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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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71 \newcommand{\pot}{\mathcal{V}}
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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}{}}
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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}{6cm}
571 IBS studies revealing controversial views\\
575 \item Topotactic transformation based on \cs
576 \item \si as supply reacting with further C in cleared volume
578 \item Serre, Reeson, Lindner ...
580 \item RT implants: highly mobile C
581 \item elevated T implants: no/low C redistribution/migration
593 Molecular dynamics (MD) simulations
602 \item Microscopic description of N particle system
603 \item Analytical interaction potential
604 \item Numerical integration using Newtons equation of motion\\
605 as a propagation rule in 6N-dimensional phase space
606 \item Observables obtained by time and/or ensemble averages
608 {\bf Details of the simulation:}
610 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
611 \item Ensemble: NpT (isothermal-isobaric)
613 \item Berendsen thermostat:
614 $\tau_{\text{T}}=100\text{ fs}$
615 \item Berendsen barostat:\\
616 $\tau_{\text{P}}=100\text{ fs}$,
617 $\beta^{-1}=100\text{ GPa}$
619 \item Erhart/Albe potential: Tersoff-like bond order potential
622 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
623 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
627 \begin{picture}(0,0)(-230,-30)
628 \includegraphics[width=5cm]{tersoff_angle.eps}
636 Density functional theory (DFT) calculations
641 Basic ingredients necessary for DFT
644 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
646 \item ... uniquely determines the ground state potential
648 \item ... minimizes the systems total energy
650 \item \underline{Born-Oppenheimer}
651 - $N$ moving electrons in an external potential of static nuclei
653 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
654 +\sum_i^N V_{\text{ext}}(r_i)
655 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
657 \item \underline{Effective potential}
658 - averaged electrostatic potential \& exchange and correlation
660 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
663 \item \underline{Kohn-Sham system}
664 - Schr\"odinger equation of N non-interacting particles
666 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
671 n(r)=\sum_i^N|\Phi_i(r)|^2
673 \item \underline{Self-consistent solution}\\
674 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
675 which in turn depends on $n(r)$
676 \item \underline{Variational principle}
677 - minimize total energy with respect to $n(r)$
685 Density functional theory (DFT) calculations
692 Details of applied DFT calculations in this work
695 \item \underline{Exchange correlation functional}
696 - approximations for the inhomogeneous electron gas
698 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
699 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
701 \item \underline{Plane wave basis set}
702 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
705 \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}}
706 \qquad ({\color{blue}300\text{ eV}})
708 \item \underline{Brillouin zone sampling} -
709 {\color{blue}$\Gamma$-point only} calculations
710 \item \underline{Pseudo potential}
711 - consider only the valence electrons
712 \item \underline{Code} - VASP 4.6
717 MD and structural optimization
720 \item MD integration: Gear predictor corrector algorithm
721 \item Pressure control: Parrinello-Rahman pressure control
722 \item Structural optimization: Conjugate gradient method
725 \begin{pspicture}(0,0)(0,0)
726 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
734 C and Si self-interstitial point defects in silicon
741 \begin{minipage}{8cm}
743 \begin{pspicture}(0,0)(7,5)
744 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
747 \item Creation of c-Si simulation volume
748 \item Periodic boundary conditions
749 \item $T=0\text{ K}$, $p=0\text{ bar}$
752 \rput(3.5,2.1){\rnode{insert}{\psframebox{
755 Insertion of interstitial C/Si atoms
758 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
761 Relaxation / structural energy minimization
764 \ncline[]{->}{init}{insert}
765 \ncline[]{->}{insert}{cool}
768 \begin{minipage}{5cm}
769 \includegraphics[width=5cm]{unit_cell_e.eps}\\
772 \begin{minipage}{9cm}
773 \begin{tabular}{l c c}
775 & size [unit cells] & \# atoms\\
777 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
778 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
782 \begin{minipage}{4cm}
