1 \chapter{Molecular dynamics 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 \section{Classical potential MD simulations}
20 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.
21 The system pressure is set to zero pressure.
22 Due to promising advantages over the Tersoff potential the bond order potential of Erhard and Albe is used.
23 A time step of one fs is set.
25 \subsection{Simulations at temperatures used in ion beam synthesis}
26 \label{subsection:initial_sims}
28 In initial simulations aiming to reproduce a precipitation process simulation volumes of $31\times 31\times 31$ unit cells are utilized.
29 Periodic boundary conditions in each direction are applied.
30 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).
31 After equilibration of the kinetic and potential energy carbon atoms are consecutively inserted.
32 The number of carbon atoms $N_{\text{Carbon}}$ necessary to form a spherical precipitate with radius $r$ is given by
34 N_{\text{Carbon}}=\frac{4}{3}\pi r^3 \cdot \frac{4}{a_{\text{SiC}}^3}
35 =\frac{16}{3} \pi \left( \frac{r}{a_{\text{SiC}}}\right)^3
36 \label{eq:md:spheric_prec}
38 with $a_{\text{SiC}}$ being the lattice constant of 3C-SiC.
39 In IBS experiments the smallest precipitates observed have radii starting from 2 nm up to 4 nm.
40 For the initial simulations a total amount of 6000 carbon atoms corresponding to a radius of approximately 3.1 nm is chosen.
41 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.
42 For reasons of simplification these regions are rectangularly shaped.
43 $V_1$ is chosen to be the total simulation volume.
44 $V_2$ approximately corresponds to the volume of a minimal 3C-SiC precipitate.
45 $V_3$ 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.
46 The two latter insertion volumes are considered since no diffusion of carbon atoms is expected within the simulated period of time at prevalent temperatures.
47 {\color{red}Todo: Refere to diffusion simulations and Mattoni paper.}
48 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
50 N_{\text{Carbon}}^{\text{3C-SiC}} =4 \left( \frac{L}{a_{\text{SiC}}}\right)^3
52 N_{\text{Silicon}}^{\text{c-Si}} =8 \left( \frac{L}{a_{\text{Si}}}\right)^3 \text{ .}
55 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.
56 Looking at the carbon concentrations simulations can be distinguished in simulations occupying low ($V_1$) and high ($V_2$, $V_3$) concentrations of carbon.
59 \begin{tabular}{l c c c}
62 & $V_1$ & $V_2$ & $V_3$ \\
64 Side length [\AA] & 168.3 & 50.0 & 49.0 \\
65 Carbon concentration [$\frac{1}{\text{c-Si unit cell}}$] & 0.20 & 7.68 & 8.16\\
70 \caption{Side lengthes of the insertion volumes $V_1$, $V_2$ and $V_3$ used for the incoorperation of 6000 carbon atoms.}
71 \label{table:md:ins_vols}
74 The insertion is realized in a way to keep the system temperature constant.
75 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.
76 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.
77 Every inserted carbon atom must exhibit a distance greater or equal than 1.5 \AA{} to present neighboured atoms to prevent too high temperatures.
78 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.
79 Figure \ref{fig:md:prec_fc} displays a flow chart of the applied steps involved in the simulation sequence.
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153 \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.}
154 \label{fig:md:prec_fc}
157 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}.
160 \includegraphics[width=12cm]{sic_prec_450_si-si_c-c.ps}
162 \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.}
163 \label{fig:md:pc_si-si_c-c}
167 \includegraphics[width=12cm]{sic_prec_450_energy.ps}
169 \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.}
170 \label{fig:md:energy_450}
172 It is easily and instantly visible that there is no significant difference among the two simulations of high carbon concentration.
173 The first C-C peak appears at about 0.15 nm, which is compareable to the nearest neighbour distance of graphite or diamond.
174 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.
175 These carbon atoms are assumed to form strong bonds.
176 This is supported by figure \ref{fig:md:energy_450} displaying the total energy of all three simulations during the whole simulation sequence.
177 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.
178 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.
180 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.
181 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.
182 Thus, such conformational chamges are very unlikely to happen.
183 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.
184 Obviously no energetically favorable relaxation is taking place at a system temperature of $450\,^{\circ}\mathrm{C}$.
