\usepackage{amssymb}\r
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% additional stuff\r
-%\usepackage{miller}\r
+\usepackage{miller}\r
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\begin{document}\r
\r
%\title{Mobility of Carbon in Silicon -- a first principles study}\r
\title{Extensive first principles study of carbon defects in silicon}\r
-\author{F. Zirkelbach} \author{B. Stritzker}\r
+\author{F. Zirkelbach}\r
+\author{B. Stritzker}\r
\affiliation{Experimentalphysik IV, Universit\"at Augsburg, 86135 Augsburg, Germany}\r
\author{K. Nordlund}\r
\affiliation{Department of Physics, University of Helsinki, 00014 Helsinki, Finland}\r
\author{J. K. N. Lindner}\r
-\author{W. G. Schmidt} \author{E. Rauls}\r
+\author{W. G. Schmidt}\r
+\author{E. Rauls}\r
\affiliation{Department Physik, Universit\"at Paderborn, 33095 Paderborn, Germany}\r
\r
\begin{abstract}\r
We present a first principles investigation of the mobility of carbon interstitials in silicon. \r
-The migration mechanism of carbon [100] dumbbell interstitials in otherwise defect-free silicon has been investigated using density functional theory calculations.\r
+The migration mechanism of carbon \hkl<1 0 0> dumbbell interstitials in otherwise defect-free silicon has been investigated using density functional theory calculations.\r
Furthermore, the influence of near-by vacancies, carbon interstitial and substitutional defects and silicon self-interstitials has been investigated systematically.\r
A long range capture radius for vacancies has been found....\r
\end{abstract}\r
Based on experimental 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.\r
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.\r
In contrast, investigations of the precipitation in strained Si$_{1-y}$C$_y$/Si heterostructures formed by molecular beam epitaxy (MBE)\cite{strane94,guedj98} suggest an initial coherent clustering of substitutional instead of interstitial C followed by a loss of coherency once the increasing strain energy surpasses the interfacial energy of an incoherent 3C-SiC precipitate in c-Si.\r
-These two different mechanisms of precipitation might be determined by the respective method of fabrication.\r
+These two different mechanisms of precipitation might be attributed to the respective method of fabrication.\r
However, in another IBS study Nejim et al. propose a topotactic transformation remaining structure and orientation likewise based on the formation of substitutional C and a concurrent reaction of the excess Si self-interstitials with further implanted C atoms in the initial state\cite{nejim95}.\r
Solving this controversy and understanding the effective underlying processes will enable significant technological progress in 3C-SiC thin film formation driving the superior polytype for potential applications in high-performance electronic device production\cite{wesch96}.\r
\r
Atomistic simulations offer a powerful tool of investigation providing detailed insight not accessible by experiment.\r
-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,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}.\r
+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,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}.\r
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.\r
% but mattoni2002 actually did a lot. maybe this should be mentioned!\r
In fact, in a combined analytical potential molecular dynamics and ab initio study\cite{mattoni2002} the interaction of substitutional C with Si self-interstitials and C interstitials is evaluated.\r
-However, investigations are, first of all, restricted to interaction chains along the $\langle 1 1 0 \rangle$ and $\langle -1 1 0 \rangle$ direction, secondly lacking combinations of C interstitials and, finally, not considering migration barriers giving further information about the probability of defect agglomeration.\r
+However, investigations are, first of all, restricted to interaction chains along the \hkl[1 1 0] and \hkl[-1 1 0] direction, secondly lacking combinations of C interstitials and, finally, not considering migration barriers giving further information about the probability of defect agglomeration.\r
\r
By first principles atomistic simulations this work aims to shed light on basic processes involved in the precipitation mechanism of SiC in Si.\r
During implantation defects such as vacancies (V), substitutional C (C$_{\text{s}}$), interstitial C (C$_{\text{i}}$) and Si self-interstitials (Si$_{\text{i}}$) are created, which play a decisive role in the precipitation process.\r
Throughout this work an energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis.\r