783 {\color{red}$\bullet$} Tetrahedral\\
784 {\color{green}$\bullet$} Hexagonal\\
785 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
786 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
787 {\color{cyan}$\bullet$} Bond-centered\\
788 {\color{black}$\bullet$} Vacancy / Substitutional
797 \begin{minipage}{9.5cm}
800 Si self-interstitial point defects in silicon\\
803 \begin{tabular}{l c c c c c}
805 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
807 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
808 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
810 \end{tabular}\\[0.2cm]
812 \begin{minipage}{4.7cm}
813 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
815 \begin{minipage}{4.7cm}
817 {\tiny nearly T $\rightarrow$ T}\\
819 \includegraphics[width=4.7cm]{nhex_tet.ps}
822 \underline{Hexagonal} \hspace{2pt}
823 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
825 \begin{minipage}{2.7cm}
826 $E_{\text{f}}^*=4.48\text{ eV}$\\
827 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
829 \begin{minipage}{0.4cm}
834 \begin{minipage}{2.7cm}
835 $E_{\text{f}}=3.96\text{ eV}$\\
836 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
839 \begin{minipage}{2.9cm}
841 \underline{Vacancy}\\
842 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
847 \begin{minipage}{3.5cm}
850 \underline{\hkl<1 1 0> dumbbell}\\
851 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
852 \underline{Tetrahedral}\\
853 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
854 \underline{\hkl<1 0 0> dumbbell}\\
855 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
867 C interstitial point defects in silicon\\[-0.1cm]
870 \begin{tabular}{l c c c c c c r}
872 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
874 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
875 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
877 \end{tabular}\\[0.1cm]
880 \begin{minipage}{2.7cm}
881 \underline{Hexagonal} \hspace{2pt}
882 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
883 $E_{\text{f}}^*=9.05\text{ eV}$\\
884 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
886 \begin{minipage}{0.4cm}
891 \begin{minipage}{2.7cm}
892 \underline{\hkl<1 0 0>}\\
893 $E_{\text{f}}=3.88\text{ eV}$\\
894 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
897 \begin{minipage}{2cm}
900 \begin{minipage}{3cm}
902 \underline{Tetrahedral}\\
903 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
908 \begin{minipage}{2.7cm}
909 \underline{Bond-centered}\\
910 $E_{\text{f}}^*=5.59\text{ eV}$\\
911 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
913 \begin{minipage}{0.4cm}
918 \begin{minipage}{2.7cm}
919 \underline{\hkl<1 1 0> dumbbell}\\
920 $E_{\text{f}}=5.18\text{ eV}$\\
921 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
924 \begin{minipage}{2cm}
927 \begin{minipage}{3cm}
929 \underline{Substitutional}\\
930 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
941 C \hkl<1 0 0> dumbbell interstitial configuration\\
945 \begin{tabular}{l c c c c c c c c}
947 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
949 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
950 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
952 \end{tabular}\\[0.2cm]
953 \begin{tabular}{l c c c c }
955 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
957 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
958 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
960 \end{tabular}\\[0.2cm]
961 \begin{tabular}{l c c c}
963 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
965 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
966 VASP & 0.109 & -0.065 & 0.174 \\
968 \end{tabular}\\[0.6cm]
971 \begin{minipage}{3.0cm}
973 \underline{Erhart/Albe}
974 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
977 \begin{minipage}{3.0cm}
980 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
984 \begin{picture}(0,0)(-185,10)
985 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
987 \begin{picture}(0,0)(-280,-150)
988 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
991 \begin{pspicture}(0,0)(0,0)
992 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
993 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
994 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
995 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1004 \begin{minipage}{8.5cm}
1007 Bond-centered interstitial configuration\\[-0.1cm]
1010 \begin{minipage}{3.0cm}
1011 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1013 \begin{minipage}{5.2cm}
1015 \item Linear Si-C-Si bond
1016 \item Si: one C \& 3 Si neighbours
1017 \item Spin polarized calculations
1018 \item No saddle point!\\
1025 \begin{minipage}[t]{6.5cm}
