\usepackage{amssymb}\r
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% additional stuff\r
-%\usepackage{miller}\r
+\usepackage{miller}\r
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\begin{document}\r
\r
%\title{Mobility of Carbon in Silicon -- a first principles study}\r
\title{Extensive first principles study of carbon defects in silicon}\r
-\author{F. Zirkelbach} \author{B. Stritzker}\r
+\author{F. Zirkelbach}\r
+\author{B. Stritzker}\r
\affiliation{Experimentalphysik IV, Universit\"at Augsburg, 86135 Augsburg, Germany}\r
\author{K. Nordlund}\r
\affiliation{Department of Physics, University of Helsinki, 00014 Helsinki, Finland}\r
\author{J. K. N. Lindner}\r
-\author{W. G. Schmidt} \author{E. Rauls}\r
+\author{W. G. Schmidt}\r
+\author{E. Rauls}\r
\affiliation{Department Physik, Universit\"at Paderborn, 33095 Paderborn, Germany}\r
\r
\begin{abstract}\r
We present a first principles investigation of the mobility of carbon interstitials in silicon. \r
-The migration mechanism of carbon [100] dumbbell interstitials in otherwise defect-free silicon has been investigated using density functional theory calculations.\r
+The migration mechanism of carbon \hkl<1 0 0> dumbbell interstitials in otherwise defect-free silicon has been investigated using density functional theory calculations.\r
Furthermore, the influence of near-by vacancies, carbon interstitial and substitutional defects and silicon self-interstitials has been investigated systematically.\r
A long range capture radius for vacancies has been found....\r
\end{abstract}\r
However, none of the mentioned studies consistently investigates entirely the relevant defect structures and reactions concentrated on the specific problem of 3C-SiC formation in C implanted Si.\r
% but mattoni2002 actually did a lot. maybe this should be mentioned!\r
In fact, in a combined analytical potential molecular dynamics and ab initio study\cite{mattoni2002} the interaction of substitutional C with Si self-interstitials and C interstitials is evaluated.\r
-However, investigations are, first of all, restricted to interaction chains along the $\langle 1 1 0 \rangle$ and $\langle -1 1 0 \rangle$ direction, secondly lacking combinations of C interstitials and, finally, not considering migration barriers giving further information about the probability of defect agglomeration.\r
+However, investigations are, first of all, restricted to interaction chains along the \hkl[1 1 0] and \hkl[-1 1 0] direction, secondly lacking combinations of C interstitials and, finally, not considering migration barriers giving further information about the probability of defect agglomeration.\r
\r
By first principles atomistic simulations this work aims to shed light on basic processes involved in the precipitation mechanism of SiC in Si.\r
During implantation defects such as vacancies (V), substitutional C (C$_{\text{s}}$), interstitial C (C$_{\text{i}}$) and Si self-interstitials (Si$_{\text{i}}$) are created, which play a decisive role in the precipitation process.\r
\r
\section{Results}\r
\r
-%After the implantation of C into Si, C interstitials are the most common defects in the Si sample.\r
-%Their mobility is the crucial quantity to be investigated.\r
-%However, the implantation process unavoidably creates a variety of further point defects, such as vacancies and silicon self-interstitials.\r
-%Already during implantation and also in the subsequent annealing process, further defects can evolve from these, like pair defects or substitutional carbon.\r
The implantation of highly energetic C atoms results in a multiplicity of possible defect configurations.\r
Next to individual Si$_{\text{i}}$, C$_{\text{i}}$, V and C$_{\text{s}}$ defects, combinations of these defects and their interaction are considered important for the problem under study.\r
