1 \chapter{Point defects in silicon}
3 Given the conversion mechnism of SiC in crystalline silicon introduced in section \ref{section:assumed_prec} the understanding of carbon and silicon interstitial point defects in c-Si is of great interest.
4 Both types of defects are examined in the following both by classical potential as well as density functional theory calculations.
6 In case of the classical potential calculations a simulation volume of nine silicon lattice constants in each direction is used.
7 Calculations are performed in an isothermal-isobaric NPT ensemble.
8 Coupling to the heat bath is achieved by the Berendsen thermostat with a time constant of 100 fs.
9 The temperature is set to zero Kelvin.
10 Pressure is controlled by a Berendsen barostat \cite{berendsen84} again using a time constant of 100 fs and a bulk modulus of 100 GPa for silicon.
11 To exclude surface effects periodic boundary conditions are applied.
13 Due to the restrictions in computer time three silicon lattice constants in each direction are considered sufficiently large enough for DFT calculations.
14 The ions are relaxed by a conjugate gradient method.
15 The cell volume and shape is allowed to change using the pressure control algorithm of Parrinello and Rahman \cite{parrinello81}.
16 Periodic boundary conditions in each direction are applied.
17 All point defects are calculated for the neutral charge state.
21 \includegraphics[width=9cm]{unit_cell_e.eps}
23 \caption[Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration.]{Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration. The black dots ({\color{black}$\bullet$}) correspond to the silicon atoms and the blue lines ({\color{blue}-}) indicate the covalent bonds of the perfect c-Si structure.}
24 \label{fig:defects:ins_pos}
27 The interstitial atom positions are displayed in figure \ref{fig:defects:ins_pos}.
28 In seperated simulation runs the silicon or carbon atom is inserted at the
30 \item tetrahedral, $\vec{r}=(0,0,0)$, ({\color{red}$\bullet$})
31 \item hexagonal, $\vec{r}=(-1/8,-1/8,1/8)$, ({\color{green}$\bullet$})
32 \item nearly \hkl<1 0 0> dumbbell, $\vec{r}=(-1/4,-1/4,-1/8)$, ({\color{yellow}$\bullet$})
33 \item nearly \hkl<1 1 0> dumbbell, $\vec{r}=(-1/8,-1/8,-1/4)$, ({\color{magenta}$\bullet$})
34 \item bond-centered, $\vec{r}=(-1/8,-1/8,-3/8)$, ({\color{cyan}$\bullet$})
36 interstitial position.
37 For the dumbbell configurations the nearest silicon atom is displaced by $(0,0,-1/8)$ and $(-1/8,-1/8,0)$ respectively of the unit cell length to avoid too high forces.
38 A vacancy or a substitutional atom is realized by removing one silicon atom and switching the type of one silicon atom respectively.
40 From an energetic point of view the free energy of formation $E_{\text{f}}$ is suitable for the characterization of defect structures.
41 For defect configurations consisting of a single atom species the formation energy is defined as
43 E_{\text{f}}=\left(E_{\text{coh}}^{\text{defect}}
44 -E_{\text{coh}}^{\text{defect-free}}\right)N
45 \label{eq:defects:ef1}
47 where $N$ and $E_{\text{coh}}^{\text{defect}}$ are the number of atoms and the cohesive energy per atom in the defect configuration and $E_{\text{coh}}^{\text{defect-free}}$ is the cohesive energy per atom of the defect-free structure.
48 The formation energy of defects consisting of two or more atom species is defined as
50 E_{\text{f}}=E-\sum_i N_i\mu_i
51 \label{eq:defects:ef2}
53 where $E$ is the free energy of the interstitial system and $N_i$ and $\mu_i$ are the amount of atoms and the chemical potential of species $i$.
54 The chemical potential is determined by the cohesive energy of the structure of the specific type in equilibrium at zero Kelvin.
55 For a defect configuration of a single atom species equation \eqref{eq:defects:ef2} is equivalent to equation \eqref{eq:defects:ef1}.
57 \section{Silicon self-interstitials}
59 Point defects in silicon have been extensively studied, both experimentally and theoretically \cite{fahey89,leung99}.
60 Quantum-mechanical total-energy calculations are an invalueable tool to investigate the energetic and structural properties of point defects since they are experimentally difficult to assess.
62 The formation energies of some of the silicon self-interstitial configurations are listed in table \ref{tab:defects:si_self} for both methods used in this work as well as results obtained by former studies \cite{leung99}.
65 \begin{tabular}{l c c c c c}
68 & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & V \\
70 Erhart/Albe MD & 3.40 & 4.48$^*$ & 5.42 & 4.39 & 3.13 \\
71 VASP & 3.77 & 3.42 & 4.41 & 3.39 & 3.63 \\
72 LDA \cite{leung99} & 3.43 & 3.31 & - & 3.31 & - \\
73 GGA \cite{leung99} & 4.07 & 3.80 & - & 3.84 & - \\
78 \caption[Formation energies of silicon self-interstitials in crystalline silicon determined by classical potential molecular dynamics and density functional calculations.]{Formation energies of silicon self-interstitials in crystalline silicon determined by classical potential molecular dynamics and density functional calculations. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal, B the bond-centered and V the vacancy interstitial configuration. The dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
79 \label{tab:defects:si_self}
81 The final configurations obtained after relaxation are presented in figure \ref{fig:defects:conf}.
87 %\begin{minipage}{5cm}
88 %\underline{\hkl<1 1 0> dumbbell}\\
89 %$E_{\text{f}}=3.39\text{ eV}$\\
90 %\includegraphics[width=3.0cm]{si_pd_vasp/110_2333.eps}
92 %\begin{minipage}{5cm}
93 %\underline{Hexagonal}\\
94 %$E_{\text{f}}=3.42\text{ eV}$\\
95 %\includegraphics[width=3.0cm]{si_pd_vasp/hex_2333.eps}
97 %\begin{minipage}{5cm}
98 %\underline{Tetrahedral}\\
99 %$E_{\text{f}}=3.77\text{ eV}$\\
100 %\includegraphics[width=3.0cm]{si_pd_vasp/tet_2333.eps}
101 %\end{minipage}\\[0.2cm]
102 %\begin{minipage}{5cm}
103 %\underline{\hkl<1 0 0> dumbbell}\\
104 %$E_{\text{f}}=4.41\text{ eV}$\\
105 %\includegraphics[width=3.0cm]{si_pd_vasp/100_2333.eps}
107 %\begin{minipage}{5cm}
108 %\underline{Vacancy}\\
109 %$E_{\text{f}}=3.63\text{ eV}$\\
110 %\includegraphics[width=3.0cm]{si_pd_vasp/vac_2333.eps}
112 %\begin{minipage}{5cm}
122 \begin{minipage}{5cm}
123 \underline{Tetrahedral}\\
124 $E_{\text{f}}=3.40\text{ eV}$\\
125 \includegraphics[width=4.0cm]{si_pd_albe/tet.eps}
127 \begin{minipage}{10cm}
128 \underline{Hexagonal}\\[0.1cm]
129 \begin{minipage}{4cm}
130 $E_{\text{f}}^*=4.48\text{ eV}$\\
131 \includegraphics[width=4.0cm]{si_pd_albe/hex_a.eps}
133 \begin{minipage}{0.8cm}
138 \begin{minipage}{4cm}
139 $E_{\text{f}}=3.96\text{ eV}$\\
140 \includegraphics[width=4.0cm]{si_pd_albe/hex.eps}
142 \end{minipage}\\[0.2cm]
143 \begin{minipage}{5cm}
144 \underline{\hkl<1 0 0> dumbbell}\\
145 $E_{\text{f}}=5.42\text{ eV}$\\
146 \includegraphics[width=4.0cm]{si_pd_albe/100.eps}
148 \begin{minipage}{5cm}
149 \underline{\hkl<1 1 0> dumbbell}\\
150 $E_{\text{f}}=4.39\text{ eV}$\\
151 \includegraphics[width=4.0cm]{si_pd_albe/110.eps}
153 \begin{minipage}{5cm}
154 \underline{Vacancy}\\
155 $E_{\text{f}}=3.13\text{ eV}$\\
156 \includegraphics[width=4.0cm]{si_pd_albe/vac.eps}
161 \caption[Relaxed silicon self-interstitial defect configurations obtained by classical potential calculations.]{Relaxed silicon self-interstitial defect configurations obtained by classical potential calculations. The silicon atoms and the bonds (only for the interstitial atom) are illustrated by yellow spheres and blue lines.}
162 \label{fig:defects:conf}
165 There are differences between the various results of the quantum-mechanical calculations but the consensus view is that the \hkl<1 1 0> dumbbell followed by the hexagonal and tetrahedral defect is the lowest in energy.
166 This is nicely reproduced by the DFT calculations performed in this work.
168 It has turned out to be very difficult to capture the results of quantum-mechanical calculations in analytical potential models.
169 Among the established analytical potentials only the EDIP \cite{bazant97,justo98} and Stillinger-Weber \cite{stillinger85} potential reproduce the correct order in energy of the defects.
170 However, these potenitals show shortcomings concerning the description of other physical properties and are unable to describe the C-C and C-Si interaction.
171 In fact the Erhart/Albe potential calculations favor the tetrahedral defect configuration.
172 The hexagonal configuration is not stable opposed to results of the authors of the potential \cite{albe_sic_pot}.
173 In the first two pico seconds while kinetic energy is decoupled from the system the Si interstitial seems to condense at the hexagonal site.
174 The formation energy of 4.48 eV is determined by this low kinetic energy configuration shortly before the relaxation process starts.
175 The Si interstitial atom then begins to slowly move towards an energetically more favorable position very close to the tetrahedral one but slightly displaced along the three coordinate axes.
176 The formation energy of 3.96 eV for this type of interstitial is equal to the result for the hexagonal one in the original work \cite{albe_sic_pot}.
177 Obviously the authors did not carefully check the relaxed results assuming a hexagonal configuration.
178 In figure \ref{fig:defects:kin_si_hex} the relaxation process is shown on the basis of the kinetic energy plot.
181 \includegraphics[width=10cm]{e_kin_si_hex.ps}
183 \caption{Kinetic energy plot of the relaxation process of the hexagonal silicon self-interstitial defect simulation using the Erhart/Albe classical potential.}
184 \label{fig:defects:kin_si_hex}
186 To exclude failures in the implementation of the potential or the MD code itself the hexagonal defect structure was double-checked with the PARCAS MD code \cite{parcas_md}.
187 The same type of interstitial arises using random insertions.
188 In addition, variations exist in which the displacement is only along two \hkl<1 0 0> axes ($E_\text{f}=3.8\text{ eV}$) or along a single \hkl<1 0 0> axes ($E_\text{f}=3.6\text{ eV}$) successively approximating the tetdrahedral configuration and formation energy.
189 The existence of these local minima located near the tetrahedral configuration seems to be an artifact of the analytical potential without physical authenticity revealing basic problems of analytical potential models for describing defect structures.
190 However, the energy barrier is small.
193 \includegraphics[width=12cm]{nhex_tet.ps}
195 \caption{Migration barrier of the tetrahedral Si self-interstitial slightly displaced along all three coordinate axes into the exact tetrahedral configuration using classical potential calculations.}
196 \label{fig:defects:nhex_tet_mig}
198 This is exemplified in figure \ref{fig:defects:nhex_tet_mig}, which shows the change in potential energy during the migration of the interstitial displaced along all three coordinate axes into the tetrahedral configuration.
199 The technique used to obtain the migration data is explained in a later section (\ref{subsection:100mig}).
200 The barrier is less than 0.2 eV.
201 Hence these artifacts should have a negligent influence in finite temperature simulations.
203 The bond-centered configuration is unstable and the \hkl<1 0 0> dumbbell interstitial is the most unfavorable configuration for both, the Erhart/Albe and VASP calculations.
205 In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, that is if the distance of two atoms is within the cutoff region $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
206 For the tetrahedral and the slightly displaced configurations four bonds to the atoms located in the center of the planes of the unit cell exist in addition to the four tetrahedral bonds.
207 The length of these bonds are, however, close to the cutoff range and thus are weak interactions not constituting actual chemical bonds.
208 The same applies to the bonds between the interstitial and the upper two atoms in the \hkl<1 1 0> dumbbell configuration.
