1 \chapter{Silicon carbide precipitation simulations}
4 The molecular dynamics (MD) technique is used to gain insight into the behavior of C existing in different concentrations in c-Si on the microscopic level at finite temperatures.
5 The simulations are restricted to classical potential simulations utilizing the analytical EA bond order potential as described in section \ref{subsection:interact_pot}.
6 Parameters are chosen according to the discussion in section \ref{section:classpotmd}.
8 At the beginning, simulations are performed, which try to mimic the conditions during IBS.
9 Results reveal limitations of the employed potential and MD in general.
10 With reference to the results of the last chapter, a workaround is discussed.
11 The approach is follwed and, finally, results gained by the MD simulations are interpreted drawing special attention to the established controversy concerning precipitation of SiC in Si.
13 \section{Simulations at temperatures used in IBS}
14 \label{section:initial_sims}
16 In initial simulations aiming to reproduce a precipitation process, simulation volumes of $31\times 31\times 31$ unit cells are utilized.
17 Periodic boundary conditions in each direction are applied.
18 The system temperature is set to $450\, ^{\circ}\mathrm{C}$, the temperature for which epitaxial growth of 3C-SiC films is achieved in IBS.
19 After equilibration of the kinetic and potential energy, C atoms are consecutively inserted.
20 The number of C atoms $N_{\text{C}}$ necessary to form a spherical precipitate with radius $r$ is given by
22 N_{\text{C}}=\frac{4}{3}\pi r^3 \cdot \frac{4}{a_{\text{SiC}}^3}
23 =\frac{16}{3} \pi \left( \frac{r}{a_{\text{SiC}}}\right)^3
24 \label{eq:md:spheric_prec}
26 with $a_{\text{SiC}}$ being the lattice constant of 3C-SiC.
27 In IBS experiments, the smallest precipitates observed have radii starting from \unit[2]{nm} up to \unit[4]{nm}.
28 For the initial simulations, a total amount of 6000 C atoms corresponding to a radius of approximately \unit[3.1]{nm} is chosen.
29 In separated simulations, the 6000 C atoms are inserted in three regions of different volume ($V_1$, $V_2$, $V_3$) within the simulation cell.
30 For reasons of simplification these regions are rectangularly shaped.
31 $V_1$ is chosen to be the total simulation volume.
32 $V_2$ approximately corresponds to the volume of a minimal 3C-SiC precipitate.
33 $V_3$ is approximately the volume containing the amount of Si atoms necessary to form such a precipitate, which is slightly smaller than $V_2$ due to the slightly lower Si density of 3C-SiC compared to c-Si.
34 The two latter insertion volumes are considered since no diffusion of C atoms is expected within the simulated period of time at prevalent temperatures.
35 This is due to the overestimated activation energy for the diffusion of a \ci \hkl<1 0 0> DB, as pointed out in section \ref{subsection:defects:mig_classical}.
36 For rectangularly shaped precipitates with side length $L$ the amount of C atoms in 3C-SiC and Si atoms in c-Si is given by
38 N_{\text{C}}^{\text{3C-SiC}} =4 \left( \frac{L}{a_{\text{SiC}}}\right)^3
40 N_{\text{Si}}^{\text{c-Si}} =8 \left( \frac{L}{a_{\text{Si}}}\right)^3 \text{ .}
43 Table \ref{table:md:ins_vols} summarizes the side length of each of the three different insertion volumes determined by equations \eqref{eq:md:n_prec} and the resulting C concentrations inside these volumes.
44 Looking at the C concentrations, simulations can be distinguished in simulations occupying low ($V_1$) and high ($V_2$, $V_3$) concentrations of C.
47 \begin{tabular}{l c c c}
50 & $V_1$ & $V_2$ & $V_3$ \\
52 Side length [\AA] & 168.3 & 50.0 & 49.0 \\
53 C concentration [$\frac{1}{\text{c-Si unit cell}}$] & 0.20 & 7.68 & 8.16\\
58 \caption{Side lengthes of the insertion volumes $V_1$, $V_2$ and $V_3$ used for the incoorperation of 6000 C atoms.}
59 \label{table:md:ins_vols}
62 The insertion is realized in a way to keep the system temperature constant.
63 In each of 600 insertion steps, 10 C atoms are inserted at random positions within the respective region, which involves an increase in potential energy that is partially transformed into kinetic energy.
64 Thus, the simulation is continued without adding more C atoms until the system temperature is equal to the chosen temperature again.
65 This is realized by the thermostat, which decouples excessive energy.
66 Every inserted C atom must exhibit a distance greater or equal to \unit[1.5]{\AA} to neighbored atoms to prevent the occurrence of too high forces.
67 Once the total amount of C is inserted, the simulation is continued for \unit[100]{ps} followed by a cooling-down process until room temperature, i.e. \unit[20]{$^{\circ}$C}, is reached.
