X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;f=posic%2Fthesis%2Fdefects.tex;h=5bec37ffa9202fc4a2d5e9066f01ba3b6a73db1d;hb=dd8959739ad96c93d2b0e735417c4dcaf97da263;hp=2206dee5c1abba36a7a0049b0260b8c9861f2f80;hpb=2fa00b0623a6e75a295de13d4b6a1b948de3680a;p=lectures%2Flatex.git

diff --git a/posic/thesis/defects.tex b/posic/thesis/defects.tex
index 2206dee..5bec37f 100644
--- a/posic/thesis/defects.tex
+++ b/posic/thesis/defects.tex
@@ -568,7 +568,7 @@ Todo: To refine the migration barrier one has to find the saddle point structure
 
 \begin{figure}[h]
 \begin{center}
-\includegraphics[width=13cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.5cm]
+\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}
@@ -595,7 +595,7 @@ In a second process 0.25 eV of energy are needed for the system to revert into a
 
 \begin{figure}[h]
 \begin{center}
-\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
+\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[1.6cm]
 \begin{picture}(0,0)(140,0)
 \includegraphics[width=2.5cm]{vasp_mig/00-1_a.eps}
 \end{picture}
@@ -620,7 +620,7 @@ The resulting migration barrier of approximately 0.9 eV is very close to the exp
 
 \begin{figure}[h]
 \begin{center}
-\includegraphics[width=13cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.5cm]
+\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}
@@ -652,4 +652,90 @@ In addition the bond-ceneterd configuration, for which spin polarized calculatio
 
 \section{Combination of point defects}
 
+The structural and energetic properties of combinations of point defects are investigated in the following.
+The focus is on combinations of the \hkl<0 0 -1> dumbbell interstitial with a second defect.
+The second defect is either another \hkl<1 0 0>-type interstitial occupying different orientations, a vacany or a substitutional carbon atom.
+Several distances of the two defects are examined.
+Investigations are restricted to quantum-mechanical calculations.
+\begin{figure}[h]
+\begin{center}
+\begin{minipage}{7.5cm}
+\includegraphics[width=7cm]{comb_pos.eps}
+\end{minipage}
+\begin{minipage}{6.0cm}
+\underline{Positions given in $a_{\text{Si}}$}\\[0.3cm]
+Initial interstitial: $\frac{1}{4}\hkl<1 1 1>$\\
+Relative silicon neighbour positions:
+\begin{enumerate}
+ \item $\frac{1}{4}\hkl<1 1 -1>$, $\frac{1}{4}\hkl<-1 -1 -1>$
+ \item $\frac{1}{2}\hkl<1 0 1>$, $\frac{1}{2}\hkl<0 1 -1>$,\\[0.2cm]
+       $\frac{1}{2}\hkl<0 -1 -1>$, $\frac{1}{2}\hkl<-1 0 -1>$
+ \item $\frac{1}{4}\hkl<1 -1 1>$, $\frac{1}{4}\hkl<-1 1 1>$
+ \item $\frac{1}{4}\hkl<-1 1 -3>$, $\frac{1}{4}\hkl<1 -1 -3>$
+ \item $\frac{1}{2}\hkl<-1 -1 0>$, $\frac{1}{2}\hkl<1 1 0>$
+\end{enumerate}
+\end{minipage}\\
+\begin{picture}(0,0)(190,20)
+\includegraphics[width=2.3cm]{100_arrow.eps}
+\end{picture}
+\begin{picture}(0,0)(220,0)
+\includegraphics[height=2.2cm]{001_arrow.eps}
+\end{picture}
+\end{center}
+\caption[\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighboured silicon atoms used for the second defect.]{\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighboured silicon atoms used for the second defect. Two possibilities exist for red numbered atoms and four possibilities exist for blue numbered atoms.}
+\label{fig:defects:pos_of_comb}
+\end{figure}
+\begin{table}[h]
+\begin{center}
+\begin{tabular}{l c c c c c}
+\hline
+\hline
+ & 1 & 2 & 3 & 4 & 5 \\
+ \hline
+ \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66\\
+ \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53\\
+ \hkl<0 -1 0> & {\color{orange}-2.39} & -2.16 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{magenta}-1.88}\\
+ \hkl<0 1 0> & {\color{cyan}-2.25} & -0.36 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{violet}-1.38}\\
+ \hkl<-1 0 0> & {\color{orange}-2.39} & -1.90 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{magenta}-1.88}\\
+ \hkl<1 0 0> & {\color{cyan}-2.25} & -0.17 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{violet}-1.38} \\
+ \hline
+ C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 \\
+ Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 \\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\caption[Energetic results of defect combinations.]{Energetic results of defect combinations. The given energies in eV are defined by equation \eqref{eq:defects:e_of_comb}. Equivalent configurations are marked by identical colors. The first column lists the types of the second defect combined with the initial \hkl<0 0 -1> dumbbell interstitial. The position index of the second defect is given in the first row according to figure \ref{fig:defects:pos_of_comb}.}
+\label{tab:defects:e_of_comb}
+\end{table}
+Figure \ref{fig:defects:pos_of_comb} shows the initial \hkl<0 0 -1> dumbbell interstitial defect and the positions of next neighboured silicon atoms used for the second defect.
+Table \ref{tab:defects:e_of_comb} summarizes energetic results obtained after relaxation of the defect combinations.
+The energy of interest $E$ is defined to be
+\begin{equation}
+E=
+E_{\text{f}}^{\text{defect combination}}-
+E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
+E_{\text{f}}^{\text{2nd defect}}
+\label{eq:defects:e_of_comb}
+\end{equation}
+with $E_{\text{f}}^{\text{defect combination}}$ being the formation energy of the defect combination, $E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}$ being the formation energy of the C \hkl<0 0 -1> dumbbell interstitial defect and $E_{\text{f}}^{\text{2nd defect}}$ being the formation energy of the second defect.
+For defects far away from each other the formation energy of the defect combination should approximately become the sum of the formation energies of the individual defects.
+The interaction of such defects is low resulting in $E=0$.
+In fact, for another \hkl<0 0 -1> dumbbell interstitial created at position $\frac{a_{\text{Si}}}{2}\hkl<3 2 3>$ relative to the initial one an energy of \ldots eV is obtained.
+Configurations wih energies greater than zero are energetically unfavorable and expose a repulsive interaction.
+These configurations are unlikely to arise or to persist for non-zero temperatures.
+Energies below zero indicate configurations favored compared to configurations in which these point defects are separated far away from each other.
+
+Investigating the first part of table \ref{tab:defects:e_of_comb}, namely the combinations with another \hkl<1 0 0>-type interstitial, most of the combinations result in energies below zero.
+Surprisingly the most favorable configurations are the ones with the second defect created at the very next silicon neighbour and a change in orientation compared to the initial one.
+\begin{figure}[h]
+\caption{}
+\label{fig:defects:comb_db_01}
+\end{figure}
+Figure \ref{} shows the structure of these two configurations.
+
+
+
+
+