At the beginning, simulations are performed, which try to mimic the conditions during IBS.
Results reveal limitations of the employed potential and MD in general.
With reference to the results of the last chapter, a workaround is discussed.
-The approach is follwed and, finally, results gained by the MD simulations are interpreted drawing special attention to the established controversy concerning precipitation of SiC in Si.
+The approach is followed and, finally, results gained by the MD simulations are interpreted drawing special attention to the established controversy concerning precipitation of SiC in Si.
\section{Simulations at temperatures used in IBS}
\label{section:initial_sims}
\hline
\end{tabular}
\end{center}
-\caption{Side lengthes of the insertion volumes $V_1$, $V_2$ and $V_3$ used for the incoorperation of 6000 C atoms.}
+\caption{Side lengths of the insertion volumes $V_1$, $V_2$ and $V_3$ used for the incorporation of 6000 C atoms.}
\label{table:md:ins_vols}
\end{table}
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\psframe*[linecolor=hb](3,11.5)(11,17)
- \rput[lt](3.2,16.8){\color{gray}INITIALIZIATION}
+ \rput[lt](3.2,16.8){\color{gray}INITIALIZATION}
\rput(7,16){\rnode{14}{\psframebox{Create $31\times 31\times 31$
unit cells of c-Si}}}
\rput(7,15){\rnode{13}{\psframebox{$T_{\text{s}}=450\,^{\circ}\mathrm{C}$,
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\rput[lt](3.2,10.8){\color{gray}CARBON INSERTION}
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+ \rput(7,10){\rnode{9}{\psframebox{Insertion of 10 C atoms}}}
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\pm1\,^{\circ}\mathrm{C}$}}}
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+ \rput(7,5.75){\rnode{5}{\psframebox{Continue for \unit[100]{ps}}}}
\ncline[]{->}{6}{5}
\trput*{\scriptsize true}
\end{figure}
It is easily and instantly visible that there is no significant difference among the two simulations of high C concentration.
Thus, in the following, the focus can indeed be directed to low ($V_1$) and high ($V_2$, $V_3$) C concentration simulations.
-The first C-C peak appears at about \unit[0.15]{nm}, which is compareable to the nearest neighbor distance of graphite or diamond.
+The first C-C peak appears at about \unit[0.15]{nm}, which is comparable to the nearest neighbor distance of graphite or diamond.
The number of C-C bonds is much smaller for $V_1$ than for $V_2$ and $V_3$ since C atoms are spread over the total simulation volume.
On average, there are only 0.2 C atoms per Si unit cell.
These C atoms are assumed to form strong bonds.
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.
In addition the abrupt increase of Si pairs at \unit[0.29]{nm} can be attributed to the Si-Si cut-off radius of \unit[0.296]{nm} as used in the present bond order potential.
The cut-off function causes artificial forces pushing the Si atoms out of the cut-off region.
-Without the abrubt increase, a maximum around \unit[0.31]{nm} gets even more conceivable.
-Analyses of randomly chosen configurations, in which distances around \unit[0.3]{nm} appear, identify \ci{} \hkl<1 0 0> DBs to be responsible for stretching the Si-Si next neighbour distance for low C concentrations, i.e. for the $V_1$ and early stages of $V_2$ and $V_3$ simulation runs.
+Without the abrupt increase, a maximum around \unit[0.31]{nm} gets even more conceivable.
+Analyses of randomly chosen configurations, in which distances around \unit[0.3]{nm} appear, identify \ci{} \hkl<1 0 0> DBs to be responsible for stretching the Si-Si next neighbor distance for low C concentrations, i.e. for the $V_1$ and early stages of $V_2$ and $V_3$ simulation runs.
This excellently agrees with the calculated value $r(13)$ in Table~\ref{tab:defects:100db_cmp} for a resulting Si-Si distance in the \ci \hkl<1 0 0> DB configuration.
\begin{figure}[tp]
The first peak observed for all insertion volumes is at approximately \unit[0.186]{nm}.
This corresponds quite well to the expected next neighbor distance of \unit[0.189]{nm} for Si and C atoms in 3C-SiC.
