2 %\documentclass[landscape,semhelv,draft]{seminar}
3 \documentclass[landscape,semhelv]{seminar}
6 \usepackage[greek,german]{babel}
7 \usepackage[latin1]{inputenc}
8 \usepackage[T1]{fontenc}
13 \usepackage{calc} % Simple computations with LaTeX variables
14 \usepackage{caption} % Improved captions
15 \usepackage{fancybox} % To have several backgrounds
17 \usepackage{fancyhdr} % Headers and footers definitions
18 \usepackage{fancyvrb} % Fancy verbatim environments
19 \usepackage{pstricks} % PSTricks with the standard color package
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}
88 \newrgbcolor{lbb}{0.75 0.8 0.88}
89 \newrgbcolor{hlbb}{0.825 0.88 0.968}
90 \newrgbcolor{lachs}{1.0 .93 .81}
93 \newcommand{\si}{Si$_{\text{i}}${}}
94 \newcommand{\ci}{C$_{\text{i}}${}}
95 \newcommand{\cs}{C$_{\text{sub}}${}}
96 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
97 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
98 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
99 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
101 % no vertical centering
112 A B C D E F G H G F E D C B A
127 Atomistic simulation studies\\[0.2cm]
133 \textsc{Frank Zirkelbach}
137 Application talk at the Max Planck Institute for Solid State Research
141 Stuttgart, November 2011
146 % no vertical centering
156 % Phase diagram of the C/Si system\\
161 \begin{minipage}{6.5cm}
162 \includegraphics[width=6.5cm]{si-c_phase.eps}
165 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
168 \begin{pspicture}(0,0)(0,0)
169 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
172 \begin{minipage}{6cm}
173 {\bf Phase diagram of the C/Si system}\\[0.2cm]
174 {\color{blue}Stoichiometric composition}
176 \item only chemical stable compound
177 \item wide band gap semiconductor\\
178 \underline{silicon carbide}, SiC
184 % motivation / properties / applications of silicon carbide
190 \begin{pspicture}(0,0)(13.5,5)
192 \psframe*[linecolor=hb](-0.2,0)(12.9,5)
194 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.2,1)(6.5,1)(6.5,3)(5.2,3)
195 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.4,0.5)(7.7,2)(7.7,2)(6.4,3.5)
197 \rput[lt](0,4.6){\color{gray}PROPERTIES}
199 \rput[lt](0.3,4){wide band gap}
200 \rput[lt](0.3,3.5){high electric breakdown field}
201 \rput[lt](0.3,3){good electron mobility}
202 \rput[lt](0.3,2.5){high electron saturation drift velocity}
203 \rput[lt](0.3,2){high thermal conductivity}
205 \rput[lt](0.3,1.5){hard and mechanically stable}
206 \rput[lt](0.3,1){chemically inert}
208 \rput[lt](0.3,0.5){radiation hardness}
210 \rput[rt](12.7,4.6){\color{gray}APPLICATIONS}
212 \rput[rt](12.5,3.85){high-temperature, high power}
213 \rput[rt](12.5,3.5){and high-frequency}
214 \rput[rt](12.5,3.15){electronic and optoelectronic devices}
216 \rput[rt](12.5,2.35){material suitable for extreme conditions}
217 \rput[rt](12.5,2){microelectromechanical systems}
218 \rput[rt](12.5,1.65){abrasives, cutting tools, heating elements}
220 \rput[rt](12.5,0.85){first wall reactor material, detectors}
221 \rput[rt](12.5,0.5){and electronic devices for space}
225 \begin{picture}(0,0)(5,-162)
226 \includegraphics[height=2.2cm]{3C_SiC_bs.eps}
228 \begin{picture}(0,0)(-120,-162)
229 \includegraphics[height=2.2cm]{nasa_600c_led.eps}
231 \begin{picture}(0,0)(-270,-162)
232 \includegraphics[height=2.2cm]{6h-sic_3c-sic.eps}
235 \begin{picture}(0,0)(10,65)
236 \includegraphics[height=2.8cm]{sic_switch.eps}
238 %\begin{picture}(0,0)(-243,65)
239 \begin{picture}(0,0)(-110,65)
240 \includegraphics[height=2.8cm]{ise_99.eps}
242 %\begin{picture}(0,0)(-135,65)
243 \begin{picture}(0,0)(-100,65)
244 \includegraphics[height=1.2cm]{infineon_schottky.eps}
246 \begin{picture}(0,0)(-233,65)
247 \includegraphics[height=2.8cm]{solar_car.eps}
257 Polytypes of SiC\\[0.4cm]
260 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
261 \begin{minipage}{1.9cm}
262 {\tiny cubic (twist)}
264 \begin{minipage}{2.9cm}
265 {\tiny hexagonal (no twist)}
268 \begin{picture}(0,0)(-150,0)
269 \includegraphics[width=7cm]{polytypes.eps}
276 \begin{tabular}{l c c c c c c}
278 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
280 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
281 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
282 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
283 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
284 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
285 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
286 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
290 \begin{pspicture}(0,0)(0,0)
291 \psellipse[linecolor=green](5.7,2.10)(0.4,0.5)
293 \begin{pspicture}(0,0)(0,0)
294 \psellipse[linecolor=green](5.6,0.92)(0.4,0.2)
296 \begin{pspicture}(0,0)(0,0)
297 \psellipse[linecolor=red](10.45,0.45)(0.4,0.2)
307 Fabrication of silicon carbide
316 \emph{Silicon carbide --- Born from the stars, perfected on earth.}
322 SiC thin films by MBE \& CVD
324 \item Much progress achieved in homo/heteroepitaxial SiC thin film growth
325 \item \underline{Commercially available} semiconductor power devices based on
