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.
+Since 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.
$V_1$ 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.
+The two latter insertion volumes are considered since no diffusion of carbon atoms is expected at this temperature.
+{\color{red}Todo: Refere to diffusion simulations and Mattoni paper.}
For rectangularly shaped precipitates with side length $L$ the amount of carbon atoms in 3C-SiC and silicon atoms in c-Si is given by
\begin{equation}
N_{\text{Carbon}}^{\text{3C-SiC}} =4 \left( \frac{L}{a_{\text{SiC}}}\right)^3
N_{\text{Silicon}}^{\text{c-Si}} =8 \left( \frac{L}{a_{\text{Si}}}\right)^3 \text{ .}
\label{eq:md:quadratic_prec2}
\end{equation}
-Table \ref{table:md:ins_vols} summarizes the side length of each of the three different insertion volumes determined by equations \eqref{eq:md:quadratic_prec} and \eqref{eq:md:quadratic_prec2} and the resulting carbon concentrations inside these volumes.
+Table \ref{table:md:ins_vols} summarizes the side length of each of the three different insertion volumes determined by equations \eqref{eq:md:quadratic_prec} and \eqref{eq:md:quadratic_prec2} and the resulting carbon concentrations inside these volumes with possible carbon diffusion being neglected.
+Looking at the carbon concentrations simulations can be distinguished in simulations occupying low ($V_1$) and high ($V_2$, $V_3$) concentrations of carbon.
\begin{table}
\begin{center}
\begin{tabular}{l c c c}
\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{fig:md:prec_fc} displays a flow chart of the applied steps involved in the simulation sequence.
-\begin{figure}
+\begin{figure}[!ht]
\begin{center}
\begin{pspicture}(0,0)(15,17)
\trput*{\scriptsize true}
\end{pspicture}
\end{center}
-\caption{Flowchart of the simulation sequence used in the molecular dnymaics simulations aiming to reproduce the precipitation process.}
+\caption[Flowchart of the simulation sequence used in molecular dnymaics simulations aiming to reproduce the precipitation process.]{Flowchart of the simulation sequence used in molecular dnymaics simulations aiming to reproduce the precipitation process. $T_{\text{s}}$ and $p_{\text{s}}$ are the preset values for the system temperature and pressure. $T_{\text{avg}}$ is the averaged actual system temperature.}
\label{fig:md:prec_fc}
\end{figure}
The radial distribution function $g(r)$ for Si-C and C-C distances is shown in figure \ref{fig:md:pc_si-si_c-c}.
\begin{figure}[!ht]
\begin{center}
- \includegraphics[width=8cm]{pc_si-c_c-c_thesis.ps}
+ \includegraphics[width=12cm]{pc_si-c_c-c_thesis.ps}
\end{center}
\caption{Radial distribution function of the Si-C and C-C distances for 6000 carbon atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of $450\,^{\circ}\mathrm{C}$.}
\label{fig:md:pc_si-si_c-c}
\end{figure}
+It is easily and instantly visible that there is no significant difference among the two simulations of high carbon concentration in the $V_2$ and $V_3$ volumes.
+
+The first C-C peak appears at about 0.15 nm, which is compareable to the nearest neighbour distance of graphite or diamond.
+The number of C-C bonds is much smaller for $V_1$ than for $V_2$ and $V_3$ since carbon atoms are spread over the total simulation volume.
+These carbon atoms are assumed to form strong bonds.
+This is supported by figure \ref{fig:md:energy_450} displaying the total energy of all three simulations during the whole simulation sequence.
+{\color{red}Todo: Add figure and check continue for 100 fs!}
+A huge decrease of the total energy during carbon insertion is observed for the simulations with high carbon concentration in contrast to the $V_1$ simulation, which shows a slight increase.
+Here is the problem.
+Hard to break this bonds again, which is necessary for the 3C-SiC conversion.
+
+The C-C peak at about 0.31 nm perfectly matches the nearest neighbour distance of two carbon atoms in the 3C-SiC lattice.
+In 3C-SiC the same distance is also expected for nearest neighbour silicon atoms.
+Figure \ref{fig:md:si-si_450} shows the radial distribution of Si-Si bonds together with a reference graph for pure c-Si.
+Indeed non-zero $g(r)$ values around 0.31 nm are observed while the amount of Si pairs at regular c-Si distances of 0.24 nm and 0.38 nm decreases.
+However, no clear peak is observed but the interval of enhanced $g(r)$ values corresponds to the width of the C-C $g(r)$ peak.
+For low concentrations of carbon, that is the $V_1$ simulation and early stages of the $V_2$ and $V_3$ simulations, analyses of configurations in which Si-Si distances around 0.3 nm appear and which are identifiable despite a high amount of disorder, which is especially observed in high concentration simulations, identify the \hkl<1 0 0> C-Si dumbbell to be responsible for stretching the Si-Si next neighbour distance.
+This excellently agrees with the calculated value $r(13)$ in table \ref{tab:defects:100db_cmp} for a resulting Si-Si distance in the \hkl<1 0 0> C-Si dumbbell configuration.
+
\subsection{Increased temperature simulations}
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