\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
+\usepackage{latexsym}
\usepackage{ae}
\usepackage{calc} % Simple computations with LaTeX variables
\usepackage{fancyvrb} % Fancy verbatim environments
\usepackage{pstricks} % PSTricks with the standard color package
+\usepackage{pstricks}
+\usepackage{pst-node}
+
+\usepackage{epic}
+\usepackage{eepic}
+
\usepackage{graphicx}
\graphicspath{{../img/}}
+\usepackage[setpagesize=false]{hyperref}
+
\usepackage{semcolor}
\usepackage{semlayer} % Seminar overlays
\usepackage{slidesec} % Seminar sections and list of slides
\begin{document}
\extraslideheight{10in}
-\slideframe{plain}
+\slideframe{none}
+
+\pagestyle{empty}
% specify width and height
\slidewidth 27.7cm
% shift it into visual area properly
\def\slideleftmargin{3.3cm}
-\def\slidetopmargin{0.0cm}
+\def\slidetopmargin{0.6cm}
\newcommand{\ham}{\mathcal{H}}
\newcommand{\pot}{\mathcal{V}}
\newcommand{\foo}{\mathcal{U}}
\newcommand{\vir}{\mathcal{W}}
+% itemize level ii
+\renewcommand\labelitemii{{\color{gray}$\bullet$}}
+
% topic
\begin{slide}
\item Integrator, potential, ensemble control
\item Simulation sequence
\end{itemize}
- \item Results gained by simulation
+ \item Simulation results
\begin{itemize}
\item Interstitials in silicon
- \item $SiC$-precipitation experiments
+ \item SiC-precipitation experiments
\end{itemize}
\item Conclusion / Outlook
\end{itemize}
% start of contents
+\begin{slide}
+
+ {\large\bf
+ Motivation / Introduction
+ }
+
+ \vspace{16pt}
+
+ Reasons for investigating C in Si:
+
+ \begin{itemize}
+ \item 3C-SiC wide band gap semiconductor formation
+ \item Strained Si (no precipitation wanted!)
+ \end{itemize}
+
+ \vspace{16pt}
+
+ Si / 3C-SiC facts:
+
+ \begin{minipage}{8cm}
+ \begin{itemize}
+ \item Unit cell:
+ \begin{itemize}
+ \item {\color{yellow}fcc} $+$
+ \item {\color{gray}fcc shifted $1/4$ of volume diagonal}
+ \end{itemize}
+ \item Lattice constants: $4a_{Si}\approx5a_{SiC}$
+ \item Silicon density:
+ \[
+ \frac{n_{SiC}}{n_{Si}}=
+ \frac{4/a_{SiC}^3}{8/a_{Si}^3}=
+ \frac{5^3}{2\cdot4^3}={\color{cyan}97,66}\,\%
+ \]
+ \end{itemize}
+ \end{minipage}
+ \hspace{8pt}
+ \begin{minipage}{4cm}
+ \includegraphics[width=4cm]{sic_unit_cell.eps}
+ \end{minipage}
+
+\end{slide}
+
+ \small
\begin{slide}
{\large\bf
\small
\vspace{6pt}
- Supposed mechanism of the conversion of heavily carbon doped Si into SiC:
+ Supposed conversion mechanism of heavily carbon doped Si into SiC:
\vspace{8pt}
Precipitation of 3C-SiC + Creation of interstitials\\
\end{minipage}
- \begin{center}
- \[5a_{SiC}=4a_{Si} \quad \Rightarrow \quad
- \frac{n_{SiC}}{n_{Si}}=\frac{\frac{4}{a_{SiC}^3}}{\frac{8}{a_{Si}^3}}=
- \frac{5^3}{2\cdot4^3}=97,66\%
- \]
- \end{center}
+ \vspace{12pt}
- Experimentally observed minimal diameter of precipitation: 4 - 5 nm
+ Experimentally observed:
+ \begin{itemize}
+ \item Minimal diameter of precipitation: 4 - 5 nm
+ \item (hkl)-planes identical for Si and SiC
+ \end{itemize}
\end{slide}
Simulation details
}
+ \vspace{12pt}
+
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-dimemnsional phase space