By comparing the resulting Si-C bonds of a \ci{} \hkl<1 0 0> DB with the C-Si distances of the low concentration simulation, it is evident that the resulting structure of the $V_1$ simulation is clearly dominated by this type of defect.
-This is not surpsising, since the \ci{} \hkl<1 0 0> DB is found to be the ground-state defect of a C interstitial in c-Si and, for the low concentration simulations, a C interstitial is expected in every fifth Si unit cell only, thus, excluding defect superposition phenomena.
+This is not surprising, since the \ci{} \hkl<1 0 0> DB is found to be the ground-state defect of a C interstitial in c-Si and, for the low concentration simulations, a C interstitial is expected in every fifth Si unit cell only, thus, excluding defect superposition phenomena.
The peak distance at \unit[0.186]{nm} and the bump at \unit[0.175]{nm} corresponds to the distance $r(3C)$ and $r(1C)$ as listed in Table~\ref{tab:defects:100db_cmp} and visualized in Fig.~\ref{fig:defects:100db_cmp}.
In addition, it can be easily identified that the \ci{} \hkl<1 0 0> DB configuration contributes to the peaks at about \unit[0.335]{nm}, \unit[0.386]{nm}, \unit[0.434]{nm}, \unit[0.469]{nm} and \unit[0.546]{nm} observed in the $V_1$ simulation.
Not only the peak locations but also the peak widths and heights become comprehensible.
The distinct peak at \unit[0.26]{nm}, which exactly matches the cut-off radius of the Si-C interaction, is again a potential artifact.
-For high C concentrations, i.e. the $V_2$ and $V_3$ simulation corresponding to a C density of about 8 atoms per c-Si unit cell, the defect concentration is likewiese increased and a considerable amount of damage is introduced in the insertion volume.
+For high C concentrations, i.e. the $V_2$ and $V_3$ simulation corresponding to a C density of about 8 atoms per c-Si unit cell, the defect concentration is likewise increased and a considerable amount of damage is introduced in the insertion volume.
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.
Short range order indeed is observed, i.e. the large amount of strong neighbored C-C bonds at \unit[0.15]{nm} as expected in graphite or diamond and Si-C bonds at \unit[0.19]{nm} as expected in SiC, but only hardly visible is the long range order.
This indicates the formation of an amorphous SiC-like phase.
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.
In molecular systems, characteristic motions take place over a wide range of time scales.
Vibrations of the covalent bond take place on the order of \unit[10$^{-14}$]{s}, of which the thermodynamic and kinetic properties are well described by MD simulations.
-To avoid dicretization errors, the integration timestep needs to be chosen smaller than the fastest vibrational frequency in the system.
+To avoid discretization errors, the integration time step needs to be chosen smaller than the fastest vibrational frequency in the system.
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.
This is due to the existence of several local minima in the free energy surface separated by large energy barriers compared to the kinetic energy of the particles, i.e. the system temperature.
-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.
+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 time step.
Hence, time scales covering the necessary amount of infrequent events to observe long-term evolution are not accessible by traditional MD simulations, which are limited to the order of nanoseconds.
New methods have been developed to bypass the time scale problem.
-The most famous appraiches are hyperdnyamics (HMD) \cite{voter97,voter97_2}, parallel replica dynamics \cite{voter98}, temperature acclerated dynamics (TAD) \cite{sorensen2000} and self-guided dynamics (SGMD) \cite{wu99}, which accelerate phase space propagation while retaining proper thermodynmic sampling.
+The most famous approaches are hyperdynamics (HMD) \cite{voter97,voter97_2}, parallel replica dynamics \cite{voter98}, temperature accelerated dynamics (TAD) \cite{sorensen2000} and self-guided dynamics (SGMD) \cite{wu99}, which accelerate phase space propagation while retaining proper thermodynamic sampling.
In addition to the time scale limitation, problems attributed to the short range potential exist.
-The sharp cut-off funtion, which limits the interacting ions to the next neighbored atoms by gradually pushing the interaction force and energy to zero between the first and second next neighbor distance, is responsible for overestimated and unphysical high forces of next neighbored atoms \cite{tang95,mattoni2007}.
