X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=posic%2Fpublications%2Femrs2012.tex;h=5a4433a603234f93901474139f8b70f23b66c116;hp=76e402ad411df899f0a5f74cb0f74c6e26eb1f71;hb=868d425d864af9b60f8f51381374a54a7d857d4e;hpb=712489af3555db2f85671f1657bf8cee74041395 diff --git a/posic/publications/emrs2012.tex b/posic/publications/emrs2012.tex index 76e402a..5a4433a 100644 --- a/posic/publications/emrs2012.tex +++ b/posic/publications/emrs2012.tex @@ -54,7 +54,7 @@ Results of atomistic simulations aimed at understanding precipitation of the highly attractive wide band gap semiconductor material silicon carbide in silicon are presented. The study involves a systematic investigation of intrinsic and carbon-related defects as well as defect combinations and defect migration by both, quantum-mechanical first-principles as well as empirical potential methods. Comparing formation and activation energies, ground-state structures of defects and defect combinations as well as energetically favorable agglomeration of defects are predicted. -Moreover, the highly accurate ab initio calculations unveil limitations of the analytical method based on a Tersoff-like bond order potential. +Moreover, the highly accurate {\em ab initio} calculations unveil limitations of the analytical method based on a Tersoff-like bond order potential. A work-around is proposed in order to subsequently apply the highly efficient technique on large structures not accessible by first-principles methods. The outcome of both types of simulation provides a basic microscopic understanding of defect formation and structural evolution particularly at non-equilibrium conditions strongly deviated from the ground state as commonly found in SiC growth processes. A possible precipitation mechanism, which conforms well to experimental findings clarifying contradictory views present in the literature is outlined. @@ -64,16 +64,33 @@ A possible precipitation mechanism, which conforms well to experimental findings \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 +% shorten +% \cite{edgar92,morkoc94,wesch96,capano97,park98}. +\cite{edgar92,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 +% shorten +% \cite{borders71,lindner99,lindner01,lindner02}. +\cite{borders71,lindner01}. 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 +% shorten +% \cite{werner96,werner97,eichhorn99,lindner99_2,koegler03} +\cite{werner96,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. +In contrast, a couple of other studies \cite{strane94,nejim95,serre95} 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. 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} and C defects and defect reactions in Si \cite{tersoff90,dal_pino93,capaz94,burnard93,leary97,capaz98,zhu98,mattoni2002,park02,jones04}. +A lot of theoretical work has been done on intrinsic point defects in Si +% shorten +% \cite{bar-yam84,bar-yam84_2,car84,batra87,bloechl93,tang97,leung99,colombo02,goedecker02,al-mushadani03,hobler05,sahli05,posselt08,ma10} +\cite{bar-yam84,car84,bloechl93,tang97,leung99,al-mushadani03,hobler05,sahli05,posselt08,ma10} +and C defects and defect reactions in Si +% shorten +%\cite{tersoff90,dal_pino93,capaz94,burnard93,leary97,capaz98,zhu98,mattoni2002,park02}. +\cite{tersoff90,dal_pino93,capaz94,burnard93,leary97,capaz98,mattoni2002}. 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. % ???? @@ -82,9 +99,9 @@ These findings are compared to empirical potential results, which, by taking int \section{Methodology} -The plane-wave based Vienna ab initio simulation package (VASP) \cite{kresse96} is used for the first-principles calculations based on density functional theory (DFT). +The plane-wave based Vienna {\em ab initio} simulation package (VASP) \cite{kresse96} is used for the first-principles calculations based on density functional theory (DFT). Exchange and correlation is taken into account by the generalized-gradient approximation \cite{perdew86,perdew92}. -Norm-conserving ultra-soft pseudopotentials \cite{hamann79} as implemented in VASP \cite{vanderbilt90} are used to describe the electron-ion interaction. +Norm-con\-ser\-ving ultra-soft pseudopotentials \cite{hamann79} as implemented in VASP \cite{vanderbilt90} are used to describe the electron-ion interaction. A kinetic energy cut-off of \unit[300]{eV} is employed. Defect structures and migration paths were modelled in cubic supercells with