Crystalline silicon and cubic silicon carbide
}
+ \vspace{8pt}
+
{\bf Lattice types and unit cells:}
\begin{itemize}
\item Crystalline silicon (c-Si) has diamond structure\\
$\Rightarrow {\color{si-yellow}\bullet}$ are Si atoms,
${\color{gray}\bullet}$ are C atoms
\end{itemize}
+ \vspace{8pt}
\begin{minipage}{8cm}
{\bf Lattice constants:}
\[
\begin{slide}
{\large\bf
- Motivation / Introduction
+ Supposed Si to 3C-SiC conversion
}
\small
\vspace{12pt}
- Experimentally observed:
+ \begin{minipage}{7cm}
+ Experimentally observed [3]:
\begin{itemize}
\item Minimal diameter of precipitation: 4 - 5 nm
\item Equal orientation of Si and SiC (hkl)-planes
\end{itemize}
+ \end{minipage}
+ \begin{minipage}{6cm}
+ \vspace{32pt}
+ \hspace{16pt}
+ {\tiny [3] J. K. N. Lindner, Appl. Phys. A 77 (2003) 27.}
+ \end{minipage}
\end{slide}
Simulation details
}
- \vspace{12pt}
+ \small
- MD basics:
+ {\bf MD basics:}
\begin{itemize}
\item Microscopic description of N particle system
\item Analytical interaction potential
\item Hamilton's equations of motion as propagation rule\\
in 6N-dimensional phase space
- \item Observables obtained by time average
+ \item Observables obtained by time or ensemble averages
\end{itemize}
-
- \vspace{12pt}
-
- Application details:
+ {\bf Application details:}
\begin{itemize}
- \item Integrator: Velocity Verlet, timestep: $1\, fs$
- \item Ensemble: NVT, Berendsen thermostat, $\tau=100.0$
- \item Potential: Tersoff-like bond order potential\\
+ \item Integrator: Velocity Verlet, timestep: $1\text{ fs}$
+ \item Ensemble: isothermal-isobaric NPT [4]
+ \begin{itemize}
+ \item Berendsen thermostat:
+ $\tau_{\text{T}}=100\text{ fs}$
+ \item Brendsen barostat:\\
+ $\tau_{\text{P}}=100\text{ fs}$,
+ $\beta^{-1}=100\text{ GPa}$
+ \end{itemize}
+ \item Potential: Tersoff-like bond order potential [5]
\[
E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
\pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
\]
- \begin{center}
- {\scriptsize P. Erhart and K. Albe. Phys. Rev. B 71 (2005) 035211}
- \end{center}
\end{itemize}
+ {\tiny
+ [4] L. Verlet, Phys. Rev. 159 (1967) 98.}\\
+ {\tiny
+ [5] P. Erhart and K. Albe, Phys. Rev. B 71 (2005) 35211.}
\begin{picture}(0,0)(-240,-70)
\includegraphics[width=5cm]{tersoff_angle.eps}
\begin{slide}
{\large\bf
- Simulation details
+ Simulation sequence
}
\vspace{8pt}
- Interstitial simulations:
+ Interstitial configurations:
\vspace{8pt}
\begin{pspicture}(0,0)(7,8)
- \rput(3.5,7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=green]{
+ \rput(3.5,7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
\parbox{7cm}{
\begin{itemize}
\item Initial configuration: $9\times9\times9$ unit cells Si
\item Periodic boundary conditions
- \item $T=0 \, K$
+ \item $T=0\text{ K}$, $p=0\text{ bar}$
\end{itemize}
}}}}
\rput(3.5,3.5){\rnode{insert}{\psframebox{
Insertion of C / Si atom:
\begin{itemize}
\item $(0,0,0)$ $\rightarrow$ {\color{red}tetrahedral}
+ (${\color{red}\triangleleft}$)
\item $(-1/8,-1/8,1/8)$ $\rightarrow$ {\color{green}hexagonal}
+ (${\color{green}\triangleright}$)
\item $(-1/8,-1/8,-1/4)$, $(-1/4,-1/4,-1/4)$\\
$\rightarrow$ {\color{magenta}110 dumbbell}
