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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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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{}}}
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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 studies\\[0.2cm]
114 \textsc{Frank Zirkelbach}
118 Application talk at the Max Planck Institute for Solid State Research
122 Stuttgart, November 2011
132 % Phase diagram of the C/Si system\\
137 \begin{minipage}{7cm}
138 \includegraphics[width=6.5cm]{si-c_phase.eps}
141 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
144 \begin{pspicture}(0,0)(0,0)
145 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
148 \begin{minipage}{6cm}
149 {\bf Phase diagram of the C/Si system}\\[0.2cm]
150 {\color{blue}Stoichiometric composition}
152 \item only chemical stable compound
153 \item wide band gap semiconductor\\
154 \underline{silicon carbide}, SiC
160 % motivation / properties / applications of silicon carbide
166 \begin{pspicture}(0,0)(13.5,5)
168 \psframe*[linecolor=hb](0,0)(13.5,5)
170 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.5,1)(7,1)(7,3)(5.5,3)
171 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.75,0.5)(8,2)(8,2)(6.75,3.5)
173 \rput[lt](0.2,4.6){\color{gray}PROPERTIES}
175 \rput[lt](0.5,4){wide band gap}
176 \rput[lt](0.5,3.5){high electric breakdown field}
177 \rput[lt](0.5,3){good electron mobility}
178 \rput[lt](0.5,2.5){high electron saturation drift velocity}
179 \rput[lt](0.5,2){high thermal conductivity}
181 \rput[lt](0.5,1.5){hard and mechanically stable}
182 \rput[lt](0.5,1){chemically inert}
184 \rput[lt](0.5,0.5){radiation hardness}
186 \rput[rt](13.3,4.6){\color{gray}APPLICATIONS}
188 \rput[rt](13,3.85){high-temperature, high power}
189 \rput[rt](13,3.5){and high-frequency}
190 \rput[rt](13,3.15){electronic and optoelectronic devices}
192 \rput[rt](13,2.35){material suitable for extreme conditions}
193 \rput[rt](13,2){microelectromechanical systems}
194 \rput[rt](13,1.65){abrasives, cutting tools, heating elements}
196 \rput[rt](13,0.85){first wall reactor material, detectors}
197 \rput[rt](13,0.5){and electronic devices for space}
201 \begin{picture}(0,0)(0,-162)
202 \includegraphics[height=2.0cm]{3C_SiC_bs.eps}
204 \begin{picture}(0,0)(-130,-162)
205 \includegraphics[height=2.0cm]{nasa_600c_led.eps}
207 \begin{picture}(0,0)(-295,-162)
208 \includegraphics[height=2.0cm]{6h-sic_3c-sic.eps}
211 \begin{picture}(0,0)(5,65)
212 \includegraphics[height=2.8cm]{sic_switch.eps}
214 \begin{picture}(0,0)(-140,65)
215 \includegraphics[height=2.8cm]{infineon_schottky.eps}
217 \begin{picture}(0,0)(-260,65)
218 \includegraphics[height=2.8cm]{ise_99.eps}
235 \begin{tabular}{l c c c c c c}
237 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
239 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
240 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
241 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
242 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
243 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
244 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
245 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
253 \begin{picture}(0,0)(-160,-155)
254 \includegraphics[width=7cm]{polytypes.eps}
256 \begin{picture}(0,0)(-10,-185)
257 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
259 \begin{picture}(0,0)(-10,-175)
260 {\tiny cubic (twist)}
262 \begin{picture}(0,0)(-60,-175)
263 {\tiny hexagonal (no twist)}
265 \begin{pspicture}(0,0)(0,0)
266 \psellipse[linecolor=green](5.7,3.03)(0.4,0.5)
268 \begin{pspicture}(0,0)(0,0)
269 \psellipse[linecolor=green](5.6,1.68)(0.4,0.2)
271 \begin{pspicture}(0,0)(0,0)
272 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
282 Fabrication of silicon carbide
289 SiC - \emph{Born from the stars, perfected on earth.}
295 Conventional thin film SiC growth:
297 \item \underline{Sublimation growth using the modified Lely method}
299 \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
300 \item Surrounded by polycrystalline SiC in a graphite crucible\\
301 at $T=2100-2400 \, ^{\circ} \text{C}$
302 \item Deposition of supersaturated vapor on cooler seed crystal
304 \item \underline{Homoepitaxial growth using CVD}
306 \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
307 \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
308 \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
310 \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
312 \item Two steps: carbonization and growth
313 \item $T=650-1050 \, ^{\circ} \text{C}$
314 \item SiC/Si lattice mismatch $\approx$ 20 \%
315 \item Quality and size not yet sufficient
319 \begin{picture}(0,0)(-280,-65)
320 \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
322 \begin{picture}(0,0)(-280,-55)
323 \begin{minipage}{5cm}
325 NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
330 \begin{picture}(0,0)(-265,-150)
331 \includegraphics[width=2.4cm]{m_lely.eps}
333 \begin{picture}(0,0)(-333,-175)
334 \begin{minipage}{5cm}
340 5. Insulation\\[-7pt]
345 \begin{picture}(0,0)(-230,-35)
347 {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
350 \begin{picture}(0,0)(-230,-10)