1026 \begin{minipage}[t]{1.2cm}
1028 {\tiny sp$^3$}\\[0.8cm]
1029 \underline{${\color{black}\uparrow}$}
1030 \underline{${\color{black}\uparrow}$}
1031 \underline{${\color{black}\uparrow}$}
1032 \underline{${\color{red}\uparrow}$}\\
1035 \begin{minipage}[t]{1.4cm}
1037 {\color{red}M}{\color{blue}O}\\[0.8cm]
1038 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1039 $\sigma_{\text{ab}}$\\[0.5cm]
1040 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1044 \begin{minipage}[t]{1.0cm}
1048 \underline{${\color{white}\uparrow\uparrow}$}
1049 \underline{${\color{white}\uparrow\uparrow}$}\\
1051 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1052 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1056 \begin{minipage}[t]{1.4cm}
1058 {\color{blue}M}{\color{green}O}\\[0.8cm]
1059 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1060 $\sigma_{\text{ab}}$\\[0.5cm]
1061 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1065 \begin{minipage}[t]{1.2cm}
1068 {\tiny sp$^3$}\\[0.8cm]
1069 \underline{${\color{green}\uparrow}$}
1070 \underline{${\color{black}\uparrow}$}
1071 \underline{${\color{black}\uparrow}$}
1072 \underline{${\color{black}\uparrow}$}\\
1080 \begin{minipage}{4.5cm}
1081 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1083 \begin{minipage}{3.5cm}
1084 {\color{gray}$\bullet$} Spin up\\
1085 {\color{green}$\bullet$} Spin down\\
1086 {\color{blue}$\bullet$} Resulting spin up\\
1087 {\color{yellow}$\bullet$} Si atoms\\
1088 {\color{red}$\bullet$} C atom
1093 \begin{minipage}{4.2cm}
1095 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1096 {\color{green}$\Box$} {\tiny unoccupied}\\
1097 {\color{red}$\bullet$} {\tiny occupied}
1106 Migration of the C \hkl<1 0 0> dumbbell interstitial
1111 {\small Investigated pathways}
1113 \begin{minipage}{8.5cm}
1114 \begin{minipage}{8.3cm}
1115 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1116 \begin{minipage}{2.4cm}
1117 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1119 \begin{minipage}{0.4cm}
1122 \begin{minipage}{2.4cm}
1123 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1125 \begin{minipage}{0.4cm}
1128 \begin{minipage}{2.4cm}
1129 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1132 \begin{minipage}{8.3cm}
1133 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1134 \begin{minipage}{2.4cm}
1135 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1137 \begin{minipage}{0.4cm}
1140 \begin{minipage}{2.4cm}
1141 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1143 \begin{minipage}{0.4cm}
1146 \begin{minipage}{2.4cm}
1147 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1150 \begin{minipage}{8.3cm}
1151 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1152 \begin{minipage}{2.4cm}
1153 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1155 \begin{minipage}{0.4cm}
1158 \begin{minipage}{2.4cm}
1159 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1161 \begin{minipage}{0.4cm}
1164 \begin{minipage}{2.4cm}
1165 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1170 \begin{minipage}{4.2cm}
1171 {\small Constrained relaxation\\
1172 technique (CRT) method}\\
1173 \includegraphics[width=4cm]{crt_orig.eps}
1175 \item Constrain diffusing atom
1176 \item Static constraints
1179 {\small Modifications}\\
1180 \includegraphics[width=4cm]{crt_mod.eps}
1182 \item Constrain all atoms
1183 \item Update individual\\
1194 Migration of the C \hkl<1 0 0> dumbbell interstitial
1200 \begin{minipage}{5.9cm}
1202 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1205 \begin{picture}(0,0)(60,0)
1206 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1208 \begin{picture}(0,0)(-5,0)
1209 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1211 \begin{picture}(0,0)(-55,0)
1212 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1214 \begin{picture}(0,0)(12.5,10)
1215 \includegraphics[width=1cm]{110_arrow.eps}
1217 \begin{picture}(0,0)(90,0)
1218 \includegraphics[height=0.9cm]{001_arrow.eps}
1224 \begin{minipage}{0.3cm}
1228 \begin{minipage}{5.9cm}
1230 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1233 \begin{picture}(0,0)(60,0)
1234 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1236 \begin{picture}(0,0)(5,0)
1237 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1239 \begin{picture}(0,0)(-55,0)
1240 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1242 \begin{picture}(0,0)(12.5,10)
1243 \includegraphics[width=1cm]{100_arrow.eps}
1245 \begin{picture}(0,0)(90,0)
1246 \includegraphics[height=0.9cm]{001_arrow.eps}
1256 \begin{minipage}{5.9cm}
1258 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1261 \begin{picture}(0,0)(60,0)
1262 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1264 \begin{picture}(0,0)(10,0)
