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3 \documentclass[landscape,semhelv]{seminar}
6 \usepackage[greek,german]{babel}
7 \usepackage[latin1]{inputenc}
8 \usepackage[T1]{fontenc}
13 \usepackage{calc} % Simple computations with LaTeX variables
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17 \usepackage{fancyhdr} % Headers and footers definitions
18 \usepackage{fancyvrb} % Fancy verbatim environments
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}
87 \newrgbcolor{hb}{0.75 0.77 0.89}
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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
153 % Phase diagram of the C/Si system\\
158 \begin{minipage}{6.5cm}
159 \includegraphics[width=6.5cm]{si-c_phase.eps}
162 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
165 \begin{pspicture}(0,0)(0,0)
166 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
169 \begin{minipage}{6cm}
170 {\bf Phase diagram of the C/Si system}\\[0.2cm]
171 {\color{blue}Stoichiometric composition}
173 \item only chemical stable compound
174 \item wide band gap semiconductor\\
175 \underline{silicon carbide}, SiC
181 % motivation / properties / applications of silicon carbide
187 \begin{pspicture}(0,0)(13.5,5)
189 \psframe*[linecolor=hb](-0.2,0)(12.9,5)
191 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.2,1)(6.5,1)(6.5,3)(5.2,3)
192 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.4,0.5)(7.7,2)(7.7,2)(6.4,3.5)
194 \rput[lt](0,4.6){\color{gray}PROPERTIES}
196 \rput[lt](0.3,4){wide band gap}
197 \rput[lt](0.3,3.5){high electric breakdown field}
198 \rput[lt](0.3,3){good electron mobility}
199 \rput[lt](0.3,2.5){high electron saturation drift velocity}
200 \rput[lt](0.3,2){high thermal conductivity}
202 \rput[lt](0.3,1.5){hard and mechanically stable}
203 \rput[lt](0.3,1){chemically inert}
205 \rput[lt](0.3,0.5){radiation hardness}
207 \rput[rt](12.7,4.6){\color{gray}APPLICATIONS}
209 \rput[rt](12.5,3.85){high-temperature, high power}
210 \rput[rt](12.5,3.5){and high-frequency}
211 \rput[rt](12.5,3.15){electronic and optoelectronic devices}
213 \rput[rt](12.5,2.35){material suitable for extreme conditions}
214 \rput[rt](12.5,2){microelectromechanical systems}
215 \rput[rt](12.5,1.65){abrasives, cutting tools, heating elements}
217 \rput[rt](12.5,0.85){first wall reactor material, detectors}
218 \rput[rt](12.5,0.5){and electronic devices for space}
222 \begin{picture}(0,0)(5,-162)
223 \includegraphics[height=2.2cm]{3C_SiC_bs.eps}
225 \begin{picture}(0,0)(-120,-162)
226 \includegraphics[height=2.2cm]{nasa_600c_led.eps}
228 \begin{picture}(0,0)(-270,-162)
229 \includegraphics[height=2.2cm]{6h-sic_3c-sic.eps}
232 \begin{picture}(0,0)(10,65)
233 \includegraphics[height=2.8cm]{sic_switch.eps}
235 %\begin{picture}(0,0)(-243,65)
236 \begin{picture}(0,0)(-110,65)
237 \includegraphics[height=2.8cm]{ise_99.eps}
239 %\begin{picture}(0,0)(-135,65)
240 \begin{picture}(0,0)(-100,65)
241 \includegraphics[height=1.2cm]{infineon_schottky.eps}
243 \begin{picture}(0,0)(-233,65)
244 \includegraphics[height=2.8cm]{solar_car.eps}
254 Polytypes of SiC\\[0.4cm]
257 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
258 \begin{minipage}{1.9cm}
259 {\tiny cubic (twist)}
261 \begin{minipage}{2.9cm}
262 {\tiny hexagonal (no twist)}
265 \begin{picture}(0,0)(-150,0)
266 \includegraphics[width=7cm]{polytypes.eps}
273 \begin{tabular}{l c c c c c c}
275 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
277 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
278 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
279 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
280 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
281 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
282 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
283 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
287 \begin{pspicture}(0,0)(0,0)
288 \psellipse[linecolor=green](5.7,2.10)(0.4,0.5)
290 \begin{pspicture}(0,0)(0,0)
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293 \begin{pspicture}(0,0)(0,0)
294 \psellipse[linecolor=red](10.45,0.45)(0.4,0.2)
305 Fabrication of silicon carbide
313 \emph{Silicon carbide --- Born from the stars, perfected on earth.}
318 SiC thin film by MBE \& CVD
320 \item Much progress achieved in homo/heteroepitaxial SiC thin film growth
321 \item \underline{Commercially available} semiconductor power devices based on
322 \underline{\foreignlanguage{greek}{a}-SiC}
323 \item Production of favored \underline{3C-SiC} material
324 \underline{less advanced}
325 \item Quality and size not yet sufficient
327 \begin{picture}(0,0)(-310,-20)
328 \includegraphics[width=2.0cm]{cree.eps}
331 Alternative method: Ion beam synthesis of SiC in Si
334 \item \underline{Sublimation growth using the modified Lely method}
336 \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
337 \item Surrounded by polycrystalline SiC in a graphite crucible\\
338 at $T=2100-2400 \, ^{\circ} \text{C}$
339 \item Deposition of supersaturated vapor on cooler seed crystal
341 \item \underline{Homoepitaxial growth using CVD}
343 \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
344 \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
345 \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
347 \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
349 \item Two steps: carbonization and growth
350 \item $T=650-1050 \, ^{\circ} \text{C}$
351 \item SiC/Si lattice mismatch $\approx$ 20 \%
352 \item Quality and size not yet sufficient
356 \begin{picture}(0,0)(-280,-65)
357 \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