+The sharp cut-off function, which limits the interacting ions to the next neighbored atoms by gradually pushing the interaction force and energy to zero between the first and second next neighbor distance, is responsible for overestimated and unphysical high forces of next neighbored atoms \cite{tang95,mattoni2007}.
This is supported by the overestimated activation energies necessary for C diffusion as investigated in section \ref{subsection:defects:mig_classical}.
-Indeed, it is not only the strong C-C bond, which is hard to break, inhibiting C diffusion and further rearrengements.
+Indeed, it is not only the strong C-C bond, which is hard to break, inhibiting C diffusion and further rearrangements.
This is also true for the low concentration simulations dominated by the occurrence of C-Si DBs spread over the whole simulation volume.
The bonds of these C-Si pairs are also affected by the cut-off artifact preventing C diffusion and agglomeration of the DBs.
This can be seen from the almost horizontal progress of the total energy graph in the continuation step of Fig.~\ref{fig:md:energy_450}, even for the low concentration simulation.
These unphysical effects inherent to this type of model potentials are solely attributed to their short range character.
While cohesive and formational energies are often well described, these effects increase for non-equilibrium structures and dynamics.
-However, since valueable insights into various physical properties can be gained using this potentials, modifications mainly affecting the cut-off were designed.
+However, since valuable insights into various physical properties can be gained using this potentials, modifications mainly affecting the cut-off were designed.
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}.
-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.
+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 challenge inherent to this study.
To conclude the obstacle needed to get passed is twofold.
The sharp cut-off of the employed bond order model potential introduces overestimated high forces between next neighbored atoms enhancing the problem of slow phase space propagation immanent to MD simulations.
This obstacle could be referred to as {\em potential enhanced slow phase space propagation}.
-Due to this, 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.
+Due to this, pushing the time scale to the limits of computational resources or applying one of the above mentioned accelerated dynamics methods exclusively will not be sufficient enough.
Instead, the approach followed in this study, is the use of higher temperatures as exploited in TAD to find transition pathways of one local energy minimum to another one more quickly.
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}.
This is justified by two reasons.
First of all, a compensation of the overestimated bond strengths due to the short range potential is expected.
Secondly, there is no conflict applying higher temperatures without the TAD corrections, since crystalline 3C-SiC is also observed for higher temperatures than \unit[450]{$^{\circ}$C} in IBS \cite{nejim95,lindner01}.
-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.
+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 occurring much more faster.
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.
-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.
+On the other hand, during implantation, the actual temperature inside the implantation volume is definitely higher than the experimentally determined temperature tapped from the surface of the sample.
\section{Increased temperature simulations}
\label{section:md:inct}
\label{eq:md:qdef}
\end{equation}
By this, bulk 3C-SiC will still result in $Q=1$ and precipitates will also reach values close to one.
-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.
+However, since the quality value does not account for bond lengths, bond angles, crystallinity or the stacking sequence, high values of $Q$ not necessarily correspond to structures close to 3C-SiC.
Structures that look promising due to high quality values need to be further investigated by other means.
\subsection{Low C concentration simulations}
\includegraphics[width=0.7\textwidth]{tot_pc_thesis.ps}\\
\includegraphics[width=0.7\textwidth]{tot_ba.ps}
\end{center}
-\caption[Si-C radial distribution and evolution of quality $Q$ for the low concentration simulations at different elevated temperatures.]{Si-C radial distribution and evolution of quality $Q$ according to equation \ref{eq:md:qdef} for the low concentration simulations at different elevated temperatures. All structures are cooled down to \degc{20}. The grey line shows resulting Si-C bonds in a configuration of \cs{} in c-Si (C$_\text{sub}$) at zero temperature. Arrows in the quality plot mark the end of C insertion and the start of the cooling down step. A fit function according to equation \eqref{eq:md:fit} shows the estimated evolution of quality in the absence of the cooling down sequence.}
+\caption[Si-C radial distribution and evolution of quality $Q$ for the low concentration simulations at different elevated temperatures.]{Si-C radial distribution and evolution of quality $Q$ according to equation \ref{eq:md:qdef} for the low concentration simulations at different elevated temperatures. All structures are cooled down to \degc{20}. The gray line shows resulting Si-C bonds in a configuration of \cs{} in c-Si (C$_\text{sub}$) at zero temperature. Arrows in the quality plot mark the end of C insertion and the start of the cooling down step. A fit function according to equation \eqref{eq:md:fit} shows the estimated evolution of quality in the absence of the cooling down sequence.}
\label{fig:md:tot_si-c_q}
\end{figure}
-Fig.~\ref{fig:md:tot_si-c_q} shows the radial distribution of Si-C bonds for different temperatures and the corresponding evolution of quality $Q$ as defined above for the low concentration simulaton.
