2 %\documentclass[landscape,semhelv,draft]{seminar}
3 \documentclass[landscape,semhelv]{seminar}
6 \usepackage[greek,german]{babel}
7 \usepackage[latin1]{inputenc}
8 \usepackage[T1]{fontenc}
13 \usepackage{calc} % Simple computations with LaTeX variables
14 \usepackage{caption} % Improved captions
15 \usepackage{fancybox} % To have several backgrounds
17 \usepackage{fancyhdr} % Headers and footers definitions
18 \usepackage{fancyvrb} % Fancy verbatim environments
19 \usepackage{pstricks} % PSTricks with the standard color package
31 \graphicspath{{../img/}}
35 \usepackage[setpagesize=false]{hyperref}
41 \usepackage{semlayer} % Seminar overlays
42 \usepackage{slidesec} % Seminar sections and list of slides
44 \input{seminar.bug} % Official bugs corrections
45 \input{seminar.bg2} % Unofficial bugs corrections
52 %\usepackage{cmbright}
53 %\renewcommand{\familydefault}{\sfdefault}
54 %\usepackage{mathptmx}
60 \extraslideheight{10in}
65 % specify width and height
70 \def\slidetopmargin{-0.15cm}
72 \newcommand{\ham}{\mathcal{H}}
73 \newcommand{\pot}{\mathcal{V}}
74 \newcommand{\foo}{\mathcal{U}}
75 \newcommand{\vir}{\mathcal{W}}
78 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
81 \renewcommand{\phi}{\varphi}
84 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
87 \newrgbcolor{si-yellow}{.6 .6 0}
88 \newrgbcolor{hb}{0.75 0.77 0.89}
89 \newrgbcolor{lbb}{0.75 0.8 0.88}
90 \newrgbcolor{hlbb}{0.825 0.88 0.968}
91 \newrgbcolor{lachs}{1.0 .93 .81}
94 \newcommand{\si}{Si$_{\text{i}}${}}
95 \newcommand{\ci}{C$_{\text{i}}${}}
96 \newcommand{\cs}{C$_{\text{sub}}${}}
97 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
98 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
99 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
100 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
102 % no vertical centering
113 A B C D E F G H G F E D C B A
128 Atomistic simulation studies\\[0.2cm]
134 \textsc{Frank Zirkelbach}
138 Application talk at the Max Planck Institute for Solid State Research
142 Stuttgart, November 2011
147 % no vertical centering
157 % Phase diagram of the C/Si system\\
162 \begin{minipage}{6.5cm}
163 \includegraphics[width=6.5cm]{si-c_phase.eps}
166 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
169 \begin{pspicture}(0,0)(0,0)
170 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
173 \begin{minipage}{6cm}
174 {\bf Phase diagram of the C/Si system}\\[0.2cm]
175 {\color{blue}Stoichiometric composition}
177 \item only chemical stable compound
178 \item wide band gap semiconductor\\
179 \underline{silicon carbide}, SiC
185 % motivation / properties / applications of silicon carbide
193 \begin{pspicture}(0,0)(13.5,5)
195 \psframe*[linecolor=hb](-0.2,0)(12.9,5)
197 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.2,1)(6.5,1)(6.5,3)(5.2,3)
198 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.4,0.5)(7.7,2)(7.7,2)(6.4,3.5)
200 \rput[lt](0,4.6){\color{gray}PROPERTIES}
202 \rput[lt](0.3,4){wide band gap}
203 \rput[lt](0.3,3.5){high electric breakdown field}
204 \rput[lt](0.3,3){good electron mobility}
205 \rput[lt](0.3,2.5){high electron saturation drift velocity}
206 \rput[lt](0.3,2){high thermal conductivity}
208 \rput[lt](0.3,1.5){hard and mechanically stable}
209 \rput[lt](0.3,1){chemically inert}
211 \rput[lt](0.3,0.5){radiation hardness}
213 \rput[rt](12.7,4.6){\color{gray}APPLICATIONS}
215 \rput[rt](12.5,3.85){high-temperature, high power}
216 \rput[rt](12.5,3.5){and high-frequency}
217 \rput[rt](12.5,3.15){electronic and optoelectronic devices}
219 \rput[rt](12.5,2.35){material suitable for extreme conditions}
220 \rput[rt](12.5,2){microelectromechanical systems}
221 \rput[rt](12.5,1.65){abrasives, cutting tools, heating elements}
223 \rput[rt](12.5,0.85){first wall reactor material, detectors}
224 \rput[rt](12.5,0.5){and electronic devices for space}
228 \begin{picture}(0,0)(5,-162)
229 \includegraphics[height=2.2cm]{3C_SiC_bs.eps}
231 \begin{picture}(0,0)(-120,-162)
232 \includegraphics[height=2.2cm]{nasa_600c_led.eps}
234 \begin{picture}(0,0)(-270,-162)
235 \includegraphics[height=2.2cm]{6h-sic_3c-sic.eps}
238 \begin{picture}(0,0)(10,65)
239 \includegraphics[height=2.8cm]{sic_switch.eps}
241 %\begin{picture}(0,0)(-243,65)
242 \begin{picture}(0,0)(-110,65)
243 \includegraphics[height=2.8cm]{ise_99.eps}
245 %\begin{picture}(0,0)(-135,65)
246 \begin{picture}(0,0)(-100,65)
247 \includegraphics[height=1.2cm]{infineon_schottky.eps}
249 \begin{picture}(0,0)(-233,65)
250 \includegraphics[height=2.8cm]{solar_car.eps}
260 Polytypes of SiC\\[0.4cm]
263 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
264 \begin{minipage}{1.9cm}
265 {\tiny cubic (twist)}
267 \begin{minipage}{2.9cm}
268 {\tiny hexagonal (no twist)}
271 \begin{picture}(0,0)(-150,0)
272 \includegraphics[width=7cm]{polytypes.eps}
279 \begin{tabular}{l c c c c c c}