359 \begin{picture}(0,0)(-280,-55)
360 \begin{minipage}{5cm}
362 NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
367 \begin{picture}(0,0)(-265,-150)
368 \includegraphics[width=2.4cm]{m_lely.eps}
370 \begin{picture}(0,0)(-333,-175)
371 \begin{minipage}{5cm}
377 5. Insulation\\[-7pt]
382 \begin{picture}(0,0)(-230,-35)
384 {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
387 \begin{picture}(0,0)(-230,-10)
389 \begin{minipage}{3cm}
390 {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
410 \item Implantation of C in Si --- Overview of experimental observations
411 \item Utilized simulation techniques and modeled problems
413 \item {\color{blue}Diploma thesis}\\
414 \underline{Monte Carlo} simulations
415 modeling the selforganization process
416 leading to periodic arrays of nanometric amorphous SiC
418 \item {\color{blue}Doctoral studies}\\
419 Classical potential \underline{molecular dynamics} simulations
421 \underline{Density functional theory} calculations
423 \ldots on defects and SiC precipitation in Si
425 \item Summary / Conclusion / Outlook
437 Fabrication of silicon carbide
442 Alternative approach:
443 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
445 \item \underline{Implantation step 1}\\
446 180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
447 $\Rightarrow$ box-like distribution of equally sized
448 and epitactically oriented SiC precipitates
450 \item \underline{Implantation step 2}\\
451 180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
452 $\Rightarrow$ destruction of SiC nanocrystals
453 in growing amorphous interface layers
454 \item \underline{Annealing}\\
455 $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
456 $\Rightarrow$ homogeneous, stoichiometric SiC layer
457 with sharp interfaces
460 \begin{minipage}{6.3cm}
461 \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
463 XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
467 \begin{minipage}{6.3cm}
470 Precipitation mechanism not yet fully understood!
472 \renewcommand\labelitemi{$\Rightarrow$}
474 \underline{Understanding the SiC precipitation}
476 \item significant technological progress in SiC thin film formation
477 \item perspectives for processes relying upon prevention of SiC precipitation
489 Supposed precipitation mechanism of SiC in Si
496 \begin{minipage}{3.8cm}
497 Si \& SiC lattice structure\\[0.2cm]
498 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
502 \begin{minipage}{3.8cm}
504 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
508 \begin{minipage}{3.8cm}
510 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
514 \begin{minipage}{4cm}
516 C-Si dimers (dumbbells)\\[-0.1cm]
517 on Si interstitial sites
521 \begin{minipage}{4.2cm}
523 Agglomeration of C-Si dumbbells\\[-0.1cm]
524 $\Rightarrow$ dark contrasts
528 \begin{minipage}{4cm}
530 Precipitation of 3C-SiC in Si\\[-0.1cm]
531 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
532 \& release of Si self-interstitials
536 \begin{minipage}{3.8cm}
538 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
542 \begin{minipage}{3.8cm}
544 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
548 \begin{minipage}{3.8cm}
550 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
554 \begin{pspicture}(0,0)(0,0)
555 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
556 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
557 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
558 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
559 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
560 $4a_{\text{Si}}=5a_{\text{SiC}}$
562 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
563 \hkl(h k l) planes match
565 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
575 Supposed precipitation mechanism of SiC in Si
582 \begin{minipage}{3.8cm}
583 Si \& SiC lattice structure\\[0.2cm]
584 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
588 \begin{minipage}{3.8cm}
590 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
594 \begin{minipage}{3.8cm}
596 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
600 \begin{minipage}{4cm}
602 C-Si dimers (dumbbells)\\[-0.1cm]
603 on Si interstitial sites
607 \begin{minipage}{4.2cm}
609 Agglomeration of C-Si dumbbells\\[-0.1cm]
610 $\Rightarrow$ dark contrasts
614 \begin{minipage}{4cm}
616 Precipitation of 3C-SiC in Si\\[-0.1cm]
617 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
618 \& release of Si self-interstitials
622 \begin{minipage}{3.8cm}
624 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
628 \begin{minipage}{3.8cm}
630 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
634 \begin{minipage}{3.8cm}
636 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
640 \begin{pspicture}(0,0)(0,0)
641 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
642 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
643 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
644 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
645 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
646 $4a_{\text{Si}}=5a_{\text{SiC}}$
648 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
649 \hkl(h k l) planes match
651 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
654 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
655 \begin{minipage}{10cm}
657 {\color{red}\bf Controversial views}
659 \item Implantations at high T (Nejim et al.)
661 \item Topotactic transformation based on \cs
662 \item \si{} as supply reacting with further C in cleared volume
664 \item Annealing behavior (Serre et al.)
