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8 \usepackage[T1]{fontenc}
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17 \usepackage{fancyhdr} % Headers and footers definitions
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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}
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53 %\usepackage{mathptmx}
59 \extraslideheight{10in}
64 % specify width and height
69 \def\slidetopmargin{-0.15cm}
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77 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
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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)
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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)
434 Selforganization of nanometric amorphous SiC lamellae
442 \item Regularly spaced, nanometric spherical\\
443 and lamellar amorphous inclusions\\
444 at the upper a/c interface
445 \item Carbon accumulation\\
451 \begin{minipage}{12cm}
452 \includegraphics[width=9cm]{../../nlsop/img/k393abild1_e_l.eps}\\
454 XTEM bright-field, \unit[180]{keV} C$^+ \rightarrow$ Si, \degc{150},
455 Dose: \unit[4.3 $\times 10^{17}$]{cm$^{-2}$}
459 \begin{picture}(0,0)(-182,-215)
460 \begin{minipage}{6.5cm}
462 \includegraphics[width=6.5cm]{../../nlsop/img/eftem.eps}\\[-0.2cm]
464 XTEM bright-field and respective EFTEM C map
478 Model displaying the formation of ordered lamellae
489 Model displaying the formation of ordered lamellae
493 \begin{minipage}{6.3cm}
496 Precipitation mechanism not yet fully understood!
498 \renewcommand\labelitemi{$\Rightarrow$}
500 \underline{Understanding the SiC precipitation}
502 \item significant technological progress in SiC thin film formation
503 \item perspectives for processes relying upon prevention of SiC precipitation
514 Supposed precipitation mechanism of SiC in Si
521 \begin{minipage}{3.8cm}
522 Si \& SiC lattice structure\\[0.2cm]
523 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
527 \begin{minipage}{3.8cm}
529 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
533 \begin{minipage}{3.8cm}
535 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
539 \begin{minipage}{4cm}
541 C-Si dimers (dumbbells)\\[-0.1cm]
542 on Si interstitial sites
546 \begin{minipage}{4.2cm}
548 Agglomeration of C-Si dumbbells\\[-0.1cm]
549 $\Rightarrow$ dark contrasts
553 \begin{minipage}{4cm}
555 Precipitation of 3C-SiC in Si\\[-0.1cm]
556 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
557 \& release of Si self-interstitials
561 \begin{minipage}{3.8cm}
563 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
567 \begin{minipage}{3.8cm}
569 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
573 \begin{minipage}{3.8cm}
575 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
579 \begin{pspicture}(0,0)(0,0)
580 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
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582 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
583 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
584 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
585 $4a_{\text{Si}}=5a_{\text{SiC}}$
587 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
588 \hkl(h k l) planes match
590 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
600 Supposed precipitation mechanism of SiC in Si
607 \begin{minipage}{3.8cm}
608 Si \& SiC lattice structure\\[0.2cm]
609 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
613 \begin{minipage}{3.8cm}
615 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
619 \begin{minipage}{3.8cm}
621 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
625 \begin{minipage}{4cm}
627 C-Si dimers (dumbbells)\\[-0.1cm]
628 on Si interstitial sites
632 \begin{minipage}{4.2cm}
634 Agglomeration of C-Si dumbbells\\[-0.1cm]
635 $\Rightarrow$ dark contrasts
639 \begin{minipage}{4cm}
641 Precipitation of 3C-SiC in Si\\[-0.1cm]
642 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
643 \& release of Si self-interstitials
647 \begin{minipage}{3.8cm}
649 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
653 \begin{minipage}{3.8cm}
655 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
659 \begin{minipage}{3.8cm}
661 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
665 \begin{pspicture}(0,0)(0,0)
666 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
667 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
668 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
669 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
670 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
671 $4a_{\text{Si}}=5a_{\text{SiC}}$
673 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
674 \hkl(h k l) planes match
676 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
679 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
680 \begin{minipage}{10cm}
682 {\color{red}\bf Controversial views}
684 \item Implantations at high T (Nejim et al.)
686 \item Topotactic transformation based on \cs
687 \item \si{} as supply reacting with further C in cleared volume
689 \item Annealing behavior (Serre et al.)
