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28 \graphicspath{{../img/}}
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38 \usepackage{semlayer} % Seminar overlays
39 \usepackage{slidesec} % Seminar sections and list of slides
41 \input{seminar.bug} % Official bugs corrections
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70 \newcommand{\ham}{\mathcal{H}}
71 \newcommand{\pot}{\mathcal{V}}
72 \newcommand{\foo}{\mathcal{U}}
73 \newcommand{\vir}{\mathcal{W}}
76 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
79 \renewcommand{\phi}{\varphi}
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89 \newrgbcolor{lachs}{1.0 .93 .81}
92 \newcommand{\si}{Si$_{\text{i}}${}}
93 \newcommand{\ci}{C$_{\text{i}}${}}
94 \newcommand{\cs}{C$_{\text{sub}}${}}
95 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
96 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
97 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
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108 Atomistic simulation study on the silicon carbide precipitation
114 \textsc{F. Zirkelbach}
118 Yet another seminar talk
122 Augsburg, 26. Mai 2011
127 % motivation / properties / applications of silicon carbide
132 \begin{pspicture}(0,0)(13.5,5)
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141 \rput[lt](0.2,4.6){\color{gray}PROPERTIES}
143 \rput[lt](0.5,4){wide band gap}
144 \rput[lt](0.5,3.5){high electric breakdown field}
145 \rput[lt](0.5,3){good electron mobility}
146 \rput[lt](0.5,2.5){high electron saturation drift velocity}
147 \rput[lt](0.5,2){high thermal conductivity}
149 \rput[lt](0.5,1.5){hard and mechanically stable}
150 \rput[lt](0.5,1){chemically inert}
152 \rput[lt](0.5,0.5){radiation hardness}
154 \rput[rt](13.3,4.6){\color{gray}APPLICATIONS}
156 \rput[rt](13,3.85){high-temperature, high power}
157 \rput[rt](13,3.5){and high-frequency}
158 \rput[rt](13,3.15){electronic and optoelectronic devices}
160 \rput[rt](13,2.35){material suitable for extreme conditions}
161 \rput[rt](13,2){microelectromechanical systems}
162 \rput[rt](13,1.65){abrasives, cutting tools, heating elements}
164 \rput[rt](13,0.85){first wall reactor material, detectors}
165 \rput[rt](13,0.5){and electronic devices for space}
169 \begin{picture}(0,0)(-3,68)
170 \includegraphics[width=2.6cm]{wide_band_gap.eps}
172 \begin{picture}(0,0)(-285,-162)
173 \includegraphics[width=3.38cm]{sic_led.eps}
175 \begin{picture}(0,0)(-195,-162)
176 \includegraphics[width=2.8cm]{6h-sic_3c-sic.eps}
178 \begin{picture}(0,0)(-313,65)
179 \includegraphics[width=2.2cm]{infineon_schottky.eps}
181 \begin{picture}(0,0)(-220,65)
182 \includegraphics[width=2.9cm]{sic_wechselrichter_ise.eps}
184 \begin{picture}(0,0)(0,-160)
185 \includegraphics[width=3.0cm]{sic_proton.eps}
187 \begin{picture}(0,0)(-60,65)
188 \includegraphics[width=3.4cm]{sic_switch.eps}
205 \begin{tabular}{l c c c c c c}
207 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
209 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
210 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
211 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
212 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
213 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
214 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
215 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
223 \begin{picture}(0,0)(-160,-155)
224 \includegraphics[width=7cm]{polytypes.eps}
226 \begin{picture}(0,0)(-10,-185)
227 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
229 \begin{picture}(0,0)(-10,-175)
230 {\tiny cubic (twist)}
232 \begin{picture}(0,0)(-60,-175)
233 {\tiny hexagonal (no twist)}
235 \begin{pspicture}(0,0)(0,0)
236 \psellipse[linecolor=green](5.7,3.03)(0.4,0.5)
238 \begin{pspicture}(0,0)(0,0)
239 \psellipse[linecolor=green](5.6,1.68)(0.4,0.2)
241 \begin{pspicture}(0,0)(0,0)
242 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
250 Fabrication of silicon carbide
257 SiC - \emph{Born from the stars, perfected on earth.}
261 Conventional thin film SiC growth:
263 \item \underline{Sublimation growth using the modified Lely method}
265 \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
266 \item Surrounded by polycrystalline SiC in a graphite crucible\\
267 at $T=2100-2400 \, ^{\circ} \text{C}$
268 \item Deposition of supersaturated vapor on cooler seed crystal
270 \item \underline{Homoepitaxial growth using CVD}
272 \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
273 \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
274 \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
276 \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
278 \item Two steps: carbonization and growth
279 \item $T=650-1050 \, ^{\circ} \text{C}$
280 \item SiC/Si lattice mismatch $\approx$ 20 \%
281 \item Quality and size not yet sufficient
285 \begin{picture}(0,0)(-280,-65)
286 \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
288 \begin{picture}(0,0)(-280,-55)
289 \begin{minipage}{5cm}
291 NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