210 A more detailed description of the chemical bonding is achieved by quantum-mechanical calculations by investigating the accumulation of negative charge between the nuclei.
211 {\color{red}Todo: Plot the electron density for these types of defect to derive conclusions of existing bonds?}
213 \section{Carbon related point defects}
215 Carbon is a common and technologically important impurity in silicon.
216 Concentrations as high as $10^{18}\text{ cm}^{-3}$ occur in Czochralski-grown silicon samples.
217 It is well established that carbon and other isovalent impurities prefer to dissolve substitutionally in silicon.
218 However, radiation damage can generate carbon interstitials \cite{watkins76} which have enough mobility at room temeprature to migrate and form defect complexes.
220 Formation energies of the most common carbon point defects in crystalline silicon are summarized in table \ref{tab:defects:c_ints} and the relaxed configurations obtained by classical potential calculations visualized in figure \ref{fig:defects:c_conf}.
221 The type of reservoir of the carbon impurity to determine the formation energy of the defect was chosen to be SiC.
222 This is consistent with the methods used in the articles \cite{tersoff90,dal_pino93}, which the results are compared to in the following.
223 Hence, the chemical potential of silicon and carbon is determined by the cohesive energy of silicon and silicon carbide.
226 \begin{tabular}{l c c c c c c}
229 & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
231 Erhart/Albe MD & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
232 %VASP & unstable & unstable & 3.15 & 3.60 & 1.39 & 4.10 \\
233 VASP & unstable & unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
234 Tersoff \cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
235 ab initio & - & - & x & - & 1.89 \cite{dal_pino93} & x+2.1 \cite{capaz94} \\
240 \caption[Formation energies of carbon point defects in crystalline silicon determined by classical potential molecular dynamics and density functional calculations.]{Formation energies of carbon point defects in crystalline silicon determined by classical potential molecular dynamics and density functional calculations. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal, B the bond-centered and S the substitutional interstitial configuration. The dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
241 \label{tab:defects:c_ints}
246 \begin{minipage}{4cm}
247 \underline{Hexagonal}\\
248 $E_{\text{f}}^*=9.05\text{ eV}$\\
249 \includegraphics[width=4.0cm]{c_pd_albe/hex.eps}
251 \begin{minipage}{0.8cm}
256 \begin{minipage}{4cm}
257 \underline{\hkl<1 0 0>}\\
258 $E_{\text{f}}=3.88\text{ eV}$\\
259 \includegraphics[width=4.0cm]{c_pd_albe/100.eps}
261 \begin{minipage}{0.5cm}
264 \begin{minipage}{5cm}
265 \underline{Tetrahedral}\\
266 $E_{\text{f}}=6.09\text{ eV}$\\
267 \includegraphics[width=4.0cm]{c_pd_albe/tet.eps}
268 \end{minipage}\\[0.2cm]
269 \begin{minipage}{4cm}
270 \underline{Bond-centered}\\
271 $E_{\text{f}}^*=5.59\text{ eV}$\\
272 \includegraphics[width=4.0cm]{c_pd_albe/bc.eps}
274 \begin{minipage}{0.8cm}
279 \begin{minipage}{4cm}
280 \underline{\hkl<1 1 0> dumbbell}\\
281 $E_{\text{f}}=5.18\text{ eV}$\\
282 \includegraphics[width=4.0cm]{c_pd_albe/110.eps}
284 \begin{minipage}{0.5cm}
287 \begin{minipage}{5cm}
288 \underline{Substitutional}\\
289 $E_{\text{f}}=0.75\text{ eV}$\\
290 \includegraphics[width=4.0cm]{c_pd_albe/sub.eps}
294 \caption[Relaxed carbon point defect configurations obtained by classical potential calculations.]{Relaxed carbon point defect configurations obtained by classical potential calculations. The silicon/carbon atoms and the bonds (only for the interstitial atom) are illustrated by yellow/grey spheres and blue lines.}
295 \label{fig:defects:c_conf}
298 Substitutional carbon in silicon is found to be the lowest configuration in energy for all potential models.
299 An experiemntal value of the formation energy of substitutional carbon was determined by a fit to solubility data yielding a concentration of $3.5 \times 10^{24} \exp{(-2.3\text{ eV}/k_{\text{B}}T)} \text{ cm}^{-3}$ \cite{bean71}.
300 However, there is no particular reason for treating the prefactor as a free parameter in the fit to the experimental data.
301 It is simply given by the atomic density of pure silicon, which is $5\times 10^{22}\text{ cm}^{-3}$.
302 Tersoff \cite{tersoff90} and Dal Pino et al. \cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from 1.6 to 1.89 eV an excellent agreement with the experimental solubility data within the entire temeprature range of the experiment is obtained.
303 This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal Pino et al. is in good agreement with the results of the quantum-mechanical calculations performed in this work.
304 Unfortunately the Erhart/Albe potential undervalues the formation energy roughly by a factor of two.
306 Except for Tersoff's tedrahedral configuration results the \hkl<1 0 0> dumbbell is the energetically most favorable interstital configuration.
307 The low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the abrupt cutoff set to 2.5 \AA{} (see ref. 11 and 13 in \cite{tersoff90}) and the real formation energy is, thus, supposed to be located between 3 and 10 eV.
308 Keeping these considerations in mind, the \hkl<1 0 0> dumbbell is the most favorable interstitial configuration for all interaction models.
309 In addition to the theoretical results compared to in table \ref{tab:defects:c_ints} there is experimental evidence of the existence of this configuration \cite{watkins76}.
310 It is frequently generated in the classical potential simulation runs in which carbon is inserted at random positions in the c-Si matrix.
311 In quantum-mechanical simulations the unstable tetrahedral and hexagonal configurations undergo a relaxation into the \hkl<1 0 0> dumbbell configuration.
312 Thus, this configuration is of great importance and discussed in more detail in section \ref{subsection:100db}.
314 The highest energy is observed for the hexagonal interstitial configuration using classical potentials.
315 Quantum-mechanical calculations reveal this configuration to be unstable, which is also reproduced by the Erhart/Albe potential.
316 In both cases a relaxation towards the \hkl<1 0 0> dumbbell configuration is observed.
317 In fact the stability of the hexagonal interstitial could not be reproduced in simulations performed in this work using the unmodifed Tersoff potential parameters.
318 Unfortunately, apart from the modified parameters, no more conditions specifying the relaxation process are given in Tersoff's study on carbon point defects in silicon \cite{tersoff90}.
320 The tetrahedral is the second most unfavorable interstitial configuration using classical potentials and keeping in mind the abrupt cutoff effect in the case of the Tersoff potential as discussed earlier.
321 Again, quantum-mechanical results reveal this configuration unstable.
322 The fact that the tetrahedral and hexagonal configurations are the two most unstable configurations in classical potential calculations and, thus, are less likely to arise in MD simulations acts in concert with the fact that these configurations are found to be unstable in the more accurate quantum-mechanical description.
324 Just as for the Si self-interstitial a carbon \hkl<1 1 0> dumbbell configuration exists.
325 For the Erhart/Albe potential the formation energy is situated in the same order as found by quantum-mechanical results.
326 Similar structures arise in both types of simulations with the silicon and carbon atom sharing a silicon lattice site aligned along \hkl<1 1 0> where the carbon atom is localized slightly closer to the next nearest silicon atom located in the opposite direction to the site-sharing silicon atom even forming a bond to the next but one silicon atom in this direction.
328 The bond-centered configuration is unstable for the Erhart/Albe potential.
329 The system moves into the \hkl<1 1 0> interstitial configuration.
330 This, like in the hexagonal case, is also true for the unmodified Tersoff potential and the given relaxation conditions.
331 Quantum-mechanical results of this configuration are discussed in more detail in section \ref{subsection:bc}.
332 In another ab inito study Capaz et al. \cite{capaz94} determined this configuration as an intermediate saddle point structure of a possible migration path, which is 2.1 eV higher than the \hkl<1 0 0> dumbbell configuration.
333 In calculations performed in this work the bond-centered configuration in fact is a real local minimum and an energy barrier is needed to reach this configuration starting from the \hkl<1 0 0> dumbbell configuration, which is discussed in section \ref{subsection:100mig}.
335 \subsection[\hkl<1 0 0> dumbbell interstitial configuration]{\boldmath\hkl<1 0 0> dumbbell interstitial configuration}
336 \label{subsection:100db}
338 As the \hkl<1 0 0> dumbbell interstitial is the lowest configuration in energy it is the most probable hence important interstitial configuration of carbon in silicon.
339 It was first identified by infra-red (IR) spectroscopy \cite{bean70} and later on by electron paramagnetic resonance (EPR) spectroscopy \cite{watkins76}.
341 Figure \ref{fig:defects:100db_cmp} schematically shows the \hkl<1 0 0> dumbbell structure and table \ref{tab:defects:100db_cmp} lists the details of the atomic displacements, distances and bond angles obtained by analytical potential and quantum-mechanical calculations.
342 For comparison, the obtained structures for both methods visualized out of the atomic position data are presented in figure \ref{fig:defects:100db_vis_cmp}.
345 \includegraphics[width=12cm]{100-c-si-db_cmp.eps}
347 \caption[Sketch of the \hkl<1 0 0> dumbbell structure.]{Sketch of the \hkl<1 0 0> dumbbell structure. Atomic displacements, distances and bond angles are listed in table \ref{tab:defects:100db_cmp}.}
348 \label{fig:defects:100db_cmp}
354 \begin{tabular}{l c c c c c c c c c}
357 & & & & \multicolumn{3}{c}{Atom 2} & \multicolumn{3}{c}{Atom 3} \\
358 & $a$ & $b$ & $|a|+|b|$ & $\Delta x$ & $\Delta y$ & $\Delta z$ & $\Delta x$ & $\Delta y$ & $\Delta z$ \\
360 Erhart/Albe & 0.084 & -0.091 & 0.175 & -0.015 & -0.015 & -0.031 & -0.014 & 0.014 & 0.020 \\
361 VASP & 0.109 & -0.065 & 0.174 & -0.011 & -0.011 & -0.024 & -0.014 & 0.014 & 0.025 \\
364 \end{tabular}\\[0.5cm]
368 \begin{tabular}{l c c c c c c c c r}
371 & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ & $a_{\text{Si}}^{\text{equi}}$\\
373 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 & 0.543 \\
374 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 & 0.548 \\
377 \end{tabular}\\[0.5cm]
381 \begin{tabular}{l c c c c }
384 & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
386 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
387 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
390 \end{tabular}\\[0.5cm]
392 \caption[Atomic displacements, distances and bond angles of the \hkl<1 0 0> dumbbell structure obtained by the Erhart/Albe potential and VASP calculations.]{Atomic displacements, distances and bond angles of the \hkl<1 0 0> dumbbell structure obtained by the Erhart/Albe potential and VASP calculations. The displacements and distances are given in nm and the angles are given in degrees. Displacements, distances and angles are schematically displayed in figure \ref{fig:defects:100db_cmp}. In addition, the equilibrium lattice constant for crystalline silicon is listed.}
393 \label{tab:defects:100db_cmp}
397 \begin{minipage}{6cm}
399 \underline{Erhart/Albe}
400 \includegraphics[width=5cm]{c_pd_albe/100_cmp.eps}
403 \begin{minipage}{6cm}
406 \includegraphics[width=5cm]{c_pd_vasp/100_cmp.eps}
410 \caption{Comparison of the visualized \hkl<1 0 0> dumbbel structures obtained by Erhart/Albe potential and VASP calculations.}
411 \label{fig:defects:100db_vis_cmp}
415 \includegraphics[height=10cm]{c_pd_vasp/eden.eps}
416 \includegraphics[height=12cm]{c_pd_vasp/100_2333_ksl.ps}
418 \caption[Charge density isosurface and Kohn-Sham levels of the C \hkl<1 0 0> dumbbell structure obtained by VASP calculations.]{Charge density isosurface and Kohn-Sham levels of the C \hkl<1 0 0> dumbbell structure obtained by VASP calculations. Yellow and grey spheres correspond to silicon and carbon atoms. The blue surface is the charge density isosurface. In the energy level diagram red and green lines and dots mark occupied and unoccupied states.}
419 \label{img:defects:charge_den_and_ksl}
421 The silicon atom numbered '1' and the C atom compose the dumbbell structure.
422 They share the lattice site which is indicated by the dashed red circle and which they are displaced from by length $a$ and $b$ respectively.