666 \item Room temperature implants $\rightarrow$ highly mobile C
667 \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
668 (indicate stable \cs{} configurations)
670 \item Strained silicon \& Si/SiC heterostructures
672 \item Coherent SiC precipitates (tensile strain)
673 \item Incoherent SiC (strain relaxation)
685 Molecular dynamics (MD) simulations
694 \item Microscopic description of N particle system
695 \item Analytical interaction potential
696 \item Numerical integration using Newtons equation of motion\\
697 as a propagation rule in 6N-dimensional phase space
698 \item Observables obtained by time and/or ensemble averages
700 {\bf Details of the simulation:}
702 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
703 \item Ensemble: NpT (isothermal-isobaric)
705 \item Berendsen thermostat:
706 $\tau_{\text{T}}=100\text{ fs}$
707 \item Berendsen barostat:\\
708 $\tau_{\text{P}}=100\text{ fs}$,
709 $\beta^{-1}=100\text{ GPa}$
711 \item Erhart/Albe potential: Tersoff-like bond order potential
714 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
715 \pot_{ij} = {\color{red}f_C(r_{ij})}
716 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
720 \begin{picture}(0,0)(-230,-30)
721 \includegraphics[width=5cm]{tersoff_angle.eps}
729 Density functional theory (DFT) calculations
734 Basic ingredients necessary for DFT
737 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
739 \item ... uniquely determines the ground state potential
741 \item ... minimizes the systems total energy
743 \item \underline{Born-Oppenheimer}
744 - $N$ moving electrons in an external potential of static nuclei
746 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
747 +\sum_i^N V_{\text{ext}}(r_i)
748 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
750 \item \underline{Effective potential}
751 - averaged electrostatic potential \& exchange and correlation
753 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
756 \item \underline{Kohn-Sham system}
757 - Schr\"odinger equation of N non-interacting particles
759 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
764 n(r)=\sum_i^N|\Phi_i(r)|^2
766 \item \underline{Self-consistent solution}\\
767 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
768 which in turn depends on $n(r)$
769 \item \underline{Variational principle}
770 - minimize total energy with respect to $n(r)$
778 Density functional theory (DFT) calculations
785 Details of applied DFT calculations in this work
788 \item \underline{Exchange correlation functional}
789 - approximations for the inhomogeneous electron gas
791 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
792 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
794 \item \underline{Plane wave basis set}
795 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
798 \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}}
799 \qquad ({\color{blue}300\text{ eV}})
801 \item \underline{Brillouin zone sampling} -
802 {\color{blue}$\Gamma$-point only} calculations
803 \item \underline{Pseudo potential}
804 - consider only the valence electrons
805 \item \underline{Code} - VASP 4.6
810 MD and structural optimization
813 \item MD integration: Gear predictor corrector algorithm
814 \item Pressure control: Parrinello-Rahman pressure control
815 \item Structural optimization: Conjugate gradient method
818 \begin{pspicture}(0,0)(0,0)
819 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
827 C and Si self-interstitial point defects in silicon
834 \begin{minipage}{8cm}
836 \begin{pspicture}(0,0)(7,5)
837 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
840 \item Creation of c-Si simulation volume
841 \item Periodic boundary conditions
842 \item $T=0\text{ K}$, $p=0\text{ bar}$
845 \rput(3.5,2.1){\rnode{insert}{\psframebox{
848 Insertion of interstitial C/Si atoms
851 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
854 Relaxation / structural energy minimization
857 \ncline[]{->}{init}{insert}
858 \ncline[]{->}{insert}{cool}
861 \begin{minipage}{5cm}
862 \includegraphics[width=5cm]{unit_cell_e.eps}\\
865 \begin{minipage}{9cm}
866 \begin{tabular}{l c c}
868 & size [unit cells] & \# atoms\\
870 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
871 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
875 \begin{minipage}{4cm}
876 {\color{red}$\bullet$} Tetrahedral\\
877 {\color{green}$\bullet$} Hexagonal\\
878 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
879 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
880 {\color{cyan}$\bullet$} Bond-centered\\
881 {\color{black}$\bullet$} Vacancy / Substitutional
890 \begin{minipage}{9.5cm}
893 Si self-interstitial point defects in silicon\\
896 \begin{tabular}{l c c c c c}
898 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
900 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
901 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
903 \end{tabular}\\[0.2cm]
905 \begin{minipage}{4.7cm}
906 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
908 \begin{minipage}{4.7cm}
910 {\tiny nearly T $\rightarrow$ T}\\
912 \includegraphics[width=4.7cm]{nhex_tet.ps}
915 \underline{Hexagonal} \hspace{2pt}
916 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
918 \begin{minipage}{2.7cm}
919 $E_{\text{f}}^*=4.48\text{ eV}$\\
920 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
922 \begin{minipage}{0.4cm}
927 \begin{minipage}{2.7cm}
928 $E_{\text{f}}=3.96\text{ eV}$\\
929 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
932 \begin{minipage}{2.9cm}
934 \underline{Vacancy}\\
935 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
940 \begin{minipage}{3.5cm}
943 \underline{\hkl<1 1 0> dumbbell}\\
944 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
945 \underline{Tetrahedral}\\
946 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
947 \underline{\hkl<1 0 0> dumbbell}\\
948 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
960 C interstitial point defects in silicon\\[-0.1cm]