+ in 6N-dimensional phase space
\item Observables obtained by time average
\end{itemize}
- \vspace{4pt}
+ \vspace{12pt}
Application details:
\begin{itemize}
- \item Integrator: velocity verlet, timestep: $1\, fs$
+ \item Integrator: Velocity Verlet, timestep: $1\, fs$
\item Ensemble control: NVT, Berendsen thermostat, $\tau=100.0$
\item Potential: Tersoff-like bond order potential\\
\[
\end{center}
\end{itemize}
+ \begin{picture}(0,0)(-240,-70)
+ \includegraphics[width=5cm]{tersoff_angle.eps}
+ \end{picture}
+
\end{slide}
\begin{slide}
Simulation details
}
+ \vspace{20pt}
+
Interstitial experiments:
+
+ \vspace{12pt}
+
\begin{itemize}
\item Initial configuration: $9\times9\times9$ unit cells Si
\item Periodic boundary conditions
\item $T=0 \, K$
\item Insertion of Si / C atom at
\begin{itemize}
- \item $(0,0,0)$ $\rightarrow$ tetrahedral
- \item $(-1/8,-1/8,1/8)$ $\rightarrow$ hexagonal
- \item $(-1/8,-1/8,-1/4)$, $(-1/4,-1/4,-1/4)$
- $\rightarrow$ 110 dumbbell
+ \item $(0,0,0)$ $\rightarrow$ {\color{red}tetrahedral}
+ \item $(-1/8,-1/8,1/8)$ $\rightarrow$ {\color{green}hexagonal}
+ \item $(-1/8,-1/8,-1/4)$, $(-1/4,-1/4,-1/4)$\\
+ $\rightarrow$ {\color{yellow}110 dumbbell}
\item random positions (critical distance check)
\end{itemize}
\item Relaxation time: $2\, ps$
+ \item Optional heating-up
\end{itemize}
+ \begin{picture}(0,0)(-210,-45)
+ \includegraphics[width=6cm]{unit_cell.eps}
+ \end{picture}
+
\end{slide}
\begin{slide}
\small
SiC precipitation experiments:
+
+ \begin{pspicture}(0,0)(12,8)
+ % nodes
+ \rput(4.5,6.5){\rnode{init}{\psframebox{\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 $600\, fs$
+ \end{itemize}
+ }}}}
+ \rput(4.5,4.5){\rnode{tc1}{\psframebox[fillstyle=solid,fillcolor=red]{
+ $T=450\pm 1\, ^{\circ}C$}}}
+ \rput(7,3.5){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=red]{
+ \parbox{3cm}{
+ Insertion of 10 atoms\\
+ at random positions}}}}
+ \rput(2,3.5){\rnode{adj1}{\psframebox[fillstyle=solid,fillcolor=red]{
+ \parbox{3.5cm}{
+ Adjusting temperature\\
+ for another $100\, fs$}}}}
+ \rput(7,2.5){\rnode{nc}{\psframebox[fillstyle=solid,fillcolor=red]{
+ $N_{atoms}=6000$}}}
+ \rput(4.5,2){\rnode{tc2}{\psframebox[fillstyle=solid,fillcolor=cyan]{
+ $T=T_{set}$}}}
+ \rput(7,1){\rnode{td}{\psframebox[fillstyle=solid,fillcolor=cyan]{
+ $T_{set}:=T_{set}-1\, ^{\circ}C$}}}
+ \rput(2,1){\rnode{adj2}{\psframebox[fillstyle=solid,fillcolor=cyan]{
+ \parbox{3.5cm}{
+ Adjusting temperature\\
+ for another $50\, fs$}}}}
+ \rput(7,0){\rnode{tc3}{\psframebox[fillstyle=solid,fillcolor=cyan]{
+ $T_{set}=20\, ^{\circ}C$}}}
+ \rput(10,0){\rnode{end}{\psframebox{End}}}
+ % help nodes
+ \rput(7,4.5){\pnode{tc1-h}}
+ \rput(2,4.5){\pnode{tc1-hh}}
+ \rput(4.5,2.5){\pnode{nc-h}}
+ \rput(9,2.5){\pnode{nc-hh}}
+ \rput(9,2){\pnode{tc2-h}}
+ \rput(2,2){\pnode{tc2-hh}}
+ \rput(4.5,0){\pnode{tc3-h}}
+ % direct lines
+ \ncline[]{->}{init}{tc1}
+ \ncline[]{->}{adj1}{tc1}
+ \ncline[]{->}{insert}{nc}
+ \ncline[]{->}{adj2}{tc2}
+ \ncline[]{->}{tc2}{td}
+ \lput*{0}{yes}
+ \ncline[]{->}{td}{tc3}
