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3 \documentclass[landscape,semhelv]{seminar}
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19 \usepackage{pstricks} % PSTricks with the standard color package
30 \graphicspath{{../img/}}
34 \usepackage[setpagesize=false]{hyperref}
40 \usepackage{semlayer} % Seminar overlays
41 \usepackage{slidesec} % Seminar sections and list of slides
43 \input{seminar.bug} % Official bugs corrections
44 \input{seminar.bg2} % Unofficial bugs corrections
51 %\usepackage{cmbright}
52 %\renewcommand{\familydefault}{\sfdefault}
53 %\usepackage{mathptmx}
59 \extraslideheight{10in}
64 % specify width and height
69 \def\slidetopmargin{-0.15cm}
71 \newcommand{\ham}{\mathcal{H}}
72 \newcommand{\pot}{\mathcal{V}}
73 \newcommand{\foo}{\mathcal{U}}
74 \newcommand{\vir}{\mathcal{W}}
77 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
80 \renewcommand{\phi}{\varphi}
83 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
86 \newrgbcolor{si-yellow}{.6 .6 0}
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89 \newrgbcolor{hlbb}{0.825 0.88 0.968}
90 \newrgbcolor{lachs}{1.0 .93 .81}
93 \newcommand{\si}{Si$_{\text{i}}${}}
94 \newcommand{\ci}{C$_{\text{i}}${}}
95 \newcommand{\cs}{C$_{\text{sub}}${}}
96 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
97 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
98 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
99 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
101 % no vertical centering
112 A B C D E F G H G F E D C B A
127 Atomistic simulation studies\\[0.2cm]
133 \textsc{Frank Zirkelbach}
137 Application talk at the Max Planck Institute for Solid State Research
141 Stuttgart, November 2011
146 % no vertical centering
156 % Phase diagram of the C/Si system\\
161 \begin{minipage}{6.5cm}
162 \includegraphics[width=6.5cm]{si-c_phase.eps}
165 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
168 \begin{pspicture}(0,0)(0,0)
169 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
172 \begin{minipage}{6cm}
173 {\bf Phase diagram of the C/Si system}\\[0.2cm]
174 {\color{blue}Stoichiometric composition}
176 \item only chemical stable compound
177 \item wide band gap semiconductor\\
178 \underline{silicon carbide}, SiC
184 % motivation / properties / applications of silicon carbide
190 \begin{pspicture}(0,0)(13.5,5)
192 \psframe*[linecolor=hb](-0.2,0)(12.9,5)
194 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.2,1)(6.5,1)(6.5,3)(5.2,3)
195 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.4,0.5)(7.7,2)(7.7,2)(6.4,3.5)
197 \rput[lt](0,4.6){\color{gray}PROPERTIES}
199 \rput[lt](0.3,4){wide band gap}
200 \rput[lt](0.3,3.5){high electric breakdown field}
201 \rput[lt](0.3,3){good electron mobility}
202 \rput[lt](0.3,2.5){high electron saturation drift velocity}
203 \rput[lt](0.3,2){high thermal conductivity}
205 \rput[lt](0.3,1.5){hard and mechanically stable}
206 \rput[lt](0.3,1){chemically inert}
208 \rput[lt](0.3,0.5){radiation hardness}
210 \rput[rt](12.7,4.6){\color{gray}APPLICATIONS}
212 \rput[rt](12.5,3.85){high-temperature, high power}
213 \rput[rt](12.5,3.5){and high-frequency}
214 \rput[rt](12.5,3.15){electronic and optoelectronic devices}
216 \rput[rt](12.5,2.35){material suitable for extreme conditions}
217 \rput[rt](12.5,2){microelectromechanical systems}
218 \rput[rt](12.5,1.65){abrasives, cutting tools, heating elements}
220 \rput[rt](12.5,0.85){first wall reactor material, detectors}
221 \rput[rt](12.5,0.5){and electronic devices for space}
225 \begin{picture}(0,0)(5,-162)
226 \includegraphics[height=2.2cm]{3C_SiC_bs.eps}
228 \begin{picture}(0,0)(-120,-162)
229 \includegraphics[height=2.2cm]{nasa_600c_led.eps}
231 \begin{picture}(0,0)(-270,-162)
232 \includegraphics[height=2.2cm]{6h-sic_3c-sic.eps}
235 \begin{picture}(0,0)(10,65)
236 \includegraphics[height=2.8cm]{sic_switch.eps}
238 %\begin{picture}(0,0)(-243,65)
239 \begin{picture}(0,0)(-110,65)
240 \includegraphics[height=2.8cm]{ise_99.eps}
242 %\begin{picture}(0,0)(-135,65)
243 \begin{picture}(0,0)(-100,65)
244 \includegraphics[height=1.2cm]{infineon_schottky.eps}
246 \begin{picture}(0,0)(-233,65)
247 \includegraphics[height=2.8cm]{solar_car.eps}
257 Polytypes of SiC\\[0.4cm]
260 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
261 \begin{minipage}{1.9cm}
262 {\tiny cubic (twist)}
264 \begin{minipage}{2.9cm}
265 {\tiny hexagonal (no twist)}
268 \begin{picture}(0,0)(-150,0)
269 \includegraphics[width=7cm]{polytypes.eps}
276 \begin{tabular}{l c c c c c c}
278 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
280 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
281 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
282 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
283 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
284 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
285 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
286 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
290 \begin{pspicture}(0,0)(0,0)
291 \psellipse[linecolor=green](5.7,2.10)(0.4,0.5)
293 \begin{pspicture}(0,0)(0,0)
294 \psellipse[linecolor=green](5.6,0.92)(0.4,0.2)
296 \begin{pspicture}(0,0)(0,0)
297 \psellipse[linecolor=red](10.45,0.45)(0.4,0.2)
307 Fabrication of silicon carbide
316 \emph{Silicon carbide --- Born from the stars, perfected on earth.}
322 SiC thin films by MBE \& CVD
