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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
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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 Introduction --- The C/Si system\\
136 \includegraphics[width=6.3cm]{si-c_phase.eps}\\
138 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
141 \begin{pspicture}(0,0)(0,0)
142 \psellipse[linecolor=red,linewidth=0.1cm](6.95,3.95)(0.5,2.8)
150 % motivation / properties / applications of silicon carbide
156 \begin{pspicture}(0,0)(13.5,5)
158 \psframe*[linecolor=hb](0,0)(13.5,5)
160 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.5,1)(7,1)(7,3)(5.5,3)
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163 \rput[lt](0.2,4.6){\color{gray}PROPERTIES}
165 \rput[lt](0.5,4){wide band gap}
166 \rput[lt](0.5,3.5){high electric breakdown field}
167 \rput[lt](0.5,3){good electron mobility}
168 \rput[lt](0.5,2.5){high electron saturation drift velocity}
169 \rput[lt](0.5,2){high thermal conductivity}
171 \rput[lt](0.5,1.5){hard and mechanically stable}
172 \rput[lt](0.5,1){chemically inert}
174 \rput[lt](0.5,0.5){radiation hardness}
176 \rput[rt](13.3,4.6){\color{gray}APPLICATIONS}
178 \rput[rt](13,3.85){high-temperature, high power}
179 \rput[rt](13,3.5){and high-frequency}
180 \rput[rt](13,3.15){electronic and optoelectronic devices}
182 \rput[rt](13,2.35){material suitable for extreme conditions}
183 \rput[rt](13,2){microelectromechanical systems}
184 \rput[rt](13,1.65){abrasives, cutting tools, heating elements}
186 \rput[rt](13,0.85){first wall reactor material, detectors}
187 \rput[rt](13,0.5){and electronic devices for space}
191 \begin{picture}(0,0)(-3,68)
192 \includegraphics[width=2.6cm]{wide_band_gap.eps}
194 \begin{picture}(0,0)(-285,-162)
195 \includegraphics[width=3.38cm]{sic_led.eps}
197 \begin{picture}(0,0)(-195,-162)
198 \includegraphics[width=2.8cm]{6h-sic_3c-sic.eps}
200 \begin{picture}(0,0)(-313,65)
201 \includegraphics[width=2.2cm]{infineon_schottky.eps}
203 \begin{picture}(0,0)(-220,65)
204 \includegraphics[width=2.9cm]{sic_wechselrichter_ise.eps}
206 \begin{picture}(0,0)(0,-160)
207 \includegraphics[width=3.0cm]{sic_proton.eps}
209 \begin{picture}(0,0)(-60,65)
210 \includegraphics[width=3.4cm]{sic_switch.eps}
225 \item Implantation of C in Si --- Overview of experimental observations
226 \item Utilized simulation techniques and modeled problems
228 \item {\color{blue}Diploma thesis}\\
229 \underline{Monte Carlo} simulations
230 modeling the selforganization process
231 leading to periodic arrays of nanometric amorphous SiC
233 \item {\color{blue}Doctoral studies}\\
234 Classical potential \underline{molecular dynamics} simulations
236 \underline{Density functional theory} calculations
238 \ldots on defects and SiC precipitation in Si
240 \item Summary / Conclusion / Outlook
264 \begin{tabular}{l c c c c c c}
266 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
268 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
269 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
270 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
271 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
272 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
273 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
274 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
282 \begin{picture}(0,0)(-160,-155)
283 \includegraphics[width=7cm]{polytypes.eps}
285 \begin{picture}(0,0)(-10,-185)
286 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
288 \begin{picture}(0,0)(-10,-175)
289 {\tiny cubic (twist)}
291 \begin{picture}(0,0)(-60,-175)
292 {\tiny hexagonal (no twist)}
294 \begin{pspicture}(0,0)(0,0)
295 \psellipse[linecolor=green](5.7,3.03)(0.4,0.5)
297 \begin{pspicture}(0,0)(0,0)
298 \psellipse[linecolor=green](5.6,1.68)(0.4,0.2)
300 \begin{pspicture}(0,0)(0,0)
301 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
309 Fabrication of silicon carbide
316 SiC - \emph{Born from the stars, perfected on earth.}
320 Conventional thin film SiC growth:
322 \item \underline{Sublimation growth using the modified Lely method}
324 \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
325 \item Surrounded by polycrystalline SiC in a graphite crucible\\
326 at $T=2100-2400 \, ^{\circ} \text{C}$
327 \item Deposition of supersaturated vapor on cooler seed crystal
329 \item \underline{Homoepitaxial growth using CVD}
331 \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
332 \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
333 \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
335 \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
337 \item Two steps: carbonization and growth
338 \item $T=650-1050 \, ^{\circ} \text{C}$
339 \item SiC/Si lattice mismatch $\approx$ 20 \%
340 \item Quality and size not yet sufficient
344 \begin{picture}(0,0)(-280,-65)
345 \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
347 \begin{picture}(0,0)(-280,-55)
348 \begin{minipage}{5cm}
350 NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
355 \begin{picture}(0,0)(-265,-150)
356 \includegraphics[width=2.4cm]{m_lely.eps}
358 \begin{picture}(0,0)(-333,-175)
359 \begin{minipage}{5cm}
365 5. Insulation\\[-7pt]
370 \begin{picture}(0,0)(-230,-35)
372 {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
375 \begin{picture}(0,0)(-230,-10)