186 The C-C peak at about 0.31 nm perfectly matches the nearest neighbour distance of two carbon atoms in the 3C-SiC lattice.
187 As can be seen from the inset this peak is also observed for the $V_1$ simulation.
188 In 3C-SiC the same distance is also expected for nearest neighbour silicon atoms.
189 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.
190 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.
191 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.
192 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.
193 The cut-off function causes artificial forces pushing the Si atoms out of the cut-off region.
194 Without the abrubt increase a maximum around 0.31 nm gets even more conceivable.
195 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.
196 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.
200 \includegraphics[width=12cm]{sic_prec_450_si-c.ps}
202 \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.}
203 \label{fig:md:pc_si-c}
205 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.
206 The first peak observed for all insertion volumes is at approximately 0.186 nm.
207 This corresponds quite well to the expected next neighbour distance of 0.189 nm for Si and C atoms in 3C-SiC.
208 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.
209 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.
210 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}.
211 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.
212 Not only the peak locations but also the peak widths and heights become comprehensible.
213 The distinct peak at 0.26 nm, which exactly matches the cut-off radius of the Si-C interaction, is again a potential artifact.
215 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.
216 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.
217 Short range order indeed is observed but only hardly visible is the long range order.
218 This indicates the formation of an amorphous SiC-like phase.
219 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}.
221 \subsection{Limitations of conventional MD and short range potentials}
222 \label{subsection:md:limit}
224 At first the formation of an amorphous SiC-like phase is unexpected since IBS experiments show crystalline 3C-SiC precipitates at prevailing temperatures.
225 On closer inspection, however, reasons become clear, which are discussed in the following.
227 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.
228 In molecular systems, characteristic motions take place over a wide range of time scales.
229 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.
230 To avoid dicretization errors the integration timestep needs to be chosen smaller than the fastest vibrational frequency in the system.
231 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.
232 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.
233 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.
234 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.
235 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.
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 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}.
239 Indeed it is not only the strong C-C bond which is hard to break inhibiting carbon diffusion and further rearrengements.
240 This is also true for the low concentration simulations dominated by the occurrence of C-Si dumbbells spread over the whole simulation volume.
241 The bonds of these C-Si pairs are also affected by the cut-off artifact preventing carbon diffusion and agglomeration of the dumbbells.
242 This can be seen from the almost horizontal progress of the total energy graph in the continuation step, even for the low concentration simulation.
243 The unphysical effects inherent to this type of model potentials are solely attributed to their short range character.
244 However, since valueable insights into various physical properties can be gained using this potentials, modifications mainly affecting the cut-off were designed.
245 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}.
246 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.
248 To conclude the obstacle needed to get passed is twofold.
249 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.
250 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.
252 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.
253 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}.
254 The TAD corrections are not applied in coming up simulations.
255 This is justified by two reasons.
256 First of all a compensation of the overestimated bond strengthes due to the short range potential is expected.
257 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}.
258 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.
259 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.
260 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.
262 \subsection{Increased temperature simulations}
263 \label{subsection:md:inct}
265 Due to the limitations of short range potentials and conventional MD as discussed above elevated temperatures are used in the following.
266 The simulation sequence and other parameters aside system temperature remain unchanged as in section \ref{subsection:initial_sims}.
267 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.
268 Temperatures ranging from $450\,^{\circ}\mathrm{C}$ up to $2050\,^{\circ}\mathrm{C}$ are used.
270 A simple quality value $Q$ is introduced, which helps to estimate the progress of structural evolution.
271 In bulk 3C-SiC every C atom has four next neighboured Si atoms and every Si atom four next neighboured C atoms.
272 The quality could be determined by counting the amount of atoms which form bonds to four atoms of the other species.
273 However, the aim of the simulation on hand is to reproduce the formation of a 3C-SiC precipitate embedded in c-Si.
274 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.
275 Thus, counting the atoms, which exhibit proper coordination is limited to the C atoms.
276 The quality value is defined to be
278 Q = \frac{\text{Amount of C atoms with 4 next neighboured Si atoms}}
279 {\text{Total amount of C atoms}} \text{ .}
282 By this, bulk 3C-SiC will still result in $Q=1$ and precipitates will also reach values close to one.
283 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.
