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17 \usepackage{fancyhdr} % Headers and footers definitions
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19 \usepackage{pstricks} % PSTricks with the standard color package
30 \graphicspath{{../img/}}
34 \usepackage[setpagesize=false]{hyperref}
40 \usepackage{semlayer} % Seminar overlays
41 \usepackage{slidesec} % Seminar sections and list of slides
43 \input{seminar.bug} % Official bugs corrections
44 \input{seminar.bg2} % Unofficial bugs corrections
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59 \extraslideheight{10in}
64 % specify width and height
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69 \def\slideleftmargin{3.3cm}
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72 \newcommand{\ham}{\mathcal{H}}
73 \newcommand{\pot}{\mathcal{V}}
74 \newcommand{\foo}{\mathcal{U}}
75 \newcommand{\vir}{\mathcal{W}}
78 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
81 \renewcommand{\phi}{\varphi}
84 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
87 \newrgbcolor{si-yellow}{.6 .6 0}
88 \newrgbcolor{hb}{0.75 0.77 0.89}
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90 \newrgbcolor{hlbb}{0.825 0.88 0.968}
91 \newrgbcolor{lachs}{1.0 .93 .81}
94 \newcommand{\si}{Si$_{\text{i}}${}}
95 \newcommand{\ci}{C$_{\text{i}}${}}
96 \newcommand{\cs}{C$_{\text{sub}}${}}
97 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
98 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
99 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
100 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
104 % no vertical centering
116 Atomistic simulation studies\\[0.2cm]
122 \textsc{Frank Zirkelbach}
126 Application talk at the Max Planck Institute for Solid State Research
130 Stuttgart, November 2011
140 % Phase diagram of the C/Si system\\
145 \begin{minipage}{7cm}
146 \includegraphics[width=6.5cm]{si-c_phase.eps}
149 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
152 \begin{pspicture}(0,0)(0,0)
153 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
156 \begin{minipage}{6cm}
157 {\bf Phase diagram of the C/Si system}\\[0.2cm]
158 {\color{blue}Stoichiometric composition}
160 \item only chemical stable compound
161 \item wide band gap semiconductor\\
162 \underline{silicon carbide}, SiC
168 % motivation / properties / applications of silicon carbide
174 \begin{pspicture}(0,0)(13.5,5)
176 \psframe*[linecolor=hb](0,0)(13.5,5)
178 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.5,1)(7,1)(7,3)(5.5,3)
179 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.75,0.5)(8,2)(8,2)(6.75,3.5)
181 \rput[lt](0.2,4.6){\color{gray}PROPERTIES}
183 \rput[lt](0.5,4){wide band gap}
184 \rput[lt](0.5,3.5){high electric breakdown field}
185 \rput[lt](0.5,3){good electron mobility}
186 \rput[lt](0.5,2.5){high electron saturation drift velocity}
187 \rput[lt](0.5,2){high thermal conductivity}
189 \rput[lt](0.5,1.5){hard and mechanically stable}
190 \rput[lt](0.5,1){chemically inert}
192 \rput[lt](0.5,0.5){radiation hardness}
194 \rput[rt](13.3,4.6){\color{gray}APPLICATIONS}
196 \rput[rt](13,3.85){high-temperature, high power}
197 \rput[rt](13,3.5){and high-frequency}
198 \rput[rt](13,3.15){electronic and optoelectronic devices}
200 \rput[rt](13,2.35){material suitable for extreme conditions}
201 \rput[rt](13,2){microelectromechanical systems}
202 \rput[rt](13,1.65){abrasives, cutting tools, heating elements}
204 \rput[rt](13,0.85){first wall reactor material, detectors}
205 \rput[rt](13,0.5){and electronic devices for space}
209 \begin{picture}(0,0)(0,-162)
210 \includegraphics[height=2.0cm]{3C_SiC_bs.eps}
212 \begin{picture}(0,0)(-130,-162)
213 \includegraphics[height=2.0cm]{nasa_600c_led.eps}
215 \begin{picture}(0,0)(-295,-162)
216 \includegraphics[height=2.0cm]{6h-sic_3c-sic.eps}
219 \begin{picture}(0,0)(5,65)
220 \includegraphics[height=2.8cm]{sic_switch.eps}
222 \begin{picture}(0,0)(-145,65)
223 \includegraphics[height=2.8cm]{infineon_schottky.eps}
225 \begin{picture}(0,0)(-260,65)
226 \includegraphics[height=2.8cm]{ise_99.eps}
243 \begin{tabular}{l c c c c c c}
245 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
247 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
248 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
249 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
250 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
251 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
252 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
253 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
261 \begin{picture}(0,0)(-160,-155)
262 \includegraphics[width=7cm]{polytypes.eps}
264 \begin{picture}(0,0)(-10,-185)
265 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
267 \begin{picture}(0,0)(-10,-175)
268 {\tiny cubic (twist)}
270 \begin{picture}(0,0)(-60,-175)
271 {\tiny hexagonal (no twist)}
273 \begin{pspicture}(0,0)(0,0)
274 \psellipse[linecolor=green](5.7,3.03)(0.4,0.5)
276 \begin{pspicture}(0,0)(0,0)
277 \psellipse[linecolor=green](5.6,1.68)(0.4,0.2)
279 \begin{pspicture}(0,0)(0,0)
280 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
290 Fabrication of silicon carbide
297 SiC - \emph{Born from the stars, perfected on earth.}
303 Conventional thin film SiC growth:
305 \item \underline{Sublimation growth using the modified Lely method}
307 \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
308 \item Surrounded by polycrystalline SiC in a graphite crucible\\
309 at $T=2100-2400 \, ^{\circ} \text{C}$
310 \item Deposition of supersaturated vapor on cooler seed crystal
312 \item \underline{Homoepitaxial growth using CVD}
314 \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
315 \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
