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16 %\title{Mobility of Carbon in Silicon -- a first principles study}
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17 \title{Extensive first principles study of carbon defects in silicon}
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18 \author{F. Zirkelbach}
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19 \author{B. Stritzker}
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20 \affiliation{Experimentalphysik IV, Universit\"at Augsburg, 86135 Augsburg, Germany}
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21 \author{K. Nordlund}
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22 \affiliation{Department of Physics, University of Helsinki, 00014 Helsinki, Finland}
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23 \author{J. K. N. Lindner}
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24 \author{W. G. Schmidt}
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26 \affiliation{Department Physik, Universit\"at Paderborn, 33095 Paderborn, Germany}
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29 We present a first principles investigation of the mobility of carbon interstitials in silicon.
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30 The migration mechanism of carbon \hkl<1 0 0> dumbbell interstitials in otherwise defect-free silicon has been investigated using density functional theory calculations.
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31 Furthermore, the influence of near-by vacancies, carbon interstitial and substitutional defects and silicon self-interstitials has been investigated systematically.
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32 A long range capture radius for vacancies has been found....
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35 \keywords{point defects, migration, interstitials, first principles calculations }
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36 \pacs{ find out later... }
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39 % --------------------------------------------------------------------------------
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40 \section{Introduction}
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42 % 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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44 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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45 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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46 However, the process of the formation of SiC precipitates in Si during C implantation is not yet fully understood.
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47 Based on experimental studies\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03} it is assumed that incorporated C atoms form C-Si dimers (dumbbells) on regular Si lattice sites.
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48 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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49 In contrast, investigations of the precipitation in strained Si$_{1-y}$C$_y$/Si heterostructures formed by molecular beam epitaxy (MBE)\cite{strane94,guedj98} suggest an initial coherent clustering of substitutional instead of interstitial C followed by a loss of coherency once the increasing strain energy surpasses the interfacial energy of an incoherent 3C-SiC precipitate in c-Si.
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50 These two different mechanisms of precipitation might be attributed to the respective method of fabrication.
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51 However, in another IBS study Nejim et al. propose a topotactic transformation remaining structure and orientation likewise based on the formation of substitutional C and a concurrent reaction of the excess Si self-interstitials with further implanted C atoms in the initial state\cite{nejim95}.
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52 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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54 Atomistic simulations offer a powerful tool of investigation providing detailed insight not accessible by experiment.
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55 A lot of theoretical work has been done on intrinsic point defects in Si\cite{bar-yam84,bar-yam84_2,car84,batra87,bloechl93,tang97,leung99,colombo02,goedecker02,al-mushadani03,hobler05,posselt08,ma10}, threshold displacement energies in Si\cite{mazzarolo01,holmstroem08} important in ion implantation, C defects and defect reactions in Si\cite{tersoff90,dal_pino93,capaz94,burnard93,leary97,capaz98,zhu98,mattoni2002,park02,jones04}, the SiC/Si interface\cite{chirita97,kitabatake93,cicero02,pizzagalli03} and defects in SiC\cite{bockstedte03,rauls03a,gao04,posselt06,gao07}.
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56 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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57 % but mattoni2002 actually did a lot. maybe this should be mentioned!
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58 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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59 However, investigations are, first of all, restricted to interaction chains along the \hkl[1 1 0] and \hkl[-1 1 0] direction, secondly lacking combinations of C interstitials and, finally, not considering migration barriers giving further information about the probability of defect agglomeration.
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61 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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62 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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63 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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65 % --------------------------------------------------------------------------------
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66 \section{Methodology}
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68 The first principles DFT calculations were performed with the plane-wave based Vienna Ab-initio Simulation Package (VASP)\cite{kresse96}.
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69 The Kohn-Sham equations were solved using the generalized-gradient XC-functional approximation proposed by Perdew and Wang (GGA-PW91)\cite{perdew86,perdew92}.
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70 The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials\cite{hamann79} as implemented in VASP\cite{vanderbilt90}.
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71 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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72 Sampling of the Brillouin zone was restricted to the $\Gamma$-point.
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73 The defect structures and the migration paths were modelled in cubic supercells containing $216\pm2$ Si atoms.
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74 The ions and cell shape were allowed to change in order to realize a constant pressure simulation.
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75 Ionic relaxation was realized by the conjugate gradient algorithm.
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76 Spin polarization has been fully accounted for.
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78 Migration and recombination pathways have been investigated utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.