a side length of \unit[1.6]{nm} containing $216$ Si atoms. These structures are large enough to restrict sampling of the Brillouin zone to the $\Gamma$-point and formation energies and structures are reasonably converged. @@ -99,10 +116,7 @@ The interaction strength, i.e. the absolute value of the binding energy, approac Within the empirical approach, defect structures are modeled in a supercell of nine Si lattice constants in each direction consisting of 5832 Si atoms. Reproducing SiC precipitation is attempted by successive insertion of 6000 C atoms to form a minimal 3C-SiC precipitate with a radius of about \unit[3.1]{nm} within the Si host consisting of 31 unit cells (238328 atoms) in each direction. -At constant temperature 10 atoms are inserted at a time. -Three different regions inside the total simulation volume are considered for a statistically distributed insertion of C atoms. -$V_1$ corresponds to the total simulation volume, $V_2$ to the size of the precipitate and $V_3$ holds the necessary amount of Si atoms of the precipitate. -After C insertion, the simulation is continued for \unit[100]{ps} and cooled down to \unit[20]{$^{\circ}$C} afterwards. +At constant temperature 10 atoms are inserted on statistically distributed positions at a time. A Tersoff-like bond order potential by Erhart and Albe (EA) \cite{albe_sic_pot} has been utilized, which accounts for nearest neighbor interactions realized by a cut-off function dropping the interaction to zero in between the first and second nearest neighbor distance. The Berendsen barostat and thermostat \cite{berendsen84} with a time constant of \unit[100]{fs} enables the isothermal-isobaric ensemble. The velocity Verlet algorithm \cite{verlet67} and a fixed time step of \unit[1]{fs} is used to integrate the equations motion. @@ -110,7 +124,7 @@ Structural relaxation of defect structures is treated by the same algorithms at \section{Defect configurations in silicon} -Table~\ref{tab:defects} summarizes the formation energies of relevant defect structures for the EA and DFT calculations, which are shown in Figs.~\ref{fig_intrinsic_def} and \ref{fig:carbon_def}. +Table~\ref{tab:defects} summarizes the formation energies of relevant defect structures for the EA and DFT calculations, which are shown in Fig.~\ref{fig:intrinsic_def} and \ref{fig:carbon_def}. \begin{table*} \centering \begin{tabular}{l c c c c c c c c c} @@ -125,54 +139,61 @@ Erhart/Albe & 4.39 & 4.48$^*$ & 3.40 & 5.42 & 3.13 & 0.75 & 3.88 & 5.18 & 5.59$^ \label{tab:defects} \end{table*} \begin{figure} +\subfloat[Intrinsic Si point defects.]{% +\begin{minipage}{0.9\columnwidth} \centering \begin{minipage}[t]{0.43\columnwidth} \centering \underline{Si$_{\text{i}}$ \hkl<1 1 0> DB}\\ -\includegraphics[width=0.9\columnwidth]{si110_bonds.eps} +\includegraphics[width=0.8\columnwidth]{si110_bonds.eps} \end{minipage} \begin{minipage}[t]{0.43\columnwidth} \centering \underline{Si$_{\text{i}}$ hexagonal}\\ -\includegraphics[width=0.9\columnwidth]{sihex_bonds.eps} +\includegraphics[width=0.8\columnwidth]{sihex_bonds.eps} \end{minipage}\\ \begin{minipage}[t]{0.43\columnwidth} \centering \underline{Si$_{\text{i}}$ tetrahedral}\\ -\includegraphics[width=0.9\columnwidth]{sitet_bonds.eps} +\includegraphics[width=0.8\columnwidth]{sitet_bonds.eps} \end{minipage} \begin{minipage}[t]{0.43\columnwidth} \centering \underline{Si$_{\text{i}}$ \hkl<1 0 0> DB}\\ -\includegraphics[width=0.9\columnwidth]{si100_bonds.eps} +\includegraphics[width=0.8\columnwidth]{si100_bonds.eps} +\end{minipage} +%\caption{Configurations of intrinsic Si point defects. Dumbbell configurations are abbreviated by DB.} \end{minipage} -\caption{Configurations of intrinsic silicon point defects. Dumbbell configurations are abbreviated by DB.} \label{fig:intrinsic_def} -\end{figure} -\begin{figure} +}\\ +\subfloat[C point defects in Si.]