352 \begin{minipage}{3cm}
353 {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
370 \item Implantation of C in Si --- Overview of experimental observations
371 \item Utilized simulation techniques and modeled problems
373 \item {\color{blue}Diploma thesis}\\
374 \underline{Monte Carlo} simulations
375 modeling the selforganization process
376 leading to periodic arrays of nanometric amorphous SiC
378 \item {\color{blue}Doctoral studies}\\
379 Classical potential \underline{molecular dynamics} simulations
381 \underline{Density functional theory} calculations
383 \ldots on defects and SiC precipitation in Si
385 \item Summary / Conclusion / Outlook
397 Fabrication of silicon carbide
402 Alternative approach:
403 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
405 \item \underline{Implantation step 1}\\
406 180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
407 $\Rightarrow$ box-like distribution of equally sized
408 and epitactically oriented SiC precipitates
410 \item \underline{Implantation step 2}\\
411 180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
412 $\Rightarrow$ destruction of SiC nanocrystals
413 in growing amorphous interface layers
414 \item \underline{Annealing}\\
415 $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
416 $\Rightarrow$ homogeneous, stoichiometric SiC layer
417 with sharp interfaces
420 \begin{minipage}{6.3cm}
421 \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
423 XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
427 \begin{minipage}{6.3cm}
430 Precipitation mechanism not yet fully understood!
432 \renewcommand\labelitemi{$\Rightarrow$}
434 \underline{Understanding the SiC precipitation}
436 \item significant technological progress in SiC thin film formation
437 \item perspectives for processes relying upon prevention of SiC precipitation
449 Supposed precipitation mechanism of SiC in Si
456 \begin{minipage}{3.8cm}
457 Si \& SiC lattice structure\\[0.2cm]
458 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
462 \begin{minipage}{3.8cm}
464 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
468 \begin{minipage}{3.8cm}
470 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
474 \begin{minipage}{4cm}
476 C-Si dimers (dumbbells)\\[-0.1cm]
477 on Si interstitial sites
481 \begin{minipage}{4.2cm}
483 Agglomeration of C-Si dumbbells\\[-0.1cm]
484 $\Rightarrow$ dark contrasts
488 \begin{minipage}{4cm}
490 Precipitation of 3C-SiC in Si\\[-0.1cm]
491 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
492 \& release of Si self-interstitials
496 \begin{minipage}{3.8cm}
498 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
502 \begin{minipage}{3.8cm}
504 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
508 \begin{minipage}{3.8cm}
510 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
514 \begin{pspicture}(0,0)(0,0)
515 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
516 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
517 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
518 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
519 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
520 $4a_{\text{Si}}=5a_{\text{SiC}}$
522 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
523 \hkl(h k l) planes match
525 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
535 Supposed precipitation mechanism of SiC in Si
542 \begin{minipage}{3.8cm}
543 Si \& SiC lattice structure\\[0.2cm]
544 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
548 \begin{minipage}{3.8cm}
550 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
554 \begin{minipage}{3.8cm}
556 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
560 \begin{minipage}{4cm}
562 C-Si dimers (dumbbells)\\[-0.1cm]
563 on Si interstitial sites
567 \begin{minipage}{4.2cm}
569 Agglomeration of C-Si dumbbells\\[-0.1cm]
570 $\Rightarrow$ dark contrasts
574 \begin{minipage}{4cm}
576 Precipitation of 3C-SiC in Si\\[-0.1cm]
577 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
578 \& release of Si self-interstitials
582 \begin{minipage}{3.8cm}
584 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
588 \begin{minipage}{3.8cm}
590 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
594 \begin{minipage}{3.8cm}
596 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
600 \begin{pspicture}(0,0)(0,0)
601 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
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603 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
604 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
605 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
606 $4a_{\text{Si}}=5a_{\text{SiC}}$
608 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
609 \hkl(h k l) planes match
611 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
614 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
615 \begin{minipage}{10cm}
617 {\color{red}\bf Controversial views}
619 \item Implantations at high T (Nejim et al.)
621 \item Topotactic transformation based on \cs
622 \item \si{} as supply reacting with further C in cleared volume
624 \item Annealing behavior (Serre et al.)