296 \begin{picture}(0,0)(-265,-150)
297 \includegraphics[width=2.4cm]{m_lely.eps}
299 \begin{picture}(0,0)(-333,-175)
300 \begin{minipage}{5cm}
306 5. Insulation\\[-7pt]
311 \begin{picture}(0,0)(-230,-35)
313 {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
316 \begin{picture}(0,0)(-230,-10)
318 \begin{minipage}{3cm}
319 {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
330 Fabrication of silicon carbide
335 Alternative approach:
336 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
338 \item \underline{Implantation step 1}\\
339 180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
340 $\Rightarrow$ box-like distribution of equally sized
341 and epitactically oriented SiC precipitates
343 \item \underline{Implantation step 2}\\
344 180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
345 $\Rightarrow$ destruction of SiC nanocrystals
346 in growing amorphous interface layers
347 \item \underline{Annealing}\\
348 $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
349 $\Rightarrow$ homogeneous, stoichiometric SiC layer
350 with sharp interfaces
353 \begin{minipage}{6.3cm}
354 \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
356 XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
360 \begin{minipage}{6.3cm}
363 Precipitation mechanism not yet fully understood!
365 \renewcommand\labelitemi{$\Rightarrow$}
367 \underline{Understanding the SiC precipitation}
369 \item significant technological progress in SiC thin film formation
370 \item perspectives for processes relying upon prevention of SiC precipitation
387 \item Supposed precipitation mechanism of SiC in Si
388 \item Utilized simulation techniques
390 \item Molecular dynamics (MD) simulations
391 \item Density functional theory (DFT) calculations
393 \item C and Si self-interstitial point defects in silicon
394 \item Silicon carbide precipitation simulations
395 \item Summary / Conclusion / Outlook
403 Supposed precipitation mechanism of SiC in Si
410 \begin{minipage}{3.8cm}
411 Si \& SiC lattice structure\\[0.2cm]
412 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
416 \begin{minipage}{3.8cm}
418 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
422 \begin{minipage}{3.8cm}
424 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
428 \begin{minipage}{4cm}
430 C-Si dimers (dumbbells)\\[-0.1cm]
431 on Si interstitial sites
435 \begin{minipage}{4.2cm}
437 Agglomeration of C-Si dumbbells\\[-0.1cm]
438 $\Rightarrow$ dark contrasts
442 \begin{minipage}{4cm}
444 Precipitation of 3C-SiC in Si\\[-0.1cm]
445 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
446 \& release of Si self-interstitials
450 \begin{minipage}{3.8cm}
452 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
456 \begin{minipage}{3.8cm}
458 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
462 \begin{minipage}{3.8cm}
464 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
468 \begin{pspicture}(0,0)(0,0)
469 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
470 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
471 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
472 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
473 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
474 $4a_{\text{Si}}=5a_{\text{SiC}}$
476 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
477 \hkl(h k l) planes match
479 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
489 Supposed precipitation mechanism of SiC in Si
496 \begin{minipage}{3.8cm}
497 Si \& SiC lattice structure\\[0.2cm]
498 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
502 \begin{minipage}{3.8cm}
504 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
508 \begin{minipage}{3.8cm}
510 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
514 \begin{minipage}{4cm}
516 C-Si dimers (dumbbells)\\[-0.1cm]
517 on Si interstitial sites
521 \begin{minipage}{4.2cm}
523 Agglomeration of C-Si dumbbells\\[-0.1cm]
524 $\Rightarrow$ dark contrasts
528 \begin{minipage}{4cm}
530 Precipitation of 3C-SiC in Si\\[-0.1cm]
531 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
532 \& release of Si self-interstitials
536 \begin{minipage}{3.8cm}
538 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
542 \begin{minipage}{3.8cm}
544 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
548 \begin{minipage}{3.8cm}
550 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
554 \begin{pspicture}(0,0)(0,0)
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558 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
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560 $4a_{\text{Si}}=5a_{\text{SiC}}$
562 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
563 \hkl(h k l) planes match
565 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
568 \rput(4.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=red]{
569 \begin{minipage}{8cm}
572 IBS studies revealing controversial views\\
576 \item Topotactic transformation based on \cs
577 \item \si as supply reacting with further C in cleared volume
579 \item Serre, Reeson, Lindner ...
