1 \chapter{Silicon carbide precipitation simulations}
3 The molecular dynamics (MD) technique is used to gain insight into the behavior of carbon existing in different concentrations in crystalline silicon on the microscopic level at finite temperatures.
4 Both, quantum-mechanical and classical potential molecular dynamics simulations are performed.
5 Since quantum-mechanical calculations are restricted to a few hundreds of atoms only small volumes composed of three unit cells in each direction and small carbon concentrations are simulated using the VASP code.
6 Thus, investigations are restricted to the diffusion process of single carbon interstitials and the agglomeration of a few dumbbell interstitials in silicon.
7 Using classical potentials volume sizes up to 31 unit cells in each direction and high carbon concentrations are realizable.
8 Simulations targeting the formation of silicon carbide precipitates are, thus, attempted in classical potential calculations only.
10 \section{Ab initio MD simulations}
12 No pressure control, since VASP does not support this feature in MD mode.
13 The time step is set to one fs.
14 Explain some more parameters that differ from the latter calculations ...
16 Molecular dynamics simulations of a single, two and ten carbon atoms in $3\times 3\times 3$ unit cells of crytsalline silicon are performed.
18 {\color{red}Todo: ... in progress ...}
20 \section{Classical potential MD simulations}
22 In contrast to the quantum-mechanical MD simulations the developed classical potential MD code is able to do constant pressure simulations using the Berendsen barostat.
23 The system pressure is set to zero pressure.
24 Due to promising advantages over the Tersoff potential the bond order potential of Erhart and Albe is used.
25 A time step of one fs is set.
27 \subsection{Simulations at temperatures used in ion beam synthesis}
28 \label{subsection:initial_sims}
30 In initial simulations aiming to reproduce a precipitation process simulation volumes of $31\times 31\times 31$ unit cells are utilized.
31 Periodic boundary conditions in each direction are applied.
32 The system temperature is set to $450\, ^{\circ}\mathrm{C}$, the temperature for which epitaxial growth of 3C-SiC films is achieved by ion beam synthesis (IBS).
33 After equilibration of the kinetic and potential energy carbon atoms are consecutively inserted.
34 The number of carbon atoms $N_{\text{Carbon}}$ necessary to form a spherical precipitate with radius $r$ is given by
36 N_{\text{Carbon}}=\frac{4}{3}\pi r^3 \cdot \frac{4}{a_{\text{SiC}}^3}
37 =\frac{16}{3} \pi \left( \frac{r}{a_{\text{SiC}}}\right)^3
38 \label{eq:md:spheric_prec}
40 with $a_{\text{SiC}}$ being the lattice constant of 3C-SiC.
41 In IBS experiments the smallest precipitates observed have radii starting from 2 nm up to 4 nm.
42 For the initial simulations a total amount of 6000 carbon atoms corresponding to a radius of approximately 3.1 nm is chosen.
43 In separated simulations these 6000 carbon atoms are inserted in three regions of different volume ($V_1$, $V_2$, $V_3$) within the simulation cell.
44 For reasons of simplification these regions are rectangularly shaped.
45 $V_1$ is chosen to be the total simulation volume.
46 $V_2$ approximately corresponds to the volume of a minimal 3C-SiC precipitate.
47 $V_3$ is approximately the volume containing the necessary amount of silicon atoms to form such a precipitate, which is slightly smaller than $V_2$ due to the slightly lower silicon density of 3C-SiC compared to c-Si.
48 The two latter insertion volumes are considered since no diffusion of carbon atoms is expected within the simulated period of time at prevalent temperatures.
49 This is due to the overestimated activation energies for carbon diffusion as pointed out in section \ref{subsection:defects:mig_classical}.
50 For rectangularly shaped precipitates with side length $L$ the amount of carbon atoms in 3C-SiC and silicon atoms in c-Si is given by
52 N_{\text{Carbon}}^{\text{3C-SiC}} =4 \left( \frac{L}{a_{\text{SiC}}}\right)^3
54 N_{\text{Silicon}}^{\text{c-Si}} =8 \left( \frac{L}{a_{\text{Si}}}\right)^3 \text{ .}
57 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 carbon concentrations inside these volumes.
58 Looking at the carbon concentrations simulations can be distinguished in simulations occupying low ($V_1$) and high ($V_2$, $V_3$) concentrations of carbon.