423 The atoms no longer have four tetrahedral bonds to the silicon atoms located on the alternating opposite edges of the cube.
424 Instead, each of the dumbbell atoms forms threefold coordinated bonds, which are located in a plane.
425 One bond is formed to the other dumbbell atom.
426 The other two bonds are bonds to the two silicon edge atoms located in the opposite direction of the dumbbell atom.
427 The distance of the two dumbbell atoms is almost the same for both types of calculations.
428 However, in the case of the VASP calculation, the dumbbell structure is pushed upwards compared to the Erhart/Albe results.
429 This is easily identified by comparing the values for $a$ and $b$ and the two structures in figure \ref{fig:defects:100db_vis_cmp}.
430 Thus, the angles of bonds of the silicon dumbbell atom ($\theta_1$ and $\theta_2$) are closer to $120^{\circ}$ signifying the predominance of $sp^2$ hybridization.
431 On the other hand, the carbon atom forms an almost collinear bond ($\theta_3$) with the two silicon edge atoms implying the predominance of $sp$ bonding.
432 This is supported by the image of the charge density isosurface in figure \ref{img:defects:charge_den_and_ksl}.
433 The two lower Si atoms are $sp^3$ hybridised and form $\sigma$ bonds to the silicon dumbbell atom.
434 The same is true for the upper two silicon atoms and the C dumbbell atom.
435 In addition the dumbbell atoms form $\pi$ bonds.
436 However, due to the increased electronegativity of the carbon atom the electron density is attracted by and thus localized around the carbon atom.
437 In the same figure the Kohn-Sham levels are shown.
438 There is no magnetization density.
439 An acceptor level arises at approximately $E_v+0.35\text{ eV}$ while a band gap of about 0.75 eV can be estimated from the Kohn-Sham level diagram for plain silicon.
441 \subsection{Bond-centered interstitial configuration}
442 \label{subsection:bc}
446 \begin{minipage}{8cm}
447 \includegraphics[width=8cm]{c_pd_vasp/bc_2333.eps}\\
450 \includegraphics[width=8cm]{c_100_mig_vasp/im_spin_diff.eps}
452 \begin{minipage}{7cm}
453 \includegraphics[width=7cm]{c_pd_vasp/bc_2333_ksl.ps}
456 \caption[Structure, charge density isosurface and Kohn-Sham level diagram of the bond-centered interstitial configuration.]{Structure, charge density isosurface and Kohn-Sham level diagram of the bond-centered interstitial configuration. Gray, green and blue surfaces mark the charge density of spin up, spin down and the resulting spin up electrons in the charge density isosurface, in which the carbon atom is represented by a red sphere. In the energy level diagram red and green lines mark occupied and unoccupied states.}
457 \label{img:defects:bc_conf}
459 In the bond-centerd insterstitial configuration the interstitial atom is located inbetween two next neighboured silicon atoms forming linear bonds.
460 In former studies this configuration is found to be an intermediate saddle point configuration determining the migration barrier of one possibe migration path of a \hkl<1 0 0> dumbbel configuration into an equivalent one \cite{capaz94}.
461 This is in agreement with results of the Erhart/Albe potential simulations which reveal this configuration to be unstable relaxing into the \hkl<1 1 0> configuration.
462 However, this fact could not be reproduced by spin polarized VASP calculations performed in this work.
463 Present results suggest this configuration to be a real local minimum.
464 In fact, an additional barrier has to be passed to reach this configuration starting from the \hkl<1 0 0> interstitital configuration, which is investigated in section \ref{subsection:100mig}.
465 After slightly displacing the carbon atom along the \hkl<1 0 0> (equivalent to a displacement along \hkl<0 1 0>), \hkl<0 0 1>, \hkl<0 0 -1> and \hkl<1 -1 0> direction the resulting structures relax back into the bond-centered configuration.
466 As we will see in later migration simulations the same would happen to structures where the carbon atom is displaced along the migration direction, which approximately is the \hkl<1 1 0> direction.
467 These relaxations indicate that the bond-cenetered configuration is a real local minimum instead of an assumed saddle point configuration.
468 Figure \ref{img:defects:bc_conf} shows the structure, the charge density isosurface and the Kohn-Sham levels of the bond-centered configuration.
469 The linear bonds of the carbon atom to the two silicon atoms indicate the $sp$ hybridization of the carbon atom.
470 Two electrons participate to the linear $\sigma$ bonds with the silicon neighbours.
471 The other two electrons constitute the $2p^2$ orbitals resulting in a net magnetization.
472 This is supported by the charge density isosurface and the Kohn-Sham levels in figure \ref{img:defects:bc_conf}.
473 The blue torus, reinforcing the assumption of the p orbital, illustrates the resulting spin up electron density.
474 In addition, the energy level diagram shows a net amount of two spin up electrons.
476 \section{Migration of the carbon interstitials}
477 \label{subsection:100mig}
479 In the following the problem of interstitial carbon migration in silicon is considered.
480 Since the carbon \hkl<1 0 0> dumbbell interstitial is the most probable hence most important configuration the migration simulations focus on this defect.
484 \begin{minipage}{15cm}
485 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
486 \begin{minipage}{4.5cm}
487 \includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
489 \begin{minipage}{0.5cm}
492 \begin{minipage}{4.5cm}
493 \includegraphics[width=4.5cm]{c_pd_vasp/bc_2333.eps}
495 \begin{minipage}{0.5cm}
498 \begin{minipage}{4.5cm}
499 \includegraphics[width=4.5cm]{c_pd_vasp/100_next_2333.eps}
502 \begin{minipage}{15cm}
503 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
504 \begin{minipage}{4.5cm}
505 \includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
507 \begin{minipage}{0.5cm}
510 \begin{minipage}{4.5cm}
511 \includegraphics[width=4.5cm]{c_pd_vasp/00-1-0-10_2333.eps}
513 \begin{minipage}{0.5cm}
516 \begin{minipage}{4.5cm}
517 \includegraphics[width=4.5cm]{c_pd_vasp/0-10_2333.eps}
520 \begin{minipage}{15cm}
521 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
522 \begin{minipage}{4.5cm}
523 \includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
525 \begin{minipage}{0.5cm}
528 \begin{minipage}{4.5cm}
529 \includegraphics[width=4.5cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
531 \begin{minipage}{0.5cm}
534 \begin{minipage}{4.5cm}
535 \includegraphics[width=4.5cm]{c_pd_vasp/0-10_ip_2333.eps}
539 \caption{Migration pathways of the carbon \hkl<1 0 0> interstitial dumbbell in silicon.}
540 \label{img:defects:c_mig_path}
542 Three different migration paths are accounted in this work, which are shown in figure \ref{img:defects:c_mig_path}.
543 The first migration investigated is a transition of a \hkl<0 0 -1> into a \hkl<0 0 1> dumbbell interstitial configuration.
544 During this migration the carbon atom is changing its silicon dumbbell partner.
545 The new partner is the one located at $\frac{a}{4}\hkl<1 1 -1>$ relative to the initial one.
546 Two of the three bonds to the next neighboured silicon atoms are preserved while the breaking of the third bond and the accompanying formation of a new bond is observed.
547 The carbon atom resides in the \hkl(1 1 0) plane.
548 This transition involves an intermediate bond-centerd configuration.
549 Results discussed in \ref{subsection:bc} indicate, that the bond-ceneterd configuration is a real local minimum.
550 Thus, the \hkl<0 0 -1> to \hkl<0 0 1> migration can be thought of a two-step mechanism in which the intermediate bond-cenetered configuration constitutes a metastable configuration.
551 Due to symmetry it is enough to consider the transition from the bond-centered to the \hkl<1 0 0> configuration or vice versa.
552 In the second path, the carbon atom is changing its silicon partner atom as in path one.
553 However, the trajectory of the carbon atom is no longer proceeding in the \hkl(1 1 0) plane.
554 The orientation of the new dumbbell configuration is transformed from \hkl<0 0 -1> to \hkl<0 -1 0>.
555 Again one bond is broken while another one is formed.
556 As a last migration path, the defect is only changing its orientation.
557 Thus, it is not responsible for long-range migration.
558 The silicon dumbbell partner remains the same.
559 The bond to the face-centered silicon atom at the bottom of the unit cell breaks and a new one is formed to the face-centered atom at the forefront of the unit cell.
563 \begin{minipage}{6cm}
564 \underline{Original}\\
565 \includegraphics[width=6cm]{crt_orig.eps}
567 \begin{minipage}{1cm}
570 \begin{minipage}{6cm}
571 \underline{Modified}\\
572 \includegraphics[width=6cm]{crt_mod.eps}
575 \caption{Schematic of the constrained relaxation technique (CRT) (left) and of the modified version (right) used to obtain migration pathways and corresponding activation energies.}
576 \label{fig:defects:crt}
578 Since the starting and final structure, which are both local minima of the potential energy surface, are known, the aim is to find the minimum energy path from one local minimum to the other one.
579 One method to find a minimum energy path is to move the diffusing atom stepwise from the starting to the final position and only allow relaxation in the plane perpendicular to the direction of the vector connecting its starting and final position.
580 This is called the constrained relaxation technique (CRT), which is schematically displayed in the left part of figure \ref{fig:defects:crt}.
581 No constraints are applied to the remaining atoms in order to allow relaxation of the surrounding lattice.
582 To prevent the remaining lattice to migrate according to the displacement of the defect an atom far away from the defect region is fixed in all three coordinate directions.
583 However, it turned out, that this method tremendously failed applying it to the present migration pathways and structures.
584 Abrupt changes in structure and free energy occured among relaxed structures of two successive displacement steps.
585 For some structures even the expected final configurations were never obtained.
586 Thus, the method mentioned above was adjusted adding further constraints in order to obtain smooth transitions, either in energy as well as structure is concerned.
587 In this new method all atoms are stepwise displaced towards their final positions.
588 Relaxation of each individual atom is only allowed in the plane perpendicular to the last individual displacement vector, as displayed in the right part of figure \ref{fig:defects:crt}.
589 The modifications used to add this feature to the VASP code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
590 Due to these constraints obtained activation energies can effectively be higher.
591 {\color{red}Todo: To refine the migration barrier one has to find the saddle point structure and recalculate the free energy of this configuration with a reduced set of constraints.}
593 \subsection{Migration barriers obtained by quantum-mechanical calculations}
595 In the following migration barriers are investigated using quantum-mechanical calculations.
596 The amount of simulated atoms is the same as for the investigation of the point defect structures.
597 Due to the time necessary for computing only ten displacement steps are used.
601 \includegraphics[width=13cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[1.5cm]
602 \begin{picture}(0,0)(150,0)
603 \includegraphics[width=2.5cm]{vasp_mig/00-1.eps}
605 \begin{picture}(0,0)(-10,0)
606 \includegraphics[width=2.5cm]{vasp_mig/bc_00-1_sp.eps}
608 \begin{picture}(0,0)(-120,0)
609 \includegraphics[width=2.5cm]{vasp_mig/bc.eps}
611 \begin{picture}(0,0)(25,20)
612 \includegraphics[width=2.5cm]{110_arrow.eps}
614 \begin{picture}(0,0)(200,0)
615 \includegraphics[height=2.2cm]{001_arrow.eps}
618 \caption[Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to bond-centered (right) transition.]{Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to bond-centered (right) transition. Bonds of the carbon atoms are illustrated by blue lines.}
619 \label{fig:defects:00-1_001_mig}
621 In figure \ref{fig:defects:00-1_001_mig} results of the \hkl<0 0 -1> to \hkl<0 0 1> migration fully described by the migration of the \hkl<0 0 -1> dumbbell to the bond-ceneterd configuration is displayed.
622 To reach the bond-centered configuration, which is 0.94 eV higher in energy than the \hkl<0 0 -1> dumbbell configuration, an energy barrier of approximately 1.2 eV, given by the saddle point structure at a displacement of 60 \%, has to be passed.