316 \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
318 \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
320 \item Two steps: carbonization and growth
321 \item $T=650-1050 \, ^{\circ} \text{C}$
322 \item SiC/Si lattice mismatch $\approx$ 20 \%
323 \item Quality and size not yet sufficient
327 \begin{picture}(0,0)(-280,-65)
328 \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
330 \begin{picture}(0,0)(-280,-55)
331 \begin{minipage}{5cm}
333 NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
338 \begin{picture}(0,0)(-265,-150)
339 \includegraphics[width=2.4cm]{m_lely.eps}
341 \begin{picture}(0,0)(-333,-175)
342 \begin{minipage}{5cm}
348 5. Insulation\\[-7pt]
353 \begin{picture}(0,0)(-230,-35)
355 {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
358 \begin{picture}(0,0)(-230,-10)
360 \begin{minipage}{3cm}
361 {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
378 \item Implantation of C in Si --- Overview of experimental observations
379 \item Utilized simulation techniques and modeled problems
381 \item {\color{blue}Diploma thesis}\\
382 \underline{Monte Carlo} simulations
383 modeling the selforganization process
384 leading to periodic arrays of nanometric amorphous SiC
386 \item {\color{blue}Doctoral studies}\\
387 Classical potential \underline{molecular dynamics} simulations
389 \underline{Density functional theory} calculations
391 \ldots on defects and SiC precipitation in Si
393 \item Summary / Conclusion / Outlook
405 Fabrication of silicon carbide
410 Alternative approach:
411 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
413 \item \underline{Implantation step 1}\\
414 180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
415 $\Rightarrow$ box-like distribution of equally sized
416 and epitactically oriented SiC precipitates
418 \item \underline{Implantation step 2}\\
419 180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
420 $\Rightarrow$ destruction of SiC nanocrystals
421 in growing amorphous interface layers
422 \item \underline{Annealing}\\
423 $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
424 $\Rightarrow$ homogeneous, stoichiometric SiC layer
425 with sharp interfaces
428 \begin{minipage}{6.3cm}
429 \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
431 XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
435 \begin{minipage}{6.3cm}
438 Precipitation mechanism not yet fully understood!
440 \renewcommand\labelitemi{$\Rightarrow$}
442 \underline{Understanding the SiC precipitation}
444 \item significant technological progress in SiC thin film formation
445 \item perspectives for processes relying upon prevention of SiC precipitation
457 Supposed precipitation mechanism of SiC in Si
464 \begin{minipage}{3.8cm}
465 Si \& SiC lattice structure\\[0.2cm]
466 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
470 \begin{minipage}{3.8cm}
472 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
476 \begin{minipage}{3.8cm}
478 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
482 \begin{minipage}{4cm}
484 C-Si dimers (dumbbells)\\[-0.1cm]
485 on Si interstitial sites
489 \begin{minipage}{4.2cm}
491 Agglomeration of C-Si dumbbells\\[-0.1cm]
492 $\Rightarrow$ dark contrasts
496 \begin{minipage}{4cm}
498 Precipitation of 3C-SiC in Si\\[-0.1cm]
499 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
500 \& release of Si self-interstitials
504 \begin{minipage}{3.8cm}
506 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
510 \begin{minipage}{3.8cm}
512 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
516 \begin{minipage}{3.8cm}
518 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
522 \begin{pspicture}(0,0)(0,0)
523 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
524 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
525 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
526 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
527 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
528 $4a_{\text{Si}}=5a_{\text{SiC}}$
530 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
531 \hkl(h k l) planes match
533 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
543 Supposed precipitation mechanism of SiC in Si
550 \begin{minipage}{3.8cm}
551 Si \& SiC lattice structure\\[0.2cm]
552 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
556 \begin{minipage}{3.8cm}
558 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
562 \begin{minipage}{3.8cm}
564 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
568 \begin{minipage}{4cm}
570 C-Si dimers (dumbbells)\\[-0.1cm]
571 on Si interstitial sites
575 \begin{minipage}{4.2cm}
577 Agglomeration of C-Si dumbbells\\[-0.1cm]
578 $\Rightarrow$ dark contrasts
582 \begin{minipage}{4cm}
584 Precipitation of 3C-SiC in Si\\[-0.1cm]
585 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
586 \& release of Si self-interstitials
590 \begin{minipage}{3.8cm}
592 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
596 \begin{minipage}{3.8cm}
598 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
602 \begin{minipage}{3.8cm}
604 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
608 \begin{pspicture}(0,0)(0,0)
609 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
610 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
611 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
612 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
613 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
614 $4a_{\text{Si}}=5a_{\text{SiC}}$
616 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
617 \hkl(h k l) planes match
619 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
622 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
623 \begin{minipage}{10cm}
625 {\color{red}\bf Controversial views}
627 \item Implantations at high T (Nejim et al.)