1265 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1267 \begin{picture}(0,0)(-60,0)
1268 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1270 \begin{picture}(0,0)(12.5,10)
1271 \includegraphics[width=1cm]{100_arrow.eps}
1273 \begin{picture}(0,0)(90,0)
1274 \includegraphics[height=0.9cm]{001_arrow.eps}
1280 \begin{minipage}{0.3cm}
1283 \begin{minipage}{6.5cm}
1286 \item Energetically most favorable path
1289 \item Activation energy: $\approx$ 0.9 eV
1290 \item Experimental values: 0.73 ... 0.87 eV
1292 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1293 \item Reorientation (path 3)
1295 \item More likely composed of two consecutive steps of type 2
1296 \item Experimental values: 0.77 ... 0.88 eV
1298 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1307 Migration of the C \hkl<1 0 0> dumbbell interstitial
1314 \begin{minipage}{6.5cm}
1317 \begin{minipage}[t]{5.9cm}
1319 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1322 \begin{pspicture}(0,0)(0,0)
1323 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1325 \begin{picture}(0,0)(60,-50)
1326 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1328 \begin{picture}(0,0)(5,-50)
1329 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1331 \begin{picture}(0,0)(-55,-50)
1332 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1334 \begin{picture}(0,0)(12.5,-40)
1335 \includegraphics[width=1cm]{110_arrow.eps}
1337 \begin{picture}(0,0)(90,-45)
1338 \includegraphics[height=0.9cm]{001_arrow.eps}
1340 \begin{pspicture}(0,0)(0,0)
1341 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1343 \begin{picture}(0,0)(60,-15)
1344 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1346 \begin{picture}(0,0)(35,-15)
1347 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1349 \begin{picture}(0,0)(-5,-15)
1350 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1352 \begin{picture}(0,0)(-55,-15)
1353 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1355 \begin{picture}(0,0)(12.5,-5)
1356 \includegraphics[width=1cm]{100_arrow.eps}
1358 \begin{picture}(0,0)(90,-15)
1359 \includegraphics[height=0.9cm]{010_arrow.eps}
1365 \begin{minipage}{5.9cm}
1368 \item Lowest activation energy: $\approx$ 2.2 eV
1369 \item 2.4 times higher than VASP
1370 \item Different pathway
1375 \begin{minipage}{6.5cm}
1378 \begin{minipage}{5.9cm}
1380 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1383 %\begin{pspicture}(0,0)(0,0)
1384 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1386 %\begin{picture}(0,0)(60,-5)
1387 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1389 %\begin{picture}(0,0)(0,-5)
1390 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1392 %\begin{picture}(0,0)(-55,-5)
1393 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1395 %\begin{picture}(0,0)(12.5,5)
1396 %\includegraphics[width=1cm]{100_arrow.eps}
1398 %\begin{picture}(0,0)(90,0)
1399 %\includegraphics[height=0.9cm]{001_arrow.eps}
1407 %\begin{minipage}{5.9cm}
1408 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1412 \begin{minipage}{5.9cm}
1413 Transition involving \ci{} \hkl<1 1 0>
1415 \item Bond-centered configuration unstable\\
1416 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1417 \item Transition minima of path 2 \& 3\\
1418 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1419 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1420 \item 2.4 - 3.4 times higher than VASP
1421 \item Rotation of dumbbell orientation
1432 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1442 E_{\text{f}}^{\text{defect combination}}-
1443 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1444 E_{\text{f}}^{\text{2nd defect}}
1450 \begin{tabular}{l c c c c c c}
1452 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1454 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1455 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1456 \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}\\
1457 \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}\\
1458 \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}\\
1459 \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}\\
1461 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1462 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1471 \begin{minipage}[t]{3.8cm}
1472 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1473 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1475 \begin{minipage}[t]{3.5cm}
1476 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1477 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1479 \begin{minipage}[t]{5.5cm}
1481 \item Restricted to VASP simulations
1482 \item $E_{\text{b}}=0$ for isolated non-interacting defects
