From: hackbard Date: Tue, 1 May 2012 20:52:46 +0000 (+0200) Subject: until c mobility X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=commitdiff_plain;h=37b4a2243e6124396d66110df1883bec7a25d441 until c mobility --- diff --git a/posic/publications/emrs2012.tex b/posic/publications/emrs2012.tex index cf4b19d..e946701 100644 --- 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). -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. -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. @@ -108,71 +108,83 @@ 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. -\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{minipage}[t]{0.48\columnwidth} +\centering +\begin{minipage}[t]{0.43\columnwidth} +\centering \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} -\begin{minipage}[t]{0.48\columnwidth} +\begin{minipage}[t]{0.43\columnwidth} +\centering \underline{Si$_{\text{i}}$ hexagonal}\\ -\includegraphics[width=\columnwidth]{sihex_bonds.eps} +\includegraphics[width=0.9\columnwidth]{sihex_bonds.eps} \end{minipage}\\ -\begin{minipage}[t]{0.48\columnwidth} +\begin{minipage}[t]{0.43\columnwidth} +\centering \underline{Si$_{\text{i}}$ tetrahedral}\\ -\includegraphics[width=\columnwidth]{sitet_bonds.eps} +\includegraphics[width=0.9\columnwidth]{sitet_bonds.eps} \end{minipage} -\begin{minipage}[t]{0.48\columnwidth} +\begin{minipage}[t]{0.43\columnwidth} +\centering \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} -\begin{minipage}[t]{0.48\columnwidth} +\centering +\begin{minipage}[t]{0.43\columnwidth} +\centering \underline{C$_{\text{s}}$} -\includegraphics[width=\columnwidth]{csub_bonds.eps} +\includegraphics[width=0.9\columnwidth]{csub_bonds.eps} \end{minipage} -\begin{minipage}[t]{0.48\columnwidth} +\begin{minipage}[t]{0.43\columnwidth} +\centering \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}\\ -\begin{minipage}[t]{0.48\columnwidth} +\begin{minipage}[t]{0.43\columnwidth} +\centering \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} -\begin{minipage}[t]{0.48\columnwidth} +\begin{minipage}[t]{0.43\columnwidth} +\centering \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} -\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. + +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{Mobility of the carbon defect} 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.