963 \begin{tabular}{l c c c c c c r}
965 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
967 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
968 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
970 \end{tabular}\\[0.1cm]
973 \begin{minipage}{2.7cm}
974 \underline{Hexagonal} \hspace{2pt}
975 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
976 $E_{\text{f}}^*=9.05\text{ eV}$\\
977 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
979 \begin{minipage}{0.4cm}
984 \begin{minipage}{2.7cm}
985 \underline{\hkl<1 0 0>}\\
986 $E_{\text{f}}=3.88\text{ eV}$\\
987 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
990 \begin{minipage}{2cm}
993 \begin{minipage}{3cm}
995 \underline{Tetrahedral}\\
996 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
1001 \begin{minipage}{2.7cm}
1002 \underline{Bond-centered}\\
1003 $E_{\text{f}}^*=5.59\text{ eV}$\\
1004 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
1006 \begin{minipage}{0.4cm}
1011 \begin{minipage}{2.7cm}
1012 \underline{\hkl<1 1 0> dumbbell}\\
1013 $E_{\text{f}}=5.18\text{ eV}$\\
1014 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
1017 \begin{minipage}{2cm}
1020 \begin{minipage}{3cm}
1022 \underline{Substitutional}\\
1023 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
1034 C \hkl<1 0 0> dumbbell interstitial configuration\\
1038 \begin{tabular}{l c c c c c c c c}
1040 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1042 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1043 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1045 \end{tabular}\\[0.2cm]
1046 \begin{tabular}{l c c c c }
1048 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1050 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1051 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1053 \end{tabular}\\[0.2cm]
1054 \begin{tabular}{l c c c}
1056 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1058 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1059 VASP & 0.109 & -0.065 & 0.174 \\
1061 \end{tabular}\\[0.6cm]
1064 \begin{minipage}{3.0cm}
1066 \underline{Erhart/Albe}
1067 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1070 \begin{minipage}{3.0cm}
1073 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1077 \begin{picture}(0,0)(-185,10)
1078 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1080 \begin{picture}(0,0)(-280,-150)
1081 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1084 \begin{pspicture}(0,0)(0,0)
1085 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1086 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1087 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1088 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1097 \begin{minipage}{8.5cm}
1100 Bond-centered interstitial configuration\\[-0.1cm]
1103 \begin{minipage}{3.0cm}
1104 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1106 \begin{minipage}{5.2cm}
1108 \item Linear Si-C-Si bond
1109 \item Si: one C \& 3 Si neighbours
1110 \item Spin polarized calculations
1111 \item No saddle point!\\
1118 \begin{minipage}[t]{6.5cm}
1119 \begin{minipage}[t]{1.2cm}
1121 {\tiny sp$^3$}\\[0.8cm]
1122 \underline{${\color{black}\uparrow}$}
1123 \underline{${\color{black}\uparrow}$}
1124 \underline{${\color{black}\uparrow}$}
1125 \underline{${\color{red}\uparrow}$}\\
1128 \begin{minipage}[t]{1.4cm}
1130 {\color{red}M}{\color{blue}O}\\[0.8cm]
1131 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1132 $\sigma_{\text{ab}}$\\[0.5cm]
1133 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1137 \begin{minipage}[t]{1.0cm}
1141 \underline{${\color{white}\uparrow\uparrow}$}
1142 \underline{${\color{white}\uparrow\uparrow}$}\\
1144 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1145 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1149 \begin{minipage}[t]{1.4cm}
1151 {\color{blue}M}{\color{green}O}\\[0.8cm]
1152 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1153 $\sigma_{\text{ab}}$\\[0.5cm]
1154 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1158 \begin{minipage}[t]{1.2cm}
1161 {\tiny sp$^3$}\\[0.8cm]
1162 \underline{${\color{green}\uparrow}$}
1163 \underline{${\color{black}\uparrow}$}
1164 \underline{${\color{black}\uparrow}$}
1165 \underline{${\color{black}\uparrow}$}\\
1173 \begin{minipage}{4.5cm}
1174 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1176 \begin{minipage}{3.5cm}
1177 {\color{gray}$\bullet$} Spin up\\
1178 {\color{green}$\bullet$} Spin down\\
1179 {\color{blue}$\bullet$} Resulting spin up\\
1180 {\color{yellow}$\bullet$} Si atoms\\
1181 {\color{red}$\bullet$} C atom
1186 \begin{minipage}{4.2cm}
1188 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1189 {\color{green}$\Box$} {\tiny unoccupied}\\
1190 {\color{red}$\bullet$} {\tiny occupied}
1199 Migration of the C \hkl<1 0 0> dumbbell interstitial
1204 {\small Investigated pathways}
1206 \begin{minipage}{8.5cm}
1207 \begin{minipage}{8.3cm}
1208 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1209 \begin{minipage}{2.4cm}
1210 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1212 \begin{minipage}{0.4cm}
1215 \begin{minipage}{2.4cm}
1216 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1218 \begin{minipage}{0.4cm}
1221 \begin{minipage}{2.4cm}
1222 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1225 \begin{minipage}{8.3cm}
1226 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1227 \begin{minipage}{2.4cm}
1228 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1230 \begin{minipage}{0.4cm}
1233 \begin{minipage}{2.4cm}
1234 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1236 \begin{minipage}{0.4cm}
1239 \begin{minipage}{2.4cm}
1240 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1243 \begin{minipage}{8.3cm}
1244 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1245 \begin{minipage}{2.4cm}
1246 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1248 \begin{minipage}{0.4cm}
1251 \begin{minipage}{2.4cm}
1252 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1254 \begin{minipage}{0.4cm}