324 \item Much progress achieved in homo/heteroepitaxial SiC thin film growth
325 \item \underline{Commercially available} semiconductor power devices based on
326 \underline{\foreignlanguage{greek}{a}-SiC}
327 \item Production of favored \underline{3C-SiC} material
328 \underline{less advanced}
329 \item Quality and size not yet sufficient
331 \begin{picture}(0,0)(-310,-20)
332 \includegraphics[width=2.0cm]{cree.eps}
337 Alternative approach:
338 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
345 \begin{minipage}{3.15cm}
347 \includegraphics[width=3cm]{imp.eps}\\
353 \begin{minipage}{3.15cm}
355 \includegraphics[width=3cm]{annealing.eps}\\
357 \unit[12]{h} annealing at \degc{1200}
362 \begin{minipage}{5.5cm}
363 \includegraphics[width=5.8cm]{ibs_3c-sic.eps}\\[-0.2cm]
366 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
378 Systematic investigation of C implantations into Si
384 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
400 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
403 \begin{pspicture}(0,0)(0,0)
404 \rput(6.0,7.0){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
405 \begin{minipage}{11cm}
406 {\color{red}Diploma thesis}\\
407 \underline{Monte Carlo} simulation modeling the selforganization process\\
408 leading to periodic arrays of nanometric amorphous SiC precipitates
412 \begin{pspicture}(0,0)(0,0)
413 \rput(6.0,-0.5){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
414 \begin{minipage}{11cm}
415 {\color{blue}Doctoral studies}\\
416 Classical potential \underline{molecular dynamics} simulations \ldots\\
417 \underline{Density functional theory} calculations \ldots\\[0.2cm]
418 \ldots on defect formation and SiC precipitation in Si
422 \begin{pspicture}(0,0)(0,0)
423 \psellipse[linecolor=red,linewidth=0.05cm](5,3.0)(0.8,1.0)
425 \begin{pspicture}(0,0)(0,0)
426 \psellipse[linecolor=blue,linewidth=0.05cm](8.2,3.2)(1.5,1.6)
436 Selforganization of nanometric amorphous SiC lamellae
439 \begin{minipage}{6cm}
440 \includegraphics[width=6cm]{}
452 Selforganization of nanometric amorphous SiC lamellae
456 \begin{minipage}{6.3cm}
459 Precipitation mechanism not yet fully understood!
461 \renewcommand\labelitemi{$\Rightarrow$}
463 \underline{Understanding the SiC precipitation}
465 \item significant technological progress in SiC thin film formation
466 \item perspectives for processes relying upon prevention of SiC precipitation
477 Supposed precipitation mechanism of SiC in Si
484 \begin{minipage}{3.8cm}
485 Si \& SiC lattice structure\\[0.2cm]
486 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
490 \begin{minipage}{3.8cm}
492 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
496 \begin{minipage}{3.8cm}
498 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
502 \begin{minipage}{4cm}
504 C-Si dimers (dumbbells)\\[-0.1cm]
505 on Si interstitial sites
509 \begin{minipage}{4.2cm}
511 Agglomeration of C-Si dumbbells\\[-0.1cm]
512 $\Rightarrow$ dark contrasts
516 \begin{minipage}{4cm}
518 Precipitation of 3C-SiC in Si\\[-0.1cm]
519 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
520 \& release of Si self-interstitials
524 \begin{minipage}{3.8cm}
526 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
530 \begin{minipage}{3.8cm}
532 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
536 \begin{minipage}{3.8cm}
538 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
542 \begin{pspicture}(0,0)(0,0)
543 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
544 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
545 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
546 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
547 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
548 $4a_{\text{Si}}=5a_{\text{SiC}}$
550 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
551 \hkl(h k l) planes match
553 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
563 Supposed precipitation mechanism of SiC in Si
570 \begin{minipage}{3.8cm}
571 Si \& SiC lattice structure\\[0.2cm]
572 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
576 \begin{minipage}{3.8cm}
578 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
582 \begin{minipage}{3.8cm}
584 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
588 \begin{minipage}{4cm}
590 C-Si dimers (dumbbells)\\[-0.1cm]
591 on Si interstitial sites
595 \begin{minipage}{4.2cm}
597 Agglomeration of C-Si dumbbells\\[-0.1cm]
598 $\Rightarrow$ dark contrasts
602 \begin{minipage}{4cm}
604 Precipitation of 3C-SiC in Si\\[-0.1cm]
605 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
606 \& release of Si self-interstitials
610 \begin{minipage}{3.8cm}
612 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
616 \begin{minipage}{3.8cm}
618 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
622 \begin{minipage}{3.8cm}
624 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
628 \begin{pspicture}(0,0)(0,0)
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632 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
633 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
634 $4a_{\text{Si}}=5a_{\text{SiC}}$
636 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
637 \hkl(h k l) planes match
639 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
642 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
643 \begin{minipage}{10cm}
645 {\color{red}\bf Controversial views}
647 \item Implantations at high T (Nejim et al.)
