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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
42 \input{seminar.bg2} % Unofficial bugs corrections
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57 \extraslideheight{10in}
62 % specify width and height
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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}
82 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
85 \newrgbcolor{si-yellow}{.6 .6 0}
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88 \newrgbcolor{hlbb}{0.825 0.88 0.968}
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}{}}
98 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
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
404 Supposed precipitation mechanism of SiC in Si
411 \begin{minipage}{3.8cm}
412 Si \& SiC lattice structure\\[0.2cm]
413 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
417 \begin{minipage}{3.8cm}
419 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
423 \begin{minipage}{3.8cm}
425 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
429 \begin{minipage}{4cm}
431 C-Si dimers (dumbbells)\\[-0.1cm]
432 on Si interstitial sites
436 \begin{minipage}{4.2cm}
438 Agglomeration of C-Si dumbbells\\[-0.1cm]
439 $\Rightarrow$ dark contrasts
443 \begin{minipage}{4cm}
445 Precipitation of 3C-SiC in Si\\[-0.1cm]
446 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
447 \& release of Si self-interstitials
451 \begin{minipage}{3.8cm}
453 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
457 \begin{minipage}{3.8cm}
459 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
463 \begin{minipage}{3.8cm}
465 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
469 \begin{pspicture}(0,0)(0,0)
470 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
471 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
472 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
473 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
474 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
475 $4a_{\text{Si}}=5a_{\text{SiC}}$
477 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
478 \hkl(h k l) planes match
480 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
490 Supposed precipitation mechanism of SiC in Si
497 \begin{minipage}{3.8cm}
498 Si \& SiC lattice structure\\[0.2cm]
499 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
503 \begin{minipage}{3.8cm}
505 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
509 \begin{minipage}{3.8cm}
511 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
515 \begin{minipage}{4cm}
517 C-Si dimers (dumbbells)\\[-0.1cm]
518 on Si interstitial sites
522 \begin{minipage}{4.2cm}
524 Agglomeration of C-Si dumbbells\\[-0.1cm]
525 $\Rightarrow$ dark contrasts
529 \begin{minipage}{4cm}
531 Precipitation of 3C-SiC in Si\\[-0.1cm]
532 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
533 \& release of Si self-interstitials
537 \begin{minipage}{3.8cm}
539 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
543 \begin{minipage}{3.8cm}
545 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
549 \begin{minipage}{3.8cm}
551 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
555 \begin{pspicture}(0,0)(0,0)
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559 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
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561 $4a_{\text{Si}}=5a_{\text{SiC}}$
563 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
564 \hkl(h k l) planes match
566 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
569 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
570 \begin{minipage}{10cm}
572 {\color{red}\bf Controversial views}
574 \item Implantations at high T (Nejim et al.)
576 \item Topotactic transformation based on \cs
577 \item \si{} as supply reacting with further C in cleared volume
579 \item Annealing behavior (Serre et al.)
581 \item Room temperature implants $\rightarrow$ highly mobile C
582 \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
583 (indicate stable \cs{} configurations)
585 \item Strained silicon \& Si/SiC heterostructures
587 \item Coherent SiC precipitates (tensile strain)
588 \item Incoherent SiC (strain relaxation)
600 Molecular dynamics (MD) simulations
609 \item Microscopic description of N particle system
610 \item Analytical interaction potential
611 \item Numerical integration using Newtons equation of motion\\
612 as a propagation rule in 6N-dimensional phase space
613 \item Observables obtained by time and/or ensemble averages
615 {\bf Details of the simulation:}
617 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
618 \item Ensemble: NpT (isothermal-isobaric)
620 \item Berendsen thermostat:
621 $\tau_{\text{T}}=100\text{ fs}$
622 \item Berendsen barostat:\\
623 $\tau_{\text{P}}=100\text{ fs}$,
624 $\beta^{-1}=100\text{ GPa}$
626 \item Erhart/Albe potential: Tersoff-like bond order potential
629 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