326 \underline{\foreignlanguage{greek}{a}-SiC}
327 \item Production of favored \underline{3C-SiC} material
328 \underline{less advanced}
329 \item Quality and size not yet sufficient
331 \begin{picture}(0,0)(-310,-20)
332 \includegraphics[width=2.0cm]{cree.eps}
337 Alternative approach:
338 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
345 \begin{minipage}{3.15cm}
347 \includegraphics[width=3cm]{imp.eps}\\
353 \begin{minipage}{3.15cm}
355 \includegraphics[width=3cm]{annealing.eps}\\
357 \unit[12]{h} annealing at \degc{1200}
362 \begin{minipage}{5.5cm}
363 \includegraphics[width=5.8cm]{ibs_3c-sic.eps}\\[-0.2cm]
366 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
378 Systematic investigation of C implantations into Si
384 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
400 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
403 \begin{pspicture}(0,0)(0,0)
404 \rput(6.0,7.0){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
405 \begin{minipage}{11cm}
406 {\color{red}Diploma thesis}\\
407 \underline{Monte Carlo} simulation modeling the selforganization process\\
408 leading to periodic arrays of nanometric amorphous SiC precipitates
412 \begin{pspicture}(0,0)(0,0)
413 \rput(6.0,-0.5){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
414 \begin{minipage}{11cm}
415 {\color{blue}Doctoral studies}\\
416 Classical potential \underline{molecular dynamics} simulations \ldots\\
417 \underline{Density functional theory} calculations \ldots\\[0.2cm]
418 \ldots on defect formation and SiC precipitation in Si
422 \begin{pspicture}(0,0)(0,0)
423 \psellipse[linecolor=red,linewidth=0.05cm](5,3.0)(0.8,1.0)
425 \begin{pspicture}(0,0)(0,0)
426 \psellipse[linecolor=blue,linewidth=0.05cm](8.2,3.2)(1.5,1.6)
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
475 Model displaying the formation of ordered lamellae
481 \includegraphics[width=8.0cm]{../../nlsop/img/modell_ng_e.eps}
487 \item Supersaturation of C in c-Si\\
488 $\rightarrow$ {\bf Carbon induced} nucleation of spherical
490 \item High interfacial energy between 3C-SiC and c-Si\\
491 $\rightarrow$ {\bf Amorphous} precipitates
492 \item \unit[20-- 30]{\%} lower silicon density of a-SiC$_x$ compared to c-Si\\
493 $\rightarrow$ {\bf Lateral strain} (black arrows)
494 \item Implantation range near surface\\
495 $\rightarrow$ {\bf Relaxation} of {\bf vertical strain component}
496 \item Reduction of the carbon supersaturation in c-Si\\
497 $\rightarrow$ {\bf Carbon diffusion} into amorphous volumina
499 \item Remaining lateral strain\\
500 $\rightarrow$ {\bf Strain enhanced} lateral amorphisation
501 \item Absence of crystalline neighbours (structural information)\\
502 $\rightarrow$ {\bf Stabilization} of amorphous inclusions
503 {\bf against recrystallization}
511 Implementation of the Monte Carlo code
517 \item \underline{Amorphization / Recrystallization}\\
518 Ion collision in discretized target determined by random numbers
519 distributed according to nuclear energy loss.
520 Amorphization/recrystallization probability:
522 p_{c \rightarrow a}(\vec{r}) = {\color{green} p_b} + {\color{blue} p_c c_C(\vec{r})} + {\color{red} \sum_{\textrm{amorphous neighbours}} \frac{p_s c_C(\vec{r'})}{(r-r')^2}}
525 \item {\color{green} $p_b$} normal `ballistic' amorphization
526 \item {\color{blue} $p_c$} carbon induced amorphization
527 \item {\color{red} $p_s$} stress enhanced amorphization
530 p_{a \rightarrow c}(\vec r) = (1 - p_{c \rightarrow a}(\vec r)) \Big(1 - \frac{\sum_{direct \, neighbours} \delta (\vec{r'})}{6} \Big) \, \textrm{,}
533 \delta (\vec r) = \left\{
535 1 & \textrm{if volume at position $\vec r$ is amorphous} \\
536 0 & \textrm{otherwise} \\
540 \item \underline{Carbon incorporation}\\
541 Incorporation volume determined according to implantation profile
542 \item \underline{Diffusion / Sputtering}
544 \item Transfer fraction of C atoms
545 of crystalline into neighbored amorphous volumes
546 \item Remove surface layer
554 \begin{minipage}{3.7cm}
563 Evolution of the \ldots
568 \item lamella precipitates
570 \ldots reproduced!\\[1.5cm]
574 Experiment \& simulation\\
575 in good agreement\\[1.0cm]
577 Simulation is able to model the whole depth region\\[1.0cm]
582 \begin{minipage}{0.4cm}
585 \begin{minipage}{8.0cm}
587 \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e_1-2.eps}\\
588 \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e2_2-2.eps}
596 Structural \& compositional details
599 \begin{minipage}[t]{7.5cm}
600 \includegraphics[height=6.5cm]{../../nlsop/img/ac_cconc_ver2_e.eps}\\
602 \begin{minipage}[t]{5.0cm}
603 \includegraphics[height=6.5cm]{../../nlsop/img/97_98_e.eps}
611 \item Fluctuation of C concentration in lamellae region
612 \item \unit[8--10]{at.\%} C saturation limit
613 within the respective conditions
614 \item Complementarily arranged and alternating sequence of layers\\
615 with a high and low amount of amorphous regions
616 \item C accumulation in the amorphous phase / Origin of stress
619 \begin{picture}(0,0)(-265,-30)
