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96 Atomistic simulation study of the silicon carbide precipitation
102 \textsc{F. Zirkelbach}
115 % motivation / properties / applications of silicon carbide
120 \begin{pspicture}(0,0)(13.5,5)
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129 \rput[lt](0.2,4.6){\color{gray}PROPERTIES}
131 \rput[lt](0.5,4){wide band gap}
132 \rput[lt](0.5,3.5){high electric breakdown field}
133 \rput[lt](0.5,3){good electron mobility}
134 \rput[lt](0.5,2.5){high electron saturation drift velocity}
135 \rput[lt](0.5,2){high thermal conductivity}
137 \rput[lt](0.5,1.5){hard and mechanically stable}
138 \rput[lt](0.5,1){chemically inert}
140 \rput[lt](0.5,0.5){radiation hardness}
142 \rput[rt](13.3,4.6){\color{gray}APPLICATIONS}
144 \rput[rt](13,3.85){high-temperature, high power}
145 \rput[rt](13,3.5){and high-frequency}
146 \rput[rt](13,3.15){electronic and optoelectronic devices}
148 \rput[rt](13,2.35){material suitable for extreme conditions}
149 \rput[rt](13,2){microelectromechanical systems}
150 \rput[rt](13,1.65){abrasives, cutting tools, heating elements}
152 \rput[rt](13,0.85){first wall reactor material, detectors}
153 \rput[rt](13,0.5){and electronic devices for space}
157 \begin{picture}(0,0)(-3,68)
158 \includegraphics[width=2.6cm]{wide_band_gap.eps}
160 \begin{picture}(0,0)(-285,-162)
161 \includegraphics[width=3.38cm]{sic_led.eps}
163 \begin{picture}(0,0)(-195,-162)
164 \includegraphics[width=2.8cm]{6h-sic_3c-sic.eps}
166 \begin{picture}(0,0)(-313,65)
167 \includegraphics[width=2.2cm]{infineon_schottky.eps}
169 \begin{picture}(0,0)(-220,65)
170 \includegraphics[width=2.9cm]{sic_wechselrichter_ise.eps}
172 \begin{picture}(0,0)(0,-160)
173 \includegraphics[width=3.0cm]{sic_proton.eps}
175 \begin{picture}(0,0)(-60,65)
176 \includegraphics[width=3.4cm]{sic_switch.eps}
190 \item Polytyps and fabrication of silicon carbide
191 \item Supposed precipitation mechanism of SiC in Si
192 \item Utilized simulation techniques
194 \item Molecular dynamics (MD) simulations
195 \item Density functional theory (DFT) calculations
197 \item C and Si self-interstitial point defects in silicon
198 \item Silicon carbide precipitation simulations
199 \item Investigation of a silicon carbide precipitate in silicon
200 \item Summary / Conclusion / Outlook
217 \begin{tabular}{l c c c c c c}
219 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
221 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
222 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
223 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
224 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
225 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
226 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
227 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
235 \begin{picture}(0,0)(-160,-155)
236 \includegraphics[width=7cm]{polytypes.eps}
238 \begin{picture}(0,0)(-10,-185)
239 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
241 \begin{picture}(0,0)(-10,-175)
242 {\tiny cubic (twist)}
244 \begin{picture}(0,0)(-60,-175)
245 {\tiny hexagonal (no twist)}
247 \begin{pspicture}(0,0)(0,0)
248 \psellipse[linecolor=green](5.7,3.03)(0.4,0.5)
250 \begin{pspicture}(0,0)(0,0)
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253 \begin{pspicture}(0,0)(0,0)
254 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
262 Fabrication of silicon carbide
269 SiC - \emph{Born from the stars, perfected on earth.}
273 Conventional thin film SiC growth:
275 \item \underline{Sublimation growth using the modified Lely method}
277 \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
278 \item Surrounded by polycrystalline SiC in a graphite crucible\\
279 at $T=2100-2400 \, ^{\circ} \text{C}$
280 \item Deposition of supersaturated vapor on cooler seed crystal
282 \item \underline{Homoepitaxial growth using CVD}
284 \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
285 \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
286 \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
287 \item High quality but limited in size of substrates
289 \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
291 \item Two steps: carbonization and growth
292 \item $T=650-1050 \, ^{\circ} \text{C}$
293 \item Quality and size not yet sufficient
297 \begin{picture}(0,0)(-280,-65)
298 \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
300 \begin{picture}(0,0)(-280,-55)
301 \begin{minipage}{5cm}
303 NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
308 \begin{picture}(0,0)(-265,-150)
309 \includegraphics[width=2.4cm]{m_lely.eps}
311 \begin{picture}(0,0)(-333,-175)
312 \begin{minipage}{5cm}
318 5. Insulation\\[-7pt]
323 \begin{picture}(0,0)(-230,-35)
325 {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
328 \begin{picture}(0,0)(-230,-10)
