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19 %\title{Mobility of Carbon in Silicon -- a first principles study}
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20 \title{First principles study of defects in carbon implanted silicon}
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21 \author{F. Zirkelbach}
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22 \author{B. Stritzker}
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23 \affiliation{Experimentalphysik IV, Universit\"at Augsburg, 86135 Augsburg, Germany}
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24 \author{K. Nordlund}
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25 \affiliation{Department of Physics, University of Helsinki, 00014 Helsinki, Finland}
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26 \author{J. K. N. Lindner}
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27 \author{W. G. Schmidt}
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29 \affiliation{Department Physik, Universit\"at Paderborn, 33095 Paderborn, Germany}
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32 A first principles investigation of the mobility of carbon and silicon interstitials in silicon is presented.
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33 The migration mechanism of a carbon \hkl<1 0 0> interstitial in otherwise defect-free silicon has been investigated using density functional theory calculations.
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34 Furthermore, the influence of a nearby vacancy, another carbon interstitial and a substitutional defect as well as a silicon self-interstitial has been investigated systematically.
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35 Interactions of various combinations of defects have been characterized including a couple of selected migration pathways within these configurations.
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36 Almost all of the investigated pairs of defects tend to agglomerate allowing for a reduction in strain.
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37 The formation of structures involving strong carbon-carbon bonds was found to occur very unlikely.
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38 In contrast, substitutional carbon was found to occur in all probability.
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39 A long range capture radius has been found for pairs of interstitial carbon as well as interstitial carbon and vacancies.
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40 A rather small capture radius has been identified for substitutional carbon and silicon self-interstitials.
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41 Based on these results conclusions regarding the precipitation mechanism of silicon carbide in bulk silicon are derived and its conformability to experimental findings is discussed.
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44 \keywords{point defects, defect clusters, migration, interstitials, ion implantation, first principles calculations}
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45 \pacs{61.72.J-,61.72.Yx,61.72.uj,66.30.J-,79.20.Rf,31.15.A-}
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48 % --------------------------------------------------------------------------------
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49 \section{Introduction}
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51 % Frank: Intro: hier kuerzer als in dem anderen Paper, dieselben (und mehr) Zitate bzgl. der Defekte (s. letzte Mail). SiC-precipitation würde ich schon erwähnen, aber nicht so detailliert.
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53 Silicon carbide (SiC) is a promising material for high-temperature, high-power and high-frequency electronic and optoelectronic devices employable under extreme conditions\cite{edgar92,morkoc94,wesch96,capano97,park98}.
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54 Ion beam synthesis (IBS) consisting of high-dose carbon implantation into crystalline silicon (c-Si) and subsequent or in situ annealing constitutes a promising technique to fabricate nano-sized precipitates and thin films of cubic SiC (3C-SiC) topotactically aligned to and embedded in the silicon host\cite{borders71,lindner99,lindner01,lindner02}.
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55 However, the process of the formation of SiC precipitates in Si during C implantation is not yet fully understood.
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56 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.
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57 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.
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58 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 precipitation by an agglomeration of substitutional instead of interstitial C followed by a loss of coherency once the increasing strain energy surpasses the interfacial energy of the incoherent 3C-SiC precipitate and c-Si.
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59 These two different mechanisms of precipitation might be attributed to the respective method of fabrication, i.e. whether it occurs inside the Si bulk or on a Si surface.
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60 However, in another IBS study Nejim et al. propose a topotactic transformation remaining structure and orientation that is likewise based on the formation of substitutional C and a concurrent reaction of the excess Si self-interstitials with further implanted C atoms\cite{nejim95}.
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61 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}.
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63 Atomistic simulations offer a powerful tool of investigation providing detailed insight not accessible by experiment.
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64 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}.
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65 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.
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66 % but mattoni2002 actually did a lot. maybe this should be mentioned!
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67 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.
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68 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 providing further information on the probability of defect agglomeration.
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70 By first principles atomistic simulations this work aims to shed light on basic processes involved in the precipitation mechanism of SiC in Si.
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71 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.
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72 In the following a systematic investigation of density functional theory (DFT) calculations of the structure, energetics and mobility of carbon defects in silicon as well as the influence of other point defects in the surrounding is presented.
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74 % --------------------------------------------------------------------------------
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75 \section{Methodology}
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77 The first principles DFT calculations were performed with the plane-wave based Vienna Ab-initio Simulation Package (VASP)\cite{kresse96}.
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78 The Kohn-Sham equations were solved using the generalized-gradient exchange-correlation (XC) functional approximation proposed by Perdew and Wang\cite{perdew86,perdew92}.
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79 The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials\cite{hamann79} as implemented in VASP\cite{vanderbilt90}.
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80 Throughout this work an energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis.
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81 Sampling of the Brillouin zone was restricted to the $\Gamma$-point.
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82 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.
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83 The ions and cell shape were allowed to change in order to realize a constant pressure simulation.
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84 Ionic relaxation was realized by the conjugate gradient algorithm.
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85 Spin polarization has been fully accounted for.
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87 Migration and recombination pathways have been investigated utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.
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88 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.