68 Fig.~\ref{fig:md:prec_fc} displays a flow chart of the applied steps involved in the simulation sequence.
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142 \caption[Flowchart of the simulation sequence used in MD simulations aiming to reproduce the precipitation process.]{Flowchart of the simulation sequence used in MD simulations aiming to reproduce the precipitation process. $T_{\text{s}}$ and $p_{\text{s}}$ are the preset values for the system temperature and pressure. $T_{\text{avg}}$ is the averaged actual system temperature.}
143 \label{fig:md:prec_fc}
146 The radial distribution function $g(r)$ for C-C and Si-Si distances is shown in Fig. \ref{fig:md:pc_si-si_c-c}.
149 \includegraphics[width=0.7\textwidth]{sic_prec_450_si-si_c-c.ps}
151 \caption[Radial distribution function of the C-C and Si-Si distances for 6000 C atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of \unit\[450\]{$^{\circ}$C} and cooled down to room temperature.]{Radial distribution function of the C-C and Si-Si distances for 6000 C atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of \unit[450]{$^{\circ}$C} and cooled down to room temperature. The bright blue graph shows the Si-Si radial distribution for pure c-Si. The insets show magnified regions of the respective type of bond.}
152 \label{fig:md:pc_si-si_c-c}
156 \includegraphics[width=0.7\textwidth]{sic_prec_450_energy.ps}
158 \caption[Total energy per atom as a function of time for the whole simulation sequence and for all three types of insertion volumes.]{Total energy per atom as a function of time for the whole simulation sequence and for all three types of insertion volumes. Arrows mark the end of C insertion and the start of the cooling process respectively.}
159 \label{fig:md:energy_450}
161 It is easily and instantly visible that there is no significant difference among the two simulations of high C concentration.
162 Thus, in the following, the focus can indeed be directed to low ($V_1$) and high ($V_2$, $V_3$) C concentration simulations.
163 The first C-C peak appears at about \unit[0.15]{nm}, which is compareable to the nearest neighbor distance of graphite or diamond.
164 The number of C-C bonds is much smaller for $V_1$ than for $V_2$ and $V_3$ since C atoms are spread over the total simulation volume.
165 On average, there are only 0.2 C atoms per Si unit cell.
166 These C atoms are assumed to form strong bonds.
167 This is supported by Fig.~\ref{fig:md:energy_450} displaying the total energy of all three simulations during the whole simulation sequence.
168 A huge decrease of the total energy during C insertion is observed for the simulations with high C concentration in contrast to the $V_1$ simulation, which shows a slight increase.
169 The difference in energy $\Delta$ growing within the C insertion process up to a value of roughly \unit[0.06]{eV} per atom persists unchanged until the end of the simulation.
170 The vast amount of strongly bonded C-C bonds in the high concentration simulations make these configurations energetically more favorable compared to the low concentration configuration.
171 However, in the same way, a lot of energy is needed to break these bonds to get out of the local energy minimum advancing towards the global minimum configuration.
172 Thus, such conformational changes are very unlikely to happen.
173 This is in accordance with the constant total energy observed in the continuation step of \unit[100]{ps} inbetween the end of C insertion and the cooling process.
174 Obviously, no energetically favorable relaxation is taking place at a system temperature of \unit[450]{$^{\circ}$C}.
176 The C-C peak at about \unit[0.31]{nm} perfectly matches the nearest neighbor distance of two C atoms in the 3C-SiC lattice.
177 As can be seen from the inset, this peak is also observed for the $V_1$ simulation.
178 Investigating the corresponding coordinates of the atoms, it turns out that concatenated and differently oriented \ci{} \hkl<1 0 0> DB interstitials constitute configurations yielding separations of C atoms by this distance.
179 In 3C-SiC, the same distance is also expected for nearest neighbor Si atoms.
180 The bottom of Fig.~\ref{fig:md:pc_si-si_c-c} shows the radial distribution of Si-Si bonds together with a reference graph for pure c-Si.
181 Indeed, non-zero $g(r)$ values around \unit[0.31]{nm} are observed while the amount of Si pairs at regular c-Si distances of \unit[0.24]{nm} and \unit[0.38]{nm} decreases.
182 However, no clear peak is observed but the interval of enhanced $g(r)$ values corresponds to the width of the C-C $g(r)$ peak.
183 In addition the abrupt increase of Si pairs at \unit[0.29]{nm} can be attributed to the Si-Si cut-off radius of \unit[0.296]{nm} as used in the present bond order potential.
184 The cut-off function causes artificial forces pushing the Si atoms out of the cut-off region.
185 Without the abrubt increase, a maximum around \unit[0.31]{nm} gets even more conceivable.
186 Analyses of randomly chosen configurations, in which distances around \unit[0.3]{nm} appear, identify \ci{} \hkl<1 0 0> DBs to be responsible for stretching the Si-Si next neighbour distance for low C concentrations, i.e. for the $V_1$ and early stages of $V_2$ and $V_3$ simulation runs.
187 This excellently agrees with the calculated value $r(13)$ in Table~\ref{tab:defects:100db_cmp} for a resulting Si-Si distance in the \ci \hkl<1 0 0> DB configuration.