377 \begin{minipage}{3cm}
378 {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
389 Fabrication of silicon carbide
394 Alternative approach:
395 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
397 \item \underline{Implantation step 1}\\
398 180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
399 $\Rightarrow$ box-like distribution of equally sized
400 and epitactically oriented SiC precipitates
402 \item \underline{Implantation step 2}\\
403 180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
404 $\Rightarrow$ destruction of SiC nanocrystals
405 in growing amorphous interface layers
406 \item \underline{Annealing}\\
407 $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
408 $\Rightarrow$ homogeneous, stoichiometric SiC layer
409 with sharp interfaces
412 \begin{minipage}{6.3cm}
413 \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
415 XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
419 \begin{minipage}{6.3cm}
422 Precipitation mechanism not yet fully understood!
424 \renewcommand\labelitemi{$\Rightarrow$}
426 \underline{Understanding the SiC precipitation}
428 \item significant technological progress in SiC thin film formation
429 \item perspectives for processes relying upon prevention of SiC precipitation
441 Supposed precipitation mechanism of SiC in Si
448 \begin{minipage}{3.8cm}
449 Si \& SiC lattice structure\\[0.2cm]
450 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
454 \begin{minipage}{3.8cm}
456 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
460 \begin{minipage}{3.8cm}
462 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
466 \begin{minipage}{4cm}
468 C-Si dimers (dumbbells)\\[-0.1cm]
469 on Si interstitial sites
473 \begin{minipage}{4.2cm}
475 Agglomeration of C-Si dumbbells\\[-0.1cm]
476 $\Rightarrow$ dark contrasts
480 \begin{minipage}{4cm}
482 Precipitation of 3C-SiC in Si\\[-0.1cm]
483 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
484 \& release of Si self-interstitials
488 \begin{minipage}{3.8cm}
490 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
494 \begin{minipage}{3.8cm}
496 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
500 \begin{minipage}{3.8cm}
502 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
506 \begin{pspicture}(0,0)(0,0)
507 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
508 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
509 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
510 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
511 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
512 $4a_{\text{Si}}=5a_{\text{SiC}}$
514 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
515 \hkl(h k l) planes match
517 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
527 Supposed precipitation mechanism of SiC in Si
534 \begin{minipage}{3.8cm}
535 Si \& SiC lattice structure\\[0.2cm]
536 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
540 \begin{minipage}{3.8cm}
542 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
546 \begin{minipage}{3.8cm}
548 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
552 \begin{minipage}{4cm}
554 C-Si dimers (dumbbells)\\[-0.1cm]
555 on Si interstitial sites
559 \begin{minipage}{4.2cm}
561 Agglomeration of C-Si dumbbells\\[-0.1cm]
562 $\Rightarrow$ dark contrasts
566 \begin{minipage}{4cm}
568 Precipitation of 3C-SiC in Si\\[-0.1cm]
569 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
570 \& release of Si self-interstitials
574 \begin{minipage}{3.8cm}
576 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
580 \begin{minipage}{3.8cm}
582 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
586 \begin{minipage}{3.8cm}
588 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
592 \begin{pspicture}(0,0)(0,0)
593 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
594 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
595 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
596 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
597 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
598 $4a_{\text{Si}}=5a_{\text{SiC}}$
600 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
601 \hkl(h k l) planes match
603 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
606 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
607 \begin{minipage}{10cm}
609 {\color{red}\bf Controversial views}
611 \item Implantations at high T (Nejim et al.)
613 \item Topotactic transformation based on \cs
614 \item \si{} as supply reacting with further C in cleared volume
616 \item Annealing behavior (Serre et al.)