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79 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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80 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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81 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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85 The implantation of highly energetic C atoms results in a multiplicity of possible defect configurations.
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86 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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87 In the following the structure and energetics of separated defects are presented.
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88 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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90 \subsection{Separated defects in silicon}
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91 % we need both: Si self-int & C int ground state configuration (for combos)
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93 Several geometries have been calculated to be stable for individual intrinsic and C related defects in Si.
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94 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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96 \begin{minipage}[t]{0.32\columnwidth}
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97 \underline{Si$_{\text{i}}$ \hkl<1 1 0> DB}\\
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98 \includegraphics[width=\columnwidth]{si110.eps}
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100 \begin{minipage}[t]{0.32\columnwidth}
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101 \underline{Si$_{\text{i}}$ hexagonal}\\
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102 \includegraphics[width=\columnwidth]{sihex.eps}
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104 \begin{minipage}[t]{0.32\columnwidth}
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105 \underline{Si$_{\text{i}}$ tetrahedral}\\
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106 \includegraphics[width=\columnwidth]{sitet.eps}
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108 \begin{minipage}[t]{0.32\columnwidth}
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109 \underline{Si$_{\text{i}}$ \hkl<1 0 0> DB}\\
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110 \includegraphics[width=\columnwidth]{si100.eps}
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112 \begin{minipage}[t]{0.32\columnwidth}
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113 \underline{Vacancy}\\
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114 \includegraphics[width=\columnwidth]{sivac.eps}
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116 \begin{minipage}[t]{0.32\columnwidth}
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117 \underline{C$_{\text{s}}$}\\
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118 \includegraphics[width=\columnwidth]{csub.eps}
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120 \begin{minipage}[t]{0.32\columnwidth}
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121 \underline{C$_{\text{i}}$ \hkl<1 0 0> DB}\\
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122 \includegraphics[width=\columnwidth]{c100.eps}
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124 \begin{minipage}[t]{0.32\columnwidth}
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125 \underline{C$_{\text{i}}$ \hkl<1 1 0> DB}\\
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126 \includegraphics[width=\columnwidth]{c110.eps}
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128 \begin{minipage}[t]{0.32\columnwidth}
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129 \underline{C$_{\text{i}}$ bond-centered}\\
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130 \includegraphics[width=\columnwidth]{cbc.eps}
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132 \caption{Configurations of silicon and carbon point defects in silicon. Silicon and carbon atoms are illustrated by yellow and grey spheres respectively. Blue lines are bonds drawn whenever considered appropriate to ease identifying defect structures for the reader. Dumbbell configurations are abbreviated by DB.}
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133 \label{fig:sep_def}
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136 \begin{ruledtabular}
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137 \begin{tabular}{l c c c c c c c c c}
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138 & Si$_{\text{i}}$ \hkl<1 1 0> DB & Si$_{\text{i}}$ H & Si$_{\text{i}}$ T & Si \hkl<1 0 0> DB & V & C$_{\text{s}}$ & C$_{\text{i}}$ \hkl<1 0 0> DB & C$_{\text{i}}$ \hkl<1 1 0> DB & C$_{\text{i}}$ BC \\
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140 Present study & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 & 1.95 & 3.72 & 4.16 & 4.66 \\
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141 \multicolumn{10}{c}{Other ab initio studies} \\
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142 Ref.\cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 & - & - & - & - \\
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143 Ref.\cite{leung99} & 3.31 & 3.31 & 3.43 & - & - & - & - & - & - \\
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144 Ref.\cite{dal_pino93,capaz94} & - & - & - & - & - & 1.89\cite{dal_pino93} & x & - & x+2.1\cite{capaz94}
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147 \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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148 \label{table:sep_eof}
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150 Results obtained by the present study compare well with results from literature\cite{leung99,al-mushadani03,dal_pino93,capaz94}.
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151 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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152 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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153 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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154 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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156 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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157 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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158 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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159 Regardless of the rather small correction due to the spin, the difference we found is much smaller (\unit[0.9]{eV}), which would nicely compare to experimental findings $(\unit[0.70-0.87]{eV})$\cite{lindner06,tipping87,song90} for the migration barrier.
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160 However, since the BC configuration constitutes a real local minimum another barrier exists which is about \unit[1.2]{eV} ($\unit[0.9]{eV}+\unit[0.3]{eV}$) in height.
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161 Indeed Capaz et al. propose another path and find it to be the lowest in energy\cite{capaz94}, in which a C$_{\text{i}}$ \hkl[0 0 -1] DB migrates into a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the next neighbored Si lattice site in \hkl[1 1 -1] direction.