691 \item Room temperature implants $\rightarrow$ highly mobile C
692 \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
693 (indicate stable \cs{} configurations)
695 \item Strained silicon \& Si/SiC heterostructures
697 \item Coherent SiC precipitates (tensile strain)
698 \item Incoherent SiC (strain relaxation)
710 Molecular dynamics (MD) simulations
719 \item Microscopic description of N particle system
720 \item Analytical interaction potential
721 \item Numerical integration using Newtons equation of motion\\
722 as a propagation rule in 6N-dimensional phase space
723 \item Observables obtained by time and/or ensemble averages
725 {\bf Details of the simulation:}
727 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
728 \item Ensemble: NpT (isothermal-isobaric)
730 \item Berendsen thermostat:
731 $\tau_{\text{T}}=100\text{ fs}$
732 \item Berendsen barostat:\\
733 $\tau_{\text{P}}=100\text{ fs}$,
734 $\beta^{-1}=100\text{ GPa}$
736 \item Erhart/Albe potential: Tersoff-like bond order potential
739 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
740 \pot_{ij} = {\color{red}f_C(r_{ij})}
741 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
745 \begin{picture}(0,0)(-230,-30)
746 \includegraphics[width=5cm]{tersoff_angle.eps}
754 Density functional theory (DFT) calculations
759 Basic ingredients necessary for DFT
762 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
764 \item ... uniquely determines the ground state potential
766 \item ... minimizes the systems total energy
768 \item \underline{Born-Oppenheimer}
769 - $N$ moving electrons in an external potential of static nuclei
771 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
772 +\sum_i^N V_{\text{ext}}(r_i)
773 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
775 \item \underline{Effective potential}
776 - averaged electrostatic potential \& exchange and correlation
778 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
781 \item \underline{Kohn-Sham system}
782 - Schr\"odinger equation of N non-interacting particles
784 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
789 n(r)=\sum_i^N|\Phi_i(r)|^2
791 \item \underline{Self-consistent solution}\\
792 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
793 which in turn depends on $n(r)$
794 \item \underline{Variational principle}
795 - minimize total energy with respect to $n(r)$
803 Density functional theory (DFT) calculations
810 Details of applied DFT calculations in this work
813 \item \underline{Exchange correlation functional}
814 - approximations for the inhomogeneous electron gas
816 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
817 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
819 \item \underline{Plane wave basis set}
820 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
823 \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}}
824 \qquad ({\color{blue}300\text{ eV}})
826 \item \underline{Brillouin zone sampling} -
827 {\color{blue}$\Gamma$-point only} calculations
828 \item \underline{Pseudo potential}
829 - consider only the valence electrons
830 \item \underline{Code} - VASP 4.6
835 MD and structural optimization
838 \item MD integration: Gear predictor corrector algorithm
839 \item Pressure control: Parrinello-Rahman pressure control
840 \item Structural optimization: Conjugate gradient method
843 \begin{pspicture}(0,0)(0,0)
844 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
852 C and Si self-interstitial point defects in silicon
859 \begin{minipage}{8cm}
861 \begin{pspicture}(0,0)(7,5)
862 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
865 \item Creation of c-Si simulation volume
866 \item Periodic boundary conditions
867 \item $T=0\text{ K}$, $p=0\text{ bar}$
870 \rput(3.5,2.1){\rnode{insert}{\psframebox{
873 Insertion of interstitial C/Si atoms
876 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
879 Relaxation / structural energy minimization
882 \ncline[]{->}{init}{insert}
883 \ncline[]{->}{insert}{cool}
886 \begin{minipage}{5cm}
887 \includegraphics[width=5cm]{unit_cell_e.eps}\\
890 \begin{minipage}{9cm}
891 \begin{tabular}{l c c}
893 & size [unit cells] & \# atoms\\
895 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
896 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
900 \begin{minipage}{4cm}
901 {\color{red}$\bullet$} Tetrahedral\\
902 {\color{green}$\bullet$} Hexagonal\\
903 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
904 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
905 {\color{cyan}$\bullet$} Bond-centered\\
906 {\color{black}$\bullet$} Vacancy / Substitutional
915 \begin{minipage}{9.5cm}
918 Si self-interstitial point defects in silicon\\
921 \begin{tabular}{l c c c c c}
923 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