Sampling of the Brillouin zone was restricted to the $\Gamma$-point.\r
The defect structures and the migration paths were modelled in cubic supercells containing $216\pm2$ Si atoms.\r
-The ions and cell shape are allowed to change in order to realize a constant pressure simulation.\r
+The ions and cell shape were allowed to change in order to realize a constant pressure simulation.\r
+Ionic relaxation was realized by the conjugate gradient algorithm.\r
Spin polarization has been fully accounted for.\r
\r
-Migration and recombination pathways have been ivestigated utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.\r
+Migration and recombination pathways have been investigated utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.\r
The defect formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ is defined by chosing 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.\r
-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, i.e. energetically favorable configurations show binding energies below zero and non-interacting isolated defects would result in a binding energy of zero.\r
+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.\r
+Accordingly, energetically favorable configurations show binding energies below zero while non-interacting isolated defects result in a binding energy of zero.\r
\r
\section{Results}\r
\r
-%After the implantation of C into Si, C interstitials are the most common defects in the Si sample.\r
-%Their mobility is the crucial quantity to be investigated.\r
-%However, the implantation process unavoidably creates a variety of further point defects, such as vacancies and silicon self-interstitials.\r
-%Already during implantation and also in the subsequent annealing process, further defects can evolve from these, like pair defects or substitutional carbon.\r
The implantation of highly energetic C atoms results in a multiplicity of possible defect configurations.\r
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.\r
In the following the structure and energetics of separated defects are presented.\r
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 transition.\r
-Fig.~\ref{fig:combos} schematically displays the positions for the initial interstitial defect (Si$_{\text{i}}$/C$_{\text{i}}$) and the neighbured defect (1-5) used for investigating defect pairs.\r
-\begin{figure}\r
-\includegraphics[width=0.5\columnwidth]{cs.eps}\r
-\caption{Positions for the initial defect (Si$_{\text{i}}$/C$_{\text{i}}$) and the neighboured defect (1-5) used for investigating defect pairs.} \r
-\label{fig:combos}\r
-\end{figure}\r
\r
\subsection{Separated defects in silicon}\r
% we need both: Si self-int & C int ground state configuration (for combos)\r
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}.\r
\begin{figure}\r
\begin{minipage}[t]{0.32\columnwidth}\r
-\underline{Si$_{\text{i}}$ $\langle 1 1 0 \rangle$ DB}\\\r
+\underline{Si$_{\text{i}}$ \hkl<1 1 0> DB}\\\r
\includegraphics[width=\columnwidth]{si110.eps}\r
\end{minipage}\r
\begin{minipage}[t]{0.32\columnwidth}\r
\includegraphics[width=\columnwidth]{sitet.eps}\r
\end{minipage}\\\r
\begin{minipage}[t]{0.32\columnwidth}\r
-\underline{Si$_{\text{i}}$ $\langle 1 0 0 \rangle$ DB}\\\r
+\underline{Si$_{\text{i}}$ \hkl<1 0 0> DB}\\\r
\includegraphics[width=\columnwidth]{si100.eps}\r
\end{minipage}\r
\begin{minipage}[t]{0.32\columnwidth}\r
\includegraphics[width=\columnwidth]{csub.eps}\r
\end{minipage}\\\r
\begin{minipage}[t]{0.32\columnwidth}\r
-\underline{C$_{\text{i}}$ $\langle 1 0 0 \rangle$ DB}\\\r
+\underline{C$_{\text{i}}$ \hkl<1 0 0> DB}\\\r
\includegraphics[width=\columnwidth]{c100.eps}\r
\end{minipage}\r
\begin{minipage}[t]{0.32\columnwidth}\r
-\underline{C$_{\text{i}}$ $\langle 1 1 0 \rangle$ DB}\\\r
+\underline{C$_{\text{i}}$ \hkl<1 1 0> DB}\\\r
\includegraphics[width=\columnwidth]{c110.eps}\r
\end{minipage}\r
\begin{minipage}[t]{0.32\columnwidth}\r
\begin{table*}\r
\begin{ruledtabular}\r
\begin{tabular}{l c c c c c c c c c}\r
- & Si$_{\text{i}}$ $\langle1 1 0\rangle$ DB & Si$_{\text{i}}$ H & Si$_{\text{i}}$ T & Si $\langle 1 0 0\rangle$ DB & V & C$_{\text{s}}$ & C$_{\text{i}}$ $\langle1 0 0\rangle$ DB & C$_{\text{i}}$ $\langle1 1 0\rangle$ DB & C$_{\text{i}}$ BC \\\r
+ & Si$_{\text{i}}$ \hkl<1 1 0> DB & Si$_{\text{i}}$ H & Si$_{\text{i}}$ T & Si \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 \\\r
\hline\r
Present study & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 & 1.95 & 3.72 & 4.16 & 4.66 \\\r
\multicolumn{10}{c}{Other ab initio studies} \\\r
Ref.\cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 & - & - & - & - \\\r
