started molecular dynamics chapter

diff --git a/posic/thesis/defects.tex b/posic/thesis/defects.tex
index e0ddc83..5465ef8 100644 (file)
--- a/posic/thesis/defects.tex
+++ b/posic/thesis/defects.tex
@@ -1133,3 +1133,16 @@ Low migration barriers are necessary to obtain this configuration and in contras
 Thus, carbon interstitials and vacancies located close together are assumed to end up in such a configuration in which the carbon atom is tetrahedrally coordinated and bound to four silicon atoms as expected in silicon carbide.
 In contrast to the above, this would suggest a silicon carbide precipitation by succesive creation of substitutional carbon instead of the agglomeration of C-Si dumbbell interstitials followed by an abrupt precipitation.
 
+{\color{red}Todo:
+C atoms in 100 DB tightend in 110 direction, which is important for stress compensation in some combined constellations.
+}
+{\color{red}Todo:
+Most of the combinations should only be thought of intermediate configurations, which need to be transformed in the later SiC precipitation process.
+}
+{\color{red}Todo:
+Better structure, better language, better methodology!
+}
+{\color{red}Todo:
+Fit of lennard-jones an other rep + attr potentials in 110 interaction data!
+}
+

diff --git a/posic/thesis/md.tex b/posic/thesis/md.tex
new file mode 100644 (file)
index 0000000..6910563
--- /dev/null
+++ b/posic/thesis/md.tex
@@ -0,0 +1,69 @@
+\chapter{Molecular dynamics simulations}
+
+The molecular dynamics (MD) technique is used to gain insight into the behavior of carbon existing in different concentrations in crystalline silicon on the microscopic level at finite temperatures.
+Both, quantum-mechanical and classical potential molecular dynamics simulations are performed.
+While quantum-mechanical calculations are restricted to a few hundreds of atoms only small volumes composed of three unit cells in each direction and small carbon concentrations are simulated using the VASP code.
+Thus, investigations are restricted to the diffusion process of single carbon interstitials and the agglomeration of a few dumbbell interstitials in silicon.
+Using classical potentials volume sizes up to 31 unit cells in each direction and high carbon concentrations are realizable.
+Simulations targeting the formation of silicon carbide precipitates are, thus, attempted in classical potential calculations only.
+
+\section{Ab initio MD simulations}
+
+Molecular dynamics simulations of a single, two and ten carbon atoms in $3\times 3\times 3$ unit cells of crytsalline silicon are performed.
+
+\section{Classical potential MD simulations}
+
+\subsection{Initial simulations}
+
+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.
+The system temperature is set to $450\, ^{\circ}\mathrm{C}$, the temperature for which epitaxial growth of 3C-SiC films is achieved by ion beam synthesis (IBS).
+After equilibration of the kinetic and potential energy carbon atoms are consecutively inserted.
+The number of carbon atoms $N_{\text{Carbon}}$ necessary to form a spherical precipitate with radius $r$ is given by
+\begin{equation}
+ N_{\text{Carbon}}=\frac{4}{3}\pi r^3 \cdot \frac{4}{a_{\text{SiC}}^3}
+                  =\frac{16}{3} \pi \left( \frac{r}{a_{\text{SiC}}}\right)^3
+\label{eq:md:spheric_prec}
+\end{equation}
+with $a_{\text{SiC}}$ being the lattice constant of 3C-SiC.
+A total amount of 6000 carbon atoms corresponds to a radius of approximately 3 nm, which is discovered to be the minimal size for precipitates in IBS experiments.
+In separated simulations these 6000 carbon atoms are inserted in three regions of different volume ($V_1$, $V_2$, $V_3$) within the simulation cell.
+For reasons of simplification these regions are rectangularly shaped.
+$V_1$ is chosen to be the total simulation volume.
+$V_2$ approximately corresponds to the volume of a minimal 3C-SiC precipitate.
+$V_3$ is approximately the volume containing the necessary amount of silicon atoms to form such a precipitate, which is slightly smaller than $V_2$ due to the slightly lower silicon density of 3C-SiC compared to c-Si.
+For rectangularly shaped precipitates with side length $L$ equation \eqref{eq:md:quadratic_prec} holds.
+\begin{equation}
+ N_{\text{Carbon}} =4 \left( \frac{L}{a_{\text{SiC}}}\right)^3
+\label{eq:md:quadratic_prec}
+\end{equation}
+Table \ref{table:md:ins_vols} summarizes the side length of each of the three different insertion volumes determined by the equations mentioned above.
+\begin{table}
+\begin{center}
+\begin{tabular}{l c c c}
+\hline
+\hline
+ & $V_1$ & $V_2$ & $V_3$ \\
+\hline
+Side length [\AA] & 168.3 & 50.0 & 49.0 \\
+\hline
+\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 carbon atoms.}
+\label{table:md:ins_vols}
+\end{table}
+The insertion is realized in a way to keep the system temperature constant.
+In each of 600 insertion steps 10 carbon atoms are inserted at random positions within the respective region, which involves an increase in kinetic energy.
+Thus, the simulation is continued without adding more carbon atoms until the system temperature is equal to the chosen temperature again, which is realized by the thermostat decoupling excessive energy.
+Every inserted carbon atom must exhibit a distance greater or equal than 1.5 \AA{} to present neighboured atoms to prevent too high temperatures.
+Once the total amount of carbon is inserted the simulation is continued for 100 ps followed by a cooling-down process until room temperature, that is  $20\, ^{\circ}\mathrm{C}$ is reached.
+Figure \ref{} displays a flow chart of the applied steps involved in the simulation sequence.
+
+The radial distribution function for Si-C and C-C distances is shown in figure \ref{}.
+
+
+\subsection{Increased temperature simulations}
+
+\subsection{Simulations close to the silicon melting point}
+

diff --git a/posic/thesis/thesis.tex b/posic/thesis/thesis.tex
index 48c2546..ca2a362 100644 (file)
--- a/posic/thesis/thesis.tex
+++ b/posic/thesis/thesis.tex
@@ -76,7 +76,8 @@
 %\include{model}
 \include{simulation}
 \include{defects}
-\include{results}
+\include{md}
+%\include{results}
 \include{summary_outlook}
 
 \appendix{}