925 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
926 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
928 \end{tabular}\\[0.2cm]
930 \begin{minipage}{4.7cm}
931 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
933 \begin{minipage}{4.7cm}
935 {\tiny nearly T $\rightarrow$ T}\\
937 \includegraphics[width=4.7cm]{nhex_tet.ps}
940 \underline{Hexagonal} \hspace{2pt}
941 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
943 \begin{minipage}{2.7cm}
944 $E_{\text{f}}^*=4.48\text{ eV}$\\
945 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
947 \begin{minipage}{0.4cm}
952 \begin{minipage}{2.7cm}
953 $E_{\text{f}}=3.96\text{ eV}$\\
954 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
957 \begin{minipage}{2.9cm}
959 \underline{Vacancy}\\
960 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
965 \begin{minipage}{3.5cm}
968 \underline{\hkl<1 1 0> dumbbell}\\
969 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
970 \underline{Tetrahedral}\\
971 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
972 \underline{\hkl<1 0 0> dumbbell}\\
973 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
985 C interstitial point defects in silicon\\[-0.1cm]
988 \begin{tabular}{l c c c c c c r}
990 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
992 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
993 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
995 \end{tabular}\\[0.1cm]
998 \begin{minipage}{2.7cm}
999 \underline{Hexagonal} \hspace{2pt}
1000 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
1001 $E_{\text{f}}^*=9.05\text{ eV}$\\
1002 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
1004 \begin{minipage}{0.4cm}
1009 \begin{minipage}{2.7cm}
1010 \underline{\hkl<1 0 0>}\\
1011 $E_{\text{f}}=3.88\text{ eV}$\\
1012 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
1015 \begin{minipage}{2cm}
1018 \begin{minipage}{3cm}
1020 \underline{Tetrahedral}\\
1021 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
1026 \begin{minipage}{2.7cm}
1027 \underline{Bond-centered}\\
1028 $E_{\text{f}}^*=5.59\text{ eV}$\\
1029 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
1031 \begin{minipage}{0.4cm}
1036 \begin{minipage}{2.7cm}
1037 \underline{\hkl<1 1 0> dumbbell}\\
1038 $E_{\text{f}}=5.18\text{ eV}$\\
1039 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
1042 \begin{minipage}{2cm}
1045 \begin{minipage}{3cm}
1047 \underline{Substitutional}\\
1048 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
1059 C \hkl<1 0 0> dumbbell interstitial configuration\\
1063 \begin{tabular}{l c c c c c c c c}
1065 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1067 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1068 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1070 \end{tabular}\\[0.2cm]
1071 \begin{tabular}{l c c c c }
1073 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1075 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1076 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1078 \end{tabular}\\[0.2cm]
1079 \begin{tabular}{l c c c}
1081 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1083 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1084 VASP & 0.109 & -0.065 & 0.174 \\
1086 \end{tabular}\\[0.6cm]
1089 \begin{minipage}{3.0cm}
1091 \underline{Erhart/Albe}
1092 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1095 \begin{minipage}{3.0cm}
1098 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1102 \begin{picture}(0,0)(-185,10)
1103 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1105 \begin{picture}(0,0)(-280,-150)
1106 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1109 \begin{pspicture}(0,0)(0,0)
1110 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1111 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1112 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1113 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1122 \begin{minipage}{8.5cm}
1125 Bond-centered interstitial configuration\\[-0.1cm]
1128 \begin{minipage}{3.0cm}
1129 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1131 \begin{minipage}{5.2cm}
1133 \item Linear Si-C-Si bond
1134 \item Si: one C \& 3 Si neighbours
1135 \item Spin polarized calculations
1136 \item No saddle point!\\
1143 \begin{minipage}[t]{6.5cm}
1144 \begin{minipage}[t]{1.2cm}
1146 {\tiny sp$^3$}\\[0.8cm]
1147 \underline{${\color{black}\uparrow}$}
1148 \underline{${\color{black}\uparrow}$}
1149 \underline{${\color{black}\uparrow}$}
1150 \underline{${\color{red}\uparrow}$}\\
1153 \begin{minipage}[t]{1.4cm}
1155 {\color{red}M}{\color{blue}O}\\[0.8cm]
1156 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1157 $\sigma_{\text{ab}}$\\[0.5cm]
1158 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1162 \begin{minipage}[t]{1.0cm}