621 \begin{minipage}{3cm}
624 Precipitation process\\
646 Model displaying the formation of ordered lamellae
650 \begin{minipage}{6.3cm}
653 Precipitation mechanism not yet fully understood!
655 \renewcommand\labelitemi{$\Rightarrow$}
657 \underline{Understanding the SiC precipitation}
659 \item significant technological progress in SiC thin film formation
660 \item perspectives for processes relying upon prevention of SiC precipitation
671 Supposed precipitation mechanism of SiC in Si
678 \begin{minipage}{3.8cm}
679 Si \& SiC lattice structure\\[0.2cm]
680 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
684 \begin{minipage}{3.8cm}
686 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
690 \begin{minipage}{3.8cm}
692 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
696 \begin{minipage}{4cm}
698 C-Si dimers (dumbbells)\\[-0.1cm]
699 on Si interstitial sites
703 \begin{minipage}{4.2cm}
705 Agglomeration of C-Si dumbbells\\[-0.1cm]
706 $\Rightarrow$ dark contrasts
710 \begin{minipage}{4cm}
712 Precipitation of 3C-SiC in Si\\[-0.1cm]
713 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
714 \& release of Si self-interstitials
718 \begin{minipage}{3.8cm}
720 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
724 \begin{minipage}{3.8cm}
726 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
730 \begin{minipage}{3.8cm}
732 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
736 \begin{pspicture}(0,0)(0,0)
737 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
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739 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
740 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
741 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
742 $4a_{\text{Si}}=5a_{\text{SiC}}$
744 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
745 \hkl(h k l) planes match
747 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
757 Supposed precipitation mechanism of SiC in Si
764 \begin{minipage}{3.8cm}
765 Si \& SiC lattice structure\\[0.2cm]
766 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
770 \begin{minipage}{3.8cm}
772 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
776 \begin{minipage}{3.8cm}
778 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
782 \begin{minipage}{4cm}
784 C-Si dimers (dumbbells)\\[-0.1cm]
785 on Si interstitial sites
789 \begin{minipage}{4.2cm}
791 Agglomeration of C-Si dumbbells\\[-0.1cm]
792 $\Rightarrow$ dark contrasts
796 \begin{minipage}{4cm}
798 Precipitation of 3C-SiC in Si\\[-0.1cm]
799 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
800 \& release of Si self-interstitials
804 \begin{minipage}{3.8cm}
806 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
810 \begin{minipage}{3.8cm}
812 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
816 \begin{minipage}{3.8cm}
818 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
822 \begin{pspicture}(0,0)(0,0)
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828 $4a_{\text{Si}}=5a_{\text{SiC}}$
830 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
831 \hkl(h k l) planes match
833 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
836 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
837 \begin{minipage}{10cm}
839 {\color{red}\bf Controversial views}
841 \item Implantations at high T (Nejim et al.)
843 \item Topotactic transformation based on \cs
844 \item \si{} as supply reacting with further C in cleared volume
846 \item Annealing behavior (Serre et al.)
848 \item Room temperature implants $\rightarrow$ highly mobile C
849 \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
850 (indicate stable \cs{} configurations)
852 \item Strained silicon \& Si/SiC heterostructures
854 \item Coherent SiC precipitates (tensile strain)
855 \item Incoherent SiC (strain relaxation)
867 Molecular dynamics (MD) simulations
876 \item Microscopic description of N particle system
877 \item Analytical interaction potential
878 \item Numerical integration using Newtons equation of motion\\
879 as a propagation rule in 6N-dimensional phase space
880 \item Observables obtained by time and/or ensemble averages
882 {\bf Details of the simulation:}
884 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
885 \item Ensemble: NpT (isothermal-isobaric)
887 \item Berendsen thermostat:
888 $\tau_{\text{T}}=100\text{ fs}$
889 \item Berendsen barostat:\\
890 $\tau_{\text{P}}=100\text{ fs}$,
891 $\beta^{-1}=100\text{ GPa}$
893 \item Erhart/Albe potential: Tersoff-like bond order potential
896 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
897 \pot_{ij} = {\color{red}f_C(r_{ij})}
898 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
902 \begin{picture}(0,0)(-230,-30)
903 \includegraphics[width=5cm]{tersoff_angle.eps}
911 Density functional theory (DFT) calculations
916 Basic ingredients necessary for DFT
919 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
921 \item ... uniquely determines the ground state potential
923 \item ... minimizes the systems total energy
925 \item \underline{Born-Oppenheimer}
926 - $N$ moving electrons in an external potential of static nuclei
928 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
929 +\sum_i^N V_{\text{ext}}(r_i)
930 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
932 \item \underline{Effective potential}
933 - averaged electrostatic potential \& exchange and correlation
935 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
938 \item \underline{Kohn-Sham system}
939 - Schr\"odinger equation of N non-interacting particles
941 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
946 n(r)=\sum_i^N|\Phi_i(r)|^2
948 \item \underline{Self-consistent solution}\\
949 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
950 which in turn depends on $n(r)$
951 \item \underline{Variational principle}
952 - minimize total energy with respect to $n(r)$
960 Density functional theory (DFT) calculations
967 Details of applied DFT calculations in this work
970 \item \underline{Exchange correlation functional}
971 - approximations for the inhomogeneous electron gas
973 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
974 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
976 \item \underline{Plane wave basis set}
977 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
980 \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}}
981 \qquad ({\color{blue}300\text{ eV}})
983 \item \underline{Brillouin zone sampling} -
984 {\color{blue}$\Gamma$-point only} calculations
985 \item \underline{Pseudo potential}
986 - consider only the valence electrons
987 \item \underline{Code} - VASP 4.6
992 MD and structural optimization
995 \item MD integration: Gear predictor corrector algorithm
996 \item Pressure control: Parrinello-Rahman pressure control
997 \item Structural optimization: Conjugate gradient method
