Due to promising advantages over the Tersoff potential the bond order potential of Erhard and Albe is used.
A time step of one fs is set.
-\subsection{Initial simulations}
+\subsection{Simulations at temperatures used in ion beam synthesis}
+\label{subsection:initial_sims}
In initial simulations aiming to reproduce a precipitation process simulation volumes of $31\times 31\times 31$ unit cells are utilized.
Periodic boundary conditions in each direction are applied.
Here is the problem.
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.
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.
-Thus, this transformation is very unlikely to happen.
+Thus, such conformational chamges are very unlikely to happen.
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.
Obviously no energetically favorable relaxation is taking place at a system temperature of $450\,^{\circ}\mathrm{C}$.
\label{fig:md:pc_si-c}
\end{figure}
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.
-The first peak observed for all insertion volumes is at approximately 0.185 nm.
+The first peak observed for all insertion volumes is at approximately 0.186 nm.
This corresponds quite well to the expected next neighbour distance of 0.189 nm for Si and C atoms in 3C-SiC.
+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.
+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.
+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}.
+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.
+Not only the peak locations but also the peak widths and heights become comprehensible.
+The distinct peak at 0.26 nm, which exactly matches the cut-off radius of the Si-C interaction, is again a potential artifact.
+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.
+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 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}.
+
+\subsection{Limitations of conventional MD and short range potentials}
+
+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.
+
+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 $10^{-14}\,\text{s}$ 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.
+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, that is 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.
+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.
+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.
+
+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 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}.
+Indeed it is not only the strong C-C bond which is hard to break inhibiting carbon diffusion and further rearrengements.
+This is also true for the low concentration simulations dominated by the occurrence of C-Si dumbbells spread over the whole simulation volume.
+The bonds of these C-Si pairs are also affected by the cut-off artifact preventing carbon diffusion and agglomeration of the dumbbells.
+This can be seen from the almost horizontal progress of the total energy graph in the continuation step, even for the low concentration simulation.
+The unphysical effects inherent to this type of model potentials are solely attributed to their short range character.
+However, since valueable 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.
+
+To conclude the obstacle needed to get passed is twofold.
+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.
+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.
+
+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.
+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}.
+The TAD corrections are not applied in coming up simulations.
+This is justified by two reasons.
+First of all a compensation of the overestimated bond strengthes 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 $450\,^{\circ}\mathrm{C}$ in IBS \cite{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.
+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.
\subsection{Increased temperature simulations}
-It is not only the C-C bonds which seem to be unbreakable.
-Also the C-Si pairs, as observed in the low concentration simulations, are stuck.
-This can be seen from the horizontal progress of the total energy graph in the continue-step.
-Higher time periods or alternatively higher temperatures to speed up the simulation are needed.
-{\color{red}Todo: Read again about the accelerated dynamics methods and maybe explain a bit more here.}
+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}.
+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 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}
+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.
+
+\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 $20\,^{\circ}\mathrm{C}$.}
+\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.
+
+TODO: mark 100-100 sub-100 and sub-sub C-C distances in plot ... then explain!
+
+\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.
+
+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 obtain a minimal and stable precipitate 5500 carbon atoms are considered necessary.
+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.
+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.
+In a second step 3C-SiC is created inside the empty sphere of radius $x$.
+The lattice constant $y$, the second unknown variable, is chosen in such a way, that the necessary amount of carbon is generated.
+This is entirely described by the system of equations \eqref{eq:md:constr_sic_01}
+\begin{equation}
+\frac{8}{a_{\text{Si}}^3}(
+\underbrace{21^3 a_{\text{Si}}^3}_{=V}
+-\frac{4}{3}\pi x^3)+
+\underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
+=21^3\cdot 8
+\label{eq:md:constr_sic_01}
+\text{ ,}
+\end{equation}
+which can be simplified to read
+\begin{equation}
+\frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500
+\Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}}
+\label{eq:md:constr_sic_02}
+\end{equation}
+and
+\begin{equation}
+%x^3=\frac{16\pi}{5500 \cdot 3}y^3=
+%\frac{16\pi}{5500 \cdot 3}\frac{5500 \cdot 3}{32 \pi}a_{\text{Si}}^3
+%\Rightarrow
+y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
+\text{ .}
+\label{eq:md:constr_sic_03}
+\end{equation}
+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.
+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.
+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}.
+\begin{table}[!ht]
+\begin{center}
+\begin{tabular}{l c c c c}
+\hline
+\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\\
+\hline
+\hline
+\end{tabular}
+\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.}
+\label{table:md:sic_prec}
+\end{center}
+\end{table}
+
+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.
+
+PC and energy of that one.
+
+Now let's see, whether annealing will lead to some energetically more favorable configurations.
+
+Estimate surface energy ...
\subsection{Simulations at temperatures exceeding the silicon melting point}
LL Cool J is hot as hell!
-
projects / lectures / latex.git / blobdiff