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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} \author{B. Stritzker}
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19 \affiliation{Experimentalphysik IV, Universit\"at Augsburg, 86135 Augsburg, Germany}
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20 \author{K. Nordlund}
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21 \affiliation{Department of Physics, University of Helsinki, 00014 Helsinki, Finland}
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22 \author{J. K. N. Lindner}
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23 \author{W. G. Schmidt} \author{E. Rauls}
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24 \affiliation{Department Physik, Universit\"at Paderborn, 33095 Paderborn, Germany}
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27 We present a first principles investigation of the mobility of carbon interstitials in silicon.
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28 The migration mechanism of carbon [100] dumbbell interstitials in otherwise defect-free silicon has been investigated using density functional theory calculations.
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29 Furthermore, the influence of near-by vacancies, carbon interstitial and substitutional defects and silicon self-interstitials has been investigated systematically.
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30 A long range capture radius for vacancies has been found....
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33 \keywords{point defects, migration, interstitials, first principles calculations }
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34 \pacs{ find out later... }
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37 % --------------------------------------------------------------------------------
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38 \section{Introduction}
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40 % 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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42 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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43 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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44 However, the process of the formation of SiC precipitates in Si during C implantation is not yet fully understood.
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45 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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46 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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47 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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48 These two different mechanisms of precipitation might be determined by the respective method of fabrication.
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49 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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50 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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52 Atomistic simulations offer a powerful tool of investigation providing detailed insight not accessible by experiment.
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53 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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54 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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55 % but mattoni2002 actually did a lot. maybe this should be mentioned!
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56 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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57 However, investigations are, first of all, restricted to interaction chains along the $\langle 1 1 0 \rangle$ and $\langle -1 1 0 \rangle$ direction, secondly lacking combinations of C interstitials and, finally, not considering migration barriers giving further information about the probability of defect agglomeration.
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59 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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60 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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61 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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63 % --------------------------------------------------------------------------------
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64 \section{Methodology}
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66 The first principles DFT calculations were performed with the plane-wave based Vienna Ab-initio Simulation Package (VASP)\cite{kresse96}.
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67 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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68 The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials\cite{hamann79} as implemented in VASP\cite{vanderbilt90}.
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69 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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70 Sampling of the Brillouin zone was restricted to the $\Gamma$-point.
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71 The defect structures and the migration paths were modelled in cubic supercells containing $216\pm2$ Si atoms.
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72 The ions and cell shape are allowed to change in order to realize a constant pressure simulation.
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73 Spin polarization has been fully accounted for.
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75 Migration and recombination pathways have been ivestigated utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.
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76 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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77 The binding energy of a defect pair is given by the difference of the formation energy of the complex and the sum of the two separated defect configurations, i.e. energetically favorable configurations show binding energies below zero and non-interacting isolated defects would result in a binding energy of zero.
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81 %After the implantation of C into Si, C interstitials are the most common defects in the Si sample.
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82 %Their mobility is the crucial quantity to be investigated.
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83 %However, the implantation process unavoidably creates a variety of further point defects, such as vacancies and silicon self-interstitials.
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84 %Already during implantation and also in the subsequent annealing process, further defects can evolve from these, like pair defects or substitutional carbon.
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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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89 Fig.~\ref{fig:combos} schematically displays the positions for the initial interstitial defect (Si$_{\text{i}}$/C$_{\text{i}}$) and the neighbured defect (1-5) used for investigating defect pairs.
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91 \includegraphics[width=0.5\columnwidth]{cs.eps}
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92 \caption{Positions for the initial defect (Si$_{\text{i}}$/C$_{\text{i}}$) and the neighboured defect (1-5) used for investigating defect pairs.}
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96 \subsection{Separated defects in silicon}
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97 % we need both: Si self-int & C int ground state configuration (for combos)
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99 Several geometries have been calculated to be stable for individual intrinsic and C related defects in Si.