281 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
283 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
284 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
285 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
286 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
287 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
288 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
289 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
293 \begin{pspicture}(0,0)(0,0)
294 \psellipse[linecolor=green](5.7,2.10)(0.4,0.5)
296 \begin{pspicture}(0,0)(0,0)
297 \psellipse[linecolor=green](5.6,0.92)(0.4,0.2)
299 \begin{pspicture}(0,0)(0,0)
300 \psellipse[linecolor=red](10.45,0.45)(0.4,0.2)
310 Fabrication of silicon carbide
319 \emph{Silicon carbide --- Born from the stars, perfected on earth.}
325 SiC thin films by MBE \& CVD
327 \item Much progress achieved in homo/heteroepitaxial SiC thin film growth
328 \item \underline{Commercially available} semiconductor power devices based on
329 \underline{\foreignlanguage{greek}{a}-SiC}
330 \item Production of favored \underline{3C-SiC} material
331 \underline{less advanced}
332 \item Quality and size not yet sufficient
334 \begin{picture}(0,0)(-310,-20)
335 \includegraphics[width=2.0cm]{cree.eps}
340 Alternative approach:
341 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
348 \begin{minipage}{3.15cm}
350 \includegraphics[width=3cm]{imp.eps}\\
356 \begin{minipage}{3.15cm}
358 \includegraphics[width=3cm]{annealing.eps}\\
360 \unit[12]{h} annealing at \degc{1200}
365 \begin{minipage}{5.5cm}
366 \includegraphics[width=5.8cm]{ibs_3c-sic.eps}\\[-0.2cm]
369 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
381 Systematic investigation of C implantations into Si
387 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
405 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
408 \begin{pspicture}(0,0)(0,0)
409 \rput(6.0,7.0){\rnode{init}{\psframebox[fillstyle=gradient,gradbegin=white,gradend=red,gradlines=1000,gradmidpoint=0.5,linestyle=none]{
410 \begin{minipage}{11cm}
411 {\color{black}Diploma thesis}\\
412 \underline{Monte Carlo} simulation modeling the selforganization process\\
413 leading to periodic arrays of nanometric amorphous SiC precipitates
417 \begin{pspicture}(0,0)(0,0)
418 \rput(6.0,-0.5){\rnode{init}{\psframebox[fillstyle=gradient,gradbegin=white,gradend=blue,gradmidpoint=0.5,gradlines=1000,linestyle=none]{
419 \begin{minipage}{11cm}
420 {\color{black}Doctoral studies}\\
421 Classical potential \underline{molecular dynamics} simulations \ldots\\
422 \underline{Density functional theory} calculations \ldots\\[0.2cm]
423 \ldots on defect formation and SiC precipitation in Si
427 \begin{pspicture}(0,0)(0,0)
428 \psellipse[linecolor=red,linewidth=0.05cm](5,3.0)(0.8,1.0)
430 \begin{pspicture}(0,0)(0,0)
431 \psellipse[linecolor=blue,linewidth=0.05cm](8.2,3.2)(1.5,1.6)
439 Selforganization of nanometric amorphous SiC lamellae
442 \begin{pspicture}(0,0)(0,0)
443 \psframebox[fillstyle=gradient,gradbegin=white,gradend=red,gradlines=1000,gradmidpoint=0.5,linestyle=none]{
444 \begin{minipage}{14cm}
456 \item Regularly spaced, nanometric spherical\\
457 and lamellar amorphous inclusions\\
458 at the upper a/c interface
459 \item Carbon accumulation\\
465 \begin{minipage}{12cm}
466 \includegraphics[width=9cm]{../../nlsop/img/k393abild1_e_l.eps}\\
468 XTEM bright-field, \unit[180]{keV} C$^+ \rightarrow$ Si, \degc{150},
469 Dose: \unit[4.3 $\times 10^{17}$]{cm$^{-2}$}
473 \begin{picture}(0,0)(-182,-215)
474 \begin{minipage}{6.5cm}
476 \includegraphics[width=6.5cm]{../../nlsop/img/eftem.eps}\\[-0.2cm]
478 XTEM bright-field and respective EFTEM C map
492 Model displaying the formation of ordered lamellae
498 \includegraphics[width=8.0cm]{../../nlsop/img/modell_ng_e.eps}
504 \item Supersaturation of C in c-Si\\
505 $\rightarrow$ {\bf Carbon induced} nucleation of spherical
507 \item High interfacial energy between 3C-SiC and c-Si\\
508 $\rightarrow$ {\bf Amorphous} precipitates
509 \item \unit[20-- 30]{\%} lower silicon density of a-SiC$_x$ compared to c-Si\\
510 $\rightarrow$ {\bf Lateral strain} (black arrows)
511 \item Implantation range near surface\\
512 $\rightarrow$ {\bf Relaxation} of {\bf vertical strain component}
513 \item Reduction of the carbon supersaturation in c-Si\\
514 $\rightarrow$ {\bf Carbon diffusion} into amorphous volumina
516 \item Remaining lateral strain\\
517 $\rightarrow$ {\bf Strain enhanced} lateral amorphisation
518 \item Absence of crystalline neighbours (structural information)\\
519 $\rightarrow$ {\bf Stabilization} of amorphous inclusions
520 {\bf against recrystallization}
528 Implementation of the Monte Carlo code
534 \item \underline{Amorphization / Recrystallization}\\
535 Ion collision in discretized target determined by random numbers
536 distributed according to nuclear energy loss.