1000 \begin{pspicture}(0,0)(0,0)
1001 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
1009 C and Si self-interstitial point defects in silicon
1016 \begin{minipage}{8cm}
1018 \begin{pspicture}(0,0)(7,5)
1019 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1022 \item Creation of c-Si simulation volume
1023 \item Periodic boundary conditions
1024 \item $T=0\text{ K}$, $p=0\text{ bar}$
1027 \rput(3.5,2.1){\rnode{insert}{\psframebox{
1030 Insertion of interstitial C/Si atoms
1033 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1036 Relaxation / structural energy minimization
1039 \ncline[]{->}{init}{insert}
1040 \ncline[]{->}{insert}{cool}
1043 \begin{minipage}{5cm}
1044 \includegraphics[width=5cm]{unit_cell_e.eps}\\
1047 \begin{minipage}{9cm}
1048 \begin{tabular}{l c c}
1050 & size [unit cells] & \# atoms\\
1052 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
1053 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
1057 \begin{minipage}{4cm}
1058 {\color{red}$\bullet$} Tetrahedral\\
1059 {\color{green}$\bullet$} Hexagonal\\
1060 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
1061 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
1062 {\color{cyan}$\bullet$} Bond-centered\\
1063 {\color{black}$\bullet$} Vacancy / Substitutional
1072 \begin{minipage}{9.5cm}
1075 Si self-interstitial point defects in silicon\\
1078 \begin{tabular}{l c c c c c}
1080 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
1082 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
1083 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
1085 \end{tabular}\\[0.2cm]
1087 \begin{minipage}{4.7cm}
1088 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
1090 \begin{minipage}{4.7cm}
1092 {\tiny nearly T $\rightarrow$ T}\\
1094 \includegraphics[width=4.7cm]{nhex_tet.ps}
1097 \underline{Hexagonal} \hspace{2pt}
1098 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
1100 \begin{minipage}{2.7cm}
1101 $E_{\text{f}}^*=4.48\text{ eV}$\\
1102 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
1104 \begin{minipage}{0.4cm}
1109 \begin{minipage}{2.7cm}
1110 $E_{\text{f}}=3.96\text{ eV}$\\
1111 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
1114 \begin{minipage}{2.9cm}
1116 \underline{Vacancy}\\
1117 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
1122 \begin{minipage}{3.5cm}
1125 \underline{\hkl<1 1 0> dumbbell}\\
1126 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
1127 \underline{Tetrahedral}\\
1128 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
1129 \underline{\hkl<1 0 0> dumbbell}\\
1130 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
1142 C interstitial point defects in silicon\\[-0.1cm]
1145 \begin{tabular}{l c c c c c c r}
1147 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
1149 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
1150 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
1152 \end{tabular}\\[0.1cm]
1155 \begin{minipage}{2.7cm}
1156 \underline{Hexagonal} \hspace{2pt}
1157 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
1158 $E_{\text{f}}^*=9.05\text{ eV}$\\
1159 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
1161 \begin{minipage}{0.4cm}
1166 \begin{minipage}{2.7cm}
1167 \underline{\hkl<1 0 0>}\\
1168 $E_{\text{f}}=3.88\text{ eV}$\\
1169 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
1172 \begin{minipage}{2cm}
1175 \begin{minipage}{3cm}
1177 \underline{Tetrahedral}\\
1178 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
1183 \begin{minipage}{2.7cm}
1184 \underline{Bond-centered}\\
1185 $E_{\text{f}}^*=5.59\text{ eV}$\\
1186 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
1188 \begin{minipage}{0.4cm}
1193 \begin{minipage}{2.7cm}
1194 \underline{\hkl<1 1 0> dumbbell}\\
1195 $E_{\text{f}}=5.18\text{ eV}$\\
1196 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
1199 \begin{minipage}{2cm}
1202 \begin{minipage}{3cm}
1204 \underline{Substitutional}\\
1205 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
1216 C \hkl<1 0 0> dumbbell interstitial configuration\\
1220 \begin{tabular}{l c c c c c c c c}
1222 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1224 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1225 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1227 \end{tabular}\\[0.2cm]
1228 \begin{tabular}{l c c c c }
1230 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1232 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1233 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1235 \end{tabular}\\[0.2cm]
1236 \begin{tabular}{l c c c}
1238 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1240 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1241 VASP & 0.109 & -0.065 & 0.174 \\
1243 \end{tabular}\\[0.6cm]
1246 \begin{minipage}{3.0cm}
1248 \underline{Erhart/Albe}
1249 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1252 \begin{minipage}{3.0cm}
1255 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1259 \begin{picture}(0,0)(-185,10)
1260 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1262 \begin{picture}(0,0)(-280,-150)
1263 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1266 \begin{pspicture}(0,0)(0,0)
1267 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1268 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1269 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1270 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1279 \begin{minipage}{8.5cm}
1282 Bond-centered interstitial configuration\\[-0.1cm]
1285 \begin{minipage}{3.0cm}
1286 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1288 \begin{minipage}{5.2cm}
1290 \item Linear Si-C-Si bond
1291 \item Si: one C \& 3 Si neighbours
1292 \item Spin polarized calculations
1293 \item No saddle point!\\
1300 \begin{minipage}[t]{6.5cm}
1301 \begin{minipage}[t]{1.2cm}
1303 {\tiny sp$^3$}\\[0.8cm]
1304 \underline{${\color{black}\uparrow}$}
1305 \underline{${\color{black}\uparrow}$}
1306 \underline{${\color{black}\uparrow}$}
1307 \underline{${\color{red}\uparrow}$}\\
1310 \begin{minipage}[t]{1.4cm}
1312 {\color{red}M}{\color{blue}O}\\[0.8cm]
1313 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1314 $\sigma_{\text{ab}}$\\[0.5cm]
1315 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1319 \begin{minipage}[t]{1.0cm}
1323 \underline{${\color{white}\uparrow\uparrow}$}
1324 \underline{${\color{white}\uparrow\uparrow}$}\\
1326 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1327 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1331 \begin{minipage}[t]{1.4cm}
1333 {\color{blue}M}{\color{green}O}\\[0.8cm]
1334 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1335 $\sigma_{\text{ab}}$\\[0.5cm]
1336 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1340 \begin{minipage}[t]{1.2cm}
1343 {\tiny sp$^3$}\\[0.8cm]
1344 \underline{${\color{green}\uparrow}$}
1345 \underline{${\color{black}\uparrow}$}
1346 \underline{${\color{black}\uparrow}$}