630 \pot_{ij} = {\color{red}f_C(r_{ij})}
631 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
635 \begin{picture}(0,0)(-230,-30)
636 \includegraphics[width=5cm]{tersoff_angle.eps}
644 Density functional theory (DFT) calculations
649 Basic ingredients necessary for DFT
652 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
654 \item ... uniquely determines the ground state potential
656 \item ... minimizes the systems total energy
658 \item \underline{Born-Oppenheimer}
659 - $N$ moving electrons in an external potential of static nuclei
661 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
662 +\sum_i^N V_{\text{ext}}(r_i)
663 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
665 \item \underline{Effective potential}
666 - averaged electrostatic potential \& exchange and correlation
668 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
671 \item \underline{Kohn-Sham system}
672 - Schr\"odinger equation of N non-interacting particles
674 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
679 n(r)=\sum_i^N|\Phi_i(r)|^2
681 \item \underline{Self-consistent solution}\\
682 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
683 which in turn depends on $n(r)$
684 \item \underline{Variational principle}
685 - minimize total energy with respect to $n(r)$
693 Density functional theory (DFT) calculations
700 Details of applied DFT calculations in this work
703 \item \underline{Exchange correlation functional}
704 - approximations for the inhomogeneous electron gas
706 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
707 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
709 \item \underline{Plane wave basis set}
710 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
713 \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}}
714 \qquad ({\color{blue}300\text{ eV}})
716 \item \underline{Brillouin zone sampling} -
717 {\color{blue}$\Gamma$-point only} calculations
718 \item \underline{Pseudo potential}
719 - consider only the valence electrons
720 \item \underline{Code} - VASP 4.6
725 MD and structural optimization
728 \item MD integration: Gear predictor corrector algorithm
729 \item Pressure control: Parrinello-Rahman pressure control
730 \item Structural optimization: Conjugate gradient method
733 \begin{pspicture}(0,0)(0,0)
734 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
742 C and Si self-interstitial point defects in silicon
749 \begin{minipage}{8cm}
751 \begin{pspicture}(0,0)(7,5)
752 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
755 \item Creation of c-Si simulation volume
756 \item Periodic boundary conditions
757 \item $T=0\text{ K}$, $p=0\text{ bar}$
760 \rput(3.5,2.1){\rnode{insert}{\psframebox{
763 Insertion of interstitial C/Si atoms
766 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
769 Relaxation / structural energy minimization
772 \ncline[]{->}{init}{insert}
773 \ncline[]{->}{insert}{cool}
776 \begin{minipage}{5cm}
777 \includegraphics[width=5cm]{unit_cell_e.eps}\\
780 \begin{minipage}{9cm}
781 \begin{tabular}{l c c}
783 & size [unit cells] & \# atoms\\
785 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
786 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
790 \begin{minipage}{4cm}
791 {\color{red}$\bullet$} Tetrahedral\\
792 {\color{green}$\bullet$} Hexagonal\\
793 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
794 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
795 {\color{cyan}$\bullet$} Bond-centered\\
796 {\color{black}$\bullet$} Vacancy / Substitutional
805 \begin{minipage}{9.5cm}
808 Si self-interstitial point defects in silicon\\
811 \begin{tabular}{l c c c c c}
813 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
815 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
816 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
818 \end{tabular}\\[0.2cm]
820 \begin{minipage}{4.7cm}
821 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
823 \begin{minipage}{4.7cm}
825 {\tiny nearly T $\rightarrow$ T}\\
827 \includegraphics[width=4.7cm]{nhex_tet.ps}
830 \underline{Hexagonal} \hspace{2pt}
831 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
833 \begin{minipage}{2.7cm}
834 $E_{\text{f}}^*=4.48\text{ eV}$\\
835 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
837 \begin{minipage}{0.4cm}
842 \begin{minipage}{2.7cm}
843 $E_{\text{f}}=3.96\text{ eV}$\\
844 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
847 \begin{minipage}{2.9cm}
849 \underline{Vacancy}\\
850 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
855 \begin{minipage}{3.5cm}
858 \underline{\hkl<1 1 0> dumbbell}\\
859 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
860 \underline{Tetrahedral}\\
861 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
862 \underline{\hkl<1 0 0> dumbbell}\\
863 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
875 C interstitial point defects in silicon\\[-0.1cm]
878 \begin{tabular}{l c c c c c c r}
880 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
