-\caption{Configurations of silicon and carbon point defects in silicon. Silicon and carbon atoms are illustrated by yellow and gray spheres respectively. Bonds are drawn whenever considered appropriate to ease identifying defect structures for the reader. Dumbbell configurations are abbreviated by DB.}
-\label{fig:sep_def}
-\end{figure}
-\begin{table*}
-\begin{ruledtabular}
-\begin{tabular}{l c c c c c c c c c}
- & Si$_{\text{i}}$ \hkl<1 1 0> DB & Si$_{\text{i}}$ H & Si$_{\text{i}}$ T & Si$_{\text{i}}$ \hkl<1 0 0> DB & V & C$_{\text{s}}$ & C$_{\text{i}}$ \hkl<1 0 0> DB & C$_{\text{i}}$ \hkl<1 1 0> DB & C$_{\text{i}}$ BC \\
-\hline
- Present study & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 & 1.95 & 3.72 & 4.16 & 4.66 \\
- \multicolumn{10}{c}{Other ab initio studies} \\
- Ref.\cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 & - & - & - & - \\
- Ref.\cite{leung99} & 3.31 & 3.31 & 3.43 & - & - & - & - & - & - \\
- Ref.\cite{dal_pino93,capaz94} & - & - & - & - & - & 1.89\cite{dal_pino93} & x & - & x+2.1\cite{capaz94}
-\end{tabular}
-\end{ruledtabular}
-\caption{Formation energies of silicon and carbon point defects in crystalline silicon given in eV. T denotes the tetrahedral, H the hexagonal and BC the bond-centered interstitial configuration. V corresponds to the vacancy configuration. Dumbbell configurations are abbreviated by DB.}
-\label{table:sep_eof}
-\end{table*}
-Results obtained by the present study compare well with results from literature\cite{leung99,al-mushadani03,dal_pino93,capaz94}.
-Regarding intrinsic defects in Si, the \hkl<1 1 0> self-interstitial dumbbell (Si$_{\text{i}}$ \hkl<1 1 0> DB) is found to be the ground state configuration closely followed by the hexagonal and tetrahedral configuration, which is consensus for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.
-In the case of a C impurity, next to the C$_{\text{s}}$ configuration, in which a C atom occupies an already vacant Si lattice site, the C \hkl<1 0 0> interstitial dumbbell (C$_{\text{i}}$ \hkl<1 0 0> DB) constitutes the energetically most favorable configuration, in which the C and Si dumbbell atoms share a regular Si lattice site.
-This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94,jones04} and experimental\cite{watkins76,song90} investigations, which all predict this configuration to be the ground state.
-%However, to our best knowledge, no energy of formation for this type of defect based on first-principles calculations has yet been explicitly stated in literature.
-However, to our best knowledge, no energy of formation for this type of defect based on first-principles calculations is available.
-
-Instead, Capaz et al.\cite{capaz94}, investigating migration pathways of the C$_{\text{i}}$ \hkl<1 0 0> DB, find this defect to be \unit[2.1]{eV} lower in energy than the bond-centered (BC) configuration.
-The BC configuration is claimed to constitute the saddle point within the C$_{\text{i}}$ \hkl[0 0 -1] DB migration path residing in the \hkl(1 1 0) plane and, thus, interpreted as the barrier of migration for the respective path.
-However, the present study indicates a local minimum state for the BC defect if spin polarized calculations are performed resulting in a net magnetization of two electrons localized in a torus around the C atom.
-Another DFT calculation without fully accounting for the electron spin results in the smearing of a single electron over two non-degenerate Kohn-Sham states and an increase of the total energy by \unit[0.3]{eV} for the BC configuration.
-Regardless of the rather small correction of \unit[0.3]{eV} due to the spin, the difference we found is much smaller (\unit[0.94]{eV}), which would nicely compare to experimentally observed migration barriers of \unit[0.70-0.87]{eV}\cite{lindner06,tipping87,song90}.