1347 \underline{${\color{black}\uparrow}$}\\
1355 \begin{minipage}{4.5cm}
1356 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1358 \begin{minipage}{3.5cm}
1359 {\color{gray}$\bullet$} Spin up\\
1360 {\color{green}$\bullet$} Spin down\\
1361 {\color{blue}$\bullet$} Resulting spin up\\
1362 {\color{yellow}$\bullet$} Si atoms\\
1363 {\color{red}$\bullet$} C atom
1368 \begin{minipage}{4.2cm}
1370 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1371 {\color{green}$\Box$} {\tiny unoccupied}\\
1372 {\color{red}$\bullet$} {\tiny occupied}
1381 Migration of the C \hkl<1 0 0> dumbbell interstitial
1386 {\small Investigated pathways}
1388 \begin{minipage}{8.5cm}
1389 \begin{minipage}{8.3cm}
1390 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1391 \begin{minipage}{2.4cm}
1392 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1394 \begin{minipage}{0.4cm}
1397 \begin{minipage}{2.4cm}
1398 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1400 \begin{minipage}{0.4cm}
1403 \begin{minipage}{2.4cm}
1404 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1407 \begin{minipage}{8.3cm}
1408 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1409 \begin{minipage}{2.4cm}
1410 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1412 \begin{minipage}{0.4cm}
1415 \begin{minipage}{2.4cm}
1416 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1418 \begin{minipage}{0.4cm}
1421 \begin{minipage}{2.4cm}
1422 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1425 \begin{minipage}{8.3cm}
1426 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1427 \begin{minipage}{2.4cm}
1428 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1430 \begin{minipage}{0.4cm}
1433 \begin{minipage}{2.4cm}
1434 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1436 \begin{minipage}{0.4cm}
1439 \begin{minipage}{2.4cm}
1440 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1445 \begin{minipage}{4.2cm}
1446 {\small Constrained relaxation\\
1447 technique (CRT) method}\\
1448 \includegraphics[width=4cm]{crt_orig.eps}
1450 \item Constrain diffusing atom
1451 \item Static constraints
1454 {\small Modifications}\\
1455 \includegraphics[width=4cm]{crt_mod.eps}
1457 \item Constrain all atoms
1458 \item Update individual\\
1469 Migration of the C \hkl<1 0 0> dumbbell interstitial
1475 \begin{minipage}{5.9cm}
1477 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1480 \begin{picture}(0,0)(60,0)
1481 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1483 \begin{picture}(0,0)(-5,0)
1484 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1486 \begin{picture}(0,0)(-55,0)
1487 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1489 \begin{picture}(0,0)(12.5,10)
1490 \includegraphics[width=1cm]{110_arrow.eps}
1492 \begin{picture}(0,0)(90,0)
1493 \includegraphics[height=0.9cm]{001_arrow.eps}
1499 \begin{minipage}{0.3cm}
1503 \begin{minipage}{5.9cm}
1505 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1508 \begin{picture}(0,0)(60,0)
1509 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1511 \begin{picture}(0,0)(5,0)
1512 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1514 \begin{picture}(0,0)(-55,0)
1515 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1517 \begin{picture}(0,0)(12.5,10)
1518 \includegraphics[width=1cm]{100_arrow.eps}
1520 \begin{picture}(0,0)(90,0)
1521 \includegraphics[height=0.9cm]{001_arrow.eps}
1531 \begin{minipage}{5.9cm}
1533 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1536 \begin{picture}(0,0)(60,0)
1537 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1539 \begin{picture}(0,0)(10,0)
1540 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1542 \begin{picture}(0,0)(-60,0)
1543 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1545 \begin{picture}(0,0)(12.5,10)
1546 \includegraphics[width=1cm]{100_arrow.eps}
1548 \begin{picture}(0,0)(90,0)
1549 \includegraphics[height=0.9cm]{001_arrow.eps}
1555 \begin{minipage}{0.3cm}
1558 \begin{minipage}{6.5cm}
1561 \item Energetically most favorable path
1564 \item Activation energy: $\approx$ 0.9 eV
1565 \item Experimental values: 0.73 ... 0.87 eV
1567 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1568 \item Reorientation (path 3)
1570 \item More likely composed of two consecutive steps of type 2
1571 \item Experimental values: 0.77 ... 0.88 eV
1573 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1582 Migration of the C \hkl<1 0 0> dumbbell interstitial
1589 \begin{minipage}{6.5cm}
1592 \begin{minipage}[t]{5.9cm}
1594 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1597 \begin{pspicture}(0,0)(0,0)
1598 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1600 \begin{picture}(0,0)(60,-50)
1601 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1603 \begin{picture}(0,0)(5,-50)
1604 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1606 \begin{picture}(0,0)(-55,-50)
1607 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1609 \begin{picture}(0,0)(12.5,-40)
1610 \includegraphics[width=1cm]{110_arrow.eps}
1612 \begin{picture}(0,0)(90,-45)
1613 \includegraphics[height=0.9cm]{001_arrow.eps}
1615 \begin{pspicture}(0,0)(0,0)
1616 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1618 \begin{picture}(0,0)(60,-15)
1619 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1621 \begin{picture}(0,0)(35,-15)
1622 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1624 \begin{picture}(0,0)(-5,-15)
1625 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1627 \begin{picture}(0,0)(-55,-15)
1628 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1630 \begin{picture}(0,0)(12.5,-5)
1631 \includegraphics[width=1cm]{100_arrow.eps}
1633 \begin{picture}(0,0)(90,-15)
1634 \includegraphics[height=0.9cm]{010_arrow.eps}
1640 \begin{minipage}{5.9cm}
1643 \item Lowest activation energy: $\approx$ 2.2 eV
1644 \item 2.4 times higher than VASP
1645 \item Different pathway
1650 \begin{minipage}{6.5cm}
1653 \begin{minipage}{5.9cm}
1655 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1658 %\begin{pspicture}(0,0)(0,0)
1659 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1661 %\begin{picture}(0,0)(60,-5)
1662 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1664 %\begin{picture}(0,0)(0,-5)
1665 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1667 %\begin{picture}(0,0)(-55,-5)
1668 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1670 %\begin{picture}(0,0)(12.5,5)
1671 %\includegraphics[width=1cm]{100_arrow.eps}
1673 %\begin{picture}(0,0)(90,0)
1674 %\includegraphics[height=0.9cm]{001_arrow.eps}
1682 %\begin{minipage}{5.9cm}
1683 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1687 \begin{minipage}{5.9cm}
1688 Transition involving \ci{} \hkl<1 1 0>
1690 \item Bond-centered configuration unstable\\
1691 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1692 \item Transition minima of path 2 \& 3\\
