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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 We investigated the migration mechanism of a carbon \hkl<1 0 0> interstitial and silicon \hkl<1 1 0> self-interstitial in otherwise defect-free silicon using density functional theory calculations.
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34 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 has been found to occur very unlikely.
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38 In contrast, substitutional carbon occurs in all probability.
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39 A long range capture radius has been observed 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 We derive conclusions on the precipitation mechanism of silicon carbide in bulk silicon and discuss conformability to experimental findings.
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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 SiC precipitation in studies on strained Si$_{1-y}$C$_y$/Si heterostructures formed by molecular beam epitaxy (MBE)\cite{strane94,guedj98} suggest an initial coherent precipitation by agglomeration of substitutional instead of interstitial C followed by the loss of coherency once the increasing strain energy surpasses the interfacial energy of the incoherent 3C-SiC precipitate and the c-Si substrate.
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59 These two different mechanisms of precipitation might be attributed to the respective method of fabrication, i.e. whether precipitation occurs inside the Si bulk or on the 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 on a microscopic level 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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73 % TODO: maybe delete: decisive role half sentence
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75 % --------------------------------------------------------------------------------
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76 \section{Methodology}
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78 The first-principles DFT calculations were performed with the plane-wave based Vienna ab initio simulation package (VASP)\cite{kresse96}.
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79 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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80 The electron-ion interaction was described by norm-conserving ultra-soft pseudopotentials\cite{hamann79} as implemented in VASP\cite{vanderbilt90}.
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81 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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82 Sampling of the Brillouin zone was restricted to the $\Gamma$-point.
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83 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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84 The ions and cell shape were allowed to change in order to realize a constant pressure simulation.
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85 Ionic relaxation was realized by the conjugate gradient algorithm.
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86 Spin polarization has been fully accounted for.
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88 Migration and recombination pathways have been investigated utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.
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89 The defect formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ is defined by choosing SiC as a particle reservoir for the C impurity, i.e. the chemical potentials are determined by the cohesive energies of a perfect Si and SiC supercell after ionic relaxation.
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90 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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91 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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95 The implantation of highly energetic C atoms results in a multiplicity of possible defect configurations.
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96 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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97 First of all, structure and energetics of separated defects are presented.
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98 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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100 \subsection{Separated defects in silicon}
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101 \label{subsection:sep_def}
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102 % we need both: Si self-int & C int ground state configuration (for combos)
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104 Several geometries have been calculated to be stable for individual intrinsic and C related defects in Si.
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105 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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107 \begin{minipage}[t]{0.32\columnwidth}
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108 \underline{Si$_{\text{i}}$ \hkl<1 1 0> DB}\\
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109 \includegraphics[width=\columnwidth]{si110.eps}
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111 \begin{minipage}[t]{0.32\columnwidth}
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112 \underline{Si$_{\text{i}}$ hexagonal}\\
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113 \includegraphics[width=\columnwidth]{sihex.eps}
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115 \begin{minipage}[t]{0.32\columnwidth}
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116 \underline{Si$_{\text{i}}$ tetrahedral}\\
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117 \includegraphics[width=\columnwidth]{sitet.eps}
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119 \begin{minipage}[t]{0.32\columnwidth}
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120 \underline{Si$_{\text{i}}$ \hkl<1 0 0> DB}\\
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121 \includegraphics[width=\columnwidth]{si100.eps}
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123 \begin{minipage}[t]{0.32\columnwidth}
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124 \underline{Vacancy}\\