1483 \item $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1484 \item Stress compensation / increase
1485 \item Most favorable: C clustering
1486 \item Unfavored: antiparallel orientations
1487 \item Indication of energetically favored\\
1492 \begin{picture}(0,0)(-295,-130)
1493 \includegraphics[width=3.5cm]{comb_pos.eps}
1501 Combinations of C-Si \hkl<1 0 0>-type interstitials
1508 Energetically most favorable combinations along \hkl<1 1 0>
1513 \begin{tabular}{l c c c c c c}
1515 & 1 & 2 & 3 & 4 & 5 & 6\\
1517 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1518 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1519 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>\\
1526 \begin{minipage}{7.0cm}
1527 \includegraphics[width=7cm]{db_along_110_cc.ps}
1529 \begin{minipage}{6.0cm}
1532 Interaction proportional to reciprocal cube of C-C distance
1534 Saturation in the immediate vicinity
1545 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1551 \begin{minipage}{3.2cm}
1552 \includegraphics[width=3cm]{sub_110_combo.eps}
1554 \begin{minipage}{7.8cm}
1555 \begin{tabular}{l c c c c c c}
1557 C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1558 \hkl<1 0 1> & \hkl<-1 0 1> \\
1560 1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1561 2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1562 3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1563 4 & \RM{4} & B & D & E & E & D \\
1564 5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1571 \begin{tabular}{l c c c c c c c c c c}
1573 Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1575 $E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1576 $E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1577 $r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1582 \begin{minipage}{6.0cm}
1583 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1585 \begin{minipage}{7cm}
1588 \item IBS: C may displace Si\\
1589 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1591 \hkl<1 1 0>-type $\rightarrow$ favored combination
1592 \renewcommand\labelitemi{$\Rightarrow$}
1593 \item Less favorable than C-Si \hkl<1 0 0> dumbbell\\
1594 ($E_{\text{f}}=3.88\text{ eV}$)
1595 \item Interaction drops quickly to zero\\
1596 (low interaction capture radius)
1605 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1612 \begin{minipage}[t]{3cm}
1613 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1614 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1616 \begin{minipage}[t]{7cm}
1619 Low activation energies\\
1620 High activation energies for reverse processes\\
1622 {\color{blue}C$_{\text{sub}}$ very stable}\\
1626 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1628 {\color{blue}Formation of SiC by successive substitution by C}
1632 \begin{minipage}[t]{3cm}
1633 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1634 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1639 \begin{minipage}{5.9cm}
1640 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1642 \begin{picture}(0,0)(70,0)
1643 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1645 \begin{picture}(0,0)(30,0)
1646 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1648 \begin{picture}(0,0)(-10,0)
1649 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1651 \begin{picture}(0,0)(-48,0)
1652 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1654 \begin{picture}(0,0)(12.5,5)
1655 \includegraphics[width=1cm]{100_arrow.eps}
1657 \begin{picture}(0,0)(97,-10)
1658 \includegraphics[height=0.9cm]{001_arrow.eps}
1664 \begin{minipage}{0.3cm}
1668 \begin{minipage}{5.9cm}
1669 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1671 \begin{picture}(0,0)(60,0)
1672 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1674 \begin{picture}(0,0)(25,0)
1675 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1677 \begin{picture}(0,0)(-20,0)
1678 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1680 \begin{picture}(0,0)(-55,0)
1681 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1683 \begin{picture}(0,0)(12.5,5)
1684 \includegraphics[width=1cm]{100_arrow.eps}
1686 \begin{picture}(0,0)(95,0)
1687 \includegraphics[height=0.9cm]{001_arrow.eps}
1699 Conclusion of defect / migration / combined defect simulations
1708 \item Accurately described by quantum-mechanical simulations
1709 \item Less accurate description by classical potential simulations
1710 \item Underestimated formation energy of \cs{} by classical approach
1711 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1716 \item C migration pathway in Si identified
1717 \item Consistent with reorientation and diffusion experiments
1720 \item Different path and ...