+ (${\color{magenta}\Box}$,$\circ$)
\item random positions (critical distance check)
\end{itemize}
}}}}
- \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=cyan]{
+ \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
\parbox{3.5cm}{
Relaxation time: $2\, ps$
}}}}
\end{pspicture}
\begin{picture}(0,0)(-210,-45)
- \includegraphics[width=6cm]{unit_cell.eps}
+ \includegraphics[width=6cm]{unit_cell_s.eps}
\end{picture}
\end{slide}
\begin{minipage}[t]{4.3cm}
\underline{Tetrahedral}\\
- $E_f=3.41\, eV$\\
+ $E_f=3.41$ eV\\
\includegraphics[width=3.8cm]{si_self_int_tetra_0.eps}
\end{minipage}
\begin{minipage}[t]{4.3cm}
\underline{110 dumbbell}\\
- $E_f=4.39\, eV$\\
+ $E_f=4.39$ eV\\
\includegraphics[width=3.8cm]{si_self_int_dumbbell_0.eps}
\end{minipage}
\begin{minipage}[t]{4.3cm}
\underline{Hexagonal} \hspace{4pt}
\href{../video/si_self_int_hexa.avi}{$\rhd$}\\
- $E_f^{\star}\approx4.48\, eV$ (unstable!)\\
+ $E_f^{\star}\approx4.48$ eV (unstable!)\\
\includegraphics[width=3.8cm]{si_self_int_hexa_0.eps}
\end{minipage}
\underline{Random insertion}
\begin{minipage}{4.3cm}
- $E_f=3.97\, eV$\\
+ $E_f=3.97$ eV\\
\includegraphics[width=3.8cm]{si_self_int_rand_397_0.eps}
\end{minipage}
\begin{minipage}{4.3cm}
- $E_f=3.75\, eV$\\
+ $E_f=3.75$ eV\\
\includegraphics[width=3.8cm]{si_self_int_rand_375_0.eps}
\end{minipage}
\begin{minipage}{4.3cm}
- $E_f=3.56\, eV$\\
+ $E_f=3.56$ eV\\
\includegraphics[width=3.8cm]{si_self_int_rand_356_0.eps}
\end{minipage}
\begin{minipage}[t]{4.3cm}
\underline{Tetrahedral}\\
- $E_f=2.67\, eV$\\
+ $E_f=2.67$ eV\\
\includegraphics[width=3.8cm]{c_in_si_int_tetra_0.eps}
\end{minipage}
\begin{minipage}[t]{4.3cm}
\underline{110 dumbbell}\\
- $E_f=1.76\, eV$\\
+ $E_f=1.76$ eV\\
\includegraphics[width=3.8cm]{c_in_si_int_dumbbell_0.eps}
\end{minipage}
\begin{minipage}[t]{4.3cm}
\underline{Hexagonal} \hspace{4pt}
\href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
- $E_f^{\star}\approx5.6\, eV$ (unstable!)\\
+ $E_f^{\star}\approx5.6$ eV (unstable!)\\
\includegraphics[width=3.8cm]{c_in_si_int_hexa_0.eps}
\end{minipage}
\footnotesize
\begin{minipage}[t]{3.3cm}
- $E_f=0.47\, eV$\\
+ $E_f=0.47$ eV\\
\includegraphics[width=3.3cm]{c_in_si_int_001db_0.eps}
\begin{picture}(0,0)(-15,-3)
- 001 dumbbell
+ 100 dumbbell
\end{picture}
\end{minipage}
\begin{minipage}[t]{3.3cm}
- $E_f=1.62\, eV$\\
+ $E_f=1.62$ eV\\
\includegraphics[width=3.2cm]{c_in_si_int_rand_162_0.eps}
\end{minipage}
\begin{minipage}[t]{3.3cm}
- $E_f=2.39\, eV$\\
+ $E_f=2.39$ eV\\
\includegraphics[width=3.1cm]{c_in_si_int_rand_239_0.eps}
\end{minipage}
\begin{minipage}[t]{3.0cm}
- $E_f=3.41\, eV$\\
+ $E_f=3.41$ eV\\
\includegraphics[width=3.3cm]{c_in_si_int_rand_341_0.eps}
\end{minipage}
\begin{slide}
{\large\bf
- Simulation details
+ Results
+ } - <100> dumbbell configuration
+
+ \vspace{8pt}
+
+ \small
+
+ \begin{minipage}{4cm}
+ \begin{itemize}
+ \item $E_f=0.47$ eV
+ \item Very often observed
+ \item Most energetically\\
+ favorable configuration
+ \item Experimental\\
+ evidence [6]
+ \end{itemize}
+ \vspace{24pt}
+ {\tiny
+ [6] G. D. Watkins and K. L. Brower,\\
+ Phys. Rev. Lett. 36 (1976) 1329.
