\underline{Augsburg}
\begin{itemize}
- \item Prof. B. Stritzker (accomodation at EP \RM{4})
+ \item Prof. B. Stritzker (accomodation at EP \RM{4} and financial support)
\item Ralf Utermann (EDV)
\end{itemize}
\item Dr. S. Sanna (VASP)
\end{itemize}
+\ldots will be written out in full soon!
\section{Molecular dynamics simulations}
\label{section:md}
-% todo
-% rewrite!
+% todo - rewrite md intro chapter
\begin{quotation}
\dq We may regard the present state of the universe as the effect of the past and the cause of the future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.\dq{}
Whether a saddle point configuration and, thus, the minimum energy path is obtained by the CRT method, needs to be verified by caculating the respective vibrational modes.
Modifications used to add the CRT feature to the VASP code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
-% todo
-% advantages of pw basis with respect to hellmann feynman forces / pulay forces
-% crt sketch needs increased text
+% todo - advantages of pw basis concenring hf forces + inc font in crt sketch
\chapter{Description of programs and tools}
\label{app:code}
+
+Programs and tools utilized within this study are introduced in the following.
+
+\section{Contents of the \textsc{posic} program suite}
+
+\begin{itemize}
+ \item mdrun
+ \item moldyn.\{c,h\}
+ \item potentials/albe.\{c,h\}
+ \item ...
+ \item ...
+\end{itemize}
+
+\section{\textsc{vasp} utilities}
+
+\begin{itemize}
+ \item mig\_fullct.sh, mig\_calc
+ \item e\_form\_tersoff, e\_fc\_tersoff
+ \item ...
+ \item ...
+ \item ...
+\end{itemize}
+
+\ldots will be written out in full soon!
\section{Form of the Tersoff potential and its derivative}
-The Tersoff potential is of the form
+The Tersoff potential \cite{tersoff_m} is of the form
\begin{eqnarray}
E & = & \sum_i E_i = \frac{1}{2} \sum_{i \ne j} V_{ij} \textrm{ ,} \\
V_{ij} & = & f_C(r_{ij}) [ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) ] \textrm{ .}
\zeta_{ij} & = & \sum_{k \ne i,j} f_C (r_{ik}) \omega_{ik} g(\theta_{ijk}) \textrm{ ,}\\
g(\theta_{ijk}) & = & 1 + c_i^2/d_i^2 - c_i^2/[d_i^2 + (h_i - \cos \theta_{ijk})^2] \textrm{ .}
\end{eqnarray}
-The cutoff function $f_C$ is taken to be
+The cut-off function $f_C$ is taken to be
\begin{equation}
f_C(r_{ij}) = \left\{
\begin{array}{ll}
\section{Implementation issues}
-As seen in the last sections the derivatives of $V_{ij}$, $V_{ji}$ and $V_{jk}$
+As seen in the last sections, the derivatives of $V_{ij}$, $V_{ji}$ and $V_{jk}$
with respect to ${\bf r}_i$ are necessary to compute the forces for atom $i$.
According to this, for every triple $(ijk)$ the derivatives of the three
potential contributions, denoted by $V_{ijk}$, $V_{jik}$ and $V_{jki}$
derivatives for each $(ijk)$ triple.
The $V_{jik}$ and $V_{jki}$ potential and its derivatives will be calculated
in subsequent loops anyways.
-To avoid multiple computation of the same potential derivatives
+To avoid multiple computation of the same potential derivatives,
the force contributions for atom $j$ and $k$ due to the $V_{ijk}$ contribution
have to be considered by calculating the derivatives of $V_{ijk}$
with respect to ${\bf r}_j$ and ${\bf r}_k$
keeping in mind that all the necessary force contributions for atom $i$
are calculated and added in subsequent loops.
-The following symmetry considerations help to obtain the
-
- \subsection{Derivative of $V_{ij}$ with respect to ${\bf r}_j$}
+\subsection{Derivative of $V_{ij}$ with respect to ${\bf r}_j$}
\begin{eqnarray}
\nabla_{{\bf r}_j} V_{ij} & = &
\nabla_{{\bf r}_j} f_A(r_{ij}) &=& - \nabla_{{\bf r}_i} f_A(r_{ij}) \\
\nabla_{{\bf r}_j} f_C(r_{ij}) &=& - \nabla_{{\bf r}_i} f_C(r_{ij})
\end{eqnarray}
-The pair contributions .... easy.