330 \begin{minipage}{3cm}
331 {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
342 Fabrication of silicon carbide
347 Alternative approach:
348 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
350 \item \underline{Implantation step 1}\\
351 180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
352 $\Rightarrow$ box-like distribution of equally sized
353 and epitactically oriented SiC precipitates
355 \item \underline{Implantation step 2}\\
356 180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
357 $\Rightarrow$ destruction of SiC nanocrystals
358 in growing amorphous interface layers
359 \item \underline{Annealing}\\
360 $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
361 $\Rightarrow$ homogeneous, stoichiometric SiC layer
362 with sharp interfaces
365 \begin{minipage}{6.3cm}
366 \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
368 XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
372 \begin{minipage}{6.3cm}
375 Precipitation mechanism not yet fully understood!
377 \renewcommand\labelitemi{$\Rightarrow$}
379 \underline{Understanding the SiC precipitation}
381 \item significant technological progress in SiC thin film formation
382 \item perspectives for processes relying upon prevention of SiC precipitation
393 Supposed precipitation mechanism of SiC in Si
400 \begin{minipage}{3.8cm}
401 Si \& SiC lattice structure\\[0.2cm]
402 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
406 \begin{minipage}{3.8cm}
408 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
412 \begin{minipage}{3.8cm}
414 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
418 \begin{minipage}{4cm}
420 C-Si dimers (dumbbells)\\[-0.1cm]
421 on Si interstitial sites
425 \begin{minipage}{4.2cm}
427 Agglomeration of C-Si dumbbells\\[-0.1cm]
428 $\Rightarrow$ dark contrasts
432 \begin{minipage}{4cm}
434 Precipitation of 3C-SiC in Si\\[-0.1cm]
435 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
436 \& release of Si self-interstitials
440 \begin{minipage}{3.8cm}
442 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
446 \begin{minipage}{3.8cm}
448 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
452 \begin{minipage}{3.8cm}
454 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
458 \begin{pspicture}(0,0)(0,0)
459 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
460 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
461 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
462 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
463 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
464 $4a_{\text{Si}}=5a_{\text{SiC}}$
466 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
467 \hkl(h k l) planes match
476 Molecular dynamics (MD) simulations
485 \item Microscopic description of N particle system
486 \item Analytical interaction potential
487 \item Numerical integration using Newtons equation of motion\\
488 as a propagation rule in 6N-dimensional phase space
489 \item Observables obtained by time and/or ensemble averages
491 {\bf Details of the simulation:}
493 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
494 \item Ensemble: NpT (isothermal-isobaric)
496 \item Berendsen thermostat:
497 $\tau_{\text{T}}=100\text{ fs}$
498 \item Berendsen barostat:\\
499 $\tau_{\text{P}}=100\text{ fs}$,
500 $\beta^{-1}=100\text{ GPa}$
502 \item Erhart/Albe potential: Tersoff-like bond order potential
505 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
506 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
510 \begin{picture}(0,0)(-230,-30)
511 \includegraphics[width=5cm]{tersoff_angle.eps}
519 Density functional theory (DFT) calculations
524 Basic ingredients necessary for DFT
527 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
529 \item ... uniquely determines the ground state potential
531 \item ... minimizes the systems total energy
533 \item \underline{Born-Oppenheimer}
534 - $N$ moving electrons in an external potential of static nuclei
536 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
537 +\sum_i^N V_{\text{ext}}(r_i)
538 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
540 \item \underline{Effective potential}
541 - averaged electrostatic potential \& exchange and correlation
543 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
546 \item \underline{Kohn-Sham system}
547 - Schr\"odinger equation of N non-interacting particles
549 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
554 n(r)=\sum_i^N|\Phi_i(r)|^2
556 \item \underline{Self-consistent solution}\\
557 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
558 which in turn depends on $n(r)$
559 \item \underline{Variational principle}
560 - minimize total energy with respect to $n(r)$
568 Density functional theory (DFT) calculations
575 Details of applied DFT calculations in this work
578 \item \underline{Exchange correlation functional}
579 - approximations for the inhomogeneous electron gas
581 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
582 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
584 \item \underline{Plane wave basis set}
585 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
588 \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}}
589 \qquad ({\color{blue}300\text{ eV}})
591 \item \underline{Brillouin zone sampling} -
592 {\color{blue}$\Gamma$-point only} calculations
593 \item \underline{Pseudo potential}
594 - consider only the valence electrons
595 \item \underline{Code} - VASP 4.6
600 MD and structural optimization
603 \item MD integration: Gear predictor corrector algorithm