618 \item Room temperature implants $\rightarrow$ highly mobile C
619 \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
620 (indicate stable \cs{} configurations)
622 \item Strained silicon \& Si/SiC heterostructures
624 \item Coherent SiC precipitates (tensile strain)
625 \item Incoherent SiC (strain relaxation)
637 Molecular dynamics (MD) simulations
646 \item Microscopic description of N particle system
647 \item Analytical interaction potential
648 \item Numerical integration using Newtons equation of motion\\
649 as a propagation rule in 6N-dimensional phase space
650 \item Observables obtained by time and/or ensemble averages
652 {\bf Details of the simulation:}
654 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
655 \item Ensemble: NpT (isothermal-isobaric)
657 \item Berendsen thermostat:
658 $\tau_{\text{T}}=100\text{ fs}$
659 \item Berendsen barostat:\\
660 $\tau_{\text{P}}=100\text{ fs}$,
661 $\beta^{-1}=100\text{ GPa}$
663 \item Erhart/Albe potential: Tersoff-like bond order potential
666 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
667 \pot_{ij} = {\color{red}f_C(r_{ij})}
668 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
672 \begin{picture}(0,0)(-230,-30)
673 \includegraphics[width=5cm]{tersoff_angle.eps}
681 Density functional theory (DFT) calculations
686 Basic ingredients necessary for DFT
689 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
691 \item ... uniquely determines the ground state potential
693 \item ... minimizes the systems total energy
695 \item \underline{Born-Oppenheimer}
696 - $N$ moving electrons in an external potential of static nuclei
698 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
699 +\sum_i^N V_{\text{ext}}(r_i)
700 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
702 \item \underline{Effective potential}
703 - averaged electrostatic potential \& exchange and correlation
705 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
708 \item \underline{Kohn-Sham system}
709 - Schr\"odinger equation of N non-interacting particles
711 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
716 n(r)=\sum_i^N|\Phi_i(r)|^2
718 \item \underline{Self-consistent solution}\\
719 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
720 which in turn depends on $n(r)$
721 \item \underline{Variational principle}
722 - minimize total energy with respect to $n(r)$
730 Density functional theory (DFT) calculations
737 Details of applied DFT calculations in this work
740 \item \underline{Exchange correlation functional}
741 - approximations for the inhomogeneous electron gas
743 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
744 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
746 \item \underline{Plane wave basis set}
747 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
750 \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}}
751 \qquad ({\color{blue}300\text{ eV}})
753 \item \underline{Brillouin zone sampling} -
754 {\color{blue}$\Gamma$-point only} calculations
755 \item \underline{Pseudo potential}
756 - consider only the valence electrons
757 \item \underline{Code} - VASP 4.6
762 MD and structural optimization
765 \item MD integration: Gear predictor corrector algorithm
766 \item Pressure control: Parrinello-Rahman pressure control
767 \item Structural optimization: Conjugate gradient method
770 \begin{pspicture}(0,0)(0,0)
771 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
779 C and Si self-interstitial point defects in silicon
786 \begin{minipage}{8cm}
788 \begin{pspicture}(0,0)(7,5)
789 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
792 \item Creation of c-Si simulation volume
793 \item Periodic boundary conditions
794 \item $T=0\text{ K}$, $p=0\text{ bar}$
797 \rput(3.5,2.1){\rnode{insert}{\psframebox{
800 Insertion of interstitial C/Si atoms
803 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
806 Relaxation / structural energy minimization
809 \ncline[]{->}{init}{insert}
810 \ncline[]{->}{insert}{cool}
813 \begin{minipage}{5cm}
814 \includegraphics[width=5cm]{unit_cell_e.eps}\\
817 \begin{minipage}{9cm}
818 \begin{tabular}{l c c}
820 & size [unit cells] & \# atoms\\
822 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
823 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
827 \begin{minipage}{4cm}
828 {\color{red}$\bullet$} Tetrahedral\\
829 {\color{green}$\bullet$} Hexagonal\\
830 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
831 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
832 {\color{cyan}$\bullet$} Bond-centered\\
833 {\color{black}$\bullet$} Vacancy / Substitutional
842 \begin{minipage}{9.5cm}
845 Si self-interstitial point defects in silicon\\
848 \begin{tabular}{l c c c c c}
850 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
852 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
853 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
855 \end{tabular}\\[0.2cm]
857 \begin{minipage}{4.7cm}
858 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
860 \begin{minipage}{4.7cm}
862 {\tiny nearly T $\rightarrow$ T}\\
864 \includegraphics[width=4.7cm]{nhex_tet.ps}
867 \underline{Hexagonal} \hspace{2pt}
868 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
870 \begin{minipage}{2.7cm}
871 $E_{\text{f}}^*=4.48\text{ eV}$\\
872 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
874 \begin{minipage}{0.4cm}
879 \begin{minipage}{2.7cm}
880 $E_{\text{f}}=3.96\text{ eV}$\\
881 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
884 \begin{minipage}{2.9cm}
886 \underline{Vacancy}\\
887 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
892 \begin{minipage}{3.5cm}
895 \underline{\hkl<1 1 0> dumbbell}\\
896 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
897 \underline{Tetrahedral}\\
898 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
899 \underline{\hkl<1 0 0> dumbbell}\\
900 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
912 C interstitial point defects in silicon\\[-0.1cm]
