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19 \usepackage{pstricks} % PSTricks with the standard color package
28 \graphicspath{{../img/}}
32 \usepackage[setpagesize=false]{hyperref}
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 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)(-3,68)
202 \includegraphics[width=2.6cm]{wide_band_gap.eps}
204 \begin{picture}(0,0)(-285,-162)
205 \includegraphics[width=3.38cm]{sic_led.eps}
207 \begin{picture}(0,0)(-195,-162)
208 \includegraphics[width=2.8cm]{6h-sic_3c-sic.eps}
210 \begin{picture}(0,0)(-313,65)
211 \includegraphics[width=2.2cm]{infineon_schottky.eps}
213 \begin{picture}(0,0)(-220,65)
214 \includegraphics[width=2.9cm]{sic_wechselrichter_ise.eps}
216 \begin{picture}(0,0)(0,-160)
217 \includegraphics[width=3.0cm]{sic_proton.eps}
219 \begin{picture}(0,0)(-60,65)
220 \includegraphics[width=3.4cm]{sic_switch.eps}
237 \begin{tabular}{l c c c c c c}
239 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
241 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
242 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
243 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
244 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
245 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
246 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
247 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
255 \begin{picture}(0,0)(-160,-155)
256 \includegraphics[width=7cm]{polytypes.eps}
258 \begin{picture}(0,0)(-10,-185)
259 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
261 \begin{picture}(0,0)(-10,-175)
262 {\tiny cubic (twist)}
264 \begin{picture}(0,0)(-60,-175)
265 {\tiny hexagonal (no twist)}
267 \begin{pspicture}(0,0)(0,0)
268 \psellipse[linecolor=green](5.7,3.03)(0.4,0.5)
270 \begin{pspicture}(0,0)(0,0)
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273 \begin{pspicture}(0,0)(0,0)
274 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
284 Fabrication of silicon carbide
291 SiC - \emph{Born from the stars, perfected on earth.}
297 Conventional thin film SiC growth:
299 \item \underline{Sublimation growth using the modified Lely method}
301 \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
302 \item Surrounded by polycrystalline SiC in a graphite crucible\\
303 at $T=2100-2400 \, ^{\circ} \text{C}$
304 \item Deposition of supersaturated vapor on cooler seed crystal
306 \item \underline{Homoepitaxial growth using CVD}
308 \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
309 \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
310 \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
312 \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
314 \item Two steps: carbonization and growth
315 \item $T=650-1050 \, ^{\circ} \text{C}$
316 \item SiC/Si lattice mismatch $\approx$ 20 \%
317 \item Quality and size not yet sufficient
321 \begin{picture}(0,0)(-280,-65)
322 \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
324 \begin{picture}(0,0)(-280,-55)
325 \begin{minipage}{5cm}
327 NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
332 \begin{picture}(0,0)(-265,-150)
333 \includegraphics[width=2.4cm]{m_lely.eps}
335 \begin{picture}(0,0)(-333,-175)
336 \begin{minipage}{5cm}
342 5. Insulation\\[-7pt]
347 \begin{picture}(0,0)(-230,-35)
349 {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
352 \begin{picture}(0,0)(-230,-10)