604 \item Pressure control: Parrinello-Rahman pressure control
605 \item Structural optimization: Conjugate gradient method
608 \begin{pspicture}(0,0)(0,0)
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617 C and Si self-interstitial point defects in silicon
624 \begin{minipage}{8cm}
626 \begin{pspicture}(0,0)(7,5)
627 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
630 \item Creation of c-Si simulation volume
631 \item Periodic boundary conditions
632 \item $T=0\text{ K}$, $p=0\text{ bar}$
635 \rput(3.5,2.1){\rnode{insert}{\psframebox{
638 Insertion of interstitial C/Si atoms
641 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
644 Relaxation / structural energy minimization
647 \ncline[]{->}{init}{insert}
648 \ncline[]{->}{insert}{cool}
651 \begin{minipage}{5cm}
652 \includegraphics[width=5cm]{unit_cell_e.eps}\\
655 \begin{minipage}{9cm}
656 \begin{tabular}{l c c}
658 & size [unit cells] & \# atoms\\
660 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
661 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
665 \begin{minipage}{4cm}
666 {\color{red}$\bullet$} Tetrahedral\\
667 {\color{green}$\bullet$} Hexagonal\\
668 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
669 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
670 {\color{cyan}$\bullet$} Bond-centered\\
671 {\color{black}$\bullet$} Vacancy / Substitutional
680 \begin{minipage}{9.5cm}
683 Si self-interstitial point defects in silicon\\
686 \begin{tabular}{l c c c c c}
688 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
690 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
691 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
693 \end{tabular}\\[0.2cm]
695 \begin{minipage}{4.7cm}
696 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
698 \begin{minipage}{4.7cm}
700 {\tiny nearly T $\rightarrow$ T}\\
702 \includegraphics[width=4.7cm]{nhex_tet.ps}
705 \underline{Hexagonal} \hspace{2pt}
706 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
708 \begin{minipage}{2.7cm}
709 $E_{\text{f}}^*=4.48\text{ eV}$\\
710 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
712 \begin{minipage}{0.4cm}
717 \begin{minipage}{2.7cm}
718 $E_{\text{f}}=3.96\text{ eV}$\\
719 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
722 \begin{minipage}{2.9cm}
724 \underline{Vacancy}\\
725 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
730 \begin{minipage}{3.5cm}
733 \underline{\hkl<1 1 0> dumbbell}\\
734 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
735 \underline{Tetrahedral}\\
736 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
737 \underline{\hkl<1 0 0> dumbbell}\\
738 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
750 C interstitial point defects in silicon\\[-0.1cm]
753 \begin{tabular}{l c c c c c c}
755 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
757 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 \\
758 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & 0.75 & 5.59$^*$ \\
760 \end{tabular}\\[0.1cm]
763 \begin{minipage}{2.7cm}
764 \underline{Hexagonal} \hspace{2pt}
765 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
766 $E_{\text{f}}^*=9.05\text{ eV}$\\
767 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
769 \begin{minipage}{0.4cm}
774 \begin{minipage}{2.7cm}
775 \underline{\hkl<1 0 0>}\\
776 $E_{\text{f}}=3.88\text{ eV}$\\
777 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
780 \begin{minipage}{2cm}
783 \begin{minipage}{3cm}
785 \underline{Tetrahedral}\\
786 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
791 \begin{minipage}{2.7cm}
792 \underline{Bond-centered}\\
793 $E_{\text{f}}^*=5.59\text{ eV}$\\
794 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
796 \begin{minipage}{0.4cm}
801 \begin{minipage}{2.7cm}
802 \underline{\hkl<1 1 0> dumbbell}\\
803 $E_{\text{f}}=5.18\text{ eV}$\\
804 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
807 \begin{minipage}{2cm}
810 \begin{minipage}{3cm}
812 \underline{Substitutional}\\
813 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
824 C \hkl<1 0 0> dumbbell interstitial configuration\\
828 \begin{tabular}{l c c c c c c c c}
830 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
832 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
833 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
835 \end{tabular}\\[0.2cm]
836 \begin{tabular}{l c c c c }
838 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
840 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
841 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
843 \end{tabular}\\[0.2cm]
844 \begin{tabular}{l c c c}
846 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
848 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
849 VASP & 0.109 & -0.065 & 0.174 \\
851 \end{tabular}\\[0.6cm]
854 \begin{minipage}{3.0cm}
856 \underline{Erhart/Albe}
857 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
860 \begin{minipage}{3.0cm}
863 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
867 \begin{picture}(0,0)(-185,10)
868 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
870 \begin{picture}(0,0)(-280,-150)
871 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
874 \begin{pspicture}(0,0)(0,0)
875 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
876 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
877 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
878 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
887 \begin{minipage}{8.5cm}
890 Bond-centered interstitial configuration\\[-0.1cm]
893 \begin{minipage}{3.0cm}