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125 \includegraphics[width=\columnwidth]{sivac.eps}
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127 \begin{minipage}[t]{0.32\columnwidth}
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128 \underline{C$_{\text{s}}$}\\
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129 \includegraphics[width=\columnwidth]{csub.eps}
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131 \begin{minipage}[t]{0.32\columnwidth}
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132 \underline{C$_{\text{i}}$ \hkl<1 0 0> DB}\\
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133 \includegraphics[width=\columnwidth]{c100.eps}
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135 \begin{minipage}[t]{0.32\columnwidth}
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136 \underline{C$_{\text{i}}$ \hkl<1 1 0> DB}\\
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137 \includegraphics[width=\columnwidth]{c110.eps}
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139 \begin{minipage}[t]{0.32\columnwidth}
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140 \underline{C$_{\text{i}}$ bond-centered}\\
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141 \includegraphics[width=\columnwidth]{cbc.eps}
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143 \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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144 \label{fig:sep_def}
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147 \begin{ruledtabular}
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148 \begin{tabular}{l c c c c c c c c c}
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149 & 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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151 Present study & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 & 1.95 & 3.72 & 4.16 & 4.66 \\
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152 \multicolumn{10}{c}{Other ab initio studies} \\
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153 Ref.\cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 & - & - & - & - \\
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154 Ref.\cite{leung99} & 3.31 & 3.31 & 3.43 & - & - & - & - & - & - \\
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155 Ref.\cite{dal_pino93,capaz94} & - & - & - & - & - & 1.89\cite{dal_pino93} & x & - & x+2.1\cite{capaz94}
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158 \caption{Formation energies of silicon and carbon point defects in crystalline silicon given in electron volt. 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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159 \label{table:sep_eof}
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161 Results obtained by the present study compare well with results from literature\cite{leung99,al-mushadani03,dal_pino93,capaz94}.
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162 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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163 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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164 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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165 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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167 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.
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168 The BC configuration is claimed to constitute the saddle point within the C$_{\text{i}}$ \hkl[0 0 -1] DB migration path residing in the \hkl(1 1 0) plane and, thus, interpreted as the barrier of migration for the respective path.
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169 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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170 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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171 Regardless of the rather small correction of \unit[0.3]{eV} due to the spin, the difference we found is much smaller (\unit[0.94]{eV}), which would nicely compare to experimentally observed migration barriers of \unit[0.70-0.87]{eV}\cite{lindner06,tipping87,song90}.
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172 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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173 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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174 Calculations in this work reinforce this path by an additional improvement of the quantitative conformance of the barrier height (\unit[0.90]{eV}) to experimental values.
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175 A more detailed description can be found in a previous study\cite{zirkelbach10a}.
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177 Next to the C$_{\text{i}}$ BC configuration the vacancy and Si$_{\text{i}}$ \hkl<1 0 0> DB have to be treated by taking into account the spin of the electrons.
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178 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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179 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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180 No other configuration, within the ones that are mentioned, is affected.
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182 Concerning the mobility of the ground state Si$_{\text{i}}$, we found an activation energy of \unit[0.67]{eV} for the transition of the Si$_{\text{i}}$ \hkl[0 1 -1] to \hkl[1 1 0] DB located at the next neighbored Si lattice site in \hkl[1 1 -1] direction.
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183 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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184 Obtained values are of the same order of magnitude than values derived from other ab initio studies\cite{bloechl93,sahli05}.
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186 \subsection{Pairs of C$_{\text{i}}$}
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188 C$_{\text{i}}$ pairs of the \hkl<1 0 0> type have been investigated in the first part.
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189 Fig.~\ref{fig:combos_ci} schematically displays the initial C$_{\text{i}}$ \hkl[0 0 -1] DB structure and various positions for the second defect (1-5) that have been used for investigating defect pairs.