1257 \begin{minipage}{2.4cm}
1258 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1263 \begin{minipage}{4.2cm}
1264 {\small Constrained relaxation\\
1265 technique (CRT) method}\\
1266 \includegraphics[width=4cm]{crt_orig.eps}
1268 \item Constrain diffusing atom
1269 \item Static constraints
1272 {\small Modifications}\\
1273 \includegraphics[width=4cm]{crt_mod.eps}
1275 \item Constrain all atoms
1276 \item Update individual\\
1287 Migration of the C \hkl<1 0 0> dumbbell interstitial
1293 \begin{minipage}{5.9cm}
1295 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1298 \begin{picture}(0,0)(60,0)
1299 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1301 \begin{picture}(0,0)(-5,0)
1302 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1304 \begin{picture}(0,0)(-55,0)
1305 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1307 \begin{picture}(0,0)(12.5,10)
1308 \includegraphics[width=1cm]{110_arrow.eps}
1310 \begin{picture}(0,0)(90,0)
1311 \includegraphics[height=0.9cm]{001_arrow.eps}
1317 \begin{minipage}{0.3cm}
1321 \begin{minipage}{5.9cm}
1323 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1326 \begin{picture}(0,0)(60,0)
1327 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1329 \begin{picture}(0,0)(5,0)
1330 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1332 \begin{picture}(0,0)(-55,0)
1333 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1335 \begin{picture}(0,0)(12.5,10)
1336 \includegraphics[width=1cm]{100_arrow.eps}
1338 \begin{picture}(0,0)(90,0)
1339 \includegraphics[height=0.9cm]{001_arrow.eps}
1349 \begin{minipage}{5.9cm}
1351 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1354 \begin{picture}(0,0)(60,0)
1355 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1357 \begin{picture}(0,0)(10,0)
1358 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1360 \begin{picture}(0,0)(-60,0)
1361 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1363 \begin{picture}(0,0)(12.5,10)
1364 \includegraphics[width=1cm]{100_arrow.eps}
1366 \begin{picture}(0,0)(90,0)
1367 \includegraphics[height=0.9cm]{001_arrow.eps}
1373 \begin{minipage}{0.3cm}
1376 \begin{minipage}{6.5cm}
1379 \item Energetically most favorable path
1382 \item Activation energy: $\approx$ 0.9 eV
1383 \item Experimental values: 0.73 ... 0.87 eV
1385 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1386 \item Reorientation (path 3)
1388 \item More likely composed of two consecutive steps of type 2
1389 \item Experimental values: 0.77 ... 0.88 eV
1391 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1400 Migration of the C \hkl<1 0 0> dumbbell interstitial
1407 \begin{minipage}{6.5cm}
1410 \begin{minipage}[t]{5.9cm}
1412 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1415 \begin{pspicture}(0,0)(0,0)
1416 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1418 \begin{picture}(0,0)(60,-50)
1419 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1421 \begin{picture}(0,0)(5,-50)
1422 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1424 \begin{picture}(0,0)(-55,-50)
1425 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1427 \begin{picture}(0,0)(12.5,-40)
1428 \includegraphics[width=1cm]{110_arrow.eps}
1430 \begin{picture}(0,0)(90,-45)
1431 \includegraphics[height=0.9cm]{001_arrow.eps}
1433 \begin{pspicture}(0,0)(0,0)
1434 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1436 \begin{picture}(0,0)(60,-15)
1437 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1439 \begin{picture}(0,0)(35,-15)
1440 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1442 \begin{picture}(0,0)(-5,-15)
1443 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1445 \begin{picture}(0,0)(-55,-15)
1446 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1448 \begin{picture}(0,0)(12.5,-5)
1449 \includegraphics[width=1cm]{100_arrow.eps}
1451 \begin{picture}(0,0)(90,-15)
1452 \includegraphics[height=0.9cm]{010_arrow.eps}
1458 \begin{minipage}{5.9cm}
1461 \item Lowest activation energy: $\approx$ 2.2 eV
1462 \item 2.4 times higher than VASP
1463 \item Different pathway
1468 \begin{minipage}{6.5cm}
1471 \begin{minipage}{5.9cm}
1473 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1476 %\begin{pspicture}(0,0)(0,0)
1477 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1479 %\begin{picture}(0,0)(60,-5)
1480 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1482 %\begin{picture}(0,0)(0,-5)
1483 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1485 %\begin{picture}(0,0)(-55,-5)
1486 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1488 %\begin{picture}(0,0)(12.5,5)
1489 %\includegraphics[width=1cm]{100_arrow.eps}
1491 %\begin{picture}(0,0)(90,0)
1492 %\includegraphics[height=0.9cm]{001_arrow.eps}
1500 %\begin{minipage}{5.9cm}
1501 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1505 \begin{minipage}{5.9cm}
1506 Transition involving \ci{} \hkl<1 1 0>
1508 \item Bond-centered configuration unstable\\
1509 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1510 \item Transition minima of path 2 \& 3\\
1511 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1512 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1513 \item 2.4 - 3.4 times higher than VASP
1514 \item Rotation of dumbbell orientation
1518 {\color{blue}Overestimated diffusion barrier}
1529 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1539 E_{\text{f}}^{\text{defect combination}}-
1540 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1541 E_{\text{f}}^{\text{2nd defect}}
1547 \begin{tabular}{l c c c c c c}
1549 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1551 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1552 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1553 \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}\\
1554 \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}\\
1555 \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}\\
1556 \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}\\