-Now having a look at $b_{ij}$.
+The pair contributions are, thus, easily obtained.
+The contribution of the bond order term is given by:
\begin{eqnarray}
\nabla_{{\bf r}_j}\cos\theta_{ijk} &=&
\nabla_{{\bf r}_j}\Big(\frac{{\bf r}_{ij}{\bf }r_{ik}}{r_{ij}r_{ik}}\Big)
\frac{\cos\theta_{ijk}}{r_{ij}^2}{\bf r}_{ij}
\end{eqnarray}
- \subsection{Derivative of $V_{ij}$ with respect to ${\bf r}_k$}
+\subsection{Derivative of $V_{ij}$ with respect to ${\bf r}_k$}
The derivative of $V_{ij}$ with respect to ${\bf r}_k$ just consists of the
single term
\nabla_{{\bf r}_k} f_A(r_{ij}) &=& 0 \\
\nabla_{{\bf r}_k} f_C(r_{ij}) &=& 0
\end{eqnarray}
-Now look at $b_{ij}$, not only angle important here!
+Concerning $b_{ij}$, in addition to the angular term, the derivative of the cut-off function has to be considered.
\begin{eqnarray}
\nabla_{{\bf r}_k}\zeta_{ij} &=&
g(\theta_{ijk})\nabla_{{\bf r}_k}f_C(r_{ik}) +
f_C(r_{ik})\nabla_{{\bf r}_k}g(\theta_{ijk}) \\
\nabla_{{\bf r}_k}f_C(r_{ik}) &=& - \nabla_{{\bf r}_i}f_C(r_{ik}) \\
\nabla_{{\bf r}_k}\cos\theta_{ijk} &=&
- \nabla_{{\bf r}_k}\Big(\frac{{\bf r}_{ij}{\bf }r_{ik}}{r_{ij}r_{ik}}\Big)
+ \nabla_{{\bf r}_k}\Big(\frac{{\bf r}_{ij}{\bf r}_{ik}}{r_{ij}r_{ik}}\Big)
\nonumber \\
&=&\frac{1}{r_{ij}r_{ik}}{\bf r}_{ij} -
\frac{\cos\theta_{ijk}}{r_{ik}^2}{\bf r}_{ik}
\subsection{Code realization}
-The implementation of the force evaluation shown in the following
-is applied to the potential designed by Erhard and Albe.
-There are slight differences comparted to the original potential by Tersoff:
+The implementation of the force evaluation shown in the following is applied to the potential designed by Erhard and Albe \cite{albe_sic_pot}.
+There are slight differences compared to the original potential by Tersoff:
\begin{itemize}
\item Difference in sign of the attractive part.
\item $c$, $d$ and $h$ values depend on atom $k$ in addition to atom $i$.
\item The exponent of the $b$ term is constantly $-\frac{1}{2}$.
\end{itemize}
These differences actually slightly ease code realization.
+The respective flow chart is displayed in Fig.~\ref{fig:flowchart}.
