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, $\vec{p}=(0,0,0)$, ({\color{red}$\bullet$})
- \item hexagonal, $\vec{p}=(-1/8,-1/8,1/8)$, ({\color{green}$\bullet$})
- \item nearly \hkl<1 0 0> dumbbell, $\vec{p}=(-1/4,-1/4,-1/8)$, ({\color{yellow}$\bullet$})
- \item nearly \hkl<1 1 0> dumbbell, $\vec{p}=(-1/8,-1/8,-1/4)$, ({\color{magenta}$\bullet$})
- \item bond-centered, $\vec{p}=(-1/8,-1/8,-3/8)$, ({\color{cyan}$\bullet$})
+ \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.
In figure \ref{fig:defects:kin_si_hex} the relaxation process is shown on the basis of the kinetic energy plot.
\begin{figure}[h]
\begin{center}
-\includegraphics[width=12cm]{e_kin_si_hex.ps}
+\includegraphics[width=10cm]{e_kin_si_hex.ps}
\end{center}
\caption{Kinetic energy plot of the relaxation process of the hexagonal silicon self-interstitial defect simulation using the Erhard/Albe classical potential.}
\label{fig:defects:kin_si_hex}
It is well established that carbon and other isovalent impurities prefer to dissolve substitutionally in silicon.
However, radiation damage can generate carbon interstitials \cite{watkins76} which have enough mobility at room temeprature to migrate and form defect complexes.
-Formation energies of the most common carbon point defects in crystalline silicon are summarized in table \ref{tab:defects:c_ints}.
+Formation energies of the most common carbon point defects in crystalline silicon are summarized in table \ref{tab:defects:c_ints} and the relaxed configurations obtained by classical potential calculations visualized in figure \ref{fig:defects:c_conf}.
The type of reservoir of the carbon impurity to determine the formation energy of the defect was 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 silicon and carbon is determined by the cohesive energy of silicon and silicon carbide.
\caption[Formation energies of carbon point defects in crystalline silicon determined by classical potential molecular dynamics and density functional calculations.]{Formation energies of carbon point defects in crystalline silicon determined by classical potential molecular dynamics and density functional calculations. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal, B the bond-centered and S the substitutional interstitial configuration. The dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
\label{tab:defects:c_ints}
\end{table}
+\begin{figure}[h]
+\begin{center}
+\begin{flushleft}
+\begin{minipage}{4cm}
+\underline{Hexagonal}\\
+$E_{\text{f}}^*=9.05\text{ eV}$\\
+\includegraphics[width=4.0cm]{c_pd_albe/hex.eps}
+\end{minipage}
+\begin{minipage}{0.8cm}
+\begin{center}
+$\Rightarrow$
+\end{center}
+\end{minipage}
+\begin{minipage}{4cm}
+\underline{\hkl<1 0 0>}\\
+$E_{\text{f}}=3.96\text{ eV}$\\
+\includegraphics[width=4.0cm]{c_pd_albe/100.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+\hfill
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{Tetrahedral}\\
+$E_{\text{f}}=6.09\text{ eV}$\\
+\includegraphics[width=4.0cm]{c_pd_albe/tet.eps}
+\end{minipage}\\[0.2cm]
+\begin{minipage}{4cm}
+\underline{Bond-centered}\\
+$E_{\text{f}}^*=5.59\text{ eV}$\\
+\includegraphics[width=4.0cm]{c_pd_albe/bc.eps}
+\end{minipage}
+\begin{minipage}{0.8cm}
+\begin{center}
+$\Rightarrow$
+\end{center}
+\end{minipage}
+\begin{minipage}{4cm}
+\underline{\hkl<1 1 0> dumbbell}\\
+$E_{\text{f}}=5.18\text{ eV}$\\
+\includegraphics[width=4.0cm]{c_pd_albe/110.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+\hfill
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{Substitutional}\\
+$E_{\text{f}}=0.75\text{ eV}$\\
+\includegraphics[width=4.0cm]{c_pd_albe/sub.eps}
+\end{minipage}
+\end{flushleft}
+\end{center}
+\caption[Relaxed carbon point defect configurations obtained by classical potential calculations.]{Relaxed carbon point defect configurations obtained by classical potential calculations. The silicon/carbon atoms and the bonds (only for the interstitial atom) are illustrated by yellow/grey spheres and blue lines.}
+\label{fig:defects:c_conf}
+\end{figure}
Substitutional carbon in silicon is found to be the lowest configuration in energy for all potential models.