{% +\begin{minipage}{0.9\columnwidth} \centering \begin{minipage}[t]{0.43\columnwidth} \centering -\underline{C$_{\text{s}}$} -\includegraphics[width=0.9\columnwidth]{csub_bonds.eps} +\underline{C$_{\text{s}}$}\\ +\includegraphics[width=0.8\columnwidth]{csub_bonds.eps} \end{minipage} \begin{minipage}[t]{0.43\columnwidth} \centering \underline{C$_{\text{i}}$ \hkl<1 0 0> DB}\\ -\includegraphics[width=0.9\columnwidth]{c100_bonds.eps} +\includegraphics[width=0.8\columnwidth]{c100_bonds.eps} \end{minipage}\\ \begin{minipage}[t]{0.43\columnwidth} \centering \underline{C$_{\text{i}}$ \hkl<1 1 0> DB}\\ -\includegraphics[width=0.9\columnwidth]{c110_bonds.eps} +\includegraphics[width=0.8\columnwidth]{c110_bonds.eps} \end{minipage} \begin{minipage}[t]{0.43\columnwidth} \centering \underline{C$_{\text{i}}$ bond-centered}\\ -\includegraphics[width=0.9\columnwidth]{cbc_bonds.eps} +\includegraphics[width=0.8\columnwidth]{cbc_bonds.eps} +\end{minipage} +%\caption{Configurations of carbon point defects in silicon. Silicon and carbon atoms are illustrated by yellow and gray spheres respectively. Dumbbell configurations are abbreviated by DB.} \end{minipage} -\caption{Configurations of carbon point defects in silicon. Silicon and carbon atoms are illustrated by yellow and gray spheres respectively. Dumbbell configurations are abbreviated by DB.} \label{fig:carbon_def} +} +\caption{Defect configurations in Si. Si and C atoms are illustrated by yellow and gray spheres respectively. Dumbbell configurations are abbreviated by DB.} \end{figure} Regarding intrinsic defects in Si, classical potential and {\em ab initio} methods predict energies of formation that are within the same order of magnitude. @@ -181,36 +202,34 @@ Instead, the tetrahedral configuration is favored, a limitation assumed to arise In the case of C impurities, although discrepancies exist, classical potential and first-principles methods depict the correct order of the formation energies. Next to the substitutional C (C$_{\text{s}}$) configuration, which is not an interstitial configuration since the C atom occupies an already vacant Si lattice site, the interstitial C (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{dal_pino93,capaz94,burnard93,leary97,jones04} and experimental \cite{watkins76,song90} investigations, which all predict this configuration to be the ground state. +This finding is in agreement with several theoretical \cite{dal_pino93,capaz94,burnard93,leary97} and experimental \cite{watkins76,song90} investigations, which all predict this configuration to be the ground state. It is worth to note that the bond-centered (BC) configuration constitutes a real local minimum in spin polarized calculations in contrast to results \cite{capaz94} without spin predicting a saddle point configuration as well as to the empirical description, which shows a relaxation into the C$_{\text{i}}$ \hkl<1 0 0> DB ground-state configuration. \section{Mobility of the carbon defect} -In the following, the migration barriers of the ground-state C defect are investigated by both, first-principles as well as the empircal method. -The migration pathways are shown in Figs.\ref{fig:vasp_mig} and \ref{fig:albe_mig} respectively. +The migration barriers of the ground-state C defect are investigated by both, first-principles as well as the empirical method. +The migration pathways are shown in Fig.~\ref{fig:mig}. \begin{figure} -\begin{center} +\subfloat[Transition path obtained by first-principles methods.]{% \includegraphics[width=\columnwidth]{path2_vasp_s.ps} -\end{center} -\caption{Migration barrier and structures of the C$_{\text{i}}$ \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition as obtained by first principles methods.} -\label{fig:vasp_mig} -\end{figure} -\begin{figure} -\begin{center} +}\\ +%\caption{Migration barrier and structures of the C$_{\text{i}}$ \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition as obtained by first principles methods.} +\subfloat[Transition involving the {\hkl[1 1 0]} DB (center) configuration within the EA description.]{% \includegraphics[width=\columnwidth]{110mig.ps} -\end{center} -\caption{Migration barrier and structures of the C$_{\text{i}}$ \hkl[0 0 -1] DB (left) to the hkl[0 -1 0] DB (right) transition involving the \hkl[1 1 0] DB (center) configuration within EA description. Migration simulations were performed utilizing time constants of \unit[1]{fs} (solid line) and \unit[100]{fs} (dashed line) for the Berendsen thermostat.} -\label{fig:albe_mig} +} +\caption{Migration barriers and structures of the C$_{\text{i}}$ \hkl[0 0 -1] DB (left) to the hkl[0 -1 0] DB (right) transition.} +%\caption{Migration barrier and structures of the C$_{\text{i}}$ \hkl[0 0 -1] DB (left) to the hkl[0 -1 0] DB (right) transition involving the \hkl[1 1 0] DB (center) configuration within EA description. Migration simulations were performed utilizing time constants of \unit[1]{fs} (solid line) and \unit[100]{fs} (dashed line) for the Berendsen thermostat.