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.
+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}
\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.
+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{zirkelbach10,zirkelbach11}.
\section{Defect combinations}
\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}.
+To overcome the drastically overestimated migration barriers of the C defect, which hamper C agglomeration, the simulation temperature is successively increased up to \unit[2050]{$^{\circ}$C}.
Fig.~\ref{fig:tot} shows the resulting radial distribution function of Si-C bonds for various elevated temperatures.
\begin{figure}
\includegraphics[width=\columnwidth]{tot_pc_thesis.ps}
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