191 \includegraphics[width=0.7\textwidth]{sic_prec_450_si-c.ps}
193 \caption{Radial distribution function of the Si-C distances for 6000 C atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of \unit[450]{$^{\circ}$C} and cooled down to room temperature. Additionally the resulting Si-C distances of a \ci{} \hkl<1 0 0> DB configuration are given.}
194 \label{fig:md:pc_si-c}
196 Fig.~\ref{fig:md:pc_si-c} displays the Si-C radial distribution function for all three insertion volumes together with the Si-C bonds as observed in a \ci{} \hkl<1 0 0> DB configuration.
197 The first peak observed for all insertion volumes is at approximately \unit[0.186]{nm}.
198 This corresponds quite well to the expected next neighbor distance of \unit[0.189]{nm} for Si and C atoms in 3C-SiC.
199 By comparing the resulting Si-C bonds of a \ci{} \hkl<1 0 0> DB with the C-Si distances of the low concentration simulation, it is evident that the resulting structure of the $V_1$ simulation is clearly dominated by this type of defect.
200 This is not surpsising, since the \ci{} \hkl<1 0 0> DB is found to be the ground-state defect of a C interstitial in c-Si and, for the low concentration simulations, a C interstitial is expected in every fifth Si unit cell only, thus, excluding defect superposition phenomena.
201 The peak distance at \unit[0.186]{nm} and the bump at \unit[0.175]{nm} corresponds to the distance $r(3C)$ and $r(1C)$ as listed in Table~\ref{tab:defects:100db_cmp} and visualized in Fig.~\ref{fig:defects:100db_cmp}.
202 In addition, it can be easily identified that the \ci{} \hkl<1 0 0> DB configuration contributes to the peaks at about \unit[0.335]{nm}, \unit[0.386]{nm}, \unit[0.434]{nm}, \unit[0.469]{nm} and \unit[0.546]{nm} observed in the $V_1$ simulation.
203 Not only the peak locations but also the peak widths and heights become comprehensible.
204 The distinct peak at \unit[0.26]{nm}, which exactly matches the cut-off radius of the Si-C interaction, is again a potential artifact.
206 For high C concentrations, i.e. the $V_2$ and $V_3$ simulation corresponding to a C density of about 8 atoms per c-Si unit cell, the defect concentration is likewiese increased and a considerable amount of damage is introduced in the insertion volume.
207 The consequential superposition of these defects and the high amounts of damage generate new displacement arrangements for the C-C as well as for the Si-C pair distances, which become hard to categorize and trace and obviously lead to a broader distribution.
208 Short range order indeed is observed, i.e. the large amount of strong neighbored C-C bonds at \unit[0.15]{nm} as expected in graphite or diamond and Si-C bonds at \unit[0.19]{nm} as expected in SiC, but only hardly visible is the long range order.
209 This indicates the formation of an amorphous SiC-like phase.
210 In fact the resulting Si-C and C-C radial distribution functions compare quite well with these obtained by cascade amorphized and melt-quenched amorphous SiC using a modified Tersoff potential \cite{gao02}.
212 In both cases, i.e. low and high C concentrations, the formation of 3C-SiC fails to appear.
213 With respect to the precipitation model, the formation of C$_{\text{i}}$ \hkl<1 0 0> DBs indeed occurs for low C concentrations.
214 However, sufficient defect agglomeration is not observed.
215 For high C concentrations, a rearrangement of the amorphous SiC structure, which is not expected at prevailing temperatures, and a transition into 3C-SiC is not observed either.
217 \section{Limitations of conventional MD and short range potentials}
218 \label{section:md:limit}
220 Results of the last section indicate possible limitations of the MD method regarding the task addressed in this study.
221 Low C concentration simulations do not reproduce the agglomeration of C$_{\text{i}}$ \hkl<1 0 0> DBs.
222 High concentration simulations result in the formation of an amorphous SiC-like phase, which is unexpected since IBS experiments show crystalline 3C-SiC precipitates at prevailing temperatures.
223 Keeping in mind the results
224 On closer inspection, however, two reasons for describing this obstacle become evident, which are discussed in the following.
226 The first reason is a general problem of MD simulations in conjunction with limitations in computer power, which results in a slow and restricted propagation in phase space.
227 In molecular systems, characteristic motions take place over a wide range of time scales.
228 Vibrations of the covalent bond take place on the order of \unit[10$^{-14}$]{s}, of which the thermodynamic and kinetic properties are well described by MD simulations.
229 To avoid dicretization errors, the integration timestep needs to be chosen smaller than the fastest vibrational frequency in the system.
230 On the other hand, infrequent processes, such as conformational changes, reorganization processes during film growth, defect diffusion and phase transitions are processes undergoing long-term evolution in the range of microseconds.
231 This is due to the existence of several local minima in the free energy surface separated by large energy barriers compared to the kinetic energy of the particles, i.e. the system temperature.
232 Thus, the average time of a transition from one potential basin to another corresponds to a great deal of vibrational periods, which in turn determine the integration timestep.
233 Hence, time scales covering the necessary amount of infrequent events to observe long-term evolution are not accessible by traditional MD simulations, which are limited to the order of nanoseconds.