Ref.\cite{leung99} & 3.31 & 3.31 & 3.43 & - & - & - & - & - & - \\\r
Ref.\cite{dal_pino93,capaz94} & - & - & - & - & - & 1.89\cite{dal_pino93} & x & - & x+2.1\cite{capaz94}\r
- %Reference & 3.40\cite{al-mushadani03}, 3.31\cite{leung99} & 3.45\cite{al-mushadani03}, 3.31\cite{leung99} & 3.43\cite{leung99} & - & 3.53\cite{al-mushadani03} & 1.89\cite{dal_pino93} & x & - & x+2.1\cite{capaz94}\r
\end{tabular}\r
\end{ruledtabular}\r
\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.}\r
\label{table:sep_eof}\r
\end{table*}\r
Results obtained by the present study compare well with results from literature\cite{leung99,al-mushadani03,dal_pino93,capaz94}.\r
-Regarding intrinsic defects in Si, the $\langle 1 1 0 \rangle$ self-interstitial dumbbell (Si$_{\text{i}}$ $\langle 1 1 0 \rangle$ DB) is found to be the ground state configuration tersely followed by the hexagonal and tetrahedral configuration, which is the consensus view for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.\r
-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$_{\text{i}}$ $\langle 1 0 0 \rangle$ interstitial dumbbell (C$_{\text{i}}$ $\langle 1 0 0 \rangle$ DB) constitutes the energetically most favorable configuration, in which the C and Si dumbbell atoms share a regular Si lattice site.\r
+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 tersely followed by the hexagonal and tetrahedral configuration, which is the consensus view for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.\r
+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.\r
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 as the ground state.\r
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.\r
\r
-Instead, Capaz et al.\cite{capaz94}, investigating migration pathways of the C$_{\text{i}}$ $\langle 1 0 0 \rangle$ DB, find this defect to be \unit[2.1]{eV} lower in energy than the bond-centered (BC) configuration, which is claimed to constitute a saddle point configuration in the migration path within the $(1 1 0)$ plane and, thus, interpreted as the barrier of migration for the respective path.\r
+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, which is claimed to constitute a saddle point configuration in the migration path within the \hkl(1 1 0) plane and, thus, interpreted as the barrier of migration for the respective path.\r
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.\r
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.\r
-Regardless of the rather small correction due to the spin, the difference we found is much smaller (\unit[0.9]{eV}), which would nicely compare to experimental findings $(\unit[0.73-0.87]{eV})$\cite{tipping87,song90} for the migration barrier.\r
+Regardless of the rather small correction due to the spin, the difference we found is much smaller (\unit[0.9]{eV}), which would nicely compare to experimental findings $(\unit[0.70-0.87]{eV})$\cite{lindner06,tipping87,song90} for the migration barrier.\r
However, since the BC configuration constitutes a real local minimum another barrier exists which is about \unit[1.2]{eV} ($\unit[0.9]{eV}+\unit[0.3]{eV}$) in height.\r
-Indeed Capaz et al. propose another path and find it to be the lowest in energy\cite{capaz94}, in which a C$_{\text{i}}$ $\langle 0 0 -1\rangle$ DB migrates into a C$_{\text{i}}$ $\langle 0 -1 0\rangle$ DB located at the next neighboured Si lattice site in $[1 1 -1]$ direction.\r
+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 into a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the next neighbored Si lattice site in \hkl[1 1 -1] direction.\r
Calculations in this work reinforce this path by an additional improvement of the quantitative conformance of the barrier height (\unit[0.9]{eV}) to experimental values.\r
A more detailed description can be found in a previous study\cite{zirkelbach10a}.\r
\r
-Next to the C BC configuration the vacancy and Si$_{\text{i}}$ $\langle 1 0 0\rangle$ DB have to be treated by taking into account the spin of the electrons.\r
+Next to the C 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.\r
For the latter two the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site and in two caps at each of the two DB atoms perpendicularly aligned to the bonds to the other two Si atoms respectively.\r
No other configuration, within the ones that are mentioned, is affected.\r
\r
-Concerning the mobility of the ground state Si$_{\text{i}}$, an activation energy shortly below \unit[0.7]{eV} was found for the migration of a Si$_{\text{i}}$ $\langle 0 1 -1\rangle$ into a $\langle 1 1 0\rangle$ DB configuration located at the next neighboured Si lattice site in $[1 1 -1]$ direction.\r
