\author{%
F. Zirkelbach\textsuperscript{\Ast,\textsf{\bfseries 1}},
- B. Stritzker\textsuperscript{\Ast,\textsf{\bfseries 1}},
- K. Nordlund\textsuperscript{\Ast,\textsf{\bfseries 2}},
- W. G. Schmidt\textsuperscript{\Ast,\textsf{\bfseries 3}},
- E. Rauls\textsuperscript{\Ast,\textsf{\bfseries 3}},
- J. K. N. Lindner\textsuperscript{\Ast,\textsf{\bfseries 3}}
+ B. Stritzker\textsuperscript{\textsf{\bfseries 1}},
+ K. Nordlund\textsuperscript{\textsf{\bfseries 2}},
+ W. G. Schmidt\textsuperscript{\textsf{\bfseries 3}},
+ E. Rauls\textsuperscript{\textsf{\bfseries 3}},
+ J. K. N. Lindner\textsuperscript{\textsf{\bfseries 3}}
}
\authorrunning{F. Zirkelbach et al.}
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}.
+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}.
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.
% ????
-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.
+In the present study, an accurate first-principles treatment is utilized to systematically investigate relevant intrinsic as well as carbon related defect structures and defect mobilities in silicon, which allow to draw conclusions on the mechanism of SiC precipitation in Si.
+These findings are compared to empirical potential results, which, by taking into account the drawbacks of the less accurate though computationally efficient method enabling molecular dynamics (MD) simulations of large structures, support and complete previous findings on SiC precipitation based on the quantum-mechanical treatment.
\section{Methodology}
-The first-principles DFT calculations were performed with the plane-wave based Vienna ab initio simulation package (VASP)\cite{kresse96}.
-The Kohn-Sham equations were solved using the generalized-gradient exchange-correlation (XC) functional approximation proposed by Perdew and Wang\cite{perdew86,perdew92}.
-The electron-ion interaction was described by norm-conserving ultra-soft pseudopotentials\cite{hamann79} as implemented in VASP\cite{vanderbilt90}.
-Throughout this work an energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis.
-To reduce the computational effort sampling of the Brillouin zone was restricted to the $\Gamma$-point, which has been shown to yield reliable results\cite{dal_pino93}.
-The defect structures and the migration paths were modelled in cubic supercells with a side length of \unit[1.6]{nm} containing $216$ Si atoms.
-Formation energies and structures are reasonably converged with respect to the system size.
-The ions and cell shape were allowed to change in order to realize a constant pressure simulation.
-The observed changes in volume were less than \unit[0.2]{\%} of the volume indicating a rather low dependence of the results on the ensemble choice.
-Ionic relaxation was realized by the conjugate gradient algorithm.
-Spin polarization has been fully accounted for.
-
-Migration and recombination pathways have been investigated utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.
-While not guaranteed to find the true minimum energy path, the method turns out to identify reasonable pathways for the investigated structures.
+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).
+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.
+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.
+The ions and cell shape are allowed to change in order to realize a constant pressure simulation realized by the conjugate gradient algorithm.
+Spin polarization is fully accounted for.
+
+Migration and recombination pathways are investigated utilizing the constraint conjugate gradient relaxation technique (CRT) \cite{kaukonen98}.
The defect formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ is defined by choosing SiC as a particle reservoir for the C impurity, i.e. the chemical potentials are determined by the cohesive energies of a perfect Si and SiC supercell after ionic relaxation.
-%In the same way defect formation energies are determined in the article\cite{dal_pino93} used for comparison.
-This corresponds to the definition utilized in another study on C defects in Si\cite{dal_pino93} that we compare our results to.
The binding energy of a defect pair is given by the difference of the formation energy of the complex and the sum of the two separated defect configurations.
-%Accordingly, energetically favorable configurations show binding energies below zero while non-interacting isolated defects result in a binding energy of zero.
Accordingly, energetically favorable configurations result in binding energies below zero while unfavorable configurations show positive values for the binding energy.
The interaction strength, i.e. the absolute value of the binding energy, approaches zero for increasingly non-interacting isolated defects.
-\section{Results}
-
-The implantation of highly energetic C atoms results in a multiplicity of possible defect configurations.
