The simulation sequence and other parameters aside system temperature remain unchanged as in section \ref{subsection:initial_sims}.
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.
Temperatures ranging from $450\,^{\circ}\mathrm{C}$ up to $2050\,^{\circ}\mathrm{C}$ are used.
-A quality value $Q$ is introduced, which is defined as
+
+A simple quality value $Q$ is introduced, which helps to estimate the progress of structural evolution.
+In bulk 3C-SiC every C atom has four next neighboured Si atoms and every Si atom four next neighboured C atoms.
+The quality could be determined by counting the amount of atoms which form bonds to four atoms of the other species.
+However, the aim of the simulation on hand is to reproduce the formation of a 3C-SiC precipitate embedded in c-Si.
+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.
+Thus, counting the atoms, which exhibit proper coordination is limited to the C atoms.
+The quality value is defined to be
\begin{equation}
Q = \frac{\text{Amount of C atoms with 4 next neighboured Si atoms}}
{\text{Total amount of C atoms}} \text{ .}
\label{eq:md:qdef}
\end{equation}
-In 3C-SiC every C atom has four next neighboured Si atoms resulting in $Q=1$.
+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.
+Structures that look promising due to high quality values need to be further investigated by other means.
-Figure ... shows the radial distribution of Si-C bonds and the corresponding quality paragraphs.
+\begin{figure}[!ht]
+\begin{center}
+\includegraphics[width=12cm]{tot_pc_thesis.ps}\\
+\includegraphics[width=12cm]{tot_ba.ps}
+\end{center}
+\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.}
+\label{fig:md:tot_si-c_q}
+\end{figure}
+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.
+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 $1650\,^{\circ}\mathrm{C}$.
+The system provides enough kinetic energy to affected atoms, which are able to escape the cut-off region.
+Another important observation in structural change is exemplified in the two shaded areas.
+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.
+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.
+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.
+Investigations of the atomic data reveal substitutional carbon to be responsible for the new Si-C bonds.
+The peak at 0.197 nm corresponds to the distance of a substitutional carbon to the next neighboured silicon atoms.
+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.
+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.
+The resulting bonds perfectly match and, thus, explain the peaks observed for the increased temperature simulations.
+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.
+
+This is also reflected in the quality values obtained for different temperatures.
+While simulations at $450\,^{\circ}\mathrm{C}$ exhibit 10 \% of fourfold coordinated carbon simulations at $2050\,^{\circ}\mathrm{C}$ exceed the 80 \% range.
+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.
+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.
+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
+\begin{equation}
+f(t)=a-\frac{b}{t} \text{ ,}
+\label{eq:md:fit}
+\end{equation}
+which results in a saturation value of 93 \%.
+Obviously the decrease in temperature accelerates the saturation and inhibits further formation of substitutional carbon.
+Conclusions drawn from investigations of the quality evolution correlate well with the findings of the radial distribution results.
+
+\begin{figure}[!ht]
+\begin{center}
+\includegraphics[width=12cm]{tot_pc2_thesis.ps}\\
+\includegraphics[width=12cm]{tot_pc3_thesis.ps}
+\end{center}
+\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.}
+\label{fig:md:tot_c-c_si-si}
+\end{figure}
+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}.
+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.
+It corresponds to the distance of second next neighboured Si atoms along a \hkl<1 1 0>-equivalent direction with substitutional C inbetween.
+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.
+
+In the upper part of figure \ref{fig:md:tot_c-c_si-si} the C-C radial distribution is shown.
+With increasing temperature a decrease of the amount of next neighboured 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.
+A slight shift towards higher distances can be observed for the maximum above 0.3 nm.
+Arrows with dashed lines mark C-C distances resulting from \hkl<1 0 0> dumbbell combinations while the arrows with the solid line mark distances arising from combinations of substitutional C.
+The continuous dashed line corresponds to the distance of a substitutional C with a next neighboured \hkl<1 0 0> dumbbell.
+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.
+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.
+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.
+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.
+Again, this is a quite promising result, since the C atoms are taking the appropriate coordination as expected in 3C-SiC.
+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.
+
+\begin{figure}[!ht]
+\begin{center}
+\includegraphics[width=12cm]{12_pc_thesis.ps}\\
+\includegraphics[width=12cm]{12_pc_c_thesis.ps}
+\end{center}
+\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 $20\,^{\circ}\mathrm{C}$.}
+\label{fig:md:12_pc}
+\end{figure}
+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.
+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 neighbours indicating an increase for the long range order.
+However this change is rather small and no significant structural change is observeable.
+As for low temperatures order in the short range exist decreasing with increasing distance.
+The increase of the amount of Si-C pairs at 0.186 nm could pe positively interpreted since this type of bond also exists in 3C-SiC.
+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.
+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.
+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.
+Investigating the atomic data indeed reveals two C atoms which are bound to and interconnect by a third C atom to be responsible for this distance.
+The C-C peak at about 0.31 nm, wich is slightly shifted to higher distances (0.317 nm) with increasing temperature 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.
+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.
+
+This said, there is clear evidence that this is amorphous SiC
+However there is no significant change in structure.
+But there is a decrease in the artifacts of the potential.
+So, first limitations might be condiered as
+Now, more temperature to increase infrequent events ...
\subsection{Constructed 3C-SiC precipitate in crystalline silicon}
projects / lectures / latex.git / blobdiff