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89 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.
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90 Accordingly, energetically favorable configurations show binding energies below zero while non-interacting isolated defects result in a binding energy of zero.
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94 The implantation of highly energetic C atoms results in a multiplicity of possible defect configurations.
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95 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.
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96 In the following the structure and energetics of separated defects are presented.
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97 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.
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99 \subsection{Separated defects in silicon}
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100 \label{subsection:sep_def}
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101 % we need both: Si self-int & C int ground state configuration (for combos)
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103 Several geometries have been calculated to be stable for individual intrinsic and C related defects in Si.
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104 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}.
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106 \begin{minipage}[t]{0.32\columnwidth}
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107 \underline{Si$_{\text{i}}$ \hkl<1 1 0> DB}\\
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108 \includegraphics[width=\columnwidth]{si110.eps}
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110 \begin{minipage}[t]{0.32\columnwidth}
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111 \underline{Si$_{\text{i}}$ hexagonal}\\
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112 \includegraphics[width=\columnwidth]{sihex.eps}
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114 \begin{minipage}[t]{0.32\columnwidth}
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115 \underline{Si$_{\text{i}}$ tetrahedral}\\
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116 \includegraphics[width=\columnwidth]{sitet.eps}
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118 \begin{minipage}[t]{0.32\columnwidth}
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119 \underline{Si$_{\text{i}}$ \hkl<1 0 0> DB}\\
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120 \includegraphics[width=\columnwidth]{si100.eps}
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122 \begin{minipage}[t]{0.32\columnwidth}
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123 \underline{Vacancy}\\
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124 \includegraphics[width=\columnwidth]{sivac.eps}
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126 \begin{minipage}[t]{0.32\columnwidth}
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127 \underline{C$_{\text{s}}$}\\
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128 \includegraphics[width=\columnwidth]{csub.eps}
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130 \begin{minipage}[t]{0.32\columnwidth}
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131 \underline{C$_{\text{i}}$ \hkl<1 0 0> DB}\\
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132 \includegraphics[width=\columnwidth]{c100.eps}
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134 \begin{minipage}[t]{0.32\columnwidth}
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135 \underline{C$_{\text{i}}$ \hkl<1 1 0> DB}\\
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136 \includegraphics[width=\columnwidth]{c110.eps}
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138 \begin{minipage}[t]{0.32\columnwidth}
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139 \underline{C$_{\text{i}}$ bond-centered}\\
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140 \includegraphics[width=\columnwidth]{cbc.eps}
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142 \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.}
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143 \label{fig:sep_def}
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146 \begin{ruledtabular}
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147 \begin{tabular}{l c c c c c c c c c}
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148 & 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 \\
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150 Present study & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 & 1.95 & 3.72 & 4.16 & 4.66 \\
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151 \multicolumn{10}{c}{Other ab initio studies} \\
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152 Ref.\cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 & - & - & - & - \\
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153 Ref.\cite{leung99} & 3.31 & 3.31 & 3.43 & - & - & - & - & - & - \\
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154 Ref.\cite{dal_pino93,capaz94} & - & - & - & - & - & 1.89\cite{dal_pino93} & x & - & x+2.1\cite{capaz94}
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157 \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.}
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158 \label{table:sep_eof}
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160 Results obtained by the present study compare well with results from literature\cite{leung99,al-mushadani03,dal_pino93,capaz94}.
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161 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}.
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162 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.
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163 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.
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164 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.
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166 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.
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167 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.
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168 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.
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169 Regardless of the rather small correction of \unit[0.3]{eV} 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.
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170 However, since the BC configuration constitutes a real local minimum another barrier exists which is about \unit[1.2]{eV} in height.
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171 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 next neighbored Si lattice site in \hkl[1 1 -1] direction.
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172 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.
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173 A more detailed description can be found in a previous study\cite{zirkelbach10a}.
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175 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.
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176 For the vacancy the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site.
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177 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.
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178 No other configuration, within the ones that are mentioned, is affected.
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180 Concerning the mobility of the ground state Si$_{\text{i}}$, an activation energy of \unit[0.67]{eV} was found for the transition of the Si$_{\text{i}}$ \hkl[0 1 -1] to the \hkl[1 1 0] DB located at the next neighbored Si lattice site in \hkl[1 1 -1] direction.
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181 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.
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182 The obtained values are within the same order of magnitude than values derived from other ab initio studies\cite{bloechl93,sahli05}.
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184 \subsection{Pairs of C$_{\text{i}}$}
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186 First of all C$_{\text{i}}$ pairs of the \hkl<1 0 0>-type have been investigated.
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187 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.
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188 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 at positions 1 to 5.