1693 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1694 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1695 \item 2.4 - 3.4 times higher than VASP
1696 \item Rotation of dumbbell orientation
1700 {\color{blue}Overestimated diffusion barrier}
1711 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1721 E_{\text{f}}^{\text{defect combination}}-
1722 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1723 E_{\text{f}}^{\text{2nd defect}}
1729 \begin{tabular}{l c c c c c c}
1731 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1733 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1734 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1735 \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}\\
1736 \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}\\
1737 \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}\\
1738 \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}\\
1740 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1741 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1750 \begin{minipage}[t]{3.8cm}
1751 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1752 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1754 \begin{minipage}[t]{3.5cm}
1755 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1756 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1758 \begin{minipage}[t]{5.5cm}
1760 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1761 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1762 \item Stress compensation / increase
1763 \item Unfavored: antiparallel orientations
1764 \item Indication of energetically favored\\
1766 \item Most favorable: C clustering
1767 \item However: High barrier ($>4\,\text{eV}$)
1768 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1773 \begin{picture}(0,0)(-295,-130)
1774 \includegraphics[width=3.5cm]{comb_pos.eps}
1782 Combinations of C-Si \hkl<1 0 0>-type interstitials
1789 Energetically most favorable combinations along \hkl<1 1 0>
1794 \begin{tabular}{l c c c c c c}
1796 & 1 & 2 & 3 & 4 & 5 & 6\\
1798 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1799 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1800 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>\\
1807 \begin{minipage}{7.0cm}
1808 \includegraphics[width=7cm]{db_along_110_cc.ps}
1810 \begin{minipage}{6.0cm}
1812 \item Interaction proportional to reciprocal cube of C-C distance
1813 \item Saturation in the immediate vicinity
1814 \renewcommand\labelitemi{$\Rightarrow$}
1815 \item Agglomeration of \ci{} expected
1816 \item Absence of C clustering
1820 Consisten with initial precipitation model
1832 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1838 %\begin{minipage}{3.2cm}
1839 %\includegraphics[width=3cm]{sub_110_combo.eps}
1841 %\begin{minipage}{7.8cm}
1842 %\begin{tabular}{l c c c c c c}
1844 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1845 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1847 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1848 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1849 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1850 %4 & \RM{4} & B & D & E & E & D \\
1851 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1858 %\begin{tabular}{l c c c c c c c c c c}
1860 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1862 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1863 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1864 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1869 \begin{minipage}{6.0cm}
1870 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1872 \begin{minipage}{7cm}
1875 \item IBS: C may displace Si\\
1876 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1878 \hkl<1 1 0>-type $\rightarrow$ favored combination
1879 \renewcommand\labelitemi{$\Rightarrow$}
1880 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1881 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1882 \item Interaction drops quickly to zero\\
1883 $\rightarrow$ low capture radius
1887 IBS process far from equilibrium\\
1888 \cs{} \& \si{} instead of thermodynamic ground state
1893 \begin{minipage}{6.5cm}
1894 \includegraphics[width=6.0cm]{162-097.ps}
1896 \item Low migration barrier
1899 \begin{minipage}{6.5cm}
1901 Ab initio MD at \degc{900}\\
1902 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1903 $t=\unit[2230]{fs}$\\
1904 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1908 Contribution of entropy to structural formation
1917 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1924 \begin{minipage}[t]{3cm}
1925 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1926 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1928 \begin{minipage}[t]{7cm}
1931 Low activation energies\\
1932 High activation energies for reverse processes\\
1934 {\color{blue}C$_{\text{sub}}$ very stable}\\
1938 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1940 {\color{blue}Formation of SiC by successive substitution by C}
1944 \begin{minipage}[t]{3cm}
1945 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1946 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1951 \begin{minipage}{5.9cm}
1952 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1954 \begin{picture}(0,0)(70,0)
1955 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1957 \begin{picture}(0,0)(30,0)
1958 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1960 \begin{picture}(0,0)(-10,0)
1961 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1963 \begin{picture}(0,0)(-48,0)
1964 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1966 \begin{picture}(0,0)(12.5,5)
1967 \includegraphics[width=1cm]{100_arrow.eps}
1969 \begin{picture}(0,0)(97,-10)
1970 \includegraphics[height=0.9cm]{001_arrow.eps}
1976 \begin{minipage}{0.3cm}
1980 \begin{minipage}{5.9cm}
1981 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1983 \begin{picture}(0,0)(60,0)
1984 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1986 \begin{picture}(0,0)(25,0)
1987 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1989 \begin{picture}(0,0)(-20,0)
1990 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1992 \begin{picture}(0,0)(-55,0)
1993 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1995 \begin{picture}(0,0)(12.5,5)
1996 \includegraphics[width=1cm]{100_arrow.eps}
1998 \begin{picture}(0,0)(95,0)
1999 \includegraphics[height=0.9cm]{001_arrow.eps}
2011 Conclusion of defect / migration / combined defect simulations
2020 \item Accurately described by quantum-mechanical simulations
2021 \item Less accurate description by classical potential simulations
2022 \item Underestimated formation energy of \cs{} by classical approach
2023 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
2028 \item C migration pathway in Si identified
2029 \item Consistent with reorientation and diffusion experiments
2032 \item Different path and ...