+ \ncline[]{->}{tc3}{end}
+ \lput*{0}{yes}
+ % lines using help nodes
+ \ncline[]{tc1}{tc1-h}
+ \lput*{0}{yes}
+ \ncline[]{->}{tc1-h}{insert}
+ \ncline[]{tc1}{tc1-hh}
+ \lput*{0}{no}
+ \ncline[]{->}{tc1-hh}{adj1}
+ \ncline[]{nc}{nc-h}
+ \lput*{0}{no}
+ \ncline[]{->}{nc-h}{tc1}
+ \ncline[]{nc}{nc-hh}
+ \ncline[]{-}{nc-hh}{tc2-h}
+ \ncline[]{->}{tc2-h}{tc2}
+ \lput*{0}{yes, {\footnotesize $T_{set}:=450\, ^{\circ}C$}}
+ \ncline[]{tc2}{tc2-hh}
+ \lput*{0}{no}
+ \ncline[]{->}{tc2-hh}{adj2}
+ \ncline[]{tc3}{tc3-h}
+ \lput*{0}{no}
+ \ncline[]{->}{tc3-h}{tc2}
+ % insertion volumes
+ \psframe[fillstyle=solid,fillcolor=white](9.5,1.3)(13.5,5.3)
+ \psframe[fillstyle=solid,fillcolor=lightgray](10,1.8)(13,4.8)
+ \psframe[fillstyle=solid,fillcolor=gray](10.5,2.3)(12.5,4.3)
+ \rput(9.75,3){\pnode{ins1}}
+ \rput(10.25,3.3){\pnode{ins2}}
+ \rput(10.75,3.6){\pnode{ins3}}
+ \ncline[]{-}{insert}{ins1}
+ \ncline[]{-}{insert}{ins2}
+ \ncline[]{-}{insert}{ins3}
+ \psframe[fillstyle=solid,fillcolor=white](9.5,7.6)(13.5,8.1)
+ \psframe[fillstyle=solid,fillcolor=lightgray](9.5,6.8)(13.5,7.3)
+ \psframe[fillstyle=solid,fillcolor=gray](9.5,6)(13.5,6.5)
+ \rput(11.5,7.85){{\tiny Simulation volume:
+ $31\times31\times31\, a^3_{Si}$}}
+ \rput(11.5,7.05){{\tiny Volume of minimal SiC precipitation}}
+ \rput(11.5,6.25){{\tiny Volume of necessary amount of Si}}
+ \end{pspicture}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Results
+ }
+
+ Si self-interstitial experiments:
+
+ {\footnotesize
+ {\bf Note:}
\begin{itemize}
- \item Initial configuration: $31\times31\times31$ unit cells Si
- \item Periodic boundary conditions
- \item $T=450\, ^{\circ}C$
- \item Steady state time: $600\, fs$
- \item C insertion steps:
- \begin{itemize}
- \item If $T=450\pm 1\, ^{\circ}C$:\\
- Insertion of 10 atoms at random positions within $V_{ins}$
- \item Otherwise: Annealing for another $100\, fs$
- \end{itemize}
- \item Annealing: ($T_a: 450\rightarrow 20 \, ^{\circ}C$)
- \begin{itemize}
- \item If $T=T_a$: Decrease $T_a$ by $1\, ^{\circ}C$
- \item Otherwise: Annealing for another $50\, fs$
- \end{itemize}
+ \item $r_{cutoff}^{Si-Si}=2.96>\frac{5.43}{2}$
+ \item Bond length near $r_{cutoff} \Rightarrow$ small bond strength
\end{itemize}
+ }
- Szenarios:
- \begin{enumerate}
- \item $V_{ins}$: total simulation volume $V$
- \item $V_{ins}$: $12\times12\times12$ SiC unit cells
- ($\sim$ volume of minimal SiC precipitation)
- \item $V_{ins}$: $9\times9\times9$ SiC unit cells
- ($\sim$ volume of necessary amount of Si)
- \end{enumerate}
+ \vspace{8pt}
+
+ \small
+
+ \begin{minipage}[t]{4.0cm}
+ \underline{Tetrahedral}
+ \begin{itemize}
+ \item $E_f=3.41\, eV$
+ \item essentialy tetrahedral\\
+ bonds
+ \end{itemize}
+ \end{minipage}
+ \hspace{0.3cm}
+ \begin{minipage}[t]{4.0cm}
+ \underline{110 dumbbell}
+ \begin{itemize}
+ \item $E_f=4.39\, eV$
+ \item essentially 4 bonds
+ \end{itemize}
+ \end{minipage}
+ \hspace{0.3cm}
+ \begin{minipage}[t]{4.0cm}
+ \underline{Hexagonal}
+ \begin{itemize}
+ \item $E_f^{\star}\approx4.48\, eV$
+ \item unstable!