61 \begin{tabular}{l c c c}
64 & $V_1$ & $V_2$ & $V_3$ \\
66 Side length [\AA] & 168.3 & 50.0 & 49.0 \\
67 Carbon concentration [$\frac{1}{\text{c-Si unit cell}}$] & 0.20 & 7.68 & 8.16\\
72 \caption{Side lengthes of the insertion volumes $V_1$, $V_2$ and $V_3$ used for the incoorperation of 6000 carbon atoms.}
73 \label{table:md:ins_vols}
76 The insertion is realized in a way to keep the system temperature constant.
77 In each of 600 insertion steps 10 carbon atoms are inserted at random positions within the respective region, which involves an increase in kinetic energy.
78 Thus, the simulation is continued without adding more carbon atoms until the system temperature is equal to the chosen temperature again, which is realized by the thermostat decoupling excessive energy.
79 Every inserted carbon atom must exhibit a distance greater or equal than 1.5 \AA{} to present neighboured atoms to prevent too high forces to occur.
80 Once the total amount of carbon is inserted the simulation is continued for 100 ps followed by a cooling-down process until room temperature, that is $20\, ^{\circ}\mathrm{C}$ is reached.
81 Figure \ref{fig:md:prec_fc} displays a flow chart of the applied steps involved in the simulation sequence.
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155 \caption[Flowchart of the simulation sequence used in molecular dnymaics simulations aiming to reproduce the precipitation process.]{Flowchart of the simulation sequence used in molecular dnymaics 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.}
156 \label{fig:md:prec_fc}
159 The radial distribution function $g(r)$ for C-C and Si-Si distances is shown in figure \ref{fig:md:pc_si-si_c-c}.
162 \includegraphics[width=12cm]{sic_prec_450_si-si_c-c.ps}
164 \caption[Radial distribution function of the C-C and Si-Si distances for 6000 carbon atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of $450\,^{\circ}\mathrm{C}$ and cooled down to room temperature.]{Radial distribution function of the C-C and Si-Si distances for 6000 carbon atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of $450\,^{\circ}\mathrm{C}$ 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.}
165 \label{fig:md:pc_si-si_c-c}
169 \includegraphics[width=12cm]{sic_prec_450_energy.ps}
171 \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 carbon insertion and the start of the cooling process respectively.}
172 \label{fig:md:energy_450}
174 It is easily and instantly visible that there is no significant difference among the two simulations of high carbon concentration.
175 The first C-C peak appears at about 0.15 nm, which is compareable to the nearest neighbour distance of graphite or diamond.
176 The number of C-C bonds is much smaller for $V_1$ than for $V_2$ and $V_3$ since carbon atoms are spread over the total simulation volume.
177 These carbon atoms are assumed to form strong bonds.
178 This is supported by figure \ref{fig:md:energy_450} displaying the total energy of all three simulations during the whole simulation sequence.
179 A huge decrease of the total energy during carbon insertion is observed for the simulations with high carbon concentration in contrast to the $V_1$ simulation, which shows a slight increase.
180 The difference in energy $\Delta$ growing within the carbon insertion process up to a value of roughly 0.06 eV per atom persists unchanged until the end of the simulation.
181 The excess amount of next neighboured strongly bounded C-C bonds in the high concentration simulations make these configurations energetically more favorable compared to the low concentration configuration.
182 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.
183 Thus, such conformational changes are very unlikely to happen.
184 This is in accordance with the constant total energy observed in the continuation step of 100 ps inbetween the end of carbon insertion and the cooling process.
185 Obviously no energetically favorable relaxation is taking place at a system temperature of $450\,^{\circ}\mathrm{C}$.
187 The C-C peak at about 0.31 nm perfectly matches the nearest neighbour distance of two carbon atoms in the 3C-SiC lattice.
188 As can be seen from the inset this peak is also observed for the $V_1$ simulation.
189 Investigating the corresponding coordinates of the atoms it turns out that concatenated and differently oriented C-Si \hkl<1 0 0> dumbbell interstitials constitute configurations yielding separations of C atoms by this distance.
190 In 3C-SiC the same distance is also expected for nearest neighbour silicon atoms.
191 The bottom of figure \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.
192 Indeed non-zero $g(r)$ values around 0.31 nm are observed while the amount of Si pairs at regular c-Si distances of 0.24 nm and 0.38 nm decreases.