537 Amorphization/recrystallization probability:
539 p_{c \rightarrow a}(\vec{r}) = {\color{green} p_b} + {\color{blue} p_c c_C(\vec{r})} + {\color{red} \sum_{\textrm{amorphous neighbours}} \frac{p_s c_C(\vec{r'})}{(r-r')^2}}
542 \item {\color{green} $p_b$} normal `ballistic' amorphization
543 \item {\color{blue} $p_c$} carbon induced amorphization
544 \item {\color{red} $p_s$} stress enhanced amorphization
547 p_{a \rightarrow c}(\vec r) = (1 - p_{c \rightarrow a}(\vec r)) \Big(1 - \frac{\sum_{direct \, neighbours} \delta (\vec{r'})}{6} \Big) \, \textrm{,}
550 \delta (\vec r) = \left\{
552 1 & \textrm{if volume at position $\vec r$ is amorphous} \\
553 0 & \textrm{otherwise} \\
557 \item \underline{Carbon incorporation}\\
558 Incorporation volume determined according to implantation profile
559 \item \underline{Diffusion / Sputtering}
561 \item Transfer fraction of C atoms
562 of crystalline into neighbored amorphous volumes
563 \item Remove surface layer
571 \begin{minipage}{3.7cm}
580 Evolution of the \ldots
585 \item lamella precipitates
587 \ldots reproduced!\\[1.5cm]
591 Experiment \& simulation\\
592 in good agreement\\[1.0cm]
594 Simulation is able to model the whole depth region\\[1.0cm]
599 \begin{minipage}{0.4cm}
602 \begin{minipage}{8.0cm}
604 \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e_1-2.eps}\\
605 \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e2_2-2.eps}
613 Structural \& compositional details
616 \begin{minipage}[t]{7.5cm}
617 \includegraphics[height=6.5cm]{../../nlsop/img/ac_cconc_ver2_e.eps}\\
619 \begin{minipage}[t]{5.0cm}
620 \includegraphics[height=6.5cm]{../../nlsop/img/97_98_e.eps}
628 \item Fluctuation of C concentration in lamellae region
629 \item \unit[8--10]{at.\%} C saturation limit
630 within the respective conditions
631 \item Complementarily arranged and alternating sequence of layers\\
632 with a high and low amount of amorphous regions
633 \item C accumulation in the amorphous phase / Origin of stress
636 \begin{picture}(0,0)(-265,-30)
638 \begin{minipage}{3cm}
641 Precipitation process\\
663 Model displaying the formation of ordered lamellae
667 \begin{minipage}{6.3cm}
670 Precipitation mechanism not yet fully understood!
672 \renewcommand\labelitemi{$\Rightarrow$}
674 \underline{Understanding the SiC precipitation}
676 \item significant technological progress in SiC thin film formation
677 \item perspectives for processes relying upon prevention of SiC precipitation
688 Supposed precipitation mechanism of SiC in Si
695 \begin{minipage}{3.8cm}
696 Si \& SiC lattice structure\\[0.2cm]
697 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
701 \begin{minipage}{3.8cm}
703 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
707 \begin{minipage}{3.8cm}
709 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
713 \begin{minipage}{4cm}
715 C-Si dimers (dumbbells)\\[-0.1cm]
716 on Si interstitial sites
720 \begin{minipage}{4.2cm}
722 Agglomeration of C-Si dumbbells\\[-0.1cm]
723 $\Rightarrow$ dark contrasts
727 \begin{minipage}{4cm}
729 Precipitation of 3C-SiC in Si\\[-0.1cm]
730 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
731 \& release of Si self-interstitials
735 \begin{minipage}{3.8cm}
737 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
741 \begin{minipage}{3.8cm}
743 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
747 \begin{minipage}{3.8cm}
749 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
753 \begin{pspicture}(0,0)(0,0)
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756 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
757 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
758 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
759 $4a_{\text{Si}}=5a_{\text{SiC}}$
761 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
762 \hkl(h k l) planes match
764 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
774 Supposed precipitation mechanism of SiC in Si
781 \begin{minipage}{3.8cm}
782 Si \& SiC lattice structure\\[0.2cm]
783 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
787 \begin{minipage}{3.8cm}
789 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
793 \begin{minipage}{3.8cm}
795 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
799 \begin{minipage}{4cm}
801 C-Si dimers (dumbbells)\\[-0.1cm]
802 on Si interstitial sites
806 \begin{minipage}{4.2cm}
808 Agglomeration of C-Si dumbbells\\[-0.1cm]
809 $\Rightarrow$ dark contrasts
813 \begin{minipage}{4cm}
815 Precipitation of 3C-SiC in Si\\[-0.1cm]
816 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
817 \& release of Si self-interstitials
821 \begin{minipage}{3.8cm}
823 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
827 \begin{minipage}{3.8cm}
829 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
833 \begin{minipage}{3.8cm}
835 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
839 \begin{pspicture}(0,0)(0,0)
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843 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
844 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
845 $4a_{\text{Si}}=5a_{\text{SiC}}$
847 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
848 \hkl(h k l) planes match
850 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
853 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
854 \begin{minipage}{10cm}
856 {\color{red}\bf Controversial views}
858 \item Implantations at high T (Nejim et al.)
860 \item Topotactic transformation based on \cs
861 \item \si{} as supply reacting with further C in cleared volume
863 \item Annealing behavior (Serre et al.)
