Thus, the simulation is continued without adding more C atoms until the system temperature is equal to the chosen temperature again.
This is realized by the thermostat, which decouples excessive energy.
Every inserted C atom must exhibit a distance greater or equal to \unit[1.5]{\AA} to neighbored atoms to prevent the occurrence of too high forces.
-Once the total amount of C is inserted, the simulation is continued for \unit[100]{ps} followed by a cooling-down process until room temperature, i.e. \unit[20]{$^{\circ}$C}, is reached.
+Once the total amount of C is inserted, the simulation is continued for \unit[100]{ps} followed by a cooling-down process until room temperature, i.e.\ \unit[20]{$^{\circ}$C}, is reached.
Fig.~\ref{fig:md:prec_fc} displays a flow chart of the applied steps involved in the simulation sequence.
\begin{figure}[tp]
\begin{center}
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 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.
+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]
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 likewise 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.
+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.
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}.
-In both cases, i.e. low and high C concentrations, the formation of 3C-SiC fails to appear.
+In both cases, i.e.\ low and high C concentrations, the formation of 3C-SiC fails to appear.
With respect to the precipitation model, the formation of C$_{\text{i}}$ \hkl<1 0 0> DBs indeed occurs for low C concentrations.
However, sufficient defect agglomeration is not observed.
For high C concentrations, a rearrangement of the amorphous SiC structure, which is not expected at prevailing temperatures, and a transition into 3C-SiC is not observed either.
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 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.
+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 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.
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.
+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}, 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.
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