649 \item Topotactic transformation based on \cs
650 \item \si{} as supply reacting with further C in cleared volume
652 \item Annealing behavior (Serre et al.)
654 \item Room temperature implants $\rightarrow$ highly mobile C
655 \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
656 (indicate stable \cs{} configurations)
658 \item Strained silicon \& Si/SiC heterostructures
660 \item Coherent SiC precipitates (tensile strain)
661 \item Incoherent SiC (strain relaxation)
673 Molecular dynamics (MD) simulations
682 \item Microscopic description of N particle system
683 \item Analytical interaction potential
684 \item Numerical integration using Newtons equation of motion\\
685 as a propagation rule in 6N-dimensional phase space
686 \item Observables obtained by time and/or ensemble averages
688 {\bf Details of the simulation:}
690 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
691 \item Ensemble: NpT (isothermal-isobaric)
693 \item Berendsen thermostat:
694 $\tau_{\text{T}}=100\text{ fs}$
695 \item Berendsen barostat:\\
696 $\tau_{\text{P}}=100\text{ fs}$,
697 $\beta^{-1}=100\text{ GPa}$
699 \item Erhart/Albe potential: Tersoff-like bond order potential
702 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
703 \pot_{ij} = {\color{red}f_C(r_{ij})}
704 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
708 \begin{picture}(0,0)(-230,-30)
709 \includegraphics[width=5cm]{tersoff_angle.eps}
717 Density functional theory (DFT) calculations
722 Basic ingredients necessary for DFT
725 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
727 \item ... uniquely determines the ground state potential
729 \item ... minimizes the systems total energy
731 \item \underline{Born-Oppenheimer}
732 - $N$ moving electrons in an external potential of static nuclei
734 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
735 +\sum_i^N V_{\text{ext}}(r_i)
736 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
738 \item \underline{Effective potential}
739 - averaged electrostatic potential \& exchange and correlation
741 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
744 \item \underline{Kohn-Sham system}
745 - Schr\"odinger equation of N non-interacting particles
747 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
752 n(r)=\sum_i^N|\Phi_i(r)|^2
754 \item \underline{Self-consistent solution}\\
755 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
756 which in turn depends on $n(r)$
757 \item \underline{Variational principle}
758 - minimize total energy with respect to $n(r)$
766 Density functional theory (DFT) calculations
773 Details of applied DFT calculations in this work
776 \item \underline{Exchange correlation functional}
777 - approximations for the inhomogeneous electron gas
779 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
780 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
782 \item \underline{Plane wave basis set}
783 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
786 \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}}
787 \qquad ({\color{blue}300\text{ eV}})
789 \item \underline{Brillouin zone sampling} -
790 {\color{blue}$\Gamma$-point only} calculations
791 \item \underline{Pseudo potential}
792 - consider only the valence electrons
793 \item \underline{Code} - VASP 4.6
798 MD and structural optimization
801 \item MD integration: Gear predictor corrector algorithm
802 \item Pressure control: Parrinello-Rahman pressure control
803 \item Structural optimization: Conjugate gradient method
806 \begin{pspicture}(0,0)(0,0)
807 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
815 C and Si self-interstitial point defects in silicon
822 \begin{minipage}{8cm}
824 \begin{pspicture}(0,0)(7,5)
825 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
828 \item Creation of c-Si simulation volume
829 \item Periodic boundary conditions
830 \item $T=0\text{ K}$, $p=0\text{ bar}$
833 \rput(3.5,2.1){\rnode{insert}{\psframebox{
836 Insertion of interstitial C/Si atoms
839 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
842 Relaxation / structural energy minimization
845 \ncline[]{->}{init}{insert}
846 \ncline[]{->}{insert}{cool}
849 \begin{minipage}{5cm}
850 \includegraphics[width=5cm]{unit_cell_e.eps}\\
853 \begin{minipage}{9cm}
854 \begin{tabular}{l c c}
856 & size [unit cells] & \# atoms\\
858 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
859 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
863 \begin{minipage}{4cm}
864 {\color{red}$\bullet$} Tetrahedral\\
865 {\color{green}$\bullet$} Hexagonal\\
866 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
867 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
868 {\color{cyan}$\bullet$} Bond-centered\\
869 {\color{black}$\bullet$} Vacancy / Substitutional
878 \begin{minipage}{9.5cm}
881 Si self-interstitial point defects in silicon\\
884 \begin{tabular}{l c c c c c}
886 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
888 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
889 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
891 \end{tabular}\\[0.2cm]
893 \begin{minipage}{4.7cm}
894 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
896 \begin{minipage}{4.7cm}
898 {\tiny nearly T $\rightarrow$ T}\\
900 \includegraphics[width=4.7cm]{nhex_tet.ps}
903 \underline{Hexagonal} \hspace{2pt}
904 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
906 \begin{minipage}{2.7cm}
907 $E_{\text{f}}^*=4.48\text{ eV}$\\
908 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
910 \begin{minipage}{0.4cm}
915 \begin{minipage}{2.7cm}
916 $E_{\text{f}}=3.96\text{ eV}$\\
917 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
920 \begin{minipage}{2.9cm}
922 \underline{Vacancy}\\
923 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
928 \begin{minipage}{3.5cm}
931 \underline{\hkl<1 1 0> dumbbell}\\
932 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
933 \underline{Tetrahedral}\\
934 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
935 \underline{\hkl<1 0 0> dumbbell}\\