581 \item RT implants: highly mobile C
582 \item elevated T implants: no/low C redistribution/migration
594 Molecular dynamics (MD) simulations
603 \item Microscopic description of N particle system
604 \item Analytical interaction potential
605 \item Numerical integration using Newtons equation of motion\\
606 as a propagation rule in 6N-dimensional phase space
607 \item Observables obtained by time and/or ensemble averages
609 {\bf Details of the simulation:}
611 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
612 \item Ensemble: NpT (isothermal-isobaric)
614 \item Berendsen thermostat:
615 $\tau_{\text{T}}=100\text{ fs}$
616 \item Berendsen barostat:\\
617 $\tau_{\text{P}}=100\text{ fs}$,
618 $\beta^{-1}=100\text{ GPa}$
620 \item Erhart/Albe potential: Tersoff-like bond order potential
623 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
624 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
628 \begin{picture}(0,0)(-230,-30)
629 \includegraphics[width=5cm]{tersoff_angle.eps}
637 Density functional theory (DFT) calculations
642 Basic ingredients necessary for DFT
645 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
647 \item ... uniquely determines the ground state potential
649 \item ... minimizes the systems total energy
651 \item \underline{Born-Oppenheimer}
652 - $N$ moving electrons in an external potential of static nuclei
654 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
655 +\sum_i^N V_{\text{ext}}(r_i)
656 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
658 \item \underline{Effective potential}
659 - averaged electrostatic potential \& exchange and correlation
661 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
664 \item \underline{Kohn-Sham system}
665 - Schr\"odinger equation of N non-interacting particles
667 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
672 n(r)=\sum_i^N|\Phi_i(r)|^2
674 \item \underline{Self-consistent solution}\\
675 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
676 which in turn depends on $n(r)$
677 \item \underline{Variational principle}
678 - minimize total energy with respect to $n(r)$
686 Density functional theory (DFT) calculations
693 Details of applied DFT calculations in this work
696 \item \underline{Exchange correlation functional}
697 - approximations for the inhomogeneous electron gas
699 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
700 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
702 \item \underline{Plane wave basis set}
703 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
706 \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}}
707 \qquad ({\color{blue}300\text{ eV}})
709 \item \underline{Brillouin zone sampling} -
710 {\color{blue}$\Gamma$-point only} calculations
711 \item \underline{Pseudo potential}
712 - consider only the valence electrons
713 \item \underline{Code} - VASP 4.6
718 MD and structural optimization
721 \item MD integration: Gear predictor corrector algorithm
722 \item Pressure control: Parrinello-Rahman pressure control
723 \item Structural optimization: Conjugate gradient method
726 \begin{pspicture}(0,0)(0,0)
727 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
735 C and Si self-interstitial point defects in silicon
742 \begin{minipage}{8cm}
744 \begin{pspicture}(0,0)(7,5)
745 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
748 \item Creation of c-Si simulation volume
749 \item Periodic boundary conditions
750 \item $T=0\text{ K}$, $p=0\text{ bar}$
753 \rput(3.5,2.1){\rnode{insert}{\psframebox{
756 Insertion of interstitial C/Si atoms
759 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
762 Relaxation / structural energy minimization
765 \ncline[]{->}{init}{insert}
766 \ncline[]{->}{insert}{cool}
769 \begin{minipage}{5cm}
770 \includegraphics[width=5cm]{unit_cell_e.eps}\\
773 \begin{minipage}{9cm}
774 \begin{tabular}{l c c}
776 & size [unit cells] & \# atoms\\
778 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
779 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
783 \begin{minipage}{4cm}
784 {\color{red}$\bullet$} Tetrahedral\\
785 {\color{green}$\bullet$} Hexagonal\\
786 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
787 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
788 {\color{cyan}$\bullet$} Bond-centered\\
789 {\color{black}$\bullet$} Vacancy / Substitutional
798 \begin{minipage}{9.5cm}
801 Si self-interstitial point defects in silicon\\
804 \begin{tabular}{l c c c c c}
806 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
808 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
809 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
811 \end{tabular}\\[0.2cm]
813 \begin{minipage}{4.7cm}
814 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
816 \begin{minipage}{4.7cm}
818 {\tiny nearly T $\rightarrow$ T}\\
820 \includegraphics[width=4.7cm]{nhex_tet.ps}
823 \underline{Hexagonal} \hspace{2pt}
824 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
826 \begin{minipage}{2.7cm}
827 $E_{\text{f}}^*=4.48\text{ eV}$\\
828 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
830 \begin{minipage}{0.4cm}
835 \begin{minipage}{2.7cm}
836 $E_{\text{f}}=3.96\text{ eV}$\\
837 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
840 \begin{minipage}{2.9cm}
842 \underline{Vacancy}\\