882 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
883 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
885 \end{tabular}\\[0.1cm]
888 \begin{minipage}{2.7cm}
889 \underline{Hexagonal} \hspace{2pt}
890 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
891 $E_{\text{f}}^*=9.05\text{ eV}$\\
892 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
894 \begin{minipage}{0.4cm}
899 \begin{minipage}{2.7cm}
900 \underline{\hkl<1 0 0>}\\
901 $E_{\text{f}}=3.88\text{ eV}$\\
902 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
905 \begin{minipage}{2cm}
908 \begin{minipage}{3cm}
910 \underline{Tetrahedral}\\
911 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
916 \begin{minipage}{2.7cm}
917 \underline{Bond-centered}\\
918 $E_{\text{f}}^*=5.59\text{ eV}$\\
919 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
921 \begin{minipage}{0.4cm}
926 \begin{minipage}{2.7cm}
927 \underline{\hkl<1 1 0> dumbbell}\\
928 $E_{\text{f}}=5.18\text{ eV}$\\
929 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
932 \begin{minipage}{2cm}
935 \begin{minipage}{3cm}
937 \underline{Substitutional}\\
938 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
949 C \hkl<1 0 0> dumbbell interstitial configuration\\
953 \begin{tabular}{l c c c c c c c c}
955 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
957 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
958 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
960 \end{tabular}\\[0.2cm]
961 \begin{tabular}{l c c c c }
963 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
965 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
966 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
968 \end{tabular}\\[0.2cm]
969 \begin{tabular}{l c c c}
971 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
973 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
974 VASP & 0.109 & -0.065 & 0.174 \\
976 \end{tabular}\\[0.6cm]
979 \begin{minipage}{3.0cm}
981 \underline{Erhart/Albe}
982 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
985 \begin{minipage}{3.0cm}
988 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
992 \begin{picture}(0,0)(-185,10)
993 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
995 \begin{picture}(0,0)(-280,-150)
996 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
999 \begin{pspicture}(0,0)(0,0)
1000 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1001 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1002 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1003 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1012 \begin{minipage}{8.5cm}
1015 Bond-centered interstitial configuration\\[-0.1cm]
1018 \begin{minipage}{3.0cm}
1019 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1021 \begin{minipage}{5.2cm}
1023 \item Linear Si-C-Si bond
1024 \item Si: one C \& 3 Si neighbours
1025 \item Spin polarized calculations
1026 \item No saddle point!\\
1033 \begin{minipage}[t]{6.5cm}
1034 \begin{minipage}[t]{1.2cm}
1036 {\tiny sp$^3$}\\[0.8cm]
1037 \underline{${\color{black}\uparrow}$}
1038 \underline{${\color{black}\uparrow}$}
1039 \underline{${\color{black}\uparrow}$}
1040 \underline{${\color{red}\uparrow}$}\\
1043 \begin{minipage}[t]{1.4cm}
1045 {\color{red}M}{\color{blue}O}\\[0.8cm]
1046 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1047 $\sigma_{\text{ab}}$\\[0.5cm]
1048 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1052 \begin{minipage}[t]{1.0cm}
1056 \underline{${\color{white}\uparrow\uparrow}$}
1057 \underline{${\color{white}\uparrow\uparrow}$}\\
1059 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1060 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1064 \begin{minipage}[t]{1.4cm}
1066 {\color{blue}M}{\color{green}O}\\[0.8cm]
1067 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1068 $\sigma_{\text{ab}}$\\[0.5cm]
1069 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1073 \begin{minipage}[t]{1.2cm}
1076 {\tiny sp$^3$}\\[0.8cm]
1077 \underline{${\color{green}\uparrow}$}
1078 \underline{${\color{black}\uparrow}$}
1079 \underline{${\color{black}\uparrow}$}
1080 \underline{${\color{black}\uparrow}$}\\
1088 \begin{minipage}{4.5cm}
1089 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1091 \begin{minipage}{3.5cm}
1092 {\color{gray}$\bullet$} Spin up\\
1093 {\color{green}$\bullet$} Spin down\\
1094 {\color{blue}$\bullet$} Resulting spin up\\
1095 {\color{yellow}$\bullet$} Si atoms\\
1096 {\color{red}$\bullet$} C atom
1101 \begin{minipage}{4.2cm}
1103 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1104 {\color{green}$\Box$} {\tiny unoccupied}\\
1105 {\color{red}$\bullet$} {\tiny occupied}
1114 Migration of the C \hkl<1 0 0> dumbbell interstitial
1119 {\small Investigated pathways}
1121 \begin{minipage}{8.5cm}
1122 \begin{minipage}{8.3cm}
1123 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1124 \begin{minipage}{2.4cm}
1125 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1127 \begin{minipage}{0.4cm}