1558 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1559 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1568 \begin{minipage}[t]{3.8cm}
1569 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1570 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1572 \begin{minipage}[t]{3.5cm}
1573 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1574 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1576 \begin{minipage}[t]{5.5cm}
1578 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1579 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1580 \item Stress compensation / increase
1581 \item Unfavored: antiparallel orientations
1582 \item Indication of energetically favored\\
1584 \item Most favorable: C clustering
1585 \item However: High barrier ($>4\,\text{eV}$)
1586 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1591 \begin{picture}(0,0)(-295,-130)
1592 \includegraphics[width=3.5cm]{comb_pos.eps}
1600 Combinations of C-Si \hkl<1 0 0>-type interstitials
1607 Energetically most favorable combinations along \hkl<1 1 0>
1612 \begin{tabular}{l c c c c c c}
1614 & 1 & 2 & 3 & 4 & 5 & 6\\
1616 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1617 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1618 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>\\
1625 \begin{minipage}{7.0cm}
1626 \includegraphics[width=7cm]{db_along_110_cc.ps}
1628 \begin{minipage}{6.0cm}
1630 \item Interaction proportional to reciprocal cube of C-C distance
1631 \item Saturation in the immediate vicinity
1632 \renewcommand\labelitemi{$\Rightarrow$}
1633 \item Agglomeration of \ci{} expected
1634 \item Absence of C clustering
1638 Consisten with initial precipitation model
1650 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1656 %\begin{minipage}{3.2cm}
1657 %\includegraphics[width=3cm]{sub_110_combo.eps}
1659 %\begin{minipage}{7.8cm}
1660 %\begin{tabular}{l c c c c c c}
1662 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1663 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1665 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1666 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1667 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1668 %4 & \RM{4} & B & D & E & E & D \\
1669 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1676 %\begin{tabular}{l c c c c c c c c c c}
1678 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1680 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1681 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1682 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1687 \begin{minipage}{6.0cm}
1688 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1690 \begin{minipage}{7cm}
1693 \item IBS: C may displace Si\\
1694 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1696 \hkl<1 1 0>-type $\rightarrow$ favored combination
1697 \renewcommand\labelitemi{$\Rightarrow$}
1698 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1699 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1700 \item Interaction drops quickly to zero\\
1701 $\rightarrow$ low capture radius
1705 IBS process far from equilibrium\\
1706 \cs{} \& \si{} instead of thermodynamic ground state
1711 \begin{minipage}{6.5cm}
1712 \includegraphics[width=6.0cm]{162-097.ps}
1714 \item Low migration barrier
1717 \begin{minipage}{6.5cm}
1719 Ab initio MD at \degc{900}\\
1720 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1721 $t=\unit[2230]{fs}$\\
1722 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1726 Contribution of entropy to structural formation
1735 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1742 \begin{minipage}[t]{3cm}
1743 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1744 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1746 \begin{minipage}[t]{7cm}
1749 Low activation energies\\
1750 High activation energies for reverse processes\\
1752 {\color{blue}C$_{\text{sub}}$ very stable}\\
1756 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1758 {\color{blue}Formation of SiC by successive substitution by C}
1762 \begin{minipage}[t]{3cm}
1763 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1764 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1769 \begin{minipage}{5.9cm}
1770 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1772 \begin{picture}(0,0)(70,0)
1773 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1775 \begin{picture}(0,0)(30,0)
1776 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1778 \begin{picture}(0,0)(-10,0)
1779 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1781 \begin{picture}(0,0)(-48,0)
1782 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1784 \begin{picture}(0,0)(12.5,5)
1785 \includegraphics[width=1cm]{100_arrow.eps}
1787 \begin{picture}(0,0)(97,-10)
1788 \includegraphics[height=0.9cm]{001_arrow.eps}
1794 \begin{minipage}{0.3cm}
1798 \begin{minipage}{5.9cm}
1799 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1801 \begin{picture}(0,0)(60,0)
1802 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1804 \begin{picture}(0,0)(25,0)
1805 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1807 \begin{picture}(0,0)(-20,0)
1808 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1810 \begin{picture}(0,0)(-55,0)
1811 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1813 \begin{picture}(0,0)(12.5,5)
1814 \includegraphics[width=1cm]{100_arrow.eps}
1816 \begin{picture}(0,0)(95,0)
1817 \includegraphics[height=0.9cm]{001_arrow.eps}
1829 Conclusion of defect / migration / combined defect simulations
1838 \item Accurately described by quantum-mechanical simulations
1839 \item Less accurate description by classical potential simulations
1840 \item Underestimated formation energy of \cs{} by classical approach
1841 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1846 \item C migration pathway in Si identified
1847 \item Consistent with reorientation and diffusion experiments
1850 \item Different path and ...