936 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
948 C interstitial point defects in silicon\\[-0.1cm]
951 \begin{tabular}{l c c c c c c r}
953 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
955 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
956 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
958 \end{tabular}\\[0.1cm]
961 \begin{minipage}{2.7cm}
962 \underline{Hexagonal} \hspace{2pt}
963 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
964 $E_{\text{f}}^*=9.05\text{ eV}$\\
965 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
967 \begin{minipage}{0.4cm}
972 \begin{minipage}{2.7cm}
973 \underline{\hkl<1 0 0>}\\
974 $E_{\text{f}}=3.88\text{ eV}$\\
975 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
978 \begin{minipage}{2cm}
981 \begin{minipage}{3cm}
983 \underline{Tetrahedral}\\
984 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
989 \begin{minipage}{2.7cm}
990 \underline{Bond-centered}\\
991 $E_{\text{f}}^*=5.59\text{ eV}$\\
992 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
994 \begin{minipage}{0.4cm}
999 \begin{minipage}{2.7cm}
1000 \underline{\hkl<1 1 0> dumbbell}\\
1001 $E_{\text{f}}=5.18\text{ eV}$\\
1002 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
1005 \begin{minipage}{2cm}
1008 \begin{minipage}{3cm}
1010 \underline{Substitutional}\\
1011 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
1022 C \hkl<1 0 0> dumbbell interstitial configuration\\
1026 \begin{tabular}{l c c c c c c c c}
1028 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1030 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1031 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1033 \end{tabular}\\[0.2cm]
1034 \begin{tabular}{l c c c c }
1036 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1038 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1039 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1041 \end{tabular}\\[0.2cm]
1042 \begin{tabular}{l c c c}
1044 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1046 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1047 VASP & 0.109 & -0.065 & 0.174 \\
1049 \end{tabular}\\[0.6cm]
1052 \begin{minipage}{3.0cm}
1054 \underline{Erhart/Albe}
1055 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1058 \begin{minipage}{3.0cm}
1061 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1065 \begin{picture}(0,0)(-185,10)
1066 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1068 \begin{picture}(0,0)(-280,-150)
1069 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1072 \begin{pspicture}(0,0)(0,0)
1073 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1074 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1075 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1076 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1085 \begin{minipage}{8.5cm}
1088 Bond-centered interstitial configuration\\[-0.1cm]
1091 \begin{minipage}{3.0cm}
1092 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1094 \begin{minipage}{5.2cm}
1096 \item Linear Si-C-Si bond
1097 \item Si: one C \& 3 Si neighbours
1098 \item Spin polarized calculations
1099 \item No saddle point!\\
1106 \begin{minipage}[t]{6.5cm}
1107 \begin{minipage}[t]{1.2cm}
1109 {\tiny sp$^3$}\\[0.8cm]
1110 \underline{${\color{black}\uparrow}$}
1111 \underline{${\color{black}\uparrow}$}
1112 \underline{${\color{black}\uparrow}$}
1113 \underline{${\color{red}\uparrow}$}\\
1116 \begin{minipage}[t]{1.4cm}
1118 {\color{red}M}{\color{blue}O}\\[0.8cm]
1119 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1120 $\sigma_{\text{ab}}$\\[0.5cm]
1121 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1125 \begin{minipage}[t]{1.0cm}
1129 \underline{${\color{white}\uparrow\uparrow}$}
1130 \underline{${\color{white}\uparrow\uparrow}$}\\
1132 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1133 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1137 \begin{minipage}[t]{1.4cm}
1139 {\color{blue}M}{\color{green}O}\\[0.8cm]
1140 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1141 $\sigma_{\text{ab}}$\\[0.5cm]
1142 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1146 \begin{minipage}[t]{1.2cm}
1149 {\tiny sp$^3$}\\[0.8cm]
1150 \underline{${\color{green}\uparrow}$}
1151 \underline{${\color{black}\uparrow}$}
1152 \underline{${\color{black}\uparrow}$}
1153 \underline{${\color{black}\uparrow}$}\\
1161 \begin{minipage}{4.5cm}
1162 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1164 \begin{minipage}{3.5cm}
1165 {\color{gray}$\bullet$} Spin up\\
1166 {\color{green}$\bullet$} Spin down\\
1167 {\color{blue}$\bullet$} Resulting spin up\\
1168 {\color{yellow}$\bullet$} Si atoms\\
1169 {\color{red}$\bullet$} C atom
1174 \begin{minipage}{4.2cm}
1176 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1177 {\color{green}$\Box$} {\tiny unoccupied}\\
1178 {\color{red}$\bullet$} {\tiny occupied}
1187 Migration of the C \hkl<1 0 0> dumbbell interstitial
1192 {\small Investigated pathways}
1194 \begin{minipage}{8.5cm}
1195 \begin{minipage}{8.3cm}
1196 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1197 \begin{minipage}{2.4cm}
1198 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1200 \begin{minipage}{0.4cm}
1203 \begin{minipage}{2.4cm}
1204 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1206 \begin{minipage}{0.4cm}
1209 \begin{minipage}{2.4cm}
1210 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1213 \begin{minipage}{8.3cm}
1214 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1215 \begin{minipage}{2.4cm}
1216 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1218 \begin{minipage}{0.4cm}
1221 \begin{minipage}{2.4cm}
1222 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1224 \begin{minipage}{0.4cm}
1227 \begin{minipage}{2.4cm}
1228 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1231 \begin{minipage}{8.3cm}
1232 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1233 \begin{minipage}{2.4cm}
1234 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1236 \begin{minipage}{0.4cm}