865 \item Room temperature implants $\rightarrow$ highly mobile C
866 \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
867 (indicate stable \cs{} configurations)
869 \item Strained silicon \& Si/SiC heterostructures
871 \item Coherent SiC precipitates (tensile strain)
872 \item Incoherent SiC (strain relaxation)
884 Molecular dynamics (MD) simulations
893 \item Microscopic description of N particle system
894 \item Analytical interaction potential
895 \item Numerical integration using Newtons equation of motion\\
896 as a propagation rule in 6N-dimensional phase space
897 \item Observables obtained by time and/or ensemble averages
899 {\bf Details of the simulation:}
901 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
902 \item Ensemble: NpT (isothermal-isobaric)
904 \item Berendsen thermostat:
905 $\tau_{\text{T}}=100\text{ fs}$
906 \item Berendsen barostat:\\
907 $\tau_{\text{P}}=100\text{ fs}$,
908 $\beta^{-1}=100\text{ GPa}$
910 \item Erhart/Albe potential: Tersoff-like bond order potential
913 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
914 \pot_{ij} = {\color{red}f_C(r_{ij})}
915 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
919 \begin{picture}(0,0)(-230,-30)
920 \includegraphics[width=5cm]{tersoff_angle.eps}
928 Density functional theory (DFT) calculations
933 Basic ingredients necessary for DFT
936 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
938 \item ... uniquely determines the ground state potential
940 \item ... minimizes the systems total energy
942 \item \underline{Born-Oppenheimer}
943 - $N$ moving electrons in an external potential of static nuclei
945 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
946 +\sum_i^N V_{\text{ext}}(r_i)
947 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
949 \item \underline{Effective potential}
950 - averaged electrostatic potential \& exchange and correlation
952 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
955 \item \underline{Kohn-Sham system}
956 - Schr\"odinger equation of N non-interacting particles
958 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
963 n(r)=\sum_i^N|\Phi_i(r)|^2
965 \item \underline{Self-consistent solution}\\
966 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
967 which in turn depends on $n(r)$
968 \item \underline{Variational principle}
969 - minimize total energy with respect to $n(r)$
977 Density functional theory (DFT) calculations
984 Details of applied DFT calculations in this work
987 \item \underline{Exchange correlation functional}
988 - approximations for the inhomogeneous electron gas
990 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
991 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
993 \item \underline{Plane wave basis set}
994 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
997 \text{Fourier series: } \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r), \quad E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}
998 \qquad ({\color{blue}300\text{ eV}})
1000 \item \underline{Brillouin zone sampling} -
1001 {\color{blue}$\Gamma$-point only} calculations
1002 \item \underline{Pseudo potential}
1003 - consider only the valence electrons
1004 \item \underline{Code} - VASP 4.6
1009 MD and structural optimization
1012 \item MD integration: Gear predictor corrector algorithm
1013 \item Pressure control: Parrinello-Rahman pressure control
1014 \item Structural optimization: Conjugate gradient method
1017 \begin{pspicture}(0,0)(0,0)
1018 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
1026 C and Si self-interstitial point defects in silicon
1033 \begin{minipage}{8cm}
1035 \begin{pspicture}(0,0)(7,5)
1036 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1039 \item Creation of c-Si simulation volume
1040 \item Periodic boundary conditions
1041 \item $T=0\text{ K}$, $p=0\text{ bar}$
1044 \rput(3.5,2.1){\rnode{insert}{\psframebox{
1047 Insertion of interstitial C/Si atoms
1050 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1053 Relaxation / structural energy minimization
1056 \ncline[]{->}{init}{insert}
1057 \ncline[]{->}{insert}{cool}
1060 \begin{minipage}{5cm}
1061 \includegraphics[width=5cm]{unit_cell_e.eps}\\
1064 \begin{minipage}{9cm}
1065 \begin{tabular}{l c c}
1067 & size [unit cells] & \# atoms\\
1069 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
1070 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
1074 \begin{minipage}{4cm}
1075 {\color{red}$\bullet$} Tetrahedral\\
1076 {\color{green}$\bullet$} Hexagonal\\
1077 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
1078 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
1079 {\color{cyan}$\bullet$} Bond-centered\\
1080 {\color{black}$\bullet$} Vacancy / Substitutional
1089 \begin{minipage}{9.5cm}
1092 Si self-interstitial point defects in silicon\\
1095 \begin{tabular}{l c c c c c}
1097 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
1099 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
1100 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
1102 \end{tabular}\\[0.2cm]
1104 \begin{minipage}{4.7cm}
1105 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
1107 \begin{minipage}{4.7cm}
1109 {\tiny nearly T $\rightarrow$ T}\\
1111 \includegraphics[width=4.7cm]{nhex_tet.ps}
1114 \underline{Hexagonal} \hspace{2pt}
1115 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
1117 \begin{minipage}{2.7cm}
1118 $E_{\text{f}}^*=4.48\text{ eV}$\\
1119 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
1121 \begin{minipage}{0.4cm}
1126 \begin{minipage}{2.7cm}
1127 $E_{\text{f}}=3.96\text{ eV}$\\
1128 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
1131 \begin{minipage}{2.9cm}
1133 \underline{Vacancy}\\
1134 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
1139 \begin{minipage}{3.5cm}
1142 \underline{\hkl<1 1 0> dumbbell}\\
1143 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
1144 \underline{Tetrahedral}\\
1145 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
1146 \underline{\hkl<1 0 0> dumbbell}\\
1147 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
1159 C interstitial point defects in silicon\\[-0.1cm]
1162 \begin{tabular}{l c c c c c c r}
1164 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
1166 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
1167 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
1169 \end{tabular}\\[0.1cm]
1172 \begin{minipage}{2.7cm}
1173 \underline{Hexagonal} \hspace{2pt}
1174 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
1175 $E_{\text{f}}^*=9.05\text{ eV}$\\
1176 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
1178 \begin{minipage}{0.4cm}
1183 \begin{minipage}{2.7cm}
1184 \underline{\hkl<1 0 0>}\\
1185 $E_{\text{f}}=3.88\text{ eV}$\\
1186 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
1189 \begin{minipage}{2cm}
1192 \begin{minipage}{3cm}
1194 \underline{Tetrahedral}\\
1195 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
1200 \begin{minipage}{2.7cm}
1201 \underline{Bond-centered}\\
1202 $E_{\text{f}}^*=5.59\text{ eV}$\\
1203 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
1205 \begin{minipage}{0.4cm}
1210 \begin{minipage}{2.7cm}
1211 \underline{\hkl<1 1 0> dumbbell}\\
1212 $E_{\text{f}}=5.18\text{ eV}$\\