623 This amount of energy is needed to break the bond of the carbon atom to the silicon atom at the bottom left.
624 In a second process 0.25 eV of energy are needed for the system to revert into a \hkl<1 0 0> configuration.
628 \includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[1.6cm]
629 \begin{picture}(0,0)(140,0)
630 \includegraphics[width=2.5cm]{vasp_mig/00-1_a.eps}
632 \begin{picture}(0,0)(20,0)
633 \includegraphics[width=2.5cm]{vasp_mig/00-1_0-10_sp.eps}
635 \begin{picture}(0,0)(-120,0)
636 \includegraphics[width=2.5cm]{vasp_mig/0-10.eps}
638 \begin{picture}(0,0)(25,20)
639 \includegraphics[width=2.5cm]{100_arrow.eps}
641 \begin{picture}(0,0)(200,0)
642 \includegraphics[height=2.2cm]{001_arrow.eps}
645 \caption[Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to the \hkl<0 -1 0> dumbbell (right) transition.]{Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to the \hkl<0 -1 0> dumbbell (right) transition. Bonds of the carbon atoms are illustrated by blue lines.}
646 \label{fig:defects:00-1_0-10_mig}
648 Figure \ref{fig:defects:00-1_0-10_mig} shows the migration barrier and structures of the \hkl<0 0 -1> to \hkl<0 -1 0> dumbbell transition.
649 The resulting migration barrier of approximately 0.9 eV is very close to the experimentally obtained values of 0.73 \cite{song90} and 0.87 eV \cite{tipping87}.
653 \includegraphics[width=13cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[1.8cm]
654 \begin{picture}(0,0)(140,0)
655 \includegraphics[width=2.2cm]{vasp_mig/00-1_b.eps}
657 \begin{picture}(0,0)(20,0)
658 \includegraphics[width=2.2cm]{vasp_mig/00-1_ip0-10_sp.eps}
660 \begin{picture}(0,0)(-120,0)
661 \includegraphics[width=2.2cm]{vasp_mig/0-10_b.eps}
663 \begin{picture}(0,0)(25,20)
664 \includegraphics[width=2.5cm]{100_arrow.eps}
666 \begin{picture}(0,0)(200,0)
667 \includegraphics[height=2.2cm]{001_arrow.eps}
670 \caption[Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to the \hkl<0 -1 0> dumbbell (right) transition in place.]{Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to the \hkl<0 -1 0> dumbbell (right) transition in place. Bonds of the carbon atoms are illustrated by blue lines.}
671 \label{fig:defects:00-1_0-10_ip_mig}
673 The third migration path in which the dumbbell is changing its orientation is shown in figure \ref{fig:defects:00-1_0-10_ip_mig}.
674 An energy barrier of roughly 1.2 eV is observed.
675 Experimentally measured activation energies for reorientation range from 0.77 eV to 0.88 eV \cite{watkins76,song90}.
676 Thus, this pathway is more likely to be composed of two consecutive steps of the second path.
678 Since the activation energy of the first and last migration path is much greater than the experimental value, the second path is identified to be responsible as a migration path for the most likely carbon interstitial in silicon explaining both, annealing and reorientation experiments.
679 The activation energy of roughly 0.9 eV nicely compares to experimental values reinforcing the correct identification of the C-Si dumbbell diffusion mechanism.
680 The theoretical description performed in this work is improved compared to a former study \cite{capaz94}, which underestimates the experimental value by 35 \%.
681 In addition the bond-ceneterd configuration, for which spin polarized calculations are necessary, is found to be a real local minimum instead of a saddle point configuration.
685 \includegraphics[width=13cm]{vasp_mig/110_mig_vasp.ps}
686 %\begin{picture}(0,0)(140,0)
687 %\includegraphics[width=2.2cm]{vasp_mig/00-1_b.eps}
689 %\begin{picture}(0,0)(20,0)
690 %\includegraphics[width=2.2cm]{vasp_mig/00-1_ip0-10_sp.eps}
692 %\begin{picture}(0,0)(-120,0)
693 %\includegraphics[width=2.2cm]{vasp_mig/0-10_b.eps}
696 \caption{Migration barriers of the \hkl<1 1 0> dumbbell to bond-centered (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) C-Si dumbbell transition.}
697 \label{fig:defects:110_mig_vasp}
699 Further migration pathways in particular those occupying other defect configurations than the \hkl<1 0 0>-type either as a transition state or a final or starting configuration are totally conceivable.
700 This is investigated in the following in order to find possible migration pathways that have an activation energy lower than the ones found up to now.
701 The next energetically favorable defect configuration is the \hkl<1 1 0> C-Si dumbbell interstitial.
702 Figure \ref{fig:defects:110_mig_vasp} shows the migration barrier of the \hkl<1 1 0> C-Si dumbbell to the bond-centered, \hkl<0 0 -1> and \hkl<0 -1 0> (in place) transition.
703 Indeed less than 0.7 eV are necessary to turn a \hkl<0 -1 0>- to a \hkl<1 1 0>-type C-Si dumbbell interstitial.
704 This transition is carried out in place, that is the Si dumbbell pair is not changed and both, the Si and C atom share the initial lattice site.
705 Thus, this transition does not contribute to long-range diffusion.
706 Once the C atom resides in the \hkl<1 1 0> interstitial configuration it can migrate into the bond-centered configuration by employing approximately 0.95 eV of activation energy, which is only slightly higher than the activation energy needed for the \hkl<0 0 -1> to \hkl<0 -1 0> pathway shown in figure \ref{fig:defects:00-1_0-10_mig}.
707 As already known from the migration of the \hkl<0 0 -1> to the bond-centered configuration as discussed in figure \ref{fig:defects:00-1_001_mig} another 0.25 eV are needed to turn back from the bond-centered to a \hkl<1 0 0>-type interstitial.
708 However, due to the fact that this migration consists of three single transitions with the second one having an activation energy slightly higher than observed for the direct transition it is considered very unlikely to occur.
709 The migration barrier of the \hkl<1 1 0> to \hkl<0 0 -1> transition, in which the C atom is changing its Si partner and, thus, moving to the neighboured lattice site is approximately 1.35 eV.
710 During this transition the C atom is escaping the \hkl(1 1 0) plane approaching the final configuration on a curved path.
711 This barrier is much higher than the ones found previously, which again make this transition very unlikely to occur.
712 For this reason the assumption that C diffusion and reorientation is achieved by transitions of the type presented in figure \ref{fig:defects:00-1_0-10_mig} is reinforced.
714 As mentioned earlier the procedure to obtain the migration barriers differs from the usually applied procedure in two ways.
715 Firstly constraints to move along the displacement direction are applied on all atoms instead of solely constraining the diffusing atom.
716 Secondly the constrainted directions are not kept constant to the initial displacement direction.
717 Instead they are updated for every displacement step.
718 These modifications to the usual procedure are applied to avoid abrupt changes in structure and free energy on the one hand and to make sure the expected final configuration is reached on the other hand.
719 Due to applying updated constraints on all atoms the obtained migration barriers and pathes might be overestimated and misguided.
720 To reinforce the applicability of the employed technique the obtained activation energies and migration pathes for the \hkl<0 0 -1> to \hkl<0 -1 0> transition are compared to two further migration calculations, which do not update the constrainted direction and which only apply updated constraints on three selected atoms, that is the diffusing C atom and the Si dumbbell pair in the initial and final configuration.
721 Results are presented in figure \ref{fig:defects:00-1_0-10_cmp}.
724 \includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_cmp.ps}
726 \caption[Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.]{Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.}
727 \label{fig:defects:00-1_0-10_cmp}
729 The method without updating the constraints but still applying them to all atoms shows a delayed crossing of the saddle point.
730 This is understandable since the update results in a more aggressive advance towards the final configuration.
731 In any case the barrier obtained is slightly higher, which means that it does not constitute an energetically more favorable pathway.
732 The method in which the constraints are only applied to the diffusing C atom and two Si atoms, ... {\color{red}Todo: does not work!} ...
734 \subsection{Migration barriers obtained by classical potential calculations}
735 \label{subsection:defects:mig_classical}
737 The same method for obtaining migration barriers and the same suggested pathways are applied to calculations employing the classical Erhart/Albe potential.
738 Since the evaluation of the classical potential and force is less computationally intensive higher amounts of steps can be used.
739 The time constant $\tau$ for the Berendsen thermostat is set to 1 fs in order to have direct velocity scaling and with the temperature set to zero Kelvin perform a steepest descent minimazation to drive the system into a local minimum.
740 However, in some cases a time constant of 100 fs resuls in lower barriers and, thus, is shown whenever appropriate.
744 \includegraphics[width=13cm]{bc_00-1.ps}\\[5.6cm]
745 \begin{pspicture}(0,0)(0,0)
746 \psframe[linecolor=red,fillstyle=none](-7,2.7)(7.2,6)
748 \begin{picture}(0,0)(140,-100)
749 \includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_00.eps}
751 \begin{picture}(0,0)(10,-100)
752 \includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_01.eps}
754 \begin{picture}(0,0)(-120,-100)
755 \includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_02.eps}
757 \begin{picture}(0,0)(25,-80)
758 \includegraphics[width=2.5cm]{110_arrow.eps}
760 \begin{picture}(0,0)(215,-100)
761 \includegraphics[height=2.2cm]{001_arrow.eps}
763 \begin{pspicture}(0,0)(0,0)
764 \psframe[linecolor=blue,fillstyle=none](-7,-0.5)(7.2,2.8)
766 \begin{picture}(0,0)(160,-10)
767 \includegraphics[width=2.2cm]{albe_mig/bc_00-1_01.eps}
769 \begin{picture}(0,0)(100,-10)
770 \includegraphics[width=2.2cm]{albe_mig/bc_00-1_02.eps}
772 \begin{picture}(0,0)(10,-10)
773 \includegraphics[width=2.2cm]{albe_mig/bc_00-1_03.eps}
775 \begin{picture}(0,0)(-120,-10)
776 \includegraphics[width=2.2cm]{albe_mig/bc_00-1_04.eps}
778 \begin{picture}(0,0)(25,10)
779 \includegraphics[width=2.5cm]{100_arrow.eps}
781 \begin{picture}(0,0)(215,-10)
782 \includegraphics[height=2.2cm]{010_arrow.eps}
785 \caption{Migration barrier and structures of the bond-centered to \hkl<0 0 -1> dumbbell transition using the classical Erhart/Albe potential.}
786 \label{fig:defects:cp_bc_00-1_mig}
787 % red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20 -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.75 -1.25 -0.25 -L -0.25 -0.25 -0.25 -r 0.6 -B 0.1
788 % blue: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20_tr100/ -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.0 -0.25 1.0 -L 0.0 -0.25 -0.25 -r 0.6 -B 0.1
790 Figure \ref{fig:defects:cp_bc_00-1_mig} shows the migration barrier and corresponding structures of the bond-centered to \hkl<0 0 -1> dumbbell transition.
791 Since the bond-centered configuration is unstable relaxing into the \hkl<1 1 0> C-Si dumbbell interstitial configuration within this potential the low kinetic energy state is used as a starting configuration.
792 Depending on the time constant activation energies of 2.4 eV and 2.2 eV respectively are obtained.
793 The migration path obtained by simulations with a time constant of 1 fs remains in the \hkl(1 1 0) plane.
794 Using 100 fs as a time constant the C atom breaks out of the \hkl(1 1 0) plane already at the beginning of the migration accompanied by a reduction in energy.
795 The energy barrier of this path is 0.2 eV lower in energy than the direct migration within the \hkl(1 1 0) plane.
796 However, the investigated pathways cover an activation energy approximately twice as high as the one obtained by quantum-mechanical calculations.
797 For the entire transition of the \hkl<0 0 -1> into the \hkl<0 0 1> configuration by passing the bond-centered configuration an additional activation energy of 0.5 eV is necessary to escape from the bond-centered and reach the \hkl<0 0 1> configuration.