843 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
848 \begin{minipage}{3.5cm}
851 \underline{\hkl<1 1 0> dumbbell}\\
852 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
853 \underline{Tetrahedral}\\
854 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
855 \underline{\hkl<1 0 0> dumbbell}\\
856 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
868 C interstitial point defects in silicon\\[-0.1cm]
871 \begin{tabular}{l c c c c c c r}
873 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
875 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
876 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
878 \end{tabular}\\[0.1cm]
881 \begin{minipage}{2.7cm}
882 \underline{Hexagonal} \hspace{2pt}
883 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
884 $E_{\text{f}}^*=9.05\text{ eV}$\\
885 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
887 \begin{minipage}{0.4cm}
892 \begin{minipage}{2.7cm}
893 \underline{\hkl<1 0 0>}\\
894 $E_{\text{f}}=3.88\text{ eV}$\\
895 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
898 \begin{minipage}{2cm}
901 \begin{minipage}{3cm}
903 \underline{Tetrahedral}\\
904 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
909 \begin{minipage}{2.7cm}
910 \underline{Bond-centered}\\
911 $E_{\text{f}}^*=5.59\text{ eV}$\\
912 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
914 \begin{minipage}{0.4cm}
919 \begin{minipage}{2.7cm}
920 \underline{\hkl<1 1 0> dumbbell}\\
921 $E_{\text{f}}=5.18\text{ eV}$\\
922 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
925 \begin{minipage}{2cm}
928 \begin{minipage}{3cm}
930 \underline{Substitutional}\\
931 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
942 C \hkl<1 0 0> dumbbell interstitial configuration\\
946 \begin{tabular}{l c c c c c c c c}
948 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
950 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
951 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
953 \end{tabular}\\[0.2cm]
954 \begin{tabular}{l c c c c }
956 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
958 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
959 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
961 \end{tabular}\\[0.2cm]
962 \begin{tabular}{l c c c}
964 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
966 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
967 VASP & 0.109 & -0.065 & 0.174 \\
969 \end{tabular}\\[0.6cm]
972 \begin{minipage}{3.0cm}
974 \underline{Erhart/Albe}
975 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
978 \begin{minipage}{3.0cm}
981 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
985 \begin{picture}(0,0)(-185,10)
986 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
988 \begin{picture}(0,0)(-280,-150)
989 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
992 \begin{pspicture}(0,0)(0,0)
993 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
994 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
995 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
996 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1005 \begin{minipage}{8.5cm}
1008 Bond-centered interstitial configuration\\[-0.1cm]
1011 \begin{minipage}{3.0cm}
1012 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1014 \begin{minipage}{5.2cm}
1016 \item Linear Si-C-Si bond
1017 \item Si: one C \& 3 Si neighbours
1018 \item Spin polarized calculations
1019 \item No saddle point!\\
1026 \begin{minipage}[t]{6.5cm}
1027 \begin{minipage}[t]{1.2cm}
1029 {\tiny sp$^3$}\\[0.8cm]
1030 \underline{${\color{black}\uparrow}$}
1031 \underline{${\color{black}\uparrow}$}
1032 \underline{${\color{black}\uparrow}$}
1033 \underline{${\color{red}\uparrow}$}\\
1036 \begin{minipage}[t]{1.4cm}
1038 {\color{red}M}{\color{blue}O}\\[0.8cm]
1039 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1040 $\sigma_{\text{ab}}$\\[0.5cm]
1041 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1045 \begin{minipage}[t]{1.0cm}
1049 \underline{${\color{white}\uparrow\uparrow}$}
1050 \underline{${\color{white}\uparrow\uparrow}$}\\
1052 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1053 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1057 \begin{minipage}[t]{1.4cm}
1059 {\color{blue}M}{\color{green}O}\\[0.8cm]
1060 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1061 $\sigma_{\text{ab}}$\\[0.5cm]
1062 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1066 \begin{minipage}[t]{1.2cm}
1069 {\tiny sp$^3$}\\[0.8cm]
1070 \underline{${\color{green}\uparrow}$}
1071 \underline{${\color{black}\uparrow}$}
1072 \underline{${\color{black}\uparrow}$}
1073 \underline{${\color{black}\uparrow}$}\\
1081 \begin{minipage}{4.5cm}
1082 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1084 \begin{minipage}{3.5cm}
1085 {\color{gray}$\bullet$} Spin up\\
1086 {\color{green}$\bullet$} Spin down\\
1087 {\color{blue}$\bullet$} Resulting spin up\\
1088 {\color{yellow}$\bullet$} Si atoms\\
1089 {\color{red}$\bullet$} C atom
1094 \begin{minipage}{4.2cm}
1096 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1097 {\color{green}$\Box$} {\tiny unoccupied}\\
1098 {\color{red}$\bullet$} {\tiny occupied}
1107 Migration of the C \hkl<1 0 0> dumbbell interstitial
1112 {\small Investigated pathways}
1114 \begin{minipage}{8.5cm}
1115 \begin{minipage}{8.3cm}
