result in separations of these defects exceeding the capture radius.
For this reason situations most likely occur in which the configuration of substitutional C can be considered without a nearby interacting Si self-interstitial and, thus, unable to form a thermodynamically more stable C-Si \hkl<1 0 0> dumbbell configuration.
}
+\label{section:defects:noneq_process_01}
The energetically most favorable configuration of the combined structures is the one with the substitutional C atom located next to the \hkl<1 1 0> interstitial along the \hkl<1 1 0> direction (configuration \RM{1}).
Compressive stress along the \hkl<1 1 0> direction originating from the Si \hkl<1 1 0> self-intesrtitial is partially compensated by tensile stress resulting from substitutional C occupying the neighboured Si lattice site.
The substitutional C is located next to the lattice site shared by the \hkl<1 1 0> Si self-interstitial along the \hkl<1 -1 0> direction.
Thus, the compressive stress along \hkl<1 1 0> of the Si \hkl<1 1 0> interstitial is not compensated but intensified by the tensile stress of the substitutional C atom, which is no longer loacted along the direction of stress.
-{\color{red}Todo: Mig of C-Si DB conf to or from C sub + Si 110 int conf.}
+{\color{red}Todo: Erhart/Albe calc for most and less favorable configuration!}
+
+{\color{red}Todo: Mig of C-Si DB conf to or from C sub + Si 110 in progress.}
+
+{\color{red}Todo: Mig of Si DB located next to a C sub (also by MD!).}
\section{Migration in systems of combined defects}
Due to this and due to the formation of new bonds, that is the bond of silicon atom number 1 to silicon atom number 5 and the bond of the carbon atom to its siliocn neighbour in the bottom left, a less steep increase of free energy is observed.
At a displacement of approximately 30 \% the bond of silicon atom number 1 to the just recently created siliocn atom is broken up again, which explains the repeated boost in energy.
Finally the system gains energy relaxing into the configuration of zero displacement.
+{\color{red}Todo: Direct migration of C in progress.}
Due to the low binding energy observed, the configuration of the vacancy created at position 3 is assumed to be stable against transition.
However, a relatively simple migration path exists, which intuitively seems to be a low energy process.
\section{Conclusions concerning the SiC conversion mechanism}
-The ground state configuration of a carbon interstitial in crystalline siliocn is found to be the C-Si \hkl<1 0 0> dumbbell interstitial configuration.
-The threefold coordinated carbon and silicon atom share a usual silicon lattice site.
-Migration simulations reveal the carbon interstitial to be mobile at prevailing implantation temperatures requireing an activation energy of approximately 0.9 eV for migration as well as reorientation processes.
+The ground state configuration of a carbon interstitial in crystalline siliocn is found to be the C-Si \hkl<1 0 0> dumbbell interstitial configuration, in which the threefold coordinated carbon and silicon atom share a usual silicon lattice site.
+This supports the assumption of C-Si \hkl<1 0 0>-type dumbbel interstitial formation in the first steps of the IBS process as proposed by the precipitation model introduced in section \ref{section:assumed_prec}.
+
+Migration simulations reveal this carbon interstitial to be mobile at prevailing implantation temperatures requireing an activation energy of approximately 0.9 eV for migration as well as reorientation processes.
+This enables possible migration of the defects to form defect agglomerates as demanded by the model.
+Unfortunately classical potential simulations show tremendously overestimated migration barriers indicating a possible failure of the necessary agglomeration of such defects.
Investigations of two carbon interstitials of the \hkl<1 0 0>-type and varying separations and orientations state an attractive interaction between these interstitials.
Depending on orientation, energetically favorable configurations are found in which these two interstitials are located close together instead of the occurernce of largely separated and isolated defects.
This is due to strain compensation enabled by the combination of such defects in certain orientations.
For dumbbells oriented along the \hkl<1 1 0> direction and the assumption that there is the possibility of free orientation, an interaction energy proportional to the reciprocal cube of the distance in the far field regime is found.
-These findings support the assumption of the C-Si dumbbell agglomeration proposed by the precipitation model introduced in section \ref{section:assumed_prec}.
+These findings support the assumption of the C-Si dumbbell agglomeration proposed by the precipitation model.
Next to the C-Si \hkl<1 0 0> dumbbell interstitial configuration, in which the C atom is sharing a Si lattice site with the corresponding Si atom the C atom could occupy the site of the Si atom, which in turn forms a Si self-interstitial.