626 \item Room temperature implants $\rightarrow$ highly mobile C
627 \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
628 (indicate stable \cs{} configurations)
630 \item Strained silicon \& Si/SiC heterostructures
632 \item Coherent SiC precipitates (tensile strain)
633 \item Incoherent SiC (strain relaxation)
645 Molecular dynamics (MD) simulations
654 \item Microscopic description of N particle system
655 \item Analytical interaction potential
656 \item Numerical integration using Newtons equation of motion\\
657 as a propagation rule in 6N-dimensional phase space
658 \item Observables obtained by time and/or ensemble averages
660 {\bf Details of the simulation:}
662 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
663 \item Ensemble: NpT (isothermal-isobaric)
665 \item Berendsen thermostat:
666 $\tau_{\text{T}}=100\text{ fs}$
667 \item Berendsen barostat:\\
668 $\tau_{\text{P}}=100\text{ fs}$,
669 $\beta^{-1}=100\text{ GPa}$
671 \item Erhart/Albe potential: Tersoff-like bond order potential
674 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
675 \pot_{ij} = {\color{red}f_C(r_{ij})}
676 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
680 \begin{picture}(0,0)(-230,-30)
681 \includegraphics[width=5cm]{tersoff_angle.eps}
689 Density functional theory (DFT) calculations
694 Basic ingredients necessary for DFT
697 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
699 \item ... uniquely determines the ground state potential
701 \item ... minimizes the systems total energy
703 \item \underline{Born-Oppenheimer}
704 - $N$ moving electrons in an external potential of static nuclei
706 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
707 +\sum_i^N V_{\text{ext}}(r_i)
708 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
710 \item \underline{Effective potential}
711 - averaged electrostatic potential \& exchange and correlation
713 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
716 \item \underline{Kohn-Sham system}
717 - Schr\"odinger equation of N non-interacting particles
719 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
724 n(r)=\sum_i^N|\Phi_i(r)|^2
726 \item \underline{Self-consistent solution}\\
727 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
728 which in turn depends on $n(r)$
729 \item \underline{Variational principle}
730 - minimize total energy with respect to $n(r)$
738 Density functional theory (DFT) calculations
745 Details of applied DFT calculations in this work
748 \item \underline{Exchange correlation functional}
749 - approximations for the inhomogeneous electron gas
751 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
752 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
754 \item \underline{Plane wave basis set}
755 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
758 \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}}
759 \qquad ({\color{blue}300\text{ eV}})
761 \item \underline{Brillouin zone sampling} -
762 {\color{blue}$\Gamma$-point only} calculations
763 \item \underline{Pseudo potential}
764 - consider only the valence electrons
765 \item \underline{Code} - VASP 4.6
770 MD and structural optimization
773 \item MD integration: Gear predictor corrector algorithm
774 \item Pressure control: Parrinello-Rahman pressure control
775 \item Structural optimization: Conjugate gradient method
778 \begin{pspicture}(0,0)(0,0)
779 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
787 C and Si self-interstitial point defects in silicon
794 \begin{minipage}{8cm}
796 \begin{pspicture}(0,0)(7,5)
797 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
800 \item Creation of c-Si simulation volume
801 \item Periodic boundary conditions
802 \item $T=0\text{ K}$, $p=0\text{ bar}$
805 \rput(3.5,2.1){\rnode{insert}{\psframebox{
808 Insertion of interstitial C/Si atoms
811 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
814 Relaxation / structural energy minimization
817 \ncline[]{->}{init}{insert}
818 \ncline[]{->}{insert}{cool}
821 \begin{minipage}{5cm}
822 \includegraphics[width=5cm]{unit_cell_e.eps}\\
825 \begin{minipage}{9cm}
826 \begin{tabular}{l c c}
828 & size [unit cells] & \# atoms\\
830 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
831 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
835 \begin{minipage}{4cm}
836 {\color{red}$\bullet$} Tetrahedral\\
837 {\color{green}$\bullet$} Hexagonal\\
838 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
839 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
840 {\color{cyan}$\bullet$} Bond-centered\\
841 {\color{black}$\bullet$} Vacancy / Substitutional
850 \begin{minipage}{9.5cm}
853 Si self-interstitial point defects in silicon\\
856 \begin{tabular}{l c c c c c}
858 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
860 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
861 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
863 \end{tabular}\\[0.2cm]
865 \begin{minipage}{4.7cm}
866 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
868 \begin{minipage}{4.7cm}
870 {\tiny nearly T $\rightarrow$ T}\\
872 \includegraphics[width=4.7cm]{nhex_tet.ps}
875 \underline{Hexagonal} \hspace{2pt}
876 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
878 \begin{minipage}{2.7cm}
879 $E_{\text{f}}^*=4.48\text{ eV}$\\
880 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
882 \begin{minipage}{0.4cm}
887 \begin{minipage}{2.7cm}
888 $E_{\text{f}}=3.96\text{ eV}$\\
889 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
892 \begin{minipage}{2.9cm}
894 \underline{Vacancy}\\
895 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
900 \begin{minipage}{3.5cm}
903 \underline{\hkl<1 1 0> dumbbell}\\
904 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
905 \underline{Tetrahedral}\\
906 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
907 \underline{\hkl<1 0 0> dumbbell}\\
908 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
920 C interstitial point defects in silicon\\[-0.1cm]