In the following the structure and energetics of separated defects are presented.\r
Fig.~\ref{fig:sep_def} shows the obtained structures while the corresponding energies of formation are summarized and compared to values from literature in Table~\ref{table:sep_eof}.\r
\begin{figure}\r
\begin{minipage}[t]{0.32\columnwidth}\r
-\underline{Si$_{\text{i}}$ $\langle 1 1 0 \rangle$ DB}\\\r
+\underline{Si$_{\text{i}}$ \hkl<1 1 0> DB}\\\r
\includegraphics[width=\columnwidth]{si110.eps}\r
\end{minipage}\r
\begin{minipage}[t]{0.32\columnwidth}\r
\includegraphics[width=\columnwidth]{sitet.eps}\r
\end{minipage}\\\r
\begin{minipage}[t]{0.32\columnwidth}\r
-\underline{Si$_{\text{i}}$ $\langle 1 0 0 \rangle$ DB}\\\r
+\underline{Si$_{\text{i}}$ \hkl<1 0 0> DB}\\\r
\includegraphics[width=\columnwidth]{si100.eps}\r
\end{minipage}\r
\begin{minipage}[t]{0.32\columnwidth}\r
\includegraphics[width=\columnwidth]{csub.eps}\r
\end{minipage}\\\r
\begin{minipage}[t]{0.32\columnwidth}\r
-\underline{C$_{\text{i}}$ $\langle 1 0 0 \rangle$ DB}\\\r
+\underline{C$_{\text{i}}$ \hkl<1 0 0> DB}\\\r
\includegraphics[width=\columnwidth]{c100.eps}\r
\end{minipage}\r
\begin{minipage}[t]{0.32\columnwidth}\r
-\underline{C$_{\text{i}}$ $\langle 1 1 0 \rangle$ DB}\\\r
+\underline{C$_{\text{i}}$ \hkl<1 1 0> DB}\\\r
\includegraphics[width=\columnwidth]{c110.eps}\r
\end{minipage}\r
\begin{minipage}[t]{0.32\columnwidth}\r
\begin{table*}\r
\begin{ruledtabular}\r
\begin{tabular}{l c c c c c c c c c}\r
- & Si$_{\text{i}}$ $\langle1 1 0\rangle$ DB & Si$_{\text{i}}$ H & Si$_{\text{i}}$ T & Si $\langle 1 0 0\rangle$ DB & V & C$_{\text{s}}$ & C$_{\text{i}}$ $\langle1 0 0\rangle$ DB & C$_{\text{i}}$ $\langle1 1 0\rangle$ DB & C$_{\text{i}}$ BC \\\r
+ & Si$_{\text{i}}$ \hkl<1 1 0> DB & Si$_{\text{i}}$ H & Si$_{\text{i}}$ T & Si \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 \\\r
\hline\r
Present study & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 & 1.95 & 3.72 & 4.16 & 4.66 \\\r
\multicolumn{10}{c}{Other ab initio studies} \\\r
Ref.\cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 & - & - & - & - \\\r
Ref.\cite{leung99} & 3.31 & 3.31 & 3.43 & - & - & - & - & - & - \\\r
Ref.\cite{dal_pino93,capaz94} & - & - & - & - & - & 1.89\cite{dal_pino93} & x & - & x+2.1\cite{capaz94}\r
- %Reference & 3.40\cite{al-mushadani03}, 3.31\cite{leung99} & 3.45\cite{al-mushadani03}, 3.31\cite{leung99} & 3.43\cite{leung99} & - & 3.53\cite{al-mushadani03} & 1.89\cite{dal_pino93} & x & - & x+2.1\cite{capaz94}\r
\end{tabular}\r
\end{ruledtabular}\r
\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.}\r
\label{table:sep_eof}\r
\end{table*}\r
Results obtained by the present study compare well with results from literature\cite{leung99,al-mushadani03,dal_pino93,capaz94}.\r
-Regarding intrinsic defects in Si, the $\langle 1 1 0 \rangle$ self-interstitial dumbbell (Si$_{\text{i}}$ $\langle 1 1 0 \rangle$ DB) is found to be the ground state configuration tersely followed by the hexagonal and tetrahedral configuration, which is the consensus view for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.\r
-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$_{\text{i}}$ $\langle 1 0 0 \rangle$ interstitial dumbbell (C$_{\text{i}}$ $\langle 1 0 0 \rangle$ DB) constitutes the energetically most favorable configuration, in which the C and Si dumbbell atoms share a regular Si lattice site.\r
+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 tersely followed by the hexagonal and tetrahedral configuration, which is the consensus view for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.\r
+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.\r
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 as the ground state.\r