Combinations of substitutional C and a \hkl<1 1 0> Si self-interstitial, which is the ground state configuration for a Si self-interstitial and, thus, assumed to be the energetically most favorable configuration for combined structures, show formation energies 0.5 eV to 1.5 eV greater than that of the C-Si \hkl<1 0 0> interstitial configuration, which remains the energetically most favorable configuration.
However, the binding energy of substitutional C and the Si self-interstitial quickly drops to zero already for short separations indicating a low interaction capture radius.
-Thus, due to missing attractive interaction forces driving the system to form C-Si \hkl<1 0 0> dumbbell interstitials substitutional C, while thermodynamically not stable, constitutes a most likely configuration occuring in IBS, a process far from equlibrium.
+Thus, due to missing attractive interaction forces driving the system to form C-Si \hkl<1 0 0> dumbbell interstitial complexes substitutional C, while thermodynamically not stable, constitutes a most likely configuration occuring in IBS, a process far from equlibrium.
Due to the low interaction capture radius substitutional C can be treated independently of the existence of separated Si self-interstitials.
This should be also true for combinations of C-Si interstitials next to a vacancy and a further separated Si self-interstitial excluded from treatment, which again is a conveivable configuration in IBS.
-By combination of the \hkl<1 0 0> dumbbell with a vacancy it is found that the configuration of substitutional carbon occupying the vacant site is the energetically most favorable configuration.
+By combination of a \hkl<1 0 0> dumbbell with a vacancy in the absence of the Si self-interstitial it is found that the configuration of substitutional carbon occupying the vacant site is the energetically most favorable configuration.
Low migration barriers are necessary to obtain this configuration and in contrast comparatively high activation energies necessary for the reverse process.
Thus, carbon interstitials and vacancies located close together are assumed to end up in such a configuration in which the carbon atom is tetrahedrally coordinated and bound to four silicon atoms as expected in silicon carbide.
-While first results point to ...
-
-In contrast to the above, this would suggest a silicon carbide precipitation by succesive creation of substitutional carbon instead of the agglomeration of C-Si dumbbell interstitials followed by an abrupt precipitation.
+While first results support the proposed precipitation model the latter suggest the formation of silicon carbide by succesive creation of substitutional carbon instead of the agglomeration of C-Si dumbbell interstitials followed by an abrupt transition.
+Prevailing conditions in the IBS process at elevated temperatures and the fact that IBS is a nonequilibrium process reinforce the possibility of formation of substitutional C instead of the thermodynamically stable C-Si dumbbell interstitials predicted by simulations at zero Kelvin.
+\label{section:defects:noneq_process_02}
-0 K simulations -> C-Si DB, however non-zero temperatures and the IBS, process far from equilibrium, so sub C should be feasible ...
-
-{\color{red}Todo: Explain that formation of SiC by substitutional C is more likely than the supposed C-Si agglomeration, at least in the absence of the accompanied Si self-interstitial.}
+{\color{blue}
+In addition, there are experimental findings, which might be exploited to reinforce the non-validity of the proposed precipitation model.
+High resolution TEM shows equal orientation of \hkl(h k l) planes of the c-Si host matrix and the 3C-SiC precipitate.
+Formation of 3C-SiC realized by successive formation of substitutional C, in which the atoms belonging to one of the two fcc lattices are substituted by C atoms perfectly conserves the \hkl(h k l) planes of the initial c-Si diamond lattice.
+Silicon self-interstitials consecutively created to the same degree are able to diffuse into the c-Si host one after another.
+Investigated combinations of C interstitials, however, result in distorted configurations, in which C atoms, which at some point will form SiC, are no longer aligned to the host.
+It is easily understandable that the mismatch in alignement will increase with increasing defect density.
+In addition, the amount of Si self-interstitials equal to the amount of agglomerated C atoms would be released all of a sudden probably not being able to diffuse into the c-Si host matrix without damaging the Si surrounding or the precipitate itself.
+In addition, IBS results in the formation of the cubic polytype of SiC only.
+As this result conforms well with the model of precipitation by substitutional C there is no obvious reason why hexagonal polytypes should not be able to form or an equal alignement would be mandatory assuming the model of precipitation by C-Si dumbbell agglomeration.
+}
-{\color{red}Todo: Si \hkl<1 1 0> migration barriers. If Si can go away fast, formation of substitutional C (and thus formation of SiC) might be a more probable process than C-Si dumbbell agglomeration.}
+{\color{red}Todo: C mobility higher than Si mobility? -> substitutional C is more likely to arise, since it migrates 'faster' to vacant sites?}
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