354 \begin{minipage}{3cm}
355 {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
372 \item Implantation of C in Si --- Overview of experimental observations
373 \item Utilized simulation techniques and modeled problems
375 \item {\color{blue}Diploma thesis}\\
376 \underline{Monte Carlo} simulations
377 modeling the selforganization process
378 leading to periodic arrays of nanometric amorphous SiC
380 \item {\color{blue}Doctoral studies}\\
381 Classical potential \underline{molecular dynamics} simulations
383 \underline{Density functional theory} calculations
385 \ldots on defects and SiC precipitation in Si
387 \item Summary / Conclusion / Outlook
399 Fabrication of silicon carbide
404 Alternative approach:
405 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
407 \item \underline{Implantation step 1}\\
408 180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
409 $\Rightarrow$ box-like distribution of equally sized
410 and epitactically oriented SiC precipitates
412 \item \underline{Implantation step 2}\\
413 180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
414 $\Rightarrow$ destruction of SiC nanocrystals
415 in growing amorphous interface layers
416 \item \underline{Annealing}\\
417 $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
418 $\Rightarrow$ homogeneous, stoichiometric SiC layer
419 with sharp interfaces
422 \begin{minipage}{6.3cm}
423 \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
425 XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
429 \begin{minipage}{6.3cm}
432 Precipitation mechanism not yet fully understood!
434 \renewcommand\labelitemi{$\Rightarrow$}
436 \underline{Understanding the SiC precipitation}
438 \item significant technological progress in SiC thin film formation
439 \item perspectives for processes relying upon prevention of SiC precipitation
451 Supposed precipitation mechanism of SiC in Si
458 \begin{minipage}{3.8cm}
459 Si \& SiC lattice structure\\[0.2cm]
460 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
464 \begin{minipage}{3.8cm}
466 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
470 \begin{minipage}{3.8cm}
472 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
476 \begin{minipage}{4cm}
478 C-Si dimers (dumbbells)\\[-0.1cm]
479 on Si interstitial sites
483 \begin{minipage}{4.2cm}
485 Agglomeration of C-Si dumbbells\\[-0.1cm]
486 $\Rightarrow$ dark contrasts
490 \begin{minipage}{4cm}
492 Precipitation of 3C-SiC in Si\\[-0.1cm]
493 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
494 \& release of Si self-interstitials
498 \begin{minipage}{3.8cm}
500 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
504 \begin{minipage}{3.8cm}
506 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
510 \begin{minipage}{3.8cm}
512 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
516 \begin{pspicture}(0,0)(0,0)
517 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
518 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
519 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
520 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
521 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
522 $4a_{\text{Si}}=5a_{\text{SiC}}$
524 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
525 \hkl(h k l) planes match
527 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
537 Supposed precipitation mechanism of SiC in Si
544 \begin{minipage}{3.8cm}
545 Si \& SiC lattice structure\\[0.2cm]
546 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
550 \begin{minipage}{3.8cm}
552 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
556 \begin{minipage}{3.8cm}
558 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
562 \begin{minipage}{4cm}
564 C-Si dimers (dumbbells)\\[-0.1cm]
565 on Si interstitial sites
569 \begin{minipage}{4.2cm}
571 Agglomeration of C-Si dumbbells\\[-0.1cm]
572 $\Rightarrow$ dark contrasts
576 \begin{minipage}{4cm}
578 Precipitation of 3C-SiC in Si\\[-0.1cm]
579 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
580 \& release of Si self-interstitials
584 \begin{minipage}{3.8cm}
586 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
590 \begin{minipage}{3.8cm}
592 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
596 \begin{minipage}{3.8cm}
598 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
602 \begin{pspicture}(0,0)(0,0)
603 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
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605 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
606 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
607 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
608 $4a_{\text{Si}}=5a_{\text{SiC}}$
610 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
611 \hkl(h k l) planes match
613 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
616 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
617 \begin{minipage}{10cm}
619 {\color{red}\bf Controversial views}
621 \item Implantations at high T (Nejim et al.)
623 \item Topotactic transformation based on \cs
624 \item \si{} as supply reacting with further C in cleared volume
626 \item Annealing behavior (Serre et al.)
628 \item Room temperature implants $\rightarrow$ highly mobile C
629 \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
630 (indicate stable \cs{} configurations)
632 \item Strained silicon \& Si/SiC heterostructures
634 \item Coherent SiC precipitates (tensile strain)
635 \item Incoherent SiC (strain relaxation)
647 Molecular dynamics (MD) simulations
656 \item Microscopic description of N particle system
657 \item Analytical interaction potential
658 \item Numerical integration using Newtons equation of motion\\
659 as a propagation rule in 6N-dimensional phase space
660 \item Observables obtained by time and/or ensemble averages