-However, since the BC configuration constitutes a real local minimum another barrier exists which is about \unit[1.2]{eV} in height.
-Indeed Capaz et al. propose another path and find it to be the lowest in energy\cite{capaz94}, in which a C$_{\text{i}}$ \hkl[0 0 -1] DB migrates to a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the neighbored Si lattice site in \hkl[1 1 -1] direction.
-Calculations in this work reinforce this path by an additional improvement of the quantitative conformance of the barrier height (\unit[0.90]{eV}) to experimental values.
-A more detailed description can be found in a previous study\cite{zirkelbach10a}.
-
-Next to the C$_{\text{i}}$ BC configuration the vacancy and Si$_{\text{i}}$ \hkl<1 0 0> DB have to be treated by taking into account the spin of the electrons.
-For the vacancy the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site.
-In the Si$_{\text{i}}$ \hkl<1 0 0> DB configuration the net spin up density is localized in two caps at each of the two DB atoms perpendicularly aligned to the bonds to the other two Si atoms respectively.
-No other configuration, within the ones that are mentioned, is affected.
-
-Concerning the mobility of the ground state Si$_{\text{i}}$, we found an activation energy of \unit[0.67]{eV} for the transition of the Si$_{\text{i}}$ \hkl[0 1 -1] to \hkl[1 1 0] DB located at the neighbored Si lattice site in \hkl[1 1 -1] direction.
-Further investigations revealed a barrier of \unit[0.94]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to Si$_{\text{i}}$ H, \unit[0.53]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to Si$_{\text{i}}$ T and \unit[0.35]{eV} for the Si$_{\text{i}}$ H to Si$_{\text{i}}$ T transition.
-%Obtained values are of the same order of magnitude than values derived from other ab initio studies\cite{bloechl93,sahli05}.
-These are of the same order of magnitude than values derived from other ab initio studies\cite{bloechl93,sahli05}.
-
-\subsection{Pairs of C$_{\text{i}}$}
-
-C$_{\text{i}}$ pairs of the \hkl<1 0 0> type have been investigated in the first part.
-Fig.~\ref{fig:combos_ci} schematically displays the initial C$_{\text{i}}$ \hkl[0 0 -1] DB structure and various positions for the second defect (1-5) that have been used for investigating defect pairs.
-Table~\ref{table:dc_c-c} summarizes resulting binding energies for the combination with a second C-Si \hkl<1 0 0> DB obtained for different orientations at positions 1 to 5.
-\begin{figure}
-\subfigure[]{\label{fig:combos_ci}\includegraphics[width=0.45\columnwidth]{combos_ci.eps}}
-\hspace{0.1cm}
-\subfigure[]{\label{fig:combos_si}\includegraphics[width=0.45\columnwidth]{combos.eps}}
-\caption{Position of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB (I) (Fig.~\ref{fig:combos_ci}) and of the lattice site chosen for the initial Si$_{\text{i}}$ \hkl<1 1 0> DB (Si$_{\text{i}}$) (Fig.~\ref{fig:combos_si}). Lattice sites for the second defect used for investigating defect pairs are numbered from 1 to 5.}
-\label{fig:combos}
-\end{figure}
-\begin{table}
-\begin{ruledtabular}
-\begin{tabular}{l c c c c c c }
- & 1 & 2 & 3 & 4 & 5 & R \\
-\hline
- \hkl[0 0 -1] & -0.08 & -1.15 & -0.08 & 0.04 & -1.66 & -0.19\\
- \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
- \hkl[0 -1 0] & -2.39 & -0.17 & -0.10 & -0.27 & -1.88 & -0.05\\
- \hkl[0 1 0] & -2.25 & -1.90 & -2.25 & -0.12 & -1.38 & -0.06\\
- \hkl[-1 0 0] & -2.39 & -0.36 & -2.25 & -0.12 & -1.88 & -0.05\\
- \hkl[1 0 0] & -2.25 & -2.16 & -0.10 & -0.27 & -1.38 & -0.06\\
-\end{tabular}
-\end{ruledtabular}
-\caption{Binding energies in eV of C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs. Equivalent configurations exhibit equal energies. Column 1 lists the orientation of the second defect, which is combined with the initial C$_{\text{i}}$ \hkl[0 0 -1] DB. The position index of the second defect is given in the first row according to Fig.~\ref{fig:combos}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect position, which is the maximum realizable defect separation distance ($\approx \unit[1.3]{nm}$) due to periodic boundary conditions.}
-\label{table:dc_c-c}
-\end{table}
-Most of the obtained configurations result in binding energies well below zero indicating a preferable agglomeration of this type of defects.