915 \begin{tabular}{l c c c c c c r}
917 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
919 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
920 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
922 \end{tabular}\\[0.1cm]
925 \begin{minipage}{2.7cm}
926 \underline{Hexagonal} \hspace{2pt}
927 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
928 $E_{\text{f}}^*=9.05\text{ eV}$\\
929 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
931 \begin{minipage}{0.4cm}
936 \begin{minipage}{2.7cm}
937 \underline{\hkl<1 0 0>}\\
938 $E_{\text{f}}=3.88\text{ eV}$\\
939 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
942 \begin{minipage}{2cm}
945 \begin{minipage}{3cm}
947 \underline{Tetrahedral}\\
948 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
953 \begin{minipage}{2.7cm}
954 \underline{Bond-centered}\\
955 $E_{\text{f}}^*=5.59\text{ eV}$\\
956 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
958 \begin{minipage}{0.4cm}
963 \begin{minipage}{2.7cm}
964 \underline{\hkl<1 1 0> dumbbell}\\
965 $E_{\text{f}}=5.18\text{ eV}$\\
966 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
969 \begin{minipage}{2cm}
972 \begin{minipage}{3cm}
974 \underline{Substitutional}\\
975 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
986 C \hkl<1 0 0> dumbbell interstitial configuration\\
990 \begin{tabular}{l c c c c c c c c}
992 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
994 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
995 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
997 \end{tabular}\\[0.2cm]
998 \begin{tabular}{l c c c c }
1000 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1002 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1003 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1005 \end{tabular}\\[0.2cm]
1006 \begin{tabular}{l c c c}
1008 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1010 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1011 VASP & 0.109 & -0.065 & 0.174 \\
1013 \end{tabular}\\[0.6cm]
1016 \begin{minipage}{3.0cm}
1018 \underline{Erhart/Albe}
1019 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1022 \begin{minipage}{3.0cm}
1025 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1029 \begin{picture}(0,0)(-185,10)
1030 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1032 \begin{picture}(0,0)(-280,-150)
1033 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1036 \begin{pspicture}(0,0)(0,0)
1037 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1038 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1039 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1040 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1049 \begin{minipage}{8.5cm}
1052 Bond-centered interstitial configuration\\[-0.1cm]
1055 \begin{minipage}{3.0cm}
1056 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1058 \begin{minipage}{5.2cm}
1060 \item Linear Si-C-Si bond
1061 \item Si: one C \& 3 Si neighbours
1062 \item Spin polarized calculations
1063 \item No saddle point!\\
1070 \begin{minipage}[t]{6.5cm}
1071 \begin{minipage}[t]{1.2cm}
1073 {\tiny sp$^3$}\\[0.8cm]
1074 \underline{${\color{black}\uparrow}$}
1075 \underline{${\color{black}\uparrow}$}
1076 \underline{${\color{black}\uparrow}$}
1077 \underline{${\color{red}\uparrow}$}\\
1080 \begin{minipage}[t]{1.4cm}
1082 {\color{red}M}{\color{blue}O}\\[0.8cm]
1083 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1084 $\sigma_{\text{ab}}$\\[0.5cm]
1085 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1089 \begin{minipage}[t]{1.0cm}
1093 \underline{${\color{white}\uparrow\uparrow}$}
1094 \underline{${\color{white}\uparrow\uparrow}$}\\
1096 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1097 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1101 \begin{minipage}[t]{1.4cm}
1103 {\color{blue}M}{\color{green}O}\\[0.8cm]
1104 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1105 $\sigma_{\text{ab}}$\\[0.5cm]
1106 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1110 \begin{minipage}[t]{1.2cm}
1113 {\tiny sp$^3$}\\[0.8cm]
1114 \underline{${\color{green}\uparrow}$}
1115 \underline{${\color{black}\uparrow}$}
1116 \underline{${\color{black}\uparrow}$}
1117 \underline{${\color{black}\uparrow}$}\\
1125 \begin{minipage}{4.5cm}
1126 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1128 \begin{minipage}{3.5cm}
1129 {\color{gray}$\bullet$} Spin up\\
1130 {\color{green}$\bullet$} Spin down\\
1131 {\color{blue}$\bullet$} Resulting spin up\\
1132 {\color{yellow}$\bullet$} Si atoms\\
1133 {\color{red}$\bullet$} C atom
1138 \begin{minipage}{4.2cm}
1140 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1141 {\color{green}$\Box$} {\tiny unoccupied}\\
1142 {\color{red}$\bullet$} {\tiny occupied}
1151 Migration of the C \hkl<1 0 0> dumbbell interstitial
1156 {\small Investigated pathways}
1158 \begin{minipage}{8.5cm}
1159 \begin{minipage}{8.3cm}
1160 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1161 \begin{minipage}{2.4cm}
1162 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1164 \begin{minipage}{0.4cm}
1167 \begin{minipage}{2.4cm}
1168 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1170 \begin{minipage}{0.4cm}
1173 \begin{minipage}{2.4cm}
1174 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1177 \begin{minipage}{8.3cm}
1178 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1179 \begin{minipage}{2.4cm}
1180 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1182 \begin{minipage}{0.4cm}
1185 \begin{minipage}{2.4cm}
1186 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1188 \begin{minipage}{0.4cm}
1191 \begin{minipage}{2.4cm}
1192 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1195 \begin{minipage}{8.3cm}
1196 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1197 \begin{minipage}{2.4cm}
1198 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1200 \begin{minipage}{0.4cm}
1203 \begin{minipage}{2.4cm}
1204 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1206 \begin{minipage}{0.4cm}