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190 Table~\ref{table:dc_c-c} summarizes resulting 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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192 \subfigure[]{\label{fig:combos_ci}\includegraphics[width=0.45\columnwidth]{combos_ci.eps}}
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194 \subfigure[]{\label{fig:combos_si}\includegraphics[width=0.45\columnwidth]{combos.eps}}
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195 \caption{Position of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB (I) (Fig.~\ref{fig:combos_ci}) and of the lattice site chosen for the initial Si$_{\text{i}}$ \hkl<1 1 0> DB (Si$_{\text{i}}$) (Fig.~\ref{fig:combos_si}). Lattice sites for the second defect used for investigating defect pairs are numbered from 1 to 5.}
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199 \begin{ruledtabular}
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200 \begin{tabular}{l c c c c c c }
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201 & 1 & 2 & 3 & 4 & 5 & R \\
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203 \hkl[0 0 -1] & -0.08 & -1.15 & -0.08 & 0.04 & -1.66 & -0.19\\
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204 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
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205 \hkl[0 -1 0] & -2.39 & -0.17 & -0.10 & -0.27 & -1.88 & -0.05\\
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206 \hkl[0 1 0] & -2.25 & -1.90 & -2.25 & -0.12 & -1.38 & -0.06\\
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207 \hkl[-1 0 0] & -2.39 & -0.36 & -2.25 & -0.12 & -1.88 & -0.05\\
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208 \hkl[1 0 0] & -2.25 & -2.16 & -0.10 & -0.27 & -1.38 & -0.06\\
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211 \caption{Binding energies in electron volt of C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs. Equivalent configurations exhibit equal energies. Column 1 lists the orientation of the second defect, which is combined with the initial C$_{\text{i}}$ \hkl[0 0 -1] DB. 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 defect separation distance ($\approx \unit[1.3]{nm}$) due to periodic boundary conditions.}
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212 \label{table:dc_c-c}
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214 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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215 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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216 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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217 Antiparallel orientations of the second defect, i.e. \hkl[0 0 1] for positions located below the \hkl(0 0 1) plane with respect to the initial one (positions 1, 2 and 4) form the energetically most unfavorable configurations.
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218 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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220 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 as-isolated DB structure, resulting in a binding energy of \unit[-2.1]{eV}.
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221 In this work we observed a further relaxation of this defect structure.
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222 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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223 Apart from that, we found a more favorable configuration 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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224 The atomic arrangement is shown in the bottom right of Fig.~\ref{fig:036-239}.
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225 The two C$_{\text{i}}$ 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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227 Investigating migration barriers allows to predict the probability of formation of defect complexes by thermally activated diffusion processes.
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228 % ground state configuration, C cluster
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229 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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230 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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231 However, a barrier height of more than \unit[4]{eV} was detected resulting in a low probability for the transition.
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232 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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233 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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234 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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235 The corresponding migration energies and atomic configurations are displayed in Fig.~\ref{fig:036-239}.
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237 \includegraphics[width=\columnwidth]{036-239.ps}
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238 \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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239 \label{fig:036-239}
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241 % strange mig from -190 -> -2.39 (barrier > 4 eV)
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242 % C-C migration -> idea:
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243 % mig from low energy confs has extremely high barrier!
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244 % low barrier only from energetically less/unfavorable confs (?)! <- prove!
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245 % => low probability of C-C clustering ?!?
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247 % should possibly be transfered to discussion section
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248 Since thermally activated C clustering is, thus, only possible by traversing energetically unfavored configurations, mass C clustering is not expected.
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249 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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250 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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251 Accordingly, lower migration barriers are expected for pathways resulting in larger separations of the C$_{\text{i}}$ DBs.
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252 % acknowledged by 188-225 (reverse order) calc
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253 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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254 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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255 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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256 In both cases the configuration yielding a binding energy of \unit[-2.25]{eV} is promising.
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257 First of all, it constitutes the second most energetically favorable structure.
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258 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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259 The migration barrier and corresponding structures are shown in Fig.~\ref{fig:188-225}.
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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 at high temperatures, accompanied by an increase in the entropic contribution to structure formation.
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267 As a result, C defect agglomeration indeed is expected, but only a low probability is assumed for C-C clustering by thermally activated processes with regard to the considered process time in IBS.
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268 % alternatively: ... considered period of time (of the IBS process).
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271 % look for precapture mechanism (local minimum in energy curve)
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272 % also: plot energy all confs with respect to C-C distance
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273 % maybe a pathway exists traversing low energy confs ?!?