\begin{figure}
\renewcommand\labelitemi{}
\renewcommand\labelitemii{}
\renewcommand\labelitemiii{}
+{\small
+\fbox{\begin{minipage}{\textwidth}
LOOP i \{
\begin{itemize}
\item // nop (only used in orig. Tersoff)
\item LOOP j \{
\begin{itemize}
\item $\zeta_{ij}=0$
- \item set $S_{ij}$ (cutoff)
+ \item set $S_{ij}$ (cut-off)
\item calculate: $r_{ij}$, $r_{ij}^2$
\item IF $r_{ij} > S_{ij}$ THEN CONTINUE
\item
\item \}
\end{itemize}
\}
-\caption{Implementation of the force evaluation for Tersoff like bond-order
- potentials using pseudocode.}
+\end{minipage}}
+}
+\caption{Flow chart of the force evaluation for Tersoff-like bond order potentials using pseudocode.}
+\label{fig:flowchart}
\end{figure}
\begin{figure}[tp]
\begin{center}
\begin{minipage}{8cm}
-\includegraphics[width=8cm]{c_pd_vasp/bc_2333.eps}\\
+\begin{center}
+\includegraphics[width=6cm]{c_pd_vasp/bc_2333.eps}\\
+\vspace*{0.2cm}
\hrule
\vspace*{0.2cm}
-\includegraphics[width=8cm]{c_100_mig_vasp/im_spin_diff.eps}
+\includegraphics[width=6cm]{c_100_mig_vasp/im_spin_diff.eps}
+\vspace*{0.2cm}
+\framebox{
+ \footnotesize
+ \begin{minipage}[t]{7.5cm}
+ \begin{minipage}[t]{1.4cm}
+ {\color{red}Si}\\
+ {\tiny sp$^3$}\\[0.8cm]
+ \underline{${\color{black}\uparrow}$}
+ \underline{${\color{black}\uparrow}$}
+ \underline{${\color{black}\uparrow}$}
+ \underline{${\color{red}\uparrow}$}\\
+ sp$^3$
+ \end{minipage}
+ \begin{minipage}[t]{1.6cm}
+ \begin{center}
+ {\color{red}M}{\color{blue}O}\\[0.8cm]
+ \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
+ $\sigma_{\text{ab}}$\\[0.5cm]
+ \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
+ $\sigma_{\text{b}}$
+ \end{center}
+ \end{minipage}
+ \begin{minipage}[t]{1.2cm}
+ \begin{center}
+ {\color{blue}C}\\
+ {\tiny sp}\\[0.2cm]
+ \underline{${\color{white}\uparrow\uparrow}$}
+ \underline{${\color{white}\uparrow\uparrow}$}\\
+ 2p\\[0.4cm]
+ \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
+ \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
+ sp
+ \end{center}
+ \end{minipage}
+ \begin{minipage}[t]{1.6cm}
+ \begin{center}
+ {\color{blue}M}{\color{green}O}\\[0.8cm]
+ \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
+ $\sigma_{\text{ab}}$\\[0.5cm]
+ \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
+ $\sigma_{\text{b}}$
+ \end{center}
+ \end{minipage}
+ \begin{minipage}[t]{1.4cm}
+ \begin{flushright}
+ {\color{green}Si}\\
+ {\tiny sp$^3$}\\[0.8cm]
+ \underline{${\color{green}\uparrow}$}
+ \underline{${\color{black}\uparrow}$}
+ \underline{${\color{black}\uparrow}$}
+ \underline{${\color{black}\uparrow}$}\\
+ sp$^3$
+ \end{flushright}
+ \end{minipage}
+ \end{minipage}
+}
+\end{center}
\end{minipage}
\begin{minipage}{7cm}
\includegraphics[width=7cm]{c_pd_vasp/bc_2333_ksl.ps}
\end{minipage}
\end{center}
-\caption[Structure, charge density isosurface and Kohn-Sham level diagram of the bond-centered interstitial configuration.]{Structure, charge density isosurface and Kohn-Sham level diagram of the bond-centered interstitial configuration. Gray, green and blue surfaces mark the charge density of spin up, spin down and the resulting spin up electrons in the charge density isosurface, in which the carbon atom is represented by a red sphere. In the energy level diagram red and green lines mark occupied and unoccupied states.}
+\caption[Structure, charge density isosurface, molecular orbital diagram and Kohn-Sham level diagram of the bond-centered interstitial configuration.]{Structure, charge density, molecular orbital diagram isosurface and Kohn-Sham level diagram of the bond-centered interstitial configuration. Gray, green and blue surfaces mark the charge density of spin up, spin down and the resulting spin up electrons in the charge density isosurface, in which the carbon atom is represented by a red sphere. In the energy level diagram red and green lines mark occupied and unoccupied states.}
\label{img:defects:bc_conf}
\end{figure}
In the BC insterstitial configuration the interstitial atom is located in between two next neighbored Si atoms forming linear bonds.
This is supported by the charge density isosurface and the Kohn-Sham levels in Fig. \ref{img:defects:bc_conf}.
The blue torus, which reinforces the assumption of the $p$ orbital, illustrates the resulting spin up electron density.