801 \includegraphics[width=13cm]{00-1_0-10.ps}\\[2.4cm]
802 \begin{pspicture}(0,0)(0,0)
803 \psframe[linecolor=red,fillstyle=none](-6,-0.5)(7.2,2.8)
805 \begin{picture}(0,0)(130,-10)
806 \includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_00.eps}
808 \begin{picture}(0,0)(0,-10)
809 \includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_min.eps}
811 \begin{picture}(0,0)(-120,-10)
812 \includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_03.eps}
814 \begin{picture}(0,0)(25,10)
815 \includegraphics[width=2.5cm]{100_arrow.eps}
817 \begin{picture}(0,0)(185,-10)
818 \includegraphics[height=2.2cm]{001_arrow.eps}
821 \caption{Migration barrier and structures of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition using the classical Erhart/Albe potential.}
822 % red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_00-1_0-10_s20 -nll -0.56 -0.56 -0.8 -fur 0.3 0.2 0 -c -0.125 -1.7 0.7 -L -0.125 -0.25 -0.25 -r 0.6 -B 0.1
823 \label{fig:defects:cp_00-1_0-10_mig}
827 \includegraphics[width=13cm]{00-1_ip0-10.ps}
829 \caption{Migration barrier of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition in place using the classical Erhart/Albe potential.}
830 \label{fig:defects:cp_00-1_ip0-10_mig}
832 Figure \ref{fig:defects:cp_00-1_0-10_mig} and \ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition, with a transition of the C atom to the neighboured lattice site in the first case and a reorientation within the same lattice site in the latter case.
833 Both pathways look similar.
834 A local minimum exists inbetween two peaks of the graph.
835 The corresponding configuration, which is illustrated for the migration simulation with a time constant of 1 fs, looks similar to the \hkl<1 1 0> configuration.
836 Indeed, this configuration is obtained by relaxation simulations without constraints of configurations near the minimum.
837 Activation energies of roughly 2.8 eV and 2.7 eV respectively are needed for migration.
839 The \hkl<1 1 0> configuration seems to play a decisive role in all migration pathways.
840 In the first migration path it is the configuration resulting from further relaxation of the rather unstable bond-centered configuration, which is fixed to be a transition point in the migration calculations.
841 The last two pathways show configurations almost identical to the \hkl<1 1 0> configuration, which constitute a local minimum within the pathway.
842 Thus, migration pathways with the \hkl<1 1 0> C-Si dumbbell interstitial configuration as a starting or final configuration are further investigated.
845 \includegraphics[width=13cm]{110_mig.ps}
847 \caption[Migration barriers of the \hkl<1 1 0> dumbbell to bond-centered (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) C-Si dumbbell transition.]{Migration barriers of the \hkl<1 1 0> dumbbell to bond-centered (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) C-Si dumbbell transition. Solid lines show results for a time constant of 1 fs and dashed lines correspond to simulations employing a time constant of 100 fs.}
848 \label{fig:defects:110_mig}
850 Figure \ref{fig:defects:110_mig} shows migration barriers of the C-Si \hkl<1 1 0> dumbbell to \hkl<0 0 -1>, \hkl<0 -1 0> (in place) and bond-centered configuration.
851 As expected there is no maximum for the transition into the bond-centered configuration.
852 As mentioned earlier the bond-centered configuration itself constitutes a saddle point configuration relaxing into the energetically more favorable \hkl<1 1 0> configuration.
853 An activation energy of 2.2 eV is necessary to reorientate the \hkl<0 0 -1> dumbbell configuration into the \hkl<1 1 0> configuration, which is 1.3 eV higher in energy.
854 Residing in this state another 0.9 eV is enough to make the C atom form a \hkl<0 0 -1> dumbbell configuration with the Si atom of the neighboured lattice site.
855 In contrast to quantum-mechanical calculations, in which the direct transition is the energetically most favorable transition and the transition composed of the intermmediate migration steps is very unlikely to occur the just presented pathway is much more supposable in classical potential simulations, since the energetically most favorable transition found so far is also composed of two migration steps with activation energies of 2.2 eV and 0.5 eV, for which the intermediate state is the bond-centered configuration, which is unstable.
856 Thus the just proposed migration path involving the \hkl<1 1 0> interstitial configuration becomes even more probable than path 1 involving the unstable bond-centered configuration.
858 Although classical potential simulations reproduce the order in energy of the \hkl<1 0 0> and \hkl<1 1 0> C-Si dumbbell interstitial configurations as obtained by more accurate quantum-mechanical calculations the obtained migration pathways and resulting activation energies differ to a great extent.
859 On the one hand the most favorable pathways differ.
860 On the other hand the activation energies obtained by classical potential simulations are tremendously overestimated by a factor of almost 2.4.
861 Thus, atomic diffusion is wrongly described in the classical potential approach.
862 The probability of already rare diffusion events is further decreased for this reason.
863 Since agglomeration of C and diffusion of Si self-interstitials are an important part of the proposed SiC precipitation mechanism a problem arises, which is formulated and discussed in more detail in section \ref{subsection:md:limit}.
865 \section{Combination of point defects}
867 The structural and energetic properties of combinations of point defects are examined in the following.
868 Investigations are restricted to quantum-mechanical calculations for two reasons.
869 First of all, as mentioned in the last section, they are far more accurate.
870 Secondly, the restrictions in size and simulation time for this type of calculation due to limited computational resources, necessitate to map the complex precipitation mechanism to a more compact and simplified modelling.
871 The investigations of defect combinations approached in the following are still feasible within the available computational power and allow to draw conclusions on some important ongoing mechanisms during SiC precipitation.
873 \subsection[Combinations with a C-Si \hkl<1 0 0>-type interstitial]{\boldmath Combinations with a C-Si \hkl<1 0 0>-type interstitial}
874 \label{subsection:defects:c-si_comb}
876 This section focuses on combinations of the \hkl<0 0 -1> dumbbell interstitial with a second defect.
877 The second defect is either another \hkl<1 0 0>-type interstitial occupying different orientations, a vacany or a substitutional carbon atom.
878 Several distances of the two defects are examined.
882 \begin{minipage}{7.5cm}
883 \includegraphics[width=7cm]{comb_pos.eps}
884 % ./visualize_contcar -w 640 -h 480 -d results/.../CONTCAR -nll -0.20 -0.20 -0.6 -fur 1.2 1.2 0.6 -c 0.5 -1.5 0.3 -L 0.5 0 0 -r 0.6 -m 3.0 0.0 0.0 0.0 3.0 0.0 0.0 0.0 3.0 -A -1 2.465
886 \begin{minipage}{6.0cm}
887 \underline{Positions given in $a_{\text{Si}}$}\\[0.3cm]
888 Initial interstitial I: $\frac{1}{4}\hkl<1 1 1>$\\
889 Relative silicon neighbour positions:
891 \item $\frac{1}{4}\hkl<1 1 -1>$, $\frac{1}{4}\hkl<-1 -1 -1>$
892 \item $\frac{1}{2}\hkl<1 0 1>$, $\frac{1}{2}\hkl<0 1 -1>$,\\[0.2cm]
893 $\frac{1}{2}\hkl<0 -1 -1>$, $\frac{1}{2}\hkl<-1 0 -1>$
894 \item $\frac{1}{4}\hkl<1 -1 1>$, $\frac{1}{4}\hkl<-1 1 1>$
895 \item $\frac{1}{4}\hkl<-1 1 -3>$, $\frac{1}{4}\hkl<1 -1 -3>$
896 \item $\frac{1}{2}\hkl<-1 -1 0>$, $\frac{1}{2}\hkl<1 1 0>$
899 \begin{picture}(0,0)(190,20)
900 \includegraphics[width=2.3cm]{100_arrow.eps}
902 \begin{picture}(0,0)(220,0)
903 \includegraphics[height=2.2cm]{001_arrow.eps}
906 \caption[\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighboured silicon atoms used for the second defect.]{\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighboured silicon atoms used for the second defect. Two possibilities exist for red numbered atoms and four possibilities exist for blue numbered atoms.}
907 \label{fig:defects:pos_of_comb}
911 \begin{tabular}{l c c c c c c}
914 & 1 & 2 & 3 & 4 & 5 & R\\
916 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
917 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
918 \hkl<0 -1 0> & {\color{orange}-2.39} & -0.17 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
919 \hkl<0 1 0> & {\color{cyan}-2.25} & -1.90 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
920 \hkl<-1 0 0> & {\color{orange}-2.39} & -0.36 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
921 \hkl<1 0 0> & {\color{cyan}-2.25} & -2.16 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
923 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
924 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
929 \caption[Energetic results of defect combinations.]{Energetic results of defect combinations. The given energies in eV are defined by equation \eqref{eq:defects:e_of_comb}. Equivalent configurations are marked by identical colors. The first column lists the types of the second defect combined with the initial \hkl<0 0 -1> dumbbell interstitial. The position index of the second defect is given in the first row according to figure \ref{fig:defects:pos_of_comb}. R is the position located at $\frac{a_{\text{Si}}}{2}\hkl<3 2 3>$ relative to the initial defect, which is the maximum realizable distance due to periodic boundary conditions.}
930 \label{tab:defects:e_of_comb}
932 Figure \ref{fig:defects:pos_of_comb} shows the initial \hkl<0 0 -1> dumbbell interstitial defect and the positions of next neighboured silicon atoms used for the second defect.
933 Table \ref{tab:defects:e_of_comb} summarizes energetic results obtained after relaxation of the defect combinations.
934 The energy of interest $E_{\text{b}}$ is defined to be
937 E_{\text{f}}^{\text{defect combination}}-
938 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
939 E_{\text{f}}^{\text{2nd defect}}
940 \label{eq:defects:e_of_comb}
942 with $E_{\text{f}}^{\text{defect combination}}$ being the formation energy of the defect combination, $E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}$ being the formation energy of the C \hkl<0 0 -1> dumbbell interstitial defect and $E_{\text{f}}^{\text{2nd defect}}$ being the formation energy of the second defect.
943 For defects far away from each other the formation energy of the defect combination should approximately become the sum of the formation energies of the individual defects without an interaction resulting in $E_{\text{b}}=0$.
944 Thus, $E_{\text{b}}$ can be best thought of a binding energy, which is required to bring the defects to infinite separation.
945 In fact, a \hkl<0 0 -1> dumbbell interstitial created at position R with a distance of $\frac{a_{\text{Si}}}{2}\hkl<3 2 3>$ ($\approx 12.8$ \AA) from the initial one results in an energy as low as -0.19 eV.
946 There is still a low interaction which is due to the equal orientation of the defects.
947 By changing the orientation of the second dumbbell interstitial to the \hkl<0 -1 0>-type the interaction is even more reduced resulting in an energy of $E_{\text{b}}=-0.05\text{ eV}$ for a distance, which is the maximum that can be realized due to periodic boundary conditions.
948 The energies obtained in the R column of table \ref{eq:defects:e_of_comb} are used as a reference to identify, whether less distanced defects of the same type are favorable or unfavorable compared to the far-off located defect.
949 Configurations wih energies greater than zero or the reference value are energetically unfavorable and expose a repulsive interaction.
950 These configurations are unlikely to arise or to persist for non-zero temperatures.
951 Energies below zero and below the reference value indicate configurations favored compared to configurations in which these point defects are separated far away from each other.
953 Investigating the first part of table \ref{tab:defects:e_of_comb}, namely the combinations with another \hkl<1 0 0>-type interstitial, most of the combinations result in energies below zero.
954 Surprisingly the most favorable configurations are the ones with the second defect created at the very next silicon neighbour and a change in orientation compared to the initial one.
955 This leads to the conclusion that an agglomeration of C-Si dumbbell interstitials as proposed by the precipitation model introduced in section \ref{section:assumed_prec} is indeed an energetically favored configuration of the system.
956 The reason for nearby interstitials being favored compared to isolated ones is most probably the reduction of strain energy enabled by combination in contrast to the strain energy created by two individual defects.
959 \begin{minipage}[t]{7cm}
960 a) \underline{$E_{\text{b}}=-2.25\text{ eV}$}
962 \includegraphics[width=6cm]{00-1dc/2-25.eps}
965 \begin{minipage}[t]{7cm}
966 b) \underline{$E_{\text{b}}=-2.39\text{ eV}$}
968 \includegraphics[width=6cm]{00-1dc/2-39.eps}
972 \caption{Relaxed structures of defect complexes obtained by creating a) \hkl<1 0 0> and b) \hkl<0 -1 0> dumbbels at position 1.}
973 \label{fig:defects:comb_db_01}
975 Figure \ref{fig:defects:comb_db_01} shows the structure of these two configurations.
976 The displayed configurations are realized by creating a \hkl<1 0 0> (a)) and \hkl<0 -1 0> (b)) dumbbell at position 1.