662 {\bf Details of the simulation:}
664 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
665 \item Ensemble: NpT (isothermal-isobaric)
667 \item Berendsen thermostat:
668 $\tau_{\text{T}}=100\text{ fs}$
669 \item Berendsen barostat:\\
670 $\tau_{\text{P}}=100\text{ fs}$,
671 $\beta^{-1}=100\text{ GPa}$
673 \item Erhart/Albe potential: Tersoff-like bond order potential
676 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
677 \pot_{ij} = {\color{red}f_C(r_{ij})}
678 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
682 \begin{picture}(0,0)(-230,-30)
683 \includegraphics[width=5cm]{tersoff_angle.eps}
691 Density functional theory (DFT) calculations
696 Basic ingredients necessary for DFT
699 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
701 \item ... uniquely determines the ground state potential
703 \item ... minimizes the systems total energy
705 \item \underline{Born-Oppenheimer}
706 - $N$ moving electrons in an external potential of static nuclei
708 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
709 +\sum_i^N V_{\text{ext}}(r_i)
710 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
712 \item \underline{Effective potential}
713 - averaged electrostatic potential \& exchange and correlation
715 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
718 \item \underline{Kohn-Sham system}
719 - Schr\"odinger equation of N non-interacting particles
721 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
726 n(r)=\sum_i^N|\Phi_i(r)|^2
728 \item \underline{Self-consistent solution}\\
729 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
730 which in turn depends on $n(r)$
731 \item \underline{Variational principle}
732 - minimize total energy with respect to $n(r)$
740 Density functional theory (DFT) calculations
747 Details of applied DFT calculations in this work
750 \item \underline{Exchange correlation functional}
751 - approximations for the inhomogeneous electron gas
753 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
754 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
756 \item \underline{Plane wave basis set}
757 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
760 \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}}
761 \qquad ({\color{blue}300\text{ eV}})
763 \item \underline{Brillouin zone sampling} -
764 {\color{blue}$\Gamma$-point only} calculations
765 \item \underline{Pseudo potential}
766 - consider only the valence electrons
767 \item \underline{Code} - VASP 4.6
772 MD and structural optimization
775 \item MD integration: Gear predictor corrector algorithm
776 \item Pressure control: Parrinello-Rahman pressure control
777 \item Structural optimization: Conjugate gradient method
780 \begin{pspicture}(0,0)(0,0)
781 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
789 C and Si self-interstitial point defects in silicon
796 \begin{minipage}{8cm}
798 \begin{pspicture}(0,0)(7,5)
799 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
802 \item Creation of c-Si simulation volume
803 \item Periodic boundary conditions
804 \item $T=0\text{ K}$, $p=0\text{ bar}$
807 \rput(3.5,2.1){\rnode{insert}{\psframebox{
810 Insertion of interstitial C/Si atoms
813 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
816 Relaxation / structural energy minimization
819 \ncline[]{->}{init}{insert}
820 \ncline[]{->}{insert}{cool}
823 \begin{minipage}{5cm}
824 \includegraphics[width=5cm]{unit_cell_e.eps}\\
827 \begin{minipage}{9cm}
828 \begin{tabular}{l c c}
830 & size [unit cells] & \# atoms\\
832 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
833 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
837 \begin{minipage}{4cm}
838 {\color{red}$\bullet$} Tetrahedral\\
839 {\color{green}$\bullet$} Hexagonal\\
840 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
841 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
842 {\color{cyan}$\bullet$} Bond-centered\\
843 {\color{black}$\bullet$} Vacancy / Substitutional
852 \begin{minipage}{9.5cm}
855 Si self-interstitial point defects in silicon\\
858 \begin{tabular}{l c c c c c}
860 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
862 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
863 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
865 \end{tabular}\\[0.2cm]
867 \begin{minipage}{4.7cm}
868 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
870 \begin{minipage}{4.7cm}
872 {\tiny nearly T $\rightarrow$ T}\\
874 \includegraphics[width=4.7cm]{nhex_tet.ps}
877 \underline{Hexagonal} \hspace{2pt}
878 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
880 \begin{minipage}{2.7cm}
881 $E_{\text{f}}^*=4.48\text{ eV}$\\
882 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
884 \begin{minipage}{0.4cm}
889 \begin{minipage}{2.7cm}
890 $E_{\text{f}}=3.96\text{ eV}$\\
891 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
894 \begin{minipage}{2.9cm}
896 \underline{Vacancy}\\
897 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
902 \begin{minipage}{3.5cm}
905 \underline{\hkl<1 1 0> dumbbell}\\
906 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
907 \underline{Tetrahedral}\\
908 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
909 \underline{\hkl<1 0 0> dumbbell}\\
910 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
922 C interstitial point defects in silicon\\[-0.1cm]
925 \begin{tabular}{l c c c c c c r}
927 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
929 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
930 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
932 \end{tabular}\\[0.1cm]
935 \begin{minipage}{2.7cm}
936 \underline{Hexagonal} \hspace{2pt}
937 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
938 $E_{\text{f}}^*=9.05\text{ eV}$\\
939 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
941 \begin{minipage}{0.4cm}