} +\label{fig:mig} \end{figure} In qualitative agreement with the results of Capaz~et~al.\ \cite{capaz94}, the lowest migration barrier of the ground-state C$_{\text{i}}$ defect within the quantum-mechanical treatment is found for the path, 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 of \unit[0.90]{eV} to experimental values (\unit[0.70-0.87]{eV}) \cite{lindner06,tipping87,song90}. +Calculations in this work reinforce this path by an additional improvement of the quantitative conformance of the barrier height of \unit[0.90]{eV} to experimental values (\unit[0.70-0.87]{eV}) \cite{song90,lindner06,tipping87}. In contrast, the empirical approach does not reproduce the same path. -Related to the instability of the BC configuration \cite{zirkelbach11}, a pathway involving the C$_{\text{i}}$ \hkl<1 1 0> DB as an intermediate configuration must be considered most plausible. -Considering a two step diffusion process and assuming equal preexponential factors, an total effective migration barrier 3.5 times higher than the one obtained by first-principles methods is obtained. -A more detailed description can be found in previous studies \cite{zirkelbach10,zirkelbach11}. +Related to the above mentioned instability of the BC configuration, a pathway involving the C$_{\text{i}}$ \hkl<1 1 0> DB as an intermediate configuration must be considered most plausible \cite{zirkelbach11}. +Considering a two step diffusion process and assuming equal preexponential factors, a total effective migration barrier 3.5 times higher than the one obtained by first-principles methods is obtained. +A more detailed description can be found in previous studies \cite{zirkelbach11,zirkelbach10}. \section{Defect combinations} @@ -235,17 +254,20 @@ The ground-state configuration is obtained for a V located right next to the C a The C atom moves towards the vacant site forming a stable C$_{\text{s}}$ configuration resulting in the release of a huge amount of energy. The second most favorable configuration is accomplished for a V located right next to the Si atom of the DB structure. This is due to the reduction of compressive strain of the Si DB atom and its two upper Si neighbors present in the isolated C$_{\text{i}}$ DB configuration. -This configuration is followed by the structure, in which the V is created at one of the neighbored lattice site below one of the Si atoms that are bound to the C atom of the initial DB. -Relaxed structures of the latter two defect combinations are shown in the bottom left of Figs.~\ref{fig:314-539} and \ref{fig:059-539} respectively together with their energetics during transition into the ground state. +This configuration is followed by the structure, in which the V is created at one of the neighbored lattice sites below one of the Si atoms that are bound to the C atom of the initial DB. +Relaxed structures of the latter two defect combinations are shown in the bottom left of Fig.~\ref{fig:314-539} and \ref{fig:059-539} respectively together with their energetics during transition into the ground state. \begin{figure} +\subfloat[V created right next to the Si atom of the initial DB. Activation energy: {\unit[0.1]{eV}}.]{% \includegraphics[width=\columnwidth]{314-539.ps} -\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created right next to the Si atom of the initial DB (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.1]{eV} is observed.} \label{fig:314-539} -\end{figure} -\begin{figure} +}\\ +%\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created right next to the Si atom of the initial DB (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.1]{eV} is observed.} +\subfloat[V created next to one of the Si atoms that is bound to the C atom of the initial DB. Activation energy: {\unit[0.6]{eV}}.]{% \includegraphics[width=\columnwidth]{059-539.ps} -\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created next to one of the Si atoms that is bound to the C atom of the initial DB (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.6]{eV} is observed.} \label{fig:059-539} +} +%\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created next to one of the Si atoms that is bound to the C atom of the initial DB (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.6]{eV} is observed.