1166 \underline{${\color{white}\uparrow\uparrow}$}
1167 \underline{${\color{white}\uparrow\uparrow}$}\\
1169 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1170 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1174 \begin{minipage}[t]{1.4cm}
1176 {\color{blue}M}{\color{green}O}\\[0.8cm]
1177 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1178 $\sigma_{\text{ab}}$\\[0.5cm]
1179 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1183 \begin{minipage}[t]{1.2cm}
1186 {\tiny sp$^3$}\\[0.8cm]
1187 \underline{${\color{green}\uparrow}$}
1188 \underline{${\color{black}\uparrow}$}
1189 \underline{${\color{black}\uparrow}$}
1190 \underline{${\color{black}\uparrow}$}\\
1198 \begin{minipage}{4.5cm}
1199 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1201 \begin{minipage}{3.5cm}
1202 {\color{gray}$\bullet$} Spin up\\
1203 {\color{green}$\bullet$} Spin down\\
1204 {\color{blue}$\bullet$} Resulting spin up\\
1205 {\color{yellow}$\bullet$} Si atoms\\
1206 {\color{red}$\bullet$} C atom
1211 \begin{minipage}{4.2cm}
1213 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1214 {\color{green}$\Box$} {\tiny unoccupied}\\
1215 {\color{red}$\bullet$} {\tiny occupied}
1224 Migration of the C \hkl<1 0 0> dumbbell interstitial
1229 {\small Investigated pathways}
1231 \begin{minipage}{8.5cm}
1232 \begin{minipage}{8.3cm}
1233 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1234 \begin{minipage}{2.4cm}
1235 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1237 \begin{minipage}{0.4cm}
1240 \begin{minipage}{2.4cm}
1241 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1243 \begin{minipage}{0.4cm}
1246 \begin{minipage}{2.4cm}
1247 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1250 \begin{minipage}{8.3cm}
1251 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1252 \begin{minipage}{2.4cm}
1253 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1255 \begin{minipage}{0.4cm}
1258 \begin{minipage}{2.4cm}
1259 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1261 \begin{minipage}{0.4cm}
1264 \begin{minipage}{2.4cm}
1265 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1268 \begin{minipage}{8.3cm}
1269 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1270 \begin{minipage}{2.4cm}
1271 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1273 \begin{minipage}{0.4cm}
1276 \begin{minipage}{2.4cm}
1277 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1279 \begin{minipage}{0.4cm}
1282 \begin{minipage}{2.4cm}
1283 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1288 \begin{minipage}{4.2cm}
1289 {\small Constrained relaxation\\
1290 technique (CRT) method}\\
1291 \includegraphics[width=4cm]{crt_orig.eps}
1293 \item Constrain diffusing atom
1294 \item Static constraints
1297 {\small Modifications}\\
1298 \includegraphics[width=4cm]{crt_mod.eps}
1300 \item Constrain all atoms
1301 \item Update individual\\
1312 Migration of the C \hkl<1 0 0> dumbbell interstitial
1318 \begin{minipage}{5.9cm}
1320 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1323 \begin{picture}(0,0)(60,0)
1324 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1326 \begin{picture}(0,0)(-5,0)
1327 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1329 \begin{picture}(0,0)(-55,0)
1330 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1332 \begin{picture}(0,0)(12.5,10)
1333 \includegraphics[width=1cm]{110_arrow.eps}
1335 \begin{picture}(0,0)(90,0)
1336 \includegraphics[height=0.9cm]{001_arrow.eps}
1342 \begin{minipage}{0.3cm}
1346 \begin{minipage}{5.9cm}
1348 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1351 \begin{picture}(0,0)(60,0)
1352 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1354 \begin{picture}(0,0)(5,0)
1355 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1357 \begin{picture}(0,0)(-55,0)
1358 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1360 \begin{picture}(0,0)(12.5,10)
1361 \includegraphics[width=1cm]{100_arrow.eps}
1363 \begin{picture}(0,0)(90,0)
1364 \includegraphics[height=0.9cm]{001_arrow.eps}
1374 \begin{minipage}{5.9cm}
1376 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1379 \begin{picture}(0,0)(60,0)
1380 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1382 \begin{picture}(0,0)(10,0)
1383 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1385 \begin{picture}(0,0)(-60,0)
1386 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1388 \begin{picture}(0,0)(12.5,10)
1389 \includegraphics[width=1cm]{100_arrow.eps}
1391 \begin{picture}(0,0)(90,0)
1392 \includegraphics[height=0.9cm]{001_arrow.eps}
1398 \begin{minipage}{0.3cm}
1401 \begin{minipage}{6.5cm}
1404 \item Energetically most favorable path
1407 \item Activation energy: $\approx$ 0.9 eV
1408 \item Experimental values: 0.73 ... 0.87 eV
1410 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1411 \item Reorientation (path 3)