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190 %\begin{minipage}{0.49\columnwidth}
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191 \subfigure[]{\label{fig:combos_ci}\includegraphics[width=0.45\columnwidth]{combos_ci.eps}}
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193 \subfigure[]{\label{fig:combos_si}\includegraphics[width=0.45\columnwidth]{combos.eps}}
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194 \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.}
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198 \begin{ruledtabular}
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199 \begin{tabular}{l c c c c c c }
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200 & 1 & 2 & 3 & 4 & 5 & R \\
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202 \hkl[0 0 -1] & -0.08 & -1.15 & -0.08 & 0.04 & -1.66 & -0.19\\
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203 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
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204 \hkl[0 -1 0] & -2.39 & -0.17 & -0.10 & -0.27 & -1.88 & -0.05\\
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205 \hkl[0 1 0] & -2.25 & -1.90 & -2.25 & -0.12 & -1.38 & -0.06\\
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206 \hkl[-1 0 0] & -2.39 & -0.36 & -2.25 & -0.12 & -1.88 & -0.05\\
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207 \hkl[1 0 0] & -2.25 & -2.16 & -0.10 & -0.27 & -1.38 & -0.06\\
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210 \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 ($\approx \unit[1.3]{nm}$) due to periodic boundary conditions.}
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211 \label{table:dc_c-c}
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213 Most of the obtained configurations result in binding energies well below zero indicating a preferable agglomeration of these type of defects.
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214 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.
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215 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.
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216 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.
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217 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.
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219 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}.
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220 In this work we found a further relaxation of this defect structure.
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221 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}.
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222 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.
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223 The atomic arrangement is shown in the bottom right of Fig.~\ref{fig:036-239}.
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224 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}.
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226 Investigating migration barriers enables to predict the probability of formation of defect complexes by thermally activated diffusion processes.
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227 % ground state configuration, C cluster
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228 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.
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229 In addition, the starting configuration exhibits a low binding energy (\unit[-1.90]{eV}) and is, thus, very likely to occur.
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230 However, a barrier height of more than \unit[4]{eV} was detected resulting in a low probability for the transition.
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231 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.
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232 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}).
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233 Starting from this configuration, an activation energy of only \unit[1.2]{eV} is necessary for the transition into the ground state configuration.
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234 The corresponding migration energies and atomic configurations are displayed in Fig.~\ref{fig:036-239}.
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236 \includegraphics[width=\columnwidth]{036-239.ps}
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237 \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.}
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238 \label{fig:036-239}
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240 % strange mig from -190 -> -2.39 (barrier > 4 eV)
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241 % C-C migration -> idea:
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242 % mig from low energy confs has extremely high barrier!
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243 % low barrier only from energetically less/unfavorable confs (?)! <- prove!
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244 % => low probability of C-C clustering ?!?
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246 % should possibly be transfered to discussion section
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247 Since thermally activated C clustering is, thus, only possible by traversing energetically unfavored configurations, mass C clustering is not expected.
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248 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.
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249 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.
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250 Thus, lower migration barriers are expected for pathways resulting in larger separations of the C$_{\text{i}}$ DBs.
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251 % calculate?!? ... hopefully acknowledged by 188-225 calc
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252 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.
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253 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.
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254 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.
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255 In both cases the configuration yielding a binding energy of \unit[-2.25]{eV} is promising.
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256 First of all, it constitutes the second most energetically favorable structure.
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257 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}).
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258 The migration barrier and correpsonding structures are shown in Fig.~\ref{fig:188-225}.
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259 % 188 - 225 transition in progress
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261 \includegraphics[width=\columnwidth]{188-225.ps}
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262 \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.}
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263 \label{fig:188-225}
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265 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.
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266 The latter is considered very important for high temperatures, which is accompanied by an increase in the entropic contribution to structure formation.
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267 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.
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269 % look for precapture mechnism (local minimum in energy curve)
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270 % also: plot energy all confs with respect to C-C distance
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271 % maybe a pathway exists traversing low energy confs ?!?
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273 % point out that configurations along 110 were extended up to the 6th NN in that direction
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274 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}.
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276 \begin{ruledtabular}
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277 \begin{tabular}{l c c c c c c }
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278 & 1 & 2 & 3 & 4 & 5 & 6 \\
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280 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
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281 C-C distance [nm] & 0.14 & 0.46 & 0.65 & 0.86 & 1.05 & 1.08
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284 \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.}
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285 \label{table:dc_110}
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287 The binding energy of these configurations with respect to the C-C distance is plotted in Fig.~\ref{fig:dc_110}.
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289 \includegraphics[width=\columnwidth]{db_along_110_cc_n.ps}
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290 \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
293 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
294 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
295 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
296 This finding, in turn, supports the previously established assumption of C agglomeration and absence of C clsutering.
\r
299 \begin{ruledtabular}
\r
300 \begin{tabular}{l c c c c c c }
\r
301 & 1 & 2 & 3 & 4 & 5 & R \\
\r
303 C$_{\text{s}}$ & 0.26$^a$/-1.28$^b$ & -0.51 & -0.93$^A$/-0.95$^B$ & -0.15 & 0.49 & -0.05\\
\r
304 V & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31
\r
307 \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
308 \label{table:dc_c-sv}
\r
311 \subsection{C$_{\text{i}}$ next to C$_{\text{s}}$}
\r
313 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
314 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
315 However, small displacements of the involved atoms near the defect result in different stable structures labeled b and B respectively.