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275 % point out that configurations along 110 were extended up to the 6th NN in that direction
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276 The binding energies of the energetically most favorable configurations with the second 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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278 \begin{ruledtabular}
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279 \begin{tabular}{l c c c c c c }
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280 & 1 & 2 & 3 & 4 & 5 & 6 \\
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282 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
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283 C-C distance [nm] & 0.14 & 0.46 & 0.65 & 0.86 & 1.05 & 1.08
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286 \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 the \hkl[1 1 0] bond chain.}
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287 \label{table:dc_110}
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289 The binding energy of these configurations with respect to the C-C distance is plotted in Fig.~\ref{fig:dc_110}.
\r
291 \includegraphics[width=\columnwidth]{db_along_110_cc_n.ps}
\r
292 \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
295 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
296 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
297 The interpolated graph suggests the disappearance of attractive interaction forces, which are proportional to the slope of the graph, in between the two lowest separation distances of the defects.
\r
298 This finding, in turn, supports the previously established assumption of C agglomeration and absence of C clustering.
\r
301 \begin{ruledtabular}
\r
302 \begin{tabular}{l c c c c c c }
\r
303 & 1 & 2 & 3 & 4 & 5 & R \\
\r
305 C$_{\text{s}}$ & 0.26$^a$/-1.28$^b$ & -0.51 & -0.93$^A$/-0.95$^B$ & -0.15 & 0.49 & -0.05\\
\r
306 V & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31
\r
309 \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
310 \label{table:dc_c-sv}
\r
313 \subsection{C$_{\text{i}}$ next to C$_{\text{s}}$}
\r
315 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
316 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
317 However, small displacements of the involved atoms near the defect result in different stable structures labeled b and B respectively.
\r
318 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
321 %./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
323 \includegraphics[width=\columnwidth]{093-095.ps}
\r
324 \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 in between 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
325 \label{fig:093-095}
\r
327 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
328 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 in between two substitutional C atoms residing on regular Si lattice sites.
\r
329 This configuration has been identified and described by spectroscopic experimental techniques\cite{song90_2} as well as theoretical studies\cite{leary97,capaz98}.
\r
330 Configuration B is found to constitute the energetically slightly more favorable configuration.
\r
331 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
332 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
333 % mattoni: A favored by 0.4 eV - NO, it is indeed B (reinforce Song and Capaz)!
\r
336 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
337 Keeping in mind the formidable agreement of the energy difference with experiment, the overestimated activation energy is quite unexpected and disappointing.
\r
338 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
339 % not satisfactory!
\r
343 \includegraphics[width=\columnwidth]{026-128.ps}
\r
344 \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
345 \label{fig:026-128}
\r
347 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
348 Nevertheless, the C and Si DB atoms remain threefold coordinated.
\r
349 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
350 Again a single bond switch, i.e. the breaking of the bond of the 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
351 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
352 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
353 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
354 % mattoni: A favored by 0.2 eV - NO! (again, missing spin polarization?)
\r
355 A net magnetization of two spin up electrons, which are equally localized as in the Si$_{\text{i}}$ \hkl<1 0 0> DB structure is observed.
\r
356 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 in between its bonds.
\r
357 Configurations a, A and B are not affected by spin polarization and show zero magnetization.
\r
358 Mattoni et~al.\cite{mattoni2002}, in contrast, find configuration b less favorable than configuration A by \unit[0.2]{eV}.
\r
359 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
360 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
361 Obviously a different energy minimum of the electronic system is obtained indicating hysteresis behavior.
\r
362 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
365 A low activation energy of \unit[0.1]{eV} is observed for the a$\rightarrow$b transition.
\r
366 Thus, configuration a is very unlikely to occur in favor of configuration b.
\r
368 % repulsive along 110
\r
369 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
370 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
371 This finding agrees well with results by Mattoni et~al.\cite{mattoni2002}.
\r
372 % all other investigated results: attractive interaction. stress compensation.
\r
373 In contrast, all other investigated configurations show attractive interactions.
\r
374 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
375 The substitution with C allows for most effective compensation of strain.
\r
376 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
377 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
378 The latter is partially compensated by the C$_{\text{s}}$ atom.
\r
379 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
381 % c agglomeration vs c clustering ... migs to b conf
\r
382 % 2 more migs: 051 -> 128 and 026! forgot why ... probably it's about probability of C clustering
\r
383 Obviously agglomeration of C$_{\text{i}}$ and C$_{\text{s}}$ is energetically favorable except for separations along one of the \hkl<1 1 0> directions.