234 New methods have been developed to bypass the time scale problem.
235 The most famous appraiches are hyperdnyamics (HMD) \cite{voter97,voter97_2}, parallel replica dynamics \cite{voter98}, temperature acclerated dynamics (TAD) \cite{sorensen2000} and self-guided dynamics (SGMD) \cite{wu99}, which accelerate phase space propagation while retaining proper thermodynmic sampling.
237 In addition to the time scale limitation, problems attributed to the short range potential exist.
238 The sharp cut-off funtion, which limits the interacting ions to the next neighbored atoms by gradually pushing the interaction force and energy to zero between the first and second next neighbor distance, is responsible for overestimated and unphysical high forces of next neighbored atoms \cite{tang95,mattoni2007}.
239 This is supported by the overestimated activation energies necessary for C diffusion as investigated in section \ref{subsection:defects:mig_classical}.
240 Indeed, it is not only the strong C-C bond, which is hard to break, inhibiting C diffusion and further rearrengements.
241 This is also true for the low concentration simulations dominated by the occurrence of C-Si DBs spread over the whole simulation volume.
242 The bonds of these C-Si pairs are also affected by the cut-off artifact preventing C diffusion and agglomeration of the DBs.
243 This can be seen from the almost horizontal progress of the total energy graph in the continuation step of Fig.~\ref{fig:md:energy_450}, even for the low concentration simulation.
244 These unphysical effects inherent to this type of model potentials are solely attributed to their short range character.
245 While cohesive and formational energies are often well described, these effects increase for non-equilibrium structures and dynamics.
246 However, since valueable insights into various physical properties can be gained using this potentials, modifications mainly affecting the cut-off were designed.
247 One possibility is to simply skip the force contributions containing the derivatives of the cut-off function, which was successfully applied to reproduce the brittle propagation of fracture in SiC at zero temperature \cite{mattoni2007}.
248 Another one is to use variable cut-off values scaled by the system volume, which properly describes thermomechanical properties of 3C-SiC \cite{tang95} but might be rather ineffective for the challange inherent to this study.
250 To conclude the obstacle needed to get passed is twofold.
251 The sharp cut-off of the employed bond order model potential introduces overestimated high forces between next neighbored atoms enhancing the problem of slow phase space propagation immanent to MD simulations, termed {\em potential enhanced slow phase space propagation} in the following.
252 Thus, pushing the time scale to the limits of computational ressources or applying one of the above mentioned accelerated dynamics methods exclusively will not be sufficient enough.
254 Instead, the approach followed in this study, is the use of higher temperatures as exploited in TAD to find transition pathways of one local energy minimum to another one more quickly.
255 Since merely increasing the temperature leads to different equilibrium kinetics than valid at low temperatures, TAD introduces basin-constrained MD allowing only those transitions that should occur at the original temperature and a properly advancing system clock \cite{sorensen2000}.
256 The TAD corrections are not applied in coming up simulations.
257 This is justified by two reasons.
258 First of all, a compensation of the overestimated bond strengths due to the short range potential is expected.
259 Secondly, there is no conflict applying higher temperatures without the TAD corrections, since crystalline 3C-SiC is also observed for higher temperatures than \unit[450]{$^{\circ}$C} in IBS \cite{nejim95,lindner01}.
260 It is therefore expected that the kinetics affecting the 3C-SiC precipitation are not much different at higher temperatures aside from the fact that it is occuring much more faster.
261 Moreover, the interest of this study is focused on structural evolution of a system far from equilibrium instead of equilibrium properties which rely upon proper phase space sampling.
262 On the other hand, during implantation, the actual temperature inside the implantation volume is definetly higher than the experimentally determined temperature tapped from the surface of the sample.
264 \section{Increased temperature simulations}
265 \label{section:md:inct}
267 Due to the limitations of short range potentials and conventional MD as discussed above, elevated temperatures are used in the following.
268 The simulation sequence and other parameters except for the system temperature remain unchanged as in section \ref{section:initial_sims}.
269 Since there is no significant difference among the $V_2$ and $V_3$ simulations only the $V_1$ and $V_2$ simulations are carried on and referred to as low C and high C concentration simulations.
270 Temperatures ranging from \degc{450} up to \unit[2050]{$^{\circ}$C} are used.
272 A simple quality value $Q$ is introduced, which helps to estimate the progress of structural evolution.
273 In bulk 3C-SiC every C atom has four next neighbored Si atoms and every Si atom four next neighbored C atoms.
274 The quality could be determined by counting the amount of atoms, which form bonds to four atoms of the other species.
275 However, the aim of the simulation is to reproduce the formation of a 3C-SiC precipitate embedded in c-Si.
276 The amount of Si atoms and, thus, the amount of Si atoms remaining in the c-Si diamond lattice is much higher than the amount of inserted C atoms.
277 Thus, counting the atoms, which exhibit proper coordination, is limited to the C atoms.
278 The quality value is defined to be
280 Q = \frac{\text{Amount of C atoms with 4 next neighbored Si atoms}}
281 {\text{Total amount of C atoms}} \text{ .}
284 By this, bulk 3C-SiC will still result in $Q=1$ and precipitates will also reach values close to one.
285 However, since the quality value does not account for bond lengthes, bond angles, crystallinity or the stacking sequence, high values of $Q$ not necessarily correspond to structures close to 3C-SiC.