\r
316 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
319 %./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
321 \includegraphics[width=\columnwidth]{093-095.ps}
\r
322 \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
323 \label{fig:093-095}
\r
325 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
326 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
327 This configuration has been identified and described by spectroscopic experimental techniques\cite{song90_2} as well as theoretical studies\cite{leary97,capaz98}.
\r
328 Configuration B is found to constitute the energetically slightly more favorable configuration.
\r
329 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
330 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
331 % mattoni: A favored by 0.4 eV - NO, it is indeed B (reinforce Song and Capaz)!
\r
334 The migration barrier was identified to be \unit[0.44]{eV}, almost three times higher than the experimental value of \unit[0.16]{eV}\cite{song90_2} estimated for the neutral charge state transition in p- and n-type Si.
\r
335 Keeping in mind the formidable agreement of the energy difference with experiment, the overestimated activation energy is quite unexpected and disappointing.
\r
336 Obviously, either the CRT algorithm fails to seize the actual saddle point structure or the influence of dopants has exceptional effect in the experimentally covered diffusion process being responsible for the low migration barrier.
\r
337 % not satisfactory!
\r
341 \includegraphics[width=\columnwidth]{026-128.ps}
\r
342 \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
343 \label{fig:026-128}
\r
345 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
346 Nevertheless, the C and Si DB atoms remain threefold coordinated.
\r
347 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
348 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
349 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
350 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
351 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
352 % mattoni: A favored by 0.2 eV - NO! (again, missing spin polarization?)
\r
353 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
354 In fact, these two configurations are very similar and are qualitatively different from the C$_{\text{i}}$ \hkl<1 0 0> DB that does not show magnetization but a nearly collinear bond of the C DB atom to its two next neighbored Si atoms while the Si DB atom approximates \unit[120]{$^{\circ}$} angles inbetween its bonds.
\r
355 Configurations a, A and B are not affected by spin polarization and show zero magnetization.
\r
356 Mattoni et~al.\cite{mattoni2002}, in contrast, find configuration b less favorable than configuration A by \unit[0.2]{eV}.
\r
357 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
358 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
359 Obviously a different energy minimum of the electronic system is obatined indicating hysteresis behavior.
\r
360 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
363 A low activation energy of \unit[0.1]{eV} is observed for the a$\rightarrow$b transition.
\r
364 Thus, configuration a is very unlikely to occur in favor of configuration b.
\r
366 % repulsive along 110
\r
367 A repulsive interaction is observed for C$_{\text{s}}$ at lattice sites along \hkl[1 1 0], i.e. positions 1 (configuration a) and 5.
\r
368 This is due to tensile strain originating from both, the C$_{\text{i}}$ DB and the C$_{\text{s}}$ atom residing within the \hkl[1 1 0] bond chain.
\r
369 This finding agrees well with results by Mattoni et~al.\cite{mattoni2002}.
\r
370 % all other investigated results: attractive interaction. stress compensation.
\r
371 In contrast, all other investigated configurations show attractive interactions.
\r
372 The most favorable configuration is found for C$_{\text{s}}$ at position 3, which corresponds to the lattice site of one of the upper next neighbored Si atoms of the DB structure that is compressively strained along \hkl[1 -1 0] and \hkl[0 0 1] by the C-Si DB.
\r
373 The substitution with C allows for most effective compensation of strain.
\r
374 This structure is followed by C$_{\text{s}}$ located at position 2, the next neighbor atom below the two Si atoms bound to the C$_{\text{i}}$ DB atom.
\r
375 As mentioned earlier these two lower Si atoms indeed experience tensile strain along the \hkl[1 1 0] bond chain, however, additional compressive strain along \hkl[0 0 1] exists.
\r
376 The latter is partially compensated by the C$_{\text{s}}$ atom.
\r
377 Yet less of compensation is realized if C$_{\text{s}}$ is located at position 4 due to a larger separation although both bottom Si atoms of the DB structure are indirectly affected, i.e. each of them is connected by another Si atom to the C atom enabling the reduction of strain along \hkl[0 0 1].
\r
379 % c agglomeration vs c clustering ... migs to b conf
\r
380 % 2 more migs: 051 -> 128 and 026! forgot why ... probably it's about probability of C clustering
\r
381 Obviously agglomeration of C$_{\text{i}}$ and C$_{\text{s}}$ is energetically favorable except for seprations along one of the \hkl<1 1 0> directions.
\r
382 The eneregtically most favorable configuration (configuration b) forms a strong but compressively strained C-C bond with a separation distance of \unit[0.142]{nm} sharing a Si lattice site.
\r
383 Again, conclusions concerning the probability of formation are drawn by investigating migration paths.
\r
384 Since C$_{\text{s}}$ is unlikely to exhibit a low activation energy for migration the focus is on C$_{\text{i}}$.
\r
385 Pathways starting from the two next most favored configurations were investigated, which show activation energies above \unit[2.2]{eV} and \unit[3.5]{eV} respectively.
\r
386 Although lower than the barriers for obtaining the ground state of two C$_{\text{i}}$ defects the activation energies are yet considered too high.
\r
387 For the same reasons as in the last subsection, structures other than the ground state configuration are, thus, assumed to arise more likely due to much lower activation energies necessary for their formation and still comparatively low binding energies.