1116 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1117 \begin{minipage}{2.4cm}
1118 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1120 \begin{minipage}{0.4cm}
1123 \begin{minipage}{2.4cm}
1124 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1126 \begin{minipage}{0.4cm}
1129 \begin{minipage}{2.4cm}
1130 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1133 \begin{minipage}{8.3cm}
1134 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1135 \begin{minipage}{2.4cm}
1136 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1138 \begin{minipage}{0.4cm}
1141 \begin{minipage}{2.4cm}
1142 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1144 \begin{minipage}{0.4cm}
1147 \begin{minipage}{2.4cm}
1148 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1151 \begin{minipage}{8.3cm}
1152 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1153 \begin{minipage}{2.4cm}
1154 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1156 \begin{minipage}{0.4cm}
1159 \begin{minipage}{2.4cm}
1160 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1162 \begin{minipage}{0.4cm}
1165 \begin{minipage}{2.4cm}
1166 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1171 \begin{minipage}{4.2cm}
1172 {\small Constrained relaxation\\
1173 technique (CRT) method}\\
1174 \includegraphics[width=4cm]{crt_orig.eps}
1176 \item Constrain diffusing atom
1177 \item Static constraints
1180 {\small Modifications}\\
1181 \includegraphics[width=4cm]{crt_mod.eps}
1183 \item Constrain all atoms
1184 \item Update individual\\
1195 Migration of the C \hkl<1 0 0> dumbbell interstitial
1201 \begin{minipage}{5.9cm}
1203 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1206 \begin{picture}(0,0)(60,0)
1207 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1209 \begin{picture}(0,0)(-5,0)
1210 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1212 \begin{picture}(0,0)(-55,0)
1213 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1215 \begin{picture}(0,0)(12.5,10)
1216 \includegraphics[width=1cm]{110_arrow.eps}
1218 \begin{picture}(0,0)(90,0)
1219 \includegraphics[height=0.9cm]{001_arrow.eps}
1225 \begin{minipage}{0.3cm}
1229 \begin{minipage}{5.9cm}
1231 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1234 \begin{picture}(0,0)(60,0)
1235 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1237 \begin{picture}(0,0)(5,0)
1238 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1240 \begin{picture}(0,0)(-55,0)
1241 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1243 \begin{picture}(0,0)(12.5,10)
1244 \includegraphics[width=1cm]{100_arrow.eps}
1246 \begin{picture}(0,0)(90,0)
1247 \includegraphics[height=0.9cm]{001_arrow.eps}
1257 \begin{minipage}{5.9cm}
1259 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1262 \begin{picture}(0,0)(60,0)
1263 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1265 \begin{picture}(0,0)(10,0)
1266 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1268 \begin{picture}(0,0)(-60,0)
1269 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1271 \begin{picture}(0,0)(12.5,10)
1272 \includegraphics[width=1cm]{100_arrow.eps}
1274 \begin{picture}(0,0)(90,0)
1275 \includegraphics[height=0.9cm]{001_arrow.eps}
1281 \begin{minipage}{0.3cm}
1284 \begin{minipage}{6.5cm}
1287 \item Energetically most favorable path
1290 \item Activation energy: $\approx$ 0.9 eV
1291 \item Experimental values: 0.73 ... 0.87 eV
1293 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1294 \item Reorientation (path 3)
1296 \item More likely composed of two consecutive steps of type 2
1297 \item Experimental values: 0.77 ... 0.88 eV
1299 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1308 Migration of the C \hkl<1 0 0> dumbbell interstitial
1315 \begin{minipage}{6.5cm}
1318 \begin{minipage}[t]{5.9cm}
1320 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1323 \begin{pspicture}(0,0)(0,0)
1324 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1326 \begin{picture}(0,0)(60,-50)
1327 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1329 \begin{picture}(0,0)(5,-50)
1330 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1332 \begin{picture}(0,0)(-55,-50)
1333 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1335 \begin{picture}(0,0)(12.5,-40)
1336 \includegraphics[width=1cm]{110_arrow.eps}
1338 \begin{picture}(0,0)(90,-45)
1339 \includegraphics[height=0.9cm]{001_arrow.eps}
1341 \begin{pspicture}(0,0)(0,0)
1342 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1344 \begin{picture}(0,0)(60,-15)
1345 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1347 \begin{picture}(0,0)(35,-15)
1348 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1350 \begin{picture}(0,0)(-5,-15)
1351 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1353 \begin{picture}(0,0)(-55,-15)
1354 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1356 \begin{picture}(0,0)(12.5,-5)
1357 \includegraphics[width=1cm]{100_arrow.eps}
1359 \begin{picture}(0,0)(90,-15)
1360 \includegraphics[height=0.9cm]{010_arrow.eps}
1366 \begin{minipage}{5.9cm}
1369 \item Lowest activation energy: $\approx$ 2.2 eV
1370 \item 2.4 times higher than VASP
1371 \item Different pathway
1376 \begin{minipage}{6.5cm}
1379 \begin{minipage}{5.9cm}
1381 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1384 %\begin{pspicture}(0,0)(0,0)