977 Structure \ref{fig:defects:comb_db_01} b) is the energetically most favorable configuration.
978 After relaxation the initial configuration is still evident.
979 As expected by the initialization conditions the two carbon atoms form a bond.
980 This bond has a length of 1.38 \AA{} close to the nex neighbour distance in diamond or graphite, which is approximately 1.54 \AA.
981 The minimum of binding energy observed for this configuration suggests prefered C clustering as a competing mechnism to the C-Si dumbbell interstitial agglomeration inevitable for the SiC precipitation.
982 {\color{red}Todo: Activation energies to obtain separated C confs FAILED (again?) - could be added in the combined defect migration chapter and mentioned here, too!}
983 However, for the second most favorable configuration, presented in figure \ref{fig:defects:comb_db_01} a), the amount of possibilities for this configuration is twice as high.
984 In this configuration the initial Si (I) and C (I) dumbbell atoms are displaced along \hkl<1 0 0> and \hkl<-1 0 0> in such a way that the Si atom is forming tetrahedral bonds with two silicon and two carbon atoms.
985 The carbon and silicon atom constituting the second defect are as well displaced in such a way, that the carbon atom forms tetrahedral bonds with four silicon neighbours, a configuration expected in silicon carbide.
986 The two carbon atoms spaced by 2.70 \AA{} do not form a bond but anyhow reside in a shorter distance as expected in silicon carbide.
987 The Si atom numbered 2 is pushed towards the carbon atom, which results in the breaking of the bond to atom 4.
988 The breaking of the $\sigma$ bond is indeed confirmed by investigating the charge density isosurface of this configuration.
989 {\color{red}Todo: Is this conf really benificial for SiC prec?}
993 \begin{minipage}[t]{5cm}
994 a) \underline{$E_{\text{b}}=-2.16\text{ eV}$}
996 \includegraphics[width=4.8cm]{00-1dc/2-16.eps}
999 \begin{minipage}[t]{5cm}
1000 b) \underline{$E_{\text{b}}=-1.90\text{ eV}$}
1002 \includegraphics[width=4.8cm]{00-1dc/1-90.eps}
1005 \begin{minipage}[t]{5cm}
1006 c) \underline{$E_{\text{b}}=-2.05\text{ eV}$}
1008 \includegraphics[width=4.8cm]{00-1dc/2-05.eps}
1012 \caption{Relaxed structures of defect complexes obtained by creating a a) \hkl<1 0 0> and b) \hkl<0 1 0> dumbbell at position 2 and a c) \hkl<0 0 1> dumbbel at position 3.}
1013 \label{fig:defects:comb_db_02}
1015 Figure \ref{fig:defects:comb_db_02} shows the next three most energetically favorable configurations.
1016 The relaxed configuration obtained by creating a second \hkl<1 0 0> dumbbell at position 2 is shown in figure \ref{fig:defects:comb_db_02} a).
1017 A binding energy of -2.16 eV is observed.
1018 After relaxation the second dumbbell is aligned along \hkl<1 1 0>.
1019 The bond of the silicon atoms 1 and 2 does not persist.
1020 Instead the silicon atom forms a bond with the initial carbon interstitial and the second carbon atom forms a bond with silicon atom 1 forming four bonds in total.
1021 The carbon atoms are spaced by 3.14 \AA, which is very close to the expected C-C next neighbour distance of 3.08 \AA{} in silicon carbide.
1022 Figure \ref{fig:defects:comb_db_02} c) displays the results of another \hkl<0 0 1> dumbbell inserted at position 3.
1023 The binding energy is -2.05 eV.
1024 Both dumbbells are tilted along the same direction remaining parallely aligned and the second dumbbell is pushed downwards in such a way, that the four dumbbell atoms form a rhomboid.
1025 Both carbon atoms form tetrahedral bonds to four silicon atoms.
1026 However, silicon atom 1 and 3, which are bound to the second carbon dumbbell interstitial are also bound to the initial carbon atom.
1027 These four atoms of the rhomboid reside in a plane and, thus, do not match the situation in silicon carbide.
1028 The carbon atoms have a distance of 2.75 \AA.
1029 In figure \ref{fig:defects:comb_db_02} b) a second \hkl<0 1 0> dumbbell is constructed at position 2.
1030 An energy of -1.90 eV is observed.
1031 The initial dumbbell and especially the carbon atom is pushed towards the silicon atom of the second dumbbell forming an additional fourth bond.
1032 Silicon atom number 1 is pulled towards the carbon atoms of the dumbbells accompanied by the disappearance of its bond to silicon number 5 as well as the bond of silicon number 5 to its next neighboured silicon atom in \hkl<1 1 -1> direction.
1033 The carbon atom of the second dumbbell forms threefold coordinated bonds to its silicon neighbours.
1034 A distance of 2.80 \AA{} is observed for the two carbon atoms.
1035 Again, the two carbon atoms and its two interconnecting silicon atoms form a rhomboid.
1036 C-C distances of 2.70 to 2.80 \AA{} seem to be characteristic for such configurations, in which the carbon atoms and the two interconnecting silicon atoms reside in a plane.
1038 Configurations obtained by adding a second dumbbell interstitial at position 4 are characterized by minimal changes from their initial creation condition during relaxation.
1039 There is a low interaction of the dumbbells, which seem to exist independent of each other.
1040 This, on the one hand, becomes evident by investigating the final structure, in which both of the dumbbells essentially retain the structure expected for a single dumbbell and on the other hand is supported by the observed binding energies which vary only slightly around zero.
1041 This low interaction is due to the larger distance and a missing direct connection by bonds along a crystallographic direction.
1042 Both carbon and silicon atoms of the dumbbells form threefold coordinated bonds to their next neighbours.
1043 The energetically most unfavorable configuration ($E_{\text{b}}=0.26\text{ eV}$) is obtained for the \hkl<0 0 1> interstitial oppositely orientated to the initial one.
1044 A dumbbell taking the same orientation as the initial one is less unfavorble ($E_{\text{b}}=0.04\text{ eV}$).
1045 Both configurations are unfavorable compared to far-off isolated dumbbells.
1046 Nonparallel orientations, that is the \hkl<0 1 0>, \hkl<0 -1 0> and its equivalents, result in binding energies of -0.12 eV and -0.27 eV, thus, constituting energetically favorable configurations.
1047 The reduction of strain energy is higher in the second case where the carbon atom of the second dumbbell is placed in the direction pointing away from the initial carbon atom.
1049 \begin{figure}[t!h!]
1051 \begin{minipage}[t]{7cm}
1052 a) \underline{$E_{\text{b}}=-1.53\text{ eV}$}
1054 \includegraphics[width=6.0cm]{00-1dc/1-53.eps}
1057 \begin{minipage}[t]{7cm}
1058 b) \underline{$E_{\text{b}}=-1.66\text{ eV}$}
1060 \includegraphics[width=6.0cm]{00-1dc/1-66.eps}
1062 \end{minipage}\\[0.2cm]
1063 \begin{minipage}[t]{7cm}
1064 c) \underline{$E_{\text{b}}=-1.88\text{ eV}$}
1066 \includegraphics[width=6.0cm]{00-1dc/1-88.eps}
1069 \begin{minipage}[t]{7cm}
1070 d) \underline{$E_{\text{b}}=-1.38\text{ eV}$}
1072 \includegraphics[width=6.0cm]{00-1dc/1-38.eps}
1076 \caption{Relaxed structures of defect complexes obtained by creating a a) \hkl<0 0 1>, a b) \hkl<0 0 -1>, a c) \hkl<0 -1 0> and a d) \hkl<1 0 0> dumbbell at position 5.}
1077 \label{fig:defects:comb_db_03}
1079 Energetically beneficial configurations of defect complexes are observed for second interstititals of all orientations placed at position 5, a position two bonds away from the initial interstitial along the \hkl<1 1 0> direction.
1080 Relaxed structures of these complexes are displayed in figure \ref{fig:defects:comb_db_03}.
1081 Figure \ref{fig:defects:comb_db_03} a) and b) show the relaxed structures of \hkl<0 0 1> and \hkl<0 0 -1> dumbbells.
1082 The upper dumbbell atoms are pushed towards each other forming fourfold coordinated bonds.
1083 While the displacements of the silicon atoms in case b) are symmetric to the \hkl(1 1 0) plane, in case a) the silicon atom of the initial dumbbel is pushed a little further in the direction of the carbon atom of the second dumbbell than the carbon atom is pushed towards the silicon atom.
1084 The bottom atoms of the dumbbells remain in threefold coordination.
1085 The symmetric configuration is energetically more favorable ($E_{\text{b}}=-1.66\text{ eV}$) since the displacements of the atoms is less than in the antiparallel case ($E_{\text{b}}=-1.53\text{ eV}$).
1086 In figure \ref{fig:defects:comb_db_03} c) and d) the nonparallel orientations, namely the \hkl<0 -1 0> and \hkl<1 0 0> dumbbells are shown.
1087 Binding energies of -1.88 eV and -1.38 eV are obtained for the relaxed structures.
1088 In both cases the silicon atom of the initial interstitial is pulled towards the near by atom of the second dumbbell so that both atoms form fourfold coordinated bonds to their next neighbours.
1089 In case c) it is the carbon and in case d) the silicon atom of the second interstitial which forms the additional bond with the silicon atom of the initial interstitial.
1090 The atom of the second dumbbell, the carbon atom of the initial dumbbell and the two interconnecting silicon atoms again reside in a plane.
1091 A typical C-C distance of 2.79 \AA{} is, thus, observed for case c).
1092 The far-off atom of the second dumbbell resides in threefold coordination.
1094 Assuming that it is possible for the system to minimize free energy by an in place reorientation of the dumbbell at any position the minimum energy orientation of dumbbells along the \hkl<1 1 0> direction and the resulting C-C distance is shown in table \ref{tab:defects:comb_db110}.
1097 \begin{tabular}{l c c c c c c}
1100 & 1 & 2 & 3 & 4 & 5 & 6\\
1102 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1103 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1104 Type & \hkl<-1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0>, \hkl<0 -1 0>\\
1109 \caption{Binding energy and type of the minimum energy configuration of an additional dumbbell with respect to the separation distance in bonds along the \hkl<1 1 0> direction and the C-C distance.}
1110 \label{tab:defects:comb_db110}
1112 \begin{figure}[t!h!]
1114 \includegraphics[width=12.5cm]{db_along_110.ps}\\
1115 \includegraphics[width=12.5cm]{db_along_110_cc.ps}
1117 \caption{Minimum binding energy of dumbbell combinations with respect to the separation distance in bonds along \hkl<1 1 0> and C-C distance.}
1118 \label{fig:defects:comb_db110}
1120 Figure \ref{fig:defects:comb_db110} shows the corresponding plot of the data including a cubic spline interplation and a suitable fitting curve.
1121 The funtion found most suitable for curve fitting is $f(x)=a/x^3$ comprising the single fit parameter $a$.
1122 Thus, far-off located dumbbells show an interaction proportional to the reciprocal cube of the distance and the amount of bonds along \hkl<1 1 0> respectively.
1123 This behavior is no longer valid for the immediate vicinity revealed by the saturating binding energy of a second dumbbell at position 1, which is ignored in the fitting procedure.
1125 \begin{figure}[t!h!]
1127 \begin{minipage}[t]{5cm}
1128 a) \underline{Pos: 1, $E_{\text{b}}=0.26\text{ eV}$}
1130 \includegraphics[width=4.8cm]{00-1dc/0-26.eps}
1133 \begin{minipage}[t]{5cm}
1134 b) \underline{Pos: 3, $E_{\text{b}}=-0.93\text{ eV}$}
1136 \includegraphics[width=4.8cm]{00-1dc/0-93.eps}
1139 \begin{minipage}[t]{5cm}
1140 c) \underline{Pos: 5, $E_{\text{b}}=0.49\text{ eV}$}
1142 \includegraphics[width=4.8cm]{00-1dc/0-49.eps}
1146 \caption{Relaxed structures of defect complexes obtained by creating a carbon substitutional at position 1 (a)), 3 (b)) and 5 (c)).}
1147 \label{fig:defects:comb_db_04}
1149 \begin{figure}[t!h!]