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162 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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163 A more detailed description can be found in a previous study\cite{zirkelbach10a}.
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165 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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166 For the latter two the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site and in two caps at each of the two DB atoms perpendicularly aligned to the bonds to the other two Si atoms respectively.
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167 No other configuration, within the ones that are mentioned, is affected.
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169 Concerning the mobility of the ground state Si$_{\text{i}}$, an activation energy shortly below \unit[0.7]{eV} was found for the migration of a Si$_{\text{i}}$ \hkl[0 1 -1] into a \hkl[1 1 0] DB configuration located at the next neighbored Si lattice site in \hkl[1 1 -1] direction.
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170 % look for values in literature for neutraly charged Si_i diffusion
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172 \subsection{Pairs of C$_{\text{i}}$}
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174 C$_{\text{i}}$ pairs of the \hkl<1 0 0>-type have been considered in the first part.
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175 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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176 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.
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178 %\begin{minipage}{0.49\columnwidth}
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179 \subfigure[]{\label{fig:combos_ci}\includegraphics[width=0.45\columnwidth]{combos_ci.eps}}
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181 \subfigure[]{\label{fig:combos_si}\includegraphics[width=0.45\columnwidth]{combos.eps}}
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182 \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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186 \begin{ruledtabular}
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187 \begin{tabular}{l c c c c c c }
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188 & 1 & 2 & 3 & 4 & 5 & R \\
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190 \hkl[0 0 -1] & -0.08 & -1.15 & -0.08 & 0.04 & -1.66 & -0.19\\
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191 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
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192 \hkl[0 -1 0] & -2.39 & -0.17 & -0.10 & -0.27 & -1.88 & -0.05\\
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193 \hkl[0 1 0] & -2.25 & -1.90 & -2.25 & -0.12 & -1.38 & -0.06\\
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194 \hkl[-1 0 0] & -2.39 & -0.36 & -2.25 & -0.12 & -1.88 & -0.05\\
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195 \hkl[1 0 0] & -2.25 & -2.16 & -0.10 & -0.27 & -1.38 & -0.06\\
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198 \caption{Binding energies of C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs in eV. Equivalent configurations exhibit equal energies. The first column lists the orientation of the defect, which is combined with the initial C$_{\text{i}}$ \hkl[0 0 -1] dumbbell. The position index of the second defect is given in the first row according to Fig.~\ref{fig:combos}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect position, which is the maximum realizable distance due to periodic boundary conditions.}
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199 \label{table:dc_c-c}
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201 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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202 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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203 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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204 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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205 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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207 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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208 In this work we found a further relaxation of this defect structure.
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209 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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210 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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211 The atomic arrangement is shown in the bottom right of Fig.~\ref{fig:036-239}.
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212 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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214 Investigating migration barriers enables to predict the probability of formation of defect complexes by thermally activated diffusion processes.
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215 % ground state configuration, C cluster
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216 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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217 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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218 However, a barrier height of more than \unit[4]{eV} was detected resulting in a low probability for the transition.
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219 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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220 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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221 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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222 The corresponding migration energies and atomic configurations are displayed in Fig.~\ref{fig:036-239}.
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224 \includegraphics[width=\columnwidth]{036-239.ps}
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225 \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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226 \label{fig:036-239}
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228 % strange mig from -190 -> -2.39 (barrier > 4 eV)
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229 % C-C migration -> idea:
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230 % mig from low energy confs has extremely high barrier!
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231 % low barrier only from energetically less/unfavorable confs (?)! <- prove!
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232 % => low probability of C-C clustering ?!?
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234 % should possibly be transfered to discussion section
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235 Since thermally activated C clustering is, thus, only possible by traversing energetically unfavored configurations, mass C clustering is not expected.
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236 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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237 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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238 Thus, lower migration barriers are expected for pathways resulting in larger separations of the C$_{\text{i}}$ DBs.
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239 % calculate?!? ... hopefully acknowledged by 188-225 calc
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240 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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241 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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242 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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243 In both cases the configuration yielding a binding energy of \unit[-2.25]{eV} is promising.
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244 First of all, it constitutes the second most energetically favorable structure.
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245 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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246 The migration barrier and correpsonding structures are shown in Fig.~\ref{fig:188-225}.