1239 \begin{minipage}{2.4cm}
1240 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1242 \begin{minipage}{0.4cm}
1245 \begin{minipage}{2.4cm}
1246 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1251 \begin{minipage}{4.2cm}
1252 {\small Constrained relaxation\\
1253 technique (CRT) method}\\
1254 \includegraphics[width=4cm]{crt_orig.eps}
1256 \item Constrain diffusing atom
1257 \item Static constraints
1260 {\small Modifications}\\
1261 \includegraphics[width=4cm]{crt_mod.eps}
1263 \item Constrain all atoms
1264 \item Update individual\\
1275 Migration of the C \hkl<1 0 0> dumbbell interstitial
1281 \begin{minipage}{5.9cm}
1283 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1286 \begin{picture}(0,0)(60,0)
1287 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1289 \begin{picture}(0,0)(-5,0)
1290 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1292 \begin{picture}(0,0)(-55,0)
1293 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1295 \begin{picture}(0,0)(12.5,10)
1296 \includegraphics[width=1cm]{110_arrow.eps}
1298 \begin{picture}(0,0)(90,0)
1299 \includegraphics[height=0.9cm]{001_arrow.eps}
1305 \begin{minipage}{0.3cm}
1309 \begin{minipage}{5.9cm}
1311 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1314 \begin{picture}(0,0)(60,0)
1315 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1317 \begin{picture}(0,0)(5,0)
1318 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1320 \begin{picture}(0,0)(-55,0)
1321 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1323 \begin{picture}(0,0)(12.5,10)
1324 \includegraphics[width=1cm]{100_arrow.eps}
1326 \begin{picture}(0,0)(90,0)
1327 \includegraphics[height=0.9cm]{001_arrow.eps}
1337 \begin{minipage}{5.9cm}
1339 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1342 \begin{picture}(0,0)(60,0)
1343 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1345 \begin{picture}(0,0)(10,0)
1346 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1348 \begin{picture}(0,0)(-60,0)
1349 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1351 \begin{picture}(0,0)(12.5,10)
1352 \includegraphics[width=1cm]{100_arrow.eps}
1354 \begin{picture}(0,0)(90,0)
1355 \includegraphics[height=0.9cm]{001_arrow.eps}
1361 \begin{minipage}{0.3cm}
1364 \begin{minipage}{6.5cm}
1367 \item Energetically most favorable path
1370 \item Activation energy: $\approx$ 0.9 eV
1371 \item Experimental values: 0.73 ... 0.87 eV
1373 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1374 \item Reorientation (path 3)
1376 \item More likely composed of two consecutive steps of type 2
1377 \item Experimental values: 0.77 ... 0.88 eV
1379 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1388 Migration of the C \hkl<1 0 0> dumbbell interstitial
1395 \begin{minipage}{6.5cm}
1398 \begin{minipage}[t]{5.9cm}
1400 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1403 \begin{pspicture}(0,0)(0,0)
1404 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1406 \begin{picture}(0,0)(60,-50)
1407 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1409 \begin{picture}(0,0)(5,-50)
1410 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1412 \begin{picture}(0,0)(-55,-50)
1413 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1415 \begin{picture}(0,0)(12.5,-40)
1416 \includegraphics[width=1cm]{110_arrow.eps}
1418 \begin{picture}(0,0)(90,-45)
1419 \includegraphics[height=0.9cm]{001_arrow.eps}
1421 \begin{pspicture}(0,0)(0,0)
1422 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1424 \begin{picture}(0,0)(60,-15)
1425 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1427 \begin{picture}(0,0)(35,-15)
1428 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1430 \begin{picture}(0,0)(-5,-15)
1431 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1433 \begin{picture}(0,0)(-55,-15)
1434 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1436 \begin{picture}(0,0)(12.5,-5)
1437 \includegraphics[width=1cm]{100_arrow.eps}
1439 \begin{picture}(0,0)(90,-15)
1440 \includegraphics[height=0.9cm]{010_arrow.eps}
1446 \begin{minipage}{5.9cm}
1449 \item Lowest activation energy: $\approx$ 2.2 eV
1450 \item 2.4 times higher than VASP
1451 \item Different pathway
1456 \begin{minipage}{6.5cm}
1459 \begin{minipage}{5.9cm}
1461 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1464 %\begin{pspicture}(0,0)(0,0)
1465 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1467 %\begin{picture}(0,0)(60,-5)
1468 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1470 %\begin{picture}(0,0)(0,-5)
1471 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1473 %\begin{picture}(0,0)(-55,-5)
1474 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1476 %\begin{picture}(0,0)(12.5,5)
1477 %\includegraphics[width=1cm]{100_arrow.eps}
1479 %\begin{picture}(0,0)(90,0)
1480 %\includegraphics[height=0.9cm]{001_arrow.eps}
1488 %\begin{minipage}{5.9cm}
1489 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1493 \begin{minipage}{5.9cm}
1494 Transition involving \ci{} \hkl<1 1 0>
1496 \item Bond-centered configuration unstable\\
1497 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1498 \item Transition minima of path 2 \& 3\\
1499 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1500 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1501 \item 2.4 - 3.4 times higher than VASP
1502 \item Rotation of dumbbell orientation
1506 {\color{blue}Overestimated diffusion barrier}
1517 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1527 E_{\text{f}}^{\text{defect combination}}-
1528 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1529 E_{\text{f}}^{\text{2nd defect}}
1535 \begin{tabular}{l c c c c c c}
1537 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1539 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1540 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1541 \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}\\
1542 \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}\\
1543 \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}\\