\r
389 \subsection{C$_{\text{i}}$ next to V}
\r
391 In the last subsection configurations of a C$_{\text{i}}$ DB with C$_{\text{s}}$ occupying a vacant site created by the implantation process have been investigated.
\r
392 Additionally, configurations might arise in IBS, in which the impinging C atom creates a vacant site near a C$_{\text{i}}$ DB but does not occupy it.
\r
393 Resulting binding energies of a C$_{\text{i}}$ DB with a nearby vacancy are listed in the second row of Table~\ref{table:dc_c-sv}.
\r
394 All investigated structures are prefered compared to isolated largely separated defects.
\r
395 In contrast to C$_{\text{s}}$ this is also valid for positions along \hkl[1 1 0] resulting in an entirely attractive interaction between defects of these types.
\r
396 Even for the largest possible distance (R) achieved in the calculations of the periodic supercell a binding energy as low as \unit[-0.31]{eV} is observed.
\r
397 The ground state configuration is obtained for a V at position 1.
\r
398 The C atom of the DB moves towards the vacant site forming a stable C$_{\text{s}}$ configuration resulting in the release of a huge amount of energy.
\r
399 The second most favored configuration is accomplished for a V located at position 3 due to the reduction of compressive strain of the Si DB atom and its two upper Si neighbors present in the C$_{\text{i}}$ DB configuration.
\r
400 This configuration is follwed by the structure, in which a vacant site is created at position 2.
\r
401 Similar to the observations for C$_{\text{s}}$ in the last subsection a reduction of strain along \hkl[0 0 1] is enabled by this configuration.
\r
402 Relaxed structures of the latter two defect combinations are shown in the bottom left of Fig.~\ref{fig:314-539} and \ref{fig:059-539} respectively together with their energetics during transition into the ground state.
\r
404 \includegraphics[width=\columnwidth]{314-539.ps}
\r
405 \caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created at position 3 (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.1]{eV} is observed.}
\r
406 \label{fig:314-539}
\r
409 \includegraphics[width=\columnwidth]{059-539.ps}
\r
410 \caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created at position 2 (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.6]{eV} is observed.}
\r
411 \label{fig:059-539}
\r
413 Activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} are observed.
\r
414 In the first case the Si and C atom of the DB move towards the vacant and initial DB lattice site respectively.
\r
415 In total three Si-Si and one more Si-C bond is formed during the transition.
\r
416 In the second case the lowest barrier is found for the migration of Si number 1 , which is substituted by the C$_{\text{i}}$ atom, towards the vacant site.
\r
417 A net amount of five Si-Si and one Si-C bond are additionally formed during the transition.
\r
418 The direct migration of the C$_{\text{i}}$ atom onto the vacant lattice site results in a somewhat higher barrier of \unit[1.0]{eV}.
\r
419 In both cases, the formation of additional bonds is responsible for the vast gain in energy rendering almost impossible the reverse processes.
\r
421 In summary, pairs of C$_{\text{i}}$ DBs and Vs, like no other before, show highly attractive interactions for all investigated combinations indpendent of orientation and separation direction of the defects.
\r
422 Furthermore, small activation energies, even for transitions into the ground state exist.
\r
423 Based on these results, a high probability for the formation of C$_{\text{s}}$ must be concluded.
\r
425 \subsection{C$_{\text{s}}$ next to Si$_{\text{i}}$}
\r
427 As shown in section~\ref{subsection:sep_def} C$_{\text{s}}$ exhibits the lowest energy of formation.
\r
428 Considering a perfect Si crystal and conservation of particles, however, the occupation of a Si lattice site by a slowed down implanted C atom is necessarily accompanied by the formation of a Si self-interstitial.
\r
429 There are good reasons for the existence of regions exhibiting such configurations with regard to the IBS process.
\r
430 Highly energetic C atoms are able to kick out a Si atom from its lattice site, resulting in a Si self-interstitial accompanied by a vacant site, which might get occupied by another C atom, which lost almost all of its kinetic energy.
\r
431 Thus, configurations of C$_{\text{s}}$ and Si self-interstitials are investigated in the following.
\r
432 The Si$_{\text{i}}$ \hkl<1 1 0> DB, which was found to exhibit the lowest energy of formation within the investigated self-interstitial configurations, is assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$.