923 \begin{tabular}{l c c c c c c r}
925 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
927 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
928 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
930 \end{tabular}\\[0.1cm]
933 \begin{minipage}{2.7cm}
934 \underline{Hexagonal} \hspace{2pt}
935 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
936 $E_{\text{f}}^*=9.05\text{ eV}$\\
937 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
939 \begin{minipage}{0.4cm}
944 \begin{minipage}{2.7cm}
945 \underline{\hkl<1 0 0>}\\
946 $E_{\text{f}}=3.88\text{ eV}$\\
947 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
950 \begin{minipage}{2cm}
953 \begin{minipage}{3cm}
955 \underline{Tetrahedral}\\
956 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
961 \begin{minipage}{2.7cm}
962 \underline{Bond-centered}\\
963 $E_{\text{f}}^*=5.59\text{ eV}$\\
964 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
966 \begin{minipage}{0.4cm}
971 \begin{minipage}{2.7cm}
972 \underline{\hkl<1 1 0> dumbbell}\\
973 $E_{\text{f}}=5.18\text{ eV}$\\
974 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
977 \begin{minipage}{2cm}
980 \begin{minipage}{3cm}
982 \underline{Substitutional}\\
983 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
994 C \hkl<1 0 0> dumbbell interstitial configuration\\
998 \begin{tabular}{l c c c c c c c c}
1000 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1002 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1003 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1005 \end{tabular}\\[0.2cm]
1006 \begin{tabular}{l c c c c }
1008 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1010 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1011 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1013 \end{tabular}\\[0.2cm]
1014 \begin{tabular}{l c c c}
1016 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1018 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1019 VASP & 0.109 & -0.065 & 0.174 \\
1021 \end{tabular}\\[0.6cm]
1024 \begin{minipage}{3.0cm}
1026 \underline{Erhart/Albe}
1027 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1030 \begin{minipage}{3.0cm}
1033 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1037 \begin{picture}(0,0)(-185,10)
1038 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1040 \begin{picture}(0,0)(-280,-150)
1041 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1044 \begin{pspicture}(0,0)(0,0)
1045 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1046 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1047 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1048 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1057 \begin{minipage}{8.5cm}
1060 Bond-centered interstitial configuration\\[-0.1cm]
1063 \begin{minipage}{3.0cm}
1064 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1066 \begin{minipage}{5.2cm}
1068 \item Linear Si-C-Si bond
1069 \item Si: one C \& 3 Si neighbours
1070 \item Spin polarized calculations
1071 \item No saddle point!\\
1078 \begin{minipage}[t]{6.5cm}
1079 \begin{minipage}[t]{1.2cm}
1081 {\tiny sp$^3$}\\[0.8cm]
1082 \underline{${\color{black}\uparrow}$}
1083 \underline{${\color{black}\uparrow}$}
1084 \underline{${\color{black}\uparrow}$}
1085 \underline{${\color{red}\uparrow}$}\\
1088 \begin{minipage}[t]{1.4cm}
1090 {\color{red}M}{\color{blue}O}\\[0.8cm]
1091 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1092 $\sigma_{\text{ab}}$\\[0.5cm]
1093 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1097 \begin{minipage}[t]{1.0cm}
1101 \underline{${\color{white}\uparrow\uparrow}$}
1102 \underline{${\color{white}\uparrow\uparrow}$}\\
1104 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1105 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1109 \begin{minipage}[t]{1.4cm}
1111 {\color{blue}M}{\color{green}O}\\[0.8cm]
1112 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1113 $\sigma_{\text{ab}}$\\[0.5cm]
1114 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1118 \begin{minipage}[t]{1.2cm}
1121 {\tiny sp$^3$}\\[0.8cm]
1122 \underline{${\color{green}\uparrow}$}
1123 \underline{${\color{black}\uparrow}$}
1124 \underline{${\color{black}\uparrow}$}
1125 \underline{${\color{black}\uparrow}$}\\
1133 \begin{minipage}{4.5cm}
1134 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1136 \begin{minipage}{3.5cm}
1137 {\color{gray}$\bullet$} Spin up\\
1138 {\color{green}$\bullet$} Spin down\\
1139 {\color{blue}$\bullet$} Resulting spin up\\
1140 {\color{yellow}$\bullet$} Si atoms\\
1141 {\color{red}$\bullet$} C atom
1146 \begin{minipage}{4.2cm}
1148 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1149 {\color{green}$\Box$} {\tiny unoccupied}\\
1150 {\color{red}$\bullet$} {\tiny occupied}
1159 Migration of the C \hkl<1 0 0> dumbbell interstitial
1164 {\small Investigated pathways}
1166 \begin{minipage}{8.5cm}
1167 \begin{minipage}{8.3cm}
1168 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1169 \begin{minipage}{2.4cm}
1170 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1172 \begin{minipage}{0.4cm}
1175 \begin{minipage}{2.4cm}
1176 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1178 \begin{minipage}{0.4cm}
1181 \begin{minipage}{2.4cm}
1182 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1185 \begin{minipage}{8.3cm}
1186 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1187 \begin{minipage}{2.4cm}
1188 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1190 \begin{minipage}{0.4cm}
1193 \begin{minipage}{2.4cm}
1194 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1196 \begin{minipage}{0.4cm}
1199 \begin{minipage}{2.4cm}
1200 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1203 \begin{minipage}{8.3cm}
1204 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1205 \begin{minipage}{2.4cm}
1206 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1208 \begin{minipage}{0.4cm}
1211 \begin{minipage}{2.4cm}
1212 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1214 \begin{minipage}{0.4cm}