894 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
896 \begin{minipage}{5.2cm}
898 \item Linear Si-C-Si bond
899 \item Si: one C \& 3 Si neighbours
900 \item Spin polarized calculations
901 \item No saddle point!\\
908 \begin{minipage}[t]{6.5cm}
909 \begin{minipage}[t]{1.2cm}
911 {\tiny sp$^3$}\\[0.8cm]
912 \underline{${\color{black}\uparrow}$}
913 \underline{${\color{black}\uparrow}$}
914 \underline{${\color{black}\uparrow}$}
915 \underline{${\color{red}\uparrow}$}\\
918 \begin{minipage}[t]{1.4cm}
920 {\color{red}M}{\color{blue}O}\\[0.8cm]
921 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
922 $\sigma_{\text{ab}}$\\[0.5cm]
923 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
927 \begin{minipage}[t]{1.0cm}
931 \underline{${\color{white}\uparrow\uparrow}$}
932 \underline{${\color{white}\uparrow\uparrow}$}\\
934 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
935 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
939 \begin{minipage}[t]{1.4cm}
941 {\color{blue}M}{\color{green}O}\\[0.8cm]
942 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
943 $\sigma_{\text{ab}}$\\[0.5cm]
944 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
948 \begin{minipage}[t]{1.2cm}
951 {\tiny sp$^3$}\\[0.8cm]
952 \underline{${\color{green}\uparrow}$}
953 \underline{${\color{black}\uparrow}$}
954 \underline{${\color{black}\uparrow}$}
955 \underline{${\color{black}\uparrow}$}\\
963 \begin{minipage}{4.5cm}
964 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
966 \begin{minipage}{3.5cm}
967 {\color{gray}$\bullet$} Spin up\\
968 {\color{green}$\bullet$} Spin down\\
969 {\color{blue}$\bullet$} Resulting spin up\\
970 {\color{yellow}$\bullet$} Si atoms\\
971 {\color{red}$\bullet$} C atom
976 \begin{minipage}{4.2cm}
978 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
979 {\color{green}$\Box$} {\tiny unoccupied}\\
980 {\color{red}$\bullet$} {\tiny occupied}
989 Migration of the C \hkl<1 0 0> dumbbell interstitial
994 {\small Investigated pathways}
996 \begin{minipage}{8.5cm}
997 \begin{minipage}{8.3cm}
998 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
999 \begin{minipage}{2.4cm}
1000 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1002 \begin{minipage}{0.4cm}
1005 \begin{minipage}{2.4cm}
1006 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1008 \begin{minipage}{0.4cm}
1011 \begin{minipage}{2.4cm}
1012 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1015 \begin{minipage}{8.3cm}
1016 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1017 \begin{minipage}{2.4cm}
1018 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1020 \begin{minipage}{0.4cm}
1023 \begin{minipage}{2.4cm}
1024 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1026 \begin{minipage}{0.4cm}
1029 \begin{minipage}{2.4cm}
1030 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1033 \begin{minipage}{8.3cm}
1034 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1035 \begin{minipage}{2.4cm}
1036 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1038 \begin{minipage}{0.4cm}
1041 \begin{minipage}{2.4cm}
1042 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1044 \begin{minipage}{0.4cm}
1047 \begin{minipage}{2.4cm}
1048 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1053 \begin{minipage}{4.2cm}
1054 {\small Constrained relaxation\\
1055 technique (CRT) method}\\
1056 \includegraphics[width=4cm]{crt_orig.eps}
1058 \item Constrain diffusing atom
1059 \item Static constraints
1062 {\small Modifications}\\
1063 \includegraphics[width=4cm]{crt_mod.eps}
1065 \item Constrain all atoms
1066 \item Update individual\\
1077 Migration of the C \hkl<1 0 0> dumbbell interstitial
1083 \begin{minipage}{5.9cm}
1085 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
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1092 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
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1095 \includegraphics[width=1cm]{vasp_mig/bc.eps}
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1098 \includegraphics[width=1cm]{110_arrow.eps}
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1101 \includegraphics[height=0.9cm]{001_arrow.eps}
1107 \begin{minipage}{0.3cm}
1111 \begin{minipage}{5.9cm}
1113 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1116 \begin{picture}(0,0)(60,0)
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1128 \begin{picture}(0,0)(90,0)
1129 \includegraphics[height=0.9cm]{001_arrow.eps}
1139 \begin{minipage}{5.9cm}
1141 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1144 \begin{picture}(0,0)(60,0)
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1147 \begin{picture}(0,0)(10,0)
1148 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
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1151 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1153 \begin{picture}(0,0)(12.5,10)
1154 \includegraphics[width=1cm]{100_arrow.eps}
1156 \begin{picture}(0,0)(90,0)
1157 \includegraphics[height=0.9cm]{001_arrow.eps}
1163 \begin{minipage}{0.3cm}
1166 \begin{minipage}{6.5cm}
1169 \item Energetically most favorable path
1172 \item Activation energy: $\approx$ 0.9 eV
1173 \item Experimental values: 0.73 ... 0.87 eV
1175 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1176 \item Reorientation (path 3)
1178 \item More likely composed of two consecutive steps of type 2
1179 \item Experimental values: 0.77 ... 0.88 eV
1181 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1190 Migration of the C \hkl<1 0 0> dumbbell interstitial