+Fig.~\ref{fig:md:tot_si-c_q} shows the radial distribution of Si-C bonds for different temperatures and the corresponding evolution of quality $Q$ as defined above for the low concentration simulation.
The first noticeable and promising change in the Si-C radial distribution is the successive decline of the artificial peak at the Si-C cut-off distance with increasing temperature up to the point of disappearance at temperatures above \degc{1650}.
Obviously, sufficient kinetic energy is provided to affected atoms that are enabled to escape the cut-off region.
Additionally, a more important structural change is observed, which is illustrated in the two shaded areas in Fig.~\ref{fig:md:tot_si-c_q}.
%
-In the grey shaded region a decrease of the peak at \unit[0.186]{nm} and of the bump at \distn{0.175} accompanied by an increase of the peak at \distn{0.197} with increasing temperature is visible.
+In the gray shaded region a decrease of the peak at \unit[0.186]{nm} and of the bump at \distn{0.175} accompanied by an increase of the peak at \distn{0.197} with increasing temperature is visible.
Similarly, the peaks at \distn{0.335} and \distn{0.386} shrink in contrast to a new peak forming at \distn{0.372} as can be seen in the yellow shaded region.
Obviously, the structure obtained from the \degc{450} simulations, which is dominated by the existence of \ci{} \hkl<1 0 0> DBs, transforms into a different structure with increasing simulation temperature.
Investigations of the atomic data reveal \cs{} to be responsible for the new Si-C bonds.
While simulations at \degc{450} exhibit \perc{10} of fourfold coordinated C, simulations at \degc{2050} exceed the \perc{80} range.
Since \cs{} has four nearest neighbored Si atoms and is the preferential type of defect in elevated temperature simulations, the increase of the quality values become evident.
The quality values at a fixed temperature increase with simulation time.
-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.
+After the end of the insertion sequence marked by the first arrow, the quality is increasing and a saturation behavior, yet before the cooling process starts, can be expected.
The evolution of the quality value of the simulation at \degc{2050} inside the range, in which the simulation is continued at constant temperature for \unit[100]{fs}, is well approximated by the simple fit function
\begin{equation}
f(t)=a-\frac{b}{t} \text{ ,}
\label{fig:md:tot_c-c_si-si}
\end{figure}
The formation of \cs{} also affects the Si-Si radial distribution displayed in the lower part of Fig.~\ref{fig:md:tot_c-c_si-si}.
-Investigating the atomic strcuture indeed shows that the peak arising at \distn{0.325} with increasing temperature is due to two Si atoms that form direct bonds to the \cs{} atom.
+Investigating the atomic structure indeed shows that the peak arising at \distn{0.325} with increasing temperature is due to two Si atoms that form direct bonds to the \cs{} atom.
The peak corresponds to the distance of next neighbored Si atoms along the \hkl<1 1 0> bond chain with C$_{\text{s}}$ in between.
Since the expected distance of these Si pairs in 3C-SiC is \distn{0.308}, the existing SiC structures embedded in the c-Si host are stretched.
In the upper part of Fig.~\ref{fig:md:tot_c-c_si-si} the C-C radial distribution is shown.
The total amount of C-C bonds with a distance inside the displayed separation range does not change significantly.
-Thus, even for elevated temperatures, agglomeration of C atoms neccessary to form a SiC precipitate does not take place within the simulated time scale.