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.\r
\r
-Instead, Capaz et al.\cite{capaz94}, investigating migration pathways of the C$_{\text{i}}$ $\langle 1 0 0 \rangle$ DB, find this defect to be \unit[2.1]{eV} lower in energy than the bond-centered (BC) configuration, which is claimed to constitute a saddle point configuration in the migration path within the $(1 1 0)$ plane and, thus, interpreted as the barrier of migration for the respective path.\r
+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, which is claimed to constitute a saddle point configuration in the migration path within the \hkl(1 1 0) plane and, thus, interpreted as the barrier of migration for the respective path.\r
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.\r
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.\r
Regardless of the rather small correction due to the spin, the difference we found is much smaller (\unit[0.9]{eV}), which would nicely compare to experimental findings $(\unit[0.70-0.87]{eV})$\cite{lindner06,tipping87,song90} for the migration barrier.\r
However, since the BC configuration constitutes a real local minimum another barrier exists which is about \unit[1.2]{eV} ($\unit[0.9]{eV}+\unit[0.3]{eV}$) in height.\r
-Indeed Capaz et al. propose another path and find it to be the lowest in energy\cite{capaz94}, in which a C$_{\text{i}}$ $\langle 0 0 -1\rangle$ DB migrates into a C$_{\text{i}}$ $\langle 0 -1 0\rangle$ DB located at the next neighboured Si lattice site in $[1 1 -1]$ direction.\r
+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 into a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the next neighboured Si lattice site in \hkl[1 1 -1] direction.\r
Calculations in this work reinforce this path by an additional improvement of the quantitative conformance of the barrier height (\unit[0.9]{eV}) to experimental values.\r
A more detailed description can be found in a previous study\cite{zirkelbach10a}.\r
\r
-Next to the C BC configuration the vacancy and Si$_{\text{i}}$ $\langle 1 0 0\rangle$ DB have to be treated by taking into account the spin of the electrons.\r
+Next to the C 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.\r
For the latter two the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site and in two caps at each of the two DB atoms perpendicularly aligned to the bonds to the other two Si atoms respectively.\r
No other configuration, within the ones that are mentioned, is affected.\r
\r
-Concerning the mobility of the ground state Si$_{\text{i}}$, an activation energy shortly below \unit[0.7]{eV} was found for the migration of a Si$_{\text{i}}$ $\langle 0 1 -1\rangle$ into a $\langle 1 1 0\rangle$ DB configuration located at the next neighboured Si lattice site in $[1 1 -1]$ direction.\r
+Concerning the mobility of the ground state Si$_{\text{i}}$, an activation energy shortly below \unit[0.7]{eV} was found for the migration of a Si$_{\text{i}}$ \hkl[0 1 -1] into a \hkl[1 1 0] DB configuration located at the next neighboured Si lattice site in \hkl[1 1 -1] direction.\r
% look for values in literature for neutraly charged Si_i diffusion\r
\r
\subsection{Pairs of C$_{\text{i}}$}\r
\r
-C$_{\text{i}}$ pairs of the $\langle 1 0 0\rangle$-type have been considered in the first part.\r
-Table~\ref{table:dc_c-c} summarizes the binding energies obtained for configurations, in which an initial C$_{\text{i}}$ $\langle 0 0 -1\rangle$ DB located at position Si$_{\text{i}}$/C$_{\text{i}}$ is combined with a defect of the same type occupying various orientations at positions 1 to 5 (see Fig.~\ref{fig:combos}).\r
+C$_{\text{i}}$ pairs of the \hkl<1 0 0>-type have been considered in the first part.\r