1217 \begin{minipage}{2.4cm}
1218 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1223 \begin{minipage}{4.2cm}
1224 {\small Constrained relaxation\\
1225 technique (CRT) method}\\
1226 \includegraphics[width=4cm]{crt_orig.eps}
1228 \item Constrain diffusing atom
1229 \item Static constraints
1232 {\small Modifications}\\
1233 \includegraphics[width=4cm]{crt_mod.eps}
1235 \item Constrain all atoms
1236 \item Update individual\\
1247 Migration of the C \hkl<1 0 0> dumbbell interstitial
1253 \begin{minipage}{5.9cm}
1255 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1258 \begin{picture}(0,0)(60,0)
1259 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1261 \begin{picture}(0,0)(-5,0)
1262 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1264 \begin{picture}(0,0)(-55,0)
1265 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1267 \begin{picture}(0,0)(12.5,10)
1268 \includegraphics[width=1cm]{110_arrow.eps}
1270 \begin{picture}(0,0)(90,0)
1271 \includegraphics[height=0.9cm]{001_arrow.eps}
1277 \begin{minipage}{0.3cm}
1281 \begin{minipage}{5.9cm}
1283 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1286 \begin{picture}(0,0)(60,0)
1287 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1289 \begin{picture}(0,0)(5,0)
1290 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1292 \begin{picture}(0,0)(-55,0)
1293 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1295 \begin{picture}(0,0)(12.5,10)
1296 \includegraphics[width=1cm]{100_arrow.eps}
1298 \begin{picture}(0,0)(90,0)
1299 \includegraphics[height=0.9cm]{001_arrow.eps}
1309 \begin{minipage}{5.9cm}
1311 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1314 \begin{picture}(0,0)(60,0)
1315 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1317 \begin{picture}(0,0)(10,0)
1318 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1320 \begin{picture}(0,0)(-60,0)
1321 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1323 \begin{picture}(0,0)(12.5,10)
1324 \includegraphics[width=1cm]{100_arrow.eps}
1326 \begin{picture}(0,0)(90,0)
1327 \includegraphics[height=0.9cm]{001_arrow.eps}
1333 \begin{minipage}{0.3cm}
1336 \begin{minipage}{6.5cm}
1339 \item Energetically most favorable path
1342 \item Activation energy: $\approx$ 0.9 eV
1343 \item Experimental values: 0.73 ... 0.87 eV
1345 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1346 \item Reorientation (path 3)
1348 \item More likely composed of two consecutive steps of type 2
1349 \item Experimental values: 0.77 ... 0.88 eV
1351 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1360 Migration of the C \hkl<1 0 0> dumbbell interstitial
1367 \begin{minipage}{6.5cm}
1370 \begin{minipage}[t]{5.9cm}
1372 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1375 \begin{pspicture}(0,0)(0,0)
1376 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1378 \begin{picture}(0,0)(60,-50)
1379 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1381 \begin{picture}(0,0)(5,-50)
1382 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1384 \begin{picture}(0,0)(-55,-50)
1385 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1387 \begin{picture}(0,0)(12.5,-40)
1388 \includegraphics[width=1cm]{110_arrow.eps}
1390 \begin{picture}(0,0)(90,-45)
1391 \includegraphics[height=0.9cm]{001_arrow.eps}
1393 \begin{pspicture}(0,0)(0,0)
1394 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1396 \begin{picture}(0,0)(60,-15)
1397 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1399 \begin{picture}(0,0)(35,-15)
1400 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1402 \begin{picture}(0,0)(-5,-15)
1403 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1405 \begin{picture}(0,0)(-55,-15)
1406 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1408 \begin{picture}(0,0)(12.5,-5)
1409 \includegraphics[width=1cm]{100_arrow.eps}
1411 \begin{picture}(0,0)(90,-15)
1412 \includegraphics[height=0.9cm]{010_arrow.eps}
1418 \begin{minipage}{5.9cm}
1421 \item Lowest activation energy: $\approx$ 2.2 eV
1422 \item 2.4 times higher than VASP
1423 \item Different pathway
1428 \begin{minipage}{6.5cm}
1431 \begin{minipage}{5.9cm}
1433 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1436 %\begin{pspicture}(0,0)(0,0)
1437 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1439 %\begin{picture}(0,0)(60,-5)
1440 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1442 %\begin{picture}(0,0)(0,-5)
1443 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1445 %\begin{picture}(0,0)(-55,-5)
1446 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1448 %\begin{picture}(0,0)(12.5,5)
1449 %\includegraphics[width=1cm]{100_arrow.eps}
1451 %\begin{picture}(0,0)(90,0)
1452 %\includegraphics[height=0.9cm]{001_arrow.eps}
1460 %\begin{minipage}{5.9cm}
1461 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1465 \begin{minipage}{5.9cm}
1466 Transition involving \ci{} \hkl<1 1 0>
1468 \item Bond-centered configuration unstable\\
1469 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1470 \item Transition minima of path 2 \& 3\\
1471 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1472 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1473 \item 2.4 - 3.4 times higher than VASP
1474 \item Rotation of dumbbell orientation
1478 {\color{blue}Overestimated diffusion barrier}
1489 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1499 E_{\text{f}}^{\text{defect combination}}-
1500 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1501 E_{\text{f}}^{\text{2nd defect}}
1507 \begin{tabular}{l c c c c c c}
1509 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1511 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1512 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1513 \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}\\
1514 \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}\\
1515 \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}\\
1516 \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}\\