1130 \begin{minipage}{2.4cm}
1131 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1133 \begin{minipage}{0.4cm}
1136 \begin{minipage}{2.4cm}
1137 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1140 \begin{minipage}{8.3cm}
1141 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1142 \begin{minipage}{2.4cm}
1143 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1145 \begin{minipage}{0.4cm}
1148 \begin{minipage}{2.4cm}
1149 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1151 \begin{minipage}{0.4cm}
1154 \begin{minipage}{2.4cm}
1155 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1158 \begin{minipage}{8.3cm}
1159 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1160 \begin{minipage}{2.4cm}
1161 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1163 \begin{minipage}{0.4cm}
1166 \begin{minipage}{2.4cm}
1167 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1169 \begin{minipage}{0.4cm}
1172 \begin{minipage}{2.4cm}
1173 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1178 \begin{minipage}{4.2cm}
1179 {\small Constrained relaxation\\
1180 technique (CRT) method}\\
1181 \includegraphics[width=4cm]{crt_orig.eps}
1183 \item Constrain diffusing atom
1184 \item Static constraints
1187 {\small Modifications}\\
1188 \includegraphics[width=4cm]{crt_mod.eps}
1190 \item Constrain all atoms
1191 \item Update individual\\
1202 Migration of the C \hkl<1 0 0> dumbbell interstitial
1208 \begin{minipage}{5.9cm}
1210 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1213 \begin{picture}(0,0)(60,0)
1214 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1216 \begin{picture}(0,0)(-5,0)
1217 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1219 \begin{picture}(0,0)(-55,0)
1220 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1222 \begin{picture}(0,0)(12.5,10)
1223 \includegraphics[width=1cm]{110_arrow.eps}
1225 \begin{picture}(0,0)(90,0)
1226 \includegraphics[height=0.9cm]{001_arrow.eps}
1232 \begin{minipage}{0.3cm}
1236 \begin{minipage}{5.9cm}
1238 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1241 \begin{picture}(0,0)(60,0)
1242 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1244 \begin{picture}(0,0)(5,0)
1245 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1247 \begin{picture}(0,0)(-55,0)
1248 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1250 \begin{picture}(0,0)(12.5,10)
1251 \includegraphics[width=1cm]{100_arrow.eps}
1253 \begin{picture}(0,0)(90,0)
1254 \includegraphics[height=0.9cm]{001_arrow.eps}
1264 \begin{minipage}{5.9cm}
1266 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1269 \begin{picture}(0,0)(60,0)
1270 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1272 \begin{picture}(0,0)(10,0)
1273 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1275 \begin{picture}(0,0)(-60,0)
1276 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1278 \begin{picture}(0,0)(12.5,10)
1279 \includegraphics[width=1cm]{100_arrow.eps}
1281 \begin{picture}(0,0)(90,0)
1282 \includegraphics[height=0.9cm]{001_arrow.eps}
1288 \begin{minipage}{0.3cm}
1291 \begin{minipage}{6.5cm}
1294 \item Energetically most favorable path
1297 \item Activation energy: $\approx$ 0.9 eV
1298 \item Experimental values: 0.73 ... 0.87 eV
1300 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1301 \item Reorientation (path 3)
1303 \item More likely composed of two consecutive steps of type 2
1304 \item Experimental values: 0.77 ... 0.88 eV
1306 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1315 Migration of the C \hkl<1 0 0> dumbbell interstitial
1322 \begin{minipage}{6.5cm}
1325 \begin{minipage}[t]{5.9cm}
1327 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1330 \begin{pspicture}(0,0)(0,0)
1331 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1333 \begin{picture}(0,0)(60,-50)
1334 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1336 \begin{picture}(0,0)(5,-50)
1337 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1339 \begin{picture}(0,0)(-55,-50)
1340 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1342 \begin{picture}(0,0)(12.5,-40)
1343 \includegraphics[width=1cm]{110_arrow.eps}
1345 \begin{picture}(0,0)(90,-45)
1346 \includegraphics[height=0.9cm]{001_arrow.eps}
1348 \begin{pspicture}(0,0)(0,0)
1349 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1351 \begin{picture}(0,0)(60,-15)
1352 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1354 \begin{picture}(0,0)(35,-15)
1355 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1357 \begin{picture}(0,0)(-5,-15)
1358 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1360 \begin{picture}(0,0)(-55,-15)
1361 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1363 \begin{picture}(0,0)(12.5,-5)
1364 \includegraphics[width=1cm]{100_arrow.eps}
1366 \begin{picture}(0,0)(90,-15)
1367 \includegraphics[height=0.9cm]{010_arrow.eps}
1373 \begin{minipage}{5.9cm}
1376 \item Lowest activation energy: $\approx$ 2.2 eV
1377 \item 2.4 times higher than VASP
1378 \item Different pathway