\r
435 \begin{ruledtabular}
\r
436 \begin{tabular}{l c c c c c c}
\r
437 & \hkl[1 1 0] & \hkl[-1 1 0] & \hkl[0 1 1] & \hkl[0 -1 1] &
\r
438 \hkl[1 0 1] & \hkl[-1 0 1] \\
\r
440 1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
\r
441 2 & \RM{2} & \RM{6} & \RM{6} & \RM{2} & \RM{8} & \RM{5} \\
\r
442 3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
\r
443 4 & \RM{4} & \RM{7} & \RM{9} & \RM{10} & \RM{10} & \RM{9} \\
\r
444 5 & \RM{5} & \RM{8} & \RM{6} & \RM{2} & \RM{6} & \RM{2} \\
\r
446 \caption{Equivalent configurations labeld \RM{1}-\RM{10} of \hkl<1 1 0>-type Si$_{\text{i}}$ DBs created at position I and C$_{\text{s}}$ created at positions 1 to 5 according to Fig.~\ref{fig:combos_si}. The respective orientation of the Si$_{\text{i}}$ DB is given in the first row.}
\r
447 \label{table:dc_si-s}
\r
451 \begin{ruledtabular}
\r
452 \begin{tabular}{l c c c c c c c c c c}
\r
453 & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & \RM{6} & \RM{7} & \RM{8} & \RM{9} & \RM{10} \\
\r
455 $E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
\r
456 $E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
\r
457 $r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
\r
459 \caption{Formation energies $E_{\text{f}}$, binding energies $E_{\text{b}}$ and C$_{\text{s}}$-Si$_{\text{i}}$ separation distances of the combinational C$_{\text{s}}$ and Si$_{\text{i}}$ configurations as defined in Table~\ref{table:dc_si-s}. Energies are given in eV while the separation is given in nm.}
\r
460 \label{table:dc_si-s_e}
\r
463 Table~\ref{table:dc_si-s} classifies equivalent configurations of \hkl<1 1 0>-type Si$_{\text{i}}$ DBs created at position I and C$_{\text{s}}$ created at positions 1 to 5 according to Fig.~\ref{fig:combos_si}.
\r
464 Corresponding formation as well as binding energies and the separation distances of the C$_{\text{s}}$ atom and the Si$_{\text{i}}$ DB lattice site are listed in Table~\ref{table:dc_si-s_e}.
\r
465 In total ten different configurations exist within the investigated range.
\r
466 Configuration \RM{1} constitutes the energetically most favorable structure exhibiting a formation energy of \unit[4.37]{eV}.
\r
467 Obviously the configuration of a \hkl[1 1 0] Si$_{\text{i}}$ DB and a next neighbored C$_{\text{s}}$ in the same direction as the alignment of the DB, as displayed in the bottom right of Fig.~\ref{fig:162-097}, enables the largest possible reduction of strain.
\r
468 The Si$_{\text{i}}$ DB atoms are displaced towards the lattice site occupied by the C$_{\text{s}}$ atom in such a way that the Si DB atom closest to the C atom does no longer form bonds to its top Si neighbors but to the second next neighbored Si atom along \hkl[1 1 0].
\r
469 However, this configuration is energetically less favorable than the \hkl<1 0 0> C$_{\text{i}}$ DB, which, thus, remains the ground state of a C atom introduced into otherwise perfect c-Si.
\r
470 The transition involving the latter two configurations is shown in Fig.~\ref{fig:dc_si-s}.
\r
472 \includegraphics[width=\columnwidth]{162-097.ps}
\r
473 \caption{Migration barrier and structures of the transition of a \hkl[1 1 0] Si$_{\text{i}}$ DB next to C$_{\text{s}}$ (right) into the C$_{\text{i}}$ \hkl[0 0 -1] DB configuration (left). An activation energy of \unit[0.12]{eV} is observed.}
\r
474 \label{fig:162-097}
\r
476 An activation energy as low as \unit[0.12]{eV} is necessary for the migration into the ground state configuration.
\r
477 Thus, the C$_{\text{i}}$ \hkl<1 0 0> DB configuration is assumed to occur more likely.
\r
478 However, only \unit[0.77]{eV} are needed for the reverse process, i.e. the formation of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB out of the ground state.
\r
479 Due to the low activation energy this process must be considered to be activated without much effort either thermally or by introduced energy of the implantation process.
\r
482 \includegraphics[width=\columnwidth]{c_sub_si110.ps}
\r
483 \caption{Binding energies of combinations of a C$_{\text{s}}$ and a Si$_{\text{i}}$ DB with respect to the separation distance. The binding energies of the defect pairs are well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}
\r
484 \label{fig:dc_si-s}
\r
486 Fig.~\ref{fig:dc_si-s} shows the binding energies of pairs of C$_{\text{s}}$ and a Si$_{\text{i}}$ \hkl<1 1 0> DB with respect to the separation distance.
\r
487 The interaction of the defects is well approximated by a Lennard-Jones 6-12 potential, which was used for curve fitting.
\r
488 The binding energy quickly drops to zero with the fit estimating almost zero interaction at \unit[0.6]{nm}.
\r
489 This indicates a low interaction capture radius of the defect pair.
\r
490 In IBS highly energetic collisions are assumed to easily produce configurations of these defects with separation distances exceeding the capture radius.
\r
491 For this reason C$_{\text{s}}$ without a nearby interacting Si$_{\text{i}}$ DB, which are, thus, unable to form the thermodynamically stable C$_{\text{i}}$ \hkl<1 0 0> DB constitutes a most likely configuration to be found in IBS.
\r
493 As mentioned above, configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ DBs might be especially important at higher temperatures due to the low activation energy necessary for its formation.
\r
494 At higher temperatures the contribution of entropy to structural formation increases, which might result in a spatial separation even for defects located within the capture radius.