-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.
-First of all, structure and energetics of separated defects are presented.
-The investigations proceed with pairs of the ground state and, thus, most probable defect configurations that are believed to be fundamental in the Si to SiC conversion.
+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.
+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.
+Structural relaxation of defect structures is treated by the same algorithms at zero temperature.
-\subsection{Separated defects in silicon}
-\label{subsection:sep_def}
-% we need both: Si self-int & C int ground state configuration (for combos)
+\section{Defect configurations in silicon}
-Several geometries have been calculated to be stable for individual intrinsic and C related defects in Si.
-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}.
+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}.
+\begin{table*}
+\centering
+\begin{tabular}{l c c c c c c c c c}
+\hline
+$E_{\text{f}}$ [eV] & 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
+VASP & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 & 1.95 & 3.72 & 4.16 & 4.66 \\
+Erhart/Albe & 4.39 & 4.48$^*$ & 3.40 & 5.42 & 3.13 & 0.75 & 3.88 & 5.18 & 5.59$^*$ \\
+\hline
+\end{tabular}
+\caption{Formation energies of C and Si point defects in c-Si given in eV. T denotes the tetrahedral, H the hexagonal and BC the bond-centered interstitial configuration. V corresponds to the vacancy. Subscript i and s indicates the interstitial and substitutional configuration. Dumbbell configurations are abbreviated by DB. Formation energies of unstable configurations are marked by an asterisk.}
+\label{tab:defects}
+\end{table*}
\begin{figure}
-\begin{minipage}[t]{0.32\columnwidth}
+\centering
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
\underline{Si$_{\text{i}}$ \hkl<1 1 0> DB}\\
-\includegraphics[width=\columnwidth]{si110.eps}
+\includegraphics[width=0.9\columnwidth]{si110_bonds.eps}
\end{minipage}
-\begin{minipage}[t]{0.32\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
\underline{Si$_{\text{i}}$ hexagonal}\\
-\includegraphics[width=\columnwidth]{sihex.eps}
-\end{minipage}
-\begin{minipage}[t]{0.32\columnwidth}
-\underline{Si$_{\text{i}}$ tetrahedral}\\
-\includegraphics[width=\columnwidth]{sitet.eps}
+\includegraphics[width=0.9\columnwidth]{sihex_bonds.eps}
\end{minipage}\\
-\begin{minipage}[t]{0.32\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
+\underline{Si$_{\text{i}}$ tetrahedral}\\
+\includegraphics[width=0.9\columnwidth]{sitet_bonds.eps}
+\end{minipage}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
\underline{Si$_{\text{i}}$ \hkl<1 0 0> DB}\\
-\includegraphics[width=\columnwidth]{si100.eps}
+\includegraphics[width=0.9\columnwidth]{si100_bonds.eps}
\end{minipage}
-\begin{minipage}[t]{0.32\columnwidth}
-\underline{Vacancy}\\
-\includegraphics[width=\columnwidth]{sivac.eps}
+\caption{Configurations of intrinsic silicon point defects. Dumbbell configurations are abbreviated by DB.}
+\label{fig:intrinsic_def}
+\end{figure}
+\begin{figure}
+\centering
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
+\underline{C$_{\text{s}}$}
+\includegraphics[width=0.9\columnwidth]{csub_bonds.eps}
\end{minipage}
-\begin{minipage}[t]{0.32\columnwidth}
-\underline{C$_{\text{s}}$}\\
-\includegraphics[width=\columnwidth]{csub.eps}
-\end{minipage}\\
-\begin{minipage}[t]{0.32\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
\underline{C$_{\text{i}}$ \hkl<1 0 0> DB}\\
-\includegraphics[width=\columnwidth]{c100.eps}
-\end{minipage}
-\begin{minipage}[t]{0.32\columnwidth}
+\includegraphics[width=0.9\columnwidth]{c100_bonds.eps}
+\end{minipage}\\
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
\underline{C$_{\text{i}}$ \hkl<1 1 0> DB}\\
-\includegraphics[width=\columnwidth]{c110.eps}
+\includegraphics[width=0.9\columnwidth]{c110_bonds.eps}
\end{minipage}
-\begin{minipage}[t]{0.32\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
\underline{C$_{\text{i}}$ bond-centered}\\
-\includegraphics[width=\columnwidth]{cbc.eps}
+\includegraphics[width=0.9\columnwidth]{cbc_bonds.eps}
\end{minipage}
-\caption{Configurations of silicon and carbon point defects in silicon. Silicon and carbon atoms are illustrated by yellow and gray spheres respectively. Bonds are drawn whenever considered appropriate to ease identifying defect structures for the reader. Dumbbell configurations are abbreviated by DB.}
-\label{fig:sep_def}
+\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}
\end{figure}
-\begin{table*}
-\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
- Present study & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 & 1.95 & 3.72 & 4.16 & 4.66 \\
- \multicolumn{10}{c}{Other ab initio studies} \\
- Ref.\cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 & - & - & - & - \\
- 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}
-\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*}
-Results obtained by the present study compare well with results from literature\cite{leung99,al-mushadani03,dal_pino93,capaz94}.