1544 \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}\\
1546 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1547 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1556 \begin{minipage}[t]{3.8cm}
1557 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1558 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1560 \begin{minipage}[t]{3.5cm}
1561 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1562 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1564 \begin{minipage}[t]{5.5cm}
1566 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1567 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1568 \item Stress compensation / increase
1569 \item Unfavored: antiparallel orientations
1570 \item Indication of energetically favored\\
1572 \item Most favorable: C clustering
1573 \item However: High barrier ($>4\,\text{eV}$)
1574 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1579 \begin{picture}(0,0)(-295,-130)
1580 \includegraphics[width=3.5cm]{comb_pos.eps}
1588 Combinations of C-Si \hkl<1 0 0>-type interstitials
1595 Energetically most favorable combinations along \hkl<1 1 0>
1600 \begin{tabular}{l c c c c c c}
1602 & 1 & 2 & 3 & 4 & 5 & 6\\
1604 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1605 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1606 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>\\
1613 \begin{minipage}{7.0cm}
1614 \includegraphics[width=7cm]{db_along_110_cc.ps}
1616 \begin{minipage}{6.0cm}
1618 \item Interaction proportional to reciprocal cube of C-C distance
1619 \item Saturation in the immediate vicinity
1620 \renewcommand\labelitemi{$\Rightarrow$}
1621 \item Agglomeration of \ci{} expected
1622 \item Absence of C clustering
1626 Consisten with initial precipitation model
1638 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1644 %\begin{minipage}{3.2cm}
1645 %\includegraphics[width=3cm]{sub_110_combo.eps}
1647 %\begin{minipage}{7.8cm}
1648 %\begin{tabular}{l c c c c c c}
1650 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1651 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1653 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1654 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1655 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1656 %4 & \RM{4} & B & D & E & E & D \\
1657 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1664 %\begin{tabular}{l c c c c c c c c c c}
1666 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1668 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1669 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1670 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1675 \begin{minipage}{6.0cm}
1676 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1678 \begin{minipage}{7cm}
1681 \item IBS: C may displace Si\\
1682 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1684 \hkl<1 1 0>-type $\rightarrow$ favored combination
1685 \renewcommand\labelitemi{$\Rightarrow$}
1686 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1687 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1688 \item Interaction drops quickly to zero\\
1689 $\rightarrow$ low capture radius
1693 IBS process far from equilibrium\\
1694 \cs{} \& \si{} instead of thermodynamic ground state
1699 \begin{minipage}{6.5cm}
1700 \includegraphics[width=6.0cm]{162-097.ps}
1702 \item Low migration barrier
1705 \begin{minipage}{6.5cm}
1707 Ab initio MD at \degc{900}\\
1708 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1709 $t=\unit[2230]{fs}$\\
1710 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1714 Contribution of entropy to structural formation
1723 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1730 \begin{minipage}[t]{3cm}
1731 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1732 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1734 \begin{minipage}[t]{7cm}
1737 Low activation energies\\
1738 High activation energies for reverse processes\\
1740 {\color{blue}C$_{\text{sub}}$ very stable}\\
1744 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1746 {\color{blue}Formation of SiC by successive substitution by C}
1750 \begin{minipage}[t]{3cm}
1751 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1752 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1757 \begin{minipage}{5.9cm}
1758 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1760 \begin{picture}(0,0)(70,0)
1761 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1763 \begin{picture}(0,0)(30,0)
1764 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1766 \begin{picture}(0,0)(-10,0)
1767 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1769 \begin{picture}(0,0)(-48,0)
1770 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1772 \begin{picture}(0,0)(12.5,5)
1773 \includegraphics[width=1cm]{100_arrow.eps}
1775 \begin{picture}(0,0)(97,-10)
1776 \includegraphics[height=0.9cm]{001_arrow.eps}
1782 \begin{minipage}{0.3cm}
1786 \begin{minipage}{5.9cm}
1787 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1789 \begin{picture}(0,0)(60,0)
1790 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1792 \begin{picture}(0,0)(25,0)
1793 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1795 \begin{picture}(0,0)(-20,0)
1796 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1798 \begin{picture}(0,0)(-55,0)
1799 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1801 \begin{picture}(0,0)(12.5,5)
1802 \includegraphics[width=1cm]{100_arrow.eps}
1804 \begin{picture}(0,0)(95,0)
1805 \includegraphics[height=0.9cm]{001_arrow.eps}
1817 Conclusion of defect / migration / combined defect simulations
1826 \item Accurately described by quantum-mechanical simulations
1827 \item Less accurate description by classical potential simulations
1828 \item Underestimated formation energy of \cs{} by classical approach
1829 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1834 \item C migration pathway in Si identified
1835 \item Consistent with reorientation and diffusion experiments
1838 \item Different path and ...
