From 8f332a5618a1846f72c914e5aaa7e960ecd540aa Mon Sep 17 00:00:00 2001 From: hackbard Date: Wed, 25 May 2011 18:03:59 +0200 Subject: [PATCH] added seminar --- posic/talks/seminar_2011.tex | 2080 ++++++++++++++++++++++++++++++++++ 1 file changed, 2080 insertions(+) create mode 100644 posic/talks/seminar_2011.tex diff --git a/posic/talks/seminar_2011.tex b/posic/talks/seminar_2011.tex new file mode 100644 index 0000000..988543e --- /dev/null +++ b/posic/talks/seminar_2011.tex @@ -0,0 +1,2080 @@ +\pdfoutput=0 +%\documentclass[landscape,semhelv,draft]{seminar} +\documentclass[landscape,semhelv]{seminar} + +\usepackage{verbatim} +\usepackage[greek,german]{babel} +\usepackage[latin1]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{amsmath} +\usepackage{latexsym} +\usepackage{ae} + +\usepackage{calc} % Simple computations with LaTeX variables +\usepackage{caption} % Improved captions +\usepackage{fancybox} % To have several backgrounds + +\usepackage{fancyhdr} % Headers and footers definitions +\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{miller} + +\usepackage[setpagesize=false]{hyperref} + +\usepackage{semcolor} +\usepackage{semlayer} % Seminar overlays +\usepackage{slidesec} % Seminar sections and list of slides + +\input{seminar.bug} % Official bugs corrections +\input{seminar.bg2} % Unofficial bugs corrections + +\articlemag{1} + +\special{landscape} + +% font +%\usepackage{cmbright} +%\renewcommand{\familydefault}{\sfdefault} +%\usepackage{mathptmx} + +\usepackage{upgreek} + +\begin{document} + +\extraslideheight{10in} +\slideframe{none} + +\pagestyle{empty} + +% specify width and height +\slidewidth 27.7cm +\slideheight 19.1cm + +% shift it into visual area properly +\def\slideleftmargin{3.3cm} +\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}$\bullet$}} + +% nice phi +\renewcommand{\phi}{\varphi} + +% roman letters +\newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}} + +% colors +\newrgbcolor{si-yellow}{.6 .6 0} +\newrgbcolor{hb}{0.75 0.77 0.89} +\newrgbcolor{lbb}{0.75 0.8 0.88} +\newrgbcolor{hlbb}{0.825 0.88 0.968} +\newrgbcolor{lachs}{1.0 .93 .81} + +% topic + +\begin{slide} +\begin{center} + + \vspace{16pt} + + {\LARGE\bf + Atomistic simulation study on the silicon carbide precipitation + in silicon + } + + \vspace{48pt} + + \textsc{F. Zirkelbach} + + \vspace{48pt} + +Yet another seminar contribution + + \vspace{08pt} + + Augsburg am 26. Mai 2011 + +\end{center} +\end{slide} + +% motivation / properties / applications of silicon carbide +\begin{slide} + +\small + +\begin{pspicture}(0,0)(13.5,5) + + + + \psframe*[linecolor=hb](0,0)(13.5,5) + + \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.5,1)(7,1)(7,3)(5.5,3) + \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.75,0.5)(8,2)(8,2)(6.75,3.5) + + \rput[lt](0.2,4.6){\color{gray}PROPERTIES} + + \rput[lt](0.5,4){wide band gap} + \rput[lt](0.5,3.5){high electric breakdown field} + \rput[lt](0.5,3){good electron mobility} + \rput[lt](0.5,2.5){high electron saturation drift velocity} + \rput[lt](0.5,2){high thermal conductivity} + + \rput[lt](0.5,1.5){hard and mechanically stable} + \rput[lt](0.5,1){chemically inert} + + \rput[lt](0.5,0.5){radiation hardness} + + \rput[rt](13.3,4.6){\color{gray}APPLICATIONS} + + \rput[rt](13,3.85){high-temperature, high power} + \rput[rt](13,3.5){and high-frequency} + \rput[rt](13,3.15){electronic and optoelectronic devices} + + \rput[rt](13,2.35){material suitable for extreme conditions} + \rput[rt](13,2){microelectromechanical systems} + \rput[rt](13,1.65){abrasives, cutting tools, heating elements} + + \rput[rt](13,0.85){first wall reactor material, detectors} + \rput[rt](13,0.5){and electronic devices for space} + +\end{pspicture} + +\begin{picture}(0,0)(-3,68) +\includegraphics[width=2.6cm]{wide_band_gap.eps} +\end{picture} +\begin{picture}(0,0)(-285,-162) +\includegraphics[width=3.38cm]{sic_led.eps} +\end{picture} +\begin{picture}(0,0)(-195,-162) +\includegraphics[width=2.8cm]{6h-sic_3c-sic.eps} +\end{picture} +\begin{picture}(0,0)(-313,65) +\includegraphics[width=2.2cm]{infineon_schottky.eps} +\end{picture} +\begin{picture}(0,0)(-220,65) +\includegraphics[width=2.9cm]{sic_wechselrichter_ise.eps} +\end{picture} +\begin{picture}(0,0)(0,-160) +\includegraphics[width=3.0cm]{sic_proton.eps} +\end{picture} +\begin{picture}(0,0)(-60,65) +\includegraphics[width=3.4cm]{sic_switch.eps} +\end{picture} + +\end{slide} + +% start of contents + +\begin{slide} + + {\large\bf + Polytypes of SiC + } + + \vspace{4cm} + + \small + +\begin{tabular}{l c c c c c c} +\hline + & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\ +\hline +Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\ +Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\ +Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\ +Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\ +Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\ +Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\ +Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\ +\hline +\end{tabular} + +{\tiny + Values for $T=300$ K +} + +\begin{picture}(0,0)(-160,-155) + \includegraphics[width=7cm]{polytypes.eps} +\end{picture} +\begin{picture}(0,0)(-10,-185) + \includegraphics[width=3.8cm]{cubic_hex.eps}\\ +\end{picture} +\begin{picture}(0,0)(-10,-175) + {\tiny cubic (twist)} +\end{picture} +\begin{picture}(0,0)(-60,-175) + {\tiny hexagonal (no twist)} +\end{picture} +\begin{pspicture}(0,0)(0,0) +\psellipse[linecolor=green](5.7,3.03)(0.4,0.5) +\end{pspicture} +\begin{pspicture}(0,0)(0,0) +\psellipse[linecolor=green](5.6,1.68)(0.4,0.2) +\end{pspicture} +\begin{pspicture}(0,0)(0,0) +\psellipse[linecolor=red](10.7,1.13)(0.4,0.2) +\end{pspicture} + +\end{slide} + +\begin{slide} + + {\large\bf + Fabrication of silicon carbide + } + + \small + + \vspace{4pt} + + SiC - \emph{Born from the stars, perfected on earth.