} +\caption{Migration barrier and structures of transitions of an initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V (left) into a C$_{\text{s}}$ configuration (right).} \end{figure} These transitions exhibit activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV}. In the first case the Si and C atom of the DB move towards the vacant and initial DB lattice site respectively. @@ -254,16 +276,14 @@ In both cases, the formation of additional bonds is responsible for the vast gai Considering the small activation energies, a high probability for the formation of stable C$_{\text{s}}$ must be concluded. In addition, it is instructive to investigate combinations of C$_{\text{s}}$ and Si$_{\text{i}}$, which can be created in IBS by highly energetic C atoms that kick out a Si atom from its lattice site, resulting in a Si self-interstitial accompanied by a vacant site, which might get occupied by another C atom that lost almost all of its kinetic energy. -Provided that the first C atom has enough kinetic energy to escape the affected region, the remaining C$_{\text{s}}$-Si$_{\text{i}}$ pair can be described as a separated defect complex. -Considering the energetically most favorable Si$_{\text{i}}$ defect, i.e. the Si$_{\text{i}}$ \hkl<1 1 0> DB, the most favorable combination is found for C$_{\text{s}}$ located right next to that DB enabling the largest possible reduction of strain. -The configuration and the transition into the ground-state configuration, i.e. the C$_{\text{i}}$ hkl<1 0 0> DB is displayed in Fig.~\ref{fig:162-097} +The most favorable configuration, which is C$_{\text{s}}$ located right next to the ground-state Si$_{\text{i}}$ defect, i.e.\ the Si$_{\text{i}}$ \hkl<1 1 0> DB, and the transition of this structure into the ground-state configuration, i.e. the C$_{\text{i}}$ \hkl<1 0 0> DB is displayed in Fig.~\ref{fig:162-097} \begin{figure} \includegraphics[width=\columnwidth]{162-097.ps} \caption{Transition of a \hkl[1 1 0] Si$_{\text{i}}$ DB next to C$_{\text{s}}$ (right) into the C$_{\text{i}}$ \hkl[0 0 -1] DB configuration (left).} \label{fig:162-097} \end{figure} Due to the low barrier of \unit[0.12]{eV}, the C$_{\text{i}}$ \hkl<1 0 0> DB configuration is very likely to occur. -However, the barrier of only \unit[0.77]{eV} for the reverse process indictaes a high probability for the the formation of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB out of the ground state, wich must be considered to be activated without much effort either thermally or by introduced energy of the implantation process. +However, the barrier of only \unit[0.77]{eV} for the reverse process indicates the possibility to form a C$_{\text{s}}$ and Si$_{\text{i}}$ DB out of the ground state activated without much effort either thermally or by introduced energy of the implantation process. \begin{figure} \includegraphics[width=\columnwidth]{c_sub_si110.ps} %\includegraphics[width=\columnwidth]{c_sub_si110_data.ps} @@ -271,83 +291,47 @@ However, the barrier of only \unit[0.77]{eV} for the reverse process indictaes a \caption{Binding energies of combinations of a C$_{\text{s}}$ and a Si$_{\text{i}}$ DB with respect to the separation distance. The interaction strength of the defect pairs are well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.} \label{fig:dc_si-s} \end{figure} -Furthermore, the interaction strength, i.e. the absolute value of the binding energy, quickly drops to zero with increasing separation distance as can be seen in Fig.~\ref{fig:dc_si-s}. -The interaction of the defects is well approximated by a Lennard-Jones (LJ) 6-12 potential, which is used for curve fitting. -Unable to model possible positive values of the binding energy, i.e. unfavorable configurations, located to the right of the minimum, the LJ fit should rather be thought as a guide for the eye describing the decrease of the interaction strength, i.e. the absolute value of the binding energy, with increasing separation distance. +Furthermore, the interaction strength quickly drops to zero with increasing separation distance as can be seen in Fig.