629 \item Topotactic transformation based on \cs
630 \item \si{} as supply reacting with further C in cleared volume
632 \item Annealing behavior (Serre et al.)
634 \item Room temperature implants $\rightarrow$ highly mobile C
635 \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
636 (indicate stable \cs{} configurations)
638 \item Strained silicon \& Si/SiC heterostructures
640 \item Coherent SiC precipitates (tensile strain)
641 \item Incoherent SiC (strain relaxation)
653 Molecular dynamics (MD) simulations
662 \item Microscopic description of N particle system
663 \item Analytical interaction potential
664 \item Numerical integration using Newtons equation of motion\\
665 as a propagation rule in 6N-dimensional phase space
666 \item Observables obtained by time and/or ensemble averages
668 {\bf Details of the simulation:}
670 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
671 \item Ensemble: NpT (isothermal-isobaric)
673 \item Berendsen thermostat:
674 $\tau_{\text{T}}=100\text{ fs}$
675 \item Berendsen barostat:\\
676 $\tau_{\text{P}}=100\text{ fs}$,
677 $\beta^{-1}=100\text{ GPa}$
679 \item Erhart/Albe potential: Tersoff-like bond order potential
682 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
683 \pot_{ij} = {\color{red}f_C(r_{ij})}
684 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
688 \begin{picture}(0,0)(-230,-30)
689 \includegraphics[width=5cm]{tersoff_angle.eps}
697 Density functional theory (DFT) calculations
702 Basic ingredients necessary for DFT
705 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
707 \item ... uniquely determines the ground state potential
709 \item ... minimizes the systems total energy
711 \item \underline{Born-Oppenheimer}
712 - $N$ moving electrons in an external potential of static nuclei
714 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
715 +\sum_i^N V_{\text{ext}}(r_i)
716 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
718 \item \underline{Effective potential}
719 - averaged electrostatic potential \& exchange and correlation
721 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
724 \item \underline{Kohn-Sham system}
725 - Schr\"odinger equation of N non-interacting particles
727 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
732 n(r)=\sum_i^N|\Phi_i(r)|^2
734 \item \underline{Self-consistent solution}\\
735 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
736 which in turn depends on $n(r)$
737 \item \underline{Variational principle}
738 - minimize total energy with respect to $n(r)$
746 Density functional theory (DFT) calculations
753 Details of applied DFT calculations in this work
756 \item \underline{Exchange correlation functional}
757 - approximations for the inhomogeneous electron gas
759 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
760 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
762 \item \underline{Plane wave basis set}
763 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
766 \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}}
767 \qquad ({\color{blue}300\text{ eV}})
769 \item \underline{Brillouin zone sampling} -
770 {\color{blue}$\Gamma$-point only} calculations
771 \item \underline{Pseudo potential}
772 - consider only the valence electrons
773 \item \underline{Code} - VASP 4.6
778 MD and structural optimization
781 \item MD integration: Gear predictor corrector algorithm
782 \item Pressure control: Parrinello-Rahman pressure control
783 \item Structural optimization: Conjugate gradient method
786 \begin{pspicture}(0,0)(0,0)
787 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
795 C and Si self-interstitial point defects in silicon
802 \begin{minipage}{8cm}
804 \begin{pspicture}(0,0)(7,5)
805 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
808 \item Creation of c-Si simulation volume
809 \item Periodic boundary conditions
810 \item $T=0\text{ K}$, $p=0\text{ bar}$
813 \rput(3.5,2.1){\rnode{insert}{\psframebox{
816 Insertion of interstitial C/Si atoms
819 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
822 Relaxation / structural energy minimization
825 \ncline[]{->}{init}{insert}
826 \ncline[]{->}{insert}{cool}
829 \begin{minipage}{5cm}
830 \includegraphics[width=5cm]{unit_cell_e.eps}\\
833 \begin{minipage}{9cm}
834 \begin{tabular}{l c c}
836 & size [unit cells] & \# atoms\\
838 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
839 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
843 \begin{minipage}{4cm}
844 {\color{red}$\bullet$} Tetrahedral\\
845 {\color{green}$\bullet$} Hexagonal\\
846 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
847 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
848 {\color{cyan}$\bullet$} Bond-centered\\
849 {\color{black}$\bullet$} Vacancy / Substitutional
858 \begin{minipage}{9.5cm}
861 Si self-interstitial point defects in silicon\\
864 \begin{tabular}{l c c c c c}
866 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
868 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
869 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
871 \end{tabular}\\[0.2cm]
873 \begin{minipage}{4.7cm}
874 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
876 \begin{minipage}{4.7cm}
878 {\tiny nearly T $\rightarrow$ T}\\
880 \includegraphics[width=4.7cm]{nhex_tet.ps}
883 \underline{Hexagonal} \hspace{2pt}
884 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
886 \begin{minipage}{2.7cm}
887 $E_{\text{f}}^*=4.48\text{ eV}$\\
888 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
890 \begin{minipage}{0.4cm}
895 \begin{minipage}{2.7cm}
896 $E_{\text{f}}=3.96\text{ eV}$\\
897 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
900 \begin{minipage}{2.9cm}
902 \underline{Vacancy}\\
903 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
908 \begin{minipage}{3.5cm}
911 \underline{\hkl<1 1 0> dumbbell}\\
912 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
913 \underline{Tetrahedral}\\
914 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
915 \underline{\hkl<1 0 0> dumbbell}\\