} + + \vspace{4pt} + + Conventional thin film SiC growth: + \begin{itemize} + \item \underline{Sublimation growth using the modified Lely method} + \begin{itemize} + \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$ + \item Surrounded by polycrystalline SiC in a graphite crucible\\ + at $T=2100-2400 \, ^{\circ} \text{C}$ + \item Deposition of supersaturated vapor on cooler seed crystal + \end{itemize} + \item \underline{Homoepitaxial growth using CVD} + \begin{itemize} + \item Step-controlled epitaxy on off-oriented 6H-SiC substrates + \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$ + \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC + \end{itemize} + \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE} + \begin{itemize} + \item Two steps: carbonization and growth + \item $T=650-1050 \, ^{\circ} \text{C}$ + \item SiC/Si lattice mismatch $\approx$ 20 \% + \item Quality and size not yet sufficient + \end{itemize} + \end{itemize} + + \begin{picture}(0,0)(-280,-65) + \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps} + \end{picture} + \begin{picture}(0,0)(-280,-55) + \begin{minipage}{5cm} + {\tiny + NASA: 6H-SiC and 3C-SiC LED\\[-7pt] + on 6H-SiC substrate + } + \end{minipage} + \end{picture} + \begin{picture}(0,0)(-265,-150) + \includegraphics[width=2.4cm]{m_lely.eps} + \end{picture} + \begin{picture}(0,0)(-333,-175) + \begin{minipage}{5cm} + {\tiny + 1. Lid\\[-7pt] + 2. Heating\\[-7pt] + 3. Source\\[-7pt] + 4. Crucible\\[-7pt] + 5. Insulation\\[-7pt] + 6. Seed crystal + } + \end{minipage} + \end{picture} + \begin{picture}(0,0)(-230,-35) + \framebox{ + {\footnotesize\color{blue}\bf Hex: micropipes along c-axis} + } + \end{picture} + \begin{picture}(0,0)(-230,-10) + \framebox{ + \begin{minipage}{3cm} + {\footnotesize\color{blue}\bf 3C-SiC fabrication\\ + less advanced} + \end{minipage} + } + \end{picture} + +\end{slide} + +\begin{slide} + + {\large\bf + Fabrication of silicon carbide + } + + \small + + Alternative approach: + Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0) + \begin{itemize} + \item \underline{Implantation step 1}\\ + 180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\ + $\Rightarrow$ box-like distribution of equally sized + and epitactically oriented SiC precipitates + + \item \underline{Implantation step 2}\\ + 180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\ + $\Rightarrow$ destruction of SiC nanocrystals + in growing amorphous interface layers + \item \underline{Annealing}\\ + $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\ + $\Rightarrow$ homogeneous, stoichiometric SiC layer + with sharp interfaces + \end{itemize} + + \begin{minipage}{6.3cm} + \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm] + {\tiny + XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0) + } + \end{minipage} +\framebox{ + \begin{minipage}{6.3cm} + \begin{center} + {\color{blue} + Precipitation mechanism not yet fully understood! + } + \renewcommand\labelitemi{$\Rightarrow$} + \small + \underline{Understanding the SiC precipitation} + \begin{itemize} + \item significant technological progress in SiC thin film formation + \item perspectives for processes relying upon prevention of SiC precipitation + \end{itemize} + \end{center} + \end{minipage} +} + +\end{slide} + +% contents + +\begin{slide} + +{\large\bf + Outline +} + + \begin{itemize} + \item Supposed precipitation mechanism of SiC in Si + \item Utilized simulation techniques + \begin{itemize} + \item Molecular dynamics (MD) simulations + \item Density functional theory (DFT) calculations + \end{itemize} + \item C and Si self-interstitial point defects in silicon + \item Silicon carbide precipitation simulations + \item Summary / Conclusion / Outlook + \end{itemize} + +\end{slide} + +\begin{slide} + + {\large\bf + Supposed precipitation mechanism of SiC in Si + } + + \scriptsize + + \vspace{0.1cm} + + \begin{minipage}{3.8cm} + Si \& SiC lattice structure\\[0.2cm] + \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm] + \hrule + \end{minipage} + \hspace{0.6cm} + \begin{minipage}{3.8cm} + \begin{center} + \includegraphics[width=3.3cm]{tem_c-si-db.eps} + \end{center} + \end{minipage} + \hspace{0.6cm} + \begin{minipage}{3.8cm} + \begin{center} + \includegraphics[width=3.3cm]{tem_3c-sic.eps} + \end{center} + \end{minipage} + + \begin{minipage}{4cm} + \begin{center} + C-Si dimers (dumbbells)\\[-0.1cm] + on Si interstitial sites + \end{center} + \end{minipage} + \hspace{0.2cm} + \begin{minipage}{4.2cm} + \begin{center} + Agglomeration of C-Si dumbbells\\[-0.1cm] + $\Rightarrow$ dark contrasts + \end{center} + \end{minipage} + \hspace{0.2cm} + \begin{minipage}{4cm} + \begin{center} + Precipitation of 3C-SiC in Si\\[-0.1cm] + $\Rightarrow$ Moir\'e fringes\\[-0.1cm] + \& release of Si self-interstitials + \end{center} + \end{minipage} + + \begin{minipage}{3.8cm} + \begin{center} + \includegraphics[width=3.3cm]{sic_prec_seq_01.eps} + \end{center} + \end{minipage} + \hspace{0.6cm} + \begin{minipage}{3.8cm} + \begin{center} + \includegraphics[width=3.3cm]{sic_prec_seq_02.eps} + \end{center} + \end{minipage} + \hspace{0.6cm} + \begin{minipage}{3.8cm} + \begin{center} + \includegraphics[width=3.3cm]{sic_prec_seq_03.eps} + \end{center} + \end{minipage} + +\begin{pspicture}(0,0)(0,0) +\psline[linewidth=4pt]{->}(8.5,2)(9.0,2) +\psellipse[linecolor=blue](11.5,5.8)(0.3,0.5) +\rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)} +\psline[linewidth=4pt]{->}(4.0,2)(4.5,2) +\rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{ + $4a_{\text{Si}}=5a_{\text{SiC}}$ + }}} +\rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{ +\hkl(h k l) planes match + }}} +\rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{ +r = 2 - 4 nm + }}} +\end{pspicture} + +\end{slide} + +\begin{slide} + + {\large\bf + Molecular dynamics (MD) simulations + } + + \vspace{12pt} + + \small + + {\bf MD basics:} + \begin{itemize} + \item Microscopic description of N particle system + \item Analytical interaction potential + \item Numerical integration using Newtons equation of motion\\ + as a propagation rule in 6N-dimensional phase space + \item Observables obtained by time and/or ensemble averages + \end{itemize} + {\bf Details of the simulation:} + \begin{itemize} + \item Integration: Velocity Verlet, timestep: $1\text{ fs}$ + \item Ensemble: NpT (isothermal-isobaric) + \begin{itemize} + \item Berendsen thermostat: + $\tau_{\text{T}}=100\text{ fs}$ + \item Berendsen barostat:\\ + $\tau_{\text{P}}=100\text{ fs}$, + $\beta^{-1}=100\text{ GPa}$ + \end{itemize} + \item Erhart/Albe potential: Tersoff-like bond order potential + \vspace*{12pt} + \[ + E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad + \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] + \] + \end{itemize} + + \begin{picture}(0,0)(-230,-30) + \includegraphics[width=5cm]{tersoff_angle.eps} + \end{picture} + +\end{slide} + +\begin{slide} + + {\large\bf + Density functional theory (DFT) calculations + } + + \small + + Basic ingredients necessary for DFT + + \begin{itemize} + \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ... + \begin{itemize} + \item ... uniquely determines the ground state potential + / wavefunctions + \item ... minimizes the systems total energy + \end{itemize} + \item \underline{Born-Oppenheimer} + - $N$ moving electrons in an external potential of static nuclei +\[ +H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2 + +\sum_i^N V_{\text{ext}}(r_i) + +\sum_{i}{init}{insert} + \ncline[]{->}{insert}{cool} + \end{pspicture} +\end{minipage} +\begin{minipage}{5cm} + \includegraphics[width=5cm]{unit_cell_e.eps}\\ +\end{minipage} + +\begin{minipage}{9cm} + \begin{tabular}{l c c} + \hline + & size [unit cells] & \# atoms\\ +\hline +VASP & $3\times 3\times 3$ & $216\pm 1$ \\ +Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\ +\hline + \end{tabular} +\end{minipage} +\begin{minipage}{4cm} +{\color{red}$\bullet$} Tetrahedral\\ +{\color{green}$\bullet$} Hexagonal\\ +{\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\ +{\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\ +{\color{cyan}$\bullet$} Bond-centered\\ +{\color{black}$\bullet$} Vacancy / Substitutional +\end{minipage} + +\end{slide} + +\begin{slide} + + \footnotesize + +\begin{minipage}{9.5cm} + + {\large\bf + Si self-interstitial point defects in silicon\\ + } + +\begin{tabular}{l c c c c c} +\hline + $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\ +\hline + VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\ + Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\ +\hline +\end{tabular}\\[0.2cm] + +\begin{minipage}{4.7cm} +\includegraphics[width=4.7cm]{e_kin_si_hex.ps} +\end{minipage} +\begin{minipage}{4.7cm} +\begin{center} +{\tiny nearly T $\rightarrow$ T}\\ +\end{center} +\includegraphics[width=4.7cm]{nhex_tet.ps} +\end{minipage}\\ + +\underline{Hexagonal} \hspace{2pt} +\href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm] +\framebox{ +\begin{minipage}{2.7cm} +$E_{\text{f}}^*=4.48\text{ eV}$\\ +\includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps} +\end{minipage} +\begin{minipage}{0.4cm} +\begin{center} +$\Rightarrow$ +\end{center} +\end{minipage} +\begin{minipage}{2.7cm} +$E_{\text{f}}=3.96\text{ eV}$\\ +\includegraphics[width=2.8cm]{si_pd_albe/hex.eps} +\end{minipage} +} +\begin{minipage}{2.9cm} +\begin{flushright} +\underline{Vacancy}\\ +\includegraphics[width=3.0cm]{si_pd_albe/vac.eps} +\end{flushright} +\end{minipage} + +\end{minipage} +\begin{minipage}{3.5cm} + +\begin{flushright} +\underline{\hkl<1 1 0> dumbbell}\\ +\includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\ +\underline{Tetrahedral}\\ +\includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\ +\underline{\hkl<1 0 0> dumbbell}\\ +\includegraphics[width=3.0cm]{si_pd_albe/100.eps} +\end{flushright} + +\end{minipage} + +\end{slide} + +\begin{slide} + +\footnotesize + + {\large\bf + C interstitial point defects in silicon\\[-0.1cm] + } + +\begin{tabular}{l c c c c c c} +\hline + $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\ +\hline + VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 \\ + Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & 0.75 & 5.59$^*$ \\ +\hline +\end{tabular}\\[0.1cm] + +\framebox{ +\begin{minipage}{2.7cm} +\underline{Hexagonal} \hspace{2pt} +\href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\ +$E_{\text{f}}^*=9.05\text{ eV}$\\ +\includegraphics[width=2.7cm]{c_pd_albe/hex.eps} +\end{minipage} +\begin{minipage}{0.4cm} +\begin{center} +$\Rightarrow$ +\end{center} +\end{minipage} +\begin{minipage}{2.7cm} +\underline{\hkl<1 0 0>}\\ +$E_{\text{f}}=3.88\text{ eV}$\\ +\includegraphics[width=2.7cm]{c_pd_albe/100.eps} +\end{minipage} +} +\begin{minipage}{2cm} +\hfill +\end{minipage} +\begin{minipage}{3cm} +\begin{flushright} +\underline{Tetrahedral}\\ +\includegraphics[width=3.0cm]{c_pd_albe/tet.eps} +\end{flushright} +\end{minipage} + +\framebox{ +\begin{minipage}{2.7cm} +\underline{Bond-centered}\\ +$E_{\text{f}}^*=5.59\text{ eV}$\\ +\includegraphics[width=2.7cm]{c_pd_albe/bc.eps} +\end{minipage} +\begin{minipage}{0.4cm} +\begin{center} +$\Rightarrow$ +\end{center} +\end{minipage} +\begin{minipage}{2.7cm} +\underline{\hkl<1 1 0> dumbbell}\\ +$E_{\text{f}}=5.18\text{ eV}$\\ +\includegraphics[width=2.7cm]{c_pd_albe/110.eps} +\end{minipage} +} +\begin{minipage}{2cm} +\hfill +\end{minipage} +\begin{minipage}{3cm} +\begin{flushright} +\underline{Substitutional}\\ +\includegraphics[width=3.0cm]{c_pd_albe/sub.eps} +\end{flushright} +\end{minipage} + +\end{slide} + +\begin{slide} + +\footnotesize + + {\large\bf\boldmath + C \hkl<1 0 0> dumbbell interstitial configuration\\ + } + +{\tiny +\begin{tabular}{l c c c c c c c c} +\hline + Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\ +\hline +Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\ +VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\ +\hline +\end{tabular}\\[0.2cm] +\begin{tabular}{l c c c c } +\hline + Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\ +\hline +Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\ +VASP & 130.7 & 114.4 & 146.0 & 107.0 \\ +\hline +\end{tabular}\\[0.2cm] +\begin{tabular}{l c c c} +\hline + Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\ +\hline +Erhart/Albe & 0.084 & -0.091 & 0.175 \\ +VASP & 0.109 & -0.065 & 0.174 \\ +\hline +\end{tabular}\\[0.6cm] +} + +\begin{minipage}{3.0cm} +\begin{center} +\underline{Erhart/Albe} +\includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps} +\end{center} +\end{minipage} +\begin{minipage}{3.0cm} +\begin{center} +\underline{VASP} +\includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps} +\end{center} +\end{minipage}\\ + +\begin{picture}(0,0)(-185,10) +\includegraphics[width=6.8cm]{100-c-si-db_cmp.eps} +\end{picture} +\begin{picture}(0,0)(-280,-150) +\includegraphics[width=3.3cm]{c_pd_vasp/eden.eps} +\end{picture} + +\begin{pspicture}(0,0)(0,0) +\psellipse[linecolor=green](5.18,5.92)(0.5,0.3) +\psellipse[linecolor=red](3.45,5.92)(1.0,0.4) +\psellipse[linecolor=blue](2.7,6.92)(0.9,0.2) +\psellipse[linecolor=blue](4.65,6.92)(0.9,0.2) +\end{pspicture} + +\end{slide} + +\begin{slide} + +\small + +\begin{minipage}{8.5cm} + + {\large\bf + Bond-centered interstitial configuration\\[-0.1cm] + } + +\begin{minipage}{3.0cm} +\includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\ +\end{minipage} +\begin{minipage}{5.2cm} +\begin{itemize} + \item Linear Si-C-Si bond + \item Si: one C \& 3 Si neighbours + \item Spin polarized calculations + \item No saddle point!