1518 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1519 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1528 \begin{minipage}[t]{3.8cm}
1529 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1530 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1532 \begin{minipage}[t]{3.5cm}
1533 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1534 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1536 \begin{minipage}[t]{5.5cm}
1538 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1539 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1540 \item Stress compensation / increase
1541 \item Unfavored: antiparallel orientations
1542 \item Indication of energetically favored\\
1544 \item Most favorable: C clustering
1545 \item However: High barrier ($>4\,\text{eV}$)
1546 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1551 \begin{picture}(0,0)(-295,-130)
1552 \includegraphics[width=3.5cm]{comb_pos.eps}
1560 Combinations of C-Si \hkl<1 0 0>-type interstitials
1567 Energetically most favorable combinations along \hkl<1 1 0>
1572 \begin{tabular}{l c c c c c c}
1574 & 1 & 2 & 3 & 4 & 5 & 6\\
1576 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1577 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1578 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>\\
1585 \begin{minipage}{7.0cm}
1586 \includegraphics[width=7cm]{db_along_110_cc.ps}
1588 \begin{minipage}{6.0cm}
1590 \item Interaction proportional to reciprocal cube of C-C distance
1591 \item Saturation in the immediate vicinity
1592 \renewcommand\labelitemi{$\Rightarrow$}
1593 \item Agglomeration of \ci{} expected
1594 \item Absence of C clustering
1598 Consisten with initial precipitation model
1610 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1616 %\begin{minipage}{3.2cm}
1617 %\includegraphics[width=3cm]{sub_110_combo.eps}
1619 %\begin{minipage}{7.8cm}
1620 %\begin{tabular}{l c c c c c c}
1622 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1623 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1625 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1626 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1627 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1628 %4 & \RM{4} & B & D & E & E & D \\
1629 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1636 %\begin{tabular}{l c c c c c c c c c c}
1638 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1640 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1641 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1642 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1647 \begin{minipage}{6.0cm}
1648 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1650 \begin{minipage}{7cm}
1653 \item IBS: C may displace Si\\
1654 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1656 \hkl<1 1 0>-type $\rightarrow$ favored combination
1657 \renewcommand\labelitemi{$\Rightarrow$}
1658 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1659 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1660 \item Interaction drops quickly to zero\\
1661 $\rightarrow$ low capture radius
1665 IBS process far from equilibrium\\
1666 \cs{} \& \si{} instead of thermodynamic ground state
1671 \begin{minipage}{6.5cm}
1672 \includegraphics[width=6.0cm]{162-097.ps}
1674 \item Low migration barrier
1677 \begin{minipage}{6.5cm}
1679 Ab initio MD at \degc{900}\\
1680 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1681 $t=\unit[2230]{fs}$\\
1682 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1686 Contribution of entropy to structural formation
1695 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1702 \begin{minipage}[t]{3cm}
1703 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1704 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1706 \begin{minipage}[t]{7cm}
1709 Low activation energies\\
1710 High activation energies for reverse processes\\
1712 {\color{blue}C$_{\text{sub}}$ very stable}\\
1716 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1718 {\color{blue}Formation of SiC by successive substitution by C}
1722 \begin{minipage}[t]{3cm}
1723 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1724 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1729 \begin{minipage}{5.9cm}
1730 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1732 \begin{picture}(0,0)(70,0)
1733 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1735 \begin{picture}(0,0)(30,0)
1736 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1738 \begin{picture}(0,0)(-10,0)
1739 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1741 \begin{picture}(0,0)(-48,0)
1742 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1744 \begin{picture}(0,0)(12.5,5)
1745 \includegraphics[width=1cm]{100_arrow.eps}
1747 \begin{picture}(0,0)(97,-10)
1748 \includegraphics[height=0.9cm]{001_arrow.eps}
1754 \begin{minipage}{0.3cm}
1758 \begin{minipage}{5.9cm}
1759 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1761 \begin{picture}(0,0)(60,0)
1762 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1764 \begin{picture}(0,0)(25,0)
1765 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1767 \begin{picture}(0,0)(-20,0)
1768 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1770 \begin{picture}(0,0)(-55,0)
1771 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1773 \begin{picture}(0,0)(12.5,5)
1774 \includegraphics[width=1cm]{100_arrow.eps}
1776 \begin{picture}(0,0)(95,0)
1777 \includegraphics[height=0.9cm]{001_arrow.eps}
1789 Conclusion of defect / migration / combined defect simulations
1798 \item Accurately described by quantum-mechanical simulations
1799 \item Less accurate description by classical potential simulations
1800 \item Underestimated formation energy of \cs{} by classical approach
1801 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1806 \item C migration pathway in Si identified
1807 \item Consistent with reorientation and diffusion experiments
1810 \item Different path and ...