916 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
928 C interstitial point defects in silicon\\[-0.1cm]
931 \begin{tabular}{l c c c c c c r}
933 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
935 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
936 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
938 \end{tabular}\\[0.1cm]
941 \begin{minipage}{2.7cm}
942 \underline{Hexagonal} \hspace{2pt}
943 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
944 $E_{\text{f}}^*=9.05\text{ eV}$\\
945 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
947 \begin{minipage}{0.4cm}
952 \begin{minipage}{2.7cm}
953 \underline{\hkl<1 0 0>}\\
954 $E_{\text{f}}=3.88\text{ eV}$\\
955 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
958 \begin{minipage}{2cm}
961 \begin{minipage}{3cm}
963 \underline{Tetrahedral}\\
964 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
969 \begin{minipage}{2.7cm}
970 \underline{Bond-centered}\\
971 $E_{\text{f}}^*=5.59\text{ eV}$\\
972 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
974 \begin{minipage}{0.4cm}
979 \begin{minipage}{2.7cm}
980 \underline{\hkl<1 1 0> dumbbell}\\
981 $E_{\text{f}}=5.18\text{ eV}$\\
982 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
985 \begin{minipage}{2cm}
988 \begin{minipage}{3cm}
990 \underline{Substitutional}\\
991 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
1002 C \hkl<1 0 0> dumbbell interstitial configuration\\
1006 \begin{tabular}{l c c c c c c c c}
1008 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1010 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1011 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1013 \end{tabular}\\[0.2cm]
1014 \begin{tabular}{l c c c c }
1016 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1018 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1019 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1021 \end{tabular}\\[0.2cm]
1022 \begin{tabular}{l c c c}
1024 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1026 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1027 VASP & 0.109 & -0.065 & 0.174 \\
1029 \end{tabular}\\[0.6cm]
1032 \begin{minipage}{3.0cm}
1034 \underline{Erhart/Albe}
1035 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1038 \begin{minipage}{3.0cm}
1041 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1045 \begin{picture}(0,0)(-185,10)
1046 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1048 \begin{picture}(0,0)(-280,-150)
1049 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1052 \begin{pspicture}(0,0)(0,0)
1053 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1054 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1055 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1056 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1065 \begin{minipage}{8.5cm}
1068 Bond-centered interstitial configuration\\[-0.1cm]
1071 \begin{minipage}{3.0cm}
1072 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1074 \begin{minipage}{5.2cm}
1076 \item Linear Si-C-Si bond
1077 \item Si: one C \& 3 Si neighbours
1078 \item Spin polarized calculations
1079 \item No saddle point!\\
1086 \begin{minipage}[t]{6.5cm}
1087 \begin{minipage}[t]{1.2cm}
1089 {\tiny sp$^3$}\\[0.8cm]
1090 \underline{${\color{black}\uparrow}$}
1091 \underline{${\color{black}\uparrow}$}
1092 \underline{${\color{black}\uparrow}$}
1093 \underline{${\color{red}\uparrow}$}\\
1096 \begin{minipage}[t]{1.4cm}
1098 {\color{red}M}{\color{blue}O}\\[0.8cm]
1099 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1100 $\sigma_{\text{ab}}$\\[0.5cm]
1101 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1105 \begin{minipage}[t]{1.0cm}
1109 \underline{${\color{white}\uparrow\uparrow}$}
1110 \underline{${\color{white}\uparrow\uparrow}$}\\
1112 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1113 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1117 \begin{minipage}[t]{1.4cm}
1119 {\color{blue}M}{\color{green}O}\\[0.8cm]
1120 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1121 $\sigma_{\text{ab}}$\\[0.5cm]
1122 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1126 \begin{minipage}[t]{1.2cm}
1129 {\tiny sp$^3$}\\[0.8cm]
1130 \underline{${\color{green}\uparrow}$}
1131 \underline{${\color{black}\uparrow}$}
1132 \underline{${\color{black}\uparrow}$}
1133 \underline{${\color{black}\uparrow}$}\\
1141 \begin{minipage}{4.5cm}
1142 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1144 \begin{minipage}{3.5cm}
1145 {\color{gray}$\bullet$} Spin up\\
1146 {\color{green}$\bullet$} Spin down\\
1147 {\color{blue}$\bullet$} Resulting spin up\\
1148 {\color{yellow}$\bullet$} Si atoms\\
1149 {\color{red}$\bullet$} C atom
1154 \begin{minipage}{4.2cm}
1156 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1157 {\color{green}$\Box$} {\tiny unoccupied}\\
1158 {\color{red}$\bullet$} {\tiny occupied}
1167 Migration of the C \hkl<1 0 0> dumbbell interstitial
1172 {\small Investigated pathways}
1174 \begin{minipage}{8.5cm}
1175 \begin{minipage}{8.3cm}
1176 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1177 \begin{minipage}{2.4cm}
1178 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1180 \begin{minipage}{0.4cm}
1183 \begin{minipage}{2.4cm}
1184 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1186 \begin{minipage}{0.4cm}
1189 \begin{minipage}{2.4cm}
1190 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1193 \begin{minipage}{8.3cm}
1194 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1195 \begin{minipage}{2.4cm}
1196 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1198 \begin{minipage}{0.4cm}
1201 \begin{minipage}{2.4cm}
1202 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1204 \begin{minipage}{0.4cm}
1207 \begin{minipage}{2.4cm}
1208 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1211 \begin{minipage}{8.3cm}
1212 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1213 \begin{minipage}{2.4cm}
1214 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1216 \begin{minipage}{0.4cm}