+Table~\ref{table:dc_c-c} summarizes the binding energies obtained for configurations, in which an initial C$_{\text{i}}$ \hkl[0 0 -1] DB located at position Si$_{\text{i}}$/C$_{\text{i}}$ is combined with a defect of the same type occupying various orientations at positions 1 to 5 (see Fig.~\ref{fig:combos}).\r
\begin{table}\r
\begin{ruledtabular}\r
\begin{tabular}{l c c c c c c }\r
& 1 & 2 & 3 & 4 & 5 & R \\\r
\hline\r
- $\langle 0 0 -1\rangle$ & -0.08 & -1.15 & -0.08 & 0.04 & -1.66 & -0.19\\\r
- $\langle 0 0 1\rangle$ & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\\r
- $\langle 0 -1 0\rangle$ & -2.39 & -0.17 & -0.10 & -0.27 & -1.88 & -0.05\\\r
- $\langle 0 1 0\rangle$ & -2.25 & -1.90 & -2.25 & -0.12 & -1.38 & -0.06\\\r
- $\langle -1 0 0\rangle$ & -2.39 & -0.36 & -2.25 & -0.12 & -1.88 & -0.05\\\r
- $\langle 1 0 0\rangle$ & -2.25 & -2.16 & -0.10 & -0.27 & -1.38 & -0.06\\\r
+ \hkl[0 0 -1] & -0.08 & -1.15 & -0.08 & 0.04 & -1.66 & -0.19\\\r
+ \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\\r
+ \hkl[0 -1 0] & -2.39 & -0.17 & -0.10 & -0.27 & -1.88 & -0.05\\\r
+ \hkl[0 1 0] & -2.25 & -1.90 & -2.25 & -0.12 & -1.38 & -0.06\\\r
+ \hkl[-1 0 0] & -2.39 & -0.36 & -2.25 & -0.12 & -1.88 & -0.05\\\r
+ \hkl[1 0 0] & -2.25 & -2.16 & -0.10 & -0.27 & -1.38 & -0.06\\\r
\end{tabular}\r
\end{ruledtabular}\r
-\caption{Binding energies of C$_{\text{i}}$ $\langle 1 0 0\rangle$-type defect pairs in eV. Equivalent configurations exhibit equal energies. The first column lists the orientation of the defect, which is combined with the initial C$_{\text{i}}$ $\langle 0 0 -1\rangle$ dumbbell. 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 $\frac{a_{\text{Si}}}{2} \langle3 2 3 \rangle$ relative to the initial defect position, which is the maximum realizable distance due to periodic boundary conditions.}\r
+\caption{Binding energies of C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs in eV. Equivalent configurations exhibit equal energies. The first column lists the orientation of the defect, which is combined with the initial C$_{\text{i}}$ \hkl[0 0 -1] dumbbell. 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 distance due to periodic boundary conditions.}\r
\label{table:dc_c-c}\r
\end{table}\r
Most of the obtained configurations result in binding energies well below zero indicating a preferable agglomeration of these type of defects.\r
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.\r
Energetically favorable and unfavorable configurations can be explained by stress compensation and increase respectively, which is due to the resulting net strain of the respective configuration of the defect combination.\r
-Antiparallel orientations of the second defect ($\langle 0 0 1\rangle$) at positions located below the (001) plane with respect to the initial one (positions 1, 2 and 4) show the energetically most unfavorable configurations.\r
+Antiparallel orientations of the second defect (\hkl[0 0 1]) at positions located below the \hkl(0 0 1) plane with respect to the initial one (positions 1, 2 and 4) show the energetically most unfavorable configurations.\r
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.\r
\r
-Mattoni et al.\cite{mattoni2002} predict the ground state configuration for a $\langle 1 0 0\rangle$ or equivalently a $\langle 0 1 0\rangle$ defect created at position 1 with both defects basically maintaining the DB structure, resulting in a binding energy of \unit[-2.1]{eV}.\r
-% in mattoni db structures are basically amintained. there is further relaxation in our case and a lower binding energy\r
+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 DB structure, resulting in a binding energy of \unit[-2.1]{eV}.\r
In this work we found a further relaxation of this defect structure.\r
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}.\r