193 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.
194 In addition the abrupt increase of Si pairs at 0.29 nm can be attributed to the Si-Si cut-off radius of 0.296 nm as used in the present bond order potential.
195 The cut-off function causes artificial forces pushing the Si atoms out of the cut-off region.
196 Without the abrubt increase a maximum around 0.31 nm gets even more conceivable.
197 For low concentrations of carbon, that is the $V_1$ simulation and early stages of the $V_2$ and $V_3$ simulations, analyses of configurations in which Si-Si distances around 0.3 nm appear and which are identifiable in regions of high disorder, which especially applies for the high concentration simulations, identify the \hkl<1 0 0> C-Si dumbbell to be responsible for stretching the Si-Si next neighbour distance.
198 This excellently agrees with the calculated value $r(13)$ in table \ref{tab:defects:100db_cmp} for a resulting Si-Si distance in the \hkl<1 0 0> C-Si dumbbell configuration.
202 \includegraphics[width=12cm]{sic_prec_450_si-c.ps}
204 \caption{Radial distribution function of the Si-C distances for 6000 carbon atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of $450\,^{\circ}\mathrm{C}$ and cooled down to room temperature together with Si-C bonds resulting in a C-Si \hkl<1 0 0> dumbbell configuration.}
205 \label{fig:md:pc_si-c}
207 Figure \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 C-Si \hkl<1 0 0> dumbbell configuration.
208 The first peak observed for all insertion volumes is at approximately 0.186 nm.
209 This corresponds quite well to the expected next neighbour distance of 0.189 nm for Si and C atoms in 3C-SiC.
210 By comparing the resulting Si-C bonds of a C-Si \hkl<1 0 0> dumbbell with the C-Si distances of the low concentration simulation it is evident that the resulting structure of the $V_1$ simulation is dominated by this type of defects.
211 This is not surpsising, since the \hkl<1 0 0> dumbbell is found to be the ground state defect of a C interstitial in c-Si and for the low concentration simulations a carbon interstitial is expected in every fifth silicon unit cell only, thus, excluding defect superposition phenomena.
212 The peak distance at 0.186 nm and the bump at 0.175 nm corresponds to the distance $r(3C)$ and $r(1C)$ as listed in table \ref{tab:defects:100db_cmp} and visualized in figure \ref{fig:defects:100db_cmp}.
213 In addition it can be easily identified that the \hkl<1 0 0> dumbbell configuration contributes to the peaks at about 0.335 nm, 0.386 nm, 0.434 nm, 0.469 nm and 0.546 nm observed in the $V_1$ simulation.
214 Not only the peak locations but also the peak widths and heights become comprehensible.
215 The distinct peak at 0.26 nm, which exactly matches the cut-off radius of the Si-C interaction, is again a potential artifact.
217 For high carbon concentrations, that is the $V_2$ and $V_3$ simulation, the defect concentration is likewiese increased and a considerable amount of damage is introduced in the insertion volume.
218 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.
219 Short range order indeed is observed but only hardly visible is the long range order.
220 This indicates the formation of an amorphous SiC-like phase.
221 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}.
223 \subsection{Limitations of conventional MD and short range potentials}
224 \label{subsection:md:limit}
226 At first the formation of an amorphous SiC-like phase is unexpected since IBS experiments show crystalline 3C-SiC precipitates at prevailing temperatures.
227 On closer inspection, however, reasons become clear, which are discussed in the following.
229 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.
230 In molecular systems, characteristic motions take place over a wide range of time scales.
231 Vibrations of the covalent bond take place on the order of $10^{-14}\,\text{s}$ of which the thermodynamic and kinetic properties are well described by MD simulations.
232 To avoid dicretization errors the integration timestep needs to be chosen smaller than the fastest vibrational frequency in the system.
233 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.
234 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, that is the system temperature.
235 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.
236 Hence, time scales covering the neccessary amount of infrequent events to observe long-term evolution are not accessible by traditional MD simulations, which are limited to the order of nanoseconds.
237 New methods have been developed to bypass the time scale problem like hyperdnyamics (HMD) \cite{voter97,voter97_2}, parallel replica dynamics \cite{voter98}, temperature acclerated dynamics (TAD) \cite{sorensen2000} and self-guided dynamics (SGMD) \cite{wu99} retaining proper thermodynmic sampling.