\\ + Real local minimum! +\end{itemize} +\end{minipage} + +\framebox{ + \tiny + \begin{minipage}[t]{6.5cm} + \begin{minipage}[t]{1.2cm} + {\color{red}Si}\\ + {\tiny sp$^3$}\\[0.8cm] + \underline{${\color{black}\uparrow}$} + \underline{${\color{black}\uparrow}$} + \underline{${\color{black}\uparrow}$} + \underline{${\color{red}\uparrow}$}\\ + sp$^3$ + \end{minipage} + \begin{minipage}[t]{1.4cm} + \begin{center} + {\color{red}M}{\color{blue}O}\\[0.8cm] + \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\ + $\sigma_{\text{ab}}$\\[0.5cm] + \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\ + $\sigma_{\text{b}}$ + \end{center} + \end{minipage} + \begin{minipage}[t]{1.0cm} + \begin{center} + {\color{blue}C}\\ + {\tiny sp}\\[0.2cm] + \underline{${\color{white}\uparrow\uparrow}$} + \underline{${\color{white}\uparrow\uparrow}$}\\ + 2p\\[0.4cm] + \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$} + \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\ + sp + \end{center} + \end{minipage} + \begin{minipage}[t]{1.4cm} + \begin{center} + {\color{blue}M}{\color{green}O}\\[0.8cm] + \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\ + $\sigma_{\text{ab}}$\\[0.5cm] + \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\ + $\sigma_{\text{b}}$ + \end{center} + \end{minipage} + \begin{minipage}[t]{1.2cm} + \begin{flushright} + {\color{green}Si}\\ + {\tiny sp$^3$}\\[0.8cm] + \underline{${\color{green}\uparrow}$} + \underline{${\color{black}\uparrow}$} + \underline{${\color{black}\uparrow}$} + \underline{${\color{black}\uparrow}$}\\ + sp$^3$ + \end{flushright} + \end{minipage} + \end{minipage} +}\\[0.1cm] + +\framebox{ +\begin{minipage}{4.5cm} +\includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps} +\end{minipage} +\begin{minipage}{3.5cm} +{\color{gray}$\bullet$} Spin up\\ +{\color{green}$\bullet$} Spin down\\ +{\color{blue}$\bullet$} Resulting spin up\\ +{\color{yellow}$\bullet$} Si atoms\\ +{\color{red}$\bullet$} C atom +\end{minipage} +} + +\end{minipage} +\begin{minipage}{4.2cm} +\begin{flushright} +\includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\ +{\color{green}$\Box$} {\tiny unoccupied}\\ +{\color{red}$\bullet$} {\tiny occupied} +\end{flushright} +\end{minipage} + +\end{slide} + +\begin{slide} + + {\large\bf\boldmath + Migration of the C \hkl<1 0 0> dumbbell interstitial + } + +\scriptsize + + {\small Investigated pathways} + +\begin{minipage}{8.5cm} +\begin{minipage}{8.3cm} +\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\ +\begin{minipage}{2.4cm} +\includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps} +\end{minipage} +\begin{minipage}{0.4cm} +$\rightarrow$ +\end{minipage} +\begin{minipage}{2.4cm} +\includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps} +\end{minipage} +\begin{minipage}{0.4cm} +$\rightarrow$ +\end{minipage} +\begin{minipage}{2.4cm} +\includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps} +\end{minipage} +\end{minipage}\\ +\begin{minipage}{8.3cm} +\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\ +\begin{minipage}{2.4cm} +\includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps} +\end{minipage} +\begin{minipage}{0.4cm} +$\rightarrow$ +\end{minipage} +\begin{minipage}{2.4cm} +\includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps} +\end{minipage} +\begin{minipage}{0.4cm} +$\rightarrow$ +\end{minipage} +\begin{minipage}{2.4cm} +\includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps} +\end{minipage} +\end{minipage}\\ +\begin{minipage}{8.3cm} +\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\ +\begin{minipage}{2.4cm} +\includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps} +\end{minipage} +\begin{minipage}{0.4cm} +$\rightarrow$ +\end{minipage} +\begin{minipage}{2.4cm} +\includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps} +\end{minipage} +\begin{minipage}{0.4cm} +$\rightarrow$ +\end{minipage} +\begin{minipage}{2.4cm} +\includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps} +\end{minipage} +\end{minipage} +\end{minipage} +\framebox{ +\begin{minipage}{4.2cm} + {\small Constrained relaxation\\ + technique (CRT) method}\\ +\includegraphics[width=4cm]{crt_orig.eps} +\begin{itemize} + \item Constrain diffusing atom + \item Static constraints +\end{itemize} +\vspace*{0.3cm} + {\small Modifications}\\ +\includegraphics[width=4cm]{crt_mod.eps} +\begin{itemize} + \item Constrain all atoms + \item Update individual\\ + constraints +\end{itemize} +\end{minipage} +} + +\end{slide} + +\begin{slide} + + {\large\bf\boldmath + Migration of the C \hkl<1 0 0> dumbbell interstitial + } + +\scriptsize + +\framebox{ +\begin{minipage}{5.9cm} +\begin{flushleft} +\includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm] +\end{flushleft} +\begin{center} +\begin{picture}(0,0)(60,0) +\includegraphics[width=1cm]{vasp_mig/00-1.eps} +\end{picture} +\begin{picture}(0,0)(-5,0) +\includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps} +\end{picture} +\begin{picture}(0,0)(-55,0) +\includegraphics[width=1cm]{vasp_mig/bc.eps} +\end{picture} +\begin{picture}(0,0)(12.5,10) +\includegraphics[width=1cm]{110_arrow.eps} +\end{picture} +\begin{picture}(0,0)(90,0) +\includegraphics[height=0.9cm]{001_arrow.eps} +\end{picture} +\end{center} +\vspace*{0.35cm} +\end{minipage} +} +\begin{minipage}{0.3cm} +\hfill +\end{minipage} +\framebox{ +\begin{minipage}{5.9cm} +\begin{flushright} +\includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm] +\end{flushright} +\begin{center} +\begin{picture}(0,0)(60,0) +\includegraphics[width=1cm]{vasp_mig/00-1_a.eps} +\end{picture} +\begin{picture}(0,0)(5,0) +\includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps} +\end{picture} +\begin{picture}(0,0)(-55,0) +\includegraphics[width=1cm]{vasp_mig/0-10.eps} +\end{picture} +\begin{picture}(0,0)(12.5,10) +\includegraphics[width=1cm]{100_arrow.eps} +\end{picture} +\begin{picture}(0,0)(90,0) +\includegraphics[height=0.9cm]{001_arrow.eps} +\end{picture} +\end{center} +\vspace*{0.3cm} +\end{minipage}\\ +} + +\vspace*{0.05cm} + +\framebox{ +\begin{minipage}{5.9cm} +\begin{flushleft} +\includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm] +\end{flushleft} +\begin{center} +\begin{picture}(0,0)(60,0) +\includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps} +\end{picture} +\begin{picture}(0,0)(10,0) +\includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps} +\end{picture} +\begin{picture}(0,0)(-60,0) +\includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps} +\end{picture} +\begin{picture}(0,0)(12.5,10) +\includegraphics[width=1cm]{100_arrow.eps} +\end{picture} +\begin{picture}(0,0)(90,0) +\includegraphics[height=0.9cm]{001_arrow.eps} +\end{picture} +\end{center} +\vspace*{0.3cm} +\end{minipage} +} +\begin{minipage}{0.3cm} +\hfill +\end{minipage} +\begin{minipage}{6.5cm} +VASP results +\begin{itemize} + \item Energetically most favorable path + \begin{itemize} + \item Path 2 + \item Activation energy: $\approx$ 0.9 eV + \item Experimental values: 0.73 ... 0.87 eV + \end{itemize} + $\Rightarrow$ {\color{blue}Diffusion} path identified! + \item Reorientation (path 3) + \begin{itemize} + \item More likely composed of two consecutive steps of type 2 + \item Experimental values: 0.77 ... 0.88 eV + \end{itemize} + $\Rightarrow$ {\color{blue}Reorientation} transition identified! +\end{itemize} +\end{minipage} + +\end{slide} + +\begin{slide} + + {\large\bf\boldmath + Migration of the C \hkl<1 0 0> dumbbell interstitial + } + +\scriptsize + +\begin{minipage}{6.5cm} + +\framebox{ +\begin{minipage}{5.9cm} +\begin{flushleft} +\includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm] +\end{flushleft} +\begin{center} +\begin{pspicture}(0,0)(0,0) +\psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7) +\end{pspicture} +\begin{picture}(0,0)(60,-50) +\includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps} +\end{picture} +\begin{picture}(0,0)(5,-50) +\includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps} +\end{picture} +\begin{picture}(0,0)(-55,-50) +\includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps} +\end{picture} +\begin{picture}(0,0)(12.5,-40) +\includegraphics[width=1cm]{110_arrow.eps} +\end{picture} +\begin{picture}(0,0)(90,-45) +\includegraphics[height=0.9cm]{001_arrow.eps} +\end{picture}\\ +\begin{pspicture}(0,0)(0,0) +\psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6) +\end{pspicture} +\begin{picture}(0,0)(60,-15) +\includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps} +\end{picture} +\begin{picture}(0,0)(35,-15) +\includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps} +\end{picture} +\begin{picture}(0,0)(-5,-15) +\includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps} +\end{picture} +\begin{picture}(0,0)(-55,-15) +\includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps} +\end{picture} +\begin{picture}(0,0)(12.5,-5) +\includegraphics[width=1cm]{100_arrow.eps} +\end{picture} +\begin{picture}(0,0)(90,-15) +\includegraphics[height=0.9cm]{010_arrow.eps} +\end{picture} +\end{center} +\end{minipage} +}\\[0.1cm] + +\begin{minipage}{5.9cm} +Erhart/Albe results +\begin{itemize} + \item Lowest activation energy: $\approx$ 2.2 eV + \item 2.4 times higher than VASP + \item Different pathway + \item Transition minima ($\rightarrow$ \hkl<1 1 0> dumbbell) +\end{itemize} +\end{minipage} + +\end{minipage} +\begin{minipage}{6.5cm} + +\framebox{ +\begin{minipage}{5.9cm} +\begin{flushright} +\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm] +\end{flushright} +\begin{center} +\begin{pspicture}(0,0)(0,0) +\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1) +\end{pspicture} +\begin{picture}(0,0)(60,-5) +\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps} +\end{picture} +\begin{picture}(0,0)(0,-5) +\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps} +\end{picture} +\begin{picture}(0,0)(-55,-5) +\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps} +\end{picture} +\begin{picture}(0,0)(12.5,5) +\includegraphics[width=1cm]{100_arrow.eps} +\end{picture} +\begin{picture}(0,0)(90,0) +\includegraphics[height=0.9cm]{001_arrow.eps} +\end{picture} +\end{center} +\vspace{0.2cm} +\end{minipage} +}\\[0.2cm] + +\framebox{ +\begin{minipage}{5.9cm} +\includegraphics[width=5.9cm]{00-1_ip0-10.ps} +\end{minipage} +} + +\end{minipage} + +\end{slide} + +\begin{slide} + + {\large\bf\boldmath + Combinations with a C-Si \hkl<1 0 0>-type interstitial + } + +\small + +\vspace*{0.1cm} + +Binding energy: +$ +E_{\text{b}}= +E_{\text{f}}^{\text{defect combination}}- +E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}- +E_{\text{f}}^{\text{2nd defect}} +$ + +\vspace*{0.1cm} + +{\scriptsize +\begin{tabular}{l c c c c c c} +\hline + $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\ + \hline + \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\ + \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\ + \hkl<0 -1 0> & {\color{orange}-2.39} & -0.17 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\ + \hkl<0 1 0> & {\color{cyan}-2.25} & -1.90 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\ + \hkl<-1 0 0> & {\color{orange}-2.39} & -0.36 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\ + \hkl<1 0 0> & {\color{cyan}-2.25} & -2.16 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\ + \hline + C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\ + Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\ +\hline +\end{tabular} +} + +\vspace*{0.3cm} + +\footnotesize + +\begin{minipage}[t]{3.8cm} +\underline{\hkl<1 