-For increasing distances of the defect pair the binding energy approaches to zero (R in Table~\ref{table:dc_c-c}) as it is expected for non-interacting isolated defects.
-Energetically favorable and unfavorable configurations can be explained by stress compensation and increase respectively based on the resulting net strain of the respective configuration of the defect combination.
-Antiparallel orientations of the second defect, i.e. \hkl[0 0 1] for positions located below the \hkl(0 0 1) plane with respect to the initial one (positions 1, 2 and 4) form the energetically most unfavorable configurations.
-In contrast, the parallel and particularly the twisted orientations constitute energetically favorable configurations, in which a vast reduction of strain is enabled by combination of these defects.
-
-Mattoni et al.\cite{mattoni2002} predict the ground state configuration for a \hkl[1 0 0] or equivalently a \hkl[0 1 0] defect created at position 1 with both defects basically maintaining the as-isolated DB structure, resulting in a binding energy of \unit[-2.1]{eV}.
-In this work we observed a further relaxation of this defect structure.
-The C atom of the second and the Si atom of the initial DB move towards each other forming a bond, which results in a somewhat lower binding energy of \unit[-2.25]{eV}.
-Apart from that, we found a more favorable configuration for the combination with a \hkl[0 -1 0] and \hkl[-1 0 0] DB respectively, which is assumed to constitute the actual ground state configuration of two C$_{\text{i}}$ DBs in Si.
-The atomic arrangement is shown in the bottom right of Fig.~\ref{fig:036-239}.
-The two C$_{\text{i}}$ atoms form a strong C-C bond, which is responsible for the large gain in energy resulting in a binding energy of \unit[-2.39]{eV}.
-
-Investigating migration barriers allows to predict the probability of formation of defect complexes by thermally activated diffusion processes.
-% ground state configuration, C cluster
-Based on the lowest energy migration path of a single C$_{\text{i}}$ DB the configuration, in which the second C$_{\text{i}}$ DB is oriented along \hkl[0 1 0] at position 2 is assumed to constitute an ideal starting point for a transition into the ground state.
-In addition, the starting configuration exhibits a low binding energy (\unit[-1.90]{eV}) and is, thus, very likely to occur.
-However, a barrier height of more than \unit[4]{eV} was detected resulting in a low probability for the transition.
-The high activation energy is attributed to the stability of such a low energy configuration, in which the C atom of the second DB is located close to the initial DB.
-Low barriers have only been identified for transitions starting from energetically less favorable configurations, e.g. the configuration of a \hkl[-1 0 0] DB located at position 2 (\unit[-0.36]{eV}).
-Starting from this configuration, an activation energy of only \unit[1.2]{eV} is necessary for the transition into the ground state configuration.
-The corresponding migration energies and atomic configurations are displayed in Fig.~\ref{fig:036-239}.
-\begin{figure}
-\includegraphics[width=\columnwidth]{036-239.ps}
-\caption{Migration barrier and structures of the transition of a C$_{\text{i}}$ \hkl[-1 0 0] DB at position 2 (left) into a C$_{\text{i}}$ \hkl[0 -1 0] DB at position 1 (right). An activation energy of \unit[1.2]{eV} is observed.}
-\label{fig:036-239}
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