1195 \begin{minipage}{6.5cm}
1198 \begin{minipage}{5.9cm}
1200 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1203 \begin{pspicture}(0,0)(0,0)
1204 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
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1207 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1209 \begin{picture}(0,0)(5,-50)
1210 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
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1213 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
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1216 \includegraphics[width=1cm]{110_arrow.eps}
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1228 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
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1231 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
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1234 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1236 \begin{picture}(0,0)(12.5,-5)
1237 \includegraphics[width=1cm]{100_arrow.eps}
1239 \begin{picture}(0,0)(90,-15)
1240 \includegraphics[height=0.9cm]{010_arrow.eps}
1246 \begin{minipage}{5.9cm}
1249 \item Lowest activation energy: $\approx$ 2.2 eV
1250 \item 2.4 times higher than VASP
1251 \item Different pathway
1252 \item Transition minima ($\rightarrow$ \hkl<1 1 0> dumbbell)
1257 \begin{minipage}{6.5cm}
1260 \begin{minipage}{5.9cm}
1262 \includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1265 \begin{pspicture}(0,0)(0,0)
1266 \psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1268 \begin{picture}(0,0)(60,-5)
1269 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1271 \begin{picture}(0,0)(0,-5)
1272 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1274 \begin{picture}(0,0)(-55,-5)
1275 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1277 \begin{picture}(0,0)(12.5,5)
1278 \includegraphics[width=1cm]{100_arrow.eps}
1280 \begin{picture}(0,0)(90,0)
1281 \includegraphics[height=0.9cm]{001_arrow.eps}
1289 \begin{minipage}{5.9cm}
1290 \includegraphics[width=5.9cm]{00-1_ip0-10.ps}
1301 Migrations involving the C \hkl<1 1 0> dumbbell interstitial
1310 \begin{minipage}{6.0cm}
1311 \includegraphics[width=6cm]{vasp_mig/110_mig_vasp.ps}
1313 \begin{minipage}{7cm}
1314 \underline{Alternative pathway and energies [eV]}\\[0.1cm]
1315 \hkl<0 -1 0> $\stackrel{0.7}{{\color{red}\longrightarrow}}$
1316 \hkl<1 1 0> $\stackrel{0.95}{{\color{blue}\longrightarrow}}$
1317 BC $\stackrel{0.25}{\longrightarrow}$ \hkl<0 0 -1>\\[0.3cm]
1318 Composed of three single transitions\\[0.3cm]
1319 Activation energy of second transition slightly\\
1320 higher than direct transition (path 2)\\[0.3cm]
1321 $\Rightarrow$ very unlikely to happen
1322 \end{minipage}\\[0.2cm]
1326 \begin{minipage}{6.0cm}
1327 \includegraphics[width=6cm]{110_mig.ps}
1329 \begin{minipage}{7cm}
1330 \underline{Alternative pathway and energies [eV]}\\[0.1cm]
1331 \hkl<0 0 -1> $\stackrel{2.2}{{\color{green}\longrightarrow}}$
1332 \hkl<1 1 0> $\stackrel{0.9}{{\color{red}\longrightarrow}}$
1333 \hkl<0 0 -1>\\[0.3cm]
1334 Composed of two single transitions\\[0.3cm]
1335 Compared to direct transition: (2.2 eV \& 0.5 eV)\\[0.3cm]
1336 $\Rightarrow$ more readily constituting a probable transition
1344 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1354 E_{\text{f}}^{\text{defect combination}}-
1355 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1356 E_{\text{f}}^{\text{2nd defect}}
1362 \begin{tabular}{l c c c c c c}
1364 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1366 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1367 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1368 \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}\\
1369 \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}\\
1370 \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}\\
1371 \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}\\
1373 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1374 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1383 \begin{minipage}[t]{3.8cm}
1384 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1385 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1387 \begin{minipage}[t]{3.5cm}
1388 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1389 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1391 \begin{minipage}[t]{5.5cm}
1393 \item Restricted to VASP simulations
1394 \item $E_{\text{b}}=0$ for isolated non-interacting defects
1395 \item $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1396 \item Stress compensation / increase
1397 \item Most favorable: C clustering
1398 \item Unfavored: antiparallel orientations
1399 \item Indication of energetically favored\\
1404 \begin{picture}(0,0)(-295,-130)
1405 \includegraphics[width=3.5cm]{comb_pos.eps}
1413 Combinations of C-Si \hkl<1 0 0>-type interstitials
1420 Energetically most favorable combinations along \hkl<1 1 0>
1425 \begin{tabular}{l c c c c c c}
1427 & 1 & 2 & 3 & 4 & 5 & 6\\
1429 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1430 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1431 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>\\
1438 \begin{minipage}{7.0cm}
1439 \includegraphics[width=7cm]{db_along_110_cc.ps}
1441 \begin{minipage}{6.0cm}
1444 Interaction proportional to reciprocal cube of C-C distance
1446 Saturation in the immediate vicinity
1457 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1463 \begin{minipage}{3.2cm}
1464 \includegraphics[width=3cm]{sub_110_combo.eps}
1466 \begin{minipage}{7.8cm}