286 Structures that look promising due to high quality values need to be further investigated by other means.
288 \subsubsection{Low C concetration simulations}
292 \includegraphics[width=0.7\textwidth]{tot_pc_thesis.ps}\\
293 \includegraphics[width=0.7\textwidth]{tot_ba.ps}
295 \caption[Si-C radial distribution and evolution of quality $Q$ for the low concentration simulations at different elevated temperatures.]{Si-C radial distribution and evolution of quality $Q$ according to equation \ref{eq:md:qdef} for the low concentration simulations at different elevated temperatures. All structures are cooled down to \degc{20}. The grey line shows resulting Si-C bonds in a configuration of \cs{} in c-Si (C$_\text{sub}$) at zero temperature. Arrows in the quality plot mark the end of C insertion and the start of the cooling down step. A fit function according to equation \eqref{eq:md:fit} shows the estimated evolution of quality in the absence of the cooling down sequence.}
296 \label{fig:md:tot_si-c_q}
298 Fig.~\ref{fig:md:tot_si-c_q} shows the radial distribution of Si-C bonds for different temperatures and the corresponding evolution of quality $Q$ as defined above for the low concentration simulaton.
299 The first noticeable and promising change in the Si-C radial distribution is the successive decline of the artificial peak at the Si-C cut-off distance with increasing temperature up to the point of disappearance at temperatures above \degc{1650}.
300 Obviously, sufficient kinetic energy is provided to affected atoms that are enabled to escape the cut-off region.
301 Additionally, a more important structural change is observed, which is illustrated in the two shaded areas in Fig.~\ref{fig:md:tot_si-c_q}.
303 In the grey shaded region a decrease of the peak at \unit[0.186]{nm} and of the bump at \distn{0.175} accompanied by an increase of the peak at \distn{0.197} with increasing temperature is visible.
304 Similarly, the peaks at \distn{0.335} and \distn{0.386} shrink in contrast to a new peak forming at \distn{0.372} as can be seen in the yellow shaded region.
305 Obviously, the structure obtained from the \degc{450} simulations, which is dominated by the existence of \ci{} \hkl<1 0 0> DBs, transforms into a different structure with increasing simulation temperature.
306 Investigations of the atomic data reveal \cs{} to be responsible for the new Si-C bonds.
307 The peak at \distn{0.197} corresponds to the distance of a \cs{} atom and its next neighbored Si atoms.
308 The one at \distn{0.372} constitutes the distance of a \cs{} atom to the second next Si neighbor along a \hkl<1 1 0> direction.
309 Comparing the radial distribution for the Si-C bonds at \degc{2050} to the resulting Si-C bonds in a configuration of a \cs{} atom in c-Si excludes all possibility of doubt.
310 The resulting bonds perfectly match and, thus, explain the peaks observed for the increased temperature simulations.
311 To conclude, by increasing the simulation temperature, the structure characterized by the \ci{} \hkl<1 0 0> DB structure transforms into a structure dominated by \cs{}.
313 This is likewise reflected in the quality values obtained for different temperatures.
314 While simulations at \degc{450} exhibit \perc{10} of fourfold coordinated C, simulations at \degc{2050} exceed the \perc{80} range.
315 Since \cs{} has four nearest neighbored Si atoms and is the preferential type of defect in elevated temperature simulations, the increase of the quality values become evident.
316 The quality values at a fixed temperature increase with simulation time.
317 After the end of the insertion sequence marked by the first arrow, the quality is increasing and a saturation behaviour, yet before the cooling process starts, can be expected.
318 The evolution of the quality value of the simulation at \degc{2050} inside the range, in which the simulation is continued at constant temperature for \unit[100]{fs}, is well approximated by the simple fit function
320 f(t)=a-\frac{b}{t} \text{ ,}
323 which results in a saturation value of \perc{93}.
324 Obviously, the decrease in temperature accelerates the saturation and inhibits further formation of \cs{}.
325 \label{subsubsection:md:ep}
326 Conclusions drawn from investigations of the quality evolution correlate well with findings of the radial distribution results.
330 \includegraphics[width=0.7\textwidth]{tot_pc2_thesis.ps}\\
331 \includegraphics[width=0.7\textwidth]{tot_pc3_thesis.ps}
333 \caption[C-C and Si-Si radial distribution for the low concentration simulations at different elevated temperatures.]{C-C and Si-Si radial distribution for the low concentration simulations at different elevated temperatures. All structures are cooled down to $20\,^{\circ}\mathrm{C}$. Arrows with dashed lines mark C-C distances of \hkl<1 0 0> DB combinations and those with solid lines mark C-C distances of combinations of substitutional C. The dashed line corresponds to the distance of a substitutional C with a next neighbored \hkl<1 0 0> DB.}
334 \label{fig:md:tot_c-c_si-si}
336 The formation of \cs{} also affects the Si-Si radial distribution displayed in the lower part of Fig.~\ref{fig:md:tot_c-c_si-si}.