In addition, the energy level diagram shows a net amount of two spin up electrons.
-% todo smaller images, therefore add mo image
\section{Migration of the carbon interstitial}
\label{subsection:100mig}
Since the \ci{} \hkl<1 0 0> DB is the most probable, hence, most important configuration, the migration of this defect atom from one site of the Si host lattice to a neighboring site is in the focus of investigation.
\begin{figure}[tp]
\begin{center}
+%
\begin{minipage}{15cm}
-\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
+\centering
+\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}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/100_next_2333.eps}
\end{minipage}
-\end{minipage}\\
+\end{minipage}\\[0.5cm]
+%
\begin{minipage}{15cm}
-\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
+\centering
+\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}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/0-10_2333.eps}
\end{minipage}
-\end{minipage}\\
+\end{minipage}\\[0.5cm]
+%
\begin{minipage}{15cm}
-\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
+\centering
+\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}
\begin{figure}[tp]
\begin{center}
-\includegraphics[width=13cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[1.5cm]
-\begin{picture}(0,0)(150,0)
-\includegraphics[width=2.5cm]{vasp_mig/00-1.eps}
-\end{picture}
-\begin{picture}(0,0)(-10,0)
-\includegraphics[width=2.5cm]{vasp_mig/bc_00-1_sp.eps}
-\end{picture}
-\begin{picture}(0,0)(-120,0)
-\includegraphics[width=2.5cm]{vasp_mig/bc.eps}
-\end{picture}
-\begin{picture}(0,0)(25,20)
-\includegraphics[width=2.5cm]{110_arrow.eps}
-\end{picture}
-\begin{picture}(0,0)(200,0)
-\includegraphics[height=2.2cm]{001_arrow.eps}
-\end{picture}
+\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 (left) to BC (right) 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}
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_0-10_vasp_s.ps}
\end{center}
-\caption[Migration barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition.]{Migration barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition. Bonds of the C atom are illustrated by blue lines. {\color{red} Prototype design, adjust related figures!}}
+\caption[Migration barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition.]{Migration barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition. Bonds of the C atom are illustrated by blue lines.}
% todo read above caption! enable [] hkls in short caption
\label{fig:defects:00-1_0-10_mig}
\end{figure}
\begin{figure}[tp]
\begin{center}
-\includegraphics[width=13cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[1.8cm]
-\begin{picture}(0,0)(140,0)
-\includegraphics[width=2.2cm]{vasp_mig/00-1_b.eps}
-\end{picture}
-\begin{picture}(0,0)(20,0)
-\includegraphics[width=2.2cm]{vasp_mig/00-1_ip0-10_sp.eps}
-\end{picture}
-\begin{picture}(0,0)(-120,0)
-\includegraphics[width=2.2cm]{vasp_mig/0-10_b.eps}
-\end{picture}
-\begin{picture}(0,0)(25,20)
-\includegraphics[width=2.5cm]{100_arrow.eps}
-\end{picture}
-\begin{picture}(0,0)(200,0)
-\includegraphics[height=2.2cm]{001_arrow.eps}
-\end{picture}
+\includegraphics[width=0.7\textwidth]{00-1_ip0-10_nosym_sp_fullct_vasp_s.ps}
\end{center}
\caption[Reorientation barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition in place.]{Reorientation barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition in place. Bonds of the carbon atoms are illustrated by blue lines.}
\label{fig:defects:00-1_0-10_ip_mig}
%\includegraphics[height=2.2cm]{010_arrow.eps}
%\end{picture}
\end{center}
-\caption[Migration barrier and structures of the \ci{} BC to \hkl<0 0 -1> DB transition using the classical EA potential.]{Migration barrier and structures of the \ci{} BC to \hkl[0 0 -1] DB transition using the classical EA potential. Two migration pathways are obtained for different time constants of the Berendsen thermostat. The lowest activation energy is \unit[2.2]{eV}. {\color{red} Prototype design, adjust related figures!}}