1467 \begin{tabular}{l c c c c c c}
1469 C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1470 \hkl<1 0 1> & \hkl<-1 0 1> \\
1472 1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1473 2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1474 3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1475 4 & \RM{4} & B & D & E & E & D \\
1476 5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1483 \begin{tabular}{l c c c c c c c c c c}
1485 Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1487 $E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1488 $E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1489 $r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1494 \begin{minipage}{6.0cm}
1495 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1497 \begin{minipage}{7cm}
1500 \item IBS: C may displace Si\\
1501 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1503 \hkl<1 1 0>-type $\rightarrow$ favored combination
1504 \renewcommand\labelitemi{$\Rightarrow$}
1505 \item Less favorable than C-Si \hkl<1 0 0> dumbbell\\
1506 ($E_{\text{f}}=3.88\text{ eV}$)
1507 \item Interaction drops quickly to zero\\
1508 (low interaction capture radius)
1517 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1524 \begin{minipage}[t]{3cm}
1525 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1526 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1528 \begin{minipage}[t]{7cm}
1531 Low activation energies\\
1532 High activation energies for reverse processes\\
1534 {\color{blue}C$_{\text{sub}}$ very stable}\\
1538 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1540 {\color{blue}Formation of SiC by successive substitution by C}
1544 \begin{minipage}[t]{3cm}
1545 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1546 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1551 \begin{minipage}{5.9cm}
1552 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1554 \begin{picture}(0,0)(70,0)
1555 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1557 \begin{picture}(0,0)(30,0)
1558 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1560 \begin{picture}(0,0)(-10,0)
1561 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1563 \begin{picture}(0,0)(-48,0)
1564 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1566 \begin{picture}(0,0)(12.5,5)
1567 \includegraphics[width=1cm]{100_arrow.eps}
1569 \begin{picture}(0,0)(97,-10)
1570 \includegraphics[height=0.9cm]{001_arrow.eps}
1576 \begin{minipage}{0.3cm}
1580 \begin{minipage}{5.9cm}
1581 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1583 \begin{picture}(0,0)(60,0)
1584 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1586 \begin{picture}(0,0)(25,0)
1587 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1589 \begin{picture}(0,0)(-20,0)
1590 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1592 \begin{picture}(0,0)(-55,0)
1593 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1595 \begin{picture}(0,0)(12.5,5)
1596 \includegraphics[width=1cm]{100_arrow.eps}
1598 \begin{picture}(0,0)(95,0)
1599 \includegraphics[height=0.9cm]{001_arrow.eps}
1611 Conclusion of defect / migration / combined defect simulations
1620 \item Accurately described by quantum-mechanical simulations
1621 \item Less correct description by classical potential simulations
1625 \item Consistent with solubility data of C in Si
1626 \item \hkl<1 0 0> C-Si dumbbell interstitial ground state configuration
1627 \item Consistent with reorientation and diffusion experiments
1628 \item C migration pathway in Si identified
1633 Concerning the precipitation mechanism
1635 \item Agglomeration of C-Si dumbbells energetically favorable
1636 \item C-Si indeed favored compared to
1637 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1638 \item Possible low interaction capture radius of
1639 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1640 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1641 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1646 {\color{blue}Some results point to a different precipitation mechanism!}
1654 Silicon carbide precipitation simulations
1660 \begin{pspicture}(0,0)(12,6.5)
1662 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1665 \item Create c-Si volume
1666 \item Periodc boundary conditions
1667 \item Set requested $T$ and $p=0\text{ bar}$
1668 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1671 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1673 Insertion of C atoms at constant T
1675 \item total simulation volume {\pnode{in1}}
1676 \item volume of minimal SiC precipitate {\pnode{in2}}
1677 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1681 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1683 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1685 \ncline[]{->}{init}{insert}
1686 \ncline[]{->}{insert}{cool}
1687 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1688 \rput(7.8,6){\footnotesize $V_1$}
1689 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1690 \rput(9.2,4.85){\tiny $V_2$}
1691 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1692 \rput(9.55,4.45){\footnotesize $V_3$}
1693 \rput(7.9,3.2){\pnode{ins1}}
1694 \rput(9.22,2.8){\pnode{ins2}}
1695 \rput(11.0,2.4){\pnode{ins3}}
1696 \ncline[]{->}{in1}{ins1}
1697 \ncline[]{->}{in2}{ins2}
1698 \ncline[]{->}{in3}{ins3}
1703 \item Restricted to classical potential simulations
1704 \item $V_2$ and $V_3$ considered due to low diffusion
1705 \item Amount of C atoms: 6000
1706 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1707 \item Simulation volume: $31\times 31\times 31$ unit cells