1151 \begin{minipage}[t]{7cm}
1152 a) \underline{Pos: 2, $E_{\text{b}}=-0.51\text{ eV}$}
1154 \includegraphics[width=6cm]{00-1dc/0-51.eps}
1157 \begin{minipage}[t]{7cm}
1158 b) \underline{Pos: 4, $E_{\text{b}}=-0.15\text{ eV}$}
1160 \includegraphics[width=6cm]{00-1dc/0-15.eps}
1164 \caption{Relaxed structures of defect complexes obtained by creating a carbon substitutional at position 2 (a)) and 4 (b)).}
1165 \label{fig:defects:comb_db_05}
1167 The second part of table \ref{tab:defects:e_of_comb} lists the energetic results of substitutional carbon and vacancy combinations with the initial \hkl<0 0 -1> dumbbell.
1168 Figures \ref{fig:defects:comb_db_04} and \ref{fig:defects:comb_db_05} show relaxed structures of substitutional carbon in combination with the \hkl<0 0 -1> dumbbell for several positions.
1169 In figure \ref{fig:defects:comb_db_04} positions 1 (a)), 3 (b)) and 5 (c)) are displayed.
1170 A substituted carbon atom at position 5 results in an energetically extremely unfavorable configuration.
1171 Both carbon atoms, the substitutional and the dumbbell atom, pull silicon atom number 1 towards their own location regarding the \hkl<1 1 0> direction.
1172 Due to this a large amount of tensile strain energy is needed, which explains the high positive value of 0.49 eV.
1173 The lowest binding energy is observed for a substitutional carbon atom inserted at position 3.
1174 The substitutional carbon atom is located above the dumbbell substituting a silicon atom which would usually be bound to and displaced along \hkl<0 0 1> and \hkl<1 1 0> by the silicon dumbbell atom.
1175 In contrast to the previous configuration strain compensation occurs resulting in a binding energy as low as -0.93 eV.
1176 Substitutional carbon at position 2 and 4, visualized in figure \ref{fig:defects:comb_db_05}, is located below the initial dumbbell.
1177 Silicon atom number 1, which is bound to the interstitial carbon atom is displaced along \hkl<0 0 -1> and \hkl<-1 -1 0>.
1178 In case a) only the first displacement is compensated by the substitutional carbon atom.
1179 This results in a somewhat higher binding energy of -0.51 eV.
1180 The binding energy gets even higher in case b) ($E_{\text{b}}=-0.15\text{ eV}$), in which the substitutional carbon is located further away from the initial dumbbell.
1181 In both cases, silicon atom number 1 is displaced in such a way, that the bond to silicon atom number 5 vanishes.
1182 In case of \ref{fig:defects:comb_db_04} a) the carbon atoms form a bond with a distance of 1.5 \AA, which is close to the C-C distance expected in diamond or graphit.
1183 Both carbon atoms are highly attracted by each other resulting in large displacements and high strain energy in the surrounding.
1184 A binding energy of 0.26 eV is observed.
1185 Substitutional carbon at positions 2, 3 and 4 are the energetically most favorable configurations and constitute promising starting points for SiC precipitation.
1186 On the one hand, C-C distances around 3.1 \AA{} exist for substitution positions 2 and 3, which are close to the C-C distance expected in silicon carbide.
1187 On the other hand stretched silicon carbide is obtained by the transition of the silicon dumbbell atom into a silicon self-interstitial located somewhere in the silicon host matrix and the transition of the carbon dumbbell atom into another substitutional atom occupying the dumbbell lattice site.
1189 \begin{figure}[t!h!]
1191 \begin{minipage}[t]{7cm}
1192 a) \underline{Pos: 2, $E_{\text{b}}=-0.59\text{ eV}$}
1194 \includegraphics[width=6.0cm]{00-1dc/0-59.eps}
1197 \begin{minipage}[t]{7cm}
1198 b) \underline{Pos: 3, $E_{\text{b}}=-3.14\text{ eV}$}
1200 \includegraphics[width=6.0cm]{00-1dc/3-14.eps}
1202 \end{minipage}\\[0.2cm]
1203 \begin{minipage}[t]{7cm}
1204 c) \underline{Pos: 4, $E_{\text{b}}=-0.54\text{ eV}$}
1206 \includegraphics[width=6.0cm]{00-1dc/0-54.eps}
1209 \begin{minipage}[t]{7cm}
1210 d) \underline{Pos: 5, $E_{\text{b}}=-0.50\text{ eV}$}
1212 \includegraphics[width=6.0cm]{00-1dc/0-50.eps}
1216 \caption{Relaxed structures of defect complexes obtained by creating vacancies at positions 2 (a)), 3 (b)), 4 (c)) and 5 (d)).}
1217 \label{fig:defects:comb_db_06}
1219 Figure \ref{fig:defects:comb_db_06} displays relaxed structures of vacancies in combination with the \hkl<0 0 -1> dumbbell interstital.
1220 The creation of a vacancy at position 1 results in a configuration of substitutional carbon on a silicon lattice site and no other remaining defects.
1221 The carbon dumbbell atom moves to position 1 where the vacancy is created and the silicon dumbbell atom recaptures the dumbbell lattice site.
1222 With a binding energy of -5.39 eV, this is the energetically most favorable configuration observed.
1223 A great amount of strain energy is reduced by removing the silicon atom at position 3, which is illustrated in figure \ref{fig:defects:comb_db_06} b).
1224 The dumbbell structure shifts towards the position of the vacancy which replaces the silicon atom usually bound to and at the same time strained by the silicon dumbbell atom.
1225 Due to the displacement into the \hkl<1 -1 0> direction the bond of the dumbbell silicon atom to the silicon atom on the top left breaks and instead forms a bond to the silicon atom located in \hkl<1 -1 1> direction which is not shown in the figure.
1226 A binding energy of -3.14 eV is obtained for this structure composing another energetically favorable configuration.
1227 A vacancy ctreated at position 2 enables a relaxation of the silicon atom number 1 mainly in \hkl<0 0 -1> direction.
1228 The bond to silicon atom number 5 breaks.
1229 Hence, the silicon dumbbell atom is not only displaced along \hkl<0 0 -1> but also and to a greater extent in \hkl<1 1 0> direction.
1230 The carbon atom is slightly displaced in \hkl<0 1 -1> direction.
1231 A binding energy of -0.59 eV indicates the occurrence of much less strain reduction compared to that in the latter configuration.
1232 Evidently this is due to a smaller displacement of silicon atom number 1, which would be directly bound to the replaced silicon atom at position 2.
1233 In the case of a vacancy created at position 4, even a slightly higher binding energy of -0.54 eV is observed, while the silicon atom at the bottom left, which is bound to the carbon dumbbell atom, is vastly displaced along \hkl<1 0 -1>.
1234 However the displacement of the carbon atom along \hkl<0 0 -1> is less than it is in the preceding configuration.
1235 Although expected due to the symmetric initial configuration silicon atom number 1 is not displaced correspondingly and also the silicon dumbbell atom is displaced to a greater extent in \hkl<-1 0 0> than in \hkl<0 -1 0> direction.
1236 The symmetric configuration is, thus, assumed to constitute a local maximum, which is driven into the present state by the conjugate gradient method used for relaxation.
1237 Figure \ref{fig:defects:comb_db_06} d) shows the relaxed structure of a vacancy created at position 5.
1238 The silicon dumbbell atom is largely displaced along \hkl<1 1 0> and somewaht less along \hkl<0 0 -1>, which corresponds to the direction towards the vacancy.
1239 The silicon dumbbell atom approaches silicon number 1.
1240 Indeed a non-zero charge density is observed inbetween these two atoms exhibiting a cylinder-like shape superposed with the charge density known from the dumbbell itself.
1241 Strain reduced by this huge displacement is partially absorbed by tensile strain on silicon atom number 1 originating from attractive forces of the carbon atom and the vacancy.
1242 A binding energy of -0.50 eV is observed.
1243 {\color{red}Todo: Jahn-Teller distortion (vacancy) $\rightarrow$ actually three possibilities. Due to the initial defect, symmetries are broken. The system should have relaxed into the minumum energy configuration!?}
1245 \subsection{Combinations of Si self-interstitials and substitutional carbon}
1247 So far the C-Si \hkl<1 0 0> interstitial was found to be the energetically most favorable configuration.
1248 In fact substitutional C exhibits a configuration more than 3 eV lower in formation energy, however, the configuration does not account for the accompanying Si self-interstitial that is generated once a C atom occupies the site of a Si atom.
1249 With regard to the IBS process, in which highly energetic C atoms enter the Si target being able to kick out Si atoms from their lattice sites, such configurations are absolutely conceivable and a significant role for the precipitation process might be attributed to them.
1250 Thus, combinations of substitutional C and an additional Si self-interstitial are examined in the following.
1251 The ground state of a single Si self-interstitial was found to be the Si \hkl<1 1 0> self-interstitial configuration.
1252 For the follwoing study the same type of self-interstitial is assumed to provide the energetically most favorable configuration in combination with substitutional C.
1256 \begin{tabular}{l c c c c c c}
1259 C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1260 \hkl<1 0 1> & \hkl<-1 0 1> \\
1262 1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1263 2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1264 3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1265 4 & \RM{4} & B & D & E & E & D \\
1266 5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1271 \caption{Equivalent configurations of \hkl<1 1 0>-type Si self-interstitials created at position I of figure \ref{fig:defects:pos_of_comb} and substitutional C created at positions 1 to 5.}
1272 \label{tab:defects:comb_csub_si110}
1276 \begin{tabular}{l c c c c c c c c c c}
1279 Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1281 $E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1282 $E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1283 $r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1288 \caption{Formation $E_{\text{f}}$ and binding $E_{\text{b}}$ energies in eV of the combinational substitutional C and Si self-interstitial configurations as defined in table \ref{tab:defects:comb_csub_si110}.}
1289 \label{tab:defects:comb_csub_si110_energy}
1291 Table \ref{tab:defects:comb_csub_si110} shows equivalent configurations of \hkl<1 1 0>-type Si self-interstitials and substitutional C.
1292 The notation of figure \ref{fig:defects:pos_of_comb} is used with the six possible Si self-interstitials created at the usual C-Si dumbbell position.
1293 Substitutional C is created at positions 1 to 5.
1294 Resulting formation and binding energies of the relaxed structures are listed in table \ref{tab:defects:comb_csub_si110_energy}.
1295 In addition the separation distance of the ssubstitutional C atom and the Si \hkl<1 1 0> dumbbell interstitial, which is defined to reside at $\frac{a_{\text{Si}}}{4} \hkl<1 1 1>$ is given.
1296 In total 10 different configurations exist within the investigated range.
1300 \includegraphics[width=12cm]{c_sub_si110.ps}
1302 \caption[Binding energy of combinations of a substitutional C and a Si \hkl<1 1 0> dumbbell self-interstitial with respect to the separation distance.]{Binding energy of combinations of a substitutional C and a Si \hkl<1 1 0> dumbbell self-interstitial with respect to the separation distance. The binding energy of the defect pair is well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}
1303 \label{fig:defects:csub_si110}
1305 According to the formation energies none of the investigated structures is energetically preferred over the C-Si \hkl<1 0 0> dumbbell interstitial, which exhibits a formation energy of 3.88 eV.
1306 Further separated defects are assumed to approximate the sum of the formation energies of the isolated single defects.
1307 This is affirmed by the plot of the binding energies with respect to the separation distance in figure \ref{fig:defects:csub_si110} approximating zero with increasing distance.
1308 Thus, the C-Si \hkl<1 0 0> dumbbell structure remains the ground state configuration of a C interstitial in c-Si with a constant number of Si atoms.
1311 However the binding energy quickly drops to zero with respect to the distance, which is reinforced by the Lennard-Jones fit estimating almost zero interaction energy already at 0.6 nm.
1312 This indicates a possibly low interaction capture radius of the defect pair.
1313 Highly energetic collisions in the IBS process might result in separations of these defects exceeding the capture radius.
1314 For this reason situations most likely occur in which the configuration of substitutional C can be considered without a nearby interacting Si self-interstitial and, thus, unable to form a thermodynamically more stable C-Si \hkl<1 0 0> dumbbell configuration.