\r
495 Indeed an ab initio molecular dynamics run at \unit[900]{$^{\circ}$C} starting from configuration \RM{1}, which -- based on the above findings -- is assumed to recombine into the ground state configuration, results in a separation of the C$_{\text{s}}$ and Si$_{\text{i}}$ DB by more than 4 next neighbor distances realized in a repeated migration mechnism of annihilating and arising Si DBs.
\r
496 The atomic configurations for two different points in time are shown in Fig.~\ref{fig:md}.
\r
497 Si atoms 1 and 2, which form the initial DB, occupy usual Si lattice sites in the final configuration while atom 3 occupies an interstitial site.
\r
499 \begin{minipage}{0.49\columnwidth}
\r
500 \includegraphics[width=\columnwidth]{md01.eps}
\r
502 \begin{minipage}{0.49\columnwidth}
\r
503 \includegraphics[width=\columnwidth]{md02.eps}\\
\r
505 \begin{minipage}{0.49\columnwidth}
\r
507 $t=\unit[2230]{fs}$
\r
510 \begin{minipage}{0.49\columnwidth}
\r
512 $t=\unit[2900]{fs}$
\r
515 \caption{Atomic configurations of an ab initio molecular dynamics run at \unit[900]{$^{\circ}$C} starting from a configuration of C$_{\text{s}}$ located next to a Si$_{\text{i}}$ DB (atoms 1 and 2). Equal atoms are marked by equal numbers. Blue lines correpsond to bonds, which are drawn for substantial atoms.}
\r
519 \section{Discussion}
\r
521 Obtained results for separated point defects in Si are in good agreement to previous theoretical work on this subject, both for intrinsic defects\cite{leung99,al-mushadani03} as well as for C point defects\cite{dal_pino93,capaz94}.
\r
522 The ground state configurations of these defects, i.e. the Si$_{\text{i}}$ \hkl<1 1 0> and C$_{\text{i}}$ \hkl<1 0 0> DB, have been reproduced and compare well to previous findings of theoretical investigations on Si$_{\text{i}}$\cite{leung99,al-mushadani03} as well as theoretical\cite{dal_pino93,capaz94,burnard93,leary97,jones04} and experimental\cite{watkins76,song90} studies on C$_{\text{i}}$.
\r
523 A quantitatively improved activation energy of \unit[0.9]{eV} for a qualitatively equal migration path based on studies by Capaz et.~al.\cite{capaz94} to experimental values\cite{song90,lindner06,tipping87} ranging from \unit[0.70-0.87]{eV} reinforce their derived mechanism of diffusion for C$_{\text{i}}$ in Si.
\r
525 The investigation of defect pairs indicates a general trend of defect agglomeration mainly driven by the potential of strain reduction.
\r
526 Obtained results for the most part compare well with results gained in previous studies\cite{leary97,capaz98,mattoni2002,liu02} and show an astonishingly good agreement with experiment\cite{song90}.
\r
527 Configurations involving two C impurities indeed exhibit the ground state for structures consisting of C-C bonds, which are responsible for the vast gain in energy.
\r
528 However, based on investigations of possible migration pathways, these structures are less likely to arise than structures, in which both C atoms are interconnected by another Si atom, which is due to high activation energies of the respective pathways or alternative pathways with less high activation energies, which, however, involve intermediate unfavorable configurations.
\r
529 Thus, agglomeration of C$_{\text{i}}$ is expected while the formation of C-C bonds is assumed to fail to appear by thermally activated diffusion processes.
\r
531 In contrast, C$_{\text{i}}$ and V were found to efficiently react with each other exhibiting activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} resulting in C$_{\text{s}}$ configurations.
\r
532 In addition, a highly attractive interaction exhibiting a large capture radius was observed, effective independent of the orientation and the direction of separation of the defects.
\r
533 Thus, the formation of C$_{\text{s}}$ is very likely to occur.
\r
534 Comparatively high energies necessary for the reverse process reveal this configuration to be extremely stable.
\r
536 Investigating configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ formation energies higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB were obtained keeping up previously derived assumptions concerning the ground state of C$_{\text{i}}$ in otherwise perfect Si.
\r
537 However, a small capture radius was identified for the respective interaction that might prevent the recombination of defects exceeding a separation of \unit[0.6]{nm} into the ground state configuration.
\r
538 In addition, a rather small activation energy of \unit[0.77]{eV} allows for the formation of a C$_{\text{s}}$-Si$_{\text{i}}$ pair originating from the C$_{\text{i}}$ \hkl<1 0 0> DB structure by thermally activated processes.
\r
539 Thus, elevated temperatures might lead to configurations of C$_{\text{s}}$ and a remaining Si atom in the near interstitial lattice, which is supported by the result of the molecular dynamics run.
\r
541 % add somewhere: nearly same energies of C_i -> Si_i + C_s, Si_i mig and C_i mig
\r
543 % add somewhere: controversy c_i vs c_s agglomeration, we suggest both!