0 0> at position 1}\\[0.1cm] +\includegraphics[width=3.5cm]{00-1dc/2-25.eps} +\end{minipage} +\begin{minipage}[t]{3.5cm} +\underline{\hkl<0 -1 0> at position 1}\\[0.1cm] +\includegraphics[width=3.2cm]{00-1dc/2-39.eps} +\end{minipage} +\begin{minipage}[t]{5.5cm} +\begin{itemize} + \item Restricted to VASP simulations + \item $E_{\text{b}}=0$ for isolated non-interacting defects + \item $E_{\text{b}} \rightarrow 0$ for increasing distance (R) + \item Stress compensation / increase + \item Most favorable: C clustering + \item Unfavored: antiparallel orientations + \item Indication of energetically favored\\ + agglomeration +\end{itemize} +\end{minipage} + +\begin{picture}(0,0)(-295,-130) +\includegraphics[width=3.5cm]{comb_pos.eps} +\end{picture} + +\end{slide} + +\begin{slide} + + {\large\bf\boldmath + Combinations of C-Si \hkl<1 0 0>-type interstitials + } + +\small + +\vspace*{0.1cm} + +Energetically most favorable combinations along \hkl<1 1 0> + +\vspace*{0.1cm} + +{\scriptsize +\begin{tabular}{l c c c c c c} +\hline + & 1 & 2 & 3 & 4 & 5 & 6\\ +\hline +$E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\ +C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\ +Type & \hkl<-1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0>, \hkl<0 -1 0>\\ +\hline +\end{tabular} +} + +\vspace*{0.3cm} + +\begin{minipage}{7.0cm} +\includegraphics[width=7cm]{db_along_110_cc.ps} +\end{minipage} +\begin{minipage}{6.0cm} +\begin{center} +{\color{blue} + Interaction proportional to reciprocal cube of C-C distance +}\\[0.2cm] + Saturation in the immediate vicinity +\end{center} +\end{minipage} + +\vspace{0.2cm} + +\end{slide} + +\begin{slide} + + {\large\bf\boldmath + Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials + } + + \scriptsize + +\begin{center} +\begin{minipage}{3.2cm} +\includegraphics[width=3cm]{sub_110_combo.eps} +\end{minipage} +\begin{minipage}{7.8cm} +\begin{tabular}{l c c c c c c} +\hline +C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> & + \hkl<1 0 1> & \hkl<-1 0 1> \\ +\hline +1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\ +2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\ +3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\ +4 & \RM{4} & B & D & E & E & D \\ +5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\ +\hline +\end{tabular} +\end{minipage} +\end{center} + +\begin{center} +\begin{tabular}{l c c c c c c c c c c} +\hline +Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\ +\hline +$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\ +$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\ +$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\ +\hline +\end{tabular} +\end{center} + +\begin{minipage}{6.0cm} +\includegraphics[width=5.8cm]{c_sub_si110.ps} +\end{minipage} +\begin{minipage}{7cm} +\small +\begin{itemize} + \item IBS: C may displace Si\\ + $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial + \item Assumption:\\ + \hkl<1 1 0>-type $\rightarrow$ favored combination + \renewcommand\labelitemi{$\Rightarrow$} + \item Less favorable than C-Si \hkl<1 0 0> dumbbell\\ + ($E_{\text{f}}=3.88\text{ eV}$) + \item Interaction drops quickly to zero\\ + (low interaction capture radius) +\end{itemize} +\end{minipage} + +\end{slide} + +\begin{slide} + + {\large\bf\boldmath + Migration in C-Si \hkl<1 0 0> and vacancy combinations + } + + \footnotesize + +\vspace{0.1cm} + +\begin{minipage}[t]{3cm} +\underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\ +\includegraphics[width=2.8cm]{00-1dc/0-59.eps} +\end{minipage} +\begin{minipage}[t]{7cm} +\vspace{0.2cm} +\begin{center} + Low activation energies\\ + High activation energies for reverse processes\\ + $\Downarrow$\\ + {\color{blue}C$_{\text{sub}}$ very stable}\\ +\vspace*{0.1cm} + \hrule +\vspace*{0.1cm} + Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\ + $\Downarrow$\\ + {\color{blue}Formation of SiC by successive substitution by C} + +\end{center} +\end{minipage} +\begin{minipage}[t]{3cm} +\underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\ +\includegraphics[width=2.8cm]{00-1dc/3-14.eps} +\end{minipage} + + +\framebox{ +\begin{minipage}{5.9cm} +\includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm] +\begin{center} +\begin{picture}(0,0)(70,0) +\includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps} +\end{picture} +\begin{picture}(0,0)(30,0) +\includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps} +\end{picture} +\begin{picture}(0,0)(-10,0) +\includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps} +\end{picture} +\begin{picture}(0,0)(-48,0) +\includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps} +\end{picture} +\begin{picture}(0,0)(12.5,5) +\includegraphics[width=1cm]{100_arrow.eps} +\end{picture} +\begin{picture}(0,0)(97,-10) +\includegraphics[height=0.9cm]{001_arrow.eps} +\end{picture} +\end{center} +\vspace{0.1cm} +\end{minipage} +} +\begin{minipage}{0.3cm} +\hfill +\end{minipage} +\framebox{ +\begin{minipage}{5.9cm} +\includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm] +\begin{center} +\begin{picture}(0,0)(60,0) +\includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps} +\end{picture} +\begin{picture}(0,0)(25,0) +\includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps} +\end{picture} +\begin{picture}(0,0)(-20,0) +\includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps} +\end{picture} +\begin{picture}(0,0)(-55,0) +\includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps} +\end{picture} +\begin{picture}(0,0)(12.5,5) +\includegraphics[width=1cm]{100_arrow.eps} +\end{picture} +\begin{picture}(0,0)(95,0) +\includegraphics[height=0.9cm]{001_arrow.eps} +\end{picture} +\end{center} +\vspace{0.1cm} +\end{minipage} +} + +\end{slide} + +\begin{slide} + + {\large\bf + Conclusion of defect / migration / combined defect simulations + } + + \small + +\vspace*{0.1cm} + +Defect structures +\begin{itemize} + \item Accurately described by quantum-mechanical simulations + \item Less accurate description by classical potential simulations +\end{itemize} +\vspace*{0.2cm} +\begin{itemize} + \item \hkl<1 0 0> C-Si dumbbell interstitial ground state configuration + \item Consistent with reorientation and diffusion experiments + \item C migration pathway in Si identified +\end{itemize} + +Concerning the precipitation mechanism +\begin{itemize} + \item Agglomeration of C-Si dumbbells energetically favorable + \item C-Si indeed favored compared to + C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial + \item Possible low interaction capture radius of + C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial + \item In absence of nearby \hkl<1 1 0> Si self-interstitial: + C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC) +\end{itemize} +\begin{center} +{\color{blue}Some results point to a different precipitation mechanism!