1383 \begin{minipage}{6.5cm}
1386 \begin{minipage}{5.9cm}
1388 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1391 %\begin{pspicture}(0,0)(0,0)
1392 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1394 %\begin{picture}(0,0)(60,-5)
1395 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1397 %\begin{picture}(0,0)(0,-5)
1398 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1400 %\begin{picture}(0,0)(-55,-5)
1401 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1403 %\begin{picture}(0,0)(12.5,5)
1404 %\includegraphics[width=1cm]{100_arrow.eps}
1406 %\begin{picture}(0,0)(90,0)
1407 %\includegraphics[height=0.9cm]{001_arrow.eps}
1415 %\begin{minipage}{5.9cm}
1416 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1420 \begin{minipage}{5.9cm}
1421 Transition involving \ci{} \hkl<1 1 0>
1423 \item Bond-centered configuration unstable\\
1424 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1425 \item Transition minima of path 2 \& 3\\
1426 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1427 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1428 \item 2.4 - 3.4 times higher than VASP
1429 \item Rotation of dumbbell orientation
1433 {\color{blue}Overestimated diffusion barrier}
1444 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1454 E_{\text{f}}^{\text{defect combination}}-
1455 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1456 E_{\text{f}}^{\text{2nd defect}}
1462 \begin{tabular}{l c c c c c c}
1464 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1466 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1467 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1468 \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}\\
1469 \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}\\
1470 \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}\\
1471 \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}\\
1473 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1474 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1483 \begin{minipage}[t]{3.8cm}
1484 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1485 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1487 \begin{minipage}[t]{3.5cm}
1488 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1489 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1491 \begin{minipage}[t]{5.5cm}
1493 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1494 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1495 \item Stress compensation / increase
1496 \item Unfavored: antiparallel orientations
1497 \item Indication of energetically favored\\
1499 \item Most favorable: C clustering
1500 \item However: High barrier ($>4\,\text{eV}$)
1501 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1506 \begin{picture}(0,0)(-295,-130)
1507 \includegraphics[width=3.5cm]{comb_pos.eps}
1515 Combinations of C-Si \hkl<1 0 0>-type interstitials
1522 Energetically most favorable combinations along \hkl<1 1 0>
1527 \begin{tabular}{l c c c c c c}
1529 & 1 & 2 & 3 & 4 & 5 & 6\\
1531 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1532 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1533 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>\\
1540 \begin{minipage}{7.0cm}
1541 \includegraphics[width=7cm]{db_along_110_cc.ps}
1543 \begin{minipage}{6.0cm}
1545 \item Interaction proportional to reciprocal cube of C-C distance
1546 \item Saturation in the immediate vicinity
1547 \renewcommand\labelitemi{$\Rightarrow$}
1548 \item Agglomeration of \ci{} expected
1549 \item Absence of C clustering
1553 Consisten with initial precipitation model
1565 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1571 %\begin{minipage}{3.2cm}
1572 %\includegraphics[width=3cm]{sub_110_combo.eps}
1574 %\begin{minipage}{7.8cm}
1575 %\begin{tabular}{l c c c c c c}
1577 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1578 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1580 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1581 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1582 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1583 %4 & \RM{4} & B & D & E & E & D \\
1584 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1591 %\begin{tabular}{l c c c c c c c c c c}
1593 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1595 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1596 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1597 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1602 \begin{minipage}{6.0cm}
1603 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1605 \begin{minipage}{7cm}
1608 \item IBS: C may displace Si\\
1609 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1611 \hkl<1 1 0>-type $\rightarrow$ favored combination
1612 \renewcommand\labelitemi{$\Rightarrow$}
1613 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1614 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1615 \item Interaction drops quickly to zero\\
1616 $\rightarrow$ low capture radius