239 In addition to the time scale limitation, problems attributed to the short range potential exist.
240 The sharp cut-off funtion, which limits the interacting ions to the next neighboured atoms by gradually pushing the interaction force and energy to zero between the first and second next neighbour distance, is responsible for overestimated and unphysical high forces of next neighboured atoms \cite{tang95,mattoni2007}.
241 This is supported by the overestimated activation energies necessary for C diffusion as investigated in section \ref{subsection:defects:mig_classical}.
242 Indeed it is not only the strong C-C bond which is hard to break inhibiting carbon diffusion and further rearrengements.
243 This is also true for the low concentration simulations dominated by the occurrence of C-Si dumbbells spread over the whole simulation volume.
244 The bonds of these C-Si pairs are also affected by the cut-off artifact preventing carbon diffusion and agglomeration of the dumbbells.
245 This can be seen from the almost horizontal progress of the total energy graph in the continuation step, even for the low concentration simulation.
246 The unphysical effects inherent to this type of model potentials are solely attributed to their short range character.
247 However, since valueable insights into various physical properties can be gained using this potentials, modifications mainly affecting the cut-off were designed.
248 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}.
249 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.
251 To conclude the obstacle needed to get passed is twofold.
252 The sharp cut-off of the used bond order model potential introduces overestimated high forces between next neighboured atoms enhancing the problem of slow phase space propagation immanent to MD simulations.
253 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.
255 Instead the first 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.
256 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}.
257 The TAD corrections are not applied in coming up simulations.
258 This is justified by two reasons.
259 First of all a compensation of the overestimated bond strengthes due to the short range potential is expected.
260 Secondly there is no conflict applying higher temperatures without the TAD corrections, since crystalline 3C-SiC is also observed for higher temperatures than $450\,^{\circ}\mathrm{C}$ in IBS \cite{lindner01}.
261 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.
262 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.
263 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.
265 \subsection{Increased temperature simulations}
266 \label{subsection:md:inct}
268 Due to the limitations of short range potentials and conventional MD as discussed above elevated temperatures are used in the following.
269 The simulation sequence and other parameters aside system temperature remain unchanged as in section \ref{subsection:initial_sims}.
270 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 refered to as low carbon and high carbon concentration simulations.
271 Temperatures ranging from $450\,^{\circ}\mathrm{C}$ up to $2050\,^{\circ}\mathrm{C}$ are used.
273 A simple quality value $Q$ is introduced, which helps to estimate the progress of structural evolution.
274 In bulk 3C-SiC every C atom has four next neighboured Si atoms and every Si atom four next neighboured C atoms.
275 The quality could be determined by counting the amount of atoms which form bonds to four atoms of the other species.
276 However, the aim of the simulation on hand is to reproduce the formation of a 3C-SiC precipitate embedded in c-Si.
277 The amount of Si atoms and, thus, the amount of Si atoms remaining in the silicon diamond lattice is much higher than the amount of inserted C atoms.
278 Thus, counting the atoms, which exhibit proper coordination is limited to the C atoms.
279 The quality value is defined to be
281 Q = \frac{\text{Amount of C atoms with 4 next neighboured Si atoms}}
282 {\text{Total amount of C atoms}} \text{ .}
285 By this, bulk 3C-SiC will still result in $Q=1$ and precipitates will also reach values close to one.
286 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.
287 Structures that look promising due to high quality values need to be further investigated by other means.
289 \subsubsection{Low carbon concetration simulations}
293 \includegraphics[width=12cm]{tot_pc_thesis.ps}\\
294 \includegraphics[width=12cm]{tot_ba.ps}
296 \caption[Si-C radial distribution and quality evolution for the low concentration simulations at different elevated temperatures.]{Si-C radial distribution and quality evolution for the low concentration simulations at different elevated temperatures. All structures are cooled down to $20\,^{\circ}\mathrm{C}$. The grey line shows resulting Si-C bonds in a configuration of substitutional C in c-Si (C$_\text{sub}$) at zero temperature. Arrows in the quality plot mark the end of carbon 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.}
297 \label{fig:md:tot_si-c_q}
299 Figure \ref{fig:md:tot_si-c_q} shows the radial distribution of Si-C bonds for different temperatures and the corresponding quality evolution as defined earlier for the low concentration simulaton, that is the $V_1$ simulation.