1219 \begin{minipage}{2.4cm}
1220 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1222 \begin{minipage}{0.4cm}
1225 \begin{minipage}{2.4cm}
1226 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1231 \begin{minipage}{4.2cm}
1232 {\small Constrained relaxation\\
1233 technique (CRT) method}\\
1234 \includegraphics[width=4cm]{crt_orig.eps}
1236 \item Constrain diffusing atom
1237 \item Static constraints
1240 {\small Modifications}\\
1241 \includegraphics[width=4cm]{crt_mod.eps}
1243 \item Constrain all atoms
1244 \item Update individual\\
1255 Migration of the C \hkl<1 0 0> dumbbell interstitial
1261 \begin{minipage}{5.9cm}
1263 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1266 \begin{picture}(0,0)(60,0)
1267 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1269 \begin{picture}(0,0)(-5,0)
1270 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1272 \begin{picture}(0,0)(-55,0)
1273 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1275 \begin{picture}(0,0)(12.5,10)
1276 \includegraphics[width=1cm]{110_arrow.eps}
1278 \begin{picture}(0,0)(90,0)
1279 \includegraphics[height=0.9cm]{001_arrow.eps}
1285 \begin{minipage}{0.3cm}
1289 \begin{minipage}{5.9cm}
1291 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1294 \begin{picture}(0,0)(60,0)
1295 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1297 \begin{picture}(0,0)(5,0)
1298 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1300 \begin{picture}(0,0)(-55,0)
1301 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1303 \begin{picture}(0,0)(12.5,10)
1304 \includegraphics[width=1cm]{100_arrow.eps}
1306 \begin{picture}(0,0)(90,0)
1307 \includegraphics[height=0.9cm]{001_arrow.eps}
1317 \begin{minipage}{5.9cm}
1319 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1322 \begin{picture}(0,0)(60,0)
1323 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1325 \begin{picture}(0,0)(10,0)
1326 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1328 \begin{picture}(0,0)(-60,0)
1329 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1331 \begin{picture}(0,0)(12.5,10)
1332 \includegraphics[width=1cm]{100_arrow.eps}
1334 \begin{picture}(0,0)(90,0)
1335 \includegraphics[height=0.9cm]{001_arrow.eps}
1341 \begin{minipage}{0.3cm}
1344 \begin{minipage}{6.5cm}
1347 \item Energetically most favorable path
1350 \item Activation energy: $\approx$ 0.9 eV
1351 \item Experimental values: 0.73 ... 0.87 eV
1353 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1354 \item Reorientation (path 3)
1356 \item More likely composed of two consecutive steps of type 2
1357 \item Experimental values: 0.77 ... 0.88 eV
1359 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1368 Migration of the C \hkl<1 0 0> dumbbell interstitial
1375 \begin{minipage}{6.5cm}
1378 \begin{minipage}[t]{5.9cm}
1380 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1383 \begin{pspicture}(0,0)(0,0)
1384 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1386 \begin{picture}(0,0)(60,-50)
1387 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1389 \begin{picture}(0,0)(5,-50)
1390 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1392 \begin{picture}(0,0)(-55,-50)
1393 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1395 \begin{picture}(0,0)(12.5,-40)
1396 \includegraphics[width=1cm]{110_arrow.eps}
1398 \begin{picture}(0,0)(90,-45)
1399 \includegraphics[height=0.9cm]{001_arrow.eps}
1401 \begin{pspicture}(0,0)(0,0)
1402 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1404 \begin{picture}(0,0)(60,-15)
1405 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1407 \begin{picture}(0,0)(35,-15)
1408 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1410 \begin{picture}(0,0)(-5,-15)
1411 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1413 \begin{picture}(0,0)(-55,-15)
1414 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1416 \begin{picture}(0,0)(12.5,-5)
1417 \includegraphics[width=1cm]{100_arrow.eps}
1419 \begin{picture}(0,0)(90,-15)
1420 \includegraphics[height=0.9cm]{010_arrow.eps}
1426 \begin{minipage}{5.9cm}
1429 \item Lowest activation energy: $\approx$ 2.2 eV
1430 \item 2.4 times higher than VASP
1431 \item Different pathway
1436 \begin{minipage}{6.5cm}
1439 \begin{minipage}{5.9cm}
1441 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1444 %\begin{pspicture}(0,0)(0,0)
1445 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1447 %\begin{picture}(0,0)(60,-5)
1448 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1450 %\begin{picture}(0,0)(0,-5)
1451 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1453 %\begin{picture}(0,0)(-55,-5)
1454 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1456 %\begin{picture}(0,0)(12.5,5)
1457 %\includegraphics[width=1cm]{100_arrow.eps}
1459 %\begin{picture}(0,0)(90,0)
1460 %\includegraphics[height=0.9cm]{001_arrow.eps}
1468 %\begin{minipage}{5.9cm}
1469 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1473 \begin{minipage}{5.9cm}
1474 Transition involving \ci{} \hkl<1 1 0>
1476 \item Bond-centered configuration unstable\\
1477 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1478 \item Transition minima of path 2 \& 3\\
1479 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1480 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1481 \item 2.4 - 3.4 times higher than VASP
1482 \item Rotation of dumbbell orientation
1486 {\color{blue}Overestimated diffusion barrier}
1497 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1507 E_{\text{f}}^{\text{defect combination}}-
1508 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1509 E_{\text{f}}^{\text{2nd defect}}
1515 \begin{tabular}{l c c c c c c}
1517 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1519 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1520 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1521 \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}\\
1522 \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}\\
1523 \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}\\