+Concerning the mobility of the ground state Si$_{\text{i}}$, an activation energy shortly below \unit[0.7]{eV} was found for the migration of a Si$_{\text{i}}$ \hkl[0 1 -1] into a \hkl[1 1 0] DB configuration located at the next neighbored Si lattice site in \hkl[1 1 -1] direction.\r
% look for values in literature for neutraly charged Si_i diffusion\r
\r
\subsection{Pairs of C$_{\text{i}}$}\r
\r
-C$_{\text{i}}$ pairs of the $\langle 1 0 0\rangle$-type have been considered in the first part.\r
-Table~\ref{table:dc_c-c} summarizes the binding energies obtained for configurations, in which an initial C$_{\text{i}}$ $\langle 0 0 -1\rangle$ DB located at position Si$_{\text{i}}$/C$_{\text{i}}$ is combined with a defect of the same type occupying various orientations at positions 1 to 5 (see Fig.~\ref{fig:combos}).\r
+C$_{\text{i}}$ pairs of the \hkl<1 0 0>-type have been considered in the first part.\r
+Fig.~\ref{fig:combos_ci} schematically displays the position of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and the various positions for the second defect (1-5) used for investigating the defect pairs.\r
+Table~\ref{table:dc_c-c} summarizes the binding energies for the combination with a second C-Si \hkl<1 0 0> DB obtained for different orientations.\r
+\begin{figure}\r
+%\begin{minipage}{0.49\columnwidth}\r
+\subfigure[]{\label{fig:combos_ci}\includegraphics[width=0.45\columnwidth]{combos_ci.eps}}\r
+\hspace{0.1cm}\r
+\subfigure[]{\label{fig:combos_si}\includegraphics[width=0.45\columnwidth]{combos.eps}}\r
+\caption{Positions of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB (I) (Fig.~\ref{fig:combos_ci}), the lattice site chosen for the initial Si$_{\text{i}}$ \hkl<1 1 0> DB (Si$_{\text{i}}$) occupying various orientations (Fig.~\ref{fig:combos_si}) and neighbored positions (1-5) for the second defect used for investigating defect pairs.} \r
+\label{fig:combos}\r
+\end{figure}\r
\begin{table}\r
\begin{ruledtabular}\r
\begin{tabular}{l c c c c c c }\r
& 1 & 2 & 3 & 4 & 5 & R \\\r
\hline\r
- $\langle 0 0 -1\rangle$ & -0.08 & -1.15 & -0.08 & 0.04 & -1.66 & -0.19\\\r
- $\langle 0 0 1\rangle$ & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\\r
- $\langle 0 -1 0\rangle$ & -2.39 & -0.17 & -0.10 & -0.27 & -1.88 & -0.05\\\r
- $\langle 0 1 0\rangle$ & -2.25 & -1.90 & -2.25 & -0.12 & -1.38 & -0.06\\\r
- $\langle -1 0 0\rangle$ & -2.39 & -0.36 & -2.25 & -0.12 & -1.88 & -0.05\\\r
- $\langle 1 0 0\rangle$ & -2.25 & -2.16 & -0.10 & -0.27 & -1.38 & -0.06\\\r
+ \hkl[0 0 -1] & -0.08 & -1.15 & -0.08 & 0.04 & -1.66 & -0.19\\\r
+ \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\\r
+ \hkl[0 -1 0] & -2.39 & -0.17 & -0.10 & -0.27 & -1.88 & -0.05\\\r
+ \hkl[0 1 0] & -2.25 & -1.90 & -2.25 & -0.12 & -1.38 & -0.06\\\r
+ \hkl[-1 0 0] & -2.39 & -0.36 & -2.25 & -0.12 & -1.88 & -0.05\\\r
+ \hkl[1 0 0] & -2.25 & -2.16 & -0.10 & -0.27 & -1.38 & -0.06\\\r
\end{tabular}\r
\end{ruledtabular}\r
-\caption{Binding energies of C$_{\text{i}}$ $\langle 1 0 0\rangle$-type defect pairs in eV. Equivalent configurations exhibit equal energies. The first column lists the orientation of the defect, which is combined with the initial C$_{\text{i}}$ $\langle 0 0 -1\rangle$ dumbbell. The position index of the second defect is given in the first row according to Fig.~\ref{fig:combos}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2} \langle3 2 3 \rangle$ relative to the initial defect position, which is the maximum realizable distance due to periodic boundary conditions.}\r
+\caption{Binding energies of C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs in eV. Equivalent configurations exhibit equal energies. The first column lists the orientation of the defect, which is combined with the initial C$_{\text{i}}$ \hkl[0 0 -1] dumbbell. The position index of the second defect is given in the first row according to Fig.~\ref{fig:combos}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect position, which is the maximum realizable distance due to periodic boundary conditions.}\r
\label{table:dc_c-c}\r
\end{table}\r
Most of the obtained configurations result in binding energies well below zero indicating a preferable agglomeration of these type of defects.\r
For increasing distances of the defect pair the binding energy approaches to zero (R in Table~\ref{table:dc_c-c}) as it is expected for non-interacting isolated defects.\r
-Energetically favorable and unfavorable configurations can be explained by stress compensation and increase respectively, which is due to the resulting net strain of the respective configuration of the defect combination.\r