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100 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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102 \begin{minipage}[t]{0.32\columnwidth}
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103 \underline{Si$_{\text{i}}$ $\langle 1 1 0 \rangle$ DB}\\
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104 \includegraphics[width=\columnwidth]{si110.eps}
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106 \begin{minipage}[t]{0.32\columnwidth}
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107 \underline{Si$_{\text{i}}$ hexagonal}\\
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108 \includegraphics[width=\columnwidth]{sihex.eps}
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110 \begin{minipage}[t]{0.32\columnwidth}
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111 \underline{Si$_{\text{i}}$ tetrahedral}\\
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112 \includegraphics[width=\columnwidth]{sitet.eps}
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114 \begin{minipage}[t]{0.32\columnwidth}
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115 \underline{Si$_{\text{i}}$ $\langle 1 0 0 \rangle$ DB}\\
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116 \includegraphics[width=\columnwidth]{si100.eps}
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118 \begin{minipage}[t]{0.32\columnwidth}
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119 \underline{Vacancy}\\
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120 \includegraphics[width=\columnwidth]{sivac.eps}
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122 \begin{minipage}[t]{0.32\columnwidth}
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123 \underline{C$_{\text{s}}$}\\
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124 \includegraphics[width=\columnwidth]{csub.eps}
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126 \begin{minipage}[t]{0.32\columnwidth}
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127 \underline{C$_{\text{i}}$ $\langle 1 0 0 \rangle$ DB}\\
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128 \includegraphics[width=\columnwidth]{c100.eps}
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130 \begin{minipage}[t]{0.32\columnwidth}
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131 \underline{C$_{\text{i}}$ $\langle 1 1 0 \rangle$ DB}\\
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132 \includegraphics[width=\columnwidth]{c110.eps}
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134 \begin{minipage}[t]{0.32\columnwidth}
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135 \underline{C$_{\text{i}}$ bond-centered}\\
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136 \includegraphics[width=\columnwidth]{cbc.eps}
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138 \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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139 \label{fig:sep_def}
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142 \begin{ruledtabular}
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143 \begin{tabular}{l c c c c c c c c c}
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144 & Si$_{\text{i}}$ $\langle1 1 0\rangle$ DB & Si$_{\text{i}}$ H & Si$_{\text{i}}$ T & Si $\langle 1 0 0\rangle$ DB & V & C$_{\text{s}}$ & C$_{\text{i}}$ $\langle1 0 0\rangle$ DB & C$_{\text{i}}$ $\langle1 1 0\rangle$ DB & C$_{\text{i}}$ BC \\
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146 Present study & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 & 1.95 & 3.72 & 4.16 & 4.66 \\
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147 \multicolumn{10}{c}{Other ab initio studies} \\
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148 Ref.\cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 & - & - & - & - \\
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149 Ref.\cite{leung99} & 3.31 & 3.31 & 3.43 & - & - & - & - & - & - \\
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150 Ref.\cite{dal_pino93,capaz94} & - & - & - & - & - & 1.89\cite{dal_pino93} & x & - & x+2.1\cite{capaz94}
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151 %Reference & 3.40\cite{al-mushadani03}, 3.31\cite{leung99} & 3.45\cite{al-mushadani03}, 3.31\cite{leung99} & 3.43\cite{leung99} & - & 3.53\cite{al-mushadani03} & 1.89\cite{dal_pino93} & x & - & x+2.1\cite{capaz94}
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154 \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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155 \label{table:sep_eof}
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157 Results obtained by the present study compare well with results from literature\cite{leung99,al-mushadani03,dal_pino93,capaz94}.
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158 Regarding intrinsic defects in Si, the $\langle 1 1 0 \rangle$ self-interstitial dumbbell (Si$_{\text{i}}$ $\langle 1 1 0 \rangle$ DB) is found to be the ground state configuration tersely followed by the hexagonal and tetrahedral configuration, which is the consensus view for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.
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159 In the case of a C impurity, next to the C$_{\text{s}}$ configuration, in which a C atom occupies an already vacant Si lattice site, the C$_{\text{i}}$ $\langle 1 0 0 \rangle$ interstitial dumbbell (C$_{\text{i}}$ $\langle 1 0 0 \rangle$ DB) constitutes the energetically most favorable configuration, in which the C and Si dumbbell atoms share a regular Si lattice site.
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160 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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161 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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163 Instead, Capaz et al.\cite{capaz94}, investigating migration pathways of the C$_{\text{i}}$ $\langle 1 0 0 \rangle$ DB, find this defect to be \unit[2.1]{eV} lower in energy than the bond-centered (BC) configuration, which is claimed to constitute a saddle point configuration in the migration path within the $(1 1 0)$ plane and, thus, interpreted as the barrier of migration for the respective path.