1524 \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}\\
1526 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1527 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1536 \begin{minipage}[t]{3.8cm}
1537 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1538 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1540 \begin{minipage}[t]{3.5cm}
1541 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1542 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1544 \begin{minipage}[t]{5.5cm}
1546 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1547 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1548 \item Stress compensation / increase
1549 \item Unfavored: antiparallel orientations
1550 \item Indication of energetically favored\\
1552 \item Most favorable: C clustering
1553 \item However: High barrier ($>4\,\text{eV}$)
1554 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1559 \begin{picture}(0,0)(-295,-130)
1560 \includegraphics[width=3.5cm]{comb_pos.eps}
1568 Combinations of C-Si \hkl<1 0 0>-type interstitials
1575 Energetically most favorable combinations along \hkl<1 1 0>
1580 \begin{tabular}{l c c c c c c}
1582 & 1 & 2 & 3 & 4 & 5 & 6\\
1584 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1585 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1586 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>\\
1593 \begin{minipage}{7.0cm}
1594 \includegraphics[width=7cm]{db_along_110_cc.ps}
1596 \begin{minipage}{6.0cm}
1598 \item Interaction proportional to reciprocal cube of C-C distance
1599 \item Saturation in the immediate vicinity
1600 \renewcommand\labelitemi{$\Rightarrow$}
1601 \item Agglomeration of \ci{} expected
1602 \item Absence of C clustering
1606 Consisten with initial precipitation model
1618 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1624 %\begin{minipage}{3.2cm}
1625 %\includegraphics[width=3cm]{sub_110_combo.eps}
1627 %\begin{minipage}{7.8cm}
1628 %\begin{tabular}{l c c c c c c}
1630 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1631 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1633 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1634 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1635 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1636 %4 & \RM{4} & B & D & E & E & D \\
1637 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1644 %\begin{tabular}{l c c c c c c c c c c}
1646 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1648 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1649 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1650 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1655 \begin{minipage}{6.0cm}
1656 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1658 \begin{minipage}{7cm}
1661 \item IBS: C may displace Si\\
1662 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1664 \hkl<1 1 0>-type $\rightarrow$ favored combination
1665 \renewcommand\labelitemi{$\Rightarrow$}
1666 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1667 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1668 \item Interaction drops quickly to zero\\
1669 $\rightarrow$ low capture radius
1673 IBS process far from equilibrium\\
1674 \cs{} \& \si{} instead of thermodynamic ground state
1679 \begin{minipage}{6.5cm}
1680 \includegraphics[width=6.0cm]{162-097.ps}
1682 \item Low migration barrier
1685 \begin{minipage}{6.5cm}
1687 Ab initio MD at \degc{900}\\
1688 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1689 $t=\unit[2230]{fs}$\\
1690 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1694 Contribution of entropy to structural formation
1703 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1710 \begin{minipage}[t]{3cm}
1711 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1712 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1714 \begin{minipage}[t]{7cm}
1717 Low activation energies\\
1718 High activation energies for reverse processes\\
1720 {\color{blue}C$_{\text{sub}}$ very stable}\\
1724 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1726 {\color{blue}Formation of SiC by successive substitution by C}
1730 \begin{minipage}[t]{3cm}
1731 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1732 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1737 \begin{minipage}{5.9cm}
1738 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1740 \begin{picture}(0,0)(70,0)
1741 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1743 \begin{picture}(0,0)(30,0)
1744 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1746 \begin{picture}(0,0)(-10,0)
1747 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1749 \begin{picture}(0,0)(-48,0)
1750 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1752 \begin{picture}(0,0)(12.5,5)
1753 \includegraphics[width=1cm]{100_arrow.eps}
1755 \begin{picture}(0,0)(97,-10)
1756 \includegraphics[height=0.9cm]{001_arrow.eps}
1762 \begin{minipage}{0.3cm}
1766 \begin{minipage}{5.9cm}
1767 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1769 \begin{picture}(0,0)(60,0)
1770 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1772 \begin{picture}(0,0)(25,0)
1773 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1775 \begin{picture}(0,0)(-20,0)
1776 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1778 \begin{picture}(0,0)(-55,0)
1779 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1781 \begin{picture}(0,0)(12.5,5)
1782 \includegraphics[width=1cm]{100_arrow.eps}
1784 \begin{picture}(0,0)(95,0)
1785 \includegraphics[height=0.9cm]{001_arrow.eps}
1797 Conclusion of defect / migration / combined defect simulations
1806 \item Accurately described by quantum-mechanical simulations
1807 \item Less accurate description by classical potential simulations
1808 \item Underestimated formation energy of \cs{} by classical approach
1809 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1814 \item C migration pathway in Si identified
1815 \item Consistent with reorientation and diffusion experiments
1818 \item Different path and ...