1716 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1721 \begin{minipage}{6.5cm}
1722 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1724 \begin{minipage}{6.5cm}
1725 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1728 \begin{minipage}{6.5cm}
1729 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1731 \begin{minipage}{6.5cm}
1733 \underline{Low C concentration ($V_1$)}\\
1734 \hkl<1 0 0> C-Si dumbbell dominated structure
1736 \item Si-C bumbs around 0.19 nm
1737 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1738 concatenated dumbbells of various orientation
1739 \item Si-Si NN distance stretched to 0.3 nm
1741 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1742 \underline{High C concentration ($V_2$, $V_3$)}\\
1743 High amount of strongly bound C-C bonds\\
1744 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1745 Only short range order observable\\
1746 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1754 Limitations of molecular dynamics and short range potentials
1761 \underline{Time scale problem of MD}\\[0.2cm]
1762 Minimize integration error\\
1763 $\Rightarrow$ discretization considerably smaller than
1764 reciprocal of fastest vibrational mode\\[0.1cm]
1765 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
1766 $\Rightarrow$ suitable choice of time step:
1767 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
1768 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
1769 Several local minima in energy surface separated by large energy barriers\\
1770 $\Rightarrow$ transition event corresponds to a multiple
1771 of vibrational periods\\
1772 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
1773 infrequent transition events\\[0.1cm]
1774 {\color{blue}Accelerated methods:}
1775 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
1779 \underline{Limitations related to the short range potential}\\[0.2cm]
1780 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
1781 and 2$^{\text{nd}}$ next neighbours\\
1782 $\Rightarrow$ overestimated unphysical high forces of next neighbours
1788 Potential enhanced problem of slow phase space propagation
1793 \underline{Approach to the (twofold) problem}\\[0.2cm]
1794 Increased temperature simulations without TAD corrections\\
1795 (accelerated methods or higher time scales exclusively not sufficient)
1797 \begin{picture}(0,0)(-260,-30)
1799 \begin{minipage}{4.2cm}
1806 \item 3C-SiC also observed for higher T
1807 \item higher T inside sample
1808 \item structural evolution vs.\\
1809 equilibrium properties
1815 \begin{picture}(0,0)(-305,-155)
1817 \begin{minipage}{2.5cm}
1821 thermodynmic sampling
1832 Increased temperature simulations at low C concentration
1837 \begin{minipage}{6.5cm}
1838 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
1840 \begin{minipage}{6.5cm}
1841 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
1844 \begin{minipage}{6.5cm}
1845 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
1847 \begin{minipage}{6.5cm}
1849 \underline{Si-C bonds:}
1851 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
1852 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
1854 \underline{Si-Si bonds:}
1855 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
1856 ($\rightarrow$ 0.325 nm)\\[0.1cm]
1857 \underline{C-C bonds:}
1859 \item C-C next neighbour pairs reduced (mandatory)
1860 \item Peak at 0.3 nm slightly shifted
1862 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
1863 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
1865 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
1867 \item Range [|-$\downarrow$]:
1868 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
1869 with nearby Si$_{\text{I}}$}
1874 \begin{picture}(0,0)(-330,-74)
1877 \begin{minipage}{1.6cm}
1880 stretched SiC\\[-0.1cm]
1892 Increased temperature simulations at high C concentration
1897 \begin{minipage}{6.5cm}
1898 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
1900 \begin{minipage}{6.5cm}
1901 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
1905 Decreasing cut-off artifact\\
1906 High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
1907 $\Rightarrow$ hard to categorize
1913 \begin{minipage}[t]{6.0cm}
1914 0.186 nm: Si-C pairs $\uparrow$\\
1915 (as expected in 3C-SiC)\\[0.2cm]
1916 0.282 nm: Si-C-C\\[0.2cm]
1917 $\approx$0.35 nm: C-Si-Si
1920 \begin{minipage}{0.2cm}
1924 \begin{minipage}[t]{6.0cm}
1925 0.15 nm: C-C pairs $\uparrow$\\
1926 (as expected in graphite/diamond)\\[0.2cm]
1927 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
1928 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
1935 {\color{red}Amorphous} SiC-like phase remains\\
1936 Slightly sharper peaks
1937 $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics}
1938 due to temperature\\[0.1cm]
1941 Continue with higher temperatures and longer time scales
1950 Valuation of a practicable temperature limit
1960 Recrystallization is a hard task!
1961 $\Rightarrow$ Avoid melting!
1970 \begin{minipage}{7.5cm}
1971 \includegraphics[width=7cm]{fe_and_t.ps}
1973 \begin{minipage}{5.5cm}
1974 \underline{Melting does not occur instantly after}\\
1975 \underline{exceeding the melting point $T_{\text{m}}=2450\text{ K}$}
1977 \item required transition enthalpy
1978 \item hysterisis behaviour
1980 \underline{Heating up c-Si by 1 K/ps}
1982 \item transition occurs at $\approx$ 3125 K
1983 \item $\Delta E=0.58\text{ eV/atom}=55.7\text{ kJ/mole}$\\
1984 (literature: 50.2 kJ/mole)