1209 \begin{minipage}{2.4cm}
1210 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1215 \begin{minipage}{4.2cm}
1216 {\small Constrained relaxation\\
1217 technique (CRT) method}\\
1218 \includegraphics[width=4cm]{crt_orig.eps}
1220 \item Constrain diffusing atom
1221 \item Static constraints
1224 {\small Modifications}\\
1225 \includegraphics[width=4cm]{crt_mod.eps}
1227 \item Constrain all atoms
1228 \item Update individual\\
1239 Migration of the C \hkl<1 0 0> dumbbell interstitial
1245 \begin{minipage}{5.9cm}
1247 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1250 \begin{picture}(0,0)(60,0)
1251 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1253 \begin{picture}(0,0)(-5,0)
1254 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1256 \begin{picture}(0,0)(-55,0)
1257 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1259 \begin{picture}(0,0)(12.5,10)
1260 \includegraphics[width=1cm]{110_arrow.eps}
1262 \begin{picture}(0,0)(90,0)
1263 \includegraphics[height=0.9cm]{001_arrow.eps}
1269 \begin{minipage}{0.3cm}
1273 \begin{minipage}{5.9cm}
1275 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1278 \begin{picture}(0,0)(60,0)
1279 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1281 \begin{picture}(0,0)(5,0)
1282 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1284 \begin{picture}(0,0)(-55,0)
1285 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1287 \begin{picture}(0,0)(12.5,10)
1288 \includegraphics[width=1cm]{100_arrow.eps}
1290 \begin{picture}(0,0)(90,0)
1291 \includegraphics[height=0.9cm]{001_arrow.eps}
1301 \begin{minipage}{5.9cm}
1303 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1306 \begin{picture}(0,0)(60,0)
1307 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1309 \begin{picture}(0,0)(10,0)
1310 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1312 \begin{picture}(0,0)(-60,0)
1313 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1315 \begin{picture}(0,0)(12.5,10)
1316 \includegraphics[width=1cm]{100_arrow.eps}
1318 \begin{picture}(0,0)(90,0)
1319 \includegraphics[height=0.9cm]{001_arrow.eps}
1325 \begin{minipage}{0.3cm}
1328 \begin{minipage}{6.5cm}
1331 \item Energetically most favorable path
1334 \item Activation energy: $\approx$ 0.9 eV
1335 \item Experimental values: 0.73 ... 0.87 eV
1337 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1338 \item Reorientation (path 3)
1340 \item More likely composed of two consecutive steps of type 2
1341 \item Experimental values: 0.77 ... 0.88 eV
1343 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1352 Migration of the C \hkl<1 0 0> dumbbell interstitial
1359 \begin{minipage}{6.5cm}
1362 \begin{minipage}[t]{5.9cm}
1364 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1367 \begin{pspicture}(0,0)(0,0)
1368 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1370 \begin{picture}(0,0)(60,-50)
1371 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1373 \begin{picture}(0,0)(5,-50)
1374 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1376 \begin{picture}(0,0)(-55,-50)
1377 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1379 \begin{picture}(0,0)(12.5,-40)
1380 \includegraphics[width=1cm]{110_arrow.eps}
1382 \begin{picture}(0,0)(90,-45)
1383 \includegraphics[height=0.9cm]{001_arrow.eps}
1385 \begin{pspicture}(0,0)(0,0)
1386 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1388 \begin{picture}(0,0)(60,-15)
1389 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1391 \begin{picture}(0,0)(35,-15)
1392 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1394 \begin{picture}(0,0)(-5,-15)
1395 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1397 \begin{picture}(0,0)(-55,-15)
1398 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1400 \begin{picture}(0,0)(12.5,-5)
1401 \includegraphics[width=1cm]{100_arrow.eps}
1403 \begin{picture}(0,0)(90,-15)
1404 \includegraphics[height=0.9cm]{010_arrow.eps}
1410 \begin{minipage}{5.9cm}
1413 \item Lowest activation energy: $\approx$ 2.2 eV
1414 \item 2.4 times higher than VASP
1415 \item Different pathway
1420 \begin{minipage}{6.5cm}
1423 \begin{minipage}{5.9cm}
1425 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1428 %\begin{pspicture}(0,0)(0,0)
1429 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1431 %\begin{picture}(0,0)(60,-5)
1432 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1434 %\begin{picture}(0,0)(0,-5)
1435 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1437 %\begin{picture}(0,0)(-55,-5)
1438 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1440 %\begin{picture}(0,0)(12.5,5)
1441 %\includegraphics[width=1cm]{100_arrow.eps}
1443 %\begin{picture}(0,0)(90,0)
1444 %\includegraphics[height=0.9cm]{001_arrow.eps}
1452 %\begin{minipage}{5.9cm}
1453 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1457 \begin{minipage}{5.9cm}
1458 Transition involving \ci{} \hkl<1 1 0>
1460 \item Bond-centered configuration unstable\\
1461 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1462 \item Transition minima of path 2 \& 3\\
1463 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1464 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1465 \item 2.4 - 3.4 times higher than VASP
1466 \item Rotation of dumbbell orientation
1470 {\color{blue}Overestimated diffusion barrier}
1481 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1491 E_{\text{f}}^{\text{defect combination}}-
1492 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1493 E_{\text{f}}^{\text{2nd defect}}
1499 \begin{tabular}{l c c c c c c}
1501 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1503 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1504 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1505 \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}\\
1506 \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}\\
1507 \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}\\
1508 \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}\\