1385 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1387 %\begin{picture}(0,0)(60,-5)
1388 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1390 %\begin{picture}(0,0)(0,-5)
1391 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1393 %\begin{picture}(0,0)(-55,-5)
1394 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1396 %\begin{picture}(0,0)(12.5,5)
1397 %\includegraphics[width=1cm]{100_arrow.eps}
1399 %\begin{picture}(0,0)(90,0)
1400 %\includegraphics[height=0.9cm]{001_arrow.eps}
1408 %\begin{minipage}{5.9cm}
1409 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1413 \begin{minipage}{5.9cm}
1414 Transition involving \ci{} \hkl<1 1 0>
1416 \item Bond-centered configuration unstable\\
1417 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1418 \item Transition minima of path 2 \& 3\\
1419 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1420 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1421 \item 2.4 - 3.4 times higher than VASP
1422 \item Rotation of dumbbell orientation
1433 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1443 E_{\text{f}}^{\text{defect combination}}-
1444 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1445 E_{\text{f}}^{\text{2nd defect}}
1451 \begin{tabular}{l c c c c c c}
1453 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1455 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1456 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1457 \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}\\
1458 \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}\\
1459 \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}\\
1460 \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}\\
1462 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1463 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1472 \begin{minipage}[t]{3.8cm}
1473 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1474 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1476 \begin{minipage}[t]{3.5cm}
1477 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1478 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1480 \begin{minipage}[t]{5.5cm}
1482 \item Restricted to VASP simulations
1483 \item $E_{\text{b}}=0$ for isolated non-interacting defects
1484 \item $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1485 \item Stress compensation / increase
1486 \item Most favorable: C clustering
1487 \item Unfavored: antiparallel orientations
1488 \item Indication of energetically favored\\
1493 \begin{picture}(0,0)(-295,-130)
1494 \includegraphics[width=3.5cm]{comb_pos.eps}
1502 Combinations of C-Si \hkl<1 0 0>-type interstitials
1509 Energetically most favorable combinations along \hkl<1 1 0>
1514 \begin{tabular}{l c c c c c c}
1516 & 1 & 2 & 3 & 4 & 5 & 6\\
1518 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1519 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1520 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>\\
1527 \begin{minipage}{7.0cm}
1528 \includegraphics[width=7cm]{db_along_110_cc.ps}
1530 \begin{minipage}{6.0cm}
1533 Interaction proportional to reciprocal cube of C-C distance
1535 Saturation in the immediate vicinity
1546 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1552 \begin{minipage}{3.2cm}
1553 \includegraphics[width=3cm]{sub_110_combo.eps}
1555 \begin{minipage}{7.8cm}
1556 \begin{tabular}{l c c c c c c}
1558 C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1559 \hkl<1 0 1> & \hkl<-1 0 1> \\
1561 1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1562 2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1563 3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1564 4 & \RM{4} & B & D & E & E & D \\
1565 5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1572 \begin{tabular}{l c c c c c c c c c c}
1574 Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1576 $E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1577 $E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1578 $r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1583 \begin{minipage}{6.0cm}
1584 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1586 \begin{minipage}{7cm}
1589 \item IBS: C may displace Si\\
1590 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1592 \hkl<1 1 0>-type $\rightarrow$ favored combination
1593 \renewcommand\labelitemi{$\Rightarrow$}
1594 \item Less favorable than C-Si \hkl<1 0 0> dumbbell\\
1595 ($E_{\text{f}}=3.88\text{ eV}$)
1596 \item Interaction drops quickly to zero\\
1597 (low interaction capture radius)
1606 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1613 \begin{minipage}[t]{3cm}
1614 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1615 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1617 \begin{minipage}[t]{7cm}
1620 Low activation energies\\
1621 High activation energies for reverse processes\\
1623 {\color{blue}C$_{\text{sub}}$ very stable}\\
1627 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1629 {\color{blue}Formation of SiC by successive substitution by C}
1633 \begin{minipage}[t]{3cm}
1634 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1635 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1640 \begin{minipage}{5.9cm}
1641 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1643 \begin{picture}(0,0)(70,0)
