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\title{First-principles and empirical potential simulation study of intrinsic
and carbon-related defects in silicon}
-%\titlerunning{}
+\titlerunning{First-principles and empirical potential simulation study
+ of intrinsic and carbon-related defects in silicon}
\author{%
F. Zirkelbach\textsuperscript{\Ast,\textsf{\bfseries 1}},
33095 Paderborn, Germany}
\received{XXXX, revised XXXX, accepted XXXX}
-p
+
\published{XXXX}
\keywords{Silicon, carbon, silicon carbide, defect formation, defect migration,
\section{Introduction}
-Silicon carbide (SiC) is a promising material for high-temperature, high-power and high-frequency electronic and optoelectronic devices, which can operate under extreme conditions\cite{edgar92,morkoc94,wesch96,capano97,park98}.
-Ion beam synthesis (IBS) consisting of high-dose carbon implantation into crystalline silicon (c-Si) and subsequent or in situ annealing is a promising technique to fabricate nano-sized precipitates and thin films of the favorable cubic SiC (3C-SiC) polytype topotactically aligned to and embedded in the silicon host\cite{borders71,lindner99,lindner01,lindner02}.
+Silicon carbide (SiC) is a promising material for high-temperature, high-power and high-frequency electronic and optoelectronic devices, which can operate under extreme conditions \cite{edgar92,morkoc94,wesch96,capano97,park98}.
+Ion beam synthesis (IBS) consisting of high-dose carbon implantation into crystalline silicon (c-Si) and subsequent or in situ annealing is a promising technique to fabricate nano-sized precipitates and thin films of the favorable cubic SiC (3C-SiC) polytype topotactically aligned to and embedded in the silicon host \cite{borders71,lindner99,lindner01,lindner02}.
However, the process of formation of SiC precipitates in Si during C implantation is not yet fully understood and controversial ideas exist in the literature.
-Based on experimental high resolution transmission electron microscopy (HREM) studies\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03} it is assumed that incorporated C atoms form C-Si dimers (dumbbells) on regular Si lattice sites.
+Based on experimental high resolution transmission electron microscopy (HREM) studies \cite{werner96,werner97,eichhorn99,lindner99_2,koegler03} it is assumed that incorporated C atoms form C-Si dimers (dumbbells) on regular Si lattice sites.
The highly mobile C interstitials agglomerate into large clusters followed by the formation of incoherent 3C-SiC nanocrystallites once a critical size of the cluster is reached.
-In contrast, a couple of other studies\cite{strane94,nejim95,guedj98} suggest initial coherent SiC formation by agglomeration of substitutional instead of interstitial C followed by the loss of coherency once the increasing strain energy surpasses the interfacial energy of the incoherent 3C-SiC precipitate and the c-Si substrate.
-
-HIER WEITER
+In contrast, a couple of other studies \cite{strane94,nejim95,guedj98} suggest initial coherent SiC formation by agglomeration of substitutional instead of interstitial C followed by the loss of coherency once the increasing strain energy surpasses the interfacial energy of the incoherent 3C-SiC precipitate and the c-Si substrate.
-Atomistic simulations offer a powerful tool of investigation on a microscopic level providing detailed insight not accessible by experiment.
-A lot of theoretical work has been done on intrinsic point defects in Si\cite{bar-yam84,bar-yam84_2,car84,batra87,bloechl93,tang97,leung99,colombo02,goedecker02,al-mushadani03,hobler05,sahli05,posselt08,ma10}, threshold displacement energies in Si\cite{mazzarolo01,holmstroem08} important in ion implantation, C defects and defect reactions in Si\cite{tersoff90,dal_pino93,capaz94,burnard93,leary97,capaz98,zhu98,mattoni2002,park02,jones04}, the SiC/Si interface\cite{chirita97,kitabatake93,cicero02,pizzagalli03} and defects in SiC\cite{bockstedte03,rauls03a,gao04,posselt06,gao07}.
+To solve this controversy and in order to understand the effective underlying processes on a microscopic level atomistic simulations are performed.
+% ????
+A lot of theoretical work has been done on intrinsic point defects in Si \cite{bar-yam84,bar-yam84_2,car84,batra87,bloechl93,tang97,leung99,colombo02,goedecker02,al-mushadani03,hobler05,sahli05,posselt08,ma10}, threshold displacement energies in Si \cite{mazzarolo01,holmstroem08} important in ion implantation, C defects and defect reactions in Si \cite{tersoff90,dal_pino93,capaz94,burnard93,leary97,capaz98,zhu98,mattoni2002,park02,jones04}, the SiC/Si interface \cite{chirita97,kitabatake93,cicero02,pizzagalli03} and defects in SiC \cite{bockstedte03,rauls03a,gao04,posselt06,gao07}.
