some on mobility ...

[lectures/latex.git] / posic / publications / emrs2012.tex

diff --git a/posic/publications/emrs2012.tex b/posic/publications/emrs2012.tex
index cf4b19d..552cff7 100644 (file)
--- a/posic/publications/emrs2012.tex
+++ b/posic/publications/emrs2012.tex
@@ -83,13 +83,13 @@ 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).
 \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).

-Exchange and correlation is taken into account by the generalized-gradient approximation as proposed by Perdew and Wang \cite{perdew86,perdew92}.
+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.
 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 has been fully accounted for.
+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.
 
 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.


@@ -108,93 +108,94 @@ The Berendsen barostat and thermostat \cite{berendsen84} with a time constant of
 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.
 
 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.
 

-\section{Results}
+\section{Defect configurations in silicon}

 
 

-\subsection{Carbon and silicon defect configurations}
-
-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{figure}

-\begin{minipage}[t]{0.48\columnwidth}
+\centering
+\begin{minipage}[t]{0.43\columnwidth}
+\centering

 \underline{Si$_{\text{i}}$ \hkl<1 1 0> DB}\\
 \underline{Si$_{\text{i}}$ \hkl<1 1 0> DB}\\

-\includegraphics[width=\columnwidth]{si110_bonds.eps}
+\includegraphics[width=0.9\columnwidth]{si110_bonds.eps}

 \end{minipage}
 \end{minipage}

-\begin{minipage}[t]{0.48\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering

 \underline{Si$_{\text{i}}$ hexagonal}\\
 \underline{Si$_{\text{i}}$ hexagonal}\\

-\includegraphics[width=\columnwidth]{sihex_bonds.eps}
+\includegraphics[width=0.9\columnwidth]{sihex_bonds.eps}

 \end{minipage}\\
 \end{minipage}\\

-\begin{minipage}[t]{0.48\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering

 \underline{Si$_{\text{i}}$ tetrahedral}\\
 \underline{Si$_{\text{i}}$ tetrahedral}\\

-\includegraphics[width=\columnwidth]{sitet_bonds.eps}
+\includegraphics[width=0.9\columnwidth]{sitet_bonds.eps}

 \end{minipage}
 \end{minipage}

-\begin{minipage}[t]{0.48\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering

 \underline{Si$_{\text{i}}$ \hkl<1 0 0> DB}\\
 \underline{Si$_{\text{i}}$ \hkl<1 0 0> DB}\\

-\includegraphics[width=\columnwidth]{si100_bonds.eps}
+\includegraphics[width=0.9\columnwidth]{si100_bonds.eps}

 \end{minipage}
 \caption{Configurations of intrinsic silicon point defects. Dumbbell configurations are abbreviated by DB.}
 \label{fig:intrinsic_def}
 \end{figure}
 \begin{figure}
 \end{minipage}
 \caption{Configurations of intrinsic silicon point defects. Dumbbell configurations are abbreviated by DB.}
 \label{fig:intrinsic_def}
 \end{figure}
 \begin{figure}

-\begin{minipage}[t]{0.48\columnwidth}
+\centering
+\begin{minipage}[t]{0.43\columnwidth}
+\centering

 \underline{C$_{\text{s}}$}
 \underline{C$_{\text{s}}$}

-\includegraphics[width=\columnwidth]{csub_bonds.eps}
+\includegraphics[width=0.9\columnwidth]{csub_bonds.eps}

 \end{minipage}
 \end{minipage}

-\begin{minipage}[t]{0.48\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering

 \underline{C$_{\text{i}}$ \hkl<1 0 0> DB}\\
 \underline{C$_{\text{i}}$ \hkl<1 0 0> DB}\\

-\includegraphics[width=\columnwidth]{c100_bonds.eps}
+\includegraphics[width=0.9\columnwidth]{c100_bonds.eps}

 \end{minipage}\\
 \end{minipage}\\

-\begin{minipage}[t]{0.48\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering

 \underline{C$_{\text{i}}$ \hkl<1 1 0> DB}\\
 \underline{C$_{\text{i}}$ \hkl<1 1 0> DB}\\

-\includegraphics[width=\columnwidth]{c110_bonds.eps}
+\includegraphics[width=0.9\columnwidth]{c110_bonds.eps}

 \end{minipage}
 \end{minipage}

-\begin{minipage}[t]{0.48\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering

 \underline{C$_{\text{i}}$ bond-centered}\\
 \underline{C$_{\text{i}}$ bond-centered}\\

-\includegraphics[width=\columnwidth]{cbc_bonds.eps}
+\includegraphics[width=0.9\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.}
 \label{fig:carbon_def}
 \end{figure}
 \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}
 \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.
 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.
 
 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.
 
 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.