1644 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1646 \begin{picture}(0,0)(30,0)
1647 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1649 \begin{picture}(0,0)(-10,0)
1650 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1652 \begin{picture}(0,0)(-48,0)
1653 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1655 \begin{picture}(0,0)(12.5,5)
1656 \includegraphics[width=1cm]{100_arrow.eps}
1658 \begin{picture}(0,0)(97,-10)
1659 \includegraphics[height=0.9cm]{001_arrow.eps}
1665 \begin{minipage}{0.3cm}
1669 \begin{minipage}{5.9cm}
1670 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1672 \begin{picture}(0,0)(60,0)
1673 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1675 \begin{picture}(0,0)(25,0)
1676 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1678 \begin{picture}(0,0)(-20,0)
1679 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1681 \begin{picture}(0,0)(-55,0)
1682 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1684 \begin{picture}(0,0)(12.5,5)
1685 \includegraphics[width=1cm]{100_arrow.eps}
1687 \begin{picture}(0,0)(95,0)
1688 \includegraphics[height=0.9cm]{001_arrow.eps}
1700 Conclusion of defect / migration / combined defect simulations
1709 \item Accurately described by quantum-mechanical simulations
1710 \item Less accurate description by classical potential simulations
1711 \item Underestimated formation energy of \cs{} by classical approach
1712 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1717 \item C migration pathway in Si identified
1718 \item Consistent with reorientation and diffusion experiments
1721 \item Different path and ...
1722 \item overestimated barrier by classical potential calculations
1725 Concerning the precipitation mechanism
1727 \item Agglomeration of C-Si dumbbells energetically favorable
1728 (stress compensation)
1729 \item C-Si indeed favored compared to
1730 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1731 \item Possible low interaction capture radius of
1732 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1733 \item Low barrier for
1734 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1735 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1736 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1739 {\color{blue}Results suggest increased participation of \cs}
1747 Silicon carbide precipitation simulations
1753 \begin{pspicture}(0,0)(12,6.5)
1755 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1758 \item Create c-Si volume
1759 \item Periodc boundary conditions
1760 \item Set requested $T$ and $p=0\text{ bar}$
1761 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1764 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1766 Insertion of C atoms at constant T
1768 \item total simulation volume {\pnode{in1}}
1769 \item volume of minimal SiC precipitate {\pnode{in2}}
1770 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1774 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1776 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1778 \ncline[]{->}{init}{insert}
1779 \ncline[]{->}{insert}{cool}
1780 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1781 \rput(7.8,6){\footnotesize $V_1$}
1782 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1783 \rput(9.2,4.85){\tiny $V_2$}
1784 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1785 \rput(9.55,4.45){\footnotesize $V_3$}
1786 \rput(7.9,3.2){\pnode{ins1}}
1787 \rput(9.22,2.8){\pnode{ins2}}
1788 \rput(11.0,2.4){\pnode{ins3}}
1789 \ncline[]{->}{in1}{ins1}
1790 \ncline[]{->}{in2}{ins2}
1791 \ncline[]{->}{in3}{ins3}
1796 \item Restricted to classical potential simulations
1797 \item $V_2$ and $V_3$ considered due to low diffusion
1798 \item Amount of C atoms: 6000
1799 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1800 \item Simulation volume: $31\times 31\times 31$ unit cells
1809 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1814 \begin{minipage}{6.5cm}
1815 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1817 \begin{minipage}{6.5cm}
1818 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1821 \begin{minipage}{6.5cm}
1822 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1824 \begin{minipage}{6.5cm}
1826 \underline{Low C concentration ($V_1$)}\\
1827 \hkl<1 0 0> C-Si dumbbell dominated structure
1829 \item Si-C bumbs around 0.19 nm
1830 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1831 concatenated dumbbells of various orientation
1832 \item Si-Si NN distance stretched to 0.3 nm
1834 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1835 \underline{High C concentration ($V_2$, $V_3$)}\\
1836 High amount of strongly bound C-C bonds\\
1837 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1838 Only short range order observable\\
1839 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1847 Limitations of molecular dynamics and short range potentials
1854 \underline{Time scale problem of MD}\\[0.2cm]
1855 Minimize integration error\\
1856 $\Rightarrow$ discretization considerably smaller than
1857 reciprocal of fastest vibrational mode\\[0.1cm]
1858 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
1859 $\Rightarrow$ suitable choice of time step:
1860 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