300 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 $1650\,^{\circ}\mathrm{C}$.
301 The system provides enough kinetic energy to affected atoms, which are able to escape the cut-off region.
302 Another important observation in structural change is exemplified in the two shaded areas.
303 In the grey shaded region a decrease of the peak at 0.186 nm and the bump at 0.175 nm and a concurrent increase of the peak at 0.197 nm with increasing temperature is visible.
304 Similarly the peaks at 0.335 nm and 0.386 nm shrink in contrast to a new peak forming at 0.372 nm as can be seen in the yellow shaded region.
305 Obviously the structure obtained from the $450\,^{\circ}\mathrm{C}$ simulations, which is dominated by the existence of \hkl<1 0 0> C-Si dumbbells transforms into a different structure with increasing simulation temperature.
306 Investigations of the atomic data reveal substitutional carbon to be responsible for the new Si-C bonds.
307 The peak at 0.197 nm corresponds to the distance of a substitutional carbon atom to the next neighboured silicon atoms.
308 The one at 0.372 nm is the distance of a substitutional carbon atom to the second next silicon neighbour along a \hkl<1 1 0> direction.
309 Comparing the radial distribution for the Si-C bonds at $2050\,^{\circ}\mathrm{C}$ to the resulting Si-C bonds in a configuration of a substitutional carbon atom in crystalline silicon 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 \hkl<1 0 0> C-Si dumbbell characterized structure transforms into a structure dominated by substitutional C.
313 This is also reflected in the quality values obtained for different temperatures.
314 While simulations at $450\,^{\circ}\mathrm{C}$ exhibit 10 \% of fourfold coordinated carbon simulations at $2050\,^{\circ}\mathrm{C}$ exceed the 80 \% range.
315 Since substitutional carbon has four next neighboured silicon 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 $2050\,^{\circ}\mathrm{C}$ inside the range in which the simulation is continued at constant temperature for 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 93 \%.
324 Obviously the decrease in temperature accelerates the saturation and inhibits further formation of substitutional carbon.
325 \label{subsubsection:md:ep}
326 Conclusions drawn from investigations of the quality evolution correlate well with the findings of the radial distribution results.
330 \includegraphics[width=12cm]{tot_pc2_thesis.ps}\\
331 \includegraphics[width=12cm]{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> dumbbell 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 neighboured \hkl<1 0 0> dumbbell.}
334 \label{fig:md:tot_c-c_si-si}
336 The formation of substitutional carbon also affects the Si-Si radial distribution displayed in the lower part of figure \ref{fig:md:tot_c-c_si-si}.
337 Investigating the atomic strcuture indeed shows that the peak arising at 0.325 nm with increasing temperature is due to two Si atoms directly bound to a C substitutional.
338 It corresponds to the distance of second next neighboured Si atoms along a \hkl<1 1 0>-equivalent direction with substitutional C inbetween.
339 Since the expected distance of these Si pairs in 3C-SiC is 0.308 nm the existing SiC structures embedded in the c-Si host are stretched.
341 In the upper part of figure \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 neighboured 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 A slight shift towards higher distances can be observed for the maximum above 0.3 nm.
347 Arrows with dashed lines mark C-C distances resulting from \hkl<1 0 0> dumbbell combinations while the arrows with solid lines mark distances arising from combinations of substitutional C.
348 The continuous dashed line corresponds to the distance of a substitutional C with a next neighboured \hkl<1 0 0> dumbbell.
349 By comparison with the radial distribution it becomes evident that the shift accompanies the advancing transformation of \hkl<1 0 0> dumbbells into substitutional C.
350 Next to combinations of two substitutional C atoms and two \hkl<1 0 0> dumbbells respectively also combinations of \hkl<1 0 0> dumbbells with a substitutional C atom arise.
351 In addition, structures form that result in distances residing inbetween the ones obtained from combinations of mixed defect types and the ones obtained by substitutional C configurations, as can be seen by quite high g(r) values to the right of the continuous dashed line and to the left of the first arrow with a solid line.
352 For the most part these structures can be identified as configurations of one substitutional C atom with either another C atom that practically occupies a Si lattice site but with a Si interstitial 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.
353 Again, this is a quite promising result, since the C atoms are taking the appropriate coordination as expected in 3C-SiC.