~\ref{fig:dc_si-s}. +The interaction of the defects is well approximated by a Lennard-Jones (LJ) 6-12 potential. +Unable to model possible positive values of the binding energy, i.e. unfavorable configurations, the LJ fit should rather be thought as a guide for the eye describing the decrease of the interaction strength, i.e. the absolute value of the binding energy, with increasing separation distance. The LJ fit estimates almost zero interaction already at \unit[0.5-0.6]{nm}, indicating a low interaction capture radius of the defect pair. In IBS separations exceeding this capture radius are easily produced. For these reasons, it must be concluded that configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ instead of the thermodynamically stable C$_{\text{i}}$ \hkl<1 0 0> DB play a decisive role in IBS, a process far from equilibrium. -Indeed, in a previous study, an ab initio molecular dynamics run at \unit[900]{$^{\circ}$C} results in a separation of the C$_{\text{s}}$ and Si$_{\text{i}}$ DB located right next to each other \cite{zirkelbach11}. +Indeed, in a previous study, an {\em ab initio} molecular dynamics run at \unit[900]{$^{\circ}$C} results in a separation of the C$_{\text{s}}$ and Si$_{\text{i}}$ DB located right next to each other \cite{zirkelbach11}. To summarize, these obtained results suggest an increased participation of C$_{\text{s}}$ already in the initial stages of precipitation under IBS conditions. \section{Large scale empirical potential MD results} +Results of the MD simulations at \unit[450]{$^{\circ}$C}, an operative and efficient temperature in IBS \cite{lindner01}, indicate the formation of C$_{\text{i}}$ \hkl<1 0 0> DBs if C is inserted into the total simulation volume. +However, no agglomeration is observed within the simulated time, which was increased up to several nanoseconds. +This is attributed to the drastically overestimated migration barrier of the C defect, which hampers C agglomeration. +To overcome this obstacle, the simulation temperature is successively increased up to \unit[2050]{$^{\circ}$C}. +Fig.~\ref{fig:tot} shows the resulting radial distribution functions of Si-C bonds for various elevated temperatures. +\begin{figure} +\includegraphics[width=\columnwidth]{tot_pc_thesis.ps} +\caption{Radial distribution function for Si-C pairs for C insertion at various elevated temperatures. Si-C distances of a single C$_{\text{s}}$ defect configuration are plotted.} +\label{fig:tot} +\end{figure} +Although not intended, a transformation from a structure dominated by C$_{\text{i}}$ into a structure consisting of C$_{\text{s}}$ with increasing temperature can clearly be observed if compared with the radial distribution of C$_{\text{s}}$ in c-Si. + +Thus, the C$_{\text{s}}$ defect and resulting stretched coherent structures of SiC, must be considered to play an important role in the IBS at elevated temperatures. +This, in fact, satisfies experimental findings of annealing experiments \cite{strane94,nejim95,serre95} and as well as the previous DFT results, which suggest C$_{\text{s}}$ to be involved at higher temperatures and in conditions that deviate the system out of the thermodynamic ground state. + \section{Summary and discussion} -Obtained results for separated point defects in Si are in good agreement to previous theoretical work on this subject, both for intrinsic defects\cite{leung99,al-mushadani03} as well as for C point defects\cite{dal_pino93,capaz94}. -The ground state configurations of these defects, i.e. the Si$_{\text{i}}$ \hkl<1 1 0> and C$_{\text{i}}$ \hkl<1 0 0> DB, have been reproduced and compare well to previous findings of theoretical investigations on Si$_{\text{i}}$\cite{leung99,al-mushadani03} as well as theoretical\cite{dal_pino93,capaz94,burnard93,leary97,jones04} and experimental\cite{watkins76,song90} studies on C$_{\text{i}}$. -A quantitatively improved activation energy of \unit[0.9]{eV} for a qualitatively equal migration path based on studies by Capaz et.~al.\cite{capaz94} to experimental values\cite{song90,lindner06,tipping87} ranging from \unit[0.70-0.87]{eV} reinforce their derived mechanism of diffusion for C$_{\text{i}}$ in Si. - -The investigation of defect pairs indicated a general trend of defect agglomeration mainly driven by the potential of strain reduction. -Obtained results for the most part compare well with results gained in previous studies\cite{leary97,capaz98,mattoni2002,liu02} and show an astonishingly good agreement with experiment\cite{song90}. -For configurations involving two C impurities the ground state configurations have been found to consist of C-C bonds, which are responsible for the vast gain in energy. -However, based on investigations of possible migration pathways, these structures are less likely to arise than structures, in