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}.
\subsection{Limitations of conventional MD and short range potentials}
+\label{subsection:md:limit}
At first the formation of an amorphous SiC-like phase is unexpected since IBS experiments show crystalline 3C-SiC precipitates at prevailing temperatures.
On closer inspection, however, reasons become clear, which are discussed in the following.
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.
\subsection{Increased temperature simulations}
+\label{subsection:md:inct}
Due to the limitations of short range potentials and conventional MD as discussed above elevated temperatures are used in the following.
The simulation sequence and other parameters aside system temperature remain unchanged as in section \ref{subsection:initial_sims}.
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.
+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.
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.
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.
+This is also true for peaks located past distances of next neighbours indicating an increase in the long range order.
However this change is rather small and no significant structural change is observeable.
-Due to the continuity of high amounts of damage investigations of atomic configurations below remain hard to identify even for the highest temperature.
-Other than in the low concentration simulations 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.
+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.
As for low temperatures order in the short range exists decreasing with increasing distance.
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.
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.
+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 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.
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 is similar for the Si-C peak at approximately 0.35 nm.
In this case, the Si and the C atom are bound to a central Si atom.
+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.
-Regarding these findings there is a clear evidence of the formation of an amorphous SiC-like phase for all high concentration simulations performed at various temperatures.
-No significant structural change is observed for elevated temperatures.
-However, with the disappearance of the peaks at the respective cut-off radii one limitation of the short range potential seems to be accomplished.
-In addition, sharper peaks in the radial distributions at distances that are also expected for a-SiC might indicate a slight acceleration of the dynamics carried out at elevated temperatures, that is an expeditious formation of a structure superiorly compareable to a-SiC.
-The increase in temperature leads to the occupation of new defect states, which is particularly evident for low carbon concentrations.
-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 enabled infrequent transitions to occur much faster, thus, leading to the intended acceleration of the dynamics and weakening of the unphysical quirks inherent to the potential.
+Regarding the outcome of both, high and low concentration simulations at increased temperatures, encouraging conclusions can be drawn.
+With the disappearance of the peaks at the respective cut-off radii one limitation of the short range potential seems to be accomplished.
+In addition, sharper peaks in the radial distributions lead to the assumption of expeditious structural formation.
+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.
+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.
{\color{red}Todo: Formation energy of C sub and nearby Si self-int, to see whether this is a preferable state!}
In the first case these occupied states would be expected to be higher in energy than the states occupied at low temperatures.
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.
{\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.}
+\subsection{Valuation of a practicable temperature limit}
+\label{subsection:md:tval}
+
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=12cm]{fe_and_t.ps}
\end{center}
-\caption{Free energy and temperature plot of plain silicon in the region around the transition temperature.}
+\caption{Free energy and temperature evolution of plain silicon at temperatures in the region around the melting transition.}
\label{fig:md:fe_and_t}
\end{figure}
-The assumed applicability 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.
+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.
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.
However, melting is not predicted to occur instantly after exceeding the melting point due to additionally required transition enthalpy and hysteresis behaviour.
To check for the possibly highest temperature at which a transition fails to appear plain silicon is heated up using a heating rate of $1\,^{\circ}\mathrm{C}/\text{ps}$.
\subsection{Constructed 3C-SiC precipitate in crystalline silicon}
-{\color{red}Todo: We want to know where we want to go ...}
-
-In the following 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 simulating the precipitation event.
-On the one hand this sheds light on characteristic values like the radial distribution function or the total amount of energy for configurations that are aimed to be reproduced by simulation possibly enabling the prediction of conditions necessary for the simulation of the precipitation process.
-On the other hand, assuming a correct alignment of the precipitate with the c-Si matrix, investigations of the behaviour of such precipitates and the surrounding can be made.
+Before proceeding with simulations at temperatrures around 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.
+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.
+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.
+Furthermore these investigations might establish the prediction of conditions necessary for the simulation of the precipitation process.
-To construct a spherical 3C-SiC precipitate in c-Si, the approach illustrated in the following is applied.
-A total simulation volume $V$ consisting of 21 unit cells of c-Si in each direction is used.
+To construct a spherical and topotactically aligned 3C-SiC precipitate in c-Si, the approach illustrated in the following is applied.