1620 IBS process far from equilibrium\\
1621 \cs{} \& \si{} instead of thermodynamic ground state
1626 \begin{minipage}{6.5cm}
1627 \includegraphics[width=6.0cm]{162-097.ps}
1629 \item Low migration barrier
1632 \begin{minipage}{6.5cm}
1634 Ab initio MD at \degc{900}\\
1635 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1636 $t=\unit[2230]{fs}$\\
1637 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1641 Contribution of entropy to structural formation
1650 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1657 \begin{minipage}[t]{3cm}
1658 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1659 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1661 \begin{minipage}[t]{7cm}
1664 Low activation energies\\
1665 High activation energies for reverse processes\\
1667 {\color{blue}C$_{\text{sub}}$ very stable}\\
1671 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1673 {\color{blue}Formation of SiC by successive substitution by C}
1677 \begin{minipage}[t]{3cm}
1678 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1679 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1684 \begin{minipage}{5.9cm}
1685 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1687 \begin{picture}(0,0)(70,0)
1688 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1690 \begin{picture}(0,0)(30,0)
1691 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1693 \begin{picture}(0,0)(-10,0)
1694 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1696 \begin{picture}(0,0)(-48,0)
1697 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1699 \begin{picture}(0,0)(12.5,5)
1700 \includegraphics[width=1cm]{100_arrow.eps}
1702 \begin{picture}(0,0)(97,-10)
1703 \includegraphics[height=0.9cm]{001_arrow.eps}
1709 \begin{minipage}{0.3cm}
1713 \begin{minipage}{5.9cm}
1714 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1716 \begin{picture}(0,0)(60,0)
1717 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1719 \begin{picture}(0,0)(25,0)
1720 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1722 \begin{picture}(0,0)(-20,0)
1723 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1725 \begin{picture}(0,0)(-55,0)
1726 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1728 \begin{picture}(0,0)(12.5,5)
1729 \includegraphics[width=1cm]{100_arrow.eps}
1731 \begin{picture}(0,0)(95,0)
1732 \includegraphics[height=0.9cm]{001_arrow.eps}
1744 Conclusion of defect / migration / combined defect simulations
1753 \item Accurately described by quantum-mechanical simulations
1754 \item Less accurate description by classical potential simulations
1755 \item Underestimated formation energy of \cs{} by classical approach
1756 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1761 \item C migration pathway in Si identified
1762 \item Consistent with reorientation and diffusion experiments
1765 \item Different path and ...
1766 \item overestimated barrier by classical potential calculations
1769 Concerning the precipitation mechanism
1771 \item Agglomeration of C-Si dumbbells energetically favorable
1772 (stress compensation)
1773 \item C-Si indeed favored compared to
1774 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1775 \item Possible low interaction capture radius of
1776 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1777 \item Low barrier for
1778 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1779 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1780 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1783 {\color{blue}Results suggest increased participation of \cs}
1791 Silicon carbide precipitation simulations
1797 \begin{pspicture}(0,0)(12,6.5)
1799 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1802 \item Create c-Si volume
1803 \item Periodc boundary conditions
1804 \item Set requested $T$ and $p=0\text{ bar}$
1805 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1808 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1810 Insertion of C atoms at constant T
1812 \item total simulation volume {\pnode{in1}}
1813 \item volume of minimal SiC precipitate {\pnode{in2}}
1814 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1818 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1820 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1822 \ncline[]{->}{init}{insert}
1823 \ncline[]{->}{insert}{cool}
1824 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1825 \rput(7.8,6){\footnotesize $V_1$}
1826 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1827 \rput(9.2,4.85){\tiny $V_2$}
1828 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1829 \rput(9.55,4.45){\footnotesize $V_3$}
1830 \rput(7.9,3.2){\pnode{ins1}}
1831 \rput(9.22,2.8){\pnode{ins2}}
1832 \rput(11.0,2.4){\pnode{ins3}}
1833 \ncline[]{->}{in1}{ins1}
1834 \ncline[]{->}{in2}{ins2}
1835 \ncline[]{->}{in3}{ins3}
1840 \item Restricted to classical potential simulations
1841 \item $V_2$ and $V_3$ considered due to low diffusion
1842 \item Amount of C atoms: 6000