which both C atoms are interconnected by another Si atom, which is due to high activation energies of the respective pathways or alternative pathways featuring less high activation energies, which, however, involve intermediate unfavorable configurations. -Thus, agglomeration of C$_{\text{i}}$ is expected while the formation of C-C bonds is assumed to fail to appear by thermally activated diffusion processes. - -In contrast, C$_{\text{i}}$ and Vs were found to efficiently react with each other exhibiting activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} resulting in stable C$_{\text{s}}$ configurations. -In addition, we observed a highly attractive interaction exhibiting a large capture radius, effective independent of the orientation and the direction of separation of the defects. -Accordingly, the formation of C$_{\text{s}}$ is very likely to occur. -Comparatively high energies necessary for the reverse process reveal this configuration to be extremely stable. - -Investigating configurations of C$_{\text{s}}$ and Si$_{\text{i}}$, formation energies higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB were obtained keeping up previously derived assumptions concerning the ground state of C$_{\text{i}}$ in otherwise perfect Si. -However, a small capture radius was identified for the respective interaction that might prevent the recombination of defects exceeding a separation of \unit[0.6]{nm} into the ground state configuration. -In addition, a rather small activation energy of \unit[0.77]{eV} allows for the formation of a C$_{\text{s}}$-Si$_{\text{i}}$ pair originating from the C$_{\text{i}}$ \hkl<1 0 0> DB structure by thermally activated processes. -Thus, elevated temperatures might lead to configurations of C$_{\text{s}}$ and a remaining Si atom in the near interstitial lattice, which is supported by the result of the molecular dynamics run. - -% add somewhere: nearly same energies of C_i -> Si_i + C_s, Si_i mig and C_i mig - -These findings allow to draw conclusions on the mechanisms involved in the process of SiC conversion in Si. -Agglomeration of C$_{\text{i}}$ is energetically favored and enabled by a low activation energy for migration. -Although ion implantation is a process far from thermodynamic equilibrium, which might result in phases not described by the Si/C phase diagram, i.e. a C phase in Si, high activation energies are believed to be responsible for a low probability of the formation of C-C clusters. - -In the context of the initially stated controversy present in the precipitation model, these findings suggest an increased participation of C$_{\text{s}}$ already in the initial stage due to its high probability of incidence. -In addition, thermally activated, C$_{\text{i}}$ might turn into C$_{\text{s}}$. -The associated emission of Si$_{\text{i}}$ serves two needs: as a vehicle for other C$_{\text{s}}$ atoms and as a supply of Si atoms needed elsewhere to form the SiC structure. -As for the vehicle, Si$_{\text{i}}$ is believed to react with C$_{\text{s}}$ turning it into highly mobile C$_{\text{i}}$ again, allowing for the rearrangement of the C atom. -The rearrangement is crucial to end up in a configuration of C atoms only occupying substitutionally the lattice sites of one of the two fcc lattices that build up the diamond lattice. -% TODO: add SiC structure info to intro -On the other hand, the conversion of some region of Si into SiC by substitutional C is accompanied by a reduction of the volume since SiC exhibits a \unit[20]{\%} smaller lattice constant than Si. -The reduction in volume is compensated by excess Si$_{\text{i}}$ serving as building blocks for the surrounding Si host or a further formation of SiC. - -We conclude that precipitation occurs by successive agglomeration of C$_{\text{s}}$. -However, the agglomeration and rearrangement of C$_{\text{s}}$ is only possible by mobile C$_{\text{i}}$, which has to be present at the same time. -Accordingly, the process is governed by both, C$_{\text{s}}$ accompanied by Si$_{\text{i}}$ as well as C$_{\text{i}}$. -It is worth to mention that there is no contradiction to results of the HREM studies\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}. -Regions showing dark contrasts in an otherwise undisturbed Si lattice are attributed to C atoms in the interstitial lattice. -However, there is no particular reason for the C species to reside in