354 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 dumbbells agglomerated.
356 To summarize, results of low concentration simulations show a phase transition in conjunction with an increase in temperature.
357 The C-Si \hkl<1 0 0> dumbbell dominated struture turns into a structure characterized by the occurence of an increasing amount of substitutional C with respect to temperature.
358 Although diamond and graphite like bonds are reduced no agglomeration of C is observed within the simulated time resulting in the formation of isolated structures of stretched SiC, which are adjusted to the c-Si host with respect to the lattice constant and alignement.
359 It would be conceivable that by agglomeration of further substitutional C atoms the interfacial energy could be overcome and a transition into an incoherent SiC precipitate could occur.
361 {\color{red}Todo: Results reinforce the assumption of an alternative precipitation model as already pointed out in the defects chapter.}
363 \subsubsection{High carbon concetration simulations}
367 \includegraphics[width=12cm]{12_pc_thesis.ps}\\
368 \includegraphics[width=12cm]{12_pc_c_thesis.ps}
370 \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 $20\,^{\circ}\mathrm{C}$.}
373 Figure \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.
374 Again, in both cases, the cut-off artifact decreases with increasing temperature.
375 Peaks that already exist for the low temperature simulations get slightly more distinct for elevated temperatures.
376 This is also true for peaks located past distances of next neighbours indicating an increase in the long range order.
377 However this change is rather small and no significant structural change is observeable.
378 Due to the continuity of high amounts of damage atomic configurations remain hard to identify even for the highest temperature.
379 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.
380 As for low temperatures order in the short range exists decreasing with increasing distance.
381 The increase of the amount of Si-C pairs at 0.186 nm could be positively interpreted since this type of bond also exists in 3C-SiC.
382 On the other hand the amount of next neighboured C atoms with a distance of approximately 0.15 nm, which is the distance of C in graphite or diamond, is likewise increased.
383 Thus, higher temperatures seem to additionally enhance a conflictive process, that is the formation of C agglomerates, instead of the desired process of 3C-SiC formation.
384 This is supported by the C-C peak at 0.252 nm, which corresponds to the second next neighbour distance in the diamond structure of elemental C.
385 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.
386 The C-C peak at about 0.31 nm, wich is slightly shifted to higher distances (0.317 nm) with increasing temperature still corresponds quite well to the next neighbour distance of C in 3C-SiC as well as a-SiC and indeed results from C-Si-C bonds.
387 The Si-C peak at 0.282 nm, 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.
388 This is similar for the Si-C peak at approximately 0.35 nm.
389 In this case, the Si and the C atom are bound to a central Si atom.
390 To summarize, the amorphous phase remains 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.
392 \subsubsection{Conclusions concerning the usage of increased temperatures}
394 Regarding the outcome of both, high and low concentration simulations at increased temperatures, encouraging conclusions can be drawn.
395 With the disappearance of the peaks at the respective cut-off radii one limitation of the short range potential seems to be accomplished.
396 In addition, sharper peaks in the radial distributions lead to the assumption of expeditious structural formation.
397 The increase in temperature leads to the occupation of new defect states, which is particularly evident but not limited to the low carbon concentration simulations.
400 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.
401 As already pointed out in section~\ref{section:defects:noneq_process_01} on page~\pageref{section:defects:noneq_process_01} and section~\ref{section:defects:noneq_process_02} on page~\pageref{section:defects:noneq_process_02} IBS is a nonequilibrium process, which might result in the formation of the thermodynamically less stable substitutional carbon and Si self-interstitital configuration.
402 Indeed 3C-SiC is metastable and if not fabricated by IBS requires strong deviation from equilibrium and/or low temperatures to stabilize the 3C polytype \cite{}.
403 In IBS highly energetic C atoms are able to generate vacant sites, which in turn can be occupied by highly mobile C atoms.
404 This is found to be favorable in the absence of the Si self-interstitial, which turned out to have a low interaction capture radius with a substitutional C atom very likely preventing the recombination into thermodynamically stable C-Si dumbbell interstitials for appropriate separations of the defect pair.
405 Results gained in this chapter show preferential occupation of Si lattice sites by substitutional C enabled by increased temperatures supporting the assumptions drawn from the defect studies of the last chapter.