+A total simulation volume $V$ consisting of 21 unit cells of c-Si in each direction is created.
To obtain a minimal and stable precipitate 5500 carbon atoms are considered necessary.
+This corresponds to a spherical 3C-SiC precipitate with a radius of approximately 3 nm.
The initial precipitate configuration is constructed in two steps.
In the first step the surrounding silicon matrix is created.
This is realized by just skipping the generation of silicon atoms inside a sphere of radius $x$, which is the first unknown variable.
\hline
& C in 3C-SiC & Si in 3C-SiC & Si in c-Si surrounding & total amount of Si\\
\hline
-Expected & 5500 & 5500 & 68588 & 74088\\
Obtained & 5495 & 5486 & 68591 & 74077\\
-Difference & 5 & 14 & -3 & 11\\
+Expected & 5500 & 5500 & 68588 & 74088\\
+Difference & -5 & -14 & 3 & -11\\
+Notation & $N^{\text{3C-SiC}}_{\text{C}}$ & $N^{\text{3C-SiC}}_{\text{Si}}$
+ & $N^{\text{c-Si}}_{\text{Si}}$ & $N^{\text{total}}_{\text{Si}}$ \\
\hline
\hline
\end{tabular}
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.
Thus, the system is equilibrated using strong coupling to the heat bath, which is set to be $20\,^{\circ}\mathrm{C}$.
-Once the main part of th excess energy is carried out previous settings for the Berendsen thermostat are restored and the system is relaxed for another 10 ps.
+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.
+
+\begin{figure}[!ht]
+\begin{center}
+\includegraphics[width=12cm]{pc_0.ps}
+\end{center}
+\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.}
+\label{fig:md:pc_sic-prec}
+\end{figure}
+Figure \ref{fig:md:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration.
+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.
+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.
+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.
+A new Si-Si peak arises at 0.307 nm, which is identical to the peak of the C-C distribution around that value.
+It corresponds to second next neighbours in 3C-SiC, which applies for Si as well as C pairs.
+The bumps of the Si-Si distribution at higher distances marked by the green arrows can be explained in the same manner.
+They correspond to the fourth and sixth next neighbour distance in 3C-SiC.
+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.
+The Si-Si and C-C peak at 0.307 nm enables the determination of the lattic constant of the embedded 3C-SiC precipitate.
+A lattice constant of 4.34 \AA{} compared to 4.36 \AA{} for bulk 3C-SiC is obtained.
+This is in accordance with the peak of Si-C pairs at a distance of 0.188 nm.
+Thus, the precipitate structure is slightly compressed compared to the bulk phase.
+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.
+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.
+
+The absence of a compression of the c-Si surrounding is due to the possibility of the system to change its volume.
+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.
+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
+\begin{equation}
+ \frac{V}{V_0}=
+ \frac{\frac{N^{\text{c-Si}}_{\text{Si}}}{8/a_{\text{c-Si of precipitate configuration}}}+
+ \frac{N^{\text{3C-SiC}}_{\text{Si}}}{4/a_{\text{3C-SiC of precipitate configuration}}}}
+ {\frac{N^{\text{total}}_{\text{Si}}}{8/a_{\text{plain c-Si}}}}
+\end{equation}
+with the notation used in table \ref{table:md:sic_prec}.
+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.
+By this a value of $a_{\text{plain c-Si}}=5.439\text{ \AA}$ is obtained.
+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.
+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 \%.
+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 $20\,^{\circ}\mathrm{C}$.
+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.
+This also explains the possibly identified slight increase of the c-Si lattice constant in the surrounding as mentioned earlier.
+As the pressure is set to zero the free energy is minimized with respect to the volume enabled by the Berendsen barostat algorithm.
+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.
+
+In the following the 3C-SiC/c-Si interface is described in further detail.
+One important size analyzing the interface is the interfacial energy.
+It is determined exactly in the same way than the formation energy as described in equation \eqref{eq:defects:ef2}.
+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
+\begin{equation}
+ E_{\text{f}}=E-
+ N^{\text{3C-SiC}}_{\text{C}} \mu_{\text{SiC}}-
+ \left(N^{\text{total}}_{\text{Si}}-N^{\text{3C-SiC}}_{\text{C}}\right)
+ \mu_{\text{Si}} \text{ ,}
+\label{eq:md:ife}
+\end{equation}
+with $E$ being the free energy of the precipitate configuration at zero temperature.