1413 \item More likely composed of two consecutive steps of type 2
1414 \item Experimental values: 0.77 ... 0.88 eV
1416 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1425 Migration of the C \hkl<1 0 0> dumbbell interstitial
1432 \begin{minipage}{6.5cm}
1435 \begin{minipage}[t]{5.9cm}
1437 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1440 \begin{pspicture}(0,0)(0,0)
1441 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1443 \begin{picture}(0,0)(60,-50)
1444 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1446 \begin{picture}(0,0)(5,-50)
1447 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1449 \begin{picture}(0,0)(-55,-50)
1450 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1452 \begin{picture}(0,0)(12.5,-40)
1453 \includegraphics[width=1cm]{110_arrow.eps}
1455 \begin{picture}(0,0)(90,-45)
1456 \includegraphics[height=0.9cm]{001_arrow.eps}
1458 \begin{pspicture}(0,0)(0,0)
1459 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1461 \begin{picture}(0,0)(60,-15)
1462 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1464 \begin{picture}(0,0)(35,-15)
1465 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1467 \begin{picture}(0,0)(-5,-15)
1468 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1470 \begin{picture}(0,0)(-55,-15)
1471 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1473 \begin{picture}(0,0)(12.5,-5)
1474 \includegraphics[width=1cm]{100_arrow.eps}
1476 \begin{picture}(0,0)(90,-15)
1477 \includegraphics[height=0.9cm]{010_arrow.eps}
1483 \begin{minipage}{5.9cm}
1486 \item Lowest activation energy: $\approx$ 2.2 eV
1487 \item 2.4 times higher than VASP
1488 \item Different pathway
1493 \begin{minipage}{6.5cm}
1496 \begin{minipage}{5.9cm}
1498 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1501 %\begin{pspicture}(0,0)(0,0)
1502 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1504 %\begin{picture}(0,0)(60,-5)
1505 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1507 %\begin{picture}(0,0)(0,-5)
1508 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1510 %\begin{picture}(0,0)(-55,-5)
1511 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1513 %\begin{picture}(0,0)(12.5,5)
1514 %\includegraphics[width=1cm]{100_arrow.eps}
1516 %\begin{picture}(0,0)(90,0)
1517 %\includegraphics[height=0.9cm]{001_arrow.eps}
1525 %\begin{minipage}{5.9cm}
1526 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1530 \begin{minipage}{5.9cm}
1531 Transition involving \ci{} \hkl<1 1 0>
1533 \item Bond-centered configuration unstable\\
1534 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1535 \item Transition minima of path 2 \& 3\\
1536 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1537 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1538 \item 2.4 - 3.4 times higher than VASP
1539 \item Rotation of dumbbell orientation
1543 {\color{blue}Overestimated diffusion barrier}
1554 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1564 E_{\text{f}}^{\text{defect combination}}-
1565 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1566 E_{\text{f}}^{\text{2nd defect}}
1572 \begin{tabular}{l c c c c c c}
1574 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1576 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1577 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1578 \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}\\
1579 \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}\\
1580 \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}\\
1581 \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}\\
1583 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1584 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1593 \begin{minipage}[t]{3.8cm}
1594 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1595 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1597 \begin{minipage}[t]{3.5cm}
1598 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1599 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1601 \begin{minipage}[t]{5.5cm}
1603 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1604 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1605 \item Stress compensation / increase
1606 \item Unfavored: antiparallel orientations
1607 \item Indication of energetically favored\\
1609 \item Most favorable: C clustering
1610 \item However: High barrier ($>4\,\text{eV}$)
1611 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1616 \begin{picture}(0,0)(-295,-130)
1617 \includegraphics[width=3.5cm]{comb_pos.eps}
1625 Combinations of C-Si \hkl<1 0 0>-type interstitials
1632 Energetically most favorable combinations along \hkl<1 1 0>
1637 \begin{tabular}{l c c c c c c}
1639 & 1 & 2 & 3 & 4 & 5 & 6\\