-Antiparallel orientations of the second defect ($\langle 0 0 1\rangle$) at positions located below the (001) plane with respect to the initial one (positions 1, 2 and 4) show the energetically most unfavorable configurations.\r
+Energetically favorable and unfavorable configurations can be explained by stress compensation and increase respectively based on the resulting net strain of the respective configuration of the defect combination.\r
+Antiparallel orientations of the second defect (\hkl[0 0 1]) at positions located below the \hkl(0 0 1) plane with respect to the initial one (positions 1, 2 and 4) show the energetically most unfavorable configurations.\r
In contrast, the parallel and particularly the twisted orientations constitute energetically favorable configurations, in which a vast reduction of strain is enabled by combination of these defects.\r
\r
-Mattoni et al.\cite{mattoni2002} predict the ground state configuration for a $\langle 1 0 0\rangle$ or equivalently a $\langle 0 1 0\rangle$ defect created at position 1 with both defects basically maintaining the DB structure, resulting in a binding energy of \unit[-2.1]{eV}.\r
-% in mattoni db structures are basically amintained. there is further relaxation in our case and a lower binding energy\r
+Mattoni et al.\cite{mattoni2002} predict the ground state configuration for a \hkl[1 0 0] or equivalently a \hkl[0 1 0] defect created at position 1 with both defects basically maintaining the DB structure, resulting in a binding energy of \unit[-2.1]{eV}.\r
In this work we found a further relaxation of this defect structure.\r
The C atom of the second and the Si atom of the initial DB move towards each other forming a bond, which results in a somewhat lower binding energy of \unit[-2.25]{eV}.\r
-Furthermore a more favorable configuration was found for the combination with a $\langle 0 -1 0\rangle$ and $\langle -1 0 0\rangle$ DB respectively, which is assumed to constitute the actual ground state configuration of two C$_{\text{i}}$ DBs in Si.\r
+Furthermore a more favorable configuration was found for the combination with a \hkl[0 -1 0] and \hkl[-1 0 0] DB respectively, which is assumed to constitute the actual ground state configuration of two C$_{\text{i}}$ DBs in Si.\r
+The atomic arrangement is shown in the bottom right of Fig.~\ref{fig:036-239}.\r
The two C atoms form a strong C-C bond, which is responsible for the large gain in energy resulting in a binding energy of \unit[-2.39]{eV}.\r
\r
-Investigating migration barriers enables to predict the probability of formation of the thermodynamic ground state defect complex by thermally activated diffusion processes.\r
-High activation energies are necessary for the migration of low energy configurations, in which the C atom of the second DB is located in the vicinity of the initial DB.\r
-The transition of the configuration, in which the second DB is of the $\langle 0 1 0\rangle$ type at position 2 (\unit[-1.90]{eV}) into a $\langle 0 1 0\rangle$-type DB at position 1 (\unit[-2.39]{eV}) for instance, revealed a barrier height of more than \unit[4]{eV}.\r
-Low barriers are only expected from energetically less favorable configurations, e.g. the configuration of the $\langle -1 0 0\rangle$ DB located at position 2.\r
-A migration barrier of \unit[?.?]{eV} \r
+Investigating migration barriers enables to predict the probability of formation of defect complexes by thermally activated diffusion processes.\r
+% ground state configuration, C cluster\r
+Based on the lowest energy migration path of a single C$_{\text{i}}$ DB the configuration, in which the second C$_{\text{i}}$ DB is oriented along \hkl[0 1 0] at position 2 is assumed to constitute an ideal starting point for a transition into the ground state.\r
+In addition, the starting configuration exhibits a low binding energy (\unit[-1.90]{eV}) and is, thus, very likely to occur.\r
+However, a barrier height of more than \unit[4]{eV} was detected resulting in a low probability for the transition.\r
+The high activation energy is attributed to the stability of such a low energy configuration, in which the C atom of the second DB is located close to the initial DB.\r
+Low barriers have only been identified for transitions starting from energetically less favorable configurations, e.g. the configuration of a \hkl[-1 0 0] DB located at position 2 (\unit[-0.36]{eV}).\r
+Starting from this configuration, an activation energy of only \unit[1.2]{eV} is necessary for the transition into the ground state configuration.\r
+The corresponding migration energies and atomic configurations are displayed in Fig.~\ref{fig:036-239}.\r