337 Investigating the atomic strcuture indeed shows that the peak arising at \distn{0.325} with increasing temperature is due to two Si atoms that form direct bonds to the \cs{} atom.
338 The peak corresponds to the distance of next neighbored Si atoms along the \hkl<1 1 0> bond chain with C$_{\text{s}}$ in between.
339 Since the expected distance of these Si pairs in 3C-SiC is \distn{0.308}, the existing SiC structures embedded in the c-Si host are stretched.
341 In the upper part of Fig.~\ref{fig:md:tot_c-c_si-si} the C-C radial distribution is shown.
342 The total amount of C-C bonds with a distance inside the displayed separation range does not change significantly.
343 Thus, even for elevated temperatures, agglomeration of C atoms neccessary to form a SiC precipitate does not take place within the simulated time scale.
344 However, with increasing temperature, a decrease of the amount of next neighbored C pairs can be observed.
345 This is a promising result gained by the high-temperature simulations since the breaking of these diomand and graphite like bonds is mandatory for the formation of 3C-SiC.
346 Obviously, acceleration of the dynamics occurred by supplying additional kinetic energy.
347 A slight shift towards higher distances can be observed for the maximum located shortly above \distn{0.3}.
348 Arrows with dashed lines mark C-C distances resulting from \ci{} \hkl<1 0 0> DB combinations while arrows with solid lines mark distances arising from combinations of \cs.
349 The continuous dashed line corresponds to the distance of \cs{} and a next neighbored \ci{} \hkl<1 0 0> DB.
351 Obviously, the shift of the peak is caused by the advancing transformation of the C$_{\text{i}}$ DB into the C$_{\text{s}}$ defect.
352 Next to combinations of two \cs{} atoms or \ci{} \hkl<1 0 0> DBs, combinations of \ci{} \hkl<1 0 0> DBs with a \cs{} atom arise.
353 In addition, structures form that result in distances residing inbetween the ones obtained from combinations of mixed defect types and the ones obtained by \cs{} configurations, as can be seen by quite high $g(r)$ values in between the continuous dashed line and the first arrow with a solid line.
354 For the most part, these structures can be identified as configurations of \cs{} with either another C atom that basically occupies a Si lattice site but is displaced by a \si{} atom residing in the very next surrounding or a C atom that nearly occupies a Si lattice site forming a defect other than the \hkl<1 0 0>-type with the Si atom.
355 Again, this is a quite promising result since the C atoms are taking the appropriate coordination as expected in 3C-SiC.
356 %However, this is contrary to the initial precipitation model proposed in section \ref{section:assumed_prec}, which assumes that the transformation into 3C-SiC takes place in a very last step once enough C-Si DBs agglomerated.
358 To summarize, results of low concentration simulations show a phase transition in conjunction with an increase in temperature.
359 The \ci{} \hkl<1 0 0> DB dominated struture turns into a structure characterized by the occurence of an increasing amount of \cs{} with respect to temperature.
360 Clearly, the high-temperature results indicate the precipitation mechanism involving an increased participation of \cs.
361 Although diamond and graphite like bonds are reduced, no agglomeration of C is observed within the simulated time.
362 Isolated structures of stretched SiC, which are adjusted to the c-Si host with respect to the lattice constant and alignement, are formed.
363 It would be conceivable that by agglomeration of further \cs{} atoms the interfacial energy could be overcome and a transition from a coherent and stretched SiC structure into an incoherent and partially strain-compensated SiC precipitate could occur.
365 \subsubsection{High C concetration simulations}
369 \includegraphics[width=0.7\textwidth]{12_pc_thesis.ps}\\
370 \includegraphics[width=0.7\textwidth]{12_pc_c_thesis.ps}
372 \caption[Si-C and C-C radial distribution for the high concentration simulations at different elevated temperatures.]{Si-C (top) and C-C (bottom) radial distribution for the high concentration simulations at different elevated temperatures. All structures are cooled down to \degc{20}.}
375 Fig.~\ref{fig:md:12_pc} displays the radial distribution for Si-C and C-C pairs obtained from high C concentration simulations at different elevated temperatures.
376 Again, in both cases, the cut-off artifact decreases with increasing temperature.
377 Peaks that already exist for the low temperature simulations get slightly more distinct for elevated temperatures.
378 This is also true for peaks located past distances of next neighbors indicating an increase in the long range order.
379 However, this change is rather small and no significant structural change is observeable.
380 Due to the continuity of high amounts of damage, atomic configurations remain hard to identify even for the highest temperature.
381 Other than in the low concentration simulation, analyzed defect structures are no longer necessarily aligned to the primarily existing but succesively disappearing c-Si host matrix inhibiting or at least hampering their identification and classification.
382 As for low temperatures, order in the short range exists decreasing with increasing separation.
383 The increase of the amount of Si-C pairs at \distn{0.186} could be positively interpreted since this type of bond also exists in 3C-SiC.
384 On the other hand, the amount of next neighbored C atoms with a distance of approximately \distn{0.15}, which is the distance of C in graphite or diamond, is likewise increased.