} +\end{center} +In progress ... +\begin{itemize} + \item \hkl<1 0 0> C-Si $\rightarrow$ + C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial + \item \hkl<1 0 0> C-Si combinations: C-C $\rightarrow$ C-...-C +\end{itemize} + + +\end{slide} + +\begin{slide} + + {\large\bf + Silicon carbide precipitation simulations + } + + \small + +{\scriptsize + \begin{pspicture}(0,0)(12,6.5) + % nodes + \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{ + \parbox{7cm}{ + \begin{itemize} + \item Create c-Si volume + \item Periodc boundary conditions + \item Set requested $T$ and $p=0\text{ bar}$ + \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$ + \end{itemize} + }}}} + \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{ + \parbox{7cm}{ + Insertion of C atoms at constant T + \begin{itemize} + \item total simulation volume {\pnode{in1}} + \item volume of minimal SiC precipitate {\pnode{in2}} + \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\ + precipitate + \end{itemize} + }}}} + \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{ + \parbox{7.0cm}{ + Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$ + }}}} + \ncline[]{->}{init}{insert} + \ncline[]{->}{insert}{cool} + \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3) + \rput(7.8,6){\footnotesize $V_1$} + \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5) + \rput(9.2,4.85){\tiny $V_2$} + \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75) + \rput(9.55,4.45){\footnotesize $V_3$} + \rput(7.9,3.2){\pnode{ins1}} + \rput(9.22,2.8){\pnode{ins2}} + \rput(11.0,2.4){\pnode{ins3}} + \ncline[]{->}{in1}{ins1} + \ncline[]{->}{in2}{ins2} + \ncline[]{->}{in3}{ins3} + \end{pspicture} +} + +\begin{itemize} + \item Restricted to classical potential simulations + \item $V_2$ and $V_3$ considered due to low diffusion + \item Amount of C atoms: 6000 + ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm) + \item Simulation volume: $31\times 31\times 31$ unit cells + (238328 Si atoms) +\end{itemize} + +\end{slide} + +\begin{slide} + + {\large\bf\boldmath + Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS + } + + \small + +\begin{minipage}{6.5cm} +\includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps} +\end{minipage} +\begin{minipage}{6.5cm} +\includegraphics[width=6.4cm]{sic_prec_450_energy.ps} +\end{minipage} + +\begin{minipage}{6.5cm} +\includegraphics[width=6.4cm]{sic_prec_450_si-c.ps} +\end{minipage} +\begin{minipage}{6.5cm} +\scriptsize +\underline{Low C concentration ($V_1$)}\\ +\hkl<1 0 0> C-Si dumbbell dominated structure +\begin{itemize} + \item Si-C bumbs around 0.19 nm + \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\ + concatenated dumbbells of various orientation + \item Si-Si NN distance stretched to 0.3 nm +\end{itemize} +{\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\ +\underline{High C concentration ($V_2$, $V_3$)}\\ +High amount of strongly bound C-C bonds\\ +Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\ +Only short range order observable\\ +{\color{blue}$\Rightarrow$ amorphous SiC-like phase} +\end{minipage} + +\end{slide} + +\begin{slide} + + {\large\bf + Limitations of molecular dynamics and short range potentials + } + +\footnotesize + +\vspace{0.2cm} + +\underline{Time scale problem of MD}\\[0.2cm] +Minimize integration error\\ +$\Rightarrow$ discretization considerably smaller than + reciprocal of fastest vibrational mode\\[0.1cm] +Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\ +$\Rightarrow$ suitable choice of time step: + $\tau=1\text{ fs}=10^{-15}\text{ s}$\\ +$\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm] +Several local minima in energy surface separated by large energy barriers\\ +$\Rightarrow$ transition event corresponds to a multiple + of vibrational periods\\ +$\Rightarrow$ phase transition made up of {\color{red}\underline{many}} + infrequent transition events\\[0.1cm] +{\color{blue}Accelerated methods:} +\underline{Temperature accelerated} MD (TAD), self-guided MD \ldots + +\vspace{0.3cm} + +\underline{Limitations related to the short range potential}\\[0.2cm] +Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$ +and 2$^{\text{nd}}$ next neighbours\\ +$\Rightarrow$ overestimated unphysical high forces of next neighbours + +\vspace{0.3cm} + +\framebox{ +\color{red} +Potential enhanced problem of slow phase space propagation +} + +\vspace{0.3cm} + +\underline{Approach to the (twofold) problem}\\[0.2cm] +Increased temperature simulations without TAD corrections\\ +(accelerated methods or higher time scales exclusively not sufficient) + +\begin{picture}(0,0)(-260,-30) +\framebox{ +\begin{minipage}{4.2cm} +\tiny +\begin{center} +\vspace{0.03cm} +\underline{IBS} +\end{center} +\begin{itemize} +\item 3C-SiC also observed for higher T +\item higher T inside sample +\item structural evolution vs.