1851 \item overestimated barrier by classical potential calculations
1854 Concerning the precipitation mechanism
1856 \item Agglomeration of C-Si dumbbells energetically favorable
1857 (stress compensation)
1858 \item C-Si indeed favored compared to
1859 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1860 \item Possible low interaction capture radius of
1861 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1862 \item Low barrier for
1863 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1864 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1865 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1868 {\color{blue}Results suggest increased participation of \cs}
1876 Silicon carbide precipitation simulations
1882 \begin{pspicture}(0,0)(12,6.5)
1884 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1887 \item Create c-Si volume
1888 \item Periodc boundary conditions
1889 \item Set requested $T$ and $p=0\text{ bar}$
1890 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1893 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1895 Insertion of C atoms at constant T
1897 \item total simulation volume {\pnode{in1}}
1898 \item volume of minimal SiC precipitate {\pnode{in2}}
1899 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1903 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1905 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1907 \ncline[]{->}{init}{insert}
1908 \ncline[]{->}{insert}{cool}
1909 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1910 \rput(7.8,6){\footnotesize $V_1$}
1911 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1912 \rput(9.2,4.85){\tiny $V_2$}
1913 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1914 \rput(9.55,4.45){\footnotesize $V_3$}
1915 \rput(7.9,3.2){\pnode{ins1}}
1916 \rput(9.22,2.8){\pnode{ins2}}
1917 \rput(11.0,2.4){\pnode{ins3}}
1918 \ncline[]{->}{in1}{ins1}
1919 \ncline[]{->}{in2}{ins2}
1920 \ncline[]{->}{in3}{ins3}
1925 \item Restricted to classical potential simulations
1926 \item $V_2$ and $V_3$ considered due to low diffusion
1927 \item Amount of C atoms: 6000
1928 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1929 \item Simulation volume: $31\times 31\times 31$ unit cells
1938 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1943 \begin{minipage}{6.5cm}
1944 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1946 \begin{minipage}{6.5cm}
1947 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1950 \begin{minipage}{6.5cm}
1951 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1953 \begin{minipage}{6.5cm}
1955 \underline{Low C concentration ($V_1$)}\\
1956 \hkl<1 0 0> C-Si dumbbell dominated structure
1958 \item Si-C bumbs around 0.19 nm
1959 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1960 concatenated dumbbells of various orientation
1961 \item Si-Si NN distance stretched to 0.3 nm
1963 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1964 \underline{High C concentration ($V_2$, $V_3$)}\\
1965 High amount of strongly bound C-C bonds\\
1966 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1967 Only short range order observable\\
1968 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1976 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1981 \begin{minipage}{6.5cm}
1982 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1984 \begin{minipage}{6.5cm}
1985 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1988 \begin{minipage}{6.5cm}
1989 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1991 \begin{minipage}{6.5cm}
1993 \underline{Low C concentration ($V_1$)}\\
1994 \hkl<1 0 0> C-Si dumbbell dominated structure
1996 \item Si-C bumbs around 0.19 nm
1997 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1998 concatenated dumbbells of various orientation
1999 \item Si-Si NN distance stretched to 0.3 nm
2001 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2002 \underline{High C concentration ($V_2$, $V_3$)}\\
2003 High amount of strongly bound C-C bonds\\
2004 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2005 Only short range order observable\\
2006 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2009 \begin{pspicture}(0,0)(0,0)
2010 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2011 \begin{minipage}{10cm}
2013 {\color{red}\bf 3C-SiC formation fails to appear}
2015 \item Low C concentration simulations
2017 \item Formation of \ci{} indeed occurs
2018 \item Agllomeration not observed
2020 \item High C concentration simulations
2022 \item Amorphous SiC-like structure\\
2023 (not expected at prevailing temperatures)
2024 \item Rearrangement and transition into 3C-SiC structure missing
2036 Limitations of molecular dynamics and short range potentials
2043 \underline{Time scale problem of MD}\\[0.2cm]
2044 Minimize integration error\\
2045 $\Rightarrow$ discretization considerably smaller than
2046 reciprocal of fastest vibrational mode\\[0.1cm]
2047 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2048 $\Rightarrow$ suitable choice of time step:
2049 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2050 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2051 Several local minima in energy surface separated by large energy barriers\\
2052 $\Rightarrow$ transition event corresponds to a multiple
2053 of vibrational periods\\
2054 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2055 infrequent transition events\\[0.1cm]
2056 {\color{blue}Accelerated methods:}
2057 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2061 \underline{Limitations related to the short range potential}\\[0.2cm]
2062 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2063 and 2$^{\text{nd}}$ next neighbours\\
2064 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2070 Potential enhanced problem of slow phase space propagation
2075 \underline{Approach to the (twofold) problem}\\[0.2cm]
2076 Increased temperature simulations without TAD corrections\\
2077 (accelerated methods or higher time scales exclusively not sufficient)
2079 \begin{picture}(0,0)(-260,-30)