284 Structures that look promising due to high quality values need to be further investigated by other means.
288 \includegraphics[width=12cm]{tot_pc_thesis.ps}\\
289 \includegraphics[width=12cm]{tot_ba.ps}
291 \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.}
292 \label{fig:md:tot_si-c_q}
294 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.
295 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 $1650\,^{\circ}\mathrm{C}$.
296 The system provides enough kinetic energy to affected atoms, which are able to escape the cut-off region.
297 Another important observation in structural change is exemplified in the two shaded areas.
298 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.
299 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.
300 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.
301 Investigations of the atomic data reveal substitutional carbon to be responsible for the new Si-C bonds.
302 The peak at 0.197 nm corresponds to the distance of a substitutional carbon to the next neighboured silicon atoms.
303 The one at 0.372 is the distance of the substitutional carbon atom to the second next silicon neighbour along the \hkl<1 1 0> direction.
304 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.
305 The resulting bonds perfectly match and, thus, explain the peaks observed for the increased temperature simulations.
306 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.
308 This is also reflected in the quality values obtained for different temperatures.
309 While simulations at $450\,^{\circ}\mathrm{C}$ exhibit 10 \% of fourfold coordinated carbon simulations at $2050\,^{\circ}\mathrm{C}$ exceed the 80 \% range.
310 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.
311 The quality values at a fixed temperature increase with simulation time.
312 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.
313 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
315 f(t)=a-\frac{b}{t} \text{ ,}
318 which results in a saturation value of 93 \%.
319 Obviously the decrease in temperature accelerates the saturation and inhibits further formation of substitutional carbon.
320 Conclusions drawn from investigations of the quality evolution correlate well with the findings of the radial distribution results.
324 \includegraphics[width=12cm]{tot_pc2_thesis.ps}\\
325 \includegraphics[width=12cm]{tot_pc3_thesis.ps}
327 \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.}
328 \label{fig:md:tot_c-c_si-si}
330 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}.
331 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.
332 It corresponds to the distance of second next neighboured Si atoms along a \hkl<1 1 0>-equivalent direction with substitutional C inbetween.
333 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.
335 In the upper part of figure \ref{fig:md:tot_c-c_si-si} the C-C radial distribution is shown.
336 With increasing temperature a decrease of the amount of next neighboured C pairs can be observed.
337 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.
338 A slight shift towards higher distances can be observed for the maximum above 0.3 nm.
339 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.
340 The continuous dashed line corresponds to the distance of a substitutional C with a next neighboured \hkl<1 0 0> dumbbell.
341 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.
342 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.
343 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.
344 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.
345 Again, this is a quite promising result, since the C atoms are taking the appropriate coordination as expected in 3C-SiC.
346 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.
348 {\color{red}Todo: Summarize again! Mention, that the agglomeration necessary in order to form 3C-SiC is missing.}
352 \includegraphics[width=12cm]{12_pc_thesis.ps}\\
353 \includegraphics[width=12cm]{12_pc_c_thesis.ps}
355 \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 $20\,^{\circ}\mathrm{C}$.}
358 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.
359 Again, in both cases, the cut-off artifact decreases with increasing temperature.
360 Peaks that already exist for the low temperature simulations get slightly more distinct for elevated temperatures.
361 This is also true for peaks located past distances of next neighbours indicating an increase in the long range order.
362 However this change is rather small and no significant structural change is observeable.
363 Due to the continuity of high amounts of damage atomic configurations remain hard to identify even for the highest temperature.
364 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.
365 As for low temperatures order in the short range exists decreasing with increasing distance.
366 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.
367 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.
368 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.
369 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.
370 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.
371 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.
372 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.
373 This is similar for the Si-C peak at approximately 0.35 nm.
374 In this case, the Si and the C atom are bound to a central Si atom.
375 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.
377 Regarding the outcome of both, high and low concentration simulations at increased temperatures, encouraging conclusions can be drawn.
378 With the disappearance of the peaks at the respective cut-off radii one limitation of the short range potential seems to be accomplished.
379 In addition, sharper peaks in the radial distributions lead to the assumption of expeditious structural formation.
380 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.