946 \begin{minipage}{2.7cm}
947 \underline{\hkl<1 0 0>}\\
948 $E_{\text{f}}=3.88\text{ eV}$\\
949 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
952 \begin{minipage}{2cm}
955 \begin{minipage}{3cm}
957 \underline{Tetrahedral}\\
958 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
963 \begin{minipage}{2.7cm}
964 \underline{Bond-centered}\\
965 $E_{\text{f}}^*=5.59\text{ eV}$\\
966 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
968 \begin{minipage}{0.4cm}
973 \begin{minipage}{2.7cm}
974 \underline{\hkl<1 1 0> dumbbell}\\
975 $E_{\text{f}}=5.18\text{ eV}$\\
976 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
979 \begin{minipage}{2cm}
982 \begin{minipage}{3cm}
984 \underline{Substitutional}\\
985 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
996 C \hkl<1 0 0> dumbbell interstitial configuration\\
1000 \begin{tabular}{l c c c c c c c c}
1002 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1004 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1005 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1007 \end{tabular}\\[0.2cm]
1008 \begin{tabular}{l c c c c }
1010 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1012 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1013 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1015 \end{tabular}\\[0.2cm]
1016 \begin{tabular}{l c c c}
1018 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1020 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1021 VASP & 0.109 & -0.065 & 0.174 \\
1023 \end{tabular}\\[0.6cm]
1026 \begin{minipage}{3.0cm}
1028 \underline{Erhart/Albe}
1029 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1032 \begin{minipage}{3.0cm}
1035 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1039 \begin{picture}(0,0)(-185,10)
1040 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1042 \begin{picture}(0,0)(-280,-150)
1043 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1046 \begin{pspicture}(0,0)(0,0)
1047 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1048 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1049 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1050 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1059 \begin{minipage}{8.5cm}
1062 Bond-centered interstitial configuration\\[-0.1cm]
1065 \begin{minipage}{3.0cm}
1066 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1068 \begin{minipage}{5.2cm}
1070 \item Linear Si-C-Si bond
1071 \item Si: one C \& 3 Si neighbours
1072 \item Spin polarized calculations
1073 \item No saddle point!\\
1080 \begin{minipage}[t]{6.5cm}
1081 \begin{minipage}[t]{1.2cm}
1083 {\tiny sp$^3$}\\[0.8cm]
1084 \underline{${\color{black}\uparrow}$}
1085 \underline{${\color{black}\uparrow}$}
1086 \underline{${\color{black}\uparrow}$}
1087 \underline{${\color{red}\uparrow}$}\\
1090 \begin{minipage}[t]{1.4cm}
1092 {\color{red}M}{\color{blue}O}\\[0.8cm]
1093 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1094 $\sigma_{\text{ab}}$\\[0.5cm]
1095 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1099 \begin{minipage}[t]{1.0cm}
1103 \underline{${\color{white}\uparrow\uparrow}$}
1104 \underline{${\color{white}\uparrow\uparrow}$}\\
1106 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1107 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1111 \begin{minipage}[t]{1.4cm}
1113 {\color{blue}M}{\color{green}O}\\[0.8cm]
1114 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1115 $\sigma_{\text{ab}}$\\[0.5cm]
1116 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1120 \begin{minipage}[t]{1.2cm}
1123 {\tiny sp$^3$}\\[0.8cm]
1124 \underline{${\color{green}\uparrow}$}
1125 \underline{${\color{black}\uparrow}$}
1126 \underline{${\color{black}\uparrow}$}
1127 \underline{${\color{black}\uparrow}$}\\
1135 \begin{minipage}{4.5cm}
1136 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1138 \begin{minipage}{3.5cm}
1139 {\color{gray}$\bullet$} Spin up\\
1140 {\color{green}$\bullet$} Spin down\\
1141 {\color{blue}$\bullet$} Resulting spin up\\
1142 {\color{yellow}$\bullet$} Si atoms\\
1143 {\color{red}$\bullet$} C atom
1148 \begin{minipage}{4.2cm}
1150 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1151 {\color{green}$\Box$} {\tiny unoccupied}\\
1152 {\color{red}$\bullet$} {\tiny occupied}
1161 Migration of the C \hkl<1 0 0> dumbbell interstitial
1166 {\small Investigated pathways}
1168 \begin{minipage}{8.5cm}
1169 \begin{minipage}{8.3cm}
1170 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1171 \begin{minipage}{2.4cm}
1172 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1174 \begin{minipage}{0.4cm}
1177 \begin{minipage}{2.4cm}
1178 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1180 \begin{minipage}{0.4cm}
1183 \begin{minipage}{2.4cm}
1184 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1187 \begin{minipage}{8.3cm}
1188 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1189 \begin{minipage}{2.4cm}
1190 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1192 \begin{minipage}{0.4cm}
1195 \begin{minipage}{2.4cm}
1196 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1198 \begin{minipage}{0.4cm}
1201 \begin{minipage}{2.4cm}
1202 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1205 \begin{minipage}{8.3cm}
1206 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1207 \begin{minipage}{2.4cm}
1208 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1210 \begin{minipage}{0.4cm}
1213 \begin{minipage}{2.4cm}
1214 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1216 \begin{minipage}{0.4cm}
1219 \begin{minipage}{2.4cm}
1220 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1225 \begin{minipage}{4.2cm}
1226 {\small Constrained relaxation\\
1227 technique (CRT) method}\\
1228 \includegraphics[width=4cm]{crt_orig.eps}
1230 \item Constrain diffusing atom
1231 \item Static constraints