1510 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1511 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1520 \begin{minipage}[t]{3.8cm}
1521 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1522 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1524 \begin{minipage}[t]{3.5cm}
1525 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1526 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1528 \begin{minipage}[t]{5.5cm}
1530 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1531 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1532 \item Stress compensation / increase
1533 \item Unfavored: antiparallel orientations
1534 \item Indication of energetically favored\\
1536 \item Most favorable: C clustering
1537 \item However: High barrier ($>4\,\text{eV}$)
1538 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1543 \begin{picture}(0,0)(-295,-130)
1544 \includegraphics[width=3.5cm]{comb_pos.eps}
1552 Combinations of C-Si \hkl<1 0 0>-type interstitials
1559 Energetically most favorable combinations along \hkl<1 1 0>
1564 \begin{tabular}{l c c c c c c}
1566 & 1 & 2 & 3 & 4 & 5 & 6\\
1568 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1569 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1570 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>\\
1577 \begin{minipage}{7.0cm}
1578 \includegraphics[width=7cm]{db_along_110_cc.ps}
1580 \begin{minipage}{6.0cm}
1582 \item Interaction proportional to reciprocal cube of C-C distance
1583 \item Saturation in the immediate vicinity
1584 \renewcommand\labelitemi{$\Rightarrow$}
1585 \item Agglomeration of \ci{} expected
1586 \item Absence of C clustering
1590 Consisten with initial precipitation model
1602 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1608 %\begin{minipage}{3.2cm}
1609 %\includegraphics[width=3cm]{sub_110_combo.eps}
1611 %\begin{minipage}{7.8cm}
1612 %\begin{tabular}{l c c c c c c}
1614 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1615 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1617 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1618 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1619 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1620 %4 & \RM{4} & B & D & E & E & D \\
1621 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1628 %\begin{tabular}{l c c c c c c c c c c}
1630 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1632 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1633 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1634 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1639 \begin{minipage}{6.0cm}
1640 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1642 \begin{minipage}{7cm}
1645 \item IBS: C may displace Si\\
1646 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1648 \hkl<1 1 0>-type $\rightarrow$ favored combination
1649 \renewcommand\labelitemi{$\Rightarrow$}
1650 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1651 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1652 \item Interaction drops quickly to zero\\
1653 $\rightarrow$ low capture radius
1657 IBS process far from equilibrium\\
1658 \cs{} \& \si{} instead of thermodynamic ground state
1663 \begin{minipage}{6.5cm}
1664 \includegraphics[width=6.0cm]{162-097.ps}
1666 \item Low migration barrier
1669 \begin{minipage}{6.5cm}
1671 Ab initio MD at \degc{900}\\
1672 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1673 $t=\unit[2230]{fs}$\\
1674 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1678 Contribution of entropy to structural formation
1687 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1694 \begin{minipage}[t]{3cm}
1695 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1696 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1698 \begin{minipage}[t]{7cm}
1701 Low activation energies\\
1702 High activation energies for reverse processes\\
1704 {\color{blue}C$_{\text{sub}}$ very stable}\\
1708 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1710 {\color{blue}Formation of SiC by successive substitution by C}
1714 \begin{minipage}[t]{3cm}
1715 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1716 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1721 \begin{minipage}{5.9cm}
1722 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1724 \begin{picture}(0,0)(70,0)
1725 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1727 \begin{picture}(0,0)(30,0)
1728 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1730 \begin{picture}(0,0)(-10,0)
1731 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1733 \begin{picture}(0,0)(-48,0)
1734 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1736 \begin{picture}(0,0)(12.5,5)
1737 \includegraphics[width=1cm]{100_arrow.eps}
1739 \begin{picture}(0,0)(97,-10)
1740 \includegraphics[height=0.9cm]{001_arrow.eps}
1746 \begin{minipage}{0.3cm}
1750 \begin{minipage}{5.9cm}
1751 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1753 \begin{picture}(0,0)(60,0)
1754 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1756 \begin{picture}(0,0)(25,0)
1757 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1759 \begin{picture}(0,0)(-20,0)
1760 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1762 \begin{picture}(0,0)(-55,0)
1763 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1765 \begin{picture}(0,0)(12.5,5)
1766 \includegraphics[width=1cm]{100_arrow.eps}
1768 \begin{picture}(0,0)(95,0)
1769 \includegraphics[height=0.9cm]{001_arrow.eps}
1781 Conclusion of defect / migration / combined defect simulations
1790 \item Accurately described by quantum-mechanical simulations
1791 \item Less accurate description by classical potential simulations
1792 \item Underestimated formation energy of \cs{} by classical approach
1793 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1798 \item C migration pathway in Si identified
1799 \item Consistent with reorientation and diffusion experiments
1802 \item Different path and ...