381 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.
382 {\color{red}Todo: Formation energy of C sub and nearby Si self-int, to see whether this is a preferable state!}
383 In the first case these occupied states would be expected to be higher in energy than the states occupied at low temperatures.
384 Since substitutional C without the presence of a Si self-interstitial is energetically more favorable than the lowest defect structure obtained without removing a Si atom, that is the \hkl<1 0 0> dumbbell interstitial, and the migration of Si self-interstitials towards the sample surface can be assumed for real life experiments \cite{}, this approach is accepted as an accelerated way of approximatively describing the structural evolution.
385 {\color{red}Todo: If C sub and Si self-int is energetically more favorable, the migration towards the surface can be kicked out. Otherwise we should actually care about removal of Si! In any way these findings suggest a different prec model.}
387 \subsection{Valuation of a practicable temperature limit}
388 \label{subsection:md:tval}
392 \includegraphics[width=12cm]{fe_and_t.ps}
394 \caption{Free energy and temperature evolution of plain silicon at temperatures in the region around the melting transition.}
395 \label{fig:md:fe_and_t}
397 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.
398 So far, the highest temperature applied corresponds to 95 \% of the absolute silicon melting temperature, which is 2450 K and specific to the Erhard/Albe potential.
399 However, melting is not predicted to occur instantly after exceeding the melting point due to additionally required transition enthalpy and hysteresis behaviour.
400 To check for the possibly highest temperature at which a transition fails to appear plain silicon is heated up using a heating rate of $1\,^{\circ}\mathrm{C}/\text{ps}$.
401 Figure \ref{fig:md:fe_and_t} shows the free energy and temperature evolution in the region around the transition temperature.
402 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.
403 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 $50.2 \text{ kJ/mole}$.
404 The late transition probably occurs due to the high heating rate and, thus, a large hysteresis behaviour extending the temperature of transition.
405 To avoid melting transitions in further simulations system temperatures well below the transition point are considered safe.
406 Thus, in the following system temperatures of 100 \% and 120 \% of the silicon melting point are used.
408 \subsection{Constructed 3C-SiC precipitate in crystalline silicon}
410 Before proceeding with simulations at temperatrures exceeding the silicon melting point a spherical 3C-SiC precipitate enclosed in a c-Si surrounding is constructed as it is expected from IBS experiments and from simulations that finally succeed in simulating the precipitation event.
411 On the one hand this sheds light on characteristic values like the radial distribution function or the total amount of free energy for such a configuration that is aimed to be reproduced by simulation.
412 On the other hand, assuming a correct alignment of the precipitate with the c-Si matrix, properties of such precipitates and the surrounding as well as the interface can be investiagted.
413 Furthermore these investigations might establish the prediction of conditions necessary for the simulation of the precipitation process.
415 To construct a spherical and topotactically aligned 3C-SiC precipitate in c-Si, the approach illustrated in the following is applied.
416 A total simulation volume $V$ consisting of 21 unit cells of c-Si in each direction is created.
417 To obtain a minimal and stable precipitate 5500 carbon atoms are considered necessary.
418 This corresponds to a spherical 3C-SiC precipitate with a radius of approximately 3 nm.
419 The initial precipitate configuration is constructed in two steps.
420 In the first step the surrounding silicon matrix is created.
421 This is realized by just skipping the generation of silicon atoms inside a sphere of radius $x$, which is the first unknown variable.
422 The silicon lattice constant $a_{\text{Si}}$ of the surrounding c-Si matrix is assumed to not alter dramatically and, thus, is used for the initial lattice creation.
423 In a second step 3C-SiC is created inside the empty sphere of radius $x$.
424 The lattice constant $y$, the second unknown variable, is chosen in such a way, that the necessary amount of carbon is generated.
425 This is entirely described by the system of equations \eqref{eq:md:constr_sic_01}
427 \frac{8}{a_{\text{Si}}^3}(
428 \underbrace{21^3 a_{\text{Si}}^3}_{=V}
429 -\frac{4}{3}\pi x^3)+
430 \underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
432 \label{eq:md:constr_sic_01}
435 which can be simplified to read
437 \frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500
438 \Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}}
439 \label{eq:md:constr_sic_02}
443 %x^3=\frac{16\pi}{5500 \cdot 3}y^3=
444 %\frac{16\pi}{5500 \cdot 3}\frac{5500 \cdot 3}{32 \pi}a_{\text{Si}}^3
446 y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
448 \label{eq:md:constr_sic_03}
450 By this means values of 2.973 nm and 4.309 \AA{} are obtained for the initial precipitate radius and lattice constant of 3C-SiC.