1721 \item overestimated barrier by classical potential calculations
1724 Concerning the precipitation mechanism
1726 \item Agglomeration of C-Si dumbbells energetically favorable
1727 (stress compensation)
1728 \item C-Si indeed favored compared to
1729 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1730 \item Possible low interaction capture radius of
1731 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1732 \item Low barrier for
1733 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1734 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1735 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1738 {\color{blue}Results suggest increased participation of \cs}
1746 Silicon carbide precipitation simulations
1752 \begin{pspicture}(0,0)(12,6.5)
1754 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1757 \item Create c-Si volume
1758 \item Periodc boundary conditions
1759 \item Set requested $T$ and $p=0\text{ bar}$
1760 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1763 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1765 Insertion of C atoms at constant T
1767 \item total simulation volume {\pnode{in1}}
1768 \item volume of minimal SiC precipitate {\pnode{in2}}
1769 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1773 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1775 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1777 \ncline[]{->}{init}{insert}
1778 \ncline[]{->}{insert}{cool}
1779 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1780 \rput(7.8,6){\footnotesize $V_1$}
1781 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1782 \rput(9.2,4.85){\tiny $V_2$}
1783 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1784 \rput(9.55,4.45){\footnotesize $V_3$}
1785 \rput(7.9,3.2){\pnode{ins1}}
1786 \rput(9.22,2.8){\pnode{ins2}}
1787 \rput(11.0,2.4){\pnode{ins3}}
1788 \ncline[]{->}{in1}{ins1}
1789 \ncline[]{->}{in2}{ins2}
1790 \ncline[]{->}{in3}{ins3}
1795 \item Restricted to classical potential simulations
1796 \item $V_2$ and $V_3$ considered due to low diffusion
1797 \item Amount of C atoms: 6000
1798 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1799 \item Simulation volume: $31\times 31\times 31$ unit cells
1808 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1813 \begin{minipage}{6.5cm}
1814 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1816 \begin{minipage}{6.5cm}
1817 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1820 \begin{minipage}{6.5cm}
1821 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1823 \begin{minipage}{6.5cm}
1825 \underline{Low C concentration ($V_1$)}\\
1826 \hkl<1 0 0> C-Si dumbbell dominated structure
1828 \item Si-C bumbs around 0.19 nm
1829 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1830 concatenated dumbbells of various orientation
1831 \item Si-Si NN distance stretched to 0.3 nm
1833 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1834 \underline{High C concentration ($V_2$, $V_3$)}\\
1835 High amount of strongly bound C-C bonds\\
1836 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1837 Only short range order observable\\
1838 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1846 Limitations of molecular dynamics and short range potentials
1853 \underline{Time scale problem of MD}\\[0.2cm]
1854 Minimize integration error\\
1855 $\Rightarrow$ discretization considerably smaller than
1856 reciprocal of fastest vibrational mode\\[0.1cm]
1857 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
1858 $\Rightarrow$ suitable choice of time step:
1859 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
1860 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
1861 Several local minima in energy surface separated by large energy barriers\\
1862 $\Rightarrow$ transition event corresponds to a multiple
1863 of vibrational periods\\
1864 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
1865 infrequent transition events\\[0.1cm]
1866 {\color{blue}Accelerated methods:}
1867 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
1871 \underline{Limitations related to the short range potential}\\[0.2cm]
1872 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
1873 and 2$^{\text{nd}}$ next neighbours\\
1874 $\Rightarrow$ overestimated unphysical high forces of next neighbours
1880 Potential enhanced problem of slow phase space propagation
1885 \underline{Approach to the (twofold) problem}\\[0.2cm]
1886 Increased temperature simulations without TAD corrections\\