+\begin{figure}\r
+\includegraphics[width=\columnwidth]{036-239.ps}\r
+\caption{Migration barrier and structures of the transition of a C$_{\text{i}}$ \hkl[-1 0 0] DB at position 2 (left) into a C$_{\text{i}}$ \hkl[0 -1 0] DB at position 1 (right). An activation energy of \unit[1.2]{eV} is observed.}\r
+\label{fig:036-239}\r
+\end{figure}\r
% strange mig from -190 -> -2.39 (barrier > 4 eV)\r
% C-C migration -> idea:\r
% mig from low energy confs has extremely high barrier!\r
% low barrier only from energetically less/unfavorable confs (?)! <- prove!\r
% => low probability of C-C clustering ?!?\r
+%\r
+% should possibly be transfered to discussion section\r
Since thermally activated C clustering is, thus, only possible by traversing energetically unfavored configurations, mass C clustering is not expected.\r
-\r
+Furthermore, the migration barrier of \unit[1.2]{eV} is still higher than the activation energy of \unit[0.9]{eV} observed for a single C$_{\text{i}}$ \hkl<1 0 0> DB in c-Si.\r
+The migration barrier of a C$_{\text{i}}$ DB in a complex system is assumed to approximate the barrier of a DB in a separated system with increasing defect separation.\r
+Thus, lower migration barriers are expected for pathways resulting in larger separations of the C$_{\text{i}}$ DBs.\r
+% calculate?!? ... hopefully acknowledged by 188-225 calc\r
+However, if the increase of separation is accompanied by an increase in binding energy, this difference is needed in addition to the activation energy for the respective migration process.\r
+Configurations, which exhibit both, a low binding energy as well as afferent transitions with low activation energies are, thus, most probable C$_{\text{i}}$ complex structures.\r
+On the other hand, if elevated temperatures enable migrations with huge activation energies, comparably small differences in configurational energy can be neglected resulting in an almost equal occupation of such configurations.\r
+In both cases the configuration yielding a binding energy of \unit[-2.25]{eV} is promising.\r
+First of all, it constitutes the second most energetically favorable structure.\r
+Secondly, a migration path with a barrier as low as \unit[0.47]{eV} exists starting from a configuration of largely separated defects exhibiting a low binding energy (\unit[-1.88]{eV}).\r
+The migration barrier and correpsonding structures are shown in Fig.~\ref{fig:188-225}.\r
+% 188 - 225 transition in progress\r
+\begin{figure}\r
+\includegraphics[width=\columnwidth]{188-225.ps}\r
+\caption{Migration barrier and structures of the transition of a C$_{\text{i}}$ \hkl[0 -1 0] DB at position 5 (left) into a C$_{\text{i}}$ \hkl[1 0 0] DB at position 1 (right). An activation energy of \unit[0.47]{eV} is observed.}\r
+\label{fig:188-225}\r
+\end{figure}\r
+Finally, this type of defect pair is represented four times (two times more often than the ground state configuration) within the systematically investigated configuration space.\r
+The latter is considered very important for high temperatures, which is accompanied by an increase in the entropic contribution to structure formation.\r
+Thus, C agglomeration indeed is expected but only a low probability is assumed for C clustering by thermally activated processes with regard to the considered period of time.\r
+% ?!?\r
% look for precapture mechnism (local minimum in energy curve)\r
+% also: plot energy all confs with respect to C-C distance\r
+% maybe a pathway exists traversing low energy confs ?!?\r
\r
-% mattoni2002: c_i c_i attraction basin not explored, too wide paramter range\r
-\r
-Energetically most favorable orientations along $[1 1 0]$ direction ...\r
-\r
-\subsection{C$_I$ next to C$_{\text{s}}$}\r
-\r
-% c_i and c_s, capaz98, mattoni2002 (restricted to 110 -110 bond chain)\r
-\r
+% point out that configurations along 110 were extended up to the 6th NN in that direction\r
+The binding energies of the energetically most favorable configurations with the seocnd DB located along the \hkl[1 1 0] direction and resulting C-C distances of the relaxed structures are summarized in Table~\ref{table:dc_110}.\r
+\begin{table}\r
+\begin{ruledtabular}\r
+\begin{tabular}{l c c c c c c }\r
+ & 1 & 2 & 3 & 4 & 5 & 6 \\\r
+\hline\r
+ $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\\r
+C-C distance [nm] & 0.14 & 0.46 & 0.65 & 0.86 & 1.05 & 1.08 \r
+\end{tabular}\r
+\end{ruledtabular}\r
+\caption{Binding energies $E_{\text{b}}$ and C-C distance of energetically most favorable C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs separated along bonds in the \hkl[1 1 0] direction.}\r