+\caption[Migration barrier and structures of the \ci{} BC to \hkl<0 0 -1> DB transition using the classical EA potential.]{Migration barrier and structures of the \ci{} BC to \hkl[0 0 -1] DB transition using the classical EA potential. Two migration pathways are obtained for different time constants of the Berendsen thermostat. The lowest activation energy is \unit[2.2]{eV}.}
\label{fig:defects:cp_bc_00-1_mig}
% red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20 -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.75 -1.25 -0.25 -L -0.25 -0.25 -0.25 -r 0.6 -B 0.1
% blue: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20_tr100/ -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.0 -0.25 1.0 -L 0.0 -0.25 -0.25 -r 0.6 -B 0.1
\begin{figure}[tp]
\begin{center}
-\includegraphics[width=13cm]{00-1_0-10.ps}\\[2.4cm]
-\begin{pspicture}(0,0)(0,0)
-\psframe[linecolor=red,fillstyle=none](-6,-0.5)(7.2,2.8)
-\end{pspicture}
-\begin{picture}(0,0)(130,-10)
-\includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_00.eps}
-\end{picture}
-\begin{picture}(0,0)(0,-10)
-\includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_min.eps}
-\end{picture}
-\begin{picture}(0,0)(-120,-10)
-\includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_03.eps}
-\end{picture}
-\begin{picture}(0,0)(25,10)
-\includegraphics[width=2.5cm]{100_arrow.eps}
-\end{picture}
-\begin{picture}(0,0)(185,-10)
-\includegraphics[height=2.2cm]{001_arrow.eps}
-\end{picture}
+\includegraphics[width=0.7\textwidth]{00-1_0-10_albe_s.ps}
\end{center}
\caption{Migration barrier and structures of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0> DB transition using the classical EA potential.}
% red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_00-1_0-10_s20 -nll -0.56 -0.56 -0.8 -fur 0.3 0.2 0 -c -0.125 -1.7 0.7 -L -0.125 -0.25 -0.25 -r 0.6 -B 0.1
\fi
-% todo
-% maybe move above stuff to conclusion chapter, at least shorten!
-% see remember in sic chapter
+% todo - sync with conclusion chapter
These findings allow to draw conclusions on the mechanisms involved in the process of SiC conversion in Si.
Agglomeration of C$_{\text{i}}$ is energetically favored and enabled by a low activation energy for migration.
\hline
\end{tabular}
\end{center}
-\caption[Properties of SiC polytypes and other semiconductor materials.]{Properties of SiC polytypes and other semiconductor materials. Doping concentrations are $10^{16}\text{ cm}^{-3}$ (A) and $10^{17}\text{ cm}^{-3}$ (B) respectively. References: \cite{wesch96,casady96,park98}. {\color{red}Todo: add more refs + check all values!}}
+\caption[Properties of SiC polytypes and other semiconductor materials.]{Properties of SiC polytypes and other semiconductor materials. Doping concentrations are $10^{16}\text{ cm}^{-3}$ (A) and $10^{17}\text{ cm}^{-3}$ (B) respectively. References: \cite{wesch96,casady96,park98}.}
\label{table:sic:properties}
\end{table}
+% todo add more refs + check all values!
Different polytypes of SiC exhibit different properties.
Some of the key properties are listed in Table~\ref{table:sic:properties} and compared to other technologically relevant semiconductor materials.
Despite the lower charge carrier mobilities for low electric fields SiC outperforms Si concerning all other properties.
High resolution transmission electron microscopy (HREM) investigations of C-implanted Si at room temperature followed by rapid thermal annealing (RTA) show the formation of C-Si dumbbell agglomerates, which are stable up to annealing temperatures of about \unit[700-800]{$^{\circ}$C}, and a transformation into 3C-SiC precipitates at higher temperatures \cite{werner96,werner97}.
The precipitates with diamateres between \unit[2]{nm} and \unit[5]{nm} are incorporated in the Si matrix without any remarkable strain fields, which is explained by the nearly equal atomic density of C-Si agglomerates and the SiC unit cell.
Implantations at \unit[500]{$^{\circ}$C} likewise suggest an initial formation of C-Si dumbbells on regular Si lattice sites, which agglomerate into large clusters \cite{lindner99_2}.