385 Thus, higher temperatures seem to additionally enhance a conflictive process, i.e. the formation of C agglomerates, obviously inconsistent with the desired process of 3C-SiC formation.
386 This is supported by the C-C peak at \distn{0.252}, which corresponds to the second next neighbor distance in the diamond structure of elemental C.
387 Investigating the atomic data indeed reveals two C atoms, which are bound to and interconnected by a third C atom, to be responsible for this distance.
388 The C-C peak at about \distn{0.31}, wich is slightly shifted to higher distances (\distn{0.317}) with increasing temperature still corresponds quite well to the next neighbor distance of C in 3C-SiC as well as a-SiC and indeed results from C-Si-C bonds.
389 The Si-C peak at \distn{0.282}, which is pronounced with increasing temperature, is constructed out of a Si atom and a C atom, which are both bound to another central C atom.
390 This is similar for the Si-C peak at approximately \distn{0.35}.
391 In this case, the Si and the C atom are bound to a central Si atom.
392 To summarize, the amorphous phase remains.
393 Though, sharper peaks in the radial distributions at distances expected for a-SiC are observed indicating a slight acceleration of the dynamics due to elevated temperatures.
395 \subsubsection{Conclusions concerning the usage of increased temperatures}
397 Regarding the outcome of both, high and low C concentration simulations at increased temperatures, encouraging conclusions can be drawn.
398 With the disappearance of the peaks at the respective cut-off radii, one limitation of the short range potential seems to be accomplished.
399 In addition, sharper peaks in the radial distribution functions lead to the assumption of expeditious structural formation.
400 The increase in temperature leads to the occupation of new defect states, which is particularly evident but not limited to the low C concentration simulations.
402 The question remains, whether these states are only occupied due to the additional supply of kinetic energy and, thus, have to be considered unnatural for temperatures applied in IBS or whether the increase in temperature indeed enables infrequent transitions to occur faster, thus, leading to the intended acceleration of the dynamics and weakening of the unphysical quirks inherent to the potential.
403 As already pointed out in section~\ref{section:defects:noneq_process_01} and section~\ref{section:defects:noneq_process_02}, IBS is a non-equilibrium process, which might result in the formation of the thermodynamically less stable \cs{} and \si{} configuration.
404 Indeed, 3C-SiC is metastable and if not fabricated by IBS requires strong deviation from equilibrium and low temperatures to stabilize the 3C polytype.
405 In IBS, highly energetic C atoms are able to generate vacant sites, which in turn can be occupied by highly mobile \ci{} atoms.
406 This is in fact found to be favorable in the absence of the \si{}, which turned out to have a low interaction capture radius with the \cs{} atom and very likely prevents the recombination into a thermodynamically stable \ci{} DB for appropriate separations of the defect pair.
407 Results gained in this chapter show preferential occupation of Si lattice sites by \cs{} enabled by increased temperatures supporting the assumptions drawn from the defect studies of the last chapter.
409 Thus, it is concluded that increased temperatures is not exclusively usefull to accelerate the dynamics approximatively describing the structural evolution.
410 Moreover it can be considered a necessary condition to deviate the system out of equilibrium enabling the formation of 3C-SiC, which is obviously realized by a successive agglomeration of \cs{}.
414 \section{Valuation of a practicable temperature limit}
415 \label{section:md:tval}
417 The assumed applicability of increased temperature simulations as discussed above and the remaining absence of either agglomeration of substitutional C in low concentration simulations or amorphous to crystalline transition in high concentration simulations suggests to further increase the system temperature.
418 So far, the highest temperature applied corresponds to 95 \% of the absolute Si melting temperature, which is 2450 K and specific to the Erhart/Albe potential.
419 However, melting is not predicted to occur instantly after exceeding the melting point due to additionally required transition enthalpy and hysteresis behaviour.
420 To check for the possibly highest temperature at which a transition fails to appear plain Si is heated up using a heating rate of $1\,^{\circ}\mathrm{C}/\text{ps}$.
421 Fig.~\ref{fig:md:fe_and_t} shows the free energy and temperature evolution in the region around the transition temperature.
422 Indeed a transition and the accompanying critical behaviour of the free energy is first observed at approximately 3125 K, which corresponds to 128 \% of the Si melting temperature.
423 The difference in free energy is 0.58 eV per atom corresponding to $55.7 \text{ kJ/mole}$, which compares quite well to the Si enthalpy of melting of $50.2 \text{ kJ/mole}$.
424 The late transition probably occurs due to the high heating rate and, thus, a large hysteresis behaviour extending the temperature of transition.
425 To avoid melting transitions in further simulations system temperatures well below the transition point are considered safe.