+ }
+ \end{minipage}
+ \begin{minipage}{8cm}
+ \includegraphics[width=9cm]{100-c-si-db_s.eps}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Simulation sequence
}
\small
\begin{pspicture}(0,0)(12,8)
% nodes
- \rput(3.5,6.5){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=green]{
+ \rput(3.5,6.5){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
\parbox{7cm}{
\begin{itemize}
\item Initial configuration: $31\times31\times31$ unit cells Si
\item Periodic boundary conditions
- \item $T=450\, ^{\circ}C$
- \item Equilibration of $E_{kin}$ and $E_{pot}$ for $600\, fs$
+ \item $T=450\, ^{\circ}\text{C}$, $p=0\text{ bar}$
+ \item Equilibration of $E_{kin}$ and $E_{pot}$
\end{itemize}
}}}}
- \rput(3.5,3.2){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=red]{
+ \rput(3.5,3.2){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
\parbox{7cm}{
- Insertion of $6000$ carbon atoms at constant\\
+ Insertion of 6000 carbon atoms at constant\\
temperature into:
\begin{itemize}
\item Total simulation volume {\pnode{in1}}
\item Volume of necessary amount of Si {\pnode{in3}}
\end{itemize}
}}}}
- \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=cyan]{
+ \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
\parbox{3.5cm}{
Cooling down to $20\, ^{\circ}C$
}}}}
\begin{slide}
{\large\bf
- Very first results of the SiC precipitation runs
- }
-
- \footnotesize
-
- \begin{minipage}[b]{6.9cm}
- \includegraphics[width=6.3cm]{../plot/sic_prec_energy.ps}
- \includegraphics[width=6.3cm]{../plot/sic_prec_temp.ps}
+ Results
+ } - SiC precipitation runs
+
+
+ \includegraphics[width=6.3cm]{pc_si-c_c-c.eps}
+ \includegraphics[width=6.3cm]{pc_si-si.eps}
+
+ \begin{minipage}[t]{6.3cm}
+ \tiny
+ \begin{itemize}
+ \item C-C peak at 0.15 nm similar to next neighbour distance of graphite
+ or diamond\\
+ $\Rightarrow$ Formation of strong C-C bonds
+ (almost only for high C concentrations)
+ \item Si-C peak at 0.19 nm similar to next neighbour distance in 3C-SiC
+ \item C-C peak at 0.31 nm equals C-C distance in 3C-SiC\\
+ (due to concatenated, differently oriented
+ <100> dumbbell interstitials)
+ \item Si-Si shows non-zero g(r) values around 0.31 nm like in 3C-SiC\\
+ and a decrease at regular distances\\
+ (no clear peak,
+ interval of enhanced g(r) corresponds to C-C peak width)
+ \end{itemize}
\end{minipage}
- \begin{minipage}[b]{5.5cm}
- \begin{itemize}
- \item {\color{red} Total simulation volume}
- \item {\color{green} Volume of minimal SiC precipitation}
- \item {\color{blue} Volume of necessary amount of Si}
- \end{itemize}
- \vspace{40pt}
- \includegraphics[width=6.3cm]{../plot/foo150.ps}
+ \begin{minipage}[t]{6.3cm}
+ \tiny
+ \begin{itemize}
+ \item Low C concentration (i.e. $V_1$):
+ The <100> dumbbell configuration
+ \begin{itemize}
+ \item is identified to stretch the Si-Si next neighbour distance
+ to 0.3 nm
+ \item is identified to contribute to the Si-C peak at 0.19 nm
+ \item explains further C-Si peaks (dashed vertical lines)
+ \end{itemize}
+ $\Rightarrow$ C atoms are first elements arranged at distances
+ expected for 3C-SiC\\
+ $\Rightarrow$ C atoms pull the Si atoms into the right
+ configuration at a later stage
+ \item High C concentration (i.e. $V_2$ and $V_3$):
+ \begin{itemize}
+ \item High amount of damage introduced into the system
+ \item Short range order observed but almost no long range order
+ \end{itemize}
+ $\Rightarrow$ Start of amorphous SiC-like phase formation\\
+ $\Rightarrow$ Higher temperatures required for proper SiC formation
+ \end{itemize}
\end{minipage}
\end{slide}
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