1641 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1642 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1643 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>\\
1650 \begin{minipage}{7.0cm}
1651 \includegraphics[width=7cm]{db_along_110_cc.ps}
1653 \begin{minipage}{6.0cm}
1655 \item Interaction proportional to reciprocal cube of C-C distance
1656 \item Saturation in the immediate vicinity
1657 \renewcommand\labelitemi{$\Rightarrow$}
1658 \item Agglomeration of \ci{} expected
1659 \item Absence of C clustering
1663 Consisten with initial precipitation model
1675 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1681 %\begin{minipage}{3.2cm}
1682 %\includegraphics[width=3cm]{sub_110_combo.eps}
1684 %\begin{minipage}{7.8cm}
1685 %\begin{tabular}{l c c c c c c}
1687 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1688 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1690 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1691 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1692 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1693 %4 & \RM{4} & B & D & E & E & D \\
1694 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1701 %\begin{tabular}{l c c c c c c c c c c}
1703 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1705 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1706 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1707 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1712 \begin{minipage}{6.0cm}
1713 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1715 \begin{minipage}{7cm}
1718 \item IBS: C may displace Si\\
1719 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1721 \hkl<1 1 0>-type $\rightarrow$ favored combination
1722 \renewcommand\labelitemi{$\Rightarrow$}
1723 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1724 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1725 \item Interaction drops quickly to zero\\
1726 $\rightarrow$ low capture radius
1730 IBS process far from equilibrium\\
1731 \cs{} \& \si{} instead of thermodynamic ground state
1736 \begin{minipage}{6.5cm}
1737 \includegraphics[width=6.0cm]{162-097.ps}
1739 \item Low migration barrier
1742 \begin{minipage}{6.5cm}
1744 Ab initio MD at \degc{900}\\
1745 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1746 $t=\unit[2230]{fs}$\\
1747 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1751 Contribution of entropy to structural formation
1760 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1767 \begin{minipage}[t]{3cm}
1768 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1769 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1771 \begin{minipage}[t]{7cm}
1774 Low activation energies\\
1775 High activation energies for reverse processes\\
1777 {\color{blue}C$_{\text{sub}}$ very stable}\\
1781 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1783 {\color{blue}Formation of SiC by successive substitution by C}
1787 \begin{minipage}[t]{3cm}
1788 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1789 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1794 \begin{minipage}{5.9cm}
1795 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1797 \begin{picture}(0,0)(70,0)
1798 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1800 \begin{picture}(0,0)(30,0)
1801 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1803 \begin{picture}(0,0)(-10,0)
1804 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1806 \begin{picture}(0,0)(-48,0)
1807 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1809 \begin{picture}(0,0)(12.5,5)
1810 \includegraphics[width=1cm]{100_arrow.eps}
1812 \begin{picture}(0,0)(97,-10)
1813 \includegraphics[height=0.9cm]{001_arrow.eps}
1819 \begin{minipage}{0.3cm}
1823 \begin{minipage}{5.9cm}
1824 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1826 \begin{picture}(0,0)(60,0)
1827 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1829 \begin{picture}(0,0)(25,0)
1830 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1832 \begin{picture}(0,0)(-20,0)
1833 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1835 \begin{picture}(0,0)(-55,0)
1836 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1838 \begin{picture}(0,0)(12.5,5)
1839 \includegraphics[width=1cm]{100_arrow.eps}
1841 \begin{picture}(0,0)(95,0)
1842 \includegraphics[height=0.9cm]{001_arrow.eps}
1854 Conclusion of defect / migration / combined defect simulations
1863 \item Accurately described by quantum-mechanical simulations
1864 \item Less accurate description by classical potential simulations
1865 \item Underestimated formation energy of \cs{} by classical approach
1866 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1871 \item C migration pathway in Si identified
1872 \item Consistent with reorientation and diffusion experiments
1875 \item Different path and ...