1213 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
1216 \begin{minipage}{2cm}
1219 \begin{minipage}{3cm}
1221 \underline{Substitutional}\\
1222 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
1233 C \hkl<1 0 0> dumbbell interstitial configuration\\
1237 \begin{tabular}{l c c c c c c c c}
1239 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1241 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1242 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1244 \end{tabular}\\[0.2cm]
1245 \begin{tabular}{l c c c c }
1247 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1249 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1250 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1252 \end{tabular}\\[0.2cm]
1253 \begin{tabular}{l c c c}
1255 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1257 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1258 VASP & 0.109 & -0.065 & 0.174 \\
1260 \end{tabular}\\[0.6cm]
1263 \begin{minipage}{3.0cm}
1265 \underline{Erhart/Albe}
1266 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1269 \begin{minipage}{3.0cm}
1272 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1276 \begin{picture}(0,0)(-185,10)
1277 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1279 \begin{picture}(0,0)(-280,-150)
1280 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1283 \begin{pspicture}(0,0)(0,0)
1284 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1285 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1286 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1287 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1296 \begin{minipage}{8.5cm}
1299 Bond-centered interstitial configuration\\[-0.1cm]
1302 \begin{minipage}{3.0cm}
1303 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1305 \begin{minipage}{5.2cm}
1307 \item Linear Si-C-Si bond
1308 \item Si: one C \& 3 Si neighbours
1309 \item Spin polarized calculations
1310 \item No saddle point!\\
1317 \begin{minipage}[t]{6.5cm}
1318 \begin{minipage}[t]{1.2cm}
1320 {\tiny sp$^3$}\\[0.8cm]
1321 \underline{${\color{black}\uparrow}$}
1322 \underline{${\color{black}\uparrow}$}
1323 \underline{${\color{black}\uparrow}$}
1324 \underline{${\color{red}\uparrow}$}\\
1327 \begin{minipage}[t]{1.4cm}
1329 {\color{red}M}{\color{blue}O}\\[0.8cm]
1330 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1331 $\sigma_{\text{ab}}$\\[0.5cm]
1332 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1336 \begin{minipage}[t]{1.0cm}
1340 \underline{${\color{white}\uparrow\uparrow}$}
1341 \underline{${\color{white}\uparrow\uparrow}$}\\
1343 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1344 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1348 \begin{minipage}[t]{1.4cm}
1350 {\color{blue}M}{\color{green}O}\\[0.8cm]
1351 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1352 $\sigma_{\text{ab}}$\\[0.5cm]
1353 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1357 \begin{minipage}[t]{1.2cm}
1360 {\tiny sp$^3$}\\[0.8cm]
1361 \underline{${\color{green}\uparrow}$}
1362 \underline{${\color{black}\uparrow}$}
1363 \underline{${\color{black}\uparrow}$}
1364 \underline{${\color{black}\uparrow}$}\\
1372 \begin{minipage}{4.5cm}
1373 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1375 \begin{minipage}{3.5cm}
1376 {\color{gray}$\bullet$} Spin up\\
1377 {\color{green}$\bullet$} Spin down\\
1378 {\color{blue}$\bullet$} Resulting spin up\\
1379 {\color{yellow}$\bullet$} Si atoms\\
1380 {\color{red}$\bullet$} C atom
1385 \begin{minipage}{4.2cm}
1387 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1388 {\color{green}$\Box$} {\tiny unoccupied}\\
1389 {\color{red}$\bullet$} {\tiny occupied}
1398 Migration of the C \hkl<1 0 0> dumbbell interstitial
1403 {\small Investigated pathways}
1405 \begin{minipage}{8.5cm}
1406 \begin{minipage}{8.3cm}
1407 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1408 \begin{minipage}{2.4cm}
1409 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1411 \begin{minipage}{0.4cm}
1414 \begin{minipage}{2.4cm}
1415 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1417 \begin{minipage}{0.4cm}
1420 \begin{minipage}{2.4cm}
1421 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1424 \begin{minipage}{8.3cm}
1425 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1426 \begin{minipage}{2.4cm}
1427 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1429 \begin{minipage}{0.4cm}
1432 \begin{minipage}{2.4cm}
1433 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1435 \begin{minipage}{0.4cm}
1438 \begin{minipage}{2.4cm}
1439 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1442 \begin{minipage}{8.3cm}
1443 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1444 \begin{minipage}{2.4cm}
1445 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1447 \begin{minipage}{0.4cm}
1450 \begin{minipage}{2.4cm}
1451 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1453 \begin{minipage}{0.4cm}
1456 \begin{minipage}{2.4cm}
1457 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1462 \begin{minipage}{4.2cm}
1463 {\small Constrained relaxation\\
1464 technique (CRT) method}\\
1465 \includegraphics[width=4cm]{crt_orig.eps}
1467 \item Constrain diffusing atom
1468 \item Static constraints
1471 {\small Modifications}\\
1472 \includegraphics[width=4cm]{crt_mod.eps}
1474 \item Constrain all atoms
1475 \item Update individual\\
1486 Migration of the C \hkl<1 0 0> dumbbell interstitial
1492 \begin{minipage}{5.9cm}
1494 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1497 \begin{picture}(0,0)(60,0)
1498 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1500 \begin{picture}(0,0)(-5,0)
1501 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1503 \begin{picture}(0,0)(-55,0)
1504 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1506 \begin{picture}(0,0)(12.5,10)
1507 \includegraphics[width=1cm]{110_arrow.eps}
1509 \begin{picture}(0,0)(90,0)
1510 \includegraphics[height=0.9cm]{001_arrow.eps}
1516 \begin{minipage}{0.3cm}
1520 \begin{minipage}{5.9cm}
1522 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1525 \begin{picture}(0,0)(60,0)
1526 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1528 \begin{picture}(0,0)(5,0)
1529 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1531 \begin{picture}(0,0)(-55,0)
1532 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1534 \begin{picture}(0,0)(12.5,10)
1535 \includegraphics[width=1cm]{100_arrow.eps}
1537 \begin{picture}(0,0)(90,0)
1538 \includegraphics[height=0.9cm]{001_arrow.eps}
1548 \begin{minipage}{5.9cm}
1550 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1553 \begin{picture}(0,0)(60,0)
1554 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1556 \begin{picture}(0,0)(10,0)
1557 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1559 \begin{picture}(0,0)(-60,0)
1560 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1562 \begin{picture}(0,0)(12.5,10)
1563 \includegraphics[width=1cm]{100_arrow.eps}
1565 \begin{picture}(0,0)(90,0)
1566 \includegraphics[height=0.9cm]{001_arrow.eps}
1572 \begin{minipage}{0.3cm}
1575 \begin{minipage}{6.5cm}
1578 \item Energetically most favorable path
1581 \item Activation energy: $\approx$ 0.9 eV
1582 \item Experimental values: 0.73 ... 0.87 eV