451 Since the generation of atoms is a discrete process with regard to the size of the volume the expected amounts of atoms are not obtained.
452 However, by applying these values the final configuration varies only slightly from the expected one by five carbon and eleven silicon atoms, as can be seen in table \ref{table:md:sic_prec}.
455 \begin{tabular}{l c c c c}
458 & C in 3C-SiC & Si in 3C-SiC & Si in c-Si surrounding & total amount of Si\\
460 Obtained & 5495 & 5486 & 68591 & 74077\\
461 Expected & 5500 & 5500 & 68588 & 74088\\
462 Difference & -5 & -14 & 3 & -11\\
463 Notation & $N^{\text{3C-SiC}}_{\text{C}}$ & $N^{\text{3C-SiC}}_{\text{Si}}$
464 & $N^{\text{c-Si}}_{\text{Si}}$ & $N^{\text{total}}_{\text{Si}}$ \\
468 \caption{Comparison of the expected and obtained amounts of Si and C atoms by applying the values from equations \eqref{eq:md:constr_sic_02} and \eqref{eq:md:constr_sic_03} in the 3C-SiC precipitate construction approach.}
469 \label{table:md:sic_prec}
473 After the initial configuration is constructed some of the atoms located at the 3C-SiC/c-Si interface show small distances, which results in high repulsive forces acting on the atoms.
474 Thus, the system is equilibrated using strong coupling to the heat bath, which is set to be $20\,^{\circ}\mathrm{C}$.
475 Once the main part of the excess energy is carried out previous settings for the Berendsen thermostat are restored and the system is relaxed for another 10 ps.
479 \includegraphics[width=12cm]{pc_0.ps}
481 \caption[Radial distribution of a 3C-SiC precipitate embeeded in c-Si at $20\,^{\circ}\mathrm{C}$.]{Radial distribution of a 3C-SiC precipitate embeeded in c-Si at $20\,^{\circ}\mathrm{C}$. The Si-Si radial distribution of plain c-Si is plotted for comparison. Green arrows mark bumps in the Si-Si distribution of the precipitate configuration, which do not exist in plain c-Si.}
482 \label{fig:md:pc_sic-prec}
484 Figure \ref{fig:md:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration.
485 The Si-Si radial distribution for both, plain c-Si and the precipitate configuration show a maximum at a distance of 0.235 nm, which is the distance of next neighboured Si atoms in c-Si.
486 Although no significant change of the lattice constant of the surrounding c-Si matrix was assumed, surprisingly there is no change at all within observational accuracy.
487 Looking closer at higher order Si-Si peaks might even allow the guess of a slight increase of the lattice constant compared to the plain c-Si structure.
488 A new Si-Si peak arises at 0.307 nm, which is identical to the peak of the C-C distribution around that value.
489 It corresponds to second next neighbours in 3C-SiC, which applies for Si as well as C pairs.
490 The bumps of the Si-Si distribution at higher distances marked by the green arrows can be explained in the same manner.
491 They correspond to the fourth and sixth next neighbour distance in 3C-SiC.
492 It is easily identifiable how these C-C peaks, which imply Si pairs at same distances inside the precipitate, contribute to the bumps observed in the Si-Si distribution.
493 The Si-Si and C-C peak at 0.307 nm enables the determination of the lattic constant of the embedded 3C-SiC precipitate.
494 A lattice constant of 4.34 \AA{} compared to 4.36 \AA{} for bulk 3C-SiC is obtained.
495 This is in accordance with the peak of Si-C pairs at a distance of 0.188 nm.
496 Thus, the precipitate structure is slightly compressed compared to the bulk phase.
497 This is a quite surprising result since due to the finite size of the c-Si surrounding a non-negligible impact of the precipitate on the materializing c-Si lattice constant especially near the precipitate could be assumed.