1843 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1844 \item Simulation volume: $31\times 31\times 31$ unit cells
1853 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1858 \begin{minipage}{6.5cm}
1859 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1861 \begin{minipage}{6.5cm}
1862 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1865 \begin{minipage}{6.5cm}
1866 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1868 \begin{minipage}{6.5cm}
1870 \underline{Low C concentration ($V_1$)}\\
1871 \hkl<1 0 0> C-Si dumbbell dominated structure
1873 \item Si-C bumbs around 0.19 nm
1874 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1875 concatenated dumbbells of various orientation
1876 \item Si-Si NN distance stretched to 0.3 nm
1878 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1879 \underline{High C concentration ($V_2$, $V_3$)}\\
1880 High amount of strongly bound C-C bonds\\
1881 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1882 Only short range order observable\\
1883 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1891 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1896 \begin{minipage}{6.5cm}
1897 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1899 \begin{minipage}{6.5cm}
1900 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1903 \begin{minipage}{6.5cm}
1904 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1906 \begin{minipage}{6.5cm}
1908 \underline{Low C concentration ($V_1$)}\\
1909 \hkl<1 0 0> C-Si dumbbell dominated structure
1911 \item Si-C bumbs around 0.19 nm
1912 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1913 concatenated dumbbells of various orientation
1914 \item Si-Si NN distance stretched to 0.3 nm
1916 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1917 \underline{High C concentration ($V_2$, $V_3$)}\\
1918 High amount of strongly bound C-C bonds\\
1919 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1920 Only short range order observable\\
1921 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1924 \begin{pspicture}(0,0)(0,0)
1925 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
1926 \begin{minipage}{10cm}
1928 {\color{red}\bf 3C-SiC formation fails to appear}
1930 \item Low C concentration simulations
1932 \item Formation of \ci{} indeed occurs
1933 \item Agllomeration not observed
1935 \item High C concentration simulations
1937 \item Amorphous SiC-like structure\\
1938 (not expected at prevailing temperatures)
1939 \item Rearrangement and transition into 3C-SiC structure missing
1951 Limitations of molecular dynamics and short range potentials
1958 \underline{Time scale problem of MD}\\[0.2cm]
1959 Minimize integration error\\
1960 $\Rightarrow$ discretization considerably smaller than
1961 reciprocal of fastest vibrational mode\\[0.1cm]
1962 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
1963 $\Rightarrow$ suitable choice of time step:
1964 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
1965 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
1966 Several local minima in energy surface separated by large energy barriers\\
1967 $\Rightarrow$ transition event corresponds to a multiple
1968 of vibrational periods\\
1969 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
1970 infrequent transition events\\[0.1cm]
1971 {\color{blue}Accelerated methods:}
1972 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
1976 \underline{Limitations related to the short range potential}\\[0.2cm]
1977 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
1978 and 2$^{\text{nd}}$ next neighbours\\
1979 $\Rightarrow$ overestimated unphysical high forces of next neighbours
1985 Potential enhanced problem of slow phase space propagation
1990 \underline{Approach to the (twofold) problem}\\[0.2cm]
1991 Increased temperature simulations without TAD corrections\\
1992 (accelerated methods or higher time scales exclusively not sufficient)
1994 \begin{picture}(0,0)(-260,-30)
1996 \begin{minipage}{4.2cm}
2003 \item 3C-SiC also observed for higher T
2004 \item higher T inside sample
2005 \item structural evolution vs.\\
2006 equilibrium properties
2012 \begin{picture}(0,0)(-305,-155)
2014 \begin{minipage}{2.5cm}
2018 thermodynmic sampling
2029 Increased temperature simulations at low C concentration
2034 \begin{minipage}{6.5cm}
2035 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2037 \begin{minipage}{6.5cm}
2038 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2041 \begin{minipage}{6.5cm}
2042 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2044 \begin{minipage}{6.5cm}
2046 \underline{Si-C bonds:}
2048 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2049 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2051 \underline{Si-Si bonds:}
2052 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2053 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2054 \underline{C-C bonds:}
2056 \item C-C next neighbour pairs reduced (mandatory)
2057 \item Peak at 0.3 nm slightly shifted