the interstitial lattice. -Contrasts are also assumed for Si$_{\text{i}}$. -Once precipitation occurs, regions of dark contrasts disappear in favor of Moir\'e patterns indicating 3C-SiC in c-Si due to the mismatch in the lattice constant. -Until then, however, these regions are either composed of stretched coherent SiC and interstitials or of already contracted incoherent SiC surrounded by Si and interstitials, where the latter is too small to be detected in HREM. -In both cases Si$_{\text{i}}$ might be attributed a third role, which is the partial compensation of tensile strain that is present either in the stretched SiC or at the interface of the contracted SiC and the Si host. - -In addition, the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate is satisfied by the mechanism of successive positioning of C$_{\text{s}}$. -In contrast, there is no obvious reason for the topotactic orientation of an agglomerate consisting exclusively of C-Si dimers, which would necessarily involve a much more profound change in structure for the transition into SiC. - -\section{Summary} - -In summary, C and Si point defects in Si, combinations of these defects and diffusion processes within such configurations have been investigated. -We have shown that C interstitials in Si tend to agglomerate, which is mainly driven by a reduction of strain. -Investigations of migration pathways, however, allow to conclude that C clustering is hindered due to high activation energies of the respective diffusion processes. -A highly attractive interaction and a large capture radius has been identified for the C$_{\text{i}}$ \hkl<1 0 0> DB and the vacancy indicating a high probability for the formation of C$_{\text{s}}$. -In contrast, a rapidly decreasing interaction with respect to the separation distance has been identified for C$_{\text{s}}$ and a Si$_{\text{i}}$ \hkl<1 1 0> DB resulting in a low probability of defects exhibiting respective separations to transform into the C$_{\text{i}}$ \hkl<1 0 0> DB, which constitutes the ground state configuration for a C atom introduced into otherwise perfect Si. -%Based on these findings conclusions on basic processes involved in the SiC precipitation in bulk Si are drawn. -Obviously, the precipitation process is governed by the formation of C$_{\text{s}}$ already in the initial stages. -Agglomeration and rearrangement of C$_{\text{s}}$, however, is only possible by mobile C$_{\text{i}}$, which, thus, needs to be present at the same time. -Si$_{\text{i}}$ constitutes the vehicle for the rearrangement of C$_{\text{s}}$. - -\section*{Acknowledgment} +Although investigations of defect combinations show the agglomeration of C$_{\text{i}}$ DBs to be energetically most favorable, configurations that may arise during IBS were presented, their dynamics indicating C$_{\text{s}}$ to play an important role particularly at high temperatures. +This is supported by the classical MD results, which show an increased participation of C$_{\text{s}}$ at increased temperatures that allow the system to deviate from the ground state. + +Based on these findings, it is concluded that in IBS at elevated temperatures, SiC conversion takes place by an initial agglomeration of C$_{\text{s}}$ into coherent, tensily strained structures of SiC followed by precipitation into incoherent SiC once a critical size is reached and the increasing strain energy of the coherent structure surpasses the interfacial energy of the incoherent precipitate. +Rearrangement of stable C$_{\text{s}}$ is enabled by excess Si$_{\text{i}}$, which not only acts as a vehicle for C but also as a supply of Si atoms needed elsewhere to form the SiC structure and to reduce possible strain at the interface of coherent SiC precipitates and the Si host. + +%It is worth to point out that the experimentally observed alignment of the \hkl(h k l) planes of precipitate and substrate is satisfied by this mechanism. +%In contrast, the topotactic orientation of the SiC precipitate originating from an agglomerate consisting exclusively of C-Si dimers would necessarily involve a much more profound change in structure. + +\begin{acknowledgement} We gratefully acknowledge financial support by the Bayerische Forschungsstiftung (Grant No. DPA-61/05) and the Deutsche Forschungsgemeinschaft (Grant No. DFG SCHM 1361/11). +\end{acknowledgement} \bibliography{../../bibdb/bibdb}{} \bibliographystyle{pss.bst}