1234 {\small Modifications}\\
1235 \includegraphics[width=4cm]{crt_mod.eps}
1237 \item Constrain all atoms
1238 \item Update individual\\
1249 Migration of the C \hkl<1 0 0> dumbbell interstitial
1255 \begin{minipage}{5.9cm}
1257 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1260 \begin{picture}(0,0)(60,0)
1261 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1263 \begin{picture}(0,0)(-5,0)
1264 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1266 \begin{picture}(0,0)(-55,0)
1267 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1269 \begin{picture}(0,0)(12.5,10)
1270 \includegraphics[width=1cm]{110_arrow.eps}
1272 \begin{picture}(0,0)(90,0)
1273 \includegraphics[height=0.9cm]{001_arrow.eps}
1279 \begin{minipage}{0.3cm}
1283 \begin{minipage}{5.9cm}
1285 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1288 \begin{picture}(0,0)(60,0)
1289 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1291 \begin{picture}(0,0)(5,0)
1292 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1294 \begin{picture}(0,0)(-55,0)
1295 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1297 \begin{picture}(0,0)(12.5,10)
1298 \includegraphics[width=1cm]{100_arrow.eps}
1300 \begin{picture}(0,0)(90,0)
1301 \includegraphics[height=0.9cm]{001_arrow.eps}
1311 \begin{minipage}{5.9cm}
1313 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1316 \begin{picture}(0,0)(60,0)
1317 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1319 \begin{picture}(0,0)(10,0)
1320 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1322 \begin{picture}(0,0)(-60,0)
1323 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1325 \begin{picture}(0,0)(12.5,10)
1326 \includegraphics[width=1cm]{100_arrow.eps}
1328 \begin{picture}(0,0)(90,0)
1329 \includegraphics[height=0.9cm]{001_arrow.eps}
1335 \begin{minipage}{0.3cm}
1338 \begin{minipage}{6.5cm}
1341 \item Energetically most favorable path
1344 \item Activation energy: $\approx$ 0.9 eV
1345 \item Experimental values: 0.73 ... 0.87 eV
1347 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1348 \item Reorientation (path 3)
1350 \item More likely composed of two consecutive steps of type 2
1351 \item Experimental values: 0.77 ... 0.88 eV
1353 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1362 Migration of the C \hkl<1 0 0> dumbbell interstitial
1369 \begin{minipage}{6.5cm}
1372 \begin{minipage}[t]{5.9cm}
1374 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1377 \begin{pspicture}(0,0)(0,0)
1378 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1380 \begin{picture}(0,0)(60,-50)
1381 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1383 \begin{picture}(0,0)(5,-50)
1384 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1386 \begin{picture}(0,0)(-55,-50)
1387 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1389 \begin{picture}(0,0)(12.5,-40)
1390 \includegraphics[width=1cm]{110_arrow.eps}
1392 \begin{picture}(0,0)(90,-45)
1393 \includegraphics[height=0.9cm]{001_arrow.eps}
1395 \begin{pspicture}(0,0)(0,0)
1396 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1398 \begin{picture}(0,0)(60,-15)
1399 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1401 \begin{picture}(0,0)(35,-15)
1402 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1404 \begin{picture}(0,0)(-5,-15)
1405 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1407 \begin{picture}(0,0)(-55,-15)
1408 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1410 \begin{picture}(0,0)(12.5,-5)
1411 \includegraphics[width=1cm]{100_arrow.eps}
1413 \begin{picture}(0,0)(90,-15)
1414 \includegraphics[height=0.9cm]{010_arrow.eps}
1420 \begin{minipage}{5.9cm}
1423 \item Lowest activation energy: $\approx$ 2.2 eV
1424 \item 2.4 times higher than VASP
1425 \item Different pathway
1430 \begin{minipage}{6.5cm}
1433 \begin{minipage}{5.9cm}
1435 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1438 %\begin{pspicture}(0,0)(0,0)
1439 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1441 %\begin{picture}(0,0)(60,-5)
1442 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1444 %\begin{picture}(0,0)(0,-5)
1445 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1447 %\begin{picture}(0,0)(-55,-5)
1448 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1450 %\begin{picture}(0,0)(12.5,5)
1451 %\includegraphics[width=1cm]{100_arrow.eps}
1453 %\begin{picture}(0,0)(90,0)
1454 %\includegraphics[height=0.9cm]{001_arrow.eps}
1462 %\begin{minipage}{5.9cm}
1463 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1467 \begin{minipage}{5.9cm}
1468 Transition involving \ci{} \hkl<1 1 0>
1470 \item Bond-centered configuration unstable\\
1471 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1472 \item Transition minima of path 2 \& 3\\
1473 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1474 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1475 \item 2.4 - 3.4 times higher than VASP
1476 \item Rotation of dumbbell orientation
1480 {\color{blue}Overestimated diffusion barrier}
1491 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1501 E_{\text{f}}^{\text{defect combination}}-
1502 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1503 E_{\text{f}}^{\text{2nd defect}}
1509 \begin{tabular}{l c c c c c c}
1511 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1513 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1514 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1515 \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}\\
1516 \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}\\
1517 \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}\\
1518 \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}\\
1520 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1521 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1530 \begin{minipage}[t]{3.8cm}