1876 \item overestimated barrier by classical potential calculations
1879 Concerning the precipitation mechanism
1881 \item Agglomeration of C-Si dumbbells energetically favorable
1882 (stress compensation)
1883 \item C-Si indeed favored compared to
1884 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1885 \item Possible low interaction capture radius of
1886 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1887 \item Low barrier for
1888 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1889 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1890 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1893 {\color{blue}Results suggest increased participation of \cs}
1901 Silicon carbide precipitation simulations
1907 \begin{pspicture}(0,0)(12,6.5)
1909 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1912 \item Create c-Si volume
1913 \item Periodc boundary conditions
1914 \item Set requested $T$ and $p=0\text{ bar}$
1915 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1918 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1920 Insertion of C atoms at constant T
1922 \item total simulation volume {\pnode{in1}}
1923 \item volume of minimal SiC precipitate {\pnode{in2}}
1924 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1928 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1930 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1932 \ncline[]{->}{init}{insert}
1933 \ncline[]{->}{insert}{cool}
1934 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1935 \rput(7.8,6){\footnotesize $V_1$}
1936 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1937 \rput(9.2,4.85){\tiny $V_2$}
1938 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1939 \rput(9.55,4.45){\footnotesize $V_3$}
1940 \rput(7.9,3.2){\pnode{ins1}}
1941 \rput(9.22,2.8){\pnode{ins2}}
1942 \rput(11.0,2.4){\pnode{ins3}}
1943 \ncline[]{->}{in1}{ins1}
1944 \ncline[]{->}{in2}{ins2}
1945 \ncline[]{->}{in3}{ins3}
1950 \item Restricted to classical potential simulations
1951 \item $V_2$ and $V_3$ considered due to low diffusion
1952 \item Amount of C atoms: 6000
1953 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1954 \item Simulation volume: $31\times 31\times 31$ unit cells
1963 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1968 \begin{minipage}{6.5cm}
1969 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1971 \begin{minipage}{6.5cm}
1972 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1975 \begin{minipage}{6.5cm}
1976 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1978 \begin{minipage}{6.5cm}
1980 \underline{Low C concentration ($V_1$)}\\
1981 \hkl<1 0 0> C-Si dumbbell dominated structure
1983 \item Si-C bumbs around 0.19 nm
1984 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1985 concatenated dumbbells of various orientation
1986 \item Si-Si NN distance stretched to 0.3 nm
1988 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1989 \underline{High C concentration ($V_2$, $V_3$)}\\
1990 High amount of strongly bound C-C bonds\\
1991 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1992 Only short range order observable\\
1993 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2001 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2006 \begin{minipage}{6.5cm}
2007 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2009 \begin{minipage}{6.5cm}
2010 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2013 \begin{minipage}{6.5cm}
2014 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2016 \begin{minipage}{6.5cm}
2018 \underline{Low C concentration ($V_1$)}\\
2019 \hkl<1 0 0> C-Si dumbbell dominated structure
2021 \item Si-C bumbs around 0.19 nm
2022 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2023 concatenated dumbbells of various orientation
2024 \item Si-Si NN distance stretched to 0.3 nm
2026 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2027 \underline{High C concentration ($V_2$, $V_3$)}\\
2028 High amount of strongly bound C-C bonds\\
2029 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2030 Only short range order observable\\
2031 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2034 \begin{pspicture}(0,0)(0,0)
2035 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2036 \begin{minipage}{10cm}
2038 {\color{red}\bf 3C-SiC formation fails to appear}
2040 \item Low C concentration simulations
2042 \item Formation of \ci{} indeed occurs
2043 \item Agllomeration not observed
2045 \item High C concentration simulations
2047 \item Amorphous SiC-like structure\\
2048 (not expected at prevailing temperatures)
2049 \item Rearrangement and transition into 3C-SiC structure missing
2061 Limitations of molecular dynamics and short range potentials
2068 \underline{Time scale problem of MD}\\[0.2cm]
2069 Minimize integration error\\
2070 $\Rightarrow$ discretization considerably smaller than
2071 reciprocal of fastest vibrational mode\\[0.1cm]
2072 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2073 $\Rightarrow$ suitable choice of time step:
2074 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2075 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2076 Several local minima in energy surface separated by large energy barriers\\
2077 $\Rightarrow$ transition event corresponds to a multiple
2078 of vibrational periods\\
2079 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2080 infrequent transition events\\[0.1cm]
2081 {\color{blue}Accelerated methods:}
2082 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2086 \underline{Limitations related to the short range potential}\\[0.2cm]
2087 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2088 and 2$^{\text{nd}}$ next neighbours\\
2089 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2095 Potential enhanced problem of slow phase space propagation
2100 \underline{Approach to the (twofold) problem}\\[0.2cm]
2101 Increased temperature simulations without TAD corrections\\
2102 (accelerated methods or higher time scales exclusively not sufficient)
2104 \begin{picture}(0,0)(-260,-30)