1887 (accelerated methods or higher time scales exclusively not sufficient)
1889 \begin{picture}(0,0)(-260,-30)
1891 \begin{minipage}{4.2cm}
1898 \item 3C-SiC also observed for higher T
1899 \item higher T inside sample
1900 \item structural evolution vs.\\
1901 equilibrium properties
1907 \begin{picture}(0,0)(-305,-155)
1909 \begin{minipage}{2.5cm}
1913 thermodynmic sampling
1924 Increased temperature simulations at low C concentration
1929 \begin{minipage}{6.5cm}
1930 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
1932 \begin{minipage}{6.5cm}
1933 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
1936 \begin{minipage}{6.5cm}
1937 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
1939 \begin{minipage}{6.5cm}
1941 \underline{Si-C bonds:}
1943 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
1944 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
1946 \underline{Si-Si bonds:}
1947 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
1948 ($\rightarrow$ 0.325 nm)\\[0.1cm]
1949 \underline{C-C bonds:}
1951 \item C-C next neighbour pairs reduced (mandatory)
1952 \item Peak at 0.3 nm slightly shifted
1954 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
1955 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
1957 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
1959 \item Range [|-$\downarrow$]:
1960 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
1961 with nearby Si$_{\text{I}}$}
1966 \begin{picture}(0,0)(-330,-74)
1969 \begin{minipage}{1.6cm}
1972 stretched SiC\\[-0.1cm]
1984 Increased temperature simulations at high C concentration
1989 \begin{minipage}{6.5cm}
1990 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
1992 \begin{minipage}{6.5cm}
1993 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
1997 Decreasing cut-off artifact\\
1998 High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
1999 $\Rightarrow$ hard to categorize
2005 \begin{minipage}[t]{6.0cm}
2006 0.186 nm: Si-C pairs $\uparrow$\\
2007 (as expected in 3C-SiC)\\[0.2cm]
2008 0.282 nm: Si-C-C\\[0.2cm]
2009 $\approx$0.35 nm: C-Si-Si
2012 \begin{minipage}{0.2cm}
2016 \begin{minipage}[t]{6.0cm}
2017 0.15 nm: C-C pairs $\uparrow$\\
2018 (as expected in graphite/diamond)\\[0.2cm]
2019 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2020 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2027 {\color{red}Amorphous} SiC-like phase remains\\
2028 Slightly sharper peaks
2029 $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics}
2030 due to temperature\\[0.1cm]
2033 Actual SiC precipitation not accessible by MD
2042 Summary and Conclusions
2050 \begin{minipage}{12.9cm}
2055 \item Point defects excellently / fairly well described
2057 \item C$_{\text{sub}}$ drastically underestimated by EA
2058 \item EA predicts correct ground state:
2059 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2060 \item Identified migration path explaining
2061 diffusion and reorientation experiments by DFT
2062 \item EA fails to describe \ci{} migration:
2063 Wrong path \& overestimated barrier
2065 \item Combinations of defects
2067 \item Agglomeration of point defects energetically favorable
2068 by compensation of stress
2069 \item Formation of C-C unlikely
2070 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2071 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2072 Low barrier (\unit[0.77]{eV}) \& low capture radius
2081 \begin{minipage}[t]{12.9cm}
2082 \underline{Pecipitation simulations}
2084 \item High C concentration $\rightarrow$ amorphous SiC like phase
2085 \item Problem of potential enhanced slow phase space propagation
2086 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2087 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2088 \item High T necessary to simulate IBS conditions (far from equilibrium)
2089 \item Precipitation by successive agglomeration of \cs (epitaxy)
2090 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2091 (stretched SiC, interface)
2110 \underline{Augsburg}
2112 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2113 \item Ralf Utermann (EDV)
2116 \underline{Helsinki}
2118 \item Prof. K. Nordlund (MD)
2123 \item Bayerische Forschungsstiftung (financial support)
2126 \underline{Paderborn}
2128 \item Prof. J. Lindner (SiC)
2129 \item Prof. G. Schmidt (DFT + financial support)
2130 \item Dr. E. Rauls (DFT + SiC)
2131 \item Dr. S. Sanna (VASP)
2138 \bf Thank you for your attention!