1531 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1532 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1534 \begin{minipage}[t]{3.5cm}
1535 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1536 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1538 \begin{minipage}[t]{5.5cm}
1540 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1541 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1542 \item Stress compensation / increase
1543 \item Unfavored: antiparallel orientations
1544 \item Indication of energetically favored\\
1546 \item Most favorable: C clustering
1547 \item However: High barrier ($>4\,\text{eV}$)
1548 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1553 \begin{picture}(0,0)(-295,-130)
1554 \includegraphics[width=3.5cm]{comb_pos.eps}
1562 Combinations of C-Si \hkl<1 0 0>-type interstitials
1569 Energetically most favorable combinations along \hkl<1 1 0>
1574 \begin{tabular}{l c c c c c c}
1576 & 1 & 2 & 3 & 4 & 5 & 6\\
1578 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1579 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1580 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>\\
1587 \begin{minipage}{7.0cm}
1588 \includegraphics[width=7cm]{db_along_110_cc.ps}
1590 \begin{minipage}{6.0cm}
1592 \item Interaction proportional to reciprocal cube of C-C distance
1593 \item Saturation in the immediate vicinity
1594 \renewcommand\labelitemi{$\Rightarrow$}
1595 \item Agglomeration of \ci{} expected
1596 \item Absence of C clustering
1600 Consisten with initial precipitation model
1612 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1618 %\begin{minipage}{3.2cm}
1619 %\includegraphics[width=3cm]{sub_110_combo.eps}
1621 %\begin{minipage}{7.8cm}
1622 %\begin{tabular}{l c c c c c c}
1624 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1625 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1627 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1628 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1629 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1630 %4 & \RM{4} & B & D & E & E & D \\
1631 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1638 %\begin{tabular}{l c c c c c c c c c c}
1640 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1642 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1643 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1644 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1649 \begin{minipage}{6.0cm}
1650 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1652 \begin{minipage}{7cm}
1655 \item IBS: C may displace Si\\
1656 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1658 \hkl<1 1 0>-type $\rightarrow$ favored combination
1659 \renewcommand\labelitemi{$\Rightarrow$}
1660 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1661 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1662 \item Interaction drops quickly to zero\\
1663 $\rightarrow$ low capture radius
1667 IBS process far from equilibrium\\
1668 \cs{} \& \si{} instead of thermodynamic ground state
1673 \begin{minipage}{6.5cm}
1674 \includegraphics[width=6.0cm]{162-097.ps}
1676 \item Low migration barrier
1679 \begin{minipage}{6.5cm}
1681 Ab initio MD at \degc{900}\\
1682 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1683 $t=\unit[2230]{fs}$\\
1684 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1688 Contribution of entropy to structural formation
1697 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1704 \begin{minipage}[t]{3cm}
1705 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1706 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1708 \begin{minipage}[t]{7cm}
1711 Low activation energies\\
1712 High activation energies for reverse processes\\
1714 {\color{blue}C$_{\text{sub}}$ very stable}\\
1718 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1720 {\color{blue}Formation of SiC by successive substitution by C}
1724 \begin{minipage}[t]{3cm}
1725 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1726 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1731 \begin{minipage}{5.9cm}
1732 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1734 \begin{picture}(0,0)(70,0)
1735 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1737 \begin{picture}(0,0)(30,0)
1738 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1740 \begin{picture}(0,0)(-10,0)
1741 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1743 \begin{picture}(0,0)(-48,0)
1744 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1746 \begin{picture}(0,0)(12.5,5)
1747 \includegraphics[width=1cm]{100_arrow.eps}
1749 \begin{picture}(0,0)(97,-10)
1750 \includegraphics[height=0.9cm]{001_arrow.eps}
1756 \begin{minipage}{0.3cm}
1760 \begin{minipage}{5.9cm}
1761 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1763 \begin{picture}(0,0)(60,0)
1764 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1766 \begin{picture}(0,0)(25,0)
1767 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1769 \begin{picture}(0,0)(-20,0)
1770 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1772 \begin{picture}(0,0)(-55,0)
1773 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1775 \begin{picture}(0,0)(12.5,5)
1776 \includegraphics[width=1cm]{100_arrow.eps}
1778 \begin{picture}(0,0)(95,0)
1779 \includegraphics[height=0.9cm]{001_arrow.eps}
1791 Conclusion of defect / migration / combined defect simulations
1800 \item Accurately described by quantum-mechanical simulations
1801 \item Less accurate description by classical potential simulations
1802 \item Underestimated formation energy of \cs{} by classical approach
1803 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1808 \item C migration pathway in Si identified
1809 \item Consistent with reorientation and diffusion experiments
1812 \item Different path and ...
