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.
\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,serre95,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.
% ????
\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.
A kinetic energy cut-off of \unit[300]{eV} is employed.
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.
\underline{Si$_{\text{i}}$ \hkl<1 0 0> DB}\\
\includegraphics[width=0.8\columnwidth]{si100_bonds.eps}
\end{minipage}
-\caption{Configurations of intrinsic silicon point defects. Dumbbell configurations are abbreviated by DB.}
+\caption{Configurations of intrinsic Si point defects. Dumbbell configurations are abbreviated by DB.}
\label{fig:intrinsic_def}
\end{figure}
\begin{figure}
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.
+In the following, 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 Figs.\ref{fig:vasp_mig} and \ref{fig:albe_mig} respectively.
\begin{figure}
\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 a high probability for the the formation of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB out of the ground state, which must be considered to be 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}
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{lindner99}, indicate the formation of C$_{\text{i}}$ \hkl<1 0 0> DBs if C is inserted into the total simulation volume.
+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.
To overcome the drastically overestimated migration barriers of the C defect, which hamper C agglomeration, the simulation temperature is successively increased up to ßunit[2050]{$^{\circ}$C}.
Fig.~\ref{fig:tot} shows the resulting radial distribution function of Si-C bonds for various elevated temperatures.
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