498 However, it seems that the size of the c-Si host matrix is chosen large enough to even find the precipitate in a compressed state.
500 The absence of a compression of the c-Si surrounding is due to the possibility of the system to change its volume.
501 Otherwise the increase of the lattice constant of the precipitate of roughly 4.31 \AA{} in the beginning up to 4.34 \AA{} in the relaxed precipitate configuration could not take place without an accompanying reduction of the lattice constant of the c-Si surrounding.
502 If the total volume is assumed to be the sum of the volumes that are composed of Si atoms forming the c-Si surrounding and Si atoms involved forming the precipitate the expected increase can be calculated by
505 \frac{\frac{N^{\text{c-Si}}_{\text{Si}}}{8/a_{\text{c-Si of precipitate configuration}}}+
506 \frac{N^{\text{3C-SiC}}_{\text{Si}}}{4/a_{\text{3C-SiC of precipitate configuration}}}}
507 {\frac{N^{\text{total}}_{\text{Si}}}{8/a_{\text{plain c-Si}}}}
509 with the notation used in table \ref{table:md:sic_prec}.
510 The lattice constant of plain c-Si at $20\,^{\circ}\mathrm{C}$ can be determined more accurately by the side lengthes of the simulation box of an equlibrated structure instead of using the radial distribution data.
511 By this a value of $a_{\text{plain c-Si}}=5.439\text{ \AA}$ is obtained.
512 The same lattice constant is assumed for the c-Si surrounding in the precipitate configuration $a_{\text{c-Si of precipitate configuration}}$ since peaks in the radial distribution match the ones of plain c-Si.
513 Using $a_{\text{3C-SiC of precipitate configuration}}=4.34\text{ \AA}$ as observed from the radial distribution finally results in an increase of the initial volume by 0.12 \%.
514 However, each side length and the total volume of the simulation box is increased by 0.20 \% and 0.61 \% respectively compared to plain c-Si at $20\,^{\circ}\mathrm{C}$.
515 Since the c-Si surrounding resides in an uncompressed state the excess increase must be attributed to relaxation of strain with the strain resulting from either the compressed precipitate or the 3C-SiC/c-Si interface region.
516 This also explains the possibly identified slight increase of the c-Si lattice constant in the surrounding as mentioned earlier.
517 As the pressure is set to zero the free energy is minimized with respect to the volume enabled by the Berendsen barostat algorithm.
518 Apparently the minimized structure with respect to the volume is a configuration of a small compressively stressed precipitate and a large amount of slightly stretched c-Si in the surrounding.
520 In the following the 3C-SiC/c-Si interface is described in further detail.
521 One important size analyzing the interface is the interfacial energy.
522 It is determined exactly in the same way than the formation energy as described in equation \eqref{eq:defects:ef2}.
523 Using the notation of table \ref{table:md:sic_prec} and assuming that the system is composed out of $N^{\text{3C-SiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
526 N^{\text{3C-SiC}}_{\text{C}} \mu_{\text{SiC}}-
527 \left(N^{\text{total}}_{\text{Si}}-N^{\text{3C-SiC}}_{\text{C}}\right)
528 \mu_{\text{Si}} \text{ ,}
531 with $E$ being the free energy of the precipitate configuration at zero temperature.
532 An interfacial energy of 2267.28 eV is obtained.
533 The amount of C atoms together with the observed lattice constant of the precipitate leads to a precipitate radius of 29.93 \AA.
534 Thus, the interface tension, given by the energy of the interface devided by the surface area of the precipitate is $20.15\,\frac{\text{eV}}{\text{nm}^2}$ or $3.23\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$.
535 This is located inside the eperimentally estimated range of $2-8\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$ \cite{taylor93}.
537 Since the precipitate configuration is artificially constructed the resulting interface does not necessarily correspond to the energetically most favorable configuration or to the configuration that is expected for an actually grown precipitate.
538 Thus annealing steps are appended to the gained structure in order to allow for a rearrangement of the atoms of the interface.
539 The precipitate structure is rapidly heated up to $2050\,^{\circ}\mathrm{C}$ with a heating rate of approximately $75\,^{\circ}\mathrm{C}/\text{ps}$.