2033 \item overestimated barrier by classical potential calculations
2036 Concerning the precipitation mechanism
2038 \item Agglomeration of C-Si dumbbells energetically favorable
2039 (stress compensation)
2040 \item C-Si indeed favored compared to
2041 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2042 \item Possible low interaction capture radius of
2043 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2044 \item Low barrier for
2045 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
2046 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
2047 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
2050 {\color{blue}Results suggest increased participation of \cs}
2058 Silicon carbide precipitation simulations
2064 \begin{pspicture}(0,0)(12,6.5)
2066 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
2069 \item Create c-Si volume
2070 \item Periodc boundary conditions
2071 \item Set requested $T$ and $p=0\text{ bar}$
2072 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
2075 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
2077 Insertion of C atoms at constant T
2079 \item total simulation volume {\pnode{in1}}
2080 \item volume of minimal SiC precipitate {\pnode{in2}}
2081 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
2085 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
2087 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
2089 \ncline[]{->}{init}{insert}
2090 \ncline[]{->}{insert}{cool}
2091 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
2092 \rput(7.8,6){\footnotesize $V_1$}
2093 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
2094 \rput(9.2,4.85){\tiny $V_2$}
2095 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
2096 \rput(9.55,4.45){\footnotesize $V_3$}
2097 \rput(7.9,3.2){\pnode{ins1}}
2098 \rput(9.22,2.8){\pnode{ins2}}
2099 \rput(11.0,2.4){\pnode{ins3}}
2100 \ncline[]{->}{in1}{ins1}
2101 \ncline[]{->}{in2}{ins2}
2102 \ncline[]{->}{in3}{ins3}
2107 \item Restricted to classical potential simulations
2108 \item $V_2$ and $V_3$ considered due to low diffusion
2109 \item Amount of C atoms: 6000
2110 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
2111 \item Simulation volume: $31\times 31\times 31$ unit cells
2120 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2125 \begin{minipage}{6.5cm}
2126 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2128 \begin{minipage}{6.5cm}
2129 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2132 \begin{minipage}{6.5cm}
2133 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2135 \begin{minipage}{6.5cm}
2137 \underline{Low C concentration ($V_1$)}\\
2138 \hkl<1 0 0> C-Si dumbbell dominated structure
2140 \item Si-C bumbs around 0.19 nm
2141 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2142 concatenated dumbbells of various orientation
2143 \item Si-Si NN distance stretched to 0.3 nm
2145 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2146 \underline{High C concentration ($V_2$, $V_3$)}\\
2147 High amount of strongly bound C-C bonds\\
2148 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2149 Only short range order observable\\
2150 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2158 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2163 \begin{minipage}{6.5cm}
2164 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2166 \begin{minipage}{6.5cm}
2167 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2170 \begin{minipage}{6.5cm}
2171 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2173 \begin{minipage}{6.5cm}
2175 \underline{Low C concentration ($V_1$)}\\
2176 \hkl<1 0 0> C-Si dumbbell dominated structure
2178 \item Si-C bumbs around 0.19 nm
2179 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2180 concatenated dumbbells of various orientation
2181 \item Si-Si NN distance stretched to 0.3 nm
2183 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2184 \underline{High C concentration ($V_2$, $V_3$)}\\
2185 High amount of strongly bound C-C bonds\\
2186 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2187 Only short range order observable\\
2188 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2191 \begin{pspicture}(0,0)(0,0)
2192 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2193 \begin{minipage}{10cm}
2195 {\color{red}\bf 3C-SiC formation fails to appear}
2197 \item Low C concentration simulations
2199 \item Formation of \ci{} indeed occurs
2200 \item Agllomeration not observed
2202 \item High C concentration simulations
2204 \item Amorphous SiC-like structure\\
2205 (not expected at prevailing temperatures)
2206 \item Rearrangement and transition into 3C-SiC structure missing
2218 Limitations of molecular dynamics and short range potentials
2225 \underline{Time scale problem of MD}\\[0.2cm]
2226 Minimize integration error\\
2227 $\Rightarrow$ discretization considerably smaller than
2228 reciprocal of fastest vibrational mode\\[0.1cm]
2229 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2230 $\Rightarrow$ suitable choice of time step:
2231 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2232 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2233 Several local minima in energy surface separated by large energy barriers\\
2234 $\Rightarrow$ transition event corresponds to a multiple
2235 of vibrational periods\\
2236 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2237 infrequent transition events\\[0.1cm]
2238 {\color{blue}Accelerated methods:}
2239 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2243 \underline{Limitations related to the short range potential}\\[0.2cm]
2244 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2245 and 2$^{\text{nd}}$ next neighbours\\
2246 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2252 Potential enhanced problem of slow phase space propagation
2257 \underline{Approach to the (twofold) problem}\\[0.2cm]
2258 Increased temperature simulations without TAD corrections\\
2259 (accelerated methods or higher time scales exclusively not sufficient)
2261 \begin{picture}(0,0)(-260,-30)