2081 \begin{minipage}{4.2cm}
2088 \item 3C-SiC also observed for higher T
2089 \item higher T inside sample
2090 \item structural evolution vs.\\
2091 equilibrium properties
2097 \begin{picture}(0,0)(-305,-155)
2099 \begin{minipage}{2.5cm}
2103 thermodynmic sampling
2114 Increased temperature simulations at low C concentration
2119 \begin{minipage}{6.5cm}
2120 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2122 \begin{minipage}{6.5cm}
2123 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2126 \begin{minipage}{6.5cm}
2127 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2129 \begin{minipage}{6.5cm}
2131 \underline{Si-C bonds:}
2133 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2134 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2136 \underline{Si-Si bonds:}
2137 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2138 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2139 \underline{C-C bonds:}
2141 \item C-C next neighbour pairs reduced (mandatory)
2142 \item Peak at 0.3 nm slightly shifted
2144 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2145 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2147 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2149 \item Range [|-$\downarrow$]:
2150 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2151 with nearby Si$_{\text{I}}$}
2156 \begin{picture}(0,0)(-330,-74)
2159 \begin{minipage}{1.6cm}
2162 stretched SiC\\[-0.1cm]
2174 Increased temperature simulations at low C concentration
2179 \begin{minipage}{6.5cm}
2180 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2182 \begin{minipage}{6.5cm}
2183 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2186 \begin{minipage}{6.5cm}
2187 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2189 \begin{minipage}{6.5cm}
2191 \underline{Si-C bonds:}
2193 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2194 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2196 \underline{Si-Si bonds:}
2197 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2198 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2199 \underline{C-C bonds:}
2201 \item C-C next neighbour pairs reduced (mandatory)
2202 \item Peak at 0.3 nm slightly shifted
2204 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2205 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2207 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2209 \item Range [|-$\downarrow$]:
2210 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2211 with nearby Si$_{\text{I}}$}
2216 %\begin{picture}(0,0)(-330,-74)
2219 %\begin{minipage}{1.6cm}
2222 %stretched SiC\\[-0.1cm]
2229 \begin{pspicture}(0,0)(0,0)
2230 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2231 \begin{minipage}{10cm}
2233 {\color{blue}\bf Stretched SiC in c-Si}
2235 \item Consistent to precipitation model involving \cs{}
2236 \item Explains annealing behavior of high/low T C implants
2238 \item Low T: highly mobiel \ci{}
2239 \item High T: stable configurations of \cs{}
2242 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2243 $\Rightarrow$ Precipitation mechanism involving \cs{}
2253 Increased temperature simulations at high C concentration
2258 \begin{minipage}{6.5cm}
2259 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2261 \begin{minipage}{6.5cm}
2262 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2270 \begin{minipage}[t]{6.0cm}
2271 0.186 nm: Si-C pairs $\uparrow$\\
2272 (as expected in 3C-SiC)\\[0.2cm]
2273 0.282 nm: Si-C-C\\[0.2cm]
2274 $\approx$0.35 nm: C-Si-Si
2277 \begin{minipage}{0.2cm}
2281 \begin{minipage}[t]{6.0cm}
2282 0.15 nm: C-C pairs $\uparrow$\\
2283 (as expected in graphite/diamond)\\[0.2cm]
2284 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2285 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2290 \item Decreasing cut-off artifact
2291 \item {\color{red}Amorphous} SiC-like phase remains
2292 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2293 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2302 High C \& small $V$ \& short $t$
2305 Slow restructuring due to strong C-C bonds
2308 High C \& low T implants
2319 Summary and Conclusions
2327 \begin{minipage}[t]{12.9cm}
2328 \underline{Pecipitation simulations}
2330 \item High C concentration $\rightarrow$ amorphous SiC like phase
2331 \item Problem of potential enhanced slow phase space propagation
2332 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2333 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2334 \item High T necessary to simulate IBS conditions (far from equilibrium)
2335 \item Precipitation by successive agglomeration of \cs (epitaxy)
2336 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2337 (stretched SiC, interface)
2345 \begin{minipage}{12.9cm}
2350 \item Point defects excellently / fairly well described
2352 \item C$_{\text{sub}}$ drastically underestimated by EA
2353 \item EA predicts correct ground state:
2354 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2355 \item Identified migration path explaining
2356 diffusion and reorientation experiments by DFT
2357 \item EA fails to describe \ci{} migration:
2358 Wrong path \& overestimated barrier
2360 \item Combinations of defects
2362 \item Agglomeration of point defects energetically favorable
2363 by compensation of stress
2364 \item Formation of C-C unlikely
2365 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2366 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2367 Low barrier (\unit[0.77]{eV}) \& low capture radius
2375 \framebox{Precipitation by successive agglomeration of \cs{}}
2393 \underline{Augsburg}
2395 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2396 \item Ralf Utermann (EDV)
2399 \underline{Helsinki}
2401 \item Prof. K. Nordlund (MD)
2406 \item Bayerische Forschungsstiftung (financial support)
2409 \underline{Paderborn}
2411 \item Prof. J. Lindner (SiC)
2412 \item Prof. G. Schmidt (DFT + financial support)
2413 \item Dr. E. Rauls (DFT + SiC)
2414 \item Dr. S. Sanna (VASP)
2421 \bf Thank you for your attention!