+ \end{itemize}
+ \end{minipage}
+
+ \vspace{8pt}
+
+ \begin{minipage}[t]{4.3cm}
+ \includegraphics[width=3.8cm]{si_self_int_tetra_0.eps}
+ \end{minipage}
+ \begin{minipage}[t]{4.3cm}
+ \includegraphics[width=3.8cm]{si_self_int_dumbbell_0.eps}
+ \end{minipage}
+ \begin{minipage}[t]{4.3cm}
+ \includegraphics[width=3.8cm]{si_self_int_hexa_0.eps}
+ \begin{center}
+ \href{../video/si_self_int_hexa.avi}{$\rhd$}
+ \end{center}
+ \end{minipage}
\end{slide}
Results
}
+ \vspace{8pt}
+
+ Si self-interstitial \underline{random insertion} experiments:
+
+ \small
+
+ \vspace{8pt}
+
+ \begin{minipage}[t]{4.0cm}
+ \begin{itemize}
+ \item $E_f=3.97\, eV$
+ \item 3 identical weak bonds
+ \item displaced in volume\\ diagonal
+ \end{itemize}
+ \end{minipage}
+ \hspace{0.3cm}
+ \begin{minipage}[t]{4.0cm}
+ \begin{itemize}
+ \item $E_f=3.75\, eV$
+ \item 4 identical weak bonds
+ \item displaced in plane\\ diagonal
+ \end{itemize}
+ \end{minipage}
+ \hspace{0.3cm}
+ \begin{minipage}[t]{4.0cm}
+ \begin{itemize}
+ \item $E_f=3.56\, eV$
+ \item single weak bond
+ \item displaced along\\ $x$-direction
+ \item closest to tetrahedral\\ configuration
+ \end{itemize}
+ \end{minipage}
+
+ \vspace{8pt}
+
+ \begin{minipage}{4.3cm}
+ \includegraphics[width=3.8cm]{si_self_int_rand_397_0.eps}
+ \end{minipage}
+ \begin{minipage}{4.3cm}
+ \includegraphics[width=3.8cm]{si_self_int_rand_375_0.eps}
+ \end{minipage}
+ \begin{minipage}{4.3cm}
+ \includegraphics[width=3.8cm]{si_self_int_rand_356_0.eps}
+ \end{minipage}
+
+ \vspace{8pt}
+
+ \begin{center}
+ {\footnotesize
+ {\bf Note:} Displacements relative to tetrahedral configuration
+ }
+ \end{center}
+
\end{slide}
\begin{slide}
Results
}
+ \vspace{8pt}
+
+ Carbon interstitial experiments:
+
+ \vspace{12pt}
+
+ \small
+
+ \begin{minipage}[t]{4.0cm}
+ \underline{Tetrahedral}
+ \begin{itemize}
+ \item $E_f=2.67\, eV$
+ \item tetrahedral bond
+ \end{itemize}
+ \end{minipage}
+ \hspace{0.3cm}
+ \begin{minipage}[t]{4.0cm}
+ \underline{110 dumbbell}
+ \begin{itemize}
+ \item $E_f=1.76\, eV$
+ \item C forms 3 bonds
+ \end{itemize}
+ \end{minipage}
+ \hspace{0.3cm}
+ \begin{minipage}[t]{4.0cm}
+ \underline{Hexagonal}
+ \begin{itemize}
+ \item $E_f^{\star}\approx5.6\, eV$
+ \item unstable!