-Furthermore a more favorable configuration was found for the combination with a $\langle 0 -1 0\rangle$ and $\langle -1 0 0\rangle$ DB respectively, which is assumed to constitute the actual ground state configuration of two C$_{\text{i}}$ DBs in Si.\r
+Furthermore a more favorable configuration was found 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.\r
The two C 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}.\r
\r
Investigating migration barriers enables to predict the probability of formation of the thermodynamic ground state defect complex by thermally activated diffusion processes.\r
High activation energies are necessary for the migration of low energy configurations, in which the C atom of the second DB is located in the vicinity of the initial DB.\r
-The transition of the configuration, in which the second DB is of the $\langle 0 1 0\rangle$ type at position 2 (\unit[-1.90]{eV}) into a $\langle 0 1 0\rangle$-type DB at position 1 (\unit[-2.39]{eV}) for instance, revealed a barrier height of more than \unit[4]{eV}.\r
-Low barriers do only exist from energetically less favorable configurations, e.g. the configuration of the $\langle -1 0 0\rangle$ DB located at position 2 (\unit[-0.36]{eV}).\r
-An activation energy of only \unit[?.?]{eV} is necessary for the transition into the ground state configuration.\r
+The transition of the configuration, in which the second DB oriented along \hkl[0 1 0] type at position 2 (\unit[-1.90]{eV}) into a \hkl[0 1 0] DB at position 1 (\unit[-2.39]{eV}) for instance, revealed a barrier height of more than \unit[4]{eV}.\r
+Low barriers do only exist from energetically less favorable configurations, e.g. the configuration of the \hkl[-1 0 0] DB located at position 2 (\unit[-0.36]{eV}).\r
+Starting from this onfiguration, an activation energy of only \unit[1.2]{eV} is necessary for the transition into the ground state configuration.\r
% strange mig from -190 -> -2.39 (barrier > 4 eV)\r
% C-C migration -> idea:\r
% mig from low energy confs has extremely high barrier!\r
% low barrier only from energetically less/unfavorable confs (?)! <- prove!\r
% => low probability of C-C clustering ?!?\r
+%\r
+% should possibly be transfered to discussion section\r
Since thermally activated C clustering is, thus, only possible by traversing energetically unfavored configurations, mass C clustering is not expected.\r
+Furthermore, the migration barrier is still higher than the activation energy observed for a single C$_{\text{i}}$ \hkl<1 0 0> DB in c-Si.\r
+The migration barrier of a C$_{\text{i}}$ DB in a complex system is assumed to approximate the barrier in a separated system with increasing defect separation distance.\r
+Thus, lower migration barriers are expected for separating C$_{\text{i}}$ DBs.\r
+% calculate?!?\r
+However, low binding energies ... and the difference needs to be overcome too.\r
+It is bound to precapture state and only \r
+However if the activation energy is $>>$ than the difference in energy of the two configurations both states are equally occupied.\r
+And at increased temperatures that enable such diffusion processes the entropy comes into play.\r
+A promising configuration ... -2.25, and the amoun tof equivalent configurations is twice as high.\r
+Thus, C agglomeration indeed is expected but only a low probability is assumed for C clustering by thermally activated processes with regard to the considered period of time.\r
% ?!?\r
% look for precapture mechnism (local minimum in energy curve)\r
% also: plot energy all confs with respect to C-C distance\r
% maybe a pathway exists traversing low energy confs ?!?\r
\r
% point out that configurations along 110 were extended up to the 6th NN in that direction\r