1861 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
1862 Several local minima in energy surface separated by large energy barriers\\
1863 $\Rightarrow$ transition event corresponds to a multiple
1864 of vibrational periods\\
1865 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
1866 infrequent transition events\\[0.1cm]
1867 {\color{blue}Accelerated methods:}
1868 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
1872 \underline{Limitations related to the short range potential}\\[0.2cm]
1873 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
1874 and 2$^{\text{nd}}$ next neighbours\\
1875 $\Rightarrow$ overestimated unphysical high forces of next neighbours
1881 Potential enhanced problem of slow phase space propagation
1886 \underline{Approach to the (twofold) problem}\\[0.2cm]
1887 Increased temperature simulations without TAD corrections\\
1888 (accelerated methods or higher time scales exclusively not sufficient)
1890 \begin{picture}(0,0)(-260,-30)
1892 \begin{minipage}{4.2cm}
1899 \item 3C-SiC also observed for higher T
1900 \item higher T inside sample
1901 \item structural evolution vs.\\
1902 equilibrium properties
1908 \begin{picture}(0,0)(-305,-155)
1910 \begin{minipage}{2.5cm}
1914 thermodynmic sampling
1925 Increased temperature simulations at low C concentration
1930 \begin{minipage}{6.5cm}
1931 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
1933 \begin{minipage}{6.5cm}
1934 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
1937 \begin{minipage}{6.5cm}
1938 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
1940 \begin{minipage}{6.5cm}
1942 \underline{Si-C bonds:}
1944 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
1945 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
1947 \underline{Si-Si bonds:}
1948 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
1949 ($\rightarrow$ 0.325 nm)\\[0.1cm]
1950 \underline{C-C bonds:}
1952 \item C-C next neighbour pairs reduced (mandatory)
1953 \item Peak at 0.3 nm slightly shifted
1955 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
1956 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
1958 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
1960 \item Range [|-$\downarrow$]:
1961 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
1962 with nearby Si$_{\text{I}}$}
1967 \begin{picture}(0,0)(-330,-74)
1970 \begin{minipage}{1.6cm}
1973 stretched SiC\\[-0.1cm]
1985 Increased temperature simulations at high C concentration
1990 \begin{minipage}{6.5cm}
1991 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
1993 \begin{minipage}{6.5cm}
1994 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
1998 Decreasing cut-off artifact\\
1999 High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2000 $\Rightarrow$ hard to categorize
2006 \begin{minipage}[t]{6.0cm}
2007 0.186 nm: Si-C pairs $\uparrow$\\
2008 (as expected in 3C-SiC)\\[0.2cm]
2009 0.282 nm: Si-C-C\\[0.2cm]
2010 $\approx$0.35 nm: C-Si-Si
2013 \begin{minipage}{0.2cm}
2017 \begin{minipage}[t]{6.0cm}
2018 0.15 nm: C-C pairs $\uparrow$\\
2019 (as expected in graphite/diamond)\\[0.2cm]
2020 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2021 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2028 {\color{red}Amorphous} SiC-like phase remains\\
2029 Slightly sharper peaks
2030 $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics}
2031 due to temperature\\[0.1cm]
2034 Actual SiC precipitation not accessible by MD
2043 Summary and Conclusions
2051 \begin{minipage}{12.9cm}
2056 \item Point defects excellently / fairly well described
2058 \item C$_{\text{sub}}$ drastically underestimated by EA
2059 \item EA predicts correct ground state:
2060 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2061 \item Identified migration path explaining
2062 diffusion and reorientation experiments by DFT
2063 \item EA fails to describe \ci{} migration:
2064 Wrong path \& overestimated barrier
2066 \item Combinations of defects
2068 \item Agglomeration of point defects energetically favorable
2069 by compensation of stress
2070 \item Formation of C-C unlikely
2071 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2072 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2073 Low barrier (\unit[0.77]{eV}) \& low capture radius
2082 \begin{minipage}[t]{12.9cm}
2083 \underline{Pecipitation simulations}
2085 \item High C concentration $\rightarrow$ amorphous SiC like phase
2086 \item Problem of potential enhanced slow phase space propagation
2087 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2088 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2089 \item High T necessary to simulate IBS conditions (far from equilibrium)
2090 \item Precipitation by successive agglomeration of \cs (epitaxy)
2091 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2092 (stretched SiC, interface)
2111 \underline{Augsburg}
2113 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2114 \item Ralf Utermann (EDV)
2117 \underline{Helsinki}
2119 \item Prof. K. Nordlund (MD)
2124 \item Bayerische Forschungsstiftung (financial support)
2127 \underline{Paderborn}
2129 \item Prof. J. Lindner (SiC)
2130 \item Prof. G. Schmidt (DFT + financial support)
2131 \item Dr. E. Rauls (DFT + SiC)
2132 \item Dr. S. Sanna (VASP)
2139 \bf Thank you for your attention!