+\label{table:dc_110}\r
+\end{table}\r
+The binding energy of these configurations with respect to the C-C distance is plotted in Fig.~\ref{fig:dc_110}\r
+\begin{figure}\r
+\includegraphics[width=\columnwidth]{db_along_110_cc_n.ps}\r
+\caption{Minimum binding energy of dumbbell combinations separated along \hkl[1 1 0] with respect to the C-C distance. The blue line is a guide for the eye and the green curve corresponds to the most suitable fit function consisting of all but the first data point.}\r
+\label{fig:dc_110}\r
+\end{figure}\r
+The interaction is found to be proportional to the reciprocal cube of the C-C distance for extended separations of the C$_{\text{i}}$ and saturates for the smallest possible separation, i.e. the ground state configuration.\r
+Not considering the previously mentioned elevated barriers for migration an attractive interaction between the C$_{\text{i}}$ defects indeed is detected with a capture radius that clearly exceeds the \unit[1]{nm} mark.\r
+The interpolated graph suggests the disappearance of attractive interaction forces, which are proportional to the slope of the graph, inbetween the two lowest separation distances of the defects.\r
+This finding, in turn, supports the previously established assumption of C agglomeration and absence of C clsutering.\r
\r
-\subsection{C$_I$ next to vacancies}\r
+\begin{table}\r
+\begin{ruledtabular}\r
+\begin{tabular}{l c c c c c c }\r
+ & 1 & 2 & 3 & 4 & 5 & R \\\r
+\hline\r
+C$_{\text{s}}$ & 0.26$^a$/-1.28$^b$ & -0.51 & -0.93$^A$/-0.95$^B$ & -0.15 & 0.49 & -0.05\\\r
+V & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\r
+\end{tabular}\r
+\end{ruledtabular}\r
+\caption{Binding energies of combinations of the C$_{\text{i}}$ \hkl[0 0 -1] defect with a substitutional C or vacancy located at positions 1 to 5 according to Fig.~\ref{fig:combos_ci}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect position, which is the maximum realizable distance due to periodic boundary conditions.}\r
+\label{table:dc_c-sv}\r
+\end{table}\r
\r
-\subsection{C$_{\text{s}}$ next to Si self interstitials}\r
+\subsection{C$_{\text{i}}$ next to C$_{\text{s}}$}\r
\r
-Non-zeor temperature, entropy, spatial separation of these defects possible, indeed observed in ab initio MD run.\r
+The first row of Table~\ref{table:dc_c-sv} lists the binding energies of C$_{\text{s}}$ next to the C$_{\text{i}}$ \hkl[0 0 -1] DB.\r
+For C$_{\text{s}}$ located at position 1 and 3 the configurations a and A correspond to the naive relaxation of the structure by substituting the Si atom by a C atom in the initial C$_{\text{i}}$ \hkl[0 0 -1] DB structure at positions 1 and 3 respectively.\r
+However, small displacements of the involved atoms near the defect result in different stable structures labeled b and B respectively.\r
+Fig.~\ref{fig:093-095} and \ref{fig:026-128} show structures A, B and a, b together with the barrier of migration for the A to B and a to b transition respectively. \r
\r
-%% Viele Bilder... da kann ich zunächst gar nicht soviel zu schreiben.... \r
+% A B\r
+%./visualize_contcar -w 640 -h 480 -d results/c_00-1_c3_csub_B -nll -0.20 -0.4 -0.1 -fur 0.9 0.6 0.9 -c 0.5 -1.5 0.375 -L 0.5 0 0.3 -r 0.6 -A -1 2.465\r
+\begin{figure}\r
+\includegraphics[width=\columnwidth]{093-095.ps}\r
+\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 3 (left) into a configuration of a twofold coordinated Si$_{\text{i}}$ located inbetween two C$_{\text{s}}$ atoms occupying the lattice sites of the initial DB and position 3 (right). An activation energy of \unit[0.44]{eV} is observed.}\r
+\label{fig:093-095}\r
+\end{figure}\r
+Configuration A consists of a C$_{\text{i}}$ \hkl[0 0 -1] DB with threefold coordinated Si and C DB atoms slightly disturbed by the C$_{\text{s}}$ at position 3, facing the Si DB atom as a next neighbor.\r
+By a single bond switch, i.e. the breaking of a Si-Si in favor of a Si-C bond, configuration B is obtained, which shows a twofold coordinated Si atom located inbetween two substitutional C atoms residing on regular Si lattice sites.\r
+This configuration has been identified and described by spectroscopic experimental techniques\cite{song90_2} as well as theoretical studies\cite{leary97,capaz98}.\r
+Configuration B is found to constitute the energetically slightly more favorable configuration.\r
+However, the gain in energy due to the significantly lower energy of a Si-C compared to a Si-Si bond turns out to be smaller than expected due to a large compensation by introduced strain as a result of the Si interstitial structure.\r