\r
384 The energetically 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
385 Again, conclusions concerning the probability of formation are drawn by investigating migration paths.
\r
386 Since C$_{\text{s}}$ is unlikely to exhibit a low activation energy for migration the focus is on C$_{\text{i}}$.
\r
387 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
388 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
389 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
391 \subsection{C$_{\text{i}}$ next to V}
\r
393 In the last subsection configurations of a C$_{\text{i}}$ DB with C$_{\text{s}}$ occupying a vacant site have been investigated.
\r
394 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
395 Resulting binding energies of a C$_{\text{i}}$ DB and a nearby vacancy are listed in the second row of Table~\ref{table:dc_c-sv}.
\r
396 All investigated structures are preferred compared to isolated largely separated defects.
\r
397 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
398 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
399 The ground state configuration is obtained for a V at position 1.
\r
400 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
401 The second most favorable 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
402 This configuration is followed by the structure, in which a vacant site is created at position 2.
\r
403 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
404 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
406 \includegraphics[width=\columnwidth]{314-539.ps}
\r
407 \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
408 \label{fig:314-539}
\r
411 \includegraphics[width=\columnwidth]{059-539.ps}
\r
412 \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
413 \label{fig:059-539}
\r
415 Activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} are observed.
\r
416 In the first case the Si and C atom of the DB move towards the vacant and initial DB lattice site respectively.
\r
417 In total three Si-Si and one more Si-C bond is formed during transition.
\r
418 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
419 A net amount of five Si-Si and one Si-C bond are additionally formed during transition.
\r
420 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
421 In both cases, the formation of additional bonds is responsible for the vast gain in energy rendering almost impossible the reverse processes.
\r
423 In summary, pairs of C$_{\text{i}}$ DBs and Vs, like no other before, show highly attractive interactions for all investigated combinations independent of orientation and separation direction of the defects.
\r
424 Furthermore, small activation energies, even for transitions into the ground state exist.
\r
425 Based on these results, a high probability for the formation of C$_{\text{s}}$ must be concluded.
\r
427 \subsection{C$_{\text{s}}$ next to Si$_{\text{i}}$}
\r
429 As shown in section~\ref{subsection:sep_def}, C$_{\text{s}}$ exhibits the lowest energy of formation.
\r
430 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
431 There are good reasons for the existence of regions exhibiting such configurations with regard to the IBS process.
\r
432 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 that lost almost all of its kinetic energy.
\r
433 %Thus, configurations of C$_{\text{s}}$ and Si self-interstitials are investigated in the following.
\r
434 Provided that the first C atom, which created the V and Si$_{\text{i}}$ pair has enough kinetic energy to escape the affected region, the C$_{\text{s}}$-Si$_{\text{i}}$ pair can be described as a separated defect complex.
\r
435 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
438 \begin{ruledtabular}
\r
439 \begin{tabular}{l c c c c c c}
\r
440 & \hkl[1 1 0] & \hkl[-1 1 0] & \hkl[0 1 1] & \hkl[0 -1 1] &
\r
441 \hkl[1 0 1] & \hkl[-1 0 1] \\
\r
443 1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
\r
444 2 & \RM{2} & \RM{6} & \RM{6} & \RM{2} & \RM{8} & \RM{5} \\
\r
445 3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
\r
446 4 & \RM{4} & \RM{7} & \RM{9} & \RM{10} & \RM{10} & \RM{9} \\
\r
447 5 & \RM{5} & \RM{8} & \RM{6} & \RM{2} & \RM{6} & \RM{2} \\
\r
449 \caption{Equivalent configurations labeled \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
450 \label{table:dc_si-s}
\r
454 \begin{ruledtabular}
\r
455 \begin{tabular}{l c c c c c c c c c c}
\r
456 & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & \RM{6} & \RM{7} & \RM{8} & \RM{9} & \RM{10} \\
\r
458 $E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
\r
459 $E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
\r
460 $r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
\r
462 \caption{Formation energies $E_{\text{f}}$, binding energies $E_{\text{b}}$ and C$_{\text{s}}$-Si$_{\text{i}}$ separation distances of configurations combining C$_{\text{s}}$ and Si$_{\text{i}}$ as defined in Table~\ref{table:dc_si-s}.}