\r
545 These findings allow to draw conclusions on the mechanisms involved in the process of SiC conversion in Si.
\r
546 Agglomeration of C$_{\text{i}}$ is energetically favored and enabled by a low activation energy for migration.
\r
547 Although ion implantation is a process far from thermodynamic equlibrium, which might result in phases not described by the Si/C phase diagram, i.e. a C phase in Si, high activation energies are believed to be responsible for a low probability of the formation of C clusters.
\r
549 Unrolling these findings on the initially stated controversy present in the precipitation model, an increased participation of C$_{\text{s}}$ already in the initial stage must be assumed due to its high probability of incidence.
\r
550 In addition, thermally activated, C$_{\text{i}}$ might turn into C$_{\text{s}}$.
\r
551 The associated emission of Si$_{\text{i}}$ serves two needs: as a vehicle for other C$_{\text{s}}$ and as a supply of Si atoms needed elsewhere to form the SiC structure.
\r
552 As for the vehicle, Si$_{\text{i}}$ is believed to react with C$_{\text{s}}$ turning it into a highly mobile C$_{\text{i}}$ again, allowing for the rearrangement of the C atom.
\r
553 The rearrangement is crucial to end up in a configuration of C atoms only occupying substitutionally the lattice sites of one of the fcc lattices that build up the diamond lattice as expected in 3C-SiC.
\r
554 On the other hand the conversion of some region of Si into SiC by substitutional C is accompanied by a reduction of the volume since SiC exhibits a \unit[20]{\%} smaller lattice constant than Si.
\r
555 The reduction in volume is compensated by excess Si$_{\text{i}}$ serving as building blocks for the surrounding Si host or a further formation of SiC.
\r
557 It is, thus, concluded that precipitation occurs by successive agglomeration of C$_{\text{s}}$.
\r
558 However, the agglomeration and rearrangement of C$_{\text{s}}$ is only possible by mobile C$_{\text{i}}$, which has to be present at the same time.
\r
559 Thus, the process is governed by both, C$_{\text{s}}$ accompanied by Si$_{\text{i}}$ as well as C$_{\text{i}}$.
\r
560 It is worth to mention that there is no contradiction to results of the HREM studies\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}.
\r
561 Regions showing dark contrasts in an otherwise undisturbed Si lattice are attributed to C atoms in the interstitial lattice.
\r
562 However, there is no particular reason for the C species to reside in the interstitial lattice.
\r
563 Contrasts are also assumed for Si$_{\text{i}}$.
\r
564 Once precipitation occurs regions of dark contrasts disappear in favor of Moir\'e patterns indicating 3C-SiC in c-Si due to the mismatch in the lattice constant.
\r
565 Until then, however, these regions are either composed of stretched coherent SiC and interstitials or of already contracted incoherent SiC surrounded by Si and interstitials too small to be detected in HREM.
\r
566 In both cases Si$_{\text{i}}$ might be attributed a third role, which is the partial compensation of tensile strain either in the stretched SiC or at the interface of the contracted SiC and the Si host.
\r
568 In addition, the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate is statisfied by the mechanism of successive positioning of C$_{\text{s}}$.
\r
569 In contrast, there is no obvious reason for the topotactic orientation of an agglomerate consisting exclusively of C-Si dimers, which would necessarily involve a much more profound change in structure for the transition into SiC.
\r
573 In summary, C and Si point defects in Si, combinations of these defects and diffusion processes within such configurations have been investigated.
\r
574 It is shown that C interstitials in Si tend to agglomerate, which is mainly driven by a reduction of strain.
\r
575 Investigations of migration pathways, however, allow to conclude that C clustering fails to appear by thermally activated processes due to high activation energies of the respective diffusion processes.
\r
576 A highly attractive interaction and a large capture radius has been identified for the C$_{\text{i}}$ \hkl<1 0 0> DB and the vacancy indicating a high probability for the formation of C$_{\text{s}}$.
\r
577 In contrast, a rapidly decreasing interaction with respect to the separation distance has been identified for C$_{\text{s}}$ and a Si$_{\text{i}}$ \hkl<1 1 0> DB resulting in a low probability of defects exhibiting respective separations to transform into the C$_{\text{i}}$ \hkl<1 0 0> DB, which constitutes the ground state configuration for a C atom introduced into otherwise perfect Si.
\r
578 Based on these findings conclusions on basic processes involved in the SiC precipitation in bulk Si are drawn.
\r
579 It is concluded that the precipitation process is governed by the formation of C$_{\text{s}}$ already in the initial stages.
\r
580 Agglomeration and rearrangement of C$_{\text{s}}$, however, is only possible by mobile C$_{\text{i}}$, which, thus, needs to be present at the same time.
\r
581 Si$_{\text{i}}$ constitutes the vehicle for the rearrangement of C$_{\text{s}}$.
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584 % ----------------------------------------------------
\r
585 \section*{Acknowledgment}
\r
586 We gratefully acknowledge financial support by the Bayerische Forschungsstiftung (DPA-61/05) and the Deutsche Forschungsgemeinschaft (DFG SCHM 1361/11).
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588 % --------------------------------- references -------------------
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590 \bibliography{../../bibdb/bibdb}{}
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591 \bibliographystyle{h-physrev3}
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projects / lectures / latex.git / blob