+Thus, even for elevated temperatures, agglomeration of C atoms necessary to form a SiC precipitate does not take place within the simulated time scale.
However, with increasing temperature, a decrease of the amount of next neighbored C pairs can be observed.
-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.
+This is a promising result gained by the high-temperature simulations since the breaking of these diamond and graphite like bonds is mandatory for the formation of 3C-SiC.
Obviously, acceleration of the dynamics occurred by supplying additional kinetic energy.
A slight shift towards higher distances can be observed for the maximum located shortly above \distn{0.3}.
Arrows with dashed lines mark C-C distances resulting from \ci{} \hkl<1 0 0> DB combinations while arrows with solid lines mark distances arising from combinations of \cs.
%However, this is contrary to the initial precipitation model proposed in section \ref{section:assumed_prec}, which assumes that the transformation into 3C-SiC takes place in a very last step once enough C-Si DBs agglomerated.
To summarize, results of low concentration simulations show a phase transition in conjunction with an increase in temperature.
-The \ci{} \hkl<1 0 0> DB dominated struture turns into a structure characterized by the occurence of an increasing amount of \cs{} with respect to temperature.
+The \ci{} \hkl<1 0 0> DB dominated structure turns into a structure characterized by the occurrence of an increasing amount of \cs{} with respect to temperature.
Clearly, the high-temperature results indicate the precipitation mechanism involving an increased participation of \cs.
Although diamond and graphite like bonds are reduced, no agglomeration of C is observed within the simulated time.
-Isolated structures of stretched SiC, which are adjusted to the c-Si host with respect to the lattice constant and alignement, are formed.
+Isolated structures of stretched SiC, which are adjusted to the c-Si host with respect to the lattice constant and alignment, are formed.
By agglomeration of \cs{} the interfacial energy could be overcome and a transition from a coherent and stretched SiC structure into an incoherent and partially strain-compensated SiC precipitate could occur.
Indeed, \si in the near surrounding is observed, which may initially compensate tensile strain in the stretched SiC structure or rearrange the \cs{} sublattice and finally serve as supply for additional C to form further SiC or compensate strain at the interface of the incoherent SiC precipitate and the Si host.
Again, in both cases, the cut-off artifact decreases with increasing temperature.
Peaks that already exist for the low temperature simulations get slightly more distinct for elevated temperatures.
This is also true for peaks located past distances of next neighbors indicating an increase in the long range order.
-However, this change is rather small and no significant structural change is observeable.
+However, this change is rather small and no significant structural change is observable.
Due to the continuity of high amounts of damage, atomic configurations remain hard to identify even for the highest temperature.
-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.
+Other than in the low concentration simulation, analyzed defect structures are no longer necessarily aligned to the primarily existing but successively disappearing c-Si host matrix inhibiting or at least hampering their identification and classification.
As for low temperatures, order in the short range exists decreasing with increasing separation.
The increase of the amount of Si-C pairs at \distn{0.186} could be positively interpreted since this type of bond also exists in 3C-SiC.
On the other hand, the amount of next neighbored C atoms with a distance of approximately \distn{0.15}, which is the distance of C in graphite or diamond, is likewise increased.
Thus, higher temperatures seem to additionally enhance a conflictive process, i.e. the formation of C agglomerates, obviously inconsistent with the desired process of 3C-SiC formation.
This is supported by the C-C peak at \distn{0.252}, which corresponds to the second next neighbor distance in the diamond structure of elemental C.
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.
-The C-C peak at about \distn{0.31}, wich is slightly shifted to higher distances (\distn{0.317}) with increasing temperature still corresponds quite well to the next neighbor distance of C in 3C-SiC as well as a-SiC and indeed results from C-Si-C bonds.
+The C-C peak at about \distn{0.31}, which is slightly shifted to higher distances (\distn{0.317}) with increasing temperature still corresponds quite well to the next neighbor distance of C in 3C-SiC as well as a-SiC and indeed results from C-Si-C bonds.