\r
463 \label{table:dc_si-s_e}
\r
466 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
467 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
468 In total ten different configurations exist within the investigated range.
\r
469 Configuration \RM{1} constitutes the energetically most favorable structure exhibiting a formation energy of \unit[4.37]{eV}.
\r
470 Obviously the configuration of a Si$_{\text{i}}$ \hkl[1 1 0] DB and a next neighbored C$_{\text{s}}$ atom along the bond chain, which has the same direction as the alignment of the DB, enables the largest possible reduction of strain.
\r
471 The relaxed structure is displayed in the bottom right of Fig.~\ref{fig:162-097}.
\r
472 Compressive strain originating from the Si$_{\text{i}}$ is compensated by tensile strain inherent to the C$_{\text{s}}$ configuration.
\r
473 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$_{\text{i}}$ 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
475 However, the 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
476 The transition involving the latter two configurations is shown in Fig.~\ref{fig:162-097}.
\r
478 \includegraphics[width=\columnwidth]{162-097.ps}
\r
479 \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} and \unit[0.77]{eV} for the reverse process is observed.}
\r
480 \label{fig:162-097}
\r
482 An activation energy as low as \unit[0.12]{eV} is necessary for the migration into the ground state configuration.
\r
483 Accordingly, the C$_{\text{i}}$ \hkl<1 0 0> DB configuration is assumed to occur more likely.
\r
484 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
485 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
488 \includegraphics[width=\columnwidth]{c_sub_si110.ps}
\r
489 \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
490 \label{fig:dc_si-s}
\r
492 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
493 The interaction of the defects is well approximated by a Lennard-Jones 6-12 potential, which was used for curve fitting.
\r
494 The binding energy quickly drops to zero.
\r
495 The Lennard-Jones fit estimates almost zero interaction already at \unit[0.6]{nm}, indicating a low interaction capture radius of the defect pair.
\r
496 In IBS highly energetic collisions are assumed to easily produce configurations of defects exhibiting separation distances exceeding the capture radius.
\r
497 For this reason C$_{\text{s}}$ without a Si$_{\text{i}}$ DB located within the immediate proximity, which is, 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
499 Similar to what was previously mentioned, configurations of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB might be particularly important at higher temperatures due to the low activation energy necessary for its formation.
\r
500 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
501 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 mechanism of annihilating and arising Si$_{\text{i}}$ DBs.
\r
502 The atomic configurations for two different points in time are shown in Fig.~\ref{fig:md}.
\r
503 Si atoms 1 and 2, which form the initial DB, occupy Si lattice sites in the final configuration while Si atom 3 is transferred from a regular lattice site into the interstitial lattice.
\r
505 \begin{minipage}{0.49\columnwidth}
\r
506 \includegraphics[width=\columnwidth]{md01.eps}
\r
508 \begin{minipage}{0.49\columnwidth}
\r
509 \includegraphics[width=\columnwidth]{md02.eps}\\
\r
511 \begin{minipage}{0.49\columnwidth}
\r
513 $t=\unit[2230]{fs}$
\r
516 \begin{minipage}{0.49\columnwidth}
\r
518 $t=\unit[2900]{fs}$
\r
521 \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$_{\text{i}}$ \hkl[1 1 0] DB (atoms 1 and 2). Equal atoms are marked by equal numbers. Bonds are drawn for substantial atoms only.}
\r
525 \section{Discussion}
\r
527 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
528 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
529 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
531 The investigation of defect pairs indicated a general trend of defect agglomeration mainly driven by the potential of strain reduction.