1813 \item overestimated barrier by classical potential calculations
1816 Concerning the precipitation mechanism
1818 \item Agglomeration of C-Si dumbbells energetically favorable
1819 (stress compensation)
1820 \item C-Si indeed favored compared to
1821 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1822 \item Possible low interaction capture radius of
1823 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1824 \item Low barrier for
1825 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1826 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1827 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1830 {\color{blue}Results suggest increased participation of \cs}
1838 Silicon carbide precipitation simulations
1844 \begin{pspicture}(0,0)(12,6.5)
1846 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1849 \item Create c-Si volume
1850 \item Periodc boundary conditions
1851 \item Set requested $T$ and $p=0\text{ bar}$
1852 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1855 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1857 Insertion of C atoms at constant T
1859 \item total simulation volume {\pnode{in1}}
1860 \item volume of minimal SiC precipitate {\pnode{in2}}
1861 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1865 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1867 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1869 \ncline[]{->}{init}{insert}
1870 \ncline[]{->}{insert}{cool}
1871 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1872 \rput(7.8,6){\footnotesize $V_1$}
1873 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1874 \rput(9.2,4.85){\tiny $V_2$}
1875 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1876 \rput(9.55,4.45){\footnotesize $V_3$}
1877 \rput(7.9,3.2){\pnode{ins1}}
1878 \rput(9.22,2.8){\pnode{ins2}}
1879 \rput(11.0,2.4){\pnode{ins3}}
1880 \ncline[]{->}{in1}{ins1}
1881 \ncline[]{->}{in2}{ins2}
1882 \ncline[]{->}{in3}{ins3}
1887 \item Restricted to classical potential simulations
1888 \item $V_2$ and $V_3$ considered due to low diffusion
1889 \item Amount of C atoms: 6000
1890 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1891 \item Simulation volume: $31\times 31\times 31$ unit cells
1900 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1905 \begin{minipage}{6.5cm}
1906 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1908 \begin{minipage}{6.5cm}
1909 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1912 \begin{minipage}{6.5cm}
1913 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1915 \begin{minipage}{6.5cm}
1917 \underline{Low C concentration ($V_1$)}\\
1918 \hkl<1 0 0> C-Si dumbbell dominated structure
1920 \item Si-C bumbs around 0.19 nm
1921 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1922 concatenated dumbbells of various orientation
1923 \item Si-Si NN distance stretched to 0.3 nm
1925 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1926 \underline{High C concentration ($V_2$, $V_3$)}\\
1927 High amount of strongly bound C-C bonds\\
1928 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1929 Only short range order observable\\
1930 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1938 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1943 \begin{minipage}{6.5cm}
1944 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1946 \begin{minipage}{6.5cm}
1947 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1950 \begin{minipage}{6.5cm}
1951 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1953 \begin{minipage}{6.5cm}
1955 \underline{Low C concentration ($V_1$)}\\
1956 \hkl<1 0 0> C-Si dumbbell dominated structure
1958 \item Si-C bumbs around 0.19 nm
1959 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1960 concatenated dumbbells of various orientation
1961 \item Si-Si NN distance stretched to 0.3 nm
1963 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1964 \underline{High C concentration ($V_2$, $V_3$)}\\
1965 High amount of strongly bound C-C bonds\\
1966 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1967 Only short range order observable\\
1968 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1971 \begin{pspicture}(0,0)(0,0)
1972 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
1973 \begin{minipage}{10cm}
1975 {\color{red}\bf 3C-SiC formation fails to appear}
1977 \item Low C concentration simulations
1979 \item Formation of \ci{} indeed occurs
1980 \item Agllomeration not observed
1982 \item High C concentration simulations
1984 \item Amorphous SiC-like structure\\
1985 (not expected at prevailing temperatures)
1986 \item Rearrangement and transition into 3C-SiC structure missing
1998 Limitations of molecular dynamics and short range potentials
2005 \underline{Time scale problem of MD}\\[0.2cm]
2006 Minimize integration error\\
2007 $\Rightarrow$ discretization considerably smaller than
2008 reciprocal of fastest vibrational mode\\[0.1cm]
2009 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2010 $\Rightarrow$ suitable choice of time step:
2011 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2012 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2013 Several local minima in energy surface separated by large energy barriers\\
2014 $\Rightarrow$ transition event corresponds to a multiple
2015 of vibrational periods\\
2016 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2017 infrequent transition events\\[0.1cm]
2018 {\color{blue}Accelerated methods:}
2019 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2023 \underline{Limitations related to the short range potential}\\[0.2cm]
2024 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2025 and 2$^{\text{nd}}$ next neighbours\\
2026 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2032 Potential enhanced problem of slow phase space propagation
2037 \underline{Approach to the (twofold) problem}\\[0.2cm]
2038 Increased temperature simulations without TAD corrections\\
2039 (accelerated methods or higher time scales exclusively not sufficient)
2041 \begin{picture}(0,0)(-260,-30)