-The agglomerates of such dimers, which do not generate lattice strain but lead to a local increase of the lattice potential \cite{werner96,wener97}, are indicated by dark contrasts and otherwise undisturbed Si lattice fringes in HREM, as can be seen in Fig.~\ref{fig:sic:hrem:c-si}.
+The agglomerates of such dimers, which do not generate lattice strain but lead to a local increase of the lattice potential \cite{werner96,werner97}, are indicated by dark contrasts and otherwise undisturbed Si lattice fringes in HREM, as can be seen in Fig.~\ref{fig:sic:hrem:c-si}.
\begin{figure}[t]
\begin{center}
\subfigure[]{\label{fig:sic:hrem:c-si}\includegraphics[width=0.25\columnwidth]{tem_c-si-db.eps}}
% strane94/guedj98: my model - c redist by si int (spe) and surface diff (mbe)
% serre95: low/high t implants -> mobile c_i / non-mobile sic precipitates
-% todo
-% own polytype stacking sequence image
+% todo - own polytype stacking sequence image
Due to restrictions by the {\textsc vasp} code, {\em ab initio} MD could only be performed at constant volume.
In MD simulations the equations of motion are integrated by a fourth order predictor corrector algorithm for a timestep of \unit[1]{fs}.
-% todo
-% All point defects are calculated for the neutral charge state.
+% todo - point defects are calculated for the neutral charge state.
Most of the parameter settings, as determined above, constitute a tradeoff regarding the tasks that need to be addressed.
These parameters include the size of the supercell, cut-off energy and $k$ point mesh.
\subsection{Energy cut-off}
To determine an appropriate cut-off energy of the plane-wave basis set a $2\times2\times2$ supercell of type 3 containing $32$ Si and $32$ C atoms in the 3C-SiC structure is equilibrated for different cut-off energies in the LDA.
-% todo
-% mention that results are within lda
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\textwidth]{sic_32pc_gamma_cutoff_lc.ps}
A nice agreement with experimental results is achieved.
Clearly, a competent parameter set is found, which is capabale of describing the C/Si system by {\em ab initio} calculations.
-% todo
-% rewrite dft chapter
-% ref for experimental values!
+% todo - ref for experimental values!
\section{Classical potential MD}
\label{section:classpotmd}
There are basically no free parameters, which could be set by the user and the properties of the potential and its parameters are well known and have been extensively tested by the authors \cite{albe_sic_pot}.
Therefore, test calculations are restricted to the time step used in the Verlet algorithm to integrate the equations of motion.
Nevertheless, a further and rather uncommon test is carried out to roughly estimate the capabilities of the EA potential regarding the description of 3C-SiC precipitation in c-Si.
-% todo
-% rather a first investigation than a test
\subsection{Time step}
Thus, the EA potential is considered an appropriate choice for the current study concerning the accurate description of the energetics of interfaces.
Furthermore, since the calculated interfacial energy is located in the lower part of the experimental range, the obtained interface structure might resemble an authentic configuration of an energetically favorable interface structure of a 3C-SiC precipitate in c-Si.
-% todo
-% nice to reproduce this value!
+% todo - nice to reproduce this value!
\subsubsection{Stability of the precipitate}
\chapter{Summary and conclusions}
\label{chapter:summary}
+{\setlength{\parindent}{0pt}
\paragraph{To summarize,}
in a short review of the C/Si compound and the fabrication of the technologically promising semiconductor SiC by IBS, two controversial assumptions of the precipitation mechanism of 3C-SiC in c-Si are elaborated.
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These propose the precipitation of SiC by agglomeration of \ci{} DBs followed by a sudden formation of SiC and otherwise a formation by successive accumulation of \cs{} via intermediate stretched SiC structures, which are coherent to the Si lattice.
To solve this controversy and contribute to the understanding of SiC precipitation in c-Si, a series of atomistic simulations is carried out.
In the first part, intrinsic and C related point defects in c-Si as well as some selected diffusion processes of the C defect are investigated by means of first-principles quatum-mechanical calculations based on DFT and classical potential calculations employing a Tersoff-like analytical bond order potential.
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Similar, implantations of an understoichiometric dose into c-Si at room temperature followed by thermal annealing result in small spherical sized C$_{\text{i}}$ agglomerates below \unit[700]{$^{\circ}$C} and SiC precipitates of the same size above \unit[700]{$^{\circ}$C}\cite{werner96} annealing temperature.