2263 \begin{minipage}{4.2cm}
2270 \item 3C-SiC also observed for higher T
2271 \item higher T inside sample
2272 \item structural evolution vs.\\
2273 equilibrium properties
2279 \begin{picture}(0,0)(-305,-155)
2281 \begin{minipage}{2.5cm}
2285 thermodynmic sampling
2296 Increased temperature simulations at low C concentration
2301 \begin{minipage}{6.5cm}
2302 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2304 \begin{minipage}{6.5cm}
2305 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2308 \begin{minipage}{6.5cm}
2309 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2311 \begin{minipage}{6.5cm}
2313 \underline{Si-C bonds:}
2315 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2316 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2318 \underline{Si-Si bonds:}
2319 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2320 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2321 \underline{C-C bonds:}
2323 \item C-C next neighbour pairs reduced (mandatory)
2324 \item Peak at 0.3 nm slightly shifted
2326 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2327 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2329 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2331 \item Range [|-$\downarrow$]:
2332 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2333 with nearby Si$_{\text{I}}$}
2338 \begin{picture}(0,0)(-330,-74)
2341 \begin{minipage}{1.6cm}
2344 stretched SiC\\[-0.1cm]
2356 Increased temperature simulations at low C concentration
2361 \begin{minipage}{6.5cm}
2362 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2364 \begin{minipage}{6.5cm}
2365 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2368 \begin{minipage}{6.5cm}
2369 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2371 \begin{minipage}{6.5cm}
2373 \underline{Si-C bonds:}
2375 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2376 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2378 \underline{Si-Si bonds:}
2379 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2380 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2381 \underline{C-C bonds:}
2383 \item C-C next neighbour pairs reduced (mandatory)
2384 \item Peak at 0.3 nm slightly shifted
2386 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2387 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2389 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2391 \item Range [|-$\downarrow$]:
2392 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2393 with nearby Si$_{\text{I}}$}
2398 %\begin{picture}(0,0)(-330,-74)
2401 %\begin{minipage}{1.6cm}
2404 %stretched SiC\\[-0.1cm]
2411 \begin{pspicture}(0,0)(0,0)
2412 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2413 \begin{minipage}{10cm}
2415 {\color{blue}\bf Stretched SiC in c-Si}
2417 \item Consistent to precipitation model involving \cs{}
2418 \item Explains annealing behavior of high/low T C implants
2420 \item Low T: highly mobiel \ci{}
2421 \item High T: stable configurations of \cs{}
2424 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2425 $\Rightarrow$ Precipitation mechanism involving \cs{}
2435 Increased temperature simulations at high C concentration
2440 \begin{minipage}{6.5cm}
2441 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2443 \begin{minipage}{6.5cm}
2444 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2452 \begin{minipage}[t]{6.0cm}
2453 0.186 nm: Si-C pairs $\uparrow$\\
2454 (as expected in 3C-SiC)\\[0.2cm]
2455 0.282 nm: Si-C-C\\[0.2cm]
2456 $\approx$0.35 nm: C-Si-Si
2459 \begin{minipage}{0.2cm}
2463 \begin{minipage}[t]{6.0cm}
2464 0.15 nm: C-C pairs $\uparrow$\\
2465 (as expected in graphite/diamond)\\[0.2cm]
2466 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2467 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2472 \item Decreasing cut-off artifact
2473 \item {\color{red}Amorphous} SiC-like phase remains
2474 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2475 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2484 High C \& small $V$ \& short $t$
2487 Slow restructuring due to strong C-C bonds
2490 High C \& low T implants
2501 Summary and Conclusions
2509 \begin{minipage}[t]{12.9cm}
2510 \underline{Pecipitation simulations}
2512 \item High C concentration $\rightarrow$ amorphous SiC like phase
2513 \item Problem of potential enhanced slow phase space propagation
2514 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2515 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2516 \item High T necessary to simulate IBS conditions (far from equilibrium)
2517 \item Precipitation by successive agglomeration of \cs (epitaxy)
2518 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2519 (stretched SiC, interface)
2527 \begin{minipage}{12.9cm}
2532 \item Point defects excellently / fairly well described
2534 \item C$_{\text{sub}}$ drastically underestimated by EA
2535 \item EA predicts correct ground state:
2536 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2537 \item Identified migration path explaining
2538 diffusion and reorientation experiments by DFT
2539 \item EA fails to describe \ci{} migration:
2540 Wrong path \& overestimated barrier
2542 \item Combinations of defects
2544 \item Agglomeration of point defects energetically favorable
2545 by compensation of stress
2546 \item Formation of C-C unlikely
2547 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2548 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2549 Low barrier (\unit[0.77]{eV}) \& low capture radius
2557 \framebox{Precipitation by successive agglomeration of \cs{}}
2575 \underline{Augsburg}
2577 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2578 \item Ralf Utermann (EDV)
2581 \underline{Helsinki}
2583 \item Prof. K. Nordlund (MD)
2588 \item Bayerische Forschungsstiftung (financial support)
2591 \underline{Paderborn}
2593 \item Prof. J. Lindner (SiC)
2594 \item Prof. G. Schmidt (DFT + financial support)
2595 \item Dr. E. Rauls (DFT + SiC)
2596 \item Dr. S. Sanna (VASP)
2603 \bf Thank you for your attention!