+ \end{itemize}
+ \end{minipage}
+
+ \vspace{8pt}
+
+ \begin{minipage}[t]{4.3cm}
+ \includegraphics[width=3.8cm]{c_in_si_int_tetra_0.eps}
+ \end{minipage}
+ \begin{minipage}[t]{4.3cm}
+ \includegraphics[width=3.8cm]{c_in_si_int_dumbbell_0.eps}
+ \end{minipage}
+ \begin{minipage}[t]{4.3cm}
+ \includegraphics[width=3.8cm]{c_in_si_int_hexa_0.eps}
+ \begin{center}
+ \href{../video/c_in_si_int_hexa.avi}{$\rhd$}
+ \end{center}
+ \end{minipage}
+
\end{slide}
\begin{slide}
Results
}
+ %\vspace{8pt}
+
+ Carbon \underline{random insertion} experiments:
+
+ %\vspace{8pt}
+
+ \footnotesize
+
+ \begin{minipage}[c]{6.3cm}
+ \begin{minipage}{3.4cm}
+ \includegraphics[width=3.3cm]{c_in_si_int_001db_0.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \begin{itemize}
+ \item $E_f=0.47\, eV$
+ \item 001 dumbbell
+ \end{itemize}
+ \end{minipage}
+ \end{minipage}
+ \begin{minipage}[c]{6.3cm}
+ \begin{minipage}{3.4cm}
+ \includegraphics[width=3.3cm]{c_in_si_int_rand_162_0.eps}
+ \end{minipage}
+ \begin{minipage}{2.8cm}
+ \begin{itemize}
+ \item $E_f=1.62\, eV$
+ \item 3 weak + strong bonds
+ \end{itemize}
+ \end{minipage}
+ \end{minipage}
+
+ \begin{minipage}[c]{6.3cm}
+ \begin{minipage}{3.4cm}
+ \includegraphics[width=3.3cm]{c_in_si_int_rand_239_0.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \begin{itemize}
+ \item $E_f=2.39\, eV$
+ \end{itemize}
+ \begin{center}
+ \href{../video/c_in_si_int_rand_239.avi}{$\rhd$}
+ \end{center}
+ \end{minipage}
+ \end{minipage}
+ \begin{minipage}[c]{6.3cm}
+ \begin{minipage}{3.4cm}
+ \includegraphics[width=3.3cm]{c_in_si_int_rand_341_0.eps}
+ \end{minipage}
+ \begin{minipage}{2.8cm}
+ \begin{itemize}
+ \item $E_f=3.41\, eV$
+ \end{itemize}
+ \begin{center}
+ \href{../video/c_in_si_int_rand_341.avi}{$\rhd$}
+ \end{center}
+ \end{minipage}
+ \end{minipage}
+
+ \vspace{4pt}
+
+ \begin{center}
+ {\bf Note:} High probability for 110 dumbbell ($1.76\, eV$) configurations!
+ \end{center}
+
\end{slide}
\begin{slide}
Results
}
+ SiC-precipitation experiments:
+
+ \begin{minipage}[t]{6.3cm}
+ %\input{../plot/sic_prec}
+ \includegraphics[width=6.0cm]{../plot/sic_prec_energy.ps}
+ \includegraphics[width=6.0cm]{../plot/sic_prec_temp.ps}
+ \end{minipage}
+ \begin{minipage}[t]{6cm}
+ \includegraphics[width=6.0cm]{../plot/sic_pc.ps}
+ \includegraphics[width=6.0cm]{../plot/sic_prec_pc.ps}
+ \end{minipage}
+
\end{slide}
\begin{slide}
Conclusion / Outlook
}
-\end{slide}
+\vspace{24pt}
+\begin{itemize}
+ \item Importance of understanding C in Si
+ \item Interstitial configurations in silicon using the Albe potential
+ \item Indication of SiC precipitation
+\end{itemize}
+
+\vspace{16pt}
+
+\begin{itemize}
+ \item Displacement and stress calculations
+ \item Diffusion dependence of temperature and carbon concentration
+ \item Analyzing results of the precipitation simulation runs
+ \item Analyzing self-designed Si/SiC interface
+\end{itemize}
+
+\end{slide}
\end{document}
projects / lectures / latex.git / blobdiff