+An interfacial energy of 2267.28 eV is obtained.
+The amount of C atoms together with the observed lattice constant of the precipitate leads to a precipitate radius of 29.93 \AA.
+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 $3.23\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$.
+This is located inside the eperimentally estimated range of $2-8\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$ \cite{taylor93}.
+
+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.
+Thus annealing steps are appended to the gained structure in order to allow for a rearrangement of the atoms of the interface.
+The precipitate structure is rapidly heated up to $2050\,^{\circ}\mathrm{C}$ with a heating rate of approximately $75\,^{\circ}\mathrm{C}/\text{ps}$.
+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.
+\begin{figure}[!ht]
+\begin{center}
+\includegraphics[width=12cm]{fe_and_t_sic.ps}
+\end{center}
+\caption{Free energy and temperature evolution of a constructed 3C-SiC precipitate embedded in c-Si at temperatures above the Si melting point.}
+\label{fig:md:fe_and_t_sic}
+\end{figure}
+Figure \ref{fig:md:fe_and_t_sic} shows the free energy and temperature evolution.
+The sudden increase of the free energy indicates possible melting occuring around 2840 K.
+\begin{figure}[!ht]
+\begin{center}
+\includegraphics[width=12cm]{pc_500-fin.ps}
+\end{center}
+\caption{Radial distribution of the constructed 3C-SiC precipitate embedded in c-Si at temperatures below and above the Si melting transition point.}
+\label{fig:md:pc_500-fin}
+\end{figure}
+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.
+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.
+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.
+This is surprising since the melting transition of plain c-Si is expected at temperatures around 3125 K, as discussed in the last section.
+Obviously the precipitate lowers the transition point of the surrounding c-Si matrix.
+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.
+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 $1\,^{\circ}\mathrm{C}/\text{ps}$.
+{\color{blue}Todo: Wait for results and then compare structure (PC) and interface energy, maybe a energetically more favorable configuration arises.}
+{\color{red}Todo: Mention the fact, that the precipitate is stable for eleveated temperatures, even for temperatures where the Si matrix is melting.}
+{\color{red}Todo: Si starts to melt at the interface, show pictures and explain, it is due to the defective interface region.}
+
+\subsection{Simulations at temperatures around the silicon melting point}
+
+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.
+A maximum temperature to avoid melting was determined in section \ref{subsection:md:tval}, which is 120 \% of the Si melting point.
+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.
+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.
+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.
+The 100 ps sequence at the respective temperature intended for the structural evolution is exchanged by a 10 ns sequence, which will hopefully result in the occurence of infrequent processes.
+The return to lower temperatures is considered seperately.
+
+\begin{figure}[!ht]
+\begin{center}
+\includegraphics[width=12cm]{fe_100.ps}
+\includegraphics[width=12cm]{q_100.ps}
+\end{center}
+\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.}
+\label{fig:md:exceed100}
+\end{figure}
+\begin{figure}[!ht]
+\begin{center}
+\includegraphics[width=12cm]{fe_120.ps}
+\includegraphics[width=12cm]{q_120.ps}
+\end{center}
+\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.}
+\label{fig:md:exceed120}
+\end{figure}
+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.
+
+{\color{red}Todo: Melting occurs, show and explain it and that it's due to the defects created.}
+
+{\color{red}Todo: Due to melting, after insertion, simulation is continued NVE, so melting hopefully will not occur, before it will be cooled down later on.}
+
+{\color{red}Todo: In additions simulations at 95 \% of the Si melting temperature are started again for longer times.}
-PC and energy of that one.
+\subsection{Further accelerated dynamics approaches}
-Now let's see, whether annealing will lead to some energetically more favorable configurations.
+{\color{red}Todo: self-guided MD!}
-Estimate surface energy ...
+{\color{red}Todo: other approaches?}
-\subsection{Simulations at temperatures exceeding the silicon melting point}
+{\color{red}
+Todo: ART MD?
+Also, how about forcing a migration of a $V_2$ configuration to a constructed prec configuration, detrmine the maximum saddle point and let the simulation run.
+}
-LL Cool J is hot as hell!
projects / lectures / latex.git / blobdiff