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164 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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165 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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166 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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167 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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168 Indeed Capaz et al. propose another path and find it to be the lowest in energy\cite{capaz94}, in which a C$_{\text{i}}$ $\langle 0 0 -1\rangle$ DB migrates into a C$_{\text{i}}$ $\langle 0 -1 0\rangle$ DB located at the next neighboured Si lattice site in $[1 1 -1]$ direction.
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169 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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170 A more detailed description can be found in a previous study\cite{zirkelbach10a}.
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172 Next to the C BC configuration the vacancy and Si$_{\text{i}}$ $\langle 1 0 0\rangle$ DB have to be treated by taking into account the spin of the electrons.
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173 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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174 No other configuration, within the ones that are mentioned, is affected.
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176 Concerning the mobility of the ground state Si$_{\text{i}}$, an activation energy shortly below \unit[0.7]{eV} was found for the migration of a Si$_{\text{i}}$ $\langle 0 1 -1\rangle$ into a $\langle 1 1 0\rangle$ DB configuration located at the next neighboured Si lattice site in $[1 1 -1]$ direction.
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177 % look for values in literature for neutraly charged Si_i diffusion
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179 \subsection{Pairs of C$_{\text{i}}$}
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181 C$_{\text{i}}$ pairs of the $\langle 1 0 0\rangle$-type have been considered in the first part.
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182 Table~\ref{table:dc_c-c} summarizes the binding energies obtained for configurations, in which an initial C$_{\text{i}}$ $\langle 0 0 -1\rangle$ DB located at position Si$_{\text{i}}$/C$_{\text{i}}$ is combined with a defect of the same type occupying various orientations at positions 1 to 5 (see Fig.~\ref{fig:combos}).
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184 \begin{ruledtabular}
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185 \begin{tabular}{l c c c c c c }
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186 & 1 & 2 & 3 & 4 & 5 & R \\
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188 $\langle 0 0 -1\rangle$ & -0.08 & -1.15 & -0.08 & 0.04 & -1.66 & -0.19\\
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189 $\langle 0 0 1\rangle$ & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
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190 $\langle 0 -1 0\rangle$ & -2.39 & -0.17 & -0.10 & -0.27 & -1.88 & -0.05\\
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191 $\langle 0 1 0\rangle$ & -2.25 & -1.90 & -2.25 & -0.12 & -1.38 & -0.06\\
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192 $\langle -1 0 0\rangle$ & -2.39 & -0.36 & -2.25 & -0.12 & -1.88 & -0.05\\
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193 $\langle 1 0 0\rangle$ & -2.25 & -2.16 & -0.10 & -0.27 & -1.38 & -0.06\\
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196 \caption{Binding energies of C$_{\text{i}}$ $\langle 1 0 0\rangle$-type defect pairs in eV. Equivalent configurations exhibit equal energies. The first column lists the orientation of the defect, which is combined with the initial C$_{\text{i}}$ $\langle 0 0 -1\rangle$ dumbbell. The position index of the second defect is given in the first row according to Fig.~\ref{fig:combos}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2} \langle3 2 3 \rangle$ relative to the initial defect position, which is the maximum realizable distance due to periodic boundary conditions.}
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197 \label{table:dc_c-c}
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199 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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200 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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201 Energetically favorable and unfavorable configurations can be explained by stress compensation and increase respectively, which is due to the resulting net strain of the respective configuration of the defect combination.
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202 Antiparallel orientations of the second defect ($\langle 0 0 1\rangle$) at positions located below the (001) plane with respect to the initial one (positions 1, 2 and 4) show the energetically most unfavorable configurations.
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203 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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205 Mattoni et al.\cite{mattoni2002} predict the ground state configuration for a $\langle 1 0 0\rangle$ or equivalently a $\langle 0 1 0\rangle$ defect created at position 1 with both defects basically maintaining the DB structure, resulting in a binding energy of \unit[-2.1]{eV}.
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206 % in mattoni db structures are basically amintained. there is further relaxation in our case and a lower binding energy
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207 In this work we found a further relaxation of this defect structure.