407 Thus, employing increased temperatures is not exclusively usefull to accelerate the dynamics approximatively describing the structural evolution.
408 Moreover it can be considered a necessary condition to deviate the system out of equilibrium enabling the formation of 3C-SiC obviously realized by a successive agglomeration of substitutional C.
411 \subsection{Valuation of a practicable temperature limit}
412 \label{subsection:md:tval}
414 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.
415 So far, the highest temperature applied corresponds to 95 \% of the absolute silicon melting temperature, which is 2450 K and specific to the Erhart/Albe potential.
416 However, melting is not predicted to occur instantly after exceeding the melting point due to additionally required transition enthalpy and hysteresis behaviour.
417 To check for the possibly highest temperature at which a transition fails to appear plain silicon is heated up using a heating rate of $1\,^{\circ}\mathrm{C}/\text{ps}$.
418 Figure \ref{fig:md:fe_and_t} shows the free energy and temperature evolution in the region around the transition temperature.
419 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 silicon melting temperature.
420 The difference in free energy is 0.58 eV per atom corresponding to $55.7 \text{ kJ/mole}$, which compares quite well to the silicon enthalpy of melting of $50.2 \text{ kJ/mole}$.
421 The late transition probably occurs due to the high heating rate and, thus, a large hysteresis behaviour extending the temperature of transition.
422 To avoid melting transitions in further simulations system temperatures well below the transition point are considered safe.
423 According to this study temperatures of 100 \% and 120 \% of the silicon melting point could be used.
424 However, defects, which are introduced due to the insertion of C atoms are known to lower the transition point.
425 Indeed simulations show melting transitions already at the melting point whenever C is inserted.
426 Thus, the system temperature of 95 \% of the silicon melting point is considered the maximum limit.
429 \includegraphics[width=12cm]{fe_and_t.ps}
431 \caption{Free energy and temperature evolution of plain silicon at temperatures in the region around the melting transition.}
432 \label{fig:md:fe_and_t}
435 \subsection{Long time scale simulations at maximum temperature}
437 As discussed in section~\ref{subsection:md:limit} and~\ref{subsection: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.
438 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.
440 A maximum temperature to avoid melting is determined in section \ref{subsection: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.
441 This value is almost equal to the temperature of $2050\,^{\circ}\mathrm{C}$ already used in former simulations.
442 Since the maximum temperature is reached the approach is reduced to the application of longer time scales.
443 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.
445 Next to the employment of longer time scales and a maximum temperature a few more changes are applied.
446 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.
447 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.
448 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.
449 The return to lower temperatures is considered seperately.
453 \includegraphics[width=12cm]{c_in_si_95_v1_si-c.ps}\\
454 \includegraphics[width=12cm]{c_in_si_95_v1_c-c.ps}
456 \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.}
457 \label{fig:md:95_long_time_v1}
461 \includegraphics[width=12cm]{c_in_si_95_v2.ps}
463 \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.}
464 \label{fig:md:95_long_time_v2}
467 Figure \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.
468 Differences are observed for both types of atom pairs indeed indicating proceeding structural changes even well beyond 100 ps of simulation time.
469 Peaks attributed to the existence of substitutional C increase and become more distinct.
470 This finding complies with the predicted increase of quality evolution as explained earlier.
471 More and more C forms tetrahedral bonds to four Si neighbours occupying vacant Si sites.
472 However, no increase of the amount of total C-C pairs within the observed region can be identified.
473 Carbon, whether substitutional or as a dumbbell does not agglomerate within the simulated period of time visible by the unchanging area beneath the graphs.
475 Figure \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.
476 There are only small changes identifiable.
477 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.
478 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.
479 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.
480 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 carbon only.
482 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.
483 Changes exist ... bridge to results after cooling down to 20 degree C.
485 {\color{red}Todo: Cooling down to $20\,^{\circ}\mathrm{C}$ by $1\,^{\circ}\mathrm{C/s}$ in progress.}
487 {\color{red}Todo: Remember NVE simulations (prevent melting).}
489 \subsection{Further accelerated dynamics approaches}
491 Since longer time scales are not sufficient \ldots
493 {\color{red}Todo: self-guided MD?}
495 {\color{red}Todo: other approaches?}
497 {\color{red}Todo: ART MD?\\
498 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?
501 \section{Conclusions concerning the SiC conversion mechanism}
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