-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 closely followed by the hexagonal and tetrahedral configuration, which is consensus for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.
-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.
-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 to be the ground state.
-%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.
-However, to our best knowledge, no energy of formation for this type of defect based on first-principles calculations is available.
-
-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.
-The BC configuration is claimed to constitute the saddle point within the C$_{\text{i}}$ \hkl[0 0 -1] DB migration path residing in the \hkl(1 1 0) plane and, thus, interpreted as the barrier of migration for the respective path.
-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.
-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.
-Regardless of the rather small correction of \unit[0.3]{eV} due to the spin, the difference we found is much smaller (\unit[0.94]{eV}), which would nicely compare to experimentally observed migration barriers of \unit[0.70-0.87]{eV}\cite{lindner06,tipping87,song90}.
-However, since the BC configuration constitutes a real local minimum another barrier exists which is about \unit[1.2]{eV} in height.
+
+Regarding intrinsic defects in Si, classical potential and {\em ab initio} methods predict energies of formation that are within the same order of magnitude.
+The EA potential does not reproduce the correct ground state, i.e. the interstitial Si (Si$_{\text{i}}$) \hkl<1 1 0> dumbbell (DB), which is consensus for Si$_{\text{i}}$ \cite{leung99,al-mushadani03}.
+Instead, the tetrahedral configuration is favored, a limitation assumed to arise due to the sharp cut-off as has already been discussed by Tersoff \cite{tersoff90}.
+
+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.
+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{Defect mobility}
+
+HIER MEHR ...
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 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 (\unit[0.90]{eV}) to experimental values.
-A more detailed description can be found in a previous study\cite{zirkelbach10a}.
-
-Next to the C$_{\text{i}}$ 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.
-For the vacancy the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site.
-In the Si$_{\text{i}}$ \hkl<1 0 0> DB configuration the net spin up density is localized in two caps at each of the two DB atoms perpendicularly aligned to the bonds to the other two Si atoms respectively.
-No other configuration, within the ones that are mentioned, is affected.
+A more detailed description can be found in a previous study \cite{zirkelbach10}.
Concerning the mobility of the ground state Si$_{\text{i}}$, we found an activation energy of \unit[0.67]{eV} for the transition of the Si$_{\text{i}}$ \hkl[0 1 -1] to \hkl[1 1 0] DB located at the neighbored Si lattice site in \hkl[1 1 -1] direction.
Further investigations revealed a barrier of \unit[0.94]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to Si$_{\text{i}}$ H, \unit[0.53]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to Si$_{\text{i}}$ T and \unit[0.35]{eV} for the Si$_{\text{i}}$ H to Si$_{\text{i}}$ T transition.
-%Obtained values are of the same order of magnitude than values derived from other ab initio studies\cite{bloechl93,sahli05}.
-These are of the same order of magnitude than values derived from other ab initio studies\cite{bloechl93,sahli05}.
+These are of the same order of magnitude as values of other {\em ab initio} studies \cite{bloechl93,sahli05}.
-\subsection{Pairs of C$_{\text{i}}$}
+\section{Defect combinations}
C$_{\text{i}}$ pairs of the \hkl<1 0 0> type have been investigated in the first part.
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.
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