1839 \item overestimated barrier by classical potential calculations
1842 Concerning the precipitation mechanism
1844 \item Agglomeration of C-Si dumbbells energetically favorable
1845 (stress compensation)
1846 \item C-Si indeed favored compared to
1847 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1848 \item Possible low interaction capture radius of
1849 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1850 \item Low barrier for
1851 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1852 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1853 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1856 {\color{blue}Results suggest increased participation of \cs}
1864 Silicon carbide precipitation simulations
1870 \begin{pspicture}(0,0)(12,6.5)
1872 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1875 \item Create c-Si volume
1876 \item Periodc boundary conditions
1877 \item Set requested $T$ and $p=0\text{ bar}$
1878 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1881 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1883 Insertion of C atoms at constant T
1885 \item total simulation volume {\pnode{in1}}
1886 \item volume of minimal SiC precipitate {\pnode{in2}}
1887 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1891 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1893 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1895 \ncline[]{->}{init}{insert}
1896 \ncline[]{->}{insert}{cool}
1897 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1898 \rput(7.8,6){\footnotesize $V_1$}
1899 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1900 \rput(9.2,4.85){\tiny $V_2$}
1901 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1902 \rput(9.55,4.45){\footnotesize $V_3$}
1903 \rput(7.9,3.2){\pnode{ins1}}
1904 \rput(9.22,2.8){\pnode{ins2}}
1905 \rput(11.0,2.4){\pnode{ins3}}
1906 \ncline[]{->}{in1}{ins1}
1907 \ncline[]{->}{in2}{ins2}
1908 \ncline[]{->}{in3}{ins3}
1913 \item Restricted to classical potential simulations
1914 \item $V_2$ and $V_3$ considered due to low diffusion
1915 \item Amount of C atoms: 6000
1916 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1917 \item Simulation volume: $31\times 31\times 31$ unit cells
1926 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1931 \begin{minipage}{6.5cm}
1932 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1934 \begin{minipage}{6.5cm}
1935 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1938 \begin{minipage}{6.5cm}
1939 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1941 \begin{minipage}{6.5cm}
1943 \underline{Low C concentration ($V_1$)}\\
1944 \hkl<1 0 0> C-Si dumbbell dominated structure
1946 \item Si-C bumbs around 0.19 nm
1947 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1948 concatenated dumbbells of various orientation
1949 \item Si-Si NN distance stretched to 0.3 nm
1951 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1952 \underline{High C concentration ($V_2$, $V_3$)}\\
1953 High amount of strongly bound C-C bonds\\
1954 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1955 Only short range order observable\\
1956 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1964 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1969 \begin{minipage}{6.5cm}
1970 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1972 \begin{minipage}{6.5cm}
1973 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1976 \begin{minipage}{6.5cm}
1977 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1979 \begin{minipage}{6.5cm}
1981 \underline{Low C concentration ($V_1$)}\\
1982 \hkl<1 0 0> C-Si dumbbell dominated structure
1984 \item Si-C bumbs around 0.19 nm
1985 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1986 concatenated dumbbells of various orientation
1987 \item Si-Si NN distance stretched to 0.3 nm
1989 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1990 \underline{High C concentration ($V_2$, $V_3$)}\\
1991 High amount of strongly bound C-C bonds\\
1992 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1993 Only short range order observable\\
1994 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1997 \begin{pspicture}(0,0)(0,0)
1998 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
1999 \begin{minipage}{10cm}
2001 {\color{red}\bf 3C-SiC formation fails to appear}
2003 \item Low C concentration simulations
2005 \item Formation of \ci{} indeed occurs
2006 \item Agllomeration not observed
2008 \item High C concentration simulations
2010 \item Amorphous SiC-like structure\\
2011 (not expected at prevailing temperatures)
2012 \item Rearrangement and transition into 3C-SiC structure missing
2024 Limitations of molecular dynamics and short range potentials
2031 \underline{Time scale problem of MD}\\[0.2cm]
2032 Minimize integration error\\
2033 $\Rightarrow$ discretization considerably smaller than
2034 reciprocal of fastest vibrational mode\\[0.1cm]
2035 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2036 $\Rightarrow$ suitable choice of time step:
2037 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2038 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2039 Several local minima in energy surface separated by large energy barriers\\
2040 $\Rightarrow$ transition event corresponds to a multiple
2041 of vibrational periods\\
2042 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2043 infrequent transition events\\[0.1cm]
2044 {\color{blue}Accelerated methods:}
2045 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2049 \underline{Limitations related to the short range potential}\\[0.2cm]
2050 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2051 and 2$^{\text{nd}}$ next neighbours\\
2052 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2058 Potential enhanced problem of slow phase space propagation
2063 \underline{Approach to the (twofold) problem}\\[0.2cm]
2064 Increased temperature simulations without TAD corrections\\
2065 (accelerated methods or higher time scales exclusively not sufficient)
2067 \begin{picture}(0,0)(-260,-30)