540 From that point on the heating rate is reduced to $1\,^{\circ}\mathrm{C}/\text{ps}$ and heating is continued to 120 \% of the Si melting temperature, that is 2940 K.
543 \includegraphics[width=12cm]{fe_and_t_sic.ps}
545 \caption{Free energy and temperature evolution of a constructed 3C-SiC precipitate embedded in c-Si at temperatures above the Si melting point.}
546 \label{fig:md:fe_and_t_sic}
548 Figure \ref{fig:md:fe_and_t_sic} shows the free energy and temperature evolution.
549 The sudden increase of the free energy indicates possible melting occuring around 2840 K.
552 \includegraphics[width=12cm]{pc_500-fin.ps}
554 \caption{Radial distribution of the constructed 3C-SiC precipitate embedded in c-Si at temperatures below and above the Si melting transition point.}
555 \label{fig:md:pc_500-fin}
557 Investigating the radial distribution function shown in figure \ref{fig:md:pc_500-fin}, which shows configurations below and above the temperature of the estimated transition, indeed supports the assumption of melting gained by the free energy plot.
558 However the precipitate itself is not involved, as can be seen from the Si-C and C-C distribution, which essentially stays the same for both temperatures.
559 Thus, it is only the c-Si surrounding undergoing a structural phase transition, which is very well reflected by the difference observed for the two Si-Si distributions.
560 This is surprising since the melting transition of plain c-Si is expected at temperatures around 3125 K, as discussed in the last section.
561 Obviously the precipitate lowers the transition point of the surrounding c-Si matrix.
562 For the rearrangement simulations temperatures well below the transition point should be used since it is very unlikely to recrystallize the molten Si surrounding properly when cooling down.
563 To play safe the precipitate configuration at 100 \% of the Si melting temperature is chosen and cooled down to $20\,^{\circ}\mathrm{C}$ with a cooling rate of $1\,^{\circ}\mathrm{C}/\text{ps}$.
564 {\color{blue}TODO: Wait for results and then compare structure (PC) and interface energy, maybe a energetically more favorable configuration arises.}
565 {\color{red}TODO: Mention the fact, that the precipitate is stable for eleveated temperatures, even for temperatures where the Si matrix is melting.}
567 \subsection{Simulations at temperatures exceeding the silicon melting point}
569 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.
570 A maximum temperature to avoid melting was determined in section \ref{subsection:md:tval}, which is 120 \% of the Si melting point.
571 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 the first MD simulations to now match the conditions given in the simulations of the self-constructed precipitate configuration for reasons of comparability.
572 To quantify, the initial simulation volume now consists of 21 Si unit cells in each direction and 5500 C atoms are inserted in either the whole volume or in a sphere with a radius of 3 nm.
573 Since the investigated temperatures exceed the Si melting point the initial Si bulk material is heated up slowly by $1\,^{\circ}\mathrm{C}/\text{ps}$ starting from $1650\,^{\circ}\mathrm{C}$ before the C insertion sequence is started.
577 \includegraphics[width=12cm]{fe_100.ps}
578 \includegraphics[width=12cm]{q_100.ps}
580 \caption[Evolution of the free energy and quality of a simulation at 100 \% of the Si melting temperature.]{Evolution of the free energy and quality of a simulation at 100 \% of the Si melting temperature. Matt colored parts of the graphs represent the C insertion sequence.}
581 \label{fig:md:exceed100}
585 \includegraphics[width=12cm]{fe_120.ps}
586 \includegraphics[width=12cm]{q_120.ps}
588 \caption[Evolution of the free energy and quality of a simulation at 120 \% of the Si melting temperature.]{Evolution of the free energy and quality of a simulation at 120 \% of the Si melting temperature. Matt colored parts of the graphs represent the C insertion sequence.}
589 \label{fig:md:exceed120}
591 Figure \ref{fig:md:exceed100} and \ref{fig:md:exceed120} show the evolution of the free energy per atom and the quality at 100 \% and 120 \% of the Si melting temperature.
595 \subsection{Further accelerated dynamics approaches}
597 {\color{red}TODO: self-guided MD!}
599 {\color{red}TODO: other approaches!}
601 {\color{red}TODO: ART MD?}
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