Since, however, the implantation temperature is considered more efficient than the postannealing temperature, SiC precipitates are expected and indeed observed for as-implanted samples \cite{lindner99,lindner01} in implantations performed at \unit[450]{$^{\circ}$C}.
-Thus, implanted C is likewise expected to occupy substitutionally regular Si lattice sites right from the start for implantations into c-Si at elevated temperatures.
+According to this, implanted C is likewise expected to occupy substitutionally regular Si lattice sites right from the start for implantations into c-Si at elevated temperatures.
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%In both cases Si$_{\text{i}}$ might be attributed a third role, which is the partial compensation of tensile strain that is present either in the stretched SiC or at the interface of the contracted SiC and the Si host.
To conclude, results of the present study indicate a precipitation of SiC in Si by successive agglomeration of \cs.
-\si{}, which is likewise existent, serves several needs: as a vehicle to rearrange the \cs{} atoms, as a building block for the surrounding Si host or further SiC and for strain compensation, either in the stretched SiC structure or at the interface of the SiC precipitate and the Si matrix.
-% todo \si reduced interfacial energy
+Elevated temperatures result in increased entropic contributions to structural formation.
+Moreover, conditions prevalent in IBS deviate the system from thermodynamic equilibrium.
+Thereby, C$_{\text{i}}$ is enabled to turn into C$_{\text{s}}$ accompanied by the emission of Si$_{\text{i}}$.
+\si{}, which is likewise existent, serves several needs: as a vehicle to rearrange the \cs{} atoms, as a building block for the surrounding Si host or further SiC and for strain compensation.
+The \si{} vehicle turns \cs{} into highly mobile \ci.
+This way, C can be easily rearranged in order to end up in a configuration of C atoms that occupy substitutionally the lattice sites of one of the fcc lattices of the diamond structure.
+Stretched SiC structures arise, which are coherently aligned to the Si matrix.
+\si{} is believed to likewise compensate the tensile strain within these structures.
+This is followed by the precipitation into incoherent 3C-SiC once the strain energy of the coherent structure surpasses the interfacial energy of the incoherent precipitate and the c-Si substrate.
+The associated volume reduction is compensated by \si{} that may serve as a supply for further SiC or as a building block for the surrounding Si host and likewise reduce existing strain in the interface region.
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-Results of the atomistic simulation study indicating the respective precipitation mechanism conform well with other experimental findings.
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-Thus, we propose an increased participation of C$_{\text{s}}$ already in the initial stages of the implantation process at temperatures above \unit[450]{$^{\circ}$C}, the temperature most applicable for the formation of SiC layers of high crystalline quality and topotactical alignment\cite{lindner99}.
-Thermally activated, C$_{\text{i}}$ is enabled to turn into C$_{\text{s}}$ accompanied by Si$_{\text{i}}$.
-The associated emission of Si$_{\text{i}}$ is needed for several reasons.
-For the agglomeration and rearrangement of C, Si$_{\text{i}}$ is needed to turn C$_{\text{s}}$ into highly mobile C$_{\text{i}}$ again.
-Since the conversion of a coherent SiC structure, i.e. C$_{\text{s}}$ occupying the Si lattice sites of one of the two fcc lattices that build up the c-Si diamond lattice, into incoherent SiC is accompanied by a reduction in volume, large amounts of strain are assumed to reside in the coherent as well as at the surface of the incoherent structure.
-Si$_{\text{i}}$ serves either as a supply of Si atoms needed in the surrounding of the contracted precipitates or as an interstitial defect minimizing the emerging strain energy of a coherent precipitate.
-The latter has been directly identified in the present simulation study, i.e. structures of two C$_{\text{s}}$ atoms and Si$_{\text{i}}$ located in the vicinity.
+Results of the atomistic simulation study that indicate the respective precipitation mechanism conform well with other experimental findings.
+By verification, the derived conclusions with respect to the precipitation mechanism are reinforced.
+Furthermore, experimental results that suggest a precipitation mechanism based on the agglomeration of \ci{} do not conflict with the proposed model of precipitation as concluded in the present study.
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