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247 % 188 - 225 transition in progress
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249 \includegraphics[width=\columnwidth]{188-225.ps}
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250 \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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251 \label{fig:188-225}
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253 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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254 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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255 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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257 % look for precapture mechnism (local minimum in energy curve)
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258 % also: plot energy all confs with respect to C-C distance
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259 % maybe a pathway exists traversing low energy confs ?!?
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261 % point out that configurations along 110 were extended up to the 6th NN in that direction
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262 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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264 \begin{ruledtabular}
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265 \begin{tabular}{l c c c c c c }
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266 & 1 & 2 & 3 & 4 & 5 & 6 \\
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268 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
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269 C-C distance [nm] & 0.14 & 0.46 & 0.65 & 0.86 & 1.05 & 1.08
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272 \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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273 \label{table:dc_110}
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275 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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277 \includegraphics[width=\columnwidth]{db_along_110_cc_n.ps}
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278 \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.}
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281 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.
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282 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.
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283 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.
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284 This finding, in turn, supports the previously established assumption of C agglomeration and absence of C clsutering.
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287 \begin{ruledtabular}
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288 \begin{tabular}{l c c c c c c }
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289 & 1 & 2 & 3 & 4 & 5 & R \\
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291 C$_{\text{s}}$ & 0.26$^a$/-1.28$^b$ & -0.51 & -0.93$^A$/-0.95$^B$ & -0.15 & 0.49 & -0.05\\
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292 V & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31
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295 \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.}
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296 \label{table:dc_c-sv}
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299 \subsection{C$_{\text{i}}$ next to C$_{\text{s}}$}
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301 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.
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302 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.
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303 However, small displacements of the involved atoms near the defect result in different stable structures labeled b and B respectively.
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304 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.
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307 %./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
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309 \includegraphics[width=\columnwidth]{093-095.ps}
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310 \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.}
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311 \label{fig:093-095}
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313 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.
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314 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.
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315 This configuration has been identified and described by spectroscopic experimental techniques\cite{song90_2} as well as theoretical studies\cite{leary97,capaz98}.
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316 Configuration B is found to constitute the energetically slightly more favorable configuration.
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317 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.
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318 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.
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319 % mattoni: A favored by 0.4 eV - NO, it is indeed B (reinforce Song and Capaz)!
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323 The migration barrier was identified to be \unit[0.44]{eV}, almost three times higher than the experimentally obtained value of \unit[0.16]{eV}\cite{song90_2}.
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324 This might be due to
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328 \includegraphics[width=\columnwidth]{026-128.ps}
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329 \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.}
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330 \label{fig:026-128}
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332 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.
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333 Nevertheless, the C and Si DB atoms remain threefold coordinated.
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334 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}).
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335 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.
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336 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.
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337 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.
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338 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}.
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339 % mattoni: A favored by 0.2 eV - NO! (again, missing spin polarization?)
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340 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.
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341 Configurations a, A and B are not affected by spin polarization and show zero magnetization.
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342 Mattoni et~al.\cite{mattoni2002}, in contrast, find configuration b less favorable than configuration A by \unit[0.2]{eV}.
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343 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.
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344 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}.
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345 Obviously a different energy minimum of the electronic system is obatined indicating hysteresis behavior.
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346 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.
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348 A low activation energy of \unit[0.1]{eV} is observed for the a$\rightarrow$b transition.
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349 Thus, configuration a is very unlikely to occur in favor of configuration b.
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350 However, migration barriers yielding
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354 % 2 more migs: 051 -> 128 and 026! forgot why ... probably it's about probability of C clustering
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356 \subsection{C$_{\text{i}}$ next to V}
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358 \subsection{C$_{\text{s}}$ next to Si$_{\text{i}}$}
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360 Non-zero temperature, entropy, spatial separation of these defects possible, indeed observed in ab initio MD run.
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362 \section{Discussion}
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364 Our calculations show that point defects which unavoidably are present after ion implantation significantly influence the mobility of implanted carbon
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365 in the silicon crystal.
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366 A large capture radius has been found for...
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367 Especially vacancies....
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371 In summary, we have shown that ...
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373 % ----------------------------------------------------
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374 \section*{Acknowledgment}
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375 We gratefully acknowledge financial support by the Bayerische Forschungsstiftung (DPA-61/05) and the Deutsche Forschungsgemeinschaft (DFG SCHM 1361/11).
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377 % --------------------------------- references -------------------
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379 \bibliography{../../bibdb/bibdb}{}
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380 \bibliographystyle{h-physrev3}
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400 %\end{thebibliography}
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projects / lectures / latex.git / blob