In fact, the same type of interstitial arises using random insertions.
In addition, variations exist, in which the displacement is only along two \hkl<1 0 0> axes ($E_\text{f}=3.8\,\text{eV}$) or along a single \hkl<1 0 0> axes ($E_\text{f}=3.6\,\text{eV}$) successively approximating the tetdrahedral configuration and formation energy.
The existence of these local minima located near the tetrahedral configuration seems to be an artifact of the analytical potential without physical authenticity revealing fundamental problems of analytical potential models for describing defect structures.
+% todo - energy barrier of what ?!?!
However, the energy barrier is small.
\begin{figure}[tp]
\begin{center}
\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 inbetween two next neighbored Si atoms forming linear bonds.
+In the BC insterstitial configuration the interstitial atom is located in between two next neighbored Si atoms forming linear bonds.
In a previous study this configuration was found to constitute an intermediate saddle point configuration determining the migration barrier of one possibe migration path of a \ci{} \hkl<1 0 0> DB configuration into an equivalent one \cite{capaz94}.
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.
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
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 cosntants look similar.
-A local minimum exists inbetween two peaks of the graph.
+A local minimum exists in between two peaks of the graph.
The corresponding configuration, which is illustrated for the results obtained for a time constant of \unit[1]{fs}, looks similar to the \ci{} \hkl<1 1 0> configuration.
Indeed, this configuration is obtained by relaxation simulations without constraints of configurations near the minimum.
Activation energies of roughly \unit[2.8]{eV} and \unit[2.7]{eV} are needed for migration.
Fig.~\ref{fig:defects:050} shows the relaxed structure of a vacancy created at position 5.
The Si DB atom is largely displaced along \hkl[1 1 0] and somewhat less along \hkl[0 0 -1], which corresponds to the direction towards the vacancy.
The \si DB atom approaches Si atom number 1.
-Indeed, a non-zero charge density is observed inbetween these two atoms exhibiting a cylinder-like shape superposed with the charge density known from the DB itself.
+Indeed, a non-zero charge density is observed in between these two atoms exhibiting a cylinder-like shape superposed with the charge density known from the DB itself.
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.
These results support the above assumptions of an increased entropic contribution to structural formation involving C$_{\text{s}}$ to a greater extent.
% link to migration of \si{}!
+% todo - make it a subsection
The possibility for separated configurations of \cs{} and \si{} becomes even more likely if one of the constituents exhibits a low barrier of migration.
In this case, the \si{} is assumed to constitute the mobile defect compared to the stable \cs{} atom.
Thus, migration paths of \si{} are investigated in the following excursus.
\section{Conclusions concerning the SiC conversion mechanism}
+\ifnum1=0
+
Obtained results for separated point defects in Si are in good agreement to previous theoretical work on this subject, both for intrinsic defects \cite{leung99,al-mushadani03} as well as for C point defects \cite{dal_pino93,capaz94}.
The ground-state configurations of these defects, i.e. the Si$_{\text{i}}$ \hkl<1 1 0> and C$_{\text{i}}$ \hkl<1 0 0> DB, are reproduced and compare well to previous findings of theoretical investigations on Si$_{\text{i}}$ \cite{leung99,al-mushadani03} as well as theoretical \cite{dal_pino93,capaz94,burnard93,leary97,jones04} and experimental \cite{watkins76,song90} studies on C$_{\text{i}}$.
A quantitatively improved activation energy of \unit[0.9]{eV} for a qualitatively equal migration path based on studies by Capaz et.~al.~\cite{capaz94} to experimental values \cite{song90,lindner06,tipping87} ranging from \unit[0.70-0.87]{eV} reinforce their derived mechanism of diffusion for C$_{\text{i}}$ in Si
Thus, elevated temperatures might lead to thermodynamically unstable configurations of C$_{\text{s}}$ and a remaining Si atom in the near interstitial lattice, which is supported by the result of an {\em ab initio} molecular dynamics run.
%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.
-% todo
-% maybe move above stuff to conclusion chapter, at least shorten!
-% see remember in sic chapter
+\fi
+
+% 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.
projects / lectures / latex.git / blobdiff