426 According to this study temperatures of 100 \% and 120 \% of the Si melting point could be used.
427 However, defects, which are introduced due to the insertion of C atoms are known to lower the transition point.
428 Indeed simulations show melting transitions already at the melting point whenever C is inserted.
429 Thus, the system temperature of 95 \% of the Si melting point is considered the maximum limit.
432 \includegraphics[width=0.7\textwidth]{fe_and_t.ps}
434 \caption{Free energy and temperature evolution of plain Si at temperatures in the region around the melting transition.}
435 \label{fig:md:fe_and_t}
438 \section{Long time scale simulations at maximum temperature}
440 As discussed in section~\ref{section:md:limit} and~\ref{section:md:inct} a further increase of the system temperature might help to overcome limitations of the short range potential and accelerate the dynamics involved in structural evolution.
441 Furthermore these results indicate that increased temperatures are necessary to drive the system out of equilibrium enabling conditions needed for the formation of a metastable cubic polytype of SiC.
443 A maximum temperature to avoid melting is determined in section \ref{section:md:tval} to be 120 \% of the Si melting point but due to defects lowering the transition point a maximum temperature of 95 \% of the Si melting temperature is considered usefull.
444 This value is almost equal to the temperature of $2050\,^{\circ}\mathrm{C}$ already used in former simulations.
445 Since the maximum temperature is reached the approach is reduced to the application of longer time scales.
446 This is considered usefull since the estimated evolution of quality in the absence of the cooling down sequence in figure~\ref{fig:md:tot_si-c_q} predicts an increase in quality and, thus, structural evolution is liekyl to occur if the simulation is proceeded at maximum temperature.
448 Next to the employment of longer time scales and a maximum temperature a few more changes are applied.
449 In the following simulations the system volume, the amount of C atoms inserted and the shape of the insertion volume are modified from the values used in first MD simulations.
450 To speed up the simulation the initial simulation volume is reduced to 21 Si unit cells in each direction and 5500 inserted C atoms in either the whole volume or in a sphere with a radius of 3 nm corresponding to the size of a precipitate consisting of 5500 C atoms.
451 The 100 ps sequence after C insertion intended for structural evolution is exchanged by a 10 ns sequence, which is hoped to result in the occurence of infrequent processes and a subsequent phase transition.
452 The return to lower temperatures is considered seperately.
456 \includegraphics[width=0.7\textwidth]{c_in_si_95_v1_si-c.ps}\\
457 \includegraphics[width=0.7\textwidth]{c_in_si_95_v1_c-c.ps}
459 \caption{Si-C (top) and C-C (bottom) radial distribution for low concentration simulations at 95 \% of the potential's Si melting point at different points in time of the simulation.}
460 \label{fig:md:95_long_time_v1}
464 \includegraphics[width=0.7\textwidth]{c_in_si_95_v2.ps}
466 \caption{Si-C and C-C radial distribution for high concentration simulations at 95 \% of the potential's Si melting point at different points in time of the simulation.}
467 \label{fig:md:95_long_time_v2}
470 Fig.~\ref{fig:md:95_long_time_v1} shows the evolution in time of the radial distribution for Si-C and C-C pairs for a low C concentration simulation.
471 Differences are observed for both types of atom pairs indeed indicating proceeding structural changes even well beyond 100 ps of simulation time.
472 Peaks attributed to the existence of substitutional C increase and become more distinct.
473 This finding complies with the predicted increase of quality evolution as explained earlier.
474 More and more C forms tetrahedral bonds to four Si neighbors occupying vacant Si sites.
475 However, no increase of the amount of total C-C pairs within the observed region can be identified.
476 C, whether substitutional or as a DB does not agglomerate within the simulated period of time visible by the unchanging area beneath the graphs.
478 Fig.~\ref{fig:md:95_long_time_v2} shows the evolution in time of the radial distribution for Si-C and C-C pairs for a high C concentration simulation.
479 There are only small changes identifiable.
480 A slight increase of the Si-C peak at approximately 0.36 nm attributed to the distance of substitutional C and the next but one Si atom along \hkl<1 1 0> is observed.
481 In the same time the C-C peak at approximately 0.32 nm corresponding to the distance of two C atoms interconnected by a Si atom along \hkl<1 1 0> slightly decreases.
482 Obviously the system preferes a slight increase of isolated substitutional C at the expense of incoherent C-Si-C precipitate configurations, which at a first glance actually appear as promising configurations in the precipitation event.
483 On second thoughts however, this process of splitting a C atom out of this structure is considered necessary in order to allow for the rearrangement of C atoms on substitutional lattice sites on the one hand and for C diffusion otherwise, which is needed to end up in a structure, in which one of the two fcc sublattices is composed out of C only.
485 For both, high and low concentration simulations the radial distribution converges as can be seen by the nearly identical graphs of the two most advanced configurations.
486 Changes exist ... bridge to results after cooling down to 20 degree C.
488 {\color{red}Todo: Cooling down to $20\,^{\circ}\mathrm{C}$ by $1\,^{\circ}\mathrm{C/s}$ in progress.}
490 % todo evtl in ausblick
492 %\subsection{Further accelerated dynamics approaches}
493 %{\color{red}Todo: self-guided MD?}
494 %{\color{red}Todo: ART MD?\\
495 %How about forcing a migration of a $V_2$ configuration to a constructed prec configuration, determine the saddle point configuration and continue the simulation from this configuration?
500 \section{Conclusions concerning the SiC conversion mechanism}
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