1803 \item overestimated barrier by classical potential calculations
1806 Concerning the precipitation mechanism
1808 \item Agglomeration of C-Si dumbbells energetically favorable
1809 (stress compensation)
1810 \item C-Si indeed favored compared to
1811 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1812 \item Possible low interaction capture radius of
1813 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1814 \item Low barrier for
1815 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1816 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1817 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1820 {\color{blue}Results suggest increased participation of \cs}
1828 Silicon carbide precipitation simulations
1834 \begin{pspicture}(0,0)(12,6.5)
1836 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1839 \item Create c-Si volume
1840 \item Periodc boundary conditions
1841 \item Set requested $T$ and $p=0\text{ bar}$
1842 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1845 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1847 Insertion of C atoms at constant T
1849 \item total simulation volume {\pnode{in1}}
1850 \item volume of minimal SiC precipitate {\pnode{in2}}
1851 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1855 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1857 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1859 \ncline[]{->}{init}{insert}
1860 \ncline[]{->}{insert}{cool}
1861 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1862 \rput(7.8,6){\footnotesize $V_1$}
1863 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1864 \rput(9.2,4.85){\tiny $V_2$}
1865 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1866 \rput(9.55,4.45){\footnotesize $V_3$}
1867 \rput(7.9,3.2){\pnode{ins1}}
1868 \rput(9.22,2.8){\pnode{ins2}}
1869 \rput(11.0,2.4){\pnode{ins3}}
1870 \ncline[]{->}{in1}{ins1}
1871 \ncline[]{->}{in2}{ins2}
1872 \ncline[]{->}{in3}{ins3}
1877 \item Restricted to classical potential simulations
1878 \item $V_2$ and $V_3$ considered due to low diffusion
1879 \item Amount of C atoms: 6000
1880 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1881 \item Simulation volume: $31\times 31\times 31$ unit cells
1890 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1895 \begin{minipage}{6.5cm}
1896 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1898 \begin{minipage}{6.5cm}
1899 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1902 \begin{minipage}{6.5cm}
1903 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1905 \begin{minipage}{6.5cm}
1907 \underline{Low C concentration ($V_1$)}\\
1908 \hkl<1 0 0> C-Si dumbbell dominated structure
1910 \item Si-C bumbs around 0.19 nm
1911 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1912 concatenated dumbbells of various orientation
1913 \item Si-Si NN distance stretched to 0.3 nm
1915 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1916 \underline{High C concentration ($V_2$, $V_3$)}\\
1917 High amount of strongly bound C-C bonds\\
1918 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1919 Only short range order observable\\
1920 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1928 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1933 \begin{minipage}{6.5cm}
1934 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1936 \begin{minipage}{6.5cm}
1937 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1940 \begin{minipage}{6.5cm}
1941 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1943 \begin{minipage}{6.5cm}
1945 \underline{Low C concentration ($V_1$)}\\
1946 \hkl<1 0 0> C-Si dumbbell dominated structure
1948 \item Si-C bumbs around 0.19 nm
1949 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1950 concatenated dumbbells of various orientation
1951 \item Si-Si NN distance stretched to 0.3 nm
1953 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1954 \underline{High C concentration ($V_2$, $V_3$)}\\
1955 High amount of strongly bound C-C bonds\\
1956 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1957 Only short range order observable\\
1958 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1961 \begin{pspicture}(0,0)(0,0)
1962 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
1963 \begin{minipage}{10cm}
1965 {\color{red}\bf 3C-SiC formation fails to appear}
1967 \item Low C concentration simulations
1969 \item Formation of \ci{} indeed occurs
1970 \item Agllomeration not observed
1972 \item High C concentration simulations
1974 \item Amorphous SiC-like structure\\
1975 (not expected at prevailing temperatures)
1976 \item Rearrangement and transition into 3C-SiC structure missing
1988 Limitations of molecular dynamics and short range potentials
1995 \underline{Time scale problem of MD}\\[0.2cm]
1996 Minimize integration error\\
1997 $\Rightarrow$ discretization considerably smaller than
1998 reciprocal of fastest vibrational mode\\[0.1cm]
1999 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2000 $\Rightarrow$ suitable choice of time step:
2001 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2002 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2003 Several local minima in energy surface separated by large energy barriers\\
2004 $\Rightarrow$ transition event corresponds to a multiple
2005 of vibrational periods\\
2006 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2007 infrequent transition events\\[0.1cm]
2008 {\color{blue}Accelerated methods:}
2009 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2013 \underline{Limitations related to the short range potential}\\[0.2cm]
2014 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2015 and 2$^{\text{nd}}$ next neighbours\\
2016 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2022 Potential enhanced problem of slow phase space propagation
2027 \underline{Approach to the (twofold) problem}\\[0.2cm]
2028 Increased temperature simulations without TAD corrections\\
2029 (accelerated methods or higher time scales exclusively not sufficient)
2031 \begin{picture}(0,0)(-260,-30)