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.
-% but mattoni2002 actually did a lot. maybe this should be mentioned!
-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.
-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 providing further information on the probability of defect agglomeration.
+% ????
-By first-principles atomistic simulations this work aims to shed light on basic processes involved in the precipitation mechanism of SiC in Si.
-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.
-In the following a systematic investigation of density functional theory (DFT) calculations of the structure, energetics and mobility of carbon defects in silicon as well as the influence of other point defects in the surrounding is presented.
-% TODO: maybe delete: decisive role half sentence
+In the present study, a highly accurate first-principles treatment is utilized to systematically investigate relevant intrinsic as well as carbon related defect structures and defect mobilities in silicon.
+These findings are compared to empirical potential results and completed by molecular dynamics simulations to draw conclusions on the SiC precipitation mechanism in Si.
-% --------------------------------------------------------------------------------
\section{Methodology}
The first-principles DFT calculations were performed with the plane-wave based Vienna ab initio simulation package (VASP)\cite{kresse96}.
\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
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*}
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}}
+\subfloat[]{\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}}
+\subfloat[]{\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[-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}
% point out that configurations along 110 were extended up to the 6th NN in that direction
The binding energies of the energetically most favorable configurations with the second 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}.
\begin{table}
-\begin{ruledtabular}
\begin{tabular}{l c c c c c c }
& 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
$E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
C-C distance [nm] & 0.14 & 0.46 & 0.65 & 0.86 & 1.05 & 1.08
\end{tabular}
-\end{ruledtabular}
\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 the \hkl[1 1 0] bond chain.}
\label{table:dc_110}
\end{table}
This finding, in turn, supports the previously established assumption of C agglomeration and absence of C clustering.
\begin{table}
-\begin{ruledtabular}
\begin{tabular}{l c c c c c c }
& 1 & 2 & 3 & 4 & 5 & R \\
\hline
C$_{\text{s}}$ & 0.26$^a$/-1.28$^b$ & -0.51 & -0.93$^A$/-0.95$^B$ & -0.15 & 0.49 & -0.05\\
V & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31
\end{tabular}
-\end{ruledtabular}
\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_ci}. 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.}
\label{table:dc_c-sv}
\end{table}
The Si$_{\text{i}}$ \hkl<1 1 0> DB, which was found to exhibit the lowest energy of formation within the investigated self-interstitial configurations, is assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$.
\begin{table}
-\begin{ruledtabular}
\begin{tabular}{l c c c c c c}
& \hkl[1 1 0] & \hkl[-1 1 0] & \hkl[0 1 1] & \hkl[0 -1 1] &
\hkl[1 0 1] & \hkl[-1 0 1] \\
\end{tabular}
\caption{Equivalent configurations labeled \RM{1}-\RM{10} of \hkl<1 1 0>-type Si$_{\text{i}}$ DBs created at position I and C$_{\text{s}}$ created at positions 1 to 5 according to Fig.~\ref{fig:combos_si}. The respective orientation of the Si$_{\text{i}}$ DB is given in the first row.}
\label{table:dc_si-s}
-\end{ruledtabular}
\end{table}
\begin{table*}
-\begin{ruledtabular}
\begin{tabular}{l c c c c c c c c c c}
& \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & \RM{6} & \RM{7} & \RM{8} & \RM{9} & \RM{10} \\
\hline
\end{tabular}
\caption{Formation energies $E_{\text{f}}$, binding energies $E_{\text{b}}$ and C$_{\text{s}}$-Si$_{\text{i}}$ separation distances of configurations combining C$_{\text{s}}$ and Si$_{\text{i}}$ as defined in Table~\ref{table:dc_si-s}.}
\label{table:dc_si-s_e}
-\end{ruledtabular}
\end{table*}
Table~\ref{table:dc_si-s} classifies equivalent configurations of \hkl<1 1 0>-type Si$_{\text{i}}$ DBs created at position I and C$_{\text{s}}$ created at positions 1 to 5 according to Fig.~\ref{fig:combos_si}.
Corresponding formation as well as binding energies and the separation distances of the C$_{\text{s}}$ atom and the Si$_{\text{i}}$ DB lattice site are listed in Table~\ref{table:dc_si-s_e}.
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