1819 \item overestimated barrier by classical potential calculations
1822 Concerning the precipitation mechanism
1824 \item Agglomeration of C-Si dumbbells energetically favorable
1825 (stress compensation)
1826 \item C-Si indeed favored compared to
1827 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1828 \item Possible low interaction capture radius of
1829 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1830 \item Low barrier for
1831 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1832 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1833 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1836 {\color{blue}Results suggest increased participation of \cs}
1844 Silicon carbide precipitation simulations
1850 \begin{pspicture}(0,0)(12,6.5)
1852 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1855 \item Create c-Si volume
1856 \item Periodc boundary conditions
1857 \item Set requested $T$ and $p=0\text{ bar}$
1858 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1861 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1863 Insertion of C atoms at constant T
1865 \item total simulation volume {\pnode{in1}}
1866 \item volume of minimal SiC precipitate {\pnode{in2}}
1867 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1871 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1873 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1875 \ncline[]{->}{init}{insert}
1876 \ncline[]{->}{insert}{cool}
1877 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1878 \rput(7.8,6){\footnotesize $V_1$}
1879 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1880 \rput(9.2,4.85){\tiny $V_2$}
1881 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1882 \rput(9.55,4.45){\footnotesize $V_3$}
1883 \rput(7.9,3.2){\pnode{ins1}}
1884 \rput(9.22,2.8){\pnode{ins2}}
1885 \rput(11.0,2.4){\pnode{ins3}}
1886 \ncline[]{->}{in1}{ins1}
1887 \ncline[]{->}{in2}{ins2}
1888 \ncline[]{->}{in3}{ins3}
1893 \item Restricted to classical potential simulations
1894 \item $V_2$ and $V_3$ considered due to low diffusion
1895 \item Amount of C atoms: 6000
1896 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1897 \item Simulation volume: $31\times 31\times 31$ unit cells
1906 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1911 \begin{minipage}{6.5cm}
1912 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1914 \begin{minipage}{6.5cm}
1915 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1918 \begin{minipage}{6.5cm}
1919 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1921 \begin{minipage}{6.5cm}
1923 \underline{Low C concentration ($V_1$)}\\
1924 \hkl<1 0 0> C-Si dumbbell dominated structure
1926 \item Si-C bumbs around 0.19 nm
1927 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1928 concatenated dumbbells of various orientation
1929 \item Si-Si NN distance stretched to 0.3 nm
1931 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1932 \underline{High C concentration ($V_2$, $V_3$)}\\
1933 High amount of strongly bound C-C bonds\\
1934 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1935 Only short range order observable\\
1936 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1944 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1949 \begin{minipage}{6.5cm}
1950 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1952 \begin{minipage}{6.5cm}
1953 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1956 \begin{minipage}{6.5cm}
1957 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1959 \begin{minipage}{6.5cm}
1961 \underline{Low C concentration ($V_1$)}\\
1962 \hkl<1 0 0> C-Si dumbbell dominated structure
1964 \item Si-C bumbs around 0.19 nm
1965 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1966 concatenated dumbbells of various orientation
1967 \item Si-Si NN distance stretched to 0.3 nm
1969 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1970 \underline{High C concentration ($V_2$, $V_3$)}\\
1971 High amount of strongly bound C-C bonds\\
1972 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1973 Only short range order observable\\
1974 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1977 \begin{pspicture}(0,0)(0,0)
1978 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
1979 \begin{minipage}{10cm}
1981 {\color{red}\bf 3C-SiC formation fails to appear}
1983 \item Low C concentration simulations
1985 \item Formation of \ci{} indeed occurs
1986 \item Agllomeration not observed
1988 \item High C concentration simulations
1990 \item Amorphous SiC-like structure\\
1991 (not expected at prevailing temperatures)
1992 \item Rearrangement and transition into 3C-SiC structure missing
2004 Limitations of molecular dynamics and short range potentials
2011 \underline{Time scale problem of MD}\\[0.2cm]
2012 Minimize integration error\\
2013 $\Rightarrow$ discretization considerably smaller than
2014 reciprocal of fastest vibrational mode\\[0.1cm]
2015 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2016 $\Rightarrow$ suitable choice of time step:
2017 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2018 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2019 Several local minima in energy surface separated by large energy barriers\\
2020 $\Rightarrow$ transition event corresponds to a multiple
2021 of vibrational periods\\
2022 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2023 infrequent transition events\\[0.1cm]
2024 {\color{blue}Accelerated methods:}
2025 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2029 \underline{Limitations related to the short range potential}\\[0.2cm]
2030 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2031 and 2$^{\text{nd}}$ next neighbours\\
2032 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2038 Potential enhanced problem of slow phase space propagation
2043 \underline{Approach to the (twofold) problem}\\[0.2cm]
2044 Increased temperature simulations without TAD corrections\\
2045 (accelerated methods or higher time scales exclusively not sufficient)
2047 \begin{picture}(0,0)(-260,-30)