-The binding energies of the energetically most favorable configurations with the seocnd DB located along the $\langle 1 1 0\rangle$ direction and resulting C-C distances of the relaxed structures are summarized in Table~\ref{table:dc_110}.\r
+The binding energies of the energetically most favorable configurations with the seocnd DB located along the \hkl[1 1 0] direction and resulting C-C distances of the relaxed structures are summarized in Table~\ref{table:dc_110}.\r
\begin{table}\r
\begin{ruledtabular}\r
\begin{tabular}{l c c c c c c }\r
C-C distance [nm] & 0.14 & 0.46 & 0.65 & 0.86 & 1.05 & 1.08 \r
\end{tabular}\r
\end{ruledtabular}\r
-\caption{Binding energies $E_{\text{b}}$ and C-C distance of energetically most favorable C$_{\text{i}}$ $\langle 1 0 0\rangle$-type defect pairs separated along bonds in $\langle 1 1 0\rangle$ direction.}\r
+\caption{Binding energies $E_{\text{b}}$ and C-C distance of energetically most favorable C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs separated along bonds in \hkl[1 1 0] direction.}\r
\label{table:dc_110}\r
\end{table}\r
The binding energy of these configurations with respect to the C-C distance is plotted in Fig.~\ref{fig:dc_110}\r
\begin{figure}\r
\includegraphics[width=\columnwidth]{db_along_110_cc_n.ps}\r
-\caption{Minimum binding energy of dumbbell combinations separated along $\langle 1 1 0\rangle$ with respect to the C-C distance. The blue line is a guide for the eye and the green curve corresponds to the most suitable fit function consisting of all but the first data point.}\r
+\caption{Minimum binding energy of dumbbell combinations separated along \hkl[1 1 0] with respect to the C-C distance. The blue line is a guide for the eye and the green curve corresponds to the most suitable fit function consisting of all but the first data point.}\r
\label{fig:dc_110}\r
\end{figure}\r
-The interaction is found to be proportional to the reciprocal cube of the C-C distance for extended separeations of the C$_{\text{i}}$ and saturates for the smallest possible separation, i.e. the ground state configuration.\r
+The interaction is found to be proportional to the reciprocal cube of the C-C distance for extended separations of the C$_{\text{i}}$ and saturates for the smallest possible separation, i.e. the ground state configuration.\r
\r
-\subsection{C$_{\text{i}}$ next to C$_{\text{s}}$}\r
-\r
-% c_i and c_s, capaz98, mattoni2002 (restricted to 110 -110 bond chain)\r
\begin{table}\r
\begin{ruledtabular}\r
\begin{tabular}{l c c c c c c }\r
& 1 & 2 & 3 & 4 & 5 & R \\\r
\hline\r
-C$_{\text{s}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\\r
-Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\r
+C$_{\text{s}}$ & 0.26$^a$/-1.28$^b$ & -0.51 & -0.93$^A$/-0.95$^B$ & -0.15 & 0.49 & -0.05\\\r
+V & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\r
\end{tabular}\r
\end{ruledtabular}\r
-\caption{Binding energies of combinations of the C$_{\text{i}}$ $[0 0 -1]$ defect with a substitutional C or vacancy located at positions 1 to 5 according to Fig.~\ref{fig:combos}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2} \langle3 2 3 \rangle$ relative to the initial defect position, which is the maximum realizable distance due to periodic boundary conditions.}\r
+\caption{Binding energies of combinations of the C$_{\text{i}}$ \hkl[0 0 -1] defect with a substitutional C or vacancy located at positions 1 to 5 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 distance due to periodic boundary conditions.}\r
\label{table:dc_c-sv}\r
\end{table}\r
-Table~\ref{table:dc_c-sv} lists the binding energies of C$_{\text{s}}$ next to the C$_{\text{i}}$ $[0 0 -1]$ DB.\r
+\r
+\subsection{C$_{\text{i}}$ next to C$_{\text{s}}$}\r
+\r