1316 \label{section:defects:noneq_process_01}
1318 The energetically most favorable configuration of the combined structures is the one with the substitutional C atom located next to the \hkl<1 1 0> interstitial along the \hkl<1 1 0> direction (configuration \RM{1}).
1319 Compressive stress along the \hkl<1 1 0> direction originating from the Si \hkl<1 1 0> self-intesrtitial is partially compensated by tensile stress resulting from substitutional C occupying the neighboured Si lattice site.
1320 In the same way the energetically most unfavorable configuration can be explained, which is configuration \RM{3}.
1321 The substitutional C is located next to the lattice site shared by the \hkl<1 1 0> Si self-interstitial along the \hkl<1 -1 0> direction.
1322 Thus, the compressive stress along \hkl<1 1 0> of the Si \hkl<1 1 0> interstitial is not compensated but intensified by the tensile stress of the substitutional C atom, which is no longer loacted along the direction of stress.
1324 {\color{red}Todo: Erhart/Albe calc for most and less favorable configuration!}
1326 {\color{red}Todo: Mig of C-Si DB conf to or from C sub + Si 110 in progress.}
1328 {\color{red}Todo: Mig of Si DB located next to a C sub (also by MD!).}
1330 \section{Migration in systems of combined defects}
1332 As already pointed out in the previous section energetic carbon atoms may kick out silicon atoms from their lattice sites during carbon implantation into crystalline silicon.
1333 However configurations might arise in which C atoms do not already occupy the vacant site but instead form a C interstitial next to the vacancy.
1334 These combinations have been investigated shortly before in the very end of section \ref{subsection:defects:c-si_comb}.
1335 In the absence of the Si self-interstitial the energetically most favorable configuration is the configuration of a substitutional carbon atom, that is the carbon atom occupying the vacant site.
1336 In addition, it is a conceivable configuration the system might experience during the silicon carbide precipitation process.
1337 Energies needed to overcome the migration barrier of the transformation into this configuration enable predictions concerning the feasibility of a silicon carbide conversion mechanism derived from these microscopic processes.
1338 This is especially important for the case, in which the vacancy is created at position 3, as displayed in figure \ref{fig:defects:comb_db_06} b).
1339 Due to the low binding energy this configuration might constitute a trap, which it is hard to escape from.
1340 However, migration simulations show that only a low amount of energy is necessary to transform the system into the energetically most favorable configuration.
1341 \begin{figure}[!t!h]
1343 \includegraphics[width=13cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[2.0cm]
1344 \begin{picture}(0,0)(170,0)
1345 \includegraphics[width=3cm]{vasp_mig/comb_2-1_init.eps}
1347 \begin{picture}(0,0)(80,0)
1348 \includegraphics[width=3cm]{vasp_mig/comb_2-1_seq_03.eps}
1350 \begin{picture}(0,0)(-10,0)
1351 \includegraphics[width=3cm]{vasp_mig/comb_2-1_seq_06.eps}
1353 \begin{picture}(0,0)(-120,0)
1354 \includegraphics[width=3cm]{vasp_mig/comb_2-1_final.eps}
1356 \begin{picture}(0,0)(25,20)
1357 \includegraphics[width=2.5cm]{100_arrow.eps}
1359 \begin{picture}(0,0)(230,0)
1360 \includegraphics[height=2.2cm]{001_arrow.eps}
1363 \caption{Transition of the configuration of the C-Si dumbbell interstitial in combination with a vacancy created at position 2 into the configuration of substitutional carbon.}
1364 \label{fig:defects:comb_mig_01}
1366 \begin{figure}[!t!h]
1368 \includegraphics[width=13cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[1.0cm]
1369 \begin{picture}(0,0)(150,0)
1370 \includegraphics[width=2cm]{vasp_mig/comb_3-1_init.eps}
1372 \begin{picture}(0,0)(60,0)
1373 \includegraphics[width=2cm]{vasp_mig/comb_3-1_seq_03.eps}
1375 \begin{picture}(0,0)(-45,0)
1376 \includegraphics[width=2cm]{vasp_mig/comb_3-1_seq_07.eps}
1378 \begin{picture}(0,0)(-130,0)
1379 \includegraphics[width=2cm]{vasp_mig/comb_3-1_final.eps}
1381 \begin{picture}(0,0)(25,20)
1382 \includegraphics[width=2.5cm]{100_arrow.eps}
1384 \begin{picture}(0,0)(230,0)
1385 \includegraphics[height=2.2cm]{001_arrow.eps}
1388 \caption{Transition of the configuration of the C-Si dumbbell interstitial in combination with a vacancy created at position 3 into the configuration of substitutional carbon.}
1389 \label{fig:defects:comb_mig_02}
1391 Figure \ref{fig:defects:comb_mig_01} and \ref{fig:defects:comb_mig_02} show the migration barriers and structures for transitions of the vacancy-interstitial configurations examined in figure \ref{fig:defects:comb_db_06} a) and b) into a configuration of substitutional carbon.
1392 If the vacancy is created at position 1 the system will end up in a configuration of substitutional C anyways.
1394 In the first case the focus is on the migration of silicon atom number 1 towards the vacant site created at position 2 while the carbon atom substitutes the site of the migrating silicon atom.
1395 An energy of 0.6 eV necessary to overcome the migration barrier is found.
1396 This energy is low enough to constitute a feasible mechanism in SiC precipitation.
1397 To reverse this process 5.4 eV are needed, which make this mechanism very unprobable.
1398 The migration path is best described by the reverse process.
1399 Starting at 100 \% energy is needed to break the bonds of silicon atom 1 to its neighboured silicon atoms and that of the carbon atom to silicon atom number 5.
1400 At a displacement of 60 \% these bonds are broken.
1401 Due to this and due to the formation of new bonds, that is the bond of silicon atom number 1 to silicon atom number 5 and the bond of the carbon atom to its siliocn neighbour in the bottom left, a less steep increase of free energy is observed.
1402 At a displacement of approximately 30 \% the bond of silicon atom number 1 to the just recently created siliocn atom is broken up again, which explains the repeated boost in energy.
1403 Finally the system gains energy relaxing into the configuration of zero displacement.
1404 {\color{red}Todo: Direct migration of C in progress.}
1406 Due to the low binding energy observed, the configuration of the vacancy created at position 3 is assumed to be stable against transition.
1407 However, a relatively simple migration path exists, which intuitively seems to be a low energy process.
1408 The migration path and the corresponding differences in free energy are displayed in figure \ref{fig:defects:comb_mig_02}.
1409 In fact, migration simulations yield a barrier as low as 0.1 eV.
1410 This energy is needed to tilt the dumbbell as the displayed structure at 30 \% displacement shows.
1411 Once this barrier is overcome, the carbon atom forms a bond to the top left silicon atom and the interstitial silicon atom capturing the vacant site is forming new tetrahedral bonds to its neighboured silicon atoms.
1412 These new bonds and the relaxation into the substitutional carbon configuration are responsible for the gain in free energy.
1413 For the reverse process approximately 2.4 eV are needed, which is 24 times higher than the forward process.
1414 Thus, substitutional carbon is assumed to be stable in contrast to the C-Si dumbbell interstitial located next to a vacancy.
1416 \section{Conclusions concerning the SiC conversion mechanism}
1418 The ground state configuration of a carbon interstitial in crystalline siliocn is found to be the C-Si \hkl<1 0 0> dumbbell interstitial configuration, in which the threefold coordinated carbon and silicon atom share a usual silicon lattice site.
1419 This supports the assumption of C-Si \hkl<1 0 0>-type dumbbel interstitial formation in the first steps of the IBS process as proposed by the precipitation model introduced in section \ref{section:assumed_prec}.
1421 Migration simulations reveal this carbon interstitial to be mobile at prevailing implantation temperatures requireing an activation energy of approximately 0.9 eV for migration as well as reorientation processes.
1422 This enables possible migration of the defects to form defect agglomerates as demanded by the model.
1423 Unfortunately classical potential simulations show tremendously overestimated migration barriers indicating a possible failure of the necessary agglomeration of such defects.
1425 Investigations of two carbon interstitials of the \hkl<1 0 0>-type and varying separations and orientations state an attractive interaction between these interstitials.
1426 Depending on orientation, energetically favorable configurations are found in which these two interstitials are located close together instead of the occurernce of largely separated and isolated defects.
1427 This is due to strain compensation enabled by the combination of such defects in certain orientations.
1428 For dumbbells oriented along the \hkl<1 1 0> direction and the assumption that there is the possibility of free orientation, an interaction energy proportional to the reciprocal cube of the distance in the far field regime is found.
1429 These findings support the assumption of the C-Si dumbbell agglomeration proposed by the precipitation model.
1431 Next to the C-Si \hkl<1 0 0> dumbbell interstitial configuration, in which the C atom is sharing a Si lattice site with the corresponding Si atom the C atom could occupy the site of the Si atom, which in turn forms a Si self-interstitial.
1432 Combinations of substitutional C and a \hkl<1 1 0> Si self-interstitial, which is the ground state configuration for a Si self-interstitial and, thus, assumed to be the energetically most favorable configuration for combined structures, show formation energies 0.5 eV to 1.5 eV greater than that of the C-Si \hkl<1 0 0> interstitial configuration, which remains the energetically most favorable configuration.
1433 However, the binding energy of substitutional C and the Si self-interstitial quickly drops to zero already for short separations indicating a low interaction capture radius.
1434 Thus, due to missing attractive interaction forces driving the system to form C-Si \hkl<1 0 0> dumbbell interstitial complexes substitutional C, while thermodynamically not stable, constitutes a most likely configuration occuring in IBS, a process far from equlibrium.
1436 Due to the low interaction capture radius substitutional C can be treated independently of the existence of separated Si self-interstitials.
1437 This should be also true for combinations of C-Si interstitials next to a vacancy and a further separated Si self-interstitial excluded from treatment, which again is a conveivable configuration in IBS.
1438 By combination of a \hkl<1 0 0> dumbbell with a vacancy in the absence of the Si self-interstitial it is found that the configuration of substitutional carbon occupying the vacant site is the energetically most favorable configuration.
1439 Low migration barriers are necessary to obtain this configuration and in contrast comparatively high activation energies necessary for the reverse process.
1440 Thus, carbon interstitials and vacancies located close together are assumed to end up in such a configuration in which the carbon atom is tetrahedrally coordinated and bound to four silicon atoms as expected in silicon carbide.
1442 While first results support the proposed precipitation model the latter suggest the formation of silicon carbide by succesive creation of substitutional carbon instead of the agglomeration of C-Si dumbbell interstitials followed by an abrupt transition.
1443 Prevailing conditions in the IBS process at elevated temperatures and the fact that IBS is a nonequilibrium process reinforce the possibility of formation of substitutional C instead of the thermodynamically stable C-Si dumbbell interstitials predicted by simulations at zero Kelvin.
1444 \label{section:defects:noneq_process_02}
1447 In addition, there are experimental findings, which might be exploited to reinforce the non-validity of the proposed precipitation model.
1448 High resolution TEM shows equal orientation of \hkl(h k l) planes of the c-Si host matrix and the 3C-SiC precipitate.
1449 Formation of 3C-SiC realized by successive formation of substitutional C, in which the atoms belonging to one of the two fcc lattices are substituted by C atoms perfectly conserves the \hkl(h k l) planes of the initial c-Si diamond lattice.
1450 Silicon self-interstitials consecutively created to the same degree are able to diffuse into the c-Si host one after another.
1451 Investigated combinations of C interstitials, however, result in distorted configurations, in which C atoms, which at some point will form SiC, are no longer aligned to the host.
1452 It is easily understandable that the mismatch in alignement will increase with increasing defect density.
1453 In addition, the amount of Si self-interstitials equal to the amount of agglomerated C atoms would be released all of a sudden probably not being able to diffuse into the c-Si host matrix without damaging the Si surrounding or the precipitate itself.
1454 In addition, IBS results in the formation of the cubic polytype of SiC only.
1455 As this result conforms well with the model of precipitation by substitutional C there is no obvious reason why hexagonal polytypes should not be able to form or an equal alignement would be mandatory assuming the model of precipitation by C-Si dumbbell agglomeration.
1458 {\color{red}Todo: C mobility higher than Si mobility? -> substitutional C is more likely to arise, since it migrates 'faster' to vacant sites?}