2049 \begin{minipage}{4.2cm}
2056 \item 3C-SiC also observed for higher T
2057 \item higher T inside sample
2058 \item structural evolution vs.\\
2059 equilibrium properties
2065 \begin{picture}(0,0)(-305,-155)
2067 \begin{minipage}{2.5cm}
2071 thermodynmic sampling
2082 Increased temperature simulations at low C concentration
2087 \begin{minipage}{6.5cm}
2088 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2090 \begin{minipage}{6.5cm}
2091 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2094 \begin{minipage}{6.5cm}
2095 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2097 \begin{minipage}{6.5cm}
2099 \underline{Si-C bonds:}
2101 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2102 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2104 \underline{Si-Si bonds:}
2105 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2106 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2107 \underline{C-C bonds:}
2109 \item C-C next neighbour pairs reduced (mandatory)
2110 \item Peak at 0.3 nm slightly shifted
2112 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2113 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2115 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2117 \item Range [|-$\downarrow$]:
2118 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2119 with nearby Si$_{\text{I}}$}
2124 \begin{picture}(0,0)(-330,-74)
2127 \begin{minipage}{1.6cm}
2130 stretched SiC\\[-0.1cm]
2142 Increased temperature simulations at low C concentration
2147 \begin{minipage}{6.5cm}
2148 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2150 \begin{minipage}{6.5cm}
2151 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2154 \begin{minipage}{6.5cm}
2155 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2157 \begin{minipage}{6.5cm}
2159 \underline{Si-C bonds:}
2161 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2162 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2164 \underline{Si-Si bonds:}
2165 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2166 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2167 \underline{C-C bonds:}
2169 \item C-C next neighbour pairs reduced (mandatory)
2170 \item Peak at 0.3 nm slightly shifted
2172 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2173 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2175 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2177 \item Range [|-$\downarrow$]:
2178 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2179 with nearby Si$_{\text{I}}$}
2184 %\begin{picture}(0,0)(-330,-74)
2187 %\begin{minipage}{1.6cm}
2190 %stretched SiC\\[-0.1cm]
2197 \begin{pspicture}(0,0)(0,0)
2198 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2199 \begin{minipage}{10cm}
2201 {\color{blue}\bf Stretched SiC in c-Si}
2203 \item Consistent to precipitation model involving \cs{}
2204 \item Explains annealing behavior of high/low T C implants
2206 \item Low T: highly mobiel \ci{}
2207 \item High T: stable configurations of \cs{}
2210 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2211 $\Rightarrow$ Precipitation mechanism involving \cs{}
2221 Increased temperature simulations at high C concentration
2226 \begin{minipage}{6.5cm}
2227 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2229 \begin{minipage}{6.5cm}
2230 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2238 \begin{minipage}[t]{6.0cm}
2239 0.186 nm: Si-C pairs $\uparrow$\\
2240 (as expected in 3C-SiC)\\[0.2cm]
2241 0.282 nm: Si-C-C\\[0.2cm]
2242 $\approx$0.35 nm: C-Si-Si
2245 \begin{minipage}{0.2cm}
2249 \begin{minipage}[t]{6.0cm}
2250 0.15 nm: C-C pairs $\uparrow$\\
2251 (as expected in graphite/diamond)\\[0.2cm]
2252 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2253 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2258 \item Decreasing cut-off artifact
2259 \item {\color{red}Amorphous} SiC-like phase remains
2260 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2261 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2270 High C \& small $V$ \& short $t$
2273 Slow restructuring due to strong C-C bonds
2276 High C \& low T implants
2287 Summary and Conclusions
2295 \begin{minipage}[t]{12.9cm}
2296 \underline{Pecipitation simulations}
2298 \item High C concentration $\rightarrow$ amorphous SiC like phase
2299 \item Problem of potential enhanced slow phase space propagation
2300 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2301 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2302 \item High T necessary to simulate IBS conditions (far from equilibrium)
2303 \item Precipitation by successive agglomeration of \cs (epitaxy)
2304 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2305 (stretched SiC, interface)
2313 \begin{minipage}{12.9cm}
2318 \item Point defects excellently / fairly well described
2320 \item C$_{\text{sub}}$ drastically underestimated by EA
2321 \item EA predicts correct ground state:
2322 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2323 \item Identified migration path explaining
2324 diffusion and reorientation experiments by DFT
2325 \item EA fails to describe \ci{} migration:
2326 Wrong path \& overestimated barrier
2328 \item Combinations of defects
2330 \item Agglomeration of point defects energetically favorable
2331 by compensation of stress
2332 \item Formation of C-C unlikely
2333 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2334 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2335 Low barrier (\unit[0.77]{eV}) \& low capture radius
2343 \framebox{Precipitation by successive agglomeration of \cs{}}
2361 \underline{Augsburg}
2363 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2364 \item Ralf Utermann (EDV)
2367 \underline{Helsinki}
2369 \item Prof. K. Nordlund (MD)
2374 \item Bayerische Forschungsstiftung (financial support)
2377 \underline{Paderborn}
2379 \item Prof. J. Lindner (SiC)
2380 \item Prof. G. Schmidt (DFT + financial support)
2381 \item Dr. E. Rauls (DFT + SiC)
2382 \item Dr. S. Sanna (VASP)
2389 \bf Thank you for your attention!