+Present results show a difference in energy of states A and B, which exactly matches the experimental value of \unit[0.02]{eV}\cite{song90_2} reinforcing qualitatively correct results of previous theoretical studies on these structures.\r
+% mattoni: A favored by 0.4 eV - NO, it is indeed B (reinforce Song and Capaz)!\r
+%\r
+% AB transition\r
+% ...\r
+The migration barrier was identified to be \unit[0.44]{eV}, almost three times higher than the experimentally obtained value of \unit[0.16]{eV}\cite{song90_2}.\r
+This might be due to\r
+\r
+% a b\r
+\begin{figure}\r
+\includegraphics[width=\columnwidth]{026-128.ps}\r
+\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 1 (left) into a C-C \hkl[1 0 0] DB occupying the lattice site at position 1 (right). An activation energy of \unit[0.1]{eV} is observed.}\r
+\label{fig:026-128}\r
+\end{figure}\r
+Configuration a is similar to configuration A except that the C$_{\text{s}}$ atom at position 1 is facing the C DB atom as a next neighbor resulting in the formation of a strong C-C bond and a much more noticeable perturbation of the DB structure.\r
+Nevertheless, the C and Si DB atoms remain threefold coordinated.\r
+Although the C-C bond exhibiting a distance of \unit[0.15]{nm} close to the distance expected in diamond or graphite should lead to a huge gain in energy, a repulsive interaction with a binding energy of \unit[0.26]{eV} is observed due to compressive strain of the Si DB atom and its top neighbors (\unit[0.230]{nm}/\unit[0.236]{nm}) along with additional tensile strain of the C$_{\text{s}}$ and its three neighboring Si atoms (\unit[0.198-0.209]{nm}/\unit[0.189]{nm}).\r
+Again a single bond switch, i.e. the breaking of the bond of a Si atom bound to the fourfold coordinated C$_{\text{s}}$ atom and the formation of a double bond between the two C atoms, results in configuration b.\r
+The two C atoms form a \hkl[1 0 0] DB sharing the initial C$_{\text{s}}$ lattice site while the initial Si DB atom occupies its previously regular lattice site.\r
+The transition is accompanied by a large gain in energy as can be seen in Fig.~\ref{fig:026-128}, making it the ground state configuration of a C$_{\text{s}}$ and C$_{\text{i}}$ DB in Si yet \unit[0.33]{eV} lower in energy than configuration B.\r
+This finding is in good agreement with a combined ab initio and experimental study of Liu et~al.\cite{liu02}, who first proposed this structure as the ground state identifying an energy difference compared to configuration B of \unit[0.2]{eV}.\r
+% mattoni: A favored by 0.2 eV - NO! (again, missing spin polarization?)\r
+A net magnetization of two spin up electrons, which are euqally localized as in the Si$_{\text{i}}$ \hkl<1 0 0> DB structure is observed.\r
+Configurations a, A and B are not affected by spin polarization and show zero magnetization.\r
+Mattoni et~al.\cite{mattoni2002}, in contrast, find configuration b less favorable than configuration A by \unit[0.2]{eV}.\r
+Next to differences in the XC-functional and plane-wave energy cut-off this discrepancy might be attributed to the missing accounting for spin polarization in their calculations, which -- as has been shown for the C$_{\text{i}}$ BC configuration -- results in an increase of configurational energy.\r
+Indeed, investigating the migration path from configurations a to b and, in doing so, reusing the wave functions of the previous migration step the final structure, i.e. configuration b, was obtained with zero magnetization and an increase in configurational energy by \unit[0.2]{eV}.\r
+Obviously a different energy minimum of the electronic system is obatined indicating hysteresis behavior.\r
+However, since the total energy is lower for the magnetic result it is believed to constitute the real, i.e. global, minimum with respect to electronic minimization.\r
+\r
+A low activation energy of \unit[0.1]{eV} is observed for the a$\rightarrow$b transition.\r
+Thus, configuration a is very unlikely to occur in favor of configuration b.\r
+However, migration barriers yielding\r
+...\r
+\r
+% mig a-b\r
+% 2 more migs: 051 -> 128 and 026! forgot why ... probably it's about probability of C clustering\r
+\r
+\subsection{C$_{\text{i}}$ next to V}\r
+\r
+\subsection{C$_{\text{s}}$ next to Si$_{\text{i}}$}\r
+\r
+Non-zero temperature, entropy, spatial separation of these defects possible, indeed observed in ab initio MD run.\r
\r
\section{Discussion}\r
+\r
Our calculations show that point defects which unavoidably are present after ion implantation significantly influence the mobility of implanted carbon \r
in the silicon crystal.\r
A large capture radius has been found for... \r
projects / lectures / latex.git / blobdiff