The Si-C peak at \distn{0.282}, which is pronounced with increasing temperature, is constructed out of a Si atom and a C atom, which are both bound to another central C atom.
This is similar for the Si-C peak at approximately \distn{0.35}.
In this case, the Si and the C atom are bound to a central Si atom.
Thus, for instance, it is not surprising that short range potentials show overestimated melting temperatures while properties of structures that are only slightly deviated from equilibrium are well described.
Due to this, increased temperatures are considered exceptionally necessary for modeling non-equilibrium processes and structures such as IBS and 3C-SiC.
-Thus, it is concluded that increased temperatures are not exclusively usefull to accelerate the dynamics approximatively describing the structural evolution.
+Thus, it is concluded that increased temperatures are not exclusively useful to accelerate the dynamics approximatively describing the structural evolution.
Moreover, it can be considered a necessary condition to deviate the system out of equilibrium enabling the formation of 3C-SiC, which is obviously realized by a successive agglomeration of \cs{}.
\ifnum1=0
As discussed in section~\ref{section:md:limit} and~\ref{section:md:inct} a further increase of the system temperature might help to overcome limitations of the short range potential and accelerate the dynamics involved in structural evolution.
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.
-A maximum temperature to avoid melting is determined in section \ref{section:md:tval} to be 120 \% of the Si melting point but due to defects lowering the transition point a maximum temperature of 95 \% of the Si melting temperature is considered usefull.
+A maximum temperature to avoid melting is determined in section \ref{section:md:tval} to be 120 \% of the Si melting point but due to defects lowering the transition point a maximum temperature of 95 \% of the Si melting temperature is considered useful.
This value is almost equal to the temperature of $2050\,^{\circ}\mathrm{C}$ already used in former simulations.
Since the maximum temperature is reached the approach is reduced to the application of longer time scales.
-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.
+This is considered useful 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 likely to occur if the simulation is proceeded at maximum temperature.
Next to the employment of longer time scales and a maximum temperature a few more changes are applied.
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.
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.
-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.
-The return to lower temperatures is considered seperately.
+The \unit[100]{ps} sequence after C insertion intended for structural evolution is exchanged by a \unit[10]{ns} sequence, which is hoped to result in the occurrence of infrequent processes and a subsequent phase transition.
+The return to lower temperatures is considered separately.
\begin{figure}[tp]
\begin{center}
\end{figure}
Fig.~\ref{fig:md:95_long_time_v1} shows the evolution in time of the radial distribution for Si-C and C-C pairs for a low C concentration simulation.
-Differences are observed for both types of atom pairs indeed indicating proceeding structural changes even well beyond 100 ps of simulation time.
+Differences are observed for both types of atom pairs indeed indicating proceeding structural changes even well beyond \unit[100]{ps} of simulation time.
Peaks attributed to the existence of substitutional C increase and become more distinct.
This finding complies with the predicted increase of quality evolution as explained earlier.
More and more C forms tetrahedral bonds to four Si neighbors occupying vacant Si sites.
There are only small changes identifiable.
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.
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.
-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.
+Obviously, the system prefers 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.
On second thoughts however, this process of splitting a C atom out of this structure is considered necessary in order to allow for the rearrangement of C atoms on substitutional lattice sites on the one hand and for C diffusion otherwise, which is needed to end up in a structure, in which one of the two fcc sublattices is composed out of C only.
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.
For the low C concentrations, time scales are still too low to observe C agglomeration sufficient for SiC precipitation, which is attributed to the slow phase space propagation inherent to MD in general.
However, a phase transition of the C$_{\text{i}}$-dominated into a clearly C$_{\text{s}}$-dominated structure is observed.
The amount of substitutionally occupied C atoms increases with increasing temperature.
-Isolated structures of stretched SiC adjusted to the c-Si host with respect to the lattice constant and alignement are formed.
+Isolated structures of stretched SiC adjusted to the c-Si host with respect to the lattice constant and alignment are formed.
Entropic contributions are assumed to be responsible for these structures at elevated temperatures that deviate from the ground state at 0 K.
Results of the MD simulations at different temperatures and C concentrations can be correlated to experimental findings.
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