2033 \begin{minipage}{4.2cm}
2040 \item 3C-SiC also observed for higher T
2041 \item higher T inside sample
2042 \item structural evolution vs.\\
2043 equilibrium properties
2049 \begin{picture}(0,0)(-305,-155)
2051 \begin{minipage}{2.5cm}
2055 thermodynmic sampling
2066 Increased temperature simulations at low C concentration
2071 \begin{minipage}{6.5cm}
2072 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2074 \begin{minipage}{6.5cm}
2075 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2078 \begin{minipage}{6.5cm}
2079 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2081 \begin{minipage}{6.5cm}
2083 \underline{Si-C bonds:}
2085 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2086 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2088 \underline{Si-Si bonds:}
2089 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2090 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2091 \underline{C-C bonds:}
2093 \item C-C next neighbour pairs reduced (mandatory)
2094 \item Peak at 0.3 nm slightly shifted
2096 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2097 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2099 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2101 \item Range [|-$\downarrow$]:
2102 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2103 with nearby Si$_{\text{I}}$}
2108 \begin{picture}(0,0)(-330,-74)
2111 \begin{minipage}{1.6cm}
2114 stretched SiC\\[-0.1cm]
2126 Increased temperature simulations at low C concentration
2131 \begin{minipage}{6.5cm}
2132 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2134 \begin{minipage}{6.5cm}
2135 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2138 \begin{minipage}{6.5cm}
2139 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2141 \begin{minipage}{6.5cm}
2143 \underline{Si-C bonds:}
2145 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2146 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2148 \underline{Si-Si bonds:}
2149 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2150 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2151 \underline{C-C bonds:}
2153 \item C-C next neighbour pairs reduced (mandatory)
2154 \item Peak at 0.3 nm slightly shifted
2156 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2157 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2159 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2161 \item Range [|-$\downarrow$]:
2162 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2163 with nearby Si$_{\text{I}}$}
2168 %\begin{picture}(0,0)(-330,-74)
2171 %\begin{minipage}{1.6cm}
2174 %stretched SiC\\[-0.1cm]
2181 \begin{pspicture}(0,0)(0,0)
2182 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2183 \begin{minipage}{10cm}
2185 {\color{blue}\bf Stretched SiC in c-Si}
2187 \item Consistent to precipitation model involving \cs{}
2188 \item Explains annealing behavior of high/low T C implants
2190 \item Low T: highly mobiel \ci{}
2191 \item High T: stable configurations of \cs{}
2194 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2195 $\Rightarrow$ Precipitation mechanism involving \cs{}
2205 Increased temperature simulations at high C concentration
2210 \begin{minipage}{6.5cm}
2211 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2213 \begin{minipage}{6.5cm}
2214 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2222 \begin{minipage}[t]{6.0cm}
2223 0.186 nm: Si-C pairs $\uparrow$\\
2224 (as expected in 3C-SiC)\\[0.2cm]
2225 0.282 nm: Si-C-C\\[0.2cm]
2226 $\approx$0.35 nm: C-Si-Si
2229 \begin{minipage}{0.2cm}
2233 \begin{minipage}[t]{6.0cm}
2234 0.15 nm: C-C pairs $\uparrow$\\
2235 (as expected in graphite/diamond)\\[0.2cm]
2236 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2237 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2242 \item Decreasing cut-off artifact
2243 \item {\color{red}Amorphous} SiC-like phase remains
2244 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2245 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2254 High C \& small $V$ \& short $t$
2257 Slow restructuring due to strong C-C bonds
2260 High C \& low T implants
2271 Summary and Conclusions
2279 \begin{minipage}[t]{12.9cm}
2280 \underline{Pecipitation simulations}
2282 \item High C concentration $\rightarrow$ amorphous SiC like phase
2283 \item Problem of potential enhanced slow phase space propagation
2284 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2285 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2286 \item High T necessary to simulate IBS conditions (far from equilibrium)
2287 \item Precipitation by successive agglomeration of \cs (epitaxy)
2288 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2289 (stretched SiC, interface)
2297 \begin{minipage}{12.9cm}
2302 \item Point defects excellently / fairly well described
2304 \item C$_{\text{sub}}$ drastically underestimated by EA
2305 \item EA predicts correct ground state:
2306 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2307 \item Identified migration path explaining
2308 diffusion and reorientation experiments by DFT
2309 \item EA fails to describe \ci{} migration:
2310 Wrong path \& overestimated barrier
2312 \item Combinations of defects
2314 \item Agglomeration of point defects energetically favorable
2315 by compensation of stress
2316 \item Formation of C-C unlikely
2317 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2318 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2319 Low barrier (\unit[0.77]{eV}) \& low capture radius
2327 \framebox{Precipitation by successive agglomeration of \cs{}}
2345 \underline{Augsburg}
2347 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2348 \item Ralf Utermann (EDV)
2351 \underline{Helsinki}
2353 \item Prof. K. Nordlund (MD)
2358 \item Bayerische Forschungsstiftung (financial support)
2361 \underline{Paderborn}
2363 \item Prof. J. Lindner (SiC)
2364 \item Prof. G. Schmidt (DFT + financial support)
2365 \item Dr. E. Rauls (DFT + SiC)
2366 \item Dr. S. Sanna (VASP)
2373 \bf Thank you for your attention!