1811 \item overestimated barrier by classical potential calculations
1814 Concerning the precipitation mechanism
1816 \item Agglomeration of C-Si dumbbells energetically favorable
1817 (stress compensation)
1818 \item C-Si indeed favored compared to
1819 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1820 \item Possible low interaction capture radius of
1821 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1822 \item Low barrier for
1823 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1824 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1825 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1828 {\color{blue}Results suggest increased participation of \cs}
1836 Silicon carbide precipitation simulations
1842 \begin{pspicture}(0,0)(12,6.5)
1844 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1847 \item Create c-Si volume
1848 \item Periodc boundary conditions
1849 \item Set requested $T$ and $p=0\text{ bar}$
1850 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1853 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1855 Insertion of C atoms at constant T
1857 \item total simulation volume {\pnode{in1}}
1858 \item volume of minimal SiC precipitate {\pnode{in2}}
1859 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1863 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1865 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1867 \ncline[]{->}{init}{insert}
1868 \ncline[]{->}{insert}{cool}
1869 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1870 \rput(7.8,6){\footnotesize $V_1$}
1871 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1872 \rput(9.2,4.85){\tiny $V_2$}
1873 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1874 \rput(9.55,4.45){\footnotesize $V_3$}
1875 \rput(7.9,3.2){\pnode{ins1}}
1876 \rput(9.22,2.8){\pnode{ins2}}
1877 \rput(11.0,2.4){\pnode{ins3}}
1878 \ncline[]{->}{in1}{ins1}
1879 \ncline[]{->}{in2}{ins2}
1880 \ncline[]{->}{in3}{ins3}
1885 \item Restricted to classical potential simulations
1886 \item $V_2$ and $V_3$ considered due to low diffusion
1887 \item Amount of C atoms: 6000
1888 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1889 \item Simulation volume: $31\times 31\times 31$ unit cells
1898 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1903 \begin{minipage}{6.5cm}
1904 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1906 \begin{minipage}{6.5cm}
1907 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1910 \begin{minipage}{6.5cm}
1911 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1913 \begin{minipage}{6.5cm}
1915 \underline{Low C concentration ($V_1$)}\\
1916 \hkl<1 0 0> C-Si dumbbell dominated structure
1918 \item Si-C bumbs around 0.19 nm
1919 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1920 concatenated dumbbells of various orientation
1921 \item Si-Si NN distance stretched to 0.3 nm
1923 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1924 \underline{High C concentration ($V_2$, $V_3$)}\\
1925 High amount of strongly bound C-C bonds\\
1926 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1927 Only short range order observable\\
1928 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1936 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1941 \begin{minipage}{6.5cm}
1942 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1944 \begin{minipage}{6.5cm}
1945 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1948 \begin{minipage}{6.5cm}
1949 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1951 \begin{minipage}{6.5cm}
1953 \underline{Low C concentration ($V_1$)}\\
1954 \hkl<1 0 0> C-Si dumbbell dominated structure
1956 \item Si-C bumbs around 0.19 nm
1957 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1958 concatenated dumbbells of various orientation
1959 \item Si-Si NN distance stretched to 0.3 nm
1961 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1962 \underline{High C concentration ($V_2$, $V_3$)}\\
1963 High amount of strongly bound C-C bonds\\
1964 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1965 Only short range order observable\\
1966 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1969 \begin{pspicture}(0,0)(0,0)
1970 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
1971 \begin{minipage}{10cm}
1973 {\color{red}\bf 3C-SiC formation fails to appear}
1975 \item Low C concentration simulations
1977 \item Formation of \ci{} indeed occurs
1978 \item Agllomeration not observed
1980 \item High C concentration simulations
1982 \item Amorphous SiC-like structure\\
1983 (not expected at prevailing temperatures)
1984 \item Rearrangement and transition into 3C-SiC structure missing
1996 Limitations of molecular dynamics and short range potentials
2003 \underline{Time scale problem of MD}\\[0.2cm]
2004 Minimize integration error\\
2005 $\Rightarrow$ discretization considerably smaller than
2006 reciprocal of fastest vibrational mode\\[0.1cm]
2007 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2008 $\Rightarrow$ suitable choice of time step:
2009 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2010 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2011 Several local minima in energy surface separated by large energy barriers\\
2012 $\Rightarrow$ transition event corresponds to a multiple
2013 of vibrational periods\\
2014 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2015 infrequent transition events\\[0.1cm]
2016 {\color{blue}Accelerated methods:}
2017 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2021 \underline{Limitations related to the short range potential}\\[0.2cm]
2022 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2023 and 2$^{\text{nd}}$ next neighbours\\
2024 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2030 Potential enhanced problem of slow phase space propagation
2035 \underline{Approach to the (twofold) problem}\\[0.2cm]
2036 Increased temperature simulations without TAD corrections\\
2037 (accelerated methods or higher time scales exclusively not sufficient)
2039 \begin{picture}(0,0)(-260,-30)