2106 \begin{minipage}{4.2cm}
2113 \item 3C-SiC also observed for higher T
2114 \item higher T inside sample
2115 \item structural evolution vs.\\
2116 equilibrium properties
2122 \begin{picture}(0,0)(-305,-155)
2124 \begin{minipage}{2.5cm}
2128 thermodynmic sampling
2139 Increased temperature simulations at low C concentration
2144 \begin{minipage}{6.5cm}
2145 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2147 \begin{minipage}{6.5cm}
2148 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2151 \begin{minipage}{6.5cm}
2152 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2154 \begin{minipage}{6.5cm}
2156 \underline{Si-C bonds:}
2158 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2159 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2161 \underline{Si-Si bonds:}
2162 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2163 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2164 \underline{C-C bonds:}
2166 \item C-C next neighbour pairs reduced (mandatory)
2167 \item Peak at 0.3 nm slightly shifted
2169 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2170 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2172 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2174 \item Range [|-$\downarrow$]:
2175 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2176 with nearby Si$_{\text{I}}$}
2181 \begin{picture}(0,0)(-330,-74)
2184 \begin{minipage}{1.6cm}
2187 stretched SiC\\[-0.1cm]
2199 Increased temperature simulations at low C concentration
2204 \begin{minipage}{6.5cm}
2205 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2207 \begin{minipage}{6.5cm}
2208 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2211 \begin{minipage}{6.5cm}
2212 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2214 \begin{minipage}{6.5cm}
2216 \underline{Si-C bonds:}
2218 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2219 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2221 \underline{Si-Si bonds:}
2222 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2223 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2224 \underline{C-C bonds:}
2226 \item C-C next neighbour pairs reduced (mandatory)
2227 \item Peak at 0.3 nm slightly shifted
2229 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2230 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2232 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2234 \item Range [|-$\downarrow$]:
2235 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2236 with nearby Si$_{\text{I}}$}
2241 %\begin{picture}(0,0)(-330,-74)
2244 %\begin{minipage}{1.6cm}
2247 %stretched SiC\\[-0.1cm]
2254 \begin{pspicture}(0,0)(0,0)
2255 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2256 \begin{minipage}{10cm}
2258 {\color{blue}\bf Stretched SiC in c-Si}
2260 \item Consistent to precipitation model involving \cs{}
2261 \item Explains annealing behavior of high/low T C implants
2263 \item Low T: highly mobiel \ci{}
2264 \item High T: stable configurations of \cs{}
2267 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2268 $\Rightarrow$ Precipitation mechanism involving \cs{}
2278 Increased temperature simulations at high C concentration
2283 \begin{minipage}{6.5cm}
2284 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2286 \begin{minipage}{6.5cm}
2287 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2295 \begin{minipage}[t]{6.0cm}
2296 0.186 nm: Si-C pairs $\uparrow$\\
2297 (as expected in 3C-SiC)\\[0.2cm]
2298 0.282 nm: Si-C-C\\[0.2cm]
2299 $\approx$0.35 nm: C-Si-Si
2302 \begin{minipage}{0.2cm}
2306 \begin{minipage}[t]{6.0cm}
2307 0.15 nm: C-C pairs $\uparrow$\\
2308 (as expected in graphite/diamond)\\[0.2cm]
2309 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2310 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2315 \item Decreasing cut-off artifact
2316 \item {\color{red}Amorphous} SiC-like phase remains
2317 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2318 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2327 High C \& small $V$ \& short $t$
2330 Slow restructuring due to strong C-C bonds
2333 High C \& low T implants
2344 Summary and Conclusions
2352 \begin{minipage}[t]{12.9cm}
2353 \underline{Pecipitation simulations}
2355 \item High C concentration $\rightarrow$ amorphous SiC like phase
2356 \item Problem of potential enhanced slow phase space propagation
2357 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2358 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2359 \item High T necessary to simulate IBS conditions (far from equilibrium)
2360 \item Precipitation by successive agglomeration of \cs (epitaxy)
2361 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2362 (stretched SiC, interface)
2370 \begin{minipage}{12.9cm}
2375 \item Point defects excellently / fairly well described
2377 \item C$_{\text{sub}}$ drastically underestimated by EA
2378 \item EA predicts correct ground state:
2379 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2380 \item Identified migration path explaining
2381 diffusion and reorientation experiments by DFT
2382 \item EA fails to describe \ci{} migration:
2383 Wrong path \& overestimated barrier
2385 \item Combinations of defects
2387 \item Agglomeration of point defects energetically favorable
2388 by compensation of stress
2389 \item Formation of C-C unlikely
2390 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2391 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2392 Low barrier (\unit[0.77]{eV}) \& low capture radius
2400 \framebox{Precipitation by successive agglomeration of \cs{}}
2418 \underline{Augsburg}
2420 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2421 \item Ralf Utermann (EDV)
2424 \underline{Helsinki}
2426 \item Prof. K. Nordlund (MD)
2431 \item Bayerische Forschungsstiftung (financial support)
2434 \underline{Paderborn}
2436 \item Prof. J. Lindner (SiC)
2437 \item Prof. G. Schmidt (DFT + financial support)
2438 \item Dr. E. Rauls (DFT + SiC)
2439 \item Dr. S. Sanna (VASP)
2446 \bf Thank you for your attention!