+The first row of Table~\ref{table:dc_c-sv} lists the binding energies of C$_{\text{s}}$ next to the C$_{\text{i}}$ \hkl[0 0 -1] DB.\r
+For C$_{\text{s}}$ located at position 1 and 3 the configurations a and A correspond to the naive relaxation of the structure by substituting a Si atom with C in the initial C$_{\text{i}}$ \hkl[0 0 -1] DB structure at positions 1 and 3 respectively.\r
+However, small displacements of the involved atoms near the defect result in different stable structures labeled b and B respectively.\r
+\r
+% A B\r
+Configuration A consists of a C$_{\text{i}}$ \hkl[0 0 -1] DB with threefold coordinated Si and C DB atoms slightly disturbed by the C$_{\text{s}}$ at position 3, facing the Si DB atom as a next neighbor.\r
+By breaking a Si-Si in favor of a Si-C bond configuration B is obtained, which shows a twofold coordinated Si atom located inbetween two substitutional C atoms residing on regular Si lattice sites.\r
+This configuration has been identified and described by spectroscopic experimental techniques\cite{song90_2} as well as theoretical studies\cite{leary97,capaz98}.\r
+Configuration B is found to constitute the energetically more favorable configuration.\r
+However, the gain in energy due to the significantly lower energy of a Si-C compared to a Si-Si bond turns out to be smaller than expected due to a large compensation by introduced strain as a result of the Si interstitial structure.\r
+Present results show a difference in energy of states A and B, which exactly matches the experimental value of \unit[0.02]{eV}\cite{song90_2} reinforcing qualitatively correct results of previous theoretical studies on these structures.\r
+% mattoni: A favored by 0.4 eV - NO, it is indeed B (reinforce Song and Capaz)!\r
+%\r
+% AB transition\r
+%Figure~\ref{fig:AB} displays the two configurations and migration barrier for the transition among the two states.\r
+\r
+% a b\r
+Configuration a is similar to configuration A except that the C$_{\text{s}}$ at position 1 is facing the C DB atom as a next neighbor resulting in the formation of a strong C-C bond and a much more noticeable perturbation of the DB structure.\r
+Nevertheless, the C and Si DB atoms remain threefold coordinated.\r
+Although the C-C bond exhibiting a distance of \unit[0.15]{nm} close to the distance expected in diamond or graphite should lead to a huge gain in energy, a repulsive interaction with a binding energy of \unit[0.26]{eV} is observed due to compressive strain of the Si DB atom and its top neighbors (\unit[0.230]{nm}/\unit[0.236]{nm}) along with additional tensile strain of the C$_{\text{s}}$ and its three neighboring Si atoms (\unit[0.198-0.209]{nm}/\unit[0.189]{nm}).\r
+\r
+...\r
+\r
+Liu et~al.\cite{liu02} propose a similar structure \unit[0.2]{eV} lower than configuration B, thus, constituting the ground state configuration.\r
+The structure labeld b indeed is the ground state configuration, in which the two C atoms form a \hkl[1 0 0] DB sharing the C$_{\text{s}}$ lattice site and the initial Si DB atom occupying the lattice site shared by the initial C$_{\text{i}}$ DB.\r
+\r
+Spin polarization for C-C Int resulting spin up electrons located as in the case of the Si 100 int.\r
+% mattoni: A favored by 0.2 eV - NO! (again, missing spin polarization?)\r
\r
\r
\subsection{C$_{\text{i}}$ next to V}\r
\r
Non-zeor temperature, entropy, spatial separation of these defects possible, indeed observed in ab initio MD run.\r
\r
-%% Viele Bilder... da kann ich zunächst gar nicht soviel zu schreiben.... \r
-\r
\section{Discussion}\r
Our calculations show that point defects which unavoidably are present after ion implantation significantly influence the mobility of implanted carbon \r
in the silicon crystal.\r
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