2043 \begin{minipage}{4.2cm}
2050 \item 3C-SiC also observed for higher T
2051 \item higher T inside sample
2052 \item structural evolution vs.\\
2053 equilibrium properties
2059 \begin{picture}(0,0)(-305,-155)
2061 \begin{minipage}{2.5cm}
2065 thermodynmic sampling
2076 Increased temperature simulations at low C concentration
2081 \begin{minipage}{6.5cm}
2082 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2084 \begin{minipage}{6.5cm}
2085 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2088 \begin{minipage}{6.5cm}
2089 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2091 \begin{minipage}{6.5cm}
2093 \underline{Si-C bonds:}
2095 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2096 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2098 \underline{Si-Si bonds:}
2099 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2100 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2101 \underline{C-C bonds:}
2103 \item C-C next neighbour pairs reduced (mandatory)
2104 \item Peak at 0.3 nm slightly shifted
2106 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2107 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2109 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2111 \item Range [|-$\downarrow$]:
2112 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2113 with nearby Si$_{\text{I}}$}
2118 \begin{picture}(0,0)(-330,-74)
2121 \begin{minipage}{1.6cm}
2124 stretched SiC\\[-0.1cm]
2136 Increased temperature simulations at low C concentration
2141 \begin{minipage}{6.5cm}
2142 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2144 \begin{minipage}{6.5cm}
2145 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2148 \begin{minipage}{6.5cm}
2149 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2151 \begin{minipage}{6.5cm}
2153 \underline{Si-C bonds:}
2155 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2156 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2158 \underline{Si-Si bonds:}
2159 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2160 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2161 \underline{C-C bonds:}
2163 \item C-C next neighbour pairs reduced (mandatory)
2164 \item Peak at 0.3 nm slightly shifted
2166 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2167 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2169 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2171 \item Range [|-$\downarrow$]:
2172 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2173 with nearby Si$_{\text{I}}$}
2178 %\begin{picture}(0,0)(-330,-74)
2181 %\begin{minipage}{1.6cm}
2184 %stretched SiC\\[-0.1cm]
2191 \begin{pspicture}(0,0)(0,0)
2192 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2193 \begin{minipage}{10cm}
2195 {\color{blue}\bf Stretched SiC in c-Si}
2197 \item Consistent to precipitation model involving \cs{}
2198 \item Explains annealing behavior of high/low T C implants
2200 \item Low T: highly mobiel \ci{}
2201 \item High T: stable configurations of \cs{}
2204 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2205 $\Rightarrow$ Precipitation mechanism involving \cs{}
2215 Increased temperature simulations at high C concentration
2220 \begin{minipage}{6.5cm}
2221 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2223 \begin{minipage}{6.5cm}
2224 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2232 \begin{minipage}[t]{6.0cm}
2233 0.186 nm: Si-C pairs $\uparrow$\\
2234 (as expected in 3C-SiC)\\[0.2cm]
2235 0.282 nm: Si-C-C\\[0.2cm]
2236 $\approx$0.35 nm: C-Si-Si
2239 \begin{minipage}{0.2cm}
2243 \begin{minipage}[t]{6.0cm}
2244 0.15 nm: C-C pairs $\uparrow$\\
2245 (as expected in graphite/diamond)\\[0.2cm]
2246 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2247 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2252 \item Decreasing cut-off artifact
2253 \item {\color{red}Amorphous} SiC-like phase remains
2254 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2255 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2264 High C \& small $V$ \& short $t$
2267 Slow restructuring due to strong C-C bonds
2270 High C \& low T implants
2281 Summary and Conclusions
2289 \begin{minipage}[t]{12.9cm}
2290 \underline{Pecipitation simulations}
2292 \item High C concentration $\rightarrow$ amorphous SiC like phase
2293 \item Problem of potential enhanced slow phase space propagation
2294 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2295 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2296 \item High T necessary to simulate IBS conditions (far from equilibrium)
2297 \item Precipitation by successive agglomeration of \cs (epitaxy)
2298 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2299 (stretched SiC, interface)
2307 \begin{minipage}{12.9cm}
2312 \item Point defects excellently / fairly well described
2314 \item C$_{\text{sub}}$ drastically underestimated by EA
2315 \item EA predicts correct ground state:
2316 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2317 \item Identified migration path explaining
2318 diffusion and reorientation experiments by DFT
2319 \item EA fails to describe \ci{} migration:
2320 Wrong path \& overestimated barrier
2322 \item Combinations of defects
2324 \item Agglomeration of point defects energetically favorable
2325 by compensation of stress
2326 \item Formation of C-C unlikely
2327 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2328 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2329 Low barrier (\unit[0.77]{eV}) \& low capture radius
2337 \framebox{Precipitation by successive agglomeration of \cs{}}
2355 \underline{Augsburg}
2357 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2358 \item Ralf Utermann (EDV)
2361 \underline{Helsinki}
2363 \item Prof. K. Nordlund (MD)
2368 \item Bayerische Forschungsstiftung (financial support)
2371 \underline{Paderborn}
2373 \item Prof. J. Lindner (SiC)
2374 \item Prof. G. Schmidt (DFT + financial support)
2375 \item Dr. E. Rauls (DFT + SiC)
2376 \item Dr. S. Sanna (VASP)
2383 \bf Thank you for your attention!