An experiemntal value of the formation energy of substitutional carbon was determined by a fit to solubility data yielding a concentration of $3.5 \times 10^{24} \exp{(-2.3\text{ eV}/k_{\text{B}}T)} \text{ cm}^{-3}$ \cite{bean71}.
However, there is no particular reason for treating the prefactor as a free parameter in the fit to the experimental data.
It is simply given by the atomic density of pure silicon, which is $5\times 10^{22}\text{ cm}^{-3}$.
-Tersoff \cite{tersoff90} and Dal Pino et. al. \cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from 1.6 to 1.89 eV an excellent agreement with the experimental solubility data within the entire temeprature range of the experiment is obtained.
-This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal Pino et. al. is in good agreement with the results of the quantum-mechanical calculations performed in this work.
+Tersoff \cite{tersoff90} and Dal Pino et al. \cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from 1.6 to 1.89 eV an excellent agreement with the experimental solubility data within the entire temeprature range of the experiment is obtained.
+This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal Pino et al. is in good agreement with the results of the quantum-mechanical calculations performed in this work.
+Unfortunately the Erhard/Albe potential undervalues the formation energy roughly by a factor of two.
Except for Tersoff's tedrahedral configuration results the \hkl<1 0 0> dumbbell is the energetically most favorable interstital configuration.
-The low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the abrupt cutoff set to 2.5 \AA (see ref. 11 and 13 in \cite{tersoff90}) and the real formation energy is, thus, supposed to be located between 3 and 10 eV.
+The low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the abrupt cutoff set to 2.5 \AA{} (see ref. 11 and 13 in \cite{tersoff90}) and the real formation energy is, thus, supposed to be located between 3 and 10 eV.
Keeping these considerations in mind, the \hkl<1 0 0> dumbbell is the most favorable interstitial configuration for all interaction models.
In addition to the theoretical results compared to in table \ref{tab:defects:c_ints} there is experimental evidence of the existence of this configuration \cite{watkins76}.
It is frequently generated in the classical potential simulation runs in which carbon is inserted at random positions in the c-Si matrix.
The highest energy is observed for the hexagonal interstitial configuration using classical potentials.
Quantum-mechanical calculations reveal this configuration to be unstable, which is also reproduced by the Erhard/Albe potential.
In both cases a relaxation towards the \hkl<1 0 0> dumbbell configuration is observed.
+In fact the stability of the hexagonal interstitial could not be reproduced in simulations performed in this work using the unmodifed Tersoff potential parameters.
+Unfortunately, apart from the modified parameters, no more conditions specifying the relaxation process are given in Tersoff's study on carbon point defects in silicon \cite{tersoff90}.
+
The tetrahedral is the second most unfavorable interstitial configuration using classical potentials and keeping in mind the abrupt cutoff effect in the case of the Tersoff potential as discussed earlier.
Again, quantum-mechanical results reveal this configuration unstable.
The fact that the tetrahedral and hexagonal configurations are the two most unstable configurations in classical potential calculations and, thus, are less likely to arise in MD simulations acts in concert with the fact that these configurations are found to be unstable in the more accurate quantum-mechanical calculations.
-Bond-centered ...
+Just as for the Si self-interstitial a carbon \hkl<1 1 0> dumbbell configuration exists.
+For the Erhard/Albe potential the formation energy is situated in the same order as found by quantum-mechanical results.