\r
532 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
533 For configurations involving two C impurities the ground state configurations have been found to to consist of C-C bonds, which are responsible for the vast gain in energy.
\r
534 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 featuring less high activation energies, which, however, involve intermediate unfavorable configurations.
\r
535 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
537 In contrast, C$_{\text{i}}$ and Vs 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 stable C$_{\text{s}}$ configurations.
\r
538 In addition, we observed a highly attractive interaction exhibiting a large capture radius, effective independent of the orientation and the direction of separation of the defects.
\r
539 Accordingly, the formation of C$_{\text{s}}$ is very likely to occur.
\r
540 Comparatively high energies necessary for the reverse process reveal this configuration to be extremely stable.
\r
542 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
543 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
544 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
545 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
547 % add somewhere: nearly same energies of C_i -> Si_i + C_s, Si_i mig and C_i mig
\r
549 These findings allow to draw conclusions on the mechanisms involved in the process of SiC conversion in Si.
\r
550 Agglomeration of C$_{\text{i}}$ is energetically favored and enabled by a low activation energy for migration.
\r
551 Although ion implantation is a process far from thermodynamic equilibrium, 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-C clusters.
\r
553 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
554 In addition, thermally activated, C$_{\text{i}}$ might turn into C$_{\text{s}}$.
\r
555 The associated emission of Si$_{\text{i}}$ serves two needs: as a vehicle for other C$_{\text{s}}$ atoms and as a supply of Si atoms needed elsewhere to form the SiC structure.
\r
556 As for the vehicle, Si$_{\text{i}}$ is believed to react with C$_{\text{s}}$ turning it into highly mobile C$_{\text{i}}$ again, allowing for the rearrangement of the C atom.
\r
557 The rearrangement is crucial to end up in a configuration of C atoms only occupying substitutionally the lattice sites of one of the two fcc lattices that build up the diamond lattice.
\r
558 % TODO: add SiC structure info to intro
\r
559 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
560 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
562 We conclude that precipitation occurs by successive agglomeration of C$_{\text{s}}$.
\r
563 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
564 Accordingly, the process is governed by both, C$_{\text{s}}$ accompanied by Si$_{\text{i}}$ as well as C$_{\text{i}}$.
\r
565 It is worth to mention that there is no contradiction to results of the HREM studies\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}.
\r
566 Regions showing dark contrasts in an otherwise undisturbed Si lattice are attributed to C atoms in the interstitial lattice.
\r
567 However, there is no particular reason for the C species to reside in the interstitial lattice.
\r
568 Contrasts are also assumed for Si$_{\text{i}}$.
\r
569 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
570 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, where the latter is too small to be detected in HREM.
\r
571 In both cases Si$_{\text{i}}$ might be attributed a third role, which is the partial compensation of tensile strain that is present either in the stretched SiC or at the interface of the contracted SiC and the Si host.
\r
573 In addition, the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate is satisfied by the mechanism of successive positioning of C$_{\text{s}}$.
\r
574 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
578 In summary, C and Si point defects in Si, combinations of these defects and diffusion processes within such configurations have been investigated.
\r
579 We have shown that C interstitials in Si tend to agglomerate, which is mainly driven by a reduction of strain.
\r
580 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
581 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
582 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
583 %Based on these findings conclusions on basic processes involved in the SiC precipitation in bulk Si are drawn.
\r
584 Obviously, the precipitation process is governed by the formation of C$_{\text{s}}$ already in the initial stages.
\r
585 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
586 Si$_{\text{i}}$ constitutes the vehicle for the rearrangement of C$_{\text{s}}$.
\r
589 % ----------------------------------------------------
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590 \section*{Acknowledgment}
\r
591 We gratefully acknowledge financial support by the Bayerische Forschungsstiftung (Grant No. DPA-61/05) and the Deutsche Forschungsgemeinschaft (Grant No. DFG SCHM 1361/11).
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593 % --------------------------------- references -------------------
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595 \bibliography{../../bibdb/bibdb}{}
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596 \bibliographystyle{h-physrev3}
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projects / lectures / latex.git / blob