1991 \begin{minipage}{4cm}
1992 Initially chosen temperatures:\\
1993 $1.0 - 1.2 \cdot T_{\text{m}}$
1996 \begin{minipage}{3cm}
2002 \begin{minipage}{5cm}
2003 Introduced C (defects)\\
2004 $\rightarrow$ reduction of transition point\\
2005 $\rightarrow$ melting already at $T_{\text{m}}$
2014 Maximum temperature used: $0.95\cdot T_{\text{m}}$
2024 Long time scale simulations at maximum temperature
2031 \underline{Differences}
2033 \item Temperature set to $0.95 \cdot T_{\text{m}}$
2034 \item Cubic insertion volume $\Rightarrow$ spherical insertion volume
2035 \item Amount of C atoms: 6000 $\rightarrow$ 5500
2036 $\Leftrightarrow r_{\text{prec}}=0.3\text{ nm}$
2037 \item Simulation volume: 21 unit cells of c-Si in each direction
2044 \begin{minipage}[t]{4.5cm}
2046 \underline{Low C concentration, Si-C}
2047 \includegraphics[width=4.5cm]{c_in_si_95_v1_si-c.ps}\\
2051 \begin{minipage}[t]{4.5cm}
2053 \underline{Low C concentration, C-C}
2054 \includegraphics[width=4.5cm]{c_in_si_95_v1_c-c.ps}\\
2059 \begin{minipage}[t]{4cm}
2061 \underline{High C concentration}
2062 \includegraphics[width=4.5cm]{c_in_si_95_v2.ps}\\
2063 No significant changes
2069 Long time scales and high temperatures most probably not sufficient enough!
2078 Investigation of a silicon carbide precipitate in silicon
2087 \begin{minipage}{5.3cm}
2089 \frac{8}{a_{\text{Si}}^3}(
2090 \underbrace{21^3 a_{\text{Si}}^3}_{=V}
2091 -\frac{4}{3}\pi x^3)+
2092 \underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
2099 \frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500
2100 \Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}}
2103 y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
2107 \begin{minipage}{0.3cm}
2110 \begin{minipage}{7.0cm}
2111 \underline{Construction}
2113 \item Simulation volume: 21$^3$ unit cells of c-Si
2114 \item Spherical topotactically aligned precipitate\\
2115 $r=3.0\text{ nm}$ $\Leftrightarrow$ $\approx$ 5500 C atoms
2116 \item Create c-Si but skipped inside sphere of radius $x$
2117 \item Create 3C-SiC inside sphere of radius $x$\\
2118 and lattice constant $y$
2119 \item Strong coupling to heat bath ($T=20\,^{\circ}\mathrm{C}$)
2125 \begin{minipage}{6.2cm}
2126 \includegraphics[width=6cm]{pc_0.ps}
2128 \begin{minipage}{6.8cm}
2131 \item Slight increase of c-Si lattice constant!
2132 \item C-C peaks (imply same distanced Si-Si peaks)
2134 \item New peak at 0.307 nm: 2$^{\text{nd}}$ NN in 3C-SiC
2135 \item Bumps ({\color{green}$\downarrow$}):
2136 4$^{\text{th}}$ and 6$^{\text{th}}$ NN
2138 \item 3C-SiC lattice constant: 4.34 \AA (bulk: 4.36 \AA)\\
2139 $\rightarrow$ compressed precipitate
2140 \item Interface tension:\\
2141 20.15 eV/nm$^2$ or $3.23 \times 10^{-4}$ J/cm$^2$\\
2142 (literature: $2 - 8 \times 10^{-4}$ J/cm$^2$)
2151 Investigation of a silicon carbide precipitate in silicon
2156 \begin{minipage}{7cm}
2157 \underline{Appended annealing steps}
2159 \item artificially constructed interface\\
2160 $\rightarrow$ allow for rearrangement of interface atoms
2161 \item check SiC stability
2163 \underline{Temperature schedule}
2165 \item rapidly heat up structure up to $2050\,^{\circ}\mathrm{C}$\\
2167 \item slow heating up to $1.2\cdot T_{\text{m}}=2940\text{ K}$
2169 $\rightarrow$ melting at around 2840 K
2170 (\href{../video/sic_prec_120.avi}{$\rhd$})
2171 \item cooling down structure at 100 \% $T_{\text{m}}$ (1 K/ps)\\
2172 $\rightarrow$ no energetically more favorable struture
2175 \begin{minipage}{6cm}
2176 \includegraphics[width=6.7cm]{fe_and_t_sic.ps}
2179 \begin{minipage}{4cm}
2180 \includegraphics[width=4cm]{sic_prec/melt_01.eps}
2182 \begin{minipage}{0.4cm}
2185 \begin{minipage}{4cm}
2186 \includegraphics[width=4cm]{sic_prec/melt_02.eps}
2188 \begin{minipage}{0.4cm}
2191 \begin{minipage}{4cm}
2192 \includegraphics[width=4cm]{sic_prec/melt_03.eps}
2200 Summary / Conclusion / Outlook
2208 \begin{minipage}{12.9cm}
2211 \item Summary \& conclusion
2213 \item Point defects excellently / fairly well described
2214 by QM / classical potential simulations
2215 \item Identified migration path explaining
2216 diffusion and reorientation experiments
2217 \item Agglomeration of point defects energetically favorable
2218 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2222 \item Discussions concerning interpretation of QM results (Paderborn)
2223 \item Compare migration barrier of
2224 \hkl<1 1 0> Si and C-Si \hkl<1 0 0> dumbbell
2225 \item Combination: Vacancy \& \hkl<1 1 0> Si self-interstitial \&
2226 C-Si \hkl<1 0 0> dumbbell (IBS)
2235 \begin{minipage}[t]{6.2cm}
2236 \underline{Pecipitation simulations}
2238 \item Summary \& conclusion
2241 $\rightarrow$ C-Si \hkl<1 0 0> dumbbell\\
2243 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2244 \item High C concentration\\
2245 $\rightarrow$ amorphous SiC like phase
2249 \item Accelerated method: self-guided MD
2250 \item Activation relaxation technique
2251 \item Constrainted transition path
2257 \begin{minipage}[t]{6.2cm}
2258 \underline{Constructed 3C-SiC precipitate}
2260 \item Summary \& conclusion
2262 \item Small / stable / compressed 3C-SiC\\
2263 precipitate in slightly stretched\\
2265 \item Interface tension matches experiemnts
2269 \item Try to improve interface
2270 \item Precipitates of different size
2292 \underline{Augsburg}
2294 \item Prof. B. Stritzker (accepting a simulator at EP \RM{4})
2295 \item Ralf Utermann (EDV)
2298 \underline{Helsinki}
2300 \item Prof. K. Nordlund (MD)
2305 \item Bayerische Forschungsstiftung (financial support)
2308 \underline{Paderborn}
2310 \item Prof. J. Lindner (SiC)
2311 \item Prof. G. Schmidt (DFT + financial support)
2312 \item Dr. E. Rauls (DFT + SiC)
2319 \bf Thank you for your attention!