1584 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1585 \item Reorientation (path 3)
1587 \item More likely composed of two consecutive steps of type 2
1588 \item Experimental values: 0.77 ... 0.88 eV
1590 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1599 Migration of the C \hkl<1 0 0> dumbbell interstitial
1606 \begin{minipage}{6.5cm}
1609 \begin{minipage}[t]{5.9cm}
1611 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1614 \begin{pspicture}(0,0)(0,0)
1615 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1617 \begin{picture}(0,0)(60,-50)
1618 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1620 \begin{picture}(0,0)(5,-50)
1621 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1623 \begin{picture}(0,0)(-55,-50)
1624 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1626 \begin{picture}(0,0)(12.5,-40)
1627 \includegraphics[width=1cm]{110_arrow.eps}
1629 \begin{picture}(0,0)(90,-45)
1630 \includegraphics[height=0.9cm]{001_arrow.eps}
1632 \begin{pspicture}(0,0)(0,0)
1633 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1635 \begin{picture}(0,0)(60,-15)
1636 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1638 \begin{picture}(0,0)(35,-15)
1639 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1641 \begin{picture}(0,0)(-5,-15)
1642 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1644 \begin{picture}(0,0)(-55,-15)
1645 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1647 \begin{picture}(0,0)(12.5,-5)
1648 \includegraphics[width=1cm]{100_arrow.eps}
1650 \begin{picture}(0,0)(90,-15)
1651 \includegraphics[height=0.9cm]{010_arrow.eps}
1657 \begin{minipage}{5.9cm}
1660 \item Lowest activation energy: $\approx$ 2.2 eV
1661 \item 2.4 times higher than VASP
1662 \item Different pathway
1667 \begin{minipage}{6.5cm}
1670 \begin{minipage}{5.9cm}
1672 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1675 %\begin{pspicture}(0,0)(0,0)
1676 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1678 %\begin{picture}(0,0)(60,-5)
1679 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1681 %\begin{picture}(0,0)(0,-5)
1682 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1684 %\begin{picture}(0,0)(-55,-5)
1685 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1687 %\begin{picture}(0,0)(12.5,5)
1688 %\includegraphics[width=1cm]{100_arrow.eps}
1690 %\begin{picture}(0,0)(90,0)
1691 %\includegraphics[height=0.9cm]{001_arrow.eps}
1699 %\begin{minipage}{5.9cm}
1700 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1704 \begin{minipage}{5.9cm}
1705 Transition involving \ci{} \hkl<1 1 0>
1707 \item Bond-centered configuration unstable\\
1708 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1709 \item Transition minima of path 2 \& 3\\
1710 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1711 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1712 \item 2.4 - 3.4 times higher than VASP
1713 \item Rotation of dumbbell orientation
1717 {\color{blue}Overestimated diffusion barrier}
1728 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1738 E_{\text{f}}^{\text{defect combination}}-
1739 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1740 E_{\text{f}}^{\text{2nd defect}}
1746 \begin{tabular}{l c c c c c c}
1748 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1750 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1751 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1752 \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}\\
1753 \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}\\
1754 \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}\\
1755 \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}\\
1757 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1758 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1767 \begin{minipage}[t]{3.8cm}
1768 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1769 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1771 \begin{minipage}[t]{3.5cm}
1772 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1773 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1775 \begin{minipage}[t]{5.5cm}
1777 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1778 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1779 \item Stress compensation / increase
1780 \item Unfavored: antiparallel orientations
1781 \item Indication of energetically favored\\
1783 \item Most favorable: C clustering
1784 \item However: High barrier ($>4\,\text{eV}$)
1785 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1790 \begin{picture}(0,0)(-295,-130)
1791 \includegraphics[width=3.5cm]{comb_pos.eps}
1799 Combinations of C-Si \hkl<1 0 0>-type interstitials
1806 Energetically most favorable combinations along \hkl<1 1 0>
1811 \begin{tabular}{l c c c c c c}
1813 & 1 & 2 & 3 & 4 & 5 & 6\\
1815 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1816 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1817 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>\\
1824 \begin{minipage}{7.0cm}
1825 \includegraphics[width=7cm]{db_along_110_cc.ps}
1827 \begin{minipage}{6.0cm}
1829 \item Interaction proportional to reciprocal cube of C-C distance
1830 \item Saturation in the immediate vicinity
1831 \renewcommand\labelitemi{$\Rightarrow$}
1832 \item Agglomeration of \ci{} expected
1833 \item Absence of C clustering
1837 Consisten with initial precipitation model
1849 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1855 %\begin{minipage}{3.2cm}
1856 %\includegraphics[width=3cm]{sub_110_combo.eps}
1858 %\begin{minipage}{7.8cm}
1859 %\begin{tabular}{l c c c c c c}
1861 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1862 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1864 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1865 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1866 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1867 %4 & \RM{4} & B & D & E & E & D \\
1868 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1875 %\begin{tabular}{l c c c c c c c c c c}
1877 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1879 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1880 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1881 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1886 \begin{minipage}{6.0cm}
1887 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1889 \begin{minipage}{7cm}
1892 \item IBS: C may displace Si\\
1893 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1895 \hkl<1 1 0>-type $\rightarrow$ favored combination
1896 \renewcommand\labelitemi{$\Rightarrow$}
1897 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1898 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1899 \item Interaction drops quickly to zero\\
1900 $\rightarrow$ low capture radius
1904 IBS process far from equilibrium\\
1905 \cs{} \& \si{} instead of thermodynamic ground state
1910 \begin{minipage}{6.5cm}
1911 \includegraphics[width=6.0cm]{162-097.ps}
1913 \item Low migration barrier
1916 \begin{minipage}{6.5cm}
1918 Ab initio MD at \degc{900}\\
1919 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1920 $t=\unit[2230]{fs}$\\
1921 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1925 Contribution of entropy to structural formation