2059 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2060 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2062 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2064 \item Range [|-$\downarrow$]:
2065 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2066 with nearby Si$_{\text{I}}$}
2071 \begin{picture}(0,0)(-330,-74)
2074 \begin{minipage}{1.6cm}
2077 stretched SiC\\[-0.1cm]
2089 Increased temperature simulations at low C concentration
2094 \begin{minipage}{6.5cm}
2095 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2097 \begin{minipage}{6.5cm}
2098 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2101 \begin{minipage}{6.5cm}
2102 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2104 \begin{minipage}{6.5cm}
2106 \underline{Si-C bonds:}
2108 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2109 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2111 \underline{Si-Si bonds:}
2112 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2113 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2114 \underline{C-C bonds:}
2116 \item C-C next neighbour pairs reduced (mandatory)
2117 \item Peak at 0.3 nm slightly shifted
2119 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2120 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2122 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2124 \item Range [|-$\downarrow$]:
2125 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2126 with nearby Si$_{\text{I}}$}
2131 %\begin{picture}(0,0)(-330,-74)
2134 %\begin{minipage}{1.6cm}
2137 %stretched SiC\\[-0.1cm]
2144 \begin{pspicture}(0,0)(0,0)
2145 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2146 \begin{minipage}{10cm}
2148 {\color{blue}\bf Stretched SiC in c-Si}
2150 \item Consistent to precipitation model involving \cs{}
2151 \item Explains annealing behavior of high/low T C implants
2153 \item Low T: highly mobiel \ci{}
2154 \item High T: stable configurations of \cs{}
2157 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2158 $\Rightarrow$ Precipitation mechanism involving \cs{}
2168 Increased temperature simulations at high C concentration
2173 \begin{minipage}{6.5cm}
2174 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2176 \begin{minipage}{6.5cm}
2177 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2185 \begin{minipage}[t]{6.0cm}
2186 0.186 nm: Si-C pairs $\uparrow$\\
2187 (as expected in 3C-SiC)\\[0.2cm]
2188 0.282 nm: Si-C-C\\[0.2cm]
2189 $\approx$0.35 nm: C-Si-Si
2192 \begin{minipage}{0.2cm}
2196 \begin{minipage}[t]{6.0cm}
2197 0.15 nm: C-C pairs $\uparrow$\\
2198 (as expected in graphite/diamond)\\[0.2cm]
2199 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2200 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2205 \item Decreasing cut-off artifact
2206 \item {\color{red}Amorphous} SiC-like phase remains
2207 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2208 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2217 High C \& small $V$ \& short $t$
2220 Slow restructuring due to strong C-C bonds
2223 High C \& low T implants
2234 Summary and Conclusions
2242 \begin{minipage}[t]{12.9cm}
2243 \underline{Pecipitation simulations}
2245 \item High C concentration $\rightarrow$ amorphous SiC like phase
2246 \item Problem of potential enhanced slow phase space propagation
2247 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2248 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2249 \item High T necessary to simulate IBS conditions (far from equilibrium)
2250 \item Precipitation by successive agglomeration of \cs (epitaxy)
2251 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2252 (stretched SiC, interface)
2260 \begin{minipage}{12.9cm}
2265 \item Point defects excellently / fairly well described
2267 \item C$_{\text{sub}}$ drastically underestimated by EA
2268 \item EA predicts correct ground state:
2269 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2270 \item Identified migration path explaining
2271 diffusion and reorientation experiments by DFT
2272 \item EA fails to describe \ci{} migration:
2273 Wrong path \& overestimated barrier
2275 \item Combinations of defects
2277 \item Agglomeration of point defects energetically favorable
2278 by compensation of stress
2279 \item Formation of C-C unlikely
2280 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2281 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2282 Low barrier (\unit[0.77]{eV}) \& low capture radius
2290 \framebox{Precipitation by successive agglomeration of \cs{}}
2308 \underline{Augsburg}
2310 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2311 \item Ralf Utermann (EDV)
2314 \underline{Helsinki}
2316 \item Prof. K. Nordlund (MD)
2321 \item Bayerische Forschungsstiftung (financial support)
2324 \underline{Paderborn}
2326 \item Prof. J. Lindner (SiC)
2327 \item Prof. G. Schmidt (DFT + financial support)
2328 \item Dr. E. Rauls (DFT + SiC)
2329 \item Dr. S. Sanna (VASP)
2336 \bf Thank you for your attention!