2041 \begin{minipage}{4.2cm}
2048 \item 3C-SiC also observed for higher T
2049 \item higher T inside sample
2050 \item structural evolution vs.\\
2051 equilibrium properties
2057 \begin{picture}(0,0)(-305,-155)
2059 \begin{minipage}{2.5cm}
2063 thermodynmic sampling
2074 Increased temperature simulations at low C concentration
2079 \begin{minipage}{6.5cm}
2080 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2082 \begin{minipage}{6.5cm}
2083 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2086 \begin{minipage}{6.5cm}
2087 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2089 \begin{minipage}{6.5cm}
2091 \underline{Si-C bonds:}
2093 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2094 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2096 \underline{Si-Si bonds:}
2097 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2098 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2099 \underline{C-C bonds:}
2101 \item C-C next neighbour pairs reduced (mandatory)
2102 \item Peak at 0.3 nm slightly shifted
2104 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2105 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2107 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2109 \item Range [|-$\downarrow$]:
2110 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2111 with nearby Si$_{\text{I}}$}
2116 \begin{picture}(0,0)(-330,-74)
2119 \begin{minipage}{1.6cm}
2122 stretched SiC\\[-0.1cm]
2134 Increased temperature simulations at low C concentration
2139 \begin{minipage}{6.5cm}
2140 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2142 \begin{minipage}{6.5cm}
2143 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2146 \begin{minipage}{6.5cm}
2147 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2149 \begin{minipage}{6.5cm}
2151 \underline{Si-C bonds:}
2153 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2154 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2156 \underline{Si-Si bonds:}
2157 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2158 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2159 \underline{C-C bonds:}
2161 \item C-C next neighbour pairs reduced (mandatory)
2162 \item Peak at 0.3 nm slightly shifted
2164 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2165 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2167 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2169 \item Range [|-$\downarrow$]:
2170 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2171 with nearby Si$_{\text{I}}$}
2176 %\begin{picture}(0,0)(-330,-74)
2179 %\begin{minipage}{1.6cm}
2182 %stretched SiC\\[-0.1cm]
2189 \begin{pspicture}(0,0)(0,0)
2190 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2191 \begin{minipage}{10cm}
2193 {\color{blue}\bf Stretched SiC in c-Si}
2195 \item Consistent to precipitation model involving \cs{}
2196 \item Explains annealing behavior of high/low T C implants
2198 \item Low T: highly mobiel \ci{}
2199 \item High T: stable configurations of \cs{}
2202 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2203 $\Rightarrow$ Precipitation mechanism involving \cs{}
2213 Increased temperature simulations at high C concentration
2218 \begin{minipage}{6.5cm}
2219 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2221 \begin{minipage}{6.5cm}
2222 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2230 \begin{minipage}[t]{6.0cm}
2231 0.186 nm: Si-C pairs $\uparrow$\\
2232 (as expected in 3C-SiC)\\[0.2cm]
2233 0.282 nm: Si-C-C\\[0.2cm]
2234 $\approx$0.35 nm: C-Si-Si
2237 \begin{minipage}{0.2cm}
2241 \begin{minipage}[t]{6.0cm}
2242 0.15 nm: C-C pairs $\uparrow$\\
2243 (as expected in graphite/diamond)\\[0.2cm]
2244 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2245 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2250 \item Decreasing cut-off artifact
2251 \item {\color{red}Amorphous} SiC-like phase remains
2252 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2253 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2262 High C \& small $V$ \& short $t$
2265 Slow restructuring due to strong C-C bonds
2268 High C \& low T implants
2279 Summary and Conclusions
2287 \begin{minipage}[t]{12.9cm}
2288 \underline{Pecipitation simulations}
2290 \item High C concentration $\rightarrow$ amorphous SiC like phase
2291 \item Problem of potential enhanced slow phase space propagation
2292 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2293 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2294 \item High T necessary to simulate IBS conditions (far from equilibrium)
2295 \item Precipitation by successive agglomeration of \cs (epitaxy)
2296 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2297 (stretched SiC, interface)
2305 \begin{minipage}{12.9cm}
2310 \item Point defects excellently / fairly well described
2312 \item C$_{\text{sub}}$ drastically underestimated by EA
2313 \item EA predicts correct ground state:
2314 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2315 \item Identified migration path explaining
2316 diffusion and reorientation experiments by DFT
2317 \item EA fails to describe \ci{} migration:
2318 Wrong path \& overestimated barrier
2320 \item Combinations of defects
2322 \item Agglomeration of point defects energetically favorable
2323 by compensation of stress
2324 \item Formation of C-C unlikely
2325 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2326 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2327 Low barrier (\unit[0.77]{eV}) \& low capture radius
2335 \framebox{Precipitation by successive agglomeration of \cs{}}
2353 \underline{Augsburg}
2355 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2356 \item Ralf Utermann (EDV)
2359 \underline{Helsinki}
2361 \item Prof. K. Nordlund (MD)
2366 \item Bayerische Forschungsstiftung (financial support)
2369 \underline{Paderborn}
2371 \item Prof. J. Lindner (SiC)
2372 \item Prof. G. Schmidt (DFT + financial support)
2373 \item Dr. E. Rauls (DFT + SiC)
2374 \item Dr. S. Sanna (VASP)
2381 \bf Thank you for your attention!