2069 \begin{minipage}{4.2cm}
2076 \item 3C-SiC also observed for higher T
2077 \item higher T inside sample
2078 \item structural evolution vs.\\
2079 equilibrium properties
2085 \begin{picture}(0,0)(-305,-155)
2087 \begin{minipage}{2.5cm}
2091 thermodynmic sampling
2102 Increased temperature simulations at low C concentration
2107 \begin{minipage}{6.5cm}
2108 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2110 \begin{minipage}{6.5cm}
2111 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2114 \begin{minipage}{6.5cm}
2115 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2117 \begin{minipage}{6.5cm}
2119 \underline{Si-C bonds:}
2121 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2122 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2124 \underline{Si-Si bonds:}
2125 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2126 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2127 \underline{C-C bonds:}
2129 \item C-C next neighbour pairs reduced (mandatory)
2130 \item Peak at 0.3 nm slightly shifted
2132 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2133 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2135 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2137 \item Range [|-$\downarrow$]:
2138 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2139 with nearby Si$_{\text{I}}$}
2144 \begin{picture}(0,0)(-330,-74)
2147 \begin{minipage}{1.6cm}
2150 stretched SiC\\[-0.1cm]
2162 Increased temperature simulations at low C concentration
2167 \begin{minipage}{6.5cm}
2168 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2170 \begin{minipage}{6.5cm}
2171 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2174 \begin{minipage}{6.5cm}
2175 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2177 \begin{minipage}{6.5cm}
2179 \underline{Si-C bonds:}
2181 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2182 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2184 \underline{Si-Si bonds:}
2185 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2186 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2187 \underline{C-C bonds:}
2189 \item C-C next neighbour pairs reduced (mandatory)
2190 \item Peak at 0.3 nm slightly shifted
2192 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2193 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2195 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2197 \item Range [|-$\downarrow$]:
2198 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2199 with nearby Si$_{\text{I}}$}
2204 %\begin{picture}(0,0)(-330,-74)
2207 %\begin{minipage}{1.6cm}
2210 %stretched SiC\\[-0.1cm]
2217 \begin{pspicture}(0,0)(0,0)
2218 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2219 \begin{minipage}{10cm}
2221 {\color{blue}\bf Stretched SiC in c-Si}
2223 \item Consistent to precipitation model involving \cs{}
2224 \item Explains annealing behavior of high/low T C implants
2226 \item Low T: highly mobiel \ci{}
2227 \item High T: stable configurations of \cs{}
2230 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2231 $\Rightarrow$ Precipitation mechanism involving \cs{}
2241 Increased temperature simulations at high C concentration
2246 \begin{minipage}{6.5cm}
2247 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2249 \begin{minipage}{6.5cm}
2250 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2258 \begin{minipage}[t]{6.0cm}
2259 0.186 nm: Si-C pairs $\uparrow$\\
2260 (as expected in 3C-SiC)\\[0.2cm]
2261 0.282 nm: Si-C-C\\[0.2cm]
2262 $\approx$0.35 nm: C-Si-Si
2265 \begin{minipage}{0.2cm}
2269 \begin{minipage}[t]{6.0cm}
2270 0.15 nm: C-C pairs $\uparrow$\\
2271 (as expected in graphite/diamond)\\[0.2cm]
2272 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2273 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2278 \item Decreasing cut-off artifact
2279 \item {\color{red}Amorphous} SiC-like phase remains
2280 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2281 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2290 High C \& small $V$ \& short $t$
2293 Slow restructuring due to strong C-C bonds
2296 High C \& low T implants
2307 Summary and Conclusions
2315 \begin{minipage}[t]{12.9cm}
2316 \underline{Pecipitation simulations}
2318 \item High C concentration $\rightarrow$ amorphous SiC like phase
2319 \item Problem of potential enhanced slow phase space propagation
2320 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2321 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2322 \item High T necessary to simulate IBS conditions (far from equilibrium)
2323 \item Precipitation by successive agglomeration of \cs (epitaxy)
2324 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2325 (stretched SiC, interface)
2333 \begin{minipage}{12.9cm}
2338 \item Point defects excellently / fairly well described
2340 \item C$_{\text{sub}}$ drastically underestimated by EA
2341 \item EA predicts correct ground state:
2342 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2343 \item Identified migration path explaining
2344 diffusion and reorientation experiments by DFT
2345 \item EA fails to describe \ci{} migration:
2346 Wrong path \& overestimated barrier
2348 \item Combinations of defects
2350 \item Agglomeration of point defects energetically favorable
2351 by compensation of stress
2352 \item Formation of C-C unlikely
2353 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2354 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2355 Low barrier (\unit[0.77]{eV}) \& low capture radius
2363 \framebox{Precipitation by successive agglomeration of \cs{}}
2381 \underline{Augsburg}
2383 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2384 \item Ralf Utermann (EDV)
2387 \underline{Helsinki}
2389 \item Prof. K. Nordlund (MD)
2394 \item Bayerische Forschungsstiftung (financial support)
2397 \underline{Paderborn}
2399 \item Prof. J. Lindner (SiC)
2400 \item Prof. G. Schmidt (DFT + financial support)
2401 \item Dr. E. Rauls (DFT + SiC)
2402 \item Dr. S. Sanna (VASP)
2409 \bf Thank you for your attention!