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208 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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209 Furthermore a more favorable configuration was found for the combination with a $\langle 0 -1 0\rangle$ and $\langle -1 0 0\rangle$ DB respectively, which is assumed to constitute the actual ground state configuration of two C$_{\text{i}}$ DBs in Si.
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210 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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212 Investigating migration barriers enables to predict the probability of formation of the thermodynamic ground state defect complex by thermally activated diffusion processes.
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213 High activation energies are necessary for the migration of low energy configurations, in which the C atom of the second DB is located in the vicinity of the initial DB.
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214 The transition of the configuration, in which the second DB is of the $\langle 0 1 0\rangle$ type at position 2 (\unit[-1.90]{eV}) into a $\langle 0 1 0\rangle$-type DB at position 1 (\unit[-2.39]{eV}) for instance, revealed a barrier height of more than \unit[4]{eV}.
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215 Low barriers do only exist from energetically less favorable configurations, e.g. the configuration of the $\langle -1 0 0\rangle$ DB located at position 2 (\unit[-0.36]{eV}).
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216 An activation energy of only \unit[?.?]{eV} is necessary for the transition into the ground state configuration.
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217 % strange mig from -190 -> -2.39 (barrier > 4 eV)
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218 % C-C migration -> idea:
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219 % mig from low energy confs has extremely high barrier!
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220 % low barrier only from energetically less/unfavorable confs (?)! <- prove!
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221 % => low probability of C-C clustering ?!?
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222 Since thermally activated C clustering is, thus, only possible by traversing energetically unfavored configurations, mass C clustering is not expected.
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224 % look for precapture mechnism (local minimum in energy curve)
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225 % also: plot energy all confs with respect to C-C distance
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226 % maybe a pathway exists traversing low energy confs ?!?
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228 The binding energies of the energetically most favorable configurations with the seocnd DB located along the $\langle 1 1 0\rangle$ direction and resulting C-C distances of the relaxed structures are summarized in Table~\ref{table:dc_110}.
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230 \begin{ruledtabular}
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231 \begin{tabular}{l c c c c c c }
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232 & 1 & 2 & 3 & 4 & 5 & 6 \\
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234 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
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235 C-C distance [nm] & 0.14 & 0.46 & 0.65 & 0.86 & 1.05 & 1.08
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238 \caption{Binding energies $E_{\text{b}}$ and C-C distance of energetically most favorable C$_{\text{i}}$ $\langle 1 0 0\rangle$-type defect pairs separated along bonds in $\langle 1 1 0\rangle$ direction.}
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239 \label{table:dc_110}
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241 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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243 \includegraphics[width=\columnwidth]{db_along_110_cc_n.ps}
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244 \caption{Minimum binding energy of dumbbell combinations separated along $\langle 1 1 0\rangle$ 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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247 The interaction is found to be proportional to the reciprocal cube of the C-C distance for extended separeations of the C$_{\text{i}}$ and saturates for the smallest possible separation, i.e. the ground state configuration.
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250 \subsection{C$_I$ next to C$_{\text{s}}$}
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252 % c_i and c_s, capaz98, mattoni2002 (restricted to 110 -110 bond chain)
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255 \subsection{C$_I$ next to V}
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257 \subsection{C$_{\text{s}}$ next to Si$_{\text{i}}$}
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259 Non-zeor temperature, entropy, spatial separation of these defects possible, indeed observed in ab initio MD run.
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261 %% Viele Bilder... da kann ich zunächst gar nicht soviel zu schreiben....
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263 \section{Discussion}
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264 Our calculations show that point defects which unavoidably are present after ion implantation significantly influence the mobility of implanted carbon
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265 in the silicon crystal.
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266 A large capture radius has been found for...
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267 Especially vacancies....
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271 In summary, we have shown that ...
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273 % ----------------------------------------------------
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274 \section*{Acknowledgment}
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275 We gratefully acknowledge financial support by the Bayerische Forschungsstiftung (DPA-61/05) and the Deutsche Forschungsgemeinschaft (DFG SCHM 1361/11).
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277 % --------------------------------- references -------------------
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279 \bibliography{../../bibdb/bibdb}{}
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280 \bibliographystyle{h-physrev3}
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300 %\end{thebibliography}
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