1934 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1941 \begin{minipage}[t]{3cm}
1942 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1943 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1945 \begin{minipage}[t]{7cm}
1948 Low activation energies\\
1949 High activation energies for reverse processes\\
1951 {\color{blue}C$_{\text{sub}}$ very stable}\\
1955 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1957 {\color{blue}Formation of SiC by successive substitution by C}
1961 \begin{minipage}[t]{3cm}
1962 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1963 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1968 \begin{minipage}{5.9cm}
1969 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1971 \begin{picture}(0,0)(70,0)
1972 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1974 \begin{picture}(0,0)(30,0)
1975 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1977 \begin{picture}(0,0)(-10,0)
1978 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1980 \begin{picture}(0,0)(-48,0)
1981 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1983 \begin{picture}(0,0)(12.5,5)
1984 \includegraphics[width=1cm]{100_arrow.eps}
1986 \begin{picture}(0,0)(97,-10)
1987 \includegraphics[height=0.9cm]{001_arrow.eps}
1993 \begin{minipage}{0.3cm}
1997 \begin{minipage}{5.9cm}
1998 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
2000 \begin{picture}(0,0)(60,0)
2001 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
2003 \begin{picture}(0,0)(25,0)
2004 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
2006 \begin{picture}(0,0)(-20,0)
2007 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
2009 \begin{picture}(0,0)(-55,0)
2010 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
2012 \begin{picture}(0,0)(12.5,5)
2013 \includegraphics[width=1cm]{100_arrow.eps}
2015 \begin{picture}(0,0)(95,0)
2016 \includegraphics[height=0.9cm]{001_arrow.eps}
2028 Conclusion of defect / migration / combined defect simulations
2037 \item Accurately described by quantum-mechanical simulations
2038 \item Less accurate description by classical potential simulations
2039 \item Underestimated formation energy of \cs{} by classical approach
2040 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
2045 \item C migration pathway in Si identified
2046 \item Consistent with reorientation and diffusion experiments
2049 \item Different path and ...
2050 \item overestimated barrier by classical potential calculations
2053 Concerning the precipitation mechanism
2055 \item Agglomeration of C-Si dumbbells energetically favorable
2056 (stress compensation)
2057 \item C-Si indeed favored compared to
2058 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2059 \item Possible low interaction capture radius of
2060 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2061 \item Low barrier for
2062 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
2063 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
2064 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
2067 {\color{blue}Results suggest increased participation of \cs}
2075 Silicon carbide precipitation simulations
2081 \begin{pspicture}(0,0)(12,6.5)
2083 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
2086 \item Create c-Si volume
2087 \item Periodc boundary conditions
2088 \item Set requested $T$ and $p=0\text{ bar}$
2089 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
2092 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
2094 Insertion of C atoms at constant T
2096 \item total simulation volume {\pnode{in1}}
2097 \item volume of minimal SiC precipitate {\pnode{in2}}
2098 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
2102 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
2104 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
2106 \ncline[]{->}{init}{insert}
2107 \ncline[]{->}{insert}{cool}
2108 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
2109 \rput(7.8,6){\footnotesize $V_1$}
2110 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
2111 \rput(9.2,4.85){\tiny $V_2$}
2112 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
2113 \rput(9.55,4.45){\footnotesize $V_3$}
2114 \rput(7.9,3.2){\pnode{ins1}}
2115 \rput(9.22,2.8){\pnode{ins2}}
2116 \rput(11.0,2.4){\pnode{ins3}}
2117 \ncline[]{->}{in1}{ins1}
2118 \ncline[]{->}{in2}{ins2}
2119 \ncline[]{->}{in3}{ins3}
2124 \item Restricted to classical potential simulations
2125 \item $V_2$ and $V_3$ considered due to low diffusion
2126 \item Amount of C atoms: 6000
2127 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
2128 \item Simulation volume: $31\times 31\times 31$ unit cells
2137 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2142 \begin{minipage}{6.5cm}
2143 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2145 \begin{minipage}{6.5cm}
2146 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2149 \begin{minipage}{6.5cm}
2150 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2152 \begin{minipage}{6.5cm}
2154 \underline{Low C concentration ($V_1$)}\\
2155 \hkl<1 0 0> C-Si dumbbell dominated structure
2157 \item Si-C bumbs around 0.19 nm
2158 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2159 concatenated dumbbells of various orientation
2160 \item Si-Si NN distance stretched to 0.3 nm
2162 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2163 \underline{High C concentration ($V_2$, $V_3$)}\\
2164 High amount of strongly bound C-C bonds\\
2165 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2166 Only short range order observable\\
2167 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2175 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2180 \begin{minipage}{6.5cm}
2181 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2183 \begin{minipage}{6.5cm}
2184 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2187 \begin{minipage}{6.5cm}
2188 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2190 \begin{minipage}{6.5cm}
2192 \underline{Low C concentration ($V_1$)}\\
2193 \hkl<1 0 0> C-Si dumbbell dominated structure
2195 \item Si-C bumbs around 0.19 nm
2196 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2197 concatenated dumbbells of various orientation
2198 \item Si-Si NN distance stretched to 0.3 nm
2200 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2201 \underline{High C concentration ($V_2$, $V_3$)}\\
2202 High amount of strongly bound C-C bonds\\
2203 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2204 Only short range order observable\\
2205 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2208 \begin{pspicture}(0,0)(0,0)
2209 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2210 \begin{minipage}{10cm}
2212 {\color{red}\bf 3C-SiC formation fails to appear}
2214 \item Low C concentration simulations
2216 \item Formation of \ci{} indeed occurs
2217 \item Agllomeration not observed
2219 \item High C concentration simulations
2221 \item Amorphous SiC-like structure\\
2222 (not expected at prevailing temperatures)
2223 \item Rearrangement and transition into 3C-SiC structure missing
2235 Limitations of molecular dynamics and short range potentials
2242 \underline{Time scale problem of MD}\\[0.2cm]
2243 Minimize integration error\\
2244 $\Rightarrow$ discretization considerably smaller than