\\ + equilibrium properties +\end{itemize} +\end{minipage} +} +\end{picture} + +\begin{picture}(0,0)(-305,-155) +\framebox{ +\begin{minipage}{2.5cm} +\tiny +\begin{center} +retain proper\\ +thermodynmic sampling +\end{center} +\end{minipage} +} +\end{picture} + +\end{slide} + +\begin{slide} + + {\large\bf + Increased temperature simulations at low C concentration + } + +\small + +\begin{minipage}{6.5cm} +\includegraphics[width=6.4cm]{tot_pc_thesis.ps} +\end{minipage} +\begin{minipage}{6.5cm} +\includegraphics[width=6.4cm]{tot_pc3_thesis.ps} +\end{minipage} + +\begin{minipage}{6.5cm} +\includegraphics[width=6.4cm]{tot_pc2_thesis.ps} +\end{minipage} +\begin{minipage}{6.5cm} +\scriptsize + \underline{Si-C bonds:} + \begin{itemize} + \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$) + \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$ + \end{itemize} + \underline{Si-Si bonds:} + {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0> + ($\rightarrow$ 0.325 nm)\\[0.1cm] + \underline{C-C bonds:} + \begin{itemize} + \item C-C next neighbour pairs reduced (mandatory) + \item Peak at 0.3 nm slightly shifted + \begin{itemize} + \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\ + $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$ + combinations (|)\\ + $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations} + ($\downarrow$) + \item Range [|-$\downarrow$]: + {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$ + with nearby Si$_{\text{I}}$} + \end{itemize} + \end{itemize} +\end{minipage} + +\begin{picture}(0,0)(-330,-74) +\color{blue} +\framebox{ +\begin{minipage}{1.6cm} +\tiny +\begin{center} +stretched SiC\\[-0.1cm] +in c-Si +\end{center} +\end{minipage} +} +\end{picture} + +\end{slide} + +\begin{slide} + + {\large\bf + Increased temperature simulations at high C concentration + } + +\footnotesize + +\begin{minipage}{6.5cm} +\includegraphics[width=6.4cm]{12_pc_thesis.ps} +\end{minipage} +\begin{minipage}{6.5cm} +\includegraphics[width=6.4cm]{12_pc_c_thesis.ps} +\end{minipage} + +\begin{center} +Decreasing cut-off artifact\\ +High amount of {\color{red}damage} \& alignement to c-Si host matrix lost +$\Rightarrow$ hard to categorize +\end{center} + +\vspace{0.1cm} + +\framebox{ +\begin{minipage}[t]{6.0cm} +0.186 nm: Si-C pairs $\uparrow$\\ +(as expected in 3C-SiC)\\[0.2cm] +0.282 nm: Si-C-C\\[0.2cm] +$\approx$0.35 nm: C-Si-Si +\end{minipage} +} +\begin{minipage}{0.2cm} +\hfill +\end{minipage} +\framebox{ +\begin{minipage}[t]{6.0cm} +0.15 nm: C-C pairs $\uparrow$\\ +(as expected in graphite/diamond)\\[0.2cm] +0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm] +0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C +\end{minipage} +} + +\vspace{0.1cm} + +\begin{center} +{\color{red}Amorphous} SiC-like phase remains\\ +Slightly sharper peaks +$\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} +due to temperature\\[0.1cm] +\framebox{ +\bf +Continue with higher temperatures and longer time scales +} +\end{center} + +\end{slide} + +\begin{slide} + + {\large\bf + Long time scale simulations at maximum temperature + } + +\small + +\vspace{0.1cm} + +\underline{Differences} +\begin{itemize} + \item Temperature set to $0.95 \cdot T_{\text{m}}$ + \item Cubic insertion volume $\Rightarrow$ spherical insertion volume + \item Amount of C atoms: 6000 $\rightarrow$ 5500 + $\Leftrightarrow r_{\text{prec}}=0.3\text{ nm}$ + \item Simulation volume: 21 unit cells of c-Si in each direction +\end{itemize} + +\footnotesize + +\vspace{0.3cm} + +\begin{minipage}[t]{4.5cm} +\begin{center} +\underline{Low C concentration, Si-C} +\includegraphics[width=4.5cm]{c_in_si_95_v1_si-c.ps}\\ +Sharper peaks! +\end{center} +\end{minipage} +\begin{minipage}[t]{4.5cm} +\begin{center} +\underline{Low C concentration, C-C} +\includegraphics[width=4.5cm]{c_in_si_95_v1_c-c.ps}\\ +Sharper peaks!\\ +No C agglomeration! +\end{center} +\end{minipage} +\begin{minipage}[t]{4cm} +\begin{center} +\underline{High C concentration} +\includegraphics[width=4.5cm]{c_in_si_95_v2.ps}\\ +No significant changes\\ +C-Si-Si $\uparrow$\\ +C-Si-C $\downarrow$ +\end{center} +\end{minipage} + +\begin{center} +\framebox{ +Long time scales and high temperatures most probably not sufficient enough! +} +\end{center} + +\end{slide} + +\begin{slide} + + {\large\bf + Summary / Conclusion / Outlook + } + + \scriptsize + +\vspace{0.1cm} + +\framebox{ +\begin{minipage}{12.9cm} + \underline{Defects} + \begin{itemize} + \item Summary \& conclusion + \begin{itemize} + \item Point defects excellently / fairly well described + by QM / classical potential simulations + \item Identified migration path explaining + diffusion and reorientation experiments + \item Agglomeration of point defects energetically favorable + \item C$_{\text{sub}}$ favored conditions (conceivable in IBS) + \end{itemize} + \item In progress + \begin{itemize} + \item Migrations separating C-C bond in \hkl<1 0 0> C-Si dumbbell + interstitial combination + \item Migration: \hkl<1 0 0> C-Si $\rightarrow$ + C$_{\text{sub}}$ \& Si \hkl<1 1 0> interstitial + \end{itemize} + \item Todo + \begin{itemize} + \item Discussions concerning interpretation of QM results (Paderborn) + \item Compare migration barrier of + \hkl<1 1 0> Si and C-Si \hkl<1 0 0> dumbbell + \item Combination: Vacancy \& \hkl<1 1 0> Si self-interstitial \& + C-Si \hkl<1 0 0> dumbbell (IBS) + \end{itemize} + \end{itemize} +\end{minipage} +} + +\vspace{0.2cm} + +\framebox{ +\begin{minipage}[t]{12.9cm} + \underline{Pecipitation simulations} + \begin{itemize} + \item Summary \& conclusion + \begin{itemize} + \item Low T + $\rightarrow$ C-Si \hkl<1 0 0> dumbbell + dominated structure + \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure + \item High C concentration + $\rightarrow$ amorphous SiC like phase + \end{itemize} + \item Todo + \begin{itemize} + \item Accelerated method: self-guided MD + \item Activation relaxation technique + \item Constrainted transition path + \end{itemize} + \end{itemize} +\end{minipage} +} + + \small + +\end{slide} + +\begin{slide} + + {\large\bf + Acknowledgements + } + + \vspace{0.1cm} + + \small + + Thanks to \ldots + + \underline{Augsburg} + \begin{itemize} + \item Prof. B. Stritzker (accepting a simulator at EP \RM{4}) + \item Ralf Utermann (EDV) + \end{itemize} + + \underline{Helsinki} + \begin{itemize} + \item Prof. K. Nordlund (MD) + \end{itemize} + + \underline{Munich} + \begin{itemize} + \item Bayerische Forschungsstiftung (financial support) + \end{itemize} + + \underline{Paderborn} + \begin{itemize} + \item Prof. J. Lindner (SiC) + \item Prof. G. Schmidt (DFT + financial support) + \item Dr. E. Rauls (DFT + SiC) + \item Dr. S. Sanna (VASP) + \end{itemize} + +\vspace{0.2cm} + +\begin{center} +\framebox{ +\bf Thank you for your attention! +} +\end{center} + +\end{slide} + +\end{document} -- 2.39.2