+Similar structures arise in both types of simulations with the silicon and carbon atom sharing a silicon lattice site aligned along \hkl[1 1 0] where the carbon atom is localized slightly closer to the next nearest silicon atom located in the opposite direction to the site-sharing silicon atom even forming a bond to the next but one silicon atom in this direction.
+
+The bond-centered configuration is unstable for the Erhard/Albe potential.
+The system moves into the \hkl<1 1 0> interstitial configuration.
+This, like in the hexagonal case, is also true for the unmodified Tersoff potential and the given relaxation conditions.
+Quantum-mechanical results of this configuration are discussed in more detail in section \ref{subsection:bc}.
+In another ab inito study Capaz et al. \cite{capaz94} determined this configuration as an intermediate saddle point structure of a possible migration path, which is 2.1 eV higher than the \hkl<1 0 0> dumbbell configuration.
+In calculations performed in this work the bond-centered configuration in fact is a real local minimum and an energy barrier is needed to reach this configuration starting from the \hkl<1 0 0> dumbbell configuration, which is discussed in section \ref{subsection:100mig}.
\subsection[\hkl<1 0 0> dumbbell interstitial configuration]{\boldmath\hkl<1 0 0> dumbbell interstitial configuration}
\label{subsection:100db}
+As the \hkl<1 0 0> dumbbell interstitial is the lowest configuration in energy it is the most probable hence important interstitial configuration of carbon in silicon.
+It was first identified by infra-red (IR) spectroscopy \cite{bean70} and later on by electron paramagnetic resonance (EPR) spectroscopy \cite{watkins76}.
+
+Figure \ref{fig:defects:100db_cmp} schematically shows the \hkl<1 0 0> dumbbell structure and table \ref{tab:defects:100db_cmp} lists the details of displacements obtained by analytical potential and quantum-mechanical calculations.
+\begin{figure}[h]
+\begin{center}
+\includegraphics[width=10cm]{100-c-si-db_cmp.eps}
+\end{center}
+\label{fig:defects:100db_cmp}
+\caption[Sketch of the \hkl<1 0 0> dumbbell structure.]{Sketch of the \hkl<1 0 0> dumbbell structure. Atomic displacements and distances are listed in table \ref{tab:defects:100db_cmp}.}
+\end{figure}
+%
+\begin{table}[h]
+\begin{center}
+\begin{tabular}{l c c c c c c c c c}
+\hline
+\hline
+ & & & & \multicolumn{3}{c}{Atom 2} & \multicolumn{3}{c}{Atom 3} \\
+ & $a$ & $b$ & $|a|+|b|$ & $\Delta x$ & $\Delta y$ & $\Delta z$ & $\Delta x$ & $\Delta y$ & $\Delta z$ \\
+\hline
+Erhard/Albe & 0.084 & -0.091 & 0.175 & -0.015 & -0.015 & -0.031 & -0.014 & 0.014 & 0.020 \\
+VASP & & & & & & & & & \\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\begin{center}
+\begin{tabular}{l c c c c c c c c}
+\hline
+\hline
+ & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$\\
+\hline
+Erhard/Albe & & & & & & & \\
+VASP & & & & & & & \\
+\hline
+\hline
+\end{tabular}\\[0.5cm]
+\end{center}
+\label{tab:defects:100db_cmp}
+\caption[Atomic displacements and distances of the \hkl<1 0 0> dumbbell structure obtained by the Erhard/Albe potential and VASP calculations.]{Atomic displacements and distances of the \hkl<1 0 0> dumbbell structure obtained by the Erhard/Albe potential and VASP calculations. The displacements and distances are given in nm and schematically displayed in figure \ref{fig:defects:100db_cmp}.}
+\end{table}
+
\subsection{Bond-centered interstitial configuration}
+\label{subsection:bc}
\section[Migration of the carbon \hkl<1 0 0> interstitial]{\boldmath Migration of the carbon \hkl<1 0 0> interstitial}
+\label{subsection:100mig}
\section{Combination of point defects}
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