2245 reciprocal of fastest vibrational mode\\[0.1cm]
2246 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2247 $\Rightarrow$ suitable choice of time step:
2248 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2249 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2250 Several local minima in energy surface separated by large energy barriers\\
2251 $\Rightarrow$ transition event corresponds to a multiple
2252 of vibrational periods\\
2253 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2254 infrequent transition events\\[0.1cm]
2255 {\color{blue}Accelerated methods:}
2256 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2260 \underline{Limitations related to the short range potential}\\[0.2cm]
2261 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2262 and 2$^{\text{nd}}$ next neighbours\\
2263 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2269 Potential enhanced problem of slow phase space propagation
2274 \underline{Approach to the (twofold) problem}\\[0.2cm]
2275 Increased temperature simulations without TAD corrections\\
2276 (accelerated methods or higher time scales exclusively not sufficient)
2278 \begin{picture}(0,0)(-260,-30)
2280 \begin{minipage}{4.2cm}
2287 \item 3C-SiC also observed for higher T
2288 \item higher T inside sample
2289 \item structural evolution vs.\\
2290 equilibrium properties
2296 \begin{picture}(0,0)(-305,-155)
2298 \begin{minipage}{2.5cm}
2302 thermodynmic sampling
2313 Increased temperature simulations at low C concentration
2318 \begin{minipage}{6.5cm}
2319 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2321 \begin{minipage}{6.5cm}
2322 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2325 \begin{minipage}{6.5cm}
2326 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2328 \begin{minipage}{6.5cm}
2330 \underline{Si-C bonds:}
2332 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2333 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2335 \underline{Si-Si bonds:}
2336 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2337 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2338 \underline{C-C bonds:}
2340 \item C-C next neighbour pairs reduced (mandatory)
2341 \item Peak at 0.3 nm slightly shifted
2343 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2344 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2346 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2348 \item Range [|-$\downarrow$]:
2349 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2350 with nearby Si$_{\text{I}}$}
2355 \begin{picture}(0,0)(-330,-74)
2358 \begin{minipage}{1.6cm}
2361 stretched SiC\\[-0.1cm]
2373 Increased temperature simulations at low C concentration
2378 \begin{minipage}{6.5cm}
2379 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2381 \begin{minipage}{6.5cm}
2382 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2385 \begin{minipage}{6.5cm}
2386 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2388 \begin{minipage}{6.5cm}
2390 \underline{Si-C bonds:}
2392 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2393 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2395 \underline{Si-Si bonds:}
2396 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2397 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2398 \underline{C-C bonds:}
2400 \item C-C next neighbour pairs reduced (mandatory)
2401 \item Peak at 0.3 nm slightly shifted
2403 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2404 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2406 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2408 \item Range [|-$\downarrow$]:
2409 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2410 with nearby Si$_{\text{I}}$}
2415 %\begin{picture}(0,0)(-330,-74)
2418 %\begin{minipage}{1.6cm}
2421 %stretched SiC\\[-0.1cm]
2428 \begin{pspicture}(0,0)(0,0)
2429 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2430 \begin{minipage}{10cm}
2432 {\color{blue}\bf Stretched SiC in c-Si}
2434 \item Consistent to precipitation model involving \cs{}
2435 \item Explains annealing behavior of high/low T C implants
2437 \item Low T: highly mobiel \ci{}
2438 \item High T: stable configurations of \cs{}
2441 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2442 $\Rightarrow$ Precipitation mechanism involving \cs{}
2452 Increased temperature simulations at high C concentration
2457 \begin{minipage}{6.5cm}
2458 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2460 \begin{minipage}{6.5cm}
2461 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2469 \begin{minipage}[t]{6.0cm}
2470 0.186 nm: Si-C pairs $\uparrow$\\
2471 (as expected in 3C-SiC)\\[0.2cm]
2472 0.282 nm: Si-C-C\\[0.2cm]
2473 $\approx$0.35 nm: C-Si-Si
2476 \begin{minipage}{0.2cm}
2480 \begin{minipage}[t]{6.0cm}
2481 0.15 nm: C-C pairs $\uparrow$\\
2482 (as expected in graphite/diamond)\\[0.2cm]
2483 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2484 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2489 \item Decreasing cut-off artifact
2490 \item {\color{red}Amorphous} SiC-like phase remains
2491 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2492 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2501 High C \& small $V$ \& short $t$
2504 Slow restructuring due to strong C-C bonds
2507 High C \& low T implants
2518 Summary and Conclusions
2526 \begin{minipage}[t]{12.9cm}
2527 \underline{Pecipitation simulations}
2529 \item High C concentration $\rightarrow$ amorphous SiC like phase
2530 \item Problem of potential enhanced slow phase space propagation
2531 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2532 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2533 \item High T necessary to simulate IBS conditions (far from equilibrium)
2534 \item Precipitation by successive agglomeration of \cs (epitaxy)
2535 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2536 (stretched SiC, interface)
2544 \begin{minipage}{12.9cm}
2549 \item Point defects excellently / fairly well described
2551 \item C$_{\text{sub}}$ drastically underestimated by EA
2552 \item EA predicts correct ground state:
2553 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2554 \item Identified migration path explaining
2555 diffusion and reorientation experiments by DFT
2556 \item EA fails to describe \ci{} migration:
2557 Wrong path \& overestimated barrier
2559 \item Combinations of defects
2561 \item Agglomeration of point defects energetically favorable
2562 by compensation of stress
2563 \item Formation of C-C unlikely
2564 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2565 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2566 Low barrier (\unit[0.77]{eV}) \& low capture radius
2574 \framebox{Precipitation by successive agglomeration of \cs{}}
2592 \underline{Augsburg}
2594 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2595 \item Ralf Utermann (EDV)
2598 \underline{Helsinki}
2600 \item Prof. K. Nordlund (MD)
2605 \item Bayerische Forschungsstiftung (financial support)
2608 \underline{Paderborn}
2610 \item Prof. J. Lindner (SiC)
2611 \item Prof. G. Schmidt (DFT + financial support)
2612 \item Dr. E. Rauls (DFT + SiC)
2613 \item Dr. S. Sanna (VASP)
2620 \bf Thank you for your attention!